[fix of dummy var; checkin of base hol and zfc
raph.levien@gmail.com**20051031184645] {
adddir ./hol
adddir ./zfc
hunk ./gh_verify.py 263
+	#print 'match_expand', target, forv
hunk ./gh_verify.py 381
-        forv = self.freevars(concl)
+        forv = self.freevars(stat)
hunk ./gh_verify.py 385
+	#print 'forv =', forv, concl
addfile ./hol/hol-zfc.gh
hunk ./hol/hol-zfc.gh 1
+# Construction of HOL in ZFC set theory
+
+import ( SET_MM_AX zfc/set_mm_ax ( ) "" )
+import ( SET_MM zfc/set_mm ( SET_MM_AX ) "" )
+
+kindbind ( class type )
+kindbind ( class val )
+kindbind ( set var )
+
+var ( wff ph ps ch th ta et)
+var ( set x y z x' y' z' x'' y'' z'' w )
+var ( type T T' T'' )
+var ( val A B C D F G G1 G2 G3 P )
+var ( var a b f g v f' )
+
+# This is a somewhat funky definition, but lets us prove the obvious
+# properties using handy theorems in set.mm
+def ( ( fst A ) ( U. ( dom ( {} A ) ) ) )
+thm ( df-fst ( ) ( )
+  ( = ( fst A ) ( U. ( dom ( {} A ) ) ) )
+  ( A {} dom U. eqid ) )
+
+def ( ( snd A ) ( U. ( dom ( `' ( {} A ) ) ) ) )
+thm ( df-snd ( ) ( )
+  ( = ( snd A ) ( U. ( dom ( `' ( {} A ) ) ) ) )
+  ( A {} `' dom U. eqid ) )
+
+# A type is a pair consisting of a set of values, and a "witness"
+# Note: if we want to add a "choice" term, then we probably want a
+# total, well-ordered relation, rather than just a witness.
+def ( ( typeset T ) ( fst T ) )
+thm ( df-typeset ( ) ( ) ( = ( typeset T ) ( fst T ) ) ( T fst eqid ) )
+def ( ( witness T ) ( snd T ) )
+thm ( df-witness ( ) ( ) ( = ( witness T ) ( snd T ) ) ( T snd eqid ) )
+
+# It's not clear that we really gain anything by having the mktype abstraction.
+def ( ( mktype A B ) ( <,> A B ) )
+thm ( df-mktype ( ) ( ) ( = ( mktype A B ) ( <,> A B ) ) ( A B <,> eqid ) )
+thm ( mktypeeq1 ( ) ( ( hyp ( = A B ) ) ) ( = ( mktype A C ) ( mktype B C ) )
+  ( hyp A B C opeq1 ax-mp ) )
+
+def ( ( istype T ) ( E. x ( E. y ( /\ ( = T ( <,> ( cv x ) ( cv y ) ) )
+                                      ( e. ( cv y ) ( cv x ) ) ) ) ) )
+
+thm (df-istype-half ((T x x' y y')) ()
+  (<-> (E. x (E. y (/\ (= T (<,> (cv x) (cv y)))
+                     (e. ( cv y) (cv x)))))
+    ( E. x' ( E. y' ( /\ ( = T ( <,> ( cv x' ) ( cv y' ) ) )
+                                          ( e. ( cv y' ) ( cv x' ) ) ) ) ) )
+  ( x cv x' cv y cv y' cv opeq12 T eqeq2d
+    y cv y' cv x cv x' cv eleq12 ancoms anbi12d
+    cbvex2v
+) )
+
+thm ( df-istype (( T x z y w) (T x' z y' w)) ( )
+  ( <-> ( istype T ) ( E. x' ( E. y' ( /\ ( = T ( <,> ( cv x' ) ( cv y' ) ) )
+                                          ( e. ( cv y' ) ( cv x' ) ) ) ) ) )
+  (x y T z w df-istype-half
+     x' y' T z w df-istype-half
+   bitr4
+))
+
+def ( ( : A T ) ( /\ ( istype T ) ( e. A ( typeset T ) ) ) )
+thm ( df-: ( ) ( )
+  ( <-> ( : A T ) ( /\ ( istype T ) ( e. A ( typeset T ) ) ) )
+  ( T istype A T typeset e. /\ pm4.2 ) )
+thm ( :eq1 ( ) ( )
+  ( -> ( = A B ) ( <-> ( : A T ) ( : B T ) ) )
+  ( A B T typeset eleq1 T istype anbi2d
+    A T df-:
+    B T df-:
+    3bitr4g
+) )
+
+# This is equal to A when A is of type T, otherwise the witness of type T
+def ( ( cast A T )
+  ( if ( e. A ( typeset T ) ) A ( witness T ) ) )
+thm ( df-cast ( ) ( )
+  ( = ( cast A T ) ( if ( e. A ( typeset T ) ) A ( witness T ) ) )
+  ( A T typeset e. A T witness if eqid ) )
+
+def ( ( var v T ) ( cast ( cv v ) T ) )
+thm ( df-var ( ) ( ) ( = ( var v T ) ( cast ( cv v ) T ) )
+  ( v cv T cast eqid ) )
+
+# A case can be made for lambda to be strict in T, i.e. equal to bottom
+# when T is not a valid type.
+def ( ( lambda x T A ) ( {<,>|} x y
+  ( /\ ( : ( cv x ) T ) ( = ( cv y ) A ) ) ) )
+
+thm (df-lambda-half ((T x y y') (A y y')) ()
+  (= ({<,>|} x y (/\ (: (cv x) T) (= (cv y) A)))
+     ({<,>|} x y' (/\ (: (cv x) T) (= (cv y') A))))
+  ( y cv y' cv A eqeq1 x cv T : anbi2d x cbvopab2v
+) )
+thm (df-lambda ((T x y z) (A y z) (T x y' z) (A y')) ( )
+  ( = ( lambda x T A ) ( {<,>|} x y'
+    ( /\ ( : ( cv x ) T ) ( = ( cv y' ) A ) ) ) )
+  (x y T A z df-lambda-half
+     x y' T A z df-lambda-half
+   eqtr4
+))
+
+# An alternate def of function application that takes on the value V
+# outside the domain of the function. We'll probably make this one
+# the default, as it allows more powerful type inferencing.
+def ( ( . F A ) ( if ( /\ ( Fun F ) ( e. A ( dom F ) ) ) ( ` F A ) ( V ) ) )
+thm ( df-app ( ) ( )
+  ( = ( . F A ) ( if ( /\ ( Fun F ) ( e. A ( dom F ) ) ) ( ` F A ) ( V ) ) )
+  ( F Fun A F dom e. /\ F A ` V if eqid ) )
+
+def ( ( false ) ( {/} ) )
+thm ( df-false ( ) ( ) ( = ( false ) ( {/} ) ) ( {/} eqid ) )
+def ( ( true ) ( {} ( {/} ) ) )
+thm ( df-true ( ) ( ) ( = ( true ) ( {} ( {/} ) ) ) ( {/} {} eqid ) )
+def ( ( bool ) ( mktype ( {,} ( false ) ( true ) ) ( false ) ) )
+thm ( df-bool ( ) ( )
+  ( = ( bool ) ( mktype ( {,} ( false ) ( true ) ) ( false ) ) )
+  ( false true {,} false mktype eqid ) )
+
+def ( ( notype ) ( mktype ( {/} ) ( {/} ) ) )
+thm ( df-notype ( ) ( ) ( = ( notype ) ( mktype ( {/} ) ( {/} ) ) )
+  ( {/} {/} mktype eqid ) )
+
+def ( ( $-> T T' )
+  ( if ( /\ ( istype T ) ( istype T' ) )
+    ( mktype ( {|} f ( :--> ( cv f ) ( typeset T ) ( typeset T' ) ) )
+          ( X. ( typeset T ) ( {} ( witness T' ) ) ) )
+    ( notype ) ) )
+
+def ( ( istrue A ) ( = A ( true ) ) )
+thm ( df-istrue ( ) ( ) ( <-> ( istrue A ) ( = A ( true ) ) ) ( A true = pm4.2 ) )
+def ( ( truthterm ph ) ( if ph ( true ) ( false ) ) )
+thm ( df-truthterm ( ) ( ) ( = ( truthterm ph ) ( if ph ( true ) ( false ) ) )
+  ( ph true false if eqid ) )
+
+def ( ( $= T )
+  ( if ( istype T )
+    ( lambda a T ( lambda b T ( truthterm ( = ( cv a ) ( cv b ) ) ) ) )
+    ( V ) ) )
+def ( ( $==> ) ( lambda a ( bool ) ( lambda b ( bool )
+  ( truthterm ( -> ( istrue ( cv a ) ) ( istrue ( cv b ) ) ) ) ) ) )
+
+# The iota operator is much more straightforward than epsilon (Hilbert-style
+# choice).
+def ( ( $iota T ) ( lambda f ( $-> T ( bool ) )
+  ( if ( E!e. x ( typeset T ) ( istrue ( . ( cv f ) ( cv x ) ) ) )
+    ( U. ( {e.|} x ( typeset T ) ( istrue  ( . ( cv f ) ( cv x ) ) ) ) )
+    ( witness T ) ) ) )
+
+# These defs are used to concisely write sequents in a form suitable
+# for manipulation by inference.
+
+def ( ( $|- G A ) ( /\ ( /\ ( : G ( bool ) ) ( : A ( bool ) ) )
+  ( -> ( istrue G ) ( istrue A ) ) ) )
+thm ( df-$|- ( ) ( )
+  ( <-> ( $|- G A ) ( /\ ( /\ ( : G ( bool ) ) ( : A ( bool ) ) )
+    ( -> ( istrue G ) ( istrue A ) ) ) )
+  ( G bool : A bool : /\ G istrue A istrue -> /\ pm4.2
+) )
+
+# "notbottom" is the predicate we're exporting for the purpose of doing
+# type inference
+def ( ( notbottom A ) ( -. ( = A ( V ) ) ) )
+thm ( df-notbottom ( ) ( )
+  ( <-> ( notbottom A ) ( -. ( = A ( V ) ) ) )
+  ( A V = -. pm4.2 ) )
+
+# Here, typeset(bool) is simply a handy example of a value which is not
+# :bool. Now that app is strict, we probably want to get rid of all
+# this special-case junk.
+def ( ( $, G1 G2 )
+  ( if ( /\ ( : G1 ( bool ) ) ( : G2 ( bool ) ) )
+    ( truthterm ( /\ ( istrue G1 ) ( istrue G2 ) ) )
+    ( typeset ( bool ) ) ) )
+thm ( df-$, ( ) ( )
+  ( = ( $, G1 G2 )
+    ( if ( /\ ( : G1 ( bool ) ) ( : G2 ( bool ) ) )
+      ( truthterm ( /\ ( istrue G1 ) ( istrue G2 ) ) )
+      ( typeset ( bool ) ) ) )
+  ( G1 bool : G2 bool : /\ G1 istrue G2 istrue /\ truthterm bool typeset if eqid
+) )
+
+# We choose om (the simple set-theoretic construction of the naturals)
+# for our ind type, but other choices would do
+def ( ( $ind ) ( mktype ( om ) ( {/} ) ) )
+thm ( df-$ind ( ) ( )
+  ( = ( $ind ) ( mktype ( om ) ( {/} ) ) )
+  ( om {/} mktype eqid ) )
+
+def ( ( $0 ) ( {/} ) )
+thm ( df-$0 ( ) ( ) ( = ( $0 ) ( {/} ) ) ( {/} eqid ) )
+
+def ( ( $suc ) ( lambda x ($ind) (suc (var x ($ind)))))
+
+# Now for our thms
+
+# Generic thms that could be moved to set.mm
+
+thm ( ifbii ( ( ph x ) ( ps x ) ( A x ) ( B x ) )
+  ( ( hyp ( <-> ph ps ) ) )
+  ( = ( if ph A B ) ( if ps A B ) )
+  ( hyp ph ps A B ifbi ax-mp ) )
+
+#thm ( ifeq1 ( ) ( )
+#  ( -> ( = A B ) ( = ( if ph A C ) ( if ph B C ) ) )
+#  (   ph A C iftrue B eqeq1d biimpd imp
+#    ph B C iftrue A B = adantr eqtrd exp
+#    ph C A iffalse ph C B iffalse eqtr3d A B = a1d pm2.61i
+#) )
+
+thm ( ifeq1i ( )
+  ( ( hyp ( = A B ) ) )
+  ( = ( if ph A C ) ( if ph B C ) )
+  ( hyp A B ph C ifeq1 ax-mp ) )
+
+#thm ( ifeq2 ( ) ( )
+#  ( -> ( = A B ) ( = ( ded ph C A ) ( ded ph C B ) ) )
+#  ( ph C A iftrue ph C B iftrue eqtr3d A B = a1d
+#      ph A C iffalse B eqeq1d biimpd imp
+#    ph B C iffalse A B = adantr eqtrd exp pm2.61i
+#) )
+
+thm ( ifeq2i ( )
+  ( ( hyp ( = A B ) ) )
+  ( = ( if ph C A ) ( if ph C B ) )
+  ( hyp A B ph C ifeq2 ax-mp ) )
+
+# similar proof as op1sta
+thm ( fstval ( ) ( ( isset.A ( e. A V ) ) ( isset.B ( e. B V ) ) )
+  ( = ( fst ( <,> A B ) ) A )
+  ( A B <,> df-fst
+    A B dmsnop unieqi isset.A unisn eqtr
+    eqtr
+) )
+
+thm ( fsteq ( ) ( )
+  ( -> ( = A B ) ( = ( fst A ) ( fst B ) ) )
+  ( A B sneq A {} B {} dmeq syl unieqd
+    A df-fst B df-fst 3eqtr4g
+) )
+
+thm ( fsteqi ( )
+  ( ( hyp ( = A B ) ) )
+  ( = ( fst A ) ( fst B ) )
+  ( hyp A B fsteq ax-mp ) )
+
+# This proof doesn't use the hypothesis, because our construction
+# of fst exists even if A is a proper class. However, we retain the
+# hypothesis so that the use of this theorem is robust even if we
+# were to switch to an ordered pair strong enough to hold classes.
+thm ( fstex ( ) ( ( hyp ( e. A ( V ) ) ) ) ( e. ( fst A ) ( V ) )
+  ( A df-fst A snex A {} V dmexg ax-mp uniex eqeltr ) )
+
+# similar proof as op2nda
+thm ( sndval ( ) ( ( isset.A ( e. A V ) ) ( isset.B ( e. B V ) ) )
+  ( = ( snd ( <,> A B ) ) B )
+  ( A B <,> df-snd
+      isset.A isset.B cnvsn dmeqi B A dmsnop eqtr
+    unieqi isset.B unisn eqtr
+    eqtr
+) )
+
+thm ( sndeq ( ) ( )
+  ( -> ( = A B ) ( = ( snd A ) ( snd B ) ) )
+  ( A B sneq A {} B {} cnveq syl A {} `' B {} `' dmeq syl unieqd
+    A df-snd B df-snd 3eqtr4g
+) )
+
+thm ( sndeqi ( )
+  ( ( hyp ( = A B ) ) )
+  ( = ( snd A ) ( snd B ) )
+  ( hyp A B sndeq ax-mp ) )
+
+# See comments for fstex
+thm ( sndex ( ) ( ( hyp ( e. A ( V ) ) ) ) ( e. ( snd A ) ( V ) )
+  ( A df-snd A snex A {} V cnvexg ax-mp A {} `' V dmexg ax-mp uniex eqeltr ) )
+
+# Move "at most one" quantifier in and out of substitution. Note how
+# much shorter the proof would be if sbi, sbal, sbex were inferences.
+thm ( sbmo ( ( w z x y ) ( ph y ) ) ( )
+  ( <-> ( [/] ( cv z ) w ( E* x ph ) ) ( E* x ( [/] ( cv z ) w ph ) ) )
+  ( x cv y cv = w ax-17 z sbf bicomi z cv w ph [/] imbi2i
+    z w ph x cv y cv = sbim bitr4 x bial
+    z w x ph x cv y cv = -> sbal bitr4 y biex
+    z w y x ph x cv y cv = -> A. sbex bitr4
+    z cv w ph [/] y ax-17 x mo2
+    ph y ax-17 x mo2 z w bisb  3bitr4 bicomi
+) )
+
+# Truth inside and outside of a substitution are equivalent.
+thm ( sbtru ( ) ( ( hyp ph ) )
+  ( [/] ( cv x ) y ph )
+  ( hyp hyp y hbth x sbf mpbir ) )
+
+thm ( sbctru ( ( A y ) ( x y ) ( ph y ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ph ) )
+  ( [/] A x ph )
+  ( hyp.1
+    hyp.2 y x sbtru
+    y cv A x ph dfsbcq mpbii
+    vtocle
+) )
+
+# Substitution theorems arguably missing from set.mm
+
+thm ( sbcf1 ( ( A y ) ( x y ) ( ph y ) )
+  ( ( hyp.1 ( -> ph ( A. x ph ) ) )
+    ( hyp.2 ( e. A ( V ) ) ) )
+  ( <-> ph ( [/] A x ph ) )
+  ( hyp.2
+
+    y cv A x ph dfsbcq
+    hyp.1 y sbf syl5bbr
+    vtocle
+) )
+
+thm ( sbcb ( ( A y ) ( x y ) ( ph y ) ( ps y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( <-> ph ps ) ) ( <-> ( [/] A x ph ) ( [/] A x ps ) ) )
+  ( hyp
+
+    y cv A x ph ps <-> dfsbcq
+    y x ph ps sbbi syl5bbr
+    y cv A x ph dfsbcq y cv A x ps dfsbcq bibi12d bitr3d
+    vtocle
+) )
+
+thm ( sbci2 ( ( A y ) ( x y ) ( ph y ) ( ps y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( -> ph ps ) ) ( -> ( [/] A x ph ) ( [/] A x ps ) ) )
+  ( hyp
+
+    y cv A x ph ps -> dfsbcq
+    y x ph ps sbim syl5bbr
+    y cv A x ph dfsbcq y cv A x ps dfsbcq imbi12d bitr3d
+    vtocle
+) )
+
+thm ( sbca ( ( A y ) ( x y ) ( ph y ) ( ps y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( /\ ph ps ) ) ( /\ ( [/] A x ph ) ( [/] A x ps ) ) )
+  ( hyp
+
+    y cv A x ph ps /\ dfsbcq
+    y x ph ps sban syl5bbr
+    y cv A x ph dfsbcq y cv A x ps dfsbcq anbi12d bitr3d
+    vtocle
+) )
+
+# "sbco" is taken
+thm ( sbcor2 ( ( A y ) ( x y ) ( ph y ) ( ps y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( \/ ph ps ) ) ( \/ ( [/] A x ph ) ( [/] A x ps ) ) )
+  ( hyp
+
+    y cv A x ph ps \/ dfsbcq
+    y x ph ps sbor syl5bbr
+    y cv A x ph dfsbcq y cv A x ps dfsbcq orbi12d bitr3d
+    vtocle
+) )
+
+thm ( sbcn2 ( ( A y ) ( x y ) ( ph y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( -. ph ) ) ( -. ( [/] A x ph ) ) )
+  ( hyp
+
+    y cv A x ph -. dfsbcq
+    y x ph sbn syl5bbr
+    y cv A x ph dfsbcq negbid bitr3d
+    vtocle
+) )
+
+thm ( sbcal2 ( ( x y z ) ( A y z ) ( ph z ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( A. y ph ) ) ( A. y ( [/] A x ph ) ) )
+  ( hyp
+
+    z cv A x y ph A. dfsbcq
+    z x y ph sbal syl5bbr
+    z cv A = y ax-17 z cv A x ph dfsbcq biald bitr3d
+    vtocle
+) )
+
+thm ( sbcex2 ( ( x y z ) ( A y z ) ( ph z ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( E. y ph ) ) ( E. y ( [/] A x ph ) ) )
+  ( hyp
+
+    z cv A x y ph E. dfsbcq
+    z x y ph sbex syl5bbr
+    z cv A = y ax-17 z cv A x ph dfsbcq biexd bitr3d
+    vtocle
+) )
+
+thm ( bisbc ( )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( <-> ph ps ) ) )
+  ( <-> ( [/] A x ph ) ( [/] A x ps ) )
+  ( hyp.1 hyp.2 x sbctru
+    hyp.1 x ph ps sbcb mpbi
+) )
+
+thm ( bisbcd ( )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( -> ph ( A. x ph ) ) )
+    ( hyp.3 ( -> ph ( <-> ps ch ) ) ) )
+  ( -> ph ( <-> ( [/] A x ps ) ( [/] A x ch ) ) )
+  ( hyp.1 hyp.3 x sbctru
+    hyp.1 x ph ps ch <-> sbci2 mpbi
+    hyp.2 hyp.1 sbcf1 bicomi
+    hyp.1 x ps ch sbcb 3imtr3
+) )
+
+thm ( hbsbc1 ( ( x y ) ( A y ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( -> ph ( A. y ph ) ) ) )
+  ( -> ( [/] A x ph ) ( A. y ( [/] A x ph ) ) )
+  ( hyp.1 hyp.2 x sbctru
+    hyp.1 x ph y ph A. sbci2 mpbi
+    hyp.1 x y ph sbcal2 sylib
+) )
+
+# If two wff's are equivalent and x is not free in one, it is not free
+# in the other. Useful for substituting definitions into fv hyps.
+thm ( bihb ( )
+  ( ( hyp.1 ( -> ph ( A. x ph ) ) )
+    ( hyp.2 ( <-> ph ps ) ) )
+  ( -> ps ( A. x ps ) )
+  ( hyp.1 hyp.2 hyp.2 x bial imbi12i mpbi ) )
+
+# A small theory of "not free in class"
+def ( ( Nfi x A ) ( A. y ( <-> ( e. ( cv y ) A ) ( A. x ( e. ( cv y ) A ) ) ) ) )
+thm (df-Nfi-half ((x y y') (A y y')) ( )
+  (<-> (A. y (<-> (e. (cv y) A) (A. x (e. (cv y) A ))))
+    (A. y' (<-> (e. (cv y') A) (A. x (e. (cv y') A )))))
+  ( y cv y' cv A eleq1
+    y cv y' cv = x ax-17 y cv y' cv A eleq1 biald bibi12d cbvalv
+) )
+
+thm ( df-Nfi ((x y z) (x y' z) (A y z) (A y')) ( )
+  ( <-> ( Nfi x A ) ( A. y' ( <-> ( e. ( cv y' ) A ) ( A. x ( e. ( cv y' ) A ) ) ) ) )
+  (y A x z df-Nfi-half
+     y' A x z df-Nfi-half
+   bitr4
+) )
+
+thm ( dvNfi ( ( A x y ) ) ( )
+  ( Nfi x A )
+  ( y cv A e. x ax-17 19.3r y ax-gen
+    x A y df-Nfi mpbir
+) )
+
+thm ( Nfiel1 ( ( x y ) ( A y ) ) ( ( hyp ( Nfi x A ) ) )
+  ( -> ( e. ( cv y ) A ) ( A. x ( e. ( cv y ) A ) ) )
+  ( hyp x A y df-Nfi mpbi a4i biimp
+) )
+
+# A little theory of explicit substitution. Likely, this should go into
+# its own module.
+
+# A note: a similar construction is used in sbab
+
+# def ( ( [:=] x A B ) ( {|} y ( [/] A x ( e. ( cv y ) B ) ) ) )
+def ( ( [:=] x A B ) ( {|} y ( e. A ( {|} x ( e. ( cv y ) B ) ) ) ) )
+thm (df-[:=]-half ( ( x y y' ) ( A y y' ) ( B y y' ) ) ( )
+  (= ({|} y (e. A ({|} x (e. (cv y) B))))
+     ({|} y' (e. A ({|} x (e. (cv y') B)))))
+  ( y cv y' cv = x ax-17 y cv y' cv B eleq1 biabd A eleq2d cbvabv
+) )
+thm (df-[:=] ((x y z) (A y z) (B y z) (x y' z) (A y') (B y')) ( )
+  ( = ( [:=] x A B ) ( {|} y' ( e. A ( {|} x ( e. ( cv y' ) B ) ) ) ) )
+  ( y A x B z df-[:=]-half
+      y' A x B z df-[:=]-half
+    eqtr4
+) )
+
+thm ( df[:=]2 ( ( x y ) ( A y ) ( B y ) ) ( )
+  ( = ( [:=] x A B ) ( {|} y ( [/] A x ( e. ( cv y ) B ) ) ) )
+  ( x A B y df-[:=]
+    A x y cv B e. df-sbc y biabi eqtr4
+) )
+
+# inference from implicit substitution to explicit substitution
+thm ( imps2exps ( ( A y ) ( B x y ) ( C x y ) )
+  ( ( hyp.1 ( e. B ( V ) ) )
+    ( hyp.2 ( -> ( = ( cv x ) B ) ( = A C ) ) ) )
+  ( = ( [:=] x B A ) C )
+  ( x B A y df-[:=]
+    hyp.1 hyp.2 A C y cv eleq2 syl elab y biabi
+    y C abid2 eqtr
+    eqtr
+) )
+
+# An inference from implicit substitution to explicit substitution, with
+# "x not free in C" as a hyp rather than a dv constraint. Note that hyp.3
+# is equivalent to ( Nfi x C ) if we choose to use that def.
+thm ( imps2expsf ( ( A y ) ( B x y ) ( C y ) )
+  ( ( hyp.1 ( e. B ( V ) ) )
+    ( hyp.2 ( -> ( = ( cv x ) B ) ( = A C ) ) )
+    ( hyp.3 ( -> ( e. ( cv y ) C ) ( A. x ( e. ( cv y ) C ) ) ) ) )
+  ( = ( [:=] x B A ) C )
+  ( x B A y df-[:=]
+    hyp.3
+    hyp.1 hyp.2 A C y cv eleq2 syl elabf y biabi
+    y C abid2 eqtr
+    eqtr
+) )
+
+# inference from explicit substitution to implicit substitution
+thm ( exps2imps ( ( A y ) ( B x y ) ( C x y ) )
+  ( ( hyp ( = ( [:=] x B A ) C ) ) )
+  ( -> ( = ( cv x ) B ) ( = A C ) )
+  ( x B y cv A e. ceqex
+
+    hyp x B A y df-[:=] eqtr3
+    B x y cv A e. clelab y biabi eqtr
+    eqabi
+
+    syl6bbr eqrd
+) )
+
+thm ( expsvar ( ( x y z ) ( A y z ) ) ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A ( cv x ) ) A )
+  ( x A x cv y df-[:=]
+    x cv z cv y cv eleq2 cbvabv A eleq2i
+    hyp z cv A y cv eleq2 elab bitr y biabi
+    y A abid2 eqtr
+    eqtr
+) )
+
+# Substitution of a variable not free does not affect a term.
+thm ( expsf ( ( x y z ) ( A y z ) ( B x y z ) ) ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A B ) B )
+  ( x A B y df-[:=]
+    y cv B e. pm4.2 x cv z cv = a1i cbvabv A eleq2i
+    hyp y cv B e. pm4.2 z cv A = a1i elab bitr y biabi
+    y B abid2 eqtr
+    eqtr
+) )
+
+# Substitution of a variable not free does not affect a term, explicit
+# "not free in" hypothesis version.
+thm ( expsff ( ( x y z ) ( A y z ) ( B y z ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( -> ( e. ( cv y ) B ) ( A. x ( e. ( cv y ) B ) ) ) ) )
+  ( = ( [:=] x A B ) B )
+  ( x A B y df-[:=]
+    y cv B e. z ax-17
+    hyp.2
+    y cv B e. pm4.2 x cv z cv = a1i cbvab A eleq2i
+    hyp.1 y cv B e. pm4.2 z cv A = a1i elab bitr y biabi
+    y B abid2 eqtr
+    eqtr
+) )
+
+# An equivalence relating explicit substitution of terms with explicit
+# substitution of wff's. Note that this proof uses df-sbc, which goes
+# against the recommendations in set.mm to use a weaker defn.
+# Probably easier to prove in terms of expssb12
+thm ( expssb1 ( ( A y ) ( B y ) ( C x y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( = ( [:=] x A B ) C ) ( [/] A x ( = B C ) ) )
+  ( C x A B [:=] eqcom
+    x A B y df-[:=] C eqeq2i bitr3
+    C y A x y cv B e. {|} e. eqab bitr
+    
+    y cv C e. x ax-17 hyp sbcf1
+    A x y cv B e. df-sbc bicomi bibi12i
+    hyp x y cv C e. y cv B e. sbcb bitr4 y bial bitr
+    hyp x y y cv C e. y cv B e. <-> sbcal2 bitr4
+    hyp C B y dfcleq C B eqcom bitr3 x bisbc bitr
+) )
+
+thm ( expssb12 ( ( A y ) ( B y ) ( C y ) ( x y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( = ( [:=] x A B ) ( [:=] x A C ) ) ( [/] A x ( = B C ) ) )
+  ( x A B [:=] x A C [:=] y dfcleq
+    x A B y df[:=]2 y cv eleq2i
+    y A x y cv B e. [/] abid bitr
+    x A C y df[:=]2 y cv eleq2i
+    y A x y cv C e. [/] abid bitr bibi12i
+    hyp x y cv B e. y cv C e. sbcb bitr4 y bial
+    hyp x y y cv B e. y cv C e. <-> sbcal2 bitr4
+    hyp B C y dfcleq x bisbc bitr4 bitr
+) )
+
+thm ( expssb2 ( ( C x ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( = C ( [:=] x A B ) ) ( [/] A x ( = C B ) ) )
+  ( hyp x C expsf x A B [:=] eqeq1i
+    hyp x C B expssb12 bitr3
+) )
+
+thm ( expssbel ( ( A y ) ( B y ) ( C y ) ( x y ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( e. ( [:=] x A B ) ( [:=] x A C ) ) ( [/] A x ( e. B C ) ) )
+  ( x A C y df[:=]2 x A B [:=] eleq2i
+    x A B [:=] y A x y cv C e. [/] clelab bitr
+    hyp y cv x B expssb2 A x y cv C e. [/] anbi1i
+    hyp x y cv B = y cv C e. sbca bitr4 y biex bitr
+    hyp x y y cv B = y cv C e. /\ sbcex2 bitr4
+    hyp B C y df-clel x bisbc bitr4
+) )
+
+thm ( expseq1 ( ( A y ) ( B y ) ( C y ) ( x y ) ) ( )
+  ( -> ( = A B ) ( = ( [:=] x A C ) ( [:=] x B C ) ) )
+  ( A B = y ax-17 A B x y cv C e. dfsbcq biabd
+    x A C y df[:=]2
+    x B C y df[:=]2 3eqtr4g
+) )
+
+thm ( expseq2i ( )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( = B C ) ) )
+  ( = ( [:=] x A B ) ( [:=] x A C ) )
+  ( hyp.1 hyp.2 x sbctru
+    hyp.1 x B C expssb12 mpbir
+) )
+
+thm ( expseq2d ( ( ph x ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( -> ph ( = B C ) ) ) )
+  ( -> ph ( = ( [:=] x A B ) ( [:=] x A C ) ) )
+  ( hyp.1 hyp.2 x sbctru hyp.1 x ph B C = sbci2 mpbi
+    ph x ax-17 hyp.1 sbcf1
+    hyp.1 x B C expssb12 3imtr4
+) )
+
+thm ( expsex ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( e. B ( V ) ) ) )
+  ( e. ( [:=] x A B ) ( V ) )
+  ( 
+    hyp.1 hyp.2 y isseti y cv B eqcom y biex mpbi x sbctru
+    hyp.1 x y B y cv = sbcex2 mpbi
+
+    hyp.1 x B y cv expssb1 y biex mpbir
+    x A B [:=] y cv eqcom y biex mpbi issetri
+) )
+
+thm ( expsex1 ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp ( e. B ( V ) ) ) )
+  ( -> ( e. A ( V ) ) ( e. ( [:=] x A B ) ( V ) ) )
+  ( A A V e. A {/} if x B expseq1 V eleq1d
+    A A V e. A {/} if V eleq1
+    {/} A V e. A {/} if V eleq1
+    0ex elimhyp
+    hyp x expsex dedth
+) )
+
+# A thm from an older set.mm, seems to be subsumed by hbsbc now
+thm ( hbsbcl ( ( x y ) ( x z ) ( y z ) ( A y ) ( A z ) ( ph z ) )
+  ( ( hbsbcl.1 ( -> ( e. ( cv y ) A ) ( A. x ( e. ( cv y ) A ) ) ) ) )
+  ( -> ( e. A ( V ) ) ( -> ( [/] A x ph ) ( A. x ( [/] A x ph ) ) ) )
+  ( z cv A x ph dfsbcq y cv z cv e. x ax-17 hbsbcl.1 hbeq z cv A x ph dfsbcq biald z x ph hbs1 syl5bi sylbird V vtocleg ) )
+
+# If x is not free in A, it is not free in [ x := A ] B.
+thm ( hbexpsc ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp ( -> ( e. ( cv y ) A ) ( A. x ( e. ( cv y ) A ) ) ) ) )
+  ( -> ( e. A ( V ) )
+       ( -> ( e. ( cv y ) ( [:=] x A B ) ) ( A. x ( e. ( cv y ) ( [:=] x A B ) ) ) ) )
+  ( hyp y cv B e. hbsbcl
+      x A B y df[:=]2 y cv eleq2i
+    y A x y cv B e. [/] abid bitr
+      x A B y df[:=]2 y cv eleq2i
+    y A x y cv B e. [/] abid bitr x bial imbi12i
+    sylibr
+) )
+
+# Same as expsfv, but with slightly weaker existence assumptions
+thm ( expsfv2 ( ( x y z ) ( A y z ) ( B y z ) ( C y z ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( e. ( [:=] x A B ) ( V ) ) )
+    ( hyp.3 ( e. ( [:=] x A C ) ( V ) ) ) )
+  ( = ( [:=] x A ( ` B C ) ) ( ` ( [:=] x A B ) ( [:=] x A C ) ) )
+  ( hyp.3
+    hyp.2
+
+    hyp.1
+    z cv C y cv fveq2
+    y cv B C fveq1 sylan9eqr x sbctru
+    hyp.1 x y cv B = z cv C = /\ y cv z cv ` B C ` = sbci2 mpbi
+    hyp.1 x y cv B = z cv C = sbca
+    hyp.1 y cv x B expssb2
+    hyp.1 z cv x C expssb2 anbi12i bitr4 bicomi
+    hyp.1 y cv z cv ` x B C ` expssb2 3imtr4
+    z cv x A C [:=] y cv fveq2
+    y cv x A B [:=] x A C [:=] fveq1 sylan9eqr eqtr3d
+    exp vtocle vtocle
+) )
+
+thm ( expsfv ( ( x y z ) ( A y z ) ( B y z ) ( C y z ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( e. B ( V ) ) )
+    ( hyp.3 ( e. C ( V ) ) ) )
+  ( = ( [:=] x A ( ` B C ) ) ( ` ( [:=] x A B ) ( [:=] x A C ) ) )
+  ( hyp.1
+    hyp.1 hyp.2 x expsex
+    hyp.1 hyp.3 x expsex
+    expsfv2
+) )
+
+# Substitution moves through abstraction when the abstraction variable is
+# distinct from both the substitution variable and its replacement.
+thm ( expsab ( ( x y ) ( A y z ) ( x z ) ( ph z ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A ( {|} y ph ) ) ( {|} y ( [/] A x ph ) ) )
+  ( x A y ph {|} z df[:=]2
+
+    hyp z y ph hbab1 x hbsbc1
+    A x ph [/] z ax-17
+
+    hyp z cv y cv = x ax-17
+    z cv y cv y ph {|} eleq1
+    y ph abid syl6bb bisbcd
+    cbvab eqtr
+) )
+
+thm ( expsopab ( ( A w y z ) ( w x y z ) ( w ph ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A ( {<,>|} y z ph ) ) ( {<,>|} y z ( [/] A x ph ) ) )
+  ( hyp y z ph w df-opab x expseq2i
+    hyp x w y z w cv y cv z cv <,> = ph /\ E. E. expsab eqtr
+
+    hyp x w cv y cv z cv <,> = ph sbca
+    w cv y cv z cv <,> = x ax-17 hyp sbcf1 A x ph [/] anbi1i bitr4 z biex bicomi
+    hyp x z w cv y cv z cv <,> = ph /\ sbcex2 bitr4 y biex
+
+    hyp x y z w cv y cv z cv <,> = ph /\ E. sbcex2 bitr4 w biabi eqtr4
+    y z A x ph [/] w df-opab eqtr4
+) )
+
+thm ( expsded ( ( x y z ) ( ph y z ) ( y z A ) ( y z B ) ( y z C ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A ( if ph B C ) ) ( if ( [/] A x ph ) ( [:=] x A B ) ( [:=] x A C ) ) )
+  ( hyp ph B C y df-if x expseq2i
+    x A y y cv B e. ph /\ y cv C e. ph -. /\ \/ {|} z df[:=]2 eqtr
+    hyp x z cv y y cv B e. ph /\ y cv C e. ph -. /\ \/ {|} expssbel
+    hyp x z cv expsf
+    hyp x y y cv B e. ph /\ y cv C e. ph -. /\ \/ expsab
+      hyp x y cv B e. ph /\ y cv C e. ph -. /\ sbcor2
+        hyp x y cv B e. ph sbca
+          hyp x y cv B expssbel
+            hyp x y cv expsf x A B [:=] eleq1i
+          bitr3 A x ph [/] anbi1i
+        bitr
+        hyp x y cv C e. ph -. sbca
+          hyp x y cv C expssbel
+            hyp x y cv expsf x A C [:=] eleq1i
+          bitr3
+          hyp x ph sbcn2
+        anbi12i bitr
+      orbi12i bitr y biabi eqtr
+    eleq12i bitr3 z biabi eqtr
+    z y y cv x A B [:=] e. A x ph [/] /\ y cv x A C [:=] e. A x ph [/] -. /\ \/ {|} abid2 eqtr
+    A x ph [/] x A B [:=] x A C [:=] y df-if eqtr4
+) )
+
+# A general inference for substitution through unary operators. Typically,
+# you'd instantiate hyp1 and hyp2 with the equality theorem for the
+# operator.
+thm ( expsoper ( ( x y ) ( A y ) ( B y ) ( C x ) ( D y ) ( E y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. ( [:=] x A B ) ( V ) ) )
+    ( hyp1 ( -> ( = ( cv y ) B ) ( = C D ) ) )
+    ( hyp2 ( -> ( = ( cv y ) ( [:=] x A B ) ) ( = C E ) ) ) )
+  ( = ( [:=] x A D ) E )
+  ( hyp.B
+
+        hyp.A y cv x B expssb2
+        hyp.A hyp1 x sbctru
+        hyp.A x y cv B = C D = sbci2 mpbi sylbi
+      hyp.A C x D expssb2 sylibr
+      hyp2
+    eqtr3d vtocle
+) )
+
+thm ( expsop1 ( ( C x ) ( x y ) ( A y ) ( B y ) ( C y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. B ( V ) ) )
+    ( hyp.C ( e. C ( V ) ) ) )
+  ( = ( [:=] x A ( <,> B C ) ) ( <,> ( [:=] x A B ) C ) )
+  ( hyp.A
+    hyp.A hyp.B x expsex
+    y visset hyp.C hyp.C B opth C eqid mpbiran2 biimpr
+    y visset hyp.C hyp.C x A B [:=] opth C eqid mpbiran2 biimpr
+    expsoper
+) )
+
+# Deduction from implicit substitution to explicit substitution.
+thm ( imps2expsd ( ( A y ) ( B x y ) ( C x y ) ( ph x y ) )
+  ( ( hyp.1 ( e. B ( V ) ) )
+    ( hyp.2 ( -> ph ( -> ( = ( cv x ) B ) ( = A C ) ) ) ) )
+  ( -> ph ( = ( [:=] x B A ) C ) )
+  ( ph x ax-17 hyp.1 sbcf1
+    x B A [:=] B x ph [/] x B A [:=] x B C [:=] if C eqeq2
+
+    hyp.1
+      ph A C iftrue eqcomd x cv B = adantr
+      hyp.2 imp eqtr3d exp
+      ph A C iffalse x cv B = a1d pm2.61i
+    imps2exps
+    hyp.1 x ph A C expsded eqtr3
+
+    dedth sylbi eqcomd
+) )
+
+# A form of the law of concretion, expressed using explicit substitution
+thm ( sbcop1 ( ( x y z ) ( A z ) ( ph z ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ph ) ( e. ( <,> A ( cv y ) ) ( {<,>|} x y ph ) ) )
+  ( hyp x y ph opabid bicomi x bisbc
+    hyp x x cv y cv <,> x y ph {<,>|} expssbel bitr4
+        hyp x visset y visset x expsop1
+      hyp x expsvar x A x cv [:=] A y cv opeq1 ax-mp eqtr
+      hyp z x y ph hbopab1 expsff
+    eleq12i bitr
+) )
+
+
+
+# An alternate def of istype that may be handier in proofs.
+# Note something really odd - it's equivalent to T e. `' E. If we put the
+# witness first and the set second, the defn would be equivalent to T e. E.
+thm ( dfistype2 ( ( T x y ) ) ( )
+  ( <-> ( istype T ) ( e. T ( {<,>|} x y ( e. ( cv y ) ( cv x ) ) ) ) )
+  ( T x y df-istype T x y y cv x cv e. elopab bitr4 ) )
+
+thm ( typeex ( ( T x y ) ) ( ( hyp ( istype T ) ) )
+  ( e. T ( V ) )
+  ( hyp T x y dfistype2 mpbi elisseti ) )
+
+
+thm ( typeseteq ( ) ( ) 
+  ( -> ( = A B ) ( = ( typeset A ) ( typeset B ) ) )
+  ( A B fsteq A df-typeset B df-typeset 3eqtr4g ) )
+
+thm ( typeseteqi ( ) ( ( hyp ( = A B ) ) )
+  ( = ( typeset A ) ( typeset B ) )
+  ( hyp A B typeseteq ax-mp ) )
+
+thm ( typesetex ( ) ( ( hyp ( istype T ) ) )
+  ( e. ( typeset T ) ( V ) )
+  ( T df-typeset hyp typeex fstex eqeltr ) )
+
+thm ( typesetval1 ( )
+  ( ( isset.A ( e. A V ) )
+    ( isset.B ( e. B V ) ) )
+  ( = ( typeset ( mktype A B ) ) A )
+  ( A B df-mktype typeseteqi
+    A B <,> df-typeset eqtr
+    isset.A isset.B fstval eqtr    
+) )
+
+# I'd like to remove the isset hyps, which can be done using the
+# "peculiar" properties of the ordered pair construction.
+thm ( typesetval ( )
+  ( ( isset.A ( e. A V ) )
+    ( isset.B ( e. B V ) )
+    ( hyp ( istype ( mktype A B ) ) ) )
+  ( = ( typeset ( mktype A B ) ) A )
+  ( A B df-mktype typeseteqi
+    A B <,> df-typeset eqtr
+    isset.A isset.B fstval eqtr    
+) )
+
+thm ( witnesseq ( ) ( ) 
+  ( -> ( = A B ) ( = ( witness A ) ( witness B ) ) )
+  ( A B sndeq A df-witness B df-witness 3eqtr4g ) )
+
+thm ( witnesseqi ( ) ( ( hyp ( = A B ) ) )
+  ( = ( witness A ) ( witness B ) )
+  ( hyp A B witnesseq ax-mp ) )
+
+thm ( witnessex ( ) ( ( hyp ( istype T ) ) )
+  ( e. ( witness T ) ( V ) )
+  ( T df-witness hyp typeex sndex eqeltr ) )
+
+thm ( witnessval1 ( )
+  ( ( isset.A ( e. A V ) )
+    ( isset.B ( e. B V ) ) )
+  ( = ( witness ( mktype A B ) ) B )
+  ( A B df-mktype witnesseqi
+    A B <,> df-witness eqtr
+    isset.A isset.B sndval eqtr    
+) )
+
+thm ( istypeeq ( ( A x y ) ( B x y ) ) ( )
+  ( -> ( = A B ) ( <-> ( istype A ) ( istype B ) ) )
+  ( A B x y y cv x cv e. {<,>|} eleq1
+    A x y dfistype2
+    B x y dfistype2
+    3bitr4g
+) )
+
+thm ( istypeeqi ( ) ( ( hyp ( = A B ) ) )
+  ( <-> ( istype A ) ( istype B ) )
+  ( hyp A B istypeeq ax-mp
+) )
+
+thm ( mktype1 ( ( T x y ) ) ( ( hyp ( istype T ) ) )
+  ( = ( mktype ( typeset T ) ( witness T ) ) T )
+  ( hyp T x y df-istype mpbi
+    T x cv y cv <,> fsteq
+    x visset y visset fstval syl6eq
+    T x cv y cv <,> sndeq
+    x visset y visset sndval syl6eq jca
+    T fst x cv T snd y cv opeq12 syl
+    T x cv y cv <,> = id eqtr4d
+    y cv x cv e. adantr
+    x y 19.23aivv ax-mp
+) )
+# Note: missing a bunch of definitional steps here
+
+thm ( witness1 ( ( T x y ) ) ( ( hyp ( istype T ) ) )
+  ( e. ( witness T ) ( typeset T ) )
+  ( hyp T x y df-istype mpbi
+    T x cv y cv <,> sndeq
+    x visset y visset sndval syl6eq
+    T x cv y cv <,> fsteq
+    x visset y visset fstval syl6eq jca
+    T snd y cv T fst x cv eleq12 syl biimpar
+    x y 19.23aivv ax-mp
+) )
+
+thm ( witness2 ( ) ( ( hyp ( istype T ) ) )
+  ( : ( witness T ) T )
+  ( hyp hyp witness1 pm3.2i T witness T df-: mpbir
+) )
+
+thm ( mktypethm ( ( A x y ) ( B x y ) )
+  ( ( hyp.1 ( e. B A ) )
+    ( hyp.2 ( e. A ( V ) ) ) )
+  ( istype ( mktype A B ) )
+  ( hyp.1
+    A B <,> x y dfistype2
+      hyp.2
+      hyp.1 elisseti
+      x cv A y cv eleq2
+      y cv B A eleq1
+    opelopab bitr mpbir
+    A B df-mktype istypeeqi mpbir
+) )
+
+thm ( boolistype ( ) ( ) ( istype ( bool ) )
+  ( df-false 0ex eqeltr true pri1
+    false true prex
+    mktypethm
+    df-bool istypeeqi mpbir
+) )
+
+thm (indistype () () (istype ($ind))
+  (  peano1 omex mktypethm
+    df-$ind istypeeqi mpbir
+))
+
+thm ( df-arrow-half ((T f g) (T' f g)) ( )
+  (=
+    ( if ( /\ ( istype T ) ( istype T' ) )
+      ( mktype ( {|} f ( :--> ( cv f ) ( typeset T ) ( typeset T' ) ) )
+     	     ( X. ( typeset T ) ( {} ( witness T' ) ) ) )
+      ( notype ) )
+    ( if ( /\ ( istype T ) ( istype T' ) )
+      ( mktype ( {|} g ( :--> ( cv g ) ( typeset T ) ( typeset T' ) ) )
+     	     ( X. ( typeset T ) ( {} ( witness T' ) ) ) )
+      ( notype ) ) )
+ ( f cv g cv T fst T' fst feq1 cbvabv T typeset T' snd {} X. mktypeeq1
+   T istype T' istype /\ notype ifeq1i
+) )
+thm ( df-arrow ((T f f') (T' f f') (T g f') (T' g f')) ()
+  ( = ( $-> T T' )
+    ( if ( /\ ( istype T ) ( istype T' ) )
+      ( mktype ( {|} g ( :--> ( cv g ) ( typeset T ) ( typeset T' ) ) )
+     	     ( X. ( typeset T ) ( {} ( witness T' ) ) ) )
+      ( notype ) ) )
+  (T T' f f' df-arrow-half
+     T T' g f' df-arrow-half
+   eqtr4
+) )
+
+thm ( arrow1 ( ( T g ) ( T' g ) )
+  ( ( hyp.1 ( istype T ) )
+    ( hyp.2 ( istype T' ) ) )
+  ( = ( $-> T T' )
+     ( mktype ( {|} g ( :--> ( cv g ) ( typeset T ) ( typeset T' ) ) )
+     	     ( X. ( typeset T ) ( {} ( witness T' ) ) ) ) )
+  ( T T' g df-arrow
+    hyp.1 hyp.2
+      T istype T' istype /\ g g cv T typeset T' typeset :--> {|}
+    T typeset T' witness {} X. mktype notype iftrue mp2an eqtr
+) )
+
+thm ( notypenotatype ( ( x y ) ) ( )
+  ( -. ( istype ( notype ) ) )
+  ( {/} noel
+
+    df-notype istypeeqi {/} {/} mktype x y dfistype2 bitr
+    {/} {/} df-mktype x y y cv x cv e. {<,>|} eleq1i bitr
+      0ex 0ex
+      x cv {/} y cv eleq2
+      y cv {/} {/} eleq1
+    opelopab bitr mtbir
+) )
+
+# with arrowistype, below, in theorem form, the implication can be
+# strenghtened to a biconditional.
+thm ( arrow2 ( ( T g ) ( T' g ) ) ( )
+  ( -> ( istype ( $-> T T' ) ) ( /\ ( istype T ) ( istype T' ) ) )
+  ( notypenotatype
+    T istype T' istype /\
+      g g cv T typeset T' typeset :--> {|}
+    T typeset T' witness {} X. mktype notype iffalse
+    T T' g df-arrow syl5req
+    notype T T' $-> istypeeq syl mtbii a3i
+) )
+
+thm ( arrow2i1 ( )
+  ( ( hyp ( istype ( $-> T T' ) ) ) )
+  ( istype T )
+  ( hyp T T' arrow2 ax-mp pm3.26i ) )
+
+thm ( arrow2i2 ( )
+  ( ( hyp ( istype ( $-> T T' ) ) ) )
+  ( istype T' )
+  ( hyp T T' arrow2 ax-mp pm3.27i ) )
+
+# Power set axiom, expressed with powerset in abstraction notation
+thm ( arrowistypelem1 ( ( x A ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( e. ( {|} x ( C_ ( cv x ) A ) ) ( V ) )
+  ( A x df-pw hyp pwex eqeltrr ) )  
+
+# The set of all functions from set A to set B is a set; incorporated into
+# new set.mm as mapex
+thm ( arrowistypelem2 ( ( A f ) ( B f ) )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.2 ( e. B ( V ) ) ) )
+  ( e. ( {|} f ( :--> ( cv f ) A B ) ) ( V ) )
+  ( hyp.1 hyp.2 xpex f arrowistypelem1
+    f cv A B fssxp 
+    f ss2abi
+    ssexi
+) )
+
+thm ( arrowistype ( ( T f ) ( T' f ) )
+  ( ( hyp.1 ( istype T ) )
+    ( hyp.2 ( istype T' ) ) )
+  ( istype ( $-> T T' ) )
+  ( hyp.2 witnessex T typeset fconst
+    hyp.2 witness1 T' witness T' typeset snssi ax-mp
+    T typeset T' witness {} X. T typeset T' witness {} T' typeset fss
+    mp2an 
+    hyp.1 typesetex T' witness snex xpex
+    f cv T typeset T' witness {} X. T typeset T' typeset feq1 elab mpbir
+    hyp.1 typesetex hyp.2 typesetex f arrowistypelem2 mktypethm
+    hyp.1 hyp.2 f arrow1 istypeeqi mpbir
+) )
+
+
+thm ( :eq1i ( ) ( ( hyp ( = A B ) ) )
+  ( <-> ( : A T ) ( : B T ) )
+  ( hyp T typeset eleq1i T istype anbi2i
+    A T df-:
+    B T df-:
+    3bitr4
+) )
+
+thm ( typeel ( ) ( ( hyp ( : A T ) ) )
+  ( e. A ( typeset T ) )
+  ( hyp A T df-: mpbi pm3.27i ) )
+
+thm ( typeistype ( ) ( ( hyp ( : A T ) ) )
+  ( istype T )
+  ( hyp A T df-: mpbi pm3.26i ) )
+
+thm ( castel ( ( T x y ) )
+  ( ( hyp ( istype T ) ) )
+  ( e. ( cast A T ) ( typeset T ) )
+  ( A T df-cast
+      
+    hyp witness1 A elimel
+    eqeltr
+) )
+
+thm ( casteq1 ( ) ( )
+  ( -> ( = A B ) ( = ( cast A T ) ( cast B T ) ) )
+  (
+    A B T typeset eleq1 A T typeset e. B T typeset e. A T witness ifbi syl
+    A B B T typeset e. T witness ifeq1 eqtrd
+    A T df-cast
+    B T df-cast 3eqtr4g
+) )
+
+thm ( casteq1i ( )
+  ( ( hyp ( = A B ) ) )
+  ( = ( cast A T ) ( cast B T ) )
+  ( hyp A B T casteq1 ax-mp ) )
+
+thm ( castt ( ) ( )
+  ( -> ( : A T ) ( = ( cast A T ) A ) )
+  ( A T df-: T istype A T typeset e. pm3.27 sylbi
+    A T typeset e. A T witness iftrue syl
+    A T df-cast syl5eq
+) )
+
+thm ( expscast ( ( T x ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( = ( [:=] x A ( cast B T ) ) ( cast ( [:=] x A B ) T ) )
+  ( hyp B T df-cast x expseq2i
+    hyp x B T typeset e. B T witness expsded eqtr
+
+      hyp x B T typeset expssbel
+      hyp x T typeset expsf x A B [:=] eleq2i bitr3
+    x A B [:=] x A T witness [:=] ifbii eqtr
+
+      hyp x T witness expsf
+    x A B [:=] T typeset e. x A B [:=] ifeq2i eqtr
+
+    x A B [:=] T df-cast eqtr4
+) )
+
+thm ( vareq1 () ()
+  (-> (= (cv x) (cv y)) (= (var x T) (var y T)))
+  ( x cv y cv T casteq1 x T df-var y T df-var 3eqtr4g
+))
+
+thm ( var: ( ) ( ( hyp ( istype T ) ) )
+  ( : ( var v T ) T )
+  ( hyp hyp v cv castel pm3.2i v cv T cast T df-: mpbir
+    v T df-var T :eq1i mpbir
+) )
+
+# Note: this should probably be used in a few places where castt is used
+# now.
+thm ( vart ( ) ( )
+  ( -> ( : ( cv x ) T ) ( = ( cv x ) ( var x T ) ) )
+  ( x cv T castt x T df-var syl5req
+) )
+
+thm ( sbeqfoo ( ( A x ) ( x y ) ) ( )
+  ( <-> ( [/] ( cv y ) x ( = ( cv x ) A ) ) ( = ( cv y ) A ) )
+  ( y cv A = x ax-17
+    x cv y cv A eqeq1
+    sbie
+) )
+
+thm ( truthterm:bool ( ) ( )
+  ( : ( truthterm ph ) ( bool ) )
+  ( boolistype
+
+    ph df-truthterm
+
+    df-true p0ex eqeltr false pri2
+    df-false 0ex eqeltr true pri1
+    ph keepel eqeltr
+
+    df-bool typeseteqi
+    false true prex df-false 0ex eqeltr typesetval1 eqtr
+    eleqtrr pm3.2i
+
+    ph truthterm bool df-: mpbir
+) )
+
+thm ( booltruth ( ) ( )
+  ( <-> ph ( istrue ( truthterm ph ) ) )
+  ( ph true false iftrue ph df-truthterm syl5eq
+    ph truthterm df-istrue sylibr
+    ph ph truthterm istrue ibib mpbi
+
+    ph true false iffalse ph df-truthterm syl5eq
+    df-false syl6eq 0nep0
+    ph truthterm {/} {/} {} eqeq1
+    df-true ph truthterm eqeq2i syl5bb
+    ph truthterm df-istrue syl5bb
+    negbid mpbiri syl
+    ph -. ph truthterm istrue -. ibib mpbi bicon4d pm2.61i
+) )
+
+# No doubt we can simplify other thms by using this
+thm ( falsenetrue ( ) ( )
+  ( -. ( = ( false ) ( true ) ) )
+  ( 0nep0 df-false df-true eqeq12i mtbir
+) )
+
+thm ( istruetruth ( )
+  ( ( hyp ( : A ( bool ) ) ) )
+  ( = A ( truthterm ( istrue A ) ) )
+  ( hyp typeel
+
+    df-bool typeseteqi
+    false true prex df-false 0ex eqeltr typesetval1 eqtr
+    eleqtr hyp typeel elisseti false true elpr mpbi
+
+      falsenetrue A false true eqeq1 mtbiri
+      A df-istrue negbii sylibr
+      A istrue true false iffalse syl
+      A istrue df-truthterm syl5req
+      A false A istrue truthterm eqeq1 mpbird
+
+      A df-istrue A istrue true false iftrue sylbir
+      A istrue df-truthterm syl5req
+      A true A istrue truthterm eqeq1 mpbird
+
+    jaoi ax-mp
+) )
+
+thm ( istrueeq ( ) ( )
+  ( -> ( = A B ) ( <-> ( istrue A ) ( istrue B ) ) )
+  ( A B true eqeq1
+    A df-istrue
+    B df-istrue 3bitr4g
+) )
+
+thm ( istrueeqi ( )
+  ( ( hyp ( = A B ) ) )
+  ( <-> ( istrue A ) ( istrue B ) )
+  ( hyp A B istrueeq ax-mp 
+) )
+
+thm ( istrueeqd ( )
+  ( ( hyp ( -> ph ( = A B ) ) ) )
+  ( -> ph ( <-> ( istrue A ) ( istrue B ) ) )
+  ( hyp A B istrueeq syl
+) )
+
+thm ( truthbi ( ) ( )
+  ( <-> ( <-> ph ps ) ( = ( truthterm ph ) ( truthterm ps ) ) )
+  ( ph ps true false ifbi ph df-truthterm ps df-truthterm 3eqtr4g
+
+    ph truthterm ps truthterm istrueeq
+    ph booltruth
+    ps booltruth 3bitr4g impbi
+) )
+
+thm ( istrueeqbi ( )
+  ( ( hyp.A ( : A ( bool ) ) )
+    ( hyp.B ( : B ( bool ) ) ) )
+  ( <-> ( = A B ) ( <-> ( istrue A ) ( istrue B ) ) )
+  ( hyp.A istruetruth hyp.B istruetruth eqeq12i
+    A istrue B istrue truthbi bitr4
+) )
+
+thm ( trueistrue ( ) ( )
+  ( istrue ( true ) )
+  ( true eqid true df-istrue mpbir
+) )
+
+thm ( nfalseistrue ( ) ( )
+  ( -. ( istrue ( false ) ) )
+  ( falsenetrue false df-istrue mtbir
+) )
+
+thm ( true:bool ( ) ( )
+  ( : ( true ) ( bool ) )
+  ( boolistype
+    df-true p0ex eqeltr false pri2
+    df-bool typeseteqi
+    false true prex df-false 0ex eqeltr typesetval1 eqtr
+    eleqtrr
+    pm3.2i true bool df-: mpbir
+) )
+
+thm ( false:bool ( ) ( )
+  ( : ( false ) ( bool ) )
+  ( boolistype
+    df-false 0ex eqeltr true pri1
+    df-bool typeseteqi
+    false true prex df-false 0ex eqeltr typesetval1 eqtr
+    eleqtrr
+    pm3.2i false bool df-: mpbir
+) )
+
+# This theorem "cheats" a bit, because it's dependent on the particular
+# behavior of istrue for non-bool args.
+thm ( istrue:booli ( )
+  ( ( hyp ( istrue A ) ) )
+  ( : A ( bool ) )
+  ( true:bool hyp A df-istrue mpbi bool :eq1i mpbir
+) )
+
+# This thm adds a substitution to the def of lambda.
+thm ( dflambda2 ( ( T v x y z ) ( A x y z ) ) ( )
+  ( = ( lambda v T A )
+     ( {<,>|} x y ( /\ ( : ( cv x ) T ) ( [/] ( cv x ) v ( = ( cv y ) A ) ) ) ) )
+  ( v T A z df-lambda
+
+    v cv T : z cv A = /\ x ax-17
+    v cv T : z cv A = /\ y ax-17
+    x cv T : v ax-17 y cv A = x ax-17 v hbsb3 hban
+    x cv T : x cv v y cv A = [/] /\ z ax-17
+
+    v cv x cv T :eq1 z cv y cv = adantr
+    z cv y cv A eqeq1
+    v x y cv A = sbequ12 sylan9bbr anbi12d
+    cbvopab eqtr
+) )
+
+# This could probably be simplified to use df-lambda instead of
+# dflambda2, but would probably require a variant of biopabdv which
+# is not in set.mm.
+thm ( abseq2 ( ( A x y ) ( B x y ) ( T v x y ) ( v x y ) ) ( )
+  ( -> ( A. v ( = A B ) ) ( = ( lambda v T A ) ( lambda v T B ) ) )
+  ( A B y cv eqeq2 v 19.20i
+    v y cv A = y cv B = x sbba4 syl x cv T : anbi2d x y biopabdv
+    v T A x y dflambda2
+    v T B x y dflambda2 3eqtr4g
+) )
+
+# The abstraction variable is not free in an abstraction.
+thm ( hbabs1 ( ( T v z ) ( A z ) ( v y ) )
+  ( )
+  ( -> ( e. ( cv y ) ( lambda v T A ) ) ( A. v ( e. ( cv y ) ( lambda v T A ) ) ) )
+  ( y v z v cv T : z cv A = /\ hbopab1
+    v T A z df-lambda y cv eleq2i
+    v T A z df-lambda y cv eleq2i v bial 3imtr4
+) )
+
+# Substitution through abstraction when the variables are identical.
+thm ( expsabs1 ( ( T v y ) ( A y ) ( v y ) ( B y ) )
+  ( ( hyp ( e. B ( V ) ) ) )
+  ( = ( [:=] v B ( lambda v T A ) ) ( lambda v T A ) )
+  ( hyp y v T A hbabs1 expsff
+) )
+
+# Substitution through abstraction when the variables are distinct.
+thm ( expsabs2 ( ( x y z ) ( B y y' z ) ( T x y y' z ) ( A y' z ) )
+  ( ( hyp ( e. B ( V ) ) ) )
+  ( = ( [:=] x B ( lambda y T A ) ) ( lambda y T ( [:=] x B A ) ) )
+  ( hyp y T A y' df-lambda x expseq2i
+    hyp x y y' y cv T : y' cv A = /\ expsopab eqtr
+    y cv T : x ax-17 hyp sbcf1
+    hyp y' cv x A expssb2 anbi12i
+    hyp x y cv T : y' cv A = sbca bitr4
+    y y' biopabi eqtr4
+    y T x B A [:=] y' df-lambda eqtr4
+) )
+
+thm ( absex ( ( A y ) ( T v y ) )
+  ( ( hyp ( istype T ) ) )
+  ( e. ( lambda v T A ) ( V ) )
+  ( v T A y df-lambda 
+    v cv T df-:
+    hyp mpbiran y cv A = anbi1i v y biopabi eqtr
+
+    hyp typesetex
+    y A moeq v cv T typeset e. a1i funopabex
+    eqeltr
+) )
+
+thm ( exps:i ( ( T x ) )
+  ( ( hyp.1 ( e. B ( V ) ) )
+    ( hyp.2 ( : A T ) ) )
+  ( : ( [:=] x B A ) T )
+  ( hyp.2 typeistype 
+
+    hyp.1 hyp.2 typeel x sbctru
+    hyp.1 x A T typeset expssbel mpbir
+    hyp.1 x T typeset expsf  x B A [:=] eleq2i mpbi
+    pm3.2i x B A [:=] T df-: mpbir
+) )
+
+# A useful inference that a function is of arrow type. Note the duplication
+# of the calculation of ( typeset ( $-> T T' ) ) with app: . Note also
+# that hyp.3 follows fairly straightforwardly from hyp.4 through fex,
+# which might allow us to eliminate the proof of absex.
+thm ( isarrow ( ( T y ) ( T' y ) ( F y ) )
+  ( ( hyp.1 ( istype T ) )
+    ( hyp.2 ( istype T' ) )
+    ( hyp.3 ( e. F ( V ) ) )
+    ( hyp.4 ( :--> F ( typeset T ) ( typeset T' ) ) ) )
+  ( : F ( $-> T T' ) )
+  ( hyp.1 hyp.2 arrowistype
+
+    hyp.4 hyp.3 y cv F T typeset T' typeset feq1 elab mpbir
+    hyp.1 hyp.2 y arrow1 typeseteqi
+    hyp.1 typesetex hyp.2 typesetex y arrowistypelem2
+    hyp.1 typesetex T' witness snex xpex typesetval1 eqtr F eleq2i mpbir
+    pm3.2i
+
+    F T T' $-> df-: mpbir
+) )
+
+thm ( abs: ( ( T x y ) ( T' x y ) ( A y ) )
+  ( ( hyp.1 ( istype T  ) )
+    ( hyp.2 ( : A T' ) ) )
+  ( : ( lambda x T A ) ( $-> T T' ) )
+  ( hyp.1
+    hyp.2 typeistype
+    hyp.1 x A absex
+
+    hyp.2 typeel x cv T typeset e. a1i rgen
+    x T A y df-lambda
+    x cv T df-:
+    hyp.1 mpbiran y cv A = anbi1i x y biopabi eqtr
+    T' typeset fopab2 mpbi
+
+    isarrow
+) )
+
+thm ( alphaconv ( ( A y z ) ( B x z ) ( T x y z ) )
+  ( ( hyp ( -> ( = ( cv x ) ( cv y ) ) ( = A B ) ) ) )
+  ( = ( lambda x T A ) ( lambda y T B ) )
+  ( x cv y cv T :eq1
+    hyp z cv eqeq2d anbi12d
+    z cbvopab1v
+
+    x T A z df-lambda
+    y T B z df-lambda 3eqtr4
+) )
+
+# Arguably, we should factor out most of this logic into a 2alpha thm.
+thm ( df-$=-half ( ( T x a y b ) ) ( )
+  ( =
+    ( if ( istype T )
+      ( lambda a T ( lambda b T ( truthterm ( = ( cv a ) ( cv b ) ) ) ) )
+      ( V ) )
+    ( if ( istype T )
+      ( lambda x T ( lambda y T ( truthterm ( = ( cv x ) ( cv y ) ) ) ) )
+      ( V ) ) )
+  ( a cv x cv y cv eqeq1 a cv y cv = x cv y cv = truthbi sylib y 19.21aiv
+    y a cv y cv = truthterm x cv y cv = truthterm T abseq2 syl
+
+    b cv y cv a cv eqeq2 a cv b cv = a cv y cv = truthbi sylib
+    T alphaconv syl5eq T alphaconv T istype V ifeq1i
+) )
+thm ( df-$= ((T x a y b) (T x' a y' b)) ( )
+  ( = ( $= T )
+    ( if ( istype T )
+      ( lambda x T ( lambda y T ( truthterm ( = ( cv x ) ( cv y ) ) ) ) )
+      ( V ) ) )
+  (T x' y' a b df-$=-half
+     T x y a b df-$=-half
+   eqtr4
+) )
+
+thm ( dfeqi ( ( T x y ) )
+  ( ( hyp ( istype T ) ) )
+  ( = ( $= T ) ( lambda x T ( lambda y T ( truthterm ( = ( cv x ) ( cv y ) ) ) ) ) )
+  ( T x y df-$=
+    hyp T istype x T y T x cv y cv = truthterm lambda lambda V iftrue eqcomd ax-mp
+    eqtr4
+) )
+
+# Useful for reverse type inference
+thm ( eqistype ( ( T x y ) ) ( )
+  ( -> ( -. ( = ( $= T ) ( V ) ) ) ( istype T ) )
+  ( T istype x T y T x cv y cv = truthterm lambda lambda V iffalse
+    T x y df-$= syl5eq con1i
+) )
+
+thm ( df-$==>-half ( ( a b x y ) ) ( )
+  (= 
+    ( lambda a ( bool ) ( lambda b ( bool )
+      ( truthterm ( -> ( istrue ( cv a ) ) ( istrue ( cv b ) ) ) ) ) )
+    ( lambda x ( bool ) ( lambda y ( bool )
+      ( truthterm ( -> ( istrue ( cv x ) ) ( istrue ( cv y ) ) ) ) ) ) )
+  ( a cv x cv istrueeq y cv istrue imbi1d
+    a cv istrue y cv istrue -> x cv istrue y cv istrue -> truthbi sylib y 19.21aiv
+    y a cv istrue y cv istrue -> truthterm x cv istrue y cv istrue -> truthterm bool abseq2 syl
+
+    b cv y cv istrueeq a cv istrue imbi2d
+    a cv istrue b cv istrue -> a cv istrue y cv istrue -> truthbi sylib
+    bool alphaconv syl5eq bool alphaconv
+) )
+thm ( df-$==> ((a b x y) (a b x' y')) ( )
+  ( = ( $==> ) ( lambda x ( bool ) ( lambda y ( bool )
+      ( truthterm ( -> ( istrue ( cv x ) ) ( istrue ( cv y ) ) ) ) ) ) )
+  (x' y' a b df-$==>-half
+     x y a b df-$==>-half
+   eqtr4
+) )
+
+
+thm ( sequent1:bool ( )
+  ( ( hyp ( $|- G A ) ) )
+  ( : G ( bool ) )
+  ( hyp G A df-$|- mpbi pm3.26i pm3.26i
+) )
+
+thm ( sequent2:bool ( )
+  ( ( hyp ( $|- G A ) ) )
+  ( : A ( bool ) )
+  ( hyp G A df-$|- mpbi pm3.26i pm3.27i
+) )
+
+thm ( sequentimp ( )
+  ( ( hyp ( $|- G A ) ) )
+  ( -> ( istrue G ) ( istrue A ) )
+  ( hyp G A df-$|- mpbi pm3.27i
+) )
+
+thm ( commaval ( )
+  ( ( hyp.1 ( : G1 ( bool ) ) )
+    ( hyp.2 ( : G2 ( bool ) ) ) )
+  ( <-> ( istrue ( $, G1 G2 ) ) ( /\ ( istrue G1 ) ( istrue G2 ) ) )
+  ( G1 G2 df-$,
+    hyp.1 hyp.2 pm3.2i G1 bool : G2 bool : /\ G1 istrue G2 istrue /\ truthterm bool typeset iftrue ax-mp eqtr
+    istrueeqi G1 istrue G2 istrue /\ booltruth bitr4
+) )
+
+thm ( comma:bool ( ) ( )
+  ( <-> ( /\ ( : G1 ( bool ) ) ( : G2 ( bool ) ) ) ( : ( $, G1 G2 ) ( bool ) ) )
+  ( G1 istrue G2 istrue /\ truthterm:bool
+    G1 bool : G2 bool : /\ G1 istrue G2 istrue /\ truthterm bool typeset iftrue
+    G1 G2 df-$, syl5req
+    G1 istrue G2 istrue /\ truthterm G1 G2 $, bool :eq1 syl mpbii
+
+    bool typeset eirr bool istype intnan bool typeset bool df-: mtbir
+    G1 bool : G2 bool : /\ G1 istrue G2 istrue /\ truthterm bool typeset iffalse
+    G1 G2 df-$, syl5req
+    bool typeset G1 G2 $, bool :eq1 syl
+    mtbii a3i impbi    
+
+) )
+
+thm ( commaval2 ( )
+  ( ( hyp ( : ( $, G1 G2 ) ( bool ) ) ) )
+  ( <-> ( istrue ( $, G1 G2 ) ) ( /\ ( istrue G1 ) ( istrue G2 ) ) )
+  ( hyp G1 G2 comma:bool mpbir pm3.26i
+    hyp G1 G2 comma:bool mpbir pm3.27i commaval
+) )
+
+# inference from comma:bool would of course be more concise; but I proved this first
+thm ( comma:booli ( )
+  ( ( hyp.1 ( : G1 ( bool ) ) )
+    ( hyp.2 ( : G2 ( bool ) ) ) )
+  ( : ( $, G1 G2 ) ( bool ) )
+  ( G1 istrue G2 istrue /\ truthterm:bool
+    G1 G2 df-$,
+    hyp.1 hyp.2 pm3.2i G1 bool : G2 bool : /\ G1 istrue G2 istrue /\ truthterm bool typeset iftrue ax-mp eqtr bool :eq1i
+    mpbir
+) )
+
+thm ( comma:bool1 ( )
+  ( ( hyp ( : ( $, G1 G2 ) ( bool ) ) ) )
+  ( : G1 ( bool ) )
+  ( hyp G1 G2 comma:bool mpbir pm3.26i
+) )
+
+thm ( comma:bool2 ( )
+  ( ( hyp ( : ( $, G1 G2 ) ( bool ) ) ) )
+  ( : G2 ( bool ) )
+  ( hyp G1 G2 comma:bool mpbir pm3.27i
+) )
+
+thm ( mksequent ( )
+  ( ( hyp.G ( : G ( bool ) ) )
+    ( hyp.A ( : A ( bool ) ) )
+    ( hyp ( -> ( istrue G ) ( istrue A ) ) ) )
+  ( $|- G A )
+  ( hyp.G hyp.A pm3.2i hyp pm3.2i
+    G A df-$|- mpbir
+) )
+
+thm ( mkanseq ( )
+ ( ( hyp.G1 ( : G1 ( bool ) ) )
+   ( hyp.G2 ( : G2 ( bool ) ) )
+   ( hyp.A ( : A ( bool ) ) )
+   ( hyp ( -> ( /\ ( istrue G1 ) ( istrue G2 ) ) ( istrue A ) ) ) )
+  ( $|- ( $, G1 G2 ) A )
+  ( hyp.G1 hyp.G2 comma:booli
+    hyp.A
+      hyp.G1 hyp.G2 commaval
+      hyp sylbi
+    mksequent
+) )
+
+thm ( mknullseq ( )
+  ( ( hyp ( istrue A ) ) )
+  ( $|- ( true ) A )
+  ( true:bool hyp istrue:booli hyp true istrue a1i mksequent ) )
+
+# Theorems about . (new version)
+
+thm ( .eq1 ( ) ( ) ( -> ( = F G ) ( = ( . F A ) ( . G A ) ) )
+  (
+      F G funeq F G dmeq A eleq2d anbi12d
+    F Fun A F dom e. /\ G Fun A G dom e. /\ F A ` V ifbi syl
+    F G A fveq1 F A ` G A ` G Fun A G dom e. /\ V ifeq1 syl eqtrd
+    F A df-app G A df-app 3eqtr4g
+) )
+
+thm ( .eq2 ( ) ( ) ( -> ( = A B ) ( = ( . C A ) ( . C B ) ) )
+  (   A B C dom eleq1 C Fun anbi2d
+    C Fun A C dom e. /\ C Fun B C dom e. /\ C A ` V ifbi syl
+    A B C fveq2 C A ` C B ` C Fun B C dom e. /\ V ifeq1 syl eqtrd
+
+    C A df-app C B df-app 3eqtr4g
+) )
+
+thm ( .eq1i ( ) ( ( hyp ( = F G ) ) ) 
+  ( = ( . F A ) ( . G A ) )
+  ( hyp F G A .eq1 ax-mp ) )
+
+thm ( .eq2i ( ) ( ( hyp ( = A B ) ) ) 
+  ( = ( . C A ) ( . C B ) )
+  ( hyp A B C .eq2 ax-mp ) )
+
+thm ( arrowi1 ( ( T x ) ( T' x ) ( F x ) )
+  ( ( hyp ( : F ( $-> T T' ) ) ) )
+  ( :--> F ( typeset T ) ( typeset T' ) )
+  (
+    hyp F T T' $-> df-: mpbi pm3.27i
+    hyp typeistype arrow2i1 hyp typeistype arrow2i2 x arrow1 typeseteqi
+    hyp typeistype arrow2i1 typesetex hyp typeistype arrow2i2 typesetex x arrowistypelem2
+    hyp typeistype arrow2i1 typesetex T' witness snex xpex typesetval1 eqtr F eleq2i mpbi
+    x cv F T typeset T' typeset feq1 x x cv T typeset T' typeset :--> {|} elabg ibi ax-mp
+) )
+
+thm ( appval ( )
+  ( ( hyp ( : F ( $-> T T' ) ) ) )
+  ( -> ( : A T ) ( = ( . F A ) ( ` F A ) ) )
+  ( A T df-: pm3.27bd
+    hyp arrowi1 F T typeset T' typeset fdm ax-mp A eleq2i sylibr
+    hyp arrowi1 F T typeset T' typeset ffun ax-mp jctil
+    F Fun A F dom e. /\ F A ` V iftrue syl
+    F A df-app syl5eq
+) )
+
+# useful for inferring the validity of functions from correct typing
+thm ( appfun ( ) ( )
+  ( -> ( e. ( . F A ) ( V ) ) ( Fun F ) )
+  (
+    nvelv F Fun A F dom e. /\ F A ` V iffalse
+    F A df-app syl5eq V eleq1d mtbiri a3i pm3.26d
+) )
+
+thm ( appfuni ( )
+  ( ( hyp  ( : ( . F A ) T ) ) )
+  ( Fun F )
+  ( hyp typeel elisseti F A appfun ax-mp
+) )
+
+thm ( appdom ( ) ( )
+  ( -> ( e. ( . F A ) ( V ) ) ( e. A ( dom F ) ) )
+  (
+    nvelv F Fun A F dom e. /\ F A ` V iffalse
+    F A df-app syl5eq V eleq1d mtbiri a3i pm3.27d
+) )
+
+# useful for doing type inference in the reverse direction
+thm ( apptype ( )
+  ( ( hyp ( : F ( $-> T T' ) ) ) )
+  ( -> ( e. ( . F A ) ( V ) ) ( : A T ) )
+  ( 
+    F A appdom
+    hyp arrowi1 F T typeset T' typeset fdm ax-mp A eleq2i sylib
+    A T df-: hyp typeistype arrow2i1 mpbiran sylibr
+) )
+
+thm ( apptypei ( )
+  ( ( hyp.1 ( : F ( $-> T T' ) ) )
+    ( hyp.2 ( : ( . F A ) T' ) ) )
+  ( : A T )
+  ( hyp.2 typeel elisseti
+    hyp.1 A apptype ax-mp
+) )
+
+# The only proper-class value of a function application is V.
+thm ( appisset ( ) ( )
+  ( <-> ( e. ( . F A ) ( V ) ) ( -. ( = ( . F A ) ( V ) ) ) )
+  ( F A ` F Fun A F dom e. /\ F A ` V if V eleq1
+    F A ` F Fun A F dom e. /\ F A ` V if V eqeq1 negbid bibi12d
+    V F Fun A F dom e. /\ F A ` V if V eleq1
+    V F Fun A F dom e. /\ F A ` V if V eqeq1 negbid bibi12d
+    F A fvex F A fvex nvelv F A ` V V eleq1 mtbiri mt2 2th
+    V eqid nvelv 2th bicon2i
+    keephyp
+    F A df-app V eleq1i
+    F A df-app V eqeq1i negbii 3bitr4
+) )
+
+thm ( app: ( ( T f ) ( T' f ) ( F f ) )
+  ( ( hyp.1 ( : F ( $-> T T' ) ) )
+    ( hyp.2 ( : A T ) ) )
+  ( : ( . F A ) T' )
+  ( hyp.1 typeistype arrow2i2
+
+      hyp.1 typeel
+      hyp.2 typeistype hyp.1 typeistype arrow2i2 f arrow1 typeseteqi
+
+      hyp.2 typeistype typesetex hyp.1 typeistype arrow2i2 typesetex f arrowistypelem2
+      hyp.2 typeistype typesetex T' witness snex xpex typesetval1 eqtr
+
+      eleqtr 
+      hyp.1 typeel elisseti
+      f cv F T typeset T' typeset feq1 elab mpbi
+
+      hyp.2 typeel
+
+    F T typeset T' typeset A  ffvrn mp2an pm3.2i
+    F A ` T' df-: mpbir
+    hyp.2 hyp.1 A appval ax-mp T' :eq1i mpbir
+) )
+
+thm ( sbc: ( ( x T ) )
+  ( ( hyp ( e. A ( V ) ) ) )
+  ( <-> ( [/] A x ( : B T ) ) ( : ( [:=] x A B ) T ) )
+  ( hyp B T df-: x bisbc
+    hyp x T istype B T typeset e. sbca bitr
+    T istype x ax-17 hyp sbcf1
+    hyp x B T typeset expssbel anbi12i bitr4
+
+    hyp x T typeset expsf x A B [:=] eleq2i T istype anbi2i bitr
+    x A B [:=] T df-: bitr4
+) )
+
+# A version of beta reduction with especially general assumptions.
+thm ( betaexps4 ( ( T v ) ( T' v y ) ( B y ) ( A y ) )
+  ( ( hyp1 ( e. B ( dom ( lambda v T' A ) ) ) )
+    ( hyp2 ( Fun ( lambda v T' A ) ) )
+    ( hyp.B ( : B T' ) ) )
+  ( = ( . ( lambda v T' A ) B ) ( [:=] v B A ) )
+  ( 
+    v T' A lambda B df-app
+      hyp2
+      hyp1
+    v T' A lambda Fun B v T' A lambda dom e. /\ v T' A lambda B ` V iftrue mp2an
+    eqtr
+
+    v T' A lambda B fvex
+      hyp2
+      hyp1
+    y visset v T' A lambda B funfvop mp2an
+    v T' A y df-lambda B y cv <,> eleq2i bitr
+    hyp1 elisseti v v cv T' : y cv A = /\ y sbcop1 bitr4
+    hyp1 elisseti v v cv T' : y cv A = sbca bitr
+        hyp.B
+      hyp1 elisseti v expsvar T' :eq1i mpbir
+      hyp1 elisseti v v cv T' sbc: mpbir
+    mpbiran
+
+    hyp1 elisseti y cv v A expssb2 bitr4 biimp
+
+    v T' A lambda B ` y cv v B A [:=] eqeq1 mpbird eqcoms
+    vtocle eqtr
+) )
+
+# Beta reduction rule expressed using explicit substitution.
+thm ( betaexps ( ( T v ) ( T' v y ) ( A y ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) ) )
+  ( = ( . ( lambda v T' A ) B ) ( [:=] v B A ) )
+  (   hyp.2 typeistype hyp.1 v abs: hyp.2 app: typeel elisseti
+    v T' A lambda B appdom ax-mp
+    hyp.2 typeistype hyp.1 v abs: hyp.2 app: appfuni
+    hyp.2
+    betaexps4
+) )
+
+# This is an older proof of the same thing
+thm ( betaexps[old] ( ( T v ) ( T' v y ) ( A y ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) ) )
+  ( = ( . ( lambda v T' A ) B ) ( [:=] v B A ) )
+  ( hyp.2 hyp.2 typeistype hyp.1 v abs: B appval ax-mp
+
+    hyp.2 typeel
+    hyp.2 typeel elisseti v expsvar
+    hyp.2 typeel elisseti v T' typeset expsf eleq12i mpbir
+    hyp.2 typeel elisseti v v cv T' typeset expssbel mpbi
+
+    hyp.2 typeel elisseti
+    hyp.1 typeel elisseti
+    v T' typeset A V y fvopab2 mpan2
+    v T' A y df-lambda 
+    v cv T' df-:
+    hyp.2 typeistype mpbiran y cv A = anbi1i v y biopabi eqtr
+
+    v cv fveq1i syl5eq
+    v sbctru
+    hyp.2 typeel elisseti v v cv T' typeset e. v T' A lambda v cv ` A = sbci2 mpbi
+    ax-mp
+
+    hyp.2 typeel elisseti v v T' A lambda v cv ` A expssb12 mpbir
+
+    hyp.2 typeel elisseti hyp.2 typeistype v A absex v visset v expsfv eqtr3
+    hyp.2 typeel elisseti v expsvar v B v T' A lambda [:=] fveq2i eqtr
+    hyp.2 typeel elisseti v T' A expsabs1 B fveq1i eqtr
+    eqtr4
+) )
+
+# a helper thm: two successive applications of beta reduction
+thm ( 2betaexp
+  ( ( B x y ) ( C x y ) ( D x y ) ( T x y ) ( T' x ) ( T'' x y ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) )
+    ( hyp.3 ( : C T'' ) ) )
+  ( = ( . ( . ( lambda x T' ( lambda y T'' A ) ) B ) C ) ( [:=] y C ( [:=] x B
+A ) ) )
+  ( hyp.3 typeistype hyp.1 y abs:
+    hyp.2 x betaexps
+
+    hyp.2 typeel elisseti x y T'' A expsabs2 eqtr
+    C .eq1i
+
+    hyp.2 typeel elisseti hyp.1 x exps:i hyp.3 y betaexps eqtr
+) )
+
+thm ( 2beta
+  ( ( B x y ) ( C x y ) ( D x y ) ( T x y ) ( T' x ) ( T'' x y ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) )
+    ( hyp.3 ( : C T'' ) )
+    ( hyp.4 ( -> ( /\ ( = ( cv x ) B ) ( = ( cv y ) C ) ) ( = A D ) ) ) )
+  ( = ( . ( . ( lambda x T' ( lambda y T'' A ) ) B ) C ) D )
+  ( hyp.1 hyp.2 hyp.3 x y 2betaexp
+
+    hyp.3 typeel elisseti
+    hyp.2 typeel elisseti hyp.4 ancoms exp imps2expsd
+    imps2exps eqtr
+) )
+
+thm ( eqval ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A T ) ) ( hyp.B ( : B T ) ) )
+  ( <-> ( istrue ( . ( . ( $= T ) A ) B ) ) ( = A B ) )
+  ( hyp.A typeistype a b dfeqi A .eq1i B .eq1i
+
+    a cv b cv = truthterm:bool
+    hyp.A
+    hyp.B
+
+    a cv A b cv B eqeq12 a cv b cv = A B = truthbi sylib
+    2beta eqtr
+
+    T $= A . B . A B = truthterm istrueeq ax-mp
+    A B = booltruth bitr4
+) )
+
+# Some duplication with eqval may be factored out here.
+thm ( eqval:bool ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A T ) ) ( hyp.B ( : B T ) ) )
+  ( : ( . ( . ( $= T ) A ) B ) ( bool ) )
+  ( A B = truthterm:bool
+
+    hyp.A typeistype a b dfeqi A .eq1i B .eq1i
+
+    a cv b cv = truthterm:bool
+    hyp.A
+    hyp.B
+
+    a cv A b cv B eqeq12 a cv b cv = A B = truthbi sylib
+    2beta eqtr bool :eq1i
+    mpbir
+) )
+
+thm ( eq: ( ( T a b ) ) ( ( hyp ( istype T ) ) )
+  ( : ( $= T ) ( $-> T ( $-> T ( bool ) ) ) )
+  ( hyp
+      hyp a cv b cv = truthterm:bool b abs:
+    a abs:
+    hyp a b dfeqi T T bool $-> $-> :eq1i mpbir
+) )
+
+thm ( appnotvi ( )
+  ( ( hyp ( : ( . F A ) T ) ) )
+  ( -. ( = F ( V ) ) )
+  ( nfunv hyp appfuni F V funeq mpbii mto
+) )
+
+# This is probably the most general form of the rule, and the others should
+# be derived in terms of this one
+thm ( appnotbottom1 ( ) ( )
+  ( -> ( notbottom ( . F A ) ) ( notbottom F ) )
+  ( F A . df-notbottom F A appisset bitr4
+    F A appfun sylbi
+    nfunv F V funeq mtbiri con2i syl
+    F df-notbottom sylibr
+) )
+
+thm ( appnotbottom2 ( ) ( )
+  ( -> ( notbottom ( . F A ) ) ( notbottom A ) )
+  ( F A . df-notbottom F A appisset bitr4
+    F A appdom sylbi
+    A F dom elisset syl nvelv A V V eleq1 mtbiri con2i syl
+    A df-notbottom sylibr
+) )
+
+# Exported as an inference rule
+thm ( appnotbottom1i ( )
+  ( ( hyp ( notbottom ( . F A ) ) ) )
+  ( notbottom F )
+  ( hyp F A appnotbottom1 ax-mp ) )
+
+thm ( appnotbottom2i ( )
+  ( ( hyp ( notbottom ( . F A ) ) ) )
+  ( notbottom A )
+  ( hyp F A appnotbottom2 ax-mp ) )
+
+# Inference rule from valid type to not-bottom.
+thm ( typenotbottomi ( )
+  ( ( hyp ( : A T ) ) )
+  ( notbottom A )
+  ( hyp typeel elisseti nvelv A V V eleq1 mtbiri con2i ax-mp
+    A df-notbottom mpbir
+) )
+
+thm ( apptypenotboti ( )
+  ( ( hyp.1 ( : F ( $-> T T' ) ) )
+    ( hyp.2 ( notbottom ( . F A ) ) ) )
+  ( : A T )
+  (     hyp.2 F A . df-notbottom mpbi
+      F A appisset mpbir
+    hyp.1 A apptype ax-mp
+) )
+
+thm ( eqistypei ( ( T a b ) )
+  ( ( hyp ( notbottom ( $= T ) ) ) )
+  ( istype T )
+  ( hyp T $= df-notbottom mpbi
+    T eqistype ax-mp
+) )
+
+thm ( eqvalistype ( ( T a b ) )
+  ( ( hyp ( : ( . ( . ( $= T ) A ) B ) ( bool ) ) ) )
+  ( istype T )
+  ( hyp typenotbottomi appnotbottom1i appnotbottom1i eqistypei
+) )
+
+thm ( eqtypei1 ( ( T a b ) )
+  ( ( hyp ( : ( . ( . ( $= T ) A ) B ) ( bool ) ) ) )
+  ( : A T )
+  ( 
+    hyp appnotvi
+
+    T $= A appisset
+    hyp eqvalistype eq: A apptype sylbir ax-mp
+    
+) )
+
+thm ( eqtypei2 ( ( T a b ) )
+  ( ( hyp ( : ( . ( . ( $= T ) A ) B ) ( bool ) ) ) )
+  ( : B T )
+  ( hyp eqvalistype eq: hyp eqtypei1 app: hyp apptypei
+) )
+
+# Useful for going from HOL to set-theoretic equality
+thm ( eqvali ()
+  ((hyp ($|- (true) (. (. ($= T) A) B))))
+  (= A B)
+  (trueistrue hyp sequentimp ax-mp
+   hyp sequent2:bool eqtypei1 hyp sequent2:bool eqtypei2 eqval mpbi
+))
+
+# Useful for going from set-theoretic to HOL equality
+thm ( eqvalr ()
+  ((hyp.A (: A T))
+   (hyp (= A B)))
+  ($|- (true) (. (. ($= T) A) B))
+  (hyp hyp.A hyp.A hyp T :eq1i mpbi eqval mpbir mknullseq
+))
+
+thm ( impval ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A ( bool ) ) ) ( hyp.B ( : B ( bool ) ) ) )
+  ( <-> ( istrue ( . ( . ( $==> ) A ) B ) ) ( -> ( istrue A ) ( istrue B ) ) )
+  ( a b df-$==> A .eq1i B .eq1i
+
+    a cv istrue b cv istrue -> truthterm:bool
+    hyp.A
+    hyp.B
+
+    a cv A istrueeq b cv B = adantr
+    b cv B istrueeq a cv A = adantl imbi12d
+    a cv istrue b cv istrue -> A istrue B istrue -> truthbi sylib
+    2beta eqtr
+
+    $==> A . B . A istrue B istrue -> truthterm istrueeq ax-mp
+    A istrue B istrue -> booltruth bitr4
+) )
+
+# a much easier proof would use imptype and app:
+thm ( impval:bool ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A ( bool ) ) ) ( hyp.B ( : B ( bool ) ) ) )
+  ( : ( . ( . ( $==> ) A ) B ) ( bool ) )
+  ( A istrue B istrue -> truthterm:bool
+
+    a b df-$==> A .eq1i B .eq1i
+
+    a cv istrue b cv istrue -> truthterm:bool
+    hyp.A
+    hyp.B
+
+    a cv A istrueeq b cv B = adantr
+    b cv B istrueeq a cv A = adantl imbi12d
+    a cv istrue b cv istrue -> A istrue B istrue -> truthbi sylib
+    2beta eqtr bool :eq1i
+    mpbir
+) )
+
+thm ( impvala ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A ( bool ) ) )
+    ( hyp.B ( : B ( bool ) ) )
+    ( hyp ( istrue ( . ( . ( $==> ) A ) B ) ) ) )
+  ( -> ( istrue A ) ( istrue B ) )
+  ( hyp hyp.A hyp.B impval mpbi
+) )
+
+thm ( impvalb ( ( A a b ) ( B a b ) ( T a b ) )
+  ( ( hyp.A ( : A ( bool ) ) )
+    ( hyp.B ( : B ( bool ) ) )
+    ( hyp ( -> ( istrue A ) ( istrue B ) ) ) )
+  ( istrue ( . ( . ( $==> ) A ) B ) )
+  ( hyp hyp.A hyp.B impval mpbir
+) )
+
+thm ( imptype ( ( a b ) ) ( )
+  ( : ( $==> ) ( $-> ( bool ) ( $-> ( bool ) ( bool ) ) ) )
+  ( boolistype
+      boolistype a cv istrue b cv istrue -> truthterm:bool b abs:
+    a abs:
+    a b df-$==> bool bool bool $-> $-> :eq1i mpbir
+) )
+
+thm ( assum_twiddle ( )
+  ( ( hyp.1 ( -> ( istrue G2 ) ( istrue G1 ) ) )
+    ( hyp.2 ( : G2 ( bool ) ) )
+    ( hyp.3 ( $|- G1 A ) ) )
+  ( $|- G2 A )
+  ( hyp.2 hyp.3 sequent2:bool hyp.1 hyp.3 sequentimp syl mksequent
+) )
+
+# The core HOL inference rules
+
+thm ( REFL ( ) ( ( hyp.A ( : A T ) ) )
+  ( $|- ( true ) ( . ( . ( $= T ) A ) A ) )
+  ( hyp.A A eqid eqvalr
+) )
+
+# Expressed in terms of explicit substitution.
+thm ( BETA_CONV_x ( ( T' x ) ( T x ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) ) )
+  ( $|- ( true ) ( . ( . ( $= T ) ( . ( lambda x T' A ) B ) ) ( [:=] x B A ) ) )
+  ( hyp.2 typeistype hyp.1 x abs: hyp.2 app:
+    hyp.1 hyp.2 x betaexps
+    eqvalr
+) )
+
+thm ( bclem_T ( )
+  ( ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( istype T )
+  ( hyp sequent2:bool typenotbottomi appnotbottom2i appnotbottom1i appnotbottom1i eqistypei
+) )
+
+thm ( bclem_T' ( )
+  ( ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( istype T' )
+  ( hyp sequent2:bool typenotbottomi appnotbottom1i appnotbottom2i appnotbottom1i appnotbottom1i eqistypei
+) )
+
+thm ( bclem_A ( )
+  ( ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( : A T )
+  ( hyp bclem_T eq:
+    hyp sequent2:bool typenotbottomi appnotbottom2i appnotbottom1i apptypenotboti
+) )
+
+thm ( bclem_B ( )
+  ( ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( : B T' )
+  ( hyp bclem_T' eq: hyp bclem_T' x var: app:
+    hyp sequent2:bool typenotbottomi appnotbottom1i appnotbottom2i apptypenotboti
+) )
+
+thm ( bclem_C ( )
+  ( ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( : C T )
+  ( hyp bclem_T eq: hyp bclem_A app:
+    hyp sequent2:bool typenotbottomi appnotbottom2i apptypenotboti
+) )
+
+thm ( bclem_x ( )
+  ( ( hyp ( : B T' ) ) )
+  ( -> ( = ( cv x ) B ) ( = ( var x T' ) B ) )
+  ( hyp x cv B T' :eq1 mpbiri
+    x cv T' castt syl x T' df-var syl5req B eqeq1d ibi
+) )
+
+# Expressed in terms of implicit substitution.
+thm ( BETA_CONV_im ( ( T' x ) ( T x ) ( B x ) ( C x ) ( G x ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) )
+    ( hyp ( $|- G ( . ( . ( $==> ) ( . ( . ( $= T' ) ( var x T' ) ) B ) )
+                          ( . ( . ( $= T ) A ) C ) ) ) ) )
+  ( $|- G ( . ( . ( $= T ) ( . ( lambda x T' A ) B ) ) C ) )
+  ( hyp sequent1:bool
+      hyp bclem_T' hyp bclem_A x abs: hyp bclem_B app:
+      hyp bclem_C
+    eqval:bool    
+
+    hyp bclem_B typeel elisseti
+      hyp sequentimp
+        hyp bclem_T' eq: hyp bclem_T' x var: app: hyp bclem_B app:
+        hyp bclem_T eq: hyp bclem_A app: hyp bclem_C app:
+      impval sylib
+        hyp bclem_T' x var: hyp bclem_B eqval
+        hyp bclem_A hyp bclem_C eqval
+      imbi12i sylib hyp bclem_B x bclem_x syl5
+    imps2expsd
+
+    hyp.1 hyp.2 x betaexps syl5eq
+
+      hyp bclem_T' hyp bclem_A x abs: hyp bclem_B app:
+      hyp bclem_C
+    eqval sylibr mksequent
+) )
+
+# A bunch of random, useful explicit substitution thms
+
+thm ( expspred ( ( x y ) ( A y ) ( B y ) ( ph x ) ( ps y ) ( ch y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. ( [:=] x A B ) ( V ) ) )
+    ( hyp1 ( -> ( = ( cv y ) B ) ( <-> ph ps ) ) )
+    ( hyp2 ( -> ( = ( cv y ) ( [:=] x A B ) ) ( <-> ph ch ) ) ) )
+  ( <-> ( [/] A x ps ) ch )
+  ( hyp.B
+
+        hyp.A y cv x B expssb2
+        hyp.A hyp1 x sbctru
+        hyp.A x y cv B = ph ps <-> sbci2 mpbi sylbi
+        hyp.A x ph ps sbcb sylib
+      ph x ax-17 hyp.A sbcf1 syl5bb
+      hyp2
+    bitr3d
+    vtocle
+) )
+
+thm ( expsfun ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. ( [:=] x A B ) ( V ) ) ) )
+  ( <-> ( [/] A x ( Fun B ) ) ( Fun ( [:=] x A B ) ) )
+  ( hyp.A
+    hyp.B
+    y cv B funeq
+    y cv x A B [:=] funeq
+    expspred
+) )
+
+thm ( expsdom ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. ( [:=] x A B ) ( V ) ) ) )
+  ( = ( [:=] x A ( dom B ) ) ( dom ( [:=] x A B ) ) )
+  (   hyp.A hyp.B
+      y cv B dmeq
+      y cv x A B [:=] dmeq
+    expsoper
+) )
+
+# This is the "calculation" form of sbca. It's a nice inference for building
+# larger substitutions from smaller ones. No mandatory hypotheses or
+# other propositional calculus steps are needed. A case can be made for
+# implementing this pattern more widely.
+thm ( sbcanc ( )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp1 ( <-> ( [/] A x ph ) ps ) )
+    ( hyp2 ( <-> ( [/] A x ch ) th ) ) )
+  ( <-> ( [/] A x ( /\ ph ch ) ) ( /\ ps th ) )
+  ( hyp.A x ph ch sbca
+    hyp1 hyp2 anbi12i bitr
+) )
+
+thm ( expsapp ( )
+  ( ( hyp.1 ( e. A ( V ) ) )
+    ( hyp.B ( e. ( [:=] x A B ) ( V ) ) )
+    ( hyp.C ( e. ( [:=] x A C ) ( V ) ) ) )
+  ( = ( [:=] x A ( . B C ) ) ( . ( [:=] x A B ) ( [:=] x A C ) ) )
+  ( hyp.1 B C df-app x expseq2i
+
+    hyp.1 x B Fun C B dom e. /\ B C ` V expsded eqtr
+
+    hyp.1
+      hyp.1 hyp.B expsfun
+        hyp.1 x C B dom expssbel
+        hyp.1 hyp.B expsdom x A C [:=] eleq2i
+      bitr3
+    sbcanc x A B C ` [:=] x A V [:=] ifbii eqtr
+
+      hyp.1 hyp.B hyp.C expsfv2
+    x A B [:=] Fun x A C [:=] x A B [:=] dom e. /\ x A V [:=] ifeq1i eqtr
+
+      hyp.1 x V expsf
+    x A B [:=] Fun x A C [:=] x A B [:=] dom e. /\ x A B [:=] x A C [:=] ` ifeq2i eqtr
+
+    x A B [:=] x A C [:=] df-app eqtr4
+) )
+
+# Calculation form of expsapp.
+thm ( expsappc ( )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. B ( V ) ) )
+    ( hyp.C ( e. C ( V ) ) )
+    ( hyp1 ( = ( [:=] x A B ) D ) )
+    ( hyp2 ( = ( [:=] x A C ) E ) ) )
+  ( = ( [:=] x A ( . B C ) ) ( . D E ) )
+  ( hyp.A hyp.A hyp.B x expsex hyp.A hyp.C x expsex expsapp
+    hyp1 x A C [:=] .eq1i eqtr
+    hyp2 D .eq2i eqtr
+) )
+
+thm ( expsistrue ( ( x y ) ( A y ) ( B y ) )
+  ( ( hyp.A ( e. A ( V ) ) )
+    ( hyp.B ( e. B ( V ) ) ) )
+  ( <-> ( [/] A x ( istrue B ) ) ( istrue ( [:=] x A B ) ) )
+  ( hyp.A
+    hyp.A hyp.B x expsex
+    y cv B istrueeq
+    y cv x A B [:=] istrueeq
+    expspred
+) )
+
+# Here are the basic substitution properties, expressed using (\x a).b
+# syntax.
+
+# identity function
+thm ( BETA1 ( ( T x ) )
+  ( ( hyp ( : A T ) ) )
+  ( $|- ( true ) ( . ( . ( $= T ) ( . ( lambda x T ( var x T ) ) A ) ) A ) )
+  ( hyp typeistype hyp typeistype x var: x abs: hyp app:
+
+    hyp typeistype x var: hyp x betaexps
+
+      hyp typeel elisseti x T df-var x expseq2i
+      hyp typeel elisseti x x cv T expscast eqtr
+      hyp typeel elisseti x expsvar T casteq1i eqtr
+      hyp A T castt ax-mp eqtr
+    eqtr
+    eqvalr
+) )
+
+# constant function
+thm ( BETA2 ( ( T x ) ( T' x ) ( B x ) )
+  ( ( hyp.1 ( : A T ) )
+    ( hyp.2 ( : B T' ) ) )
+  ( $|- ( true ) ( . ( . ( $= T' ) ( . ( lambda x T B ) A ) ) B ) )
+  ( hyp.1 typeistype hyp.2 x abs: hyp.1 app:
+
+    hyp.2 hyp.1 x betaexps
+
+    hyp.1 typeel elisseti x B expsf eqtr
+      
+    eqvalr
+) )
+
+thm ( betaexps2lem1 ( )
+  ( ( hyp ( : ( . A B ) T' ) ) )
+  ( e. B ( dom A ) )
+  ( hyp typeel elisseti A B appdom ax-mp
+) )
+
+thm ( betaexps2lem2 ( ( B y ) ( A y ) ( T x y ) )
+  ( ( hyp ( : ( . ( lambda x T A ) B ) T' ) ) )
+  ( : B T )
+  (
+    hyp betaexps2lem1
+    hyp betaexps2lem1 elisseti x T A lambda y eldm2 mpbi
+
+    x T A y df-lambda B y cv <,> eleq2i y biex mpbi
+    hyp betaexps2lem1 elisseti x x cv T : y cv A = /\ y sbcop1 y biex mpbir
+    
+    hyp betaexps2lem1 elisseti x x cv T : y cv A = sbca y biex mpbi
+    y B x x cv T : [/]  B x y cv A = [/] 19.42v mpbi
+    pm3.26i
+
+    hyp betaexps2lem1 elisseti x x cv T sbc: mpbi
+    hyp betaexps2lem1 elisseti x expsvar T :eq1i mpbi
+) )
+
+# A version of betaexps with a different type assumption - this time,
+# on the result of the application, rather than the inputs.
+thm ( betaexps2 ( ( T v ) ( T' v ) )
+  ( ( hyp ( : ( . ( lambda v T' A ) B ) T ) ) )
+  ( = ( . ( lambda v T' A ) B ) ( [:=] v B A ) )
+  ( hyp betaexps2lem1
+    hyp appfuni
+    hyp betaexps2lem2
+    betaexps4
+) )
+
+# A version of betaexps with yet another type assumption - this time,
+# on the result of the reduction.
+thm ( betaexps3 ( ( T x y ) ( T' x ) ( A y ) ( B y ) )
+  ( ( hyp ( : ( [:=] x B A ) T' ) )
+    ( hyp.B ( : B T ) ) )
+  ( = ( . ( lambda x T A ) B ) ( [:=] x B A ) )
+  (   hyp typeel elisseti y isseti
+
+      x T A y df-lambda
+      x T A lambda x y x cv T : y cv A = /\ {<,>|} dmeq ax-mp
+      x y x cv T : y cv A = /\ dmopab eqtr B eleq2i
+      B x y x cv T : y cv A = /\ E. df-sbc
+      bitr4
+      hyp.B typeel elisseti x y x cv T : y cv A = /\ sbcex2 bitr
+
+        hyp.B typeel elisseti x x cv T : y cv A = sbca
+          hyp.B typeel elisseti x x cv T sbc:
+        hyp.B typeel elisseti y cv x A expssb2 bicomi anbi12i
+        hyp.B hyp.B typeel elisseti x expsvar T :eq1i mpbir mpbiran bitr
+      y biex bitr
+    mpbir
+
+      x y x cv T : y cv A = /\ funopab
+        y A moeq
+      x cv T : moani mpgbir
+      x T A y df-lambda x T A lambda x y x cv T : y cv A = /\ {<,>|} funeq ax-mp
+    mpbir
+    hyp.B
+    betaexps4
+) )
+
+# A handy inference for creating simple equality sequents
+thm ( BETA3lem1 ( )
+  ( ( hyp1 ( = A B ) )
+    ( hyp2 ( : B T ) ) )
+  ( $|- ( true ) ( . ( . ( $= T ) A ) B ) )
+  ( hyp2 hyp1 T :eq1i mpbir
+    hyp1
+    
+    eqvalr
+) )
+
+thm ( BETA3lem2 ( ( T x ) ( T' x ) ( T'' x ) )
+  ( ( hyp.1 ( $|- ( true ) ( . ( . ( $= ( $-> T T' ) )
+      ( . ( lambda x T'' F ) A ) ) G ) ) )
+    ( hyp.2 ( $|- ( true ) ( . ( . ( $= T )
+      ( . ( lambda x T'' B ) A ) ) C ) ) ) )
+  ( = ( [:=] x A ( . F B ) ) ( . G C ) )
+  ( 
+    hyp.1 sequent2:bool eqtypei1 betaexps2lem1 elisseti
+      hyp.1 sequent2:bool eqtypei1 betaexps2
+    hyp.1 sequent2:bool eqtypei1 typeel elisseti eqeltrr
+      hyp.2 sequent2:bool eqtypei1 betaexps2
+    hyp.2 sequent2:bool eqtypei1 typeel elisseti eqeltrr
+    expsapp
+
+    hyp.1 sequent2:bool eqtypei1 betaexps2
+    hyp.1 eqvali eqtr3
+    x A B [:=] .eq1i eqtr
+
+    hyp.2 sequent2:bool eqtypei1 betaexps2
+    hyp.2 eqvali eqtr3
+    G .eq2i eqtr
+
+) )
+
+# application
+thm ( BETA3 ( ( T x ) ( T' x ) ( T'' x ) )
+  ( ( hyp.1 ( $|- ( true ) ( . ( . ( $= ( $-> T T' ) )
+      ( . ( lambda x T'' F ) A ) ) G ) ) )
+    ( hyp.2 ( $|- ( true ) ( . ( . ( $= T )
+      ( . ( lambda x T'' B ) A ) ) C ) ) ) )
+  ( $|- ( true ) ( . ( . ( $= T' )
+      ( . ( lambda x T'' ( . F B ) ) A ) ) ( . G C ) ) )
+  ( 
+      hyp.1 sequent2:bool eqtypei2
+      hyp.2 sequent2:bool eqtypei2
+    app:
+
+    hyp.1 hyp.2 BETA3lem2 T' :eq1i mpbir
+    hyp.1 sequent2:bool eqtypei1 betaexps2lem2
+    betaexps3
+    hyp.1 hyp.2 BETA3lem2 eqtr
+
+      hyp.1 sequent2:bool eqtypei2
+      hyp.2 sequent2:bool eqtypei2
+    app:
+    BETA3lem1
+) )
+
+thm ( BETA4lem ( ( T x y ) ( T' x y ) ( T'' x y ) ( A y ) )
+  ( ( hyp ( $|- ( true ) ( . ( . ( $= T )
+      ( . ( lambda x T'' B ) A ) ) C ) ) )
+    ( hyp.T' ( istype T' ) ) )
+  ( = ( [:=] x A ( lambda y T' B ) ) ( lambda y T' C ) )
+  (
+      hyp sequent2:bool eqtypei1 betaexps2lem2 typeel elisseti
+    x y T' B expsabs2
+
+      y x A B [:=] C T' abseq2
+    hyp sequent2:bool eqtypei1 betaexps2
+    hyp eqvali eqtr3 mpg
+    eqtr
+) )
+
+# abstraction
+thm ( BETA4 ( ( T x y ) ( T' x y ) ( T'' x y ) ( A y ) )
+  ( ( hyp ( $|- ( true ) ( . ( . ( $= T )
+      ( . ( lambda x T'' B ) A ) ) C ) ) )
+    ( hyp.T' ( istype T' ) ) )
+  ( $|- ( true ) ( . ( . ( $= ( $-> T' T ) )
+      ( . ( lambda x T'' ( lambda y T' B ) ) A ) ) ( lambda y T' C ) ) )
+  (
+    hyp.T' hyp sequent2:bool eqtypei2 y abs:
+    hyp hyp.T' y BETA4lem T' T $-> :eq1i mpbir    
+    hyp sequent2:bool eqtypei1 betaexps2lem2
+    betaexps3
+
+    hyp hyp.T' y BETA4lem eqtr
+
+    hyp.T' hyp sequent2:bool eqtypei2 y abs:
+    BETA3lem1
+) )
+
+thm ( ASSUME ( )
+  ( ( hyp ( : A ( bool ) ) ) )
+  ( $|- A A )
+  ( hyp hyp A istrue id mksequent
+) )
+
+thm ( SUBST_app ( )
+  ( ( hyp.1 ( $|- G1 ( . ( . ( $= ( $-> T T' ) ) A ) B ) ) )
+    ( hyp.2 ( $|- G2 ( . ( . ( $= T ) C ) D ) ) ) )
+  ( $|- ( $, G1 G2 ) ( . ( . ( $= T' ) ( . A C ) ) ( . B D ) ) )
+  ( hyp.1 sequent1:bool
+    hyp.2 sequent1:bool
+
+    hyp.1 sequent2:bool eqtypei1 hyp.2 sequent2:bool eqtypei1 app:
+    hyp.1 sequent2:bool eqtypei2 hyp.2 sequent2:bool eqtypei2 app: eqval:bool
+
+    hyp.1 sequentimp
+    hyp.1 sequent2:bool eqtypei1 hyp.1 sequent2:bool eqtypei2 eqval sylib
+    A B C .eq1 syl
+
+    hyp.2 sequentimp
+    hyp.2 sequent2:bool eqtypei1  hyp.2 sequent2:bool eqtypei2 eqval sylib
+    C D B .eq2 syl sylan9eq
+
+    hyp.1 sequent2:bool eqtypei1 hyp.2 sequent2:bool eqtypei1 app:
+    hyp.1 sequent2:bool eqtypei2 hyp.2 sequent2:bool eqtypei2 app: eqval sylibr mkanseq
+) )
+
+thm ( ABS ( ( G x ) ( T x ) ( T' x ) )
+  ( ( hyp.3 ( istype T' ) )
+    ( hyp.4 ( $|- G  ( . ( . ( $= T ) A ) B ) ) ) )
+  ( $|- G
+    ( . ( . ( $= ( $-> T' T ) ) ( lambda x T' A ) ) ( lambda x T' B ) ) )
+  ( hyp.4 sequent1:bool
+        hyp.3 hyp.4 sequent2:bool eqtypei1 x abs:
+        hyp.3 hyp.4 sequent2:bool eqtypei2 x abs:
+      eqval:bool
+
+      hyp.4 sequentimp
+      hyp.4 sequent2:bool eqtypei1  hyp.4 sequent2:bool eqtypei2 eqval sylib x 19.21aiv
+      x A B T' abseq2 syl
+        hyp.3 hyp.4 sequent2:bool eqtypei1 x abs:
+        hyp.3 hyp.4 sequent2:bool eqtypei2 x abs:
+      eqval sylibr
+    mksequent
+) )
+
+thm ( DISCH ( )
+  ( ( hyp ( $|- ( $, G1 A ) B ) ) )
+  ( $|- G1 ( . ( . ( $==> ) A ) B ) )
+  ( hyp sequent1:bool G1 A comma:bool mpbir pm3.26i
+
+      hyp sequent1:bool G1 A comma:bool mpbir pm3.27i
+    hyp sequent2:bool impval:bool
+
+      hyp sequent1:bool commaval2
+      hyp sequentimp sylbir exp
+
+        hyp sequent1:bool G1 A comma:bool mpbir pm3.27i
+      hyp sequent2:bool impval sylibr
+    mksequent
+) )
+
+# This form writes the union of the assumptions as an explicit term in
+# the statement proved, suitable for subsequent manipulation by inference.
+# There are, of course, other ways to do this.
+thm ( MP ( )
+  ( ( hyp.2 ( $|- G1 ( . ( . ( $==> ) A ) B ) ) )
+    ( hyp.3 ( $|- G2 A ) ) )
+  ( $|- ( $, G1 G2 ) B )
+  ( hyp.2 sequent1:bool hyp.3 sequent1:bool
+    imptype hyp.3 sequent2:bool app: hyp.2 sequent2:bool apptypei
+    hyp.2 sequentimp hyp.3 sequent2:bool
+      imptype hyp.3 sequent2:bool app: hyp.2 sequent2:bool apptypei
+    impval sylib
+    hyp.3 sequentimp syl5 imp mkanseq
+) )
+
+# A derived rule in HOL, but core here, as it's used to implement
+# SUBST.
+thm ( EQ_MP ( )
+  ( ( hyp.2 ( $|- G1 ( . ( . ( $= ( bool ) ) A ) B ) ) )
+    ( hyp.3 ( $|- G2 A ) ) )
+  ( $|- ( $, G1 G2 ) B )
+  ( hyp.2 sequent1:bool hyp.3 sequent1:bool
+    boolistype eq: hyp.3 sequent2:bool app: hyp.2 sequent2:bool apptypei
+    hyp.2 sequentimp hyp.3 sequent2:bool
+      boolistype eq: hyp.3 sequent2:bool app: hyp.2 sequent2:bool apptypei
+    eqval sylib istrueeqd biimpd
+    hyp.3 sequentimp syl5 imp mkanseq
+) )
+
+# This is a derived rule in HOL, but is a core rule here as it's used
+# to manipulate assumptions.
+
+thm ( ADD_ASSUM ( )
+  ( ( hyp.1 ( $|- G1 A ) )
+    ( hyp.2 ( : G2 ( bool ) ) ) )
+  ( $|- ( $, G1 G2 ) A )
+  ( hyp.1 sequent1:bool
+    hyp.2
+    hyp.1 sequent2:bool
+    hyp.1 sequentimp G2 istrue adantr
+    mkanseq
+) )
+
+# One of the basic assumption manipulation rules
+thm ( ASSUM_t ( )
+ ( ( hyp ( $|- ( $, G ( true ) ) A ) ) )
+ ( $|- G A )
+ (
+   trueistrue hyp sequent1:bool commaval2 biimpr mpan2
+   hyp sequent1:bool G true comma:bool mpbir pm3.26i
+   hyp assum_twiddle
+) )
+
+thm ( ASSUM_rot ( )
+  ( ( hyp ( $|- ( $, ( $, G1 G2 ) G3 ) A ) ) )
+  ( $|- ( $, ( $, G2 G3 ) G1 ) A )
+  ( 
+    hyp sequent1:bool commaval2
+      hyp sequent1:bool comma:bool1 commaval2
+    G3 istrue anbi1i bitr
+      G1 istrue G2 istrue G3 istrue anass G1 istrue G2 istrue G3 istrue /\ ancom bitr
+    bitr
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool2
+    comma:booli
+    hyp sequent1:bool comma:bool1 comma:bool1
+    comma:booli commaval2
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool2
+    comma:booli commaval2 G1 istrue anbi1i bitr
+    bitr4 biimpr
+
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool2
+    comma:booli
+    hyp sequent1:bool comma:bool1 comma:bool1
+    comma:booli
+
+    hyp assum_twiddle
+) )
+
+thm ( ASSUM_swap ( )
+  ( ( hyp ( $|- ( $, ( $, G1 G2 ) G3 ) A ) ) )
+  ( $|- ( $, ( $, G2 G1 ) G3 ) A )
+  ( 
+      hyp sequent1:bool commaval2
+      hyp sequent1:bool comma:bool1 commaval2
+    G3 istrue anbi1i bitr
+      G1 istrue G2 istrue ancom G3 istrue anbi1i
+    bitr
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool1 comma:bool1
+    comma:booli
+    hyp sequent1:bool comma:bool2
+    comma:booli commaval2
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool1 comma:bool1
+    comma:booli commaval2 G3 istrue anbi1i bitr
+    bitr4 biimpr
+
+      hyp sequent1:bool comma:bool1 comma:bool2
+      hyp sequent1:bool comma:bool1 comma:bool1
+    comma:booli
+    hyp sequent1:bool comma:bool2
+    comma:booli
+
+    hyp assum_twiddle
+) )
+
+thm ( ASSUM_absorb ( )
+  ( ( hyp ( $|- ( $, G1 ( $, G2 G2 ) ) A ) ) )
+  ( $|- ( $, G1 G2 ) A )
+  ( 
+      hyp sequent1:bool commaval2
+      hyp sequent1:bool comma:bool2 commaval2
+    G1 istrue anbi2i bitr
+      G2 istrue anidm G1 istrue anbi2i
+    bitr
+      hyp sequent1:bool comma:bool1
+      hyp sequent1:bool comma:bool2 comma:bool1
+    comma:booli commaval2
+    bitr4 biimpr
+
+      hyp sequent1:bool comma:bool1
+      hyp sequent1:bool comma:bool2 comma:bool1
+    comma:booli
+
+    hyp assum_twiddle
+) )
+
+thm ( betanotboti_lem1 ( )
+  ( ( hyp ( notbottom ( . ( lambda x T A ) B ) ) ) )
+  ( e. B ( dom ( lambda x T A ) ) )
+  ( hyp
+    x T A lambda B . df-notbottom mpbi
+    x T A lambda B appisset mpbir
+    x T A lambda B appdom ax-mp
+) )
+
+thm ( betanotboti ( ( T x y ) ( A y ) ( B y ) )
+  ( ( hyp ( notbottom ( . ( lambda x T A ) B ) ) ) )
+  ( : B T )
+  ( hyp betanotboti_lem1
+      x T A y df-lambda
+      x T A lambda x y x cv T : y cv A = /\ {<,>|} dmeq ax-mp
+      x y x cv T : y cv A = /\ dmopab eqtr B eleq2i
+    mpbi
+      B x y x cv T : y cv A = /\ E. df-sbc mpbir
+    hyp betanotboti_lem1 elisseti x y x cv T : y cv A = /\ sbcex2 mpbi
+    hyp betanotboti_lem1 elisseti x x cv T : y cv A = sbca y biex mpbi
+    B x x cv T : [/] B x y cv A = [/] pm3.26 y 19.23aiv ax-mp
+    hyp betanotboti_lem1 elisseti x x cv T sbc: mpbi
+    hyp betanotboti_lem1 elisseti x expsvar T :eq1i mpbi
+) )
+
+# Some basic defs that could be in their own module, but are convenient here.
+
+def ( ( $! T )
+  ( lambda f ( $-> T ( bool ) )
+    ( . ( . ( $= ( $-> T ( bool ) ) ) ( var f ( $-> T ( bool ) ) ) )
+        ( lambda x T ( true ) ) ) ) )
+
+thm (.eq1d ()
+  ((hyp (-> ph (= F G))))
+  (-> ph (= (. F A) (. G A)))
+  (hyp F G A .eq1 syl))
+
+thm (.eq2d ()
+  ((hyp (-> ph (= A B))))
+  (-> ph (= (. F A) (. F B)))
+  (hyp A B F .eq2 syl))
+
+thm (df-$!-half ((T f g x y)) ()
+  (= (lambda f ($-> T (bool)) (. (. ($= ($-> T (bool))) (var f ($-> T (bool))))
+                                 (lambda x T (true))))
+     (lambda g ($-> T (bool)) (. (. ($= ($-> T (bool))) (var g ($-> T (bool))))
+                                 (lambda y T (true)))))
+  (
+   f g T bool $-> vareq1
+   T bool $-> $= .eq2d
+   x T true lambda .eq1d
+     true eqid x cv y cv = a1i T alphaconv
+     T bool $-> $= g T bool $-> var . .eq2i
+   syl6eq
+   T bool $-> alphaconv
+))
+
+thm (df-$! ((T f' x' g y) (T f x g y))
+  ((hyp (istype T )))
+  ($|- ( true ) ( . ( . ( $= ( $-> ( $-> T ( bool ) ) ( bool ) ) ) ( $! T ) )
+    ( lambda f ( $-> T ( bool ) )
+      ( . ( . ( $= ( $-> T ( bool ) ) ) ( var f ( $-> T ( bool ) ) ) )
+          ( lambda x T ( true ) ) ) ) ) )
+  (
+   hyp boolistype arrowistype
+     hyp boolistype arrowistype eq:
+       hyp boolistype arrowistype f' var:
+     app:
+       hyp true:bool x' abs:
+     app:
+   f' abs:
+
+     f' T x' g y df-$!-half
+     f T x g y df-$!-half eqtr4
+   eqvalr
+))
+
+# Relating the universal quantifiers of HOL and ZFC. The initial
+# beta-reduction of the $! application is pretty painful here - it's
+# likely that it could be redone in HOL more easily.
+thm ( forall-$! ( ( T f x ) ( T y ) ( x y ) ( A y ) )
+  ( ( hyp.T ( istype T ) )
+    ( hyp.A ( : A ( bool ) ) ) )
+  ( <-> ( istrue ( . ( $! T ) ( lambda x T A ) ) )
+        ( A. x ( -> ( e. ( cv x ) ( typeset T ) ) ( istrue A ) ) ) )
+  ( trueistrue hyp.T f x df-$! sequentimp ax-mp
+      hyp.T f x df-$! sequent2:bool eqtypei1
+      hyp.T f x df-$! sequent2:bool eqtypei2
+    eqval mpbi
+    x T A lambda .eq1i
+
+          hyp.T boolistype arrowistype eq:
+        hyp.T boolistype arrowistype f var: app:
+      hyp.T true:bool x abs: app:
+      hyp.T hyp.A x abs:
+    f betaexps eqtr
+
+      hyp.T x A absex
+        hyp.T boolistype arrowistype eq:
+        hyp.T boolistype arrowistype f var:
+      app: typeel elisseti
+      hyp.T true:bool x abs: typeel elisseti
+        hyp.T x A absex
+        hyp.T boolistype arrowistype eq: typeel elisseti
+        hyp.T boolistype arrowistype f var: typeel elisseti
+        hyp.T x A absex f T bool $-> $= expsf
+          hyp.T x A absex f T bool $-> df-var f expseq2i
+          hyp.T x A absex f f cv T bool $-> expscast eqtr
+          hyp.T x A absex f expsvar T bool $-> casteq1i eqtr
+          hyp.T hyp.A x abs: x T A lambda T bool $-> castt ax-mp
+        eqtr
+      expsappc
+      hyp.T x A absex f x T true lambda expsf
+    expsappc eqtr istrueeqi
+
+    hyp.T hyp.A x abs: hyp.T true:bool x abs: eqval bitr
+
+      hyp.T hyp.A x abs: arrowi1 x T A lambda T typeset bool typeset ffn ax-mp
+      hyp.T true:bool x abs: arrowi1 x T true lambda T typeset bool typeset ffn ax-mp
+      x T A lambda T typeset x T true lambda T typeset y eqfnfv
+    mp2an T typeset eqid mpbiran
+
+    y T typeset x T A lambda y cv ` x T true lambda y cv ` = df-ral bitr
+
+      y cv T df-: hyp.T mpbiran
+          y T vart y cv y T var x T A lambda .eq2 syl 
+        hyp.T hyp.A x abs: y cv appval eqtr3d
+            y T vart y cv y T var x T true lambda .eq2 syl
+          hyp.T true:bool x abs: y cv appval eqtr3d
+            true:bool hyp.T y var: x betaexps    
+          hyp.T y var: typeel elisseti x true expsf eqtr
+        syl5eqr
+      eqeq12d  x T A lambda y T var . df-istrue
+      hyp.A hyp.T y var: x betaexps istrueeqi bitr3
+      hyp.T y var: typeel elisseti hyp.A typeel elisseti x expsistrue bitr4 syl5bbr
+    sylbir pm5.74i y bial bitr4
+
+    bitr
+    x cv T typeset e. A istrue -> y ax-17 x sb8
+
+        y visset x expsvar
+        y visset x T typeset expsf
+      eleq12i
+      y visset x x cv T typeset expssbel
+    bitr3 y cv x A istrue [/] imbi1i
+
+    y visset x x cv T typeset e. A istrue sbci2 bitr4
+
+    y cv T df-: hyp.T mpbiran
+      y T vart y cv y T var x A expseq1 syl
+      istrueeqd
+        y visset hyp.A typeel elisseti x expsistrue
+        hyp.T y var: typeel elisseti hyp.A typeel elisseti x expsistrue
+      bibi12i sylibr
+   sylbir pm5.74i bitr3 y bial bitr bitr4
+) )
+
+# We adapt the ZFC gen rule to HOL in this proof, but it could probably
+# be done a completely different way, proved entirely in HOL.
+thm ( gen-$! ( ( T x ) )
+  ( ( hyp.T ( istype T ) )
+    ( hyp ( istrue A ) ) )
+  ( istrue ( . ( $! T ) ( lambda x T A ) ) )
+  ( hyp x cv T typeset e. a1i x ax-gen
+    hyp.T hyp istrue:booli x forall-$! mpbir
+) )
+
+thm ( $!-type ((T x y))
+  ((hyp (istype T)))
+  (: ($! T) ($-> ($-> T (bool)) (bool)))
+  ( hyp y x df-$! sequent2:bool eqtypei1
+))
+
+def (($F)
+  (. ($! (bool)) (lambda x (bool) (var x (bool)))))
+
+thm (df-$F ((x y) (x' y)) ()
+  ($|- (true) (. (. ($= (bool)) ($F)) (. ($! (bool)) (lambda x' (bool) (var x' (bool))))))
+  (
+   boolistype $!-type boolistype boolistype x var: x abs: app:
+     x y bool vareq1
+     bool alphaconv
+     bool $! .eq2i
+       x' y bool vareq1
+       bool alphaconv
+       bool $! .eq2i
+     eqtr4
+   eqvalr
+))
+
+thm ($F:bool () ()
+  (: ($F) (bool))
+  ( x df-$F sequent2:bool eqtypei1 ))
+
+# If we have one counterexample, then the universal quantification is
+# also false.
+thm (refute ((x A))
+  ((hyp.1 (e. A V))
+   (hyp.2 (-> (= (cv x) A) (-. ph))))
+  (-. (A. x ph))
+  ( hyp.1 x isseti hyp.2 x 19.22i ax-mp
+    x ph exnal mpbi
+))
+
+# the jc ... sylibr sequence should arguably be its own propcalc theorem,
+# and "eleq1a ax-mp" arguably should be as well (it frequently appears as
+# "eleq1 mpbii" in set.mm).
+thm ($F-false () ()
+  (= ($F) (false))
+  ( x df-$F eqvali
+
+    false:bool typeel elisseti
+
+      false:bool typeel false bool typeset x cv eleq1a ax-mp
+
+      nfalseistrue    
+        false:bool x cv false bool :eq1 mpbiri x bool vart syl
+        x cv false = id eqtr3d
+        istrueeqd
+      mtbiri
+      jc x bool var istrue pm4.13 x cv bool typeset e. imbi2i negbii sylibr
+    refute
+
+    boolistype boolistype x var: x forall-$!
+    mtbir
+    nfalseistrue 2th bicon4i
+
+      boolistype $!-type boolistype boolistype x var: x abs: app:
+      false:bool
+    istrueeqbi mpbir eqtr
+))
+
+def (($~) (lambda x (bool) (. (. ($==>) (var x (bool))) ($F))))
+
+thm (df-$~ ((x' y) (x y)) ()
+  ($|- (true) (. (. ($= ($-> (bool) (bool))) ($~))
+    (lambda x (bool) (. (. ($==>) (var x (bool))) ($F)))))
+  (
+   boolistype imptype boolistype x' var: app: $F:bool app: x' abs:
+     x' y bool vareq1 $==> .eq2d $F .eq1d bool alphaconv
+       x y bool vareq1 $==> .eq2d $F .eq1d bool alphaconv
+     eqtr4
+   eqvalr
+))
+
+thm ($~: () ()
+  (: ($~) ($-> (bool) (bool)))
+  ( x df-$~ sequent2:bool eqtypei1
+))
+
+# dup'd from hol-zfc-derived
+thm ( BETA31 ( ( T'' x ) ( T x ) ( B x ) ( T' x ) )
+  ( ( hyp ( $|- ( true ) ( . ( . ( $= ( $-> T T' ) )
+      ( . ( lambda x T'' F ) A ) ) F' ) ) )
+    ( hyp.B ( : B T ) ) )
+  ( $|- ( true ) ( . ( . ( $= T' )
+      ( . ( lambda x T'' ( . F B ) ) A ) ) ( . F' B ) ) )
+  ( hyp
+      hyp sequent2:bool eqtypei1 typenotbottomi betanotboti
+    hyp.B x BETA2
+    BETA3
+) )
+
+thm ( BETA32 ( ( T'' x ) ( T x ) ( F x ) ( T' x ) )
+  ( ( hyp ( $|- ( true ) ( . ( . ( $= T )
+      ( . ( lambda x T'' B ) A ) ) C ) ) )
+    ( hyp.F ( : F ( $-> T T' ) ) ) )
+  ( $|- ( true ) ( . ( . ( $= T' )
+      ( . ( lambda x T'' ( . F B ) ) A ) ) ( . F C ) ) )
+  (
+      hyp sequent2:bool eqtypei1 typenotbottomi betanotboti
+    hyp.F x BETA2
+    hyp BETA3
+) )
+
+thm (not-$~lem ()
+  ((hyp (: A (bool))))
+  (<-> (istrue (. (. ($==>) A) ($F))) (-. (istrue A)))
+  ( hyp $F:bool impval
+    A istrue $F istrue pm4.1 bitr
+    nfalseistrue $F-false istrueeqi negbii mpbir A istrue -. a1bi bitr4
+))
+
+thm (not-$~ ()
+  ((hyp (: A (bool))))
+  (<-> (istrue (. ($~) A)) (-. (istrue A)))
+  ( x df-$~ eqvali A .eq1i
+
+    hyp x BETA1 imptype BETA32 $F:bool BETA31 eqvali
+    eqtr istrueeqi
+
+    hyp not-$~lem bitr
+))
+
+def (($\/) (lambda x (bool) (lambda y (bool)
+  (. ($! (bool)) (lambda z (bool)
+    (. (. ($==>) (. (. ($==>) (var x (bool))) (var z (bool))))
+      (. (. ($==>) (. (. ($==>) (var y (bool))) (var z (bool))))
+        (var z (bool)))))))))
+
+thm (alcases ((x ch) (x et))
+  ((hyp1 (-> ph (<-> ps ch)))
+   (hyp2 (-> th (<-> ps et)))
+   (hyp3 (E. x ph))
+   (hyp4 (E. x th)))
+   (<-> (A. x (-> (\/ ph th) ps)) (/\ ch et))
+  ( ph th ps jaob
+    hyp1 pm5.74i hyp2 pm5.74i anbi12i bitr x bial
+    x ph ch -> th et -> 19.26 bitr
+    x ph ch 19.23v hyp3 ch a1bi bitr4 
+    x th et 19.23v hyp4 et a1bi bitr4 anbi12i bitr
+))
+
+thm (.eq12d ()
+  ((hyp1 (-> ph (= F G)))
+   (hyp2 (-> ph (= A B))))
+  (-> ph (= (. F A) (. G B)))
+  (hyp1 A .eq1d hyp2 G .eq2d eqtrd))
+
+thm (abseq2i ((T v))
+  ((hyp (= A B)))
+  (= (lambda v T A) (lambda v T B))
+  (hyp v ax-gen
+     v A B T abseq2
+   ax-mp
+))
+
+thm (abseq2d ((ph v) (T v))
+  ((hyp (-> ph (= A B))))
+  (-> ph (= (lambda v T A) (lambda v T B)))
+  (hyp v 19.21aiv
+     v A B T abseq2
+   syl
+))
+
+thm (df-$\/-half ((x y z x' y' z')) ()
+  (=
+    (lambda x (bool) (lambda y (bool)
+      (. ($! (bool)) (lambda z (bool)
+        (. (. ($==>) (. (. ($==>) (var x (bool))) (var z (bool))))
+          (. (. ($==>) (. (. ($==>) (var y (bool))) (var z (bool))))
+            (var z (bool))))))))
+    (lambda x' (bool) (lambda y' (bool)
+      (. ($! (bool)) (lambda z' (bool)
+        (. (. ($==>) (. (. ($==>) (var x' (bool))) (var z' (bool))))
+          (. (. ($==>) (. (. ($==>) (var y' (bool))) (var z' (bool))))
+            (var z' (bool)))))))))
+  (
+   y y' bool vareq1 $==> .eq2d z' bool var .eq1d $==> .eq2d z' bool var .eq1d
+   $==> $==> x bool var . z' bool var . . .eq2d
+   z' bool abseq2d
+
+     z z' bool vareq1 $==> x bool var . .eq2d $==> .eq2d
+       z z' bool vareq1 $==> y bool var . .eq2d $==> .eq2d
+         z z' bool vareq1
+       .eq12d
+     .eq12d bool alphaconv
+
+   syl5eq
+   bool $! .eq2d bool alphaconv
+   x bool abseq2i
+
+     x x' bool vareq1 $==> .eq2d z' bool var .eq1d
+     $==> .eq2d
+     $==> $==> y' bool var . z' bool var . . z' bool var . .eq1d
+     z' bool abseq2d
+     bool $! .eq2d
+     y' bool abseq2d
+     bool alphaconv
+
+   eqtr
+))
+
+thm (df-$\/ ((x y z x'' y'' z'') (x' y' z' x'' y'' z'')) ()
+  ($|- (true) (. (. ($= ($-> (bool) ($-> (bool) (bool)))) ($\/))
+    (lambda x (bool) (lambda y (bool)
+      (. ($! (bool)) (lambda z (bool)
+        (. (. ($==>) (. (. ($==>) (var x (bool))) (var z (bool))))
+          (. (. ($==>) (. (. ($==>) (var y (bool))) (var z (bool))))
+            (var z (bool))))))))))
+  (
+   boolistype boolistype
+       boolistype $!-type
+         boolistype
+           imptype imptype boolistype x' var: app: boolistype z' var: app: app:
+             imptype
+             imptype boolistype y' var: app: boolistype z' var: app: app:
+           boolistype z' var: app: app:
+       z' abs: app:
+   y' abs: x' abs:
+   
+     x' y' z' x'' y'' z'' df-$\/-half
+       x y z x'' y'' z'' df-$\/-half
+     eqtr4
+
+   eqvalr
+))
+
+thm ($\/: ((x y z)) ()
+  (: ($\/) ($-> (bool) ($-> (bool) (bool))))
+  ( x y z df-$\/ sequent2:bool eqtypei1
+))
+
+# The axioms of the theory bool, which make use of the defs above.
+
+thm ( IMP_ANTISYM_AX ( ) ( )
+  ( $|- ( true )
+    ( . ( $! ( bool ) ) ( lambda a ( bool )
+      ( . ( $! ( bool ) ) ( lambda b ( bool )
+        ( . ( . ( $==> ) ( . ( . ( $==> ) ( var a ( bool ) ) ) ( var b ( bool ) ) ) )
+           ( . ( . ( $==> ) ( . ( . ( $==> ) ( var b ( bool ) ) ) ( var a ( bool ) ) ) )
+             ( . ( . ( $= ( bool ) ) ( var a ( bool ) ) ) ( var b ( bool ) ) ) ) ) ) ) ) ) )
+  ( boolistype
+    boolistype
+
+    a bool var istrue b bool var istrue bi3
+
+      imptype boolistype a var: app: boolistype b var: app:
+        imptype boolistype b var: app: boolistype a var: app:
+        boolistype eq: boolistype a var: app: boolistype b var: app:
+      impval:bool
+    impval
+
+    boolistype a var: boolistype b var: impval
+
+      imptype boolistype b var: app: boolistype a var: app:
+      boolistype eq: boolistype a var: app: boolistype b var: app:
+    impval
+      boolistype b var: boolistype a var: impval
+      boolistype a var: boolistype b var: eqval
+    boolistype a var: boolistype b var: istrueeqbi bitr imbi12i
+    bitr imbi12i bitr
+
+    mpbir
+    b gen-$!
+    a gen-$!
+    mknullseq
+) )
+
+thm (df-$suc ((x y) (x' y)) ()
+  (= ($suc) (lambda x' ($ind) (suc (var x' ($ind)))))
+  (
+   x y $ind vareq1
+   x $ind var y $ind var suceq syl
+   $ind alphaconv
+     x' y $ind vareq1
+     x' $ind var y $ind var suceq syl
+     $ind alphaconv
+   eqtr4
+))
+
+thm (typesetind () ()
+  (= (typeset ($ind)) (om))
+  ( df-$ind typeseteqi omex 0ex typesetval1 eqtr
+))
+
+thm ($0:$ind () ()
+  (: ($0) ($ind))
+  (indistype df-$0 peano1 eqeltr typesetind eleqtrr pm3.2i $0 $ind df-: mpbir
+))
+
+thm ($suc: () ()
+  (: ($suc) ($-> ($ind) ($ind)))
+  ( indistype indistype
+      indistype x var: typeel
+      typesetind eleqtr
+      x $ind var peano2 ax-mp
+    typesetind eleqtrr
+    pm3.2i x $ind var suc $ind df-: mpbir x abs:
+    x df-$suc $ind $ind $-> :eq1i mpbir
+))
+
+thm ($suc-suc ((A x))
+  ((hyp (: A ($ind))))
+  (= (. ($suc) A) (suc A))
+  (x df-$suc A .eq1i
+     $suc: x df-$suc $ind $ind $-> :eq1i mpbi hyp app: betaexps2
+       hyp typeel elisseti
+         hyp x cv A $ind :eq1 mpbiri x $ind vart syl x cv A = id eqtr3d
+     x $ind var A suceq syl imps2exps eqtr
+   eqtr
+))
+
+thm ( nsuconto () ()
+  ($|- (true) (. ($! ($ind)) (lambda x ($ind)
+    (. ($~) (. (. ($= ($ind)) (. ($suc) (var x ($ind)))) ($0))))))
+  ( x $ind var nsuceq0
+    df-$0 x $ind var suc eqeq2i negbii mpbir
+      indistype eq: $suc: indistype x var: app: app: $0:$ind app: not-$~
+        $suc: indistype x var: app: $0:$ind eqval
+      indistype x var: $suc-suc $0 eqeq1i bitr negbii bitr mpbir
+    x cv $ind typeset e. a1i x ax-gen
+
+    indistype
+      $~:
+      indistype eq: $suc: indistype x var: app: app: $0:$ind app:
+    app:
+    x forall-$! mpbir mknullseq
+))
+
+thm ( suc1to1 () ()
+  ($|- (true)
+    (. ($! ($ind)) (lambda x ($ind)
+      (. ($! ($ind)) (lambda y ($ind)
+        (. (. ($==>) (. (. ($= ($ind)) (. ($suc) (var x ($ind)))) (. ($suc) (var y ($ind)))))
+          (. (. ($= ($ind)) (var x ($ind))) (var y ($ind)))))))))
+  ( indistype indistype
+
+      $suc: indistype x var: app:
+    $suc: indistype y var: app: eqval
+      indistype x var: $suc-suc
+    indistype y var: $suc-suc eqeq12i bitr
+
+      indistype x var: typeel
+    typesetind eleqtr
+      indistype y var: typeel
+    typesetind eleqtr
+
+    x $ind var y $ind var peano4 mp2an bitr
+    indistype x var: indistype y var: eqval bitr4 biimp
+
+      indistype eq: $suc: indistype x var: app: app: $suc: indistype y var: app: app:
+      indistype eq: indistype x var: app: indistype y var: app:
+    impval mpbir
+
+    y gen-$! x gen-$! mknullseq
+))
+
+thm ( etacore ((T f x y) (T' f x y) (A x y))
+  ((hyp (: A ($-> T T'))))
+  (= A (lambda x T (. A (var x T))))
+  ( hyp arrowi1
+    A T typeset T' typeset ffn ax-mp
+    A T typeset x y fnopabfv mpbi
+
+    x T A x T var . y df-lambda 
+          hyp x cv appval
+          x T vart x cv x T var A .eq2 syl
+        eqtr3d
+      y cv eqeq2d
+      pm5.32i
+        x cv T df-: hyp typeistype arrow2i1 mpbiran
+      y cv A x cv ` = anbi1i
+    bitr3 x y biopabi
+    eqtr eqtr4
+))
+
+thm ( ETA_AX ((T f x) (T' f x))
+  ((hyp (istype ($-> T T'))))
+  ($|- (true) (. ($! ($-> T T')) (lambda f ($-> T T')
+    (. (. ($= ($-> T T')) (lambda x T (. (var f ($-> T T')) (var x T))))
+                          (var f ($-> T T'))))))
+  ( hyp
+    trueistrue
+      hyp arrow2i1 hyp f var: hyp arrow2i1 x var: app: x abs:
+      hyp f var: x etacore eqcomi
+    eqvalr sequentimp ax-mp f gen-$! mknullseq
+))
+
+thm (forall-bool ((A x))
+  ((hyp (: A ($-> (bool) (bool)))))
+  (<-> (istrue (. ($! (bool)) A))
+       (/\ (istrue (. A (false))) (istrue (. A (true)))))
+  ( hyp x etacore bool $! .eq2i istrueeqi
+
+    boolistype hyp boolistype x var: app: x forall-$!
+
+          df-bool typeseteqi
+        false true prex df-false 0ex eqeltr typesetval1 eqtr x cv eleq2i
+      x visset false true elpr bitr
+    A x bool var . istrue imbi1i x bial
+
+          false:bool x cv false bool :eq1 mpbiri x bool vart syl
+        x cv false = id eqtr3d
+      x bool var false A .eq2 syl istrueeqd
+          true:bool x cv true bool :eq1 mpbiri x bool vart syl
+        x cv true = id eqtr3d
+      x bool var true A .eq2 syl istrueeqd
+      false:bool typeel elisseti x isseti
+      true:bool typeel elisseti x isseti
+    alcases bitr bitr
+
+    bitr
+))
+
+thm (or-$\/lem ()
+  ((hyp1 (: A (bool)))
+   (hyp2 (istrue B)))
+  (istrue (. (. ($==>) A) B))
+  ( hyp2 A istrue a1i
+    hyp1 hyp2 istrue:booli impval mpbir
+))
+
+# We probably don't need both this one and not-$~lem; they're the same
+# modulo false/$F
+thm (or-$\/lem2 ()
+  ((hyp (: A (bool))))
+  (<-> (istrue (. (. ($==>) A) (false))) (-. (istrue A)))
+  ( hyp false:bool impval
+    A istrue false istrue pm4.1 bitr
+    nfalseistrue A istrue -. a1bi bitr4
+))
+
+thm (or-$\/ ((A x y z) (B x y z))
+  ((hyp1 (: A (bool)))
+   (hyp2 (: B (bool))))
+  (<-> (istrue (. (. ($\/) A) B)) (\/ (istrue A) (istrue B)))
+  ( x y z df-$\/ eqvali A .eq1i B .eq1i
+
+      hyp1 x BETA1 imptype BETA32 boolistype z var: BETA31
+      imptype BETA32
+        imptype imptype boolistype y var: app: boolistype z var: app: app:
+      boolistype z var: app: BETA31
+      boolistype z BETA4 boolistype $!-type BETA32
+    boolistype y BETA4 eqvali B .eq1i eqtr
+
+    hyp2 y BETA1 imptype BETA32 boolistype z var: BETA31
+    imptype BETA32 boolistype z var: BETA31
+    imptype imptype hyp1 app: boolistype z var: app: app: BETA32
+    boolistype z BETA4 boolistype $!-type BETA32
+    eqvali eqtr istrueeqi
+
+    boolistype imptype
+      imptype hyp1 app: boolistype z var: app: app:
+      imptype
+        imptype hyp2 app: boolistype z var: app: app:
+      boolistype z var: app:
+    app: z abs: forall-bool
+
+    false:bool z BETA1 imptype hyp1 app: BETA32 imptype BETA32
+    false:bool z BETA1 imptype hyp2 app: BETA32 imptype BETA32
+    false:bool z BETA1 BETA3 BETA3 eqvali istrueeqi
+
+      imptype hyp1 app: false:bool app:
+      imptype imptype hyp2 app: false:bool app: app: false:bool app:
+    impval
+      hyp1 or-$\/lem2
+        imptype hyp2 app: false:bool app: or-$\/lem2
+      hyp2 or-$\/lem2 bicon2i bitr4
+    imbi12i bitr
+    A istrue B istrue df-or bitr4
+    bitr
+
+      imptype hyp1 app: true:bool app:
+        imptype hyp2 app: true:bool app:
+        trueistrue
+      or-$\/lem
+    or-$\/lem
+
+    true:bool z BETA1 imptype hyp1 app: BETA32 imptype BETA32
+    true:bool z BETA1 imptype hyp2 app: BETA32 imptype BETA32
+    true:bool z BETA1 BETA3 BETA3 eqvali istrueeqi
+    mpbir
+      z bool
+        $==> $==> A . z bool var . .
+        $==> $==> B . z bool var . . z bool var . .
+      lambda false . istrue
+    biantru bitr3 bitr4
+    bitr
+))
+
+thm (BOOL_CASES () ()
+  ($|- (true) (. ($! (bool)) (lambda x (bool)
+    (. (. ($\/) (. (. ($= (bool)) (var x (bool))) (false)))
+      (. (. ($= (bool)) (var x (bool))) (true))))))
+  (
+      trueistrue false:bool REFL sequentimp ax-mp
+        false:bool false:bool eqval:bool
+        false:bool true:bool eqval:bool
+      or-$\/ biimpr orci
+    ax-mp
+
+      false:bool x BETA1 boolistype eq: BETA32 false:bool BETA31
+    $\/: BETA32
+      false:bool x BETA1 boolistype eq: BETA32 true:bool BETA31
+    BETA3 eqvali istrueeqi
+    mpbir
+
+      trueistrue true:bool REFL sequentimp ax-mp
+        true:bool false:bool eqval:bool
+        true:bool true:bool eqval:bool
+      or-$\/ biimpr olci
+    ax-mp
+
+      true:bool x BETA1 boolistype eq: BETA32 false:bool BETA31
+    $\/: BETA32
+      true:bool x BETA1 boolistype eq: BETA32 true:bool BETA31
+    BETA3 eqvali istrueeqi
+    mpbir pm3.2i
+
+    boolistype
+      $\/:
+        boolistype eq: boolistype x var: app: false:bool app: app:
+        boolistype eq: boolistype x var: app: true:bool app:
+      app:
+    x abs: forall-bool mpbir mknullseq
+))
+
+# mechanism for creating new types
+
+def ((newtype A B) (mktype ({e.|} x (dom A) (istrue (. A (cv x)))) B))
+
+thm (df-newtype ((A x y) (A x' y)) ()
+  (= (newtype A B) (mktype ({e.|} x' (dom A) (istrue (. A (cv x')))) B))
+  (
+   x cv y cv A .eq2 istrueeqd A dom cbvrabv B mktypeeq1
+     x' cv y cv A .eq2 istrueeqd A dom cbvrabv B mktypeeq1
+   eqtr4
+))
+
+thm (newtypeistype ((A x) (B x))
+  ((hyp.1 (: A ($-> T (bool))))
+   (hyp.2 ($|- (true) (. A B))))
+  (istype (newtype A B))
+  ( 
+      hyp.2 sequent2:bool betaexps2lem1
+      trueistrue hyp.2 sequentimp ax-mp
+      x cv B A .eq2 istrueeqd A dom elrab
+    biimpr mp2an
+
+      hyp.1 typeel elisseti A V dmexg ax-mp
+    x A x cv . istrue rabex
+    mktypethm
+
+    A B x df-newtype istypeeqi mpbir
+))
+
+# basic iota set theory -- move down when done
+
+# euuni, reuuni
+
+# reucl
+#thm (iota1 () () (-> (E!e. x A ph) (e. (U. ({e.|} x A ph)) A)) ())
+
+thm (cbvreu ((A x y))
+  ((hyp1 (-> ph (A. y ph)))
+   (hyp2 (-> ps (A. x ps)))
+   (hyp3 (-> (= (cv x) (cv y)) (<-> ph ps))))
+  (<-> (E!e. x A ph) (E!e. y A ps))
+  (   x cv A e. y ax-17 hyp1 hban
+      y cv A e. x ax-17 hyp2 hban
+      x cv y cv A eleq1 hyp3 anbi12d
+    cbveu
+    x A ph df-reu
+    y A ps df-reu 3bitr4
+))
+
+thm (df-iota-half ((T f x f' x')) ()
+ (=
+    (lambda f ($-> T (bool))
+      (if (E!e. x (typeset T) (istrue (. (cv f) (cv x))))
+        (U. ({e.|} x (typeset T) (istrue (. (cv f) (cv x)))))
+        (witness T)))
+    (lambda f' ($-> T (bool))
+      (if (E!e. x' (typeset T) (istrue (. (cv f') (cv x'))))
+        (U. ({e.|} x' (typeset T) (istrue (. (cv f') (cv x')))))
+        (witness T))))
+  ( f cv f' cv x cv .eq1 istrueeqd x T typeset bireudv
+      f' cv x cv . istrue x' ax-17
+      f' cv x' cv . istrue x ax-17
+      x cv x' cv f' cv .eq2 istrueeqd
+    T typeset cbvreu syl6bb
+      x T typeset f cv x cv . istrue E!e.
+      x' T typeset f' cv x' cv . istrue E!e.
+      x T typeset f cv x cv . istrue {e.|} U.
+      T witness
+    ifbi syl
+
+    f cv f' cv x cv .eq1 istrueeqd x T typeset birabsdv
+    x cv x' cv f' cv .eq2 istrueeqd T typeset cbvrabv syl6eq unieqd
+      x T typeset f cv x cv . istrue {e.|} U.
+      x' T typeset f' cv x' cv . istrue {e.|} U.
+      x' T typeset f' cv x' cv . istrue E!e.
+      T witness
+    ifeq1 syl eqtrd
+
+    T bool $-> alphaconv
+))
+
+thm (df-iota ((T f' x' g y) (T f x g y)) ()
+ (= ( $iota T ) ( lambda f ( $-> T ( bool ) )
+  ( if ( E!e. x ( typeset T ) ( istrue ( . ( cv f ) ( cv x ) ) ) )
+    ( U. ( {e.|} x ( typeset T ) ( istrue  ( . ( cv f ) ( cv x ) ) ) ) )
+    (witness T))))
+  (
+   f' T x' g y df-iota-half   
+     f T x g y df-iota-half   
+   eqtr4
+))
+
+thm (iota: ((T f x))
+  ((hyp (istype T)))
+  (: ($iota T) ($-> ($-> T (bool)) T))
+  ( hyp boolistype arrowistype
+
+    hyp
+      x T typeset f cv x cv . istrue reucl
+
+        x T typeset f cv x cv . istrue E!e.
+        x T typeset f cv x cv . istrue {e.|} U.
+        T witness
+      iftrue eqcomd T typeset eleq1d
+    mpbid
+
+      hyp witness1
+        x T typeset f cv x cv . istrue E!e.
+        x T typeset f cv x cv . istrue {e.|} U.
+        T witness
+      iffalse eqcomd T typeset eleq1d
+    mpbii pm2.61i
+
+        x T typeset f cv x cv . istrue E!e.
+        x T typeset f cv x cv . istrue {e.|} U.
+        T witness
+      if T df-: biimpr
+    mp2an f abs:
+    T f x df-iota T bool $-> T $-> :eq1i mpbir
+))
+
+# especially handy for doing substitutions over ded terms
+thm (ifbieq1d ()
+  ((hyp1 (-> ph (<-> ps ch)))
+   (hyp2 (-> ph (= A B))))
+  (-> ph (= (if ps A C) (if ch B C)))
+  ( hyp1 ps ch A C ifbi syl
+    hyp2 A B ch C ifeq1 syl eqtrd
+))
+
+thm ( exists-$! ((T x))
+  ( ( hyp.T ( istype T ) )
+    ( hyp.A ( : A ( bool ) ) ) )
+  (<-> (istrue (. ($~) (. ($! T) (lambda x T (. ($~) A)))))
+       (E.e. x (typeset T) (istrue A)))
+  ( hyp.T $!-type hyp.T $~: hyp.A app: x abs: app: not-$~
+    hyp.T $~: hyp.A app: x forall-$!
+      x T typeset A istrue -. df-ral
+    hyp.A not-$~ x cv T typeset e. imbi2i x bial bitr4
+    bitr4
+    negbii bitr
+
+    x T typeset A istrue dfrex2 bitr4
+))
+
+# equivalence between E* x. f(x) in set theory and ZFC notations
+thm (iota_ax_lem1 ((T x y))
+  ((hyp (istype T)))
+  (<-> (istrue (. ($! T) (lambda x T (. ($! T) (lambda y T
+          (. (. ($==>)
+            (. (var f ($-> T (bool))) (var x T)))
+            (. (. ($==>) (. (var f ($-> T (bool))) (var y T)))
+              (. (. ($= T) (var x T)) (var y T)))))))))
+    (A.e. x (typeset T) (A.e. y (typeset T)
+      (-> (istrue (. (var f ($-> T (bool))) (cv x)))
+        (-> (istrue (. (var f ($-> T (bool))) (cv y)))
+          (= (cv x) (cv y)))))))
+  (
+    hyp hyp $!-type hyp
+      hyp boolistype arrowistype f var: hyp x var: app:
+      hyp boolistype arrowistype f var: hyp y var: app:
+      hyp x var: hyp y var: eqval:bool impval:bool impval:bool
+    y abs: app: x forall-$!
+
+    x cv T df-: hyp mpbiran
+    y cv T df-: hyp mpbiran anbi12i
+
+      x T vart x cv x T var f T bool $-> var .eq2 syl istrueeqd y cv T : adantr
+
+      y T vart y cv y T var f T bool $-> var .eq2 syl istrueeqd x cv T : adantl
+
+      x T vart y cv eqeq1d y T vart x T var eqeq2d sylan9bb
+      hyp x var: hyp y var: eqval syl6bbr
+    imbi12d
+
+    hyp boolistype arrowistype f var: hyp y var: app:
+    hyp x var: hyp y var: eqval:bool impval
+    syl6bbr imbi12d
+
+      hyp boolistype arrowistype f var: hyp x var: app:
+      hyp boolistype arrowistype f var: hyp y var: app:
+      hyp x var: hyp y var: eqval:bool impval:bool impval
+    syl6bbr sylbir
+    exp pm5.74d y bialdv
+
+    hyp
+      hyp boolistype arrowistype f var: hyp x var: app:
+      hyp boolistype arrowistype f var: hyp y var: app:
+      hyp x var: hyp y var: eqval:bool impval:bool impval:bool y
+    forall-$! syl6bbr
+    pm5.74i x bial
+
+    bitr4
+
+    x T typeset y T typeset
+    f T bool $-> var x cv . istrue f T bool $-> var y cv . istrue
+    x cv y cv = -> -> A.e. df-ral
+
+    y T typeset f T bool $-> var x cv . istrue f T bool $-> var y cv . istrue
+    x cv y cv = -> -> df-ral
+    x cv T typeset e. imbi2i x bial
+    bitr bitr4
+))
+
+
+# The core of the iota axiom
+thm (iota_ax_lem2 ((T g x y) (A g x y))
+  ((hyp (istype T))
+   (hyp.A (: A ($-> T (bool)))))
+  (-> (E.e. x (typeset T) (istrue (. A (cv x))))
+    (-> (A.e. x (typeset T) (A.e. y (typeset T)
+       (-> (/\ (istrue (. A (cv x)))
+               ([/] (cv y) x (istrue (. A (cv x)))))
+           (= (cv x) (cv y)))))
+       (istrue (. A (. ($iota T) A)))))
+  ( 
+    x T typeset A x cv . istrue y reu2
+
+        x T typeset A x cv . istrue E!e.
+        x T typeset A x cv . istrue {e.|} U.
+        T witness
+      iftrue
+
+    T g x df-iota A .eq1i
+
+      hyp iota: T g x df-iota T bool $-> T $-> :eq1i mpbi
+      hyp.A app: betaexps2
+
+      hyp.A typeel elisseti
+
+        g cv A x cv .eq1 istrueeqd x T typeset bireudv
+        g cv A x cv .eq1 istrueeqd x T typeset birabsdv unieqd
+      T witness ifbieq1d imps2exps eqtr
+    eqtr syl5req
+
+      hyp iota: hyp.A app: typeel
+        x cv T $iota A . A .eq2 istrueeqd
+      T typeset reuuni2
+    mpan mpbird sylbir exp
+))
+
+thm (iota_ax_lem3 ()
+  ((hyp1 (: A (bool)))
+   (hyp2 (: B (bool)))
+   (hyp3 (: C (bool)))
+   (hyp4 (-> (istrue A) (-> (istrue B) (istrue C)))))
+  (istrue (. (. ($==>) A) (. (. ($==>) B) C)))
+  ( hyp4
+    hyp1 hyp2 hyp3 impval:bool impval
+    hyp2 hyp3 impval A istrue imbi2i bitr mpbir
+))
+
+thm (iota_ax ((T f x y))
+  ((hyp (istype T)))
+  ($|- (true) (. ($! ($-> T (bool))) (lambda f ($-> T (bool))
+    (. ( . ($==>)
+      (. ($~) (. ($! T) (lambda x T (. ($~) (. (var f ($-> T (bool))) (var x T)))))))
+      (. (. ($==>)
+        (. ($! T) (lambda x T (. ($! T) (lambda y T
+          (. (. ($==>)
+            (. (var f ($-> T (bool))) (var x T)))
+            (. (. ($==>) (. (var f ($-> T (bool))) (var y T)))
+              (. (. ($= T) (var x T)) (var y T)))))))))
+        (. (var f ($-> T (bool))) (. ($iota T) (var f ($-> T (bool))))))))))
+  ( hyp boolistype arrowistype
+
+      $~:
+        hyp $!-type hyp
+        $~: hyp boolistype arrowistype f var: hyp x var: app: app:
+      x abs: app: app:
+
+      hyp $!-type hyp hyp $!-type hyp
+        imptype hyp boolistype arrowistype f var: hyp x var: app: app:
+        imptype hyp boolistype arrowistype f var: hyp y var: app: app:
+        hyp eq: hyp x var: app: hyp y var: app: app: app:
+      y abs: app: x abs: app:
+
+        hyp boolistype arrowistype f var:
+      hyp iota: hyp boolistype arrowistype f var: app: app:
+
+    hyp hyp boolistype arrowistype f var: x y iota_ax_lem2
+
+    hyp hyp boolistype arrowistype f var: hyp x var: app: x exists-$!
+    x cv T df-: hyp mpbiran x T vart sylbir
+    x cv x T var f T bool $-> var .eq2 istrueeqd syl birexa
+    bitr4
+
+    hyp x y f iota_ax_lem1
+        f T bool $-> var x cv . istrue
+        y cv x f T bool $-> var x cv . istrue [/]
+        x cv y cv =
+      impexp
+        f T bool $-> var y cv . istrue x ax-17
+        x cv y cv f T bool $-> var .eq2 istrueeqd sbie
+      x cv y cv = imbi1i f T bool $-> var x cv . istrue imbi2i
+    bitr y T typeset biral x T typeset biral
+    bitr4
+
+    f T bool $-> var T $iota f T bool $-> var . . istrue imbi1i imbi12i
+    mpbir
+
+    iota_ax_lem3
+    f gen-$! mknullseq
+))
addfile ./zfc/set_mm.gh
hunk ./zfc/set_mm.gh 1
+# Converted from set.mm - Version of 22-Oct-2005
+
+# Original by Norm Megill
+
+# This file (specifically, the version of this file with the above date)
+# is placed in the public domain per the Creative Commons Public Domain
+# Dedication.  http://creativecommons.org/licenses/publicdomain/
+
+import (SET_MM set_mm () "")
+var (wff ph)
+var (wff ps)
+var (wff ch)
+var (wff th)
+var (wff ta)
+thm (a1i () ((a1i.1 ph)) (-> ps ph) (a1i.1 ph ps ax-1 ax-mp))
+thm (a2i () ((a2i.1 (-> ph (-> ps ch)))) (-> (-> ph ps) (-> ph ch)) (a2i.1 ph ps ch ax-2 ax-mp))
+thm (syl () ((syl.1 (-> ph ps)) (syl.2 (-> ps ch))) (-> ph ch) (syl.1 syl.2 ph a1i a2i ax-mp))
+thm (com12 () ((com12.1 (-> ph (-> ps ch)))) (-> ps (-> ph ch)) (ps ph ax-1 com12.1 a2i syl))
+thm (a1d () ((a1d.1 (-> ph ps))) (-> ph (-> ch ps)) (a1d.1 ch a1i com12))
+thm (a2d () ((a2d.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (-> ps ch) (-> ps th))) (a2d.1 ps ch th ax-2 syl))
+thm (imim2 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps))) (ph ps -> ch ax-1 a2d))
+thm (imim1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch))) (ps ch ph imim2 com12))
+thm (imim2i () ((imim1i.1 (-> ph ps))) (-> (-> ch ph) (-> ch ps)) (imim1i.1 ch a1i a2i))
+thm (imim1i () ((imim2i.1 (-> ph ps))) (-> (-> ps ch) (-> ph ch)) (imim2i.1 ph ps ch imim1 ax-mp))
+thm (imim12i () ((imim12i.1 (-> ph ps)) (imim12i.2 (-> ch th))) (-> (-> ps ch) (-> ph th)) (imim12i.2 ps imim2i imim12i.1 th imim1i syl))
+thm (3syl () ((3syl.1 (-> ph ps)) (3syl.2 (-> ps ch)) (3syl.3 (-> ch th))) (-> ph th) (3syl.1 3syl.2 syl 3syl.3 syl))
+thm (syl5 () ((syl5.1 (-> ph (-> ps ch))) (syl5.2 (-> th ps))) (-> ph (-> th ch)) (syl5.1 syl5.2 ch imim1i syl))
+thm (syl6 () ((syl6.1 (-> ph (-> ps ch))) (syl6.2 (-> ch th))) (-> ph (-> ps th)) (syl6.1 syl6.2 ps imim2i syl))
+thm (syl7 () ((syl7.1 (-> ph (-> ps (-> ch th)))) (syl7.2 (-> ta ch))) (-> ph (-> ps (-> ta th))) (syl7.1 syl7.2 th imim1i syl6))
+thm (syl8 () ((syl8.1 (-> ph (-> ps (-> ch th)))) (syl8.2 (-> th ta))) (-> ph (-> ps (-> ch ta))) (syl8.1 syl8.2 ch imim2i syl6))
+thm (imim2d () ((imim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (-> th ps) (-> th ch))) (imim1d.1 th a1d a2d))
+thm (syld () ((syld.1 (-> ph (-> ps ch))) (syld.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (syld.1 syld.2 ps imim2d a2i ax-mp))
+thm (imim1d () ((imim2d.1 (-> ph (-> ps ch)))) (-> ph (-> (-> ch th) (-> ps th))) (imim2d.1 ps ch th imim1 syl))
+thm (imim12d () ((imim12d.1 (-> ph (-> ps ch))) (imim12d.2 (-> ph (-> th ta)))) (-> ph (-> (-> ch th) (-> ps ta))) (imim12d.1 th imim1d imim12d.2 ps imim2d syld))
+thm (pm2.04 () () (-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph ps ch ax-2 ps ph ax-1 syl5))
+thm (pm2.83 () () (-> (-> ph (-> ps ch)) (-> (-> ph (-> ch th)) (-> ph (-> ps th)))) (ps ch th imim1 ph imim2i a2d))
+thm (com23 () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> ch (-> ps th))) (com3.1 ps ch th pm2.04 syl))
+thm (com13 () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ch (-> ps (-> ph th))) (com3.1 com12 com23 com12))
+thm (com3l () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ps (-> ch (-> ph th))) (com3.1 com23 com13))
+thm (com3r () ((com3.1 (-> ph (-> ps (-> ch th))))) (-> ch (-> ph (-> ps th))) (com3.1 com3l com3l))
+thm (com34 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> ps (-> th (-> ch ta)))) (com4.1 ch th ta pm2.04 syl6))
+thm (com24 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> th (-> ch (-> ps ta)))) (com4.1 com34 com23 com34))
+thm (com14 () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> th (-> ps (-> ch (-> ph ta)))) (com4.1 com34 com13 com34))
+thm (com4l () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ps (-> ch (-> th (-> ph ta)))) (com4.1 com14 com3l))
+thm (com4t () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ch (-> th (-> ph (-> ps ta)))) (com4.1 com4l com4l))
+thm (com4r () ((com4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> th (-> ph (-> ps (-> ch ta)))) (com4.1 com4t com4l))
+thm (a1dd () ((a1dd.1 (-> ph (-> ps ch)))) (-> ph (-> ps (-> th ch))) (a1dd.1 th a1d com23))
+thm (mp2 () ((mp2.1 ph) (mp2.2 ps) (mp2.3 (-> ph (-> ps ch)))) ch (mp2.2 mp2.1 mp2.3 ax-mp ax-mp))
+thm (mpi () ((mpi.1 ps) (mpi.2 (-> ph (-> ps ch)))) (-> ph ch) (mpi.1 mpi.2 com12 ax-mp))
+thm (mpii () ((mpii.1 ch) (mpii.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpii.1 mpii.2 com23 mpi))
+thm (mpd () ((mpd.1 (-> ph ps)) (mpd.2 (-> ph (-> ps ch)))) (-> ph ch) (mpd.1 mpd.2 a2i ax-mp))
+thm (mpdd () ((mpdd.1 (-> ph (-> ps ch))) (mpdd.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpdd.1 mpdd.2 a2d mpd))
+thm (mpid () ((mpid.1 (-> ph ch)) (mpid.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpid.1 ps a1d mpid.2 mpdd))
+thm (mpdi () ((mpdi.1 (-> ps ch)) (mpdi.2 (-> ph (-> ps (-> ch th))))) (-> ph (-> ps th)) (mpdi.1 mpdi.2 com12 mpid com12))
+thm (mpcom () ((mpcom.1 (-> ps ph)) (mpcom.2 (-> ph (-> ps ch)))) (-> ps ch) (mpcom.1 mpcom.2 com12 mpd))
+thm (syldd () ((syldd.1 (-> ph (-> ps (-> ch th)))) (syldd.2 (-> ph (-> ps (-> th ta))))) (-> ph (-> ps (-> ch ta))) (syldd.1 syldd.2 th ta ch imim2 syl6 mpdd))
+thm (sylcom () ((sylcom.1 (-> ph (-> ps ch))) (sylcom.2 (-> ps (-> ch th)))) (-> ph (-> ps th)) (sylcom.1 com12 sylcom.2 syld com12))
+thm (syl5com () ((syl5com.2 (-> ph (-> ps ch))) (syl5com.1 (-> th ps))) (-> th (-> ph ch)) (syl5com.1 ph a1d syl5com.2 sylcom))
+thm (syl6com () ((syl6com.1 (-> ph (-> ps ch))) (syl6com.2 (-> ch th))) (-> ps (-> ph th)) (syl6com.1 syl6com.2 syl6 com12))
+thm (syli () ((syli.1 (-> ps (-> ph ch))) (syli.2 (-> ch (-> ph th)))) (-> ps (-> ph th)) (syli.1 syli.2 com12 sylcom))
+thm (syl5d () ((syl5d.1 (-> ph (-> ps (-> ch th)))) (syl5d.2 (-> ph (-> ta ch)))) (-> ph (-> ps (-> ta th))) (syl5d.1 syl5d.2 th imim1d syld))
+thm (syl6d () ((syl6d.1 (-> ph (-> ps (-> ch th)))) (syl6d.2 (-> ph (-> th ta)))) (-> ph (-> ps (-> ch ta))) (syl6d.1 syl6d.2 ch imim2d syld))
+thm (syl9 () ((syl9.1 (-> ph (-> ps ch))) (syl9.2 (-> th (-> ch ta)))) (-> ph (-> th (-> ps ta))) (syl9.2 ph a1i syl9.1 syl5d))
+thm (syl9r () ((syl9r.1 (-> ph (-> ps ch))) (syl9r.2 (-> th (-> ch ta)))) (-> th (-> ph (-> ps ta))) (syl9r.1 syl9r.2 syl9 com12))
+thm (id () () (-> ph ph) (ph ph ax-1 ph ph ph ph -> -> ax-1 mpi))
+thm (id1 () () (-> ph ph) (ph ph ax-1 ph ph ph -> ax-1 ph ph ph -> ph ax-2 ax-mp ax-mp))
+thm (idd () () (-> ph (-> ps ps)) (ps id ph a1i))
+thm (pm2.27 () () (-> ph (-> (-> ph ps) ps)) (ph ps -> id com12))
+thm (pm2.43 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph ps pm2.27 a2i))
+thm (pm2.43i () ((pm2.43i.1 (-> ph (-> ph ps)))) (-> ph ps) (pm2.43i.1 ph ps pm2.43 ax-mp))
+thm (pm2.43d () ((pm2.43d.1 (-> ph (-> ps (-> ps ch))))) (-> ph (-> ps ch)) (pm2.43d.1 ps ch pm2.43 syl))
+thm (pm2.43a () ((pm2.43a.1 (-> ps (-> ph (-> ps ch))))) (-> ps (-> ph ch)) (pm2.43a.1 com23 pm2.43i))
+thm (pm2.43b () ((pm2.43a.1 (-> ps (-> ph (-> ps ch))))) (-> ph (-> ps ch)) (pm2.43a.1 pm2.43a com12))
+thm (sylc () ((sylc.1 (-> ph (-> ps ch))) (sylc.2 (-> th ph)) (sylc.3 (-> th ps))) (-> th ch) (sylc.3 sylc.2 sylc.1 syl mpd))
+thm (pm2.86 () () (-> (-> (-> ph ps) (-> ph ch)) (-> ph (-> ps ch))) (ps ph ax-1 ph ch -> imim1i com23))
+thm (pm2.86i () ((pm2.86i.1 (-> (-> ph ps) (-> ph ch)))) (-> ph (-> ps ch)) (pm2.86i.1 ph ps ch pm2.86 ax-mp))
+thm (pm2.86d () ((pm2.86d.1 (-> ph (-> (-> ps ch) (-> ps th))))) (-> ph (-> ps (-> ch th))) (pm2.86d.1 ps ch th pm2.86 syl))
+thm (loolin () () (-> (-> (-> ph ps) (-> ps ph)) (-> ps ph)) (ps ph ax-1 ps ph -> imim1i pm2.43d))
+thm (loowoz () () (-> (-> (-> ph ps) (-> ph ch)) (-> (-> ps ph) (-> ps ch))) (ps ph ax-1 ph ch -> imim1i a2d))
+thm (xxxid () () (-> ph ph) (ph id))
+thm (a3i () ((a3i.1 (-> (-. ph) (-. ps)))) (-> ps ph) (a3i.1 ph ps ax-3 ax-mp))
+thm (a3d () ((a3d.1 (-> ph (-> (-. ps) (-. ch))))) (-> ph (-> ch ps)) (a3d.1 ps ch ax-3 syl))
+thm (pm2.21 () () (-> (-. ph) (-> ph ps)) (ph -. ps -. ax-1 a3d))
+thm (pm2.24 () () (-> ph (-> (-. ph) ps)) (ph ps pm2.21 com12))
+thm (pm2.21i () ((pm2.21i.1 (-. ph))) (-> ph ps) (pm2.21i.1 ps -. a1i a3i))
+thm (pm2.21d () ((pm2.21d.1 (-> ph (-. ps)))) (-> ph (-> ps ch)) (pm2.21d.1 ch -. a1d a3d))
+thm (pm2.18 () () (-> (-> (-. ph) ph) ph) (ph ph -. ph -> -. pm2.21 a2i a3d pm2.43i))
+thm (peirce () () (-> (-> (-> ph ps) ph) ph) (ph ps pm2.21 ph imim1i ph pm2.18 syl))
+thm (looinv () () (-> (-> (-> ph ps) ps) (-> (-> ps ph) ph)) (ph ps -> ps ph imim1 ph ps peirce syl6))
+thm (nega () () (-> (-. (-. ph)) ph) (ph -. ph pm2.21 ph pm2.18 syl))
+thm (negb () () (-> ph (-. (-. ph))) (ph -. nega a3i))
+thm (pm2.01 () () (-> (-> ph (-. ph)) (-. ph)) (ph nega ph -. imim1i ph -. pm2.18 syl))
+thm (pm2.01d () ((pm2.01d.1 (-> ph (-> ps (-. ps))))) (-> ph (-. ps)) (pm2.01d.1 ps pm2.01 syl))
+thm (con2 () () (-> (-> ph (-. ps)) (-> ps (-. ph))) (ph nega ps -. imim1i a3d))
+thm (con2d () ((con2d.1 (-> ph (-> ps (-. ch))))) (-> ph (-> ch (-. ps))) (con2d.1 ps ch con2 syl))
+thm (con1 () () (-> (-> (-. ph) ps) (-> (-. ps) ph)) (ps negb ph -. imim2i a3d))
+thm (con1d () ((con1d.1 (-> ph (-> (-. ps) ch)))) (-> ph (-> (-. ch) ps)) (con1d.1 ps ch con1 syl))
+thm (con3 () () (-> (-> ph ps) (-> (-. ps) (-. ph))) (ps negb ph imim2i con2d))
+thm (con3d () ((con3d.1 (-> ph (-> ps ch)))) (-> ph (-> (-. ch) (-. ps))) (con3d.1 ps ch con3 syl))
+thm (con1i () ((con1.a (-> (-. ph) ps))) (-> (-. ps) ph) (con1.a ps negb syl a3i))
+thm (con2i () ((con2.a (-> ph (-. ps)))) (-> ps (-. ph)) (ph nega con2.a syl a3i))
+thm (con3i () ((con3.a (-> ph ps))) (-> (-. ps) (-. ph)) (ph nega con3.a syl con1i))
+thm (pm2.36 () () (-> (-> ps ch) (-> (-> (-. ph) ps) (-> (-. ch) ph))) (ps ch ph -. imim2 ph ch con1 syl6))
+thm (pm2.37 () () (-> (-> ps ch) (-> (-> (-. ps) ph) (-> (-. ph) ch))) (ps ph con1 ch imim1d com12))
+thm (pm2.38 () () (-> (-> ps ch) (-> (-> (-. ps) ph) (-> (-. ch) ph))) (ps ch con3 ph imim1d))
+thm (pm2.5 () () (-> (-. (-> ph ps)) (-> (-. ph) ps)) (ph ps pm2.21 con3i ps pm2.21d))
+thm (pm2.51 () () (-> (-. (-> ph ps)) (-> ph (-. ps))) (ps ph ax-1 con3i ph a1d))
+thm (pm2.52 () () (-> (-. (-> ph ps)) (-> (-. ph) (-. ps))) (ps ph ax-1 con3i ph -. a1d))
+thm (pm2.521 () () (-> (-. (-> ph ps)) (-> ps ph)) (ph ps pm2.52 a3d))
+thm (pm2.21ni () ((pm2.21ni.1 ph)) (-> (-. ph) ps) (pm2.21ni.1 ps -. a1i con1i))
+thm (mto () ((mto.1 (-. ps)) (mto.2 (-> ph ps))) (-. ph) (mto.1 mto.2 con3i ax-mp))
+thm (mtoi () ((mtoi.1 (-. ch)) (mtoi.2 (-> ph (-> ps ch)))) (-> ph (-. ps)) (mtoi.1 mtoi.2 con3d mpi))
+thm (mtod () ((mtod.1 (-> ph (-. ch))) (mtod.2 (-> ph (-> ps ch)))) (-> ph (-. ps)) (mtod.1 mtod.2 con3d mpd))
+thm (mt2 () ((mt2.1 ps) (mt2.2 (-> ph (-. ps)))) (-. ph) (mt2.1 mt2.2 con2i ax-mp))
+thm (mt2i () ((mt2i.1 ch) (mt2i.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (mt2i.1 mt2i.2 con2d mpi))
+thm (mt2d () ((mt2d.1 (-> ph ch)) (mt2d.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (mt2d.1 mt2d.2 con2d mpd))
+thm (mt3 () ((mt3.1 (-. ps)) (mt3.2 (-> (-. ph) ps))) ph (mt3.1 mt3.2 con1i ax-mp))
+thm (mt3i () ((mt3i.1 (-. ch)) (mt3i.2 (-> ph (-> (-. ps) ch)))) (-> ph ps) (mt3i.1 mt3i.2 con1d mpi))
+thm (mt3d () ((mt3d.1 (-> ph (-. ch))) (mt3d.2 (-> ph (-> (-. ps) ch)))) (-> ph ps) (mt3d.1 mt3d.2 con1d mpd))
+thm (nsyl () ((nsyl.1 (-> ph (-. ps))) (nsyl.2 (-> ch ps))) (-> ph (-. ch)) (nsyl.1 nsyl.2 con3i syl))
+thm (nsyld () ((nsyld.1 (-> ph (-> ps (-. ch)))) (nsyld.2 (-> ph (-> ta ch)))) (-> ph (-> ps (-. ta))) (nsyld.1 nsyld.2 con3d syld))
+thm (nsyl2 () ((nsyl2.1 (-> ph (-. ps))) (nsyl2.2 (-> (-. ch) ps))) (-> ph ch) (nsyl2.1 nsyl2.2 con1i syl))
+thm (nsyl3 () ((nsyl3.1 (-> ph (-. ps))) (nsyl3.2 (-> ch ps))) (-> ch (-. ph)) (nsyl3.2 nsyl3.1 con2i syl))
+thm (nsyl4 () ((nsyl4.1 (-> ph ps)) (nsyl4.2 (-> (-. ph) ch))) (-> (-. ch) ps) (nsyl4.2 con1i nsyl4.1 syl))
+thm (nsyli () ((nsyli.1 (-> ph (-> ps ch))) (nsyli.2 (-> th (-. ch)))) (-> ph (-> th (-. ps))) (nsyli.1 con3d nsyli.2 syl5))
+thm (pm3.2im () () (-> ph (-> ps (-. (-> ph (-. ps))))) (ph ps -. pm2.27 con2d))
+thm (mth8 () () (-> ph (-> (-. ps) (-. (-> ph ps)))) (ph ps pm2.27 con3d))
+thm (pm2.61 () () (-> (-> ph ps) (-> (-> (-. ph) ps) ps)) (ph ps ps -. imim2 ps pm2.18 syl6 ph ps con1 syl5))
+thm (pm2.61i () ((pm2.61i.1 (-> ph ps)) (pm2.61i.2 (-> (-. ph) ps))) ps (pm2.61i.1 pm2.61i.2 ph ps pm2.61 mp2))
+thm (pm2.61d () ((pm2.61d.1 (-> ph (-> ps ch))) (pm2.61d.2 (-> ph (-> (-. ps) ch)))) (-> ph ch) (pm2.61d.1 com12 pm2.61d.2 com12 pm2.61i))
+thm (pm2.61d1 () ((pm2.61d1.1 (-> ph (-> ps ch))) (pm2.61d1.2 (-> (-. ps) ch))) (-> ph ch) (pm2.61d1.1 pm2.61d1.2 ph a1i pm2.61d))
+thm (pm2.61d2 () ((pm2.61d2.1 (-> ph (-> (-. ps) ch))) (pm2.61d2.2 (-> ps ch))) (-> ph ch) (pm2.61d2.2 ph a1i pm2.61d2.1 pm2.61d))
+thm (pm2.61ii () ((pm2.61ii.1 (-> (-. ph) (-> (-. ps) ch))) (pm2.61ii.2 (-> ph ch)) (pm2.61ii.3 (-> ps ch))) ch (pm2.61ii.2 pm2.61ii.1 pm2.61ii.3 pm2.61d2 pm2.61i))
+thm (pm2.61nii () ((pm2.61nii.1 (-> ph (-> ps ch))) (pm2.61nii.2 (-> (-. ph) ch)) (pm2.61nii.3 (-> (-. ps) ch))) ch (pm2.61nii.1 com12 pm2.61nii.2 pm2.61d1 pm2.61nii.3 pm2.61i))
+thm (pm2.61iii () ((pm2.61iii.1 (-> (-. ph) (-> (-. ps) (-> (-. ch) th)))) (pm2.61iii.2 (-> ph th)) (pm2.61iii.3 (-> ps th)) (pm2.61iii.4 (-> ch th))) th (pm2.61iii.2 ch -. a1d ps -. a1d pm2.61iii.1 pm2.61i pm2.61iii.3 pm2.61iii.4 pm2.61ii))
+thm (pm2.6 () () (-> (-> (-. ph) ps) (-> (-> ph ps) ps)) (ph ps pm2.61 com12))
+thm (pm2.65 () () (-> (-> ph ps) (-> (-> ph (-. ps)) (-. ph))) (ph ps pm3.2im a2i con2d))
+thm (pm2.65i () ((pm2.65i.1 (-> ph ps)) (pm2.65i.2 (-> ph (-. ps)))) (-. ph) (pm2.65i.2 pm2.65i.1 nsyl ph pm2.01 ax-mp))
+thm (pm2.65d () ((pm2.65d.1 (-> ph (-> ps ch))) (pm2.65d.2 (-> ph (-> ps (-. ch))))) (-> ph (-. ps)) (ps ch pm2.65 pm2.65d.1 pm2.65d.2 sylc))
+thm (ja () ((ja.1 (-> (-. ph) ch)) (ja.2 (-> ps ch))) (-> (-> ph ps) ch) (ph ps pm2.27 ja.2 syl6 ja.1 ph ps -> a1d pm2.61i))
+thm (jc () ((jc.1 (-> ph ps)) (jc.2 (-> ph ch))) (-> ph (-. (-> ps (-. ch)))) (ps ch pm3.2im jc.1 jc.2 sylc))
+thm (pm3.26im () () (-> (-. (-> ph (-. ps))) ph) (ph ps -. pm2.21 con1i))
+thm (pm3.27im () () (-> (-. (-> ph (-. ps))) ps) (ps -. ph ax-1 con1i))
+thm (impt () () (-> (-> ph (-> ps ch)) (-> (-. (-> ph (-. ps))) ch)) (ps ch con3 ph imim2i com23 con1d))
+thm (expt () () (-> (-> (-. (-> ph (-. ps))) ch) (-> ph (-> ps ch))) (ph ps pm3.2im ch imim1d com12))
+thm (impi () ((impi.1 (-> ph (-> ps ch)))) (-> (-. (-> ph (-. ps))) ch) (impi.1 ph ps ch impt ax-mp))
+thm (expi () ((expi.1 (-> (-. (-> ph (-. ps))) ch))) (-> ph (-> ps ch)) (expi.1 ph ps ch expt ax-mp))
+thm (bijust () () (-. (-> (-> ph ph) (-. (-> ph ph)))) (ph id ph ph -> pm2.01 mt2))
+thm (biigb () () (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (ph ps df-bi ch th ax-1 ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. -> ax-1 ph ps <-> ph ps -> ps ph -> -. -> -. df-bi ph ps <-> ph ps -> ps ph -> -. -> -. <-> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. -> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. -> -. ch th ch -> -> -. -. ax-1 ax-mp ch th ch -> -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. -> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. -> ax-3 ax-mp ph ps <-> ph ps -> ps ph -> -. -> -. <-> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. -> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. -> ch th ch -> -> -. -> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. ax-1 ax-mp ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. -> ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> -. -> ch th ch -> -> -. ax-2 ax-mp ax-mp ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> -. -> -. ph ps <-> ph ps -> ps ph -> -. -> -. <-> -> ch th ch -> -> ax-3 ax-mp ax-mp ax-mp))
+thm (bi1 () () (-> (<-> ph ps) (-> ph ps)) (ph ps df-bi ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> pm3.26im ax-mp ph ps -> ps ph -> pm3.26im syl))
+thm (bi2 () () (-> (<-> ph ps) (-> ps ph)) (ph ps df-bi ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> pm3.26im ax-mp ph ps -> ps ph -> pm3.27im syl))
+thm (bi3 () () (-> (-> ph ps) (-> (-> ps ph) (<-> ph ps))) (ph ps df-bi ph ps <-> ph ps -> ps ph -> -. -> -. -> ph ps -> ps ph -> -. -> -. ph ps <-> -> pm3.27im ax-mp expi))
+thm (biimp () ((biimp.1 (<-> ph ps))) (-> ph ps) (biimp.1 ph ps bi1 ax-mp))
+thm (biimpr () ((biimpr.1 (<-> ph ps))) (-> ps ph) (biimpr.1 ph ps bi2 ax-mp))
+thm (biimpd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ph (-> ps ch)) (biimpd.1 ps ch bi1 syl))
+thm (biimprd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ph (-> ch ps)) (biimpd.1 ps ch bi2 syl))
+thm (biimpcd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ps (-> ph ch)) (biimpd.1 biimpd com12))
+thm (biimprcd () ((biimpd.1 (-> ph (<-> ps ch)))) (-> ch (-> ph ps)) (biimpd.1 biimprd com12))
+thm (impbi () ((impbi.1 (-> ph ps)) (impbi.2 (-> ps ph))) (<-> ph ps) (impbi.1 impbi.2 ph ps bi3 mp2))
+thm (bii () () (<-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (ph ps bi1 ph ps bi2 jc ph ps bi3 impi impbi))
+thm (bi2.04 () () (<-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph ps ch pm2.04 ps ph ch pm2.04 impbi))
+thm (pm4.13 () () (<-> ph (-. (-. ph))) (ph negb ph nega impbi))
+thm (pm4.8 () () (<-> (-> ph (-. ph)) (-. ph)) (ph pm2.01 ph -. ph ax-1 impbi))
+thm (pm4.81 () () (<-> (-> (-. ph) ph) ph) (ph pm2.18 ph ph pm2.24 impbi))
+thm (pm4.1 () () (<-> (-> ph ps) (-> (-. ps) (-. ph))) (ph ps con3 ps ph ax-3 impbi))
+thm (bi2.03 () () (<-> (-> ph (-. ps)) (-> ps (-. ph))) (ph ps con2 ps ph con2 impbi))
+thm (bi2.15 () () (<-> (-> (-. ph) ps) (-> (-. ps) ph)) (ph ps con1 ps ph con1 impbi))
+thm (pm5.4 () () (<-> (-> ph (-> ph ps)) (-> ph ps)) (ph ps pm2.43 ph ps -> ph ax-1 impbi))
+thm (imdi () () (<-> (-> ph (-> ps ch)) (-> (-> ph ps) (-> ph ch))) (ph ps ch ax-2 ph ps ch pm2.86 impbi))
+thm (pm5.41 () () (<-> (-> (-> ph ps) (-> ph ch)) (-> ph (-> ps ch))) (ph ps ch pm2.86 ph ps ch ax-2 impbi))
+thm (pm4.2 () () (<-> ph ph) (ph id ph id impbi))
+thm (pm4.2i () () (-> ph (<-> ps ps)) (ps pm4.2 ph a1i))
+thm (bicomi () ((bicom.1 (<-> ph ps))) (<-> ps ph) (bicom.1 biimpr bicom.1 biimp impbi))
+thm (bitr () ((bitr.1 (<-> ph ps)) (bitr.2 (<-> ps ch))) (<-> ph ch) (bitr.1 biimp bitr.2 biimp syl bitr.2 biimpr bitr.1 biimpr syl impbi))
+thm (bitr2 () ((bitr2.1 (<-> ph ps)) (bitr2.2 (<-> ps ch))) (<-> ch ph) (bitr2.1 bitr2.2 bitr bicomi))
+thm (bitr3 () ((bitr3.1 (<-> ps ph)) (bitr3.2 (<-> ps ch))) (<-> ph ch) (bitr3.1 bicomi bitr3.2 bitr))
+thm (bitr4 () ((bitr4.1 (<-> ph ps)) (bitr4.2 (<-> ch ps))) (<-> ph ch) (bitr4.1 bitr4.2 bicomi bitr))
+thm (3bitr () ((3bitr.1 (<-> ph ps)) (3bitr.2 (<-> ps ch)) (3bitr.3 (<-> ch th))) (<-> ph th) (3bitr.1 3bitr.2 3bitr.3 bitr bitr))
+thm (3bitr3 () ((3bitr3.1 (<-> ph ps)) (3bitr3.2 (<-> ph ch)) (3bitr3.3 (<-> ps th))) (<-> ch th) (3bitr3.2 3bitr3.1 bitr3 3bitr3.3 bitr))
+thm (3bitr3r () ((3bitr3.1 (<-> ph ps)) (3bitr3.2 (<-> ph ch)) (3bitr3.3 (<-> ps th))) (<-> th ch) (3bitr3.3 3bitr3.1 3bitr3.2 bitr3 bitr3))
+thm (3bitr4 () ((3bitr4.1 (<-> ph ps)) (3bitr4.2 (<-> ch ph)) (3bitr4.3 (<-> th ps))) (<-> ch th) (3bitr4.2 3bitr4.1 3bitr4.3 bitr4 bitr))
+thm (3bitr4r () ((3bitr4.1 (<-> ph ps)) (3bitr4.2 (<-> ch ph)) (3bitr4.3 (<-> th ps))) (<-> th ch) (3bitr4.2 3bitr4.1 3bitr4.3 bitr4 bitr2))
+thm (imbi2i () ((bi.a (<-> ph ps))) (<-> (-> ch ph) (-> ch ps)) (bi.a biimp ch imim2i bi.a biimpr ch imim2i impbi))
+thm (imbi1i () ((bi.a (<-> ph ps))) (<-> (-> ph ch) (-> ps ch)) (bi.a biimpr ch imim1i bi.a biimp ch imim1i impbi))
+thm (negbii () ((bi.a (<-> ph ps))) (<-> (-. ph) (-. ps)) (bi.a biimpr con3i bi.a biimp con3i impbi))
+thm (imbi12i () ((bi2im.1 (<-> ph ps)) (bi2im.2 (<-> ch th))) (<-> (-> ph ch) (-> ps th)) (bi2im.2 ph imbi2i bi2im.1 th imbi1i bitr))
+thm (mpbi () ((mpbi.min ph) (mpbi.maj (<-> ph ps))) ps (mpbi.min mpbi.maj biimp ax-mp))
+thm (mpbir () ((mpbir.min ps) (mpbir.maj (<-> ph ps))) ph (mpbir.min mpbir.maj biimpr ax-mp))
+thm (mtbi () ((mtbi.1 (-. ph)) (mtbi.2 (<-> ph ps))) (-. ps) (mtbi.1 mtbi.2 negbii mpbi))
+thm (mtbir () ((mtbir.1 (-. ps)) (mtbir.2 (<-> ph ps))) (-. ph) (mtbir.1 mtbir.2 negbii mpbir))
+thm (mpbii () ((mpbii.min ps) (mpbii.maj (-> ph (<-> ps ch)))) (-> ph ch) (mpbii.min mpbii.maj biimpd mpi))
+thm (mpbiri () ((mpbiri.min ch) (mpbiri.maj (-> ph (<-> ps ch)))) (-> ph ps) (mpbiri.min mpbiri.maj biimprd mpi))
+thm (mpbid () ((mpbid.min (-> ph ps)) (mpbid.maj (-> ph (<-> ps ch)))) (-> ph ch) (mpbid.min mpbid.maj biimpd mpd))
+thm (mpbird () ((mpbird.min (-> ph ch)) (mpbird.maj (-> ph (<-> ps ch)))) (-> ph ps) (mpbird.min mpbird.maj biimprd mpd))
+thm (a1bi () ((a1bi.1 ph)) (<-> ps (-> ph ps)) (ps ph ax-1 a1bi.1 ph ps pm2.27 ax-mp impbi))
+thm (sylib () ((sylib.1 (-> ph ps)) (sylib.2 (<-> ps ch))) (-> ph ch) (sylib.1 sylib.2 biimp syl))
+thm (sylbi () ((sylbi.1 (<-> ph ps)) (sylbi.2 (-> ps ch))) (-> ph ch) (sylbi.1 biimp sylbi.2 syl))
+thm (sylibr () ((sylibr.1 (-> ph ps)) (sylibr.2 (<-> ch ps))) (-> ph ch) (sylibr.1 sylibr.2 biimpr syl))
+thm (sylbir () ((sylbir.1 (<-> ps ph)) (sylbir.2 (-> ps ch))) (-> ph ch) (sylbir.1 biimpr sylbir.2 syl))
+thm (sylibd () ((sylibd.1 (-> ph (-> ps ch))) (sylibd.2 (-> ph (<-> ch th)))) (-> ph (-> ps th)) (sylibd.1 sylibd.2 biimpd syld))
+thm (sylbid () ((sylbid.1 (-> ph (<-> ps ch))) (sylbid.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (sylbid.1 biimpd sylbid.2 syld))
+thm (sylibrd () ((sylibrd.1 (-> ph (-> ps ch))) (sylibrd.2 (-> ph (<-> th ch)))) (-> ph (-> ps th)) (sylibrd.1 sylibrd.2 biimprd syld))
+thm (sylbird () ((sylbird.1 (-> ph (<-> ch ps))) (sylbird.2 (-> ph (-> ch th)))) (-> ph (-> ps th)) (sylbird.1 biimprd sylbird.2 syld))
+thm (syl5ib () ((syl5ib.1 (-> ph (-> ps ch))) (syl5ib.2 (<-> th ps))) (-> ph (-> th ch)) (syl5ib.1 syl5ib.2 biimp syl5))
+thm (syl5ibr () ((syl5ibr.1 (-> ph (-> ps ch))) (syl5ibr.2 (<-> ps th))) (-> ph (-> th ch)) (syl5ibr.1 syl5ibr.2 biimpr syl5))
+thm (syl5bi () ((syl5bi.1 (-> ph (<-> ps ch))) (syl5bi.2 (-> th ps))) (-> ph (-> th ch)) (syl5bi.1 biimpd syl5bi.2 syl5))
+thm (syl5bir () ((syl5bir.1 (-> ph (<-> ps ch))) (syl5bir.2 (-> th ch))) (-> ph (-> th ps)) (syl5bir.1 biimprd syl5bir.2 syl5))
+thm (syl6ib () ((syl6ib.1 (-> ph (-> ps ch))) (syl6ib.2 (<-> ch th))) (-> ph (-> ps th)) (syl6ib.1 syl6ib.2 biimp syl6))
+thm (syl6ibr () ((syl6ibr.1 (-> ph (-> ps ch))) (syl6ibr.2 (<-> th ch))) (-> ph (-> ps th)) (syl6ibr.1 syl6ibr.2 biimpr syl6))
+thm (syl6bi () ((syl6bi.1 (-> ph (<-> ps ch))) (syl6bi.2 (-> ch th))) (-> ph (-> ps th)) (syl6bi.1 biimpd syl6bi.2 syl6))
+thm (syl6bir () ((syl6bir.1 (-> ph (<-> ch ps))) (syl6bir.2 (-> ch th))) (-> ph (-> ps th)) (syl6bir.1 biimprd syl6bir.2 syl6))
+thm (bisyl7 () ((bisyl7.1 (-> ph (-> ps (-> ch th)))) (bisyl7.2 (<-> ta ch))) (-> ph (-> ps (-> ta th))) (bisyl7.1 bisyl7.2 biimp syl7))
+thm (bisyl8 () ((bisyl8.1 (-> ph (-> ps (-> ch th)))) (bisyl8.2 (<-> th ta))) (-> ph (-> ps (-> ch ta))) (bisyl8.1 bisyl8.2 biimp syl8))
+thm (3imtr3 () ((3imtr3.1 (-> ph ps)) (3imtr3.2 (<-> ph ch)) (3imtr3.3 (<-> ps th))) (-> ch th) (3imtr3.2 3imtr3.1 sylbir 3imtr3.3 sylib))
+thm (3imtr4 () ((3imtr4.1 (-> ph ps)) (3imtr4.2 (<-> ch ph)) (3imtr4.3 (<-> th ps))) (-> ch th) (3imtr4.2 3imtr4.1 sylbi 3imtr4.3 sylibr))
+thm (bicon1i () ((bicon1.1 (<-> (-. ph) ps))) (<-> (-. ps) ph) (bicon1.1 biimp con1i bicon1.1 biimpr con2i impbi))
+thm (bicon2i () ((bicon2.1 (<-> ph (-. ps)))) (<-> ps (-. ph)) (bicon2.1 bicomi bicon1i bicomi))
+thm (pm4.64 () () (<-> (-> (-. ph) ps) (\/ ph ps)) (ph ps df-or bicomi))
+thm (pm2.54 () () (-> (-> (-. ph) ps) (\/ ph ps)) (ph ps df-or biimpr))
+thm (pm4.63 () () (<-> (-. (-> ph (-. ps))) (/\ ph ps)) (ph ps df-an bicomi))
+thm (dfor2 () () (<-> (\/ ph ps) (-> (-> ph ps) ps)) (ph ps df-or ph ps pm2.6 ph ps pm2.21 ps imim1i impbi bitr))
+thm (ori () ((ori.1 (\/ ph ps))) (-> (-. ph) ps) (ori.1 ph ps df-or mpbi))
+thm (orri () ((orri.1 (-> (-. ph) ps))) (\/ ph ps) (orri.1 ph ps df-or mpbir))
+thm (ord () ((ord.1 (-> ph (\/ ps ch)))) (-> ph (-> (-. ps) ch)) (ord.1 ps ch df-or sylib))
+thm (orrd () ((orrd.1 (-> ph (-> (-. ps) ch)))) (-> ph (\/ ps ch)) (orrd.1 ps ch df-or sylibr))
+thm (imor () () (<-> (-> ph ps) (\/ (-. ph) ps)) (ph pm4.13 ps imbi1i ph -. ps df-or bitr4))
+thm (pm4.62 () () (<-> (-> ph (-. ps)) (\/ (-. ph) (-. ps))) (ph ps -. imor))
+thm (pm4.66 () () (<-> (-> (-. ph) (-. ps)) (\/ ph (-. ps))) (ph ps -. pm4.64))
+thm (iman () () (<-> (-> ph ps) (-. (/\ ph (-. ps)))) (ps pm4.13 ph imbi2i ph ps -. df-an bicon2i bitr))
+thm (annim () () (<-> (/\ ph (-. ps)) (-. (-> ph ps))) (ph ps iman bicon2i))
+thm (pm4.61 () () (<-> (-. (-> ph ps)) (/\ ph (-. ps))) (ph ps annim bicomi))
+thm (pm4.65 () () (<-> (-. (-> (-. ph) ps)) (/\ (-. ph) (-. ps))) (ph -. ps pm4.61))
+thm (pm4.67 () () (<-> (-. (-> (-. ph) (-. ps))) (/\ (-. ph) ps)) (ph -. ps pm4.63))
+thm (imnan () () (<-> (-> ph (-. ps)) (-. (/\ ph ps))) (ph ps df-an bicon2i))
+thm (oridm () () (<-> (\/ ph ph) ph) (ph ph df-or ph ph pm2.24 ph pm2.18 impbi bitr4))
+thm (pm4.25 () () (<-> ph (\/ ph ph)) (ph oridm bicomi))
+thm (pm1.2 () () (-> (\/ ph ph) ph) (ph oridm biimp))
+thm (orcom () () (<-> (\/ ph ps) (\/ ps ph)) (ph ps bi2.15 ph ps df-or ps ph df-or 3bitr4))
+thm (pm1.4 () () (-> (\/ ph ps) (\/ ps ph)) (ph ps orcom biimp))
+thm (pm2.62 () () (-> (\/ ph ps) (-> (-> ph ps) ps)) (ph ps dfor2 biimp))
+thm (pm2.621 () () (-> (-> ph ps) (-> (\/ ph ps) ps)) (ph ps pm2.62 com12))
+thm (pm2.68 () () (-> (-> (-> ph ps) ps) (\/ ph ps)) (ph ps dfor2 biimpr))
+thm (orel1 () () (-> (-. ph) (-> (\/ ph ps) ps)) (ph ps df-or biimp com12))
+thm (orel2 () () (-> (-. ph) (-> (\/ ps ph) ps)) (ph ps orel1 ps ph orcom syl5ib))
+thm (pm2.25 () () (\/ ph (-> (\/ ph ps) ps)) (ph ps orel1 orri))
+thm (pm2.53 () () (-> (\/ ph ps) (-> (-. ph) ps)) (ph ps orel1 com12))
+thm (orbi2i () ((bi.oa (<-> ph ps))) (<-> (\/ ch ph) (\/ ch ps)) (bi.oa ch -. imbi2i ch ph df-or ch ps df-or 3bitr4))
+thm (orbi1i () ((bi.oa (<-> ph ps))) (<-> (\/ ph ch) (\/ ps ch)) (ph ch orcom bi.oa ch orbi2i ch ps orcom 3bitr))
+thm (orbi12i () ((bi2or.1 (<-> ph ps)) (bi2or.2 (<-> ch th))) (<-> (\/ ph ch) (\/ ps th)) (bi2or.2 ph orbi2i bi2or.1 th orbi1i bitr))
+thm (or12 () () (<-> (\/ ph (\/ ps ch)) (\/ ps (\/ ph ch))) (ps -. ph -. ch bi2.04 ph ch df-or ps -. imbi2i ps ch df-or ph -. imbi2i 3bitr4r ph ps ch \/ df-or ps ph ch \/ df-or 3bitr4))
+thm (pm1.5 () () (-> (\/ ph (\/ ps ch)) (\/ ps (\/ ph ch))) (ph ps ch or12 biimp))
+thm (orass () () (<-> (\/ (\/ ph ps) ch) (\/ ph (\/ ps ch))) (ch ph ps or12 ph ps \/ ch orcom ps ch orcom ph orbi2i 3bitr4))
+thm (pm2.31 () () (-> (\/ ph (\/ ps ch)) (\/ (\/ ph ps) ch)) (ph ps ch orass biimpr))
+thm (pm2.32 () () (-> (\/ (\/ ph ps) ch) (\/ ph (\/ ps ch))) (ph ps ch orass biimp))
+thm (or23 () () (<-> (\/ (\/ ph ps) ch) (\/ (\/ ph ch) ps)) (ps ch orcom ph orbi2i ph ps ch orass ph ch ps orass 3bitr4))
+thm (or4 () () (<-> (\/ (\/ ph ps) (\/ ch th)) (\/ (\/ ph ch) (\/ ps th))) (ps ch th or12 ph orbi2i ph ps ch th \/ orass ph ch ps th \/ orass 3bitr4))
+thm (or42 () () (<-> (\/ (\/ ph ps) (\/ ch th)) (\/ (\/ ph ch) (\/ th ps))) (ph ps ch th or4 ps th orcom ph ch \/ orbi2i bitr))
+thm (orordi () () (<-> (\/ ph (\/ ps ch)) (\/ (\/ ph ps) (\/ ph ch))) (ph oridm ps ch \/ orbi1i ph ph ps ch or4 bitr3))
+thm (orordir () () (<-> (\/ (\/ ph ps) ch) (\/ (\/ ph ch) (\/ ps ch))) (ch oridm ph ps \/ orbi2i ph ps ch ch or4 bitr3))
+thm (olc () () (-> ph (\/ ps ph)) (ph ps -. ax-1 orrd))
+thm (pm2.07 () () (-> ph (\/ ph ph)) (ph ph olc))
+thm (orc () () (-> ph (\/ ph ps)) (ph ps pm2.24 orrd))
+thm (orci () ((orci.1 (-> (\/ ph ps) ch))) (-> ph ch) (ph ps orc orci.1 syl))
+thm (olci () ((olci.1 (-> (\/ ph ps) ch))) (-> ps ch) (ps ph olc olci.1 syl))
+thm (pm2.45 () () (-> (-. (\/ ph ps)) (-. ph)) (ph ps orc con3i))
+thm (pm2.46 () () (-> (-. (\/ ph ps)) (-. ps)) (ps ph olc con3i))
+thm (pm2.47 () () (-> (-. (\/ ph ps)) (\/ (-. ph) ps)) (ph ps pm2.45 ph -. ps orc syl))
+thm (pm2.48 () () (-> (-. (\/ ph ps)) (\/ ph (-. ps))) (ph ps pm2.46 ph -. a1d orrd))
+thm (pm2.49 () () (-> (-. (\/ ph ps)) (\/ (-. ph) (-. ps))) (ph ps pm2.46 ph -. -. a1d orrd))
+thm (pm2.67 () () (-> (-> (\/ ph ps) ps) (-> ph ps)) (ph ps orc ps imim1i))
+thm (pm3.2 () () (-> ph (-> ps (/\ ph ps))) (ph ps df-an biimpr expi))
+thm (pm3.21 () () (-> ph (-> ps (/\ ps ph))) (ps ph pm3.2 com12))
+thm (pm3.2i () ((pm3.2i.1 ph) (pm3.2i.2 ps)) (/\ ph ps) (pm3.2i.1 pm3.2i.2 ph ps pm3.2 mp2))
+thm (pm3.37 () () (-> (-> (/\ ph ps) ch) (-> (/\ ph (-. ch)) (-. ps))) (ps ph pm3.21 ch imim1d com12 ph ch iman syl6ib con2d))
+thm (pm3.43i () () (-> (-> ph ps) (-> (-> ph ch) (-> ph (/\ ps ch)))) (ps ch pm3.2 ph imim2i a2d))
+thm (jca () ((jca.1 (-> ph ps)) (jca.2 (-> ph ch))) (-> ph (/\ ps ch)) (jca.1 jca.2 jc ps ch df-an sylibr))
+thm (jcai () ((jcai.1 (-> ph ps)) (jcai.2 (-> ph (-> ps ch)))) (-> ph (/\ ps ch)) (jcai.1 jcai.1 jcai.2 mpd jca))
+thm (jctl () ((jctl.1 ps)) (-> ph (/\ ps ph)) (jctl.1 ph a1i ph id jca))
+thm (jctr () ((jctr.1 ps)) (-> ph (/\ ph ps)) (ph id jctr.1 ph a1i jca))
+thm (jctil () ((jctil.1 (-> ph ps)) (jctil.2 ch)) (-> ph (/\ ch ps)) (jctil.2 ph a1i jctil.1 jca))
+thm (jctir () ((jctir.1 (-> ph ps)) (jctir.2 ch)) (-> ph (/\ ps ch)) (jctir.1 jctir.2 ph a1i jca))
+thm (ancl () () (-> (-> ph ps) (-> ph (/\ ph ps))) (ph ps pm3.2 a2i))
+thm (ancr () () (-> (-> ph ps) (-> ph (/\ ps ph))) (ph ps pm3.21 a2i))
+thm (ancli () ((ancli.1 (-> ph ps))) (-> ph (/\ ph ps)) (ph id ancli.1 jca))
+thm (ancri () ((ancri.1 (-> ph ps))) (-> ph (/\ ps ph)) (ancri.1 ph id jca))
+thm (ancld () ((ancld.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ps ch))) (ancld.1 ps ch ancl syl))
+thm (ancrd () ((ancrd.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ch ps))) (ancrd.1 ps ch ancr syl))
+thm (anc2l () () (-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ph ch)))) (ph ch pm3.2 ps imim2d a2i))
+thm (anc2r () () (-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ch ph)))) (ph ch pm3.21 ps imim2d a2i))
+thm (anc2li () ((anc2li.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ph ch))) (anc2li.1 ph ps ch anc2l ax-mp))
+thm (anc2ri () ((anc2ri.1 (-> ph (-> ps ch)))) (-> ph (-> ps (/\ ch ph))) (anc2ri.1 ph ps ch anc2r ax-mp))
+thm (anor () () (<-> (/\ ph ps) (-. (\/ (-. ph) (-. ps)))) (ph ps df-an ph ps -. imor negbii bitr))
+thm (ianor () () (<-> (-. (/\ ph ps)) (\/ (-. ph) (-. ps))) (ph ps anor negbii ph -. ps -. \/ pm4.13 bitr4))
+thm (ioran () () (<-> (-. (\/ ph ps)) (/\ (-. ph) (-. ps))) (ph pm4.13 ps pm4.13 orbi12i negbii ph -. ps -. anor bitr4))
+thm (pm4.52 () () (<-> (/\ ph (-. ps)) (-. (\/ (-. ph) ps))) (ph ps -. anor ps pm4.13 ph -. orbi2i negbii bitr4))
+thm (pm4.53 () () (<-> (-. (/\ ph (-. ps))) (\/ (-. ph) ps)) (ph ps pm4.52 bicon2i bicomi))
+thm (pm4.54 () () (<-> (/\ (-. ph) ps) (-. (\/ ph (-. ps)))) (ph -. ps anor ph pm4.13 ps -. orbi1i negbii bitr4))
+thm (pm4.55 () () (<-> (-. (/\ (-. ph) ps)) (\/ ph (-. ps))) (ph ps pm4.54 bicon2i bicomi))
+thm (pm4.56 () () (<-> (/\ (-. ph) (-. ps)) (-. (\/ ph ps))) (ph ps ioran bicomi))
+thm (oran () () (<-> (\/ ph ps) (-. (/\ (-. ph) (-. ps)))) (ph ps \/ pm4.13 ph ps ioran negbii bitr))
+thm (pm4.57 () () (<-> (-. (/\ (-. ph) (-. ps))) (\/ ph ps)) (ph ps oran bicomi))
+thm (pm3.1 () () (-> (/\ ph ps) (-. (\/ (-. ph) (-. ps)))) (ph ps anor biimp))
+thm (pm3.11 () () (-> (-. (\/ (-. ph) (-. ps))) (/\ ph ps)) (ph ps anor biimpr))
+thm (pm3.12 () () (\/ (\/ (-. ph) (-. ps)) (/\ ph ps)) (ph ps pm3.11 orri))
+thm (pm3.13 () () (-> (-. (/\ ph ps)) (\/ (-. ph) (-. ps))) (ph ps pm3.11 con1i))
+thm (pm3.14 () () (-> (\/ (-. ph) (-. ps)) (-. (/\ ph ps))) (ph ps pm3.1 con2i))
+thm (pm3.26 () () (-> (/\ ph ps) ph) (ph ps df-an ph ps pm3.26im sylbi))
+thm (pm3.26i () ((pm3.26i.1 (/\ ph ps))) ph (pm3.26i.1 ph ps pm3.26 ax-mp))
+thm (pm3.26d () ((pm3.26d.1 (-> ph (/\ ps ch)))) (-> ph ps) (pm3.26d.1 ps ch pm3.26 syl))
+thm (pm3.26bd () ((pm3.26bd.1 (<-> ph (/\ ps ch)))) (-> ph ps) (pm3.26bd.1 biimp pm3.26d))
+thm (pm3.27 () () (-> (/\ ph ps) ps) (ph ps df-an ph ps pm3.27im sylbi))
+thm (pm3.27i () ((pm3.27i.1 (/\ ph ps))) ps (pm3.27i.1 ph ps pm3.27 ax-mp))
+thm (pm3.27d () ((pm3.27d.1 (-> ph (/\ ps ch)))) (-> ph ch) (pm3.27d.1 ps ch pm3.27 syl))
+thm (pm3.27bd () ((pm3.27bd.1 (<-> ph (/\ ps ch)))) (-> ph ch) (pm3.27bd.1 biimp pm3.27d))
+thm (pm3.41 () () (-> (-> ph ch) (-> (/\ ph ps) ch)) (ph ps pm3.26 ch imim1i))
+thm (pm3.42 () () (-> (-> ps ch) (-> (/\ ph ps) ch)) (ph ps pm3.27 ch imim1i))
+thm (anclb () () (<-> (-> ph ps) (-> ph (/\ ph ps))) (ph ps ancl ph ps pm3.27 ph imim2i impbi))
+thm (ancrb () () (<-> (-> ph ps) (-> ph (/\ ps ph))) (ph ps ancr ps ph pm3.26 ph imim2i impbi))
+thm (pm3.4 () () (-> (/\ ph ps) (-> ph ps)) (ph ps pm3.27 ph a1d))
+thm (pm4.45im () () (<-> ph (/\ ph (-> ps ph))) (ph ps ax-1 ancli ph ps ph -> pm3.26 impbi))
+thm (anim12i () ((anim12i.1 (-> ph ps)) (anim12i.2 (-> ch th))) (-> (/\ ph ch) (/\ ps th)) (ph ch pm3.26 anim12i.1 syl ph ch pm3.27 anim12i.2 syl jca))
+thm (anim1i () ((anim1i.1 (-> ph ps))) (-> (/\ ph ch) (/\ ps ch)) (anim1i.1 ch id anim12i))
+thm (anim2i () ((anim1i.1 (-> ph ps))) (-> (/\ ch ph) (/\ ch ps)) (ch id anim1i.1 anim12i))
+thm (orim12i () ((orim12i.1 (-> ph ps)) (orim12i.2 (-> ch th))) (-> (\/ ph ch) (\/ ps th)) (orim12i.1 con3i orim12i.2 con3i anim12i con3i ph ch oran ps th oran 3imtr4))
+thm (orim1i () ((orim1i.1 (-> ph ps))) (-> (\/ ph ch) (\/ ps ch)) (orim1i.1 ch id orim12i))
+thm (orim2i () ((orim1i.1 (-> ph ps))) (-> (\/ ch ph) (\/ ch ps)) (ch id orim1i.1 orim12i))
+thm (pm2.3 () () (-> (\/ ph (\/ ps ch)) (\/ ph (\/ ch ps))) (ps ch pm1.4 ph orim2i))
+thm (jao () () (-> (-> ph ps) (-> (-> ch ps) (-> (\/ ph ch) ps))) (ph ps con3 ps -. ph -. ch -. pm3.43i ps ph -. ch -. /\ con1 syl6 ph ch oran bisyl7 ch ps con3 syl5 syl))
+thm (jaoi () ((jaoi.1 (-> ph ps)) (jaoi.2 (-> ch ps))) (-> (\/ ph ch) ps) (jaoi.1 jaoi.2 ph ps ch jao mp2))
+thm (pm2.41 () () (-> (\/ ps (\/ ph ps)) (\/ ph ps)) (ps ph olc ph ps \/ id jaoi))
+thm (pm2.42 () () (-> (\/ (-. ph) (-> ph ps)) (-> ph ps)) (ph ps pm2.21 ph ps -> id jaoi))
+thm (pm2.4 () () (-> (\/ ph (\/ ph ps)) (\/ ph ps)) (ph ps orc ph ps \/ id jaoi))
+thm (pm4.44 () () (<-> ph (\/ ph (/\ ph ps))) (ph ph ps /\ orc ph id ph ps pm3.26 jaoi impbi))
+thm (pm5.63 () () (<-> (\/ ph ps) (\/ ph (/\ (-. ph) ps))) (ph ps pm2.53 ancld orrd ph ps pm2.24 ph -. ps pm3.4 jaoi orrd impbi))
+thm (impexp () () (<-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (ph ps df-an ch imbi1i ph ps ch expt ph ps ch impt impbi bitr))
+thm (pm3.3 () () (-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (ph ps ch impexp biimp))
+thm (pm3.31 () () (-> (-> ph (-> ps ch)) (-> (/\ ph ps) ch)) (ph ps ch impexp biimpr))
+thm (imp () ((imp.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) ch) (ph ps df-an imp.1 impi sylbi))
+thm (impcom () ((imp.1 (-> ph (-> ps ch)))) (-> (/\ ps ph) ch) (imp.1 com12 imp))
+thm (pm4.14 () () (<-> (-> (/\ ph ps) ch) (-> (/\ ph (-. ch)) (-. ps))) (ph ps ch impexp ph ps ch bi2.04 bitr ph ch iman ps imbi2i ps ph ch -. /\ bi2.03 3bitr))
+thm (pm4.15 () () (<-> (-> (/\ ph ps) (-. ch)) (-> (/\ ps ch) (-. ph))) (ph ps ch -. impexp ps ch imnan ph imbi2i ph ps ch /\ bi2.03 3bitr))
+thm (pm4.78 () () (<-> (\/ (-> ph ps) (-> ph ch)) (-> ph (\/ ps ch))) (ph ps -. ph ch -> impexp ph ps annim ph ch -> imbi1i ps -. ph ch bi2.04 ph imbi2i ph ps -. ch -> pm5.4 bitr 3bitr3 ph ps -> ph ch -> df-or ps ch df-or ph imbi2i 3bitr4))
+thm (pm4.79 () () (<-> (\/ (-> ps ph) (-> ch ph)) (-> (/\ ps ch) ph)) (ph -. ps -. ch -. pm4.78 ps ch ianor ph -. imbi2i bitr4 ps ph pm4.1 ch ph pm4.1 orbi12i ps ch /\ ph pm4.1 3bitr4))
+thm (pm4.87 () () (/\ (/\ (<-> (-> (/\ ph ps) ch) (-> ph (-> ps ch))) (<-> (-> ph (-> ps ch)) (-> ps (-> ph ch)))) (<-> (-> ps (-> ph ch)) (-> (/\ ps ph) ch))) (ph ps ch impexp ph ps ch bi2.04 pm3.2i ps ph ch pm3.31 ps ph ch pm3.3 impbi pm3.2i))
+thm (pm3.33 () () (-> (/\ (-> ph ps) (-> ps ch)) (-> ph ch)) (ph ps ch imim1 imp))
+thm (pm3.34 () () (-> (/\ (-> ps ch) (-> ph ps)) (-> ph ch)) (ps ch ph imim2 imp))
+thm (pm3.35 () () (-> (/\ ph (-> ph ps)) ps) (ph ps pm2.27 imp))
+thm (pm5.31 () () (-> (/\ ch (-> ph ps)) (-> ph (/\ ps ch))) (ch ps pm3.21 ph imim2d imp))
+thm (imp3a () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (/\ ps ch) th)) (imp3.1 ps ch th impexp sylibr))
+thm (imp31 () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> (/\ (/\ ph ps) ch) th) (imp3.1 imp imp))
+thm (imp32 () ((imp3.1 (-> ph (-> ps (-> ch th))))) (-> (/\ ph (/\ ps ch)) th) (imp3.1 imp3a imp))
+thm (imp4a () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> ps (-> (/\ ch th) ta))) (imp4.1 ch th ta impexp syl6ibr))
+thm (imp4b () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph ps) (-> (/\ ch th) ta)) (imp4.1 imp4a imp))
+thm (imp4c () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> (/\ (/\ ps ch) th) ta)) (imp4.1 imp3a imp3a))
+thm (imp4d () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> ph (-> (/\ ps (/\ ch th)) ta)) (imp4.1 imp4a imp3a))
+thm (imp41 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ (/\ ph ps) ch) th) ta) (imp4.1 imp imp31))
+thm (imp42 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ ph (/\ ps ch)) th) ta) (imp4.1 imp32 imp))
+thm (imp43 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\ ph ps) (/\ ch th)) ta) (imp4.1 imp4b imp))
+thm (imp44 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph (/\ (/\ ps ch) th)) ta) (imp4.1 imp4c imp))
+thm (imp45 () ((imp4.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ ph (/\ ps (/\ ch th))) ta) (imp4.1 imp4d imp))
+thm (exp () ((exp.1 (-> (/\ ph ps) ch))) (-> ph (-> ps ch)) (ph ps df-an exp.1 sylbir expi))
+thm (expcom () ((exp.1 (-> (/\ ph ps) ch))) (-> ps (-> ph ch)) (exp.1 exp com12))
+thm (exp3a () ((exp3a.1 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps (-> ch th))) (exp3a.1 ps ch th impexp sylib))
+thm (exp31 () ((exp31.1 (-> (/\ (/\ ph ps) ch) th))) (-> ph (-> ps (-> ch th))) (exp31.1 exp exp))
+thm (exp32 () ((exp32.1 (-> (/\ ph (/\ ps ch)) th))) (-> ph (-> ps (-> ch th))) (exp32.1 exp exp3a))
+thm (exp4a () ((exp4a.1 (-> ph (-> ps (-> (/\ ch th) ta))))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4a.1 ch th ta impexp syl6ib))
+thm (exp4b () ((exp4b.1 (-> (/\ ph ps) (-> (/\ ch th) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4b.1 exp3a exp))
+thm (exp4c () ((exp4c.1 (-> ph (-> (/\ (/\ ps ch) th) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4c.1 exp3a exp3a))
+thm (exp4d () ((exp4d.1 (-> ph (-> (/\ ps (/\ ch th)) ta)))) (-> ph (-> ps (-> ch (-> th ta)))) (exp4d.1 exp3a exp4a))
+thm (exp41 () ((exp41.1 (-> (/\ (/\ (/\ ph ps) ch) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp41.1 exp exp31))
+thm (exp42 () ((exp42.1 (-> (/\ (/\ ph (/\ ps ch)) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp42.1 exp31 exp3a))
+thm (exp43 () ((exp43.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp43.1 exp exp4b))
+thm (exp44 () ((exp44.1 (-> (/\ ph (/\ (/\ ps ch) th)) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp44.1 exp32 exp3a))
+thm (exp45 () ((exp45.1 (-> (/\ ph (/\ ps (/\ ch th))) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (exp45.1 exp32 exp4a))
+thm (impac () ((impac.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) (/\ ch ps)) (impac.1 ancrd imp))
+thm (adantl () ((adantl.1 (-> ph ps))) (-> (/\ ch ph) ps) (adantl.1 ch a1i imp))
+thm (adantr () ((adantr.1 (-> ph ps))) (-> (/\ ph ch) ps) (adantr.1 ch a1d imp))
+thm (adantld () ((adantld.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ th ps) ch)) (adantld.1 th a1d imp3a))
+thm (adantrd () ((adantrd.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ ps th) ch)) (adantrd.1 ps th pm3.26 syl5))
+thm (adantll () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ th ph) ps) ch) (adant2.1 exp th adantl imp))
+thm (adantlr () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ (/\ ph th) ps) ch) (adant2.1 exp th adantr imp))
+thm (adantrl () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ ph (/\ th ps)) ch) (adant2.1 exp th adantld imp))
+thm (adantrr () ((adant2.1 (-> (/\ ph ps) ch))) (-> (/\ ph (/\ ps th)) ch) (adant2.1 exp th adantrd imp))
+thm (adantlll () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ (/\ ta ph) ps) ch) th) (adantl2.1 exp31 ta a1i imp41))
+thm (adantllr () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ (/\ ph ta) ps) ch) th) (adantl2.1 exp31 ta a1d imp41))
+thm (adantlrl () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph (/\ ta ps)) ch) th) (adantl2.1 exp31 ta a1d imp42))
+thm (adantlrr () ((adantl2.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph (/\ ps ta)) ch) th) (adantl2.1 exp31 ta a1dd imp42))
+thm (adantrll () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ (/\ ta ps) ch)) th) (adantr2.1 exp32 ta a1d imp44))
+thm (adantrlr () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ (/\ ps ta) ch)) th) (adantr2.1 exp32 ta a1dd imp44))
+thm (adantrrl () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ ps (/\ ta ch))) th) (adantr2.1 exp32 ta a1dd imp45))
+thm (adantrrr () ((adantr2.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\ ps (/\ ch ta))) th) (adantr2.1 ta a1d exp32 imp45))
+thm (ad2antll () ((ad2ant.1 (-> ph ps))) (-> (/\ (/\ ph ch) th) ps) (ad2ant.1 ch adantr th adantr))
+thm (ad2antlr () ((ad2ant.1 (-> ph ps))) (-> (/\ (/\ ch ph) th) ps) (ad2ant.1 ch adantl th adantr))
+thm (ad2antrl () ((ad2ant.1 (-> ph ps))) (-> (/\ ch (/\ ph th)) ps) (ad2ant.1 th adantr ch adantl))
+thm (ad2antrr () ((ad2ant.1 (-> ph ps))) (-> (/\ ch (/\ th ph)) ps) (ad2ant.1 th adantl ch adantl))
+thm (biimpa () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ph ps) ch) (biimpa.1 biimpd imp))
+thm (biimpar () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ph ch) ps) (biimpa.1 biimprd imp))
+thm (biimpac () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ps ph) ch) (biimpa.1 biimpcd imp))
+thm (biimparc () ((biimpa.1 (-> ph (<-> ps ch)))) (-> (/\ ch ph) ps) (biimpa.1 biimprcd imp))
+thm (jaob () () (<-> (-> (\/ ph ch) ps) (/\ (-> ph ps) (-> ch ps))) (ph ch orc ps imim1i ch ph olc ps imim1i jca ph ps ch jao imp impbi))
+thm (pm4.77 () () (<-> (/\ (-> ps ph) (-> ch ph)) (-> (\/ ps ch) ph)) (ps ch ph jaob bicomi))
+thm (jaod () ((jaod.1 (-> ph (-> ps ch))) (jaod.2 (-> ph (-> th ch)))) (-> ph (-> (\/ ps th) ch)) (ps ch th jao jaod.1 jaod.2 sylc))
+thm (jaodan () ((jaodan.1 (-> (/\ ph ps) ch)) (jaodan.2 (-> (/\ ph th) ch))) (-> (/\ ph (\/ ps th)) ch) (jaodan.1 exp jaodan.2 exp jaod imp))
+thm (jaao () ((jaao.1 (-> ph (-> ps ch))) (jaao.2 (-> th (-> ta ch)))) (-> (/\ ph th) (-> (\/ ps ta) ch)) (jaao.1 th adantr jaao.2 ph adantl jaod))
+thm (pm2.63 () () (-> (\/ ph ps) (-> (\/ (-. ph) ps) ps)) (ph ps pm2.53 ph ps \/ ps idd jaod))
+thm (pm2.64 () () (-> (\/ ph ps) (-> (\/ ph (-. ps)) ph)) (ph ph ps \/ ax-1 ps ph orel2 jaoi com12))
+thm (pm3.44 () () (-> (/\ (-> ps ph) (-> ch ph)) (-> (\/ ps ch) ph)) (ps ch ph jaob biimpr))
+thm (pm4.43 () () (<-> ph (/\ (\/ ph ps) (\/ ph (-. ps)))) (ph ps orc ph ps -. orc jca ph ps pm2.64 imp impbi))
+thm (anidm () () (<-> (/\ ph ph) ph) (ph ph pm3.26 ph ph pm3.2 pm2.43i impbi))
+thm (pm4.24 () () (<-> ph (/\ ph ph)) (ph anidm bicomi))
+thm (anidms () ((anidms.1 (-> (/\ ph ph) ps))) (-> ph ps) (anidms.1 exp pm2.43i))
+thm (ancom () () (<-> (/\ ph ps) (/\ ps ph)) (ph ps pm3.27 ph ps pm3.26 jca ps ph pm3.27 ps ph pm3.26 jca impbi))
+thm (ancoms () ((ancoms.1 (-> (/\ ph ps) ch))) (-> (/\ ps ph) ch) (ps ph ancom ancoms.1 sylbi))
+thm (ancomsd () ((ancomsd.1 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> (/\ ch ps) th)) (ancomsd.1 ch ps ancom syl5ib))
+thm (pm3.22 () () (-> (/\ ph ps) (/\ ps ph)) (ph ps ancom biimp))
+thm (anass () () (<-> (/\ (/\ ph ps) ch) (/\ ph (/\ ps ch))) (ph ps ch -. impexp ps ch imnan ph imbi2i bitr negbii ph ps /\ ch df-an ph ps ch /\ df-an 3bitr4))
+thm (anasss () ((anasss.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ph (/\ ps ch)) th) (anasss.1 exp31 imp32))
+thm (anassrs () ((anassrs.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ (/\ ph ps) ch) th) (anassrs.1 exp32 imp31))
+thm (imdistan () () (<-> (-> ph (-> ps ch)) (-> (/\ ph ps) (/\ ph ch))) (ph ps ch anc2l imp3a ph ch pm3.27 ph ps /\ imim2i exp3a impbi))
+thm (imdistani () ((imdistani.1 (-> ph (-> ps ch)))) (-> (/\ ph ps) (/\ ph ch)) (imdistani.1 anc2li imp))
+thm (imdistanri () ((imdistanri.1 (-> ph (-> ps ch)))) (-> (/\ ps ph) (/\ ch ph)) (imdistanri.1 com12 impac))
+thm (imdistand () ((imdistand.1 (-> ph (-> ps (-> ch th))))) (-> ph (-> (/\ ps ch) (/\ ps th))) (imdistand.1 ps ch th imdistan sylib))
+thm (pm5.3 () () (<-> (-> (/\ ph ps) ch) (-> (/\ ph ps) (/\ ph ch))) (ph ps ch pm3.3 imdistand ph ch pm3.27 ph ps /\ imim2i impbi))
+thm (pm5.61 () () (<-> (/\ (\/ ph ps) (-. ps)) (/\ ph (-. ps))) (ps ph orel2 imdistanri ph ps orc ps -. anim1i impbi))
+thm (sylan () ((sylan.1 (-> (/\ ph ps) ch)) (sylan.2 (-> th ph))) (-> (/\ th ps) ch) (sylan.2 sylan.1 exp syl imp))
+thm (sylanb () ((sylan.1 (-> (/\ ph ps) ch)) (sylanb.2 (<-> th ph))) (-> (/\ th ps) ch) (sylan.1 sylanb.2 biimp sylan))
+thm (sylanbr () ((sylan.1 (-> (/\ ph ps) ch)) (sylanbr.2 (<-> ph th))) (-> (/\ th ps) ch) (sylan.1 sylanbr.2 biimpr sylan))
+thm (sylan2 () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2.2 (-> th ps))) (-> (/\ ph th) ch) (sylan.1 exp sylan2.2 syl5 imp))
+thm (sylan2b () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2b.2 (<-> th ps))) (-> (/\ ph th) ch) (sylan.1 sylan2b.2 biimp sylan2))
+thm (sylan2br () ((sylan.1 (-> (/\ ph ps) ch)) (sylan2br.2 (<-> ps th))) (-> (/\ ph th) ch) (sylan.1 sylan2br.2 biimpr sylan2))
+thm (syl2an () ((sylan.1 (-> (/\ ph ps) ch)) (syl2an.2 (-> th ph)) (syl2an.3 (-> ta ps))) (-> (/\ th ta) ch) (sylan.1 syl2an.2 sylan syl2an.3 sylan2))
+thm (syl2anb () ((sylan.1 (-> (/\ ph ps) ch)) (syl2anb.2 (<-> th ph)) (syl2anb.3 (<-> ta ps))) (-> (/\ th ta) ch) (sylan.1 syl2anb.2 sylanb syl2anb.3 sylan2b))
+thm (syl2anbr () ((sylan.1 (-> (/\ ph ps) ch)) (syl2anbr.2 (<-> ph th)) (syl2anbr.3 (<-> ps ta))) (-> (/\ th ta) ch) (sylan.1 syl2anbr.2 sylanbr syl2anbr.3 sylan2br))
+thm (syland () ((syland.1 (-> ph (-> (/\ ps ch) th))) (syland.2 (-> ph (-> ta ps)))) (-> ph (-> (/\ ta ch) th)) (syland.2 syland.1 exp3a syld imp3a))
+thm (sylan2d () ((sylan2d.1 (-> ph (-> (/\ ps ch) th))) (sylan2d.2 (-> ph (-> ta ch)))) (-> ph (-> (/\ ps ta) th)) (sylan2d.1 ancomsd sylan2d.2 syland ancomsd))
+var (wff et)
+thm (syl2and () ((syl2and.1 (-> ph (-> (/\ ps ch) th))) (syl2and.2 (-> ph (-> ta ps))) (syl2and.3 (-> ph (-> et ch)))) (-> ph (-> (/\ ta et) th)) (syl2and.1 syl2and.3 sylan2d syl2and.2 syland))
+thm (sylanl1 () ((sylanl1.1 (-> (/\ (/\ ph ps) ch) th)) (sylanl1.2 (-> ta ph))) (-> (/\ (/\ ta ps) ch) th) (sylanl1.1 sylanl1.2 ps anim1i sylan))
+thm (sylanl2 () ((sylanl2.1 (-> (/\ (/\ ph ps) ch) th)) (sylanl2.2 (-> ta ps))) (-> (/\ (/\ ph ta) ch) th) (sylanl2.1 sylanl2.2 ph anim2i sylan))
+thm (sylanr1 () ((sylanr1.1 (-> (/\ ph (/\ ps ch)) th)) (sylanr1.2 (-> ta ps))) (-> (/\ ph (/\ ta ch)) th) (sylanr1.1 sylanr1.2 ch anim1i sylan2))
+thm (sylanr2 () ((sylanr2.1 (-> (/\ ph (/\ ps ch)) th)) (sylanr2.2 (-> ta ch))) (-> (/\ ph (/\ ps ta)) th) (sylanr2.1 sylanr2.2 ps anim2i sylan2))
+thm (sylani () ((sylani.1 (-> ph (-> (/\ ps ch) th))) (sylani.2 (-> ta ps))) (-> ph (-> (/\ ta ch) th)) (sylani.1 sylani.2 ph a1i syland))
+thm (sylan2i () ((sylan2i.1 (-> ph (-> (/\ ps ch) th))) (sylan2i.2 (-> ta ch))) (-> ph (-> (/\ ps ta) th)) (sylan2i.1 sylan2i.2 ph a1i sylan2d))
+thm (syl2ani () ((syl2ani.1 (-> ph (-> (/\ ps ch) th))) (syl2ani.2 (-> ta ps)) (syl2ani.3 (-> et ch))) (-> ph (-> (/\ ta et) th)) (syl2ani.1 syl2ani.3 sylan2i syl2ani.2 sylani))
+thm (syldan () ((syldan.1 (-> (/\ ph ps) ch)) (syldan.2 (-> (/\ ph ch) th))) (-> (/\ ph ps) th) (syldan.1 exp syldan.2 exp syld imp))
+thm (sylan9 () ((sylan9.1 (-> ph (-> ps ch))) (sylan9.2 (-> th (-> ch ta)))) (-> (/\ ph th) (-> ps ta)) (sylan9.1 th adantr sylan9.2 ph adantl syld))
+thm (sylan9r () ((sylan9r.1 (-> ph (-> ps ch))) (sylan9r.2 (-> th (-> ch ta)))) (-> (/\ th ph) (-> ps ta)) (sylan9r.1 sylan9r.2 syl9r imp))
+thm (sylanc () ((sylanc.1 (-> (/\ ph ps) ch)) (sylanc.2 (-> th ph)) (sylanc.3 (-> th ps))) (-> th ch) (sylanc.1 exp sylanc.2 sylanc.3 sylc))
+thm (sylancb () ((sylancb.1 (-> (/\ ph ps) ch)) (sylancb.2 (<-> th ph)) (sylancb.3 (<-> th ps))) (-> th ch) (sylancb.1 sylancb.2 sylancb.3 syl2anb anidms))
+thm (sylancbr () ((sylancbr.1 (-> (/\ ph ps) ch)) (sylancbr.2 (<-> ph th)) (sylancbr.3 (<-> ps th))) (-> th ch) (sylancbr.1 sylancbr.2 sylancbr.3 syl2anbr anidms))
+thm (pm2.61an1 () ((pm2.61an1.1 (-> (/\ ph ps) ch)) (pm2.61an1.2 (-> (/\ (-. ph) ps) ch))) (-> ps ch) (pm2.61an1.1 exp pm2.61an1.2 exp pm2.61i))
+thm (pm2.61an2 () ((pm2.61an2.1 (-> (/\ ph ps) ch)) (pm2.61an2.2 (-> (/\ ph (-. ps)) ch))) (-> ph ch) (pm2.61an2.1 exp pm2.61an2.2 exp pm2.61d))
+thm (abai () () (<-> (/\ ph ps) (/\ ph (-> ph ps))) (ph ps pm3.26 ph ps pm3.4 jca ph ph ps -> pm3.26 ph ps pm3.35 jca impbi))
+thm (anbi2i () ((bi.aa (<-> ph ps))) (<-> (/\ ch ph) (/\ ch ps)) (bi.aa biimp ch anim2i bi.aa biimpr ch anim2i impbi))
+thm (anbi1i () ((bi.aa (<-> ph ps))) (<-> (/\ ph ch) (/\ ps ch)) (ph ch ancom bi.aa ch anbi2i ch ps ancom 3bitr))
+thm (anbi12i () ((bi2an.1 (<-> ph ps)) (bi2an.2 (<-> ch th))) (<-> (/\ ph ch) (/\ ps th)) (bi2an.1 ch anbi1i bi2an.2 ps anbi2i bitr))
+thm (pm5.53 () () (<-> (-> (\/ (\/ ph ps) ch) th) (/\ (/\ (-> ph th) (-> ps th)) (-> ch th))) (ph ps \/ ch th jaob ph ps th jaob ch th -> anbi1i bitr))
+thm (an12 () () (<-> (/\ ph (/\ ps ch)) (/\ ps (/\ ph ch))) (ph ps ancom ch anbi1i ph ps ch anass ps ph ch anass 3bitr3))
+thm (an23 () () (<-> (/\ (/\ ph ps) ch) (/\ (/\ ph ch) ps)) (ps ch ancom ph anbi2i ph ps ch anass ph ch ps anass 3bitr4))
+thm (an1s () ((an1s.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ps (/\ ph ch)) th) (ps ph ch an12 an1s.1 sylbi))
+thm (an1rs () ((an1rs.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\ ph ch) ps) th) (ph ch ps an23 an1rs.1 sylbi))
+thm (anabs1 () () (<-> (/\ (/\ ph ps) ph) (/\ ph ps)) (ph ps /\ ph pm3.26 ph ps /\ id ph ps pm3.26 jca impbi))
+thm (anabs5 () () (<-> (/\ ph (/\ ph ps)) (/\ ph ps)) (ph ps /\ ph ancom ph ps anabs1 bitr3))
+thm (anabs7 () () (<-> (/\ ps (/\ ph ps)) (/\ ph ps)) (ps ph ps /\ pm3.27 ph ps pm3.27 ph ps /\ id jca impbi))
+thm (anabsi5 () ((anabsi5.1 (-> ph (-> (/\ ph ps) ch)))) (-> (/\ ph ps) ch) (anabsi5.1 ps adantr pm2.43i))
+thm (anabsi6 () ((anabsi6.1 (-> ph (-> (/\ ps ph) ch)))) (-> (/\ ph ps) ch) (anabsi6.1 ancomsd anabsi5))
+thm (anabsi7 () ((anabsi7.1 (-> ps (-> (/\ ph ps) ch)))) (-> (/\ ph ps) ch) (anabsi7.1 exp3a pm2.43b imp))
+thm (anabsi8 () ((anabsi8.1 (-> ps (-> (/\ ps ph) ch)))) (-> (/\ ph ps) ch) (anabsi8.1 anabsi5 ancoms))
+thm (anabss1 () ((anabss1.1 (-> (/\ (/\ ph ps) ph) ch))) (-> (/\ ph ps) ch) (anabss1.1 ps adantrr anidms))
+thm (anabss3 () ((anabss3.1 (-> (/\ (/\ ph ps) ps) ch))) (-> (/\ ph ps) ch) (anabss3.1 ph adantrl anidms))
+thm (anabss4 () ((anabss4.1 (-> (/\ (/\ ps ph) ps) ch))) (-> (/\ ph ps) ch) (anabss4.1 anabss1 ancoms))
+thm (anabss5 () ((anabss5.1 (-> (/\ ph (/\ ph ps)) ch))) (-> (/\ ph ps) ch) (anabss5.1 ps adantlr anidms))
+thm (anabss7 () ((anabss7.1 (-> (/\ ps (/\ ph ps)) ch))) (-> (/\ ph ps) ch) (anabss7.1 exp anabsi7))
+thm (anabsan () ((anabsan.1 (-> (/\ (/\ ph ph) ps) ch))) (-> (/\ ph ps) ch) (anabsan.1 an1rs anabss1))
+thm (anabsan2 () ((anabsan2.1 (-> (/\ ph (/\ ps ps)) ch))) (-> (/\ ph ps) ch) (anabsan2.1 anassrs anabss3))
+thm (an4 () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ ph ch) (/\ ps th))) (ps ch th an12 ph anbi2i ph ps ch th /\ anass ph ch ps th /\ anass 3bitr4))
+thm (an42 () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ ph ch) (/\ th ps))) (ph ps ch th an4 ps th ancom ph ch /\ anbi2i bitr))
+thm (an4s () ((an4s.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> (/\ (/\ ph ch) (/\ ps th)) ta) (ph ch ps th an4 an4s.1 sylbi))
+thm (an42s () ((an41r3s.1 (-> (/\ (/\ ph ps) (/\ ch th)) ta))) (-> (/\ (/\ ph ch) (/\ th ps)) ta) (ph ps ch th an42 an41r3s.1 sylbir))
+thm (anandi () () (<-> (/\ ph (/\ ps ch)) (/\ (/\ ph ps) (/\ ph ch))) (ph anidm ps ch /\ anbi1i ph ph ps ch an4 bitr3))
+thm (anandir () () (<-> (/\ (/\ ph ps) ch) (/\ (/\ ph ch) (/\ ps ch))) (ch anidm ph ps /\ anbi2i ph ps ch ch an4 bitr3))
+thm (anandis () ((anandis.1 (-> (/\ (/\ ph ps) (/\ ph ch)) ta))) (-> (/\ ph (/\ ps ch)) ta) (anandis.1 an4s anabsan))
+thm (anandirs () ((anandirs.1 (-> (/\ (/\ ph ch) (/\ ps ch)) ta))) (-> (/\ (/\ ph ps) ch) ta) (anandirs.1 an4s anabsan2))
+thm (bi () () (<-> (<-> ph ps) (/\ (-> ph ps) (-> ps ph))) (ph ps bii ph ps -> ps ph -> df-an bitr4))
+thm (impbid () ((impbid.1 (-> ph (-> ps ch))) (impbid.2 (-> ph (-> ch ps)))) (-> ph (<-> ps ch)) (impbid.1 impbid.2 jca ps ch bi sylibr))
+thm (bicom () () (<-> (<-> ph ps) (<-> ps ph)) (ph ps -> ps ph -> ancom ph ps bi ps ph bi 3bitr4))
+thm (bicomd () ((bicomd.1 (-> ph (<-> ps ch)))) (-> ph (<-> ch ps)) (bicomd.1 ps ch bicom sylib))
+thm (pm4.11 () () (<-> (<-> ph ps) (<-> (-. ph) (-. ps))) (ph ps pm4.1 ps ph pm4.1 anbi12i ph ps bi ps -. ph -. bi 3bitr4 ps -. ph -. bicom bitr))
+thm (bicon4i () ((bicon4.1 (<-> (-. ph) (-. ps)))) (<-> ph ps) (bicon4.1 ph ps pm4.11 mpbir))
+thm (bicon4d () ((bicon4d.1 (-> ph (<-> (-. ps) (-. ch))))) (-> ph (<-> ps ch)) (bicon4d.1 ps ch pm4.11 sylibr))
+thm (bicon2 () () (<-> (<-> ph (-. ps)) (<-> ps (-. ph))) (ph ps bi2.03 ps ph bi2.15 anbi12i ph ps -. bi ps ph -. bi 3bitr4))
+thm (bicon2d () ((bicon2d.1 (-> ph (<-> ps (-. ch))))) (-> ph (<-> ch (-. ps))) (bicon2d.1 ch ps bicon2 sylibr))
+thm (bicon1d () ((bicon1d.1 (-> ph (<-> (-. ps) ch)))) (-> ph (<-> (-. ch) ps)) (bicon1d.1 bicomd bicon2d bicomd))
+thm (bitrd () ((bitrd.1 (-> ph (<-> ps ch))) (bitrd.2 (-> ph (<-> ch th)))) (-> ph (<-> ps th)) (bitrd.1 biimpd bitrd.2 sylibd bitrd.2 biimprd bitrd.1 sylibrd impbid))
+thm (bitr2d () ((bitr2d.1 (-> ph (<-> ps ch))) (bitr2d.2 (-> ph (<-> ch th)))) (-> ph (<-> th ps)) (bitr2d.1 bitr2d.2 bitrd bicomd))
+thm (bitr3d () ((bitr3d.1 (-> ph (<-> ps ch))) (bitr3d.2 (-> ph (<-> ps th)))) (-> ph (<-> ch th)) (bitr3d.1 bicomd bitr3d.2 bitrd))
+thm (bitr4d () ((bitr4d.1 (-> ph (<-> ps ch))) (bitr4d.2 (-> ph (<-> th ch)))) (-> ph (<-> ps th)) (bitr4d.1 bitr4d.2 bicomd bitrd))
+thm (syl5bb () ((syl5bb.1 (-> ph (<-> ps ch))) (syl5bb.2 (<-> th ps))) (-> ph (<-> th ch)) (syl5bb.2 ph a1i syl5bb.1 bitrd))
+thm (syl5rbb () ((syl5rbb.1 (-> ph (<-> ps ch))) (syl5rbb.2 (<-> th ps))) (-> ph (<-> ch th)) (syl5rbb.1 syl5rbb.2 syl5bb bicomd))
+thm (syl5bbr () ((syl5bbr.1 (-> ph (<-> ps ch))) (syl5bbr.2 (<-> ps th))) (-> ph (<-> th ch)) (syl5bbr.1 syl5bbr.2 bicomi syl5bb))
+thm (syl5rbbr () ((syl5rbbr.1 (-> ph (<-> ps ch))) (syl5rbbr.2 (<-> ps th))) (-> ph (<-> ch th)) (syl5rbbr.1 syl5rbbr.2 bicomi syl5rbb))
+thm (syl6bb () ((syl6bb.1 (-> ph (<-> ps ch))) (syl6bb.2 (<-> ch th))) (-> ph (<-> ps th)) (syl6bb.1 syl6bb.2 ph a1i bitrd))
+thm (syl6rbb () ((syl6rbb.1 (-> ph (<-> ps ch))) (syl6rbb.2 (<-> ch th))) (-> ph (<-> th ps)) (syl6rbb.1 syl6rbb.2 syl6bb bicomd))
+thm (syl6bbr () ((syl6bbr.1 (-> ph (<-> ps ch))) (syl6bbr.2 (<-> th ch))) (-> ph (<-> ps th)) (syl6bbr.1 syl6bbr.2 bicomi syl6bb))
+thm (syl6rbbr () ((syl6rbbr.1 (-> ph (<-> ps ch))) (syl6rbbr.2 (<-> th ch))) (-> ph (<-> th ps)) (syl6rbbr.1 syl6rbbr.2 bicomi syl6rbb))
+thm (sylan9bb () ((sylan9bb.1 (-> ph (<-> ps ch))) (sylan9bb.2 (-> th (<-> ch ta)))) (-> (/\ ph th) (<-> ps ta)) (sylan9bb.1 th adantr sylan9bb.2 ph adantl bitrd))
+thm (sylan9bbr () ((sylan9bbr.1 (-> ph (<-> ps ch))) (sylan9bbr.2 (-> th (<-> ch ta)))) (-> (/\ th ph) (<-> ps ta)) (sylan9bbr.1 sylan9bbr.2 sylan9bb ancoms))
+thm (3imtr3d () ((3imtr3d.1 (-> ph (-> ps ch))) (3imtr3d.2 (-> ph (<-> ps th))) (3imtr3d.3 (-> ph (<-> ch ta)))) (-> ph (-> th ta)) (3imtr3d.2 3imtr3d.1 3imtr3d.3 sylibd sylbird))
+thm (3imtr4d () ((3imtr4d.1 (-> ph (-> ps ch))) (3imtr4d.2 (-> ph (<-> th ps))) (3imtr4d.3 (-> ph (<-> ta ch)))) (-> ph (-> th ta)) (3imtr4d.2 3imtr4d.1 3imtr4d.3 sylibrd sylbid))
+thm (3bitrd () ((3bitrd.1 (-> ph (<-> ps ch))) (3bitrd.2 (-> ph (<-> ch th))) (3bitrd.3 (-> ph (<-> th ta)))) (-> ph (<-> ps ta)) (3bitrd.1 3bitrd.2 bitrd 3bitrd.3 bitrd))
+thm (3bitr3d () ((3bitr3d.1 (-> ph (<-> ps ch))) (3bitr3d.2 (-> ph (<-> ps th))) (3bitr3d.3 (-> ph (<-> ch ta)))) (-> ph (<-> th ta)) (3bitr3d.2 3bitr3d.1 bitr3d 3bitr3d.3 bitrd))
+thm (3bitr4d () ((3bitr4d.1 (-> ph (<-> ps ch))) (3bitr4d.2 (-> ph (<-> th ps))) (3bitr4d.3 (-> ph (<-> ta ch)))) (-> ph (<-> th ta)) (3bitr4d.2 3bitr4d.1 3bitr4d.3 bitr4d bitrd))
+thm (3imtr3g () ((3imtr3g.1 (-> ph (-> ps ch))) (3imtr3g.2 (<-> ps th)) (3imtr3g.3 (<-> ch ta))) (-> ph (-> th ta)) (3imtr3g.1 imp 3imtr3g.2 ph anbi2i 3imtr3g.3 3imtr3 exp))
+thm (3imtr4g () ((3imtr4g.1 (-> ph (-> ps ch))) (3imtr4g.2 (<-> th ps)) (3imtr4g.3 (<-> ta ch))) (-> ph (-> th ta)) (3imtr4g.1 3imtr4g.2 bicomi 3imtr4g.3 bicomi 3imtr3g))
+thm (3bitr3g () ((3bitr3g.1 (-> ph (<-> ps ch))) (3bitr3g.2 (<-> ps th)) (3bitr3g.3 (<-> ch ta))) (-> ph (<-> th ta)) (3bitr3g.1 3bitr3g.2 syl5bbr 3bitr3g.3 syl6bb))
+thm (3bitr4g () ((3bitr4g.1 (-> ph (<-> ps ch))) (3bitr4g.2 (<-> th ps)) (3bitr4g.3 (<-> ta ch))) (-> ph (<-> th ta)) (3bitr4g.1 3bitr4g.2 syl5bb 3bitr4g.3 syl6bbr))
+thm (prth () () (-> (/\ (-> ph ps) (-> ch th)) (-> (/\ ph ch) (/\ ps th))) (ps th pm3.2 ch imim2d ph imim2i com23 imp4b))
+thm (pm3.48 () () (-> (/\ (-> ph ps) (-> ch th)) (-> (\/ ph ch) (\/ ps th))) (ph ps -> ch th -> pm3.26 con3d ph ps -> ch th -> pm3.27 imim12d ph ch df-or ps th df-or 3imtr4g))
+thm (anim12d () ((anim12d.1 (-> ph (-> ps ch))) (anim12d.2 (-> ph (-> th ta)))) (-> ph (-> (/\ ps th) (/\ ch ta))) (ps ch th ta prth anim12d.1 anim12d.2 sylanc))
+thm (anim1d () ((anim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ ps th) (/\ ch th))) (anim1d.1 ph th idd anim12d))
+thm (anim2d () ((anim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (/\ th ps) (/\ th ch))) (ph th idd anim1d.1 anim12d))
+thm (pm3.45 () () (-> (-> ph ps) (-> (/\ ph ch) (/\ ps ch))) (ph ps -> id ch anim1d))
+thm (im2anan9 () ((im2an9.1 (-> ph (-> ps ch))) (im2an9.2 (-> th (-> ta et)))) (-> (/\ ph th) (-> (/\ ps ta) (/\ ch et))) (im2an9.1 th adantr im2an9.2 ph adantl anim12d))
+thm (im2anan9r () ((im2an9.1 (-> ph (-> ps ch))) (im2an9.2 (-> th (-> ta et)))) (-> (/\ th ph) (-> (/\ ps ta) (/\ ch et))) (im2an9.1 th adantl im2an9.2 ph adantr anim12d))
+thm (orim12d () ((orim12d.1 (-> ph (-> ps ch))) (orim12d.2 (-> ph (-> th ta)))) (-> ph (-> (\/ ps th) (\/ ch ta))) (ps ch th ta pm3.48 orim12d.1 orim12d.2 sylanc))
+thm (orim1d () ((orim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (\/ ps th) (\/ ch th))) (orim1d.1 ph th idd orim12d))
+thm (orim2d () ((orim1d.1 (-> ph (-> ps ch)))) (-> ph (-> (\/ th ps) (\/ th ch))) (ph th idd orim1d.1 orim12d))
+thm (orim2 () () (-> (-> ps ch) (-> (\/ ph ps) (\/ ph ch))) (ps ch -> id ph orim2d))
+thm (pm2.73 () () (-> (-> ph ps) (-> (\/ (\/ ph ps) ch) (\/ ps ch))) (ph ps pm2.621 ch orim1d))
+thm (pm2.74 () () (-> (-> ps ph) (-> (\/ (\/ ph ps) ch) (\/ ph ch))) (ps ph orel2 ch orim1d ph ch orc ph ps \/ ch \/ a1d ja))
+thm (pm2.75 () () (-> (\/ ph ps) (-> (\/ ph (-> ps ch)) (\/ ph ch))) (ph ch orc ph ps ch -> \/ a1d ps ch pm2.27 ph orim2d jaoi))
+thm (pm2.76 () () (-> (\/ ph (-> ps ch)) (-> (\/ ph ps) (\/ ph ch))) (ph ps ch pm2.75 com12))
+thm (pm2.8 () () (-> (\/ ps ch) (-> (\/ (-. ch) th) (\/ ps th))) (ps th orc ch -. th \/ a1d ch ps pm2.24 th orim1d jaoi))
+thm (pm2.81 () () (-> (-> ps (-> ch th)) (-> (\/ ph ps) (-> (\/ ph ch) (\/ ph th)))) (ps ch th -> ph orim2 ph ch th pm2.76 syl6))
+thm (pm2.82 () () (-> (\/ (\/ ph ps) ch) (-> (\/ (\/ ph (-. ch)) th) (\/ (\/ ph ps) th))) (ph ps \/ ph ch -. \/ ax-1 ch ps pm2.24 ph orim2d jaoi th orim1d))
+thm (pm2.85 () () (-> (-> (\/ ph ps) (\/ ph ch)) (\/ ph (-> ps ch))) (ph ps \/ ph ch \/ imor ph ps pm2.48 ph ch \/ orim1i sylbi ps ch imor ph orbi2i ph ps -. ch orordi bitr2 sylib))
+thm (pm3.2ni () ((pm3.2ni.1 (-. ph)) (pm3.2ni.2 (-. ps))) (-. (\/ ph ps)) (pm3.2ni.1 pm3.2ni.2 pm3.2i ph ps ioran mpbir))
+thm (orabs () () (<-> ph (/\ (\/ ph ps) ph)) (ph ps orc ancri ph ps \/ ph pm3.27 impbi))
+thm (oranabs () () (<-> (/\ (\/ ph (-. ps)) ps) (/\ ph ps)) (ph ps pm3.2 ps ph ps /\ pm2.21 jaoi imp ph ps -. orc ps anim1i impbi))
+thm (pm5.74 () () (<-> (-> ph (<-> ps ch)) (<-> (-> ph ps) (-> ph ch))) (ps ch bi1 ph imim2i a2d ps ch bi2 ph imim2i a2d impbid ph ps -> ph ch -> bi1 pm2.86d ph ps -> ph ch -> bi2 pm2.86d anim12d ph pm4.24 ps ch bi 3imtr4g impbi))
+thm (pm5.74i () ((pm5.74i.1 (-> ph (<-> ps ch)))) (<-> (-> ph ps) (-> ph ch)) (pm5.74i.1 ph ps ch pm5.74 mpbi))
+thm (pm5.74d () ((pm5.74d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (-> ps ch) (-> ps th))) (pm5.74d.1 ps ch th pm5.74 sylib))
+thm (pm5.74ri () ((pm5.74ri.1 (<-> (-> ph ps) (-> ph ch)))) (-> ph (<-> ps ch)) (pm5.74ri.1 ph ps ch pm5.74 mpbir))
+thm (pm5.74rd () ((pm5.74rd.1 (-> ph (<-> (-> ps ch) (-> ps th))))) (-> ph (-> ps (<-> ch th))) (pm5.74rd.1 ps ch th pm5.74 sylibr))
+thm (mpbidi () ((mpbidi.min (-> th (-> ph ps))) (mpbidi.maj (-> ph (<-> ps ch)))) (-> th (-> ph ch)) (mpbidi.min mpbidi.maj pm5.74i sylib))
+thm (ibib () () (<-> (-> ph ps) (-> ph (<-> ph ps))) (ph ps pm3.4 ph ps pm3.26 ps a1d impbid exp ph ps bi1 com12 impbid pm5.74i))
+thm (ibibr () () (<-> (-> ph ps) (-> ph (<-> ps ph))) (ph ps ibib ph ps bicom ph imbi2i bitr))
+thm (ibi () ((ibi.1 (-> ph (<-> ph ps)))) (-> ph ps) (ibi.1 biimpd pm2.43i))
+thm (ibir () ((ibir.1 (-> ph (<-> ps ph)))) (-> ph ps) (ibir.1 bicomd ibi))
+thm (ibd () ((ibd.1 (-> ph (-> ps (<-> ps ch))))) (-> ph (-> ps ch)) (ibd.1 ps ch ibib sylibr))
+thm (pm5.501 () () (-> ph (<-> ps (<-> ph ps))) (ph ps ibib pm5.74ri))
+thm (ordi () () (<-> (\/ ph (/\ ps ch)) (/\ (\/ ph ps) (\/ ph ch))) (ps ch pm3.26 ph orim2i ps ch pm3.27 ph orim2i jca ph ps df-or ph -. ps ch pm3.43i ph ch df-or ph ps ch /\ df-or 3imtr4g sylbi imp impbi))
+thm (ordir () () (<-> (\/ (/\ ph ps) ch) (/\ (\/ ph ch) (\/ ps ch))) (ch ph ps ordi ph ps /\ ch orcom ph ch orcom ps ch orcom anbi12i 3bitr4))
+thm (jcab () () (<-> (-> ph (/\ ps ch)) (/\ (-> ph ps) (-> ph ch))) (ph -. ps ch ordi ph ps ch /\ imor ph ps imor ph ch imor anbi12i 3bitr4))
+thm (pm4.76 () () (<-> (/\ (-> ph ps) (-> ph ch)) (-> ph (/\ ps ch))) (ph ps ch jcab bicomi))
+thm (jcad () ((jcad.1 (-> ph (-> ps ch))) (jcad.2 (-> ph (-> ps th)))) (-> ph (-> ps (/\ ch th))) (jcad.1 imp jcad.2 imp jca exp))
+thm (jctild () ((jctild.1 (-> ph (-> ps ch))) (jctild.2 (-> ph th))) (-> ph (-> ps (/\ th ch))) (jctild.2 ps a1d jctild.1 jcad))
+thm (jctird () ((jctird.1 (-> ph (-> ps ch))) (jctird.2 (-> ph th))) (-> ph (-> ps (/\ ch th))) (jctird.1 jctird.2 ps a1d jcad))
+thm (pm3.43 () () (-> (/\ (-> ph ps) (-> ph ch)) (-> ph (/\ ps ch))) (ph ps ch jcab biimpr))
+thm (andi () () (<-> (/\ ph (\/ ps ch)) (\/ (/\ ph ps) (/\ ph ch))) (ph -. ps -. ch -. ordi ps ch ioran ph -. orbi2i ph ps ianor ph ch ianor anbi12i 3bitr4 negbii ph ps ch \/ anor ph ps /\ ph ch /\ oran 3bitr4))
+thm (andir () () (<-> (/\ (\/ ph ps) ch) (\/ (/\ ph ch) (/\ ps ch))) (ch ph ps andi ph ps \/ ch ancom ph ch ancom ps ch ancom orbi12i 3bitr4))
+thm (orddi () () (<-> (\/ (/\ ph ps) (/\ ch th)) (/\ (/\ (\/ ph ch) (\/ ph th)) (/\ (\/ ps ch) (\/ ps th)))) (ph ps ch th /\ ordir ph ch th ordi ps ch th ordi anbi12i bitr))
+thm (anddi () () (<-> (/\ (\/ ph ps) (\/ ch th)) (\/ (\/ (/\ ph ch) (/\ ph th)) (\/ (/\ ps ch) (/\ ps th)))) (ph ps ch th \/ andir ph ch th andi ps ch th andi orbi12i bitr))
+thm (bibi2i () ((bibi.a (<-> ph ps))) (<-> (<-> ch ph) (<-> ch ps)) (ch ph bi bibi.a ch imbi1i ch ph -> anbi2i bibi.a ch imbi2i ps ch -> anbi1i ch ps bi bitr4 3bitr))
+thm (bibi1i () ((bibi.a (<-> ph ps))) (<-> (<-> ph ch) (<-> ps ch)) (ph ch bicom bibi.a ch bibi2i ch ps bicom 3bitr))
+thm (bibi12i () ((bibi.a (<-> ph ps)) (bibi12.2 (<-> ch th))) (<-> (<-> ph ch) (<-> ps th)) (bibi12.2 ph bibi2i bibi.a th bibi1i bitr))
+thm (negbid () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-. ps) (-. ch))) (bid.1 ps ch pm4.11 sylib))
+thm (imbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-> th ps) (-> th ch))) (bid.1 th a1d pm5.74d))
+thm (imbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (-> ps th) (-> ch th))) (bid.1 negbid th -. imbi2d ps th pm4.1 ch th pm4.1 3bitr4g))
+thm (orbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (\/ th ps) (\/ th ch))) (bid.1 th -. imbi2d th ps df-or th ch df-or 3bitr4g))
+thm (orbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (\/ ps th) (\/ ch th))) (bid.1 th orbi2d ps th orcom ch th orcom 3bitr4g))
+thm (anbi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\ th ps) (/\ th ch))) (bid.1 biimpd th anim2d bid.1 biimprd th anim2d impbid))
+thm (anbi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (/\ ps th) (/\ ch th))) (bid.1 th anbi2d ps th ancom ch th ancom 3bitr4g))
+thm (bibi2d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (<-> th ps) (<-> th ch))) (bid.1 th imbi2d ps th -> anbi1d bid.1 th imbi1d th ch -> anbi2d bitrd th ps bi th ch bi 3bitr4g))
+thm (bibi1d () ((bid.1 (-> ph (<-> ps ch)))) (-> ph (<-> (<-> ps th) (<-> ch th))) (bid.1 th bibi2d ps th bicom ch th bicom 3bitr4g))
+thm (orbi1 () () (-> (<-> ph ps) (<-> (\/ ph ch) (\/ ps ch))) (ph ps <-> id ch orbi1d))
+thm (anbi1 () () (-> (<-> ph ps) (<-> (/\ ph ch) (/\ ps ch))) (ph ps <-> id ch anbi1d))
+thm (pm4.22 () () (-> (/\ (<-> ph ps) (<-> ps ch)) (<-> ph ch)) (ph ps <-> id ch bibi1d biimpar))
+thm (imbi1 () () (-> (<-> ph ps) (<-> (-> ph ch) (-> ps ch))) (ph ps <-> id ch imbi1d))
+thm (imbi2 () () (-> (<-> ph ps) (<-> (-> ch ph) (-> ch ps))) (ph ps <-> ch ax-1 pm5.74d))
+thm (bibi1 () () (-> (<-> ph ps) (<-> (<-> ph ch) (<-> ps ch))) (ph ps <-> id ch bibi1d))
+thm (imbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (-> ps th) (-> ch ta))) (bi12d.1 th imbi1d bi12d.2 ch imbi2d bitrd))
+thm (orbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (\/ ps th) (\/ ch ta))) (bi12d.1 th orbi1d bi12d.2 ch orbi2d bitrd))
+thm (anbi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (/\ ps th) (/\ ch ta))) (bi12d.1 th anbi1d bi12d.2 ch anbi2d bitrd))
+thm (bibi12d () ((bi12d.1 (-> ph (<-> ps ch))) (bi12d.2 (-> ph (<-> th ta)))) (-> ph (<-> (<-> ps th) (<-> ch ta))) (bi12d.1 th bibi1d bi12d.2 ch bibi2d bitrd))
+thm (pm4.39 () () (-> (/\ (<-> ph ch) (<-> ps th)) (<-> (\/ ph ps) (\/ ch th))) (ph ch <-> ps th <-> pm3.26 ph ch <-> ps th <-> pm3.27 orbi12d))
+thm (pm4.38 () () (-> (/\ (<-> ph ch) (<-> ps th)) (<-> (/\ ph ps) (/\ ch th))) (ph ch <-> ps th <-> pm3.26 ph ch <-> ps th <-> pm3.27 anbi12d))
+thm (bi2anan9 () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ ph th) (<-> (/\ ps ta) (/\ ch et))) (bi2an9.1 ta anbi1d bi2an9.2 ch anbi2d sylan9bb))
+thm (bi2anan9r () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ th ph) (<-> (/\ ps ta) (/\ ch et))) (bi2an9.1 bi2an9.2 bi2anan9 ancoms))
+thm (bi2bian9 () ((bi2an9.1 (-> ph (<-> ps ch))) (bi2an9.2 (-> th (<-> ta et)))) (-> (/\ ph th) (<-> (<-> ps ta) (<-> ch et))) (bi2an9.1 th adantr bi2an9.2 ph adantl bibi12d))
+thm (pm4.71 () () (<-> (-> ph ps) (<-> ph (/\ ph ps))) (ph ps ancl ph ps pm3.26 ph ps -> a1i impbid ph ph ps /\ bi1 ph ps pm3.27 syl6 impbi))
+thm (pm4.71r () () (<-> (-> ph ps) (<-> ph (/\ ps ph))) (ph ps pm4.71 ph ps ancom ph bibi2i bitr))
+thm (pm4.71i () ((pm4.71i.1 (-> ph ps))) (<-> ph (/\ ph ps)) (pm4.71i.1 ph ps pm4.71 mpbi))
+thm (pm4.71ri () ((pm4.71ri.1 (-> ph ps))) (<-> ph (/\ ps ph)) (pm4.71ri.1 ph ps pm4.71r mpbi))
+thm (pm4.71rd () ((pm4.71rd.1 (-> ph (-> ps ch)))) (-> ph (<-> ps (/\ ch ps))) (pm4.71rd.1 ps ch pm4.71r sylib))
+thm (pm4.45 () () (<-> ph (/\ ph (\/ ph ps))) (ph ps orc pm4.71i))
+thm (pm4.72 () () (<-> (-> ph ps) (<-> ps (\/ ph ps))) (ps ph olc ph ps -> a1i ph ps pm2.621 impbid ps ph ps \/ bi2 ph ps pm2.67 syl impbi))
+thm (iba () () (-> ph (<-> ps (/\ ps ph))) (ph ps ancrb pm5.74ri))
+thm (ibar () () (-> ph (<-> ps (/\ ph ps))) (ph ps anclb pm5.74ri))
+thm (pm5.32 () () (<-> (-> ph (<-> ps ch)) (<-> (/\ ph ps) (/\ ph ch))) (ps ch pm4.11 ph imbi2i ph ps -. ch -. pm5.74 ph ps -. -> ph ch -. -> pm4.11 3bitr ph ps df-an ph ch df-an bibi12i bitr4))
+thm (pm5.32i () ((pm5.32i.1 (-> ph (<-> ps ch)))) (<-> (/\ ph ps) (/\ ph ch)) (pm5.32i.1 ph ps ch pm5.32 mpbi))
+thm (pm5.32ri () ((pm5.32i.1 (-> ph (<-> ps ch)))) (<-> (/\ ps ph) (/\ ch ph)) (pm5.32i.1 pm5.32i ps ph ancom ch ph ancom 3bitr4))
+thm (pm5.32d () ((pm5.32d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (/\ ps ch) (/\ ps th))) (pm5.32d.1 ps ch th pm5.32 sylib))
+thm (pm5.32rd () ((pm5.32d.1 (-> ph (-> ps (<-> ch th))))) (-> ph (<-> (/\ ch ps) (/\ th ps))) (pm5.32d.1 pm5.32d ch ps ancom th ps ancom 3bitr4g))
+thm (pm5.33 () () (<-> (/\ ph (-> ps ch)) (/\ ph (-> (/\ ph ps) ch))) (ph ps ibar ch imbi1d pm5.32i))
+thm (pm5.36 () () (<-> (/\ ph (<-> ph ps)) (/\ ps (<-> ph ps))) (ph ps <-> id pm5.32ri))
+thm (pm5.42 () () (<-> (-> ph (-> ps ch)) (-> ph (-> ps (/\ ph ch)))) (ph ch ibar ps imbi2d pm5.74i))
+thm (oibabs () () (<-> (<-> ph ps) (-> (\/ ph ps) (<-> ph ps))) (ph ps <-> ph ps \/ ax-1 ph ps orc ph ps <-> imim1i ibd ps ph olc ph ps <-> imim1i ps ph ibib ps ph bicom ps imbi2i bitr sylibr impbid impbi))
+thm (exmid () () (\/ ph (-. ph)) (ph -. id orri))
+thm (pm2.1 () () (\/ (-. ph) ph) (ph nega orri))
+thm (pm2.13 () () (\/ ph (-. (-. (-. ph)))) (ph -. negb orri))
+thm (pm3.24 () () (-. (/\ ph (-. ph))) (ph -. exmid ph ph -. ianor mpbir))
+thm (pm2.26 () () (\/ (-. ph) (-> (-> ph ps) ps)) (ph nega ps imim1i com12 orri))
+thm (pm5.18 () () (<-> (<-> ph ps) (-. (<-> ph (-. ps)))) (ph ps bicom ph ps -. bicom ps ph pm2.61 ps ph pm2.65 ph ps con2 syl5 anim12d ps -. ph bi syl5ib ph ps annim syl6ib com12 ps ph -> ph ps -> imnan sylib ps ph bi negbii sylibr ps ph bi negbii ps ph pm2.5 ps ph annim ph ps -. pm2.21 ps adantl sylbir jca ph ps annim ph ps -. ax-1 ps -. adantr sylbir ph ps pm2.51 jca jaoi ps ph -> ph ps -> ianor ps -. ph bi 3imtr4 sylbi impbi bitr bicon2i bitr))
+thm (nbbn () () (<-> (<-> (-. ph) ps) (-. (<-> ph ps))) (ph -. ps bicom ph ps bicom ps ph pm5.18 bitr bicon2i bitr))
+thm (pm5.11 () () (\/ (-> ph ps) (-> (-. ph) ps)) (ph ps pm2.5 orri))
+thm (pm5.12 () () (\/ (-> ph ps) (-> ph (-. ps))) (ph ps pm2.51 orri))
+thm (pm5.13 () () (\/ (-> ph ps) (-> ps ph)) (ph ps pm2.521 orri))
+thm (pm5.14 () () (\/ (-> ph ps) (-> ps ch)) (ps ph ax-1 con3i ch pm2.21d orri))
+thm (pm5.15 () () (\/ (<-> ph ps) (<-> ph (-. ps))) (ph ps pm5.18 biimpr con1i orri))
+thm (pm5.16 () () (-. (/\ (<-> ph ps) (<-> ph (-. ps)))) (ph ps pm5.18 biimp ph ps <-> ph ps -. <-> pm4.62 mpbi ph ps <-> ph ps -. <-> ianor mpbir))
+thm (pm5.17 () () (<-> (/\ (\/ ph ps) (-. (/\ ph ps))) (<-> ph (-. ps))) (ph ps orcom ps ph df-or bitr ph ps imnan bicomi anbi12i ps -. ph bi bitr4 ps -. ph bicom bitr))
+thm (pm5.19 () () (-. (<-> ph (-. ph))) (ph pm4.2 ph ph pm5.18 mpbi))
+thm (dfbi () () (<-> (<-> ph ps) (\/ (/\ ph ps) (/\ (-. ph) (-. ps)))) (ph ps pm5.18 ph ps imnan ps ph bi2.15 ph -. ps iman bitr anbi12i ph ps -. bi ph ps /\ ph -. ps -. /\ ioran 3bitr4r bicon1i bitr))
+thm (xor () () (<-> (-. (<-> ph ps)) (\/ (/\ ph (-. ps)) (/\ ps (-. ph)))) (ph -. ps dfbi ph ps nbbn ps ph -. ancom ph pm4.13 ps -. anbi1i orbi12i ps ph -. /\ ph ps -. /\ orcom bitr3 3bitr3))
+thm (pm5.24 () () (<-> (-. (\/ (/\ ph ps) (/\ (-. ph) (-. ps)))) (\/ (/\ ph (-. ps)) (/\ ps (-. ph)))) (ph ps dfbi negbii ph ps xor bitr3))
+thm (xor2 () () (<-> (-. (<-> ph ps)) (/\ (\/ ph ps) (-. (/\ ph ps)))) (ph ps xor ph ps /\ ph -. ps -. /\ ioran ph ps pm5.24 ph ps oran ph ps /\ -. anbi2i ph ps /\ -. ph ps \/ ancom bitr3 3bitr3 bitr))
+thm (pm5.55 () () (\/ (<-> (\/ ph ps) ph) (<-> (\/ ph ps) ps)) (ps ph pm5.13 ps ph pm4.72 ps ph orcom ph bibi2i ph ph ps \/ bicom 3bitr ph ps pm4.72 ps ph ps \/ bicom bitr orbi12i mpbi))
+thm (pm5.1 () () (-> (/\ ph ps) (<-> ph ps)) (ph ps pm5.501 biimpa))
+thm (pm5.21 () () (-> (/\ (-. ph) (-. ps)) (<-> ph ps)) (ph -. ps -. pm5.1 bicon4d))
+thm (pm5.21ni () ((pm5.21ni.1 (-> ph ps)) (pm5.21ni.2 (-> ch ps))) (-> (-. ps) (<-> ph ch)) (ph ch pm5.21 pm5.21ni.1 con3i pm5.21ni.2 con3i sylanc))
+thm (pm5.21nii () ((pm5.21ni.1 (-> ph ps)) (pm5.21ni.2 (-> ch ps)) (pm5.21nii.3 (-> ps (<-> ph ch)))) (<-> ph ch) (pm5.21nii.3 pm5.21ni.1 pm5.21ni.2 pm5.21ni pm2.61i))
+thm (pm5.35 () () (-> (/\ (-> ph ps) (-> ph ch)) (-> ph (<-> ps ch))) (ph ps -> ph ch -> pm5.1 pm5.74rd))
+thm (pm5.54 () () (\/ (<-> (/\ ph ps) ph) (<-> (/\ ph ps) ps)) (ph ps /\ ph pm5.1 anabss1 ps ph iba bicomd pm5.21ni orri))
+thm (elimant () () (-> (/\ (-> ph ps) (-> (-> ps ch) (-> ph th))) (-> ph (-> ch th))) (ph ps -> id impac ps ph pm5.1 syl ch imbi1d ph th -> imbi1d biimpd exp com23 imp ph ch th imdi syl6ibr pm2.43d))
+thm (baib () ((baib.1 (<-> ph (/\ ps ch)))) (-> ps (<-> ph ch)) (ps ch ibar baib.1 syl6rbbr))
+thm (baibr () ((baibr.1 (<-> ph (/\ ps ch)))) (-> ps (<-> ch ph)) (baibr.1 baib bicomd))
+thm (pm5.44 () () (-> (-> ph ps) (<-> (-> ph ch) (-> ph (/\ ps ch)))) (ph ps ch jcab baibr))
+thm (msca () ((msca.1 (-> ph (-> ps ch))) (msca.2 (-> th (-> ps (-. ch))))) (-> ph (-> ps (-. th))) (ph ps pm3.27 msca.1 imp jc msca.2 nsyl exp))
+thm (orcana () () (<-> (-> ph (\/ ps ch)) (-> (/\ ph (-. ps)) ch)) (ps ch df-or ph imbi2i ph ps -. ch impexp bitr4))
+thm (pm5.6 () () (<-> (-> (/\ ph (-. ps)) ch) (-> ph (\/ ps ch))) (ph ps ch orcana bicomi))
+thm (orbidi () () (<-> (\/ ph (<-> ps ch)) (<-> (\/ ph ps) (\/ ph ch))) (ph ch orc ph ps \/ a1d ph ps orc ph ch \/ a1d impbid ps ch <-> id ph orbi2d jaoi ph ps ch pm2.85 ph ch ps pm2.85 anim12i ph ps \/ ph ch \/ bi ps ch bi ph orbi2i ph ps ch -> ch ps -> ordi bitr 3imtr4 impbi))
+thm (biass () () (<-> (<-> (<-> ph ps) ch) (<-> ph (<-> ps ch))) (ph ps ch /\ ps -. ch -. /\ andi ps ch dfbi ph anbi2i ph ps ch anass ph ps -. ch -. anass orbi12i 3bitr4 ph -. ps ch -. /\ ps -. ch /\ andi ps ch xor ch ps -. ancom ps ch -. /\ orbi2i bitr ph -. anbi2i ph -. ps ch -. anass ph -. ps -. ch anass orbi12i 3bitr4 orbi12i ph ps ch <-> dfbi ph ps <-> ch dfbi ph ps dfbi ch anbi1i ph ps /\ ph -. ps -. /\ ch andir bitr ph ps xor ps ph -. ancom ph ps -. /\ orbi2i bitr ch -. anbi1i ph ps -. /\ ph -. ps /\ ch -. andir bitr orbi12i ph ps /\ ch /\ ph -. ps -. /\ ch /\ ph ps -. /\ ch -. /\ ph -. ps /\ ch -. /\ or42 3bitr 3bitr4r))
+thm (biluk () () (<-> (<-> ph ps) (<-> (<-> ch ps) (<-> ph ch))) (ph ps bicom ch bibi1i ps ph ch biass bitr ph ps <-> ch ps ph ch <-> <-> biass mpbi ch ps ph ch <-> biass bitr4))
+thm (pm5.7 () () (<-> (<-> (\/ ph ch) (\/ ps ch)) (\/ ch (<-> ph ps))) (ch ph ps orbidi ch ph orcom ch ps orcom bibi12i bitr2))
+thm (bigolden () () (<-> (<-> (/\ ph ps) ph) (<-> ps (\/ ph ps))) (ph ps pm4.71 ph ps pm4.72 ph ph ps /\ bicom 3bitr3r))
+thm (pm5.71 () () (-> (-> ps (-. ch)) (<-> (/\ (\/ ph ps) ch) (/\ ph ch))) (ps ph orel2 ph ps orc ps -. a1i impbid ch anbi1d ch ph ps \/ ph <-> pm2.21 pm5.32rd ja))
+thm (pm5.75 () () (-> (/\ (-> ch (-. ps)) (<-> ph (\/ ps ch))) (<-> (/\ ph (-. ps)) ch)) (ph ps ch \/ bi1 ph ps ch pm5.6 sylibr ch ps -. -> adantl ph ps ch \/ bi2 ch ps olc ph ps olc imim12i syl ch ps ph pm5.6 sylibr exp3a a2d impcom ch ps -. -> ph ps ch \/ <-> pm3.26 jcad impbid))
+thm (nan () () (<-> (-> ph (-. (/\ ps ch))) (-> (/\ ph ps) (-. ch))) (ph ps ch -. impexp ps ch imnan ph imbi2i bitr2))
+thm (orcanai () ((orcanai.1 (-> ph (\/ ps ch)))) (-> (/\ ph (-. ps)) ch) (orcanai.1 ord imp))
+thm (intnan () ((intnan.1 (-. ph))) (-. (/\ ps ph)) (intnan.1 ps ph pm3.27 mto))
+thm (intnanr () ((intnan.1 (-. ph))) (-. (/\ ph ps)) (intnan.1 ph ps pm3.26 mto))
+thm (intnand () ((intnand.1 (-> ph (-. ps)))) (-> ph (-. (/\ ch ps))) (intnand.1 ch ps pm3.27 nsyl))
+thm (intnanrd () ((intnand.1 (-> ph (-. ps)))) (-> ph (-. (/\ ps ch))) (intnand.1 ps ch pm3.26 nsyl))
+thm (mpan () ((mpan.1 ph) (mpan.2 (-> (/\ ph ps) ch))) (-> ps ch) (mpan.1 mpan.2 exp ax-mp))
+thm (mpan2 () ((mpan2.1 ps) (mpan2.2 (-> (/\ ph ps) ch))) (-> ph ch) (mpan2.1 mpan2.2 exp mpi))
+thm (mp2an () ((mp2an.1 ph) (mp2an.2 ps) (mp2an.3 (-> (/\ ph ps) ch))) ch (mp2an.2 mp2an.1 mp2an.3 mpan ax-mp))
+thm (mpani () ((mpani.1 ps) (mpani.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ch th)) (mpani.1 mpani.2 exp3a mpi))
+thm (mpan2i () ((mpan2i.1 ch) (mpan2i.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps th)) (mpan2i.1 mpan2i.2 exp3a mpii))
+thm (mp2ani () ((mp2ani.1 ps) (mp2ani.2 ch) (mp2ani.3 (-> ph (-> (/\ ps ch) th)))) (-> ph th) (mp2ani.2 mp2ani.1 mp2ani.3 mpani mpi))
+thm (mpand () ((mpand.1 (-> ph ps)) (mpand.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ch th)) (mpand.1 mpand.2 exp3a mpd))
+thm (mpan2d () ((mpan2d.1 (-> ph ch)) (mpan2d.2 (-> ph (-> (/\ ps ch) th)))) (-> ph (-> ps th)) (mpan2d.1 mpan2d.2 exp3a mpid))
+thm (mp2and () ((mp2and.1 (-> ph ps)) (mp2and.2 (-> ph ch)) (mp2and.3 (-> ph (-> (/\ ps ch) th)))) (-> ph th) (mp2and.2 mp2and.1 mp2and.3 mpand mpd))
+thm (mpdan () ((mpdan.1 (-> ph ps)) (mpdan.2 (-> (/\ ph ps) ch))) (-> ph ch) (mpdan.1 mpdan.2 exp mpd))
+thm (mpancom () ((mpancom.1 (-> ps ph)) (mpancom.2 (-> (/\ ph ps) ch))) (-> ps ch) (mpancom.1 mpancom.2 ancoms mpdan))
+thm (mpanl1 () ((mpanl1.1 ph) (mpanl1.2 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ps ch) th) (mpanl1.1 mpanl1.2 exp mpan imp))
+thm (mpanl2 () ((mpanl2.1 ps) (mpanl2.2 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ ph ch) th) (mpanl2.1 mpanl2.2 exp mpan2 imp))
+thm (mpanl12 () ((mpanl12.1 ph) (mpanl12.2 ps) (mpanl12.3 (-> (/\ (/\ ph ps) ch) th))) (-> ch th) (mpanl12.2 mpanl12.1 mpanl12.3 mpanl1 mpan))
+thm (mpanr1 () ((mpanr1.1 ps) (mpanr1.2 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph ch) th) (mpanr1.1 mpanr1.2 exp mpani imp))
+thm (mpanr2 () ((mpanr2.1 ch) (mpanr2.2 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph ps) th) (mpanr2.1 mpanr2.2 exp mpan2i imp))
+thm (mpanlr1 () ((mpanlr1.1 ps) (mpanlr1.2 (-> (/\ (/\ ph (/\ ps ch)) th) ta))) (-> (/\ (/\ ph ch) th) ta) (mpanlr1.1 mpanlr1.2 exp mpanr1 imp))
+thm (mtt () () (-> (-. ph) (<-> (-. ps) (-> ps ph))) (ps ph pm2.21 ph -. a1i ps ph con3 com12 impbid))
+thm (mt2bi () ((mt2bi.1 ph)) (<-> (-. ps) (-> ps (-. ph))) (ps ph -. pm2.21 mt2bi.1 ps ph con2 mpi impbi))
+thm (mtbid () ((mtbid.min (-> ph (-. ps))) (mtbid.maj (-> ph (<-> ps ch)))) (-> ph (-. ch)) (mtbid.min mtbid.maj biimprd mtod))
+thm (mtbird () ((mtbird.min (-> ph (-. ch))) (mtbird.maj (-> ph (<-> ps ch)))) (-> ph (-. ps)) (mtbird.min mtbird.maj biimpd mtod))
+thm (mtbii () ((mtbii.min (-. ps)) (mtbii.maj (-> ph (<-> ps ch)))) (-> ph (-. ch)) (mtbii.min mtbii.maj biimprd mtoi))
+thm (mtbiri () ((mtbiri.min (-. ch)) (mtbiri.maj (-> ph (<-> ps ch)))) (-> ph (-. ps)) (mtbiri.min mtbiri.maj biimpd mtoi))
+thm (2th () ((2th.1 ph) (2th.2 ps)) (<-> ph ps) (2th.2 ph a1i 2th.1 ps a1i impbi))
+thm (2false () ((2false.1 (-. ph)) (2false.2 (-. ps))) (<-> ph ps) (2false.1 2false.2 ph ps pm5.21 mp2an))
+thm (tbt () ((tbt.1 ph)) (<-> ps (<-> ps ph)) (tbt.1 ps a1i ps a1d ps ph ax-1 impbid tbt.1 ps ph bi2 mpi impbi))
+thm (nbn () ((nbn.1 (-. ph))) (<-> (-. ps) (<-> ps ph)) (ps ph pm2.21 nbn.1 ps -. a1i ps pm2.21d impbid nbn.1 ps ph bi1 mtoi impbi))
+thm (biantru () ((biantru.1 ph)) (<-> ps (/\ ps ph)) (biantru.1 ph ps iba ax-mp))
+thm (biantrur () ((biantrur.1 ph)) (<-> ps (/\ ph ps)) (biantrur.1 ph ps ibar ax-mp))
+thm (biantrud () ((biantrud.1 (-> ph ps))) (-> ph (<-> ch (/\ ch ps))) (biantrud.1 ch anim2i expcom ch ps pm3.26 ph a1i impbid))
+thm (biantrurd () ((biantrud.1 (-> ph ps))) (-> ph (<-> ch (/\ ps ch))) (biantrud.1 ch biantrud ch ps ancom syl6bb))
+thm (mpbiran () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbiran.2 ps)) (<-> ph ch) (mpbiran.1 mpbiran.2 ch biantrur bitr4))
+thm (mpbiran2 () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbiran2.2 ch)) (<-> ph ps) (mpbiran.1 mpbiran2.2 ps biantru bitr4))
+thm (mpbir2an () ((mpbiran.1 (<-> ph (/\ ps ch))) (mpbir2an.2 ps) (mpbir2an.3 ch)) ph (mpbir2an.3 mpbiran.1 mpbir2an.2 mpbiran mpbir))
+thm (biimt () () (-> ph (<-> ps (-> ph ps))) (ps ph ax-1 ph a1i ph ps pm2.27 impbid))
+thm (pm5.5 () () (-> ph (<-> (-> ph ps) ps)) (ph ps biimt bicomd))
+thm (pm5.62 () () (<-> (\/ (/\ ph ps) (-. ps)) (\/ ph (-. ps))) (ph ps ps -. ordir ps exmid mpbiran2))
+thm (biort () () (-> ph (<-> ph (\/ ph ps))) (ph ps orc ph a1d ph ph ps \/ ax-1 impbid))
+thm (biorf () () (-> (-. ph) (<-> ps (\/ ph ps))) (ph -. ps biimt ph ps df-or syl6bbr))
+thm (biorfi () ((biorfi.1 (-. ph))) (<-> ps (\/ ps ph)) (biorfi.1 ph ps biorf ax-mp ph ps orcom bitr))
+thm (bianfi () ((bianfi.1 (-. ph))) (<-> ph (/\ ps ph)) (bianfi.1 ps ph /\ pm2.21i ps ph pm3.27 impbi))
+thm (bianfd () ((bianfd.1 (-> ph (-. ps)))) (-> ph (<-> ps (/\ ps ch))) (bianfd.1 ps ch /\ pm2.21d ps ch pm3.26 ph a1i impbid))
+thm (pm4.82 () () (<-> (/\ (-> ph ps) (-> ph (-. ps))) (-. ph)) (ph ps pm2.65 imp ph ps pm2.21 ph ps -. pm2.21 jca impbi))
+thm (pm4.83 () () (<-> (/\ (-> ph ps) (-> (-. ph) ps)) ps) (ph exmid ps a1bi ph ph -. ps jaob bitr2))
+thm (pclem6 () () (-> (<-> ph (/\ ps (-. ph))) (-. ps)) (ph ps ph -. /\ bi1 ps ph -. pm3.27 syl6 pm2.01d ph ps ph -. /\ bi2 exp3a com23 ps ph con3 syli mpd))
+thm (biantr () () (-> (/\ (<-> ph ps) (<-> ch ps)) (<-> ph ch)) (ch ps <-> id ph bibi2d biimparc))
+thm (bimsc1 () () (-> (/\ (-> ph ps) (<-> ch (/\ ps ph))) (<-> ch ph)) (ch ps ph /\ <-> id ph ps pm4.71r biimp bicomd sylan9bbr))
+thm (ecase2d () ((ecase2d.1 (-> ph ps)) (ecase2d.2 (-> ph (-. (/\ ps ch)))) (ecase2d.3 (-> ph (-. (/\ ps th)))) (ecase2d.4 (-> ph (\/ ta (\/ ch th))))) (-> ph ta) (ecase2d.1 ecase2d.2 ps ch imnan sylibr mpd ecase2d.1 ecase2d.3 ps th imnan sylibr mpd jca ch th ioran sylibr ecase2d.4 ta ch th \/ orcom sylib ord mpd))
+thm (ecase3 () ((ecase3.1 (-> ph ch)) (ecase3.2 (-> ps ch)) (ecase3.3 (-> (-. (\/ ph ps)) ch))) ch (ph ps ioran ecase3.3 sylbir exp ecase3.1 ecase3.2 pm2.61ii))
+thm (ecase () ((ecase.1 (-> (-. ph) ch)) (ecase.2 (-> (-. ps) ch)) (ecase.3 (-> (/\ ph ps) ch))) ch (ecase.3 exp ecase.1 ecase.2 pm2.61nii))
+thm (ecase3d () ((ecase3d.1 (-> ph (-> ps th))) (ecase3d.2 (-> ph (-> ch th))) (ecase3d.3 (-> ph (-> (-. (\/ ps ch)) th)))) (-> ph th) (ecase3d.1 com12 ecase3d.2 com12 ecase3d.3 com12 ecase3))
+thm (caselem () () (<-> (/\ (\/ ph ps) (\/ ch th)) (\/ (\/ (/\ ph ch) (/\ ps ch)) (\/ (/\ ph th) (/\ ps th)))) (ph ps \/ ch th andi ph ps ch andir ph ps th andir orbi12i bitr))
+thm (ccase () ((ccase.1 (-> (/\ ph ps) ta)) (ccase.2 (-> (/\ ch ps) ta)) (ccase.3 (-> (/\ ph th) ta)) (ccase.4 (-> (/\ ch th) ta))) (-> (/\ (\/ ph ch) (\/ ps th)) ta) (ph ch ps th caselem ccase.1 ccase.2 jaoi ccase.3 ccase.4 jaoi jaoi sylbi))
+thm (ccased () ((ccased.1 (-> ph (-> (/\ ps ch) et))) (ccased.2 (-> ph (-> (/\ th ch) et))) (ccased.3 (-> ph (-> (/\ ps ta) et))) (ccased.4 (-> ph (-> (/\ th ta) et)))) (-> ph (-> (/\ (\/ ps th) (\/ ch ta)) et)) (ccased.1 ccased.2 jaod ccased.3 ccased.4 jaod jaod ps th ch ta caselem syl5ib))
+thm (ccase2 () ((ccase2.1 (-> (/\ ph ps) ta)) (ccase2.2 (-> ch ta)) (ccase2.3 (-> th ta))) (-> (/\ (\/ ph ch) (\/ ps th)) ta) (ccase2.1 ccase2.2 ps adantr ccase2.3 ph adantl ccase2.3 ch adantl ccase))
+thm (4cases () ((4cases.1 (-> (/\ ph ps) ch)) (4cases.2 (-> (/\ ph (-. ps)) ch)) (4cases.3 (-> (/\ (-. ph) ps) ch)) (4cases.4 (-> (/\ (-. ph) (-. ps)) ch))) ch (4cases.1 4cases.3 pm2.61an1 4cases.2 4cases.4 pm2.61an1 pm2.61i))
+thm (niabn () ((niabn.1 ph)) (-> (-. ps) (<-> (/\ ch ps) (-. ph))) (ch ps pm3.27 niabn.1 ps pm2.21ni pm5.21ni))
+thm (dedlem0a () () (-> ph (<-> ps (-> (-> ch ph) (/\ ps ph)))) (ps ch ph -> ax-1 ph a1i ph ch ax-1 ps imim1i com12 impbid ph ps iba ch ph -> imbi2d bitrd))
+thm (dedlem0b () () (-> (-. ph) (<-> ps (-> (-> ps ph) (/\ ch ph)))) (ph ch ph /\ pm2.21 ps imim2d com23 ps ph pm2.21 ch ph pm3.27 imim12i con1d com12 impbid))
+thm (dedlema () () (-> ph (<-> ps (\/ (/\ ps ph) (/\ ch (-. ph))))) (ps ch ph -. /\ orc ph a1i ph ps idd ph ps pm2.24 ch adantld jaod impbid ph ps iba ch ph -. /\ orbi1d bitrd))
+thm (dedlemb () () (-> (-. ph) (<-> ch (\/ (/\ ps ph) (/\ ch (-. ph))))) (ch ph -. /\ ps ph /\ olc expcom ph ps ch -> pm2.21 com23 imp3a ch ph -. pm3.26 ph -. a1i jaod impbid))
+thm (elimh () ((elimh.1 (-> (<-> ph (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> ch ta))) (elimh.2 (-> (<-> ps (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> th ta))) (elimh.3 th)) ta (ch ph ps dedlema elimh.1 syl ibi elimh.3 ch ps ph dedlemb elimh.2 syl mpbii pm2.61i))
+thm (dedt () ((dedt.1 (-> (<-> ph (\/ (/\ ph ch) (/\ ps (-. ch)))) (<-> th ta))) (dedt.2 ta)) (-> ch th) (ch ph ps dedlema dedt.2 dedt.1 mpbiri syl))
+thm (con3th () () (-> (-> ph ps) (-> (-. ps) (-. ph))) (ps ps ph ps -> /\ ph ph ps -> -. /\ \/ <-> id negbid ph -. imbi1d ps ps ph ps -> /\ ph ph ps -> -. /\ \/ <-> id ph imbi2d ph ps ph ps -> /\ ph ph ps -> -. /\ \/ <-> id ph imbi2d ph id elimh con3i dedt))
+thm (consensus () () (<-> (\/ (\/ (/\ ph ps) (/\ (-. ph) ch)) (/\ ps ch)) (\/ (/\ ph ps) (/\ (-. ph) ch))) (ph ps /\ ph -. ch /\ \/ id ph ps ch dedlema biimpd ch adantrd ph ch ps dedlemb biimpd ps adantld pm2.61i ps ph ancom ch ph -. ancom orbi12i sylib jaoi ph ps /\ ph -. ch /\ \/ ps ch /\ orc impbi))
+thm (pm4.42 () () (<-> ph (\/ (/\ ph ps) (/\ ph (-. ps)))) (ps ph ph dedlema ps ph ph dedlemb pm2.61i))
+thm (ninba () ((ninba.1 ph)) (-> (-. ps) (<-> (-. ph) (/\ ch ps))) (ninba.1 ps ch niabn bicomd))
+thm (prlem1 () ((prlem1.1 (-> ph (<-> et ch))) (prlem1.2 (-> ps (-. th)))) (-> ph (-> ps (-> (\/ (/\ ps ch) (/\ th ta)) et))) (prlem1.1 biimprcd ps adantl ps a1dd th et pm2.24 prlem1.2 syl5 ta adantr ph a1d jaoi com3l))
+thm (prlem2 () () (<-> (\/ (/\ ph ps) (/\ ch th)) (/\ (\/ ph ch) (\/ (/\ ph ps) (/\ ch th)))) (ph ch orabs ps anbi1i ph ch \/ ph ps anass bitr ch ph orabs ch ph orcom ch anbi1i bitr th anbi1i ph ch \/ ch th anass bitr orbi12i ph ch \/ ph ps /\ ch th /\ andi bitr4))
+thm (oplem1 () ((oplem1.1 (-> ph (\/ ps ch))) (oplem1.2 (-> ph (\/ th ta))) (oplem1.3 (<-> ps th)) (oplem1.4 (-> ch (<-> th ta)))) (-> ph ps) (oplem1.1 ord oplem1.2 ord oplem1.3 negbii syl5ib jcad oplem1.4 oplem1.3 syl5bb biimpar syl6 ps pm2.18 syl))
+thm (rnlem () () (<-> (/\ (/\ ph ps) (/\ ch th)) (/\ (/\ (/\ ph ch) (/\ ps th)) (/\ (/\ ph th) (/\ ps ch)))) (ph ps ch th /\ anandir ph ch th anandi ps ch th anandi anbi12i ps ch /\ ps th /\ ancom ph ch /\ ph th /\ /\ anbi2i ph ch /\ ph th /\ ps th /\ ps ch /\ an4 bitr 3bitr))
+thm (3orass () () (<-> (\/\/ ph ps ch) (\/ ph (\/ ps ch))) (ph ps ch df-3or ph ps ch orass bitr))
+thm (3anass () () (<-> (/\/\ ph ps ch) (/\ ph (/\ ps ch))) (ph ps ch df-3an ph ps ch anass bitr))
+thm (3anrot () () (<-> (/\/\ ph ps ch) (/\/\ ps ch ph)) (ph ps ch /\ ancom ph ps ch 3anass ps ch ph df-3an 3bitr4))
+thm (3orrot () () (<-> (\/\/ ph ps ch) (\/\/ ps ch ph)) (ph ps ch \/ orcom ph ps ch 3orass ps ch ph df-3or 3bitr4))
+thm (3ancoma () () (<-> (/\/\ ph ps ch) (/\/\ ps ph ch)) (ph ps ancom ch anbi1i ph ps ch df-3an ps ph ch df-3an 3bitr4))
+thm (3ancomb () () (<-> (/\/\ ph ps ch) (/\/\ ph ch ps)) (ph ps ch 3ancoma ps ph ch 3anrot bitr))
+thm (3anrev () () (<-> (/\/\ ph ps ch) (/\/\ ch ps ph)) (ph ps ch 3ancoma ch ps ph 3anrot bitr4))
+thm (3simpa () () (-> (/\/\ ph ps ch) (/\ ph ps)) (ph ps ch df-3an pm3.26bd))
+thm (3simpb () () (-> (/\/\ ph ps ch) (/\ ph ch)) (ph ps ch 3ancomb ph ch ps 3simpa sylbi))
+thm (3simpc () () (-> (/\/\ ph ps ch) (/\ ps ch)) (ph ps ch 3anrot ps ch ph 3simpa sylbi))
+thm (3simp1 () () (-> (/\/\ ph ps ch) ph) (ph ps ch 3simpa pm3.26d))
+thm (3simp2 () () (-> (/\/\ ph ps ch) ps) (ph ps ch 3simpa pm3.27d))
+thm (3simp3 () () (-> (/\/\ ph ps ch) ch) (ph ps ch 3simpc pm3.27d))
+thm (3simp1i () ((3simp1i.1 (/\/\ ph ps ch))) ph (3simp1i.1 ph ps ch 3simp1 ax-mp))
+thm (3simp2i () ((3simp1i.1 (/\/\ ph ps ch))) ps (3simp1i.1 ph ps ch 3simp2 ax-mp))
+thm (3simp3i () ((3simp1i.1 (/\/\ ph ps ch))) ch (3simp1i.1 ph ps ch 3simp3 ax-mp))
+thm (3simp1d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph ps) (3simp1d.1 ps ch th 3simp1 syl))
+thm (3simp2d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph ch) (3simp1d.1 ps ch th 3simp2 syl))
+thm (3simp3d () ((3simp1d.1 (-> ph (/\/\ ps ch th)))) (-> ph th) (3simp1d.1 ps ch th 3simp3 syl))
+thm (3adant1 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ th ph ps) ch) (th ph ps 3simpc 3adant.1 syl))
+thm (3adant2 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ ph th ps) ch) (ph th ps 3simpb 3adant.1 syl))
+thm (3adant3 () ((3adant.1 (-> (/\ ph ps) ch))) (-> (/\/\ ph ps th) ch) (ph ps th 3simpa 3adant.1 syl))
+thm (3ad2ant1 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ph ps th) ch) (3ad2ant.1 th adantr ps 3adant2))
+thm (3ad2ant2 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ps ph th) ch) (3ad2ant.1 th adantr ps 3adant1))
+thm (3ad2ant3 () ((3ad2ant.1 (-> ph ch))) (-> (/\/\ ps th ph) ch) (3ad2ant.1 th adantl ps 3adant1))
+thm (3adantl1 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ta ph ps) ch) th) (3adantl.1 exp ta 3adant1 imp))
+thm (3adantl2 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ph ta ps) ch) th) (3adantl.1 exp ta 3adant2 imp))
+thm (3adantl3 () ((3adantl.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\ (/\/\ ph ps ta) ch) th) (3adantl.1 exp ta 3adant3 imp))
+thm (3adantr1 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ta ps ch)) th) (3adantr.1 ancoms ta 3adantl1 ancoms))
+thm (3adantr2 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ps ta ch)) th) (3adantr.1 ancoms ta 3adantl2 ancoms))
+thm (3adantr3 () ((3adantr.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\ ph (/\/\ ps ch ta)) th) (3adantr.1 ancoms ta 3adantl3 ancoms))
+thm (3mix1 () () (-> ph (\/\/ ph ps ch)) (ph ps ch \/ orc ph ps ch 3orass sylibr))
+thm (3mix2 () () (-> ph (\/\/ ps ph ch)) (ph ch ps 3mix1 ps ph ch 3orrot sylibr))
+thm (3mix3 () () (-> ph (\/\/ ps ch ph)) (ph ps ch 3mix1 ph ps ch 3orrot sylib))
+thm (3pm3.2i () ((3pm3.2i.1 ph) (3pm3.2i.2 ps) (3pm3.2i.3 ch)) (/\/\ ph ps ch) (ph ps ch df-3an 3pm3.2i.1 3pm3.2i.2 pm3.2i 3pm3.2i.3 mpbir2an))
+thm (3jca () ((3jca.1 (-> ph ps)) (3jca.2 (-> ph ch)) (3jca.3 (-> ph th))) (-> ph (/\/\ ps ch th)) (3jca.1 3jca.2 jca 3jca.3 jca ps ch th df-3an sylibr))
+thm (3jcad () ((3jcad.1 (-> ph (-> ps ch))) (3jcad.2 (-> ph (-> ps th))) (3jcad.3 (-> ph (-> ps ta)))) (-> ph (-> ps (/\/\ ch th ta))) (3jcad.1 imp 3jcad.2 imp 3jcad.3 imp 3jca exp))
+thm (3anim123i () ((3anim123i.1 (-> ph ps)) (3anim123i.2 (-> ch th)) (3anim123i.3 (-> ta et))) (-> (/\/\ ph ch ta) (/\/\ ps th et)) (3anim123i.1 3anim123i.2 anim12i 3anim123i.3 anim12i ph ch ta df-3an ps th et df-3an 3imtr4))
+thm (3anbi123i () ((bi3.1 (<-> ph ps)) (bi3.2 (<-> ch th)) (bi3.3 (<-> ta et))) (<-> (/\/\ ph ch ta) (/\/\ ps th et)) (bi3.1 bi3.2 anbi12i bi3.3 anbi12i ph ch ta df-3an ps th et df-3an 3bitr4))
+thm (3orbi123i () ((bi3.1 (<-> ph ps)) (bi3.2 (<-> ch th)) (bi3.3 (<-> ta et))) (<-> (\/\/ ph ch ta) (\/\/ ps th et)) (bi3.1 bi3.2 orbi12i bi3.3 orbi12i ph ch ta df-3or ps th et df-3or 3bitr4))
+thm (3imp () ((3imp.1 (-> ph (-> ps (-> ch th))))) (-> (/\/\ ph ps ch) th) (ph ps ch df-3an 3imp.1 imp31 sylbi))
+thm (3impa () ((3impa.1 (-> (/\ (/\ ph ps) ch) th))) (-> (/\/\ ph ps ch) th) (3impa.1 exp31 3imp))
+thm (3impb () ((3impb.1 (-> (/\ ph (/\ ps ch)) th))) (-> (/\/\ ph ps ch) th) (3impb.1 exp32 3imp))
+thm (3exp () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> ph (-> ps (-> ch th))) (ph ps ch df-3an 3exp.1 sylbir exp31))
+thm (3expa () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\ (/\ ph ps) ch) th) (3exp.1 3exp imp31))
+thm (3expb () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\ ph (/\ ps ch)) th) (3exp.1 3exp imp32))
+thm (3com12 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ps ph ch) th) (3exp.1 3exp com12 3imp))
+thm (3com13 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ch ps ph) th) (ch ps ph 3anrev 3exp.1 sylbi))
+thm (3com23 () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ph ch ps) th) (3exp.1 3exp com23 3imp))
+thm (3coml () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ps ch ph) th) (3exp.1 3com23 3com13))
+thm (3comr () ((3exp.1 (-> (/\/\ ph ps ch) th))) (-> (/\/\ ch ph ps) th) (3exp.1 3coml 3coml))
+thm (3imp1 () ((3imp1.1 (-> ph (-> ps (-> ch (-> th ta)))))) (-> (/\ (/\/\ ph ps ch) th) ta) (3imp1.1 3imp imp))
+thm (3exp1 () ((3exp1.1 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> ph (-> ps (-> ch (-> th ta)))) (3exp1.1 exp 3exp))
+var (wff ze)
+thm (syl3an1 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1.2 (-> ta ph))) (-> (/\/\ ta ps ch) th) (syl3an.1 3expb syl3an1.2 sylan 3impb))
+thm (syl3an2 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2.2 (-> ta ps))) (-> (/\/\ ph ta ch) th) (syl3an.1 3exp syl3an2.2 syl5 3imp))
+thm (syl3an3 () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3.2 (-> ta ch))) (-> (/\/\ ph ps ta) th) (syl3an.1 3exp syl3an3.2 syl7 3imp))
+thm (syl3an1b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1b.2 (<-> ta ph))) (-> (/\/\ ta ps ch) th) (syl3an.1 syl3an1b.2 biimp syl3an1))
+thm (syl3an2b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2b.2 (<-> ta ps))) (-> (/\/\ ph ta ch) th) (syl3an.1 syl3an2b.2 biimp syl3an2))
+thm (syl3an3b () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3b.2 (<-> ta ch))) (-> (/\/\ ph ps ta) th) (syl3an.1 syl3an3b.2 biimp syl3an3))
+thm (syl3an1br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an1br.2 (<-> ph ta))) (-> (/\/\ ta ps ch) th) (syl3an.1 syl3an1br.2 biimpr syl3an1))
+thm (syl3an2br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an2br.2 (<-> ps ta))) (-> (/\/\ ph ta ch) th) (syl3an.1 syl3an2br.2 biimpr syl3an2))
+thm (syl3an3br () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an3br.2 (<-> ch ta))) (-> (/\/\ ph ps ta) th) (syl3an.1 syl3an3br.2 biimpr syl3an3))
+thm (syl3an () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3an.2 (-> ta ph)) (syl3an.3 (-> et ps)) (syl3an.4 (-> ze ch))) (-> (/\/\ ta et ze) th) (syl3an.2 syl3an.3 syl3an.4 3anim123i syl3an.1 syl))
+thm (syl3anb () ((syl3an.1 (-> (/\/\ ph ps ch) th)) (syl3anb.2 (<-> ta ph)) (syl3anb.3 (<-> et ps)) (syl3anb.4 (<-> ze ch))) (-> (/\/\ ta et ze) th) (syl3anb.2 syl3anb.3 syl3anb.4 3anbi123i syl3an.1 sylbi))
+thm (syl3anl1 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an11.2 (-> et ph))) (-> (/\ (/\/\ et ps ch) th) ta) (syl3anl1.1 exp syl3an11.2 syl3an1 imp))
+thm (syl3anl2 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an12.2 (-> et ps))) (-> (/\ (/\/\ ph et ch) th) ta) (syl3anl1.1 exp syl3an12.2 syl3an2 imp))
+thm (syl3anl3 () ((syl3anl1.1 (-> (/\ (/\/\ ph ps ch) th) ta)) (syl3an13.2 (-> et ch))) (-> (/\ (/\/\ ph ps et) th) ta) (syl3anl1.1 exp syl3an13.2 syl3an3 imp))
+thm (syl3anc () ((syl3anc.1 (-> (/\/\ ph ps ch) th)) (syl3anc.2 (-> ta ph)) (syl3anc.3 (-> ta ps)) (syl3anc.4 (-> ta ch))) (-> ta th) (syl3anc.2 syl3anc.3 syl3anc.4 3jca syl3anc.1 syl))
+thm (3impdi () ((3impdi.1 (-> (/\ (/\ ph ps) (/\ ph ch)) th))) (-> (/\/\ ph ps ch) th) (3impdi.1 anandis 3impb))
+thm (3impdir () ((3impdir.1 (-> (/\ (/\ ph ps) (/\ ch ps)) th))) (-> (/\/\ ph ch ps) th) (3impdir.1 anandirs 3impa))
+thm (3jao () () (-> (/\/\ (-> ph ps) (-> ch ps) (-> th ps)) (-> (\/\/ ph ch th) ps)) (ph ps ch jao ph ch \/ ps th jao syl6 3imp ph ch th df-3or syl5ib))
+thm (3jaoi () ((3jaoi.1 (-> ph ps)) (3jaoi.2 (-> ch ps)) (3jaoi.3 (-> th ps))) (-> (\/\/ ph ch th) ps) (3jaoi.1 3jaoi.2 3jaoi.3 3pm3.2i ph ps ch th 3jao ax-mp))
+thm (3jaod () ((3jaod.1 (-> ph (-> ps ch))) (3jaod.2 (-> ph (-> th ch))) (3jaod.3 (-> ph (-> ta ch)))) (-> ph (-> (\/\/ ps th ta) ch)) (ps ch th ta 3jao 3jaod.1 3jaod.2 3jaod.3 syl3anc))
+thm (3jaoian () ((3jaoian.1 (-> (/\ ph ps) ch)) (3jaoian.2 (-> (/\ th ps) ch)) (3jaoian.3 (-> (/\ ta ps) ch))) (-> (/\ (\/\/ ph th ta) ps) ch) (3jaoian.1 exp 3jaoian.2 exp 3jaoian.3 exp 3jaoi imp))
+thm (3jaodan () ((3jaodan.1 (-> (/\ ph ps) ch)) (3jaodan.2 (-> (/\ ph th) ch)) (3jaodan.3 (-> (/\ ph ta) ch))) (-> (/\ ph (\/\/ ps th ta)) ch) (3jaodan.1 exp 3jaodan.2 exp 3jaodan.3 exp 3jaod imp))
+thm (syl3an9b () ((syl3an9b.1 (-> ph (<-> ps ch))) (syl3an9b.2 (-> th (<-> ch ta))) (syl3an9b.3 (-> et (<-> ta ze)))) (-> (/\/\ ph th et) (<-> ps ze)) (syl3an9b.1 syl3an9b.2 sylan9bb syl3an9b.3 sylan9bb 3impa))
+thm (3orbi123d () ((bi3d.1 (-> ph (<-> ps ch))) (bi3d.2 (-> ph (<-> th ta))) (bi3d.3 (-> ph (<-> et ze)))) (-> ph (<-> (\/\/ ps th et) (\/\/ ch ta ze))) (bi3d.1 bi3d.2 orbi12d bi3d.3 orbi12d ps th et df-3or ch ta ze df-3or 3bitr4g))
+thm (3anbi123d () ((bi3d.1 (-> ph (<-> ps ch))) (bi3d.2 (-> ph (<-> th ta))) (bi3d.3 (-> ph (<-> et ze)))) (-> ph (<-> (/\/\ ps th et) (/\/\ ch ta ze))) (bi3d.1 bi3d.2 anbi12d bi3d.3 anbi12d ps th et df-3an ch ta ze df-3an 3bitr4g))
+thm (3anim123d () ((im3d.1 (-> ph (-> ps ch))) (im3d.2 (-> ph (-> th ta))) (im3d.3 (-> ph (-> et ze)))) (-> ph (-> (/\/\ ps th et) (/\/\ ch ta ze))) (im3d.1 im3d.2 anim12d im3d.3 anim12d ps th et df-3an ch ta ze df-3an 3imtr4g))
+thm (3orim123d () ((im3d.1 (-> ph (-> ps ch))) (im3d.2 (-> ph (-> th ta))) (im3d.3 (-> ph (-> et ze)))) (-> ph (-> (\/\/ ps th et) (\/\/ ch ta ze))) (im3d.1 im3d.2 orim12d im3d.3 orim12d ps th et df-3or ch ta ze df-3or 3imtr4g))
+thm (an6 () () (<-> (/\ (/\/\ ph ps ch) (/\/\ th ta et)) (/\/\ (/\ ph th) (/\ ps ta) (/\ ch et))) (ph ps ch df-3an th ta et df-3an anbi12i ph ps /\ ch th ta /\ et an4 ph ps th ta an4 ch et /\ anbi1i 3bitr ph th /\ ps ta /\ ch et /\ df-3an bitr4))
+thm (mp3an1 () ((mp3an1.1 ph) (mp3an1.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ps ch) th) (mp3an1.1 mp3an1.2 3expb mpan))
+thm (mp3an2 () ((mp3an2.1 ps) (mp3an2.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ph ch) th) (mp3an2.1 mp3an2.2 3expa mpanl2))
+thm (mp3an3 () ((mp3an3.1 ch) (mp3an3.2 (-> (/\/\ ph ps ch) th))) (-> (/\ ph ps) th) (mp3an3.1 mp3an3.2 3expa mpan2))
+thm (mp3an12 () ((mp3an12.1 ph) (mp3an12.2 ps) (mp3an12.3 (-> (/\/\ ph ps ch) th))) (-> ch th) (mp3an12.2 mp3an12.1 mp3an12.3 mp3an1 mpan))
+thm (mp3an13 () ((mp3an13.1 ph) (mp3an13.2 ch) (mp3an13.3 (-> (/\/\ ph ps ch) th))) (-> ps th) (mp3an13.1 mp3an13.2 mp3an13.3 mp3an3 mpan))
+thm (mp3an23 () ((mp3an23.1 ps) (mp3an23.2 ch) (mp3an23.3 (-> (/\/\ ph ps ch) th))) (-> ph th) (mp3an23.1 mp3an23.2 mp3an23.3 mp3an3 mpan2))
+thm (mp3an1i () ((mp3an1i.1 ps) (mp3an1i.2 (-> ph (-> (/\/\ ps ch th) ta)))) (-> ph (-> (/\ ch th) ta)) (mp3an1i.1 mp3an1i.2 com12 mp3an1 com12))
+thm (mp3anl1 () ((mp3anl1.1 ph) (mp3anl1.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ps ch) th) ta) (mp3anl1.1 mp3anl1.2 exp mp3an1 imp))
+thm (mp3anl2 () ((mp3anl2.1 ps) (mp3anl2.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ph ch) th) ta) (mp3anl2.1 mp3anl2.2 exp mp3an2 imp))
+thm (mp3anl3 () ((mp3anl3.1 ch) (mp3anl3.2 (-> (/\ (/\/\ ph ps ch) th) ta))) (-> (/\ (/\ ph ps) th) ta) (mp3anl3.1 mp3anl3.2 exp mp3an3 imp))
+thm (mp3an () ((mp3an.1 ph) (mp3an.2 ps) (mp3an.3 ch) (mp3an.4 (-> (/\/\ ph ps ch) th))) th (mp3an.2 mp3an.3 mp3an.1 mp3an.4 mp3an1 mp2an))
+thm (biimp3a () ((biimp3a.1 (-> (/\ ph ps) (<-> ch th)))) (-> (/\/\ ph ps ch) th) (biimp3a.1 biimpa 3impa))
+thm (ecase23d () ((ecase23d.1 (-> ph (-. ch))) (ecase23d.2 (-> ph (-. th))) (ecase23d.3 (-> ph (\/\/ ps ch th)))) (-> ph ps) (ecase23d.1 ecase23d.2 jca ch th ioran sylibr ecase23d.3 ps ch th 3orass sylib ord mt3d))
+thm (3ecase () ((3ecase.1 (-> (-. ph) th)) (3ecase.2 (-> (-. ps) th)) (3ecase.3 (-> (-. ch) th)) (3ecase.4 (-> (/\/\ ph ps ch) th))) th (3ecase.4 3exp 3ecase.1 ch a1d ps a1d pm2.61i 3ecase.2 3ecase.3 pm2.61nii))
+thm (meredith () () (-> (-> (-> (-> (-> ph ps) (-> (-. ch) (-. th))) ch) ta) (-> (-> ta ph) (-> th ph))) (ch ch -. ph -. th -. -> -> ax-3 ph ps pm2.21 ch -. th -. -> imim1i com23 syl5 ta imim1i con3d ch -. ph -. th -. -> pm2.27 impi com12 ta -. imim2d com12 a2d ta ph con3 syl5 ph th ax-3 syl6 syl))
+thm (merlem1 () () (-> (-> (-> ch (-> (-. ph) ps)) ta) (-> ph ta)) (ph -. ps ta -. ch -. -> ph -. ps -> -. ta meredith ph -. ps -> ta -. ch -. -> -. ph -. ps -> -. -. -> ta ch ta ph -. -> ph -. ps -> -. ph -. -> -> meredith ax-mp ta ph -. ph -. ps -> ph ch ph -. ps -> -> meredith ax-mp))
+thm (merlem2 () () (-> (-> (-> ph ph) ch) (-> th ch)) (ch ch -> ph th -. ph merlem1 ch ch ph th ph ph -> meredith ax-mp))
+thm (merlem3 () () (-> (-> (-> ps ch) ph) (-> ch ph)) (ch -. ch -. ch -. -> ph ph -> merlem2 ch -. ch -. -> ph ph -> ch -. ch -. -> -> ch ph -> ps -. ps -. -> -> ps -> merlem2 ax-mp ch ph ps ps ph ph -> ch -. ch -. -> -> meredith ax-mp ph ph ch ch ps ch -> meredith ax-mp))
+thm (merlem4 () () (-> ta (-> (-> ta ph) (-> th ph))) (ph ph th th ta meredith ph ph -> th -. th -. -> -> th -> ta ta ph -> th ph -> -> merlem3 ax-mp))
+thm (merlem5 () () (-> (-> ph ps) (-> (-. (-. ph)) ps)) (ps ps ps ps ps meredith ps ps ps ph -. -. ph meredith ph ps -> ph -. ps ps ps -> ps -. ps -. -> -> ps -> ps -> ps ps -> ps ps -> -> -> -. merlem1 ph ps -> ph -. -. ps -> -> ps ps -> ps -. ps -. -> -> ps -> ps -> ps ps -> ps ps -> -> -> -. -> ph -. ps ps -> ps -. ps -. -> -> ps -> ps -> ps ps -> ps ps -> -> -> -. -> -> ph ps ps -> ps -. ph -. -. -. -> -> ps -> merlem4 ax-mp ph ps -> ph -. -. ps -> -> ps ps -> ps -. ps -. -> -> ps -> ps -> ps ps -> ps ps -> -> -> -. ph ps ps -> ps -. ps -. -> -> ps -> ps -> ps ps -> ps ps -> -> -> ps ps -> ps -. ph -. -. -. -> -> ps -> ph -> meredith ax-mp ax-mp ax-mp))
+thm (merlem6 () () (-> ch (-> (-> (-> ps ch) ph) (-> th ph))) (ps ch -> ph th merlem4 ps ch ps ch -> ph -> th ph -> -> merlem3 ax-mp))
+thm (merlem7 () () (-> ph (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th))) (ps ch -> th ch ta -> th -. ps -. -> -> merlem4 ch ta -> th -. ps -. -> -> th -> ps ch -> th -> ph -. ch -. merlem6 ch ta th ps ps ch -> th -> ch ta -> th -. ps -. -> -> th -> -> ph -. -> ch -. ph -. -> -> meredith ax-mp ps ch -> th -> ch ta -> th -. ps -. -> -> th -> -> ph -. ch ph ps ch -> meredith ax-mp ax-mp))
+thm (merlem8 () () (-> (-> (-> ps ch) th) (-> (-> (-> ch ta) (-> (-. th) (-. ps))) th)) (ph ph ph ph ph meredith ph ph -> ph -. ph -. -> -> ph -> ph -> ph ph -> ph ph -> -> -> ps ch th ta merlem7 ax-mp))
+thm (merlem9 () () (-> (-> (-> ph ps) (-> ch (-> th (-> ps ta)))) (-> et (-> ch (-> th (-> ps ta))))) (th ps ta -> -> ch et -. ps -. merlem6 th ps ta -> ch th ps ta -> -> -> et -. -> ps -. et -. -> -> ch th ps ta -> -> -> et -. -> ps -. et -. -> -> -. th -. -> -. ph -. -> merlem8 ax-mp ps ta ch th ps ta -> -> -> et -. -> ps -. et -. -> -> -. th -. -> ph ch th ps ta -> -> -> et -. -> ps -. et -. -> -> meredith ax-mp ch th ps ta -> -> -> et -. ps et ph ps -> meredith ax-mp))
+thm (merlem10 () () (-> (-> ph (-> ph ps)) (-> th (-> ph ps))) (ph ph ph ph ph meredith ph ps -> ph ph th ph meredith ph ps -> ph -> ph -. th -. -> -> ph -> ph ph ph ps -> -> th ps ph ph -> ph -. ph -. -> -> ph -> ph -> ph ph -> ph ph -> -> -> merlem9 ax-mp ax-mp))
+thm (merlem11 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph ph ph ph ph meredith ph ps ph ph ps -> -> merlem10 ph ph ps -> -> ph ps -> ph ph -> ph -. ph -. -> -> ph -> ph -> ph ph -> ph ph -> -> -> merlem10 ax-mp ax-mp))
+thm (merlem12 () () (-> (-> (-> th (-> (-. (-. ch)) ch)) ph) ph) (ch ch merlem5 ch ch -. -. ch -> th merlem2 ax-mp th ch -. -. ch -> -> ph th ch -. -. ch -> -> ph -> merlem4 ax-mp th ch -. -. ch -> -> ph -> ph merlem11 ax-mp))
+thm (merlem13 () () (-> (-> ph ps) (-> (-> (-> th (-> (-. (-. ch)) ch)) (-. (-. ph))) ps)) (th ch th ch -. -. ch -> -> ph -. -. -> -. merlem12 th ch ph -. -. merlem12 th ch -. -. ch -> -> ph -. -. -> ph -. -. merlem5 ax-mp th ch -. -. ch -> -> ph -. -. -> -. -. ph -. -. -> th ch -. -. ch -> -> ph -. -. -> -. ps -> th ch -. -. ch -> -> ph -. -. -> -. th ch -. -. ch -> -> merlem6 ax-mp th ch -. -. ch -> -> ph -. -. -> -. ps th ch -. -. ch -> -> ph -. -. -> -. ph -. th ch -. -. ch -> -> th ch -. -. ch -> -> ph -. -. -> -. -> meredith ax-mp ax-mp ph -. th ch -. -. ch -> -> ph -. -. -> -. -> ps ps -> ph ps ps -> ph -. th ch -. -. ch -> -> ph -. -. -> -. -> -> ph -> merlem6 ax-mp ps ps -> ph -. th ch -. -. ch -> -> ph -. -. -> -. -> -> ph -> ph merlem11 ax-mp ps ps ph th ch -. -. ch -> -> ph -. -. -> ph meredith ax-mp))
+thm (luk-1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch))) (ch ch ph -. -. ph ps meredith ph ps ch ch -> ph -. merlem13 ph ps -> ch ch -> ph -. -. -. ph -. -> -> ph -. -. -> ps -> ps ch -> ph ch -> -> ph -> ph ps -> -. merlem13 ax-mp ps ch -> ph ch -> -> ph ph ps -> -. -. ph ps -> ch ch -> ph -. -. -. ph -. -> -> ph -. -. -> ps -> meredith ax-mp ax-mp))
+thm (luk-2 () () (-> (-> (-. ph) ph) ph) (ph ph -. ph -> -. merlem5 ph ph -. ph -> -. -> ph -. -. ph -. ph -> -. -> -> ph -. ph ph -. ph -> -. -> ph -. -. ph -. ph -> -. -> -> ph -. -> merlem4 ax-mp ph ph -. ph -> -. -> ph -. -. ph -. ph -> -. -> -> ph -. -> ph -. merlem11 ax-mp ph ph -. ph -> -. ph -. ph -. ph -> ph -. meredith ax-mp ph -. ph -> ph merlem11 ax-mp))
+thm (luk-3 () () (-> ph (-> (-. ph) ps)) (ph -. ps merlem11 ph -. ph ps ph -. ps -> merlem1 ax-mp))
+thm (luklem1 () ((luklem1.1 (-> ph ps)) (luklem1.2 (-> ps ch))) (-> ph ch) (luklem1.2 luklem1.1 ph ps ch luk-1 ax-mp ax-mp))
+thm (luklem2 () () (-> (-> ph (-. ps)) (-> (-> (-> ph ch) th) (-> ps th))) (ph ps -. ch luk-1 ps ch luk-3 ps ps -. ch -> ph ch -> luk-1 ax-mp luklem1 ps ph ch -> th luk-1 luklem1))
+thm (luklem3 () () (-> ph (-> (-> (-> (-. ph) ps) ch) (-> th ch))) (ph th -. luk-3 ph -. th ps ch luklem2 luklem1))
+thm (luklem4 () () (-> (-> (-> (-> (-. ph) ph) ph) ps) ps) (ph -. ph -> ph -> luk-2 ph luk-2 ph -. ph -> ph -> ph -. ph -> ph -> ph -. ph -> ph -> ps -. luklem3 ax-mp ax-mp ps -. ph -. ph -> ph -> ps luk-1 ax-mp ps luk-2 luklem1))
+thm (luklem5 () () (-> ph (-> ps ph)) (ph ph ph ps luklem3 ph ps ph -> luklem4 luklem1))
+thm (luklem6 () () (-> (-> ph (-> ph ps)) (-> ph ps)) (ph ph ps -> ps luk-1 ph ps -> -. ps -. luklem5 ps -. ph ps -> ps ps luklem2 ps ph ps -> ps -> luklem4 luklem1 luklem1 ph ps -> -. ph ps -> ps -> ph ps -> luk-1 ax-mp ph ps -> ps -> ph ps -> -> ph ps -> -. ph ps -> -> ph ps -> luk-1 ax-mp ph ps -> ph ps -> ps -> ph ps -> -> ph ps -> -> luklem4 ax-mp luklem1))
+thm (luklem7 () () (-> (-> ph (-> ps ch)) (-> ps (-> ph ch))) (ph ps ch -> ch luk-1 ps ps ch -> luklem5 ps ch -> ps ch luk-1 luklem1 ps ch -> ch luklem6 luklem1 ps ps ch -> ch -> ph ch -> luk-1 ax-mp luklem1))
+thm (luklem8 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps))) (ch ph ps luk-1 ch ph -> ph ps -> ch ps -> luklem7 ax-mp))
+thm (ax1 () () (-> ph (-> ps ph)) (ph ps luklem5))
+thm (ax2 () () (-> (-> ph (-> ps ch)) (-> (-> ph ps) (-> ph ch))) (ph ps ch luklem7 ps ph ch -> ph luklem8 ph ch luklem6 ph ph ch -> -> ph ch -> ph ps -> luklem8 ax-mp luklem1 luklem1))
+thm (ax3 () () (-> (-> (-. ph) (-. ps)) (-> ps ph)) (ph -. ps ph ph luklem2 ph ps ph -> luklem4 luklem1))
+var (set x)
+var (set y)
+var (set z)
+var (set w)
+var (set v)
+var (set u)
+thm (ax46 () () (-> (-> (A. x (-. (A. x ph))) (A. x ph)) ph) (x ph ax-6 x ph ax-4 ja))
+thm (ax4 () () (-> (A. x ph) ph) (x ph A. x x ph A. -. A. ax-1 x ph ax46 syl))
+thm (ax6 () () (-> (-. (A. x (-. (A. x ph)))) ph) (x x ph A. -. A. x ph A. pm2.21 x ph ax46 syl))
+thm (a4i () ((a4i.1 (A. x ph))) ph (a4i.1 x ph ax-4 ax-mp))
+thm (gen2 () ((gen2.1 ph)) (A. x (A. y ph)) (gen2.1 y ax-gen x ax-gen))
+thm (a4s () ((a4s.1 (-> ph ps))) (-> (A. x ph) ps) (x ph ax-4 a4s.1 syl))
+thm (a4sd () ((a4sd.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) ch)) (a4sd.1 x ps ax-4 syl5))
+thm (mpg () ((mpg.1 (-> (A. x ph) ps)) (mpg.2 ph)) ps (mpg.2 x ax-gen mpg.1 ax-mp))
+thm (mpgbi () ((mpgbi.1 (<-> (A. x ph) ps)) (mpgbi.2 ph)) ps (mpgbi.1 biimp mpgbi.2 mpg))
+thm (mpgbir () ((mpgbir.1 (<-> ph (A. x ps))) (mpgbir.2 ps)) ph (mpgbir.1 biimpr mpgbir.2 mpg))
+thm (a5i () ((a5i.1 (-> (A. x ph) ps))) (-> (A. x ph) (A. x ps)) (x ph ps ax-5 a5i.1 mpg))
+thm (a6e () () (-> (E. x (A. x ph)) ph) (x x ph A. df-ex x ph ax-6 sylbi))
+thm (a7s () ((a7s.1 (-> (A. x (A. y ph)) ps))) (-> (A. y (A. x ph)) ps) (y x ph ax-7 a7s.1 syl))
+thm (19.20 () () (-> (A. x (-> ph ps)) (-> (A. x ph) (A. x ps))) (x ph ax-4 ps imim1i x a4s a5i x ph ps ax-5 syl))
+thm (19.20i () ((19.20i.1 (-> ph ps))) (-> (A. x ph) (A. x ps)) (19.20i.1 x a4s a5i))
+thm (19.20i2 () ((19.20i.1 (-> ph ps))) (-> (A. x (A. y ph)) (A. x (A. y ps))) (19.20i.1 y 19.20i x 19.20i))
+thm (19.20ii () ((19.20ii.1 (-> ph (-> ps ch)))) (-> (A. x ph) (-> (A. x ps) (A. x ch))) (19.20ii.1 x 19.20i x ps ch 19.20 syl))
+thm (19.20d () ((19.20d.1 (-> ph (A. x ph))) (19.20d.2 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) (A. x ch))) (19.20d.1 19.20d.2 x 19.20ii syl))
+thm (19.15 () () (-> (A. x (<-> ph ps)) (<-> (A. x ph) (A. x ps))) (ph ps bi1 x 19.20ii ph ps bi2 x 19.20ii impbid))
+thm (bial () ((bial.1 (<-> ph ps))) (<-> (A. x ph) (A. x ps)) (x ph ps 19.15 bial.1 mpg))
+thm (bi2al () ((bial.1 (<-> ph ps))) (<-> (A. x (A. y ph)) (A. x (A. y ps))) (bial.1 y bial x bial))
+thm (hbth () ((hbth.1 ph)) (-> ph (A. x ph)) (hbth.1 x ax-gen ph a1i))
+thm (hba1 () () (-> (A. x ph) (A. x (A. x ph))) (x ph A. id a5i))
+thm (hbne () ((hb.1 (-> ph (A. x ph)))) (-> (-. ph) (A. x (-. ph))) (hb.1 con3i x 19.20i x ph ax-6 nsyl4))
+thm (hbal () ((hb.1 (-> ph (A. x ph)))) (-> (A. y ph) (A. x (A. y ph))) (hb.1 y 19.20i y x ph ax-7 syl))
+thm (hbex () ((hb.1 (-> ph (A. x ph)))) (-> (E. y ph) (A. x (E. y ph))) (hb.1 hbne y hbal hbne y ph df-ex y ph df-ex x bial 3imtr4))
+thm (hbim () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (-> ph ps) (A. x (-> ph ps))) (hb.1 hbne ph ps pm2.21 x 19.20i syl hb.2 ps ph ax-1 x 19.20i syl ja))
+thm (hbor () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (\/ ph ps) (A. x (\/ ph ps))) (hb.1 hbne hb.2 hbim ph ps df-or ph ps df-or x bial 3imtr4))
+thm (hban () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (/\ ph ps) (A. x (/\ ph ps))) (hb.1 hb.2 hbne hbim hbne ph ps df-an ph ps df-an x bial 3imtr4))
+thm (hbbi () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps)))) (-> (<-> ph ps) (A. x (<-> ph ps))) (hb.1 hb.2 hbim hb.2 hb.1 hbim hban ph ps bi ph ps bi x bial 3imtr4))
+thm (hb3or () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps))) (hb.3 (-> ch (A. x ch)))) (-> (\/\/ ph ps ch) (A. x (\/\/ ph ps ch))) (hb.1 hb.2 hbor hb.3 hbor ph ps ch df-3or ph ps ch df-3or x bial 3imtr4))
+thm (hb3an () ((hb.1 (-> ph (A. x ph))) (hb.2 (-> ps (A. x ps))) (hb.3 (-> ch (A. x ch)))) (-> (/\/\ ph ps ch) (A. x (/\/\ ph ps ch))) (hb.1 hb.2 hban hb.3 hban ph ps ch df-3an ph ps ch df-3an x bial 3imtr4))
+thm (hbn1 () () (-> (-. (A. x ph)) (A. x (-. (A. x ph)))) (x ph hba1 hbne))
+thm (hbe1 () () (-> (E. x ph) (A. x (E. x ph))) (x ph -. hbn1 x ph df-ex x ph df-ex x bial 3imtr4))
+thm (hbnt () () (-> (A. x (-> ph (A. x ph))) (-> (-. ph) (A. x (-. ph)))) (ph x ph A. con3 x 19.20ii x ph ax-6 con1i syl5))
+thm (ax6-2 () () (-> (-. (A. x ph)) (A. x (-. (A. x ph)))) (x ph hbn1))
+thm (modal-5 () () (-> (-. (A. x (-. ph))) (A. x (-. (A. x (-. ph))))) (x ph -. hbn1))
+thm (modal-b () () (-> ph (A. x (-. (A. x (-. ph))))) (x ph -. ax-6 a3i))
+thm (19.8a () () (-> ph (E. x ph)) (x ph -. ax-4 con2i x ph df-ex sylibr))
+thm (19.2 () () (-> (A. x ph) (E. x ph)) (ph x 19.8a x a4s))
+thm (19.3r () ((19.3r.1 (-> ph (A. x ph)))) (<-> ph (A. x ph)) (19.3r.1 x ph ax-4 impbi))
+thm (alcom () () (<-> (A. x (A. y ph)) (A. y (A. x ph))) (x y ph ax-7 y x ph ax-7 impbi))
+thm (alnex () () (<-> (A. x (-. ph)) (-. (E. x ph))) (x ph df-ex bicon2i))
+thm (alex () () (<-> (A. x ph) (-. (E. x (-. ph)))) (ph pm4.13 x bial x ph -. alnex bitr))
+thm (19.9r () ((19.9r.1 (-> ph (A. x ph)))) (<-> ph (E. x ph)) (ph x 19.8a x ph df-ex 19.9r.1 con3i x 19.20i con3i x ph ax-6 syl sylbi impbi))
+thm (19.9t () () (-> (A. x (-> ph (A. x ph))) (-> (E. x ph) ph)) (x ph hbnt con1d x ph df-ex syl5ib))
+thm (19.9d () ((19.9d.1 (-> ps (A. x ps))) (19.9d.2 (-> ps (-> ph (A. x ph))))) (-> ps (-> (E. x ph) ph)) (19.9d.1 19.9d.2 x 19.20i x ph 19.9t 3syl))
+thm (exnal () () (<-> (E. x (-. ph)) (-. (A. x ph))) (x ph alex bicon2i))
+thm (19.22 () () (-> (A. x (-> ph ps)) (-> (E. x ph) (E. x ps))) (ph ps con3 x 19.20ii con3d x ph df-ex x ps df-ex 3imtr4g))
+thm (19.22i () ((19.22i.1 (-> ph ps))) (-> (E. x ph) (E. x ps)) (x ph ps 19.22 19.22i.1 mpg))
+thm (19.22i2 () ((19.22i.1 (-> ph ps))) (-> (E. x (E. y ph)) (E. x (E. y ps))) (19.22i.1 y 19.22i x 19.22i))
+thm (alinexa () () (<-> (A. x (-> ph (-. ps))) (-. (E. x (/\ ph ps)))) (ph ps imnan x bial x ph ps /\ alnex bitr))
+thm (exanali () () (<-> (E. x (/\ ph (-. ps))) (-. (A. x (-> ph ps)))) (ph ps iman x bial x ph ps -. /\ alnex bitr bicon2i))
+thm (alexn () () (<-> (A. x (E. y (-. ph))) (-. (E. x (A. y ph)))) (y ph exnal x bial x y ph A. alnex bitr))
+thm (excomim () () (-> (E. x (E. y ph)) (E. y (E. x ph))) (ph x 19.8a x y 19.22i2 x ph hbe1 y hbex 19.9r sylibr))
+thm (excom () () (<-> (E. x (E. y ph)) (E. y (E. x ph))) (x y ph excomim y x ph excomim impbi))
+thm (19.12 () () (-> (E. x (A. y ph)) (A. y (E. x ph))) (y ph hba1 x hbex y ph ax-4 x 19.22i y 19.20i syl))
+thm (19.16 () ((19.16.1 (-> ph (A. x ph)))) (-> (A. x (<-> ph ps)) (<-> ph (A. x ps))) (x ph ps 19.15 19.16.1 19.3r syl5bb))
+thm (19.17 () ((19.17.1 (-> ps (A. x ps)))) (-> (A. x (<-> ph ps)) (<-> (A. x ph) ps)) (x ph ps 19.15 19.17.1 19.3r syl6bbr))
+thm (19.18 () () (-> (A. x (<-> ph ps)) (<-> (E. x ph) (E. x ps))) (ph ps bi1 x 19.20i x ph ps 19.22 syl ph ps bi2 x 19.20i x ps ph 19.22 syl impbid))
+thm (biex () ((biex.1 (<-> ph ps))) (<-> (E. x ph) (E. x ps)) (x ph ps 19.18 biex.1 mpg))
+thm (bi2ex () ((bi2ex.1 (<-> ph ps))) (<-> (E. x (E. y ph)) (E. x (E. y ps))) (bi2ex.1 y biex x biex))
+thm (bi3ex () ((bi3ex.1 (<-> ph ps))) (<-> (E. x (E. y (E. z ph))) (E. x (E. y (E. z ps)))) (bi3ex.1 z biex x y bi2ex))
+thm (exancom () () (<-> (E. x (/\ ph ps)) (E. x (/\ ps ph))) (ph ps ancom x biex))
+thm (19.19 () ((19.19.1 (-> ph (A. x ph)))) (-> (A. x (<-> ph ps)) (<-> ph (E. x ps))) (x ph ps 19.18 19.19.1 19.9r syl5bb))
+thm (19.21 () ((19.21.1 (-> ph (A. x ph)))) (<-> (A. x (-> ph ps)) (-> ph (A. x ps))) (x ph ps 19.20 19.21.1 syl5 19.21.1 x ps hba1 hbim x ps ax-4 ph imim2i x 19.20i syl impbi))
+thm (19.21-2 () ((19.21-2.1 (-> ph (A. x ph))) (19.21-2.2 (-> ph (A. y ph)))) (<-> (A. x (A. y (-> ph ps))) (-> ph (A. x (A. y ps)))) (19.21-2.2 ps 19.21 x bial 19.21-2.1 y ps A. 19.21 bitr))
+thm (stdpc5 () ((stdpc5.1 (-> ph (A. x ph)))) (-> (A. x (-> ph ps)) (-> ph (A. x ps))) (stdpc5.1 ps 19.21 biimp))
+thm (19.21ai () ((19.21ai.1 (-> ph (A. x ph))) (19.21ai.2 (-> ph ps))) (-> ph (A. x ps)) (19.21ai.1 19.21ai.2 x 19.20i syl))
+thm (19.21ad () ((19.21ad.1 (-> ph (A. x ph))) (19.21ad.2 (-> ps (A. x ps))) (19.21ad.3 (-> ph (-> ps ch)))) (-> ph (-> ps (A. x ch))) (19.21ad.1 19.21ad.2 hban 19.21ad.3 imp 19.21ai exp))
+thm (19.21bi () ((19.21bi.1 (-> ph (A. x ps)))) (-> ph ps) (19.21bi.1 x ps ax-4 syl))
+thm (19.21bbi () ((19.21bbi.1 (-> ph (A. x (A. y ps))))) (-> ph ps) (19.21bbi.1 19.21bi 19.21bi))
+thm (19.22d () ((19.22d.1 (-> ph (A. x ph))) (19.22d.2 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) (E. x ch))) (19.22d.1 19.22d.2 19.21ai x ps ch 19.22 syl))
+thm (19.23 () ((19.23.1 (-> ps (A. x ps)))) (<-> (A. x (-> ph ps)) (-> (E. x ph) ps)) (x ph ps 19.22 19.23.1 19.9r syl6ibr x ph hbe1 19.23.1 hbim ph x 19.8a ps imim1i 19.21ai impbi))
+thm (19.23ai () ((19.23ai.1 (-> ps (A. x ps))) (19.23ai.2 (-> ph ps))) (-> (E. x ph) ps) (19.23ai.2 x 19.22i 19.23ai.1 19.9r sylibr))
+thm (19.23bi () ((19.23bi.1 (-> (E. x ph) ps))) (-> ph ps) (ph x 19.8a 19.23bi.1 syl))
+thm (19.23ad () ((19.23ad.1 (-> ph (A. x ph))) (19.23ad.2 (-> ch (A. x ch))) (19.23ad.3 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) ch)) (19.23ad.1 19.23ad.3 19.21ai 19.23ad.2 ps 19.23 sylib))
+thm (19.26 () () (<-> (A. x (/\ ph ps)) (/\ (A. x ph) (A. x ps))) (ph ps pm3.26 x 19.20i ph ps pm3.27 x 19.20i jca ph ps pm3.2 x 19.20ii imp impbi))
+thm (19.26-2 () () (<-> (A. x (A. y (/\ ph ps))) (/\ (A. x (A. y ph)) (A. x (A. y ps)))) (y ph ps 19.26 x bial x y ph A. y ps A. 19.26 bitr))
+thm (19.27 () ((19.27.1 (-> ps (A. x ps)))) (<-> (A. x (/\ ph ps)) (/\ (A. x ph) ps)) (x ph ps 19.26 19.27.1 19.3r x ph A. anbi2i bitr4))
+thm (19.28 () ((19.28.1 (-> ph (A. x ph)))) (<-> (A. x (/\ ph ps)) (/\ ph (A. x ps))) (x ph ps 19.26 19.28.1 19.3r x ps A. anbi1i bitr4))
+thm (19.29 () () (-> (/\ (A. x ph) (E. x ps)) (E. x (/\ ph ps))) (x ph ps -. 19.20 x ps alnex syl6ib con3i x ph A. x ps E. df-an x ph ps -. -> exnal 3imtr4 ph ps df-an x biex sylibr))
+thm (19.29r () () (-> (/\ (E. x ph) (A. x ps)) (E. x (/\ ph ps))) (x ps ph 19.29 x ph E. x ps A. ancom x ph ps exancom 3imtr4))
+thm (19.29r2 () () (-> (/\ (E. x (E. y ph)) (A. x (A. y ps))) (E. x (E. y (/\ ph ps)))) (x y ph E. y ps A. 19.29r y ph ps 19.29r x 19.22i syl))
+thm (19.29x () () (-> (/\ (E. x (A. y ph)) (A. x (E. y ps))) (E. x (E. y (/\ ph ps)))) (x y ph A. y ps E. 19.29r y ph ps 19.29 x 19.22i syl))
+thm (19.35 () () (<-> (E. x (-> ph ps)) (-> (A. x ph) (E. x ps))) (x ph ps -. 19.26 ph ps annim x bial x ph A. x ps -. A. df-an 3bitr3 bicon2i x ps df-ex x ph A. imbi2i x ph ps -> df-ex 3bitr4r))
+thm (19.35i () ((19.35i.1 (E. x (-> ph ps)))) (-> (A. x ph) (E. x ps)) (19.35i.1 x ph ps 19.35 mpbi))
+thm (19.35ri () ((19.35ri.1 (-> (A. x ph) (E. x ps)))) (E. x (-> ph ps)) (19.35ri.1 x ph ps 19.35 mpbir))
+thm (19.36 () ((19.36.1 (-> ps (A. x ps)))) (<-> (E. x (-> ph ps)) (-> (A. x ph) ps)) (x ph ps 19.35 19.36.1 19.9r x ph A. imbi2i bitr4))
+thm (19.36i () ((19.36i.1 (-> ps (A. x ps))) (19.36i.2 (E. x (-> ph ps)))) (-> (A. x ph) ps) (19.36i.2 19.36i.1 ph 19.36 mpbi))
+thm (19.37 () ((19.37.1 (-> ph (A. x ph)))) (<-> (E. x (-> ph ps)) (-> ph (E. x ps))) (x ph ps 19.35 19.37.1 19.3r x ps E. imbi1i bitr4))
+thm (19.38 () () (-> (-> (E. x ph) (A. x ps)) (A. x (-> ph ps))) (x ph hbe1 x ps hba1 hbim ph x 19.8a x ps ax-4 imim12i 19.21ai))
+thm (19.39 () () (-> (-> (E. x ph) (E. x ps)) (E. x (-> ph ps))) (x ph 19.2 x ps E. imim1i x ph ps 19.35 sylibr))
+thm (19.24 () () (-> (-> (A. x ph) (A. x ps)) (E. x (-> ph ps))) (x ps 19.2 x ph A. imim2i x ph ps 19.35 sylibr))
+thm (19.25 () () (-> (A. y (E. x (-> ph ps))) (-> (E. y (A. x ph)) (E. y (E. x ps)))) (x ph ps 19.35 biimp y 19.20i y x ph A. x ps E. 19.22 syl))
+thm (19.30 () () (-> (A. x (\/ ph ps)) (\/ (A. x ph) (E. x ps))) (x ps -. ph 19.20 ph ps orcom ps ph df-or bitr x bial x ph A. x ps -. A. -. orcom x ps df-ex x ph A. orbi2i x ps -. A. x ph A. imor 3bitr4 3imtr4))
+thm (19.32 () ((19.32.1 (-> ph (A. x ph)))) (<-> (A. x (\/ ph ps)) (\/ ph (A. x ps))) (19.32.1 hbne ps 19.21 ph ps df-or x bial ph x ps A. df-or 3bitr4))
+thm (19.31 () ((19.31.1 (-> ps (A. x ps)))) (<-> (A. x (\/ ph ps)) (\/ (A. x ph) ps)) (19.31.1 ph 19.32 ph ps orcom x bial x ph A. ps orcom 3bitr4))
+thm (19.43 () () (<-> (E. x (\/ ph ps)) (\/ (E. x ph) (E. x ps))) (ph ps ioran x bial x ph -. ps -. 19.26 x ph alnex x ps alnex anbi12i 3bitr negbii x ph ps \/ df-ex x ph E. x ps E. oran 3bitr4))
+thm (19.44 () ((19.44.1 (-> ps (A. x ps)))) (<-> (E. x (\/ ph ps)) (\/ (E. x ph) ps)) (x ph ps 19.43 19.44.1 19.9r x ph E. orbi2i bitr4))
+thm (19.45 () ((19.45.1 (-> ph (A. x ph)))) (<-> (E. x (\/ ph ps)) (\/ ph (E. x ps))) (x ph ps 19.43 19.45.1 19.9r x ps E. orbi1i bitr4))
+thm (19.33 () () (-> (\/ (A. x ph) (A. x ps)) (A. x (\/ ph ps))) (ph ps orc x 19.20i ps ph olc x 19.20i jaoi))
+thm (19.33b () () (-> (-. (/\ (E. x ph) (E. x ps))) (<-> (A. x (\/ ph ps)) (\/ (A. x ph) (A. x ps)))) (x ph E. x ps E. ianor x ph alnex x ps alnex orbi12i bitr4 ph ps biorf x 19.20i x ps ph ps \/ 19.15 syl x ps A. x ph A. olc syl6bir ps ph biorf ps ph orcom syl6bb x 19.20i x ph ph ps \/ 19.15 syl x ph A. x ps A. orc syl6bir jaoi sylbi x ph ps 19.33 x ph E. x ps E. /\ -. a1i impbid))
+thm (19.34 () () (-> (\/ (A. x ph) (E. x ps)) (E. x (\/ ph ps))) (x ph 19.2 x ps E. orim1i x ph ps 19.43 sylibr))
+thm (19.40 () () (-> (E. x (/\ ph ps)) (/\ (E. x ph) (E. x ps))) (ph ps pm3.26 x 19.22i ph ps pm3.27 x 19.22i jca))
+thm (19.41 () ((19.41.1 (-> ps (A. x ps)))) (<-> (E. x (/\ ph ps)) (/\ (E. x ph) ps)) (x ph ps /\ df-ex 19.41.1 hbne ph -. 19.31 ph ps ianor x bial x ph E. ps ianor x ph alnex ps -. orbi1i bitr4 3bitr4 bicon2i bitr4))
+thm (19.42 () ((19.42.1 (-> ph (A. x ph)))) (<-> (E. x (/\ ph ps)) (/\ ph (E. x ps))) (19.42.1 ps 19.41 x ph ps exancom ph x ps E. ancom 3bitr4))
+thm (alrot4 () () (<-> (A. x (A. y (A. z (A. w ph)))) (A. z (A. w (A. x (A. y ph))))) (y z w ph A. alcom y w ph alcom z bial bitr x bial x z w y ph A. A. alcom x w y ph A. alcom z bial 3bitr))
+thm (excom13 () () (<-> (E. x (E. y (E. z ph))) (E. z (E. y (E. x ph)))) (x y z ph E. excom x z ph excom y biex y z x ph E. excom 3bitr))
+thm (exrot3 () () (<-> (E. x (E. y (E. z ph))) (E. y (E. z (E. x ph)))) (x y z ph excom13 z y x ph E. excom bitr))
+thm (exrot4 () () (<-> (E. x (E. y (E. z (E. w ph)))) (E. z (E. w (E. x (E. y ph))))) (y z w ph excom13 x biex x w z y ph E. excom13 bitr))
+thm (nex () ((nex.1 (-. ph))) (-. (E. x ph)) (x ph alnex nex.1 mpgbi))
+thm (nexd () ((nexd.1 (-> ph (A. x ph))) (nexd.2 (-> ph (-. ps)))) (-> ph (-. (E. x ps))) (nexd.1 nexd.2 19.21ai x ps alnex sylib))
+thm (hbim1 () ((hbim1.1 (-> ph (A. x ph))) (hbim1.2 (-> ph (-> ps (A. x ps))))) (-> (-> ph ps) (A. x (-> ph ps))) (hbim1.2 a2i hbim1.1 ps 19.21 sylibr))
+thm (biald () ((biald.1 (-> ph (A. x ph))) (biald.2 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x ps) (A. x ch))) (biald.1 biald.2 19.21ai x ps ch 19.15 syl))
+thm (biexd () ((biexd.1 (-> ph (A. x ph))) (biexd.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x ps) (E. x ch))) (biexd.1 biexd.2 19.21ai x ps ch 19.18 syl))
+thm (exan () ((exan.1 (/\ (E. x ph) ps))) (E. x (/\ ph ps)) (x ph hbe1 ps 19.27 exan.1 x ph E. ps ancom mpbi mpgbi x ps ph 19.29 ax-mp x ps ph exancom mpbi))
+thm (albi () () (<-> (A. x (<-> ph ps)) (/\ (A. x (-> ph ps)) (A. x (-> ps ph)))) (ph ps bi x bial x ph ps -> ps ph -> 19.26 bitr))
+thm (2albi () () (<-> (A. x (A. y (<-> ph ps))) (/\ (A. x (A. y (-> ph ps))) (A. x (A. y (-> ps ph))))) (y ph ps albi x bial x y ph ps -> A. y ps ph -> A. 19.26 bitr))
+thm (hbnd () ((hbnd.1 (-> ph (A. x ph))) (hbnd.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (-. ps) (A. x (-. ps)))) (hbnd.1 hbnd.2 19.21ai x ps hbnt syl))
+thm (hbimd () ((hbimd.1 (-> ph (A. x ph))) (hbimd.2 (-> ph (-> ps (A. x ps)))) (hbimd.3 (-> ph (-> ch (A. x ch))))) (-> ph (-> (-> ps ch) (A. x (-> ps ch)))) (hbimd.1 hbimd.2 hbnd ps ch pm2.21 x 19.20i syl6com hbimd.3 ch ps ax-1 x 19.20i syl6com ja com12))
+thm (hband () ((hband.1 (-> ph (-> ps (A. x ps)))) (hband.2 (-> ph (-> ch (A. x ch))))) (-> ph (-> (/\ ps ch) (A. x (/\ ps ch)))) (hband.1 hband.2 anim12d x ps ch 19.26 syl6ibr))
+thm (hbbid () ((hbbid.1 (-> ph (A. x ph))) (hbbid.2 (-> ph (-> ps (A. x ps)))) (hbbid.3 (-> ph (-> ch (A. x ch))))) (-> ph (-> (<-> ps ch) (A. x (<-> ps ch)))) (hbbid.1 hbbid.2 hbbid.3 hbimd hbbid.1 hbbid.3 hbbid.2 hbimd anim12d ps ch bi x ps ch albi 3imtr4g))
+thm (hbald () ((hbald.1 (-> ph (A. y ph))) (hbald.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (A. y ps) (A. x (A. y ps)))) (hbald.1 hbald.2 19.20d y x ps ax-7 syl6))
+thm (hbexd () ((hbexd.1 (-> ph (A. y ph))) (hbexd.2 (-> ph (-> ps (A. x ps))))) (-> ph (-> (E. y ps) (A. x (E. y ps)))) (hbexd.1 hbexd.2 19.22d y x ps 19.12 syl6))
+thm (19.21g () () (-> (A. x (-> ph (A. x ph))) (<-> (A. x (-> ph ps)) (-> ph (A. x ps)))) (x ph ps 19.20 ph imim2d com12 x a4s x ph x ph A. -> hba1 x ph x ph A. -> ax-4 x ps hba1 x ph x ph A. -> A. a1i hbimd x ps ax-4 ph imim2i x 19.20i syl6 impbid))
+thm (exintr () () (-> (A. x (-> ph ps)) (-> (E. x ph) (E. x (/\ ph ps)))) (x ph ps -> hba1 ph ps ancl x a4s 19.22d))
+thm (aaan () ((aaan.1 (-> ph (A. y ph))) (aaan.2 (-> ps (A. x ps)))) (<-> (A. x (A. y (/\ ph ps))) (/\ (A. x ph) (A. y ps))) (aaan.1 ps 19.28 x bial aaan.2 y hbal ph 19.27 bitr))
+thm (eeor () ((eeor.1 (-> ph (A. y ph))) (eeor.2 (-> ps (A. x ps)))) (<-> (E. x (E. y (\/ ph ps))) (\/ (E. x ph) (E. y ps))) (eeor.1 ps 19.45 x biex eeor.2 y hbex ph 19.44 bitr))
+thm (qexmid () () (E. x (-> ph (A. x ph))) (x ph A. x 19.8a 19.35ri))
+thm (ax9 () () (-> (A. x (-> (= (cv x) (cv y)) (A. x ph))) ph) (x y ax-9 x x cv y cv = df-ex mpbir x x cv y cv = x ph A. 19.22 mpi x ph a6e syl))
+thm (ax9a () () (-. (A. x (-. (= (cv x) (cv y))))) (x y x x cv y cv = -. A. -. ax9 x x cv y cv = -. ax-6 a3i mpg))
+thm (a9e () () (E. x (= (cv x) (cv y))) (x y ax9a x x cv y cv = df-ex mpbir))
+thm (equid () () (= (cv x) (cv x)) (x x x ax-12 pm2.43i x 19.20i x x x cv x cv = ax9 syl x x cv x cv = ax-6 pm2.61i))
+thm (stdpc6 () () (A. x (= (cv x) (cv x))) (x equid x ax-gen))
+thm (equcomi () () (-> (= (cv x) (cv y)) (= (cv y) (cv x))) (x equid x y x ax-8 mpi))
+thm (equcom () () (<-> (= (cv x) (cv y)) (= (cv y) (cv x))) (x y equcomi y x equcomi impbi))
+thm (equcoms () ((equcoms.1 (-> (= (cv x) (cv y)) ph))) (-> (= (cv y) (cv x)) ph) (y x equcomi equcoms.1 syl))
+thm (equtr () () (-> (= (cv x) (cv y)) (-> (= (cv y) (cv z)) (= (cv x) (cv z)))) (y x z ax-8 equcoms))
+thm (equtrr () () (-> (= (cv x) (cv y)) (-> (= (cv z) (cv x)) (= (cv z) (cv y)))) (z x y equtr com12))
+thm (equtr2 () () (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv z))) (= (cv x) (cv y))) (x z y equtr y z equcomi syl5 imp))
+thm (equequ1 () () (-> (= (cv x) (cv y)) (<-> (= (cv x) (cv z)) (= (cv y) (cv z)))) (x y z ax-8 x y z equtr impbid))
+thm (equequ2 () () (-> (= (cv x) (cv y)) (<-> (= (cv z) (cv x)) (= (cv z) (cv y)))) (x y z equtrr y x z equtrr equcoms impbid))
+thm (elequ1 () () (-> (= (cv x) (cv y)) (<-> (e. (cv x) (cv z)) (e. (cv y) (cv z)))) (x y z ax-13 y x z ax-13 equcoms impbid))
+thm (elequ2 () () (-> (= (cv x) (cv y)) (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (x y z ax-14 y x z ax-14 equcoms impbid))
+thm (alequcom () () (-> (A. x (= (cv x) (cv y))) (A. y (= (cv y) (cv x)))) (x y x cv y cv = ax-10 pm2.43i x y equcomi y 19.20i syl))
+thm (alequcoms () ((alequcoms.1 (-> (A. x (= (cv x) (cv y))) ph))) (-> (A. y (= (cv y) (cv x))) ph) (y x alequcom alequcoms.1 syl))
+thm (nalequcoms () ((nalequcoms.1 (-> (-. (A. x (= (cv x) (cv y)))) ph))) (-> (-. (A. y (= (cv y) (cv x)))) ph) (x y alequcom nalequcoms.1 nsyl4 con1i))
+thm (hbae () () (-> (A. x (= (cv x) (cv y))) (A. z (A. x (= (cv x) (cv y))))) (z x y ax-12 x x cv y cv = ax-4 syl7 x z x cv y cv = ax-10 alequcoms y z x cv y cv = ax-10 x y x cv y cv = ax-10 pm2.43i syl5 alequcoms pm2.61ii a5i x z x cv y cv = ax-7 syl))
+thm (hbaes () ((hbalequs.1 (-> (A. z (A. x (= (cv x) (cv y)))) ph))) (-> (A. x (= (cv x) (cv y))) ph) (x y z hbae hbalequs.1 syl))
+thm (hbnae () () (-> (-. (A. x (= (cv x) (cv y)))) (A. z (-. (A. x (= (cv x) (cv y)))))) (x y z hbae hbne))
+thm (hbnaes () ((hbnalequs.1 (-> (A. z (-. (A. x (= (cv x) (cv y))))) ph))) (-> (-. (A. x (= (cv x) (cv y)))) ph) (x y z hbnae hbnalequs.1 syl))
+thm (equs3 () () (<-> (E. x (/\ (= (cv x) (cv y)) ph)) (-. (A. x (-> (= (cv x) (cv y)) (-. ph))))) (x x cv y cv = ph alinexa bicon2i))
+thm (equs4 () () (-> (A. x (-> (= (cv x) (cv y)) ph)) (E. x (/\ (= (cv x) (cv y)) ph))) (x x cv y cv = ph -> ax-4 x x cv y cv = ph -. -> ax-4 msca x x cv y cv = ph -. -> hbn1 syl6 a5i x y x x cv y cv = ph -. -> A. -. ax9 syl x y ph equs3 sylibr))
+thm (equs5 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> (E. x (/\ (= (cv x) (cv y)) ph)) (A. x (-> (= (cv x) (cv y)) ph)))) (x y x hbnae x x cv y cv = ph -. -> hbn1 x y ph -. ax-11 ph x x cv y cv = ph -. -> A. con1 syl6 com23 19.21ad x y ph equs3 syl5ib))
+thm (equsal () ((equsal.1 (-> ps (A. x ps))) (equsal.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) (cv y)) ph)) ps) (equsal.2 equsal.1 19.3r syl6bb pm5.74i x bial equsal.1 x ps A. x cv y cv = ax-1 a5i syl x y ps ax9 impbi bitr4))
+thm (equsex () ((equsex.1 (-> ps (A. x ps))) (equsex.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) (cv y)) ph)) ps) (x x cv y cv = ph -. -> exnal x cv y cv = ph df-an x biex equsex.1 hbne equsex.2 negbid equsal bicon2i 3bitr4))
+thm (dral1 () ((dral1.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (A. x ph) (A. y ps))) (dral1.1 biimpd x 19.20ii hbaes x y ps ax-10 syld dral1.1 biimprd y 19.20ii hbaes y x ph ax-10 alequcoms syld impbid))
+thm (dral2 () ((dral2.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (A. z ph) (A. z ps))) (x y z hbae dral2.1 biald))
+thm (drex1 () ((drex1.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (E. x ph) (E. y ps))) (drex1.1 negbid dral1 negbid x ph df-ex y ps df-ex 3bitr4g))
+thm (drex2 () ((drex2.1 (-> (A. x (= (cv x) (cv y))) (<-> ph ps)))) (-> (A. x (= (cv x) (cv y))) (<-> (E. z ph) (E. z ps))) (x y z hbae drex2.1 biexd))
+thm (a4a () ((a4a.1 (-> ps (A. x ps))) (a4a.2 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) ps) (a4a.2 a4a.1 syl6com x 19.20i x y ps ax9 syl))
+thm (a4c () ((a4c.1 (-> ph (A. x ph))) (a4c.2 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ph (E. x ps)) (a4c.1 hbne a4c.2 con3d a4a con2i x ps df-ex sylibr))
+thm (a4c1 () ((a4c1.1 (-> ch (A. x ch))) (a4c1.2 (-> ch (-> ph (A. x ph)))) (a4c1.3 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ch (-> ph (E. x ps))) (a4c1.1 ph adantr a4c1.2 imp jca x ch ph 19.26 sylibr a4c1.3 ch adantld a4c exp))
+thm (cbv1 () ((cbv1.1 (-> ph (-> ps (A. y ps)))) (cbv1.2 (-> ph (-> ch (A. x ch)))) (cbv1.3 (-> ph (-> (= (cv x) (cv y)) (-> ps ch))))) (-> (A. x (A. y ph)) (-> (A. x ps) (A. y ch))) (cbv1.1 y a4s x 19.20ii x y ps ax-7 syl6 cbv1.3 com23 cbv1.2 syl6d x 19.20ii x y ch ax9 syl6 y 19.20ii a7s syld))
+thm (cbv2 () ((cbv2.1 (-> ph (-> ps (A. y ps)))) (cbv2.2 (-> ph (-> ch (A. x ch)))) (cbv2.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> (A. x (A. y ph)) (<-> (A. x ps) (A. y ch))) (cbv2.1 cbv2.2 cbv2.3 ps ch bi1 syl6 cbv1 cbv2.2 cbv2.1 cbv2.3 ps ch bi2 syl6 y x equcomi syl5 cbv1 a7s impbid))
+thm (cbv3 () ((cbv3.1 (-> ph (A. y ph))) (cbv3.2 (-> ps (A. x ps))) (cbv3.3 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) (A. y ps)) (cbv3.1 ph imim2i cbv3.2 ph ph -> a1i cbv3.3 ph ph -> a1i cbv1 ph id y ax-gen mpg))
+thm (cbval () ((cbval.1 (-> ph (A. y ph))) (cbval.2 (-> ps (A. x ps))) (cbval.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x ph) (A. y ps)) (cbval.1 ph imim2i cbval.2 ph ph -> a1i cbval.3 ph ph -> a1i cbv2 ph id y ax-gen mpg))
+thm (cbvex () ((cbvex.1 (-> ph (A. y ph))) (cbvex.2 (-> ps (A. x ps))) (cbvex.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x ph) (E. y ps)) (cbvex.1 hbne cbvex.2 hbne cbvex.3 negbid cbval negbii x ph df-ex y ps df-ex 3bitr4))
+thm (chvar () ((chv2.1 (-> ps (A. x ps))) (chv2.2 (-> (= (cv x) (cv y)) (<-> ph ps))) (chv2.3 ph)) ps (chv2.1 chv2.2 biimpd a4a chv2.3 mpg))
+thm (equvini () () (-> (= (cv x) (cv y)) (E. z (/\ (= (cv x) (cv z)) (= (cv z) (cv y))))) (z y a9e z equid z cv y cv = jctl z 19.22i ax-mp z z cv x cv = hba1 z x z ax-8 z a4s z cv y cv = anim1d 19.22d mpi z x a9e z x equcomi z equid jctir z 19.22i ax-mp z z cv y cv = hba1 z cv y cv = z cv z cv = ax-1 z a4s x cv z cv = anim2d 19.22d mpi jaoi x cv y cv = a1d z z cv x cv = A. z z cv y cv = A. ioran z x z hbnae z y z hbnae hban z x y ax-12 imp x z y ax-8 anc2li equcoms a4c1 sylbi pm2.61i))
+thm (hbequid () () (-> (= (cv x) (cv x)) (A. x (= (cv x) (cv x)))) (x x x ax-12 pm2.43i x a4s x x x cv x cv = A. -. hbn1 x x cv x cv = ax-6 19.21ai x cv x cv = a1d pm2.61i))
+thm (sbimi () ((sbimi.1 (-> ph ps))) (-> ([/] (cv y) x ph) ([/] (cv y) x ps)) (sbimi.1 x cv y cv = imim2i sbimi.1 x cv y cv = anim2i x 19.22i anim12i y x ph df-sb y x ps df-sb 3imtr4))
+thm (bisb () ((bisb.1 (<-> ph ps))) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps)) (bisb.1 biimp y x sbimi bisb.1 biimpr y x sbimi impbi))
+thm (drsb1 () () (-> (A. x (= (cv x) (cv y))) (<-> ([/] (cv z) x ph) ([/] (cv z) y ph))) (x y z equequ1 x a4s ph imbi1d x y z equequ1 x a4s ph anbi1d drex1 anbi12d z x ph df-sb z y ph df-sb 3bitr4g))
+thm (sb1 () () (-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (y x ph df-sb pm3.27bd))
+thm (sb2 () () (-> (A. x (-> (= (cv x) (cv y)) ph)) ([/] (cv y) x ph)) (x x cv y cv = ph -> ax-4 x y ph equs4 jca y x ph df-sb sylibr))
+thm (sb3 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> (E. x (/\ (= (cv x) (cv y)) ph)) ([/] (cv y) x ph))) (x y ph equs5 x y ph sb2 syl6))
+thm (sb4 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph equs5 y x ph sb1 syl5))
+thm (sb4b () () (-> (-. (A. x (= (cv x) (cv y)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph sb4 x y ph sb2 x x cv y cv = A. -. a1i impbid))
+thm (sbequ1 () () (-> (= (cv x) (cv y)) (-> ph ([/] (cv y) x ph))) (x cv y cv = ph pm3.4 x cv y cv = ph /\ x 19.8a jca y x ph df-sb sylibr exp))
+thm (sbequ2 () () (-> (= (cv x) (cv y)) (-> ([/] (cv y) x ph) ph)) (x cv y cv = ph -> x x cv y cv = ph /\ E. pm3.26 com12 y x ph df-sb syl5ib))
+thm (dfsb2 () () (<-> ([/] (cv y) x ph) (\/ (/\ (= (cv x) (cv y)) ph) (A. x (-> (= (cv x) (cv y)) ph)))) (x y ph sbequ2 x a4s x x cv y cv = ax-4 jctild x cv y cv = ph /\ x x cv y cv = ph -> A. orc syl6 x y ph sb4 x x cv y cv = ph -> A. x cv y cv = ph /\ olc syl6 pm2.61i x y ph sbequ1 imp x y ph sb2 jaoi impbi))
+thm (stdpc7 () () (-> (= (cv x) (cv y)) (-> ([/] (cv x) y ph) ph)) (y x ph sbequ2 equcoms))
+thm (sbequ12 () () (-> (= (cv x) (cv y)) (<-> ph ([/] (cv y) x ph))) (x y ph sbequ1 x y ph sbequ2 impbid))
+thm (sbequ12r () () (-> (= (cv x) (cv y)) (<-> ([/] (cv x) y ph) ph)) (y x ph sbequ12 x y equcom x cv y ph [/] ph bicom 3imtr4))
+thm (sbequ12a () () (-> (= (cv x) (cv y)) (<-> ([/] (cv y) x ph) ([/] (cv x) y ph))) (x y ph sbequ12 y x ph sbequ12 equcoms bitr3d))
+thm (sbid () () (<-> ([/] (cv x) x ph) ph) (x equid x x ph sbequ12 ax-mp bicomi))
+thm (stdpc4 () () (-> (A. x ph) ([/] (cv y) x ph)) (ph x cv y cv = ax-1 x 19.20i x y ph sb2 syl))
+thm (sbf () ((sbf.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ph) ph) (y x ph sb1 sbf.1 x cv y cv = 19.41 sylib pm3.27d sbf.1 x ph y stdpc4 syl impbi))
+thm (sbf2 () () (<-> ([/] (cv y) x (A. x ph)) (A. x ph)) (x ph hba1 y sbf))
+thm (sb6x () ((sb6x.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (sb6x.1 y sbf sb6x.1 ph x cv y cv = ax-1 19.21ai sylbi x y ph sb2 impbi))
+thm (sb6y () ((sb6y.1 (-> ph (A. y ph)))) (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (x y ph sbequ2 x a4s y x ph ax-10 alequcoms ph x cv y cv = ax-1 x 19.20i syl6 sb6y.1 syl5 syld x y ph sb4 pm2.61i x y ph sb2 impbi))
+thm (hbsb2 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph)))) (x y ph sb4 x y ph sb2 a5i syl6))
+thm (hbs1f () ((hbs1f.1 (-> ph (A. x ph)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (y x ph sb1 hbs1f.1 x cv y cv = 19.41 sylib pm3.27d hbs1f.1 x ph y stdpc4 a5i 3syl))
+thm (hbsb3 () ((hbsb3.1 (-> ph (A. y ph)))) (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (x y ph sbequ2 x a4s y x ph ax-10 alequcoms hbsb3.1 syl5 syld x y ph sbequ1 x 19.20ii syld x y ph hbsb2 pm2.61i))
+thm (sbequi () () (-> (= (cv x) (cv y)) (-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (z x ph hbsb2 x y z equvini x z ph stdpc7 z y ph sbequ1 sylan9 z 19.22i syl z x cv z ph [/] y cv z ph [/] 19.35 sylib sylan9 z y z hbnae z y ph hbsb2 19.9d syl9 exp com23 z x ph sbequ2 z a4s x cv y cv = adantr x y ph sbequ1 x z y ph drsb1 biimpd alequcoms sylan9r syld exp z y x ph drsb1 biimpd x y ph stdpc7 sylan9 z y ph sbequ1 z a4s x cv y cv = adantr syld exp pm2.61ii))
+thm (sbequ () () (-> (= (cv x) (cv y)) (<-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (x y z ph sbequi y x z ph sbequi equcoms impbid))
+thm (drsb2 () () (-> (A. x (= (cv x) (cv y))) (<-> ([/] (cv x) z ph) ([/] (cv y) z ph))) (x y z ph sbequi x a4s y x z ph sbequi equcoms x a4s impbid))
+thm (sbn1 () () (-> ([/] (cv y) x (-. ph)) (-. ([/] (cv y) x ph))) (x y ph -. sbequ2 x y ph sbequ2 nsyld x a4s x y ph -. sb4 y x ph sb1 x y ph equs3 sylib con2i syl6 pm2.61i))
+thm (sbn2 () () (-> (-. ([/] (cv y) x ph)) ([/] (cv y) x (-. ph))) (x y ph sbequ1 con3d com12 x y ph -. -. sb2 ph pm4.13 y x bisb sylibr con3i x y ph -. equs3 sylibr jca y x ph -. df-sb sylibr))
+thm (sbn () () (<-> ([/] (cv y) x (-. ph)) (-. ([/] (cv y) x ph))) (y x ph sbn1 y x ph sbn2 impbi))
+thm (sb5y () ((sb5y.1 (-> ph (A. y ph)))) (<-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (sb5y.1 hbne x sb6y y x ph sbn bitr3 bicon2i x y ph equs3 bitr4))
+thm (sbi1 () () (-> ([/] (cv y) x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x y ph ps -> sbequ2 x y ph sbequ2 syl5d x y ps sbequ1 syl6d x a4s x y ph ps -> sb4 x cv y cv = ph ps ax-2 x 19.20ii x y ps sb2 syl6 syl6 x y ph sb4 syl5d pm2.61i))
+thm (sbi2 () () (-> (-> ([/] (cv y) x ph) ([/] (cv y) x ps)) ([/] (cv y) x (-> ph ps))) (y x ph sbn ph ps pm2.21 y x sbimi sylbir ps ph ax-1 y x sbimi ja))
+thm (sbim () () (<-> ([/] (cv y) x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x ph ps sbi1 y x ph ps sbi2 impbi))
+thm (sbor () () (<-> ([/] (cv y) x (\/ ph ps)) (\/ ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x ph -. ps sbim y x ph sbn y cv x ps [/] imbi1i bitr ph ps df-or y x bisb y cv x ph [/] y cv x ps [/] df-or 3bitr4))
+thm (sb19.21 () ((sb19.21.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x (-> ph ps)) (-> ph ([/] (cv y) x ps))) (y x ph ps sbim sb19.21.1 y sbf y cv x ps [/] imbi1i bitr))
+thm (sban () () (<-> ([/] (cv y) x (/\ ph ps)) (/\ ([/] (cv y) x ph) ([/] (cv y) x ps))) (y x ph ps -. -> sbn y x ph ps -. sbim y x ps sbn y cv x ph [/] imbi2i bitr negbii bitr ph ps df-an y x bisb y cv x ph [/] y cv x ps [/] df-an 3bitr4))
+thm (sbbi () () (<-> ([/] (cv y) x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (ph ps bi y x bisb y x ph ps sbim y x ps ph sbim anbi12i y x ph ps -> ps ph -> sban y cv x ph [/] y cv x ps [/] bi 3bitr4 bitr))
+thm (sblbis () ((sblbis.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ch ph)) (<-> ([/] (cv y) x ch) ps)) (y x ch ph sbbi sblbis.1 y cv x ch [/] bibi2i bitr))
+thm (sbrbis () ((sbrbis.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ([/] (cv y) x ch))) (y x ph ch sbbi sbrbis.1 y cv x ch [/] bibi1i bitr))
+thm (sbrbif () ((sbrbif.1 (-> ch (A. x ch))) (sbrbif.2 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ch)) (sbrbif.2 ch sbrbis sbrbif.1 y sbf ps bibi2i bitr))
+thm (sbea4 () () (-> ([/] (cv y) x ph) (E. x ph)) (x ph -. y stdpc4 y x ph sbn sylib con2i x ph df-ex sylibr))
+thm (sbia4 () () (-> (A. x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x ph ps -> y stdpc4 y x ph ps sbim sylib))
+thm (sbba4 () () (-> (A. x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))) (x ph ps <-> y stdpc4 y x ph ps sbbi sylib))
+thm (bisbd () ((bisbd.1 (-> ph (A. x ph))) (bisbd.2 (-> ph (<-> ps ch)))) (-> ph (<-> ([/] (cv y) x ps) ([/] (cv y) x ch))) (bisbd.1 bisbd.2 19.21ai x ps ch y sbba4 syl))
+thm (sbequ5 () () (<-> ([/] (cv w) z (A. x (= (cv x) (cv y)))) (A. x (= (cv x) (cv y)))) (x y z hbae w sbf))
+thm (sbequ6 () () (<-> ([/] (cv w) z (-. (A. x (= (cv x) (cv y))))) (-. (A. x (= (cv x) (cv y))))) (x y z hbnae w sbf))
+thm (sbt () ((sbt.1 ph)) ([/] (cv y) x ph) (x y ph sb2 sbt.1 x cv y cv = a1i mpg))
+thm (equsb1 () () ([/] (cv y) x (= (cv x) (cv y))) (x y x cv y cv = sb2 x cv y cv = id mpg))
+thm (equsb2 () () ([/] (cv y) x (= (cv y) (cv x))) (x y y cv x cv = sb2 x y equcomi mpg))
+thm (sbequ8 () () (<-> ([/] (cv y) x ph) ([/] (cv y) x (-> (= (cv x) (cv y)) ph))) (y x equsb1 y cv x ph [/] a1bi y x x cv y cv = ph sbim bitr4))
+thm (sbied () ((sbied.1 (-> ph (A. x ph))) (sbied.2 (-> ph (-> ch (A. x ch)))) (sbied.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> ([/] (cv y) x ps) ch)) (sbied.1 sbied.3 ps ch bi1 syl6 imp3a x 19.20i x x cv y cv = ps /\ ch 19.22 syl y x ps sb1 syl5 sbied.2 x 19.20i x ch hba1 ch 19.23 sylib x ch ax-4 syl6 syld syl sbied.1 sbied.2 x a4s sbied.3 ps ch bi2 syl6 com23 x 19.20ii x y ps sb2 syl6 syld syl impbid))
+thm (sbie () ((sbie.1 (-> ps (A. x ps))) (sbie.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> ([/] (cv y) x ph) ps) (ph id ph id x hbth sbie.1 ph ph -> a1i sbie.2 ph ph -> a1i sbied ax-mp))
+thm (hbsb4 () ((hbsb4.1 (-> ph (A. z ph)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph)))) (z x y equequ1 z a4s dral1 negbid x y ph hbsb2 x z y cv x ph [/] ax-10 alequcoms syl9r sylbid x y z hbae x x cv y cv = ax-4 z 19.20i x y ph sbequ2 z a4s x y ph sbequ1 z 19.20ii hbsb4.1 syl5 syld 3syl z z cv x cv = A. -. z z cv y cv = A. -. /\ a1d x y ph sb4 z x x hbnae z y x hbnae hban z x z hbnae z y z hbnae hban z x y ax-12 imp hbsb4.1 z z cv x cv = A. -. z z cv y cv = A. -. /\ a1i hbimd 19.20d x y ph sb2 z 19.20i a7s syl6 syl9 pm2.61i exp pm2.61i))
+thm (hbsb4t () () (-> (A. x (A. z (-> ph (A. z ph)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph))))) (z ph ax-4 ph z ph A. -> biantru ph z ph A. bi bitr4 x z bi2al x ph z ph A. y sbba4 z a4s z x ph z ph A. <-> A. hba1 x ph z ph A. y sbba4 z a4s biald imbi12d a7s sylbi z ph hba1 y x hbsb4 syl5bir))
+thm (dvelimf2 () ((dvelimf2.1 (-> ph (A. x ph))) (dvelimf2.2 (-> ps (A. z ps))) (dvelimf2.3 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (z x z z cv y cv = ph -> A. ax-10 alequcoms z z cv y cv = ph -> hba1 syl5 x x cv y cv = A. -. a1d x z z hbnae x y z hbnae hban x z x hbnae x y x hbnae hban x z y ax-12 imp dvelimf2.1 x x cv z cv = A. -. x x cv y cv = A. -. /\ a1i hbimd hbald exp pm2.61i dvelimf2.2 dvelimf2.3 equsal dvelimf2.2 dvelimf2.3 equsal x bial 3imtr3g))
+thm (dvelimf () ((dvelimf.1 (-> ph (A. x ph))) (dvelimf.2 (-> ps (A. z ps))) (dvelimf.3 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (dvelimf.1 y z hbsb4 dvelimf.2 dvelimf.3 sbie dvelimf.2 dvelimf.3 sbie x bial 3imtr3g))
+thm (dvelimdf () ((ddelimdf.1 (-> ph (A. x ph))) (ddelimdf.2 (-> ph (A. z ph))) (ddelimdf.3 (-> ph (-> ps (A. x ps)))) (ddelimdf.4 (-> ph (-> ch (A. z ch)))) (ddelimdf.5 (-> ph (-> (= (cv z) (cv y)) (<-> ps ch))))) (-> ph (-> (-. (A. x (= (cv x) (cv y)))) (-> ch (A. x ch)))) (ddelimdf.2 ddelimdf.1 19.21ai ddelimdf.3 z x 19.20i2 z x ps y hbsb4t 3syl imp ddelimdf.2 ddelimdf.4 ddelimdf.5 sbied x x cv y cv = A. -. adantr ddelimdf.1 ddelimdf.2 ddelimdf.4 ddelimdf.5 sbied biald x x cv y cv = A. -. adantr 3imtr3d exp))
+thm (sbco () () (<-> ([/] (cv y) x ([/] (cv x) y ph)) ([/] (cv y) x ph)) (y x equsb2 y x ph sbequ12 bicomd y x sbimi ax-mp y x x cv y ph [/] ph sbbi mpbi))
+thm (sbid2 () ((sbid2.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) x ([/] (cv x) y ph)) ph) (y x ph sbco sbid2.1 y sbf bitr))
+thm (sbidm () () (<-> ([/] (cv y) x ([/] (cv y) x ph)) ([/] (cv y) x ph)) (x y y cv x ph [/] sbequ12 bicomd x a4s x y x hbnae x y ph hbsb2 x cv y cv = y cv x ph [/] pm4.2i x x cv y cv = A. -. a1i sbied pm2.61i))
+thm (sbco2 () ((sbco2.1 (-> ph (A. z ph)))) (<-> ([/] (cv y) z ([/] (cv z) x ph)) ([/] (cv y) x ph)) (x y z z cv x ph [/] sbequ sbco2.1 x sbid2 syl5bbr x y ph sbequ12 bitr3d x a4s x y x hbnae sbco2.1 x hbsb3 y z hbsb4 x y z z cv x ph [/] sbequ sbco2.1 x sbid2 syl5bbr x x cv y cv = A. -. a1i sbied bicomd pm2.61i))
+thm (sbco2d () ((sbco2d.1 (-> ph (A. x ph))) (sbco2d.2 (-> ph (A. z ph))) (sbco2d.3 (-> ph (-> ps (A. z ps))))) (-> ph (<-> ([/] (cv y) z ([/] (cv z) x ps)) ([/] (cv y) x ps))) (sbco2d.2 sbco2d.3 hbim1 y x sbco2 sbco2d.1 z ps sb19.21 y z bisb sbco2d.2 y z cv x ps [/] sb19.21 bitr sbco2d.1 y ps sb19.21 3bitr3 pm5.74ri))
+thm (sbco3 () () (<-> ([/] (cv z) y ([/] (cv y) x ph)) ([/] (cv z) x ([/] (cv x) y ph))) (x y z y cv x ph [/] drsb1 x y ph sbequ12a x 19.20i x y cv x ph [/] x cv y ph [/] z sbba4 syl bitr3d x y y hbnae x y x hbnae x y ph hbsb2 z sbco2d x y ph sbco z x bisb syl5rbbr pm2.61i))
+thm (sbcom () () (<-> ([/] (cv y) z ([/] (cv y) x ph)) ([/] (cv y) x ([/] (cv y) z ph))) (x z y y cv x ph [/] drsb1 x z x hbae x z y ph drsb1 y bisbd bitr3d x x cv y cv = A. -. z z cv y cv = A. -. /\ adantr x z z hbnae x y z hbnae hban x z x hbnae x y x hbnae hban x z y ax-12 imp x 19.20i x z cv y cv = x cv y cv = ph -> 19.21g 3syl biald z z cv y cv = A. -. adantrr x z x hbnae z y x hbnae hban x z z hbnae z y z hbnae hban z x y ax-12 nalequcoms imp z 19.20i z x cv y cv = z cv y cv = ph -> 19.21g 3syl z cv y cv = x cv y cv = ph bi2.04 z bial syl5bb biald z x z cv y cv = x cv y cv = ph -> -> alcom syl5bb x x cv y cv = A. -. adantrl bitr3d z y y cv x ph [/] sb4b x y z hbnae x y ph sb4b z cv y cv = imbi2d biald sylan9bbr x x cv z cv = A. -. adantl x y y cv z ph [/] sb4b z y x hbnae z y ph sb4b x cv y cv = imbi2d biald sylan9bb x x cv z cv = A. -. adantl 3bitr4d pm2.61an1 exp x y z hbae x y ph sbequ12 x a4s y bisbd x y y cv z ph [/] sbequ12 x a4s bitr3d z y y cv x ph [/] sbequ12 z a4s z y x hbae z y ph sbequ12 z a4s y bisbd bitr3d pm2.61ii))
+thm (sb5rf () ((sb5rf.1 (-> ph (A. y ph)))) (<-> ph (E. y (/\ (= (cv y) (cv x)) ([/] (cv y) x ph)))) (sb5rf.1 x sbid2 x y y cv x ph [/] sb1 sylbir sb5rf.1 y x ph sbequ12r biimpa 19.23ai impbi))
+thm (sb6rf () ((sb5rf.1 (-> ph (A. y ph)))) (<-> ph (A. y (-> (= (cv y) (cv x)) ([/] (cv y) x ph)))) (sb5rf.1 x y ph sbequ1 equcoms com12 19.21ai y x y cv x ph [/] sb2 x y ph sbco sylib sb5rf.1 x sbf sylib impbi))
+thm (sb8 () ((sb8.1 (-> ph (A. y ph)))) (<-> (A. x ph) (A. y ([/] (cv y) x ph))) (sb8.1 x hbal x ph y stdpc4 19.21ai sb8.1 x hbsb3 y hbal y y cv x ph [/] x stdpc4 sb8.1 x sbid2 sylib 19.21ai impbi))
+thm (sb8e () ((sb8e.1 (-> ph (A. y ph)))) (<-> (E. x ph) (E. y ([/] (cv y) x ph))) (sb8e.1 hbne x sb8 y x ph sbn y bial bitr negbii x ph df-ex y y cv x ph [/] df-ex 3bitr4))
+thm (sb9i () () (-> (A. x ([/] (cv x) y ph)) (A. y ([/] (cv y) x ph))) (y x y ph drsb1 y x y ph drsb2 bitr3d dral1 biimprd y x ph hbsb2 x 19.20ii hbnaes x x cv y ph [/] y stdpc4 y x ph sbco sylib y 19.20i a7s syl6 pm2.61i))
+thm (sb9 () () (<-> (A. x ([/] (cv x) y ph)) (A. y ([/] (cv y) x ph))) (x y ph sb9i y x ph sb9i impbi))
+thm (ax17eq ((x z) (y z)) () (-> (= (cv x) (cv y)) (A. z (= (cv x) (cv y)))) (z x y ax-12 z x x cv y cv = ax-16 z y x cv y cv = ax-16 pm2.61ii))
+thm (ax17el ((x z) (y z)) () (-> (e. (cv x) (cv y)) (A. z (e. (cv x) (cv y)))) (z x y ax-15 z x x cv y cv e. ax-16 z y x cv y cv e. ax-16 pm2.61ii))
+thm (ax11a ((x y)) () (-> (= (cv x) (cv y)) (-> ph (A. x (-> (= (cv x) (cv y)) ph)))) (x y x cv y cv = ph -> ax-16 ph x cv y cv = ax-1 syl5 x cv y cv = a1d x y ph ax-11 pm2.61i))
+thm (equid2 ((x y)) () (= (cv x) (cv x)) (y x a9e x cv x cv = y ax-17 y x x ax-8 pm2.43i 19.23ai ax-mp))
+thm (a4b ((ps x)) ((a4b.1 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> (A. x ph) ps) (ps x ax-17 a4b.1 a4a))
+thm (a4b1 ((ps x)) ((a4b1.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (A. x ph) ps) (a4b1.1 biimpd a4b))
+thm (a4w ((ph x)) ((a4w.1 (-> (= (cv x) (cv y)) (-> ph ps)))) (-> ph (E. x ps)) (ph x ax-17 a4w.1 a4c))
+thm (a4w1 ((ps x)) ((a4w1.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (a4w1.2 ps)) (E. x ph) (a4w1.2 a4w1.1 biimprd a4w ax-mp))
+thm (equvin ((x z) (y z)) () (<-> (= (cv x) (cv y)) (E. z (/\ (= (cv x) (cv z)) (= (cv z) (cv y))))) (x y z equvini x cv y cv = z ax-17 x z y equtr imp 19.23ai impbi))
+thm (a16g ((x y)) () (-> (A. x (= (cv x) (cv y))) (-> ph (A. z ph))) (x y z hbae x z ax9a x y x cv z cv = -. ax-16 mt3i x z equcomi syl 19.21ai x y ph ax-16 z z cv x cv = A. ph pm4.2i dral1 biimprd syl9r mpcom))
+thm (a16gb ((x y)) () (-> (A. x (= (cv x) (cv y))) (<-> ph (A. z ph))) (x y ph z a16g z ph ax-4 x x cv y cv = A. a1i impbid))
+thm (bialdv ((ph x)) ((bialdv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x ps) (A. x ch))) (ph x ax-17 bialdv.1 biald))
+thm (biexdv ((ph x)) ((bialdv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x ps) (E. x ch))) (ph x ax-17 bialdv.1 biexd))
+thm (bi2aldv ((ph x) (ph y)) ((bi2aldv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A. x (A. y ps)) (A. x (A. y ch)))) (bi2aldv.1 y bialdv x bialdv))
+thm (bi2exdv ((ph x) (ph y)) ((bi2aldv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y ps)) (E. x (E. y ch)))) (bi2aldv.1 y biexdv x biexdv))
+thm (bi3exdv ((ph x) (ph y) (ph z)) ((bi3exdv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y (E. z ps))) (E. x (E. y (E. z ch))))) (bi3exdv.1 z biexdv x y bi2exdv))
+thm (bi4exdv ((ph x) (ph y) (ph z) (ph w)) ((bi4exdv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E. x (E. y (E. z (E. w ps)))) (E. x (E. y (E. z (E. w ch)))))) (bi4exdv.1 z w bi2exdv x y bi2exdv))
+thm (19.9rv ((ph x)) () (<-> ph (E. x ph)) (ph x ax-17 19.9r))
+thm (19.21v ((ph x)) () (<-> (A. x (-> ph ps)) (-> ph (A. x ps))) (ph x ax-17 ps 19.21))
+thm (19.21aiv ((ph x)) ((19.21aiv.1 (-> ph ps))) (-> ph (A. x ps)) (ph x ax-17 19.21aiv.1 19.21ai))
+thm (19.21aivv ((ph x) (ph y)) ((19.21aivv.1 (-> ph ps))) (-> ph (A. x (A. y ps))) (19.21aivv.1 y 19.21aiv x 19.21aiv))
+thm (19.21adv ((ph x) (ps x)) ((19.21adv.1 (-> ph (-> ps ch)))) (-> ph (-> ps (A. x ch))) (ph x ax-17 ps x ax-17 19.21adv.1 19.21ad))
+thm (19.20dv ((ph x)) ((19.20dv.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x ps) (A. x ch))) (ph x ax-17 19.20dv.1 19.20d))
+thm (19.22dv ((ph x)) ((19.20dv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) (E. x ch))) (ph x ax-17 19.20dv.1 19.22d))
+thm (19.20dvv ((ph x) (ph y)) ((19.20dvv.1 (-> ph (-> ps ch)))) (-> ph (-> (A. x (A. y ps)) (A. x (A. y ch)))) (19.20dvv.1 y 19.20dv x 19.20dv))
+thm (19.22dvv ((ph x) (ph y)) ((19.20dvv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x (E. y ps)) (E. x (E. y ch)))) (19.20dvv.1 y 19.22dv x 19.22dv))
+thm (19.23v ((ps x)) () (<-> (A. x (-> ph ps)) (-> (E. x ph) ps)) (ps x ax-17 ph 19.23))
+thm (19.23vv ((ps x) (ps y)) () (<-> (A. x (A. y (-> ph ps))) (-> (E. x (E. y ph)) ps)) (y ph ps 19.23v x bial x y ph E. ps 19.23v bitr))
+thm (19.23aiv ((ps x)) ((19.23aiv.1 (-> ph ps))) (-> (E. x ph) ps) (ps x ax-17 19.23aiv.1 19.23ai))
+thm (19.23aivv ((ps x) (ps y)) ((19.23aivv.1 (-> ph ps))) (-> (E. x (E. y ph)) ps) (19.23aivv.1 y 19.23aiv x 19.23aiv))
+thm (19.23adv ((ch x) (ph x)) ((19.23adv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x ps) ch)) (ph x ax-17 ch x ax-17 19.23adv.1 19.23ad))
+thm (19.23advv ((ch x) (ph x) (ch y) (ph y)) ((19.23advv.1 (-> ph (-> ps ch)))) (-> ph (-> (E. x (E. y ps)) ch)) (19.23advv.1 y 19.23adv x 19.23adv))
+thm (19.27v ((ps x)) () (<-> (A. x (/\ ph ps)) (/\ (A. x ph) ps)) (ps x ax-17 ph 19.27))
+thm (19.28v ((ph x)) () (<-> (A. x (/\ ph ps)) (/\ ph (A. x ps))) (ph x ax-17 ps 19.28))
+thm (19.36v ((ps x)) () (<-> (E. x (-> ph ps)) (-> (A. x ph) ps)) (ps x ax-17 ph 19.36))
+thm (19.36aiv ((ps x)) ((19.36aiv.1 (E. x (-> ph ps)))) (-> (A. x ph) ps) (ps x ax-17 19.36aiv.1 19.36i))
+thm (19.12vv ((x y) (ps x) (ph y)) () (<-> (E. x (A. y (-> ph ps))) (A. y (E. x (-> ph ps)))) (y ph ps 19.21v x biex x ph y ps A. 19.36v x ph ps 19.36v y bial y x ph A. ps 19.21v bitr2 3bitr))
+thm (19.37v ((ph x)) () (<-> (E. x (-> ph ps)) (-> ph (E. x ps))) (ph x ax-17 ps 19.37))
+thm (19.37aiv ((ph x)) ((19.37aiv.1 (E. x (-> ph ps)))) (-> ph (E. x ps)) (19.37aiv.1 x ph ps 19.37v mpbi))
+thm (19.41v ((ps x)) () (<-> (E. x (/\ ph ps)) (/\ (E. x ph) ps)) (ps x ax-17 ph 19.41))
+thm (19.41vv ((ps x) (ps y)) () (<-> (E. x (E. y (/\ ph ps))) (/\ (E. x (E. y ph)) ps)) (y ph ps 19.41v x biex x y ph E. ps 19.41v bitr))
+thm (19.41vvv ((ps x) (ps y) (ps z)) () (<-> (E. x (E. y (E. z (/\ ph ps)))) (/\ (E. x (E. y (E. z ph))) ps)) (y z ph ps 19.41vv x biex x y z ph E. E. ps 19.41v bitr))
+thm (19.42v ((ph x)) () (<-> (E. x (/\ ph ps)) (/\ ph (E. x ps))) (ph x ax-17 ps 19.42))
+thm (exdistr ((ph y)) () (<-> (E. x (E. y (/\ ph ps))) (E. x (/\ ph (E. y ps)))) (y ph ps 19.42v x biex))
+thm (19.42vv ((ph x) (ph y)) () (<-> (E. x (E. y (/\ ph ps))) (/\ ph (E. x (E. y ps)))) (x y ph ps exdistr x ph y ps E. 19.42v bitr))
+thm (exdistr2 ((ph y) (ph z)) () (<-> (E. x (E. y (E. z (/\ ph ps)))) (E. x (/\ ph (E. y (E. z ps))))) (y z ph ps 19.42vv x biex))
+thm (3exdistr ((ph y) (ph z) (ps z)) () (<-> (E. x (E. y (E. z (/\/\ ph ps ch)))) (E. x (/\ ph (E. y (/\ ps (E. z ch)))))) (ph ps ch 3anass z biex z ph ps ch /\ 19.42v z ps ch 19.42v ph anbi2i 3bitr y biex y ph ps z ch E. /\ 19.42v bitr x biex))
+thm (4exdistr ((ph y) (ph z) (ph w) (ps z) (ps w) (ch w)) () (<-> (E. x (E. y (E. z (E. w (/\ (/\ ph ps) (/\ ch th)))))) (E. x (/\ ph (E. y (/\ ps (E. z (/\ ch (E. w th)))))))) (ph ps ch th /\ anass w biex w ph ps ch th /\ /\ 19.42v w ps ch th /\ 19.42v ph anbi2i w ch th 19.42v ps anbi2i ph anbi2i 3bitr bitr z biex z ph ps ch w th E. /\ /\ 19.42v z ps ch w th E. /\ 19.42v ph anbi2i 3bitr y biex y ph ps z ch w th E. /\ E. /\ 19.42v bitr x biex))
+thm (cbvalv ((ph y) (ps x)) ((cbvalv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A. x ph) (A. y ps)) (ph y ax-17 ps x ax-17 cbvalv.1 cbval))
+thm (cbvexv ((ph y) (ps x)) ((cbvalv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E. x ph) (E. y ps)) (ph y ax-17 ps x ax-17 cbvalv.1 cbvex))
+thm (cbval2 ((x y) (y z) (w x) (w z)) ((cbval2.1 (-> ph (A. z ph))) (cbval2.2 (-> ph (A. w ph))) (cbval2.3 (-> ps (A. x ps))) (cbval2.4 (-> ps (A. y ps))) (cbval2.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (A. x (A. y ph)) (A. z (A. w ps))) (cbval2.1 y hbal cbval2.3 w hbal x cv z cv = w ax-17 cbval2.2 hban x cv z cv = y ax-17 cbval2.4 hban cbval2.5 expcom pm5.32d cbval x cv z cv = y ax-17 ph 19.28 x cv z cv = w ax-17 ps 19.28 3bitr3 x cv z cv = y ph A. w ps A. pm5.32 mpbir cbval))
+thm (cbvex2 ((x y) (y z) (w x) (w z)) ((cbval2.1 (-> ph (A. z ph))) (cbval2.2 (-> ph (A. w ph))) (cbval2.3 (-> ps (A. x ps))) (cbval2.4 (-> ps (A. y ps))) (cbval2.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. x (E. y ph)) (E. z (E. w ps))) (cbval2.1 y hbex cbval2.3 w hbex x cv z cv = w ax-17 cbval2.2 hban x cv z cv = y ax-17 cbval2.4 hban cbval2.5 expcom pm5.32d cbvex x cv z cv = y ax-17 ph 19.42 x cv z cv = w ax-17 ps 19.42 3bitr3 x cv z cv = y ph E. w ps E. pm5.32 mpbir cbvex))
+thm (cbval2v ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (w x) (y z)) ((cbval2v.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (A. x (A. y ph)) (A. z (A. w ps))) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbval2v.1 cbval2))
+thm (cbvex2v ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (w x) (y z)) ((cbval2v.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. x (E. y ph)) (E. z (E. w ps))) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbval2v.1 cbvex2))
+thm (cbvald ((ph x) (ch x)) ((cbvald.1 (-> ph (A. y ph))) (cbvald.2 (-> ph (-> ps (A. y ps)))) (cbvald.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> (A. x ps) (A. y ch))) (cbvald.1 cbvald.2 hbim1 ph ch -> x ax-17 cbvald.3 com12 pm5.74d cbval x ph ps 19.21v cbvald.1 ch 19.21 3bitr3 pm5.74ri))
+thm (cbvexd ((ph x) (ch x)) ((cbvald.1 (-> ph (A. y ph))) (cbvald.2 (-> ph (-> ps (A. y ps)))) (cbvald.3 (-> ph (-> (= (cv x) (cv y)) (<-> ps ch))))) (-> ph (<-> (E. x ps) (E. y ch))) (cbvald.1 cbvald.1 cbvald.2 hbnd cbvald.3 ps ch pm4.11 syl6ib cbvald negbid x ps df-ex y ch df-ex 3bitr4g))
+var (set f)
+var (set g)
+thm (cbvex4v ((u v) (ph v) (ph u) (f g) (f ph) (g ph) (x y) (ch x) (ch y) (w z) (ch z) (ch w) (ps x) (ps y) (f ps) (g ps) (x z) (w x) (u x) (u z) (u w) (y z) (w y) (v y) (v z) (v w) (g z) (f w)) ((cbvex4v.1 (-> (/\ (= (cv x) (cv v)) (= (cv y) (cv u))) (<-> ph ps))) (cbvex4v.2 (-> (/\ (= (cv z) (cv f)) (= (cv w) (cv g))) (<-> ps ch)))) (<-> (E. x (E. y (E. z (E. w ph)))) (E. v (E. u (E. f (E. g ch))))) (cbvex4v.1 z w bi2exdv cbvex2v cbvex4v.2 cbvex2v v u bi2ex bitr))
+thm (eeanv ((ph y) (ps x)) () (<-> (E. x (E. y (/\ ph ps))) (/\ (E. x ph) (E. y ps))) (x y ph ps exdistr ps x ax-17 y hbex ph 19.41 bitr))
+thm (eeeanv ((y z) (ph y) (ph z) (x z) (ps x) (ps z) (x y) (ch x) (ch y)) () (<-> (E. x (E. y (E. z (/\/\ ph ps ch)))) (/\/\ (E. x ph) (E. y ps) (E. z ch))) (y z ph ps ch /\ 19.42vv y z ps ch eeanv ph anbi2i bitr x biex x ph y ps E. z ch E. /\ 19.41v bitr ph ps ch 3anass x y z bi3ex x ph E. y ps E. z ch E. 3anass 3bitr4))
+thm (ee4anv ((ph z) (ph w) (ps x) (ps y) (y z) (w x)) () (<-> (E. x (E. y (E. z (E. w (/\ ph ps))))) (/\ (E. x (E. y ph)) (E. z (E. w ps)))) (y z w ph ps /\ E. excom x biex y w ph ps eeanv x z bi2ex x z y ph E. w ps E. eeanv 3bitr))
+thm (nexdv ((ph x)) ((nexdv.1 (-> ph (-. ps)))) (-> ph (-. (E. x ps))) (ph x ax-17 nexdv.1 nexd))
+thm (chvarv ((ps x)) ((chv.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (chv.2 ph)) ps (chv.1 a4b1 chv.2 mpg))
+thm (cleljust ((x z) (y z)) () (<-> (e. (cv x) (cv y)) (E. z (/\ (= (cv z) (cv x)) (e. (cv z) (cv y))))) (x cv y cv e. z ax-17 z x y elequ1 equsex bicomi))
+thm (hbs1 ((x y)) () (-> ([/] (cv y) x ph) (A. x ([/] (cv y) x ph))) (x y y cv x ph [/] ax-16 x y ph hbsb2 pm2.61i))
+thm (hbsb ((y z)) ((hbsb.1 (-> ph (A. z ph)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph))) (z y y cv x ph [/] ax-16 hbsb.1 y x hbsb4 pm2.61i))
+thm (sbcom2 ((x z) (w x) (y z)) () (<-> ([/] (cv w) z ([/] (cv y) x ph)) ([/] (cv y) x ([/] (cv w) z ph))) (z x x cv y cv = z cv w cv = ph -> -> alcom x cv y cv = z cv w cv = ph bi2.04 x bial x z cv w cv = x cv y cv = ph -> 19.21v bitr z bial z x cv y cv = z cv w cv = ph -> 19.21v x bial 3bitr3 x x cv y cv = A. -. z z cv w cv = A. -. /\ a1i z w y cv x ph [/] sb4b x y ph sb4b z cv w cv = imbi2d z bialdv sylan9bbr x y w cv z ph [/] sb4b z w ph sb4b x cv y cv = imbi2d x bialdv sylan9bb 3bitr4d exp x y z hbae x y ph sbequ12 x a4s w bisbd x y w cv z ph [/] sbequ12 x a4s bitr3d z w y cv x ph [/] sbequ12 z a4s z w x hbae z w ph sbequ12 z a4s y bisbd bitr3d pm2.61ii))
+thm (sb5 ((x y)) () (<-> ([/] (cv y) x ph) (E. x (/\ (= (cv x) (cv y)) ph))) (y x ph hbs1 x y ph sbequ12 equsex bicomi))
+thm (sb6 ((x y)) () (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ph))) (y x ph hbs1 x y ph sbequ12 equsal bicomi))
+thm (2sb5 ((x y) (x z) (y z) (w y)) () (<-> ([/] (cv z) x ([/] (cv w) y ph)) (E. x (E. y (/\ (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))) (z x w cv y ph [/] sb5 y x cv z cv = y cv w cv = ph /\ 19.42v x cv z cv = y cv w cv = ph anass y biex w y ph sb5 x cv z cv = anbi2i 3bitr4r x biex bitr))
+thm (2sb6 ((x y) (x z) (y z) (w y)) () (<-> ([/] (cv z) x ([/] (cv w) y ph)) (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))) (z x w cv y ph [/] sb6 y x cv z cv = y cv w cv = ph -> 19.21v x cv z cv = y cv w cv = ph impexp y bial w y ph sb6 x cv z cv = imbi2i 3bitr4r x bial bitr))
+thm (sb6a ((x y)) () (<-> ([/] (cv y) x ph) (A. x (-> (= (cv x) (cv y)) ([/] (cv x) y ph)))) (y x ph sb6 y x ph sbequ12 equcoms pm5.74i x bial bitr))
+thm (2sb5rf ((x y) (w x) (y z) (w z)) ((2sb5rf.1 (-> ph (A. z ph))) (2sb5rf.2 (-> ph (A. w ph)))) (<-> ph (E. z (E. w (/\ (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) ([/] (cv z) x ([/] (cv w) y ph)))))) (2sb5rf.1 x sb5rf w z cv x cv = w cv y cv = w cv y z cv x ph [/] [/] /\ 19.42v z x w y ph sbcom2 z cv x cv = w cv y cv = /\ anbi2i z cv x cv = w cv y cv = w cv y z cv x ph [/] [/] anass bitr w biex 2sb5rf.2 z x hbsb y sb5rf z cv x cv = anbi2i 3bitr4r z biex bitr))
+thm (2sb6rf ((x y) (w x) (y z) (w z)) ((2sb5rf.1 (-> ph (A. z ph))) (2sb5rf.2 (-> ph (A. w ph)))) (<-> ph (A. z (A. w (-> (/\ (= (cv z) (cv x)) (= (cv w) (cv y))) ([/] (cv z) x ([/] (cv w) y ph)))))) (2sb5rf.1 x sb6rf w z cv x cv = w cv y cv = w cv y z cv x ph [/] [/] -> 19.21v z x w y ph sbcom2 z cv x cv = w cv y cv = /\ imbi2i z cv x cv = w cv y cv = w cv y z cv x ph [/] [/] impexp bitr w bial 2sb5rf.2 z x hbsb y sb6rf z cv x cv = imbi2i 3bitr4r z bial bitr))
+thm (sb7 ((x z) (y z) (ph z)) () (<-> ([/] (cv y) x ph) (E. z (/\ (= (cv z) (cv y)) (E. x (/\ (= (cv x) (cv z)) ph))))) (z x ph sb5 y z bisb ph z ax-17 y x sbco2 y z x x cv z cv = ph /\ E. sb5 3bitr3))
+thm (sb10f ((x y)) ((sb10f.1 (-> ph (A. x ph)))) (<-> ([/] (cv y) z ph) (E. x (/\ (= (cv x) (cv y)) ([/] (cv x) z ph)))) (sb10f.1 y z hbsb x y z ph sbequ equsex bicomi))
+thm (sbid2v ((ph x)) () (<-> ([/] (cv y) x ([/] (cv x) y ph)) ph) (ph x ax-17 y sbid2))
+thm (sbelx ((x y) (ph x)) () (<-> ph (E. x (/\ (= (cv x) (cv y)) ([/] (cv x) y ph)))) (y x ph sbid2v y x x cv y ph [/] sb5 bitr3))
+thm (sbel2x ((x y) (x z) (w x) (y z) (w y) (w z) (ph x) (ph y)) () (<-> ph (E. x (E. y (/\ (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ([/] (cv y) w ([/] (cv x) z ph)))))) (x cv z ph [/] y w sbelx x cv z cv = anbi2i x biex ph x z sbelx x y x cv z cv = y cv w cv = y cv w x cv z ph [/] [/] /\ exdistr 3bitr4 x cv z cv = y cv w cv = y cv w x cv z ph [/] [/] anass x y bi2ex bitr4))
+thm (sbal1 ((x y)) () (-> (-. (A. x (= (cv x) (cv z)))) (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph)))) (y z x ph A. sbequ12 y a4s y z ph sbequ12 y a4s x 19.20i hbaes x ph z cv y ph [/] 19.15 syl bitr3d x x cv z cv = A. -. a1d x ph hba1 z y hbsb4 x ph ax-4 z y sbimi x 19.20i syl6 y y cv z cv = A. -. adantl y z ph sb4 x 19.20ii hbnaes x y y cv z cv = ph -> ax-7 syl6 x y y cv z cv = ax-16 x x cv z cv = A. -. a1d x y z ax-12 pm2.61i x y cv z cv = ph 19.20 syl9 y 19.20ii y z x ph A. sb2 syl6 hbnaes sylan9 impbid exp pm2.61i))
+thm (sbal ((x y) (x z)) () (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph))) (x z ph x a16gb z y sbimi z y x z sbequ5 z y ph x ph A. sbbi 3imtr3 x z z cv y ph [/] x a16gb bitr3d x z y ph sbal1 pm2.61i))
+thm (sbex ((x y) (x z)) () (<-> ([/] (cv z) y (E. x ph)) (E. x ([/] (cv z) y ph))) (z y x ph -. A. sbn z y x ph -. sbal z y ph sbn x bial bitr negbii bitr x ph df-ex z y bisb x z cv y ph [/] df-ex 3bitr4))
+thm (sbalv ((x z) (y z)) ((sbalv.1 (<-> ([/] (cv y) x ph) ps))) (<-> ([/] (cv y) x (A. z ph)) (A. z ps)) (y x z ph sbal sbalv.1 z bial bitr))
+thm (exsb ((x y) (ph y)) () (<-> (E. x ph) (E. y (A. x (-> (= (cv x) (cv y)) ph)))) (ph y ax-17 x sb8e y x ph sb6 y biex bitr))
+thm (2exsb ((x y) (x z) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) ph)))))) (y ph w exsb x biex x w y y cv w cv = ph -> A. excom bitr x y y cv w cv = ph -> A. z exsb x cv z cv = y cv w cv = ph impexp y bial y x cv z cv = y cv w cv = ph -> 19.21v bitr2 x bial z biex bitr w biex w z x y x cv z cv = y cv w cv = /\ ph -> A. A. excom 3bitr))
+thm (dvelim ((ps z)) ((ddelim.1 (-> ph (A. x ph))) (ddelim.2 (-> (= (cv z) (cv y)) (<-> ph ps)))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))) (ddelim.1 ps z ax-17 ddelim.2 dvelimf))
+thm (dveeq1 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv y) (cv z)) (A. x (= (cv y) (cv z))))) (w cv z cv = x ax-17 w y z equequ1 dvelim))
+thm (dveeq2 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv z) (cv y)) (A. x (= (cv z) (cv y))))) (z cv w cv = x ax-17 w y z equequ2 dvelim))
+thm (dveel1 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (e. (cv y) (cv z)) (A. x (e. (cv y) (cv z))))) (w cv z cv e. x ax-17 w y z elequ1 dvelim))
+thm (dveel2 ((w z) (w x) (x z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (e. (cv z) (cv y)) (A. x (e. (cv z) (cv y))))) (z cv w cv e. x ax-17 w y z elequ2 dvelim))
+thm (sbal2 ((y z) (x z)) () (-> (-. (A. x (= (cv x) (cv y)))) (<-> ([/] (cv z) y (A. x ph)) (A. x ([/] (cv z) y ph)))) (x y y hbnae x y z dveeq1 x 19.20i hbnaes x y cv z cv = ph 19.21g syl biald y x y cv z cv = ph -> alcom syl5rbbr z y x ph A. sb6 z y ph sb6 x bial 3bitr4g))
+thm (ax15 ((w y) (w z) (w x)) () (-> (-. (A. z (= (cv z) (cv x)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> (e. (cv x) (cv y)) (A. z (e. (cv x) (cv y)))))) (z z cv y cv = hbn1 z y w dveel2 hbim1 w x y elequ1 z z cv y cv = A. -. imbi2d dvelim z z cv y cv = hbn1 x cv y cv e. 19.21 syl6ib pm2.86d))
+thm (euf ((x y) (x z) (y z) (ph z)) ((euf.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E. y (A. x (<-> ph (= (cv x) (cv y)))))) (x ph z df-eu euf.1 x cv z cv = y ax-17 hbbi x hbal x ph x cv y cv = <-> A. z ax-17 z y x equequ2 ph bibi2d x bialdv cbvex bitr))
+thm (bieud ((x y) (ph y) (ps y) (ch y)) ((bieud.1 (-> ph (A. x ph))) (bieud.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E! x ps) (E! x ch))) (bieud.1 bieud.2 x cv y cv = bibi1d biald y biexdv x ps y df-eu x ch y df-eu 3bitr4g))
+thm (bieudv ((ph x)) ((bieudv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E! x ps) (E! x ch))) (ph x ax-17 bieudv.1 bieud))
+thm (bieu () ((bieu.1 (<-> ph ps))) (<-> (E! x ph) (E! x ps)) (x equid x equid x hbth bieu.1 x cv x cv = a1i bieud ax-mp))
+thm (hbeu1 ((x y) (ph y)) () (-> (E! x ph) (A. x (E! x ph))) (x ph x cv y cv = <-> hba1 y hbex x ph y df-eu x ph y df-eu x bial 3imtr4))
+thm (hbeu ((y z) (x z) (ph z)) ((hbeu.1 (-> ph (A. x ph)))) (-> (E! y ph) (A. x (E! y ph))) (y x y ph y cv z cv = <-> A. ax-10 alequcoms y ph y cv z cv = <-> hba1 syl5 x y y hbnae x y x hbnae hbeu.1 x x cv y cv = A. -. a1i x y z dveeq1 hbbid hbald pm2.61i z hbex y ph z df-eu y ph z df-eu x bial 3imtr4))
+thm (sb8eu ((x y) (x z) (y z) (ph z)) ((sb8eu.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E! y ([/] (cv y) x ph))) (sb8eu.1 x cv z cv = y ax-17 hbbi x sb8 y cv z cv = x ax-17 x y z equequ1 sbie ph sblbis y bial bitr z biex x ph z df-eu y y cv x ph [/] z df-eu 3bitr4))
+thm (cbveu ((x y)) ((cbveu.1 (-> ph (A. y ph))) (cbveu.2 (-> ps (A. x ps))) (cbveu.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E! x ph) (E! y ps)) (cbveu.1 x sb8eu cbveu.2 cbveu.3 sbie y bieu bitr))
+thm (eu1 ((x y)) ((eu1.1 (-> ph (A. y ph)))) (<-> (E! x ph) (E. x (/\ ph (A. y (-> ([/] (cv y) x ph) (= (cv x) (cv y))))))) (y x ph hbs1 y euf eu1.1 x sb8eu x y equcom y cv x ph [/] imbi2i y bial eu1.1 x sb6rf anbi12i ph y y cv x ph [/] x cv y cv = -> A. ancom y y cv x ph [/] y cv x cv = albi 3bitr4 x biex 3bitr4))
+thm (mo ((x y) (x z) (y z) (ph z)) ((mo.1 (-> ph (A. y ph)))) (<-> (E. y (A. x (-> ph (= (cv x) (cv y))))) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))))) (mo.1 x cv z cv = y ax-17 hbim x hbal x ph x cv y cv = -> A. z ax-17 z y x equequ2 ph imbi2d x bialdv cbvex mo.1 x cv z cv = y ax-17 hbim y x ph hbs1 y cv z cv = x ax-17 hbim x y ph sbequ2 x y z ax-8 imim12d cbv3 ancli mo.1 x cv z cv = y ax-17 hbim y x ph hbs1 y cv z cv = x ax-17 hbim aaan sylibr ph x cv z cv = y cv x ph [/] y cv z cv = prth x z y equtr2 syl6 x y 19.20i2 syl z 19.23aiv sylbir x y cv x ph [/] ph x cv y cv = -> 19.20 y 19.20i a7s y x y cv x ph [/] A. x ph x cv y cv = -> A. 19.22 syl mo.1 x hbsb3 y 19.22i syl5com ph y cv x ph [/] x cv y cv = impexp ph y cv x ph [/] x cv y cv = bi2.04 bitr x y bi2al syl5ib y y cv x ph [/] alnex mo.1 x hbsb3 hbne mo.1 hbne x y ph sbequ1 equcoms con3d cbv3 ph x cv y cv = pm2.21 x 19.20i x ph x cv y cv = -> A. y 19.8a 3syl sylbir x y ph y cv x ph [/] /\ x cv y cv = -> A. A. a1d pm2.61i impbi))
+thm (euex ((x y) (ph y)) () (-> (E! x ph) (E. x ph)) (ph y ax-17 x eu1 x ph y y cv x ph [/] x cv y cv = -> A. 19.40 sylbi pm3.26d))
+thm (eumo0 ((x y)) ((eumo0.1 (-> ph (A. y ph)))) (-> (E! x ph) (E. y (A. x (-> ph (= (cv x) (cv y)))))) (eumo0.1 x euf ph x cv y cv = bi1 x 19.20i y 19.22i sylbi))
+thm (eu2 ((x y)) ((eu2.1 (-> ph (A. y ph)))) (<-> (E! x ph) (/\ (E. x ph) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))))) (x ph euex eu2.1 x eumo0 eu2.1 x mo sylib jca x ph y ph y cv x ph [/] /\ x cv y cv = -> A. 19.29r ph y cv x ph [/] x cv y cv = impexp y bial eu2.1 y cv x ph [/] x cv y cv = -> 19.21 bitr ph anbi2i ph y y cv x ph [/] x cv y cv = -> A. abai bitr4 x biex sylib eu2.1 x eu1 sylibr impbi))
+thm (eu3 ((x y)) ((eu3.1 (-> ph (A. y ph)))) (<-> (E! x ph) (/\ (E. x ph) (E. y (A. x (-> ph (= (cv x) (cv y))))))) (eu3.1 x eu2 eu3.1 x mo x ph E. anbi2i bitr4))
+thm (euor () ((euor.1 (-> ph (A. x ph)))) (-> (/\ (-. ph) (E! x ps)) (E! x (\/ ph ps))) (euor.1 hbne ph ps biorf bieud biimpa))
+thm (euorv ((ph x)) () (-> (/\ (-. ph) (E! x ps)) (E! x (\/ ph ps))) (ph x ax-17 ps euor))
+thm (mo2 ((x y)) ((mo2.1 (-> ph (A. y ph)))) (<-> (E* x ph) (E. y (A. x (-> ph (= (cv x) (cv y)))))) (x ph df-mo x ph alnex ph x cv y cv = pm2.21 x 19.20i x ph x cv y cv = -> A. y 19.8a syl sylbir mo2.1 x eumo0 ja mo2.1 x eu3 biimpr expcom impbi bitr))
+thm (mo3 ((x y)) ((mo3.1 (-> ph (A. y ph)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y)))))) (mo3.1 x mo2 mo3.1 x mo bitr))
+thm (mo4f ((x y) (ph y)) ((mo4f.1 (-> ps (A. x ps))) (mo4f.2 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y)))))) (ph y ax-17 x mo3 mo4f.1 mo4f.2 sbie ph anbi2i x cv y cv = imbi1i x y bi2al bitr))
+thm (mo4 ((x y) (ph y) (ps x)) ((mo4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y)))))) (ps x ax-17 mo4.1 mo4f))
+thm (bimod () ((bimod.1 (-> ph (A. x ph))) (bimod.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E* x ps) (E* x ch))) (bimod.1 bimod.2 biexd bimod.1 bimod.2 bieud imbi12d x ps df-mo x ch df-mo 3bitr4g))
+thm (bimo () ((bimo.1 (<-> ps ch))) (<-> (E* x ps) (E* x ch)) (x equid x equid x hbth bimo.1 x cv x cv = a1i bimod ax-mp))
+thm (hbmo1 () () (-> (E* x ph) (A. x (E* x ph))) (x ph hbe1 x ph hbeu1 hbim x ph df-mo x ph df-mo x bial 3imtr4))
+thm (hbmo () ((hbmo.1 (-> ph (A. x ph)))) (-> (E* y ph) (A. x (E* y ph))) (hbmo.1 y hbex hbmo.1 y hbeu hbim y ph df-mo y ph df-mo x bial 3imtr4))
+thm (cbvmo ((x y)) ((cbvmo.1 (-> ph (A. y ph))) (cbvmo.2 (-> ps (A. x ps))) (cbvmo.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E* x ph) (E* y ps)) (cbvmo.1 cbvmo.2 cbvmo.3 cbvex cbvmo.1 cbvmo.2 cbvmo.3 cbveu imbi12i x ph df-mo y ps df-mo 3bitr4))
+thm (eu5 ((x y) (ph y)) () (<-> (E! x ph) (/\ (E. x ph) (E* x ph))) (ph y ax-17 x eu3 ph y ax-17 x mo2 x ph E. anbi2i bitr4))
+thm (eu4 ((x y) (ph y) (ps x)) ((eu4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E! x ph) (/\ (E. x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y))))))) (x ph eu5 eu4.1 mo4 x ph E. anbi2i bitr))
+thm (eumo () () (-> (E! x ph) (E* x ph)) (x ph eu5 pm3.27bd))
+thm (eumoi () ((eumoi.1 (E! x ph))) (E* x ph) (eumoi.1 x ph eumo ax-mp))
+thm (exmoeu () () (<-> (E. x ph) (-> (E* x ph) (E! x ph))) (x ph df-mo biimp com12 x ph df-mo biimpr x ph euex imim12i x ph E. x ph E! peirce syl impbi))
+thm (exmoeu2 () () (-> (E. x ph) (<-> (E* x ph) (E! x ph))) (x ph eu5 baibr))
+thm (moabs () () (<-> (E* x ph) (-> (E. x ph) (E* x ph))) (x ph E. x ph E! pm5.4 x ph df-mo x ph E. imbi2i x ph df-mo 3bitr4r))
+thm (exmo () () (\/ (E. x ph) (E* x ph)) (x ph E. x ph E! pm2.21 x ph df-mo sylibr orri))
+thm (immo ((x y) (ph y) (ps y)) () (-> (A. x (-> ph ps)) (-> (E* x ps) (E* x ph))) (ph ps x cv y cv = imim1 x 19.20ii y 19.22dv ps y ax-17 x mo2 ph y ax-17 x mo2 3imtr4g))
+thm (moimv ((x y) (ph x) (ph y) (ps y)) () (-> (E* x (-> ph ps)) (-> ph (E* x ps))) (ps ph ax-1 ph a1i x cv y cv = imim1d x 19.20dv y 19.22dv ph ps -> y ax-17 x mo2 ps y ax-17 x mo2 3imtr4g com12))
+thm (euimmo () () (-> (A. x (-> ph ps)) (-> (E! x ps) (E* x ph))) (x ph ps immo x ps eumo syl5))
+thm (euim () () (-> (/\ (E. x ph) (A. x (-> ph ps))) (-> (E! x ps) (E! x ph))) (x ph ps euimmo x ph E. adantl x ph exmoeu2 x ph ps -> A. adantr sylibd))
+thm (moan () () (-> (E* x ph) (E* x (/\ ps ph))) (x ps ph /\ ph immo ps ph pm3.27 mpg))
+thm (moani () ((moani.1 (E* x ph))) (E* x (/\ ps ph)) (moani.1 x ph ps moan ax-mp))
+thm (moor () () (-> (E* x (\/ ph ps)) (E* x ph)) (x ph ph ps \/ immo ph ps orc mpg))
+thm (mooran1 () () (-> (\/ (E* x ph) (E* x ps)) (E* x (/\ ph ps))) (x ph ps moan ps ph ancom x bimo sylib x ps ph moan jaoi))
+thm (mooran2 () () (-> (E* x (\/ ph ps)) (/\ (E* x ph) (E* x ps))) (x ph ps moor ph ps orcom x bimo x ps ph moor sylbi jca))
+thm (moanim ((x y) (ph y) (ps y)) ((moanim.1 (-> ph (A. x ph)))) (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))) (ph ps x cv y cv = impexp x bial moanim.1 ps x cv y cv = -> 19.21 bitr y biex ph ps /\ y ax-17 x mo2 ps y ax-17 x mo2 ph imbi2i y ph x ps x cv y cv = -> A. 19.37v bitr4 3bitr4))
+thm (euan () ((moanim.1 (-> ph (A. x ph)))) (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))) (moanim.1 ps 19.42 moanim.1 ps moanim ph anbi2i ph x ps E* abai bitr4 anbi12i x ph ps /\ E. ph x ph ps /\ E* anass ph ph x ps E. x ps E* an4 3bitr4 x ph ps /\ eu5 ph ps anabs1 x biex moanim.1 ph ps /\ 19.41 bitr3 x ph ps /\ E* anbi1i bitr ph pm4.24 x ps eu5 anbi12i 3bitr4))
+thm (moanimv ((ph x)) () (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))) (ph x ax-17 ps moanim))
+thm (euanv ((ph x)) () (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))) (ph x ax-17 ps euan))
+thm (mopick ((x y) (ph y) (ps y)) () (-> (/\ (E* x ph) (E. x (/\ ph ps))) (-> ph ps)) (ph ps /\ y ax-17 y x ph hbs1 y x ps hbs1 hban x y ph sbequ12 x y ps sbequ12 anbi12d cbvex x y ps sbequ2 ph y cv x ph [/] /\ imim2i exp3a com4t imp ph y ax-17 x mo3 y ph y cv x ph [/] /\ x cv y cv = -> ax-4 x a4s sylbi syl5 y 19.23aiv sylbi impcom))
+thm (eupick () () (-> (/\ (E! x ph) (E. x (/\ ph ps))) (-> ph ps)) (x ph ps mopick x ph eumo sylan))
+thm (eupickb () () (-> (/\/\ (E! x ph) (E! x ps) (E. x (/\ ph ps))) (<-> ph ps)) (x ph ps eupick x ps E! 3adant2 x ph E! x ps E! x ph ps /\ E. 3simpc ph ps pm3.22 x 19.22i x ps E! anim2i x ps ph eupick 3syl impbid))
+thm (mopick2 () () (-> (/\/\ (E* x ph) (E. x (/\ ph ps)) (E. x (/\ ph ch))) (E. x (/\/\ ph ps ch))) (ph ps pm3.26 x 19.22i x ph E* x ph ch /\ E. 3ad2ant2 x ph hbmo1 x ph ps /\ hbe1 x ph ch /\ hbe1 hb3an x ph ps mopick x ph ch mopick anim12i x ph E* x ph ps /\ E. x ph ch /\ E. 3anass x ph E* x ph ps /\ E. x ph ch /\ E. anandi bitr ph ps ch jcab 3imtr4 ancld ph ps ch 3anass syl6ibr 19.22d mpd))
+thm (moexex () ((moexex.1 (-> ph (A. y ph)))) (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))) (x ph hbmo1 x y ps E* hba1 x ph ps /\ hbe1 y hbmo hbim hbim moexex.1 moexex.1 x hbmo x ph ps mopick exp com3r 19.21ad y x ph ps /\ E. ps immo x a4sd syl6 19.23ai moexex.1 x hbex ph ps pm3.26 x 19.22i 19.23ai con3i y x ph ps /\ E. exmo ori syl x y ps E* A. a1d x ph E* a1d pm2.61i imp))
+thm (moexexv ((ph y)) () (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))) (ph y ax-17 x ps moexex))
+thm (2moex () () (-> (E* x (E. y ph)) (A. y (E* x ph))) (y ph hbe1 x hbmo x ph y ph E. immo ph y 19.8a mpg 19.21ai))
+thm (2euex () () (-> (E! x (E. y ph)) (E. y (E! x ph))) (x y ph E. eu5 y ph hbe1 x hbmo x ph E. 19.41 biimpr x y ph excom sylanb x y ph 2moex 19.21bi x ph E. anim2i x ph eu5 sylibr y 19.22i syl sylbi))
+thm (2eumo () () (-> (E! x (E* y ph)) (E* x (E! y ph))) (x y ph E! y ph E* euimmo y ph eumo mpg))
+thm (2eu2ex () () (-> (E! x (E! y ph)) (E. x (E. y ph))) (x y ph E! euex y ph euex x 19.22i syl))
+thm (2moswap () () (-> (A. x (E* y ph)) (-> (E* x (E. y ph)) (E* y (E. x ph)))) (y ph hbe1 x ph moexex expcom ph y 19.8a pm4.71ri x biex y bimo syl6ibr))
+thm (2euswap () () (-> (A. x (E* y ph)) (-> (E! x (E. y ph)) (E! y (E. x ph)))) (x y ph excomim x y ph E* A. a1i x y ph 2moswap anim12d x y ph E. eu5 y x ph E. eu5 3imtr4g))
+thm (2exeu () () (-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y ph))) (x ph hbe1 y hbmo y ph E. 19.41 y ph x ph E. immo ph x 19.8a mpg y ph E. anim2i x 19.22i sylbir y x ph excom sylanb x y ph E. y ph E* /\ y ph E. immo y ph E. y ph E* pm3.26 mpg x y ph E. E. adantl anim12i ancoms x y ph E. eu5 y x ph E. eu5 anbi12i x y ph E! eu5 y ph eu5 x biex y ph eu5 x bimo anbi12i bitr 3imtr4))
+thm (2mo ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ([/] (cv z) x ([/] (cv w) y ph))) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (v z x equequ2 u w y equequ2 bi2anan9 ph imbi2d x y bi2aldv cbvex2v ph x cv v cv = y cv u cv = /\ -> z ax-17 ph x cv v cv = y cv u cv = /\ -> w ax-17 z x w cv y ph [/] hbs1 z cv v cv = w cv u cv = /\ x ax-17 hbim w y ph hbs1 z x hbsb z cv v cv = w cv u cv = /\ y ax-17 hbim y w ph sbequ12 x z w cv y ph [/] sbequ12 sylan9bbr x z v equequ1 y w u equequ1 bi2anan9 imbi12d cbval2 biimp ancli y z w ph x cv v cv = y cv u cv = /\ -> z cv x w cv y ph [/] [/] z cv v cv = w cv u cv = /\ -> /\ A. alcom ph x cv v cv = y cv u cv = /\ -> w ax-17 w y ph hbs1 z x hbsb z cv v cv = w cv u cv = /\ y ax-17 hbim aaan z bial bitr x bial y ph x cv v cv = y cv u cv = /\ -> A. z ax-17 z x w cv y ph [/] hbs1 z cv v cv = w cv u cv = /\ x ax-17 hbim w hbal aaan bitr sylibr ph x cv v cv = y cv u cv = /\ z cv x w cv y ph [/] [/] z cv v cv = w cv u cv = /\ prth x v z equtr2 y u w equtr2 anim12i an4s syl6 z w 19.20i2 x y 19.20i2 syl v u 19.23aivv sylbir x y z w z cv x w cv y ph [/] [/] ph x cv z cv = y cv w cv = /\ -> -> alrot4 y z cv x w cv y ph [/] [/] ph x cv z cv = y cv w cv = /\ -> 19.20 x 19.20ii z w 19.20i2 sylbi w x y z cv x w cv y ph [/] [/] A. A. x y ph x cv z cv = y cv w cv = /\ -> A. A. 19.22 z 19.20i z w x y z cv x w cv y ph [/] [/] A. A. E. w x y ph x cv z cv = y cv w cv = /\ -> A. A. E. 19.22 3syl z x w cv y ph [/] hbs1 w y ph hbs1 z x hbsb 19.21ai z w 19.22i2 syl5com ph z cv x w cv y ph [/] [/] x cv z cv = y cv w cv = /\ impexp ph z cv x w cv y ph [/] [/] x cv z cv = y cv w cv = /\ bi2.04 bitr z w bi2al x y bi2al syl5ib w z cv x w cv y ph [/] [/] alnex z bial z w z cv x w cv y ph [/] [/] E. alnex bitr ph -. z ax-17 ph -. w ax-17 z x w cv y ph [/] hbs1 hbne w y ph hbs1 z x hbsb hbne y w ph sbequ12 x z w cv y ph [/] sbequ12 sylan9bbr negbid cbval2 biimpr ph x cv z cv = y cv w cv = /\ pm2.21 x y 19.20i2 w x y ph x cv z cv = y cv w cv = /\ -> A. A. E. z 19.8a 19.23bi 3syl sylbir x y z w ph z cv x w cv y ph [/] [/] /\ x cv z cv = y cv w cv = /\ -> A. A. A. A. a1d pm2.61i impbi))
+thm (2mos ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (x z) (w x) (y z) (w y)) ((2mos.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ps) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (z w x y ph 2mo ps x ax-17 x z y ax17eq w ph sb19.21 x cv z cv = ps -> y ax-17 2mos.1 expcom pm5.74d sbie bitr3 pm5.74ri sbie ph anbi2i x cv z cv = y cv w cv = /\ imbi1i z w bi2al x y bi2al bitr))
+thm (2eu1 () () (-> (A. x (E* y ph)) (<-> (E! x (E! y ph)) (/\ (E! x (E. y ph)) (E! y (E. x ph))))) (x y ph E! eu5 y ph eu5 x biex y ph eu5 x bimo anbi12i bitr pm3.27bd x x y ph E* A. y ph E. /\ y ph E. y ph E* /\ immo x y ph E* ax-4 y ph E. anim2i ancoms mpg x y ph E* hba1 y ph E. moanim sylib ancrd x y ph 2moswap com12 imdistani syl6 syl x y ph 2eu2ex x y ph 2eu2ex x y ph excom sylib jca jctild x y ph E. eu5 y x ph E. eu5 anbi12i x y ph E. E. x y ph E. E* y x ph E. E. y x ph E. E* an4 bitr syl6ibr com12 x y ph 2exeu x y ph E* A. a1i impbid))
+thm (2eu2 () () (-> (E! y (E. x ph)) (<-> (E! x (E! y ph)) (E! x (E. y ph)))) (y x ph E. eumo y x ph 2moex x y ph 2eu1 x y ph E. E! y x ph E. E! pm3.26 syl6bi 3syl x y ph 2exeu expcom impbid))
+thm (2eu3 () () (-> (A. x (A. y (\/ (E* x ph) (E* y ph)))) (<-> (/\ (E! x (E! y ph)) (E! y (E! x ph))) (/\ (E! x (E. y ph)) (E! y (E. x ph))))) (y ph hbmo1 x ph E* 19.31 x bial x ph hbmo1 y hbal y ph E* 19.32 bitr y x ph 2eu1 biimpd y x ph E. E! x y ph E. E! ancom syl6ib x y ph E! E! adantld x y ph 2eu1 biimpd y x ph E! E! adantrd jaoi x y ph 2exeu y x ph 2exeu ancoms jca y x ph E* A. x y ph E* A. \/ a1i impbid sylbi))
+thm (2eu4 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) (y ph E. z ax-17 x eu3 x ph E. w ax-17 y eu3 anbi12i x y ph E. E. z x y ph E. x cv z cv = -> A. E. y x ph E. E. w y x ph E. y cv w cv = -> A. E. an4 y x ph excom x y ph E. E. anbi2i x y ph E. E. anidm bitr x y ph y cv w cv = -> A. hba1 19.3r x y ph x cv z cv = -> A. A. anbi2i ph x cv z cv = y cv w cv = jcab y bial y ph x cv z cv = -> ph y cv w cv = -> 19.26 bitr x bial x y ph x cv z cv = -> A. y ph y cv w cv = -> A. 19.26 bitr x y ph x cv z cv = -> A. x y ph y cv w cv = -> A. A. 19.26 3bitr4 y y ph x cv z cv = -> A. x ph y cv w cv = -> A. 19.26 y y ph x cv z cv = -> A. ax-4 y ph x cv z cv = -> hba1 impbi y x ph y cv w cv = -> alcom anbi12i bitr x bial bitr4 y ph x cv z cv = 19.23v x ph y cv w cv = 19.23v anbi12i x y bi2al y ph hbe1 x cv z cv = y ax-17 hbim x ph hbe1 y cv w cv = x ax-17 hbim aaan 3bitr z w bi2ex z w x y ph E. x cv z cv = -> A. y x ph E. y cv w cv = -> A. eeanv bitr2 anbi12i 3bitr))
+thm (2eu5 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E! y ph)) (A. x (E* y ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) (y x ph E. eumo x y ph E. E! adantl y x ph 2moex syl pm4.71i x y ph 2eu1 pm5.32ri bitr4 x y ph z w 2eu4 bitr3))
+thm (2eu6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E. z (E. w (A. x (A. y (<-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))) (x y ph z w 2eu4 z w z cv x w cv y ph [/] [/] v u z cv x w cv y ph [/] [/] v cv z u cv w z cv x w cv y ph [/] [/] [/] [/] /\ z cv v cv = w cv u cv = /\ -> A. A. 19.29r2 ph z ax-17 ph w ax-17 z x w cv y ph [/] hbs1 w y ph hbs1 z x hbsb y w ph sbequ12 x z w cv y ph [/] sbequ12 sylan9bbr cbvex2 z v x equequ2 w u y equequ2 bi2anan9 ph imbi2d x y bi2aldv cbvex2v ph x cv v cv = y cv u cv = /\ -> z ax-17 ph x cv v cv = y cv u cv = /\ -> w ax-17 z x w cv y ph [/] hbs1 z cv v cv = w cv u cv = /\ x ax-17 hbim w y ph hbs1 z x hbsb z cv v cv = w cv u cv = /\ y ax-17 hbim y w ph sbequ12 x z w cv y ph [/] sbequ12 sylan9bbr x z v equequ1 y w u equequ1 bi2anan9 imbi12d cbval2 v u bi2ex v u z w z cv x w cv y ph [/] [/] 2mo 3bitr anbi12i x y ph x cv z cv = y cv w cv = /\ 2albi x y ph x cv z cv = y cv w cv = /\ -> A. A. x y x cv z cv = y cv w cv = /\ ph -> A. A. ancom bitr z w bi2ex z x equcom w y equcom anbi12i z cv x w cv y ph [/] [/] ph /\ imbi2i z cv x w cv y ph [/] [/] ph x cv z cv = y cv w cv = /\ impexp bitr x y bi2al z cv x w cv y ph [/] [/] ph /\ z cv x cv = w cv y cv = /\ -> v ax-17 z cv x w cv y ph [/] [/] ph /\ z cv x cv = w cv y cv = /\ -> u ax-17 z x w cv y ph [/] hbs1 z x w cv y ph [/] hbs1 u w hbsb v z hbsb hban z cv v cv = w cv u cv = /\ x ax-17 hbim w y ph hbs1 z x hbsb w y ph hbs1 z x hbsb u w hbsb v z hbsb hban z cv v cv = w cv u cv = /\ y ax-17 hbim y u ph sbequ12 x v u cv y ph [/] sbequ12 sylan9bbr u cv w w cv y ph [/] [/] z ax-17 v x sbco2 z x u w w cv y ph [/] sbcom2 v z bisb ph w ax-17 u y sbco2 v x bisb 3bitr3r syl6bb z cv x w cv y ph [/] [/] anbi2d x v z equequ2 y u w equequ2 bi2anan9 imbi12d cbval2 z x w cv y ph [/] hbs1 w y ph hbs1 z x hbsb ph x cv z cv = y cv w cv = /\ -> 19.21-2 3bitr3 z cv x w cv y ph [/] [/] anbi2i z cv x w cv y ph [/] [/] x y ph x cv z cv = y cv w cv = /\ -> A. A. abai bitr4 z x w y ph 2sb6 x y ph x cv z cv = y cv w cv = /\ -> A. A. anbi1i bitr z w bi2ex bitr4 3imtr4 ph x cv z cv = y cv w cv = /\ bi2 x y 19.20i2 z w 19.22i2 x y ph z w 2exsb sylibr ph x cv z cv = y cv w cv = /\ bi1 x y 19.20i2 z w 19.22i2 jca impbi bitr))
+thm (2eu7 () () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y (/\ (E. x ph) (E. y ph))))) (x ph hbe1 y hbeu y ph E. euan x ph E. y ph E. ancom y bieu y ph hbe1 x ph E. euan y ph E. y x ph E. E! ancom 3bitr x bieu x y ph E. E! y x ph E. E! ancom 3bitr4r))
+thm (2eu8 () () (<-> (E! x (E! y (/\ (E. x ph) (E. y ph)))) (E! x (E! y (/\ (E! x ph) (E. y ph))))) (x y ph 2eu2 pm5.32i x ph hbeu1 y hbeu y ph E. euan x ph E! y ph E. ancom y bieu y ph hbe1 x ph E! euan y ph E. y x ph E! E! ancom 3bitr x bieu x y ph E. E! y x ph E! E! ancom 3bitr4r x y ph 2eu7 3bitr3r))
+thm (exists1 ((x y)) () (<-> (E! x (= (cv x) (cv x))) (A. x (= (cv x) (cv y)))) (x x cv x cv = y df-eu x equid x cv y cv = tbt x cv y cv = x cv x cv = bicom bitr x bial y biex bicomi x y y hbae 19.9r bicomi bitr bitr))
+thm (exists2 ((x y)) () (-> (/\ (E. x ph) (E. x (-. ph))) (-. (E! x (= (cv x) (cv x))))) (x y exists1 ph pm3.24 x y ph ax-16 a5i x ph 19.9t syl x y ph -. ax-16 a5i x ph -. 19.9t syl anim12d mtoi sylbi con2i))
+var (set t)
+thm (axun ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (x y w z ax-un w x y elequ2 w x z elequ1 anbi12d cbvexv y cv x cv e. imbi1i y bial x biex mpbi))
+thm (axpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (A. y (-> (A. x (-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (x y w z ax-pow w x y elequ1 w x z elequ1 imbi12d cbvalv y cv x cv e. imbi1i y bial x biex mpbi))
+thm (axreg ((x y) (x z) (y z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (x y z ax-reg 19.23bi))
+thm (axinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (x y w z ax-inf w y x elequ1 w y z elequ1 z cv x cv e. anbi1d z biexdv imbi12d cbvalv y cv x cv e. anbi2i x biex mpbi))
+thm (axac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. x (A. y (A. z (-> (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x y z w v u t ax-ac v w u equequ2 t u cv z cv e. z cv t cv e. /\ u cv t cv e. t cv x cv e. /\ /\ E. bibi2d t w z elequ2 u cv z cv e. anbi2d t w u elequ2 t w x elequ1 anbi12d anbi12d cbvexv u cv w cv = bibi1i syl6bb u bialdv u y z elequ1 z cv w cv e. anbi1d u y w elequ1 w cv x cv e. anbi1d anbi12d w biexdv u y w equequ1 bibi12d cbvalv syl6bb cbvexv y cv z cv e. z cv w cv e. /\ imbi2i y z bi2al x biex mpbi))
+thm (axext ((x y) (x z) (y z)) () (E. z (-> (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y))) (= (cv x) (cv y)))) (z x y ax-ext z z cv x cv e. z cv y cv e. <-> x cv y cv = 19.36v mpbir))
+thm (zfext2 ((x z) (w z) (w x) (y z) (w y)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))) (w x a9e z w y ax-ext w x z elequ2 z cv y cv e. bibi1d z bialdv w x y equequ1 imbi12d mpbii w 19.23aiv ax-mp))
+thm (bm1.1 ((x y) (x z) (y z) (ph z)) ((bm1.1.1 (-> ph (A. x ph)))) (-> (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (E! x (A. y (<-> (e. (cv y) (cv x)) ph)))) (y y cv x cv e. ph <-> y cv z cv e. ph <-> 19.26 y cv x cv e. ph y cv z cv e. biantr y 19.20i y x z ax-ext syl sylbir y cv z cv e. x ax-17 bm1.1.1 hbbi y hbal x z y elequ2 ph bibi1d y bialdv sbie sylan2b x z gen2 x y y cv x cv e. ph <-> A. E. jctr y y cv x cv e. ph <-> A. z ax-17 x eu2 sylibr))
+var (class A)
+var (class B)
+var (class C)
+var (class D)
+var (class R)
+var (class S)
+var (class T)
+thm (abid () () (<-> (e. (cv x) ({|} x ph)) ph) (x x ph df-clab x ph sbid bitr))
+thm (hbab1 ((x y)) () (-> (e. (cv y) ({|} x ph)) (A. x (e. (cv y) ({|} x ph)))) (y x ph hbs1 y x ph df-clab y x ph df-clab x bial 3imtr4))
+thm (hbab ((x z)) ((hbab.1 (-> ph (A. x ph)))) (-> (e. (cv z) ({|} y ph)) (A. x (e. (cv z) ({|} y ph)))) (x z z cv y ph [/] ax-16 hbab.1 z y hbsb4 pm2.61i z y ph df-clab z y ph df-clab x bial 3imtr4))
+thm (dfcleq ((A x) (B x) (x y) (x z) (y z)) () (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))) (x y z ax-ext A B df-cleq))
+thm (eqrd ((A x) (B x) (ph x)) ((eqrd.1 (-> ph (<-> (e. (cv x) A) (e. (cv x) B))))) (-> ph (= A B)) (eqrd.1 x 19.21aiv A B x dfcleq sylibr))
+thm (eqri ((A x) (B x)) ((eqri.1 (<-> (e. (cv x) A) (e. (cv x) B)))) (= A B) (A B x dfcleq eqri.1 mpgbir))
+thm (eqid ((A x)) () (= A A) (x cv A e. pm4.2 eqri))
+thm (eqcom ((A x) (B x)) () (<-> (= A B) (= B A)) (x cv A e. x cv B e. bicom x bial A B x dfcleq B A x dfcleq 3bitr4))
+thm (eqcoms () ((eqcoms.1 (-> (= A B) ph))) (-> (= B A) ph) (B A eqcom eqcoms.1 sylbi))
+thm (eqcomi () ((eqcomi.1 (= A B))) (= B A) (eqcomi.1 A B eqcom mpbi))
+thm (eqcomd () ((eqcomd.1 (-> ph (= A B)))) (-> ph (= B A)) (eqcomd.1 A B eqcom sylib))
+thm (eqeq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (= A C) (= B C))) (A B x dfcleq biimp 19.21bi x cv C e. bibi1d x bialdv A C x dfcleq B C x dfcleq 3bitr4g))
+thm (eqeq1i () ((eqeq1i.1 (= A B))) (<-> (= A C) (= B C)) (eqeq1i.1 A B C eqeq1 ax-mp))
+thm (eqeq1d () ((eqeq1d.1 (-> ph (= A B)))) (-> ph (<-> (= A C) (= B C))) (eqeq1d.1 A B C eqeq1 syl))
+thm (eqeq2 () () (-> (= A B) (<-> (= C A) (= C B))) (A B C eqeq1 C A eqcom C B eqcom 3bitr4g))
+thm (eqeq2i () ((eqeq2i.1 (= A B))) (<-> (= C A) (= C B)) (eqeq2i.1 A B C eqeq2 ax-mp))
+thm (eqeq2d () ((eqeq2d.1 (-> ph (= A B)))) (-> ph (<-> (= C A) (= C B))) (eqeq2d.1 A B C eqeq2 syl))
+thm (eqeq12 () () (-> (/\ (= A B) (= C D)) (<-> (= A C) (= B D))) (A B C eqeq1 C D B eqeq2 sylan9bb))
+thm (eqeq12i () ((eqeq12i.1 (= A B)) (eqeq12i.2 (= C D))) (<-> (= A C) (= B D)) (eqeq12i.1 C eqeq1i eqeq12i.2 B eqeq2i bitr))
+thm (eqeq12d () ((eqeq12d.1 (-> ph (= A B))) (eqeq12d.2 (-> ph (= C D)))) (-> ph (<-> (= A C) (= B D))) (eqeq12d.1 C eqeq1d eqeq12d.2 B eqeq2d bitrd))
+thm (eqeqan12d () ((eqeqan12d.1 (-> ph (= A B))) (eqeqan12d.2 (-> ps (= C D)))) (-> (/\ ph ps) (<-> (= A C) (= B D))) (eqeqan12d.1 ps adantr eqeqan12d.2 ph adantl eqeq12d))
+thm (eqeqan12rd () ((eqeqan12rd.1 (-> ph (= A B))) (eqeqan12rd.2 (-> ps (= C D)))) (-> (/\ ps ph) (<-> (= A C) (= B D))) (eqeqan12rd.1 eqeqan12rd.2 eqeqan12d ancoms))
+thm (eqtrt () () (-> (/\ (= A B) (= B C)) (= A C)) (A B C eqeq1 biimpar))
+thm (eqtr2t () () (-> (/\ (= A B) (= A C)) (= B C)) (B A C eqtrt A B eqcom sylanb))
+thm (eqtr3t () () (-> (/\ (= A C) (= B C)) (= A B)) (A C B eqtrt B C eqcom sylan2b))
+thm (eqtr () ((eqtr.1 (= A B)) (eqtr.2 (= B C))) (= A C) (eqtr.1 eqtr.2 A eqeq2i mpbi))
+thm (eqtr2 () ((eqtr2.1 (= A B)) (eqtr2.2 (= B C))) (= C A) (eqtr2.1 eqtr2.2 eqtr eqcomi))
+thm (eqtr3 () ((eqtr3.1 (= A B)) (eqtr3.2 (= A C))) (= B C) (eqtr3.1 eqcomi eqtr3.2 eqtr))
+thm (eqtr4 () ((eqtr4.1 (= A B)) (eqtr4.2 (= C B))) (= A C) (eqtr4.1 eqtr4.2 eqcomi eqtr))
+thm (3eqtr () ((3eqtr.1 (= A B)) (3eqtr.2 (= B C)) (3eqtr.3 (= C D))) (= A D) (3eqtr.1 3eqtr.2 3eqtr.3 eqtr eqtr))
+thm (3eqtr3 () ((3eqtr3.1 (= A B)) (3eqtr3.2 (= A C)) (3eqtr3.3 (= B D))) (= C D) (3eqtr3.2 3eqtr3.1 3eqtr3.3 eqtr eqtr3))
+thm (3eqtr3r () ((3eqtr3.1 (= A B)) (3eqtr3.2 (= A C)) (3eqtr3.3 (= B D))) (= D C) (3eqtr3.3 3eqtr3.1 3eqtr3.2 eqtr3 eqtr3))
+thm (3eqtr4 () ((3eqtr4.1 (= A B)) (3eqtr4.2 (= C A)) (3eqtr4.3 (= D B))) (= C D) (3eqtr4.2 3eqtr4.1 3eqtr4.3 eqtr4 eqtr))
+thm (3eqtr4r () ((3eqtr4.1 (= A B)) (3eqtr4.2 (= C A)) (3eqtr4.3 (= D B))) (= D C) (3eqtr4.2 3eqtr4.1 3eqtr4.3 eqtr4 eqtr2))
+thm (eqtrd () ((eqtrd.1 (-> ph (= A B))) (eqtrd.2 (-> ph (= B C)))) (-> ph (= A C)) (eqtrd.1 eqtrd.2 A eqeq2d mpbid))
+thm (eqtr2d () ((eqtr2d.1 (-> ph (= A B))) (eqtr2d.2 (-> ph (= B C)))) (-> ph (= C A)) (eqtr2d.1 eqtr2d.2 eqtrd eqcomd))
+thm (eqtr3d () ((eqtr3d.1 (-> ph (= A B))) (eqtr3d.2 (-> ph (= A C)))) (-> ph (= B C)) (eqtr3d.1 eqcomd eqtr3d.2 eqtrd))
+thm (eqtr4d () ((eqtr4d.1 (-> ph (= A B))) (eqtr4d.2 (-> ph (= C B)))) (-> ph (= A C)) (eqtr4d.1 eqtr4d.2 eqcomd eqtrd))
+thm (3eqtrd () ((3eqtrd.1 (-> ph (= A B))) (3eqtrd.2 (-> ph (= B C))) (3eqtrd.3 (-> ph (= C D)))) (-> ph (= A D)) (3eqtrd.1 3eqtrd.2 3eqtrd.3 eqtrd eqtrd))
+thm (3eqtr3d () ((3eqtr3d.1 (-> ph (= A B))) (3eqtr3d.2 (-> ph (= A C))) (3eqtr3d.3 (-> ph (= B D)))) (-> ph (= C D)) (3eqtr3d.2 3eqtr3d.1 3eqtr3d.3 eqtrd eqtr3d))
+thm (3eqtr4d () ((3eqtr4d.1 (-> ph (= A B))) (3eqtr4d.2 (-> ph (= C A))) (3eqtr4d.3 (-> ph (= D B)))) (-> ph (= C D)) (3eqtr4d.2 3eqtr4d.1 3eqtr4d.3 eqtr4d eqtrd))
+thm (3eqtr4rd () ((3eqtr4d.1 (-> ph (= A B))) (3eqtr4d.2 (-> ph (= C A))) (3eqtr4d.3 (-> ph (= D B)))) (-> ph (= D C)) (3eqtr4d.3 3eqtr4d.1 eqtr4d 3eqtr4d.2 eqtr4d))
+thm (syl5eq () ((syl5eq.1 (-> ph (= A B))) (syl5eq.2 (= C A))) (-> ph (= C B)) (syl5eq.2 ph a1i syl5eq.1 eqtrd))
+thm (syl5req () ((syl5req.1 (-> ph (= A B))) (syl5req.2 (= C A))) (-> ph (= B C)) (syl5req.1 syl5req.2 syl5eq eqcomd))
+thm (syl5eqr () ((syl5eqr.1 (-> ph (= A B))) (syl5eqr.2 (= A C))) (-> ph (= C B)) (syl5eqr.1 syl5eqr.2 eqcomi syl5eq))
+thm (syl5reqr () ((syl5reqr.1 (-> ph (= A B))) (syl5reqr.2 (= A C))) (-> ph (= B C)) (syl5reqr.1 syl5reqr.2 eqcomi syl5req))
+thm (syl6eq () ((syl6eq.1 (-> ph (= A B))) (syl6eq.2 (= B C))) (-> ph (= A C)) (syl6eq.1 syl6eq.2 ph a1i eqtrd))
+thm (syl6req () ((syl6req.1 (-> ph (= A B))) (syl6req.2 (= B C))) (-> ph (= C A)) (syl6req.1 syl6req.2 syl6eq eqcomd))
+thm (syl6eqr () ((syl6eqr.1 (-> ph (= A B))) (syl6eqr.2 (= C B))) (-> ph (= A C)) (syl6eqr.1 syl6eqr.2 eqcomi syl6eq))
+thm (syl6reqr () ((syl6reqr.1 (-> ph (= A B))) (syl6reqr.2 (= C B))) (-> ph (= C A)) (syl6reqr.1 syl6reqr.2 eqcomi syl6req))
+thm (sylan9eq () ((sylan9eq.1 (-> ph (= A B))) (sylan9eq.2 (-> ps (= B C)))) (-> (/\ ph ps) (= A C)) (sylan9eq.1 ps adantr sylan9eq.2 ph adantl eqtrd))
+thm (sylan9eqr () ((sylan9eqr.1 (-> ph (= A B))) (sylan9eqr.2 (-> ps (= B C)))) (-> (/\ ps ph) (= A C)) (sylan9eqr.1 sylan9eqr.2 sylan9eq ancoms))
+thm (3eqtr3g () ((3eqtr3g.1 (-> ph (= A B))) (3eqtr3g.2 (= A C)) (3eqtr3g.3 (= B D))) (-> ph (= C D)) (3eqtr3g.1 3eqtr3g.2 syl5eqr 3eqtr3g.3 syl6eq))
+thm (3eqtr4g () ((3eqtr4g.1 (-> ph (= A B))) (3eqtr4g.2 (= C A)) (3eqtr4g.3 (= D B))) (-> ph (= C D)) (3eqtr4g.1 3eqtr4g.2 syl5eq 3eqtr4g.3 syl6eqr))
+var (class F)
+var (class G)
+thm (eq2tr () ((eq2tr.1 (-> (= A C) (= D F))) (eq2tr.2 (-> (= B D) (= C G)))) (<-> (/\ (= A C) (= B F)) (/\ (= B D) (= A G))) (A C = B D = ancom eq2tr.1 B eqeq2d pm5.32i eq2tr.2 A eqeq2d pm5.32i 3bitr3))
+thm (eleq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. A C) (e. B C))) (A B x cv eqeq2 x cv C e. anbi1d x biexdv A C x df-clel B C x df-clel 3bitr4g))
+thm (eleq2 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. C A) (e. C B))) (A B x dfcleq biimp 19.21bi x cv C = anbi2d x biexdv C A x df-clel C B x df-clel 3bitr4g))
+thm (eleq12 () () (-> (/\ (= A B) (= C D)) (<-> (e. A C) (e. B D))) (A B C eleq1 C D B eleq2 sylan9bb))
+thm (eleq1i () ((eleq1i.1 (= A B))) (<-> (e. A C) (e. B C)) (eleq1i.1 A B C eleq1 ax-mp))
+thm (eleq2i () ((eleq1i.1 (= A B))) (<-> (e. C A) (e. C B)) (eleq1i.1 A B C eleq2 ax-mp))
+thm (eleq12i () ((eleq1i.1 (= A B)) (eleq12i.2 (= C D))) (<-> (e. A C) (e. B D)) (eleq12i.2 A eleq2i eleq1i.1 D eleq1i bitr))
+thm (eleq1d () ((eleq1d.1 (-> ph (= A B)))) (-> ph (<-> (e. A C) (e. B C))) (eleq1d.1 A B C eleq1 syl))
+thm (eleq2d () ((eleq1d.1 (-> ph (= A B)))) (-> ph (<-> (e. C A) (e. C B))) (eleq1d.1 A B C eleq2 syl))
+thm (eleq12d () ((eleq1d.1 (-> ph (= A B))) (eleq12d.2 (-> ph (= C D)))) (-> ph (<-> (e. A C) (e. B D))) (eleq12d.2 A eleq2d eleq1d.1 D eleq1d bitrd))
+thm (eleq1a () () (-> (e. A B) (-> (= C A) (e. C B))) (C A B eleq1 biimprcd))
+thm (eqeltr () ((eqeltr.1 (= A B)) (eqeltr.2 (e. B C))) (e. A C) (eqeltr.2 eqeltr.1 C eleq1i mpbir))
+thm (eqeltrr () ((eqeltrr.1 (= A B)) (eqeltrr.2 (e. A C))) (e. B C) (eqeltrr.1 eqcomi eqeltrr.2 eqeltr))
+thm (eleqtr () ((eleqtr.1 (e. A B)) (eleqtr.2 (= B C))) (e. A C) (eleqtr.1 eleqtr.2 A eleq2i mpbi))
+thm (eleqtrr () ((eleqtrr.1 (e. A B)) (eleqtrr.2 (= C B))) (e. A C) (eleqtrr.1 eleqtrr.2 eqcomi eleqtr))
+thm (eqeltrd () ((eqeltrd.1 (-> ph (= A B))) (eqeltrd.2 (-> ph (e. B C)))) (-> ph (e. A C)) (eqeltrd.2 eqeltrd.1 C eleq1d mpbird))
+thm (eqeltrrd () ((eqeltrrd.1 (-> ph (= A B))) (eqeltrrd.2 (-> ph (e. A C)))) (-> ph (e. B C)) (eqeltrrd.1 eqcomd eqeltrrd.2 eqeltrd))
+thm (eleqtrd () ((eleqtrd.1 (-> ph (e. A B))) (eleqtrd.2 (-> ph (= B C)))) (-> ph (e. A C)) (eleqtrd.1 eleqtrd.2 A eleq2d mpbid))
+thm (eleqtrrd () ((eleqtrrd.1 (-> ph (e. A B))) (eleqtrrd.2 (-> ph (= C B)))) (-> ph (e. A C)) (eleqtrrd.1 eleqtrrd.2 eqcomd eleqtrd))
+thm (syl6eleqr () ((syl6eleqr.1 (-> ph (e. A B))) (syl6eleqr.2 (= C B))) (-> ph (e. A C)) (syl6eleqr.1 syl6eleqr.2 A eleq2i sylibr))
+thm (cleqf ((A y) (B y) (x y)) ((cleqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cleqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))) (A B y dfcleq x cv A e. x cv B e. <-> y ax-17 cleqf.1 cleqf.2 hbbi x cv y cv A eleq1 x cv y cv B eleq1 bibi12d cbval bitr4))
+thm (nelneq () () (-> (/\ (e. A C) (-. (e. B C))) (-. (= A B))) (A B C eleq1 biimpcd con3d imp))
+thm (nelneq2 () () (-> (/\ (e. A B) (-. (e. A C))) (-. (= B C))) (B C A eleq2 biimpcd con3d imp))
+thm (hblem ((A y) (x y) (x z) (y z)) ((hblem.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv z) A) (A. x (e. (cv z) A))) (y cv z cv A eleq1 y cv z cv A eleq1 x bialdv imbi12d hblem.1 chvarv))
+thm (hbeq ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((hbeq.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbeq.2 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (= A B) (A. x (= A B))) (hbeq.1 w hblem hbeq.2 w hblem hbbi w hbal A B w dfcleq A B w dfcleq x bial 3imtr4))
+thm (hbel ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((hbel.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbel.2 (-> (e. (cv z) B) (A. x (e. (cv z) B))))) (-> (e. A B) (A. x (e. A B))) (y cv w cv e. x ax-17 hbel.1 hbeq hbel.2 w hblem hban w hbex A B w df-clel A B w df-clel x bial 3imtr4))
+thm (hbeleq ((x y) (x z) (y z) (A y) (A z)) ((hbeleq.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (= (cv y) A) (A. x (= (cv y) A))) (z cv y cv e. x ax-17 y cv z cv e. x ax-17 hbeleq.1 hbel hbeq))
+thm (eqab ((A x) (x y) (A y) (ph y)) () (<-> (= A ({|} x ph)) (A. x (<-> (e. (cv x) A) ph))) (y cv A e. x ax-17 y x ph hbab1 cleqf x ph abid x cv A e. bibi2i x bial bitr))
+thm (eqabr ((A x)) () (<-> (= ({|} x ph) A) (A. x (<-> ph (e. (cv x) A)))) (A x ph eqab x ph {|} A eqcom ph x cv A e. bicom x bial 3bitr4))
+thm (eqabi () ((eqabi.1 (= A ({|} x ph)))) (<-> (e. (cv x) A) ph) (eqabi.1 x cv eleq2i x ph abid bitr))
+thm (eqabri () ((eqabri.1 (= ({|} x ph) A))) (<-> ph (e. (cv x) A)) (x ph abid eqabri.1 x cv eleq2i bitr3))
+thm (eqabd () ((eqabd.1 (-> ph (= A ({|} x ps))))) (-> ph (<-> (e. (cv x) A) ps)) (eqabd.1 x cv eleq2d x ps abid syl6bb))
+thm (eq2ab ((ph y) (ps y) (x y)) () (<-> (= ({|} x ph) ({|} x ps)) (A. x (<-> ph ps))) (y x ph hbab1 y x ps hbab1 cleqf x ph abid x ps abid bibi12i x bial bitr))
+thm (biabri ((A x)) ((biabri.1 (<-> (e. (cv x) A) ph))) (= A ({|} x ph)) (A x ph eqab biabri.1 mpgbir))
+thm (biabi () ((biabi.1 (<-> ph ps))) (= ({|} x ph) ({|} x ps)) (x ph ps eq2ab biabi.1 mpgbir))
+thm (biabd () ((biabd.1 (-> ph (A. x ph))) (biabd.2 (-> ph (<-> ps ch)))) (-> ph (= ({|} x ps) ({|} x ch))) (biabd.1 biabd.2 19.21ai x ps ch eq2ab sylibr))
+thm (biabdv ((ph x)) ((biabdv.1 (-> ph (<-> ps ch)))) (-> ph (= ({|} x ps) ({|} x ch))) (ph x ax-17 biabdv.1 biabd))
+thm (biabrdv ((A x) (ph x)) ((biabrdv.1 (-> ph (<-> (e. (cv x) A) ps)))) (-> ph (= A ({|} x ps))) (biabrdv.1 x 19.21aiv A x ps eqab sylibr))
+thm (biabldv ((A x) (ph x)) ((biabldv.1 (-> ph (<-> ps (e. (cv x) A))))) (-> ph (= ({|} x ps) A)) (biabldv.1 x 19.21aiv x ps A eqabr sylibr))
+thm (abid2 ((A x)) () (= ({|} x (e. (cv x) A)) A) (x cv A e. pm4.2 biabri eqcomi))
+thm (clelab ((A x) (x y) (A y) (ph y)) () (<-> (e. A ({|} x ph)) (E. x (/\ (= (cv x) A) ph))) (y x ph df-clab y cv A = anbi2i y biex A x ph {|} y df-clel x cv A = ph /\ y ax-17 y cv A = x ax-17 y x ph hbs1 hban x cv y cv A eqeq1 x y ph sbequ12 anbi12d cbvex 3bitr4))
+thm (sbab ((A z) (x z) (y z)) () (-> (= (cv x) (cv y)) (= A ({|} z ([/] (cv y) x (e. (cv z) A))))) (x y z cv A e. sbequ12 biabrdv))
+thm (sbabel ((v w) (A w) (A v) (w x) (x z) (v x) (v z) (y z) (v y) (ph v)) ((sbabel.1 (-> (e. (cv w) A) (A. x (e. (cv w) A))))) (<-> ([/] (cv y) x (e. ({|} z ph) A)) (e. ({|} z ([/] (cv y) x ph)) A)) (y x v v cv z ph {|} = v cv A e. /\ sbex y x v cv z ph {|} = v cv A e. sban w cv v cv e. x ax-17 sbabel.1 hbel y sbf y cv x v cv z ph {|} = [/] anbi2i bicomi y x z z cv v cv e. ph <-> sbal y x z cv v cv e. ph sbbi z cv v cv e. x ax-17 y sbf y cv x ph [/] bibi1i bicomi bitr4 z bial bitr v cv z ph eqab y x bisb v cv z y cv x ph [/] eqab 3bitr4 v cv A e. anbi1i bitr3 bitr v biex bitr z ph {|} A v df-clel y x bisb z y cv x ph [/] {|} A v df-clel 3bitr4))
+thm (neeq1 () () (-> (= A B) (<-> (=/= A C) (=/= B C))) (A B C eqeq1 negbid A C df-ne B C df-ne 3bitr4g))
+thm (neeq2 () () (-> (= A B) (<-> (=/= C A) (=/= C B))) (A B C eqeq2 negbid C A df-ne C B df-ne 3bitr4g))
+thm (neeq1i () ((neeq1i.1 (= A B))) (<-> (=/= A C) (=/= B C)) (neeq1i.1 A B C neeq1 ax-mp))
+thm (neeq2i () ((neeq1i.1 (= A B))) (<-> (=/= C A) (=/= C B)) (neeq1i.1 A B C neeq2 ax-mp))
+thm (neeq1d () ((neeq1d.1 (-> ph (= A B)))) (-> ph (<-> (=/= A C) (=/= B C))) (neeq1d.1 A B C neeq1 syl))
+thm (neeq2d () ((neeq1d.1 (-> ph (= A B)))) (-> ph (<-> (=/= C A) (=/= C B))) (neeq1d.1 A B C neeq2 syl))
+thm (eqneqi () ((eqneqi.1 (<-> (= A B) (= C D)))) (<-> (=/= A B) (=/= C D)) (eqneqi.1 negbii A B df-ne C D df-ne 3bitr4))
+thm (eqneqd () ((eqneqd.1 (-> ph (<-> (= A B) (= C D))))) (-> ph (<-> (=/= A B) (=/= C D))) (eqneqd.1 negbid A B df-ne C D df-ne 3bitr4g))
+thm (necom () () (<-> (=/= A B) (=/= B A)) (A B eqcom eqneqi))
+thm (neleq1 () () (-> (= A B) (<-> (e/ A C) (e/ B C))) (A B C eleq1 negbid A C df-nel B C df-nel 3bitr4g))
+thm (neleq2 () () (-> (= A B) (<-> (e/ C A) (e/ C B))) (A B C eleq2 negbid C A df-nel C B df-nel 3bitr4g))
+thm (ralnex () () (<-> (A.e. x A (-. ph)) (-. (E.e. x A ph))) (x x cv A e. ph alinexa x A ph -. df-ral x A ph df-rex negbii 3bitr4))
+thm (rexnal () () (<-> (E.e. x A (-. ph)) (-. (A.e. x A ph))) (x x cv A e. ph exanali x A ph -. df-rex x A ph df-ral negbii 3bitr4))
+thm (dfral2 () () (<-> (A.e. x A ph) (-. (E.e. x A (-. ph)))) (x A ph rexnal bicon2i))
+thm (dfrex2 () () (<-> (E.e. x A ph) (-. (A.e. x A (-. ph)))) (x A ph ralnex bicon2i))
+thm (biralda () ((biralda.1 (-> ph (A. x ph))) (biralda.2 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (biralda.1 biralda.2 exp pm5.74d biald x A ps df-ral x A ch df-ral 3bitr4g))
+thm (birexda () ((biralda.1 (-> ph (A. x ph))) (biralda.2 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (biralda.1 biralda.2 exp pm5.32d biexd x A ps df-rex x A ch df-rex 3bitr4g))
+thm (biraldva ((ph x)) ((biraldva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 biraldva.1 biralda))
+thm (birexdva ((ph x)) ((biraldva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 biraldva.1 birexda))
+thm (birald () ((birald.1 (-> ph (A. x ph))) (birald.2 (-> ph (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (birald.1 birald.2 x cv A e. adantr biralda))
+thm (birexd () ((birald.1 (-> ph (A. x ph))) (birald.2 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (birald.1 birald.2 x cv A e. adantr birexda))
+thm (biraldv ((ph x)) ((biraldv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 biraldv.1 A birald))
+thm (birexdv ((ph x)) ((biraldv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 biraldv.1 A birexd))
+thm (biraldv2 ((ph x)) ((biraldv2.1 (-> ph (<-> (-> (e. (cv x) A) ps) (-> (e. (cv x) B) ch))))) (-> ph (<-> (A.e. x A ps) (A.e. x B ch))) (biraldv2.1 x bialdv x A ps df-ral x B ch df-ral 3bitr4g))
+thm (birexdv2 ((ph x)) ((birexdv2.1 (-> ph (<-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) B) ch))))) (-> ph (<-> (E.e. x A ps) (E.e. x B ch))) (birexdv2.1 x biexdv x A ps df-rex x B ch df-rex 3bitr4g))
+thm (biral () ((biral.1 (<-> ph ps))) (<-> (A.e. x A ph) (A.e. x A ps)) (ph pm4.2 ph pm4.2 x hbth biral.1 ph ph <-> a1i A birald ax-mp))
+thm (birex () ((biral.1 (<-> ph ps))) (<-> (E.e. x A ph) (E.e. x A ps)) (ph pm4.2 ph pm4.2 x hbth biral.1 ph ph <-> a1i A birexd ax-mp))
+thm (bi2ral () ((biral.1 (<-> ph ps))) (<-> (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps))) (biral.1 y B biral x A biral))
+thm (bi2rex () ((biral.1 (<-> ph ps))) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))) (biral.1 y B birex x A birex))
+thm (biral2 () ((biral2.1 (<-> (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps)))) (<-> (A.e. x A ph) (A.e. x B ps)) (biral2.1 x bial x A ph df-ral x B ps df-ral 3bitr4))
+thm (birex2 () ((birex2.1 (<-> (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps)))) (<-> (E.e. x A ph) (E.e. x B ps)) (birex2.1 x biex x A ph df-rex x B ps df-rex 3bitr4))
+thm (birala () ((birala.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. x A ps)) (birala.1 pm5.74i x bial x A ph df-ral x A ps df-ral 3bitr4))
+thm (birexa () ((birala.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. x A ps)) (birala.1 pm5.32i birex2))
+thm (bi2rexa ((x y) (A y)) ((bi2rexa.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> ph ps)))) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))) (bi2rexa.1 birexdva birexa))
+thm (r2al ((x y) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A. x (A. y (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (x A y B ph A.e. df-ral y x cv A e. y cv B e. ph -> 19.21v x cv A e. y cv B e. ph impexp y bial y B ph df-ral x cv A e. imbi2i 3bitr4 x bial bitr4))
+thm (bi2ralda ((x y) (A y)) ((bi2ralda.1 (-> ph (A. x ph))) (bi2ralda.2 (-> ph (A. y ph))) (bi2ralda.3 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))) (bi2ralda.1 bi2ralda.2 x cv A e. y ax-17 hban bi2ralda.3 anassrs biralda biralda))
+thm (bi2raldva ((x y) (ph x) (ph y) (A y)) ((bi2raldva.1 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))) (ph x ax-17 ph y ax-17 bi2raldva.1 bi2ralda))
+thm (bi2rexdva ((x y) (ph x) (ph y) (A y)) ((bi2raldva.1 (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch)))) (-> ph (<-> (E.e. x A (E.e. y B ps)) (E.e. x A (E.e. y B ch)))) (bi2raldva.1 anassrs birexdva birexdva))
+thm (ralinexa () () (<-> (A.e. x A (-> ph (-. ps))) (-. (E.e. x A (/\ ph ps)))) (ph ps imnan x A biral x A ph ps /\ ralnex bitr))
+thm (rexanali () () (<-> (E.e. x A (/\ ph (-. ps))) (-. (A.e. x A (-> ph ps)))) (ph ps annim x A birex x A ph ps -> rexnal bitr))
+thm (risset ((A x) (B x)) () (<-> (e. A B) (E.e. x B (= (cv x) A))) (x x cv B e. x cv A = exancom x B x cv A = df-rex A B x df-clel 3bitr4r))
+thm (hbral ((x y)) ((hbral.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbral.2 (-> ph (A. x ph)))) (-> (A.e. y A ph) (A. x (A.e. y A ph))) (hbral.1 hbral.2 hbim y hbal y A ph df-ral y A ph df-ral x bial 3imtr4))
+thm (hbra1 () () (-> (A.e. x A ph) (A. x (A.e. x A ph))) (x x cv A e. ph -> hba1 x A ph df-ral x A ph df-ral x bial 3imtr4))
+thm (hbrex ((x y)) ((hbrex.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbrex.2 (-> ph (A. x ph)))) (-> (E.e. y A ph) (A. x (E.e. y A ph))) (hbrex.1 hbrex.2 hban y hbex y A ph df-rex y A ph df-rex x bial 3imtr4))
+thm (hbre1 () () (-> (E.e. x A ph) (A. x (E.e. x A ph))) (x x cv A e. ph /\ hbe1 x A ph df-rex x A ph df-rex x bial 3imtr4))
+thm (r3al ((x y) (x z) (y z) (A y) (A z) (B z)) () (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A. x (A. y (A. z (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph))))) (x A y z y cv B e. z cv C e. /\ ph -> A. A. df-ral y B z C ph r2al x A biral x cv A e. y cv B e. z cv C e. 3anass ph imbi1i x cv A e. y cv B e. z cv C e. /\ ph impexp bitr z bial z x cv A e. y cv B e. z cv C e. /\ ph -> 19.21v bitr y bial y x cv A e. z y cv B e. z cv C e. /\ ph -> A. 19.21v bitr x bial 3bitr4))
+thm (r2ex ((x y) (A y)) () (<-> (E.e. x A (E.e. y B ph)) (E. x (E. y (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (x A y B ph E.e. df-rex y x cv A e. y cv B e. ph /\ 19.42v x cv A e. y cv B e. ph anass y biex y B ph df-rex x cv A e. anbi2i 3bitr4 x biex bitr4))
+thm (rexex () () (-> (E.e. x A ph) (E. x ph)) (x A ph df-rex x cv A e. ph pm3.27 x 19.22i sylbi))
+thm (ra4 () () (-> (A.e. x A ph) (-> (e. (cv x) A) ph)) (x A ph df-ral x x cv A e. ph -> ax-4 sylbi))
+thm (ra4e () () (-> (/\ (e. (cv x) A) ph) (E.e. x A ph)) (x cv A e. ph /\ x 19.8a x A ph df-rex sylibr))
+thm (ra42 () () (-> (A.e. x A (A.e. y B ph)) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (x A y B ph A.e. ra4 y B ph ra4 syl6 imp3a))
+thm (rspec () ((rspec.1 (A.e. x A ph))) (-> (e. (cv x) A) ph) (rspec.1 x A ph ra4 ax-mp))
+thm (rgen () ((rgen.1 (-> (e. (cv x) A) ph))) (A.e. x A ph) (x A ph df-ral rgen.1 mpgbir))
+thm (rgen2 ((y z) (A y) (A z) (x z)) ((rgen2.1 (-> (/\ (e. (cv x) A) (e. (cv y) A)) ph))) (A.e. x A (A.e. y A ph)) (rgen2.1 exp y ax-gen y x y hbae y cv x cv A eleq1 y a4s y cv A e. ph -> imbi1d biald mpbiri y cv A e. ph pm2.43 y 19.20i y y cv A e. ph -> A. x cv A e. ax-1 3syl z cv A e. y ax-17 z cv x cv A eleq1 dvelim rgen2.1 exp y 19.20i syl6 pm2.61i y A ph df-ral sylibr rgen))
+thm (mprg () ((mprg.1 (-> (A.e. x A ph) ps)) (mprg.2 (-> (e. (cv x) A) ph))) ps (mprg.2 rgen mprg.1 ax-mp))
+thm (mprgbir () ((mprgbir.1 (<-> ph (A.e. x A ps))) (mprgbir.2 (-> (e. (cv x) A) ps))) ph (mprgbir.2 rgen mprgbir.1 mpbir))
+thm (r19.20 () () (-> (A.e. x A (-> ph ps)) (-> (A.e. x A ph) (A.e. x A ps))) (x A ph ps -> df-ral x cv A e. ph ps ax-2 x 19.20ii sylbi x A ph df-ral x A ps df-ral 3imtr4g))
+thm (r19.20i2 () ((r19.20i2.1 (-> (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps)))) (-> (A.e. x A ph) (A.e. x B ps)) (r19.20i2.1 x 19.20i x A ph df-ral x B ps df-ral 3imtr4))
+thm (r19.20i () ((r19.20i.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (A.e. x A ph) (A.e. x A ps)) (r19.20i.1 a2i r19.20i2))
+thm (r19.20si () ((r19.20si.1 (-> ph ps))) (-> (A.e. x A ph) (A.e. x A ps)) (r19.20si.1 x cv A e. a1i r19.20i))
+thm (r19.20da () ((r19.20da.1 (-> ph (A. x ph))) (r19.20da.2 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (r19.20da.1 r19.20da.2 exp a2d 19.20d x A ps df-ral x A ch df-ral 3imtr4g))
+thm (r19.20dva ((ph x)) ((r19.20dva.1 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (ph x ax-17 r19.20dva.1 r19.20da))
+thm (r19.20sdv ((ph x)) ((r19.20sdv.1 (-> ph (-> ps ch)))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))) (r19.20sdv.1 x cv A e. adantr r19.20dva))
+thm (r19.20dv2 ((ph x)) ((r19.20dv2.1 (-> ph (-> (-> (e. (cv x) A) ps) (-> (e. (cv x) B) ch))))) (-> ph (-> (A.e. x A ps) (A.e. x B ch))) (r19.20dv2.1 x 19.20dv x A ps df-ral x B ch df-ral 3imtr4g))
+thm (r19.21ai () ((r19.21ai.1 (-> ph (A. x ph))) (r19.21ai.2 (-> ph (-> (e. (cv x) A) ps)))) (-> ph (A.e. x A ps)) (r19.21ai.1 r19.21ai.2 19.21ai x A ps df-ral sylibr))
+thm (r19.21aiv ((ph x)) ((r19.21aiv.1 (-> ph (-> (e. (cv x) A) ps)))) (-> ph (A.e. x A ps)) (ph x ax-17 r19.21aiv.1 r19.21ai))
+thm (r19.21v ((ph x)) () (<-> (A.e. x A (-> ph ps)) (-> ph (A.e. x A ps))) (x cv A e. ph ps bi2.04 x bial x ph x cv A e. ps -> 19.21v bitr x A ph ps -> df-ral x A ps df-ral ph imbi2i 3bitr4))
+thm (r19.21ad () ((r19.21ad.1 (-> ph (A. x ph))) (r19.21ad.2 (-> ps (A. x ps))) (r19.21ad.3 (-> ph (-> ps (-> (e. (cv x) A) ch))))) (-> ph (-> ps (A.e. x A ch))) (r19.21ad.1 r19.21ad.2 r19.21ad.3 19.21ad x A ch df-ral syl6ibr))
+thm (r19.21adv ((ph x) (ps x)) ((r19.21adv.1 (-> ph (-> ps (-> (e. (cv x) A) ch))))) (-> ph (-> ps (A.e. x A ch))) (ph x ax-17 ps x ax-17 r19.21adv.1 r19.21ad))
+thm (r19.21aivv ((x y) (ph x) (ph y) (A y)) ((r19.21aivv.1 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) ps)))) (-> ph (A.e. x A (A.e. y B ps))) (r19.21aivv.1 exp3a r19.21adv r19.21aiv))
+thm (r19.21advv ((x y) (ph x) (ph y) (ps x) (ps y) (A y)) ((r19.21advv.1 (-> ph (-> ps (-> (/\ (e. (cv x) A) (e. (cv y) B)) ch))))) (-> ph (-> ps (A.e. x A (A.e. y B ch)))) (r19.21advv.1 imp r19.21aivv exp))
+thm (rgen2a ((x y) (A y)) ((rgen2a.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph))) (A.e. x A (A.e. y B ph)) (rgen2a.1 exp r19.21aiv rgen))
+thm (rgen3 ((y z) (A y) (A z) (x z)) ((rgen3.1 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) ph))) (A.e. x A (A.e. y A (A.e. z A ph))) (rgen3.1 3exp imp r19.21aiv rgen2))
+thm (r19.21bi () ((r19.21bi.1 (-> ph (A.e. x A ps)))) (-> (/\ ph (e. (cv x) A)) ps) (r19.21bi.1 x A ps df-ral sylib 19.21bi imp))
+thm (rspec2 () ((rspec2.1 (A.e. x A (A.e. y B ph)))) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph) (rspec2.1 rspec r19.21bi))
+thm (rspec3 () ((rspec3.1 (A.e. x A (A.e. y B (A.e. z C ph))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph) (rspec3.1 rspec2 r19.21bi 3impa))
+thm (r19.21be () ((r19.21be.1 (-> ph (A.e. x A ps)))) (A.e. x A (-> ph ps)) (r19.21be.1 r19.21bi expcom rgen))
+thm (nrex () ((nrex.1 (-> (e. (cv x) A) (-. ps)))) (-. (E.e. x A ps)) (nrex.1 rgen x A ps ralnex mpbi))
+thm (nrexdv ((ph x)) ((nrexdv.1 (-> (/\ ph (e. (cv x) A)) (-. ps)))) (-> ph (-. (E.e. x A ps))) (nrexdv.1 exp r19.21aiv x A ps ralnex sylib))
+thm (r19.22 () () (-> (A.e. x A (-> ph ps)) (-> (E.e. x A ph) (E.e. x A ps))) (x cv A e. ph ps imdistan x bial x x cv A e. ph /\ x cv A e. ps /\ 19.22 sylbi x A ph ps -> df-ral x A ph df-rex x A ps df-rex imbi12i 3imtr4))
+thm (r19.22i () ((r19.22i.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) (E.e. x A ps)) (x A ph ps r19.22 r19.22i.1 mprg))
+thm (r19.22i2 () ((r19.22i2.1 (-> (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps)))) (-> (E.e. x A ph) (E.e. x B ps)) (r19.22i2.1 x 19.22i x A ph df-rex x B ps df-rex 3imtr4))
+thm (r19.22si () ((r19.22si.1 (-> ph ps))) (-> (E.e. x A ph) (E.e. x A ps)) (r19.22si.1 x cv A e. a1i r19.22i))
+thm (r19.22d () ((r19.22d.1 (-> ph (A. x ph))) (r19.22d.2 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22d.1 r19.22d.2 r19.21ai x A ps ch r19.22 syl))
+thm (r19.22dv2 ((ph x)) ((r19.22dv2.1 (-> ph (-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) B) ch))))) (-> ph (-> (E.e. x A ps) (E.e. x B ch))) (r19.22dv2.1 x 19.22dv x A ps df-rex x B ch df-rex 3imtr4g))
+thm (r19.22dv ((ph x)) ((r19.22dv.1 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (ph x ax-17 r19.22dv.1 r19.22d))
+thm (r19.22sdv ((ph x)) ((r19.22sdv.1 (-> ph (-> ps ch)))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22sdv.1 x cv A e. a1d r19.22dv))
+thm (r19.22dva ((ph x)) ((r19.22dva.1 (-> (/\ ph (e. (cv x) A)) (-> ps ch)))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))) (r19.22dva.1 exp r19.22dv))
+thm (r19.12 ((x y) (A y) (B x)) () (-> (E.e. x A (A.e. y B ph)) (A.e. y B (E.e. x A ph))) (x cv A e. y ax-17 y B ph hbra1 hbrex x A y B ph A.e. E.e. y cv B e. ax-1 r19.21ai y B ph ra4 com12 x cv A e. a1d r19.22dv r19.20i syl))
+thm (r19.23v ((ps x)) () (<-> (A.e. x A (-> ph ps)) (-> (E.e. x A ph) ps)) (x cv A e. ph ps impexp x bial x A ph df-rex ps imbi1i x x cv A e. ph /\ ps 19.23v bitr4 x A ph ps -> df-ral 3bitr4r))
+thm (r19.23ai () ((r19.23ai.1 (-> ps (A. x ps))) (r19.23ai.2 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) ps) (x A ph df-rex r19.23ai.1 r19.23ai.2 imp 19.23ai sylbi))
+thm (r19.23aiv ((ps x)) ((r19.23aiv.1 (-> (e. (cv x) A) (-> ph ps)))) (-> (E.e. x A ph) ps) (ps x ax-17 r19.23aiv.1 r19.23ai))
+thm (r19.23ad () ((r19.23ad.1 (-> ph (A. x ph))) (r19.23ad.2 (-> ch (A. x ch))) (r19.23ad.3 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) ch)) (r19.23ad.1 r19.23ad.2 r19.23ad.3 imp3a 19.23ad x A ps df-rex syl5ib))
+thm (r19.23adv ((ph x) (ch x)) ((r19.23adv.1 (-> ph (-> (e. (cv x) A) (-> ps ch))))) (-> ph (-> (E.e. x A ps) ch)) (r19.23adv.1 imp3a x 19.23adv x A ps df-rex syl5ib))
+thm (r19.23aivv ((x y) (ps x) (ps y) (A y)) ((r19.23aivv.1 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (-> ph ps)))) (-> (E.e. x A (E.e. y B ph)) ps) (r19.23aivv.1 exp r19.23adv r19.23aiv))
+thm (r19.23advv ((x y) (ph x) (ph y) (ch x) (ch y) (A y)) ((r19.23advv.1 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) (-> ps ch))))) (-> ph (-> (E.e. x A (E.e. y B ps)) ch)) (r19.23advv.1 exp3a imp r19.23adv exp r19.23adv))
+thm (r19.26 () () (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) (A.e. x A ps))) (x cv A e. ph ps jcab x bial x x cv A e. ph -> x cv A e. ps -> 19.26 bitr x A ph ps /\ df-ral x A ph df-ral x A ps df-ral anbi12i 3bitr4))
+thm (r19.26-2 () () (<-> (A.e. x A (A.e. y B (/\ ph ps))) (/\ (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps)))) (y B ph ps r19.26 x A biral x A y B ph A.e. y B ps A.e. r19.26 bitr))
+thm (r19.26m () () (<-> (A. x (/\ (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps))) (/\ (A.e. x A ph) (A.e. x B ps))) (x x cv A e. ph -> x cv B e. ps -> 19.26 x A ph df-ral x B ps df-ral anbi12i bitr4))
+thm (r19.15 () () (-> (A.e. x A (<-> ph ps)) (<-> (A.e. x A ph) (A.e. x A ps))) (x A ph ps <-> hbra1 x A ph ps <-> A.e. id 19.21ai x A ph ps <-> ra4 imp biralda))
+thm (r19.27av ((ps x)) () (-> (/\ (A.e. x A ph) ps) (A.e. x A (/\ ph ps))) (x cv A e. ph pm2.27 ps anim1d com12 x 19.20i x A ph df-ral ps anbi1i x x cv A e. ph -> ps 19.27v bitr4 x A ph ps /\ df-ral 3imtr4))
+thm (r19.28av ((ph x)) () (-> (/\ ph (A.e. x A ps)) (A.e. x A (/\ ph ps))) (x A ps ph r19.27av ph x A ps A.e. ancom ph ps ancom x A biral 3imtr4))
+thm (r19.29 () () (-> (/\ (A.e. x A ph) (E.e. x A ps)) (E.e. x A (/\ ph ps))) (x x cv A e. ph -> x cv A e. ps /\ 19.29 x cv A e. ph ps anandi x cv A e. ph abai x cv A e. ps /\ anbi1i x cv A e. x cv A e. ph -> ps anandi bitr4 x cv A e. x cv A e. ph -> ps an12 3bitr x biex sylibr x A ph df-ral x A ps df-rex anbi12i x A ph ps /\ df-rex 3imtr4))
+thm (r19.29r () () (-> (/\ (E.e. x A ph) (A.e. x A ps)) (E.e. x A (/\ ph ps))) (x A ps ph r19.29 x A ph E.e. x A ps A.e. ancom ph ps ancom x A birex 3imtr4))
+thm (r19.32v ((ph x)) () (<-> (A.e. x A (\/ ph ps)) (\/ ph (A.e. x A ps))) (x A ph -. ps r19.21v ph ps df-or x A biral ph x A ps A.e. df-or 3bitr4))
+thm (r19.35 () () (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) (E.e. x A ps))) (x A ph ps -. r19.26 ph ps annim x A biral x A ph A.e. x A ps -. A.e. df-an 3bitr3 bicon2i x A ps dfrex2 x A ph A.e. imbi2i x A ph ps -> dfrex2 3bitr4r))
+thm (r19.36av ((ps x)) () (-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps)) (x A ph ps r19.35 x cv A e. ps idd r19.23aiv x A ph A.e. imim2i sylbi))
+thm (r19.37av ((ph x)) () (-> (E.e. x A (-> ph ps)) (-> ph (E.e. x A ps))) (x A ph ps r19.35 ph x cv A e. ax-1 r19.21aiv x A ps E.e. imim1i sylbi))
+thm (r19.40 () () (-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) (E.e. x A ps))) (ph ps pm3.26 x A r19.22si ph ps pm3.27 x A r19.22si jca))
+thm (r19.41v ((ps x)) () (<-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) ps)) (x cv A e. ph ps anass x biex x x cv A e. ph /\ ps 19.41v bitr3 x A ph ps /\ df-rex x A ph df-rex ps anbi1i 3bitr4))
+thm (r19.42v ((ph x)) () (<-> (E.e. x A (/\ ph ps)) (/\ ph (E.e. x A ps))) (x A ps ph r19.41v ph ps ancom x A birex ph x A ps E.e. ancom 3bitr4))
+thm (r19.43 () () (<-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) (E.e. x A ps))) (x cv A e. ph ps andi x biex x x cv A e. ph /\ x cv A e. ps /\ 19.43 bitr x A ph ps \/ df-rex x A ph df-rex x A ps df-rex orbi12i 3bitr4))
+thm (r19.44av ((ps x)) () (-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) ps)) (x A ph ps r19.43 x cv A e. ps idd r19.23aiv x A ph E.e. orim2i sylbi))
+thm (r19.45av ((ph x)) () (-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps))) (x A ph ps r19.43 x cv A e. ph idd r19.23aiv x A ps E.e. orim1i sylbi))
+thm (hbreu1 () () (-> (E!e. x A ph) (A. x (E!e. x A ph))) (x x cv A e. ph /\ hbeu1 x A ph df-reu x A ph df-reu x bial 3imtr4))
+thm (rabid () () (<-> (e. (cv x) ({e.|} x A ph)) (/\ (e. (cv x) A) ph)) (x A ph df-rab eqabi))
+thm (rabid2 ((A x)) () (<-> (= A ({e.|} x A ph)) (A.e. x A ph)) (x cv A e. ph pm4.71 x bial A x x cv A e. ph /\ eqab bitr4 x A ph df-ral x A ph df-rab A eqeq2i 3bitr4r))
+thm (rabswap () () (= ({e.|} x A (e. (cv x) B)) ({e.|} x B (e. (cv x) A))) (x cv A e. x cv B e. ancom x biabi x A x cv B e. df-rab x B x cv A e. df-rab 3eqtr4))
+thm (hbrab1 ((x y)) () (-> (e. (cv y) ({e.|} x A ph)) (A. x (e. (cv y) ({e.|} x A ph)))) (y x x cv A e. ph /\ hbab1 x A ph df-rab y cv eleq2i x A ph df-rab y cv eleq2i x bial 3imtr4))
+thm (hbrab ((x y) (x z) (y z) (A z)) ((hbrab.1 (-> ph (A. x ph))) (hbrab.2 (-> (e. (cv z) A) (A. x (e. (cv z) A))))) (-> (e. (cv z) ({e.|} y A ph)) (A. x (e. (cv z) ({e.|} y A ph)))) (z cv y cv e. x ax-17 hbrab.2 hbel hbrab.1 hban z y hbab y A ph df-rab z cv eleq2i y A ph df-rab z cv eleq2i x bial 3imtr4))
+thm (ralcom ((x y) (B x) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A.e. y B (A.e. x A ph))) (x cv A e. y cv B e. ancom ph imbi1i x y bi2al x y y cv B e. x cv A e. /\ ph -> alcom bitr x A y B ph r2al y B x A ph r2al 3bitr4))
+thm (rexcom ((x y) (B x) (A y)) () (<-> (E.e. x A (E.e. y B ph)) (E.e. y B (E.e. x A ph))) (x cv A e. y cv B e. ancom ph anbi1i x y bi2ex x y y cv B e. x cv A e. /\ ph /\ excom bitr x A y B ph r2ex y B x A ph r2ex 3bitr4))
+thm (ralcom2 ((y z) (A y) (A z) (x z) (A x)) () (-> (A.e. x A (A.e. y A ph)) (A.e. y A (A.e. x A ph))) (x x cv A e. x x cv A e. ph -> A. -> A. id x y x hbae x cv y cv A eleq1 x a4s ph imbi1d dral1 x cv A e. imbi2d biald x cv y cv A eleq1 x a4s x x cv A e. ph -> A. imbi1d dral1 imbi12d mpbii x y x hbnae y hbal x y y hbnae z cv A e. y ax-17 z cv x cv A eleq1 dvelim nalequcoms y y cv A e. ph -> hba1 x x cv y cv = A. -. a1i hbimd y a4s hbald z cv A e. x ax-17 z cv y cv A eleq1 dvelim y y cv A e. ph -> ax-4 x cv A e. imim2i com23 x 19.20ii syl9 y 19.20ii syld hbnaes pm2.61i x A y A ph A.e. df-ral y A ph df-ral x cv A e. imbi2i x bial bitr y A x A ph A.e. df-ral x A ph df-ral y cv A e. imbi2i y bial bitr 3imtr4))
+thm (ralcom3 () () (<-> (A.e. x A (-> (e. (cv x) B) ph)) (A.e. x B (-> (e. (cv x) A) ph))) (x cv A e. x cv B e. ph pm2.04 r19.20i2 x cv B e. x cv A e. ph pm2.04 r19.20i2 impbi))
+thm (reeanv ((ph y) (ps x) (x y) (A y) (B x)) () (<-> (E.e. x A (E.e. y B (/\ ph ps))) (/\ (E.e. x A ph) (E.e. y B ps))) (x cv A e. y cv B e. ph ps /\ anass x cv A e. y cv B e. ph ps an4 bitr3 x y bi2ex x y x cv A e. y cv B e. ph ps /\ /\ exdistr x y x cv A e. ph /\ y cv B e. ps /\ eeanv 3bitr3 y B ph ps /\ df-rex x A birex x A y y cv B e. ph ps /\ /\ E. df-rex bitr x A ph df-rex y B ps df-rex anbi12i 3bitr4))
+thm (bireudva ((ph x)) ((bireudva.1 (-> (/\ ph (e. (cv x) A)) (<-> ps ch)))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))) (bireudva.1 exp pm5.32d x bieudv x A ps df-reu x A ch df-reu 3bitr4g))
+thm (bireudv ((ph x)) ((bireudv.1 (-> ph (<-> ps ch)))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))) (bireudv.1 x cv A e. adantr bireudva))
+thm (bireua () ((bireua.1 (-> (e. (cv x) A) (<-> ph ps)))) (<-> (E!e. x A ph) (E!e. x A ps)) (bireua.1 pm5.32i x bieu x A ph df-reu x A ps df-reu 3bitr4))
+thm (bireu () ((bireu.1 (<-> ph ps))) (<-> (E!e. x A ph) (E!e. x A ps)) (bireu.1 x cv A e. a1i bireua))
+thm (raleqf ((A y) (B y) (x y)) ((raleqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))) (raleqf.1 raleqf.2 hbeq A B x cv eleq2 ph imbi1d biald x A ph df-ral x B ph df-ral 3bitr4g))
+thm (rexeqf ((A y) (B y) (x y)) ((raleqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ph))) (raleqf.1 raleqf.2 hbeq A B x cv eleq2 ph anbi1d biexd x A ph df-rex x B ph df-rex 3bitr4g))
+thm (reueqf ((A y) (B y) (x y)) ((raleqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (raleqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))) (raleqf.1 raleqf.2 hbeq A B x cv eleq2 ph anbi1d bieud x A ph df-reu x B ph df-reu 3bitr4g))
+thm (raleq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))) (y cv A e. x ax-17 y cv B e. x ax-17 ph raleqf))
+thm (rexeq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ph))) (y cv A e. x ax-17 y cv B e. x ax-17 ph rexeqf))
+thm (reueq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))) (y cv A e. x ax-17 y cv B e. x ax-17 ph reueqf))
+thm (raleqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ps))) (A B x ph raleq raleqd.1 x B biraldv bitrd))
+thm (rexeqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ps))) (A B x ph rexeq raleqd.1 x B birexdv bitrd))
+thm (reueqd ((A x) (B x)) ((raleqd.1 (-> (= A B) (<-> ph ps)))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ps))) (A B x ph reueq raleqd.1 x B bireudv bitrd))
+thm (cbvralf ((A z) (x y) (x z) (y z)) ((cbvralf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (cbvralf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (cbvralf.3 (-> ph (A. y ph))) (cbvralf.4 (-> ps (A. x ps))) (cbvralf.5 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) (z cv x cv e. y ax-17 cbvralf.2 hbel cbvralf.3 hbim z cv y cv e. x ax-17 cbvralf.1 hbel cbvralf.4 hbim x cv y cv A eleq1 cbvralf.5 imbi12d cbval x A ph df-ral y A ps df-ral 3bitr4))
+thm (cbvral ((x y) (x z) (A x) (y z) (A y) (A z)) ((cbvral.1 (-> ph (A. y ph))) (cbvral.2 (-> ps (A. x ps))) (cbvral.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) (z cv A e. x ax-17 z cv A e. y ax-17 cbvral.1 cbvral.2 cbvral.3 cbvralf))
+thm (cbvrex ((x y) (A x) (A y)) ((cbvral.1 (-> ph (A. y ph))) (cbvral.2 (-> ps (A. x ps))) (cbvral.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. y A ps)) (x cv A e. y ax-17 cbvral.1 hban y cv A e. x ax-17 cbvral.2 hban x cv y cv A eleq1 cbvral.3 anbi12d cbvex x A ph df-rex y A ps df-rex 3bitr4))
+thm (cbvralv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (A.e. x A ph) (A.e. y A ps)) (ph y ax-17 ps x ax-17 cbvralv.1 A cbvral))
+thm (cbvrexv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E.e. x A ph) (E.e. y A ps)) (ph y ax-17 ps x ax-17 cbvralv.1 A cbvrex))
+thm (cbvreuv ((ph y) (ps x) (x y) (A x) (A y)) ((cbvralv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E!e. x A ph) (E!e. y A ps)) (x cv A e. ph /\ y ax-17 y cv A e. ps /\ x ax-17 x cv y cv A eleq1 cbvralv.1 anbi12d cbveu x A ph df-reu y A ps df-reu 3bitr4))
+thm (cbvral2v ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (ph z) (ps y) (ch x) (ch w)) ((cbvral2v.1 (-> (= (cv x) (cv z)) (<-> ph ch))) (cbvral2v.2 (-> (= (cv y) (cv w)) (<-> ch ps)))) (<-> (A.e. x A (A.e. y B ph)) (A.e. z A (A.e. w B ps))) (cbvral2v.1 y B biraldv A cbvralv cbvral2v.2 B cbvralv z A biral bitr))
+thm (cbvral3v ((ph w) (ps z) (v x) (ch x) (ch v) (u y) (th y) (th u) (w x) (A x) (A w) (x y) (B x) (w y) (v y) (B y) (v w) (B w) (B v) (x z) (u x) (C x) (y z) (C y) (w z) (v z) (u z) (C z) (u w) (C w) (u v) (C v) (C u)) ((cbvral3v.1 (-> (= (cv x) (cv w)) (<-> ph ch))) (cbvral3v.2 (-> (= (cv y) (cv v)) (<-> ch th))) (cbvral3v.3 (-> (= (cv z) (cv u)) (<-> th ps)))) (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A.e. w A (A.e. v B (A.e. u C ps)))) (cbvral3v.1 z C biraldv y B biraldv A cbvralv cbvral3v.2 cbvral3v.3 B C cbvral2v w A biral bitr))
+thm (reurex () () (-> (E!e. x A ph) (E.e. x A ph)) (x x cv A e. ph /\ euex x A ph df-reu x A ph df-rex 3imtr4))
+thm (reu2 ((x y) (A x) (A y) (ph y)) () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))))) (x cv A e. ph /\ y ax-17 x eu2 x A ph df-reu x A ph df-rex x A y A ph y cv x ph [/] /\ x cv y cv = -> A.e. df-ral y x cv A e. y cv A e. ph y cv x ph [/] /\ x cv y cv = -> -> 19.21v y cv A e. x ax-17 y x ph hbs1 hban x cv y cv A eleq1 x y ph sbequ12 anbi12d sbie x cv A e. ph /\ anbi2i x cv A e. ph y cv A e. y cv x ph [/] an4 bitr x cv y cv = imbi1i x cv A e. y cv A e. /\ ph y cv x ph [/] /\ x cv y cv = impexp x cv A e. y cv A e. ph y cv x ph [/] /\ x cv y cv = -> impexp 3bitr y bial y A ph y cv x ph [/] /\ x cv y cv = -> df-ral x cv A e. imbi2i 3bitr4 x bial bitr4 anbi12i 3bitr4))
+thm (reu5 () () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (E* x (/\ (e. (cv x) A) ph)))) (x x cv A e. ph /\ eu5 x A ph df-reu x A ph df-rex x x cv A e. ph /\ E* anbi1i 3bitr4))
+thm (reu4 ((x y) (A x) (A y) (ph y) (ps x)) ((reu4.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ps) (= (cv x) (cv y))))))) (x A ph y reu2 ps x ax-17 reu4.1 sbie ph anbi2i x cv y cv = imbi1i x A y A bi2ral x A ph E.e. anbi2i bitr))
+thm (2reuswap ((x y) (A x) (A y)) () (-> (A.e. x A (E* y (/\ (e. (cv y) A) ph))) (-> (E!e. x A (E.e. y A ph)) (E!e. y A (E.e. x A ph)))) (x A y y cv A e. ph /\ E* df-ral y x cv A e. y cv A e. ph /\ moanimv x bial bitr4 x y x cv A e. y cv A e. ph /\ /\ 2euswap x A y A ph E.e. df-reu y A x cv A e. ph /\ df-rex y A x cv A e. ph r19.42v y cv A e. x cv A e. ph an12 y biex 3bitr3 x bieu bitr y A x A ph E.e. df-reu x A y cv A e. ph r19.42v x A y cv A e. ph /\ df-rex bitr3 y bieu bitr 3imtr4g sylbi))
+thm (birabi () ((birabi.1 (-> (e. (cv x) A) (<-> ps ch)))) (= ({e.|} x A ps) ({e.|} x A ch)) (birabi.1 pm5.32i x biabi x A ps df-rab x A ch df-rab 3eqtr4))
+thm (birabdv ((ph x)) ((birabdv.1 (-> ph (-> (e. (cv x) A) (<-> ps ch))))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))) (birabdv.1 pm5.32d x biabdv x A ps df-rab x A ch df-rab 3eqtr4g))
+thm (birabsdv ((ph x)) ((birabsdv.1 (-> ph (<-> ps ch)))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))) (birabsdv.1 x cv A e. a1d birabdv))
+thm (rabeqf ((A y) (B y) (x y)) ((rabeqf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (rabeqf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))) (rabeqf.1 rabeqf.2 hbeq A B x cv eleq2 ph anbi1d biabd x A ph df-rab x B ph df-rab 3eqtr4g))
+thm (rabeq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))) (y cv A e. x ax-17 y cv B e. x ax-17 ph rabeqf))
+thm (eqrabi () ((eqrabi.1 (= A ({e.|} x B ph)))) (<-> (e. (cv x) A) (/\ (e. (cv x) B) ph)) (eqrabi.1 x cv eleq2i x B ph rabid bitr))
+thm (visset () () (e. (cv x) (V)) (x cv eqid x df-v eqabi mpbir))
+thm (isset ((A x)) () (<-> (e. A (V)) (E. x (= (cv x) A))) (A V x df-clel x visset x cv A = biantru x biex bitr4))
+thm (isseti ((A x)) ((isseti.1 (e. A (V)))) (E. x (= (cv x) A)) (isseti.1 A x isset mpbi))
+thm (issetri ((A x)) ((issetri.1 (E. x (= (cv x) A)))) (e. A (V)) (issetri.1 A x isset mpbir))
+thm (elisset ((A x) (B x)) () (-> (e. A B) (e. A (V))) (A B x df-clel x x cv A = x cv B e. 19.40 sylbi pm3.26d A x isset sylibr))
+thm (elisseti () ((elisseti.1 (e. A B))) (e. A (V)) (elisseti.1 A B elisset ax-mp))
+thm (elex ((A x)) () (-> (e. A B) (E. x (= (cv x) A))) (A B elisset A x isset sylib))
+thm (ralv () () (<-> (A.e. x (V) ph) (A. x ph)) (x V ph df-ral x visset ph a1bi x bial bitr4))
+thm (rexv () () (<-> (E.e. x (V) ph) (E. x ph)) (x V ph df-rex x visset ph biantrur x biex bitr4))
+thm (rabab () () (= ({e.|} x (V) ph) ({|} x ph)) (x V ph df-rab x cv V e. ph pm3.27 x visset ph jctl impbi x biabi eqtr))
+thm (ralcom4 ((x y) (A y)) () (<-> (A.e. x A (A. y ph)) (A. y (A.e. x A ph))) (y V x A ph ralcom y x A ph A.e. ralv y ph ralv x A biral 3bitr3r))
+thm (rexcom4 ((x y) (A y)) () (<-> (E.e. x A (E. y ph)) (E. y (E.e. x A ph))) (y V x A ph rexcom y x A ph E.e. rexv y ph rexv x A birex 3bitr3r))
+thm (ceqsalg ((A x)) ((ceqsalg.1 (-> ps (A. x ps))) (ceqsalg.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (A. x (-> (= (cv x) A) ph)) ps)) (ceqsalg.2 biimpd a2i x 19.20i ceqsalg.1 x cv A = 19.23 sylib A B x elex syl5com ceqsalg.1 ceqsalg.2 biimprcd 19.21ai A B e. a1i impbid))
+thm (ceqsal ((A x)) ((ceqsal.1 (-> ps (A. x ps))) (ceqsal.2 (e. A (V))) (ceqsal.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) A) ph)) ps) (ceqsal.2 ceqsal.1 ceqsal.3 V ceqsalg ax-mp))
+thm (ceqsalv ((A x) (ps x)) ((ceqsalv.1 (e. A (V))) (ceqsalv.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A. x (-> (= (cv x) A) ph)) ps) (ps x ax-17 ceqsalv.1 ceqsalv.2 ceqsal))
+thm (gencl ((ps x)) ((gencl.1 (<-> th (E. x (/\ ch (= A B))))) (gencl.2 (-> (= A B) (<-> ph ps))) (gencl.3 (-> ch ph))) (-> th ps) (gencl.1 gencl.2 gencl.3 syl5bi impcom x 19.23aiv sylbi))
+thm (2gencl ((x y) (R x) (ps x) (C y) (S y) (ch y)) ((2gencl.1 (<-> (e. C S) (E. x (/\ (e. (cv x) R) (= A C))))) (2gencl.2 (<-> (e. D S) (E. y (/\ (e. (cv y) R) (= B D))))) (2gencl.3 (-> (= A C) (<-> ph ps))) (2gencl.4 (-> (= B D) (<-> ps ch))) (2gencl.5 (-> (/\ (e. (cv x) R) (e. (cv y) R)) ph))) (-> (/\ (e. C S) (e. D S)) ch) (2gencl.2 2gencl.4 C S e. imbi2d 2gencl.1 2gencl.3 y cv R e. imbi2d 2gencl.5 exp gencl com12 gencl impcom))
+thm (3gencl ((x y) (x z) (y z) (D y) (D z) (F z) (R x) (R y) (S y) (S z) (ps x) (ch y) (th z)) ((3gencl.1 (<-> (e. D S) (E. x (/\ (e. (cv x) R) (= A D))))) (3gencl.2 (<-> (e. F S) (E. y (/\ (e. (cv y) R) (= B F))))) (3gencl.3 (<-> (e. G S) (E. z (/\ (e. (cv z) R) (= C G))))) (3gencl.4 (-> (= A D) (<-> ph ps))) (3gencl.5 (-> (= B F) (<-> ps ch))) (3gencl.6 (-> (= C G) (<-> ch th))) (3gencl.7 (-> (/\/\ (e. (cv x) R) (e. (cv y) R) (e. (cv z) R)) ph))) (-> (/\/\ (e. D S) (e. F S) (e. G S)) th) (3gencl.3 3gencl.6 D S e. F S e. /\ imbi2d 3gencl.1 3gencl.2 3gencl.4 z cv R e. imbi2d 3gencl.5 z cv R e. imbi2d 3gencl.7 3exp imp 2gencl com12 gencl com12 exp 3imp))
+thm (cgsex2g ((x y) (ps x) (ps y) (A x) (A y) (B x) (B y)) ((cgsex2g.1 (-> (/\ (= (cv x) A) (= (cv y) B)) ch)) (cgsex2g.2 (-> ch (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (<-> (E. x (E. y (/\ ch ph))) ps)) (cgsex2g.2 biimpa x y 19.23aivv A C e. B D e. /\ a1i cgsex2g.2 biimprcd ancld x y 19.22dvv A C x elex B D y elex anim12i x y x cv A = y cv B = eeanv sylibr cgsex2g.1 x y 19.22i2 syl syl5com impbid))
+thm (cgsex4g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ps z) (ps w)) ((cgsex4g.1 (-> (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) ch)) (cgsex4g.2 (-> ch (<-> ph ps)))) (-> (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) (<-> (E. x (E. y (E. z (E. w (/\ ch ph))))) ps)) (cgsex4g.2 biimpa z w 19.23aivv x y 19.23aivv A R e. B S e. /\ C R e. D S e. /\ /\ a1i cgsex4g.2 biimprcd ancld z w 19.22dvv x y 19.22dvv A R x elex B S y elex anim12i x y x cv A = y cv B = eeanv sylibr C R z elex D S w elex anim12i z w z cv C = w cv D = eeanv sylibr anim12i x y z w x cv A = y cv B = /\ z cv C = w cv D = /\ ee4anv sylibr cgsex4g.1 z w 19.22i2 x y 19.22i2 syl syl5com impbid))
+thm (ceqsex ((A x)) ((ceqsex.1 (-> ps (A. x ps))) (ceqsex.2 (e. A (V))) (ceqsex.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) A) ph)) ps) (ceqsex.1 ceqsex.3 biimpa 19.23ai ceqsex.2 x isseti ceqsex.1 ceqsex.3 biimprcd 19.21ai x x cv A = ph exintr syl mpi impbi))
+thm (ceqsexv ((A x) (ps x)) ((ceqsexv.1 (e. A (V))) (ceqsexv.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E. x (/\ (= (cv x) A) ph)) ps) (ps x ax-17 ceqsexv.1 ceqsexv.2 ceqsex))
+thm (gencbvex ((ps x) (ph y) (th x) (ch y) (A y)) ((gencbvex.1 (e. A (V))) (gencbvex.2 (-> (= A (cv y)) (<-> ph ps))) (gencbvex.3 (-> (= A (cv y)) (<-> ch th))) (gencbvex.4 (<-> th (E. x (/\ ch (= A (cv y))))))) (<-> (E. x (/\ ch ph)) (E. y (/\ th ps))) (x y y cv A = th ps /\ /\ excom gencbvex.1 gencbvex.3 gencbvex.2 anbi12d bicomd eqcoms ceqsexv x biex x y cv A = E. th ps anass gencbvex.4 gencbvex.3 pm5.32i A y cv = ch ancom A y cv eqcom th anbi1i 3bitr3 x biex x y cv A = th 19.41v 3bitr ps anbi1i x y cv A = th ps /\ 19.41v 3bitr4r y biex 3bitr3))
+thm (gencbval ((ps x) (ph y) (th x) (ch y) (A y)) ((gencbval.1 (e. A (V))) (gencbval.2 (-> (= A (cv y)) (<-> ph ps))) (gencbval.3 (-> (= A (cv y)) (<-> ch th))) (gencbval.4 (<-> th (E. x (/\ ch (= A (cv y))))))) (<-> (A. x (-> ch ph)) (A. y (-> th ps))) (gencbval.1 gencbval.2 negbid gencbval.3 gencbval.4 gencbvex x ch ph exanali y th ps exanali 3bitr3 bicon4i))
+thm (vtoclf ((A x)) ((vtoclf.1 (-> ps (A. x ps))) (vtoclf.2 (e. A (V))) (vtoclf.3 (-> (= (cv x) A) (<-> ph ps))) (vtoclf.4 ph)) ps (vtoclf.1 vtoclf.2 x isseti vtoclf.3 biimpd x 19.22i ax-mp 19.36i vtoclf.4 mpg))
+thm (vtocl ((A x) (ps x)) ((vtocl.1 (e. A (V))) (vtocl.2 (-> (= (cv x) A) (<-> ph ps))) (vtocl.3 ph)) ps (ps x ax-17 vtocl.1 vtocl.2 vtocl.3 vtoclf))
+thm (vtocl2 ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((vtocl2.1 (e. A (V))) (vtocl2.2 (e. B (V))) (vtocl2.3 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps))) (vtocl2.4 ph)) ps (vtocl2.1 x isseti vtocl2.2 y isseti x y x cv A = y cv B = eeanv vtocl2.3 biimpd x y 19.22i2 sylbir mp2an y ph ps 19.36v x biex x y ph A. ps 19.36v bitr mpbi vtocl2.4 y ax-gen mpg))
+thm (vtocl3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (ps x) (ps y) (ps z)) ((vtocl3.1 (e. A (V))) (vtocl3.2 (e. B (V))) (vtocl3.3 (e. C (V))) (vtocl3.4 (-> (/\/\ (= (cv x) A) (= (cv y) B) (= (cv z) C)) (<-> ph ps))) (vtocl3.5 ph)) ps (vtocl3.1 x isseti vtocl3.2 y isseti vtocl3.3 z isseti x y z x cv A = y cv B = z cv C = eeeanv vtocl3.4 biimpd z 19.22i x y 19.22i2 sylbir mp3an z ph ps 19.36v y biex y z ph A. ps 19.36v bitr x biex x y z ph A. A. ps 19.36v bitr mpbi vtocl3.5 y z gen2 mpg))
+thm (vtoclb ((A x) (ch x) (th x)) ((vtoclb.1 (e. A (V))) (vtoclb.2 (-> (= (cv x) A) (<-> ph ch))) (vtoclb.3 (-> (= (cv x) A) (<-> ps th))) (vtoclb.4 (<-> ph ps))) (<-> ch th) (vtoclb.1 vtoclb.2 vtoclb.3 bibi12d vtoclb.4 vtocl))
+thm (vtoclgf ((A y) (x y)) ((vtoclgf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (vtoclgf.2 (-> ps (A. x ps))) (vtoclgf.3 (-> (= (cv x) A) (<-> ph ps))) (vtoclgf.4 ph)) (-> (e. A B) ps) (A B elisset A y isset vtoclgf.1 hbeleq x cv A = y ax-17 y cv x cv A eqeq1 cbvex bitr vtoclgf.2 vtoclgf.4 vtoclgf.3 mpbii 19.23ai sylbi syl))
+thm (vtoclg ((x y) (A x) (A y) (ps x)) ((vtoclg.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclg.2 ph)) (-> (e. A B) ps) (y cv A e. x ax-17 ps x ax-17 vtoclg.1 vtoclg.2 B vtoclgf))
+thm (vtoclbg ((A x) (ch x) (th x)) ((vtoclbg.1 (-> (= (cv x) A) (<-> ph ch))) (vtoclbg.2 (-> (= (cv x) A) (<-> ps th))) (vtoclbg.3 (<-> ph ps))) (-> (e. A B) (<-> ch th)) (vtoclbg.1 vtoclbg.2 bibi12d vtoclbg.3 B vtoclg))
+thm (vtocl2gf ((A x) (y z) (A y) (A z) (B y) (B z) (x z)) ((vtocl2gf.1 (-> ps (A. x ps))) (vtocl2gf.2 (-> ch (A. y ch))) (vtocl2gf.3 (-> (= (cv x) A) (<-> ph ps))) (vtocl2gf.4 (-> (= (cv y) B) (<-> ps ch))) (vtocl2gf.5 ph)) (-> (/\ (e. A C) (e. B D)) ch) (z cv B e. y ax-17 A V e. y ax-17 vtocl2gf.2 hbim vtocl2gf.4 A V e. imbi2d z cv A e. x ax-17 vtocl2gf.1 vtocl2gf.3 vtocl2gf.5 V vtoclgf D vtoclgf impcom A C elisset sylan))
+thm (vtocl2g ((A x) (A y) (B y) (ps x) (ch y)) ((vtocl2g.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl2g.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl2g.3 ph)) (-> (/\ (e. A C) (e. B D)) ch) (ps x ax-17 ch y ax-17 vtocl2g.1 vtocl2g.2 vtocl2g.3 C D vtocl2gf))
+thm (vtoclga ((A x) (B x) (ps x)) ((vtoclga.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclga.2 (-> (e. (cv x) B) ph))) (-> (e. A B) ps) (x cv A B eleq1 vtoclga.1 imbi12d vtoclga.2 B vtoclg pm2.43i))
+thm (vtocl2ga ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y) (ps x) (ch y)) ((vtocl2ga.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl2ga.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl2ga.3 (-> (/\ (e. (cv x) C) (e. (cv y) D)) ph))) (-> (/\ (e. A C) (e. B D)) ch) (vtocl2ga.2 A C e. imbi2d vtocl2ga.1 y cv D e. imbi2d vtocl2ga.3 exp vtoclga com12 vtoclga impcom))
+thm (vtocl3ga ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (D x) (D y) (D z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch y) (th z)) ((vtocl3ga.1 (-> (= (cv x) A) (<-> ph ps))) (vtocl3ga.2 (-> (= (cv y) B) (<-> ps ch))) (vtocl3ga.3 (-> (= (cv z) C) (<-> ch th))) (vtocl3ga.4 (-> (/\/\ (e. (cv x) D) (e. (cv y) R) (e. (cv z) S)) ph))) (-> (/\/\ (e. A D) (e. B R) (e. C S)) th) (vtocl3ga.2 A D e. imbi2d vtocl3ga.3 A D e. imbi2d vtocl3ga.1 y cv R e. z cv S e. /\ imbi2d vtocl3ga.4 3exp imp3a vtoclga com12 vtocl2ga exp com3r 3imp))
+thm (vtocleg ((A x) (ph x)) ((vtocleg.1 (-> (= (cv x) A) ph))) (-> (e. A B) ph) (A B x elex vtocleg.1 x 19.23aiv syl))
+thm (vtocle ((A x) (ph x)) ((vtocle.1 (e. A (V))) (vtocle.2 (-> (= (cv x) A) ph))) ph (vtocle.1 vtocle.2 V vtocleg ax-mp))
+thm (vtoclef ((A x)) ((vtoclef.1 (-> ph (A. x ph))) (vtoclef.2 (e. A (V))) (vtoclef.3 (-> (= (cv x) A) ph))) ph (vtoclef.2 x isseti vtoclef.1 vtoclef.3 19.23ai ax-mp))
+thm (vtoclri ((A x) (B x) (ps x)) ((vtoclri.1 (-> (= (cv x) A) (<-> ph ps))) (vtoclri.2 (A.e. x B ph))) (-> (e. A B) ps) (vtoclri.1 vtoclri.2 rspec vtoclga))
+thm (cla4gf ((x y) (A y)) ((cla4gf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cla4gf.2 (-> ps (A. x ps))) (cla4gf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A. x ph) ps)) (A B elisset A y isset cla4gf.1 hbeleq x cv A = y ax-17 y cv x cv A eqeq1 cbvex bitr cla4gf.3 biimpd x 19.22i sylbi cla4gf.2 ph 19.36 sylib syl))
+thm (cla4egf ((x y) (A y)) ((cla4gf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (cla4gf.2 (-> ps (A. x ps))) (cla4gf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> ps (E. x ph))) (cla4gf.1 cla4gf.2 hbne cla4gf.3 negbid B cla4gf con2d x ph df-ex syl6ibr))
+thm (cla4gv ((ps x) (x y) (A x) (A y)) ((cla4gv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A. x ph) ps)) (y cv A e. x ax-17 ps x ax-17 cla4gv.1 B cla4gf))
+thm (cla4egv ((ps x) (x y) (A x) (A y)) ((cla4gv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> ps (E. x ph))) (y cv A e. x ax-17 ps x ax-17 cla4gv.1 B cla4egf))
+thm (cla4e2gv ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2gv.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (-> ps (E. x (E. y ph)))) (cla4e2gv.1 biimprcd x y 19.22dvv A C x elex B D y elex anim12i x y x cv A = y cv B = eeanv sylibr syl5com))
+thm (cla42gv ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2gv.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (-> (A. x (A. y ph)) ps)) (cla4e2gv.1 negbid C D cla4e2gv y ph exnal x biex x y ph A. exnal bitr2 syl6ibr a3d))
+thm (cla4v ((A x) (ps x)) ((cla4v.1 (e. A (V))) (cla4v.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (A. x ph) ps) (cla4v.1 cla4v.2 V cla4gv ax-mp))
+thm (cla4ev ((A x) (ps x)) ((cla4v.1 (e. A (V))) (cla4v.2 (-> (= (cv x) A) (<-> ph ps)))) (-> ps (E. x ph)) (cla4v.1 cla4v.2 negbid cla4v con2i x ph df-ex sylibr))
+thm (cla4e2v ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((cla4e2v.1 (e. A (V))) (cla4e2v.2 (e. B (V))) (cla4e2v.3 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> ps (E. x (E. y ph))) (cla4e2v.1 cla4e2v.2 cla4e2v.3 V V cla4e2gv mp2an))
+thm (rcla4 ((x y) (A x) (A y) (B x)) ((rcla4.1 (-> ps (A. x ps))) (rcla4.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A.e. x B ph) ps)) (x B ph df-ral y cv A e. x ax-17 A B e. x ax-17 rcla4.1 hbim x cv A B eleq1 rcla4.2 imbi12d B cla4gf pm2.43b sylbi com12))
+thm (rcla4v ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (-> (A.e. x B ph) ps)) (ps x ax-17 rcla4v.1 B rcla4))
+thm (rcla4va ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (e. A B) (A.e. x B ph)) ps) (rcla4v.1 B rcla4v imp))
+thm (rcla4ev ((A x) (B x) (ps x)) ((rcla4v.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (/\ (e. A B) ps) (E.e. x B ph)) (x cv A B eleq1 rcla4v.1 anbi12d B cla4egv anabsi5 x B ph df-rex sylibr))
+thm (rcla42v ((x y) (A x) (A y) (C x) (D x) (B y) (D y) (ch x) (ps y)) ((rcla42v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla42v.2 (-> (= (cv y) B) (<-> ch ps)))) (-> (/\ (e. A C) (e. B D)) (-> (A.e. x C (A.e. y D ph)) ps)) (rcla42v.1 y D biraldv C rcla4v rcla42v.2 D rcla4v sylan9))
+thm (rcla42ev ((x y) (A x) (A y) (C x) (D x) (B y) (D y) (ch x) (ps y)) ((rcla42v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla42v.2 (-> (= (cv y) B) (<-> ch ps)))) (-> (/\ (/\ (e. A C) (e. B D)) ps) (E.e. x C (E.e. y D ph))) (rcla42v.2 D rcla4ev A C e. anim2i anassrs rcla42v.1 y D birexdv C rcla4ev syl))
+thm (rcla43v ((ps z) (ch x) (th y) (x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (R x) (S x) (S y) (T x) (T y) (T z)) ((rcla43v.1 (-> (= (cv x) A) (<-> ph ch))) (rcla43v.2 (-> (= (cv y) B) (<-> ch th))) (rcla43v.3 (-> (= (cv z) C) (<-> th ps)))) (-> (/\/\ (e. A R) (e. B S) (e. C T)) (-> (A.e. x R (A.e. y S (A.e. z T ph))) ps)) (rcla43v.1 z T biraldv rcla43v.2 z T biraldv R S rcla42v rcla43v.3 T rcla4v sylan9 3impa))
+thm (eqvinc ((A x) (x y) (x z) (A y) (A z) (y z) (B x) (B y) (B z)) ((eqvinc.1 (e. A (V)))) (<-> (= A B) (E. x (/\ (= (cv x) A) (= (cv x) B)))) (eqvinc.1 A B V eleq1 mpbii x visset x cv B V eleq1 mpbii x cv A = adantl x 19.23aiv z cv B A eqeq2 z cv B x cv eqeq2 x cv A = anbi2d x biexdv eqvinc.1 y cv A z cv eqeq1 y cv A x cv eqeq1 A x cv eqcom syl6bb x cv z cv = anbi1d x biexdv y z x equvin vtoclb V vtoclbg pm5.21nii))
+thm (eqvincf ((A y) (B y) (x y)) ((eqvincf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (eqvincf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (eqvincf.3 (e. A (V)))) (<-> (= A B) (E. x (/\ (= (cv x) A) (= (cv x) B)))) (eqvincf.3 B y eqvinc eqvincf.1 hbeleq eqvincf.2 hbeleq hban x cv A = x cv B = /\ y ax-17 y cv x cv A eqeq1 y cv x cv B eqeq1 anbi12d cbvex bitr))
+thm (alexeq ((A x) (x y) (A y) (ph y)) ((alexeq.1 (e. A (V)))) (<-> (A. x (-> (= (cv x) A) ph)) (E. x (/\ (= (cv x) A) ph))) (alexeq.1 y cv A x cv eqeq2 ph imbi1d x bialdv y cv A x cv eqeq2 ph anbi1d x biexdv x y ph equs4 x y x hbae x x cv y cv = ph -> hba1 x y x cv y cv = ph -> ax-16 x cv y cv = ph pm3.4 syl5 19.23ad x y ph equs5 pm2.61i impbi vtoclb))
+thm (ceqex ((A x) (x y) (A y) (ph y)) () (-> (= (cv x) A) (<-> ph (E. x (/\ (= (cv x) A) ph)))) (x cv A = x 19.8a A x isset sylibr y cv A x cv eqeq2 y cv A x cv eqeq2 ph anbi1d x biexdv ph bibi2d imbi12d x cv y cv = ph /\ x 19.8a exp x x cv y cv = ph -> ax-4 com12 y visset x ph alexeq syl5ibr impbid V vtoclg mpcom))
+thm (ceqsexg ((x y) (A x) (A y)) ((ceqsexg.1 (-> ps (A. x ps))) (ceqsexg.2 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)) (y cv A e. x ax-17 x x cv A = ph /\ hbe1 ceqsexg.1 hbbi x A ph ceqex ceqsexg.2 bibi12d ph pm4.2 B vtoclgf))
+thm (ceqsexgv ((A x) (ps x)) ((ceqsexgv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)) (ps x ax-17 ceqsexgv.1 B ceqsexg))
+thm (ceqsrexv ((A x) (B x) (ps x)) ((ceqsrexv.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (E.e. x B (/\ (= (cv x) A) ph)) ps)) (x cv A B eleq1 ceqsrexv.1 anbi12d B ceqsexgv A B e. ps ibar bitr4d x B x cv A = ph /\ df-rex x cv A = x cv B e. ph an12 x biex bitr4 syl5bb))
+thm (clel2 ((A x) (B x)) ((clel2.1 (e. A (V)))) (<-> (e. A B) (A. x (-> (= (cv x) A) (e. (cv x) B)))) (clel2.1 x cv A B eleq1 ceqsalv bicomi))
+thm (clel3g ((A x) (B x)) () (-> (e. B C) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x)))))) (x cv B A eleq2 C ceqsexgv bicomd))
+thm (clel3 ((A x) (B x)) ((clel3.1 (e. B (V)))) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x))))) (clel3.1 B V A x clel3g ax-mp))
+thm (clel4 ((A x) (B x)) ((clel4.1 (e. B (V)))) (<-> (e. A B) (A. x (-> (= (cv x) B) (e. A (cv x))))) (clel4.1 x cv B A eleq2 ceqsalv bicomi))
+thm (elabf ((ps y) (A x) (x y) (A y) (ph y)) ((elabf.1 (-> ps (A. x ps))) (elabf.2 (e. A (V))) (elabf.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) (y cv A e. x ax-17 y x ph hbab1 hbel elabf.1 hbbi elabf.2 x cv A x ph {|} eleq1 x ph abid syl5bbr elabf.3 bitr3d vtoclef))
+thm (elab ((ps x) (A x)) ((elab.1 (e. A (V))) (elab.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) (ps x ax-17 elab.1 elab.2 elabf))
+thm (elabgf ((A y) (ph z) (x y) (x z) (y z)) ((elabgf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (elabgf.2 (-> ps (A. x ps))) (elabgf.3 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (e. A ({|} x ph)) ps)) (elabgf.1 elabgf.1 z x ph hbab1 hbel elabgf.2 hbbi x cv A x ph {|} eleq1 elabgf.3 bibi12d x ph abid B vtoclgf))
+thm (elabg ((ps x) (x y) (A x) (A y)) ((elabg.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A B) (<-> (e. A ({|} x ph)) ps)) (y cv A e. x ax-17 ps x ax-17 elabg.1 B elabgf))
+thm (elab2g ((ps x) (A x)) ((elab2g.1 (-> (= (cv x) A) (<-> ph ps))) (elab2g.2 (= B ({|} x ph)))) (-> (e. A C) (<-> (e. A B) ps)) (elab2g.1 C elabg elab2g.2 A eleq2i syl5bb))
+thm (elab2 ((ps x) (A x)) ((elab2.1 (e. A (V))) (elab2.2 (-> (= (cv x) A) (<-> ph ps))) (elab2.3 (= B ({|} x ph)))) (<-> (e. A B) ps) (elab2.1 elab2.2 elab2.3 V elab2g ax-mp))
+thm (elab3g ((ps x) (A x)) ((elab3g.1 (-> ps (e. A (V)))) (elab3g.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({|} x ph)) ps) (elab3g.2 x ph {|} elabg ibi elab3g.1 elab3g.2 V elabg syl biimprd pm2.43i impbi))
+thm (elrabf ((x y) (A y) (B y)) ((elrabf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (elrabf.2 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (elrabf.3 (-> ps (A. x ps))) (elrabf.4 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ps)) (A x B ph {e.|} elisset A B elisset ps adantr elrabf.1 elrabf.1 elrabf.2 hbel elrabf.3 hban x cv A B eleq1 elrabf.4 anbi12d V elabgf x B ph df-rab A eleq2i syl5bb pm5.21nii))
+thm (elrab ((ps x) (x y) (A x) (A y) (B x) (B y)) ((elrab.1 (-> (= (cv x) A) (<-> ph ps)))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ps)) (y cv A e. x ax-17 y cv B e. x ax-17 ps x ax-17 elrab.1 elrabf))
+thm (cbvab ((x y) (x z) (y z) (ph z) (ps z)) ((cbvab.1 (-> ph (A. y ph))) (cbvab.2 (-> ps (A. x ps))) (cbvab.3 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({|} x ph) ({|} y ps)) (cbvab.1 z x hbab z y ps hbab1 cleqf cbvab.2 y visset cbvab.3 elabf y ps abid bitr4 mpgbir))
+thm (cbvabv ((x y) (ph y) (ps x)) ((cbvabv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({|} x ph) ({|} y ps)) (ph y ax-17 ps x ax-17 cbvabv.1 cbvab))
+thm (cbvrab ((x y) (x z) (y z) (A z)) ((cbvrab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (cbvrab.2 (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (cbvrab.3 (-> ph (A. y ph))) (cbvrab.4 (-> ps (A. x ps))) (cbvrab.5 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({e.|} x A ph) ({e.|} y A ps)) (z cv x cv e. y ax-17 cbvrab.2 hbel cbvrab.3 hban z cv y cv e. x ax-17 cbvrab.1 hbel cbvrab.4 hban x cv y cv A eleq1 cbvrab.5 anbi12d cbvab x A ph df-rab y A ps df-rab 3eqtr4))
+thm (cbvrabv ((x y) (x z) (A x) (y z) (A y) (A z) (ph y) (ps x)) ((cbvrabv.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (= ({e.|} x A ph) ({e.|} y A ps)) (z cv A e. x ax-17 z cv A e. y ax-17 ph y ax-17 ps x ax-17 cbvrabv.1 cbvrab))
+thm (eueq ((x y) (A x) (A y)) () (<-> (e. A (V)) (E! x (= (cv x) A))) (x cv A y cv eqtr3t x y gen2 x x cv A = E. biantru A x isset x cv y cv A eqeq1 eu4 3bitr4))
+thm (eueq1 ((A x)) ((eueq1.1 (e. A (V)))) (E! x (= (cv x) A)) (eueq1.1 A x eueq mpbi))
+thm (eueq2 ((ph x) (A x) (B x)) ((eueq2.1 (e. A (V))) (eueq2.2 (e. B (V)))) (E! x (\/ (/\ ph (= (cv x) A)) (/\ (-. ph) (= (cv x) B)))) (ph -. x ph x cv A = /\ euorv ph negb eueq2.1 x eueq1 x ph x cv A = euanv biimpr mpan2 sylanc ph negb x cv B = bianfd ph x cv A = /\ orbi2d ph -. ph x cv A = /\ orcom syl5bb x bieudv mpbid eueq2.2 x eueq1 x ph -. x cv B = euanv biimpr mpan2 ph x ph -. x cv B = /\ euorv mpdan ph -. id x cv A = bianfd ph -. x cv B = /\ orbi1d x bieudv mpbid pm2.61i))
+thm (eueq3 ((ph x) (ps x) (A x) (B x) (C x)) ((eueq3.1 (e. A (V))) (eueq3.2 (e. B (V))) (eueq3.3 (e. C (V))) (eueq3.4 (-. (/\ ph ps)))) (E! x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)))) (ph ps \/ -. ps \/ x ph x cv A = /\ euorv ph ps pm2.45 eueq3.4 ph ps imnan mpbir con2i jaoi con2i eueq3.1 x eueq1 x ph x cv A = euanv biimpr mpan2 sylanc ph ps \/ negb orci x cv B = bianfd eueq3.4 ph ps imnan mpbir x cv C = bianfd orbi12d ph x cv A = /\ orbi2d ph ps \/ -. ps \/ ph x cv A = /\ orcom ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ 3orass 3bitr4g x bieudv mpbid ph ph ps \/ -. \/ x ps x cv C = /\ euorv eueq3.4 ph ps imnan mpbir ph ps pm2.46 jaoi con2i eueq3.3 x eueq1 x ps x cv C = euanv biimpr mpan2 sylanc eueq3.4 ph ps imnan mpbir con2i x cv A = bianfd ph ps \/ negb olci x cv B = bianfd orbi12d ps x cv C = /\ orbi1d ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ df-3or syl6bbr x bieudv mpbid eueq3.2 x eueq1 x ph ps \/ -. x cv B = euanv biimpr mpan2 ph ps \/ x ph ps \/ -. x cv B = /\ euorv mpdan ph ps pm2.45 x cv A = bianfd ph ps pm2.46 x cv C = bianfd ph ps \/ -. x cv B = /\ orbi1d ps x cv C = /\ ph ps \/ -. x cv B = /\ orcom syl6bb orbi12d ph ps ph ps \/ -. x cv B = /\ orass ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ 3orass 3bitr4g x bieudv mpbid ecase3))
+thm (moeq ((A x)) () (E* x (= (cv x) A)) (A x isset A x eueq bitr3 biimp x x cv A = df-mo mpbir))
+thm (moeq3 ((x y) (ph x) (ph y) (ps x) (ps y) (A x) (A y) (B x) (B y) (C x) (C y)) ((moeq3.1 (e. B (V))) (moeq3.2 (e. C (V))) (moeq3.3 (-. (/\ ph ps)))) (E* x (\/\/ (/\ ph (= (cv x) A)) (/\ (-. (\/ ph ps)) (= (cv x) B)) (/\ ps (= (cv x) C)))) (y cv A x cv eqeq2 ph anbi2d y cv A = ph ps \/ -. x cv B = /\ pm4.2i y cv A = ps x cv C = /\ pm4.2i 3orbi123d x bieudv y visset moeq3.1 moeq3.2 moeq3.3 x eueq3 V vtoclg x ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ \/\/ eumo syl y visset moeq3.1 moeq3.2 moeq3.3 x eueq3 A V e. x cv y cv = pm2.21 x visset x cv A V eleq1 mpbii syl5 ph anim2d ph ps \/ -. x cv B = /\ ps x cv C = /\ \/ orim1d ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ 3orass ph x cv y cv = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ 3orass 3imtr4g x 19.21aiv x ph x cv A = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ \/\/ ph x cv y cv = /\ ph ps \/ -. x cv B = /\ ps x cv C = /\ \/\/ euimmo syl mpi pm2.61i))
+thm (mosub ((x y) (A x) (A y)) ((mosub.1 (E* x ph))) (E* x (E. y (/\ (= (cv y) A) ph))) (y A moeq mosub.1 y ax-gen y y cv A = x ph moexexv mp2an))
+thm (mo2icl ((x y) (A x) (A y) (ph y)) () (-> (A. x (-> ph (= (cv x) A))) (E* x ph)) (y cv A x cv eqeq2 ph imbi2d x bialdv x ph E* imbi1d x ph x cv y cv = -> A. y 19.8a ph y ax-17 x mo2 sylibr V vtoclg x visset x cv A V eleq1 mpbii ph imim2i con3d com12 x 19.20dv x ph alnex x ph exmo ori sylbi syl6 pm2.61i))
+thm (euxfr2 ((ph x) (A x) (x y)) ((euxfr2.1 (e. A (V))) (euxfr2.2 (E* y (= (cv x) A)))) (<-> (E! x (E. y (/\ (= (cv x) A) ph))) (E! y ph)) (x y x cv A = ph /\ 2euswap euxfr2.2 ph moani ph x cv A = ancom y bimo mpbi mpg y x x cv A = ph /\ 2euswap x A moeq ph moani ph x cv A = ancom x bimo mpbi mpg impbi euxfr2.1 x cv A = ph pm4.2i ceqsexv y bieu bitr))
+thm (euxfr ((ps x) (ph y) (A x) (x y)) ((euxfr.1 (e. A (V))) (euxfr.2 (E! y (= (cv x) A))) (euxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E! x ph) (E! y ps)) (euxfr.2 y x cv A = euex ax-mp ph biantrur y x cv A = ph 19.41v bitr4 euxfr.3 pm5.32i y biex bitr x bieu euxfr.1 euxfr.2 eumoi ps euxfr2 bitr))
+thm (ru ((x y)) () (e/ ({|} x (e/ (cv x) (cv x))) (V)) (y cv y cv e. pm5.19 x cv y cv y cv eleq1 x cv y cv = id x cv y cv = id eleq12d negbid x cv x cv df-nel syl5bb bibi12d a4b1 mto y cv x x cv x cv e/ eqab mtbir y nex x x cv x cv e/ {|} y isset mtbir x x cv x cv e/ {|} V df-nel mpbir))
+thm (vsbcint ((ps x) (ps y) (x y) (A x)) ((vsbcint.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (= (cv y) A) (<-> ([/] (cv y) x ph) ps)) (y visset x cv y cv A eqeq1 ceqsexv y x ph hbs1 ps x ax-17 hbbi x y ph sbequ12 bicomd vsbcint.1 sylan9bb 19.23ai sylbir))
+thm (sbralie ((x y) (x z) (y z) (ph y) (ph z) (ps x) (ps z)) ((sbralie.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (<-> ([/] (cv x) y (A.e. x (cv y) ph)) (A.e. y (cv x) ps)) (ph z ax-17 z x ph hbs1 x z ph sbequ12 y cv cbvral x y bisb z x cv z cv x ph [/] A.e. y ax-17 y cv x cv z z cv x ph [/] raleq sbie bitr z cv x ph [/] y ax-17 y z z cv x ph [/] hbs1 z y z cv x ph [/] sbequ12 x cv cbvral ph z ax-17 y x sbco2 ps x ax-17 sbralie.1 sbie bitr y x cv biral 3bitr))
+thm (dfsbcq () () (-> (= A B) (<-> ([/] A x ph) ([/] B x ph))) (A B x ph {|} eleq1 A x ph df-sbc B x ph df-sbc 3bitr4g))
+thm (sbceq1 () () (-> (= (cv x) A) (<-> ph ([/] A x ph))) (x cv A x ph dfsbcq x ph sbid syl5bbr))
+thm (a4sbc ((ph y) (A y) (x y)) () (-> (e. A B) (-> (A. x ph) ([/] A x ph))) (y cv A x ph dfsbcq x ph y stdpc4 syl5bi B vtocleg))
+thm (hbsbcg ((x y) (x z) (y z) (A y) (A z) (ph z) (B y)) ((hbsbcg.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. A B) (-> ([/] A x ph) (A. x ([/] A x ph)))) (z cv A x ph dfsbcq y cv z cv e. x ax-17 hbsbcg.1 hbeq z cv A x ph dfsbcq biald z x ph hbs1 syl5bi sylbird B vtocleg))
+thm (hbsbc ((B x) (x y) (A y) (B y)) ((hbsbcg.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (-> (e. A B) ([/] A x ph)) (A. x (-> (e. A B) ([/] A x ph)))) (hbsbcg.1 y cv B e. x ax-17 hbel hbsbcg.1 B ph hbsbcg hbim1))
+thm (hbsbcv ((A x) (x y) (A y)) ((hbsbcv.1 (e. A (V)))) (-> ([/] A x ph) (A. x ([/] A x ph))) (y cv A e. x ax-17 V ph hbsbc hbsbcv.1 A x ph [/] a1bi hbsbcv.1 A x ph [/] a1bi x bial 3imtr4))
+thm (sbcco ((x z) (A z) (y z) (ph y) (ph z)) () (-> (e. A B) (<-> ([/] A y ([/] (cv y) x ph)) ([/] A x ph))) (z cv A y y cv x ph [/] dfsbcq z cv A x ph dfsbcq ph y ax-17 z x sbco2 B vtoclbg))
+thm (sbcco2 ((x y) (ph y) (A y)) ((sbcco2.1 (-> (= (cv x) (cv y)) (= A B)))) (<-> ([/] (cv x) y ([/] B x ph)) ([/] A x ph)) (A x ph [/] y ax-17 sbcco2.1 y cv x cv eqcom B A eqcom 3imtr4 B A x ph dfsbcq syl sbie))
+thm (sbc5g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph)))) (y cv A e. x ax-17 V ph hbsbc x A ph sbceq1 A V e. imbi2d V ceqsexg A V e. ph biimt x cv A = anbi2d x biexdv A V e. x x cv A = ph /\ E. biimt bitr3d bitr3d pm5.74rd A B elisset A B elisset sylc))
+thm (sbc6g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))) (y cv A e. x ax-17 V ph hbsbc x A ph sbceq1 A V e. imbi2d V ceqsalg A V e. ph biimt x cv A = imbi2d x bialdv A V e. x x cv A = ph -> A. biimt bitr3d bitr3d pm5.74rd A B elisset A B elisset sylc))
+thm (sbc5 ((A x)) ((sbc5.1 (e. A (V)))) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (sbc5.1 A V x ph sbc5g ax-mp))
+thm (sbc6 ((A x)) ((sbc6.1 (e. A (V)))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph))) (sbc6.1 x ph hbsbcv sbc6.1 x A ph sbceq1 ceqsal bicomi))
+thm (sbc2or ((A x)) () (\/ (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))) (A V x ph sbc5g A x ph [/] x x cv A = ph /\ E. <-> A x ph [/] x x cv A = ph -> A. <-> orc syl A x ph [/] x x cv A = ph /\ E. pm5.15 x x cv A = ph /\ E. -. x x cv A = ph -> A. pm5.1 x visset x cv A V eleq1 mpbii ph adantr con3i x nexdv x visset x cv A V eleq1 mpbii con3i ph pm2.21d x 19.21aiv sylanc A x ph [/] bibi2d A x ph [/] x x cv A = ph /\ E. <-> orbi2d mpbii pm2.61i))
+thm (sbcie ((A x) (ps x)) ((sbcie.1 (e. A (V))) (sbcie.2 (-> (= (cv x) A) (<-> ph ps)))) (<-> ([/] A x ph) ps) (sbcie.1 x ph sbc5 sbcie.2 biimpa x 19.23aiv sylbi sbcie.2 biimprcd x 19.21aiv sbcie.1 x ph sbc6 sylibr impbi))
+thm (elrabsf ((w z) (A z) (A w) (y z) (w y) (B y) (B z) (B w) (ph z) (x y) (x z) (w x)) ((elrabsf.1 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (e. A ({e.|} x B ph)) (/\ (e. A B) ([/] A x ph))) (elrabsf.1 y cv B e. z ax-17 ph z ax-17 z x ph hbs1 x z ph sbequ12 cbvrab A eleq2i w cv A e. z ax-17 w cv B e. z ax-17 w cv A e. z ax-17 V z cv x ph [/] hbsbc z A z cv x ph [/] sbceq1 z cv A = z 19.8a A z isset sylibr A V e. A z z cv x ph [/] [/] biimt syl bitrd elrabf A B elisset A V e. A z z cv x ph [/] [/] biimt syl pm5.32i bitr4 A B z x ph sbcco pm5.32i 3bitr))
+thm (elabs2 ((x y)) () (<-> (e. A ({|} x ph)) (/\ (e. A (V)) ([/] A x ph))) (x V ph df-rab x visset ph biantrur x biabi eqtr4 A eleq2i y cv V e. x ax-17 A ph elrabsf bitr3))
+thm (elabs () ((elabs.1 (e. A (V)))) (<-> (e. A ({|} x ph)) ([/] A x ph)) (A x ph elabs2 elabs.1 mpbiran))
+thm (sbcn ((x y) (A y) (ph y)) () (-> (e. A B) (<-> ([/] A x (-. ph)) (-. ([/] A x ph)))) (y cv A x ph -. dfsbcq y cv A x ph dfsbcq negbid y x ph sbn B vtoclbg))
+thm (sbcim ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ([/] A x ph) ([/] A x ps)))) (y cv A x ph ps -> dfsbcq y cv A x ph dfsbcq y cv A x ps dfsbcq imbi12d y x ph ps sbim B vtoclbg))
+thm (sbcan ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (/\ ph ps)) (/\ ([/] A x ph) ([/] A x ps)))) (y cv A x ph ps /\ dfsbcq y cv A x ph dfsbcq y cv A x ps dfsbcq anbi12d y x ph ps sban B vtoclbg))
+thm (sbcor ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (\/ ph ps)) (\/ ([/] A x ph) ([/] A x ps)))) (y cv A x ph ps \/ dfsbcq y cv A x ph dfsbcq y cv A x ps dfsbcq orbi12d y x ph ps sbor B vtoclbg))
+thm (sbcbi ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (<-> ph ps)) (<-> ([/] A x ph) ([/] A x ps)))) (y cv A x ph ps <-> dfsbcq y cv A x ph dfsbcq y cv A x ps dfsbcq bibi12d y x ph ps sbbi B vtoclbg))
+thm (sbcal ((x z) (A x) (A z) (x y) (y z) (ph z)) () (-> (e. A B) (<-> ([/] A y (A. x ph)) (A. x ([/] A y ph)))) (z cv A y x ph A. dfsbcq z cv A y ph dfsbcq x bialdv z y x ph sbal B vtoclbg))
+thm (sbcex ((x z) (A x) (A z) (x y) (y z) (ph z)) () (-> (e. A B) (<-> ([/] A y (E. x ph)) (E. x ([/] A y ph)))) (z cv A y x ph E. dfsbcq z cv A y ph dfsbcq x biexdv z y x ph sbex B vtoclbg))
+thm (sbcel1 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> ([/] A x (e. (cv x) B)) (e. A B))) (A C elisset A V x elex y cv A e. x ax-17 V x cv B e. hbsbc A V e. A B e. -> x ax-17 hbbi x A x cv B e. sbceq1 x cv A B eleq1 bitr3d A V e. imbi2d 19.23ai pm5.74rd mpcom syl))
+thm (sbcel2 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (<-> ([/] B x (e. A (cv x))) (e. A B))) (B C elisset B V x elex y cv B e. x ax-17 V A x cv e. hbsbc B V e. A B e. -> x ax-17 hbbi x B A x cv e. sbceq1 x cv B A eleq2 bitr3d B V e. imbi2d 19.23ai pm5.74rd mpcom syl))
+thm (bisbcdv ((ph x)) ((bisbcdv.1 (-> ph (<-> ps ch)))) (-> (/\ (e. A B) ph) (<-> ([/] A x ps) ([/] A x ch))) (A B x ps ch <-> a4sbc bisbcdv.1 x 19.21aiv syl5 A B x ps ch sbcbi sylibd imp))
+thm (sbcgf ((A y) (ph y) (x y)) ((sbcgf.1 (-> ph (A. x ph)))) (-> (e. A B) (<-> ([/] A x ph) ph)) (A B y x ph sbcco x equid sbcgf.1 y sbf x cv x cv = a1i A B y bisbcdv mpan2 A B y ph sbc5g A B y elex ph biantrurd y y cv A = ph 19.41v syl6rbbr 3bitrd bitr3d))
+thm (sbc19.21g () ((sbc19.21g.1 (-> ph (A. x ph)))) (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ph ([/] A x ps)))) (A B x ph ps sbcim sbc19.21g.1 A B sbcgf A x ps [/] imbi1d bicomd bitr4d))
+thm (ra4sbcf ((A y) (x y) (B x) (B y)) ((ra4sbcf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. A B) (-> (A.e. x B ph) ([/] A x ph))) (x B ph df-ral ra4sbcf.1 ra4sbcf.1 B ph hbsbc x cv A B eleq1 x A ph sbceq1 imbi12d B cla4gf pm2.43b sylbi com12))
+thm (ra5 () ((ra5.1 (-> ph (A. x ph)))) (-> (A.e. x A (-> ph ps)) (-> ph (A.e. x A ps))) (x A ph ps -> df-ral x cv A e. ph ps bi2.04 x bial bitr ra5.1 x cv A e. ps -> stdpc5 sylbi x A ps df-ral syl6ibr))
+thm (axrep ((ph w) (x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph))))))) (y visset w cv y cv x cv eleq2 y ph A. anbi1d x biexdv z cv x cv e. bibi2d z bialdv x biexdv x y z ph z cv y cv = -> A. E. A. imbi2d y ph ax-4 z cv y cv = imim1i z 19.20i y 19.22i x 19.20i x y z ph w ax-rep syl z cv y cv e. x ax-17 x x cv w cv e. y ph A. /\ hbe1 hbbi z hbal z cv x cv e. y ax-17 x cv w cv e. y ax-17 y ph hba1 hban x hbex hbbi z hbal y cv x cv z cv eleq2 x x cv w cv e. y ph A. /\ E. bibi1d z bialdv cbvex sylib vtocl 19.35ri))
+thm (axrep2 ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv w)) (A. y ph))))))) (y z ph z cv y cv = -> A. hbe1 z cv x cv e. y ax-17 x cv w cv e. y ax-17 y ph hba1 hban x hbex hbbi z hbal hbim x hbex w visset y w x elequ2 y ph A. anbi1d x biexdv z cv x cv e. bibi2d z bialdv y z ph z cv y cv = -> A. E. imbi2d x biexdv x y z ph axrep vtoclf))
+thm (zfrep2 ((x y) (x z) (w x) (y z) (w y) (w z)) ((zfrep2.1 (-> ph (A. z ph)))) (-> (A. x (E. z (A. y (-> ph (= (cv y) (cv z)))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))) (x z y ph w axrep2 19.35i y cv x cv e. z ax-17 x cv w cv e. z ax-17 z ph hba1 hban x hbex hbbi y hbal y cv z cv e. x ax-17 x x cv w cv e. ph /\ hbe1 hbbi y hbal x cv z cv = y ax-17 x z y elequ2 z ph ax-4 zfrep2.1 impbi x cv w cv e. anbi2i x biex x cv z cv = a1i bibi12d biald cbvex sylib))
+thm (zfrep3 ((x y) (x z) (w x) (y z) (w y) (w z)) ((zfrep3.1 (-> ph (A. z ph)))) (-> (A. x (-> (e. (cv x) (cv w)) (E. z (A. y (-> ph (= (cv y) (cv z))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))) (z x cv w cv e. y ph y cv z cv = -> A. 19.37v x cv w cv e. ph y cv z cv = impexp y bial y x cv w cv e. ph y cv z cv = -> 19.21v bitr2 z biex bitr3 x bial x cv w cv e. z ax-17 zfrep3.1 hban x y w zfrep2 sylbi x cv w cv e. ph anabs5 x biex y cv z cv e. bibi2i y bial z biex sylib))
+thm (zfrepclf ((y z) (A y) (v y) (A z) (v z) (A v) (ph z) (ph v) (A w) (v w) (x y) (x z) (v x) (w x)) ((zfrepclf.1 (-> (e. (cv w) A) (A. x (e. (cv w) A)))) (zfrepclf.2 (e. A (V))) (zfrepclf.3 (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z)))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))) (zfrepclf.2 w cv v cv e. x ax-17 zfrepclf.1 hbeq v cv A x cv eleq2 zfrepclf.3 syl6bi 19.21ai ph z ax-17 x v y zfrep3 syl w cv v cv e. x ax-17 zfrepclf.1 hbeq v cv A x cv eleq2 ph anbi1d biexd y cv z cv e. bibi2d y bialdv z biexdv mpbid vtocle))
+thm (zfrep3cl ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) ((zfrep3cl.1 (e. A (V))) (zfrep3cl.2 (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z)))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))) (y cv A e. x ax-17 zfrep3cl.1 zfrep3cl.2 zfrepclf))
+thm (zfrep4 ((ph y) (ph z) (y z) (ps z) (x y) (x z)) ((zfrep4.1 (e. ({|} x ph) (V))) (zfrep4.2 (-> ph (E. z (A. y (-> ps (= (cv y) (cv z)))))))) (e. ({|} y (E. x (/\ ph ps))) (V)) (x ph abid ps anbi1i x biex y biabi y x ph hbab1 zfrep4.1 x ph abid zfrep4.2 sylbi zfrepclf z cv y x x cv x ph {|} e. ps /\ E. eqab z biex mpbir issetri eqeltrr))
+thm (zfaus ((x y) (A x) (x z) (A y) (y z) (A z) (ph y) (ph z)) ((zfaus.1 (e. A (V)))) (E. y (A. x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph)))) (zfaus.1 y z a9e y z x equtr y x equcomi syl6 ph adantrd x 19.21aiv y 19.22i ax-mp z cv A e. a1i zfrep3cl z cv x cv = z cv A e. ph an12 z biex x visset z cv x cv A eleq1 ph anbi1d ceqsexv bitr3 x cv y cv e. bibi2i x bial y biex mpbi))
+thm (bm1.3ii ((ph x) (x z) (ph z) (x y) (y z)) ((bm1.3ii.1 (E. x (A. y (-> ph (e. (cv y) (cv x))))))) (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (bm1.3ii.1 x z y elequ2 ph imbi2d y bialdv cbvexv mpbi z visset x y ph zfaus pm3.2i exan x y ph y cv z cv e. -> A. y y cv x cv e. y cv z cv e. ph /\ <-> A. 19.42v y ph y cv z cv e. -> y cv x cv e. y cv z cv e. ph /\ <-> 19.26 ph y cv z cv e. y cv x cv e. bimsc1 y 19.20i sylbir x 19.22i sylbir z 19.23aiv ax-mp))
+thm (nalset ((x y) (x z) (y z)) () (-. (E. x (A. y (e. (cv y) (cv x))))) (x y y cv x cv e. alexn x visset y z z cv z cv e. -. zfaus z y y elequ1 z y x elequ1 z y z elequ1 z y y elequ2 bitrd negbid anbi12d bibi12d a4b1 y cv y cv e. y cv x cv e. pclem6 syl y 19.22i ax-mp mpgbi))
+thm (nvelv ((x y)) () (-. (e. (V) (V))) (x y nalset y visset y cv x cv e. tbt y bial x cv V y dfcleq bitr4 x biex mtbi V x isset mtbir))
+thm (vnex () () (-. (E. x (= (cv x) (V)))) (nvelv V x isset mtbi))
+thm (dfin5 ((A x) (B x)) () (= (i^i A B) ({e.|} x A (e. (cv x) B))) (A B x df-in x A x cv B e. df-rab eqtr4))
+thm (dfss () () (<-> (C_ A B) (= A (i^i A B))) (A B df-ss A B i^i A eqcom bitr))
+thm (dfdif2 ((A x) (B x)) () (= (\ A B) ({e.|} x A (-. (e. (cv x) B)))) (A B x df-dif x A x cv B e. -. df-rab eqtr4))
+thm (eldif ((A x) (B x) (C x)) () (<-> (e. A (\ B C)) (/\ (e. A B) (-. (e. A C)))) (A B C \ elisset A B elisset A C e. -. adantr x cv A B eleq1 x cv A C eleq1 negbid anbi12d B C x df-dif V elab2g pm5.21nii))
+thm (dfss2 ((A x) (B x)) () (<-> (C_ A B) (A. x (-> (e. (cv x) A) (e. (cv x) B)))) (A B dfss A B x df-in A eqeq2i A x x cv A e. x cv B e. /\ eqab 3bitr x cv A e. x cv B e. pm4.71 x bial bitr4))
+thm (dfss3 ((A x) (B x)) () (<-> (C_ A B) (A.e. x A (e. (cv x) B))) (A B x dfss2 x A x cv B e. df-ral bitr4))
+thm (dfss2f ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (C_ A B) (A. x (-> (e. (cv x) A) (e. (cv x) B)))) (A B y dfss2 x cv A e. x cv B e. -> y ax-17 dfss2f.1 dfss2f.2 hbim x cv y cv A eleq1 x cv y cv B eleq1 imbi12d cbval bitr4))
+thm (dfss3f ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (<-> (C_ A B) (A.e. x A (e. (cv x) B))) (dfss2f.1 dfss2f.2 dfss2f x A x cv B e. df-ral bitr4))
+thm (hbss ((A y) (B y) (x y)) ((dfss2f.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (dfss2f.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (C_ A B) (A. x (C_ A B))) (x x cv A e. x cv B e. -> hba1 dfss2f.1 dfss2f.2 dfss2f dfss2f.1 dfss2f.2 dfss2f x bial 3imtr4))
+thm (ssel ((A x) (B x) (C x)) () (-> (C_ A B) (-> (e. C A) (e. C B))) (A B x dfss2 biimp 19.21bi x cv C = anim2d x 19.22dv C A x df-clel C B x df-clel 3imtr4g))
+thm (ssel2 () () (-> (/\ (C_ A B) (e. C A)) (e. C B)) (A B C ssel imp))
+thm (sseli () ((sseli.1 (C_ A B))) (-> (e. C A) (e. C B)) (sseli.1 A B C ssel ax-mp))
+thm (sselii () ((sseli.1 (C_ A B)) (sselii.2 (e. C A))) (e. C B) (sselii.2 sseli.1 C sseli ax-mp))
+thm (sseld () ((sseld.1 (-> ph (C_ A B)))) (-> ph (-> (e. C A) (e. C B))) (sseld.1 A B C ssel syl))
+thm (sseldd () ((sseld.1 (-> ph (C_ A B))) (sseldd.2 (-> ph (e. C A)))) (-> ph (e. C B)) (sseldd.2 sseld.1 C sseld mpd))
+thm (ssriv ((A x) (B x)) ((ssriv.1 (-> (e. (cv x) A) (e. (cv x) B)))) (C_ A B) (A B x dfss2 ssriv.1 mpgbir))
+thm (ssrdv ((A x) (B x) (ph x)) ((ssrdv.1 (-> ph (-> (e. (cv x) A) (e. (cv x) B))))) (-> ph (C_ A B)) (ssrdv.1 x 19.21aiv A B x dfss2 sylibr))
+thm (sstr2 ((A x) (B x) (C x)) () (-> (C_ A B) (-> (C_ B C) (C_ A C))) (A B x dfss2 x cv A e. x cv B e. x cv C e. imim1 x 19.20ii B C x dfss2 A C x dfss2 3imtr4g sylbi))
+thm (sstr () () (-> (/\ (C_ A B) (C_ B C)) (C_ A C)) (A B C sstr2 imp))
+thm (sstri () ((sstri.1 (C_ A B)) (sstri.2 (C_ B C))) (C_ A C) (sstri.1 sstri.2 A B C sstr2 mp2))
+thm (sstrd () ((sstrd.1 (-> ph (C_ A B))) (sstrd.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (A B C sstr sstrd.1 sstrd.2 sylanc))
+thm (sylan9ss () ((sylan9ss.1 (-> ph (C_ A B))) (sylan9ss.2 (-> ps (C_ B C)))) (-> (/\ ph ps) (C_ A C)) (sylan9ss.1 ps adantr sylan9ss.2 ph adantl sstrd))
+thm (sylan9ssr () ((sylan9ssr.1 (-> ph (C_ A B))) (sylan9ssr.2 (-> ps (C_ B C)))) (-> (/\ ps ph) (C_ A C)) (sylan9ssr.1 sylan9ssr.2 sylan9ss ancoms))
+thm (eqss ((A x) (B x)) () (<-> (= A B) (/\ (C_ A B) (C_ B A))) (x x cv A e. x cv B e. albi A B x dfcleq A B x dfss2 B A x dfss2 anbi12i 3bitr4))
+thm (eqssi () ((eqssi.1 (C_ A B)) (eqssi.2 (C_ B A))) (= A B) (A B eqss eqssi.1 eqssi.2 mpbir2an))
+thm (eqssd () ((eqssd.1 (-> ph (C_ A B))) (eqssd.2 (-> ph (C_ B A)))) (-> ph (= A B)) (eqssd.1 eqssd.2 jca A B eqss sylibr))
+thm (ssid () () (C_ A A) (A eqid A A eqss mpbi pm3.26i))
+thm (ssv ((A x)) () (C_ A (V)) (x cv A elisset ssriv))
+thm (sseq1 () () (-> (= A B) (<-> (C_ A C) (C_ B C))) (B A C sstr2 A B C sstr2 anim12i B A eqss A C C_ B C C_ bi 3imtr4 eqcoms))
+thm (sseq2 () () (-> (= A B) (<-> (C_ C A) (C_ C B))) (C A B sstr2 com12 C B A sstr2 com12 anim12i A B eqss C A C_ C B C_ bi 3imtr4))
+thm (sseq12 () () (-> (/\ (= A B) (= C D)) (<-> (C_ A C) (C_ B D))) (A B C sseq1 C D B sseq2 sylan9bb))
+thm (sseq1i () ((sseq1i.1 (= A B))) (<-> (C_ A C) (C_ B C)) (sseq1i.1 A B C sseq1 ax-mp))
+thm (sseq2i () ((sseq1i.1 (= A B))) (<-> (C_ C A) (C_ C B)) (sseq1i.1 A B C sseq2 ax-mp))
+thm (sseq12i () ((sseq1i.1 (= A B)) (sseq12i.2 (= C D))) (<-> (C_ A C) (C_ B D)) (sseq1i.1 C sseq1i sseq12i.2 B sseq2i bitr))
+thm (sseq1d () ((sseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C_ A C) (C_ B C))) (sseq1d.1 A B C sseq1 syl))
+thm (sseq2d () ((sseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C_ C A) (C_ C B))) (sseq1d.1 A B C sseq2 syl))
+thm (sseq12d () ((sseq1d.1 (-> ph (= A B))) (sseq12d.2 (-> ph (= C D)))) (-> ph (<-> (C_ A C) (C_ B D))) (sseq1d.1 C sseq1d sseq12d.2 B sseq2d bitrd))
+thm (eqsstr () ((eqsstr.1 (= A B)) (eqsstr.2 (C_ B C))) (C_ A C) (eqsstr.2 eqsstr.1 C sseq1i mpbir))
+thm (eqsstr3 () ((eqsstr3.1 (= B A)) (eqsstr3.2 (C_ B C))) (C_ A C) (eqsstr3.1 eqcomi eqsstr3.2 eqsstr))
+thm (sseqtr () ((sseqtr.1 (C_ A B)) (sseqtr.2 (= B C))) (C_ A C) (sseqtr.1 sseqtr.2 A sseq2i mpbi))
+thm (sseqtr4 () ((sseqtr4.1 (C_ A B)) (sseqtr4.2 (= C B))) (C_ A C) (sseqtr4.1 sseqtr4.2 eqcomi sseqtr))
+thm (eqsstrd () ((eqsstrd.1 (-> ph (= A B))) (eqsstrd.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (eqsstrd.2 eqsstrd.1 C sseq1d mpbird))
+thm (eqsstr3d () ((eqsstr3d.1 (-> ph (= B A))) (eqsstr3d.2 (-> ph (C_ B C)))) (-> ph (C_ A C)) (eqsstr3d.1 eqcomd eqsstr3d.2 eqsstrd))
+thm (sseqtrd () ((sseqtrd.1 (-> ph (C_ A B))) (sseqtrd.2 (-> ph (= B C)))) (-> ph (C_ A C)) (sseqtrd.1 sseqtrd.2 A sseq2d mpbid))
+thm (sseqtr4d () ((sseqtr4d.1 (-> ph (C_ A B))) (sseqtr4d.2 (-> ph (= C B)))) (-> ph (C_ A C)) (sseqtr4d.1 sseqtr4d.2 eqcomd sseqtrd))
+thm (3sstr3 () ((3sstr3.1 (C_ A B)) (3sstr3.2 (= A C)) (3sstr3.3 (= B D))) (C_ C D) (3sstr3.1 3sstr3.2 B sseq1i 3sstr3.3 C sseq2i bitr2 mpbir))
+thm (3sstr4 () ((3sstr4.1 (C_ A B)) (3sstr4.2 (= C A)) (3sstr4.3 (= D B))) (C_ C D) (3sstr4.1 3sstr4.2 eqcomi 3sstr4.3 eqcomi 3sstr3))
+thm (3sstr3g () ((3sstr3g.1 (-> ph (C_ A B))) (3sstr3g.2 (= A C)) (3sstr3g.3 (= B D))) (-> ph (C_ C D)) (3sstr3g.1 3sstr3g.2 3sstr3g.3 sseq12i sylib))
+thm (3sstr4g () ((3sstr4g.1 (-> ph (C_ A B))) (3sstr4g.2 (= C A)) (3sstr4g.3 (= D B))) (-> ph (C_ C D)) (3sstr4g.1 3sstr4g.2 eqcomi 3sstr4g.3 eqcomi 3sstr3g))
+thm (3sstr3d () ((3sstr3d.1 (-> ph (C_ A B))) (3sstr3d.2 (-> ph (= A C))) (3sstr3d.3 (-> ph (= B D)))) (-> ph (C_ C D)) (3sstr3d.1 3sstr3d.2 3sstr3d.3 sseq12d mpbid))
+thm (3sstr4d () ((3sstr4d.1 (-> ph (C_ A B))) (3sstr4d.2 (-> ph (= C A))) (3sstr4d.3 (-> ph (= D B)))) (-> ph (C_ C D)) (3sstr4d.1 3sstr4d.2 eqcomd 3sstr4d.3 eqcomd 3sstr3d))
+thm (syl5ss () ((syl5ss.1 (-> ph (C_ A B))) (syl5ss.2 (= C A))) (-> ph (C_ C B)) (syl5ss.1 syl5ss.2 B sseq1i sylibr))
+thm (syl5ssr () ((syl5ssr.1 (-> ph (C_ A B))) (syl5ssr.2 (= A C))) (-> ph (C_ C B)) (syl5ssr.1 syl5ssr.2 eqcomi syl5ss))
+thm (syl6ss () ((syl6ss.1 (-> ph (C_ A B))) (syl6ss.2 (= B C))) (-> ph (C_ A C)) (syl6ss.1 syl6ss.2 A sseq2i sylib))
+thm (syl6ssr () ((syl6ssr.1 (-> ph (C_ A B))) (syl6ssr.2 (= C B))) (-> ph (C_ A C)) (syl6ssr.1 syl6ssr.2 eqcomi syl6ss))
+thm (eqimss () () (-> (= A B) (C_ A B)) (A ssid A B A sseq2 mpbii))
+thm (eqimss2 () () (-> (= B A) (C_ A B)) (A B eqimss eqcoms))
+thm (nss ((A x) (B x)) () (<-> (-. (C_ A B)) (E. x (/\ (e. (cv x) A) (-. (e. (cv x) B))))) (x x cv A e. x cv B e. -> exnal x cv A e. x cv B e. annim x biex A B x dfss2 negbii 3bitr4r))
+thm (ss2ab ((ph y) (ps y) (x y)) () (<-> (C_ ({|} x ph) ({|} x ps)) (A. x (-> ph ps))) (y x ph hbab1 y x ps hbab1 dfss2f x ph abid x ps abid imbi12i x bial bitr))
+thm (ss2abi () ((ss2abi.1 (-> ph ps))) (C_ ({|} x ph) ({|} x ps)) (x ph ps ss2ab ss2abi.1 mpgbir))
+thm (ss2rab () () (<-> (C_ ({e.|} x A ph) ({e.|} x A ps)) (A.e. x A (-> ph ps))) (x A ph df-rab x A ps df-rab sseq12i x x cv A e. ph /\ x cv A e. ps /\ ss2ab x A ph ps -> df-ral x cv A e. ph ps imdistan x bial bitr2 3bitr))
+thm (ss2rabi () ((ss2rabi.1 (-> (e. (cv x) A) (-> ph ps)))) (C_ ({e.|} x A ph) ({e.|} x A ps)) (x A ph ps ss2rab ss2rabi.1 mprgbir))
+thm (rabss ((A x) (B x)) () (-> (C_ A B) (C_ ({e.|} x A ph) ({e.|} x B ph))) (A B x dfss2 x cv A e. x cv B e. ph pm3.45 x 19.20i x x cv A e. ph /\ x cv B e. ph /\ ss2ab sylibr x A ph df-rab x B ph df-rab sseq12i sylibr sylbi))
+thm (ssab ((A x)) () (C_ ({|} x (/\ (e. (cv x) A) ph)) A) (x cv A e. ph pm3.26 x ss2abi x A abid2 sseqtr))
+thm (ssrab ((A x)) () (C_ ({e.|} x A ph) A) (x A ph df-rab x A ph ssab eqsstr))
+thm (dfpss2 () () (<-> (C: A B) (/\ (C_ A B) (-. (= A B)))) (A B df-pss A B df-ne A B C_ anbi2i bitr))
+thm (dfpss3 () () (<-> (C: A B) (/\ (C_ A B) (-. (C_ B A)))) (A B eqss negbii A B C_ anbi2i A B dfpss2 A B C_ B A C_ anclb A B C_ B A C_ iman A B C_ A B C_ B A C_ /\ iman 3bitr3 bicon4i 3bitr4))
+thm (psseq1 () () (-> (= A B) (<-> (C: A C) (C: B C))) (A B C sseq1 A B C neeq1 anbi12d A C df-pss B C df-pss 3bitr4g))
+thm (psseq2 () () (-> (= A B) (<-> (C: C A) (C: C B))) (A B C sseq2 A B C neeq2 anbi12d C A df-pss C B df-pss 3bitr4g))
+thm (psseq1i () ((psseq1i.1 (= A B))) (<-> (C: A C) (C: B C)) (psseq1i.1 A B C psseq1 ax-mp))
+thm (psseq2i () ((psseq1i.1 (= A B))) (<-> (C: C A) (C: C B)) (psseq1i.1 A B C psseq2 ax-mp))
+thm (psseq12i () ((psseq1i.1 (= A B)) (psseq12i.2 (= C D))) (<-> (C: A C) (C: B D)) (psseq1i.1 C psseq1i psseq12i.2 B psseq2i bitr))
+thm (psseq1d () ((psseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C: A C) (C: B C))) (psseq1d.1 A B C psseq1 syl))
+thm (psseq2d () ((psseq1d.1 (-> ph (= A B)))) (-> ph (<-> (C: C A) (C: C B))) (psseq1d.1 A B C psseq2 syl))
+thm (psseq12d () ((psseq1d.1 (-> ph (= A B))) (psseq12d.2 (-> ph (= C D)))) (-> ph (<-> (C: A C) (C: B D))) (psseq1d.1 C psseq1d psseq12d.2 B psseq2d bitrd))
+thm (pssss () () (-> (C: A B) (C_ A B)) (A B df-pss pm3.26bd))
+thm (pssssd () ((pssssd.1 (-> ph (C: A B)))) (-> ph (C_ A B)) (pssssd.1 A B pssss syl))
+thm (sspss () () (<-> (C_ A B) (\/ (C: A B) (= A B))) (A B dfpss2 biimpr exp con1d orrd A B pssss A B eqimss jaoi impbi))
+thm (pssirr () () (-. (C: A A)) (A A C_ pm3.24 A A dfpss3 mtbir))
+thm (pssn2lp () () (-. (/\ (C: A B) (C: B A))) (A B C_ B A C_ /\ pm3.24 A B dfpss3 B A dfpss3 anbi12i A B C_ B A C_ -. B A C_ A B C_ -. an42 bitr A B C_ -. B A C_ -. orc B A C_ -. adantr A B C_ B A C_ ianor sylibr A B C_ B A C_ /\ anim2i sylbi mto))
+thm (sspsstri () () (<-> (\/ (C_ A B) (C_ B A)) (\/\/ (C: A B) (= A B) (C: B A))) (A B sspss B A sspss B A eqcom B A C: orbi2i bitr orbi12i A B C: B A C: A B = orordir A B C: B A C: A B = or23 A B C: A B = B A C: df-3or bitr4 bitr3 bitr))
+thm (ssnpss () () (-> (C_ A B) (-. (C: B A))) (A B sspss A B pssn2lp A B C: B A C: imnan mpbir A pssirr A B A psseq1 mtbii jaoi sylbi))
+thm (psstr () () (-> (/\ (C: A B) (C: B C)) (C: A C)) (A B pssss B C pssss sylan9ss C B pssn2lp A C B psseq1 B C C: anbi1d mtbiri con2i jca A C dfpss2 sylibr))
+thm (sspsstr () () (-> (/\ (C_ A B) (C: B C)) (C: A C)) (A B C psstr exp A B C psseq1 biimprd jaoi imp A B sspss sylanb))
+thm (psssstr () () (-> (/\ (C: A B) (C_ B C)) (C: A C)) (A B C psstr exp B C A psseq2 biimpcd jaod imp B C sspss sylan2b))
+thm (difeq1 ((A x) (B x) (C x)) () (-> (= A B) (= (\ A C) (\ B C))) (A B x cv eleq2 x cv C e. -. anbi1d x biabdv A C x df-dif B C x df-dif 3eqtr4g))
+thm (difeq2 ((A x) (B x) (C x)) () (-> (= A B) (= (\ C A) (\ C B))) (A B x cv eleq2 negbid x cv C e. anbi2d x biabdv C A x df-dif C B x df-dif 3eqtr4g))
+thm (difeq1i () ((difeq1i.1 (= A B))) (= (\ A C) (\ B C)) (difeq1i.1 A B C difeq1 ax-mp))
+thm (difeq2i () ((difeq1i.1 (= A B))) (= (\ C A) (\ C B)) (difeq1i.1 A B C difeq2 ax-mp))
+thm (difeq12i () ((difeq1i.1 (= A B)) (difeq12i.2 (= C D))) (= (\ A C) (\ B D)) (difeq1i.1 C difeq1i difeq12i.2 B difeq2i eqtr))
+thm (difeq1d () ((difeq1d.1 (-> ph (= A B)))) (-> ph (= (\ A C) (\ B C))) (difeq1d.1 A B C difeq1 syl))
+thm (difeq2d () ((difeq1d.1 (-> ph (= A B)))) (-> ph (= (\ C A) (\ C B))) (difeq1d.1 A B C difeq2 syl))
+thm (difeqri ((A x) (B x) (C x)) ((difeqri.1 (<-> (/\ (e. (cv x) A) (-. (e. (cv x) B))) (e. (cv x) C)))) (= (\ A B) C) (A B x df-dif difeqri.1 bicomi biabri eqtr4))
+thm (hbdif ((A y) (B y) (x y)) ((hbdif.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbdif.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (\ A B)) (A. x (e. (cv y) (\ A B)))) (hbdif.1 hbdif.2 hbne hban y cv A B eldif y cv A B eldif x bial 3imtr4))
+thm (eldifi () () (-> (e. A (\ B C)) (e. A B)) (A B C eldif pm3.26bd))
+thm (eldifn () () (-> (e. A (\ B C)) (-. (e. A C))) (A B C eldif pm3.27bd))
+thm (elndif () () (-> (e. A B) (-. (e. A (\ C B)))) (A C B eldifn con2i))
+thm (neldif () () (-> (/\ (e. A B) (-. (e. A (\ B C)))) (e. A C)) (A B C eldif biimpr exp con1d imp))
+thm (difdif ((A x) (B x)) () (= (\ A (\ B A)) A) (x cv B A eldif negbii x cv B e. x cv A e. iman bitr4 x cv A e. anbi2i x cv A e. x cv B e. pm4.45im bitr4 difeqri))
+thm (difss ((A x) (B x)) () (C_ (\ A B) A) (x cv A B eldifi ssriv))
+thm (ddif ((A x)) () (= (\ (V) (\ (V) A)) A) (x cv V A eldif x visset mpbiran bicon2i x visset x cv V A \ e. -. biantrur bitr2 difeqri))
+thm (ssconb ((A x) (B x) (C x)) () (-> (/\ (C_ A C) (C_ B C)) (<-> (C_ A (\ C B)) (C_ B (\ C A)))) (x cv A e. x cv C e. -> x cv B e. x cv C e. -> pm5.1 A C x cv ssel B C x cv ssel syl2an x cv A e. x cv B e. bi2.03 A C C_ B C C_ /\ a1i anbi12d x cv A e. x cv C e. x cv B e. -. jcab x cv B e. x cv C e. x cv A e. -. jcab 3bitr4g x cv C B eldif x cv A e. imbi2i x cv C A eldif x cv B e. imbi2i 3bitr4g x bialdv A C B \ x dfss2 B C A \ x dfss2 3bitr4g))
+thm (sscon ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (\ C B) (\ C A))) (A B x cv ssel con3d x cv C e. anim2d x cv C B eldif x cv C A eldif 3imtr4g ssrdv))
+thm (ssdif ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (\ A C) (\ B C))) (A B x cv ssel x cv C e. -. anim1d x cv A C eldif x cv B C eldif 3imtr4g ssrdv))
+thm (elun ((A x) (B x) (C x)) () (<-> (e. A (u. B C)) (\/ (e. A B) (e. A C))) (A B C u. elisset A B elisset A C elisset jaoi x cv A B eleq1 x cv A C eleq1 orbi12d B C x df-un V elab2g pm5.21nii))
+thm (uneqri ((A x) (B x) (C x)) ((uneqri.1 (<-> (\/ (e. (cv x) A) (e. (cv x) B)) (e. (cv x) C)))) (= (u. A B) C) (x cv A B elun uneqri.1 bitr eqri))
+thm (unidm ((A x)) () (= (u. A A) A) (x cv A e. oridm uneqri))
+thm (uncom ((A x) (B x)) () (= (u. A B) (u. B A)) (x cv A e. x cv B e. orcom x cv A B elun x cv B A elun 3bitr4 eqri))
+thm (uneq1 ((A x) (B x) (C x)) () (-> (= A B) (= (u. A C) (u. B C))) (A B x cv eleq2 x cv C e. orbi1d x cv A C elun x cv B C elun 3bitr4g eqrd))
+thm (uneq2 () () (-> (= A B) (= (u. C A) (u. C B))) (A B C uneq1 C A uncom C B uncom 3eqtr4g))
+thm (uneq1i () ((uneq1i.1 (= A B))) (= (u. A C) (u. B C)) (uneq1i.1 A B C uneq1 ax-mp))
+thm (uneq2i () ((uneq1i.1 (= A B))) (= (u. C A) (u. C B)) (uneq1i.1 A B C uneq2 ax-mp))
+thm (uneq12i () ((uneq1i.1 (= A B)) (uneq12i.2 (= C D))) (= (u. A C) (u. B D)) (uneq1i.1 C uneq1i uneq12i.2 B uneq2i eqtr))
+thm (uneq1d () ((uneq1d.1 (-> ph (= A B)))) (-> ph (= (u. A C) (u. B C))) (uneq1d.1 A B C uneq1 syl))
+thm (uneq2d () ((uneq1d.1 (-> ph (= A B)))) (-> ph (= (u. C A) (u. C B))) (uneq1d.1 A B C uneq2 syl))
+thm (uneq12d () ((uneq1d.1 (-> ph (= A B))) (uneq12d.2 (-> ph (= C D)))) (-> ph (= (u. A C) (u. B D))) (uneq1d.1 C uneq1d uneq12d.2 B uneq2d eqtrd))
+thm (uneq12 () () (-> (/\ (= A B) (= C D)) (= (u. A C) (u. B D))) (A B C uneq1 C D B uneq2 sylan9eq))
+thm (hbun ((A y) (B y) (x y)) ((hbun.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbun.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (u. A B)) (A. x (e. (cv y) (u. A B)))) (hbun.1 hbun.2 hbor y cv A B elun y cv A B elun x bial 3imtr4))
+thm (unass ((A x) (B x) (C x)) () (= (u. (u. A B) C) (u. A (u. B C))) (x cv A e. x cv B e. x cv C e. orass x cv A B elun x cv C e. orbi1i x cv B C elun x cv A e. orbi2i 3bitr4 x cv A B C u. elun bitr4 uneqri))
+thm (un12 () () (= (u. A (u. B C)) (u. B (u. A C))) (A B uncom C uneq1i A B C unass B A C unass 3eqtr3))
+thm (un23 () () (= (u. (u. A B) C) (u. (u. A C) B)) (B C uncom A uneq2i A B C unass A C B unass 3eqtr4))
+thm (un4 () () (= (u. (u. A B) (u. C D)) (u. (u. A C) (u. B D))) (B C D un12 A uneq2i A B C D u. unass A C B D u. unass 3eqtr4))
+thm (unundi () () (= (u. A (u. B C)) (u. (u. A B) (u. A C))) (A unidm B C u. uneq1i A A B C un4 eqtr3))
+thm (unundir () () (= (u. (u. A B) C) (u. (u. A C) (u. B C))) (C unidm A B u. uneq2i A B C C un4 eqtr3))
+thm (ssun1 ((A x) (B x)) () (C_ A (u. A B)) (x cv A e. x cv B e. orc x cv A B elun sylibr ssriv))
+thm (ssun2 () () (C_ A (u. B A)) (A B ssun1 A B uncom sseqtr))
+thm (ssun3 () () (-> (C_ A B) (C_ A (u. B C))) (B C ssun1 A B B C u. sstr2 mpi))
+thm (ssun4 () () (-> (C_ A B) (C_ A (u. C B))) (B C ssun2 A B C B u. sstr2 mpi))
+thm (elun1 () () (-> (e. A B) (e. A (u. B C))) (B C ssun1 A sseli))
+thm (elun2 () () (-> (e. A B) (e. A (u. C B))) (B C ssun2 A sseli))
+thm (unss1 ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (u. A C) (u. B C))) (x cv A e. x cv B e. -> id x cv C e. orim1d x cv A C elun x cv B C elun 3imtr4g x 19.20i A B x dfss2 A C u. B C u. x dfss2 3imtr4))
+thm (ssequn1 ((A x) (B x)) () (<-> (C_ A B) (= (u. A B) B)) (A B x df-un B eqeq2i A B u. B eqcom x cv A e. x cv B e. pm4.72 x bial A B x dfss2 B x x cv A e. x cv B e. \/ eqab 3bitr4 3bitr4r))
+thm (unss2 () () (-> (C_ A B) (C_ (u. C A) (u. C B))) (A B C unss1 C A uncom C B uncom 3sstr4g))
+thm (unss12 () () (-> (/\ (C_ A B) (C_ C D)) (C_ (u. A C) (u. B D))) (A B C unss1 C D C_ adantr C D B unss2 A B C_ adantl sstrd))
+thm (ssequn2 () () (<-> (C_ A B) (= (u. B A) B)) (A B ssequn1 A B uncom B eqeq1i bitr))
+thm (unss ((A x) (B x) (C x)) () (<-> (/\ (C_ A C) (C_ B C)) (C_ (u. A B) C)) (x cv A B elun x cv C e. imbi1i x cv A e. x cv B e. x cv C e. jaob bitr x bial A B u. C x dfss2 A C x dfss2 B C x dfss2 anbi12i x x cv A e. x cv C e. -> x cv B e. x cv C e. -> 19.26 bitr4 3bitr4r))
+thm (unssi () ((unssi.1 (C_ A C)) (unssi.2 (C_ B C))) (C_ (u. A B) C) (unssi.1 unssi.2 pm3.2i A C B unss mpbi))
+thm (ssun () () (-> (\/ (C_ A B) (C_ A C)) (C_ A (u. B C))) (A B C ssun3 A C B ssun4 jaoi))
+thm (elin ((A x) (B x) (C x)) () (<-> (e. A (i^i B C)) (/\ (e. A B) (e. A C))) (A B C i^i elisset A C elisset A B e. adantl x cv A B eleq1 x cv A C eleq1 anbi12d B C x df-in V elab2g pm5.21nii))
+thm (incom ((A x) (B x)) () (= (i^i A B) (i^i B A)) (x cv A e. x cv B e. ancom x cv A B elin x cv B A elin 3bitr4 eqri))
+thm (ineqri ((A x) (B x) (C x)) ((ineqri.1 (<-> (/\ (e. (cv x) A) (e. (cv x) B)) (e. (cv x) C)))) (= (i^i A B) C) (x cv A B elin ineqri.1 bitr eqri))
+thm (ineq1 ((A x) (B x) (C x)) () (-> (= A B) (= (i^i A C) (i^i B C))) (A B x cv eleq2 x cv C e. anbi1d x cv A C elin x cv B C elin 3bitr4g eqrd))
+thm (ineq2 () () (-> (= A B) (= (i^i C A) (i^i C B))) (A B C ineq1 C A incom C B incom 3eqtr4g))
+thm (ineq12 () () (-> (/\ (= A B) (= C D)) (= (i^i A C) (i^i B D))) (A B C ineq1 C D B ineq2 sylan9eq))
+thm (ineq1i () ((ineq1i.1 (= A B))) (= (i^i A C) (i^i B C)) (ineq1i.1 A B C ineq1 ax-mp))
+thm (ineq2i () ((ineq1i.1 (= A B))) (= (i^i C A) (i^i C B)) (ineq1i.1 A B C ineq2 ax-mp))
+thm (ineq12i () ((ineq1i.1 (= A B)) (ineq12i.2 (= C D))) (= (i^i A C) (i^i B D)) (ineq1i.1 C ineq1i ineq12i.2 B ineq2i eqtr))
+thm (ineq1d () ((ineq1d.1 (-> ph (= A B)))) (-> ph (= (i^i A C) (i^i B C))) (ineq1d.1 A B C ineq1 syl))
+thm (ineq2d () ((ineq1d.1 (-> ph (= A B)))) (-> ph (= (i^i C A) (i^i C B))) (ineq1d.1 A B C ineq2 syl))
+thm (ineq12d () ((ineq1d.1 (-> ph (= A B))) (ineq12d.2 (-> ph (= C D)))) (-> ph (= (i^i A C) (i^i B D))) (ineq1d.1 C ineq1d ineq12d.2 B ineq2d eqtrd))
+thm (hbin ((A y) (B y) (x y)) ((hbin.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbin.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (i^i A B)) (A. x (e. (cv y) (i^i A B)))) (hbin.1 hbin.2 hban y cv A B elin y cv A B elin x bial 3imtr4))
+thm (birabrdv ((ph x) (A x) (B x)) ((birabrdv.1 (-> ph (-> (e. (cv x) B) (<-> (e. (cv x) A) ch))))) (-> ph (= (i^i B A) ({e.|} x B ch))) (birabrdv.1 pm5.32d x cv B A elin syl5bb biabrdv x B ch df-rab syl6eqr))
+thm (inidm ((A x)) () (= (i^i A A) A) (x cv A e. anidm ineqri))
+thm (inass ((A x) (B x) (C x)) () (= (i^i (i^i A B) C) (i^i A (i^i B C))) (x cv A e. x cv B e. x cv C e. anass x cv B C elin x cv A e. anbi2i bitr4 x cv A B elin x cv C e. anbi1i x cv A B C i^i elin 3bitr4 ineqri))
+thm (in12 () () (= (i^i A (i^i B C)) (i^i B (i^i A C))) (A B incom C ineq1i A B C inass B A C inass 3eqtr3))
+thm (in23 () () (= (i^i (i^i A B) C) (i^i (i^i A C) B)) (B C incom A ineq2i A B C inass A C B inass 3eqtr4))
+thm (in4 () () (= (i^i (i^i A B) (i^i C D)) (i^i (i^i A C) (i^i B D))) (B C D in12 A ineq2i A B C D i^i inass A C B D i^i inass 3eqtr4))
+thm (inindi () () (= (i^i A (i^i B C)) (i^i (i^i A B) (i^i A C))) (A inidm B C i^i ineq1i A A B C in4 eqtr3))
+thm (inindir () () (= (i^i (i^i A B) C) (i^i (i^i A C) (i^i B C))) (C inidm A B i^i ineq2i A B C C in4 eqtr3))
+thm (sseqin2 () () (<-> (C_ A B) (= (i^i B A) A)) (A B df-ss A B incom A eqeq1i bitr))
+thm (inss1 ((A x) (B x)) () (C_ (i^i A B) A) (x cv A B elin pm3.26bd ssriv))
+thm (inss2 () () (C_ (i^i A B) B) (B A incom B A inss1 eqsstr3))
+thm (ssin () () (<-> (/\ (C_ A B) (C_ A C)) (C_ A (i^i B C))) (A B i^i A A C i^i A ineq12 A B C inindi syl5eq A inidm syl6eq A B df-ss A C df-ss anbi12i A B C i^i df-ss 3imtr4 B C inss1 A B C i^i B sstr2 mpi B C inss2 A B C i^i C sstr2 mpi jca impbi))
+thm (ssini () ((ssini.1 (C_ A B)) (ssini.2 (C_ A C))) (C_ A (i^i B C)) (ssini.1 ssini.2 pm3.2i A B C ssin mpbi))
+thm (ssrin ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (i^i A C) (i^i B C))) (x cv A e. x cv B e. x cv C e. pm3.45 x cv A C elin x cv B C elin 3imtr4g x 19.20i A B x dfss2 A C i^i B C i^i x dfss2 3imtr4))
+thm (sslin () () (-> (C_ A B) (C_ (i^i C A) (i^i C B))) (A B C ssrin C A incom C B incom 3sstr4g))
+thm (ss2in () () (-> (/\ (C_ A B) (C_ C D)) (C_ (i^i A C) (i^i B D))) (A B C ssrin C D B sslin sylan9ss))
+thm (ssinss1 () () (-> (C_ A C) (C_ (i^i A B) C)) (A B inss1 A B i^i A C sstr2 ax-mp))
+thm (nssinpss () () (<-> (-. (C_ A B)) (C: (i^i A B) A)) (A B inss1 A A B i^i C_ -. biantrur A ssid A B C_ biantrur A A B ssin bitr negbii A B i^i A dfpss3 3bitr4))
+thm (nsspssun () () (<-> (-. (C_ A B)) (C: B (u. A B))) (B A ssun2 A B u. B C_ -. biantrur B ssid A B C_ biantru A B B unss bitr negbii B A B u. dfpss3 3bitr4))
+thm (dfss4 ((A x) (B x)) () (<-> (C_ A B) (= (\ B (\ B A)) A)) (A B sseqin2 x cv B e. x cv A e. abai x cv B e. x cv A e. iman x cv B e. anbi2i bitr x cv B A elin x cv B B A \ eldif x cv B A eldif negbii x cv B e. anbi2i bitr 3bitr4 eqri A eqeq1i bitr))
+thm (dfun2 ((A x) (B x)) () (= (u. A B) (\ (V) (\ (\ (V) A) B))) (x cv V A eldif x visset mpbiran x cv B e. -. anbi1i x cv V A \ B eldif x cv A e. x cv B e. ioran 3bitr4 bicon2i x cv A B elun x cv V V A \ B \ eldif x visset mpbiran 3bitr4 eqri))
+thm (dfin2 ((A x) (B x)) () (= (i^i A B) (\ A (\ (V) B))) (x cv V B eldif x visset mpbiran bicon2i x cv A e. anbi2i x cv A B elin x cv A V B \ eldif 3bitr4 eqri))
+thm (difin ((A x) (B x)) () (= (\ A (i^i A B)) (\ A B)) (x cv A e. x cv B e. -. abai x cv A e. x cv B e. imnan x cv A e. anbi2i bitr x cv A B eldif x cv A A B i^i eldif x cv A B elin negbii x cv A e. anbi2i bitr 3bitr4r eqri))
+thm (dfun3 () () (= (u. A B) (\ (V) (i^i (\ (V) A) (\ (V) B)))) (A B dfun2 V A \ V B \ dfin2 B ddif V A \ difeq2i eqtr2 V difeq2i eqtr))
+thm (dfin3 () () (= (i^i A B) (\ (V) (u. (\ (V) A) (\ (V) B)))) (A V B \ \ ddif V A \ V B \ dfun2 A ddif V B \ difeq1i V difeq2i eqtr V difeq2i A B dfin2 3eqtr4r))
+thm (dfin4 () () (= (i^i A B) (\ A (\ A B))) (A B inss1 A B i^i A dfss4 mpbi A B difin A difeq2i eqtr3))
+thm (invdif () () (= (i^i A (\ (V) B)) (\ A B)) (A V B \ dfin2 B ddif A difeq2i eqtr))
+thm (indif () () (= (i^i A (\ A B)) (\ A B)) (A A B \ dfin4 A B dfin4 A difeq2i A B difin eqtr3 eqtr))
+thm (indi ((A x) (B x) (C x)) () (= (i^i A (u. B C)) (u. (i^i A B) (i^i A C))) (x cv A e. x cv B e. x cv C e. andi x cv B C elun x cv A e. anbi2i x cv A B elin x cv A C elin orbi12i 3bitr4 x cv A B C u. elin x cv A B i^i A C i^i elun 3bitr4 eqri))
+thm (undi ((A x) (B x) (C x)) () (= (u. A (i^i B C)) (i^i (u. A B) (u. A C))) (x cv A e. x cv B e. x cv C e. ordi x cv B C elin x cv A e. orbi2i x cv A B elun x cv A C elun anbi12i 3bitr4 x cv A B C i^i elun x cv A B u. A C u. elin 3bitr4 eqri))
+thm (indir () () (= (i^i (u. A B) C) (u. (i^i A C) (i^i B C))) (C A B indi A B u. C incom A C incom B C incom uneq12i 3eqtr4))
+thm (undir () () (= (u. (i^i A B) C) (i^i (u. A C) (u. B C))) (C A B undi A B i^i C uncom A C uncom B C uncom A C u. C A u. B C u. C B u. ineq12 mp2an 3eqtr4))
+thm (unineq ((A x) (B x) (C x)) () (<-> (/\ (= (u. A C) (u. B C)) (= (i^i A C) (i^i B C))) (= A B)) (x cv C e. x cv A e. iba x cv C e. x cv B e. iba bibi12d A C i^i B C i^i x cv eleq2 x cv A C elin x cv B C elin 3bitr3g syl5bir A C u. B C u. = adantld x cv C e. x cv A e. biorf x cv C e. x cv B e. biorf bibi12d A C uncom B C uncom eqeq12i C A u. C B u. x cv eleq2 sylbi x cv C A elun x cv C B elun 3bitr3g syl5bir A C i^i B C i^i = adantrd pm2.61i eqrd A B C uneq1 A B C ineq1 jca impbi))
+thm (difundi () () (= (\ A (u. B C)) (i^i (\ A B) (\ A C))) (B C dfun3 A difeq2i A V B \ V C \ inindi A V B \ V C \ i^i dfin2 A B invdif A C invdif ineq12i 3eqtr3 eqtr))
+thm (difundir () () (= (\ (u. A B) C) (u. (\ A C) (\ B C))) (A B V C \ indir A B u. C invdif A C invdif B C invdif uneq12i 3eqtr3))
+thm (difindi () () (= (\ A (i^i B C)) (u. (\ A B) (\ A C))) (B C dfin3 A difeq2i A V B \ V C \ indi A V B \ V C \ u. dfin2 A B invdif A C invdif uneq12i 3eqtr3 eqtr))
+thm (difindir () () (= (\ (i^i A B) C) (i^i (\ A C) (\ B C))) (A B V C \ inindir A B i^i C invdif A C invdif B C invdif ineq12i 3eqtr3))
+thm (undm () () (= (\ (V) (u. A B)) (i^i (\ (V) A) (\ (V) B))) (V A B difundi))
+thm (indm () () (= (\ (V) (i^i A B)) (u. (\ (V) A) (\ (V) B))) (V A B difindi))
+thm (difun1 () () (= (\ A (u. B C)) (\ (\ A B) C)) (A V B \ V C \ inass A V B \ i^i C invdif B C undm A ineq2i A B C u. invdif eqtr3 3eqtr3 A B invdif C difeq1i eqtr3))
+thm (dif23 () () (= (\ (\ A B) C) (\ (\ A C) B)) (B C uncom A difeq2i A B C difun1 A C B difun1 3eqtr3))
+thm (symdif1 () () (= (u. (\ A B) (\ B A)) (\ (u. A B) (i^i A B))) (A B A B i^i difundir A B difin A B incom B difeq2i B A difin eqtr uneq12i eqtr2))
+thm (symdif2 ((A x) (B x)) () (= (u. (\ A B) (\ B A)) ({|} x (-. (<-> (e. (cv x) A) (e. (cv x) B))))) (x cv A B \ B A \ elun x cv A B eldif x cv A e. pm4.13 x cv B e. -. anbi1i bitr x cv B A eldif x cv B e. x cv A e. -. ancom bitr orbi12i x cv A e. -. -. x cv B e. -. /\ x cv A e. -. x cv B e. /\ orcom x cv A e. -. x cv B e. dfbi bitr4 x cv A e. x cv B e. nbbn bitr 3bitr biabri))
+thm (unab ((x y) (ph y) (ps y)) () (= (u. ({|} x ph) ({|} x ps)) ({|} x (\/ ph ps))) (y x ph df-clab y x ps df-clab orbi12i y x ph ps sbor bitr4 y cv x ph {|} x ps {|} elun y x ph ps \/ df-clab 3bitr4 eqri))
+thm (inab ((x y) (ph y) (ps y)) () (= (i^i ({|} x ph) ({|} x ps)) ({|} x (/\ ph ps))) (y x ph df-clab y x ps df-clab anbi12i y x ph ps sban bitr4 y cv x ph {|} x ps {|} elin y x ph ps /\ df-clab 3bitr4 eqri))
+thm (difab ((x y) (ph y) (ps y)) () (= (\ ({|} x ph) ({|} x ps)) ({|} x (/\ ph (-. ps)))) (y x ps sbn y x ps -. df-clab y x ps df-clab negbii 3bitr4 y visset y cv x ps {|} e. -. biantrur bitr2 difeqri x ph {|} ineq2i x ph {|} x ps {|} invdif x ph ps -. inab 3eqtr3))
+thm (unrab () () (= (u. ({e.|} x A ph) ({e.|} x A ps)) ({e.|} x A (\/ ph ps))) (x x cv A e. ph /\ x cv A e. ps /\ unab x cv A e. ph ps andi x biabi eqcomi eqtr x A ph df-rab x A ps df-rab uneq12i x A ph ps \/ df-rab 3eqtr4))
+thm (difrab () () (= (\ ({e.|} x A ph) ({e.|} x A ps)) ({e.|} x A (/\ ph (-. ps)))) (x x cv A e. ph /\ x cv A e. ps /\ difab x cv A e. ph ps -. anass x cv A e. ps pm3.27 con3i x cv A e. ph /\ anim2i x cv A e. ps pm3.2 ph adantr con3d imdistani impbi bitr3 x biabi eqtr4 x A ph df-rab x A ps df-rab difeq12i x A ph ps -. /\ df-rab 3eqtr4))
+thm (dfrab2 ((A x)) () (= ({e.|} x A ph) (i^i ({|} x ph) A)) (x A ph df-rab x x cv A e. ph inab x A abid2 x ph {|} ineq1i eqtr3 A x ph {|} incom 3eqtr))
+thm (inex1 ((A x) (A y) (x y) (B x) (B y)) ((inex1.1 (e. A (V)))) (e. (i^i A B) (V)) (inex1.1 x y y cv B e. zfaus x cv A B i^i y dfcleq y cv A B elin y cv x cv e. bibi2i y bial bitr x biex mpbir issetri))
+thm (inex2 () ((inex2.1 (e. A (V)))) (e. (i^i B A) (V)) (B A incom inex2.1 B inex1 eqeltr))
+thm (inex1g ((A x) (B x)) () (-> (e. A C) (e. (i^i A B) (V))) (x cv A B ineq1 V eleq1d x visset B inex1 C vtoclg))
+thm (ssex () ((ssex.1 (e. B (V)))) (-> (C_ A B) (e. A (V))) (A B df-ss ssex.1 A inex2 A B i^i A V eleq1 mpbii sylbi))
+thm (ssexi () ((ssexi.1 (e. B (V))) (ssexi.2 (C_ A B))) (e. A (V)) (ssexi.2 ssexi.1 A ssex ax-mp))
+thm (ssexg ((A x) (B x)) () (-> (e. B C) (-> (C_ A B) (e. A (V)))) (x cv B A sseq2 A V e. imbi1d x visset A ssex C vtoclg))
+thm (difexg () () (-> (e. A C) (e. (\ A B) (V))) (A B difss A C A B \ ssexg mpi))
+thm (zfausab ((A x)) ((zfausab.1 (e. A (V)))) (e. ({|} x (/\ (e. (cv x) A) ph)) (V)) (zfausab.1 x A ph ssab ssexi))
+thm (rabexg ((A x)) () (-> (e. A B) (e. ({e.|} x A ph) (V))) (x A ph ssrab A B x A ph {e.|} ssexg mpi))
+thm (rabex ((A x)) ((rabex.1 (e. A (V)))) (e. ({e.|} x A ph) (V)) (rabex.1 A V x ph rabexg ax-mp))
+thm (reupick ((A x) (B x)) () (-> (/\ (/\ (C_ A B) (/\ (E.e. x A ph) (E!e. x B ph))) ph) (<-> (e. (cv x) A) (e. (cv x) B))) (A B x cv ssel x A ph E.e. x B ph E!e. /\ ph ad2antll A B x cv ssel ancrd ph anim1d x cv B e. x cv A e. ph an23 syl6ib x 19.22dv x x cv B e. ph /\ x cv A e. eupick exp syl9 com23 imp32 x A ph df-rex x B ph df-reu anbi12i sylan2b exp3a com23 imp impbid))
+thm (dfnul2 () () (= ({/}) ({|} x (-. (= (cv x) (cv x))))) (df-nul x cv eleq2i x cv V V eldif x cv eqid x cv V e. pm3.24 2th bicon2i 3bitr biabri))
+thm (dfnul3 () () (= ({/}) ({e.|} x A (-. (e. (cv x) A)))) (x cv A e. pm3.24 x cv eqid 2th bicon1i x biabi x dfnul2 x A x cv A e. -. df-rab 3eqtr4))
+thm (noel ((A x)) () (-. (e. A ({/}))) (x cv eqid x dfnul2 eqabi bicon2i mpbi x cv A {/} eleq1 mtbii V vtocleg A {/} elisset con3i pm2.61i))
+thm (n0i () () (-> (e. B A) (-. (= A ({/})))) (B noel A {/} B eleq2 mtbiri con2i))
+thm (n0f ((x y) (A y)) ((nnullf.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (<-> (-. (= A ({/}))) (E. x (e. (cv x) A))) (nnullf.1 y cv {/} e. x ax-17 cleqf x cv noel x cv A e. nbn x bial bitr4 negbii x x cv A e. df-ex bitr4))
+thm (n0 ((x y) (A x) (A y)) () (<-> (-. (= A ({/}))) (E. x (e. (cv x) A))) (y cv A e. x ax-17 n0f))
+thm (abn0 ((x y) (ph y)) () (<-> (-. (= ({|} x ph) ({/}))) (E. x ph)) (x ph {|} y n0 y x ph hbab1 x cv x ph {|} e. y ax-17 y cv x cv x ph {|} eleq1 cbvex x ph abid x biex 3bitr))
+thm (rex0 () () (-. (E.e. x ({/}) ph)) (x cv noel ph -. pm2.21i nrex))
+thm (rabn0 () () (<-> (-. (= ({e.|} x A ph) ({/}))) (E.e. x A ph)) (x x cv A e. ph /\ abn0 x A ph df-rab {/} eqeq1i negbii x A ph df-rex 3bitr4))
+thm (rab0 () () (= ({e.|} x ({/}) ph) ({/})) (x cv noel ph intnanr x nex x {/} ph rabn0 x {/} ph df-rex bitr bicon1i mpbi))
+thm (eq0 ((A x)) () (<-> (= A ({/})) (A. x (-. (e. (cv x) A)))) (A x n0 x x cv A e. df-ex bitr bicon4i))
+thm (0el ((x y) (A x) (A y)) () (<-> (e. ({/}) A) (E.e. x A (A. y (-. (e. (cv y) (cv x)))))) ({/} A x risset x cv y eq0 x A birex bitr))
+thm (un0 ((A x)) () (= (u. A ({/})) A) (x cv noel x cv A e. biorfi bicomi uneqri))
+thm (in0 ((A x)) () (= (i^i A ({/})) ({/})) (x cv noel x cv A e. bianfi bicomi ineqri))
+thm (inv () () (= (i^i A (V)) A) (A V inss1 A ssid A ssv ssini eqssi))
+thm (unv () () (= (u. A (V)) (V)) (A V u. ssv V A ssun2 eqssi))
+thm (0ss ((A x)) () (C_ ({/}) A) (x cv noel x cv A e. pm2.21i ssriv))
+thm (ss0b () () (<-> (C_ A ({/})) (= A ({/}))) (A 0ss A {/} eqss biimpr mpan2 A {/} eqimss impbi))
+thm (ss0 () () (-> (C_ A ({/})) (= A ({/}))) (A ss0b biimp))
+thm (un00 () () (<-> (/\ (= A ({/})) (= B ({/}))) (= (u. A B) ({/}))) (A {/} B {/} uneq12 {/} un0 syl6eq A B ssun1 A B u. {/} A sseq2 mpbii A ss0b sylib B A ssun2 A B u. {/} B sseq2 mpbii B ss0b sylib jca impbi))
+thm (vss () () (<-> (C_ (V) A) (= A (V))) (A ssv V A C_ jctl A V eqss sylibr A V eqimss2 impbi))
+thm (0pss () () (<-> (C: ({/}) A) (-. (= A ({/})))) ({/} A dfpss2 A 0ss mpbiran {/} A eqcom negbii bitr))
+thm (npss0 () () (-. (C: A ({/}))) ({/} eqid A {/} pssss A ss0 A {/} {/} psseq1 3syl ibi {/} 0pss sylib mt2))
+thm (pssv () () (<-> (C: A (V)) (-. (= A (V)))) (A V dfpss2 A ssv mpbiran))
+thm (disj ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (A.e. x A (-. (e. (cv x) B)))) (A B x df-in {/} eqeq1i x x cv A e. x cv B e. /\ {/} eqabr x cv A e. x cv B e. imnan x cv noel x cv A e. x cv B e. /\ nbn bitr2 x bial 3bitr x A x cv B e. -. df-ral bitr4))
+thm (disj1 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (A. x (-> (e. (cv x) A) (-. (e. (cv x) B))))) (A B x disj x A x cv B e. -. df-ral bicomi bitr4))
+thm (disj2 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (C_ A (\ (V) B))) (x visset x cv B e. -. biantrur x cv V B eldif bitr4 x cv A e. imbi2i x bial A B x disj1 A V B \ x dfss2 3bitr4))
+thm (disj3 ((A x) (B x)) () (<-> (= (i^i A B) ({/})) (= A (\ A B))) (x cv A e. x cv B e. -. pm4.71 x cv A B eldif x cv A e. bibi2i bitr4 x bial A B x disj1 A A B \ x dfcleq 3bitr4))
+thm (disj4 () () (<-> (= (i^i A B) ({/})) (-. (C: (\ A B) A))) (A B disj3 A A B \ eqcom A B \ A dfpss2 A B difss mpbiran bicon2i 3bitr))
+thm (disjpss () () (-> (/\ (= (i^i A B) ({/})) (-. (= B ({/})))) (C: A (u. A B))) (A B i^i {/} B sseq2 B ssid B A C_ biantru B A B ssin bitr syl5bb B ss0 syl6bi con3d imp B A nsspssun B A uncom A psseq2i bitr sylib))
+thm (undisj1 () () (<-> (/\ (= (i^i A C) ({/})) (= (i^i B C) ({/}))) (= (i^i (u. A B) C) ({/}))) (A C i^i B C i^i un00 A B C indir {/} eqeq1i bitr4))
+thm (undisj2 () () (<-> (/\ (= (i^i A B) ({/})) (= (i^i A C) ({/}))) (= (i^i A (u. B C)) ({/}))) (A B i^i A C i^i un00 A B C indi {/} eqeq1i bitr4))
+thm (ssindif0 () () (<-> (C_ A B) (= (i^i A (\ (V) B)) ({/}))) (A V B \ disj2 B ddif A sseq2i bitr2))
+thm (inelcm () () (-> (/\ (e. A B) (e. A C)) (-. (= (i^i B C) ({/})))) (A B C elin A B C i^i n0i sylbir))
+thm (minel () () (-> (/\ (e. A B) (= (i^i C B) ({/}))) (-. (e. A C))) (A C B inelcm con2i A C e. A B e. imnan sylibr con2d impcom))
+thm (undif4 ((A x) (B x) (C x)) () (-> (= (i^i A C) ({/})) (= (u. A (\ B C)) (\ (u. A B) C))) (x cv A e. x cv C e. -. pm2.61 x cv C e. -. x cv A e. -. ax-1 x cv A e. x cv C e. -. -> a1i impbid x cv A e. x cv C e. -. df-or syl5bb x cv A e. x cv B e. \/ anbi2d x cv B C eldif x cv A e. orbi2i x cv A e. x cv B e. x cv C e. -. ordi bitr x cv A B elun x cv C e. -. anbi1i 3bitr4g x cv A B C \ elun x cv A B u. C eldif 3bitr4g x 19.20i A C x disj1 A B C \ u. A B u. C \ x dfcleq 3imtr4))
+thm (0ex () () (e. ({/}) (V)) (x cv in0 x visset {/} inex1 eqeltrr))
+thm (zfnul ((x y) (x z) (y z)) () (E. x (A. y (-. (e. (cv y) (cv x))))) (z visset x y y cv z cv e. -. zfaus y cv z cv e. pm3.24 y cv x cv e. y cv z cv e. y cv z cv e. -. /\ <-> id mtbiri y 19.20i x 19.22i ax-mp))
+thm (class2set ((A x)) () (e. ({e.|} x A (e. A (V))) (V)) (A V x A V e. rabexg 0ex A V e. -. x cv A e. pm3.26 nrexdv x A A V e. rabn0 bicon1i sylib V eleq1d mpbiri pm2.61i))
+thm (ssdif0 ((A x) (B x)) () (<-> (C_ A B) (= (\ A B) ({/}))) (x cv A e. x cv B e. iman x cv A B eldif negbii bitr4 x bial A B x dfss2 A B \ x eq0 3bitr4))
+thm (vdif0 () () (<-> (= A (V)) (= (\ (V) A) ({/}))) (A vss V A ssdif0 bitr3))
+thm (pssdifn0 () () (-> (/\ (C_ A B) (-. (= A B))) (-. (= (\ B A) ({/})))) (A B eqss biimpr exp B A ssdif0 syl5ibr con3d imp))
+thm (ssnelpss () () (-> (C_ A B) (-> (/\ (e. C B) (-. (e. C A))) (C: A B))) (A B dfpss2 baibr C B A nelneq2 B A eqcom negbii sylib syl5bi))
+thm (pssnel ((A x) (B x)) () (-> (C: A B) (E. x (/\ (e. (cv x) B) (-. (e. (cv x) A))))) (A B dfpss2 A B pssdifn0 sylbi B A \ x n0 sylib x cv B A eldif x biex sylib))
+thm (difin0ss ((A x) (B x) (C x)) () (-> (= (i^i (\ A B) C) ({/})) (-> (C_ C A) (C_ C B))) (A B \ C i^i x eq0 x cv A e. x cv B e. annim x cv C e. anbi2i x cv C e. x cv A e. x cv B e. -. /\ ancom bitr3 negbii x cv C e. x cv A e. x cv B e. -> iman x cv A B \ C elin x cv A B eldif x cv C e. anbi1i bitr negbii 3bitr4 x cv C e. x cv A e. x cv B e. ax-2 sylbir x 19.20ii sylbi C A x dfss2 C B x dfss2 3imtr4g))
+thm (inssdif0 ((A x) (B x) (C x)) () (<-> (C_ (i^i A B) C) (= (i^i A (\ B C)) ({/}))) (x cv A e. x cv B e. x cv C e. impexp x cv B e. x cv C e. iman x cv A e. imbi2i x cv A e. x cv B e. x cv C e. -. /\ imnan 3bitr x cv A B elin x cv C e. imbi1i x cv A B C \ elin x cv B C eldif x cv A e. anbi2i bitr negbii 3bitr4 x bial A B i^i C x dfss2 A B C \ i^i x eq0 3bitr4))
+thm (difid () () (= (\ A A) ({/})) (A ssid A A ssdif0 mpbi))
+thm (dif0 () () (= (\ A ({/})) A) (A difid A difeq2i A A difdif eqtr3))
+thm (0dif () () (= (\ ({/}) A) ({/})) ({/} A difss {/} A \ ss0 ax-mp))
+thm (difdisj () () (= (i^i A (\ B A)) ({/})) (A B inss1 A B A inssdif0 mpbi))
+thm (difin0 () () (= (\ (i^i A B) B) ({/})) (A B B difindir B difid A B \ ineq2i A B \ in0 3eqtr))
+thm (undifv () () (= (u. A (\ (V) A)) (V)) (A V A \ dfun3 V A \ V difdisj V difeq2i V dif0 3eqtr))
+thm (undif1 () () (= (u. (\ A B) B) (u. A B)) (A B invdif B uneq1i A V B \ B undir V B \ B uncom B undifv eqtr A B u. ineq2i A B u. inv 3eqtr eqtr3))
+thm (undif2 () () (= (u. A (\ B A)) (u. A B)) (A B A \ uncom B A undif1 B A uncom 3eqtr))
+thm (difun2 () () (= (\ (u. A B) B) (\ A B)) (A B B difundir B difid A B \ uneq2i A B \ un0 3eqtr))
+thm (ssundif () () (<-> (C_ A B) (= (u. A (\ B A)) B)) (A B ssequn1 A B undif2 B eqeq1i bitr4))
+thm (difdifdir () () (= (\ (\ A B) C) (\ (\ A C) (\ B C))) (C A difdisj C A C \ incom eqtr3 A C \ V B \ i^i uneq2i A C \ B invdif A C \ V B \ i^i un0 A B C dif23 3eqtr4r A C \ V B \ C indi 3eqtr4 B V C \ indm B C invdif V difeq2i C ddif V B \ uneq2i 3eqtr3r A C \ ineq2i A C \ B C \ invdif 3eqtr))
+thm (r19.2z ((A x)) () (-> (-. (= A ({/}))) (-> (A.e. x A ph) (E.e. x A ph))) (x A ph df-ral x x cv A e. ph exintr sylbi A x n0 x A ph df-rex 3imtr4g com12))
+thm (r19.3rzv ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> ph (A.e. x A ph))) (A x n0 x x cv A e. E. ph biimt sylbi x A ph df-ral x x cv A e. ph 19.23v bitr syl6bbr))
+thm (r19.9rzv ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> ph (E.e. x A ph))) (A ph -. x r19.3rzv bicomd bicon2d x A ph dfrex2 syl6bbr))
+thm (r19.28zv ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> (A.e. x A (/\ ph ps)) (/\ ph (A.e. x A ps)))) (A ph x r19.3rzv x A ps A.e. anbi1d x A ph ps r19.26 syl6rbbr))
+thm (r19.45zv ((A x) (ph x)) () (-> (-. (= A ({/}))) (<-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps)))) (A ph x r19.9rzv x A ps E.e. orbi1d x A ph ps r19.43 syl6rbbr))
+thm (r19.27zv ((A x) (ps x)) () (-> (-. (= A ({/}))) (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) ps))) (A ps x r19.3rzv x A ph A.e. anbi2d x A ph ps r19.26 syl6rbbr))
+thm (r19.36zv ((A x) (ps x)) () (-> (-. (= A ({/}))) (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps))) (A ps x r19.9rzv x A ph A.e. imbi2d bicomd x A ph ps r19.35 syl5bb))
+thm (rzal ((A x)) () (-> (= A ({/})) (A.e. x A ph)) (A {/} x cv eleq2 x cv noel ph pm2.21i syl6bi r19.21aiv))
+thm (ralidm ((A x)) () (<-> (A.e. x A (A.e. x A ph)) (A.e. x A ph)) (x A x A ph A.e. A.e. x A ph A.e. pm5.1 A x x A ph A.e. rzal A x ph rzal sylanc A x n0 x x cv A e. E. x A ph A.e. biimt x A x A ph A.e. df-ral x A ph hbra1 x cv A e. 19.23 bitr syl6rbbr sylbi pm2.61i))
+thm (raaan ((ph y) (ps x) (x y) (A x) (A y)) () (<-> (A.e. x A (A.e. y A (/\ ph ps))) (/\ (A.e. x A ph) (A.e. y A ps))) (x A y A ph ps /\ A.e. A.e. x A ph A.e. y A ps A.e. /\ pm5.1 A x y A ph ps /\ A.e. rzal A x ph rzal A y ps rzal jca sylanc A ph y r19.3rzv y A ps A.e. anbi1d y A ph ps r19.26 syl6rbbr x A biraldv A y A ps A.e. x r19.3rzv x A ph A.e. anbi2d x A ph y A ps A.e. r19.26 syl6rbbr bitrd pm2.61i))
+thm (ifeq1 ((ph x) (A x) (B x) (C x)) () (-> (= A B) (= (if ph A C) (if ph B C))) (A B x cv eleq2 ph anbi1d x cv C e. ph -. /\ orbi1d x biabdv ph A C x df-if ph B C x df-if 3eqtr4g))
+thm (ifeq2 ((ph x) (A x) (B x) (C x)) () (-> (= A B) (= (if ph C A) (if ph C B))) (A B x cv eleq2 ph -. anbi1d x cv C e. ph /\ orbi2d x biabdv ph C A x df-if ph C B x df-if 3eqtr4g))
+thm (iftrue ((ph x) (A x) (B x)) () (-> ph (= (if ph A B) A)) (ph x cv A e. x cv B e. dedlema biabrdv ph A B x df-if syl6reqr))
+thm (iffalse ((ph x) (A x) (B x)) () (-> (-. ph) (= (if ph A B) B)) (ph x cv B e. x cv A e. dedlemb biabrdv ph A B x df-if syl6reqr))
+thm (ifeq12 () () (-> (/\ (= A B) (= C D)) (= (if ph A C) (if ph B D))) (A B ph C ifeq1 C D ph B ifeq2 sylan9eq))
+thm (ifeq1d () ((ifeq1d.1 (-> ph (= A B)))) (-> ph (= (if ps A C) (if ps B C))) (ifeq1d.1 A B ps C ifeq1 syl))
+thm (ifeq2d () ((ifeq1d.1 (-> ph (= A B)))) (-> ph (= (if ps C A) (if ps C B))) (ifeq1d.1 A B ps C ifeq2 syl))
+thm (ifbi () () (-> (<-> ph ps) (= (if ph A B) (if ps A B))) (ph ps dfbi ph A B iftrue ps A B iftrue eqcomd sylan9eq ph A B iffalse ps A B iffalse eqcomd sylan9eq jaoi sylbi))
+thm (ifbid () ((ifbid.1 (-> ph (<-> ps ch)))) (-> ph (= (if ps A B) (if ch A B))) (ifbid.1 ps ch A B ifbi syl))
+thm (hbif ((x y) (x z) (y z) (A y) (A z) (B y) (B z) (ph z)) ((hbif.1 (-> ph (A. x ph))) (hbif.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbif.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (if ph A B)) (A. x (e. (cv y) (if ph A B)))) (y cv z cv e. x ax-17 hbif.2 hbel hbif.1 hban y cv z cv e. x ax-17 hbif.3 hbel hbif.1 hbne hban hbor y z hbab ph A B z df-if y cv eleq2i ph A B z df-if y cv eleq2i x bial 3imtr4))
+thm (elimif () ((sbif.1 (-> (= (if ph A B) A) (<-> ps ch))) (sbif.2 (-> (= (if ph A B) B) (<-> ps th)))) (<-> ps (\/ (/\ ph ch) (/\ (-. ph) th))) (ph exmid ps biantrur ph ph -. ps andir ph A B iftrue sbif.1 syl pm5.32i ph A B iffalse sbif.2 syl pm5.32i orbi12i 3bitr))
+thm (ifid () () (= (if ph A A) A) (ph A A iftrue ph A A iffalse pm2.61i))
+thm (eqif () () (<-> (= A (if ph B C)) (\/ (/\ ph (= A B)) (/\ (-. ph) (= A C)))) (ph B C if B A eqeq2 ph B C if C A eqeq2 elimif))
+thm (elif () () (<-> (e. A (if ph B C)) (\/ (/\ ph (e. A B)) (/\ (-. ph) (e. A C)))) (ph B C if B A eleq2 ph B C if C A eleq2 elimif))
+thm (ifel () () (<-> (e. (if ph A B) C) (\/ (/\ ph (e. A C)) (/\ (-. ph) (e. B C)))) (ph A B if A C eleq1 ph A B if B C eleq1 elimif))
+thm (ifcl () () (-> (/\ (e. A C) (e. B C)) (e. (if ph A B) C)) (ph A B iftrue C eleq1d biimpar B C e. adantrr ph A B iffalse C eleq1d biimpar A C e. adantrl pm2.61an1))
+thm (dedth () ((dedth.1 (-> (= A (if ph A B)) (<-> ps ch))) (dedth.2 ch)) (-> ph ps) (dedth.2 ph A B iftrue eqcomd dedth.1 syl mpbiri))
+thm (dedth2v () ((dedth2v.1 (-> (= A (if ph A C)) (<-> ps ch))) (dedth2v.2 (-> (= B (if ph B D)) (<-> ch th))) (dedth2v.3 th)) (-> ph ps) (dedth2v.3 ph A C iftrue eqcomd dedth2v.1 syl ph B D iftrue eqcomd dedth2v.2 syl bitrd mpbiri))
+thm (dedth3v () ((dedth3v.1 (-> (= A (if ph A D)) (<-> ps ch))) (dedth3v.2 (-> (= B (if ph B R)) (<-> ch th))) (dedth3v.3 (-> (= C (if ph C S)) (<-> th ta))) (dedth3v.4 ta)) (-> ph ps) (dedth3v.4 ph A D iftrue eqcomd dedth3v.1 syl ph B R iftrue eqcomd dedth3v.2 syl ph C S iftrue eqcomd dedth3v.3 syl 3bitrd mpbiri))
+thm (dedth2h () ((dedth2h.1 (-> (= A (if ph A C)) (<-> ch th))) (dedth2h.2 (-> (= B (if ps B D)) (<-> th ta))) (dedth2h.3 ta)) (-> (/\ ph ps) ch) (dedth2h.1 ps imbi2d dedth2h.2 dedth2h.3 dedth dedth imp))
+thm (dedth3h () ((dedth3h.1 (-> (= A (if ph A D)) (<-> th ta))) (dedth3h.2 (-> (= B (if ps B R)) (<-> ta et))) (dedth3h.3 (-> (= C (if ch C S)) (<-> et ze))) (dedth3h.4 ze)) (-> (/\/\ ph ps ch) th) (dedth3h.1 ps ch /\ imbi2d dedth3h.2 dedth3h.3 dedth3h.4 dedth2h dedth exp3a 3imp))
+var (wff si)
+var (wff rh)
+thm (dedth4h () ((dedth4h.1 (-> (= A (if ph A R)) (<-> ta et))) (dedth4h.2 (-> (= B (if ps B S)) (<-> et ze))) (dedth4h.3 (-> (= C (if ch C F)) (<-> ze si))) (dedth4h.4 (-> (= D (if th D G)) (<-> si rh))) (dedth4h.5 rh)) (-> (/\ (/\ ph ps) (/\ ch th)) ta) (dedth4h.1 ch th /\ imbi2d dedth4h.2 ch th /\ imbi2d dedth4h.3 dedth4h.4 dedth4h.5 dedth2h dedth2h imp))
+thm (elimhyp () ((elimhyp.1 (-> (= A (if ph A B)) (<-> ph ps))) (elimhyp.2 (-> (= B (if ph A B)) (<-> ch ps))) (elimhyp.3 ch)) ps (ph A B iftrue eqcomd elimhyp.1 syl ibi elimhyp.3 ph A B iffalse eqcomd elimhyp.2 syl mpbii pm2.61i))
+thm (elimhyp2v () ((elimhyp2v.1 (-> (= A (if ph A C)) (<-> ph ch))) (elimhyp2v.2 (-> (= B (if ph B D)) (<-> ch th))) (elimhyp2v.3 (-> (= C (if ph A C)) (<-> ta et))) (elimhyp2v.4 (-> (= D (if ph B D)) (<-> et th))) (elimhyp2v.5 ta)) th (ph A C iftrue eqcomd elimhyp2v.1 syl ph B D iftrue eqcomd elimhyp2v.2 syl bitrd ibi elimhyp2v.5 ph A C iffalse eqcomd elimhyp2v.3 syl ph B D iffalse eqcomd elimhyp2v.4 syl bitrd mpbii pm2.61i))
+thm (elimhyp3v () ((elimhyp3v.1 (-> (= A (if ph A D)) (<-> ph ch))) (elimhyp3v.2 (-> (= B (if ph B R)) (<-> ch th))) (elimhyp3v.3 (-> (= C (if ph C S)) (<-> th ta))) (elimhyp3v.4 (-> (= D (if ph A D)) (<-> et ze))) (elimhyp3v.5 (-> (= R (if ph B R)) (<-> ze si))) (elimhyp3v.6 (-> (= S (if ph C S)) (<-> si ta))) (elimhyp3v.7 et)) ta (ph A D iftrue eqcomd elimhyp3v.1 syl ph B R iftrue eqcomd elimhyp3v.2 syl ph C S iftrue eqcomd elimhyp3v.3 syl 3bitrd ibi elimhyp3v.7 ph A D iffalse eqcomd elimhyp3v.4 syl ph B R iffalse eqcomd elimhyp3v.5 syl ph C S iffalse eqcomd elimhyp3v.6 syl 3bitrd mpbii pm2.61i))
+thm (elimhyp4v () ((elimhyp4v.1 (-> (= A (if ph A D)) (<-> ph ch))) (elimhyp4v.2 (-> (= B (if ph B R)) (<-> ch th))) (elimhyp4v.3 (-> (= C (if ph C S)) (<-> th ta))) (elimhyp4v.4 (-> (= F (if ph F G)) (<-> ta ps))) (elimhyp4v.5 (-> (= D (if ph A D)) (<-> et ze))) (elimhyp4v.6 (-> (= R (if ph B R)) (<-> ze si))) (elimhyp4v.7 (-> (= S (if ph C S)) (<-> si rh))) (elimhyp4v.8 (-> (= G (if ph F G)) (<-> rh ps))) (elimhyp4v.9 et)) ps (ph A D iftrue eqcomd elimhyp4v.1 syl ph B R iftrue eqcomd elimhyp4v.2 syl bitrd ph C S iftrue eqcomd elimhyp4v.3 syl ph F G iftrue eqcomd elimhyp4v.4 syl 3bitrd ibi elimhyp4v.9 ph A D iffalse eqcomd elimhyp4v.5 syl ph B R iffalse eqcomd elimhyp4v.6 syl bitrd ph C S iffalse eqcomd elimhyp4v.7 syl ph F G iffalse eqcomd elimhyp4v.8 syl 3bitrd mpbii pm2.61i))
+thm (elimel () ((elimel.1 (e. B C))) (e. (if (e. A C) A B) C) (A A C e. A B if C eleq1 B A C e. A B if C eleq1 elimel.1 elimhyp))
+thm (keephyp () ((keephyp.1 (-> (= A (if ph A B)) (<-> ps th))) (keephyp.2 (-> (= B (if ph A B)) (<-> ch th))) (keephyp.3 ps) (keephyp.4 ch)) th (keephyp.3 ph A B iftrue eqcomd keephyp.1 syl mpbii keephyp.4 ph A B iffalse eqcomd keephyp.2 syl mpbii pm2.61i))
+thm (keephyp2v () ((keephyp2v.1 (-> (= A (if ph A C)) (<-> ps ch))) (keephyp2v.2 (-> (= B (if ph B D)) (<-> ch th))) (keephyp2v.3 (-> (= C (if ph A C)) (<-> ta et))) (keephyp2v.4 (-> (= D (if ph B D)) (<-> et th))) (keephyp2v.5 ps) (keephyp2v.6 ta)) th (keephyp2v.5 ph A C iftrue eqcomd keephyp2v.1 syl ph B D iftrue eqcomd keephyp2v.2 syl bitrd mpbii keephyp2v.6 ph A C iffalse eqcomd keephyp2v.3 syl ph B D iffalse eqcomd keephyp2v.4 syl bitrd mpbii pm2.61i))
+thm (keephyp3v () ((keephyp3v.1 (-> (= A (if ph A D)) (<-> rh ch))) (keephyp3v.2 (-> (= B (if ph B R)) (<-> ch th))) (keephyp3v.3 (-> (= C (if ph C S)) (<-> th ta))) (keephyp3v.4 (-> (= D (if ph A D)) (<-> et ze))) (keephyp3v.5 (-> (= R (if ph B R)) (<-> ze si))) (keephyp3v.6 (-> (= S (if ph C S)) (<-> si ta))) (keephyp3v.7 rh) (keephyp3v.8 et)) ta (keephyp3v.7 ph A D iftrue eqcomd keephyp3v.1 syl ph B R iftrue eqcomd keephyp3v.2 syl ph C S iftrue eqcomd keephyp3v.3 syl 3bitrd mpbii keephyp3v.8 ph A D iffalse eqcomd keephyp3v.4 syl ph B R iffalse eqcomd keephyp3v.5 syl ph C S iffalse eqcomd keephyp3v.6 syl 3bitrd mpbii pm2.61i))
+thm (keepel () ((keepel.1 (e. A C)) (keepel.2 (e. B C))) (e. (if ph A B) C) (A ph A B if C eleq1 B ph A B if C eleq1 keepel.1 keepel.2 keephyp))
+thm (ifex () ((dedex.1 (e. A (V))) (dedex.2 (e. B (V)))) (e. (if ph A B) (V)) (dedex.1 dedex.2 ph keepel))
+thm (pweq ((A x) (B x)) () (-> (= A B) (= (P~ A) (P~ B))) (A B x cv sseq2 x biabdv A x df-pw B x df-pw 3eqtr4g))
+thm (elpw ((A x) (B x)) ((elpw.1 (e. A (V)))) (<-> (e. A (P~ B)) (C_ A B)) (elpw.1 x cv A B P~ eleq1 x cv A B sseq1 B x df-pw eqabi vtoclb))
+thm (elpwg ((A x) (B x)) () (-> (e. A C) (<-> (e. A (P~ B)) (C_ A B))) (x cv A B P~ eleq1 x cv A B sseq1 bibi12d x visset B elpw C vtoclg))
+thm (elpw2g () () (-> (e. B C) (<-> (e. A (P~ B)) (C_ A B))) (A B P~ B elpwg ibi B C e. a1i A B C_ id B C A ssexg A V B elpwg syl6 pm5.74d mpbiri impbid))
+thm (hbpw ((y z) (A y) (A z) (x y) (x z)) ((hbpw.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (P~ A)) (A. x (e. (cv y) (P~ A)))) (z cv y cv e. x ax-17 y cv z cv e. x ax-17 hbpw.1 hbel hbss y visset A elpw y visset A elpw x bial 3imtr4))
+thm (pwid () ((pwid.1 (e. A (V)))) (e. A (P~ A)) (A ssid pwid.1 A elpw mpbir))
+thm (pwex ((A x) (A y) (A z) (x y) (x z) (y z)) ((zfpowcl.1 (e. A (V)))) (e. (P~ A) (V)) (zfpowcl.1 z cv A pweq V eleq1d x y z axpow y cv z cv x dfss2 y cv x cv e. imbi1i y bial x biex mpbir bm1.3ii z cv y df-pw x cv eqeq2i x cv y y cv z cv C_ eqab bitr x biex mpbir issetri vtocl))
+thm (pwexg ((A x)) () (-> (e. A B) (e. (P~ A) (V))) (x cv A pweq V eleq1d x visset pwex B vtoclg))
+thm (sneq ((A x) (B x)) () (-> (= A B) (= ({} A) ({} B))) (A B x cv eqeq2 x biabdv A x df-sn B x df-sn 3eqtr4g))
+thm (sneqi () ((sneqi.1 (= A B))) (= ({} A) ({} B)) (sneqi.1 A B sneq ax-mp))
+thm (sneqd () ((sneqd.1 (-> ph (= A B)))) (-> ph (= ({} A) ({} B))) (sneqd.1 A B sneq syl))
+thm (dfsn2 () () (= ({} A) ({,} A A)) (A A df-pr A {} unidm eqtr2))
+thm (elsn ((A x)) () (<-> (e. (cv x) ({} A)) (= (cv x) A)) (A x df-sn eqabi))
+thm (dfpr2 ((A x) (B x)) () (= ({,} A B) ({|} x (\/ (= (cv x) A) (= (cv x) B)))) (A B df-pr x cv A {} B {} elun x A elsn x B elsn orbi12i bitr biabri eqtr))
+thm (elprg ((A x) (B x) (C x)) () (-> (e. A D) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C)))) (x cv A B eqeq1 x cv A C eqeq1 orbi12d B C x dfpr2 D elab2g))
+thm (elpr () ((elpr.1 (e. A (V)))) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C))) (elpr.1 A V B C elprg ax-mp))
+thm (elpr2 () ((elpr2.1 (e. B (V))) (elpr2.2 (e. C (V)))) (<-> (e. A ({,} B C)) (\/ (= A B) (= A C))) (A B C {,} B C elprg ibi elpr2.1 A B V eleq1 mpbiri elpr2.2 A C V eleq1 mpbiri jaoi A V B C elprg syl ibir impbi))
+thm (hbpr ((A y) (B y) (x y)) ((hbpr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hppr.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) ({,} A B)) (A. x (e. (cv y) ({,} A B)))) (hbpr.1 hbeleq hppr.2 hbeleq hbor y visset A B elpr y visset A B elpr x bial 3imtr4))
+thm (elsncg () () (-> (e. A C) (<-> (e. A ({} B)) (= A B))) (A C B B elprg B dfsn2 eqcomi A eleq2i A B = oridm 3bitr3g))
+thm (elsnc () ((elsnc.1 (e. A (V)))) (<-> (e. A ({} B)) (= A B)) (elsnc.1 A V B elsncg ax-mp))
+thm (elsni () () (-> (e. A ({} B)) (= A B)) (A B {} B elsncg ibi))
+thm (snidg () () (-> (e. A B) (e. A ({} A))) (A eqid A B A elsncg mpbiri))
+thm (snidb () () (<-> (e. A (V)) (e. A ({} A))) (A V snidg A A {} elisset impbi))
+thm (snid () ((snid.1 (e. A (V)))) (e. A ({} A)) (snid.1 A snidb mpbi))
+thm (elsnc2g () () (-> (e. B C) (<-> (e. A ({} B)) (= A B))) (A B elsni B C e. a1i A B B {} eleq1 B C snidg syl5bir com12 impbid))
+thm (elsnc2 () ((elsnc2.1 (e. B (V)))) (<-> (e. A ({} B)) (= A B)) (elsnc2.1 B V A elsnc2g ax-mp))
+thm (hbsn ((A y) (x y)) ((hbsn.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) ({} A)) (A. x (e. (cv y) ({} A)))) (hbsn.1 hbsn.1 hbpr A dfsn2 y cv eleq2i A dfsn2 y cv eleq2i x bial 3imtr4))
+thm (eltp () ((eltp.1 (e. A (V)))) (<-> (e. A ({,,} B C D)) (\/\/ (= A B) (= A C) (= A D))) (B C D df-tp A eleq2i A B C {,} D {} elun eltp.1 B C elpr eltp.1 D elsnc orbi12i 3bitr A B = A C = A D = df-3or bitr4))
+thm (dftp2 ((A x) (B x) (C x)) () (= ({,,} A B C) ({|} x (\/\/ (= (cv x) A) (= (cv x) B) (= (cv x) C)))) (x visset A B C eltp biabri))
+thm (disjsn ((A x) (B x)) () (<-> (= (i^i A ({} B)) ({/})) (-. (e. B A))) (B noel A B {} i^i {/} B eleq2 mtbiri B A snidg ancli B A B {} elin sylibr nsyl x cv B A eleq1 biimpcd x B elsn syl5ib con3d com12 x 19.21aiv A B {} x disj1 sylibr impbi))
+thm (disjsn2 () () (-> (-. (= A B)) (= (i^i ({} A) ({} B)) ({/}))) (B A elsni eqcomd con3i A {} B disjsn sylibr))
+thm (snprc ((A x)) () (<-> (-. (e. A (V))) (= ({} A) ({/}))) (x A elsn x biex A {} x n0 A x isset 3bitr4 bicon1i))
+thm (prprc () () (-> (-. (e. A (V))) (= ({,} A B) ({} B))) (A snprc A {} {/} B {} uneq1 A B df-pr {/} B {} uncom B {} un0 eqtr2 3eqtr4g sylbi))
+thm (prcom () () (= ({,} A B) ({,} B A)) (A {} B {} uncom A B df-pr B A df-pr 3eqtr4))
+thm (pri1 () ((pri1.1 (e. A (V)))) (e. A ({,} A B)) (A eqid A A = A B = orc ax-mp pri1.1 A B elpr mpbir))
+thm (pri2 () ((pri2.1 (e. B (V)))) (e. B ({,} A B)) (pri2.1 A pri1 B A prcom eleqtr))
+thm (tpi1 () ((tpi1.1 (e. A (V)))) (e. A ({,,} A B C)) (tpi1.1 B pri1 A A B {,} C {} elun1 A B C df-tp syl6eleqr ax-mp))
+thm (tpi2 () ((tpi2.1 (e. B (V)))) (e. B ({,,} A B C)) (tpi2.1 A pri2 B A B {,} C {} elun1 A B C df-tp syl6eleqr ax-mp))
+thm (tpi3 () ((tpi3.1 (e. C (V)))) (e. C ({,,} A B C)) (tpi3.1 snid C C {} A B {,} elun2 A B C df-tp syl6eleqr ax-mp))
+thm (snnz () ((snnz.1 (e. A (V)))) (-. (= ({} A) ({/}))) (snnz.1 snid A A {} n0i ax-mp))
+thm (prnz () ((prnz.1 (e. A (V)))) (-. (= ({,} A B) ({/}))) (prnz.1 B pri1 A A B {,} n0i ax-mp))
+thm (tpnz () ((tpnz.1 (e. A (V)))) (-. (= ({,,} A B C) ({/}))) (tpnz.1 B C tpi1 A A B C {,,} n0i ax-mp))
+thm (snss ((A x) (B x)) ((snss.1 (e. A (V)))) (<-> (e. A B) (C_ ({} A) B)) (x A elsn x cv B e. imbi1i x bial A {} B x dfss2 snss.1 B x clel2 3bitr4r))
+thm (snssg ((A x) (B x)) () (-> (e. A C) (<-> (e. A B) (C_ ({} A) B))) (x cv A B eleq1 x cv A sneq B sseq1d x visset B snss C vtoclbg))
+thm (snssi () () (-> (e. A B) (C_ ({} A) B)) (A B B snssg ibi))
+thm (sssn ((A x) (B x)) () (<-> (C_ A ({} B)) (\/ (= A ({/})) (= A ({} B)))) (A B {} x cv ssel x cv B elsni syl6 x cv B A eleq1 syl6 ibd x 19.23adv A x n0 syl5ib B A snssi syl6 anc2li A B {} eqss syl6ibr orrd B {} 0ss A {/} B {} sseq1 mpbiri A B {} eqimss jaoi impbi))
+thm (snsspr () () (C_ ({} A) ({,} A B)) (A eqid A B = pm2.21ni orri A V A B elprg mpbiri A A B {,} snssi syl A snprc biimp A B {,} 0ss A V e. -. a1i eqsstrd pm2.61i))
+thm (prss ((A x) (B x) (C x)) ((prss.1 (e. A (V))) (prss.2 (e. B (V)))) (<-> (/\ (e. A C) (e. B C)) (C_ ({,} A B) C)) (A C x cv eleq1a B C x cv eleq1a jaao x visset A B elpr syl5ib ssrdv prss.1 B pri1 A B {,} C A ssel mpi prss.2 A pri2 A B {,} C B ssel mpi jca impbi))
+thm (tpss ((A x) (B x) (C x) (D x)) ((tpss.1 (e. A (V))) (tpss.2 (e. B (V))) (tpss.3 (e. C (V)))) (<-> (/\/\ (e. A D) (e. B D) (e. C D)) (C_ ({,,} A B C) D)) (x cv A = x cv D e. x cv B = x cv C = 3jao A D x cv eleq1a B D x cv eleq1a C D x cv eleq1a syl3an x visset A B C eltp syl5ib ssrdv tpss.1 B C tpi1 A B C {,,} D A ssel mpi tpss.2 A C tpi2 A B C {,,} D B ssel mpi tpss.3 A B tpi3 A B C {,,} D C ssel mpi 3jca impbi))
+thm (sneqr () ((sneqr.1 (e. A (V)))) (-> (= ({} A) ({} B)) (= A B)) (sneqr.1 snid A {} B {} A eleq2 mpbii sneqr.1 B elsnc sylib))
+thm (snsspw ((A x)) () (C_ ({} A) (P~ A)) (x cv A eqimss x A elsn A x df-pw eqabi 3imtr4 ssriv))
+thm (prsspw ((A x) (B x) (C x)) ((prsspw.1 (e. A (V))) (prsspw.2 (e. B (V)))) (<-> (C_ ({,} A B) (P~ C)) (/\ (C_ A C) (C_ B C))) (A B {,} C P~ x dfss2 x visset A B elpr x visset C elpw imbi12i x cv A = x cv B = x cv C C_ jaob bitr x bial x x cv A = x cv C C_ -> x cv B = x cv C C_ -> 19.26 prsspw.1 x cv A C sseq1 ceqsalv prsspw.2 x cv B C sseq1 ceqsalv anbi12i bitr 3bitr))
+thm (snex ((A x)) () (e. ({} A) (V)) (x cv A sneq V eleq1d x visset pwex x cv snsspw ssexi V vtoclg A snprc 0ex A {} {/} V eleq1 mpbiri sylbi pm2.61i))
+thm (el ((x y)) () (E. y (e. (cv x) (cv y))) (x visset snid x cv snex y cv x cv {} x cv eleq2 cla4ev ax-mp))
+thm (snelpw () ((snelpw.1 (e. A (V)))) (<-> (e. A B) (e. ({} A) (P~ B))) (snelpw.1 B snss A snex B elpw bitr4))
+thm (rext ((x y) (x z) (y z)) () (-> (A. z (-> (e. (cv x) (cv z)) (e. (cv y) (cv z)))) (= (cv x) (cv y))) (x visset snid x cv snex z cv x cv {} x cv eleq2 z cv x cv {} y cv eleq2 imbi12d cla4v mpi y x cv elsn y x equcomi sylbi syl))
+thm (sspwb ((A x) (B x)) () (<-> (C_ A B) (C_ (P~ A) (P~ B))) (x cv A B sstr2 com12 x visset A elpw x visset B elpw 3imtr4g ssrdv A P~ B P~ x cv {} ssel x cv snex A elpw x visset A snss bitr4 x cv snex B elpw x visset B snss bitr4 3imtr3g ssrdv impbi))
+thm (ssextss ((A x) (B x)) () (<-> (C_ A B) (A. x (-> (C_ (cv x) A) (C_ (cv x) B)))) (A B sspwb A P~ B P~ x dfss2 x visset A elpw x visset B elpw imbi12i x bial 3bitr))
+thm (ssext ((A x) (B x)) () (<-> (= A B) (A. x (<-> (C_ (cv x) A) (C_ (cv x) B)))) (A B x ssextss B A x ssextss anbi12i A B eqss x x cv A C_ x cv B C_ albi 3bitr4))
+thm (nssss ((A x) (B x)) () (<-> (-. (C_ A B)) (E. x (/\ (C_ (cv x) A) (-. (C_ (cv x) B))))) (x x cv A C_ x cv B C_ -> exnal x cv A C_ x cv B C_ annim x biex A B x ssextss negbii 3bitr4r))
+thm (pweqb () () (<-> (= A B) (= (P~ A) (P~ B))) (A B sspwb B A sspwb anbi12i A B eqss A P~ B P~ eqss 3bitr4))
+thm (moabex ((x y) (ph y)) () (-> (E* x ph) (e. ({|} x ph) (V))) (ph y ax-17 x mo2 y cv x df-sn y cv snex eqeltrr x cv y cv = ph pm3.26 x ss2abi ssexi x ph x cv y cv = -> hba1 ph x cv y cv = pm4.71 biimp x a4s ph x cv y cv = ancom syl6bb biabd V eleq1d mpbiri y 19.23aiv sylbi))
+thm (euabex () () (-> (E! x ph) (e. ({|} x ph) (V))) (x ph eumo x ph moabex syl))
+thm (preq1 () () (-> (= A B) (= ({,} A C) ({,} B C))) (A B sneq C {} uneq1d A C df-pr B C df-pr 3eqtr4g))
+thm (preq2 () () (-> (= A B) (= ({,} C A) ({,} C B))) (A B C preq1 C A prcom C B prcom 3eqtr4g))
+thm (preqr1 () ((preqr1.1 (e. A (V))) (preqr1.2 (e. B (V)))) (-> (= ({,} A C) ({,} B C)) (= A B)) (preqr1.1 C pri1 A C {,} B C {,} A eleq2 mpbii preqr1.1 B C elpr sylib preqr1.2 C pri1 A C {,} B C {,} B eleq2 mpbiri preqr1.2 A C elpr sylib A B eqcom A C B eqeq2 oplem1))
+thm (prer2 () ((prer2.1 (e. A (V))) (prer2.2 (e. B (V)))) (-> (= ({,} C A) ({,} C B)) (= A B)) (C A prcom C B prcom eqeq12i prer2.1 prer2.2 C preqr1 sylbi))
+thm (preq12b () ((preq12b.1 (e. A (V))) (preq12b.2 (e. B (V))) (preq12b.3 (e. C (V))) (preq12b.4 (e. D (V)))) (<-> (= ({,} A B) ({,} C D)) (\/ (/\ (= A C) (= B D)) (/\ (= A D) (= B C)))) (preq12b.1 B pri1 A B {,} C D {,} A eleq2 mpbii preq12b.1 C D elpr sylib A C B preq1 C D {,} eqeq1d preq12b.2 preq12b.4 C prer2 syl6bi com12 ancld C D prcom A B {,} eqeq2i A D B preq1 D C {,} eqeq1d preq12b.2 preq12b.3 D prer2 syl6bi com12 sylbi ancld orim12d mpd A C B preq1 B D C preq2 sylan9eq A D B preq1 D B prcom syl6eq B C D preq1 sylan9eq jaoi impbi))
+thm (prel12 () ((preq12b.1 (e. A (V))) (preq12b.2 (e. B (V))) (preq12b.3 (e. C (V))) (preq12b.4 (e. D (V)))) (-> (-. (= A B)) (<-> (= ({,} A B) ({,} C D)) (/\ (e. A ({,} C D)) (e. B ({,} C D))))) (preq12b.1 B pri1 A B {,} C D {,} A eleq2 mpbii preq12b.2 A pri2 A B {,} C D {,} B eleq2 mpbii jca A B = -. a1i B D A eqeq2 negbid A D = A C = orel2 syl6bi com3l imp ancrd B C A eqeq2 negbid A C = A D = orel1 syl6bi com3l imp ancrd orim12d preq12b.2 C D elpr B C = B D = orcom bitr preq12b.1 preq12b.2 preq12b.3 preq12b.4 preq12b 3imtr4g exp preq12b.1 C D elpr syl5ib imp3a impbid))
+thm (opeq1 () () (-> (= A B) (= (<,> A C) (<,> B C))) (A B C preq1 A C {,} B C {,} A {} preq2 syl A B sneq A {} B {} B C {,} preq1 syl eqtrd A C df-op B C df-op 3eqtr4g))
+thm (opeq2 () () (-> (= A B) (= (<,> C A) (<,> C B))) (A B C preq2 C A {,} C B {,} C {} preq2 syl C A df-op C B df-op 3eqtr4g))
+thm (opeq12 () () (-> (/\ (= A C) (= B D)) (= (<,> A B) (<,> C D))) (A C B opeq1 B D C opeq2 sylan9eq))
+thm (hbop ((A y) (B y) (x y)) ((hbop.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbop.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (<,> A B)) (A. x (e. (cv y) (<,> A B)))) (hbop.1 hbsn hbop.1 hbop.2 hbpr hbpr A B df-op y cv eleq2i A B df-op y cv eleq2i x bial 3imtr4))
+thm (nnullss ((x y) (A x) (A y)) () (-> (-. (= A ({/}))) (E. x (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))))) (A y n0 y visset A snss y visset snnz y cv snex x cv y cv {} A sseq1 x cv y cv {} {/} eqeq1 negbid anbi12d cla4ev mpan2 sylbi y 19.23aiv sylbi))
+thm (exss ((x y) (x z) (A x) (y z) (A y) (A z) (ph y) (ph z)) () (-> (E.e. x A ph) (E. y (/\ (C_ (cv y) A) (E.e. x (cv y) ph)))) (x A ph df-rab {/} eqeq1i negbii x A ph rabn0 x x cv A e. ph /\ {|} z n0 3bitr3 z cv snex y cv z cv {} A sseq1 y cv z cv {} x ph rexeq anbi12d cla4ev z visset x x cv A e. ph /\ {|} snss x A ph ssab z cv {} x x cv A e. ph /\ {|} A sstr2 mpi sylbi z cv x x cv A e. [/] z cv x ph [/] pm3.27 z x equsb1 x z cv elsn z x bisb mpbir jctil z x x cv A e. ph /\ df-clab z x x cv A e. ph sban bitr x z cv {} ph df-rab z cv eleq2i z x x cv z cv {} e. ph /\ df-clab z x x cv z cv {} e. ph sban 3bitr 3imtr4 z cv x z cv {} ph {e.|} n0i syl x z cv {} ph rabn0 sylib sylanc z 19.23aiv sylbi))
+thm (pw0 () () (= (P~ ({/})) ({} ({/}))) ({/} x df-pw eqabi x cv {/} eqss x cv 0ss mpbiran2 bitr4 biabri {/} x df-sn eqtr4))
+thm (pwpw0 ((x y)) () (= (P~ ({} ({/}))) ({,} ({/}) ({} ({/})))) (x cv {/} {} y dfss2 y {/} elsn y cv x cv e. imbi2i y bial bitr y y cv x cv e. y cv {/} = exintr x cv y n0 {/} x cv y df-clel 0ex x cv snss y y cv {/} = y cv x cv e. exancom 3bitr3 3imtr4g sylbi anc2li x cv {/} {} eqss syl6ibr orrd {/} {} 0ss x cv {/} {/} {} sseq1 mpbiri x cv {/} {} eqimss jaoi impbi x biabi {/} {} x df-pw {/} {/} {} x dfpr2 3eqtr4))
+thm (pwv () () (= (P~ (V)) (V)) (x cv ssv x visset V elpw mpbir x visset 2th eqri))
+thm (p0ex () () (e. ({} ({/})) (V)) ({/} snex))
+thm (pp0ex () () (e. ({,} ({/}) ({} ({/}))) (V)) (pwpw0 p0ex pwex eqeltrr))
+thm (0nep0 () () (-. (= ({/}) ({} ({/})))) (0ex snnz {/} {} {/} eqcom mtbi))
+thm (0inp0 () () (-> (= A ({/})) (-. (= A ({} ({/}))))) (0nep0 A {/} {/} {} eqeq1 mtbiri))
+thm (dtru ((x y)) () (-. (A. x (= (cv x) (cv y)))) (y cv 0inp0 p0ex x cv {/} {} y cv eqeq2 negbid cla4ev syl 0ex x cv {/} y cv eqeq2 negbid cla4ev pm2.61i x y cv x cv = exnal y cv x cv eqcom x bial negbii bitr mpbi))
+thm (dtrucor ((x y)) ((dtrucor.1 (= (cv x) (cv y)))) (=/= (cv x) (cv y)) (x y dtru x cv y cv =/= pm2.21i dtrucor.1 mpg))
+thm (zfpair ((x z) (w x) (v x) (w z) (v z) (v w) (y z) (w y) (v y)) () (e. ({,} (cv x) (cv y)) (V)) (x cv y cv w dfpr2 z z cv {/} = w cv x cv = /\ z cv {/} {} = w cv y cv = /\ 19.43 z cv {/} = w cv x cv = z cv {/} {} = w cv y cv = prlem2 z biex z z cv {/} = w cv x cv = 19.41v 0ex z isseti mpbiran z z cv {/} {} = w cv y cv = 19.41v p0ex z isseti mpbiran orbi12i 3bitr3r w biabi {/} {/} {} z dfpr2 pp0ex eqeltrr v x w equequ2 z cv 0inp0 w cv y cv = prlem1 w 19.21adv a4w v y w equequ2 z cv 0inp0 con2i w cv x cv = prlem1 z cv {/} = w cv x cv = /\ z cv {/} {} = w cv y cv = /\ orcom bisyl7 w 19.21adv a4w jaoi zfrep4 eqeltr eqeltr))
+thm (prex ((x y) (A x) (A y) (B x) (B y)) () (e. ({,} A B) (V)) (x cv A B preq1 V eleq1d y cv B x cv preq2 V eleq1d x y zfpair V vtoclg syl5bi V vtocleg A B prprc B snex A B {,} B {} V eleq1 mpbiri syl B A prprc A B prcom syl5eq A snex A B {,} A {} V eleq1 mpbiri syl pm2.61nii))
+thm (opex () () (e. (<,> A B) (V)) (A B df-op A {} A B {,} prex eqeltr))
+thm (elop () ((elop.1 (e. A (V)))) (<-> (e. A (<,> B C)) (\/ (= A ({} B)) (= A ({,} B C)))) (B C df-op A eleq2i elop.1 B {} B C {,} elpr bitr))
+thm (opi1 () () (e. ({} A) (<,> A B)) (A snex A B {,} pri1 A B df-op eleqtrr))
+thm (opi2 () () (e. ({,} A B) (<,> A B)) (A B prex A {} pri2 A B df-op eleqtrr))
+thm (opnz () () (-. (= (<,> A B) ({/}))) (A B opi1 A {} A B <,> n0i ax-mp))
+thm (opth () ((opth.1 (e. A (V))) (opth.2 (e. B (V))) (opth.3 (e. D (V)))) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D))) (C D opi1 A B <,> C D <,> C {} eleq2 mpbiri C snex A B elop sylib opth.1 snid C {} A {} A eleq2 mpbiri opth.1 B pri1 C {} A B {,} A eleq2 mpbiri jaoi syl opth.1 C elsnc sylib A B <,> C D <,> C B <,> eqeq1 A C B opeq1 syl5bi C D df-op C B df-op eqeq12i C D prex C B prex C {} prer2 sylbi opth.3 opth.2 C prer2 eqcomd syl syl6 jcai A C B D opeq12 impbi))
+thm (opthg ((A x) (B x) (C x) (D x)) ((opthg.1 (e. A (V))) (opthg.2 (e. B (V)))) (-> (e. D R) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D)))) (x cv D C opeq2 A B <,> eqeq2d x cv D B eqeq2 A C = anbi2d opthg.1 opthg.2 x visset C opth R vtoclbg))
+thm (opthgg ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y) (T x) (T y)) () (-> (/\/\ (e. A R) (e. B S) (e. D T)) (<-> (= (<,> A B) (<,> C D)) (/\ (= A C) (= B D)))) (x cv A y cv opeq1 C D <,> eqeq1d x cv A C eqeq1 y cv D = anbi1d bibi12d D T e. imbi2d y cv B A opeq2 C D <,> eqeq1d y cv B D eqeq1 A C = anbi2d bibi12d D T e. imbi2d x visset y visset D T C opthg R S vtocl2g exp 3imp))
+thm (otthg () ((otthg.1 (e. A (V))) (otthg.2 (e. B (V))) (otthg.3 (e. R (V)))) (-> (/\ (e. D F) (e. S G)) (<-> (= (<,> (<,> A B) R) (<,> (<,> C D) S)) (/\/\ (= A C) (= B D) (= R S)))) (A B opex otthg.3 S G C D <,> opthg otthg.1 otthg.2 D F C opthg R S = anbi1d sylan9bbr A C = B D = R S = df-3an syl6bbr))
+thm (eqvinop ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((eqvinop.1 (e. B (V))) (eqvinop.2 (e. C (V)))) (<-> (= A (<,> B C)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (= (<,> (cv x) (cv y)) (<,> B C)))))) (x visset y visset eqvinop.2 B opth A x cv y cv <,> = anbi2i A x cv y cv <,> = x cv B = y cv C = /\ ancom x cv B = y cv C = A x cv y cv <,> = anass 3bitr y biex y x cv B = y cv C = A x cv y cv <,> = /\ 19.42v eqvinop.2 y cv C x cv opeq2 A eqeq2d ceqsexv x cv B = anbi2i 3bitr x biex eqvinop.1 x cv B C opeq1 A eqeq2d ceqsexv bitr2))
+thm (copsexg ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (ph z) (ph w)) () (-> (= A (<,> (cv x) (cv y))) (<-> ph (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) ph))))) (x visset y visset A z w eqvinop z cv w cv <,> x cv y cv <,> eqcom x visset y visset w visset z cv opth bitr y w cv ph ceqex x z cv y y cv w cv = ph /\ E. ceqex sylan9bbr sylbi z cv w cv <,> x cv y cv <,> eqcom x visset y visset w visset z cv opth bitr ph anbi1i x cv z cv = y cv w cv = ph anass bitr y biex y x cv z cv = y cv w cv = ph /\ 19.42v bitr x biex syl6bbr A z cv w cv <,> x cv y cv <,> eqeq1 A z cv w cv <,> x cv y cv <,> eqeq1 ph anbi1d x y bi2exdv ph bibi2d imbi12d mpbiri z cv w cv <,> x cv y cv <,> = adantr z w 19.23aivv sylbi pm2.43i))
+thm (copsex2g ((x y) (ps x) (ps y) (A x) (A y) (B x) (B y)) ((copsex2g.1 (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)))) (-> (/\ (e. A C) (e. B D)) (<-> (E. x (E. y (/\ (= (<,> A B) (<,> (cv x) (cv y))) ph))) ps)) (x y x cv A = y cv B = eeanv x y A B <,> x cv y cv <,> = ph /\ E. hbe1 ps x ax-17 hbbi y A B <,> x cv y cv <,> = ph /\ hbe1 x hbex ps y ax-17 hbbi x cv A y cv B opeq12 A B <,> x y ph copsexg eqcoms syl copsex2g.1 bitr3d 19.23ai 19.23ai sylbir A C x elex B D y elex syl2an))
+thm (copsex4g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ps z) (ps w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w)) ((copsex4g.1 (-> (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) (<-> ph ps)))) (-> (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) (<-> (E. x (E. y (E. z (E. w (/\ (/\ (= (<,> A B) (<,> (cv x) (cv y))) (= (<,> C D) (<,> (cv z) (cv w)))) ph))))) ps)) (x visset y visset B S A opthg A B <,> x cv y cv <,> eqcom syl5bb A R e. adantl z visset w visset D S C opthg C D <,> z cv w cv <,> eqcom syl5bb C R e. adantl bi2anan9 ph anbi1d x y z w bi4exdv x cv A = y cv B = /\ z cv C = w cv D = /\ /\ id copsex4g.1 R S cgsex4g bitrd))
+thm (opprc1 () () (-> (-. (e. A (V))) (= (<,> A B) ({,} ({/}) ({} B)))) (A snprc A {} {/} A B {,} preq1 sylbi A B prprc A B {,} B {} {/} preq2 syl eqtrd A B df-op syl5eq))
+thm (opprc1b ((A x) (B x)) () (<-> (-. (e. A (V))) (e. ({/}) (<,> A B))) (0ex B {} pri1 A B opprc1 {/} eleq2d mpbiri x cv A B opeq1 {/} eleq2d negbid x visset snnz x cv {} {/} eqcom mtbi x visset B prnz x cv B {,} {/} eqcom mtbi pm3.2ni 0ex x cv B elop mtbir V vtoclg con2i impbi))
+thm (opprc2 () () (-> (-. (e. B (V))) (= (<,> A B) (<,> A A))) (B A prprc B A prcom A dfsn2 3eqtr3g A B {,} A A {,} A {} preq2 syl A B df-op A A df-op 3eqtr4g))
+thm (opprc3 () () (<-> (/\ (-. (e. A (V))) (-. (e. B (V)))) (= (<,> A B) ({} ({/})))) (B A opprc2 A A opprc1 A snprc A {} {/} {/} preq2 sylbi eqtrd sylan9eqr {/} dfsn2 syl6eqr 0ex snid A B <,> {/} {} {/} eleq2 mpbiri A B opprc1b sylibr A B opprc1 {/} {} eqeq1d B snex 0ex {/} prer2 {/} dfsn2 {/} B {} {,} eqeq2i B snprc 3imtr4 syl6bi anc2li mpcom impbi))
+thm (opth2 ((A x) (B x) (C x) (D x)) ((opth2.1 (e. B (V))) (opth2.2 (e. D (V)))) (-> (= (<,> A B) (<,> C D)) (= B D)) (x cv A B opeq1 C D <,> eqeq1d B D = imbi1d x visset opth2.1 opth2.2 C opth pm3.27bd V vtoclg {/} A B <,> C D <,> nelneq2 A B opprc1b C D opprc1b bicon1i bicomi syl2anb B D = pm2.21d A B opprc1 C D opprc1 eqeqan12d B snex D snex {/} prer2 opth2.1 D sneqr syl syl6bi pm2.61an2 pm2.61i))
+thm (moop2 ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w) (x z)) () (E* x (= A (<,> B (cv x)))) (A B x cv <,> z y cv x z cv B e. [/] {|} y cv <,> eqtr2t x visset y visset B z y cv x z cv B e. [/] {|} opth2 syl x y gen2 w cv A e. x ax-17 y x z cv B e. hbs1 w z hbab w cv y cv e. x ax-17 hbop hbeq x y B z sbab B z y cv x z cv B e. [/] {|} x cv opeq1 syl x cv y cv z y cv x z cv B e. [/] {|} opeq2 eqtrd A eqeq2d mo4f mpbir))
+thm (mosubop ((x y) (x z) (A x) (y z) (A y) (A z)) ((mosubop.1 (E* x ph))) (E* x (E. y (E. z (/\ (= A (<,> (cv y) (cv z))) ph)))) (y z A y cv z cv <,> = ph /\ E. hbe1 x hbmo z A y cv z cv <,> = ph /\ hbe1 y hbex x hbmo mosubop.1 A y cv z cv <,> = x ax-17 A y z ph copsexg bimod mpbii 19.23ai 19.23ai A y cv z cv <,> = ph pm3.26 y z 19.22i2 x 19.23aiv con3i x y z A y cv z cv <,> = ph /\ E. E. exmo ori syl pm2.61i))
+thm (euop2 ((ph x) (A x) (x y)) () (<-> (E! x (E. y (/\ (= (cv x) (<,> A (cv y))) ph))) (E! y ph)) (A y cv opex y x cv A moop2 ph euxfr2))
+thm (eusn ((x y) (x z) (y z) (ph y)) () (<-> (E! x ph) (E. x (= ({|} x ph) ({} (cv x))))) (x ph y cv {} eqabr x y cv elsn ph bibi2i x bial bitr y biex x ph {|} x cv {} = y ax-17 y x ph hbab1 z cv y cv {} e. x ax-17 hbeq x cv y cv sneq x ph {|} eqeq2d cbvex x ph y df-eu 3bitr4r))
+thm (opthwiener () ((opthw.1 (e. A (V))) (opthw.2 (e. B (V)))) (<-> (= ({,} ({,} ({} A) ({/})) ({} ({} B))) ({,} ({,} ({} C) ({/})) ({} ({} D)))) (/\ (= A C) (= B D))) (A {} {/} {,} B {} {} {,} C {} {/} {,} D {} {} {,} = id B {} snex A {} {/} {,} pri2 A {} {/} {,} B {} {} {,} C {} {/} {,} D {} {} {,} B {} {} eleq2 mpbii B {} snex C {} {/} {,} D {} {} elpr sylib 0ex C {} pri2 opthw.2 snnz 0ex B {} elsnc {/} B {} eqcom bitr mtbir {/} C {} {/} {,} B {} {} nelneq2 mp2an C {} {/} {,} B {} {} eqcom mtbi B {} {} C {} {/} {,} = B {} {} D {} {} = biorf ax-mp sylibr B {} {} D {} {} C {} {/} {,} preq2 syl eqtr4d A {} {/} prex C {} {/} prex B {} {} preqr1 syl A snex C snex {/} preqr1 syl opthw.1 C sneqr syl B {} snex A {} {/} {,} pri2 A {} {/} {,} B {} {} {,} C {} {/} {,} D {} {} {,} B {} {} eleq2 mpbii B {} snex C {} {/} {,} D {} {} elpr sylib 0ex C {} pri2 opthw.2 snnz 0ex B {} elsnc {/} B {} eqcom bitr mtbir {/} C {} {/} {,} B {} {} nelneq2 mp2an C {} {/} {,} B {} {} eqcom mtbi B {} {} C {} {/} {,} = B {} {} D {} {} = biorf ax-mp sylibr B snex D {} sneqr opthw.2 D sneqr 3syl jca A C sneq A {} C {} {/} preq1 A {} {/} {,} C {} {/} {,} B {} {} preq1 3syl B D sneq B {} D {} sneq B {} {} D {} {} C {} {/} {,} preq2 3syl sylan9eq impbi))
+thm (pwin ((A x) (B x)) () (= (P~ (i^i A B)) (i^i (P~ A) (P~ B))) (x cv A B ssin x visset A elpw x visset B elpw anbi12i x visset A B i^i elpw 3bitr4 ineqri eqcomi))
+thm (pwunss ((A x) (B x)) () (C_ (u. (P~ A) (P~ B)) (P~ (u. A B))) (x cv A B ssun x cv A P~ B P~ elun x visset A elpw x visset B elpw orbi12i bitr x visset A B u. elpw 3imtr4 ssriv))
+thm (pwssun ((A x) (A y) (x y) (B x) (B y)) () (<-> (\/ (C_ A B) (C_ B A)) (C_ (P~ (u. A B)) (u. (P~ A) (P~ B)))) (A B C_ B A C_ orcom B A ssequn2 A B u. A pweq A B u. P~ A P~ eqimss syl sylbi A B ssequn1 A B u. B pweq A B u. P~ B P~ eqimss syl sylbi orim12i sylbi A B u. P~ A P~ B P~ ssun syl A B u. P~ A P~ B P~ u. x cv y cv {,} ssel x cv {} A y cv {} B unss12 x visset A snss y visset B snss syl2anb x y zfpair A B u. elpw x cv y cv df-pr A B u. sseq1i bitr2 sylib syl5 exp3a com23 imp31 x cv y cv {,} A P~ B P~ elun sylib x y zfpair A elpw x visset y visset A prss bitr4 pm3.27bd x y zfpair B elpw x visset y visset B prss bitr4 pm3.26bd orim12i syl ord exp com23 imp ssrdv exp31 y cv A e. A B C_ bi2.15 syl6ib com23 imp ssrdv exp orrd impbi))
+thm (pwun () () (<-> (\/ (C_ A B) (C_ B A)) (= (P~ (u. A B)) (u. (P~ A) (P~ B)))) (A B pwunss A B u. P~ A P~ B P~ u. C_ biantru A B pwssun A B u. P~ A P~ B P~ u. eqss 3bitr4))
+thm (dfuni2 ((x y) (A x) (A y)) () (= (U. A) ({|} x (E.e. y A (e. (cv x) (cv y))))) (A x y df-uni y x cv y cv e. y cv A e. exancom y A x cv y cv e. df-rex bitr4 x biabi eqtr))
+thm (eluni ((A x) (x y) (A y) (B x) (B y)) () (<-> (e. A (U. B)) (E. x (/\ (e. A (cv x)) (e. (cv x) B)))) (A B U. elisset A x cv elisset x cv B e. adantr x 19.23aiv y cv A x cv eleq1 x cv B e. anbi1d x biexdv B y x df-uni V elab2g pm5.21nii))
+thm (eluni2 ((A x) (B x)) () (<-> (e. A (U. B)) (E.e. x B (e. A (cv x)))) (x A x cv e. x cv B e. exancom A B x eluni x B A x cv e. df-rex 3bitr4))
+thm (elunii ((A x) (B x) (C x)) () (-> (/\ (e. A B) (e. B C)) (e. A (U. C))) (x cv B A eleq2 x cv B C eleq1 anbi12d C cla4egv anabsi7 A C x eluni sylibr))
+thm (hbuni ((y z) (A y) (A z) (x y) (x z)) ((hbuni.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (U. A)) (A. x (e. (cv y) (U. A)))) (y cv z cv e. x ax-17 y cv z cv e. x ax-17 hbuni.1 hbel hban z hbex y cv A z eluni y cv A z eluni x bial 3imtr4))
+thm (eluniab ((A x) (x y) (A y) (ph y)) () (<-> (e. A (U. ({|} x ph))) (E. x (/\ (e. A (cv x)) ph))) (A x ph {|} y eluni A y cv e. x ax-17 y x ph hbab1 hban A x cv e. ph /\ y ax-17 y cv x cv A eleq2 y cv x cv x ph {|} eleq1 anbi12d x ph abid A x cv e. anbi2i syl6bb cbvex bitr))
+thm (unieq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= (U. A) (U. B))) (A B y cv eleq2 x cv y cv e. anbi2d y biexdv x biabdv A x y df-uni B x y df-uni 3eqtr4g))
+thm (unieqi () ((unieqi.1 (= A B))) (= (U. A) (U. B)) (unieqi.1 A B unieq ax-mp))
+thm (unieqd () ((unieqd.1 (-> ph (= A B)))) (-> ph (= (U. A) (U. B))) (unieqd.1 A B unieq syl))
+thm (unpr ((x y) (A x) (A y) (B x) (B y)) ((unpr.1 (e. A (V))) (unpr.2 (e. B (V)))) (= (U. ({,} A B)) (u. A B)) (y x cv y cv e. y cv A = /\ x cv y cv e. y cv B = /\ 19.43 y visset A B elpr x cv y cv e. anbi2i x cv y cv e. y cv A = y cv B = andi bitr y biex unpr.1 x cv y clel3 y y cv A = x cv y cv e. exancom bitr unpr.2 x cv y clel3 y y cv B = x cv y cv e. exancom bitr orbi12i 3bitr4r x biabi A B x df-un A B {,} x y df-uni 3eqtr4r))
+thm (unop () () (= (U. (<,> A B)) ({,} A B)) (A B df-op unieqi A snex A B prex unpr A B snsspr A {} A B {,} ssequn1 mpbi 3eqtr))
+thm (unisn () ((unisn.1 (e. A (V)))) (= (U. ({} A)) A) (A dfsn2 unieqi unisn.1 unisn.1 unpr A unidm 3eqtr))
+thm (unisng ((A x)) () (-> (e. A B) (= (U. ({} A)) A)) (x cv A sneq unieqd x cv A = id eqeq12d x visset unisn B vtoclg))
+thm (uniun ((x y) (A x) (A y) (B x) (B y)) () (= (U. (u. A B)) (u. (U. A) (U. B))) (y x cv y cv e. y cv A e. /\ x cv y cv e. y cv B e. /\ 19.43 y cv A B elun x cv y cv e. anbi2i x cv y cv e. y cv A e. y cv B e. andi bitr y biex x cv A y eluni x cv B y eluni orbi12i 3bitr4 x cv A B u. y eluni x cv A U. B U. elun 3bitr4 eqri))
+thm (uniin ((x y) (A x) (A y) (B x) (B y)) () (C_ (U. (i^i A B)) (i^i (U. A) (U. B))) (y x cv y cv e. y cv A e. /\ x cv y cv e. y cv B e. /\ 19.40 x cv A B i^i y eluni y cv A B elin x cv y cv e. anbi2i x cv y cv e. y cv A e. y cv B e. anandi bitr y biex bitr x cv A U. B U. elin x cv A y eluni x cv B y eluni anbi12i bitr 3imtr4 ssriv))
+thm (uniss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (U. A) (U. B))) (A B y cv ssel x cv y cv e. anim2d y 19.22dv x 19.21aiv x y x cv y cv e. y cv A e. /\ E. y x cv y cv e. y cv B e. /\ E. ss2ab sylibr A x y df-uni B x y df-uni 3sstr4g))
+thm (ssuni ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (C_ A B) (e. B C)) (C_ A (U. C))) (x cv B A sseq2 A C U. C_ imbi1d y cv x cv e. x cv C e. /\ x 19.8a expcom y cv C x eluni syl6ibr y cv A e. imim2d y 19.20dv A x cv y dfss2 A C U. y dfss2 3imtr4g vtoclga impcom))
+thm (uni0 () () (= (U. ({/})) ({/})) ({/} {} 0ss {/} {/} {} uniss ax-mp 0ex unisn sseqtr {/} U. ss0 ax-mp))
+thm (uni0b ((x y) (A x) (A y)) () (<-> (= (U. A) ({/})) (C_ A ({} ({/})))) (x {/} elsn x A biral A {/} {} x dfss3 A U. y n0 x A y y cv x cv e. rexcom4 x cv y n0 x A birex y cv A x eluni2 y biex 3bitr4r x A x cv {/} = rexnal 3bitr bicon4i 3bitr4r))
+thm (elssuni () () (-> (e. A B) (C_ A (U. B))) (A ssid A A B ssuni mpan))
+thm (unissb ((x y) (A x) (A y) (B x) (B y)) () (<-> (C_ (U. A) B) (A.e. x A (C_ (cv x) B))) (A U. B y dfss2 y cv A x eluni y cv B e. imbi1i x y cv x cv e. x cv A e. /\ y cv B e. 19.23v bitr4 y bial y x y cv x cv e. x cv A e. /\ y cv B e. -> alcom y x cv A e. y cv x cv e. y cv B e. -> 19.21v y cv x cv e. x cv A e. y cv B e. impexp y cv x cv e. x cv A e. y cv B e. bi2.04 bitr y bial x cv B y dfss2 x cv A e. imbi2i 3bitr4 x bial bitr 3bitr x A x cv B C_ df-ral bitr4))
+thm (uniss2 ((x y) (A x) (A y) (B x) (B y)) () (-> (A.e. x A (E.e. y B (C_ (cv x) (cv y)))) (C_ (U. A) (U. B))) (x cv y cv B ssuni expcom r19.23aiv x A r19.20si A B U. x unissb sylibr))
+thm (unidif ((x y) (A x) (A y) (B x) (B y)) () (-> (A.e. x A (E.e. y (\ A B) (C_ (cv x) (cv y)))) (= (U. (\ A B)) (U. A))) (x A y A B \ uniss2 A B difss A B \ A uniss ax-mp jctil A B \ U. A U. eqss sylibr))
+thm (unidif0 () () (= (U. (\ A ({} ({/})))) (U. A)) (A {/} {} \ {/} {} uniun A {/} {} undif1 A {/} {} uncom eqtr2 unieqi 0ex unisn A {/} {} \ U. uneq2i A {/} {} \ U. un0 eqtr2 3eqtr4r {/} {} A uniun 0ex unisn A U. uneq1i 3eqtr {/} A U. uncom A U. un0 3eqtr))
+thm (ssunieq ((A x) (B x)) () (-> (/\ (e. A B) (A.e. x B (C_ (cv x) A))) (= A (U. B))) (A B elssuni B A x unissb biimpr anim12i A B U. eqss sylibr))
+thm (unisseq ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (= A (U. ({e.|} x B (C_ (cv x) A))))) (A ssid A B e. jctr x cv A A sseq1 B elrab sylibr x cv y cv A sseq1 B elrab pm3.27bd rgen A x B x cv A C_ {e.|} y ssunieq mpan2 syl))
+thm (uniex ((x y) (x z) (A x) (y z) (A y) (A z)) ((uniex.1 (e. A (V)))) (e. (U. A) (V)) (uniex.1 z cv A unieq V eleq1d x y z axun y cv z cv x eluni y cv x cv e. imbi1i y bial x biex mpbir bm1.3ii x cv z cv U. y dfcleq x biex mpbir issetri vtocl))
+thm (uniexg ((A x)) () (-> (e. A B) (e. (U. A) (V))) (x cv A unieq V eleq1d x visset uniex B vtoclg))
+thm (unex () ((unex.1 (e. A (V))) (unex.2 (e. B (V)))) (e. (u. A B) (V)) (unex.1 unex.2 unpr A B prex uniex eqeltrr))
+thm (unexb ((x y) (A x) (A y) (B x) (B y)) () (<-> (/\ (e. A (V)) (e. B (V))) (e. (u. A B) (V))) (x cv A y cv uneq1 V eleq1d y cv B A uneq2 V eleq1d x visset y visset unex V V vtocl2g A B ssun1 A B u. V A ssexg mpi B A ssun2 A B u. V B ssexg mpi jca impbi))
+thm (difex2 () () (-> (e. B C) (<-> (e. A (V)) (e. (\ A B) (V)))) (A V B difexg B C e. a1i B C elisset A B \ V e. anim1i ancoms B A B \ unexb sylib B A undif2 V eleq1i sylib A B ssun2 B A u. V A ssexg mpi syl expcom impbid))
+thm (tpex () () (e. ({,,} A B C) (V)) (A B C df-tp A B prex C snex unex eqeltr))
+thm (opeluu () () (-> (e. (<,> (cv x) (cv y)) A) (/\ (e. (cv x) (U. (U. A))) (e. (cv y) (U. (U. A))))) (x cv y cv opi2 x cv y cv {,} x cv y cv <,> A elunii mpan x visset y cv pri1 x cv x cv y cv {,} A U. elunii mpan syl x cv y cv opi2 x cv y cv {,} x cv y cv <,> A elunii mpan y visset x cv pri2 y cv x cv y cv {,} A U. elunii mpan syl jca))
+thm (euuni ((x y) (x z) (y z) (ph y) (ph z)) () (-> (E! x ph) (<-> ph (= (U. ({|} x ph)) (cv x)))) (x ph euabex x ph {|} V uniexg syl x ph {|} U. y eueq y cv x ph {|} U. eqcom y bieu z x ph hbab1 hbuni z cv y cv e. x ax-17 hbeq x ph {|} U. x cv = y ax-17 y cv x cv x ph {|} U. eqeq2 cbveu 3bitr sylib x ph eusn x visset snid x ph {|} x cv {} x cv eleq2 mpbiri x ph abid sylib x ph {|} x cv {} unieq x visset unisn syl6eq jca x 19.22i sylbi x ph x ph {|} U. x cv = eupickb 3exp imp3a mp2and))
+thm (reuuni1 () () (-> (/\ (e. (cv x) A) (E!e. x A ph)) (<-> ph (= (U. ({e.|} x A ph)) (cv x)))) (x x cv A e. ph /\ euuni biimpd exp3a impcom x x cv A e. ph /\ euuni x cv A e. ph pm3.27 syl6bir x cv A e. adantl impbid x A ph df-rab unieqi x cv eqeq1i syl6bbr x A ph df-reu sylan2b))
+thm (reuuni2f ((ph y) (x y) (A x) (A y) (B y)) ((reuuni2f.1 (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (reuuni2f.2 (-> (e. B A) (-> ps (A. x ps)))) (reuuni2f.3 (-> (= (cv x) B) (<-> ph ps)))) (-> (/\ (e. B A) (E!e. x A ph)) (<-> ps (= (U. ({e.|} x A ph)) B))) (reuuni2f.1 reuuni2f.1 y cv A e. x ax-17 hbel reuuni2f.1 y cv A e. x ax-17 hbel x A ph hbreu1 B A e. a1i reuuni2f.1 y cv A e. x ax-17 hbel reuuni2f.2 y x A ph hbrab1 hbuni reuuni2f.1 hbeq B A e. a1i hbbid hbimd hbim1 x cv B A eleq1 reuuni2f.3 x cv B x A ph {e.|} U. eqeq2 bibi12d x A ph E!e. imbi2d imbi12d x A ph reuuni1 exp A vtoclgf pm2.43i imp))
+thm (reuuni2 ((ph y) (x y) (A x) (A y) (B x) (B y) (ps x)) ((reuuni2.1 (-> (= (cv x) B) (<-> ph ps)))) (-> (/\ (e. B A) (E!e. x A ph)) (<-> ps (= (U. ({e.|} x A ph)) B))) (y cv B e. x ax-17 ps x ax-17 B A e. a1i reuuni2.1 reuuni2f))
+thm (reucl ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) (e. (U. ({e.|} x A ph)) A)) (x x cv A e. ph /\ eusn y x x cv A e. ph /\ hbab1 hbuni y cv A e. x ax-17 hbel x x cv A e. ph /\ {|} x cv {} unieq x visset unisn syl6req x visset snid x x cv A e. ph /\ {|} x cv {} x cv eleq2 mpbiri x x cv A e. ph /\ abid sylib pm3.26d eqeltrrd 19.23ai sylbi x A ph df-reu x A ph df-rab unieqi A eleq1i 3imtr4))
+thm (reuuni3 ((x y) (A x) (A y) (ph y) (ps x) (ch x)) ((reuuni3.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (reuuni3.2 (-> (= (cv x) (U. ({e.|} y A ps))) (<-> ph ch)))) (-> (E!e. x A ph) ch) (x A ph reucl reuuni3.1 A cbvrabv unieqi A eleq1i sylib reuuni3.1 A cbvrabv unieqi reuuni3.2 A reuuni2 mpbiri mpancom))
+thm (reuuni4 ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) ([/] (U. ({e.|} x A ph)) x ph)) (x A ph reucl x A ph reurex x A ph hbreu1 y x A ph hbrab1 hbuni V ph hbsbc x A ph reuuni1 x x A ph {e.|} U. ph sbceq1 eqcoms syl6bi ph x A ph {e.|} U. x ph [/] ibib sylibr expcom x A ph {e.|} U. V e. a1i com4l r19.23ad mpd x A ph {e.|} U. A elisset syl5 mpd))
+thm (reucl2 ((x y) (A x) (A y) (ph y)) () (-> (E!e. x A ph) (e. (U. ({e.|} x A ph)) ({e.|} x A ph))) (x A ph reucl x A ph reuuni4 jca y cv A e. x ax-17 x A ph {e.|} U. ph elrabsf sylibr))
+thm (ralxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((ralxfr.1 (-> (e. (cv y) B) (e. A B))) (ralxfr.2 (-> (e. (cv x) B) (E.e. y B (= (cv x) A)))) (ralxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (A.e. x B ph) (A.e. y B ps)) (ralxfr.1 ralxfr.3 B rcla4v syl com12 r19.21aiv y B ps hbra1 ph y ax-17 y B ps ra4 ralxfr.3 biimprcd syl6 r19.23ad ralxfr.2 syl5 r19.21aiv impbi))
+thm (rexxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((ralxfr.1 (-> (e. (cv y) B) (e. A B))) (ralxfr.2 (-> (e. (cv x) B) (E.e. y B (= (cv x) A)))) (ralxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E.e. x B ph) (E.e. y B ps)) (ralxfr.1 ralxfr.2 ralxfr.3 negbid ralxfr negbii x B ph dfrex2 y B ps dfrex2 3bitr4))
+thm (rabxfr ((A x) (B z) (C z) (x y) (x z) (D x) (y z) (D y) (D z) (ph y) (ph z) (ps x) (ps z)) ((rabxfr.1 (-> (e. (cv z) B) (A. y (e. (cv z) B)))) (rabxfr.2 (-> (e. (cv z) C) (A. y (e. (cv z) C)))) (rabxfr.3 (-> (e. (cv y) D) (e. A D))) (rabxfr.4 (-> (= (cv x) A) (<-> ph ps))) (rabxfr.5 (-> (= (cv y) B) (= A C)))) (-> (e. B D) (<-> (e. C ({e.|} x D ph)) (e. B ({e.|} y D ps)))) (rabxfr.3 y cv D e. A D e. ibibr mpbi ps anbi1d rabxfr.4 D elrab y D ps rabid 3bitr4g birabi B eleq2i rabxfr.1 z cv D e. y ax-17 rabxfr.2 z cv x D ph {e.|} e. y ax-17 hbel rabxfr.5 x D ph {e.|} eleq1d elrabf rabxfr.1 z cv D e. y ax-17 rabxfr.1 z y D ps hbrab1 hbel y cv B y D ps {e.|} eleq1 elrabf 3bitr3 B D e. C x D ph {e.|} e. B y D ps {e.|} e. pm5.32 mpbir))
+thm (reuxfr2 ((ph x) (A x) (x y) (B x) (B y)) ((reuxfr2.1 (-> (e. (cv y) B) (e. A B))) (reuxfr2.2 (-> (e. (cv x) B) (E* y (/\ (e. (cv y) B) (= (cv x) A)))))) (<-> (E!e. x B (E.e. y B (/\ (= (cv x) A) ph))) (E!e. y B ph)) (x B y x cv A = ph /\ 2reuswap reuxfr2.2 y y cv B e. x cv A = /\ ph moan syl ph y cv B e. x cv A = /\ ancom y cv B e. x cv A = ph anass bitr y bimo sylib mprg y B x x cv A = ph /\ 2reuswap x A moeq x cv B e. ph /\ moani x cv B e. ph /\ x cv A = ancom x cv A = x cv B e. ph an12 bitr x bimo mpbi y cv B e. a1i mprg impbi reuxfr2.1 x cv A = ph pm4.2i B ceqsrexv syl bireua bitr))
+thm (reuxfr ((ps x) (ph y) (A x) (x y) (B x) (B y)) ((reuxfr.1 (-> (e. (cv y) B) (e. A B))) (reuxfr.2 (-> (e. (cv x) B) (E!e. y B (= (cv x) A)))) (reuxfr.3 (-> (= (cv x) A) (<-> ph ps)))) (<-> (E!e. x B ph) (E!e. y B ps)) (reuxfr.2 y B x cv A = reurex syl ph biantrurd y B x cv A = ph r19.41v reuxfr.3 pm5.32i y B birex bitr3 syl6bb bireua reuxfr.1 reuxfr.2 y B x cv A = df-reu y y cv B e. x cv A = /\ eumo sylbi syl ps reuxfr2 bitr))
+thm (reuhyp ((A x) (B y) (C y) (x y)) ((reuhyp.1 (-> (e. (cv x) C) (e. B C))) (reuhyp.2 (-> (/\ (e. (cv x) C) (e. (cv y) C)) (<-> (= (cv x) A) (= (cv y) B))))) (-> (e. (cv x) C) (E!e. y C (= (cv x) A))) (reuhyp.1 B C elisset syl B y eueq sylib y cv B C eleq1 reuhyp.1 syl5bir com12 pm4.71rd reuhyp.2 exp pm5.32d bitr4d y bieudv mpbid y C x cv A = df-reu sylibr))
+thm (reuunixfr ((B x) (x z) (C x) (C z) (x y) (A x) (y z) (A y) (A z) (ph y) (ph z) (ps x) (ps z)) ((reuunixfr.1 (-> (e. (cv z) C) (A. y (e. (cv z) C)))) (reuunixfr.2 (-> (e. (cv y) A) (e. B A))) (reuunixfr.3 (-> (e. (U. ({e.|} y A ps)) A) (e. C A))) (reuunixfr.4 (-> (= (cv x) B) (<-> ph ps))) (reuunixfr.5 (-> (= (cv y) (U. ({e.|} y A ps))) (= B C))) (reuunixfr.6 (-> (e. (cv x) A) (E!e. y A (= (cv x) B))))) (-> (E!e. x A ph) (= (U. ({e.|} x A ph)) C)) (reuunixfr.2 reuunixfr.6 reuunixfr.4 reuxfr y A ps reucl2 y A ps reucl z y A ps hbrab1 hbuni reuunixfr.1 reuunixfr.2 reuunixfr.4 reuunixfr.5 rabxfr syl mpbird sylbi z cv C e. x ax-17 z cv C e. x ax-17 z x A ph hbrab1 hbel C A e. a1i x cv C x A ph {e.|} eleq1 reuuni2f reuunixfr.2 reuunixfr.6 reuunixfr.4 reuxfr y A ps reucl reuunixfr.3 syl sylbi x A ph rabid baibr bireua biimp sylanc mpbid x A ph rabid baib birabi unieqi syl5eqr))
+thm (unipw ((A x) (A y) (x y)) () (= (U. (P~ A)) A) (x cv A P~ y eluni y visset A elpw y cv A x cv ssel sylbi impcom y 19.23aiv sylbi ssriv x visset snid x cv snex y cv x cv {} x cv eleq2 y cv x cv {} A P~ eleq1 anbi12d cla4ev mpan x visset A snelpw x cv A P~ y eluni 3imtr4 ssriv eqssi))
+thm (pwuni ((A x)) () (C_ A (P~ (U. A))) (x cv A elssuni x visset A U. elpw sylibr ssriv))
+thm (uniexb () () (<-> (e. A (V)) (e. (U. A) (V))) (A V uniexg A pwuni A U. V pwexg A U. P~ V A ssexg syl mpi impbi))
+thm (pwexb () () (<-> (e. A (V)) (e. (P~ A) (V))) (A P~ uniexb A unipw V eleq1i bitr2))
+thm (univ () () (= (U. (V)) (V)) (pwv unieqi V unipw eqtr3))
+thm (dfint2 ((x y) (A x) (A y)) () (= (|^| A) ({|} x (A.e. y A (e. (cv x) (cv y))))) (A x y df-int y A x cv y cv e. df-ral x biabi eqtr4))
+thm (inteq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= (|^| A) (|^| B))) (A B y x cv y cv e. raleq x biabdv A x y dfint2 B x y dfint2 3eqtr4g))
+thm (inteqi () ((inteqi.1 (= A B))) (= (|^| A) (|^| B)) (inteqi.1 A B inteq ax-mp))
+thm (inteqd () ((inteqd.1 (-> ph (= A B)))) (-> ph (= (|^| A) (|^| B))) (inteqd.1 A B inteq syl))
+thm (elint ((A x) (x y) (A y) (B x) (B y)) ((elint.1 (e. A (V)))) (<-> (e. A (|^| B)) (A. x (-> (e. (cv x) B) (e. A (cv x))))) (elint.1 y cv A x cv eleq1 x cv B e. imbi2d x bialdv B y x df-int elab2))
+thm (elint2 ((A x) (B x)) ((elint2.1 (e. A (V)))) (<-> (e. A (|^| B)) (A.e. x B (e. A (cv x)))) (elint2.1 B x elint x B A x cv e. df-ral bitr4))
+thm (elintg ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> (e. A (|^| B)) (A.e. x B (e. A (cv x))))) (y cv A B |^| eleq1 y cv A x cv eleq1 x B biraldv bibi12d y visset B x elint2 C vtoclg))
+thm (elinti ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (e. A (|^| B)) (-> (e. C B) (e. A C))) (x cv A y cv eleq1 y cv B e. imbi2d y bialdv x visset B y elint biimp vtoclga y cv C B eleq1 y cv C A eleq2 imbi12d B cla4gv pm2.43b syl))
+thm (hbint ((y z) (A y) (A z) (x y) (x z)) ((hbint.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (|^| A)) (A. x (e. (cv y) (|^| A)))) (y cv z cv e. x ax-17 hbint.1 hbel y cv z cv e. x ax-17 hbim z hbal y visset A z elint y visset A z elint x bial 3imtr4))
+thm (elintab ((A x) (A y) (x y) (ph y)) ((inteqab.1 (e. A (V)))) (<-> (e. A (|^| ({|} x ph))) (A. x (-> ph (e. A (cv x))))) (inteqab.1 x ph {|} y elint y x ph hbab1 A y cv e. x ax-17 hbim ph A x cv e. -> y ax-17 y cv x cv x ph {|} eleq1 x ph abid syl6bb y cv x cv A eleq2 imbi12d cbval bitr))
+thm (elintrab ((A x)) ((inteqab.1 (e. A (V)))) (<-> (e. A (|^| ({e.|} x B ph))) (A.e. x B (-> ph (e. A (cv x))))) (inteqab.1 x x cv B e. ph /\ elintab x cv B e. ph A x cv e. impexp x bial bitr x B ph df-rab inteqi A eleq2i x B ph A x cv e. -> df-ral 3bitr4))
+thm (int0 ((x y)) () (= (|^| ({/})) (V)) (y cv noel x cv y cv e. pm2.21i y ax-gen x cv eqid 2th x biabi {/} x y df-int x df-v 3eqtr4))
+thm (intss1 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (C_ (|^| B) A)) (y cv A B eleq1 y cv A x cv eleq2 imbi12d B cla4gv pm2.43a x visset B y elint syl5ib ssrdv))
+thm (ssint ((x y) (A x) (A y) (B x) (B y)) () (<-> (C_ A (|^| B)) (A.e. x B (C_ A (cv x)))) (A B |^| y dfss3 y visset B x elint2 y A biral y A x B y cv x cv e. ralcom A x cv y dfss3 x B biral bitr4 3bitr))
+thm (ssintub ((x y) (A x) (A y) (B x) (B y)) () (C_ A (|^| ({e.|} x B (C_ A (cv x))))) (A x B A x cv C_ {e.|} y ssint x cv y cv A sseq2 B elrab pm3.27bd mprgbir))
+thm (intmin ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A B) (= A (|^| ({e.|} x B (C_ A (cv x)))))) (A ssid x cv A A sseq2 x cv A y cv eleq2 imbi12d B rcla4v mpii y visset x B A x cv C_ elintrab syl5ib ssrdv A x B ssintub jctil A x B A x cv C_ {e.|} |^| eqss sylibr))
+thm (intss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (|^| B) (|^| A))) (y cv A e. y cv B e. x cv y cv e. imim1 y 19.20ii x visset B y elint x visset A y elint 3imtr4g x 19.21aiv A B y dfss2 B |^| A |^| x dfss2 3imtr4))
+thm (intmin2 ((A x)) ((intmin2.1 (e. A (V)))) (= A (|^| ({|} x (C_ A (cv x))))) (intmin2.1 A V x intmin ax-mp x V A x cv C_ df-rab x visset A x cv C_ biantrur x biabi eqtr4 inteqi eqtr))
+thm (intmin3 ((A x) (ps x)) ((intmin3.2 (-> (= (cv x) A) (<-> ph ps))) (intmin3.3 ps)) (-> (e. A B) (C_ (|^| ({|} x ph)) A)) (intmin3.3 intmin3.2 B elabg mpbiri A x ph {|} intss1 syl))
+thm (int0el () () (-> (e. ({/}) A) (= (|^| A) ({/}))) ({/} A intss1 A |^| 0ss {/} A e. a1i eqssd))
+thm (intex ((A x)) () (<-> (-. (= A ({/}))) (e. (|^| A) (V))) (A x n0 x cv A intss1 x visset A |^| ssex syl x 19.23aiv sylbi nvelv A {/} inteq int0 syl6eq V eleq1d mtbiri con2i impbi))
+thm (intnex () () (<-> (-. (e. (|^| A) (V))) (= (|^| A) (V))) (A intex bicon1i A {/} inteq int0 syl6eq sylbi nvelv A |^| V V eleq1 mtbiri impbi))
+thm (intexab () () (<-> (E. x ph) (e. (|^| ({|} x ph)) (V))) (x ph abn0 x ph {|} intex bitr3))
+thm (intexrab () () (<-> (E.e. x A ph) (e. (|^| ({e.|} x A ph)) (V))) (x x cv A e. ph /\ intexab x A ph df-rex x A ph df-rab inteqi V eleq1i 3bitr4))
+thm (intabs ((x y) (A x) (ph y) (ps x) (ch x)) ((intabs.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (intabs.2 (-> (= (cv x) (|^| ({|} y ps))) (<-> ph ch))) (intabs.3 (/\ (C_ (|^| ({|} y ps)) A) ch))) (= (|^| ({|} x (/\ (C_ (cv x) A) ph))) (|^| ({|} x ph))) (x cv y ps {|} |^| A sseq1 intabs.2 anbi12d intabs.3 V intmin3 y ps {|} intnex x x cv A C_ ph /\ {|} |^| ssv y ps {|} |^| V x x cv A C_ ph /\ {|} |^| sseq2 mpbiri sylbi pm2.61i intabs.1 cbvabv inteqi eqcomi sseqtr x cv A C_ ph pm3.27 x ss2abi x x cv A C_ ph /\ {|} x ph {|} intss ax-mp eqssi))
+thm (intun ((x y) (A x) (A y) (B x) (B y)) () (= (|^| (u. A B)) (i^i (|^| A) (|^| B))) (y y cv A e. x cv y cv e. -> y cv B e. x cv y cv e. -> 19.26 y cv A B elun x cv y cv e. imbi1i y cv A e. y cv B e. x cv y cv e. jaob bitr y bial x visset A y elint x visset B y elint anbi12i 3bitr4 x visset A B u. y elint x cv A |^| B |^| elin 3bitr4 eqri))
+thm (intpr ((x y) (A x) (A y) (B x) (B y)) ((intpr.1 (e. A (V))) (intpr.2 (e. B (V)))) (= (|^| ({,} A B)) (i^i A B)) (y y cv A = x cv y cv e. -> y cv B = x cv y cv e. -> 19.26 y visset A B elpr x cv y cv e. imbi1i y cv A = y cv B = x cv y cv e. jaob bitr y bial intpr.1 x cv y clel4 intpr.2 x cv y clel4 anbi12i 3bitr4 x visset A B {,} y elint x cv A B elin 3bitr4 eqri))
+thm (intsn () ((intsn.1 (e. A (V)))) (= (|^| ({} A)) A) (A dfsn2 inteqi intsn.1 intsn.1 intpr A inidm 3eqtr))
+thm (op1stb () ((intsn.1 (e. A (V)))) (= (|^| (|^| (<,> A B))) A) (A B df-op inteqi A snex A B prex intpr A B snsspr A {} A B {,} df-ss mpbi 3eqtr inteqi intsn.1 intsn eqtr))
+thm (intunsn () ((intunsn.1 (e. B (V)))) (= (|^| (u. A ({} B))) (i^i (|^| A) B)) (A B {} intun intunsn.1 intsn A |^| ineq2i eqtr))
+thm (eliun ((x y) (A x) (A y) (B y) (C y)) () (<-> (e. A (U_ x B C)) (E.e. x B (e. A C))) (A x B C U_ elisset A C elisset x cv B e. a1i r19.23aiv y cv A C eleq1 x B birexdv x B C y df-iun V elab2g pm5.21nii))
+thm (eliin ((x y) (A x) (A y) (B y) (C y)) () (-> (e. A D) (<-> (e. A (|^|_ x B C)) (A.e. x B (e. A C)))) (y cv A C eleq1 x B biraldv x B C y df-iin D elab2g))
+thm (iunconst ((x y) (A x) (A y) (B x) (B y)) () (-> (-. (= A ({/}))) (= (U_ x A B) B)) (A x n0 x x cv A e. E. y cv B e. ibar sylbi y cv x A B eliun x A y cv B e. df-rex x x cv A e. y cv B e. 19.41v 3bitr syl6rbbr eqrd))
+thm (iuniin ((x y) (y z) (A y) (A z) (x z) (B x) (B z) (C z)) () (C_ (U_ x A (|^|_ y B C)) (|^|_ y B (U_ x A C))) (x A y B z cv C e. r19.12 z cv x A y B C |^|_ eliun z visset z cv V y B C eliin ax-mp x A birex bitr z visset z cv V y B x A C U_ eliin ax-mp z cv x A C eliun y B biral bitr 3imtr4 ssriv))
+thm (iunss1 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (C_ A B) (C_ (U_ x A C) (U_ x B C))) (A B x cv ssel y cv C e. anim1d r19.22dv2 y 19.21aiv y x A y cv C e. E.e. x B y cv C e. E.e. ss2ab sylibr x A C y df-iun x B C y df-iun 3sstr4g))
+thm (iuneq1 ((A x) (B x)) () (-> (= A B) (= (U_ x A C) (U_ x B C))) (A B x C iunss1 B A x C iunss1 anim12i A B eqss x A C U_ x B C U_ eqss 3imtr4))
+thm (iineq1 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (= A B) (= (|^|_ x A C) (|^|_ x B C))) (A B x y cv C e. raleq y biabdv x A C y df-iin x B C y df-iin 3eqtr4g))
+thm (ss2iun ((x y) (A y) (B y) (C y)) () (-> (A.e. x A (C_ B C)) (C_ (U_ x A B) (U_ x A C))) (x A B C C_ hbra1 x A B C C_ ra4 B C y cv ssel syl6 r19.22d y cv x A B eliun y cv x A C eliun 3imtr4g ssrdv))
+thm (iuneq2 () () (-> (A.e. x A (= B C)) (= (U_ x A B) (U_ x A C))) (x A B C ss2iun x A C B ss2iun anim12i B C eqss x A biral x A B C C_ C B C_ r19.26 bitr x A B U_ x A C U_ eqss 3imtr4))
+thm (iineq2 ((x y) (A y) (B y) (C y)) () (-> (A.e. x A (= B C)) (= (|^|_ x A B) (|^|_ x A C))) (x A B C = hbra1 x A B C = ra4 B C y cv eleq2 syl6 imp biralda y biabdv x A B y df-iin x A C y df-iin 3eqtr4g))
+thm (iuneq2i () ((iuneq2i.1 (-> (e. (cv x) A) (= B C)))) (= (U_ x A B) (U_ x A C)) (x A B C iuneq2 iuneq2i.1 mprg))
+thm (iineq2i () ((iuneq2i.1 (-> (e. (cv x) A) (= B C)))) (= (|^|_ x A B) (|^|_ x A C)) (x A B C iineq2 iuneq2i.1 mprg))
+thm (iuneq2dv ((ph x)) ((iuneq2dv.1 (-> ph (-> (e. (cv x) A) (= B C))))) (-> ph (= (U_ x A B) (U_ x A C))) (iuneq2dv.1 r19.21aiv x A B C iuneq2 syl))
+thm (iineq2dv ((ph x)) ((iuneq2dv.1 (-> ph (-> (e. (cv x) A) (= B C))))) (-> ph (= (|^|_ x A B) (|^|_ x A C))) (iuneq2dv.1 r19.21aiv x A B C iineq2 syl))
+thm (hbiu1 ((y z) (A y) (A z) (B y) (B z) (x y) (x z)) () (-> (e. (cv y) (U_ x A B)) (A. x (e. (cv y) (U_ x A B)))) (x A z cv B e. hbre1 y z hbab x A B z df-iun y cv eleq2i x A B z df-iun y cv eleq2i x bial 3imtr4))
+thm (hbii1 ((y z) (A y) (A z) (B y) (B z) (x y) (x z)) () (-> (e. (cv y) (|^|_ x A B)) (A. x (e. (cv y) (|^|_ x A B)))) (x A z cv B e. hbra1 y z hbab x A B z df-iin y cv eleq2i x A B z df-iin y cv eleq2i x bial 3imtr4))
+thm (dfiun2g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w)) () (-> (A. x (e. B C)) (= (U_ x A B) (U. ({|} y (E.e. x A (= (cv y) B)))))) (x B C e. hba1 B C w cv z clel3g x a4s z z cv B = w cv z cv e. exancom syl6bb x cv A e. anbi2d z x cv A e. w cv z cv e. z cv B = /\ 19.42v syl6bbr biexd x z x cv A e. w cv z cv e. z cv B = /\ /\ excom x w cv z cv e. x cv A e. z cv B = /\ 19.42v x cv A e. w cv z cv e. z cv B = an12 x biex z visset y cv z cv B eqeq1 x A birexdv elab x A z cv B = df-rex bitr w cv z cv e. anbi2i 3bitr4 z biex bitr syl6bb x A w cv B e. df-rex syl5bb w biabdv x A B w df-iun y x A y cv B = E.e. {|} w z df-uni 3eqtr4g))
+thm (dfiun2 ((x y) (A x) (A y) (B y)) ((dfiun2.1 (e. B (V)))) (= (U_ x A B) (U. ({|} y (E.e. x A (= (cv y) B))))) (x B V A y dfiun2g dfiun2.1 mpg))
+thm (dfiin2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((dfiun2.1 (e. B (V)))) (= (|^|_ x A B) (|^| ({|} y (E.e. x A (= (cv y) B))))) (x A w cv B e. df-ral dfiun2.1 w cv z clel4 x cv A e. imbi2i z x cv A e. z cv B = w cv z cv e. -> 19.21v bitr4 x bial x z x cv A e. z cv B = w cv z cv e. -> -> alcom bitr x cv A e. z cv B = w cv z cv e. impexp x bial x x cv A e. z cv B = /\ w cv z cv e. 19.23v bitr3 z visset y cv z cv B eqeq1 x A birexdv elab x A z cv B = df-rex bitr w cv z cv e. imbi1i bitr4 z bial 3bitr w biabi x A B w df-iin y x A y cv B = E.e. {|} w z df-int 3eqtr4))
+thm (cbviunv ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C x) (C z)) ((cbviunv.1 (-> (= (cv x) (cv y)) (= B C)))) (= (U_ x A B) (U_ y A C)) (cbviunv.1 z cv eleq2d A cbvrexv z biabi x A B z df-iun y A C z df-iun 3eqtr4))
+thm (iunss ((x y) (C x) (C y) (A y) (B y)) () (<-> (C_ (U_ x A B) C) (A.e. x A (C_ B C))) (B C y dfss2 x A biral x A y y cv B e. y cv C e. -> A. df-ral x cv A e. y cv B e. y cv C e. impexp y bial y x cv A e. y cv B e. y cv C e. -> 19.21v bitr2 x bial 3bitr x x cv A e. y cv B e. /\ y cv C e. 19.23v y cv x A B eliun x A y cv B e. df-rex bitr y cv C e. imbi1i bitr4 y bial x y x cv A e. y cv B e. /\ y cv C e. -> alcom x A B U_ C y dfss2 3bitr4 bitr2))
+thm (ssiun ((x y) (C x) (C y) (A y) (B y)) () (-> (E.e. x A (C_ C B)) (C_ C (U_ x A B))) (x A C B C_ df-rex y cv C e. y cv B e. pm3.35 x cv A e. anim2i exp32 com23 imp C B y cv ssel sylan2 x 19.22i y 19.21aiv y cv x A B eliun x A y cv B e. df-rex bitr2 y cv C e. imbi2i y bial x y cv C e. x cv A e. y cv B e. /\ 19.37v y bial C x A B U_ y dfss2 3bitr4 sylib sylbi))
+thm (ssiun2 ((A y) (B y) (x y)) () (-> (e. (cv x) A) (C_ B (U_ x A B))) (x A y cv B e. ra4e y cv x A B eliun sylibr exp ssrdv))
+thm (ssiun2s ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((ssiun2s.1 (-> (= (cv x) C) (= B D)))) (-> (e. C A) (C_ D (U_ x A B))) (y cv C e. x ax-17 C A e. x ax-17 y cv D e. x ax-17 y x A B hbiu1 hbss hbim x cv C A eleq1 ssiun2s.1 x A B U_ sseq1d imbi12d x A B ssiun2 A vtoclgf pm2.43i))
+thm (iunss2 ((x y) (B x) (C y) (D x)) () (-> (A.e. x A (E.e. y B (C_ C D))) (C_ (U_ x A C) (U_ y B D))) (y B C D ssiun x A r19.20si x A C y B D U_ iunss sylibr))
+thm (iunrab ((y z) (A y) (A z) (x y) (x z) (B x) (B y) (B z) (ph z)) () (= (U_ x A ({e.|} y B ph)) ({e.|} y B (E.e. x A ph))) (z B x A z cv y ph [/] E.e. df-rab x cv B e. y ax-17 x cv B e. z ax-17 x A ph E.e. z ax-17 x cv A e. y ax-17 z y ph hbs1 hbrex y z ph sbequ12 x A birexdv cbvrab z cv x A y B ph {e.|} eliun x cv B e. y ax-17 z cv ph elrabsf x A birex x A z cv B e. z cv y ph [/] r19.42v 3bitr biabri 3eqtr4r))
+thm (iunab ((A y) (x y)) () (= (U_ x A ({|} y ph)) ({|} y (E.e. x A ph))) (x A y V ph iunrab y ph rabab x cv A e. a1i iuneq2i y x A ph E.e. rabab 3eqtr3))
+thm (iunxdif2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((iunxdif2.1 (-> (= (cv x) (cv y)) (= C D)))) (-> (A.e. x A (E.e. y (\ A B) (C_ C D))) (= (U_ y (\ A B) D) (U_ x A C))) (x A y A B \ C D iunss2 A B difss A B \ A y D iunss1 ax-mp iunxdif2.1 A cbviunv eqcomi sseqtr jctil y A B \ D U_ x A C U_ eqss sylibr))
+thm (ssiin ((x y) (C x) (C y) (A x) (A y) (B y)) () (<-> (C_ C (|^|_ x A B)) (A.e. x A (C_ C B))) (y visset y cv V x A B eliin ax-mp y cv C e. imbi2i x A y cv C e. y cv B e. r19.21v bitr4 x A y cv C e. y cv B e. -> df-ral bitr y bial C x A B |^|_ y dfss2 C B y dfss2 x A biral x A y y cv C e. y cv B e. -> A. df-ral y x cv A e. y cv C e. y cv B e. -> 19.21v x bial x y x cv A e. y cv C e. y cv B e. -> -> alcom bitr3 3bitr 3bitr4))
+thm (iinss ((x y) (C x) (C y) (A x) (A y) (B y)) () (-> (E.e. x A (C_ B C)) (C_ (|^|_ x A B) C)) (x y x cv A e. y cv B e. y cv C e. -> /\ 19.12 x A y y cv B e. y cv C e. -> A. df-rex y x cv A e. y cv B e. y cv C e. -> 19.28v x biex bitr4 x A y cv B e. y cv C e. -> df-rex y bial 3imtr4 x A y cv B e. y cv C e. r19.36av y visset y cv V x A B eliin ax-mp syl5ib y 19.20i syl B C y dfss2 x A birex x A B |^|_ C y dfss2 3imtr4))
+thm (uniiun ((x y) (A x) (A y)) () (= (U. A) (U_ x A (cv x))) (A y x dfuni2 x A x cv y df-iun eqtr4))
+thm (intiin ((x y) (A x) (A y)) () (= (|^| A) (|^|_ x A (cv x))) (A y x dfint2 x A x cv y df-iin eqtr4))
+thm (iunid ((x y) (A x) (A y)) () (= (U_ x A ({} (cv x))) A) (y cv x A x cv {} eliun x A y cv x cv {} e. df-rex x cv A e. y cv x cv {} e. ancom y x cv elsn y x equcom bitr x cv A e. anbi1i bitr x biex y cv A e. x ax-17 x cv y cv A eleq1 equsex 3bitr bitr eqri))
+thm (iun0 ((x y) (A y)) () (= (U_ x A ({/})) ({/})) (y cv x A {/} eliun y cv noel x cv A e. a1i nrex y cv noel 2false bitr eqri))
+thm (0iun ((x y) (A y)) () (= (U_ x ({/}) A) ({/})) (y cv x {/} A eliun x y cv A e. rex0 y cv noel 2false bitr eqri))
+thm (iunn0 ((x y) (A x) (A y) (B y)) () (<-> (E.e. x A (-. (= B ({/})))) (-. (= (U_ x A B) ({/})))) (B y n0 x A birex x A y y cv B e. E. df-rex x y x cv A e. y cv B e. /\ excom x y x cv A e. y cv B e. exdistr y cv x A B eliun x A y cv B e. df-rex bitr2 y biex 3bitr3 x A B U_ y n0 bitr4 3bitr))
+thm (iunin2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (U_ x A (i^i B C)) (i^i B (U_ x A C))) (x A y cv B e. y cv C e. r19.42v y cv B C elin x A birex y cv x A C eliun y cv B e. anbi2i 3bitr4 y cv x A B C i^i eliun y cv B x A C U_ elin 3bitr4 eqri))
+thm (iinun2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (|^|_ x A (u. B C)) (u. B (|^|_ x A C))) (x A y cv B e. y cv C e. r19.32v y cv B C elun x A biral y visset y cv V x A C eliin ax-mp y cv B e. orbi2i 3bitr4 y visset y cv V x A B C u. eliin ax-mp y cv B x A C |^|_ elun 3bitr4 eqri))
+thm (iundif2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (= (U_ x A (\ B C)) (\ B (|^|_ x A C))) (x A y cv B e. y cv C e. -. r19.42v y cv B C eldif x A birex y visset y cv V x A C eliin ax-mp negbii x A y cv C e. rexnal bitr4 y cv B e. anbi2i 3bitr4 y cv x A B C \ eliun y cv B x A C |^|_ eldif 3bitr4 eqri))
+thm (iindif2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (-. (= A ({/}))) (= (|^|_ x A (\ B C)) (\ B (U_ x A C)))) (A x y cv B e. y cv C e. -. r19.28zv y cv B C eldif x A biral y cv x A C eliun negbii x A y cv C e. ralnex bitr4 y cv B e. anbi2i 3bitr4g y visset y cv V x A B C \ eliin ax-mp y cv B x A C U_ eldif 3bitr4g eqrd))
+thm (iunxsn ((x y) (A x) (A y) (B y) (C x) (C y)) ((iunxsn.1 (e. A (V))) (iunxsn.2 (-> (= (cv x) A) (= B C)))) (= (U_ x ({} A) B) C) (y cv x A {} B eliun x A {} y cv B e. df-rex x A elsn y cv B e. anbi1i x biex iunxsn.1 iunxsn.2 y cv eleq2d ceqsexv bitr bitr bitr eqri))
+thm (iunun ((x y) (A y) (B y) (C y)) () (= (U_ x A (u. B C)) (u. (U_ x A B) (U_ x A C))) (x A B C u. y df-iun y cv B C elun x A birex x A y cv B e. y cv C e. r19.43 bitr y biabi y x A y cv B e. E.e. x A y cv C e. E.e. unab eqcomi eqtr x A B y df-iun x A C y df-iun uneq12i eqcomi eqtr eqtr))
+thm (iunxun ((x y) (A y) (B y) (C y)) () (= (U_ x (u. A B) C) (u. (U_ x A C) (U_ x B C))) (x A B u. y cv C e. df-rex x cv A B elun y cv C e. anbi1i x cv A e. x cv B e. y cv C e. andir bitr x biex x x cv A e. y cv C e. /\ x cv B e. y cv C e. /\ 19.43 y cv x A C eliun x A y cv C e. df-rex bitr y cv x B C eliun x B y cv C e. df-rex bitr orbi12i bitr4 3bitr y cv x A B u. C eliun y cv x A C U_ x B C U_ elun 3bitr4 eqri))
+thm (iinuni ((x y) (A x) (A y) (B x) (B y)) () (= (u. A (|^| B)) (|^|_ x B (u. A (cv x)))) (x B y cv A e. y cv x cv e. r19.32v y cv A x cv elun x B biral y visset B x elint2 y cv A e. orbi2i 3bitr4r y cv A B |^| elun y visset y cv V x B A x cv u. eliin ax-mp 3bitr4 eqri))
+thm (iununi ((x y) (A x) (A y) (B x) (B y)) () (<-> (-> (= B ({/})) (= A ({/}))) (= (u. A (U. B)) (U_ x B (u. A (cv x))))) (B {/} = A {/} = imor B x y cv A e. y cv x cv e. r19.45zv y cv A n0i con2i y cv A e. y cv x cv e. biorf x B birexdv y cv A e. x B y cv x cv e. E.e. biorf bitr3d syl jaoi bicomd sylbi y cv A x cv elun x B birex syl6bbr y cv A B U. elun y cv B x eluni2 y cv A e. orbi2i bitr y cv x B A x cv u. eliun 3bitr4g eqrd A B U. u. x B A x cv u. U_ y cv eleq2 y cv B x eluni y cv A e. orbi2i y cv A B U. elun y cv A e. x ax-17 y cv x cv e. x cv B e. /\ 19.45 3bitr4 y cv x B A x cv u. eliun x B y cv A x cv u. e. df-rex bitr 3bitr3g biimpd x y cv A e. y cv x cv e. x cv B e. /\ \/ x cv B e. y cv A x cv u. e. /\ 19.39 y cv A e. y cv x cv e. x cv B e. /\ orc x cv B e. y cv A x cv u. e. pm3.26 imim12i x 19.22i 3syl x y cv A e. x cv B e. 19.37v sylib y 19.23adv A y n0 B x n0 3imtr4g a3d impbi))
+thm (iinpw ((x y) (A x) (A y)) () (= (P~ (|^| A)) (|^|_ x A (P~ (cv x)))) (y cv A x ssint y visset x cv elpw x A biral bitr4 y visset A |^| elpw y visset y cv V x A x cv P~ eliin ax-mp 3bitr4 eqri))
+thm (iunpwss ((x y) (A x) (A y)) () (C_ (U_ x A (P~ (cv x))) (P~ (U. A))) (x A y cv x cv ssiun y cv x A x cv P~ eliun y visset x cv elpw x A birex bitr y visset A U. elpw A x uniiun y cv sseq2i bitr 3imtr4 ssriv))
+thm (iunpw ((x y) (A x) (A y)) ((iunpw.1 (e. A (V)))) (<-> (E.e. x A (= (cv x) (U. A))) (= (P~ (U. A)) (U_ x A (P~ (cv x))))) (x cv A U. y cv sseq2 biimprcd x A r19.22sdv com12 x A y cv x cv ssiun A x uniiun syl6ssr x A x cv A U. = E.e. a1i impbid y visset A U. elpw y cv x A x cv P~ eliun x cv y df-pw eqabi x A birex bitr 3bitr4g eqrd A U. ssid A U. P~ x A x cv P~ U_ A U. eleq2 iunpw.1 uniex A U. elpw syl5bbr mpbii A U. x A x cv P~ eliun sylib x cv A elssuni iunpw.1 uniex x cv elpw biimp anim12i x cv A U. eqss sylibr exp r19.22i syl impbi))
+thm (dftr2 ((x y) (A x) (A y)) () (<-> (Tr A) (A. x (A. y (-> (/\ (e. (cv x) (cv y)) (e. (cv y) A)) (e. (cv x) A))))) (A U. A x dfss2 A df-tr y x cv y cv e. y cv A e. /\ x cv A e. 19.23v x cv A y eluni x cv A e. imbi1i bitr4 x bial 3bitr4))
+thm (dftr5 ((x y) (A x) (A y)) () (<-> (Tr A) (A.e. x A (A.e. y (cv x) (e. (cv y) A)))) (A y x dftr2 y x y cv x cv e. x cv A e. /\ y cv A e. -> alcom y cv x cv e. x cv A e. y cv A e. impexp y bial y x cv x cv A e. y cv A e. -> df-ral bitr4 y x cv x cv A e. y cv A e. r19.21v bitr x bial x A y x cv y cv A e. A.e. df-ral bitr4 3bitr))
+thm (dftr3 ((x y) (A x) (A y)) () (<-> (Tr A) (A.e. x A (C_ (cv x) A))) (A x y dftr5 x cv A y dfss3 x A biral bicomi bitr))
+thm (dftr4 ((A x)) () (<-> (Tr A) (C_ A (P~ A))) (x visset A elpw x A biral A A P~ x dfss3 A x dftr3 3bitr4r))
+thm (treq () () (-> (= A B) (<-> (Tr A) (Tr B))) (A B unieq A sseq1d A B B U. sseq2 bitrd A df-tr B df-tr 3bitr4g))
+thm (trel ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (Tr A) (-> (/\ (e. B C) (e. C A)) (e. B A))) (x cv B C eleq1 x cv B A eleq1 C A e. imbi2d imbi12d A Tr imbi2d y cv C x cv eleq2 y cv C A eleq1 x cv A e. imbi1d imbi12d A Tr imbi2d A x y dftr2 biimp 19.21bbi exp3a A vtoclg com4l C A e. x cv A e. pm2.43 syl6 C vtoclg pm2.43b imp3a))
+thm (trel3 () () (-> (Tr A) (-> (/\/\ (e. B C) (e. C D) (e. D A)) (e. B A))) (A C D trel B C e. anim2d B C e. C D e. D A e. 3anass syl5ib A B C trel syld))
+thm (trss ((A x) (B x)) () (-> (Tr A) (-> (e. B A) (C_ B A))) (x cv B A eleq1 x cv B A sseq1 imbi12d A Tr imbi2d A x dftr3 x A x cv A C_ ra4 sylbi A vtoclg pm2.43b))
+thm (trin ((A x) (B x)) () (-> (/\ (Tr A) (Tr B)) (Tr (i^i A B))) (A x cv trss B x cv trss im2anan9 x cv A B elin syl5ib x cv A B ssin syl6ib r19.21aiv A B i^i x dftr3 sylibr))
+thm (tr0 () () (Tr ({/})) ({/} P~ 0ss {/} dftr4 mpbir))
+thm (trv () () (Tr (V)) (V U. ssv V df-tr mpbir))
+thm (breq () () (-> (= R S) (<-> (br A R B) (br A S B))) (R S A B <,> eleq2 A R B df-br A S B df-br 3bitr4g))
+thm (breq1 () () (-> (= A B) (<-> (br A R C) (br B R C))) (A B C opeq1 R eleq1d A R C df-br B R C df-br 3bitr4g))
+thm (breq2 () () (-> (= A B) (<-> (br C R A) (br C R B))) (A B C opeq2 R eleq1d C R A df-br C R B df-br 3bitr4g))
+thm (breq12 () () (-> (/\ (= A B) (= C D)) (<-> (br A R C) (br B R D))) (A B R C breq1 C D B R breq2 sylan9bb))
+thm (breqi () ((breqi.1 (= R S))) (<-> (br A R B) (br A S B)) (breqi.1 R S A B breq ax-mp))
+thm (breq1i () ((breq1i.1 (= A B))) (<-> (br A R C) (br B R C)) (breq1i.1 A B R C breq1 ax-mp))
+thm (breq2i () ((breq1i.1 (= A B))) (<-> (br C R A) (br C R B)) (breq1i.1 A B C R breq2 ax-mp))
+thm (breq12i () ((breq1i.1 (= A B)) (breq12i.2 (= C D))) (<-> (br A R C) (br B R D)) (breq1i.1 R C breq1i breq12i.2 B R breq2i bitr))
+thm (breq1d () ((breq1d.1 (-> ph (= A B)))) (-> ph (<-> (br A R C) (br B R C))) (breq1d.1 A B R C breq1 syl))
+thm (breq2d () ((breq1d.1 (-> ph (= A B)))) (-> ph (<-> (br C R A) (br C R B))) (breq1d.1 A B C R breq2 syl))
+thm (breq12d () ((breq1d.1 (-> ph (= A B))) (breq12d.2 (-> ph (= C D)))) (-> ph (<-> (br A R C) (br B R D))) (breq1d.1 R C breq1d breq12d.2 B R breq2d bitrd))
+thm (breqan12d () ((breq1d.1 (-> ph (= A B))) (breqan12i.2 (-> ps (= C D)))) (-> (/\ ph ps) (<-> (br A R C) (br B R D))) (A B C D R breq12 breq1d.1 breqan12i.2 syl2an))
+thm (breqan12rd () ((breq1d.1 (-> ph (= A B))) (breqan12i.2 (-> ps (= C D)))) (-> (/\ ps ph) (<-> (br A R C) (br B R D))) (breq1d.1 breqan12i.2 R breqan12d ancoms))
+thm (eqbrtr () ((eqbrtr.1 (= A B)) (eqbrtr.2 (br B R C))) (br A R C) (eqbrtr.2 eqbrtr.1 R C breq1i mpbir))
+thm (eqbrtrd () ((eqbrtrd.1 (-> ph (= A B))) (eqbrtrd.2 (-> ph (br B R C)))) (-> ph (br A R C)) (eqbrtrd.2 eqbrtrd.1 R C breq1d mpbird))
+thm (eqbrtrr () ((eqbrtrr.1 (= A B)) (eqbrtrr.2 (br A R C))) (br B R C) (eqbrtrr.1 eqcomi eqbrtrr.2 eqbrtr))
+thm (eqbrtrrd () ((eqbrtrrd.1 (-> ph (= A B))) (eqbrtrrd.2 (-> ph (br A R C)))) (-> ph (br B R C)) (eqbrtrrd.1 eqcomd eqbrtrrd.2 eqbrtrd))
+thm (breqtr () ((breqtr.1 (br A R B)) (breqtr.2 (= B C))) (br A R C) (breqtr.1 breqtr.2 A R breq2i mpbi))
+thm (breqtrd () ((breqtrd.1 (-> ph (br A R B))) (breqtrd.2 (-> ph (= B C)))) (-> ph (br A R C)) (breqtrd.1 breqtrd.2 A R breq2d mpbid))
+thm (breqtrr () ((breqtrr.1 (br A R B)) (breqtrr.2 (= C B))) (br A R C) (breqtrr.1 breqtrr.2 eqcomi breqtr))
+thm (breqtrrd () ((breqtrrd.1 (-> ph (br A R B))) (breqtrrd.2 (-> ph (= C B)))) (-> ph (br A R C)) (breqtrrd.1 breqtrrd.2 eqcomd breqtrd))
+thm (3brtr3 () ((3brtr3.1 (br A R B)) (3brtr3.2 (= A C)) (3brtr3.3 (= B D))) (br C R D) (3brtr3.2 3brtr3.1 eqbrtrr 3brtr3.3 breqtr))
+thm (3brtr4 () ((3brtr4.1 (br A R B)) (3brtr4.2 (= C A)) (3brtr4.3 (= D B))) (br C R D) (3brtr4.2 3brtr4.1 eqbrtr 3brtr4.3 breqtrr))
+thm (3brtr3d () ((3brtr3d.1 (-> ph (br A R B))) (3brtr3d.2 (-> ph (= A C))) (3brtr3d.3 (-> ph (= B D)))) (-> ph (br C R D)) (3brtr3d.1 3brtr3d.2 3brtr3d.3 R breq12d mpbid))
+thm (3brtr4d () ((3brtr4d.1 (-> ph (br A R B))) (3brtr4d.2 (-> ph (= C A))) (3brtr4d.3 (-> ph (= D B)))) (-> ph (br C R D)) (3brtr4d.1 3brtr4d.2 3brtr4d.3 R breq12d mpbird))
+thm (3brtr3g () ((3brtr3g.1 (-> ph (br A R B))) (3brtr3g.2 (= A C)) (3brtr3g.3 (= B D))) (-> ph (br C R D)) (3brtr3g.1 3brtr3g.2 3brtr3g.3 R breq12i sylib))
+thm (3brtr4g () ((3brtr4g.1 (-> ph (br A R B))) (3brtr4g.2 (= C A)) (3brtr4g.3 (= D B))) (-> ph (br C R D)) (3brtr4g.1 3brtr4g.2 3brtr4g.3 R breq12i sylibr))
+thm (syl5eqbr () ((syl5eqbr.1 (-> ph (br A R B))) (syl5eqbr.2 (= C A))) (-> ph (br C R B)) (syl5eqbr.1 syl5eqbr.2 B eqid 3brtr4g))
+thm (syl5eqbrr () ((syl5eqbrr.1 (-> ph (br A R B))) (syl5eqbrr.2 (= A C))) (-> ph (br C R B)) (syl5eqbrr.1 syl5eqbrr.2 B eqid 3brtr3g))
+thm (syl5breq () ((syl5breq.1 (-> ph (= A B))) (syl5breq.2 (br C R A))) (-> ph (br C R B)) (syl5breq.2 syl5breq.1 C R breq2d mpbii))
+thm (syl5breqr () ((syl5breqr.1 (-> ph (= B A))) (syl5breqr.2 (br C R A))) (-> ph (br C R B)) (syl5breqr.1 eqcomd syl5breqr.2 syl5breq))
+thm (syl6eqbr () ((syl6eqbr.1 (-> ph (= A B))) (syl6eqbr.2 (br B R C))) (-> ph (br A R C)) (syl6eqbr.2 syl6eqbr.1 R C breq1d mpbiri))
+thm (syl6breq () ((syl6breq.1 (-> ph (br A R B))) (syl6breq.2 (= B C))) (-> ph (br A R C)) (syl6breq.1 A eqid syl6breq.2 3brtr3g))
+thm (syl6breqr () ((syl6breqr.1 (-> ph (br A R B))) (syl6breqr.2 (= C B))) (-> ph (br A R C)) (syl6breqr.1 syl6breqr.2 eqcomi syl6breq))
+thm (ssbrd () ((ssbrd.1 (-> ph (C_ A B)))) (-> ph (-> (br C A D) (br C B D))) (ssbrd.1 C D <,> sseld C A D df-br C B D df-br 3imtr4g))
+thm (hbbr ((A y) (B y) (R y) (x y)) ((hbbr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbbr.2 (-> (e. (cv y) R) (A. x (e. (cv y) R)))) (hbbr.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (br A R B) (A. x (br A R B))) (hbbr.1 hbbr.3 hbop hbbr.2 hbel A R B df-br A R B df-br x bial 3imtr4))
+thm (brab1 ((x z) (y z) (R z)) () (<-> (br (cv x) R (cv y)) (e. (cv x) ({|} z (br (cv z) R (cv y))))) (x visset z cv x cv R y cv breq1 elab bicomi))
+thm (brprc () () (-> (-. (e. B (V))) (<-> (br A R B) (br A R A))) (B A opprc2 R eleq1d A R B df-br A R A df-br 3bitr4g))
+thm (opabid ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph z) (ph w) (ph v)) () (<-> (e. (<,> (cv x) (cv y)) ({<,>|} x y ph)) ph) (x cv y cv <,> z x y z cv x cv y cv <,> = ph /\ E. E. clelab x y ph z df-opab x cv y cv <,> eleq2i z z cv x cv y cv <,> = z cv w cv v cv <,> = /\ v cv y w cv x ph [/] [/] 19.41v z cv x cv y cv <,> = z cv w cv v cv <,> = v cv y w cv x ph [/] [/] anass z biex x cv y cv <,> w cv v cv <,> eqcom x cv y cv opex w cv v cv <,> z eqvinc w visset v visset y visset x cv opth 3bitr3 v cv y w cv x ph [/] [/] anbi1i 3bitr3r w v bi2ex ph w v x y sbel2x z w v z cv x cv y cv <,> = z cv w cv v cv <,> = v cv y w cv x ph [/] [/] /\ /\ E. excom z w v z cv x cv y cv <,> = z cv w cv v cv <,> = v cv y w cv x ph [/] [/] /\ exdistr2 z v z cv x cv y cv <,> = z cv w cv v cv <,> = v cv y w cv x ph [/] [/] /\ /\ excom w biex 3bitr3 3bitr4 y z cv x cv y cv <,> = ph /\ E. w ax-17 z cv w cv y cv <,> = x ax-17 w x ph hbs1 hban y hbex x cv w cv y cv opeq1 z cv eqeq2d x w ph sbequ12 anbi12d y biexdv cbvex z cv w cv y cv <,> = w cv x ph [/] /\ v ax-17 z cv w cv v cv <,> = y ax-17 v y w cv x ph [/] hbs1 hban y cv v cv w cv opeq2 z cv eqeq2d y v w cv x ph [/] sbequ12 anbi12d cbvex w biex bitr z cv x cv y cv <,> = anbi2i z biex bitr4 3bitr4))
+thm (opabss ((x z) (R x) (R z) (y z) (R y)) () (C_ ({<,>|} x y (br (cv x) R (cv y))) R) (x y x cv R y cv br z df-opab z cv x cv y cv <,> R eleq1 biimpar x cv R y cv df-br sylan2b x y 19.23aivv z ss2abi eqsstr z R abid2 sseqtr))
+thm (biopabd ((x z) (y z) (ph z) (ps z) (ch z)) ((biopabd.1 (-> ph (A. x ph))) (biopabd.2 (-> ph (A. y ph))) (biopabd.3 (-> ph (<-> ps ch)))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))) (ph z ax-17 biopabd.1 biopabd.2 biopabd.3 z cv x cv y cv <,> = anbi2d biexd biexd biabd x y ps z df-opab x y ch z df-opab 3eqtr4g))
+thm (biopabdv ((x y) (ph x) (ph y)) ((biopabdv.1 (-> ph (<-> ps ch)))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))) (ph x ax-17 ph y ax-17 biopabdv.1 biopabd))
+thm (biopabi ((x y) (x z) (y z) (ph z) (ps z)) ((biopab.1 (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} x y ps)) (z cv eqid biopab.1 z cv z cv = a1i x y biopabdv ax-mp))
+thm (cbvopab ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph v) (ps v)) ((cbvopab.1 (-> ph (A. z ph))) (cbvopab.2 (-> ph (A. w ph))) (cbvopab.3 (-> ps (A. x ps))) (cbvopab.4 (-> ps (A. y ps))) (cbvopab.5 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z w ps)) (v cv x cv y cv <,> = z ax-17 cbvopab.1 hban v cv x cv y cv <,> = w ax-17 cbvopab.2 hban v cv z cv w cv <,> = x ax-17 cbvopab.3 hban v cv z cv w cv <,> = y ax-17 cbvopab.4 hban x cv z cv y cv w cv opeq12 v cv eqeq2d cbvopab.5 anbi12d cbvex2 v biabi x y ph v df-opab z w ps v df-opab 3eqtr4))
+thm (cbvopabv ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps y)) ((cbvopabv.1 (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z w ps)) (ph z ax-17 ph w ax-17 ps x ax-17 ps y ax-17 cbvopabv.1 cbvopab))
+thm (cbvopab1 ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((cbvopab1.1 (-> ph (A. z ph))) (cbvopab1.2 (-> ps (A. x ps))) (cbvopab1.3 (-> (= (cv x) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z y ps)) (w cv x cv y cv <,> = z ax-17 cbvopab1.1 hban y hbex w cv z cv y cv <,> = x ax-17 cbvopab1.2 hban y hbex x cv z cv y cv opeq1 w cv eqeq2d cbvopab1.3 anbi12d y biexdv cbvex w biabi x y ph w df-opab z y ps w df-opab 3eqtr4))
+thm (cbvopab1s ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (= ({<,>|} x y ph) ({<,>|} z y ([/] (cv z) x ph))) (y w cv x cv y cv <,> = ph /\ E. z ax-17 w cv z cv y cv <,> = x ax-17 z x ph hbs1 hban y hbex x cv z cv y cv opeq1 w cv eqeq2d x z ph sbequ12 anbi12d y biexdv cbvex w biabi x y ph w df-opab z y z cv x ph [/] w df-opab 3eqtr4))
+thm (cbvopab1v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps w)) ((cbvopab1v.1 (-> (= (cv x) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} z y ps)) (x cv z cv y cv opeq1 w cv eqeq2d cbvopab1v.1 anbi12d y biexdv cbvexv w biabi x y ph w df-opab z y ps w df-opab 3eqtr4))
+thm (cbvopab2v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps y) (ps w)) ((cbvopab2v.1 (-> (= (cv y) (cv z)) (<-> ph ps)))) (= ({<,>|} x y ph) ({<,>|} x z ps)) (y cv z cv x cv opeq2 w cv eqeq2d cbvopab2v.1 anbi12d cbvexv x biex w biabi x y ph w df-opab x z ps w df-opab 3eqtr4))
+thm (elopab ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) () (<-> (e. A ({<,>|} x y ph)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) ph)))) (A x y ph {<,>|} elisset x cv y cv opex A x cv y cv <,> V eleq1 mpbiri ph adantr x y 19.23aivv z cv A x y ph {<,>|} eleq1 z cv A x cv y cv <,> eqeq1 ph anbi1d x y bi2exdv x y ph z df-opab eqabi V vtoclbg pm5.21nii))
+thm (hbopab ((x y) (x z) (w x) (y z) (w y) (w z)) ((hbopab.1 (-> ph (A. z ph)))) (-> (e. (cv w) ({<,>|} x y ph)) (A. z (e. (cv w) ({<,>|} x y ph)))) (w cv x cv y cv <,> = z ax-17 hbopab.1 hban y hbex x hbex w cv x y ph elopab w cv x y ph elopab z bial 3imtr4))
+thm (hbopab1 ((x y) (w x) (w y) (x z) (ph w)) () (-> (e. (cv z) ({<,>|} x y ph)) (A. x (e. (cv z) ({<,>|} x y ph)))) (x y w cv x cv y cv <,> = ph /\ E. hbe1 z w hbab x y ph w df-opab z cv eleq2i x y ph w df-opab z cv eleq2i x bial 3imtr4))
+thm (hbopab2 ((x y) (w x) (w y) (y z) (ph w)) () (-> (e. (cv z) ({<,>|} x y ph)) (A. y (e. (cv z) ({<,>|} x y ph)))) (y w cv x cv y cv <,> = ph /\ hbe1 x hbex z w hbab x y ph w df-opab z cv eleq2i x y ph w df-opab z cv eleq2i y bial 3imtr4))
+thm (opabsb ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph v)) () (<-> (e. (<,> (cv z) (cv w)) ({<,>|} x y ph)) ([/] (cv w) y ([/] (cv z) x ph))) (y w a9e v cv z cv w cv <,> e. y ax-17 v x y ph hbopab2 hbel w y z cv x ph [/] hbs1 hbbi x z a9e y cv w cv = x ax-17 v cv z cv w cv <,> e. x ax-17 v x y ph hbopab1 hbel z x ph hbs1 w y hbsb hbbi hbim x cv z cv y cv w cv opeq12 x y ph {<,>|} eleq1d x y ph opabid syl5bbr x z ph sbequ12 y w z cv x ph [/] sbequ12 sylan9bb bitr3d exp 19.23ai ax-mp 19.23ai ax-mp))
+thm (opelopabg ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (ch x) (ch y) (ph z)) ((opelopabg.1 (-> (= (cv x) A) (<-> ph ps))) (opelopabg.2 (-> (= (cv y) B) (<-> ps ch)))) (-> (/\ (e. A C) (e. B D)) (<-> (e. (<,> A B) ({<,>|} x y ph)) ch)) (A C x elex B D y elex anim12i x y x cv A = y cv B = eeanv sylibr z cv A B <,> e. x ax-17 z x y ph hbopab1 hbel ch x ax-17 hbbi z cv A B <,> e. y ax-17 z x y ph hbopab2 hbel ch y ax-17 hbbi x cv A y cv B opeq12 x y ph {<,>|} eleq1d x y ph opabid syl5bbr opelopabg.1 opelopabg.2 sylan9bb bitr3d 19.23ai 19.23ai syl))
+thm (brabg ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopabg.1 (-> (= (cv x) A) (<-> ph ps))) (opelopabg.2 (-> (= (cv y) B) (<-> ps ch))) (brabg.5 (= R ({<,>|} x y ph)))) (-> (/\ (e. A C) (e. B D)) (<-> (br A R B) ch)) (opelopabg.1 opelopabg.2 C D opelopabg A R B df-br brabg.5 A B <,> eleq2i bitr syl5bb))
+thm (opelopab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopab.1 (e. A (V))) (opelopab.2 (e. B (V))) (opelopab.3 (-> (= (cv x) A) (<-> ph ps))) (opelopab.4 (-> (= (cv y) B) (<-> ps ch)))) (<-> (e. (<,> A B) ({<,>|} x y ph)) ch) (opelopab.1 opelopab.2 opelopab.3 opelopab.4 V V opelopabg mp2an))
+thm (brab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((opelopab.1 (e. A (V))) (opelopab.2 (e. B (V))) (opelopab.3 (-> (= (cv x) A) (<-> ph ps))) (opelopab.4 (-> (= (cv y) B) (<-> ps ch))) (brab.5 (= R ({<,>|} x y ph)))) (<-> (br A R B) ch) (opelopab.1 opelopab.2 opelopab.3 opelopab.4 brab.5 V V brabg mp2an))
+thm (ssopab2 ((ph z) (ps z) (x y) (x z) (y z)) () (<-> (C_ ({<,>|} x y ph) ({<,>|} x y ps)) (A. x (A. y (-> ph ps)))) (z x y ph hbopab1 z x y ps hbopab1 hbss z x y ph hbopab2 z x y ps hbopab2 hbss x cv y cv opex z isseti z cv x y ph copsexg z cv x y ps copsexg imbi12d z x y z cv x cv y cv <,> = ph /\ E. E. x y z cv x cv y cv <,> = ps /\ E. E. ss2ab z x y z cv x cv y cv <,> = ph /\ E. E. x y z cv x cv y cv <,> = ps /\ E. E. -> ax-4 sylbi syl5bir x y ph z df-opab x y ps z df-opab sseq12i syl5ib z 19.23aiv ax-mp 19.21ai 19.21ai x y ph ps -> A. hba1 y ph ps -> hba1 y ph ps -> ax-4 z cv x cv y cv <,> = anim2d 19.22d x a4s 19.22d z 19.21aiv z x y z cv x cv y cv <,> = ph /\ E. E. x y z cv x cv y cv <,> = ps /\ E. E. ss2ab sylibr x y ph z df-opab x y ps z df-opab 3sstr4g impbi))
+thm (ssopab2i ((x y)) ((ssopab2i.1 (-> ph ps))) (C_ ({<,>|} x y ph) ({<,>|} x y ps)) (x y ph ps ssopab2 ssopab2i.1 y ax-gen mpgbir))
+thm (unopab ((x z) (y z) (ph z) (ps z)) () (= (u. ({<,>|} x y ph) ({<,>|} x y ps)) ({<,>|} x y (\/ ph ps))) (z x y z cv x cv y cv <,> = ph /\ E. E. x y z cv x cv y cv <,> = ps /\ E. E. unab x y z cv x cv y cv <,> = ph /\ E. y z cv x cv y cv <,> = ps /\ E. 19.43 z cv x cv y cv <,> = ph ps andi y biex y z cv x cv y cv <,> = ph /\ z cv x cv y cv <,> = ps /\ 19.43 bitr2 x biex bitr3 z biabi eqtr x y ph z df-opab x y ps z df-opab uneq12i x y ph ps \/ z df-opab 3eqtr4))
+thm (epelc ((x y) (A x) (A y) (B x) (B y)) ((epelc.1 (e. A (V))) (epelc.2 (e. B (V)))) (<-> (br A (E) B) (e. A B)) (epelc.1 epelc.2 x cv A y cv eleq1 y cv B A eleq2 x y df-eprel brab))
+thm (epel () () (<-> (br (cv x) (E) (cv y)) (e. (cv x) (cv y))) (x visset y visset epelc))
+thm (ideqg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (<-> (br A (I) B) (= A B))) (x cv A y cv eqeq1 y cv B A eqeq2 x y df-id C D brabg))
+thm (ideq () ((ideq.1 (e. A (V))) (ideq.2 (e. B (V)))) (<-> (br A (I) B) (= A B)) (ideq.1 ideq.2 A V B V ideqg mp2an))
+thm (poss ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Po R B) (Po R A))) (A B x cv ssel A B y cv ssel anim12d A B z cv ssel anim12d x cv A e. y cv A e. z cv A e. df-3an x cv B e. y cv B e. z cv B e. df-3an 3imtr4g x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ imim1d z 19.20dv y 19.20dv x 19.20dv x B y B z B x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ r3al x A y A z A x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ r3al 3imtr4g R B x y z df-po R A x y z df-po 3imtr4g))
+thm (poeq1 ((x y) (x z) (R x) (y z) (R y) (R z) (S x) (S y) (S z) (A x) (A y) (A z)) () (-> (= R S) (<-> (Po R A) (Po S A))) (R S x cv x cv breq negbid R S x cv y cv breq R S y cv z cv breq anbi12d R S x cv z cv breq imbi12d anbi12d z A biraldv y A biraldv x A biraldv R A x y z df-po S A x y z df-po 3bitr4g))
+thm (poeq2 () () (-> (= A B) (<-> (Po R A) (Po R B))) (A B R poss B A R poss anim12i A B eqss R B Po R A Po bi 3imtr4 bicomd))
+thm (pocl ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) () (-> (Po R A) (-> (/\/\ (e. B A) (e. C A) (e. D A)) (/\ (-. (br B R B)) (-> (/\ (br B R C) (br C R D)) (br B R D))))) (x cv B = id x cv B = id R breq12d negbid x cv B R y cv breq1 y cv R z cv br anbi1d x cv B R z cv breq1 imbi12d anbi12d R A Po imbi2d y cv C B R breq2 y cv C R z cv breq1 anbi12d B R z cv br imbi1d B R B br -. anbi2d R A Po imbi2d z cv D C R breq2 B R C br anbi2d z cv D B R breq2 imbi12d B R B br -. anbi2d R A Po imbi2d R A x y z df-po x A y A z A x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ r3al bitr biimp 19.21bbi 19.21bi com12 vtocl3ga com12))
+thm (poirr () () (-> (/\ (Po R A) (e. B A)) (-. (br B R B))) (R A B B B pocl imp pm3.26d B A e. anidm B A e. B A e. B A e. df-3an B A e. B A e. anabs1 bitr2 bitr3 sylan2b))
+thm (potr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))) (R A B C D pocl imp pm3.27d))
+thm (po2nr () () (-> (/\ (Po R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))) (R A B poirr C A e. adantrr R A B C B potr B A e. C A e. B A e. df-3an sylan2br exp44 com34 pm2.43d imp32 mtod))
+thm (po3nr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))) (R A B D po2nr C A e. 3adantr2 R A B C D potr D R B br anim1d B R C br C R D br D R B br df-3an syl5ib mtod))
+thm (po0 ((x y) (x z) (R x) (y z) (R y) (R z)) () (Po R ({/})) (R {/} x y z df-po x cv noel y {/} z {/} x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ A.e. A.e. pm2.21i mprgbir))
+thm (sopo ((x y) (R x) (R y) (A x) (A y)) () (-> (Or R A) (Po R A)) (R A x y df-so pm3.26bd))
+thm (soss ((x y) (R x) (R y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (-> (Or R B) (Or R A))) (A B R poss A B x cv ssel A B y cv ssel anim12d x cv R y cv br x cv y cv = y cv R x cv br \/\/ imim1d y 19.20dv x 19.20dv x B y B x cv R y cv br x cv y cv = y cv R x cv br \/\/ r2al x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ r2al 3imtr4g anim12d R B x y df-so R A x y df-so 3imtr4g))
+thm (soeq1 ((x y) (R x) (R y) (S x) (S y) (A x) (A y)) () (-> (= R S) (<-> (Or R A) (Or S A))) (R S A poeq1 R S x cv y cv breq R S = x cv y cv = pm4.2i R S y cv x cv breq 3orbi123d y A biraldv x A biraldv anbi12d R A x y df-so S A x y df-so 3bitr4g))
+thm (soeq2 () () (-> (= A B) (<-> (Or R A) (Or R B))) (A B R soss B A R soss anim12i A B eqss R B Or R A Or bi 3imtr4 bicomd))
+thm (sonr () () (-> (/\ (Or R A) (e. B A)) (-. (br B R B))) (R A B poirr R A sopo sylan))
+thm (sotr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))) (R A B C D potr R A sopo sylan))
+thm (solin ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (R x) (R y)) () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (\/\/ (br B R C) (= B C) (br C R B))) (x cv B R y cv breq1 x cv B y cv eqeq1 x cv B y cv R breq2 3orbi123d R A Or imbi2d y cv C B R breq2 y cv C B eqeq2 y cv C R B breq1 3orbi123d R A Or imbi2d R A x y df-so x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ ra42 R A Po adantl sylbi com12 vtocl2ga impcom))
+thm (so2nr () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))) (R A B C po2nr R A sopo sylan))
+thm (so3nr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))) (R A B C D po3nr R A sopo sylan))
+thm (sotric () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (br B R C) (-. (\/ (= B C) (br C R B))))) (B C B R breq2 negbid R A B sonr syl5bi com12 C A e. adantrr R A B C so2nr B R C br C R B br imnan sylibr con2d jaod R A B C solin B R C br B C = C R B br 3orass B R C br B C = C R B br \/ df-or bitr sylib impbid bicon2d))
+thm (sotrieq () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (-. (\/ (br B R C) (br C R B))))) (B C B R breq2 negbid R A B sonr syl5bi B C C R breq2 negbid R A C sonr syl5bir anim12d com12 anandis R A B C sotric bicon2d biimpar ord con1d exp imp3a impbid B R C br C R B br ioran syl6bbr))
+thm (sotrieq2 () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (/\ (-. (br B R C)) (-. (br C R B))))) (R A B C sotrieq B R C br C R B br ioran syl6bb))
+thm (itlso ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) ((itlso.1 (-> (e. (cv x) A) (-. (br (cv x) R (cv x))))) (itlso.2 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (itlso.3 (-> (/\ (e. (cv x) A) (e. (cv y) A)) (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x)))))) (Or R A) (R A x y df-so itlso.1 y cv A e. z cv A e. 3ad2ant1 itlso.2 jca rgen3 R A x y z df-po mpbir itlso.3 rgen2 mpbir2an))
+thm (so ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) ((so.1 (-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) (/\ (<-> (br (cv x) R (cv y)) (-. (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))))) (Or R A) (x cv eqid x cv x cv = x cv R x cv br orc ax-mp y cv x cv A eleq1 x cv A e. anbi2d y cv x cv x cv eqeq2 y cv x cv R x cv breq1 orbi12d y cv x cv x cv R breq2 negbid bibi12d imbi12d x cv A e. y cv A e. y cv A e. 3anass y cv A e. anidm x cv A e. anbi2i bitr2 z cv y cv = x cv A e. pm4.2i z cv y cv = y cv A e. pm4.2i z cv y cv A eleq1 3anbi123d x cv R y cv br x cv y cv = y cv R x cv br \/ -. <-> imbi1d so.1 pm3.26d chvarv sylbi bicon2d chvarv mpbii anidms so.1 pm3.27d x cv A e. y cv A e. y cv A e. 3anass y cv A e. anidm x cv A e. anbi2i bitr2 z cv y cv = x cv A e. pm4.2i z cv y cv = y cv A e. pm4.2i z cv y cv A eleq1 3anbi123d x cv R y cv br x cv y cv = y cv R x cv br \/ -. <-> imbi1d so.1 pm3.26d chvarv sylbi bicon2d biimprd x cv R y cv br x cv y cv = y cv R x cv br 3orass x cv R y cv br x cv y cv = y cv R x cv br \/ df-or bitr sylibr itlso))
+thm (so0 ((x y) (R x) (R y)) () (Or R ({/})) (R {/} x y df-so R po0 mpbiran x cv noel y {/} x cv R y cv br x cv y cv = y cv R x cv br \/\/ A.e. pm2.21i mprgbir))
+thm (supeq1 ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= B C) (= (sup B A R) (sup C A R))) (B C y x cv R y cv br -. raleq B C z y cv R z cv br rexeq y cv R x cv br imbi2d y A biraldv anbi12d x A birabsdv unieqd B A R x y z df-sup C A R x y z df-sup 3eqtr4g))
+thm (supmo ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w)) ((supmo.1 (Or R A))) (E* x (/\ (e. (cv x) A) (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))))) (x cv w cv A eleq1 x cv w cv R y cv breq1 negbid y B biraldv x cv w cv y cv R breq2 z B y cv R z cv br E.e. imbi1d y A biraldv anbi12d anbi12d mo4 y cv x cv R w cv breq1 y cv x cv R z cv breq1 z B birexdv imbi12d A rcla4va con3d imp y cv z cv x cv R breq2 negbid B cbvralv z B x cv R z cv br ralnex bitr sylan2b an1rs y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. adantlrr y B w cv R y cv br -. A.e. adantrl w cv A e. adantrl y cv w cv R x cv breq1 y cv w cv R z cv breq1 z B birexdv imbi12d A rcla4v impcom con3d imp y cv z cv w cv R breq2 negbid B cbvralv z B w cv R z cv br ralnex bitr sylan2b anasss y A y cv R w cv br z B y cv R z cv br E.e. -> A.e. adantrrr y B x cv R y cv br -. A.e. adantll x cv A e. adantll jca supmo.1 R A x cv w cv sotrieq2 mpan y B w cv R y cv br -. A.e. y A y cv R w cv br z B y cv R z cv br E.e. -> A.e. /\ adantrr y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ adantlr mpbird w ax-gen mpgbir))
+thm (supex ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (e. (sup B A R) (V)) (B A R x y z df-sup x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ df-rab unieqi eqtr supmo.1 x y B z supmo x x cv A e. y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ /\ moabex ax-mp uniex eqeltr))
+thm (supeu ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (E!e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z)))))))) (supmo.1 x y B z supmo x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ E.e. jctr x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ reu5 sylibr))
+thm (supcl ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (e. (sup B A R) A)) (supmo.1 x y B z supeu x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ reucl syl B A R x y z df-sup A eleq1i sylibr))
+thm (supub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w) (C w)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (e. C B) (-. (br (sup B A R) R C)))) (w cv C B A R sup R breq2 negbid x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ E.e. imbi2d y cv w cv B A R sup R breq2 negbid B rcla4v B A R x y z df-sup eqcomi x cv B A R sup R y cv breq1 negbid y B biraldv x cv B A R sup y cv R breq2 z B y cv R z cv br E.e. imbi1d y A biraldv anbi12d A reuuni2 supmo.1 x y B z supcl supmo.1 x y B z supeu sylanc mpbiri pm3.26d syl5 vtoclga com12))
+thm (suplub ((C z) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (R x) (R y) (R z) (R w) (B x) (B y) (B z) (B w) (C w)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (/\ (e. C A) (br C R (sup B A R))) (E.e. z B (br C R (cv z))))) (w cv C R B A R sup breq1 w cv C R z cv breq1 z B birexdv imbi12d x A y B x cv R y cv br -. A.e. y A y cv R x cv br z B y cv R z cv br E.e. -> A.e. /\ E.e. imbi2d y cv w cv R B A R sup breq1 y cv w cv R z cv breq1 z B birexdv imbi12d A rcla4v B A R x y z df-sup eqcomi x cv B A R sup R y cv breq1 negbid y B biraldv x cv B A R sup y cv R breq2 z B y cv R z cv br E.e. imbi1d y A biraldv anbi12d A reuuni2 supmo.1 x y B z supcl supmo.1 x y B z supeu sylanc mpbiri pm3.27d syl5 vtoclga com12 imp3a))
+thm (supnub ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((supmo.1 (Or R A))) (-> (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (-> (/\ (e. C A) (A.e. z B (-. (br C R (cv z))))) (-. (br C R (sup B A R))))) (supmo.1 x y B z C suplub exp3a imp z B C R z cv br dfrex2 syl6ib con2d exp imp3a))
+thm (supeui ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (E!e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))) (sup.2 sup.1 x y B z supeu ax-mp))
+thm (supcli ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (e. (sup B A R) A) (sup.2 sup.1 x y B z supcl ax-mp))
+thm (supubi ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (e. C B) (-. (br (sup B A R) R C))) (sup.2 sup.1 x y B z C supub ax-mp))
+thm (suplubi ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (/\ (e. C A) (br C R (sup B A R))) (E.e. z B (br C R (cv z)))) (sup.2 sup.1 x y B z C suplub ax-mp))
+thm (supnubi ((C z) (x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((sup.1 (Or R A)) (sup.2 (E.e. x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))) (-> (/\ (e. C A) (A.e. z B (-. (br C R (cv z))))) (-. (br C R (sup B A R)))) (sup.2 sup.1 x y B z C supnub ax-mp))
+thm (supsn ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (R x) (R y) (R z)) ((supsn.1 (Or R A))) (-> (e. B A) (= (sup ({} B) A R) B)) (y cv B elsni y cv B B R breq2 negbid supsn.1 R A B sonr mpan syl5bir syl com12 r19.21aiv z cv B y cv R breq2 B {} rcla4ev B A snidg sylan exp y cv A e. a1d r19.21aiv jca y cv B elsni y cv B B R breq2 negbid supsn.1 R A B sonr mpan syl5bir syl com12 r19.21aiv z cv B y cv R breq2 B {} rcla4ev B A snidg sylan exp y cv A e. a1d r19.21aiv jca x cv B R y cv breq1 negbid y B {} biraldv x cv B y cv R breq2 z B {} y cv R z cv br E.e. imbi1d y A biraldv anbi12d A rcla4ev mpdan supsn.1 x y B {} z supmo jctir x A y B {} x cv R y cv br -. A.e. y A y cv R x cv br z B {} y cv R z cv br E.e. -> A.e. /\ reu5 sylibr x cv B R y cv breq1 negbid y B {} biraldv x cv B y cv R breq2 z B {} y cv R z cv br E.e. imbi1d y A biraldv anbi12d A reuuni2 mpdan mpbid B {} A R x y z df-sup syl5eq))
+thm (fri ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((fri.1 (e. B (V)))) (-> (/\ (Fr R A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (A.e. y B (-. (br (cv y) R (cv x)))))) (R A z x y df-fr fri.1 z cv B A sseq1 z cv B {/} eqeq1 negbid anbi12d z cv B y y cv R x cv br -. raleq x rexeqd imbi12d cla4v sylbi imp))
+thm (dffr2 ((x y) (x z) (w x) (v x) (R x) (y z) (w y) (v y) (R y) (w z) (v z) (R z) (v w) (R w) (R v) (A x) (A w) (A v)) () (<-> (Fr R A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) ({|} z (br (cv z) R (cv y)))) ({/})))))) (R A x v w df-fr x cv z z cv R y cv br {|} w disj w visset z cv w cv R y cv breq1 elab negbii w x cv biral bitr y x cv birex y cv v cv w cv R breq2 negbid w x cv biraldv x cv cbvrexv bitr x cv A C_ x cv {/} = -. /\ imbi2i x bial bitr4))
+thm (frc ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((frc.1 (e. B (V)))) (-> (Fr R A) (-> (/\ (C_ B A) (-. (= B ({/})))) (E.e. x B (= (i^i B ({|} y (br (cv y) R (cv x)))) ({/}))))) (R A z x y dffr2 frc.1 z cv B A sseq1 z cv B {/} eqeq1 negbid anbi12d z cv B y y cv R x cv br {|} ineq1 {/} eqeq1d x rexeqd imbi12d cla4v sylbi))
+thm (frss ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Fr R B) (Fr R A))) (x cv A B sstr2 com12 x cv {/} = -. anim1d y x cv x cv z z cv R y cv br {|} i^i {/} = E.e. imim1d x 19.20dv R B x y z dffr2 R A x y z dffr2 3imtr4g))
+thm (freq1 ((x y) (x z) (R x) (y z) (R y) (R z) (S x) (S y) (S z) (A x) (A y) (A z)) () (-> (= R S) (<-> (Fr R A) (Fr S A))) (R S z cv y cv breq negbid z x cv biraldv y x cv birexdv x cv A C_ x cv {/} = -. /\ imbi2d x bialdv R A x y z df-fr S A x y z df-fr 3bitr4g))
+thm (freq2 () () (-> (= A B) (<-> (Fr R A) (Fr R B))) (A B R frss B A R frss anim12i A B eqss R B Fr R A Fr bi 3imtr4 bicomd))
+thm (frirr ((x y) (x z) (R x) (y z) (R y) (R z) (A y) (A z)) () (-> (/\ (Fr R A) (e. (cv x) A)) (-. (br (cv x) R (cv x)))) (x visset snnz x cv snex R A y z frc exp3a x visset A snss syl5ib mpii y x cv elsn y cv x cv z cv R breq2 z biabdv x cv {} ineq2d {/} eqeq1d x visset z cv x cv R x cv breq1 elab biimpr x visset snid jctil x cv x cv {} z z cv R x cv br {|} elin sylibr x cv x cv {} z z cv R x cv br {|} i^i n0i syl con2i syl6bi sylbi r19.23aiv syl6 imp))
+thm (fr2nr ((w z) (R z) (R w) (x z) (w x) (y z) (w y) (A z) (A w)) () (-> (/\ (Fr R A) (/\ (e. (cv x) A) (e. (cv y) A))) (-. (/\ (br (cv x) R (cv y)) (br (cv y) R (cv x))))) (y visset x cv prnz y x zfpair R A w z frc w cv y cv z cv R breq2 z biabdv y cv x cv {,} ineq2d {/} eqeq1d negbid x R y z brab1 x visset y cv pri2 x cv y cv x cv {,} z z cv R y cv br {|} inelcm mpan sylbi syl5bir com12 w cv x cv z cv R breq2 z biabdv y cv x cv {,} ineq2d {/} eqeq1d negbid y R x z brab1 y visset x cv pri1 y cv y cv x cv {,} z z cv R x cv br {|} inelcm mpan sylbi syl5bir com12 jaao w visset y cv x cv elpr syl5ib con3i expi r19.23aiv syl6 mpan2i y visset x visset A prss syl5ib ancomsd imp))
+thm (fr3nr ((v w) (w x) (v x) (w y) (v y) (w z) (v z) (R w) (R v) (A w) (A v)) () (-> (/\ (Fr R A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-. (/\/\ (br (cv x) R (cv y)) (br (cv y) R (cv z)) (br (cv z) R (cv x))))) (y visset z cv x cv tpnz y cv z cv x cv tpex R A v w frc v cv y cv = y cv z cv x cv {,,} w w cv R v cv br {|} i^i {/} = -. v cv z cv = v cv x cv = 3jao v cv y cv w cv R breq2 w biabdv y cv z cv x cv {,,} ineq2d {/} eqeq1d negbid x R y w brab1 x visset y cv z cv tpi3 x cv y cv z cv x cv {,,} w w cv R y cv br {|} inelcm mpan sylbi syl5bir com12 v cv z cv w cv R breq2 w biabdv y cv z cv x cv {,,} ineq2d {/} eqeq1d negbid y R z w brab1 y visset z cv x cv tpi1 y cv y cv z cv x cv {,,} w w cv R z cv br {|} inelcm mpan sylbi syl5bir com12 v cv x cv w cv R breq2 w biabdv y cv z cv x cv {,,} ineq2d {/} eqeq1d negbid z R x w brab1 z visset y cv x cv tpi2 z cv y cv z cv x cv {,,} w w cv R x cv br {|} inelcm mpan sylbi syl5bir com12 syl3an v visset y cv z cv x cv eltp syl5ib con3i expi r19.23aiv syl6 mpan2i y visset z visset x visset A tpss syl5ib x cv A e. y cv A e. z cv A e. 3anrot syl5ib imp))
+thm (fr0 ((x y) (x z) (R x) (y z) (R y) (R z)) () (Fr R ({/})) (R {/} x y z dffr2 x cv ss0 x cv {/} C_ x cv {/} = iman mpbi y x cv x cv z z cv R y cv br {|} i^i {/} = E.e. pm2.21i mpgbir))
+thm (efrirr ((A x)) () (-> (Fr (E) A) (-. (e. A A))) (x cv A E freq2 x cv A x cv eleq1 x cv A A eleq2 bitrd negbid imbi12d E x cv x frirr exp x x epel negbii syl6ib pm2.01d A vtoclg com12 pm2.01d))
+thm (efrn2lp () () (-> (/\ (Fr (E) A) (/\ (e. (cv x) A) (e. (cv y) A))) (-. (/\ (e. (cv x) (cv y)) (e. (cv y) (cv x))))) (E A x y fr2nr x y epel y x epel anbi12i negbii sylib))
+thm (epne3 () () (-> (/\ (Fr (E) A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-. (/\/\ (e. (cv x) (cv y)) (e. (cv y) (cv z)) (e. (cv z) (cv x))))) (E A x y z fr3nr x y epel y z epel z x epel 3anbi123i negbii sylib))
+thm (tz7.2 () () (-> (/\ (/\ (Tr A) (Fr (E) A)) (e. B A)) (/\ (C_ B A) (-. (= B A)))) (A B trss B A A eleq1 negbid A efrirr syl5bir com12 con2d anim12i B A e. B A C_ B A = -. jcab sylibr imp))
+thm (dfepfr ((x y) (x z) (y z) (A x) (A z)) () (<-> (Fr (E) A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) (cv y)) ({/})))))) (E A x y z dffr2 z y epel z biabi z y cv abid2 eqtr x cv ineq2i {/} eqeq1i y x cv birex x cv A C_ x cv {/} = -. /\ imbi2i x bial bitr))
+thm (epfrc ((x y) (A x) (A y) (B x) (B y)) ((epfrc.1 (e. B (V)))) (-> (/\ (Fr (E) A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (epfrc.1 E A x y frc imp y x epel y biabi y x cv abid2 eqtr2 B ineq2i {/} eqeq1i x B birex sylibr))
+thm (dfwe2 ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) () (<-> (We R A) (/\ (Fr R A) (A.e. x A (A.e. y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))))))) (R A df-we R A x cv y cv solin exp x y 19.21aivv x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ r2al sylibr R A Fr anim2i R A x y fr2nr z cv A e. 3adantr3 x cv z cv y cv R breq2 biimprd x cv R y cv br anim2d impcom nsyl x cv R y cv br y cv R z cv br /\ x cv z cv = imnan sylibr R A x y z fr3nr x cv R y cv br y cv R z cv br z cv R x cv br df-3an negbii sylib x cv R y cv br y cv R z cv br /\ z cv R x cv br imnan sylibr jcad x cv z cv = z cv R x cv br ioran syl6ibr x cv R z cv br imim1d x cv R z cv br x cv z cv = z cv R x cv br 3orass x cv R z cv br x cv z cv = z cv R x cv br \/ orcom x cv z cv = z cv R x cv br \/ x cv R z cv br df-or 3bitr syl5ib R A x frirr y cv A e. adantrr z cv A e. 3adantr3 jctild exp a2d z 19.20dv y 19.20dv x 19.20dv x A y A z A x cv R z cv br x cv z cv = z cv R x cv br \/\/ r3al x A y A z A x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ r3al 3imtr4g y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ ralidm y cv z cv x cv R breq2 y cv z cv x cv eqeq2 y cv z cv R x cv breq1 3orbi123d A cbvralv y A biral bitr3 x A biral R A x y z df-po 3imtr4g ancrd R A x y df-so syl6ibr imdistani impbi bitr))
+thm (wess () () (-> (C_ A B) (-> (We R B) (We R A))) (A B R frss A B R soss anim12d R B df-we R A df-we 3imtr4g))
+thm (weeq1 () () (-> (= R S) (<-> (We R A) (We S A))) (R S A freq1 R S A soeq1 anbi12d R A df-we S A df-we 3bitr4g))
+thm (weeq2 () () (-> (= A B) (<-> (We R A) (We R B))) (A B R freq2 A B R soeq2 anbi12d R A df-we R B df-we 3bitr4g))
+thm (wefr () () (-> (We R A) (Fr R A)) (R A df-we pm3.26bd))
+thm (weso () () (-> (We R A) (Or R A)) (R A df-we pm3.27bd))
+thm (wecmpep () () (-> (/\ (We (E) A) (/\ (e. (cv x) A) (e. (cv y) A))) (\/\/ (e. (cv x) (cv y)) (= (cv x) (cv y)) (e. (cv y) (cv x)))) (E A x cv y cv solin x y epel x cv y cv = pm4.2 y x epel 3orbi123i sylib E A weso sylan))
+thm (wetrep () () (-> (/\ (We (E) A) (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A))) (-> (/\ (e. (cv x) (cv y)) (e. (cv y) (cv z))) (e. (cv x) (cv z)))) (E A x cv y cv z cv sotr E A weso sylan x y epel y z epel anbi12i x z epel 3imtr3g))
+thm (wefrc ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (We (E) A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (B A E wess x cv y cv B ineq2 {/} eqeq1d B rcla4ev exp E B We adantl B y cv inss1 y visset B inex2 B x epfrc E B wefr sylan exp32 mpi x cv B y cv elin B y cv i^i x cv i^i {/} = anbi1i x cv B e. x cv y cv e. B y cv i^i x cv i^i {/} = anass bitr birex2 syl6ib y cv B e. adantr B z x y wetrep exp3a y cv B e. z cv B e. x cv B e. df-3an y cv B e. z cv B e. x cv B e. 3anrot bitr3 sylan2b exp44 imp com34 imp3a z cv B x cv elin syl5ib imp4a com23 r19.21adv B x cv i^i y cv z dfss3 syl6ibr B x cv i^i y cv dfss B x cv y cv in23 B x cv i^i eqeq2i bitr biimp {/} eqeq1d biimprd syl6 exp3a imp4a r19.22dv syld pm2.61d exp y 19.23adv B y n0 syl5ib syl6com imp32))
+thm (we0 () () (We R ({/})) (R {/} df-we R fr0 R so0 mpbir2an))
+thm (wereu ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((wereu.1 (e. B (V)))) (-> (/\ (We R A) (/\ (C_ B A) (-. (= B ({/}))))) (E!e. x B (A.e. y B (-. (br (cv y) R (cv x)))))) (wereu.1 R A x y fri R A wefr sylan R B x cv z cv solin R B weso sylan x cv R z cv br x cv z cv = z cv R x cv br df-3or x cv R z cv br x cv z cv = z cv R x cv br or23 x cv R z cv br z cv R x cv br \/ x cv z cv = df-or 3bitr sylib x cv R z cv br z cv R x cv br ioran syl5ibr B A R wess impcom sylan y cv x cv R z cv breq1 negbid B rcla4v y cv z cv R x cv breq1 negbid B rcla4v im2anan9 ancomsd imp syl5 exp4b pm2.43d B {/} = -. adantrr x z 19.21aivv x B z B y B y cv R x cv br -. A.e. y B y cv R z cv br -. A.e. /\ x cv z cv = -> r2al sylibr jca x cv z cv y cv R breq2 negbid y B biraldv B reu4 sylibr))
+thm (ordeq () () (-> (= A B) (<-> (Ord A) (Ord B))) (A B treq A B E weeq2 anbi12d A df-ord B df-ord 3bitr4g))
+thm (elong ((A x)) () (-> (e. A B) (<-> (e. A (On)) (Ord A))) (x cv A ordeq x df-on B elab2g))
+thm (elon () ((elon.1 (e. A (V)))) (<-> (e. A (On)) (Ord A)) (elon.1 A V elong ax-mp))
+thm (eloni () () (-> (e. A (On)) (Ord A)) (A On elong ibi))
+thm (elon2 () () (<-> (e. A (On)) (/\ (Ord A) (e. A (V)))) (A eloni A On elisset jca A V elong biimparc impbi))
+thm (limeq () () (-> (= A B) (<-> (Lim A) (Lim B))) (A B ordeq A B {/} eqeq1 negbid A B unieq A eqeq2d A B B U. eqeq1 bitrd 3anbi123d A df-lim B df-lim 3bitr4g))
+thm (ordwe () () (-> (Ord A) (We (E) A)) (A df-ord pm3.27bd))
+thm (ordtr () () (-> (Ord A) (Tr A)) (A df-ord pm3.26bd))
+thm (ordfr () () (-> (Ord A) (Fr (E) A)) (A ordwe E A wefr syl))
+thm (ordelss () () (-> (/\ (Ord A) (e. B A)) (C_ B A)) (A B trss imp A ordtr sylan))
+thm (trssord () () (-> (/\/\ (Tr A) (C_ A B) (Ord B)) (Ord A)) (A B E wess imp B ordwe sylan2 A Tr anim2i A Tr A B C_ B Ord 3anass A df-ord 3imtr4))
+thm (ordeirr () () (-> (Ord A) (-. (e. A A))) (A ordfr A efrirr syl))
+thm (nordeq () () (-> (/\ (Ord A) (e. B A)) (-. (= A B))) (A B A eleq1 negbid A ordeirr syl5bi com12 con2d imp))
+thm (ordn2lp () () (-> (Ord A) (-. (/\ (e. A B) (e. B A)))) (A ordeirr A ordtr A A B trel syl mtod))
+thm (tz7.5 ((A x) (B x)) () (-> (/\ (Ord A) (/\ (C_ B A) (-. (= B ({/}))))) (E.e. x B (= (i^i B (cv x)) ({/})))) (A B x wefrc A ordwe sylan))
+thm (ordelord ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (Ord A) (e. B A)) (Ord B)) (x cv B A eleq1 A Ord anbi2d x cv B ordeq imbi12d A z y x wetrep A ordwe sylan A Ord x cv A e. pm3.26 z cv y cv e. y cv x cv e. /\ adantr A ordtr A z cv y cv x cv trel3 syl x cv A e. z cv y cv e. y cv x cv e. 3anrot x cv A e. z cv y cv e. y cv x cv e. 3anass bitr3 syl5ibr exp3a imp31 A ordtr A y cv x cv trel syl exp3a com23 imp31 z cv y cv e. adantrl A Ord x cv A e. pm3.27 z cv y cv e. y cv x cv e. /\ adantr 3jca sylanc exp pm2.43d z y 19.21aivv x cv z y dftr2 sylibr A ordtr A x cv trss syl x cv A E wess A ordwe syl5com syld imp jca x cv df-ord sylibr A vtoclg anabsi7))
+thm (ordelon () () (-> (/\ (Ord A) (e. B A)) (e. B (On))) (A B ordelord B A elong A Ord adantl mpbird))
+thm (onelon () () (-> (/\ (e. A (On)) (e. B A)) (e. B (On))) (A B ordelon A eloni sylan))
+thm (tz7.7 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Ord A) (Tr B)) (<-> (e. B A) (/\ (C_ B A) (-. (= B A))))) (A B tz7.2 A ordtr A ordfr jca sylan exp B Tr adantr A ordtr A x cv trss x cv A B eldifi syl5 A B x cv difin0ss com12 syl6 syl B Tr B A C_ ad2antll imp32 y cv x cv B eleq1 biimpcd x cv A B eldifn nsyli imp B A C_ adantll A Ord B Tr /\ adantl B x cv y cv trel exp3a com23 imp x cv A B eldifn nsyli exp B A C_ adantld imp32 A Ord adantll A y x wecmpep A ordwe B A y cv ssel2 x cv A B eldifi anim12i syl2an B Tr adantlr ecase23d exp44 com34 imp31 ssrdv A B \ x cv i^i {/} = adantrr eqssd x cv A B eldifi A Ord B Tr /\ B A C_ /\ A B \ x cv i^i {/} = ad2antrl eqeltrrd exp32 r19.23adv A B difss A A B \ x tz7.5 mpanr1 syl5 exp4b com23 B Tr adantrd pm2.43i B A pssdifn0 syl7 exp4a pm2.43d imp3a impbid))
+thm (ordelssne () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (/\ (C_ A B) (-. (= A B))))) (B A tz7.7 A ordtr sylan2 ancoms))
+thm (ordelpss () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (C: A B))) (A B ordelssne A B dfpss2 syl6bbr))
+thm (ordsseleq () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (\/ (e. A B) (= A B)))) (A B ordelssne biimprd exp3a A B e. A B = orcom A B = A B e. df-or bitr syl6ibr A B ordelssne A B C_ A B = -. pm3.26 syl6bi A B eqimss jctir A B e. A B = A B C_ jaob sylibr impbid))
+thm (ordin () () (-> (/\ (Ord A) (Ord B)) (Ord (i^i A B))) (A B inss2 A B i^i B trssord mp3an2 A B trin A ordtr B ordtr syl2an sylan anabss3))
+thm (onin () () (-> (/\ (e. A (On)) (e. B (On))) (e. (i^i A B) (On))) (A B ordin A eloni B eloni syl2an A On e. B On e. pm3.26 A On B inex1g A B i^i V elong 3syl mpbird))
+thm (ordtri3or () () (-> (/\ (Ord A) (Ord B)) (\/\/ (e. A B) (= A B) (e. B A))) (A B ordin A B i^i ordeirr syl A B i^i A B elin A B incom B eleq1i A B i^i A e. anbi2i bitr negbii A B i^i A e. B A i^i B e. ianor bitr sylib A B inss1 A B i^i A ordsseleq mpbii A B ordin sylan anabss1 ord A B df-ss syl6ibr B A inss1 B A i^i B ordsseleq mpbii B A ordin sylan anabss4 ord B A df-ss syl6ibr orim12d mpd A B ordsseleq B A ordsseleq ancoms orbi12d mpbid A B e. A B = B A e. df-3or A B = oridm A B eqcom A B = orbi2i bitr3 A B e. B A e. \/ orbi2i A B e. A B = B A e. or23 A B e. A B = B A e. B A = or4 3bitr4 bitr sylibr))
+thm (ordtri1 () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (-. (e. B A)))) (A B ordsseleq A B ordn2lp A B e. B A e. imnan sylibr A B B eleq2 negbid B ordeirr syl5bir com12 jaao A B ordtri3or A B e. A B = B A e. df-3or A B e. A B = \/ B A e. orcom B A e. A B e. A B = \/ df-or 3bitr sylib impbid bitrd))
+thm (ontri1 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (-. (e. B A)))) (A B ordtri1 A eloni B eloni syl2an))
+thm (ordtri2 () () (-> (/\ (Ord A) (Ord B)) (<-> (e. A B) (-. (\/ (= A B) (e. B A))))) (B A ordsseleq B A eqcom B A e. orbi2i B A e. A B = orcom bitr syl6bb B A ordtri1 bitr3d ancoms bicon2d))
+thm (ordtri3 () () (-> (/\ (Ord A) (Ord B)) (<-> (= A B) (-. (\/ (e. A B) (e. B A))))) (A B A eleq2 negbid A ordeirr syl5bi A B B eleq2 negbid B ordeirr syl5bir anim12d A B e. B A e. ioran syl6ibr com12 A B ordtri3or A B e. A B = B A e. df-3or A B e. A B = B A e. or23 A B e. B A e. \/ A B = df-or 3bitr sylib impbid))
+thm (ordtri4 () () (-> (/\ (Ord A) (Ord B)) (<-> (= A B) (/\ (C_ A B) (-. (e. A B))))) (A B ordtri3 A B ordtri1 A B e. -. anbi1d B A e. -. A B e. -. ancom A B e. B A e. ioran bitr4 syl6bb bitr4d))
+thm (orddisj () () (-> (Ord A) (= (i^i A ({} A)) ({/}))) (A ordeirr A A disjsn sylibr))
+thm (onfr ((x y) (x z) (y z)) () (Fr (E) (On)) (On x z dfepfr z cv y cv x cv ineq2 {/} eqeq1d x cv rcla4ev expcom x cv On C_ a1d x cv On y cv ssel y visset elon syl6ib x cv y cv inss2 x visset y cv inex1 y cv z epfrc exp mpani y cv Tr z ax-17 z x cv x cv z cv i^i {/} = hbre1 x cv y cv inss1 z cv sseli y cv z cv trss x cv y cv inss2 z cv sseli syl5 z cv y cv sseqin2 y cv z cv i^i z cv x cv ineq2 x cv y cv z cv inass syl5req sylbi {/} eqeq1d biimprcd sylan9 imp anim12i exp32 pm2.43b exp com23 z x cv x cv z cv i^i {/} = ra4e syl8 r19.23ad sylan9 y cv ordfr y cv ordtr sylanc syl6 com3r pm2.61i y 19.23adv x cv y n0 syl5ib imp mpgbir))
+thm (ordon ((x y)) () (Ord (On)) (On df-ord On x dftr3 x cv y cv ordelord x visset elon sylanb exp y visset elon syl6ibr ssrdv mprgbir E On x y dfwe2 onfr x cv y cv ordtri3or x y epel x cv y cv = pm4.2 y x epel 3orbi123i sylibr x cv eloni y cv eloni syl2an rgen2 mpbir2an mpbir2an))
+thm (epweon () () (We (E) (On)) (ordon On ordwe ax-mp))
+thm (onprc () () (-. (e. (On) (V))) (ordon On ordeirr ax-mp ordon On V elong mpbiri mto))
+thm (ordeleqon () () (<-> (Ord A) (\/ (e. A (On)) (= A (On)))) (onprc On A elisset mto ordon A On ordtri3or mpan2 A On e. A On = On A e. df-3or sylib ord mt3i A eloni ordon A On ordeq mpbiri jaoi impbi))
+thm (ordsson () () (-> (Ord A) (C_ A (On))) (ordon A On ordelssne mpan2 A On C_ A On = -. pm3.26 syl6bi A ordeleqon biimp ord A On eqimss syl6 pm2.61d))
+thm (onsst () () (-> (e. A (On)) (C_ A (On))) (A eloni A ordsson syl))
+thm (tfi ((A x)) () (-> (/\ (C_ A (On)) (A.e. x (On) (-> (C_ (cv x) A) (e. (cv x) A)))) (= A (On))) (x cv On A eldifn x cv On e. x cv A C_ x cv A e. -> -> adantl On A x cv difin0ss x cv onsst syl5com x cv A e. imim1d a2i x cv On A eldifi syl5 imp mtod exp x 19.20i x On x cv A C_ x cv A e. -> df-ral x On A \ On A \ x cv i^i {/} = -. df-ral 3imtr4 x On A \ On A \ x cv i^i {/} = ralnex sylib On A ssdif0 negbii On A difss ordon On On A \ x tz7.5 mpan mpan sylbi nsyl2 A On C_ anim2i A On eqss sylibr))
+thm (tfis ((ph y) (y z) (ph z) (x y) (x z)) ((tfis.1 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ([/] (cv y) x ph)) ph)))) (-> (e. (cv x) (On)) ph) (x On ph ssrab z cv On e. x ax-17 y cv z cv e. x ax-17 y x ph hbs1 hbral z x ph hbs1 hbim hbim x cv z cv On eleq1 x cv z cv y y cv x ph [/] raleq x z ph sbequ12 imbi12d imbi12d tfis.1 chvar z cv x On ph {e.|} y dfss3 z cv On e. x ax-17 y cv ph elrabsf pm3.27bd y z cv r19.20si sylbi syl5 anc2li z cv On e. x ax-17 z cv ph elrabsf syl6ibr rgen x On ph {e.|} z tfi mp2an eqcomi eqrabi pm3.27bd))
+thm (tfis2f ((ph y) (x y)) ((tfis2f.1 (-> ps (A. x ps))) (tfis2f.2 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis2f.3 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. (cv x) (On)) ph) (tfis2f.3 tfis2f.1 tfis2f.2 sbie y x cv biral syl5ib tfis))
+thm (tfis2 ((ps x) (ph y) (x y)) ((tfis2.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis2.2 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. (cv x) (On)) ph) (ps x ax-17 tfis2.1 tfis2.2 tfis2f))
+thm (tfis3 ((ps x) (ph y) (ch x) (A x) (x y)) ((tfis3.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (tfis3.2 (-> (= (cv x) A) (<-> ph ch))) (tfis3.3 (-> (e. (cv x) (On)) (-> (A.e. y (cv x) ps) ph)))) (-> (e. A (On)) ch) (tfis3.2 tfis3.1 tfis3.3 tfis2 vtoclga))
+thm (ssorduni ((x y) (A x) (A y)) () (-> (C_ A (On)) (Ord (U. A))) (ordon A U. On trssord 3exp mpii A On y cv ssel y cv eloni y cv ordtr y cv x cv trss 3syl syl6 y cv A e. x cv y cv e. x cv y cv C_ anc2r syl x cv y cv A ssuni syl8 r19.23adv x cv A y eluni2 syl5ib r19.21aiv A U. x dftr3 sylibr A On y cv ssel y cv eloni y cv x cv ordelord exp x visset elon syl6ibr syl syl6 r19.23adv x cv A y eluni2 syl5ib ssrdv sylc))
+thm (onunit () () (-> (e. A B) (-> (C_ A (On)) (e. (U. A) (On)))) (A B uniexg A U. V elong syl A ssorduni syl5bir))
+thm (onuni () ((onuni.1 (e. A (V)))) (-> (C_ A (On)) (e. (U. A) (On))) (onuni.1 A V onunit ax-mp))
+thm (uniord () () (-> (Ord A) (Ord (U. A))) (A ordsson A ssorduni syl))
+thm (onelpsst () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (/\ (C_ A B) (-. (= A B))))) (A B ordelssne A eloni B eloni syl2an))
+thm (onsseleq () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (\/ (e. A B) (= A B)))) (A B ordsseleq A eloni B eloni syl2an))
+thm (onelsst () () (-> (e. A (On)) (-> (e. B A) (C_ B A))) (A eloni A ordtr A B trss 3syl))
+thm (ordtr1 () () (-> (Ord C) (-> (/\ (e. A B) (e. B C)) (e. A C))) (C ordtr C A B trel syl))
+thm (ordtr2 () () (-> (/\ (Ord A) (Ord C)) (-> (/\ (C_ A B) (e. B C)) (e. A C))) (A B ordsseleq biimpd C A B ordtr1 exp3a B C A eleq1a com12 C Ord a1i jaod com23 imp syl9 exp C B ordelord syl5 pm2.43d exp3a imp com23 imp3a))
+thm (ontr1 () () (-> (e. C (On)) (-> (/\ (e. A B) (e. B C)) (e. A C))) (C eloni C A B ordtr1 syl))
+thm (ontr2 () () (-> (/\ (e. A (On)) (e. C (On))) (-> (/\ (C_ A B) (e. B C)) (e. A C))) (A C B ordtr2 A eloni C eloni syl2an))
+thm (ordunidif ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Ord A) (e. B A)) (= (U. (\ A B)) (U. A))) (A B ordelon B x cv onelsst syl A B ordelon B A B eldif biimpr exp B eloni B ordeirr syl syl5 A Ord adantl mpd jctild x cv A e. adantr y cv B x cv sseq2 A B \ rcla4ev syl6 x cv A B eldif biimpr x cv ssid jctir exp y cv x cv x cv sseq2 A B \ rcla4ev syl6 A Ord B A e. /\ adantl pm2.61d exp r19.21aiv x A y B unidif syl))
+thm (onint ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (e. (|^| A) A)) (A On z cv ssel x cv z cv ontri1 x cv z cv y cv ssel syl6bir exp sylan9 com4r imp31 r19.20dva A x cv z disj y visset A z elint2 3imtr4g A On x cv ssel imdistani sylan2 exp32 com4l imp32 ssrdv x cv A intss1 A On C_ A x cv i^i {/} = ad2antrl eqssd A eleq1d biimpd exp32 com34 pm2.43d r19.23adv ordon On A x tz7.5 mpan syl5 anabsi5))
+thm (onint0 () () (-> (C_ A (On)) (<-> (= (|^| A) ({/})) (e. ({/}) A))) (A onint 0ex A |^| {/} V eleq1 mpbiri A intex sylibr sylan2 A |^| {/} A eleq1 A On C_ adantl mpbid exp A int0el A On C_ a1i impbid))
+thm (onssmin ((x y) (A x) (A y)) () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (E.e. x A (A.e. y A (C_ (cv x) (cv y))))) (A onint y cv A intss1 rgen jctir x cv A |^| y cv sseq1 y A biraldv A rcla4ev syl))
+thm (onminsb ((x y) (ph y)) ((onminsb.1 (-> ps (A. x ps))) (onminsb.2 (-> (= (cv x) (|^| ({e.|} x (On) ph))) (<-> ph ps)))) (-> (E.e. x (On) ph) ps) (x On ph rabn0 x On ph ssrab x On ph {e.|} onint mpan sylbir y x On ph hbrab1 hbint y cv On e. x ax-17 onminsb.1 onminsb.2 elrabf pm3.27bd syl))
+thm (onminesb ((x y)) () (-> (E.e. x (On) ph) ([/] (|^| ({e.|} x (On) ph)) x ph)) (x On ph rabn0 x On ph ssrab x On ph {e.|} onint mpan sylbir y cv On e. x ax-17 x On ph {e.|} |^| ph elrabsf pm3.27bd syl))
+thm (onintss ((ps x) (A x)) ((onintss.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A (On)) (-> ps (C_ (|^| ({e.|} x (On) ph)) A))) (onintss.1 On elrab A x On ph {e.|} intss1 sylbir exp))
+thm (oninton () () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (e. (|^| A) (On))) (A onint exp A On A |^| ssel syld imp))
+thm (onintrab () () (<-> (e. (|^| ({e.|} x (On) ph)) (V)) (e. (|^| ({e.|} x (On) ph)) (On))) (x On ph {e.|} intex x On ph ssrab x On ph {e.|} oninton mpan sylbir x On ph {e.|} |^| On elisset impbi))
+thm (onintrab2 () () (<-> (E.e. x (On) ph) (e. (|^| ({e.|} x (On) ph)) (On))) (x On ph intexrab x ph onintrab bitr))
+thm (onnmin () () (-> (/\ (C_ A (On)) (e. B A)) (-. (e. B (|^| A)))) (B A intss1 A On C_ adantl A |^| B ontri1 A oninton B A n0i sylan2 A On B ssel2 sylanc mpbid))
+thm (onnminsb ((A x) (ps x)) ((onnminsb.1 (-> (= (cv x) A) (<-> ph ps)))) (-> (e. A (On)) (-> (e. A (|^| ({e.|} x (On) ph))) (-. ps))) (onnminsb.1 On elrab x On ph ssrab x On ph {e.|} A onnmin mpan sylbir exp con2d))
+thm (oneqmini ((A x) (B x)) () (-> (C_ B (On)) (-> (/\ (e. A B) (A.e. x A (-. (e. (cv x) B)))) (= A (|^| B)))) (B On A ssel B On x cv ssel anim12d A x cv ontri1 syl6 exp3a imp pm5.74d x cv B e. x cv A e. bi2.03 syl6bb biraldv2 A B x ssint syl5bb biimprd exp imp3a A B intss1 B On C_ a1i x A x cv B e. -. A.e. adantrd jcad A B |^| eqss syl6ibr))
+thm (oneqmin ((A x) (B x)) () (-> (/\ (C_ B (On)) (-. (= B ({/})))) (<-> (= A (|^| B)) (/\ (e. A B) (A.e. x A (-. (e. (cv x) B)))))) (A B |^| B eleq1 B onint syl5bir com12 A B |^| x cv eleq2 biimpd B x cv onnmin exp con2d syl9r r19.21adv B {/} = -. adantr jcad B A x oneqmini B {/} = -. adantr impbid))
+thm (bm2.5ii ((x y) (A x) (A y)) ((bm2.5ii.1 (e. A (V)))) (-> (C_ A (On)) (= (U. A) (|^| ({e.|} x (On) (A.e. y A (C_ (cv y) (cv x))))))) (bm2.5ii.1 onuni A U. On x intmin A x cv y unissb x cv On e. a1i birabi inteqi syl6eq syl))
+thm (onminex ((x y) (ph y) (ps x)) ((onminex.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (E.e. x (On) ph) (E.e. x (On) (/\ ph (A.e. y (cv x) (-. ps))))) (y x x cv On e. ph /\ hbab1 hbint y x x cv On e. ph /\ hbab1 hbint y x x cv On e. ph /\ hbab1 hbel y x x cv On e. ph /\ hbab1 hbint ps -. x ax-17 hbim y hbal hban x cv x x cv On e. ph /\ {|} |^| x x cv On e. ph /\ {|} eleq1 x cv x x cv On e. ph /\ {|} |^| y cv eleq2 ps -. imbi1d y bialdv anbi12d x x cv On e. ph /\ {|} cla4egf anabsi5 x On ph ssab x x cv On e. ph /\ {|} onint mpan x On ph ssab x x cv On e. ph /\ {|} oninton mpan x x cv On e. ph /\ {|} |^| y cv onelon exp syl y visset x cv y cv On eleq1 onminex.1 anbi12d elab x On ph ssab x x cv On e. ph /\ {|} y cv onnmin mpan sylbir exp con2d syli y 19.21aiv sylanc x x cv On e. ph /\ abn0 x x cv On e. ph /\ abid bicomi y x cv ps -. df-ral anbi12i x cv On e. ph y x cv ps -. A.e. anass bitr3 x biex 3imtr3 x On ph df-rex x On ph y x cv ps -. A.e. /\ df-rex 3imtr4))
+thm (ord0 () () (Ord ({/})) ({/} df-ord tr0 E we0 mpbir2an))
+thm (0elon () () (e. ({/}) (On)) (ord0 0ex elon mpbir))
+thm (ord0eln0 () () (-> (Ord A) (<-> (e. ({/}) A) (-. (= A ({/}))))) ({/} A n0i A Ord a1i ord0 A noel A {/} ordtri2 bicon2d mpbiri mpan2 ord impbid))
+thm (on0eln0 () () (-> (e. A (On)) (<-> (e. ({/}) A) (-. (= A ({/}))))) (A eloni A ord0eln0 syl))
+thm (dflim2 () () (<-> (Lim A) (/\/\ (Ord A) (e. ({/}) A) (= A (U. A)))) (A df-lim A ord0eln0 A A U. = anbi1d pm5.32i A Ord {/} A e. A A U. = 3anass A Ord A {/} = -. A A U. = 3anass 3bitr4 bitr4))
+thm (inton () () (= (|^| (On)) ({/})) (0elon On int0el ax-mp))
+thm (nlim0 () () (-. (Lim ({/}))) ({/} eqid {/} df-lim {/} Ord {/} {/} = -. {/} {/} U. = 3simp2 sylbi mt2))
+thm (limord () () (-> (Lim A) (Ord A)) (A df-lim A Ord A {/} = -. A A U. = 3simp1 sylbi))
+thm (limuni () () (-> (Lim A) (= A (U. A))) (A df-lim A Ord A {/} = -. A A U. = 3simp3 sylbi))
+thm (0ellim () () (-> (Lim A) (e. ({/}) A)) (nlim0 A {/} limeq mtbiri con2i A limord A ord0eln0 syl mpbird))
+thm (limelon () () (-> (/\ (e. A B) (Lim A)) (e. A (On))) (A B elong A limord syl5bir imp))
+thm (onne0 () () (-. (= (On) ({/}))) (0elon {/} On n0i ax-mp))
+thm (suceq () () (-> (= A B) (= (suc A) (suc B))) (A B sneq A uneq2d A B B {} uneq1 eqtrd A df-suc B df-suc 3eqtr4g))
+thm (elsuci () () (-> (e. A (suc B)) (\/ (e. A B) (= A B))) (B df-suc A eleq2i A B B {} elun bitr A B elsni A B e. orim2i sylbi))
+thm (elsucg () () (-> (e. A C) (<-> (e. A (suc B)) (\/ (e. A B) (= A B)))) (A C B elsncg A B e. orbi2d B df-suc A eleq2i A B B {} elun bitr syl5bb))
+thm (elsuc2g () () (-> (e. B C) (<-> (e. A (suc B)) (\/ (e. A B) (= A B)))) (B C A elsnc2g A B e. orbi2d A B B {} elun syl5bb B df-suc A eleq2i syl5bb))
+thm (elsuc () ((elsuc.1 (e. A (V)))) (<-> (e. A (suc B)) (\/ (e. A B) (= A B))) (elsuc.1 A V B elsucg ax-mp))
+thm (elsuc2 () ((elsuc.1 (e. A (V)))) (<-> (e. B (suc A)) (\/ (e. B A) (= B A))) (elsuc.1 A V B elsuc2g ax-mp))
+thm (hbsuc ((A y) (x y)) ((hbsuc.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (suc A)) (A. x (e. (cv y) (suc A)))) (hbsuc.1 hbsuc.1 hbeleq hbor y visset A elsuc y visset A elsuc x bial 3imtr4))
+thm (elelsuc () () (-> (e. A B) (e. A (suc B))) (A B e. A B = orc A B B elsucg mpbird))
+thm (sucel ((x y) (A x) (A y) (B x) (B y)) () (<-> (e. (suc A) B) (E.e. x B (A. y (<-> (e. (cv y) (cv x)) (\/ (e. (cv y) A) (= (cv y) A)))))) (A suc B x risset x cv A suc y dfcleq y visset A elsuc y cv x cv e. bibi2i y bial bitr x B birex bitr))
+thm (suc0 () () (= (suc ({/})) ({} ({/}))) ({/} df-suc {/} {/} {} uncom {/} {} un0 3eqtr))
+thm (sucprc () () (-> (-. (e. A (V))) (= (suc A) A)) (A snprc A {} {/} A uneq2 sylbi A df-suc syl5eq A un0 syl6eq))
+thm (sucon () () (= (suc (On)) (On)) (onprc On sucprc ax-mp))
+thm (unisuc () ((unisuc.1 (e. A (V)))) (<-> (Tr A) (= (U. (suc A)) A)) (A U. A ssequn1 A df-tr A df-suc unieqi A A {} uniun unisuc.1 unisn A U. uneq2i 3eqtr A eqeq1i 3bitr4))
+thm (sssucid () () (C_ A (suc A)) (A A {} ssun1 A df-suc sseqtr4))
+thm (sucexb () () (<-> (e. A (V)) (e. (suc A) (V))) (A A {} unexb A snex A V e. biantru A df-suc V eleq1i 3bitr4))
+thm (sucexg () () (-> (e. A B) (e. (suc A) (V))) (A B elisset A sucexb sylib))
+thm (sucex () ((sucex.1 (e. A (V)))) (e. (suc A) (V)) (sucex.1 A V sucexg ax-mp))
+thm (sucid () ((sucid.1 (e. A (V)))) (e. A (suc A)) (sucid.1 snid A df-suc A eleq2i A A A {} elun bitr biimpr olci ax-mp))
+thm (sucidg ((A x)) () (-> (e. A B) (e. A (suc A))) (x cv A = id x cv A suceq eleq12d x visset sucid B vtoclg))
+thm (nsuceq0 () () (-. (= (suc A) ({/}))) (A noel A suc {/} A eleq2 A V sucidg syl5bi com12 mtoi A sucprc {/} eqeq1d 0ex A {/} V eleq1 mpbiri syl6bi con3d pm2.43i pm2.61i))
+thm (eqelsuc () ((eqelsuc.1 (e. A (V)))) (-> (= A B) (e. A (suc B))) (eqelsuc.1 sucid A B suceq A eleq2d mpbii))
+thm (trsuc () () (-> (/\ (Tr A) (e. (suc B) A)) (e. B A)) (A B B suc trel exp3a B V sucidg syl5com B sucprc A eleq1d biimpd A Tr a1d pm2.61i imp))
+thm (trsucss () () (-> (Tr A) (-> (e. B (suc A)) (C_ B A))) (A B trss B A eqimss A Tr a1i jaod B A elsuci syl5))
+thm (ordsssuc () () (-> (/\ (e. A (On)) (Ord B)) (<-> (C_ A B) (e. A (suc B)))) (A B ordsseleq A eloni sylan A On B elsucg B Ord adantr bitr4d))
+thm (onsssuc () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (e. A (suc B)))) (A B ordsssuc B eloni sylan2))
+thm (ordsssuc2 () () (-> (/\ (Ord A) (e. B (On))) (<-> (C_ A B) (e. A (suc B)))) (B On A ssexg A V elong syl6 imp A B onsssuc biimpd exp com3l imp sylbird exp com3r imp A B suc elong A B onsssuc biimprd exp com3r sylbird com3l imp impbid))
+thm (onmindif ((A x) (B x)) () (-> (/\ (C_ A (On)) (e. B (On))) (e. B (|^| (\ A (suc B))))) (x cv B ontri1 x cv B onsssuc bitr3d bicon1d A On x cv ssel2 sylan biimpd exp31 com23 imp4b x cv A B suc eldif syl5ib r19.21aiv B On A B suc \ x elintg A On C_ adantl mpbird))
+thm (onmindif2 ((A x)) () (-> (/\ (C_ A (On)) (-. (= A ({/})))) (e. (|^| A) (|^| (\ A ({} (|^| A)))))) (A x cv onnmin A {/} = -. adantlr A |^| x cv ontri1 A |^| x cv onsseleq bitr3d A oninton x cv A e. adantr A On x cv ssel2 A {/} = -. adantlr sylanc mpbid ord x A |^| elsn x cv A |^| eqcom bitr syl6ibr con1d exp imp3a x cv A A |^| {} eldif syl5ib r19.21aiv A intex A |^| V A A |^| {} \ x elintg sylbi A On C_ adantl mpbird))
+thm (suceloni ((A x)) () (-> (e. A (On)) (e. (suc A) (On))) (ordon A suc On trssord 3exp mpii A x cv onelsst x A elsn x cv A eqimss sylbi A On e. a1i orim12d A df-suc x cv eleq2i x cv A A {} elun bitr2 x cv A C_ oridm 3imtr3g A sssucid x cv A A suc sstr2 mpi syl6 r19.21aiv A suc x dftr3 sylibr A onsst A On snssi jca A On A {} unss sylib A df-suc syl5ss sylc A On sucexg A suc V elong syl mpbird))
+thm (ordnbtwn () () (-> (Ord A) (-. (/\ (e. A B) (e. B (suc A))))) (A B ordn2lp A ordeirr jca A B e. B A e. /\ A A e. ioran sylibr B A elsuci A B e. anim2i A B e. B A e. B A = andi sylib B A A eleq2 biimpac A B e. B A e. /\ orim2i syl nsyl))
+thm (onnbtwn () () (-> (e. A (On)) (-. (/\ (e. A B) (e. B (suc A))))) (A eloni A B ordnbtwn syl))
+thm (ordsuc () () (<-> (Ord A) (Ord (suc A))) (A V elong A suceloni A suc eloni syl syl6bir A suc A ordelord exp A V sucidg syl5com impbid A sucprc eqcomd A A suc ordeq syl pm2.61i))
+thm (ordpwsuc ((A x)) () (-> (Ord A) (= (i^i (P~ A) (On)) (suc A))) (x cv A ordsssuc expcom pm5.32d x cv On e. x cv A suc e. pm3.27 A Ord a1i A ordsuc A suc x cv ordelon exp sylbi ancrd impbid bitrd x cv A P~ On elin x visset A elpw x cv On e. anbi1i x cv A C_ x cv On e. ancom 3bitr syl5bb eqrd))
+thm (onpwsuc () () (-> (e. A (On)) (= (i^i (P~ A) (On)) (suc A))) (A eloni A ordpwsuc syl))
+thm (sucelon () () (<-> (e. A (On)) (e. (suc A) (On))) (A ordsuc A sucexb anbi12i A elon2 A suc elon2 3bitr4))
+thm (ordsucss () () (-> (Ord B) (-> (e. A B) (C_ (suc A) B))) (A B ordnbtwn A B e. B A suc e. imnan sylibr B Ord adantr A suc B ordtri1 A ordsuc sylanb sylibrd B A ordelord sylan exp31 pm2.43b pm2.43b))
+thm (sucssel () () (-> (e. A C) (-> (C_ (suc A) B) (e. A B))) (A suc B A ssel A C sucidg syl5com))
+thm (ordelsuc () () (-> (/\ (e. A C) (Ord B)) (<-> (e. A B) (C_ (suc A) B))) (B A ordsucss A C e. adantl A C B sucssel B Ord adantr impbid))
+thm (onsucmin ((A x)) () (-> (e. A (On)) (= (suc A) (|^| ({e.|} x (On) (e. A (cv x)))))) (A sucelon A suc On x intmin sylbi A On x cv ordelsuc x cv eloni sylan2 exp birabdv inteqd eqtr4d))
+thm (ordsucelsuc () () (-> (Ord B) (<-> (e. A B) (e. (suc A) (suc B)))) (A suc B ordsseleq A ordsuc sylanb A V e. adantl B A ordsucss A V e. A Ord ad2antrr A V B sucssel A Ord B Ord /\ adantr impbid A sucexb A suc V B elsucg sylbi A Ord B Ord /\ adantr 3bitr4d exp A B elisset A suc B suc elisset A sucexb sylibr pm5.21ni A Ord B Ord /\ a1d pm2.61i biimpd B A ordelord sylan exp31 pm2.43a pm2.43d A suc B ordsseleq A ordsuc sylanb A V e. adantl B A ordsucss A V e. A Ord ad2antrr A V B sucssel A Ord B Ord /\ adantr impbid A sucexb A suc V B elsucg sylbi A Ord B Ord /\ adantr 3bitr4d exp A B elisset A suc B suc elisset A sucexb sylibr pm5.21ni A Ord B Ord /\ a1d pm2.61i biimprd B suc A suc ordelord A ordsuc sylibr B ordsuc sylanb sylan exp31 pm2.43a pm2.43d impbid))
+thm (ordsucsssuc () () (-> (/\ (Ord A) (Ord B)) (<-> (C_ A B) (C_ (suc A) (suc B)))) (A B ordsucelsuc negbid B Ord adantr A B ordtri1 A suc B suc ordtri1 A ordsuc B ordsuc syl2anb 3bitr4d))
+thm (orddif () () (-> (Ord A) (= A (\ (suc A) ({} A)))) (A orddisj A A {} disj3 A df-suc A {} difeq1i A A {} difun2 eqtr A eqeq2i bitr4 sylib))
+thm (orduniss () () (-> (Ord A) (C_ (U. A) A)) (A ordtr A df-tr sylib))
+thm (ordtri2or () () (-> (/\ (Ord A) (Ord B)) (\/ (e. A B) (C_ B A))) (A B ordtri3or A B e. A B = B A e. 3orass sylib B A ordsseleq ancoms B A e. B A = orcom B A eqcom B A e. orbi1i bitr syl6bb A B e. orbi2d mpbird))
+thm (ordtri2or2 () () (-> (/\ (Ord A) (Ord B)) (\/ (C_ A B) (C_ B A))) (A B ordtri2or B A ordelss exp B A C_ orim1d A Ord adantl mpd))
+thm (ordssun () () (-> (/\ (Ord B) (Ord C)) (<-> (C_ A (u. B C)) (\/ (C_ A B) (C_ A C)))) (B C ordtri2or2 B C ssequn1 B C u. C A sseq2 sylbi A C C_ A B C_ olc syl6bi C B ssequn2 B C u. B A sseq2 sylbi A B C_ A C C_ orc syl6bi jaoi syl A B C ssun B Ord C Ord /\ a1i impbid))
+thm (ordequn () () (-> (/\ (Ord B) (Ord C)) (-> (= A (u. B C)) (\/ (= A B) (= A C)))) (B C ordtri2or2 B C ssequn1 B C u. C A eqeq2 sylbi A C = A B = olc syl6bi C B ssequn2 B C u. B A eqeq2 sylbi A B = A C = orc syl6bi jaoi syl))
+thm (ordun () () (-> (/\ (Ord A) (Ord B)) (Ord (u. A B))) (A B u. eqid A B A B u. ordequn mpi A B u. A ordeq biimprcd A B u. B ordeq biimprcd jaao mpd))
+thm (ordsucun ((A x) (B x)) () (-> (/\ (Ord A) (Ord B)) (= (suc (u. A B)) (u. (suc A) (suc B)))) (A B x cv ordssun x cv On e. adantl x cv A B u. ordsssuc A B ordun sylan2 x cv A ordsssuc B Ord adantrr x cv B ordsssuc A Ord adantrl orbi12d 3bitr3d x cv A suc B suc elun syl6bbr expcom pm5.32d A B ordun A B u. ordsuc A B u. suc x cv ordelon exp sylbi syl pm4.71rd A suc B suc ordun A suc B suc u. x cv ordelon exp syl A ordsuc B ordsuc syl2anb pm4.71rd 3bitr4d eqrd))
+thm (ordunisssuc ((A x) (B x)) () (-> (/\ (C_ A (On)) (Ord B)) (<-> (C_ (U. A) B) (C_ A (suc B)))) (x cv B ordsssuc A On x cv ssel2 sylan an1rs biraldva A B x unissb A B suc x dfss3 3bitr4g))
+thm (onsucuni () () (-> (C_ A (On)) (C_ A (suc (U. A)))) (A ssorduni A U. ssid A A U. ordunisssuc mpbii mpdan))
+thm (ordsucuni () () (-> (Ord A) (C_ A (suc (U. A)))) (A ordsson A onsucuni syl))
+thm (orduniorsuc () () (-> (Ord A) (\/ (= A (U. A)) (= A (suc (U. A))))) (A orduniss A uniord A U. A ordelssne mpancom biimprd mpand A A U. ordsucss syld A ordsucuni jctild A A U. eqcom negbii A A U. suc eqss 3imtr4g orrd))
+thm (unon ((x y)) () (= (U. (On)) (On)) (x cv On y eluni2 y cv x cv onelon exp r19.23aiv sylbi x cv suceloni x visset sucid jctil x cv x cv suc On elunii syl impbi eqri))
+thm (ordunisuc ((A x)) () (-> (Ord A) (= (U. (suc A)) A)) (A ordeleqon x cv A suceq unieqd x cv A = id eqeq12d x cv eloni x cv ordtr syl x visset unisuc sylib vtoclga onprc A On V eleq1 mtbiri A sucprc syl unieqd unon A On unieq A On = id eqeq12d mpbiri eqtrd jaoi sylbi))
+thm (orduniss2 ((A x)) () (-> (Ord A) (= (U. ({e.|} x (On) (C_ (cv x) A))) A)) (A ordpwsuc x On x cv A C_ df-rab x x cv On e. x cv A C_ inab x x cv On e. {|} x x cv A C_ {|} incom eqtr3 A x df-pw eqcomi x On abid2 ineq12i 3eqtr syl5eq unieqd A ordunisuc eqtrd))
+thm (onsucuni2 () () (-> (/\ (e. A (On)) (= A (suc B))) (= A (suc (U. A)))) (A B suc On eleq1 B sucelon syl6bbr biimpac B eloni B ordeirr syl B suc B B eleq2 B On sucidg syl5bi com12 mtod B eloni B ordunisuc syl B suc eqeq2d mtbird A B suc = adantl A B suc = id A B suc unieq eqeq12d B On e. adantr mtbird exp A On e. adantl mpd A eloni A orduniorsuc ord syl A B suc = adantr mpd))
+thm (0elsuc () () (-> (Ord A) (e. ({/}) (suc A))) (A ordsuc A nsuceq0 A suc ord0eln0 mpbiri sylbi))
+thm (suc11 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (suc A) (suc B)) (= A B))) (A eloni A B ordn2lp A B e. B A e. ianor sylib syl B On e. adantr A On B suc sucssel A suc B suc eqimss syl5 A B elsuci ord com12 syl9 B On A suc sucssel A suc B suc eqimss2 syl5 B A elsuci ord com12 B A eqcom syl6ib syl9 jaao mpd A B suceq A On e. B On e. /\ a1i impbid))
+thm (limon () () (Lim (On)) (ordon onne0 unon eqcomi 3pm3.2i On df-lim mpbir))
+thm (onord () ((on.1 (e. A (On)))) (Ord A) (on.1 A eloni ax-mp))
+thm (ontrc () ((on.1 (e. A (On)))) (Tr A) (on.1 onord A ordtr ax-mp))
+thm (oneirr () ((on.1 (e. A (On)))) (-. (e. A A)) (on.1 onord A ordeirr ax-mp))
+thm (onel () ((on.1 (e. A (On)))) (-> (e. B A) (e. B (On))) (on.1 A B onelon mpan))
+thm (onss () ((on.1 (e. A (On)))) (C_ A (On)) (on.1 A onsst ax-mp))
+thm (onelss () ((on.1 (e. A (On)))) (-> (e. B A) (C_ B A)) (on.1 A B onelsst ax-mp))
+thm (onssneli () ((on.1 (e. A (On)))) (-> (C_ A B) (-. (e. B A))) (on.1 B onel B eloni B ordeirr 3syl A B B ssel com12 mtod con2i))
+thm (onssneli2 () ((on.1 (e. A (On)))) (-> (C_ B A) (-. (e. A B))) (on.1 oneirr B A A ssel mtoi))
+thm (onelin () ((on.1 (e. A (On)))) (-> (e. B A) (= B (i^i B A))) (on.1 B onelss B A dfss sylib))
+thm (onelun () ((on.1 (e. A (On)))) (-> (e. B A) (= (u. A B) A)) (on.1 B onelss B A ssequn2 sylib))
+thm (onsuc () ((on.1 (e. A (On)))) (e. (suc A) (On)) (on.1 A suceloni ax-mp))
+thm (onunisuc () ((on.1 (e. A (On)))) (= (U. (suc A)) A) (on.1 ontrc on.1 elisseti unisuc mpbi))
+thm (onuniorsuc () ((on.1 (e. A (On)))) (\/ (= A (U. A)) (= A (suc (U. A)))) (on.1 onord A orduniorsuc ax-mp))
+thm (onuninsuc ((A x)) ((on.1 (e. A (On)))) (<-> (= A (U. A)) (-. (E.e. x (On) (= A (suc (cv x)))))) (on.1 oneirr A A U. = id x cv df-suc A eqeq2i A x cv x cv {} u. unieq sylbi x cv x cv {} uniun x visset unisn x cv U. uneq2i eqtr syl6eq on.1 A x cv suc On eleq1 mpbii ordon On ordtr ax-mp On x cv trsuc mpan x cv eloni x cv ordtr syl x cv df-tr sylib 3syl x cv U. x cv ssequn1 sylib eqtrd sylan9eqr x visset sucid A x cv suc x cv eleq2 mpbiri A A U. = adantr eqeltrd mto A x cv suc = A A U. = imnan mpbir x cv On e. a1i r19.23aiv on.1 onuniorsuc ori on.1 onss on.1 elisseti onuni ax-mp jctil x cv A U. suceq A eqeq2d On rcla4ev syl impbi bicon2i))
+thm (onssel () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (<-> (C_ A B) (\/ (e. A B) (= A B))) (on.1 on.2 A B onsseleq mp2an))
+thm (onun () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (e. (u. A B) (On)) (on.2 onord on.1 onord B A ordtri2or mp2an on.1 B onelun on.1 A B u. A On eleq1 mpbiri syl A B ssequn1 on.2 A B u. B On eleq1 mpbiri sylbi jaoi ax-mp))
+thm (onsucss () ((on.1 (e. A (On))) (on.2 (e. B (On)))) (<-> (e. A B) (C_ (suc A) B)) (on.1 on.2 onord A On B ordelsuc mp2an))
+thm (nlimsuc () ((nlimsuc.1 (e. A (V)))) (-. (Lim (suc A))) (A ordsuc A suc ordeirr sylbi A suc A suc U. A suc eleq1 nlimsuc.1 sucid A ordtr nlimsuc.1 unisuc sylib A suc eleq1d mpbiri syl5bir com12 mtod A suc limuni nsyl A suc Ord A suc {/} = -. A suc A suc U. = 3simp1 A suc df-lim A ordsuc 3imtr4 con3i pm2.61i))
+thm (unizlim () () (-> (Ord A) (<-> (= A (U. A)) (\/ (= A ({/})) (Lim A)))) (A df-lim biimpr 3exp com23 imp orrd exp uni0 eqcomi A {/} = id A {/} unieq eqeq12d mpbiri A limuni jaoi A Ord a1i impbid))
+thm (orduninsuc ((A x)) () (-> (Ord A) (<-> (= A (U. A)) (-. (E.e. x (On) (= A (suc (cv x))))))) (A ordeleqon A A On e. A {/} if = id A A On e. A {/} if unieq eqeq12d A A On e. A {/} if x cv suc eqeq1 x On birexdv negbid bibi12d 0elon A elimel x onuninsuc dedth unon eqcomi onprc x visset sucex On x cv suc V eleq1 mpbiri mto x cv On e. a1i nrex 2th A On = id A On unieq eqeq12d A On x cv suc eqeq1 x On birexdv negbid bibi12d mpbiri jaoi sylbi))
+thm (ordunisuc2 ((A x)) () (-> (Ord A) (<-> (= A (U. A)) (A.e. x A (e. (suc (cv x)) A)))) (A x orduninsuc A x cv suc ordtri3 x cv suceloni x cv suc eloni syl sylan2 bicon2d x cv A onnbtwn x cv A e. A x cv suc e. imnan sylibr con2d x cv A e. x cv suc A e. pm2.21 syl6 A Ord adantl x cv suc A e. x cv A e. ax-1 A Ord x cv On e. /\ a1i jaod x cv A ordtri2or x cv eloni sylan ancoms x cv A e. A x cv C_ orcom sylib x cv A e. x cv suc A e. -> adantr A x cv ordsssuc2 biimpd x cv A e. x cv suc A e. -> adantr A Ord x cv On e. /\ x cv A e. x cv suc A e. -> pm3.27 orim12d mpd exp impbid bitr3d exp pm5.74d x cv On e. x cv A e. pm3.27 A Ord a1i A x cv ordelon exp ancrd impbid x cv suc A e. imbi1d x cv On e. x cv A e. x cv suc A e. impexp syl5bbr bitrd biraldv2 x On A x cv suc = ralnex syl5bbr bitrd))
+thm (ordzsl ((A x)) () (<-> (Ord A) (\/\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (Lim A))) (A x orduninsuc biimprd A unizlim sylibd orrd A {/} = x On A x cv suc = E.e. A Lim 3orass A {/} = x On A x cv suc = E.e. A Lim or12 bitr sylibr ord0 A {/} ordeq mpbiri A x cv suc On eleq1 x cv suceloni syl5bir A eloni syl6com r19.23aiv A limord 3jaoi impbi))
+thm (onzsl ((A x)) () (<-> (e. A (On)) (\/\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))) (/\ (e. A (V)) (Lim A)))) (A On elisset pm4.71ri A V elong pm5.32i A V e. A {/} = x On A x cv suc = E.e. \/ A Lim andi 0ex A {/} V eleq1 mpbiri x visset sucex A x cv suc V eleq1 mpbiri x cv On e. a1i r19.23aiv jaoi pm4.71ri A V e. A Lim /\ orbi1i bitr4 A x ordzsl A {/} = x On A x cv suc = E.e. A Lim df-3or bitr A V e. anbi2i A {/} = x On A x cv suc = E.e. A V e. A Lim /\ df-3or 3bitr4 3bitr))
+thm (dflim3 ((A x)) () (<-> (Lim A) (/\ (Ord A) (-. (\/ (= A ({/})) (E.e. x (On) (= A (suc (cv x)))))))) (A limord nlim0 A {/} limeq mtbiri con2i A limuni A limord A x orduninsuc syl mpbid jca A {/} = x On A x cv suc = E.e. ioran sylibr jca A x ordzsl biimp A {/} = x On A x cv suc = E.e. A Lim df-3or sylib orcanai impbi))
+thm (dflim4 ((A x)) () (<-> (Lim A) (/\/\ (Ord A) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A)))) (A dflim2 A x ordunisuc2 {/} A e. anbi2d pm5.32i A Ord {/} A e. A A U. = 3anass A Ord {/} A e. x A x cv suc A e. A.e. 3anass 3bitr4 bitr))
+thm (limsuc ((A x) (B x)) () (-> (Lim A) (<-> (e. B A) (e. (suc B) A))) (A x dflim4 x cv B suceq A eleq1d A rcla4v com12 A Ord {/} A e. 3ad2ant3 sylbi A limord A ordtr A B trsuc exp 3syl impbid))
+thm (limsssuc ((A x) (B x)) () (-> (Lim A) (<-> (C_ A B) (C_ A (suc B)))) (B sssucid A B B suc sstr2 mpi A Lim a1i x cv B A eleq1 biimpcd A B limsuc biimpa A B suc ordtri1 A limord B A e. adantr A B ordelord A limord sylan B ordsuc sylib sylanc bicon2d mpbid exp sylan9r con2d exp com23 imp31 A B suc x cv ssel2 x visset B elsuc sylib ord con1d A Lim adantll mpd exp ssrdv exp impbid))
+thm (nlimon ((x y)) () (= ({e.|} x (On) (\/ (= (cv x) ({/})) (E.e. y (On) (= (cv x) (suc (cv y)))))) ({e.|} x (On) (-. (Lim (cv x))))) (x cv eloni x cv y dflim3 baib bicon2d syl birabi))
+thm (limuni2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (-. (= A ({/}))) (A.e. x A (Lim (cv x)))) (Lim (U. A))) (x cv z cv limeq A rcla4v z visset z cv V limelon mpan syl6com ssrdv A ssorduni syl A {/} = -. adantl A z n0 x cv z cv limeq A rcla4v {/} z cv A elunii expcom z cv 0ellim syl5 syld z 19.23aiv sylbi imp x cv z cv limeq A rcla4v com12 z cv y cv limsuc z cv A e. anbi1d y cv suc z cv A elunii syl6bi exp3a com3r sylcom r19.23adv y cv A z eluni2 syl5ib r19.21aiv A {/} = -. adantl 3jca A U. y dflim4 sylibr))
+thm (on0eqelt () () (-> (e. A (On)) (\/ (= A ({/})) (e. ({/}) A))) (A 0ss 0elon {/} A onsseleq mpan mpbii {/} A eqcom {/} A e. orbi2i {/} A e. A {/} = orcom bitr sylib))
+thm (snsn0non () () (-. (e. ({} ({} ({/}))) (On))) (0ex snnz 0ex {/} {} elsnc {/} {/} {} eqcom bitr mtbir 0ex snid {/} {} {/} {} {} {/} ssel mpi mto p0ex snid {/} {} {} {/} {} onelsst mpi mto))
+thm (dfom2 ((x y) (x z) (y z)) () (= (om) ({e.|} x (On) (C_ (suc (cv x)) ({e.|} y (On) (-. (Lim (cv y))))))) (x visset elon z z cv Lim x cv z cv e. -> A. anbi1i z cv x cv onsssuc z cv x cv ontri1 bitr3d ancoms y cv z cv limeq negbid On elrab x cv On e. z cv On e. /\ a1i imbi12d exp pm5.74d z visset z cv V limelon mpan pm4.71ri x cv z cv e. imbi1i z cv On e. z cv Lim x cv z cv e. impexp z cv On e. z cv Lim -. ibar x cv z cv e. -. imbi2d z cv Lim x cv z cv e. pm4.1 syl5bb pm5.74i 3bitr syl6rbbr z cv On e. z cv x cv suc e. pm3.27 x cv On e. a1i x cv suceloni x cv suc z cv onelon exp syl ancrd impbid z cv y On y cv Lim -. {e.|} e. imbi1d z cv On e. z cv x cv suc e. z cv y On y cv Lim -. {e.|} e. impexp syl5bbr bitrd z bialdv x cv suc y On y cv Lim -. {e.|} z dfss2 syl6bbr pm5.32i bitr3 x biabi x z df-om x On x cv suc y On y cv Lim -. {e.|} C_ df-rab 3eqtr4))
+thm (elom ((A x) (A y) (x y)) ((elom.1 (e. A (V)))) (<-> (e. A (om)) (/\ (Ord A) (A. x (-> (Lim (cv x)) (e. A (cv x)))))) (elom.1 y cv A ordeq y cv A x cv eleq1 x cv Lim imbi2d x bialdv anbi12d y x df-om elab2))
+thm (elomg ((A x) (A y) (x y)) () (-> (e. A B) (<-> (e. A (om)) (/\ (Ord A) (A. x (-> (Lim (cv x)) (e. A (cv x))))))) (y cv A om eleq1 y cv A ordeq y cv A x cv eleq1 x cv Lim imbi2d x bialdv anbi12d y visset x elom B vtoclbg))
+thm (omsson ((x y)) () (C_ (om) (On)) (x visset y elom pm3.26bd x visset elon sylibr ssriv))
+thm (limomss ((x y) (A x) (A y)) () (-> (Lim A) (C_ (om) A)) (A limord A ordeleqon y cv A limeq y cv A x cv eleq2 imbi12d On cla4gv x visset y elom pm3.27bd syl5 com23 imp ssrdv exp omsson A On om sseq2 mpbiri A Lim a1d jaoi sylbi mpcom))
+thm (nnont () () (-> (e. A (om)) (e. A (On))) (omsson A sseli))
+thm (nnon () ((nnon.1 (e. A (om)))) (e. A (On)) (nnon.1 A nnont ax-mp))
+thm (nnord () () (-> (e. A (om)) (Ord A)) (A nnont A eloni syl))
+thm (ordom ((x y) (x z) (y z)) () (Ord (om)) (om y x dftr2 x cv y cv ordelord x cv nnord sylan ancoms z cv y cv x cv trel exp3a com12 z cv limord z cv ordtr syl syl5 a2d z 19.20dv x visset z elom pm3.27bd syl5 imp jca y visset z elom sylibr x ax-gen mpgbir omsson ordon om On trssord mp3an))
+thm (elnn () () (-> (/\ (e. A B) (e. B (om))) (e. A (om))) (ordom om ordtr ax-mp om A B trel ax-mp))
+thm (omon () () (\/ (e. (om) (On)) (= (om) (On))) (ordom om ordeleqon mpbi))
+thm (nnlim ((A x)) () (-> (e. A (om)) (-. (Lim A))) (A nnord A ordeirr syl A om x elomg ibi pm3.27d x cv A limeq x cv A A eleq2 imbi12d om cla4gv mpd mtod))
+thm (omssnlim ((x y)) () (C_ (om) ({e.|} x (On) (-. (Lim (cv x))))) (y cv nnont y cv nnlim jca x cv y cv limeq negbid On elrab sylibr ssriv))
+thm (limom () () (Lim (om)) (ordom om ordeleqon ordom om ordeirr ax-mp om On x elomg ordom x cv om ordtri1 om Lim -. adantr x cv om ordsseleq biimpd x cv nnlim om Lim -. a1i x cv om limeq biimpd con3d com12 jaod sylan9 sylbird a3d mpanl2 x cv limord sylan exp pm2.43b x 19.21aiv ordom jctil syl5bir mt3i limon om On limeq mpbiri jaoi sylbi ax-mp))
+thm (peano2b () () (<-> (e. A (om)) (e. (suc A) (om))) (limom om A limsuc ax-mp))
+thm (nnsuc ((A x)) () (-> (/\ (e. A (om)) (-. (= A ({/})))) (E.e. x (om) (= A (suc (cv x))))) (A nnlim A {/} = -. adantr A x orduninsuc A {/} = -. adantr A df-lim biimpr 3exp imp sylbird A nnord sylan mt3d A x cv suc om eleq1 biimpcd x cv peano2b syl6ibr ancrd x cv On e. adantld x 19.22dv x On A x cv suc = df-rex x om A x cv suc = df-rex 3imtr4g A {/} = -. adantr mpd))
+thm (peano1 () () (e. ({/}) (om)) (limom om 0ellim ax-mp))
+thm (peano2 () () (-> (e. A (om)) (e. (suc A) (om))) (A peano2b biimp))
+thm (peano3 () () (-> (e. A (om)) (-. (= (suc A) ({/})))) (A nsuceq0 A om e. a1i))
+thm (peano4 () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (= (suc A) (suc B)) (= A B))) (A B suc11 A nnont B nnont syl2an))
+thm (peano5 ((x y) (A x) (A y)) () (-> (/\ (e. ({/}) A) (A.e. x (om) (-> (e. (cv x) A) (e. (suc (cv x)) A)))) (C_ (om) A)) (y cv om A eldifn {/} A e. x om x cv A e. x cv suc A e. -> A.e. /\ adantl y cv x nnsuc y cv om A eldifi {/} A e. adantl y cv {/} om A \ eleq1 biimpcd con3d {/} A om elndif syl5com imp sylanc x om x cv A e. x cv suc A e. -> A.e. adantlr om A \ y cv i^i {/} = adantr x om x cv A e. x cv suc A e. -> hbra1 y cv om A \ e. om A \ y cv i^i {/} = /\ x ax-17 hban y cv A e. x ax-17 x om x cv A e. x cv suc A e. -> ra4 x visset sucid y cv x cv suc x cv eleq2 mpbiri y cv x cv suc om eleq1 x cv peano2b syl6bbr x cv om A neldif x cv y cv om A \ minel sylan2 exp32 syl6bi mpid y cv om A eldifi syl5 imp3a x cv suc A y cv eleq1a com12 imim12d com13 sylan9 r19.23ad exp32 {/} A e. a1i imp41 mpd exp mtod nrexdv om A difss ordom om om A \ y tz7.5 mpan mpan nsyl2 om A ssdif0 sylibr))
+thm (nn0suc ((A x)) () (-> (e. A (om)) (\/ (= A ({/})) (E.e. x (om) (= A (suc (cv x)))))) (A x nnsuc exp orrd))
+thm (find ((A x)) ((find.1 (/\/\ (C_ A (om)) (e. ({/}) A) (A.e. x A (e. (suc (cv x)) A))))) (= A (om)) (find.1 3simp1i find.1 x cv suc A e. x cv om e. ax-1 x A r19.20si x A om x cv suc A e. ralcom3 sylib {/} A e. anim2i A om C_ anim2i A om C_ {/} A e. x A x cv suc A e. A.e. 3anass A om C_ {/} A e. x om x cv A e. x cv suc A e. -> A.e. 3anass 3imtr4 ax-mp A x peano5 A om C_ 3adant1 ax-mp eqssi))
+thm (finds ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((finds.1 (-> (= (cv x) ({/})) (<-> ph ps))) (finds.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (finds.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (finds.4 (-> (= (cv x) A) (<-> ph ta))) (finds.5 ps) (finds.6 (-> (e. (cv y) (om)) (-> ch th)))) (-> (e. A (om)) ta) (finds.5 0ex finds.1 elab mpbir finds.6 y visset finds.2 elab y visset sucex finds.3 elab 3imtr4g rgen x ph {|} y peano5 mp2an A sseli finds.4 om elabg mpbid))
+thm (findsg ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((findsg.1 (-> (= (cv x) B) (<-> ph ps))) (findsg.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (findsg.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (findsg.4 (-> (= (cv x) A) (<-> ph ta))) (findsg.5 (-> (e. B (om)) ps)) (findsg.6 (-> (/\ (/\ (e. (cv y) (om)) (e. B (om))) (C_ B (cv y))) (-> ch th)))) (-> (/\ (/\ (e. A (om)) (e. B (om))) (C_ B A)) ta) (x cv {/} B sseq2 B {/} = adantl B {/} x cv eqeq2 findsg.1 syl6bir imp imbi12d x cv {/} B sseq2 ph imbi1d B ss0 con3i ph ps <-> pm2.21d pm5.74d sylan9bbr pm2.61an1 B om e. imbi2d x cv y cv B sseq2 findsg.2 imbi12d B om e. imbi2d x cv y cv suc B sseq2 findsg.3 imbi12d B om e. imbi2d x cv A B sseq2 findsg.4 imbi12d B om e. imbi2d findsg.5 B {/} C_ a1d y visset sucex B x eqvinc findsg.1 findsg.5 syl5bir findsg.3 biimpd sylan9r x 19.23aiv sylbi eqcoms B y cv suc C_ imim2i B y cv C_ ch -> a1d com4r y cv om e. adantl B y cv onsssuc B y cv suc onelpsst y cv suceloni sylan2 bitrd B nnont y cv nnont syl2an ancoms findsg.6 exp th B y cv suc C_ ax-1 syl8 a2d com23 sylbird B y cv suc C_ B y cv suc = annim syl5ibr pm2.61d exp a2d finds imp31))
+thm (finds2 ((x y) (ta x) (ta y) (ps x) (ch x) (th x) (ph y)) ((finds2.1 (-> (= (cv x) ({/})) (<-> ph ps))) (finds2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (finds2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (finds2.4 (-> ta ps)) (finds2.5 (-> (e. (cv y) (om)) (-> ta (-> ch th))))) (-> (e. (cv x) (om)) (-> ta ph)) (finds2.4 0ex finds2.1 ta imbi2d elab mpbir finds2.5 a2d y visset finds2.2 ta imbi2d elab y visset sucex finds2.3 ta imbi2d elab 3imtr4g rgen x ta ph -> {|} y peano5 mp2an x cv sseli x ta ph -> abid sylib))
+thm (findes ((x y) (x z) (y z) (ph y) (ph z)) ((findes.1 ([/] ({/}) x ph)) (findes.2 (-> (e. (cv x) (om)) (-> ph ([/] (suc (cv x)) x ph))))) (-> (e. (cv x) (om)) ph) (z cv {/} x ph dfsbcq z y x ph sbequ z cv y cv suc x ph dfsbcq z x ph sbequ12r findes.1 y cv om e. x ax-17 y x ph hbs1 y visset sucex x ph hbsbcv hbim hbim x cv y cv om eleq1 x y ph sbequ12 x cv y cv suceq x cv suc y cv suc x ph dfsbcq syl imbi12d imbi12d findes.2 chvar finds))
+thm (tfinds ((x y) (x z) (y z) (A x) (ps x) (ch x) (ch z) (th x) (ta x) (ph y) (ph z)) ((tfinds.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds.4 (-> (= (cv x) A) (<-> ph ta))) (tfinds.5 ps) (tfinds.6 (-> (e. (cv y) (On)) (-> ch th))) (tfinds.7 (-> (Lim (cv x)) (-> (A.e. y (cv x) ch) ph)))) (-> (e. A (On)) ta) (tfinds.2 tfinds.4 x cv eloni x cv df-lim biimpr 3exp x cv x cv U. = x cv Lim con3 syl6 com23 x cv y orduninsuc biimprd con1d syl6d x cv {/} = y On x cv y cv suc = E.e. df-or syl6ibr syl tfinds.5 tfinds.1 mpbiri y x cv ch A.e. a1d x cv On e. a1d x cv On e. y ax-17 y x cv ch hbra1 ph y ax-17 hbim hbim x cv y cv suc z z cv x ph [/] raleq y z x ph sbequ ch x ax-17 tfinds.2 sbie syl5bbr x cv cbvralv ph z ax-17 z x ph hbs1 x z ph sbequ12 y cv suc cbvral 3bitr4g biimpd tfinds.6 y visset sucid tfinds.2 y cv suc rcla4v ax-mp syl5 sylan9r tfinds.3 y cv On e. adantl sylibrd x cv On e. a1d exp r19.23ai jaoi syl6 pm2.43a tfinds.7 pm2.61d2 tfis3))
+thm (tfindsg ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((tfindsg.1 (-> (= (cv x) B) (<-> ph ps))) (tfindsg.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfindsg.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfindsg.4 (-> (= (cv x) A) (<-> ph ta))) (tfindsg.5 (-> (e. B (On)) ps)) (tfindsg.6 (-> (/\ (/\ (e. (cv y) (On)) (e. B (On))) (C_ B (cv y))) (-> ch th))) (tfindsg.7 (-> (/\ (/\ (Lim (cv x)) (e. B (On))) (C_ B (cv x))) (-> (A.e. y (cv x) (-> (C_ B (cv y)) ch)) ph)))) (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ B A)) ta) (x cv {/} B sseq2 B {/} = adantl B {/} x cv eqeq2 tfindsg.1 syl6bir imp imbi12d x cv {/} B sseq2 ph imbi1d B ss0 con3i ph ps <-> pm2.21d pm5.74d sylan9bbr pm2.61an1 B On e. imbi2d x cv y cv B sseq2 tfindsg.2 imbi12d B On e. imbi2d x cv y cv suc B sseq2 tfindsg.3 imbi12d B On e. imbi2d x cv A B sseq2 tfindsg.4 imbi12d B On e. imbi2d tfindsg.5 B {/} C_ a1d y visset sucex B x eqvinc tfindsg.1 tfindsg.5 syl5bir tfindsg.3 biimpd sylan9r x 19.23aiv sylbi eqcoms B y cv suc C_ imim2i B y cv C_ ch -> a1d com4r y cv On e. adantl B y cv onsssuc B y cv suc onelpsst y cv suceloni sylan2 bitrd ancoms tfindsg.6 exp th B y cv suc C_ ax-1 syl8 a2d com23 sylbird B y cv suc C_ B y cv suc = annim syl5ibr pm2.61d exp a2d B On e. B y cv C_ ch -> pm2.27 y x cv r19.20sdv x cv Lim B x cv C_ ad2antlr tfindsg.7 syld exp31 com3l com4t tfinds imp31))
+thm (tfindsg2 ((A x) (x y) (B x) (B y) (ps x) (ch x) (th x) (ta x) (ph y)) ((tfindsg2.1 (-> (= (cv x) (suc B)) (<-> ph ps))) (tfindsg2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfindsg2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfindsg2.4 (-> (= (cv x) A) (<-> ph ta))) (tfindsg2.5 (-> (e. B (On)) ps)) (tfindsg2.6 (-> (/\ (e. (cv y) (On)) (e. B (cv y))) (-> ch th))) (tfindsg2.7 (-> (/\ (Lim (cv x)) (e. B (cv x))) (-> (A.e. y (cv x) (-> (e. B (cv y)) ch)) ph)))) (-> (/\ (e. A (On)) (e. B A)) ta) (A B onelon B sucelon sylib A eloni A B ordsucss syl imp tfindsg2.1 tfindsg2.2 tfindsg2.3 tfindsg2.4 B sucelon tfindsg2.5 sylbir B On y cv ordelsuc y cv eloni sylan2 ancoms tfindsg2.6 exp B On e. adantr sylbird B sucelon sylan2br imp tfindsg2.7 exp B On e. adantr B On x cv ordelsuc x cv eloni sylan2 B On y cv ordelsuc x cv y cv onelon y cv eloni syl sylan2 anassrs ch imbi1d biraldva ph imbi1d imbi12d x visset x cv V limelon mpan sylan2 ancoms mpbid B sucelon sylan2br imp tfindsg exp31 imp3a B A e. adantr mp2and))
+thm (tfindes ((x y) (x z) (y z) (ph y) (ph z)) ((tfindes.1 ([/] ({/}) x ph)) (tfindes.2 (-> (e. (cv x) (On)) (-> ph ([/] (suc (cv x)) x ph)))) (tfindes.3 (-> (Lim (cv y)) (-> (A.e. x (cv y) ph) ([/] (cv y) x ph))))) (-> (e. (cv x) (On)) ph) (y cv {/} x ph dfsbcq y z x ph sbequ y cv z cv suc x ph dfsbcq y x ph sbequ12r tfindes.1 z cv On e. x ax-17 z x ph hbs1 z visset sucex x ph hbsbcv hbim hbim x cv z cv On eleq1 x z ph sbequ12 x cv z cv suceq x cv suc z cv suc x ph dfsbcq syl imbi12d imbi12d tfindes.2 chvar tfindes.3 ph z ax-17 z x ph hbs1 x z ph sbequ12 y cv cbvral syl5ibr tfinds))
+thm (tfinds2 ((x y) (ta x) (ta y) (ps x) (ch x) (th x) (ph y)) ((tfinds2.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds2.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds2.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds2.4 (-> ta ps)) (tfinds2.5 (-> (e. (cv y) (On)) (-> ta (-> ch th)))) (tfinds2.6 (-> (Lim (cv x)) (-> ta (-> (A.e. y (cv x) ch) ph))))) (-> (e. (cv x) (On)) (-> ta ph)) (tfinds2.4 0ex tfinds2.1 ta imbi2d sbcie mpbir tfinds2.5 a2d x y sbimi x visset x cv V y On sbcel1 ax-mp x y ta ch -> ta th -> sbim 3imtr3 x visset tfinds2.2 bicomd equcoms ta imbi2d sbcie y visset sucex tfinds2.3 ta imbi2d sbcie x y bisb x cv y cv suceq ta ph -> sbcco2 bitr3 3imtr3g tfinds2.6 a2d y x cv ta ch r19.21v syl5ib y x sbimi y cv Lim x ax-17 x cv y cv limeq sbie y x y x cv ta ch -> A.e. ta ph -> sbim 3imtr3 tfinds2.2 bicomd equcoms ta imbi2d sbralie syl5ibr tfindes))
+thm (tfinds3 ((ph y) (A x) (ps x) (ch x) (th x) (ta x) (x y) (et x) (et y)) ((tfinds3.1 (-> (= (cv x) ({/})) (<-> ph ps))) (tfinds3.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (tfinds3.3 (-> (= (cv x) (suc (cv y))) (<-> ph th))) (tfinds3.4 (-> (= (cv x) A) (<-> ph ta))) (tfinds3.5 (-> et ps)) (tfinds3.6 (-> (e. (cv y) (On)) (-> et (-> ch th)))) (tfinds3.7 (-> (Lim (cv x)) (-> et (-> (A.e. y (cv x) ch) ph))))) (-> (e. A (On)) (-> et ta)) (tfinds3.1 et imbi2d tfinds3.2 et imbi2d tfinds3.3 et imbi2d tfinds3.4 et imbi2d tfinds3.5 tfinds3.6 a2d tfinds3.7 a2d y x cv et ch r19.21v syl5ib tfinds))
+thm (ssnlim ((A x)) () (-> (/\ (Ord A) (C_ A ({e.|} x (On) (-. (Lim (cv x)))))) (C_ A (om))) (limom A x On x cv Lim -. {e.|} om ssel x cv om limeq negbid On elrab pm3.27bd syl6 mt2i A Ord adantl ordom A om ordtri1 mpan2 A x On x cv Lim -. {e.|} C_ adantr mpbird))
+var (set h)
+var (set j)
+var (set k)
+var (set m)
+var (set n)
+var (class H)
+var (class J)
+var (class K)
+var (class M)
+var (class N)
+thm (xpeq1 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (X. A C) (X. B C))) (A B x cv eleq2 y cv C e. anbi1d x y biopabdv A C x y df-xp B C x y df-xp 3eqtr4g))
+thm (xpeq2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (X. C A) (X. C B))) (A B y cv eleq2 x cv C e. anbi2d x y biopabdv C A x y df-xp C B x y df-xp 3eqtr4g))
+thm (elxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (/\ (e. (cv x) B) (e. (cv y) C)))))) (B C x y df-xp A eleq2i A x y x cv B e. y cv C e. /\ elopab bitr))
+thm (elxp2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E.e. x B (E.e. y C (= A (<,> (cv x) (cv y)))))) (y C x cv B e. A x cv y cv <,> = /\ df-rex y C x cv B e. A x cv y cv <,> = r19.42v x cv B e. y cv C e. A x cv y cv <,> = anass A x cv y cv <,> = x cv B e. y cv C e. /\ ancom y cv C e. x cv B e. A x cv y cv <,> = an12 3bitr4r y biex 3bitr3 x biex x B y C A x cv y cv <,> = E.e. df-rex A B C x y elxp 3bitr4r))
+thm (hbxp ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (x y) (x z) (w x)) ((hbxp.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbxp.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (X. A B)) (A. x (e. (cv y) (X. A B)))) (y cv z cv w cv <,> = x ax-17 y cv z cv e. x ax-17 hbxp.1 hbel y cv w cv e. x ax-17 hbxp.2 hbel hban hban w hbex z hbex y cv A B z w elxp y cv A B z w elxp x bial 3imtr4))
+thm (opelxpex ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (e. (<,> A B) (X. C D)) (e. A (V))) (A B <,> C D x y elxp x visset A B <,> x cv y cv <,> {/} eleq2 A B opprc1b x cv y cv opprc1b 3bitr4g bicon4d mpbiri x cv C e. y cv D e. /\ adantr x y 19.23aivv sylbi))
+thm (brrelex () () (-> (/\ (Rel R) (br A R B)) (e. A (V))) (R df-rel R V V X. A B <,> ssel A R B df-br syl5ib A B V V opelxpex syl6 sylbi imp))
+thm (brrelexi () ((brrelexi.1 (Rel R))) (-> (br A R B) (e. A (V))) (brrelexi.1 R A B brrelex mpan))
+thm (nprrel () ((nprrel.1 (Rel R)) (nprrel.2 (-. (e. A (V))))) (-. (br A R B)) (nprrel.2 nprrel.1 A B brrelexi mto))
+thm (fconstopab ((x y) (A x) (A y) (B x) (B y)) () (= (X. A ({} B)) ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))) (A B {} x y df-xp y B elsn x cv A e. anbi2i x y biopabi eqtr))
+thm (vtoclr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (-> (e. C D) (-> (/\ (br A R B) (br B R C)) (br A R C))) (C D elisset x cv A R y cv breq1 y cv R C br anbi1d x cv A R C breq1 imbi12d C V e. imbi2d y cv B A R breq2 y cv B R C breq1 anbi12d A R C br imbi1d C V e. imbi2d z cv C y cv R breq2 x cv R y cv br anbi2d z cv C x cv R breq2 imbi12d vtoclr.2 V vtoclg V V vtocl2g vtoclr.1 A B brrelexi vtoclr.1 B C brrelexi syl2an pm2.43b syl))
+thm (vtoclrbr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) (vtoclrbr.3 (br (cv x) R (cv x)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (vtoclr.1 vtoclr.2 C V A B vtoclr C A R brprc x cv A R x cv breq1 x cv A A R breq2 bitrd vtoclrbr.3 V vtoclg syl5bir vtoclr.1 A B brrelexi syl5 B R C br adantrd pm2.61i))
+thm (vtoclibr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((vtoclr.1 (Rel R)) (vtoclr.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))) (vtoclibr.3 (-. (br (cv x) R (cv x))))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (x cv B R x cv breq1 x cv B B R breq2 bitrd negbid vtoclibr.3 V vtoclg vtoclr.1 B B brrelexi con3i pm2.61i C B R brprc mtbiri a3i vtoclr.1 vtoclr.2 C V A B vtoclr syl anabsi7))
+thm (opelxp ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((opelxp.1 (e. B (V)))) (<-> (e. (<,> A B) (X. C D)) (/\ (e. A C) (e. B D))) (A B C D opelxpex A C elisset B D e. adantr z cv A B opeq1 C D X. eleq1d z cv A C eleq1 B D e. anbi1d z cv B <,> x cv y cv <,> eqcom x visset y visset opelxp.1 z cv opth bitr x cv C e. y cv D e. /\ anbi1i x cv z cv = y cv B = x cv C e. y cv D e. an4 bitr y biex y x cv z cv = x cv C e. /\ y cv B = y cv D e. /\ 19.42v bitr x biex x x cv z cv = x cv C e. /\ y y cv B = y cv D e. /\ E. 19.41v bitr z cv B <,> C D x y elxp z cv C x df-clel B D y df-clel anbi12i 3bitr4 V vtoclbg pm5.21nii))
+thm (brxp () ((opelxp.1 (e. B (V)))) (<-> (br A (X. C D) B) (/\ (e. A C) (e. B D))) (A C D X. B df-br opelxp.1 A C D opelxp bitr))
+thm (opelxpg ((A x) (B x) (C x) (D x)) () (-> (e. B R) (<-> (e. (<,> A B) (X. C D)) (/\ (e. A C) (e. B D)))) (x cv B A opeq2 C D X. eleq1d x cv B D eleq1 A C e. anbi2d x visset A C D opelxp R vtoclbg))
+thm (opelxpi () () (-> (/\ (e. A C) (e. B D)) (e. (<,> A B) (X. C D))) (B D A C D opelxpg biimprd anabsi7))
+thm (ralxp ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (ph y) (ph z) (ps x)) ((ralxp.1 (-> (= (cv x) (<,> (cv y) (cv z))) (<-> ph ps)))) (<-> (A.e. x (X. A B) ph) (A.e. y A (A.e. z B ps))) (ralxp.1 A B X. rcla4v com12 z visset y cv A B opelxp syl5ibr r19.21aivv x cv A B y z elxp x cv y cv z cv <,> = y cv A e. z cv B e. /\ pm3.26 y z 19.22i2 sylbi y A z B ps A.e. hbra1 x cv A B X. e. ph -> y ax-17 hbim y cv A e. z ax-17 z B ps hbra1 hbral x cv A B X. e. ph -> z ax-17 hbim x cv y cv z cv <,> A B X. eleq1 z visset y cv A B opelxp syl6bb ralxp.1 imbi12d y A z B ps ra42 syl5bir 19.23ai 19.23ai syl pm2.43b r19.21aiv impbi))
+thm (opthprc ((A x) (B x) (C x) (D x)) () (<-> (= (u. (X. A ({} ({/}))) (X. B ({} ({} ({/}))))) (u. (X. C ({} ({/}))) (X. D ({} ({} ({/})))))) (/\ (= A C) (= B D))) (A {/} {} X. B {/} {} {} X. u. C {/} {} X. D {/} {} {} X. u. x cv {/} <,> eleq2 0ex x cv A {/} {} opelxp 0ex snid mpbiran2 0ex x cv B {/} {} {} opelxp 0nep0 0ex {/} {} elsnc mtbir x cv B e. bianfi bitr4 orbi12i x cv {/} <,> A {/} {} X. B {/} {} {} X. elun 0nep0 0ex {/} {} elsnc mtbir x cv A e. biorfi 3bitr4r 0ex x cv C {/} {} opelxp 0ex snid mpbiran2 0ex x cv D {/} {} {} opelxp 0nep0 0ex {/} {} elsnc mtbir x cv D e. bianfi bitr4 orbi12i x cv {/} <,> C {/} {} X. D {/} {} {} X. elun 0nep0 0ex {/} {} elsnc mtbir x cv C e. biorfi 3bitr4r 3bitr4g eqrd A {/} {} X. B {/} {} {} X. u. C {/} {} X. D {/} {} {} X. u. x cv {/} {} <,> eleq2 p0ex x cv A {/} {} opelxp 0nep0 p0ex {/} elsnc {/} {} {/} eqcom bitr mtbir x cv A e. bianfi bitr4 p0ex x cv B {/} {} {} opelxp p0ex snid mpbiran2 orbi12i x cv {/} {} <,> A {/} {} X. B {/} {} {} X. elun 0nep0 p0ex {/} elsnc {/} {} {/} eqcom bitr mtbir {/} {} {/} {} e. x cv B e. biorf ax-mp 3bitr4r p0ex x cv C {/} {} opelxp 0nep0 p0ex {/} elsnc {/} {} {/} eqcom bitr mtbir x cv C e. bianfi bitr4 p0ex x cv D {/} {} {} opelxp p0ex snid mpbiran2 orbi12i x cv {/} {} <,> C {/} {} X. D {/} {} {} X. elun 0nep0 p0ex {/} elsnc {/} {} {/} eqcom bitr mtbir {/} {} {/} {} e. x cv D e. biorf ax-mp 3bitr4r 3bitr4g eqrd jca A {/} {} X. C {/} {} X. B {/} {} {} X. D {/} {} {} X. uneq12 A C {/} {} xpeq1 B D {/} {} {} xpeq1 syl2an impbi))
+thm (brelg () ((brelg.1 (C_ R (X. C D)))) (-> (e. B S) (-> (br A R B) (/\ (e. A C) (e. B D)))) (B S A C D opelxpg brelg.1 A B <,> sseli syl5bi A R B df-br syl5ib))
+thm (brel () ((brel.1 (e. B (V))) (brel.2 (C_ R (X. C D)))) (-> (br A R B) (/\ (e. A C) (e. B D))) (brel.1 brel.2 B V A brelg ax-mp))
+thm (elxp3 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (E. x (E. y (/\ (= (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) (X. B C)))))) (A B C x y elxp x cv y cv <,> A eqcom y visset x cv B C opelxp anbi12i x y bi2ex bitr4))
+thm (xpundi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (= (X. A (u. B C)) (u. (X. A B) (X. A C))) (y cv B C elun x cv A e. anbi2i x cv A e. y cv B e. y cv C e. andi bitr x y biopabi x y x cv A e. y cv B e. /\ x cv A e. y cv C e. /\ unopab eqtr4 A B C u. x y df-xp A B x y df-xp A C x y df-xp uneq12i 3eqtr4))
+thm (xpundir ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (= (X. (u. A B) C) (u. (X. A C) (X. B C))) (x cv A B elun y cv C e. anbi1i x cv A e. x cv B e. y cv C e. andir bitr x y biopabi x y x cv A e. y cv C e. /\ x cv B e. y cv C e. /\ unopab eqtr4 A B u. C x y df-xp A C x y df-xp B C x y df-xp uneq12i 3eqtr4))
+thm (xpun () () (= (X. (u. A B) (u. C D)) (u. (u. (X. A C) (X. A D)) (u. (X. B C) (X. B D)))) (A B u. C D xpundi A B C xpundir A B D xpundir uneq12i A C X. B C X. A D X. B D X. un4 3eqtr))
+thm (elvv ((x y) (A x) (A y)) () (<-> (e. A (X. (V) (V))) (E. x (E. y (= A (<,> (cv x) (cv y)))))) (A V V x y elxp x visset y visset pm3.2i A x cv y cv <,> = biantru x y bi2ex bitr4))
+thm (xpss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (X. A B) (X. (V) (V))) (z cv x cv y cv <,> = x cv A e. y cv B e. /\ pm3.26 x y 19.22i2 z cv A B x y elxp z cv x y elvv 3imtr4 ssriv))
+thm (brinxp () () (-> (/\ (e. A C) (e. B D)) (<-> (br A R B) (br A (i^i R (X. C D)) B))) (A C e. B D e. /\ A R B br ibar A C B D opelxpi A C e. B D e. /\ A B <,> C D X. e. ibib mpbi A R B df-br A C e. B D e. /\ a1i anbi12d A R C D X. i^i B df-br A B <,> R C D X. elin A B <,> R e. A B <,> C D X. e. ancom 3bitr syl6bbr bitrd))
+thm (weinxp ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z)) () (<-> (We R A) (We (i^i R (X. A A)) A)) (z cv A x cv ssel z cv A y cv ssel anim12d z cv {/} = -. adantr y cv A x cv A R brinxp ancoms syl6 exp3a imp31 negbid biraldva birexdva pm5.74i z bial R A z x y df-fr R A A X. i^i A z x y df-fr 3bitr4 x cv A y cv A R brinxp x cv A e. y cv A e. /\ x cv y cv = pm4.2i y cv A x cv A R brinxp ancoms 3orbi123d pm5.74i x y bi2al x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ r2al x A y A x cv R A A X. i^i y cv br x cv y cv = y cv R A A X. i^i x cv br \/\/ r2al 3bitr4 anbi12i R A x y dfwe2 R A A X. i^i A x y dfwe2 3bitr4))
+thm (opabssxp ((x y) (A x) (A y) (B x) (B y)) () (C_ ({<,>|} x y (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (X. A B)) (x cv A e. y cv B e. /\ ph pm3.26 x y ssopab2i A B x y df-xp sseqtr4))
+thm (optocl ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ps x) (ps y)) ((optocl.1 (= D (X. B C))) (optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (optocl.3 (-> (/\ (e. (cv x) B) (e. (cv y) C)) ph))) (-> (e. A D) ps) (optocl.1 A eleq2i A B C x y elxp3 optocl.2 y visset x cv B C opelxp optocl.3 sylbi syl5bi imp x y 19.23aivv sylbi sylbi))
+thm (2optocl ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ch z) (ch w) (R z) (R w)) ((2optocl.1 (= R (X. C D))) (2optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (2optocl.3 (-> (= (<,> (cv z) (cv w)) B) (<-> ps ch))) (2optocl.4 (-> (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (/\ (e. (cv z) C) (e. (cv w) D))) ph))) (-> (/\ (e. A R) (e. B R)) ch) (2optocl.1 2optocl.3 A R e. imbi2d 2optocl.1 2optocl.2 z cv C e. w cv D e. /\ imbi2d 2optocl.4 exp optocl com12 optocl impcom))
+thm (3optocl ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B z) (B w) (B v) (B u) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (F x) (F y) (F z) (F w) (F v) (F u) (R z) (R w) (R v) (R u) (ps x) (ps y) (ch z) (ch w) (th v) (th u)) ((3optocl.1 (= R (X. D F))) (3optocl.2 (-> (= (<,> (cv x) (cv y)) A) (<-> ph ps))) (3optocl.3 (-> (= (<,> (cv z) (cv w)) B) (<-> ps ch))) (3optocl.4 (-> (= (<,> (cv v) (cv u)) C) (<-> ch th))) (3optocl.5 (-> (/\/\ (/\ (e. (cv x) D) (e. (cv y) F)) (/\ (e. (cv z) D) (e. (cv w) F)) (/\ (e. (cv v) D) (e. (cv u) F))) ph))) (-> (/\/\ (e. A R) (e. B R) (e. C R)) th) (3optocl.1 3optocl.4 A R e. B R e. /\ imbi2d 3optocl.1 3optocl.2 v cv D e. u cv F e. /\ imbi2d 3optocl.3 v cv D e. u cv F e. /\ imbi2d 3optocl.5 3expa exp 2optocl com12 optocl impcom 3impa))
+thm (opbrop ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S x) (S y) (S z) (S w) (S v) (S u) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u)) ((opbrop.1 (-> (/\ (/\ (= (cv z) A) (= (cv w) B)) (/\ (= (cv v) C) (= (cv u) D))) (<-> ph ps))) (opbrop.2 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) ph)))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) ps)) (opbrop.1 S S copsex4g A B <,> S S X. e. C D <,> S S X. e. /\ anbi2d A B opex C D opex x cv A B <,> S S X. eleq1 y cv S S X. e. anbi1d x cv A B <,> z cv w cv <,> eqeq1 y cv v cv u cv <,> = anbi1d ph anbi1d z w v u bi4exdv anbi12d y cv C D <,> S S X. eleq1 A B <,> S S X. e. anbi2d y cv C D <,> v cv u cv <,> eqeq1 A B <,> z cv w cv <,> = anbi2d ph anbi1d z w v u bi4exdv anbi12d opbrop.2 brab syl5bb A S B S opelxpi C S D S opelxpi anim12i ps biantrurd bitr4d))
+thm (cbvop ((ph y) (ph z) (y z) (ps x) (x y) (x z) (A x) (A y) (A z) (B x) (B y) (B z)) ((cbvop.1 (-> (= (cv x) (<,> (cv y) (cv z))) (<-> ph ps)))) (<-> (E.e. x (X. A B) ph) (E.e. y A (E.e. z B ps))) (y z x cv y cv z cv <,> = E. E. x cv A B X. e. ph anass x cv A B y z elxp3 y cv z cv <,> x cv A B X. eleq1 pm5.32i x cv y cv z cv <,> eqcom x cv A B X. e. anbi1i bitr4 y z bi2ex y z x cv y cv z cv <,> = x cv A B X. e. 19.41vv 3bitr ph anbi1i y z x cv y cv z cv <,> = x cv A B X. e. ph /\ 19.41vv 3bitr4 x biex x y z x cv y cv z cv <,> = x cv A B X. e. ph /\ /\ exrot3 y cv z cv opex x cv y cv z cv <,> A B X. eleq1 cbvop.1 anbi12d ceqsexv y z bi2ex 3bitr x A B X. ph df-rex y A z B ps r2ex z visset y cv A B opelxp ps anbi1i y z bi2ex bitr4 3bitr4))
+thm (xp0r ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (X. ({/}) A) ({/})) (z cv {/} A x y elxp x cv noel x cv {/} e. y cv A e. pm3.26 z cv x cv y cv <,> = adantl mto y nex x nex z cv noel 2false bitr eqri))
+thm (0nelxp ((x y) (A x) (A y) (B x) (B y)) () (-. (e. ({/}) (X. A B))) (x cv {} noel x cv y cv opi1 {/} x cv y cv <,> x cv {} eleq2 mpbiri mto x cv A e. y cv B e. /\ intnanr y nex x nex {/} A B x y elxp mtbir))
+thm (onxpdisj ((x y) (x z) (y z)) () (= (i^i (On) (X. (V) (V))) ({/})) (On V V X. x disj x cv on0eqelt V V 0nelxp x cv {/} V V X. eleq1 mtbiri x cv y z elvv y visset y cv z cv opprc1b bicon1i mpbir x cv y cv z cv <,> {/} eleq2 mtbiri y z 19.23aivv sylbi con2i jaoi syl mprgbir))
+thm (releq () () (-> (= A B) (<-> (Rel A) (Rel B))) (A B V V X. sseq1 A df-rel B df-rel 3bitr4g))
+thm (hbrel ((A y) (x y)) ((hbrel.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (Rel A) (A. x (Rel A))) (hbrel.1 y cv V V X. e. x ax-17 hbss A df-rel A df-rel x bial 3imtr4))
+thm (ssrel () () (-> (C_ A B) (-> (Rel B) (Rel A))) (A B V V X. sstr2 B df-rel A df-rel 3imtr4g))
+thm (relss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (Rel A) (<-> (C_ A B) (A. x (A. y (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))))) (A B x cv y cv <,> ssel A Rel a1i y 19.21adv x 19.21adv A df-rel A V V X. z cv ssel sylbi z cv x y elvv syl6ib x cv y cv <,> A e. x cv y cv <,> B e. -> id z cv x cv y cv <,> = anim2d z cv x cv y cv <,> B eleq1 biimpar syl6 z cv x cv y cv <,> A eleq1 pm5.32i syl5ib exp3a y 19.20i y z cv x cv y cv <,> = z cv A e. z cv B e. -> 19.23v sylib x 19.20i x y z cv x cv y cv <,> = E. z cv A e. z cv B e. -> 19.23v sylib syl9 z cv A e. z cv B e. pm2.43 syl6 z 19.21adv A B z dfss2 syl6ibr impbid))
+thm (relssi ((x y) (A x) (A y) (B x) (B y)) ((relssi.1 (Rel A)) (relssi.2 (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))) (C_ A B) (relssi.1 A B x y relss ax-mp relssi.2 y ax-gen mpgbir))
+thm (relssdv ((x y) (A x) (A y) (B x) (B y) (ph x) (ph y)) ((relssdv.1 (-> ph (Rel A))) (relssdv.2 (-> ph (-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B))))) (-> ph (C_ A B)) (relssdv.2 x y 19.21aivv relssdv.1 A B x y relss syl mpbird))
+thm (cleqrel ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (Rel B)) (<-> (= A B) (A. x (A. y (<-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))))) (A B x y relss B A x y relss bi2anan9 A B eqss x y x cv y cv <,> A e. x cv y cv <,> B e. 2albi 3bitr4g))
+thm (cleqreli ((x y) (A x) (A y) (B x) (B y)) ((cleqreli.1 (Rel A)) (cleqreli.2 (Rel B)) (cleqreli.3 (<-> (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv y)) B)))) (= A B) (cleqreli.1 cleqreli.2 A B x y cleqrel mp2an cleqreli.3 y ax-gen mpgbir))
+thm (relsn () ((relsn.1 (e. A (V)))) (Rel ({} (<,> A B))) (relsn.1 A V B V opelxpi mpan relsn.1 A V V opelxp relsn.1 relsn.1 mpbir2an B A opprc2 V V X. eleq1d mpbiri pm2.61i A B <,> V V X. snssi ax-mp A B <,> {} df-rel mpbir))
+thm (relxp () () (Rel (X. A B)) (A B xpss A B X. df-rel mpbir))
+thm (ssxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (/\ (C_ A B) (C_ C D)) (C_ (X. A C) (X. B D))) (A C relxp A B C_ C D C_ /\ a1i x cv A e. x cv B e. y cv C e. y cv D e. prth y visset x cv A C opelxp y visset x cv B D opelxp 3imtr4g A B x cv ssel C D y cv ssel syl2an relssdv))
+thm (xpex ((x y) (A x) (A y) (B x) (B y)) ((xpex.1 (e. A (V))) (xpex.2 (e. B (V)))) (e. (X. A B) (V)) (xpex.1 xpex.2 unex pwex pwex A B relxp y visset x cv A B opelxp x cv A snssi x cv {} A B ssun3 syl x cv snex A B u. elpw sylibr y cv B e. adantr x cv A snssi x cv {} A B ssun3 syl y cv B snssi y cv {} B A ssun4 syl anim12i x cv {} A B u. y cv {} unss sylib x cv y cv df-pr syl5ss x y zfpair A B u. elpw sylibr jca x cv {} x cv y cv {,} prex A B u. P~ elpw x cv y cv df-op A B u. P~ P~ eleq1i x cv snex x y zfpair A B u. P~ prss 3bitr4r sylib sylbi relssi ssexi))
+thm (xpexg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (e. (X. A B) (V))) (x cv A y cv xpeq1 V eleq1d y cv B A xpeq2 V eleq1d x visset y visset xpex C D vtocl2g))
+thm (relun () () (<-> (Rel (u. A B)) (/\ (Rel A) (Rel B))) (A V V X. B unss A df-rel B df-rel anbi12i A B u. df-rel 3bitr4r))
+thm (relin () () (-> (Rel A) (Rel (i^i A B))) (A B inss1 A B i^i A ssrel ax-mp))
+thm (reldif () () (-> (Rel A) (Rel (\ A B))) (A B difss A B \ A ssrel ax-mp))
+thm (reluni ((x y) (A x) (A y)) () (<-> (Rel (U. A)) (A.e. x A (Rel (cv x)))) (x A y cv x cv e. y cv V V X. e. r19.23v y cv A x eluni2 y cv V V X. e. imbi1i bitr4 y bial x cv df-rel x cv V V X. y dfss2 bitr x A biral x A y y cv x cv e. y cv V V X. e. -> ralcom4 bitr A U. df-rel A U. V V X. y dfss2 bitr 3bitr4r))
+thm (relopab ((x y)) () (Rel ({<,>|} x y ph)) (x visset y visset pm3.2i ph a1i x y ssopab2i V V x y df-xp sseqtr4 x y ph {<,>|} df-rel mpbir))
+thm (inopab ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps z) (ps w)) () (= (i^i ({<,>|} x y ph) ({<,>|} x y ps)) ({<,>|} x y (/\ ph ps))) (x y ph relopab x y ph {<,>|} x y ps {<,>|} relin ax-mp x y ph ps /\ relopab w y z cv x ph [/] z cv x ps [/] sban z x ph ps sban w y bisb z w x y ph opabsb z w x y ps opabsb anbi12i 3bitr4r z cv w cv <,> x y ph {<,>|} x y ps {<,>|} elin z w x y ph ps /\ opabsb 3bitr4 cleqreli))
+thm (inxp ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (= (i^i (X. A B) (X. C D)) (X. (i^i A C) (i^i B D))) (A B relxp A B X. C D X. relin ax-mp A C i^i B D i^i relxp x cv A e. y cv B e. x cv C e. y cv D e. an4 y visset x cv A B opelxp y visset x cv C D opelxp anbi12i x cv A C elin y cv B D elin anbi12i 3bitr4 x cv y cv <,> A B X. C D X. elin y visset x cv A C i^i B D i^i opelxp 3bitr4 cleqreli))
+thm (xpindi () () (= (X. A (i^i B C)) (i^i (X. A B) (X. A C))) (A B A C inxp A inidm A A i^i A B C i^i xpeq1 ax-mp eqtr2))
+thm (xpindir () () (= (X. (i^i A B) C) (i^i (X. A C) (X. B C))) (A C B C inxp C inidm C C i^i C A B i^i xpeq2 ax-mp eqtr2))
+thm (rel0 () () (Rel ({/})) (V V X. 0ss {/} df-rel mpbir))
+thm (reli ((x y)) () (Rel (I)) (x y x cv y cv = relopab x y df-id I x y x cv y cv = {<,>|} releq ax-mp mpbir))
+thm (rele ((x y)) () (Rel (E)) (x y x cv y cv e. relopab x y df-eprel E x y x cv y cv e. {<,>|} releq ax-mp mpbir))
+thm (coeq1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= A B) (= (o. A C) (o. B C))) (A B z cv y cv breq x cv C z cv br anbi2d z biexdv x y biopabdv A C x y z df-co B C x y z df-co 3eqtr4g))
+thm (coeq2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (= A B) (= (o. C A) (o. C B))) (A B x cv z cv breq z cv C y cv br anbi1d z biexdv x y biopabdv C A x y z df-co C B x y z df-co 3eqtr4g))
+thm (coeq1i () ((coeq1i.1 (= A B))) (= (o. A C) (o. B C)) (coeq1i.1 A B C coeq1 ax-mp))
+thm (coeq2i () ((coeq1i.1 (= A B))) (= (o. C A) (o. C B)) (coeq1i.1 A B C coeq2 ax-mp))
+thm (coeq1d () ((coeq1d.1 (-> ph (= A B)))) (-> ph (= (o. A C) (o. B C))) (coeq1d.1 A B C coeq1 syl))
+thm (coeq2d () ((coeq1d.1 (-> ph (= A B)))) (-> ph (= (o. C A) (o. C B))) (coeq1d.1 A B C coeq2 syl))
+thm (hbco ((y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B y) (B z) (B w) (B v) (x y) (x z) (w x) (v x)) ((hbco.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbco.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (o. A B)) (A. x (e. (cv y) (o. A B)))) (y cv z cv e. x ax-17 hbco.2 y cv v cv e. x ax-17 hbbr y cv v cv e. x ax-17 hbco.1 y cv w cv e. x ax-17 hbbr hban v hbex y z w hbopab A B z w v df-co y cv eleq2i A B z w v df-co y cv eleq2i x bial 3imtr4))
+thm (opelco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((opelco.1 (e. A (V))) (opelco.2 (e. B (V)))) (<-> (e. (<,> A B) (o. C D)) (E. x (/\ (br A D (cv x)) (br (cv x) C B)))) (C D y z x df-co A B <,> eleq2i opelco.1 opelco.2 y cv A D x cv breq1 x cv C z cv br anbi1d x biexdv z cv B x cv C breq2 A D x cv br anbi2d x biexdv opelopab bitr))
+thm (brco ((A x) (B x) (C x) (D x)) ((opelco.1 (e. A (V))) (opelco.2 (e. B (V)))) (<-> (br A (o. C D) B) (E. x (/\ (br A D (cv x)) (br (cv x) C B)))) (A C D o. B df-br opelco.1 opelco.2 C D x opelco bitr))
+thm (opelcog ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) () (-> (/\ (e. A R) (e. B S)) (<-> (e. (<,> A B) (o. C D)) (E. x (/\ (e. (<,> A (cv x)) D) (e. (<,> (cv x) B) C))))) (y cv A z cv opeq1 C D o. eleq1d y cv A D x cv breq1 x cv C z cv br anbi1d x biexdv bibi12d z cv B A opeq2 C D o. eleq1d z cv B x cv C breq2 A D x cv br anbi2d x biexdv bibi12d y visset z visset C D x opelco R S vtocl2g A D x cv df-br x cv C B df-br anbi12i x biex syl6bb))
+thm (cnvss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (`' A) (`' B))) (A B y cv x cv <,> ssel y cv A x cv df-br y cv B x cv df-br 3imtr4g x y 19.21aivv x y y cv A x cv br y cv B x cv br ssopab2 sylibr A x y df-cnv B x y df-cnv 3sstr4g))
+thm (cnveq () () (-> (= A B) (= (`' A) (`' B))) (A B cnvss B A cnvss anim12i A B eqss A `' B `' eqss 3imtr4))
+thm (elcnv ((x y) (A x) (A y) (R x) (R y)) () (<-> (e. A (`' R)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (br (cv y) R (cv x)))))) (R x y df-cnv A eleq2i A x y y cv R x cv br elopab bitr))
+thm (elcnv2 ((x y) (A x) (A y) (R x) (R y)) () (<-> (e. A (`' R)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) (e. (<,> (cv y) (cv x)) R))))) (A R x y elcnv y cv R x cv df-br A x cv y cv <,> = anbi2i x y bi2ex bitr))
+thm (hbcnv ((y z) (w y) (A y) (w z) (A z) (A w) (x y) (x z) (w x)) ((hbcnv.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (`' A)) (A. x (e. (cv y) (`' A)))) (y cv z cv w cv <,> = x ax-17 y cv w cv e. x ax-17 hbcnv.1 y cv z cv e. x ax-17 hbbr hban w hbex z hbex y cv A z w elcnv y cv A z w elcnv x bial 3imtr4))
+thm (opelcnvg ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (-> (/\ (e. A C) (e. B D)) (<-> (e. (<,> A B) (`' R)) (e. (<,> B A) R))) (x cv A y cv R breq2 y cv B R A breq1 C D opelopabg R x y df-cnv A B <,> eleq2i syl5bb B R A df-br syl6bb))
+thm (brcnvg () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (`' R) B) (br B R A))) (A C B D R opelcnvg A R `' B df-br B R A df-br 3bitr4g))
+thm (opelcnv () ((opelcnv.1 (e. A (V))) (opelcnv.2 (e. B (V)))) (<-> (e. (<,> A B) (`' R)) (e. (<,> B A) R)) (opelcnv.1 opelcnv.2 A V B V R opelcnvg mp2an))
+thm (brcnv () ((opelcnv.1 (e. A (V))) (opelcnv.2 (e. B (V)))) (<-> (br A (`' R) B) (br B R A)) (opelcnv.1 opelcnv.2 A V B V R brcnvg mp2an))
+thm (cnvco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (`' (o. A B)) (o. (`' B) (`' A))) (x cv A B o. y cv df-br x visset y visset A B z opelco x cv B z cv br z cv A y cv br ancom y visset z visset A brcnv z visset x visset B brcnv anbi12i bitr4 z biex 3bitr y x biopabi A B o. y x df-cnv B `' A `' y x z df-co 3eqtr4))
+thm (cnvuni ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (= (`' (U. A)) (U_ x A (`' (cv x)))) (y cv A U. z w elcnv2 w cv z cv <,> A x eluni2 y cv z cv w cv <,> = anbi2i x A y cv z cv w cv <,> = w cv z cv <,> x cv e. r19.42v bitr4 z w bi2ex x A z w y cv z cv w cv <,> = w cv z cv <,> x cv e. /\ E. rexcom4 x A w y cv z cv w cv <,> = w cv z cv <,> x cv e. /\ rexcom4 z biex bitr2 y cv x cv z w elcnv2 x A birex bitr4 3bitr y cv x A x cv `' eliun bitr4 eqri))
+thm (dfdm3 ((x y) (A x) (A y)) () (= (dom A) ({|} x (E. y (e. (<,> (cv x) (cv y)) A)))) (A x y df-dm x cv A y cv df-br y biex x biabi eqtr))
+thm (dfrn2 ((x y) (A x) (A y)) () (= (ran A) ({|} y (E. x (br (cv x) A (cv y))))) (A df-rn A `' y x df-dm y visset x visset A brcnv x biex y biabi 3eqtr))
+thm (dfrn3 ((x y) (A x) (A y)) () (= (ran A) ({|} y (E. x (e. (<,> (cv x) (cv y)) A)))) (A y x dfrn2 x cv A y cv df-br x biex y biabi eqtr))
+thm (dfdm4 ((x y) (A x) (A y)) () (= (dom A) (ran (`' A))) (y visset x visset A brcnv y biex x biabi A `' x y dfrn2 A x y df-dm 3eqtr4r))
+thm (dfdmf ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v)) ((dfdmf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dfdmf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (= (dom A) ({|} x (E. y (e. (<,> (cv x) (cv y)) A)))) (A w v dfdm3 z cv w cv v cv <,> e. y ax-17 dfdmf.2 hbel w cv y cv <,> A e. v ax-17 v cv y cv w cv opeq2 A eleq1d cbvex w biabi z cv w cv y cv <,> e. x ax-17 dfdmf.1 hbel y hbex y x cv y cv <,> A e. E. w ax-17 w cv x cv y cv opeq1 A eleq1d y biexdv cbvab 3eqtr))
+thm (eldm ((x y) (A x) (A y) (B x) (B y)) ((eldm.1 (e. A (V)))) (<-> (e. A (dom B)) (E. y (br A B (cv y)))) (eldm.1 x cv A B y cv breq1 y biexdv B x y df-dm elab2))
+thm (eldm2 ((A y) (B y)) ((eldm.1 (e. A (V)))) (<-> (e. A (dom B)) (E. y (e. (<,> A (cv y)) B))) (eldm.1 B y eldm A B y cv df-br y biex bitr))
+thm (eldmg ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> (e. A (dom B)) (E. y (e. (<,> A (cv y)) B)))) (x cv A B dom eleq1 x cv A y cv opeq1 B eleq1d y biexdv x visset B y eldm2 C vtoclbg))
+thm (dmss ((x y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (C_ (dom A) (dom B))) (A B x cv y cv <,> ssel y 19.22dv x visset A y eldm2 x visset B y eldm2 3imtr4g ssrdv))
+thm (dmeq () () (-> (= A B) (= (dom A) (dom B))) (A B dmss B A dmss anim12i A B eqss A dom B dom eqss 3imtr4))
+thm (dmeqi () ((dmeqi.1 (= A B))) (= (dom A) (dom B)) (dmeqi.1 A B dmeq ax-mp))
+thm (dmeqd () ((dmeqd.1 (-> ph (= A B)))) (-> ph (= (dom A) (dom B))) (dmeqd.1 A B dmeq syl))
+thm (opeldm ((A y) (B y) (C y)) ((opeldm.1 (e. A (V)))) (-> (e. (<,> A B) C) (e. A (dom C))) (y cv B A opeq2 C eleq1d V cla4egv opeldm.1 C y eldm2 syl6ibr B A opprc2 C eleq1d opeldm.1 y cv A A opeq2 C eleq1d cla4ev opeldm.1 C y eldm2 sylibr syl6bi pm2.61i))
+thm (breldm () ((breldm.1 (e. A (V)))) (-> (br A R B) (e. A (dom R))) (A R B df-br breldm.1 B R opeldm sylbi))
+thm (dmun ((x y) (A x) (A y) (B x) (B y)) () (= (dom (u. A B)) (u. (dom A) (dom B))) (x cv y cv <,> A B elun y biex y x cv y cv <,> A e. x cv y cv <,> B e. 19.43 bitr x visset A B u. y eldm2 x cv A dom B dom elun x visset A y eldm2 x visset B y eldm2 orbi12i bitr 3bitr4 eqri))
+thm (dmin ((x y) (A x) (A y) (B x) (B y)) () (C_ (dom (i^i A B)) (i^i (dom A) (dom B))) (y x cv y cv <,> A e. x cv y cv <,> B e. 19.40 x visset A B i^i y eldm2 x cv y cv <,> A B elin y biex bitr x cv A dom B dom elin x visset A y eldm2 x visset B y eldm2 anbi12i bitr 3imtr4 ssriv))
+thm (dmuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (dom (U. A)) (U_ x A (dom (cv x)))) (y cv z cv <,> A x eluni z biex z x y cv z cv <,> x cv e. x cv A e. /\ excom z y cv z cv <,> x cv e. E. x cv A e. ancom z y cv z cv <,> x cv e. x cv A e. 19.41v y visset x cv z eldm2 x cv A e. anbi2i 3bitr4 x biex 3bitr x A y cv x cv dom e. df-rex bitr4 y visset A U. z eldm2 y cv x A x cv dom eliun 3bitr4 eqri))
+thm (dmopab ((x y) (x z) (y z) (ph z)) () (= (dom ({<,>|} x y ph)) ({|} x (E. y ph))) (z x y ph hbopab1 z x y ph hbopab2 dfdmf x y ph opabid y biex x biabi eqtr))
+thm (dmopabss ((x y) (A x) (A y)) () (C_ (dom ({<,>|} x y (/\ (e. (cv x) A) ph))) A) (x y x cv A e. ph /\ dmopab y x cv A e. ph 19.42v x biabi x A y ph E. ssab eqsstr eqsstr))
+thm (dmopab3 ((x y) (A x) (A y)) () (<-> (A.e. x A (E. y ph)) (= (dom ({<,>|} x y (/\ (e. (cv x) A) ph))) A)) (x A y ph E. df-ral x cv A e. y ph E. pm4.71 x bial A x x cv A e. y ph E. /\ eqab A x x cv A e. y ph E. /\ {|} eqcom x y x cv A e. ph /\ dmopab y x cv A e. ph 19.42v x biabi eqtr A eqeq1i bitr4 bitr3 3bitr))
+thm (dm0 ((x y)) () (= (dom ({/})) ({/})) (x cv y cv <,> noel y nex x cv eqid x cv x cv = negb ax-mp 2false x biabi {/} x y dfdm3 x dfnul2 3eqtr4))
+thm (dmsn0 ((x y)) () (= (dom ({} ({/}))) ({/})) (x cv y cv opnz x cv y cv opex {/} elsnc mtbir y nex x cv eqid x cv x cv = negb ax-mp 2false x biabi {/} {} x y dfdm3 x dfnul2 3eqtr4))
+thm (dmsnsn0 ((x y)) () (= (dom ({} ({} ({/})))) ({/})) (y visset x cv V e. -. a1i orri x cv V e. y cv V e. oran mpbi x cv y cv opprc3 mtbi x cv y cv opex {/} {} elsnc mtbir y nex x cv eqid x cv x cv = negb ax-mp 2false x biabi {/} {} {} x y dfdm3 x dfnul2 3eqtr4))
+thm (dmi ((x y)) () (= (dom (I)) (V)) (y x a9e x visset y visset ideq x cv y cv eqcom bitr y biex mpbir x cv eqid 2th x biabi I x y df-dm x df-v 3eqtr4))
+thm (dmv () () (= (dom (V)) (V)) (V dom ssv dmi I ssv I V dmss ax-mp eqsstr3 eqssi))
+thm (dmsnop ((x y) (A x) (A y) (B x) (B y)) () (= (dom ({} (<,> A B))) ({} A)) (x visset y visset B V A opthg x cv y cv opex A B <,> elsnc syl5bb y biexdv y x cv A = y cv B = 19.42v syl6bb B y isset y y cv B = E. x cv A = iba sylbi bitr4d x biabdv A B <,> {} x y dfdm3 A x df-sn 3eqtr4g B A opprc2 A B <,> A A <,> sneq A B <,> {} A A <,> {} dmeq 3syl x visset y visset A V A opthg x cv y cv opex A A <,> elsnc syl5bb y biexdv y x cv A = y cv A = 19.42v syl6bb A y isset y y cv A = E. x cv A = iba sylbi bitr4d x biabdv A A <,> {} x y dfdm3 A x df-sn 3eqtr4g A V e. -. anidm A A opprc3 bitr3 A A <,> {/} {} sneq dmeqd sylbi dmsnsn0 syl6eq A snprc biimp eqtr4d pm2.61i syl6eq pm2.61i))
+thm (dmsnsnsn () () (= (dom ({} ({} ({} A)))) ({} A)) (A dfsn2 A {} A A {,} A {} preq2 ax-mp A {} dfsn2 A A df-op 3eqtr4r sneqi dmeqi A A dmsnop eqtr3))
+thm (dm0rn0 ((x y) (A x) (A y)) () (<-> (= (dom A) ({/})) (= (ran A) ({/}))) (x y x cv A y cv br excom negbii x y x cv A y cv br E. alnex y x x cv A y cv br E. alnex 3bitr4 x cv noel y x cv A y cv br E. nbn x bial y cv noel x x cv A y cv br E. nbn y bial 3bitr3 x y x cv A y cv br E. {/} eqabr y x x cv A y cv br E. {/} eqabr 3bitr4 A x y df-dm {/} eqeq1i A y x dfrn2 {/} eqeq1i 3bitr4))
+thm (reldm0 ((x y) (A x) (A y)) () (-> (Rel A) (<-> (= A ({/})) (= (dom A) ({/})))) (rel0 A {/} x y cleqrel mpan2 A dom x eq0 x visset A y eldm2 negbii y x cv y cv <,> A e. alnex x cv y cv <,> noel x cv y cv <,> A e. nbn y bial bitr3 bitr x bial bitr2 syl6bb))
+thm (dmexg ((x y) (A x) (A y)) () (-> (e. A B) (e. (dom A) (V))) (A B uniexg A U. V uniexg A x y dfdm3 x y A opeluu pm3.26d y 19.23aiv x ss2abi x A U. U. abid2 sseqtr eqsstr A U. U. V A dom ssexg mpi 3syl))
+thm (dmxp ((x y) (A x) (A y) (B x) (B y)) () (-> (-. (= B ({/}))) (= (dom (X. A B)) A)) (B y n0 biimp x cv A e. a1d r19.21aiv x A y y cv B e. dmopab3 A B x y df-xp dmeqi A eqeq1i bitr4 sylib))
+thm (dmxpid () () (= (dom (X. A A)) A) (A {/} A xpeq1 A xp0r syl6eq dmeqd dm0 syl6eq A {/} = id eqtr4d A A dmxp pm2.61i))
+thm (elreldm ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (e. B A)) (e. (|^| (|^| B)) (dom A))) (A df-rel A V V X. B ssel sylbi B x y elvv syl6ib B x cv y cv <,> A eleq1 x visset y cv A opeldm syl6bi B x cv y cv <,> inteq inteqd x visset y cv op1stb syl6eq A dom eleq1d sylibrd x y 19.23aivv syli imp))
+thm (rneq () () (-> (= A B) (= (ran A) (ran B))) (A B cnveq dmeqd A df-rn B df-rn 3eqtr4g))
+thm (rneqi () ((rneqi.1 (= A B))) (= (ran A) (ran B)) (rneqi.1 A B rneq ax-mp))
+thm (rneqd () ((rneqd.1 (-> ph (= A B)))) (-> ph (= (ran A) (ran B))) (rneqd.1 A B rneq syl))
+thm (rnss () () (-> (C_ A B) (C_ (ran A) (ran B))) (A B cnvss A `' B `' dmss syl A df-rn B df-rn 3sstr4g))
+thm (brelrn () ((brelrn.1 (e. A (V))) (brelrn.2 (e. B (V)))) (-> (br A C B) (e. B (ran C))) (brelrn.2 brelrn.1 C brcnv brelrn.2 C `' A breldm sylbir C df-rn syl6eleqr))
+thm (opelrn () ((brelrn.1 (e. A (V))) (brelrn.2 (e. B (V)))) (-> (e. (<,> A B) C) (e. B (ran C))) (A C B df-br brelrn.1 brelrn.2 C brelrn sylbir))
+thm (dfrnf ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v)) ((dfrnf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dfrnf.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (= (ran A) ({|} y (E. x (e. (<,> (cv x) (cv y)) A)))) (A w v dfrn3 z cv v cv w cv <,> e. x ax-17 dfrnf.1 hbel x cv w cv <,> A e. v ax-17 v cv x cv w cv opeq1 A eleq1d cbvex w biabi z cv x cv w cv <,> e. y ax-17 dfrnf.2 hbel x hbex x x cv y cv <,> A e. E. w ax-17 w cv y cv x cv opeq2 A eleq1d x biexdv cbvab 3eqtr))
+thm (elrn ((x y) (A x) (A y) (B x) (B y)) ((elrn.1 (e. A (V)))) (<-> (e. A (ran B)) (E. x (e. (<,> (cv x) A) B))) (elrn.1 y cv A x cv opeq2 B eleq1d x biexdv B y x dfrn3 elab2))
+thm (elrn2 ((A x) (B x)) ((elrn.1 (e. A (V)))) (<-> (e. A (ran B)) (E. x (br (cv x) B A))) (elrn.1 B x elrn x cv B A df-br x biex bitr4))
+thm (hbrn ((x y) (x z) (w x) (y z) (w y) (w z) (A y) (A z) (A w)) ((hbrn.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (ran A)) (A. x (e. (cv y) (ran A)))) (w cv z cv y cv <,> e. x ax-17 y cv z cv e. x ax-17 hbrn.1 hbel hbel z hbex y visset A z elrn y visset A z elrn x bial 3imtr4))
+thm (hbdm ((x y) (A y)) ((hbdm.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (dom A)) (A. x (e. (cv y) (dom A)))) (hbdm.1 hbcnv hbrn A dfdm4 y cv eleq2i A dfdm4 y cv eleq2i x bial 3imtr4))
+thm (rnopab ((x y) (x z) (y z) (ph z)) () (= (ran ({<,>|} x y ph)) ({|} y (E. x ph))) (z x y ph hbopab1 z x y ph hbopab2 dfrnf x y ph opabid x biex y biabi eqtr))
+thm (rn0 () () (= (ran ({/})) ({/})) (dm0 {/} dm0rn0 mpbi))
+thm (relrn0 () () (-> (Rel A) (<-> (= A ({/})) (= (ran A) ({/})))) (A reldm0 A dm0rn0 syl6bb))
+thm (rnexg ((x y) (A x) (A y)) () (-> (e. A B) (e. (ran A) (V))) (A B uniexg A U. V uniexg A y x dfrn3 x y A opeluu pm3.27d x 19.23aiv y ss2abi y A U. U. abid2 sseqtr eqsstr A U. U. V A ran ssexg mpi 3syl))
+thm (dmco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (dom (o. A B)) (dom B)) (x cv A B o. y cv df-br x visset y visset A B z opelco bitr y biex y z x cv B z cv br z cv A y cv br /\ excom bitr y x cv B z cv br z cv A y cv br 19.42v pm3.26bd z 19.22i sylbi x ss2abi A B o. x y df-dm B x z df-dm 3sstr4))
+thm (rnco () () (C_ (ran (o. A B)) (ran A)) (B `' A `' dmco A B o. df-rn A B cnvco dmeqi eqtr A df-rn 3sstr4))
+thm (dmcosseq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ (ran B) (dom A)) (= (dom (o. A B)) (dom B))) (x x cv B z cv br hbe1 y z cv A y cv br E. x ax-17 hbim z hbal z x x cv B z cv br E. y z cv A y cv br E. -> hba1 x cv B z cv br x 19.8a y z cv A y cv br E. imim1i ancld z a4s 19.22d 19.21ai x cv B z cv br y z cv A y cv br E. pm3.26 z 19.22i x ax-gen jctil x z x cv B z cv br y z cv A y cv br E. /\ E. z x cv B z cv br E. albi sylibr z visset B x elrn2 z visset A y eldm imbi12i z bial x visset A B o. y eldm2 x visset y visset A B z opelco y biex y z x cv B z cv br z cv A y cv br /\ excom y x cv B z cv br z cv A y cv br 19.42v z biex bitr 3bitr x visset B z eldm bibi12i x bial 3imtr4 B ran A dom z dfss2 A B o. dom B dom x dfcleq 3imtr4))
+thm (dmcoeq () () (-> (= (dom A) (ran B)) (= (dom (o. A B)) (dom B))) (A dom B ran eqimss2 B A dmcosseq syl))
+thm (rncoeq () () (-> (= (dom A) (ran B)) (= (ran (o. A B)) (ran A))) (B `' A `' dmcoeq A dom B ran eqcom B df-rn A dfdm4 eqeq12i bitr A B o. df-rn A B cnvco dmeqi eqtr A df-rn eqeq12i 3imtr4))
+thm (reseq1 () () (-> (= A B) (= (|` A C) (|` B C))) (A B C V X. ineq1 A C df-res B C df-res 3eqtr4g))
+thm (reseq2 () () (-> (= A B) (= (|` C A) (|` C B))) (A B V xpeq1 C ineq2d C A df-res C B df-res 3eqtr4g))
+thm (hbres ((A y) (B y) (x y)) ((hbres.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbres.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (|` A B)) (A. x (e. (cv y) (|` A B)))) (hbres.1 hbres.2 y cv V e. x ax-17 hbxp hbin A B df-res y cv eleq2i A B df-res y cv eleq2i x bial 3imtr4))
+thm (res0 () () (= (|` A ({/})) ({/})) (A {/} df-res V xp0r A ineq2i A in0 3eqtr))
+thm (opelres () ((opelres.1 (e. B (V)))) (<-> (e. (<,> A B) (|` C D)) (/\ (e. (<,> A B) C) (e. A D))) (C D df-res A B <,> eleq2i A B <,> C D V X. elin opelres.1 A D V opelxp opelres.1 mpbiran2 A B <,> C e. anbi2i 3bitr))
+thm (opres () ((opres.1 (e. B (V)))) (-> (e. A D) (<-> (e. (<,> A B) (|` C D)) (e. (<,> A B) C))) (opres.1 A C D opelres pm3.26bd A D e. a1i opres.1 A C D opelres biimpr expcom impbid))
+thm (resieq ((A x) (B x) (C x)) () (-> (/\ (e. B A) (e. C A)) (<-> (br B (|` (I) A) C) (= B C))) (x cv C B I A |` breq2 x cv C B eqeq2 bibi12d B A e. imbi2d x visset B A I opres x visset B A x cv V ideqg mpan2 B I x cv df-br syl5bbr bitrd B I A |` x cv df-br syl5bb A vtoclg impcom))
+thm (resundi () () (= (|` A (u. B C)) (u. (|` A B) (|` A C))) (B C V xpundir A ineq2i A B V X. C V X. indi eqtr A B C u. df-res A B df-res A C df-res uneq12i 3eqtr4))
+thm (resundir () () (= (|` (u. A B) C) (u. (|` A C) (|` B C))) (A B C V X. indir A B u. C df-res A C df-res B C df-res uneq12i 3eqtr4))
+thm (dmres ((x y) (A x) (A y) (B x) (B y)) () (= (dom (|` A B)) (i^i B (dom A))) (y visset x cv A B opelres y biex x visset A B |` y eldm2 x visset A y eldm2 x cv B e. anbi1i y x cv y cv <,> A e. x cv B e. 19.41v bitr4 3bitr4r ineqri A dom B incom eqtr3))
+thm (ssdmres () () (<-> (C_ A (dom B)) (= (dom (|` B A)) A)) (A B dom df-ss B A dmres A eqeq1i bitr4))
+thm (dmresexg () () (-> (e. B C) (e. (dom (|` A B)) (V))) (B C A dom inex1g A B dmres V eleq1i sylibr))
+thm (resss () () (C_ (|` A B) A) (A B df-res A B V X. inss1 eqsstr))
+thm (rescom () () (= (|` (|` A B) C) (|` (|` A C) B)) (A B V X. C V X. in23 A B df-res C V X. ineq1i A C df-res B V X. ineq1i 3eqtr4 A B |` C df-res A C |` B df-res 3eqtr4))
+thm (ssres () () (-> (C_ A B) (C_ (|` A C) (|` B C))) (A B C V X. ssrin A C df-res B C df-res 3sstr4g))
+thm (ssres2 () () (-> (C_ A B) (C_ (|` C A) (|` C B))) (V ssid A B V V ssxp mpan2 A V X. B V X. C sslin syl C A df-res C B df-res 3sstr4g))
+thm (relres () () (Rel (|` A B)) (A B df-res A B V X. inss2 B V xpss sstri eqsstr A B |` df-rel mpbir))
+thm (resabs1 () () (-> (C_ B C) (= (|` (|` A C) B) (|` A B))) (B C sseqin2 C B i^i B V xpeq1 sylbi C V B V inxp V inidm V V i^i V C B i^i xpeq2 ax-mp eqtr2 syl5reqr A ineq2d A C V X. B V X. inass syl6reqr A C |` B df-res A C df-res B V X. ineq1i eqtr A B df-res 3eqtr4g))
+thm (resabs2 () () (-> (C_ B C) (= (|` (|` A B) C) (|` A B))) (V ssid B C V V ssxp B V X. C V X. dfss sylib mpan2 A ineq2d A B V X. C V X. inass syl6reqr A B |` C df-res A B df-res C V X. ineq1i eqtr A B df-res 3eqtr4g))
+thm (residm () () (= (|` (|` A B) B) (|` A B)) (B ssid B B A resabs2 ax-mp))
+thm (resima () () (= (" (|` A B) B) (" A B)) (A B residm rneqi A B |` B df-ima A B df-ima 3eqtr4))
+thm (relssres ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (Rel A) (C_ (dom A) B)) (= (|` A B) A)) (A Rel A dom B C_ pm3.26 A dom B x cv ssel x visset y cv A opeldm syl5 ancld y visset x cv A B opelres syl6ibr A Rel adantl relssdv A B resss jctil A B |` A eqss sylibr))
+thm (resexg () () (-> (e. A C) (e. (|` A B) (V))) (A C B V X. inex1g A B df-res V eleq1i sylibr))
+thm (resopab ((x y) (A x) (A y)) () (= (|` ({<,>|} x y ph) A) ({<,>|} x y (/\ (e. (cv x) A) ph))) (x y ph {<,>|} A df-res A V x y df-xp y visset x cv A e. biantru x y biopabi eqtr4 x y ph {<,>|} ineq2i x y ph {<,>|} x y x cv A e. {<,>|} incom eqtr x y x cv A e. ph inopab 3eqtr))
+thm (iss ((x y) (A x) (A y)) () (<-> (C_ A (I)) (= A (|` (I) (dom A)))) (A I x cv y cv <,> ssel x cv I y cv df-br x visset y visset ideq bitr3 syl6ib pm4.71rd A I x cv y cv <,> ssel x cv I y cv df-br x visset y visset ideq bitr3 syl6ib pm4.71rd x cv y cv eqcom x cv y cv <,> A e. anbi1i syl6bb y biexdv x visset y cv x cv x cv opeq2 A eleq1d ceqsexv syl6bb x visset A y eldm2 syl5bb x cv y cv = anbi2d x cv y cv x cv opeq2 A eleq1d pm5.32i syl6bb bitr4d y visset x cv I A dom opelres x cv I y cv df-br x visset y visset ideq bitr3 x cv A dom e. anbi1i bitr2 syl6bb x y 19.21aivv reli A I ssrel mpi I A dom relres A I A dom |` x y cleqrel mpan2 syl mpbird I A dom resss A I A dom |` I sseq1 mpbiri impbi))
+thm (dmresi () () (= (dom (|` (I) A)) A) (A ssv dmi sseqtr4 A I ssdmres mpbi))
+thm (resid () () (-> (Rel A) (= (|` A (V)) A)) (A dom ssv A V relssres mpan2))
+thm (imaeq1 () () (-> (= A B) (= (" A C) (" B C))) (A B C reseq1 rneqd A C df-ima B C df-ima 3eqtr4g))
+thm (imaeq2 () () (-> (= A B) (= (" C A) (" C B))) (A B C reseq2 rneqd C A df-ima C B df-ima 3eqtr4g))
+thm (dfima2 ((x y) (A x) (A y) (B x) (B y)) () (= (" A B) ({|} y (E.e. x B (br (cv x) A (cv y))))) (A B df-ima A B |` y x dfrn3 A B df-res x cv y cv <,> eleq2i x cv y cv <,> A B V X. elin x cv B e. x cv A y cv br ancom x cv A y cv df-br y visset x cv B e. biantru y visset x cv B V opelxp bitr4 anbi12i bitr2 3bitr x biex x B x cv A y cv br df-rex bitr4 y biabi 3eqtr))
+thm (dfima3 ((x y) (A x) (A y) (B x) (B y)) () (= (" A B) ({|} y (E. x (/\ (e. (cv x) B) (e. (<,> (cv x) (cv y)) A))))) (A B y x dfima2 x B x cv A y cv br df-rex x cv A y cv df-br x cv B e. anbi2i x biex bitr y biabi eqtr))
+thm (elima ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E.e. x C (br (cv x) B A))) (elima.1 y cv A x cv B breq2 x C birexdv B C y x dfima2 elab2))
+thm (elima2 ((A x) (B x) (C x)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E. x (/\ (e. (cv x) C) (br (cv x) B A)))) (elima.1 B C x elima x C x cv B A br df-rex bitr))
+thm (elima3 ((A x) (B x) (C x)) ((elima.1 (e. A (V)))) (<-> (e. A (" B C)) (E. x (/\ (e. (cv x) C) (e. (<,> (cv x) A) B)))) (elima.1 B C x elima2 x cv B A df-br x cv C e. anbi2i x biex bitr))
+thm (hbima ((y z) (A y) (A z) (B y) (B z) (x y) (x z) (w x) (w y) (w z)) ((hbima.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbima.2 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (" A B)) (A. x (e. (cv y) (" A B)))) (w cv z cv e. x ax-17 hbima.2 hbel w cv z cv e. x ax-17 w cv y cv e. x ax-17 hbop hbima.1 hbel hban z hbex y visset A B z elima3 y visset A B z elima3 x bial 3imtr4))
+thm (imadmrn ((x y) (A x) (A y)) () (= (" A (dom A)) (ran A)) (x visset y cv A opeldm pm4.71i x cv y cv <,> A e. x cv A dom e. ancom bitr2 x biex y biabi A A dom y x dfima3 A y x dfrn3 3eqtr4))
+thm (imassrn ((x y) (A x) (A y) (B x) (B y)) () (C_ (" A B) (ran A)) (x cv B e. x cv y cv <,> A e. pm3.27 x 19.22i y ss2abi A B y x dfima3 A y x dfrn3 3sstr4))
+thm (imaexg () () (-> (e. A C) (e. (" A B) (V))) (A C rnexg A B imassrn A ran V A B " ssexg mpi syl))
+thm (imai ((x y) (A x) (A y)) () (= (" (I) A) A) (I A y x dfima3 x cv I y cv df-br x visset y visset ideq bitr3 x cv A e. anbi2i x cv A e. x cv y cv = ancom bitr x biex y visset x cv y cv A eleq1 ceqsexv bitr y biabi y A abid2 3eqtr))
+thm (rnresi () () (= (ran (|` (I) A)) A) (I A df-ima A imai eqtr3))
+thm (ima0 () () (= (" A ({/})) ({/})) (A {/} df-ima A res0 rneqi rn0 3eqtr))
+thm (imasng ((x y) (A y) (A x) (R y) (R x)) () (-> (e. A B) (= (" R ({} A)) ({|} y (br A R (cv y))))) (A B elisset x cv A R y cv breq1 V ceqsexgv x A {} x cv R y cv br df-rex x A elsn x cv R y cv br anbi1i x biex bitr syl5bb y biabdv R A {} y x dfima2 syl5eq syl))
+thm (relimasn ((A y) (R y)) () (-> (Rel R) (= (" R ({} A)) ({|} y (br A R (cv y))))) (A snprc A {} {/} R imaeq2 sylbi R ima0 syl6eq R Rel adantl R A y cv brrelex exp con3d imp y nexdv y A R y cv br abn0 bicon1i sylib eqtr4d exp A V R y imasng pm2.61d2))
+thm (elimasn ((A x) (B x) (C x)) ((elimasn.1 (e. B (V))) (elimasn.2 (e. C (V)))) (<-> (e. C (" A ({} B))) (e. (<,> B C) A)) (elimasn.2 A B {} x elima3 x B elsn x cv C <,> A e. anbi1i x biex elimasn.1 x cv B C opeq1 A eleq1d ceqsexv 3bitr))
+thm (args ((x y) (F x) (F y)) () (= ({|} x (E. y (= (" F ({} (cv x))) ({} (cv y))))) ({|} x (E! y (br (cv x) F (cv y))))) (x visset x cv V F y imasng ax-mp y cv {} eqeq1i y biex y x cv F y cv br eusn bitr4 x biabi))
+thm (eliniseg ((A x) (B x) (C x)) ((eliniseg.1 (e. C (V)))) (-> (e. B D) (<-> (e. C (" (`' A) ({} B))) (br C A B))) (x cv B sneq x cv {} B {} A `' imaeq2 syl C eleq2d x cv B C A breq2 bibi12d x visset eliniseg.1 A `' elimasn x cv A `' C df-br x visset eliniseg.1 A brcnv bitr3 bitr D vtoclg))
+thm (iniseg ((A x) (B x)) () (-> (e. B C) (= (" (`' A) ({} B)) ({|} x (br (cv x) A B)))) (B C elisset x visset B V A eliniseg biabrdv syl))
+thm (dffr3 ((x y) (x z) (R x) (y z) (R y) (R z) (A x)) () (<-> (Fr R A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (= (i^i (cv x) (" (`' R) ({} (cv y)))) ({/})))))) (R A x y z dffr2 y visset y cv V R z iniseg ax-mp x cv ineq2i {/} eqeq1i y x cv birex x cv A C_ x cv {/} = -. /\ imbi2i x bial bitr4))
+thm (imass1 () () (-> (C_ A B) (C_ (" A C) (" B C))) (A B C ssres A C |` B C |` rnss syl A C df-ima B C df-ima 3sstr4g))
+thm (imass2 () () (-> (C_ A B) (C_ (" C A) (" C B))) (A B C ssres2 C A |` C B |` rnss syl C A df-ima C B df-ima 3sstr4g))
+thm (ndmima () () (-> (-. (e. A (dom B))) (= (" B ({} A)) ({/}))) (B dom A disjsn biimpr B A {} dmres A {} B dom incom eqtr syl5eq B A {} |` dm0rn0 sylib B A {} df-ima syl5eq))
+thm (relcnv ((x y) (A x) (A y)) () (Rel (`' A)) (x y y cv A x cv br relopab A x y df-cnv A `' x y y cv A x cv br {<,>|} releq ax-mp mpbir))
+thm (cotr ((x y) (x z) (R x) (y z) (R y) (R z)) () (<-> (C_ (o. R R) R) (A. x (A. y (A. z (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))))) (x z y x cv R y cv br y cv R z cv br /\ E. {<,>|} R x cv z cv <,> ssel x cv R z cv df-br syl6ibr x z y x cv R y cv br y cv R z cv br /\ E. opabid syl5ibr R R x z y df-co R sseq1i y x cv R y cv br y cv R z cv br /\ x cv R z cv br 19.23v 3imtr4 z 19.21aiv y z x cv R y cv br y cv R z cv br /\ x cv R z cv br -> alcom sylibr x 19.21aiv x z y x cv R y cv br y cv R z cv br /\ E. x cv R z cv br ssopab2 y z x cv R y cv br y cv R z cv br /\ x cv R z cv br -> alcom y x cv R y cv br y cv R z cv br /\ x cv R z cv br 19.23v z bial bitr x bial bitr4 x z R opabss x z y x cv R y cv br y cv R z cv br /\ E. {<,>|} x z x cv R z cv br {<,>|} R sstr2 mpi sylbir R R x z y df-co syl5ss impbi))
+thm (cnvsym ((x y) (R x) (R y)) () (<-> (C_ (`' R) R) (A. x (A. y (-> (br (cv x) R (cv y)) (br (cv y) R (cv x)))))) (R y x df-cnv R sseq1i y x x cv R y cv br {<,>|} R y cv x cv <,> ssel y cv R x cv df-br syl6ibr y x x cv R y cv br opabid syl5ibr sylbi x y 19.21aivv y x x cv R y cv br y cv R x cv br ssopab2 y x x cv R y cv br y cv R x cv br -> alcom bitr y x R opabss y x x cv R y cv br {<,>|} y x y cv R x cv br {<,>|} R sstr2 mpi sylbir R y x df-cnv syl5ss impbi))
+thm (intasym ((x y) (R x) (R y)) () (<-> (C_ (i^i R (`' R)) (I)) (A. x (A. y (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv x))) (= (cv x) (cv y)))))) (R R `' inss2 R relcnv R R `' i^i R `' ssrel mp2 R R `' i^i I x y relss ax-mp x cv R y cv df-br x visset y visset R brcnv x cv R `' y cv df-br bitr3 anbi12i x cv y cv <,> R R `' elin bitr4 x visset y visset ideq x cv I y cv df-br bitr3 imbi12i x y bi2al bitr4))
+thm (intirr ((x y) (R x) (R y)) () (<-> (= (i^i R (I)) ({/})) (A. x (-. (br (cv x) R (cv x))))) (R I inss2 reli R I i^i I ssrel mp2 rel0 R I i^i {/} x y cleqrel mp2an x cv R x cv df-br x visset y cv x cv x cv opeq2 R eleq1d ceqsexv bitr4 x cv y cv <,> noel x cv y cv <,> R I i^i e. nbn bicon1i x visset y visset ideq x cv I y cv df-br x cv y cv eqcom 3bitr3r x cv y cv <,> R e. anbi2i y cv x cv = x cv y cv <,> R e. ancom x cv y cv <,> R I elin 3bitr4r bitr2 y biex y x cv y cv <,> R I i^i e. x cv y cv <,> {/} e. <-> exnal 3bitr bicon2i x bial bitr))
+thm (soirri () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S)))) (-. (br A R A)) (soi.2 R S A sonr mpan A S e. adantl soi.1 soi.3 A brel con3i pm2.61i))
+thm (sotri () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S))) (sotri.4 (e. B (V))) (sotri.5 (e. C (V)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (A S e. B S e. C S e. /\/\ id 3exp B S e. a1d imp43 sotri.4 soi.3 A brel sotri.5 soi.3 B brel syl2an soi.2 R S A B C sotr mpan mpcom))
+thm (son2lpi () ((soi.1 (e. A (V))) (soi.2 (Or R S)) (soi.3 (C_ R (X. S S))) (son2lpi.4 (e. B (V)))) (-. (/\ (br A R B) (br B R A))) (soi.1 soi.2 soi.3 soirri soi.1 soi.2 soi.3 son2lpi.4 soi.1 sotri mto))
+thm (cnvopab ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (= (`' ({<,>|} x y ph)) ({<,>|} y x ph)) (x y ph {<,>|} relcnv y x ph relopab w visset z visset x y ph {<,>|} opelcnv y cv z cv w cv <,> e. x ax-17 z x y ph hbopab1 hbel y cv w cv z cv <,> e. x ax-17 z y x ph hbopab2 hbel hbbi x cv z cv w cv opeq1 x y ph {<,>|} eleq1d x cv z cv w cv opeq2 y x ph {<,>|} eleq1d bibi12d z cv x cv w cv <,> e. y ax-17 z x y ph hbopab2 hbel z cv w cv x cv <,> e. y ax-17 z y x ph hbopab1 hbel hbbi y cv w cv x cv opeq2 x y ph {<,>|} eleq1d y cv w cv x cv opeq1 y x ph {<,>|} eleq1d bibi12d x y ph opabid y x ph opabid bitr4 chvar chvar bitr cleqreli))
+thm (cnv0 ((x y)) () (= (`' ({/})) ({/})) ({/} relcnv rel0 x visset y visset {/} opelcnv x cv y cv <,> noel y cv x cv <,> noel 2false bitr4 cleqreli))
+thm (cnvi ((x y)) () (= (`' (I)) (I)) (I relcnv reli x cv y cv eqcom x cv I y cv df-br x visset y visset ideq bitr3 x visset y visset I brcnv x cv I `' y cv df-br y visset x visset ideq 3bitr3 3bitr4r cleqreli))
+thm (op1sta () ((op1sta.1 (e. A (V)))) (= (U. (dom ({} (<,> A B)))) A) (A B dmsnop unieqi op1sta.1 unisn eqtr))
+thm (cnvsn ((x y) (A x) (A y) (B x) (B y)) ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (`' ({} (<,> A B))) ({} (<,> B A))) (A B <,> {} relcnv cnvsn.2 A relsn y cv B = x cv A = ancom y cv x cv opex B A <,> elsnc y visset x visset cnvsn.1 B opth bitr y visset x visset A B <,> {} opelcnv x cv y cv opex A B <,> elsnc x visset y visset cnvsn.2 A opth 3bitr 3bitr4r cleqreli))
+thm (rnsnop () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (ran ({} (<,> A B))) ({} B)) (A B <,> {} df-rn cnvsn.1 cnvsn.2 cnvsn dmeqi B A dmsnop eqtr eqtr))
+thm (op2ndb () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (|^| (|^| (|^| (`' ({} (<,> A B)))))) B) (cnvsn.1 cnvsn.2 cnvsn inteqi B A opex intsn eqtr inteqi inteqi cnvsn.2 A op1stb eqtr))
+thm (op2nda () ((cnvsn.1 (e. A (V))) (cnvsn.2 (e. B (V)))) (= (U. (ran ({} (<,> A B)))) B) (A B <,> {} df-rn cnvsn.1 cnvsn.2 cnvsn dmeqi eqtr unieqi cnvsn.2 A op1sta eqtr))
+thm (elxp4 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (/\ (= A (<,> (U. (dom ({} A))) (U. (ran ({} A))))) (/\ (e. (U. (dom ({} A))) B) (e. (U. (ran ({} A))) C)))) (A B C x y elxp A x cv y cv <,> sneq rneqd unieqd x visset y visset op2nda syl6req pm4.71ri x cv B e. y cv C e. /\ anbi1i y cv A {} ran U. = A x cv y cv <,> = x cv B e. y cv C e. /\ anass bitr y biex A snex A {} V rnexg ax-mp uniex y cv A {} ran U. x cv opeq2 A eqeq2d y cv A {} ran U. C eleq1 x cv B e. anbi2d anbi12d ceqsexv bitr A x cv A {} ran U. <,> sneq dmeqd unieqd x visset A {} ran U. op1sta syl6req pm4.71ri x cv B e. A {} ran U. C e. /\ anbi1i x cv A {} dom U. = A x cv A {} ran U. <,> = x cv B e. A {} ran U. C e. /\ anass 3bitr x biex A snex A {} V dmexg ax-mp uniex x cv A {} dom U. A {} ran U. opeq1 A eqeq2d x cv A {} dom U. B eleq1 A {} ran U. C e. anbi1d anbi12d ceqsexv 3bitr))
+thm (elxp5 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (<-> (e. A (X. B C)) (/\ (= A (<,> (|^| (|^| A)) (U. (ran ({} A))))) (/\ (e. (|^| (|^| A)) B) (e. (U. (ran ({} A))) C)))) (A B C x y elxp A x cv y cv <,> sneq rneqd unieqd x visset y visset op2nda syl6req pm4.71ri x cv B e. y cv C e. /\ anbi1i y cv A {} ran U. = A x cv y cv <,> = x cv B e. y cv C e. /\ anass bitr y biex A snex A {} V rnexg ax-mp uniex y cv A {} ran U. x cv opeq2 A eqeq2d y cv A {} ran U. C eleq1 x cv B e. anbi2d anbi12d ceqsexv bitr A x cv A {} ran U. <,> inteq inteqd x visset A {} ran U. op1stb syl6req pm4.71ri x cv B e. A {} ran U. C e. /\ anbi1i x cv A |^| |^| = A x cv A {} ran U. <,> = x cv B e. A {} ran U. C e. /\ anass 3bitr x biex x visset x cv A |^| |^| V eleq1 mpbii A x cv A {} ran U. <,> = x cv B e. A {} ran U. C e. /\ /\ adantr x 19.23aiv A |^| |^| B elisset A {} ran U. C e. adantr A A |^| |^| A {} ran U. <,> = adantl x cv A |^| |^| A {} ran U. opeq1 A eqeq2d x cv A |^| |^| B eleq1 A {} ran U. C e. anbi1d anbi12d V ceqsexgv pm5.21nii 3bitr))
+thm (cnvun ((x y) (A x) (A y) (B x) (B y)) () (= (`' (u. A B)) (u. (`' A) (`' B))) (A B u. relcnv A `' B `' relun A relcnv B relcnv mpbir2an y cv x cv <,> A B elun x visset y visset A opelcnv x visset y visset B opelcnv orbi12i bitr4 x visset y visset A B u. opelcnv x cv y cv <,> A `' B `' elun 3bitr4 cleqreli))
+thm (cnvin ((x y) (A x) (A y) (B x) (B y)) () (= (`' (i^i A B)) (i^i (`' A) (`' B))) (A B i^i relcnv A relcnv A `' B `' relin ax-mp y cv x cv <,> A B elin x visset y visset A opelcnv x visset y visset B opelcnv anbi12i bitr4 x visset y visset A B i^i opelcnv x cv y cv <,> A `' B `' elin 3bitr4 cleqreli))
+thm (rnun () () (= (ran (u. A B)) (u. (ran A) (ran B))) (A B cnvun dmeqi A `' B `' dmun eqtr A B u. df-rn A df-rn B df-rn uneq12i 3eqtr4))
+thm (rnin () () (C_ (ran (i^i A B)) (i^i (ran A) (ran B))) (A B cnvin dmeqi A `' B `' dmin eqsstr A B i^i df-rn A df-rn B df-rn ineq12i 3sstr4))
+thm (rnuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (ran (U. A)) (U_ x A (ran (cv x)))) (y cv z cv <,> A x eluni y biex y x y cv z cv <,> x cv e. x cv A e. /\ excom y y cv z cv <,> x cv e. E. x cv A e. ancom y y cv z cv <,> x cv e. x cv A e. 19.41v z visset x cv y elrn x cv A e. anbi2i 3bitr4 x biex 3bitr x A z cv x cv ran e. df-rex bitr4 z visset A U. y elrn z cv x A x cv ran eliun 3bitr4 eqri))
+thm (imaun () () (= (" A (u. B C)) (u. (" A B) (" A C))) (A B C resundi rneqi A B |` A C |` rnun eqtr A B C u. df-ima A B df-ima A C df-ima uneq12i 3eqtr4))
+thm (dminss ((x y) (A x) (A y) (R x) (R y)) () (C_ (i^i (dom R) A) (" (`' R) (" R A))) (x cv A e. x cv R y cv br /\ x 19.8a ancoms y visset R A x elima2 sylibr x cv R y cv br x cv A e. pm3.26 y visset x visset R brcnv sylibr jca y 19.22i x visset R y eldm x cv A e. anbi1i x cv R dom A elin y x cv R y cv br x cv A e. 19.41v 3bitr4 x visset R `' R A " y elima2 3imtr4 ssriv))
+thm (imainss ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (C_ (i^i (" R A) B) (" R (i^i A (" (`' R) B)))) (y cv B e. y cv R `' x cv br /\ y 19.8a y visset x visset R brcnv sylan2br ancoms x cv A e. anim2i x cv R y cv br y cv B e. pm3.26 x cv A e. adantl jca anassrs x cv A R `' B " elin x visset R `' B y elima2 x cv A e. anbi2i bitr x cv R y cv br anbi1i sylibr x 19.22i y visset R A x elima2 y cv B e. anbi1i y cv R A " B elin x x cv A e. x cv R y cv br /\ y cv B e. 19.41v 3bitr4 y visset R A R `' B " i^i x elima2 3imtr4 ssriv))
+thm (imaiun ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (" A (U. B)) (U_ x B (" A (cv x)))) (z cv B x eluni z cv y cv <,> A e. anbi1i z biex x z cv x cv e. x cv B e. /\ z cv y cv <,> A e. 19.41v z cv x cv e. x cv B e. z cv y cv <,> A e. anass z cv x cv e. x cv B e. z cv y cv <,> A e. an12 bitr x biex bitr3 z biex z x x cv B e. z cv x cv e. z cv y cv <,> A e. /\ /\ excom x z x cv B e. z cv x cv e. z cv y cv <,> A e. /\ exdistr 3bitr y visset A x cv z elima3 x B birex x B z z cv x cv e. z cv y cv <,> A e. /\ E. df-rex bitr2 3bitr y visset A B U. z elima3 y cv x B A x cv " eliun 3bitr4 eqri))
+thm (cnvxp ((x y) (A x) (A y) (B x) (B y)) () (= (`' (X. A B)) (X. B A)) (A B X. relcnv B A relxp x visset y visset A B X. opelcnv y cv A e. x cv B e. ancom x visset y cv A B opelxp y visset x cv B A opelxp 3bitr4 bitr cleqreli))
+thm (xp0 () () (= (X. A ({/})) ({/})) (A xp0r {/} A X. {/} cnveq ax-mp {/} A cnvxp cnv0 3eqtr3))
+thm (xpdisj1 () () (-> (= (i^i A B) ({/})) (= (i^i (X. A C) (X. B D)) ({/}))) (A B i^i {/} C D i^i xpeq1 C D i^i xp0r syl6eq A C B D inxp syl5eq))
+thm (xpdisj2 () () (-> (= (i^i A B) ({/})) (= (i^i (X. C A) (X. D B)) ({/}))) (A B i^i {/} C D i^i xpeq2 C D i^i xp0 syl6eq C A D B inxp syl5eq))
+thm (xpsndisj () () (-> (-. (= B D)) (= (i^i (X. A ({} B)) (X. C ({} D))) ({/}))) (B D disjsn2 B {} D {} A C xpdisj2 syl))
+thm (resdisj () () (-> (= (i^i A B) ({/})) (= (|` (|` C A) B) ({/}))) (A B V V xpdisj1 C ineq2d C in0 syl6eq C A |` B df-res C A df-res B V X. ineq1i C A V X. B V X. inass 3eqtr syl5eq))
+thm (rnxp () () (-> (-. (= A ({/}))) (= (ran (X. A B)) B)) (A B dmxp A B X. df-rn A B cnvxp dmeqi eqtr syl5eq))
+thm (relco ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (Rel (o. A B)) (x y z x cv B z cv br z cv A y cv br /\ E. relopab A B x y z df-co A B o. x y z x cv B z cv br z cv A y cv br /\ E. {<,>|} releq ax-mp mpbir))
+thm (cores ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (C_ (ran B) C) (= (o. (|` A C) B) (o. A B))) (x cv B z cv br z cv C e. pm3.26 z cv A y cv br anim1i B ran C C_ a1i B ran C z cv ssel x visset z visset B brelrn syl5 ancld z cv A y cv br anim1d impbid y visset z cv A C opelres z cv A C |` y cv df-br z cv A y cv df-br z cv C e. anbi1i 3bitr4 z cv A y cv br z cv C e. ancom bitr x cv B z cv br anbi2i x cv B z cv br z cv C e. z cv A y cv br anass bitr4 syl5bb z biexdv x visset y visset A C |` B z opelco x visset y visset A B z opelco 3bitr4g x y 19.21aivv A C |` B relco A B relco A C |` B o. A B o. x y cleqrel mp2an sylibr))
+thm (dfrel2 ((x y) (R x) (R y)) () (<-> (Rel R) (= (`' (`' R)) R)) (R `' relcnv x visset y visset R `' opelcnv y visset x visset R opelcnv bitr x y gen2 R `' `' R x y cleqrel mpbiri mpan R `' relcnv R `' `' R releq mpbii impbi))
+thm (cnvcnv () () (= (`' (`' A)) (i^i A (X. (V) (V)))) (A V V X. cnvin A V V X. i^i `' A `' V V X. `' i^i cnveq ax-mp A V V X. inss2 A V V X. i^i df-rel mpbir A V V X. i^i dfrel2 mpbi A `' V V X. `' cnvin A `' relcnv A `' `' df-rel mpbi V V relxp V V X. dfrel2 mpbi sseqtr4 A `' `' V V X. `' `' dfss mpbi eqtr4 3eqtr3r))
+thm (cnvcnvss () () (C_ (`' (`' A)) A) (A cnvcnv A V V X. inss1 eqsstr))
+thm (co02 ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (o. A ({/})) ({/})) (A {/} relco rel0 x cv z cv <,> noel x cv {/} z cv df-br mtbir z cv A y cv br intnanr z nex x visset y visset A {/} z opelco mtbir x cv y cv <,> noel 2false cleqreli))
+thm (co01 () () (= (o. ({/}) A) ({/})) (cnv0 A `' coeq2i A `' co02 eqtr2 cnv0 {/} A cnvco 3eqtr4 {/} `' {/} A o. `' cnveq ax-mp rel0 {/} dfrel2 mpbi {/} A relco {/} A o. dfrel2 mpbi 3eqtr3r))
+thm (coi1 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (Rel A) (= (o. A (I)) A)) (A I relco x visset y visset A I z opelco x visset z visset ideq x cv z cv eqcom bitr z cv A y cv br anbi1i z biex x visset z cv x cv A y cv breq1 ceqsexv 3bitr x cv A y cv df-br bitr x y gen2 A I o. A x y cleqrel mpbiri mpan))
+thm (coi2 () () (-> (Rel A) (= (o. (I) A) A)) (A dfrel2 cnvi A `' `' A I `' coeq2 I `' I A coeq1 sylan9eq mpan2 sylbi A `' I cnvco A relcnv A `' coi1 ax-mp A `' I o. A `' cnveq ax-mp eqtr3 syl5reqr A dfrel2 biimp eqtrd))
+thm (coass ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) () (= (o. (o. A B) C) (o. A (o. B C))) (A B o. C relco A B C o. relco z w x cv C z cv br z cv B w cv br w cv A y cv br /\ /\ excom x cv C z cv br z cv B w cv br w cv A y cv br anass w z bi2ex bitr4 z cv A B o. y cv df-br z visset y visset A B w opelco bitr x cv C z cv br anbi2i z biex x visset y visset A B o. C z opelco w x cv C z cv br z cv B w cv br w cv A y cv br /\ 19.42v z biex 3bitr4 x cv B C o. w cv df-br x visset w visset B C z opelco bitr w cv A y cv br anbi1i w biex x visset y visset A B C o. w opelco z x cv C z cv br z cv B w cv br /\ w cv A y cv br 19.41v w biex 3bitr4 3bitr4 cleqreli))
+thm (relssdr ((x y) (A x) (A y)) () (-> (Rel A) (C_ A (X. (dom A) (ran A)))) (A Rel id x cv y cv <,> A e. y 19.8a x cv y cv <,> A e. x 19.8a jca y visset x cv A dom A ran opelxp x visset A y eldm2 y visset A x elrn anbi12i bitr sylibr A Rel a1i relssdv))
+thm (cnvexg () () (-> (e. A B) (e. (`' A) (V))) (A relcnv A `' relssdr ax-mp A B rnexg A df-rn V eleq1i sylib A B dmexg A dfdm4 V eleq1i sylib jca A `' dom V A `' ran V xpexg A `' dom A `' ran X. V A `' ssexg 3syl mpi))
+thm (cnvex () ((cnvex.1 (e. A (V)))) (e. (`' A) (V)) (cnvex.1 A V cnvexg ax-mp))
+thm (cnvpo ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z)) () (<-> (Po R A) (Po (`' R) A)) (z A x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> r19.26 x A biral x A z A x cv R x cv br -. A.e. z A x cv R y cv br y cv R z cv br /\ x cv R z cv br -> A.e. r19.26 x A x cv R x cv br -. ralidm x A x cv R x cv br -. A.e. x A z A x cv R x cv br -. A.e. A.e. pm5.1 A x x cv R x cv br -. rzal A x z A x cv R x cv br -. A.e. rzal sylanc A x cv R x cv br -. z r19.3rzv x A biraldv pm2.61i bitr2 x A z A x cv R y cv br y cv R z cv br /\ x cv R z cv br -> A.e. A.e. anbi1i 3bitr x A x A x cv R x cv br -. A.e. z A x cv R y cv br y cv R z cv br /\ x cv R z cv br -> A.e. r19.26 bitr4 z A z cv R `' z cv br -. z cv R `' y cv br y cv R `' x cv br /\ z cv R `' x cv br -> r19.26 z cv x cv = id z cv x cv = id R breq12d z visset z visset R brcnv syl5bb negbid A cbvralv z visset y visset R brcnv y visset x visset R brcnv anbi12i y cv R z cv br x cv R y cv br ancom bitr z visset x visset R brcnv imbi12i z A biral anbi12i bitr2 x A biral x A z A z cv R `' z cv br -. z cv R `' y cv br y cv R `' x cv br /\ z cv R `' x cv br -> /\ ralcom 3bitr y A biral x A y A z A x cv R x cv br -. x cv R y cv br y cv R z cv br /\ x cv R z cv br -> /\ A.e. ralcom z A y A x A z cv R `' z cv br -. z cv R `' y cv br y cv R `' x cv br /\ z cv R `' x cv br -> /\ A.e. ralcom 3bitr4 R A x y z df-po R `' A z y x df-po 3bitr4))
+thm (cnvso ((x y) (A x) (A y) (R x) (R y)) () (<-> (Or R A) (Or (`' R) A)) (R A cnvpo x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ ralcom y visset x visset R brcnv y cv x cv eqcom x visset y visset R brcnv 3orbi123i y A x A bi2ral bitr4 anbi12i R A x y df-so R `' A y x df-so 3bitr4))
+thm (coexg () () (-> (/\ (e. A C) (e. B D)) (e. (o. A B) (V))) (A B relco A B o. relssdr A B dmco A B rnco A B o. dom B dom A B o. ran A ran ssxp mp2an A B o. A B o. dom A B o. ran X. B dom A ran X. sstr2 mpi syl ax-mp B D dmexg A C rnexg anim12i ancoms B dom V A ran V xpexg B dom A ran X. V A B o. ssexg 3syl mpi))
+thm (coex () ((coex.1 (e. A (V))) (coex.2 (e. B (V)))) (e. (o. A B) (V)) (coex.1 coex.2 A V B V coexg mp2an))
+thm (dmco2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (dom (o. A B)) (" (`' B) (dom A))) (x visset y visset A B z opelco y biex y z x cv B z cv br z cv A y cv br /\ excom y x cv B z cv br z cv A y cv br 19.42v z visset x visset B opelcnv x cv B z cv df-br bitr4 A z y df-dm eqabi anbi12i z cv x cv <,> B `' e. z cv A dom e. ancom bitr3 bitr z biex 3bitr x visset A B o. y eldm2 x visset B `' A dom z elima3 3bitr4 eqri))
+thm (dffun2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (A. y (A. z (-> (/\ (br (cv x) A (cv y)) (br (cv x) A (cv z))) (= (cv y) (cv z)))))))) (A df-fun y z df-id A A `' o. sseq2i A A `' y z x df-co y z y cv z cv = {<,>|} sseq1i y z x y cv A `' x cv br x cv A z cv br /\ E. y cv z cv = ssopab2 3bitr y visset x visset A brcnv x cv A z cv br anbi1i x biex y cv z cv = imbi1i x x cv A y cv br x cv A z cv br /\ y cv z cv = 19.23v bitr4 z bial z x x cv A y cv br x cv A z cv br /\ y cv z cv = -> alcom bitr y bial y x z x cv A y cv br x cv A z cv br /\ y cv z cv = -> A. alcom 3bitr A Rel anbi2i bitr))
+thm (dffun3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E. z (A. y (-> (br (cv x) A (cv y)) (= (cv y) (cv z)))))))) (A x y z dffun2 y cv z cv x cv A breq2 mo4 x cv A y cv br z ax-17 y mo2 bitr3 x bial A Rel anbi2i bitr))
+thm (dffun4 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (A. y (A. z (-> (/\ (e. (<,> (cv x) (cv y)) A) (e. (<,> (cv x) (cv z)) A)) (= (cv y) (cv z)))))))) (A x y z dffun2 x cv A y cv df-br x cv A z cv df-br anbi12i y cv z cv = imbi1i z bial x y bi2al A Rel anbi2i bitr))
+thm (dffun5 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E. z (A. y (-> (e. (<,> (cv x) (cv y)) A) (= (cv y) (cv z)))))))) (A x z y dffun3 x cv A y cv df-br y cv z cv = imbi1i y bial z biex x bial A Rel anbi2i bitr))
+thm (dffunmof ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A z) (A w) (A v) (A u)) ((dffunmof.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (dffunmof.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (<-> (Fun A) (/\ (Rel A) (A. x (E* y (br (cv x) A (cv y)))))) (A w u v dffun3 z cv w cv e. y ax-17 dffunmof.2 z cv v cv e. y ax-17 hbbr w cv A y cv br v ax-17 v cv y cv w cv A breq2 cbvmo w bial w cv A v cv br u ax-17 v mo2 w bial z cv w cv e. x ax-17 dffunmof.1 z cv y cv e. x ax-17 hbbr y hbmo y x cv A y cv br E* w ax-17 w cv x cv = y ax-17 w cv x cv A y cv breq1 bimod cbval 3bitr3r A Rel anbi2i bitr4))
+thm (dffunmo ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (Fun A) (/\ (Rel A) (A. x (E* y (br (cv x) A (cv y)))))) (z cv A e. x ax-17 z cv A e. y ax-17 dffunmof))
+thm (funmo ((x y) (A x) (A y)) () (-> (Fun A) (E* y (br (cv x) A (cv y)))) (A x y dffunmo pm3.27bd 19.21bi))
+thm (funrel () () (-> (Fun A) (Rel A)) (A df-fun pm3.26bd))
+thm (funss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Fun B) (Fun A))) (A B ssrel B funrel syl5 A B x cv y cv <,> ssel y cv z cv = imim1d y 19.20dv z 19.22dv x 19.20dv B x z y dffun5 pm3.27bd syl5 jcad A x z y dffun5 syl6ibr))
+thm (funeq () () (-> (= A B) (<-> (Fun A) (Fun B))) (B A funss A B funss anim12i ancoms A B eqss A Fun B Fun bi 3imtr4))
+thm (hbfun ((F y) (x y)) ((hbfun.1 (-> (e. (cv y) F) (A. x (e. (cv y) F))))) (-> (Fun F) (A. x (Fun F))) (hbfun.1 hbrel hbfun.1 hbfun.1 hbcnv hbco y cv I e. x ax-17 hbss hban F df-fun F df-fun x bial 3imtr4))
+thm (funeu ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (/\ (Fun F) (br (cv x) F (cv y))) (E! y (br (cv x) F (cv y)))) (x cv F y cv br y 19.8a F x z y dffun3 pm3.27bd 19.21bi anim12i x cv F y cv br z ax-17 y eu3 sylibr ancoms))
+thm (funeu2 ((x y) (F x) (F y)) () (-> (/\ (Fun F) (e. (<,> (cv x) (cv y)) F)) (E! y (e. (<,> (cv x) (cv y)) F))) (F x y funeu x cv F y cv df-br F Fun anbi2i x cv F y cv df-br y bieu 3imtr3))
+thm (dffun6 ((x y) (A x) (A y)) () (<-> (Fun A) (/\ (Rel A) (A.e. x (dom A) (E* y (br (cv x) A (cv y)))))) (A x y dffunmo y x cv A y cv br moabs x visset A y eldm y x cv A y cv br E* imbi1i bitr4 x bial x A dom y x cv A y cv br E* df-ral bitr4 A Rel anbi2i bitr))
+thm (dffun7 ((x y) (A x) (A y)) () (<-> (Fun A) (/\ (Rel A) (A.e. x (dom A) (E! y (br (cv x) A (cv y)))))) (A funrel A Fun y ax-17 y x cv y cv <,> A e. hbeu1 A x y funeu2 exp 19.23ad x visset A y eldm2 x cv A y cv df-br y bieu 3imtr4g r19.21aiv jca y x cv A y cv br eumo x A dom r19.20si A Rel anim2i A x y dffun6 sylibr impbi))
+thm (funfn () () (<-> (Fun A) (Fn A (dom A))) (A dom eqid A Fun biantru A A dom df-fn bitr4))
+thm (funsn ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((funsn.1 (e. A (V))) (funsn.2 (e. B (V)))) (Fun ({} (<,> A B))) (A B <,> {} x y z dffun4 funsn.1 B relsn y cv B z cv eqtr3t x cv y cv opex A B <,> elsnc x visset y visset funsn.2 A opth pm3.27bd sylbi x cv z cv opex A B <,> elsnc x visset z visset funsn.2 A opth pm3.27bd sylbi syl2an z ax-gen x y gen2 mpbir2an))
+thm (fun0 () () (Fun ({/})) ({/} {/} <,> {} 0ss 0ex 0ex funsn {/} {/} {/} <,> {} funss mp2))
+thm (funi () () (Fun (I)) (I df-fun reli I relcnv I `' coi2 ax-mp cnvi eqtr I ssid eqsstr mpbir2an))
+thm (nfunv () () (-. (Fun (V))) (V V 0nelxp 0ex V df-rel biimp {/} sseld mpi mto V funrel mto))
+thm (funopab ((x y) (x z) (y z) (ph z)) () (<-> (Fun ({<,>|} x y ph)) (A. x (E* y ph))) (z x y ph hbopab1 z x y ph hbopab2 dffunmof x y ph relopab mpbiran x cv x y ph {<,>|} y cv df-br x y ph opabid bitr y bimo x bial bitr))
+thm (funopabeq ((x y) (A y)) () (Fun ({<,>|} x y (= (cv y) A))) (x y y cv A = funopab y A moeq mpgbir))
+thm (funco ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (Fun F) (Fun G)) (Fun (o. F G))) (z x cv G z cv br y z cv F y cv br moexexv G z x funmo F z y dffunmo pm3.27bd syl2an ancoms x visset y visset F G z brco y bimo sylibr x 19.21aiv F G relco jctil F G o. x y dffunmo sylibr))
+thm (funres () () (-> (Fun F) (Fun (|` F A))) (F A resss F A |` F funss ax-mp))
+thm (funssres ((x y) (F x) (F y) (G x) (G y)) () (-> (/\ (Fun F) (C_ G F)) (= (|` F (dom G)) G)) (G F x cv y cv <,> ssel x visset y cv G opeldm G F C_ a1i jcad F Fun adantl y x cv y cv <,> F e. x cv y cv <,> G e. eupick F x y funeu2 G F x cv y cv <,> ssel ancrd y 19.22dv x visset G y eldm2 syl5ib imp syl2an exp43 com23 imp com34 pm2.43d imp3a impbid y visset x cv F G dom opelres syl6rbbr x y 19.21aivv G F ssrel F funrel syl5com imp F G dom relres jctil F G dom |` G x y cleqrel syl mpbird))
+thm (fun2ssres () () (-> (/\ (/\ (Fun F) (C_ G F)) (C_ A (dom G))) (= (|` F A) (|` G A))) (F G funssres F G dom |` G A reseq1 syl eqcomd A G dom F resabs1 sylan9eq eqcomd))
+thm (funun ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (/\ (Fun F) (Fun G)) (= (i^i (dom F) (dom G)) ({/}))) (Fun (u. F G))) (F funrel G funrel anim12i F G relun sylibr F dom G dom i^i {/} = adantr F dom G dom x disj1 biimp 19.21bi x cv F dom e. x cv G dom e. imnan sylib x visset y cv F opeldm x visset z cv G opeldm anim12i nsyl x cv y cv <,> F e. x cv z cv <,> G e. /\ x cv y cv <,> F e. x cv z cv <,> F e. /\ orel2 syl F dom G dom x disj1 biimp 19.21bi con2d x cv G dom e. x cv F dom e. imnan sylib x visset y cv G opeldm x visset z cv F opeldm anim12i nsyl x cv y cv <,> G e. x cv z cv <,> F e. /\ x cv y cv <,> G e. x cv z cv <,> G e. /\ orel1 syl orim12d F Fun G Fun /\ adantl x cv y cv <,> F G elun x cv z cv <,> F G elun anbi12i x cv y cv <,> F e. x cv y cv <,> G e. x cv z cv <,> F e. x cv z cv <,> G e. anddi bitr syl5ib F x y z dffun4 pm3.27bd 19.21bi 19.21bbi G x y z dffun4 pm3.27bd 19.21bi 19.21bbi jaao F dom G dom i^i {/} = adantr syld z 19.21aiv x y 19.21aivv jca F G u. x y z dffun4 sylibr))
+thm (funcnvcnv () () (-> (Fun A) (Fun (`' (`' A)))) (A cnvcnvss A `' `' A funss ax-mp))
+thm (funcnv2 ((x y) (A x) (A y)) () (<-> (Fun (`' A)) (A. y (E* x (br (cv x) A (cv y))))) (A `' y x dffunmo A relcnv y x y cv A `' x cv br E* A. biantrur y visset x visset A brcnv x bimo y bial bitr3 bitr))
+thm (funcnv ((x y) (A x) (A y)) () (<-> (Fun (`' A)) (A.e. y (ran A) (E* x (br (cv x) A (cv y))))) (x visset y visset A brelrn pm4.71ri x bimo x y cv A ran e. x cv A y cv br moanimv bitr y bial A y x funcnv2 y A ran x x cv A y cv br E* df-ral 3bitr4))
+thm (fun2cnv ((x y) (A x) (A y)) () (<-> (Fun (`' (`' A))) (A. x (E* y (br (cv x) A (cv y))))) (A `' x y funcnv2 y visset x visset A brcnv y bimo x bial bitr))
+thm (fununi ((f g) (f x) (f y) (f z) (f w) (f v) (A f) (g x) (g y) (g z) (g w) (g v) (A g) (x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (A.e. f A (/\ (Fun (cv f)) (A.e. g A (\/ (C_ (cv f) (cv g)) (C_ (cv g) (cv f)))))) (Fun (U. A))) (f cv funrel g A f cv g cv C_ g cv f cv C_ \/ A.e. adantr f A r19.20si A f reluni sylibr f cv Fun g A f cv g cv C_ g cv f cv C_ \/ r19.28av f A r19.20si w cv v cv x cv y cv <,> ssel x cv z cv <,> v cv e. anim1d v cv x y z dffun4 pm3.27bd y z x cv y cv <,> v cv e. x cv z cv <,> v cv e. /\ y cv z cv = -> A. ax-4 x a4s z x cv y cv <,> v cv e. x cv z cv <,> v cv e. /\ y cv z cv = -> ax-4 3syl syl9r w cv Fun adantl v cv w cv x cv z cv <,> ssel x cv y cv <,> w cv e. anim2d w cv x y z dffun4 pm3.27bd y z x cv y cv <,> w cv e. x cv z cv <,> w cv e. /\ y cv z cv = -> A. ax-4 x a4s z x cv y cv <,> w cv e. x cv z cv <,> w cv e. /\ y cv z cv = -> ax-4 3syl syl9r v cv Fun adantr jaod imp v A r19.20si w A r19.20si f cv w cv funeq f cv w cv g cv sseq1 f cv w cv g cv sseq2 orbi12d anbi12d g cv v cv w cv sseq2 g cv v cv w cv sseq1 orbi12d w cv Fun anbi2d A A cbvral2v f A g A f cv Fun f cv g cv C_ g cv f cv C_ \/ /\ ralcom g cv w cv f cv sseq1 g cv w cv f cv sseq2 orbi12d f cv g cv C_ g cv f cv C_ orcom syl5bb f cv Fun anbi2d f cv v cv funeq f cv v cv w cv sseq2 f cv v cv w cv sseq1 orbi12d anbi12d A A cbvral2v bitr anbi12i f A g A f cv Fun f cv g cv C_ g cv f cv C_ \/ /\ A.e. A.e. anidm w cv Fun v cv Fun w cv v cv C_ v cv w cv C_ \/ anandir w A v A bi2ral w A v A w cv Fun w cv v cv C_ v cv w cv C_ \/ /\ v cv Fun w cv v cv C_ v cv w cv C_ \/ /\ r19.26-2 bitr2 3bitr3 x cv y cv <,> A w eluni x cv z cv <,> A v eluni anbi12i w v x cv y cv <,> w cv e. w cv A e. /\ x cv z cv <,> v cv e. v cv A e. /\ eeanv x cv y cv <,> w cv e. w cv A e. x cv z cv <,> v cv e. v cv A e. an4 x cv y cv <,> w cv e. x cv z cv <,> v cv e. /\ w cv A e. v cv A e. /\ ancom bitr w v bi2ex bitr3 bitr y cv z cv = imbi1i v w cv A e. v cv A e. /\ x cv y cv <,> w cv e. x cv z cv <,> v cv e. /\ /\ y cv z cv = 19.23v w bial w cv A e. v cv A e. /\ x cv y cv <,> w cv e. x cv z cv <,> v cv e. /\ y cv z cv = impexp w v bi2al w A v A x cv y cv <,> w cv e. x cv z cv <,> v cv e. /\ y cv z cv = -> r2al bitr4 w v w cv A e. v cv A e. /\ x cv y cv <,> w cv e. x cv z cv <,> v cv e. /\ /\ E. y cv z cv = 19.23v 3bitr3r bitr 3imtr4 z 19.21aiv x y 19.21aivv syl jca A U. x y z dffun4 sylibr))
+thm (funcnvuni ((f g) (f x) (f y) (f z) (f w) (f v) (A f) (g x) (g y) (g z) (g w) (g v) (A g) (x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (A.e. f A (/\ (Fun (`' (cv f))) (A.e. g A (\/ (C_ (cv f) (cv g)) (C_ (cv g) (cv f)))))) (Fun (`' (U. A)))) (f cv v cv cnveq f cv `' v cv `' funeq syl f cv v cv g cv sseq1 f cv v cv g cv sseq2 orbi12d g A biraldv anbi12d A rcla4v z cv v cv `' funeq biimprcd g A v cv g cv C_ g cv v cv C_ \/ A.e. adantr g cv x cv v cv sseq2 g cv x cv v cv sseq1 orbi12d A rcla4v z cv v cv `' w cv x cv `' sseq12 ancoms w cv x cv `' z cv v cv `' sseq12 orbi12d v cv x cv cnvss x cv v cv cnvss orim12i syl5bir com12 exp3a syl6com r19.23adv com23 w 19.21adv v cv `' Fun adantl jcad syl6com r19.23adv x cv v cv cnveq z cv eqeq2d A cbvrexv syl5ib z 19.21aiv z y x A y cv x cv `' = E.e. {|} z cv Fun w y x A y cv x cv `' = E.e. {|} z cv w cv C_ w cv z cv C_ \/ A.e. /\ df-ral z visset y cv z cv x cv `' eqeq1 x A birexdv elab w y x A y cv x cv `' = E.e. {|} z cv w cv C_ w cv z cv C_ \/ df-ral w visset y cv w cv x cv `' eqeq1 x A birexdv elab z cv w cv C_ w cv z cv C_ \/ imbi1i w bial bitr z cv Fun anbi2i imbi12i z bial bitr2 sylib z y x A y cv x cv `' = E.e. {|} w fununi syl A x cnvuni x visset cnvex x A y dfiun2 eqtr A U. `' y x A y cv x cv `' = E.e. {|} U. funeq ax-mp sylibr))
+thm (fun11uni ((f g) (A f) (A g)) () (-> (A.e. f A (/\ (/\ (Fun (cv f)) (Fun (`' (cv f)))) (A.e. g A (\/ (C_ (cv f) (cv g)) (C_ (cv g) (cv f)))))) (/\ (Fun (U. A)) (Fun (`' (U. A))))) (f cv Fun f cv `' Fun pm3.26 g A f cv g cv C_ g cv f cv C_ \/ A.e. anim1i f A r19.20si f A g fununi syl f cv Fun f cv `' Fun pm3.27 g A f cv g cv C_ g cv f cv C_ \/ A.e. anim1i f A r19.20si f A g funcnvuni syl jca))
+thm (funin ((x y) (F x) (F y) (G x) (G y)) () (-> (Fun F) (Fun (i^i F G))) (F G relin y x cv F y cv br x cv y cv <,> G e. moan x cv y cv <,> G e. x cv F y cv br ancom x cv y cv <,> F G elin x cv F G i^i y cv df-br x cv F y cv df-br x cv y cv <,> G e. anbi1i 3bitr4 bitr4 y bimo sylib x 19.20i anim12i F x y dffunmo F G i^i x y dffunmo 3imtr4))
+thm (funres11 () () (-> (Fun (`' F)) (Fun (`' (|` F A)))) (F A resss F A |` F cnvss ax-mp F A |` `' F `' funss ax-mp))
+thm (funcnvres () () (-> (Fun (`' F)) (= (`' (|` F A)) (|` (`' F) (" F A)))) (F A resss F A |` F cnvss ax-mp F `' F A |` `' funssres mpan2 F A df-ima F A |` df-rn eqtr F A " F A |` `' dom F `' reseq2 ax-mp syl5req))
+thm (funcnvres2 () () (-> (Fun F) (= (`' (|` (`' F) A)) (|` F (" (`' F) A)))) (F funcnvcnv F `' A funcnvres syl F funrel F dfrel2 sylib F `' `' F F `' A " reseq1 syl eqtrd))
+thm (funimacnv () () (-> (Fun F) (= (" F (" (`' F) A)) (i^i A (ran F)))) (F funcnvcnv F `' A funcnvres syl F funrel F dfrel2 sylib F `' `' F F `' A " reseq1 syl eqtrd rneqd F F `' A " df-ima syl6reqr F `' A |` dfdm4 F `' A dmres F df-rn A ineq2i eqtr4 eqtr3 syl6eq))
+thm (funimass1 () () (-> (/\ (Fun F) (C_ A (ran F))) (-> (C_ (" (`' F) A) B) (C_ A (" F B)))) (F A funimacnv A F ran dfss biimp eqcomd sylan9eq F B " sseq1d F `' A " B F imass2 syl5bi))
+thm (funimass2 () () (-> (Fun F) (-> (C_ A (" (`' F) B)) (C_ (" F A) B))) (F B funimacnv F A " sseq2d B F ran inss1 F A " B F ran i^i B sstr2 mpi syl6bi A F `' B " F imass2 syl5))
+thm (imadif ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (Fun (`' F)) (= (" F (\ A B)) (\ (" F A) (" F B)))) (F `' Fun x ax-17 x x cv F y cv br x cv B e. -. /\ hbe1 hban x x cv F y cv br x cv B e. -. mopick F `' x y funmo y visset x visset F brcnv x bimo sylib sylan con2d x cv B e. x cv F y cv br imnan sylib 19.21ai exp x x cv F y cv br x cv B e. -. exancom x x cv B e. x cv F y cv br /\ alnex 3imtr3g x x cv A e. x cv F y cv br /\ E. anim2d x cv A e. x cv B e. -. x cv F y cv br anandir x biex x x cv A e. x cv F y cv br /\ x cv B e. -. x cv F y cv br /\ 19.40 sylbi syl5 x x cv A e. x cv F y cv br /\ x cv B e. x cv F y cv br /\ -. 19.29r x x cv B e. x cv F y cv br /\ alnex sylan2br x cv A e. x cv F y cv br /\ x cv B e. -. x cv F y cv br -. andi x cv B e. x cv F y cv br ianor x cv A e. x cv F y cv br /\ anbi2i x cv A e. x cv B e. -. x cv F y cv br an23 x cv F y cv br pm3.24 x cv A e. intnan x cv A e. x cv F y cv br x cv F y cv br -. anass mtbir x cv A e. x cv F y cv br /\ x cv B e. -. /\ biorfi bitr 3bitr4 x biex sylib F `' Fun a1i impbid x cv A B eldif x cv F y cv br anbi1i x biex y visset F A x elima2 y visset F B x elima2 negbii anbi12i 3bitr4g y visset F A B \ x elima2 y cv F A " F B " eldif 3bitr4g eqrd))
+thm (funimaexg ((B w) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. B C) (-> (Fun A) (e. (" A B) (V)))) (w cv B A imaeq2 V eleq1d A Fun imbi2d A x z y dffun5 pm3.27bd x cv y cv <,> A e. z ax-17 x y w zfrep2 A w cv " z isset A w cv y x dfima3 z cv eqeq2i z cv y x x cv w cv e. x cv y cv <,> A e. /\ E. eqab bitr z biex bitr sylibr syl C vtoclg))
+thm (funimaex () ((zfrep5.1 (e. B (V)))) (-> (Fun A) (e. (" A B) (V))) (zfrep5.1 B V A funimaexg ax-mp))
+thm (resfunexg () () (-> (e. B C) (-> (Fun A) (e. (|` A B) (V)))) (B C A dmresexg A Fun adantr B C A funimaexg A B df-ima V eleq1i syl6ib imp jca A B |` dom V A B |` ran V xpexg A B relres A B |` relssdr ax-mp A B |` dom A B |` ran X. V A B |` ssexg mpi 3syl exp))
+thm (fneq1 () () (-> (= F G) (<-> (Fn F A) (Fn G A))) (F G funeq F G dmeq A eqeq1d anbi12d F A df-fn G A df-fn 3bitr4g))
+thm (fneq2 () () (-> (= A B) (<-> (Fn F A) (Fn F B))) (A B F dom eqeq2 F Fun anbi2d F A df-fn F B df-fn 3bitr4g))
+thm (hbfn ((F y) (A y) (x y)) ((hbfn.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbfn.2 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (Fn F A) (A. x (Fn F A))) (hbfn.1 hbfun hbfn.1 hbdm hbfn.2 hbeq hban F A df-fn F A df-fn x bial 3imtr4))
+thm (fnfun () () (-> (Fn F A) (Fun F)) (F A df-fn pm3.26bd))
+thm (fnrel () () (-> (Fn F A) (Rel F)) (F A fnfun F funrel syl))
+thm (fndm () () (-> (Fn F A) (= (dom F) A)) (F A df-fn pm3.27bd))
+thm (funfni () ((funfni.1 (-> (/\ (Fun F) (e. B (dom F))) ph))) (-> (/\ (Fn F A) (e. B A)) ph) (funfni.1 F A fnfun B A e. adantr F A fndm B eleq2d biimpar sylanc))
+thm (fndmu () () (-> (/\ (Fn F A) (Fn F B)) (= A B)) (F A fndm eqcomd F B fndm sylan9eq))
+thm (fnbr () ((fnbr.1 (e. B (V))) (fnbr.2 (e. C (V)))) (-> (/\ (Fn F A) (br B F C)) (e. B A)) (F A fndm B eleq2d biimpa fnbr.1 F C breldm sylan2))
+thm (fnop () ((fnbr.1 (e. B (V))) (fnbr.2 (e. C (V)))) (-> (/\ (Fn F A) (e. (<,> B C) F)) (e. B A)) (fnbr.1 fnbr.2 F A fnbr B F C df-br sylan2br))
+thm (fneu ((x y) (F x) (F y) (B x) (B y) (A x)) () (-> (/\ (Fn F A) (e. B A)) (E! y (br B F (cv y)))) (x cv B A eleq1 F A Fn anbi2d x cv B F y cv breq1 y bieudv imbi12d F A fndm x cv eleq2d x visset F y eldm syl5bbr F Fun y ax-17 y x cv F y cv br hbeu1 hbim F x y funeu expcom 19.23ai F A fnfun syl5 syl6bir pm2.43a imp A vtoclg anabsi7))
+thm (fneu2 ((F y) (B y)) () (-> (/\ (Fn F A) (e. B A)) (E! y (e. (<,> B (cv y)) F))) (F A B y fneu B F y cv df-br y bieu sylib))
+thm (fnun () () (-> (/\ (/\ (Fn F A) (Fn G B)) (= (i^i A B) ({/}))) (Fn (u. F G) (u. A B))) (F dom A G dom B ineq12 {/} eqeq1d F Fun G Fun /\ anbi2d F G funun syl6bir F dom A G dom B uneq12 F G dmun syl5eq jctird F G u. A B u. df-fn syl6ibr exp3a impcom an4s F A df-fn G B df-fn syl2anb imp))
+thm (fnresdm () () (-> (Fn F A) (= (|` F A) F)) (F A relssres F A fnrel F A fndm F dom A eqimss syl sylanc))
+thm (fnresdisj () () (-> (Fn F A) (<-> (= (i^i A B) ({/})) (= (|` F B) ({/})))) (F A fndm B ineq1d F B dmres B F dom incom eqtr syl5eq {/} eqeq1d F B relres F B |` reldm0 ax-mp syl5rbb))
+thm (2elresin () () (-> (/\ (Fn F A) (Fn G B)) (<-> (/\ (e. (<,> (cv x) (cv y)) F) (e. (<,> (cv x) (cv z)) G)) (/\ (e. (<,> (cv x) (cv y)) (|` F (i^i A B))) (e. (<,> (cv x) (cv z)) (|` G (i^i A B)))))) (x visset y visset F A fnop x visset z visset G B fnop anim12i F A Fn G B Fn x cv y cv <,> F e. x cv z cv <,> G e. an4 x cv A B elin 3imtr4 y visset x cv A B i^i F opres z visset x cv A B i^i G opres anbi12d biimprd syl exp pm2.43d F A B i^i resss x cv y cv <,> sseli G A B i^i resss x cv z cv <,> sseli anim12i F A Fn G B Fn /\ a1i impbid))
+thm (fnssres () () (-> (/\ (Fn F A) (C_ B A)) (Fn (|` F B) B)) (F A fnfun F B funres syl B A C_ adantr F A fndm B sseq2d biimpar B F ssdmres sylib jca F B |` B df-fn sylibr))
+thm (fnresin1 () () (-> (Fn F A) (Fn (|` F (i^i A B)) (i^i A B))) (A B inss1 F A A B i^i fnssres mpan2))
+thm (fnresin2 () () (-> (Fn F A) (Fn (|` F (i^i B A)) (i^i B A))) (B A inss2 F A B A i^i fnssres mpan2))
+thm (fnresi () () (Fn (|` (I) A) A) (I A |` A df-fn funi I A funres ax-mp A ssv dmi sseqtr4 A I ssdmres mpbi mpbir2an))
+thm (fnima () () (-> (Fn F A) (= (" F A) (ran F))) (F A fndm F dom A F imaeq2 F imadmrn syl5reqr syl))
+thm (fn0 ((x y) (F x) (F y)) () (<-> (Fn F ({/})) (= F ({/}))) (F {/} fndm x cv noel F dom {/} x cv eleq2 mtbiri x visset F y eldm2 negbii sylib y x cv y cv <,> F e. alnex sylibr 19.21bi x cv y cv <,> noel jctir x cv y cv <,> F e. x cv y cv <,> {/} e. pm5.21 3syl x y 19.21aivv F {/} fnrel rel0 jctir F {/} x y cleqrel syl mpbird {/} {/} df-fn fun0 dm0 mpbir2an F {/} {/} fneq1 mpbiri impbi))
+thm (fnex () () (-> (e. A B) (-> (Fn F A) (e. F (V)))) (F A fndm B eleq1d biimprd A B F funimaexg F A fnfun syl5com F A fndm F dom A F imaeq2 syl F imadmrn syl5eqr V eleq1d sylibrd jcad F dom B F ran V xpexg syl6 F dom F ran X. V F ssexg F A fnrel F relssdr syl syl5com syld com12))
+thm (funex () () (-> (e. (dom F) B) (-> (Fun F) (e. F (V)))) (F dom B F fnex F F dom df-fn F dom eqid mpbiran2 syl5ibr))
+thm (funopabex ((x y) (A x) (A y)) ((funopabex.1 (e. A (V))) (funopabex.2 (-> (e. (cv x) A) (E* y ph)))) (e. ({<,>|} x y (/\ (e. (cv x) A) ph)) (V)) (funopabex.1 x y A ph dmopabss ssexi x y x cv A e. ph /\ funopab funopabex.2 y x cv A e. ph moanimv mpbir mpgbir x y x cv A e. ph /\ {<,>|} V funex mp2))
+thm (funopabex2 ((x y) (A x) (A y) (B y)) ((funopabex2.1 (e. A (V)))) (e. ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) (V)) (funopabex2.1 y B moeq x cv A e. a1i funopabex))
+thm (funrnex () () (-> (e. (dom F) B) (-> (Fun F) (e. (ran F) (V)))) (F B funex F V rnexg syl6))
+thm (zfrep6 ((ph z) (ph w) (ph v) (w z) (v z) (v w) (x y) (x z) (w x) (v x) (y z) (w y) (v y)) () (-> (A.e. x (cv z) (E! y ph)) (E. w (A.e. x (cv z) (E.e. y (cv w) ph)))) (v cv x y x cv z cv e. ph /\ {<,>|} ran e. w ax-17 x z cv y x y x cv z cv e. ph /\ {<,>|} ran ph E.e. A.e. w ax-17 w x y x cv z cv e. ph /\ hbopab1 hbrn hbeleq v cv w cv e. y ax-17 v x y x cv z cv e. ph /\ hbopab2 hbrn ph rexeqf z cv birald V cla4egf x y x cv z cv e. ph /\ {<,>|} V funrnex z visset y ph euex x z cv r19.20si z cv x y ph E. rabid2 sylibr y x cv z cv e. ph 19.42v x biabi x y x cv z cv e. ph /\ dmopab x z cv y ph E. df-rab 3eqtr4 syl6reqr V eleq1d mpbiri y ph eumo x cv z cv e. imim2i y x cv z cv e. ph moanimv sylibr x 19.20i x z cv y ph E! df-ral x y x cv z cv e. ph /\ funopab 3imtr4 sylc x z cv y ph E! hbra1 y ph euex x z cv r19.20si z cv x y ph E. rabid2 sylibr y x cv z cv e. ph 19.42v x biabi x y x cv z cv e. ph /\ dmopab x z cv y ph E. df-rab 3eqtr4 syl6reqr x cv eleq2d x y x cv z cv e. ph /\ opabid x visset y visset x y x cv z cv e. ph /\ {<,>|} opelrn sylbir exp impac y 19.22i x y x cv z cv e. ph /\ dmopab eqabi y x y x cv z cv e. ph /\ {<,>|} ran ph df-rex 3imtr4 syl6bir r19.21ai sylc))
+thm (fnopabg ((x y) (x z) (A x) (y z) (A y) (A z) (F z) (w x) (w y) (w z)) ((fnopabg.1 (= F ({<,>|} x y (/\ (e. (cv x) A) ph))))) (<-> (A.e. x A (E! y ph)) (Fn F A)) (x A y ph E! hbra1 x A y ph E! ra4 y ph eumo x cv A e. imim2i y x cv A e. ph moanimv sylibr syl 19.21ai x y x cv A e. ph /\ funopab sylibr y ph euex x A r19.20si x A y ph dmopab3 sylib jca x y x cv A e. ph /\ {<,>|} A df-fn sylibr fnopabg.1 F x y x cv A e. ph /\ {<,>|} A fneq1 ax-mp sylibr z x y x cv A e. ph /\ hbopab1 fnopabg.1 z cv eleq2i fnopabg.1 z cv eleq2i x bial 3imtr4 z cv A e. x ax-17 hbfn F A x cv z fneu2 w cv x cv z cv <,> e. y ax-17 z x y x cv A e. ph /\ hbopab2 fnopabg.1 z cv eleq2i fnopabg.1 z cv eleq2i y bial 3imtr4 hbel x cv y cv <,> F e. z ax-17 z cv y cv x cv opeq2 F eleq1d cbveu sylib fnopabg.1 x cv y cv <,> eleq2i x y x cv A e. ph /\ opabid bitr y bieu y x cv A e. ph euanv bitr pm3.27bd syl exp r19.21ai impbi))
+thm (fnopab ((x y) (A x) (A y)) ((fnopab.1 (-> (e. (cv x) A) (E! y ph))) (fnopab.2 (= F ({<,>|} x y (/\ (e. (cv x) A) ph))))) (Fn F A) (fnopab.1 rgen fnopab.2 fnopabg mpbi))
+thm (fnopab2 ((x y) (A x) (A y) (B y)) ((fnopab2.1 (e. B (V))) (fnopab2.2 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (Fn F A) (fnopab2.1 y eueq1 x cv A e. a1i fnopab2.2 fnopab))
+thm (dmopab2 ((x y) (A x) (A y) (B y)) ((fnopab2.1 (e. B (V))) (fnopab2.2 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (= (dom F) A) (fnopab2.1 fnopab2.2 fnopab2 F A fndm ax-mp))
+thm (feq1 () () (-> (= F G) (<-> (:--> F A B) (:--> G A B))) (F G A fneq1 F G rneq B sseq1d anbi12d F A B df-f G A B df-f 3bitr4g))
+thm (feq2 () () (-> (= A B) (<-> (:--> F A C) (:--> F B C))) (A B F fneq2 F ran C C_ anbi1d F A C df-f F B C df-f 3bitr4g))
+thm (feq3 () () (-> (= A B) (<-> (:--> F C A) (:--> F C B))) (A B F ran sseq2 F C Fn anbi2d F C A df-f F C B df-f 3bitr4g))
+thm (hbf ((F y) (A y) (B y) (x y)) ((hbf.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbf.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbf.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (:--> F A B) (A. x (:--> F A B))) (hbf.1 hbf.2 hbfn hbf.1 hbrn hbf.3 hbss hban F A B df-f F A B df-f x bial 3imtr4))
+thm (ffn () () (-> (:--> F A B) (Fn F A)) (F A B df-f pm3.26bd))
+thm (fnf () () (<-> (Fn F A) (:--> F A (V))) (F ran ssv F A V df-f biimpr mpan2 F A V ffn impbi))
+thm (ffun () () (-> (:--> F A B) (Fun F)) (F A B ffn F A fnfun syl))
+thm (frel () () (-> (:--> F A B) (Rel F)) (F A B ffn F A fnrel syl))
+thm (fdm () () (-> (:--> F A B) (= (dom F) A)) (F A B ffn F A fndm syl))
+thm (frn () () (-> (:--> F A B) (C_ (ran F) B)) (F A B df-f pm3.27bd))
+thm (fnfrn () () (<-> (Fn F A) (:--> F A (ran F))) (F ran ssid F A Fn biantru F A F ran df-f bitr4))
+thm (fss () () (-> (/\ (:--> F A B) (C_ B C)) (:--> F A C)) (F A B ffn B C C_ adantr F A B frn F ran B C sstr2 syl imp jca F A C df-f sylibr))
+thm (fco () () (-> (/\ (:--> F B C) (:--> G A B)) (:--> (o. F G) A C)) (F G funco F B C ffun G A B ffun syl2an F B C fdm G ran sseq2d G A B frn syl5bir imp G F dmcosseq syl G A B fdm F B C :--> adantl eqtrd jca F G o. A df-fn sylibr F G rnco F G o. ran F ran C sstr2 F B C frn syl5 ax-mp G A B :--> adantr jca F G o. A C df-f sylibr))
+thm (fssxp () () (-> (:--> F A B) (C_ F (X. A B))) (F A B frn A ssid A A F ran B ssxp mpan syl F A B fdm F dom A F ran xpeq1 F dom F ran X. A F ran X. A B X. sseq1 3syl mpbird F A B frel F relssdr F F dom F ran X. A B X. sstr2 3syl mpd))
+thm (opelf () ((opelf.1 (e. D (V)))) (-> (/\ (:--> F A B) (e. (<,> C D) F)) (/\ (e. C A) (e. D B))) (F A B fssxp C D <,> sseld opelf.1 C A B opelxp syl6ib imp))
+thm (fun () () (-> (/\ (/\ (:--> F A C) (:--> G B D)) (= (i^i A B) ({/}))) (:--> (u. F G) (u. A B) (u. C D))) (F A G B fnun expcom F ran C G ran D unss12 F G rnun syl5ss A B i^i {/} = a1i anim12d F A C df-f G B D df-f anbi12i F A Fn F ran C C_ G B Fn G ran D C_ an4 bitr F G u. A B u. C D u. df-f 3imtr4g impcom))
+thm (fssres () () (-> (/\ (:--> F A B) (C_ C A)) (:--> (|` F C) C B)) (F A C fnssres F C resss F C |` F rnss ax-mp F C |` ran F ran B sstr mpan anim12i an1rs F A B df-f sylanb F C |` C B df-f sylibr))
+thm (fssres2 () () (-> (/\ (:--> (|` F A) A B) (C_ C A)) (:--> (|` F C) C B)) (F A |` A B C fssres C A F resabs1 F A |` C |` F C |` C B feq1 syl F A |` A B :--> adantl mpbid))
+thm (fcoi1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (-> (:--> F A B) (= (o. F (|` (I) A)) F)) (F A B ffn x visset y visset F A fnop exp pm4.71rd x visset y visset x cv V y cv V F I A |` z opelcog mp2an z visset x cv I A opelres x visset z visset ideq x cv I z cv df-br x cv z cv eqcom 3bitr3 x cv A e. anbi1i bitr z cv y cv <,> F e. anbi1i z cv x cv = x cv A e. z cv y cv <,> F e. anass bitr z biex x visset z cv x cv y cv opeq1 F eleq1d x cv A e. anbi2d ceqsexv 3bitr syl6rbbr x y 19.21aivv F A fnrel F I A |` relco F I A |` o. F x y cleqrel mpan syl mpbird syl))
+thm (fcoi2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (-> (:--> F A B) (= (o. (|` (I) B) F) F)) (y visset F A B x cv opelf pm3.27d exp x cv y cv <,> F e. y cv B e. pm4.71 sylib x visset y visset x cv V y cv V I B |` F z opelcog mp2an y visset z cv I B opelres z cv I y cv df-br z visset y visset ideq bitr3 z cv B e. anbi1i bitr x cv z cv <,> F e. anbi2i x cv z cv <,> F e. z cv y cv = z cv B e. an12 bitr z biex y visset z cv y cv x cv opeq2 F eleq1d z cv y cv B eleq1 anbi12d ceqsexv 3bitr syl6rbbr x y 19.21aivv F A B frel I B |` F relco I B |` F o. F x y cleqrel mpan syl mpbird))
+thm (feu ((F y) (A y) (B y) (C y)) () (-> (/\ (:--> F A B) (e. C A)) (E!e. y B (e. (<,> C (cv y)) F))) (F A C y fneu2 F A B ffn sylan y visset F A B C opelf pm3.27d exp pm4.71rd y bieudv C A e. adantr mpbid y B C y cv <,> F e. df-reu sylibr))
+thm (fcnvres ((x y) (F x) (F y) (A x) (A y) (B x) (B y)) () (-> (:--> F A B) (= (`' (|` F A)) (|` (`' F) B))) (y visset F A B x cv opelf pm3.26d exp x cv y cv <,> F e. x cv A e. pm4.71 sylib y visset x visset F A |` opelcnv y visset x cv F A opelres bitr syl6bbr y visset F A B x cv opelf pm3.27d exp x cv y cv <,> F e. y cv B e. pm4.71 sylib x visset y cv F `' B opelres y visset x visset F opelcnv y cv B e. anbi1i bitr syl6bbr bitr3d y x 19.21aivv F A |` relcnv F `' B relres F A |` `' F `' B |` y x cleqrel mp2an sylibr))
+thm (fint ((A x) (B x) (F x)) ((fint.1 (-. (= B ({/}))))) (<-> (:--> F A (|^| B)) (A.e. x B (:--> F A (cv x)))) (fint.1 B F A Fn x r19.3rzv ax-mp F ran B x ssint anbi12i x B F A Fn F ran x cv C_ r19.26 bitr4 F A B |^| df-f F A x cv df-f x B biral 3bitr4))
+thm (fin () () (<-> (:--> F A (i^i B C)) (/\ (:--> F A B) (:--> F A C))) (F A Fn anidm F ran B C ssin anbi12i F A Fn F A Fn F ran B C_ F ran C C_ an4 bitr3 F A B C i^i df-f F A B df-f F A C df-f anbi12i 3bitr4))
+thm (fex () () (-> (e. A C) (-> (:--> F A B) (e. F (V)))) (A C F fnex F A B ffn syl5))
+thm (f0 () () (:--> ({/}) ({/}) A) ({/} {/} A df-f {/} eqid {/} fn0 mpbir rn0 A 0ss eqsstr mpbir2an))
+thm (f00 () () (<-> (:--> F A ({/})) (/\ (= F ({/})) (= A ({/})))) (F A {/} ffun F A {/} frn F ran ss0 syl F dm0rn0 sylibr jca F {/} df-fn sylibr F fn0 sylib F A {/} fdm F A {/} frn F ran ss0 syl F dm0rn0 sylibr eqtr3d jca {/} f0 F {/} A {/} feq1 A {/} {/} {/} feq2 sylan9bb mpbiri impbi))
+thm (fconst ((x y) (A x) (A y) (B x) (B y)) ((fconst.1 (e. B (V)))) (:--> (X. A ({} B)) A ({} B)) (B {} f0 A {/} B {} xpeq1 B {} xp0r syl6eq A B {} X. {/} A B {} feq1 syl A {/} {/} B {} feq2 bitrd mpbiri A B {} dmxp A B {} X. df-rn A B {} cnvxp dmeqi eqtr syl5eq A B {} X. ran B {} eqimss syl A B {} X. A df-fn A B {} X. x y dffunmo A B {} relxp y B moeq x cv A e. moani y visset x cv A B {} brxp y B elsn x cv A e. anbi2i bitr y bimo mpbir x ax-gen mpbir2an fconst.1 snnz B {} A dmxp ax-mp mpbir2an jctil A B {} X. A B {} df-f sylibr pm2.61i))
+thm (fconstg ((A x) (B x)) () (-> (e. B C) (:--> (X. A ({} B)) A ({} B))) (x cv B sneq x cv {} B {} A xpeq2 syl x cv B sneq jca A x cv {} X. A B {} X. A x cv {} feq1 x cv {} B {} A B {} X. A feq3 sylan9bb syl x visset A fconst C vtoclg))
+thm (f1eq1 () () (-> (= F G) (<-> (:-1-1-> F A B) (:-1-1-> G A B))) (F G A B feq1 F G cnveq F `' G `' funeq syl anbi12d F A B df-f1 G A B df-f1 3bitr4g))
+thm (f1eq2 () () (-> (= A B) (<-> (:-1-1-> F A C) (:-1-1-> F B C))) (A B F C feq2 F `' Fun anbi1d F A C df-f1 F B C df-f1 3bitr4g))
+thm (f1eq3 () () (-> (= A B) (<-> (:-1-1-> F C A) (:-1-1-> F C B))) (A B F C feq3 F `' Fun anbi1d F C A df-f1 F C B df-f1 3bitr4g))
+thm (hbf1 ((F y) (A y) (B y) (x y)) ((hbf1.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbf1.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbf1.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (:-1-1-> F A B) (A. x (:-1-1-> F A B))) (hbf1.1 hbf1.2 hbf1.3 hbf hbf1.1 hbcnv hbfun hban F A B df-f1 F A B df-f1 x bial 3imtr4))
+thm (f11 ((x y) (F x) (F y)) () (<-> (:-1-1-> F A B) (/\ (:--> F A B) (A. y (E* x (br (cv x) F (cv y)))))) (F A B df-f1 F `' y x dffunmo F relcnv mpbiran y visset x visset F brcnv x bimo y bial bitr F A B :--> anbi2i bitr))
+thm (f1f () () (-> (:-1-1-> F A B) (:--> F A B)) (F A B df-f1 pm3.26bd))
+thm (f1cnv () () (<-> (:-1-1-> (`' (`' A)) (dom A) (V)) (/\ (Fun (`' A)) (Fun (`' (`' A))))) (A `' `' A dom V df-f1 A `' `' A dom fnf A `' `' A dom df-fn A dfdm4 A `' df-rn eqtr2 mpbiran2 bitr3 A relcnv A `' dfrel2 mpbi A `' `' `' A `' funeq ax-mp anbi12i A `' `' Fun A `' Fun ancom 3bitr))
+thm (f1co () () (-> (/\ (:-1-1-> F B C) (:-1-1-> G A B)) (:-1-1-> (o. F G) A C)) (F B C G A fco G `' F `' funco F G cnvco F G o. `' G `' F `' o. funeq ax-mp sylibr ancoms anim12i an4s F B C df-f1 G A B df-f1 anbi12i F G o. A C df-f1 3imtr4))
+thm (foeq1 () () (-> (= F G) (<-> (:-onto-> F A B) (:-onto-> G A B))) (F G A fneq1 F G rneq B eqeq1d anbi12d F A B df-fo G A B df-fo 3bitr4g))
+thm (foeq2 () () (-> (= A B) (<-> (:-onto-> F A C) (:-onto-> F B C))) (A B F fneq2 F ran C = anbi1d F A C df-fo F B C df-fo 3bitr4g))
+thm (foeq3 () () (-> (= A B) (<-> (:-onto-> F C A) (:-onto-> F C B))) (A B F ran eqeq2 F C Fn anbi2d F C A df-fo F C B df-fo 3bitr4g))
+thm (hbfo ((F y) (A y) (B y) (x y)) ((hbfo.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbfo.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbfo.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (:-onto-> F A B) (A. x (:-onto-> F A B))) (hbfo.1 hbfo.2 hbfn hbfo.1 hbrn hbfo.3 hbeq hban F A B df-fo F A B df-fo x bial 3imtr4))
+thm (fof () () (-> (:-onto-> F A B) (:--> F A B)) (F ran B eqimss F A Fn anim2i F A B df-fo F A B df-f 3imtr4))
+thm (forn () () (-> (:-onto-> F A B) (= (ran F) B)) (F A B df-fo pm3.27bd))
+thm (foima () () (-> (:-onto-> F A B) (= (" F A) B)) (F A B fof F A B fdm F dom A F imaeq2 3syl F imadmrn syl5reqr F A B forn eqtrd))
+thm (fnforn () () (<-> (Fn F A) (:-onto-> F A (ran F))) (F ran eqid F A Fn biantru F A F ran df-fo bitr4))
+thm (funforn () () (<-> (Fun A) (:-onto-> A (dom A) (ran A))) (A dom eqid A Fun biantru A A dom df-fn bitr4 A A dom fnforn bitr))
+thm (fornex () () (-> (e. A C) (-> (:-onto-> F A B) (e. B (V)))) (F A B fof F A B ffun F C funrnex com12 3syl F A B fof F A B fdm syl C eleq1d F A B forn V eleq1d 3imtr3d com12))
+thm (fores () () (-> (/\ (Fun F) (C_ A (dom F))) (:-onto-> (|` F A) A (" F A))) (F A funres A F dom C_ anim1i F A |` A df-fn F A |` A F A " df-fo F A df-ima eqcomi mpbiran2 A F ssdmres F A |` Fun anbi2i 3bitr4 sylibr))
+thm (f1oeq1 () () (-> (= F G) (<-> (:-1-1-onto-> F A B) (:-1-1-onto-> G A B))) (F G A B f1eq1 F G A B foeq1 anbi12d F A B df-f1o G A B df-f1o 3bitr4g))
+thm (f1oeq2 () () (-> (= A B) (<-> (:-1-1-onto-> F A C) (:-1-1-onto-> F B C))) (A B F C f1eq2 A B F C foeq2 anbi12d F A C df-f1o F B C df-f1o 3bitr4g))
+thm (f1oeq3 () () (-> (= A B) (<-> (:-1-1-onto-> F C A) (:-1-1-onto-> F C B))) (A B F C f1eq3 A B F C foeq3 anbi12d F C A df-f1o F C B df-f1o 3bitr4g))
+thm (hbf1o ((F y) (A y) (B y) (x y)) ((hbf1o.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbf1o.2 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbf1o.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (:-1-1-onto-> F A B) (A. x (:-1-1-onto-> F A B))) (hbf1o.1 hbf1o.2 hbf1o.3 hbf1 hbf1o.1 hbf1o.2 hbf1o.3 hbfo hban F A B df-f1o F A B df-f1o x bial 3imtr4))
+thm (f1of1 () () (-> (:-1-1-onto-> F A B) (:-1-1-> F A B)) (F A B df-f1o pm3.26bd))
+thm (f1of () () (-> (:-1-1-onto-> F A B) (:--> F A B)) (F A B f1of1 F A B f1f syl))
+thm (f1ofn () () (-> (:-1-1-onto-> F A B) (Fn F A)) (F A B f1of F A B ffn syl))
+thm (f1ofun () () (-> (:-1-1-onto-> F A B) (Fun F)) (F A B f1ofn F A fnfun syl))
+thm (f1orel () () (-> (:-1-1-onto-> F A B) (Rel F)) (F A B f1ofun F funrel syl))
+thm (f1o2 () () (<-> (:-1-1-onto-> F A B) (/\/\ (Fn F A) (Fun (`' F)) (= (ran F) B))) (F A B df-f1 pm3.27bd F A B df-fo biimp anim12i F ran B eqimss F A Fn anim2i F A B df-f sylibr F `' Fun anim1i F A B df-f1 sylibr ancoms F A B df-fo biimpr F `' Fun adantl jca impbi F `' Fun F A Fn F ran B = an12 bitr F A B df-f1o F A Fn F `' Fun F ran B = 3anass 3bitr4))
+thm (f1o3 () () (<-> (:-1-1-onto-> F A B) (/\ (:-onto-> F A B) (Fun (`' F)))) (F A B :--> F `' Fun F A Fn F ran B = /\ an23 F A B df-f1 F A B df-fo anbi12i F ran B eqimss F A Fn anim2i F A B df-f sylibr pm4.71ri F `' Fun anbi1i 3bitr4 F A B df-f1o F A B df-fo F `' Fun anbi1i 3bitr4))
+thm (f1ofo () () (-> (:-1-1-onto-> F A B) (:-onto-> F A B)) (F A B f1o3 pm3.26bd))
+thm (f1o4 () () (<-> (:-1-1-onto-> F A B) (/\ (Fn F A) (Fn (`' F) B))) (F A Fn F `' Fun F ran B = 3anass F A B f1o2 F `' B df-fn F df-rn B eqeq1i F `' Fun anbi2i bitr4 F A Fn anbi2i 3bitr4))
+thm (f1o5 () () (<-> (:-1-1-onto-> F A B) (/\ (:-1-1-> F A B) (= (ran F) B))) (F A B df-f1o F A B df-fo F A B :-1-1-> anbi2i F A B :-1-1-> F A Fn F ran B = an12 F A B f1f F A B ffn syl F ran B = adantr pm4.71ri bitr4 3bitr))
+thm (f1orn () () (<-> (:-1-1-onto-> F A (ran F)) (/\ (Fn F A) (Fun (`' F)))) (F A Fn F `' Fun F ran F ran = df-3an F A F ran f1o2 F ran eqid F A Fn F `' Fun /\ biantru 3bitr4))
+thm (f1f1orn () () (-> (:-1-1-> F A B) (:-1-1-onto-> F A (ran F))) (F A B f1f F A B ffn syl F A B df-f1 pm3.27bd jca F A f1orn sylibr))
+thm (f1ocnv () () (-> (:-1-1-onto-> F A B) (:-1-1-onto-> (`' F) B A)) (F df-rn B eqeq1i F `' Fun anbi2i F `' B df-fn bitr4 biimp F A fnfun F funcnvcnv syl F A fndm F dfdm4 syl5eqr jca anim12i ancoms F A B f1o2 F A Fn F `' Fun F ran B = 3anass bitr F `' B A f1o2 F `' B Fn F `' `' Fun F `' ran A = 3anass bitr 3imtr4))
+thm (f1ocnvb () () (-> (Rel F) (<-> (:-1-1-onto-> F A B) (:-1-1-onto-> (`' F) B A))) (F A B f1ocnv F Rel a1i F dfrel2 F `' `' F A B f1oeq1 sylbi F `' B A f1ocnv syl5bi impbid))
+thm (f1ores () () (-> (/\ (:-1-1-> F A B) (C_ C A)) (:-1-1-onto-> (|` F C) C (" F C))) (F C fores F A B ffun C A C_ adantr F A B fdm C sseq2d biimpar sylanc F C funres11 anim12i an1rs F A B df-f1 C A C_ anbi1i F C |` C F C " f1o3 3imtr4))
+thm (f1imacnv () () (-> (/\ (:-1-1-> F A B) (C_ C A)) (= (" (`' F) (" F C)) C)) (F A B df-f1 pm3.27bd C A C_ adantr F C funcnvres F C |` `' F `' F C " |` F C " imaeq1 3syl F A B C f1ores F C |` C F C " f1ocnv F C |` `' F C " C f1of F C |` `' F C " C fdm F C |` `' dom F C " F C |` `' imaeq2 3syl F C |` `' imadmrn syl5reqr F C |` `' F C " C f1ofo F C |` `' F C " C forn syl eqtrd 3syl eqtr3d F `' F C " resima syl5eqr))
+thm (f1oun () () (-> (/\ (/\ (:-1-1-onto-> F A B) (:-1-1-onto-> G C D)) (/\ (= (i^i A C) ({/})) (= (i^i B D) ({/})))) (:-1-1-onto-> (u. F G) (u. A C) (u. B D))) (F A G C fnun exp F `' B G `' D fnun F G cnvun F G u. `' F `' G `' u. B D u. fneq1 ax-mp sylibr exp im2anan9 an4s F A B f1o4 G C D f1o4 syl2anb F G u. A C u. B D u. f1o4 syl6ibr imp))
+thm (f1oco () () (-> (/\ (:-1-1-onto-> F B C) (:-1-1-onto-> G A B)) (:-1-1-onto-> (o. F G) A C)) (F B Fn F `' Fun F ran C = G A Fn G `' Fun G ran B = an6 F G funco G `' F `' funco F G cnvco F G o. `' G `' F `' o. funeq ax-mp sylibr ancoms anim12i F G dmcoeq A eqeq1d biimpar F ran C = adantrr F G rncoeq C eqeq1d biimpar G dom A = adantrl jca F dom B G ran eqtr3t sylan an42s anim12i F B Fn G A Fn /\ F `' Fun G `' Fun /\ F ran C = G ran B = /\ 3anass F B df-fn G A df-fn anbi12i F Fun F dom B = G Fun G dom A = an4 bitr F `' Fun G `' Fun /\ F ran C = G ran B = /\ /\ anbi1i F Fun G Fun /\ F dom B = G dom A = /\ F `' Fun G `' Fun /\ F ran C = G ran B = /\ an4 3bitr F G o. A Fn F G o. `' Fun F G o. ran C = 3anass F G o. A df-fn F G o. `' Fun F G o. ran C = /\ anbi1i F G o. Fun F G o. dom A = F G o. `' Fun F G o. ran C = an4 3bitr 3imtr4 sylbi F B C f1o2 G A B f1o2 anbi12i F G o. A C f1o2 3imtr4))
+thm (f1ococnv2 () () (-> (:-1-1-onto-> F A B) (= (o. F (`' F)) (|` (I) B))) (F A B f1of F A B ffun F df-fun pm3.27bd 3syl F F `' o. iss sylib F A B f1of F A B fdm syl F A B f1ocnv F `' B A df-f1o pm3.27bd F `' B A forn 3syl eqtr4d F F `' dmcoeq syl F A B f1ocnv F `' B A f1of F `' B A fdm 3syl eqtrd F F `' o. dom B I reseq2 syl eqtrd))
+thm (f1ococnv1 () () (-> (:-1-1-onto-> F A B) (= (o. (`' F) F) (|` (I) A))) (F A B f1orel F dfrel2 sylib F `' coeq2d F A B f1ocnv F `' B A f1ococnv2 syl eqtr3d))
+thm (f1dmex () () (-> (e. B C) (-> (:-1-1-> F A B) (e. A (V)))) (B C F ran ssexg F A B f1f F A B frn syl syl5 F ran V F `' A fornex F A B f1f1orn F A F ran f1ocnv F `' F ran A f1ofo 3syl syl5 syli))
+thm (ffoss ((F x) (A x) (B x)) ((f11o.1 (e. F (V)))) (<-> (:--> F A B) (E. x (/\ (:-onto-> F A (cv x)) (C_ (cv x) B)))) (F A B df-f F A fnforn F ran B C_ anbi1i bitr f11o.1 F V rnexg ax-mp x cv F ran F A foeq3 x cv F ran B sseq1 anbi12d cla4ev sylbi F A x cv B fss F A x cv fof sylan x 19.23aiv impbi))
+thm (f11o ((F x) (A x) (B x)) ((f11o.1 (e. F (V)))) (<-> (:-1-1-> F A B) (E. x (/\ (:-1-1-onto-> F A (cv x)) (C_ (cv x) B)))) (f11o.1 A B x ffoss F `' Fun anbi1i F A B df-f1 F A x cv f1o3 x cv B C_ anbi1i F A x cv :-onto-> F `' Fun x cv B C_ an23 bitr x biex x F A x cv :-onto-> x cv B C_ /\ F `' Fun 19.41v bitr 3bitr4))
+thm (f10 () () (:-1-1-> ({/}) ({/}) A) ({/} {/} A df-f1 A f0 fun0 cnv0 {/} `' {/} funeq ax-mp mpbir mpbir2an))
+thm (f1o00 () () (<-> (:-1-1-onto-> F ({/}) A) (/\ (= F ({/})) (= A ({/})))) (F {/} A f1o4 F fn0 biimp F `' A Fn adantr F fn0 F {/} cnveq cnv0 syl6eq sylbi F `' {/} A fneq1 syl biimpa {/} A fndm syl dm0 syl5reqr jca F fn0 biimpr A {/} = adantr {/} eqid {/} fn0 mpbir F {/} cnveq cnv0 syl6eq F `' {/} A fneq1 syl A {/} {/} fneq2 sylan9bb mpbiri jca impbi bitr))
+thm (f1o0 () () (:-1-1-onto-> ({/}) ({/}) ({/})) ({/} {/} f1o00 {/} eqid {/} eqid mpbir2an))
+thm (f1oi () () (:-1-1-onto-> (|` (I) A) A A) (I A |` A A f1o3 I A |` A A df-fo A fnresi A rnresi mpbir2an funi cnvi I `' I funeq ax-mp mpbir I A funres11 ax-mp mpbir2an))
+thm (f1ovi () () (:-1-1-onto-> (I) (V) (V)) (V f1oi reli I resid ax-mp I V |` I V V f1oeq1 ax-mp mpbi))
+thm (f1osn () ((f1osn.1 (e. A (V))) (f1osn.2 (e. B (V)))) (:-1-1-onto-> ({} (<,> A B)) ({} A) ({} B)) (A B <,> {} A {} B {} f1o4 A B <,> {} A {} df-fn f1osn.1 f1osn.2 funsn A B dmsnop mpbir2an B A <,> {} B {} df-fn f1osn.2 f1osn.1 funsn B A dmsnop mpbir2an f1osn.1 f1osn.2 cnvsn A B <,> {} `' B A <,> {} B {} fneq1 ax-mp mpbir mpbir2an))
+thm (fv2 ((x y) (A x) (A y) (F x) (F y)) ((fv2.1 (e. A (V)))) (= (` F A) (U. ({|} x (A. y (<-> (br A F (cv y)) (= (cv y) (cv x))))))) (F A x df-fv F A {} " x cv {} y dfcleq F A {} y x dfima2 eqabi x A {} x cv F y cv br df-rex bitr x A elsn x cv F y cv br anbi1i x biex fv2.1 x cv A F y cv breq1 ceqsexv 3bitr y x cv elsn bibi12i y bial bitr x biabi unieqi eqtr))
+thm (fvprc ((A x) (F x)) () (-> (-. (e. A (V))) (= (` F A) ({/}))) (x visset snnz A snprc A {} {/} F imaeq2 sylbi F ima0 syl6eq x cv {} eqeq1d {/} x cv {} eqcom syl6bb mtbiri x nexdv x F A {} " x cv {} = abn0 bicon1i sylib unieqd F A x df-fv syl5eq uni0 syl6eq))
+thm (elfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) ((elfv.1 (e. B (V)))) (<-> (e. A (` F B)) (E. x (/\ (e. A (cv x)) (A. y (<-> (br B F (cv y)) (= (cv y) (cv x))))))) (elfv.1 F x y fv2 A eleq2i A x y B F y cv br y cv x cv = <-> A. eluniab bitr))
+thm (fveq1 ((A x) (F x) (G x)) () (-> (= F G) (= (` F A) (` G A))) (F G A {} imaeq1 x cv {} eqeq1d x biabdv unieqd F A x df-fv G A x df-fv 3eqtr4g))
+thm (fveq2 ((A x) (B x) (F x)) () (-> (= A B) (= (` F A) (` F B))) (A B sneq A {} B {} F imaeq2 syl x cv {} eqeq1d x biabdv unieqd F A x df-fv F B x df-fv 3eqtr4g))
+thm (fveq1i () ((fveq1i.1 (= F G))) (= (` F A) (` G A)) (fveq1i.1 F G A fveq1 ax-mp))
+thm (fveq1d () ((fveq1d.1 (-> ph (= F G)))) (-> ph (= (` F A) (` G A))) (fveq1d.1 F G A fveq1 syl))
+thm (fveq2i () ((fveq2i.1 (= A B))) (= (` F A) (` F B)) (fveq2i.1 A B F fveq2 ax-mp))
+thm (fveq2d () ((fveq2d.1 (-> ph (= A B)))) (-> ph (= (` F A) (` F B))) (fveq2d.1 A B F fveq2 syl))
+thm (hbfv ((y z) (F y) (F z) (A y) (A z) (x y) (x z)) ((hbfv.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbfv.2 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (` F A)) (A. x (e. (cv y) (` F A)))) (hbfv.1 hbfv.2 hbsn hbima y cv z cv {} e. x ax-17 hbeq y z hbab hbuni F A z df-fv y cv eleq2i F A z df-fv y cv eleq2i x bial 3imtr4))
+thm (fvex ((A x) (F x)) () (e. (` F A) (V)) (F A x df-fv F A {} " x cv {} unieq x visset unisn syl6req x ax-gen x F A {} " U. moeq x F A {} " x cv {} = x cv F A {} " U. = immo mp2 x F A {} " x cv {} = moabex ax-mp uniex eqeltr))
+thm (fv3 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((fv3.1 (e. A (V)))) (= (` F A) ({|} x (/\ (E. y (/\ (e. (cv x) (cv y)) (br A F (cv y)))) (E! y (br A F (cv y)))))) (fv3.1 x cv F z y elfv A F y cv br y cv z cv = bi2 y 19.20i z visset y cv z cv A F breq2 ceqsalv sylib x cv z cv e. anim2i z 19.22i z cv y cv x cv eleq2 z cv y cv A F breq2 anbi12d cbvexv sylib z x cv z cv e. y A F y cv br y cv z cv = <-> A. 19.40 pm3.27d y A F y cv br z df-eu sylibr jca y A F y cv br hbeu1 x cv z cv e. y ax-17 y A F y cv br y cv z cv = <-> hba1 hban z hbex hbim A F y cv br y cv z cv = bi1 y z x ax-14 syl6 com23 imp3a y a4s anc2ri com12 z 19.22dv y A F y cv br z df-eu syl5ib 19.23ai imp impbi bitr biabri))
+thm (fvres ((F x) (A x) (B x)) () (-> (e. A B) (= (` (|` F B) A) (` F A))) (A B snssi A {} B F resabs1 F B |` A {} |` F A {} |` rneq 3syl F B |` A {} df-ima F A {} df-ima 3eqtr4g x cv {} eqeq1d x biabdv unieqd F B |` A x df-fv F A x df-fv 3eqtr4g))
+thm (funssfv () () (-> (/\ (/\ (Fun F) (C_ G F)) (e. A (dom G))) (= (` F A) (` G A))) (F G funssres A fveq1d eqcomd A G dom F fvres sylan9eq eqcomd))
+thm (tz6.12-1 ((y z) (F y) (F z) (A y) (A z)) ((tz6.12.1 (e. A (V)))) (-> (/\ (br A F (cv y)) (E! y (br A F (cv y)))) (= (` F A) (cv y))) (tz6.12.1 F z y fv3 eqabi y z cv y cv e. A F y cv br exancom y A F y cv br E! anbi1i y A F y cv br z cv y cv e. /\ E. y A F y cv br E! ancom bitr y A F y cv br z cv y cv e. eupick sylbi sylbi com12 y A F y cv br E! adantr z cv y cv e. A F y cv br /\ y 19.8a y A F y cv br E! anim1i anasss tz6.12.1 F z y fv3 eqabi sylibr expcom impbid eqrd))
+thm (tz6.12 ((F y) (A y)) ((tz6.12.1 (e. A (V)))) (-> (/\ (e. (<,> A (cv y)) F) (E! y (e. (<,> A (cv y)) F))) (= (` F A) (cv y))) (A F y cv df-br A F y cv df-br y bieu anbi12i tz6.12.1 F y tz6.12-1 sylbir))
+thm (tz6.12f ((x y) (x z) (w x) (y z) (w y) (w z) (F z) (F w)) ((tz6.12f.1 (-> (e. (cv w) F) (A. y (e. (cv w) F))))) (-> (/\ (e. (<,> (cv x) (cv y)) F) (E! y (e. (<,> (cv x) (cv y)) F))) (= (` F (cv x)) (cv y))) (x cv y cv <,> F e. y x cv y cv <,> F e. E! /\ F x cv ` y cv = -> z ax-17 z cv y cv x cv opeq2 F eleq1d w cv x cv z cv <,> e. y ax-17 tz6.12f.1 hbel x cv y cv <,> F e. z ax-17 z cv y cv x cv opeq2 F eleq1d cbveu z cv y cv = a1i anbi12d z cv y cv F x cv ` eqeq2 imbi12d x visset z F tz6.12 chvar))
+thm (tz6.12-2 ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) () (-> (-. (E! y (br A F (cv y)))) (= (` F A) ({/}))) (y A F y cv br E! -. z ax-17 x y A F y cv br E! {|} z eq0 z visset x cv z cv = y A F y cv br E! pm4.2i elab negbii z bial bitr2 sylib F A ` sseq2d z cv A F fveq2 z cv A F y cv breq1 y bieudv x biabdv sseq12d z visset F x y fv3 y x cv y cv e. z cv F y cv br /\ E. y z cv F y cv br E! pm3.27 x ss2abi eqsstr V vtoclg syl5bi F A ` ss0 syl6com A F fvprc y A F y cv br E! -. a1d pm2.61i))
+thm (tz6.12c ((F y) (A y)) ((tz6.12c.1 (e. A (V)))) (-> (E! y (br A F (cv y))) (<-> (= (` F A) (cv y)) (br A F (cv y)))) (y A F y cv br euex y A F y cv br hbeu1 A F F A ` br y ax-17 hbim tz6.12c.1 F y tz6.12-1 expcom F A ` y cv A F breq2 biimprd syli com12 19.23ai mpcom F A ` y cv A F breq2 biimpcd syl tz6.12c.1 F y tz6.12-1 expcom impbid))
+thm (tz6.12i ((F y) (A y) (B y)) ((tz6.12i.1 (e. A (V)))) (-> (-. (= B ({/}))) (-> (= (` F A) B) (br A F B))) (F A fvex F A ` B V eleq1 mpbii y cv B F A ` eqeq2 y cv B {/} eqeq1 negbid y cv B A F breq2 imbi12d imbi12d F A ` y cv {/} eqeq1 negbid tz6.12i.1 y F tz6.12c y A F tz6.12-2 nsyl4 biimpd syl6bir pm2.43a V vtoclg mpcom com12))
+thm (ndmfv ((x y) (A x) (A y) (F x) (F y)) () (-> (-. (e. A (dom F))) (= (` F A) ({/}))) (x cv A F dom eleq1 x cv A F y cv breq1 y biexdv x visset F y eldm V vtoclbg y A F y cv br euex syl5bir con3d y A F tz6.12-2 syl6 A F fvprc A F dom e. -. a1d pm2.61i))
+thm (ndmfvrcl () ((ndmfvrcl.1 (= (dom F) S)) (ndmfvrcl.2 (-. (e. ({/}) S)))) (-> (e. (` F A) S) (e. A S)) (ndmfvrcl.2 A F ndmfv S eleq1d mtbiri a3i ndmfvrcl.1 A eleq2i sylib))
+thm (nfvres () () (-> (-. (e. A B)) (= (` (|` F B) A) ({/}))) (F B dmres A eleq2i A B F dom elin bitr pm3.26bd con3i A F B |` ndmfv syl))
+thm (fveqres () () (-> (= (` F A) (` G A)) (= (` (|` F B) A) (` (|` G B) A))) (A B F fvres A B G fvres eqeq12d biimprd A B F nfvres A B G nfvres eqtr4d F A ` G A ` = a1d pm2.61i))
+thm (funbrfv ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((funbrfv.1 (e. B (V)))) (-> (Fun F) (-> (br A F B) (= (` F A) B))) (F A B brrelex F funrel sylan funbrfv.1 x cv A F y cv breq1 F Fun anbi2d x cv A F fveq2 y cv eqeq1d imbi12d y cv B A F breq2 F Fun anbi2d y cv B F A ` eqeq2 imbi12d x visset F y tz6.12-1 F x y funeu sylan2 anabss7 V V vtocl2g mpan2 mpcom exp))
+thm (funfvopi () ((funopfv.1 (e. B (V)))) (-> (Fun F) (-> (e. (<,> A B) F) (= (` F A) B))) (funopfv.1 F A funbrfv A F B df-br syl5ibr))
+thm (funopfvg ((A x) (B x) (F x)) () (-> (/\ (e. B C) (Fun F)) (-> (e. (<,> A B) F) (= (` F A) B))) (x cv B A opeq2 F eleq1d x cv B F A ` eqeq2 imbi12d F Fun imbi2d x visset F A funfvopi C vtoclg imp))
+thm (fnfvbr ((x y) (F x) (F y) (A x) (B x) (B y) (C x)) ((fnfvbr.1 (e. C (V)))) (-> (/\ (Fn F A) (e. B A)) (<-> (= (` F B) C) (br B F C))) (fnfvbr.1 x cv C F B ` eqeq2 x cv C B F breq2 bibi12d F A Fn B A e. /\ imbi2d F A B x fneu y cv B F x cv breq1 x bieudv y cv B F fveq2 x cv eqeq1d y cv B F x cv breq1 bibi12d imbi12d y visset x F tz6.12c A vtoclg F A Fn adantl mpd vtocl))
+thm (fnfvop () ((fnfvbr.1 (e. C (V)))) (-> (/\ (Fn F A) (e. B A)) (<-> (= (` F B) C) (e. (<,> B C) F))) (fnfvbr.1 F A B fnfvbr B F C df-br syl6bb))
+thm (funfvop () ((funfvop.1 (e. B (V)))) (-> (/\ (Fun F) (e. A (dom F))) (<-> (= (` F A) B) (e. (<,> A B) F))) (funfvop.1 F F dom A fnfvop F F dom df-fn F dom eqid mpbiran2 sylanbr))
+thm (fnopabfv ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F y) (F z) (F w)) () (<-> (Fn F A) (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x))))))) (z visset w visset F A fnop exp pm4.71rd w visset F A z cv fnfvop w cv F z cv ` eqcom syl5bb exp pm5.32d bitr4d z visset w visset x cv z cv A eleq1 x cv z cv F fveq2 y cv eqeq2d anbi12d y cv w cv F z cv ` eqeq1 z cv A e. anbi2d opelopab syl6bbr z w 19.21aivv F A fnrel x y x cv A e. y cv F x cv ` = /\ relopab jctir F x y x cv A e. y cv F x cv ` = /\ {<,>|} z w cleqrel syl mpbird F x cv fvex x y x cv A e. y cv F x cv ` = /\ {<,>|} eqid fnopab2 F x y x cv A e. y cv F x cv ` = /\ {<,>|} A fneq1 mpbiri impbi))
+thm (fvelima ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (Fun F) (e. A (" F B))) (E.e. x B (= (` F (cv x)) A))) (y cv A F B " eleq1 y cv A F x cv ` eqeq2 x B birexdv imbi12d F Fun imbi2d y visset F x cv funfvopi x cv F y cv df-br syl5ib x B r19.22sdv y visset F B x elima syl5ib F B " vtoclg pm2.43b imp))
+thm (fnrnfv ((x y) (A x) (A y) (F x) (F y)) () (-> (Fn F A) (= (ran F) ({|} y (E.e. x A (= (cv y) (` F (cv x))))))) (F A fndm x cv eleq2d x visset y cv F opeldm syl5bi pm4.71rd y visset F A x cv fnfvop F x cv ` y cv eqcom syl5rbbr exp pm5.32d bitrd x biexdv x A y cv F x cv ` = df-rex syl6bbr y biabdv F y x dfrn3 syl5eq))
+thm (fniunfv ((x y) (A x) (A y) (F x) (F y)) () (-> (Fn F A) (= (U_ x A (` F (cv x))) (U. (ran F)))) (F A y x fnrnfv unieqd F x cv fvex x A y dfiun2 syl6reqr))
+thm (fnsnfv ((A y) (B y) (F y)) () (-> (/\ (Fn F A) (e. B A)) (= ({} (` F B)) (" F ({} B)))) (y visset F A B fnfvbr y cv F B ` eqcom syl5bb y biabdv F B ` y df-sn F A Fn B A e. /\ a1i F A fnrel F B y relimasn syl B A e. adantr 3eqtr4d))
+thm (funfv () () (-> (Fun F) (= (` F A) (U. (" F ({} A))))) (F F dom A fnsnfv F F dom df-fn F dom eqid mpbiran2 sylanbr unieqd F A fvex unisn syl5eqr exp A F ndmfv A F ndmima unieqd uni0 syl6eq eqtr4d pm2.61d1))
+thm (funfv2 ((A y) (F y)) () (-> (Fun F) (= (` F A) (U. ({|} y (br A F (cv y)))))) (F A funfv F funrel F A y relimasn syl unieqd eqtrd))
+thm (dmfco ((y z) (A y) (A z) (F y) (F z) (G y) (G z)) () (-> (/\ (Fun G) (e. A (dom G))) (<-> (e. A (dom (o. F G))) (e. (` G A) (dom F)))) (A G dom F G o. y eldmg y visset A G dom y cv V F G z opelcog mpan2 y biexdv bitrd G Fun adantl z visset G A funfvop z cv G A ` eqcom syl5bb z cv y cv <,> F e. anbi1d z biexdv G A fvex z cv G A ` y cv opeq1 F eleq1d ceqsexv syl5bbr y biexdv G A fvex F y eldm2 syl5bb bitr4d))
+thm (fvco ((A z) (F z) (G z)) () (-> (/\ (/\ (Fun F) (Fun G)) (e. A (dom G))) (= (` (o. F G) A) (` F (` G A)))) (G A F dmfco F Fun anbi2d F G A ` fvex A G dom F G A ` ` V F G z opelcog mpan2 G Fun adantl z visset G A funfvop z cv G A ` eqcom syl5bb z cv F G A ` ` <,> F e. anbi1d z biexdv G A fvex z cv G A ` F G A ` ` opeq1 F eleq1d ceqsexv syl5bbr bitr4d F G A ` ` eqid F G A ` fvex F G A ` funfvop mpbii syl5bir sylbid exp4b com3r imp41 F G A ` fvex F G o. A funfvop F G funco sylan A G dom e. adantlr mpbird exp A F G o. ndmfv G Fun A G dom e. /\ adantl G A F dmfco negbid G A ` F ndmfv syl6bi imp eqtr4d exp F Fun adantll pm2.61d))
+thm (fvco2 () () (-> (/\ (/\ (Fun F) (Fn G A)) (e. C A)) (= (` (o. F G) C) (` F (` G C)))) (F G C fvco exp31 com3l imp A funfni exp com3r imp31))
+thm (fvco3 () () (-> (/\ (/\ (Fun F) (:--> G A B)) (e. C A)) (= (` (o. F G) C) (` F (` G C)))) (F G A C fvco2 G A B ffn sylanl2))
+thm (fvopab3 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ch x) (ch y)) ((fvopab3.1 (e. B (V))) (fvopab3.2 (-> (= (cv x) A) (<-> ph ps))) (fvopab3.3 (-> (= (cv y) B) (<-> ps ch))) (fvopab3.4 (-> (e. (cv x) C) (E! y ph))) (fvopab3.5 (= F ({<,>|} x y (/\ (e. (cv x) C) ph))))) (-> (e. A C) (<-> (= (` F A) B) ch)) (fvopab3.1 x cv A C eleq1 fvopab3.2 anbi12d fvopab3.3 A C e. anbi2d C V opelopabg mpan2 fvopab3.4 fvopab3.5 fnopab fvopab3.1 F C A fnfvop mpan fvopab3.5 A B <,> eleq2i syl6bb A C e. ch ibar 3bitr4d))
+thm (fvopab3ig ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (ch x) (ch y)) ((fvopab3ig.1 (-> (= (cv x) A) (<-> ph ps))) (fvopab3ig.2 (-> (= (cv y) B) (<-> ps ch))) (fvopab3ig.3 (-> (e. (cv x) C) (E* y ph))) (fvopab3ig.4 (= F ({<,>|} x y (/\ (e. (cv x) C) ph))))) (-> (/\ (e. A C) (e. B D)) (-> ch (= (` F A) B))) (x cv A C eleq1 fvopab3ig.1 anbi12d fvopab3ig.2 A C e. anbi2d C D opelopabg biimpar exp43 pm2.43a imp x y x cv C e. ph /\ funopab fvopab3ig.3 y x cv C e. ph moanimv mpbir mpgbir B D x y x cv C e. ph /\ {<,>|} A funopfvg mpan2 A C e. adantl syld fvopab3ig.4 A fveq1i B eqeq1i syl6ibr))
+thm (fvopab4g ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((fvopab4g.1 (-> (= (cv x) A) (= B C))) (fvopab4g.2 (= F ({<,>|} x y (/\ (e. (cv x) D) (= (cv y) B)))))) (-> (/\ (e. A D) (e. C R)) (= (` F A) C)) (C eqid fvopab4g.1 y cv eqeq2d y cv C C eqeq1 y B moeq x cv D e. a1i fvopab4g.2 R fvopab3ig mpi))
+thm (fvopab4 ((x y) (A x) (A y) (B y) (C x) (C y) (D x) (D y)) ((fvopab4g.1 (-> (= (cv x) A) (= B C))) (fvopab4g.2 (= F ({<,>|} x y (/\ (e. (cv x) D) (= (cv y) B))))) (fvopab4.3 (e. C (V)))) (-> (e. A D) (= (` F A) C)) (fvopab4.3 fvopab4g.1 fvopab4g.2 V fvopab4g mpan2))
+thm (fvopabg ((x y) (A x) (A y) (B y) (C x) (C y)) ((fvopabg.1 (-> (= (cv x) A) (= B C)))) (-> (/\ (e. A D) (e. C R)) (= (` ({<,>|} x y (= (cv y) B)) A) C)) (fvopabg.1 x visset y cv B = biantrur x y biopabi R fvopab4g A D elisset sylan))
+thm (fvopabn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((fvopabg.1 (-> (= (cv x) A) (= B C)))) (-> (-. (e. C (V))) (= (` ({<,>|} x y (= (cv y) B)) A) ({/}))) (z visset snnz z cv A w cv opeq1 x y y cv B = {<,>|} eleq1d V ceqsexgv z A elsn z cv w cv <,> x y y cv B = {<,>|} e. anbi1i z biex syl5bb w visset fvopabg.1 y cv eqeq2d y cv w cv C eqeq1 V V opelopabg mpan2 bitrd w biabdv w visset w cv C V eleq1 mpbii w 19.23aiv con3i w w cv C = abn0 bicon1i sylib sylan9eq x y y cv B = {<,>|} A {} w z dfima3 syl5eq z cv {} eqeq1d {/} z cv {} eqcom syl6bb mtbiri z 19.21aiv z x y y cv B = {<,>|} A {} " z cv {} = alnex z x y y cv B = {<,>|} A {} " z cv {} = abn0 bicon1i bitr sylib unieqd x y y cv B = {<,>|} A z df-fv syl5eq uni0 syl6eq exp A x y y cv B = {<,>|} fvprc C V e. -. a1d pm2.61i))
+thm (fvopabgf ((w z) (v z) (A z) (v w) (A w) (A v) (w y) (v y) (u y) (B y) (u w) (B w) (u v) (B v) (B u) (C z) (C w) (C v) (x y) (w x) (v x) (u x) (x z)) ((fvopabgf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabgf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabgf.3 (-> (= (cv x) A) (= B C)))) (-> (/\ (e. A D) (e. C R)) (= (` ({<,>|} x y (= (cv y) B)) A) C)) (z cv w cv e. x ax-17 fvopabgf.1 w visset eqvincf w x u cv B e. hbs1 v u hbab fvopabgf.2 hbeq x w B u sbab eqcomd fvopabgf.3 sylan9eq 19.23ai sylbi D R v fvopabg y cv B = w ax-17 y cv B = v ax-17 w x u cv B e. hbs1 v u hbab hbeleq v cv u w cv x u cv B e. [/] {|} = y ax-17 y cv v cv = id x w B u sbab eqeqan12rd cbvopab A fveq1i syl5eq))
+thm (fvopabnf ((w z) (v z) (A z) (v w) (A w) (A v) (w y) (v y) (u y) (B y) (u w) (B w) (u v) (B v) (B u) (C z) (C w) (C v) (x y) (w x) (v x) (u x) (x z)) ((fvopabgf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabgf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabgf.3 (-> (= (cv x) A) (= B C)))) (-> (-. (e. C (V))) (= (` ({<,>|} x y (= (cv y) B)) A) ({/}))) (z cv w cv e. x ax-17 fvopabgf.1 w visset eqvincf w x u cv B e. hbs1 v u hbab fvopabgf.2 hbeq x w B u sbab eqcomd fvopabgf.3 sylan9eq 19.23ai sylbi v fvopabn y cv B = w ax-17 y cv B = v ax-17 w x u cv B e. hbs1 v u hbab hbeleq v cv u w cv x u cv B e. [/] {|} = y ax-17 y cv v cv = id x w B u sbab eqeqan12rd cbvopab A fveq1i syl5eq))
+thm (fvopabf ((A z) (B y) (C z) (x y) (x z)) ((fvopabf.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (fvopabf.2 (-> (e. (cv z) C) (A. x (e. (cv z) C)))) (fvopabf.3 (e. A (V))) (fvopabf.4 (e. C (V))) (fvopabf.5 (-> (= (cv x) A) (= B C)))) (= (` ({<,>|} x y (= (cv y) B)) A) C) (fvopabf.3 fvopabf.4 fvopabf.1 fvopabf.2 fvopabf.5 V V y fvopabgf mp2an))
+thm (fvopab ((x z) (A x) (A z) (B y) (C x) (C z) (x y) (y z)) ((fvopab.1 (e. A (V))) (fvopab.2 (e. C (V))) (fvopab.3 (-> (= (cv x) A) (= B C)))) (= (` ({<,>|} x y (= (cv y) B)) A) C) (z cv A e. x ax-17 z cv C e. x ax-17 fvopab.1 fvopab.2 fvopab.3 y fvopabf))
+thm (fvopab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (/\ (e. (cv x) A) (e. B C)) (= (` ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B))) (cv x)) B)) (B C y elex x cv A e. y ax-17 z x y x cv A e. y cv B = /\ hbopab2 z cv x cv e. y ax-17 hbfv z cv B e. y ax-17 hbeq hbim y visset y cv B V eleq1 mpbii z x y x cv A e. y cv B = /\ hbopab2 x tz6.12f x y x cv A e. y cv B = /\ opabid sylanbr y x cv A e. y cv B = euanv x y x cv A e. y cv B = /\ opabid y bieu B y eueq x cv A e. anbi2i 3bitr4r sylan2b exp43 pm2.43a mpdi com12 y cv B x y x cv A e. y cv B = /\ {<,>|} x cv ` eqeq2 sylibd 19.23ai syl impcom))
+thm (fvsn () ((fvsn.1 (e. A (V))) (fvsn.2 (e. B (V)))) (= (` ({} (<,> A B)) A) B) (fvsn.1 fvsn.2 funsn A B opex snid fvsn.2 A B <,> {} A funfvopi mp2))
+thm (eqfnfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (G x) (G y)) () (-> (/\ (Fn F A) (Fn G B)) (<-> (= F G) (/\ (= A B) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) (F dom A G dom B eqeq12 F G dmeq syl5bi F A fndm G B fndm syl2an F G x cv fveq1 x cv A e. a1d r19.21aiv F A Fn G B Fn /\ a1i jcad y visset F A x cv fnfvop G A Fn adantlr y visset G A x cv fnfvop F A Fn adantll bibi12d F x cv ` G x cv ` y cv eqeq1 syl5bi exp a2d com3r F A fndm x cv eleq2d x visset y cv F opeldm syl5bi con3d G A Fn adantr G A fndm x cv eleq2d x visset y cv G opeldm syl5bi con3d F A Fn adantl jcad x cv y cv <,> F e. x cv y cv <,> G e. pm5.21 x cv A e. F x cv ` G x cv ` = -> a1d syl6com pm2.61i y 19.21adv x 19.20dv x A F x cv ` G x cv ` = df-ral syl5ib F G x y cleqrel F A fnrel G A fnrel syl2an sylibrd A B G fneq2 biimparc sylan2 exp32 imp4b impbid))
+thm (cleqfvf ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F y) (F z) (G y) (G z)) ((cleqfvf.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (cleqfvf.2 (-> (e. (cv y) G) (A. x (e. (cv y) G))))) (-> (/\ (Fn F A) (Fn G B)) (<-> (= F G) (/\ (= A B) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) (F A G B z eqfnfv cleqfvf.1 y cv z cv e. x ax-17 hbfv cleqfvf.2 y cv z cv e. x ax-17 hbfv hbeq F x cv ` G x cv ` = z ax-17 z cv x cv F fveq2 z cv x cv G fveq2 eqeq12d A cbvral A B = anbi2i F G = bibi2i sylib))
+thm (fvreseq ((B x) (F x) (G x)) () (-> (/\ (/\ (Fn F A) (Fn G A)) (C_ B A)) (<-> (= (|` F B) (|` G B)) (A.e. x B (= (` F (cv x)) (` G (cv x)))))) (F A B fnssres G A B fnssres anim12i anandirs F B |` B G B |` B x eqfnfv x cv B F fvres x cv B G fvres eqeq12d birala B eqid x B F B |` x cv ` G B |` x cv ` = A.e. biantrur bitr3 syl6bbr syl))
+thm (fvelrn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) () (-> (Fn F A) (<-> (e. B (ran F)) (E.e. x A (= (` F (cv x)) B)))) (y cv B F ran eleq1 y cv B F x cv ` eqeq2 x A birexdv bibi12d F A Fn imbi2d x visset y visset F A fnbr x visset F y tz6.12-1 exp F x y funeu F A fnfun sylan syl5 anabsi7 jca exp y visset F A x cv fnfvbr biimpd exp imp3a impbid x biexdv y visset F x elrn2 x A F x cv ` y cv = df-rex 3bitr4g V vtoclg B F ran elisset F x cv fvex F x cv ` B V eleq1 mpbii x cv A e. a1i r19.23aiv pm5.21ni F A Fn a1d pm2.61i))
+thm (elrnopab ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (F z) (w x) (w z) (C w) (C x) (C z) (F w)) ((elrnopab.1 (e. B (V))) (elrnopab.2 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) B)))))) (<-> (e. C (ran F)) (E.e. x A (= C B))) (elrnopab.1 y eueq1 x cv A e. a1i elrnopab.2 fnopab F A C z fvelrn w x y x cv A e. y cv B = /\ hbopab1 elrnopab.2 w cv eleq2i elrnopab.2 w cv eleq2i x bial 3imtr4 w cv z cv e. x ax-17 hbfv w cv C e. x ax-17 hbeq F x cv ` C = z ax-17 z cv x cv F fveq2 C eqeq1d A cbvrex elrnopab.1 x A B V y fvopab2 mpan2 elrnopab.2 x cv fveq1i syl5eq C eqeq1d B C eqcom syl6bb birexa bitr syl6bb ax-mp))
+thm (chfnrn ((x y) (A x) (A y) (F x) (F y)) () (-> (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) (cv x)))) (C_ (ran F) (U. A))) (F A y cv x fvelrn biimpd x A F x cv ` x cv e. hbra1 x A F x cv ` x cv e. ra4 F x cv ` y cv x cv eleq1 biimpcd syl6 r19.22d sylan9 y cv A x eluni2 syl6ibr ssrdv))
+thm (funopfv ((F x) (A x)) () (-> (/\ (Fun F) (e. A (dom F))) (e. (<,> A (` F A)) F)) (F A fvex x isseti x visset F A funfvop F A ` x cv A opeq2 F eleq1d biimprcd syl6bi pm2.43d x cv F A ` eqcom syl5ib x 19.23adv mpi))
+thm (fnopfv () () (-> (/\ (Fn F A) (e. B A)) (e. (<,> B (` F B)) F)) (F B funopfv A funfni))
+thm (fvrn ((x y) (F x) (F y) (A x)) () (-> (/\ (Fun F) (e. A (dom F))) (e. (` F A) (ran F))) (x cv A F dom eleq1 F Fun anbi2d x cv A F fveq2 F ran eleq1d imbi12d F x cv funopfv x visset y cv x cv F x cv ` opeq1 F eleq1d cla4ev syl F x cv fvex F y elrn sylibr F dom vtoclg anabsi7))
+thm (fnfvrn () () (-> (/\ (Fn F A) (e. B A)) (e. (` F B) (ran F))) (F B fvrn A funfni))
+thm (ffvrn () () (-> (/\ (:--> F A B) (e. C A)) (e. (` F C) B)) (F A C fnfvrn F A B ffn sylan F A B frn F C ` sseld C A e. adantr mpd))
+thm (ffvrni () ((ffvrni.1 (:--> F A B))) (-> (e. C A) (e. (` F C) B)) (ffvrni.1 F A B C ffvrn mpan))
+thm (fopab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C y) (C z)) ((fopab2.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C)))))) (<-> (A.e. x A (e. C B)) (:--> F A B)) (C B elisset C y eueq sylib x A r19.20si fopab2.1 fnopabg sylib x A C B e. hbra1 y cv B e. x ax-17 x A C B e. ra4 C B y cv eleq1a x cv A e. imim2i imp3a syl 19.23ad x y x cv A e. y cv C = /\ rnopab eqabi syl5ib y 19.21aiv z x y x cv A e. y cv C = /\ hbopab2 hbrn z cv B e. y ax-17 dfss2f sylibr fopab2.1 rneqi syl5ss jca F A B df-f sylibr F A B fdm x A y y cv C = dmopab3 C y isset x A biral fopab2.1 dmeqi A eqeq1i 3bitr4r sylib z x y x cv A e. y cv C = /\ hbopab1 z cv A e. x ax-17 z cv B e. x ax-17 hbf fopab2.1 F x y x cv A e. y cv C = /\ {<,>|} A B feq1 ax-mp fopab2.1 F x y x cv A e. y cv C = /\ {<,>|} A B feq1 ax-mp x bial 3imtr4 F A B x cv ffvrn C V e. adantr x A C V y fvopab2 fopab2.1 x cv fveq1i syl5eq B eleq1d F A B :--> adantll mpbid exp r19.20da mpd impbi))
+thm (fopab ((x y) (A x) (A y) (B x) (B y) (C y)) ((fopab2.1 (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) C))))) (fopab.2 (-> (e. (cv x) A) (e. C B)))) (:--> F A B) (fopab.2 rgen fopab2.1 B fopab2 mpbi))
+thm (ffnfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (<-> (:--> F A B) (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B)))) (F A B ffn F A B x cv ffvrn exp r19.21aiv jca F A Fn x A F x cv ` B e. A.e. pm3.26 F A y cv x fvelrn biimpd x A F x cv ` B e. hbra1 y cv B e. x ax-17 x A F x cv ` B e. ra4 F x cv ` y cv B eleq1 biimpcd syl6 r19.23ad sylan9 r19.21aiv F ran B y dfss3 sylibr jca F A B df-f sylibr impbi))
+thm (fnfvrnss ((A x) (B x) (F x)) () (-> (/\ (Fn F A) (A.e. x A (e. (` F (cv x)) B))) (C_ (ran F) B)) (F A B x ffnfv F A B frn sylbir))
+thm (fopabfv ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (<-> (:--> F A B) (/\ (= F ({<,>|} x y (/\ (e. (cv x) A) (= (cv y) (` F (cv x)))))) (A.e. x A (e. (` F (cv x)) B)))) (F A B x ffnfv F A x y fnopabfv x A F x cv ` B e. A.e. anbi1i bitr))
+thm (fsn ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) ((fsn.1 (e. A (V))) (fsn.2 (e. B (V)))) (<-> (:--> F ({} A) ({} B)) (= F ({} (<,> A B)))) (y visset F A {} B {} x cv opelf x A elsn y B elsn anbi12i sylib exp x cv A y cv B opeq12 F eleq1d fsn.1 snid F A {} B {} A y feu mpan2 fsn.2 y eueq1 A B <,> F e. biantru y A B <,> F e. y cv B = euanv y cv B A opeq2 F eleq1d pm5.32i y B elsn A y cv <,> F e. anbi1i A B <,> F e. y cv B = ancom 3bitr4r y bieu y B {} A y cv <,> F e. df-reu bitr4 bitr3 bitr sylibr syl5bir com12 impbid x cv y cv opex A B <,> elsnc x visset y visset fsn.2 A opth bitr2 syl6bb x y 19.21aivv F A {} B {} frel fsn.1 B relsn jctir F A B <,> {} x y cleqrel syl mpbird fsn.1 fsn.2 f1osn F A B <,> {} A {} B {} f1oeq1 mpbiri F A {} B {} f1of syl impbi))
+thm (fsn2 () ((fsn2.1 (e. A (V)))) (<-> (:--> F ({} A) B) (/\ (e. (` F A) B) (= F ({} (<,> A (` F A)))))) (fsn2.1 snid F A {} B A ffvrn mpan2 F A {} B ffn F A {} fnfrn biimp F A {} fndm F dom A {} F imaeq2 syl F imadmrn syl5eqr fsn2.1 snid F A {} A fnsnfv mpan2 eqtr4d F ran F A ` {} F A {} feq3 syl mpbid syl jca F A {} F A ` {} B fss ancoms F A ` B snssi sylan impbi fsn2.1 F A fvex F fsn F A ` B e. anbi2i bitr))
+thm (fnressn ((A x) (B x) (F x)) () (-> (/\ (Fn F A) (e. B A)) (= (|` F ({} B)) ({} (<,> B (` F B))))) (x cv B sneq x cv {} B {} F reseq2 syl x cv B F fveq2 x cv B F x cv ` F B ` opeq12 mpdan sneqd eqeq12d F A Fn imbi2d F A x cv {} fnssres x visset A snss sylan2b F x cv {} |` x cv {} fnf x visset F x cv {} |` V fsn2 F x cv {} |` x cv fvex F x cv {} |` x cv F x cv {} |` x cv ` <,> {} = biantrur x visset snid x cv x cv {} F fvres ax-mp F x cv {} |` x cv ` F x cv ` x cv opeq2 ax-mp sneqi F x cv {} |` eqeq2i bitr3 3bitr sylib expcom vtoclga impcom))
+thm (fressnfv ((A x) (B x) (C x) (F x)) () (-> (/\ (Fn F A) (e. B A)) (<-> (:--> (|` F ({} B)) ({} B) C) (e. (` F B) C))) (x cv B sneq x cv {} B {} F reseq2 F x cv {} |` F B {} |` x cv {} C feq1 syl x cv {} B {} F B {} |` C feq2 bitrd syl x cv B F fveq2 C eleq1d bibi12d F A Fn imbi2d F A x cv fnressn x visset snid x cv x cv {} F fvres ax-mp F x cv {} |` x cv ` F x cv ` x cv opeq2 ax-mp sneqi F x cv {} |` eqeq2i F x cv {} |` x cv F x cv {} |` x cv ` <,> {} = F x cv {} |` x cv ` C e. iba x visset snid x cv x cv {} F fvres ax-mp C eleq1i syl5rbbr x visset F x cv {} |` C fsn2 syl5bb sylbir syl expcom vtoclga impcom))
+thm (fvconst () () (-> (/\ (:--> F A ({} B)) (e. C A)) (= (` F C) B)) (F A B {} C ffvrn F C ` B elsni syl))
+thm (fvi ((A x)) () (-> (e. A B) (= (` (I) A) A)) (x cv A I fveq2 x cv A = id eqeq12d I V df-fn funi dmi mpbir2an x visset x cv eqid x visset x visset ideq mpbir x cv I x cv df-br mpbi x visset I V x cv fnfvop mpbiri mp2an B vtoclg))
+thm (fvresi () () (-> (e. B A) (= (` (|` (I) A) B) B)) (B A I fvres B A fvi eqtrd))
+thm (fconst2 ((A x) (B x) (F x)) ((fconst2.1 (e. B (V)))) (<-> (:--> F A ({} B)) (= F (X. A ({} B)))) (F A B x cv fvconst fconst2.1 A fconst A B {} X. A B x cv fvconst mpan F A B {} :--> adantl eqtr4d exp r19.21aiv A eqid jctil F A B {} ffn fconst2.1 A fconst A B {} X. A B {} ffn ax-mp F A A B {} X. A x eqfnfv mpan2 syl mpbird fconst2.1 A fconst F A B {} X. A B {} feq1 mpbiri impbi))
+thm (fvconst2 () ((fconst2.1 (e. B (V)))) (-> (e. C A) (= (` (X. A ({} B)) C) B)) (fconst2.1 A fconst A B {} X. A B C fvconst mpan))
+thm (fvconst2g ((A x) (B x) (C x)) () (-> (/\ (e. B D) (e. C A)) (= (` (X. A ({} B)) C) B)) (x cv B sneq x cv {} B {} A xpeq2 syl C fveq1d x cv B = id eqeq12d C A e. imbi2d x visset C A fvconst2 D vtoclg imp))
+thm (fconstfv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (<-> (:--> F A ({} B)) (/\ (Fn F A) (A.e. x A (= (` F (cv x)) B)))) (F A B {} ffn F A B x cv fvconst exp r19.21aiv jca A {/} F fneq2 F fn0 syl6bb B {} f0 F {/} {/} B {} feq1 mpbiri syl6bi A {/} F B {} feq2 sylibrd x A F x cv ` B = A.e. adantrd F A y cv z fvelrn x cv z cv F fveq2 B eqeq1d A rcla4v impcom y cv eqeq1d birexdva A B y cv = z r19.9rzv bicomd sylan9bbr sylan9bbr y B elsn y cv B eqcom bitr2 syl6bb eqrd an1rs exp31 imdistand F A B {} df-fo F A B {} fof sylbir syl6 pm2.61i impbi))
+thm (funfvima () () (-> (/\ (Fun F) (e. B (dom F))) (-> (e. B A) (e. (` F B) (" F A)))) (B A F fvres F A |` ran eleq1d F A df-ima F B ` eleq2i syl6rbbr F A |` B fvrn F A funres sylan syl5bir com12 exp F A dmres B eleq2i B A F dom elin bitr syl5ibr exp3a com12 imp3a pm2.43b))
+thm (funfvima2 () () (-> (/\ (Fun F) (C_ A (dom F))) (-> (e. B A) (e. (` F B) (" F A)))) (F B A funfvima exp com23 a2d A F dom B ssel syl5 imp))
+thm (funfvima3 ((A x) (F x) (G x)) () (-> (/\ (Fun F) (C_ F G)) (-> (e. A (dom F)) (e. (` F A) (" G ({} A))))) (F G A F A ` <,> ssel F A funopfv syl5 imp x cv A sneq x cv {} A {} G imaeq2 syl F A ` eleq2d x cv A F A ` opeq1 G eleq1d x visset F A fvex G elimasn F dom vtoclbg F G C_ F Fun ad2antrr mpbird exp32 impcom))
+thm (fvclss ((x y) (F x) (F y)) () (C_ ({|} y (E. x (= (cv y) (` F (cv x))))) (u. (ran F) ({} ({/})))) (x visset y cv F tz6.12i y cv F x cv ` eqcom syl5ib x 19.22dv y visset F x elrn2 syl6ibr com12 con1d y {/} elsn syl6ibr orrd y ss2abi F ran {/} {} y df-un sseqtr4))
+thm (fvclex ((x y) (F x) (F y)) ((fvclex.1 (e. F (V)))) (e. ({|} y (E. x (= (cv y) (` F (cv x))))) (V)) (fvclex.1 F V rnexg ax-mp p0ex unex y x F fvclss ssexi))
+thm (fvresex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y)) ((fvresex.1 (e. A (V)))) (e. ({|} y (E. x (= (cv y) (` (|` F A) (cv x))))) (V)) (x visset F x cv fvex z cv x cv F fveq2 w fvopab z w w cv F z cv ` = {<,>|} x cv F A fveqres ax-mp y cv eqeq2i x biex y biabi fvresex.1 z w F z cv ` funopabeq A V z w w cv F z cv ` = {<,>|} resfunexg mp2 y x fvclex eqeltrr))
+thm (abrexexlem1 ((x y) (F x) (F y) (A x) (A y)) ((abrexexlem1.1 (e. A (V)))) (e. ({|} y (E.e. x A (= (cv y) (` F (cv x))))) (V)) (abrexexlem1.1 y x F fvresex x A y cv F x cv ` = df-rex x cv A F fvres y cv eqeq2d biimpar x 19.22i sylbi y ss2abi ssexi))
+thm (abrexexlem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((abrexexlem2.1 (e. A (V))) (abrexexlem2.2 (e. B (V)))) (e. ({|} y (E.e. x A (= (cv y) B))) (V)) (x visset y cv B = biantrur x y biopabi x cv fveq1i x visset abrexexlem2.2 x V B V y fvopab2 mp2an eqtr y cv eqeq2i x A birex y cv x y y cv B = {<,>|} x cv ` = z ax-17 w cv y cv e. x ax-17 w x y y cv B = hbopab1 w cv z cv e. x ax-17 hbfv hbeq x cv z cv x y y cv B = {<,>|} fveq2 y cv eqeq2d A cbvrex bitr3 y biabi z A y cv x y y cv B = {<,>|} z cv ` = E.e. w ax-17 z cv A e. y ax-17 x cv w cv e. y ax-17 w x y y cv B = hbopab2 w cv z cv e. y ax-17 hbfv hbeq hbrex y cv w cv x y y cv B = {<,>|} z cv ` eqeq1 z A birexdv cbvab eqtr abrexexlem2.1 w z x y y cv B = {<,>|} abrexexlem1 eqeltr))
+thm (abrexex ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((abrexex.1 (e. A (V)))) (e. ({|} y (E.e. x A (= (cv y) B))) (V)) (abrexex.1 z B class2set y x abrexexlem2 y visset y cv B V eleq1 mpbii B V e. z cv B e. ax-1 r19.21aiv B z B V e. rabid2 sylibr y cv eqeq2d syl ibi x A r19.22si y ss2abi ssexi))
+thm (abrexexg ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (e. ({|} y (E.e. x A (= (cv y) B))) (V))) (z cv A x y cv B = rexeq y biabdv V eleq1d z visset y x B abrexex C vtoclg))
+thm (iunexg ((x y) (A x) (A y) (B y)) () (-> (/\ (e. A C) (A. x (e. B D))) (e. (U_ x A B) (V))) (x B D A y dfiun2g A C e. adantl A C y x B abrexexg y x A y cv B = E.e. {|} V uniexg syl x B D e. A. adantr eqeltrd))
+thm (iunex ((A x)) ((iunex.1 (e. A (V))) (iunex.2 (e. B (V)))) (e. (U_ x A B) (V)) (iunex.1 iunex.2 x ax-gen A V x B V iunexg mp2an))
+thm (abrexex2 ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) ((abrexex2.1 (e. A (V))) (abrexex2.2 (e. ({|} y ph) (V)))) (e. ({|} y (E.e. x A ph)) (V)) (x A ph E.e. z ax-17 x cv A e. y ax-17 z y ph hbs1 hbrex y z ph sbequ12 x A birexdv cbvab z y ph df-clab x A birex z biabi eqtr4 x A y ph {|} z df-iun abrexex2.1 abrexex2.2 x iunex eqeltrr eqeltr))
+thm (f1fv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z)) () (<-> (:-1-1-> F A B) (/\ (:--> F A B) (A.e. x A (A.e. y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))))))) (F A B z x f11 F A B ffn F A fndm x cv eleq2d x visset F z cv breldm syl5bi F A fndm y cv eleq2d y visset F z cv breldm syl5bi anim12d ancrd x cv A e. y cv A e. /\ x cv F z cv br y cv F z cv br /\ pm3.27 F A Fn a1i impbid z visset F A x cv fnfvbr z cv F x cv ` eqcom syl5bb z visset F A y cv fnfvbr z cv F y cv ` eqcom syl5bb bi2anan9 anandis exp pm5.32d bitr4d x cv y cv = imbi1d x cv A e. y cv A e. /\ z cv F x cv ` = z cv F y cv ` = /\ x cv y cv = impexp syl6bb z bialdv z x cv A e. y cv A e. /\ z cv F x cv ` = z cv F y cv ` = /\ x cv y cv = -> 19.21v z z cv F x cv ` = z cv F y cv ` = /\ x cv y cv = 19.23v F x cv fvex F y cv ` z eqvinc x cv y cv = imbi1i bitr4 x cv A e. y cv A e. /\ imbi2i bitr syl6bb x y bi2aldv x cv y cv F z cv breq1 mo4 z bial z x y x cv F z cv br y cv F z cv br /\ x cv y cv = -> A. alcom z y x cv F z cv br y cv F z cv br /\ x cv y cv = -> alcom x bial 3bitr x A y A F x cv ` F y cv ` = x cv y cv = -> r2al 3bitr4g syl pm5.32i bitr))
+thm (f1fvf ((x y) (w x) (v x) (A x) (w y) (v y) (A y) (v w) (A w) (A v) (B x) (B y) (B w) (B v) (w z) (v z) (F z) (F w) (F v) (x z) (y z)) ((f1fvf.1 (-> (e. (cv z) F) (A. x (e. (cv z) F)))) (f1fvf.2 (-> (e. (cv z) F) (A. y (e. (cv z) F))))) (<-> (:-1-1-> F A B) (/\ (:--> F A B) (A.e. x A (A.e. y A (-> (= (` F (cv x)) (` F (cv y))) (= (cv x) (cv y))))))) (F A B w v f1fv f1fvf.2 z cv w cv e. y ax-17 hbfv f1fvf.2 z cv v cv e. y ax-17 hbfv hbeq w cv v cv = y ax-17 hbim F w cv ` F y cv ` = v ax-17 w cv y cv = v ax-17 hbim v cv y cv F fveq2 F w cv ` eqeq2d v cv y cv w cv eqeq2 imbi12d A cbvral w A biral y cv A e. x ax-17 f1fvf.1 z cv w cv e. x ax-17 hbfv f1fvf.1 z cv y cv e. x ax-17 hbfv hbeq w cv y cv = x ax-17 hbim hbral y A F x cv ` F y cv ` = x cv y cv = -> A.e. w ax-17 w cv x cv F fveq2 F y cv ` eqeq1d w cv x cv y cv eqeq1 imbi12d y A biraldv A cbvral bitr F A B :--> anbi2i bitr))
+thm (f1fveq ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y) (F x) (F y)) () (-> (/\ (:-1-1-> F A B) (/\ (e. C A) (e. D A))) (<-> (= (` F C) (` F D)) (= C D))) (x cv C F fveq2 F y cv ` eqeq1d x cv C y cv eqeq1 imbi12d F A B :-1-1-> imbi2d y cv D F fveq2 F C ` eqeq2d y cv D C eqeq2 imbi12d F A B :-1-1-> imbi2d F A B x y f1fv pm3.27bd x A y A F x cv ` F y cv ` = x cv y cv = -> ra42 syl com12 vtocl2ga impcom C D F fveq2 F A B :-1-1-> C A e. D A e. /\ /\ a1i impbid))
+thm (f1ocnvfv1 () () (-> (/\ (:-1-1-onto-> F A B) (e. C A)) (= (` (`' F) (` F C)) C)) (F A B f1ococnv1 C fveq1d C A e. adantr F `' F A B C fvco3 F A B f1ocnv F `' B A f1ofun syl F A B f1of jca sylan C A fvresi F A B :-1-1-onto-> adantl 3eqtr3d))
+thm (f1ocnvfv2 () () (-> (/\ (:-1-1-onto-> F A B) (e. C B)) (= (` F (` (`' F) C)) C)) (F A B f1ococnv2 C fveq1d C B e. adantr F F `' B A C fvco3 F A B f1ofun F A B f1ocnv F `' B A f1of syl jca sylan C B fvresi F A B :-1-1-onto-> adantl 3eqtr3d))
+thm (f1ocnvfv () () (-> (/\ (:-1-1-onto-> F A B) (e. C A)) (-> (= (` F C) D) (= (` (`' F) D) C))) (F A B C f1ocnvfv1 F `' D ` eqeq2d D F C ` F `' fveq2 eqcoms syl5bi))
+thm (f1ocnvfvb () () (-> (/\ (:-1-1-onto-> F A B) (/\ (e. C A) (e. D B))) (<-> (= (` F C) D) (= (` (`' F) D) C))) (F A B C D f1ocnvfv D B e. adantrr F A B D f1ocnvfv2 F C ` eqeq2d C F `' D ` F fveq2 eqcoms syl5bi C A e. adantrl impbid))
+thm (cbvfo ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (ph y) (ps x)) ((cbvfo.1 (-> (= (` F (cv x)) (cv y)) (<-> ph ps)))) (-> (:-onto-> F A B) (<-> (A.e. x A ph) (A.e. y B ps))) (F A B fof F A B ffun x visset F y cv breldm F Fun a1i y visset F x cv funbrfv jcad x 19.22dv y visset F x elrn2 syl5ib x x cv F dom e. ph -> hba1 ps x ax-17 cbvfo.1 biimpcd x cv F dom e. imim2i imp3a x a4s 19.23ad syl9 y 19.21adv x visset y visset F brelrn F Fun a1i y visset F x cv funbrfv jcad y 19.22dv x visset F y eldm syl5ib y y cv F ran e. ps -> hba1 ph y ax-17 cbvfo.1 biimprcd y cv F ran e. imim2i imp3a y a4s 19.23ad syl9 x 19.21adv impbid x F dom ph df-ral y F ran ps df-ral 3bitr4g 3syl F A B fof F A B fdm F dom A x ph raleq 3syl F A B forn F ran B y ps raleq syl 3bitr3d))
+thm (cbvexfo ((x y) (A x) (A y) (B x) (B y) (F x) (F y) (ph y) (ps x)) ((cbvfo.1 (-> (= (` F (cv x)) (cv y)) (<-> ph ps)))) (-> (:-onto-> F A B) (<-> (E.e. x A ph) (E.e. y B ps))) (cbvfo.1 negbid A B cbvfo negbid x A ph dfrex2 y B ps dfrex2 3bitr4g))
+thm (isoeq1 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (G x) (G y) (R x) (R y) (S x) (S y)) () (-> (= H G) (<-> (Isom H R S A B) (Isom G R S A B))) (H G A B f1oeq1 H G x cv fveq1 H G y cv fveq1 S breq12d x cv R y cv br bibi2d y A biraldv x A biraldv anbi12d H R S A B x y df-iso G R S A B x y df-iso 3bitr4g))
+thm (isoeq2 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (R x) (R y) (S x) (S y) (T x) (T y)) () (-> (= R T) (<-> (Isom H R S A B) (Isom H T S A B))) (R T x cv y cv breq H x cv ` S H y cv ` br bibi1d y A biraldv x A biraldv H A B :-1-1-onto-> anbi2d H R S A B x y df-iso H T S A B x y df-iso 3bitr4g))
+thm (isoeq3 ((x y) (A x) (A y) (B x) (B y) (H x) (H y) (R x) (R y) (S x) (S y) (T x) (T y)) () (-> (= S T) (<-> (Isom H R S A B) (Isom H R T A B))) (S T H x cv ` H y cv ` breq x cv R y cv br bibi2d y A biraldv x A biraldv H A B :-1-1-onto-> anbi2d H R S A B x y df-iso H R T A B x y df-iso 3bitr4g))
+thm (isoeq4 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (H x) (H y) (R x) (R y) (S x) (S y)) () (-> (= A C) (<-> (Isom H R S A B) (Isom H R S C B))) (A C H B f1oeq2 A C y x cv R y cv br H x cv ` S H y cv ` br <-> raleq x raleqd anbi12d H R S A B x y df-iso H R S C B x y df-iso 3bitr4g))
+thm (isoeq5 ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (H x) (H y) (R x) (R y) (S x) (S y)) () (-> (= B C) (<-> (Isom H R S A B) (Isom H R S A C))) (B C H A f1oeq3 x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. anbi1d H R S A B x y df-iso H R S A C x y df-iso 3bitr4g))
+thm (hbiso ((y z) (w y) (H y) (w z) (H z) (H w) (R y) (R z) (R w) (S y) (S z) (S w) (A y) (A z) (A w) (B y) (B z) (B w) (x y) (x z) (w x)) ((hbiso.1 (-> (e. (cv y) H) (A. x (e. (cv y) H)))) (hbiso.2 (-> (e. (cv y) R) (A. x (e. (cv y) R)))) (hbiso.3 (-> (e. (cv y) S) (A. x (e. (cv y) S)))) (hbiso.4 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbiso.5 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (Isom H R S A B) (A. x (Isom H R S A B))) (hbiso.1 hbiso.4 hbiso.5 hbf1o y cv z cv e. x ax-17 hbiso.4 hbel y cv w cv e. x ax-17 hbiso.4 hbel y cv z cv e. x ax-17 hbiso.2 y cv w cv e. x ax-17 hbbr hbiso.1 y cv z cv e. x ax-17 hbfv hbiso.3 hbiso.1 y cv w cv e. x ax-17 hbfv hbbr hbbi hbral hbral hban H R S A B z w df-iso H R S A B z w df-iso x bial 3imtr4))
+thm (isof1o ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y)) () (-> (Isom H R S A B) (:-1-1-onto-> H A B)) (H R S A B x y df-iso pm3.26bd))
+thm (isorel ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (C x) (C y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (/\ (e. C A) (e. D A))) (<-> (br C R D) (br (` H C) S (` H D)))) (x cv C R y cv breq1 x cv C H fveq2 S H y cv ` breq1d bibi12d y cv D C R breq2 y cv D H fveq2 H C ` S breq2d bibi12d A A rcla42v H R S A B x y df-iso pm3.27bd syl5com imp))
+thm (isoid ((x y) (A x) (A y) (R x) (R y)) () (Isom (|` (I) A) R R A A) (I A |` R R A A x y df-iso A f1oi x cv A I fvres x visset x cv V fvi ax-mp syl6eq y cv A I fvres y visset y cv V fvi ax-mp syl6eq R breqan12d bicomd rgen2 mpbir2an))
+thm (isocnv ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (Isom (`' H) S R B A)) (H A B f1ocnv x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. adantr H A B z cv f1ocnvfv2 w cv B e. adantrr H A B w cv f1ocnvfv2 z cv B e. adantrl S breq12d x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. adantlr x cv H `' z cv ` R y cv breq1 x cv H `' z cv ` H fveq2 S H y cv ` breq1d bibi12d H `' z cv ` R y cv br H H `' z cv ` ` S H y cv ` br bicom syl6bb y cv H `' w cv ` H fveq2 H H `' z cv ` ` S breq2d y cv H `' w cv ` H `' z cv ` R breq2 bibi12d A A rcla42v imp H `' B A z cv ffvrn H `' B A w cv ffvrn anim12i anandis sylan an1rs H A B f1ocnv H `' B A f1of syl sylanl1 bitr3d exp32 r19.21adv r19.21aiv jca H R S A B x y df-iso H `' S R B A z w df-iso 3imtr4))
+thm (isotr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (T x) (T y) (T z) (T w) (T v) (T u) (G x) (G y) (G z) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u)) () (-> (/\ (Isom H R S A B) (Isom G S T B C)) (Isom (o. G H) R T A C)) (G B C :-1-1-onto-> v B u B v cv S u cv br G v cv ` T G u cv ` br <-> A.e. A.e. pm3.26 H A B :-1-1-onto-> z A w A z cv R w cv br H z cv ` S H w cv ` br <-> A.e. A.e. pm3.26 anim12i ancoms G B C H A f1oco syl H A B f1of H A B x cv ffvrn exp H A B y cv ffvrn exp anim12d syl z A w A z cv R w cv br H z cv ` S H w cv ` br <-> A.e. A.e. adantr v cv H x cv ` S u cv breq1 v cv H x cv ` G fveq2 T G u cv ` breq1d bibi12d u cv H y cv ` H x cv ` S breq2 u cv H y cv ` G fveq2 G H x cv ` ` T breq2d bibi12d B B rcla42v com12 G B C :-1-1-onto-> adantl sylan9 imp z cv x cv R w cv breq1 z cv x cv H fveq2 S H w cv ` breq1d bibi12d w cv y cv x cv R breq2 w cv y cv H fveq2 H x cv ` S breq2d bibi12d A A rcla42v impcom H A B :-1-1-onto-> adantll G B C :-1-1-onto-> v B u B v cv S u cv br G v cv ` T G u cv ` br <-> A.e. A.e. /\ adantlr G H A B x cv fvco3 y cv A e. adantrr G H A B y cv fvco3 x cv A e. adantrl T breq12d G B C f1ofun H A B f1of anim12i sylan G B C :-1-1-onto-> v B u B v cv S u cv br G v cv ` T G u cv ` br <-> A.e. A.e. pm3.26 H A B :-1-1-onto-> z A w A z cv R w cv br H z cv ` S H w cv ` br <-> A.e. A.e. pm3.26 anim12i ancoms sylan 3bitr4d exp32 r19.21adv r19.21aiv jca H R S A B z w df-iso G S T B C v u df-iso anbi12i G H o. R T A C x y df-iso 3imtr4))
+thm (isotrALT ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (T x) (T y) (T z) (T w) (G x) (G y) (G z) (G w) (H x) (H y) (H z) (H w)) () (-> (/\ (Isom H R S A B) (Isom G S T B C)) (Isom (o. G H) R T A C)) (G B C :-1-1-onto-> z B w B z cv S w cv br G z cv ` T G w cv ` br <-> A.e. A.e. pm3.26 H A B :-1-1-onto-> x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. pm3.26 anim12i ancoms G B C H A f1oco syl H A B :-1-1-onto-> x ax-17 x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. hbra1 hban G B C :-1-1-onto-> z B w B z cv S w cv br G z cv ` T G w cv ` br <-> A.e. A.e. /\ x ax-17 hban H A B :-1-1-onto-> y ax-17 x cv A e. y ax-17 y A x cv R y cv br H x cv ` S H y cv ` br <-> hbra1 hbral hban G B C :-1-1-onto-> z B w B z cv S w cv br G z cv ` T G w cv ` br <-> A.e. A.e. /\ y ax-17 hban x cv A e. y ax-17 H A B f1of H A B x cv ffvrn exp H A B y cv ffvrn exp anim12d syl x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. adantr z cv H x cv ` S w cv breq1 z cv H x cv ` G fveq2 T G w cv ` breq1d bibi12d w cv H y cv ` H x cv ` S breq2 w cv H y cv ` G fveq2 G H x cv ` ` T breq2d bibi12d B B rcla42v com12 G B C :-1-1-onto-> adantl sylan9 imp x A y A x cv R y cv br H x cv ` S H y cv ` br <-> ra42 imp H A B :-1-1-onto-> adantll G B C :-1-1-onto-> z B w B z cv S w cv br G z cv ` T G w cv ` br <-> A.e. A.e. /\ adantlr G H A B x cv fvco3 y cv A e. adantrr G H A B y cv fvco3 x cv A e. adantrl T breq12d G B C f1ofun H A B f1of anim12i sylan G B C :-1-1-onto-> z B w B z cv S w cv br G z cv ` T G w cv ` br <-> A.e. A.e. pm3.26 H A B :-1-1-onto-> x A y A x cv R y cv br H x cv ` S H y cv ` br <-> A.e. A.e. pm3.26 anim12i ancoms sylan 3bitr4d exp32 r19.21ad r19.21ai jca H R S A B x y df-iso G S T B C z w df-iso anbi12i G H o. R T A C x y df-iso 3imtr4))
+thm (isomin ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (C x) (C y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (/\ (C_ C A) (e. D A))) (<-> (= (i^i C (" (`' R) ({} D))) ({/})) (= (i^i (" H C) (" (`' S) ({} (` H D)))) ({/})))) (C A x cv ssel H R S A B isof1o H A B f1ofn y visset H A x cv fnfvbr exp 3syl syl9r imp31 birexdva y visset H C x elima syl6rbbr H D fvex y visset H D ` V S eliniseg ax-mp H R S A B Isom C A C_ /\ a1i anbi12d y cv H C " S `' H D ` {} " elin x C H x cv ` y cv = y cv S H D ` br r19.41v 3bitr4g D A e. adantrr C A x cv ssel x visset D A R eliniseg H R S A B Isom x cv A e. ad2antrr H R S A B x cv D isorel bitrd H x cv ` y cv S H D ` breq1 biimpar syl5bir exp32 syl9r com34 imp32 r19.22dv sylbid x cv C R `' D {} " elin x biex C R `' D {} " i^i x n0 x C x cv R `' D {} " e. df-rex 3bitr4 syl6ibr y 19.23adv H C " S `' H D ` {} " i^i y n0 syl5ib a3d C A x cv ssel com12 H x cv C funfvima A funfni exp com13 syld com13 imp D A e. adantrr H R S A B isof1o H A B f1ofn syl sylan x cv R `' D {} " e. adantrd C A x cv ssel x visset D A R eliniseg H R S A B Isom x cv A e. ad2antrr H R S A B x cv D isorel biimpd H D fvex H x cv fvex H D ` V S eliniseg ax-mp syl6ibr sylbid exp32 syl9r com34 imp32 imp3a jcad x cv C R `' D {} " elin H x cv ` H C " S `' H D ` {} " elin 3imtr4g H x cv ` H C " S `' H D ` {} " i^i n0i syl6 x 19.23adv C R `' D {} " i^i x n0 syl5ib a3d impbid))
+thm (isoini ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (e. D A)) (= (" H (i^i A (" (`' R) ({} D)))) (i^i B (" (`' S) ({} (` H D)))))) (H R S A B isof1o H A B f1ofo H A B forn y cv eleq2d 3syl H R S A B isof1o H A B f1ofn H A y cv x fvelrn 3syl bitr3d H D fvex y visset H D ` V S eliniseg ax-mp H R S A B Isom a1i anbi12d D A e. adantr x visset D A R eliniseg x cv A e. anbi2d x cv A R `' D {} " elin syl5bb x cv H y cv br anbi1d x cv A e. x cv R D br x cv H y cv br anass syl6bb H R S A B Isom adantl H R S A B x cv D isorel y visset H A x cv fnfvbr bicomd H R S A B isof1o H A B f1ofn syl sylan D A e. adantrr anbi12d H x cv ` S H D ` br H x cv ` y cv = ancom H x cv ` y cv S H D ` breq1 pm5.32i bitr syl6bb exp32 com23 imp pm5.32d bitrd birexdv2 x A H x cv ` y cv = y cv S H D ` br r19.41v syl6bb bitr4d y cv B S `' H D ` {} " elin syl5bb biabrdv H A R `' D {} " i^i y x dfima2 syl6reqr))
+thm (isofrlem ((x y) (w x) (A x) (w y) (A y) (A w) (x z) (B x) (y z) (B y) (w z) (B z) (B w) (R x) (R y) (R w) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (-> (Fr S B) (Fr R A))) (H R S A B isof1o H A B f1ofun x visset H funimaex z cv H x cv " B sseq1 z cv H x cv " {/} eqeq1 negbid anbi12d z cv H x cv " S `' w cv {} " ineq1 {/} eqeq1d w rexeqd imbi12d V cla4gv syl 3syl S B z w dffr3 syl5ib H R S A B isof1o H A B f1ofn x cv A y cv ssel H y cv x cv funfvima A funfni H y cv ` H x cv " n0i syl6 exp sylan9r pm2.43d y 19.23adv x cv y n0 syl5ib exp imp3a syl H A B f1ofo H x cv imassrn H A B forn H x cv " sseq2d mpbii syl jctild syl syl5d H w cv x cv y fvelima H R S A B isof1o x cv A C_ adantr H A B f1ofun syl w cv H x cv " e. H x cv " S `' w cv {} " i^i {/} = pm3.26 syl2an H R S A B x cv y cv isomin x cv A y cv ssel imdistani sylan2 H y cv ` w cv sneq H y cv ` {} w cv {} S `' imaeq2 syl H x cv " ineq2d {/} eqeq1d sylan9bb w cv H x cv " e. H x cv " S `' w cv {} " i^i {/} = pm3.27 syl5bir exp42 imp com3l com4t imp r19.22dv mpd exp32 r19.23adv exp x cv {/} = -. adantrd a2d syld x 19.21adv R A x y dffr3 syl6ibr))
+thm (isofr () () (-> (Isom H R S A B) (<-> (Fr R A) (Fr S B))) (H R S A B isocnv H `' S R B A isofrlem syl H R S A B isofrlem impbid))
+thm (isowe ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (R x) (R y) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (<-> (We R A) (We S B))) (H R S A B isofr H R S A B x cv y cv isorel H A B x cv y cv f1fveq H R S A B isof1o H A B f1of1 syl sylan bicomd H R S A B y cv x cv isorel x cv A e. y cv A e. ancom sylan2b 3orbi123d exp32 r19.21adv y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ H x cv ` S H y cv ` br H x cv ` H y cv ` = H y cv ` S H x cv ` br \/\/ r19.15 syl6 r19.21aiv x A y A x cv R y cv br x cv y cv = y cv R x cv br \/\/ A.e. y A H x cv ` S H y cv ` br H x cv ` H y cv ` = H y cv ` S H x cv ` br \/\/ A.e. r19.15 syl H R S A B isof1o H A B f1ofo H y cv ` w cv H x cv ` S breq2 H y cv ` w cv H x cv ` eqeq2 H y cv ` w cv S H x cv ` breq1 3orbi123d A B cbvfo x A biraldv H x cv ` z cv S w cv breq1 H x cv ` z cv w cv eqeq1 H x cv ` z cv w cv S breq2 3orbi123d w B biraldv A B cbvfo bitrd 3syl bitrd anbi12d R A x y dfwe2 S B z w dfwe2 3bitr4g))
+thm (f1oiso ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B v) (B u) (H x) (H y) (H z) (H w) (H v) (H u) (R x) (R y) (R z) (R w) (R v) (R u) (S v) (S u)) () (-> (/\ (:-1-1-onto-> H A B) (= S ({<,>|} z w (E.e. x A (E.e. y A (/\ (/\ (= (cv z) (` H (cv x))) (= (cv w) (` H (cv y)))) (br (cv x) R (cv y)))))))) (Isom H R S A B)) (H A B :-1-1-onto-> S z w x A y A z cv H x cv ` = w cv H y cv ` = /\ x cv R y cv br /\ E.e. E.e. {<,>|} = pm3.26 S z w x A y A z cv H x cv ` = w cv H y cv ` = /\ x cv R y cv br /\ E.e. E.e. {<,>|} H v cv ` H u cv ` <,> eleq2 H A B v cv x cv f1fveq v cv x cv eqcom syl6bb anassrs H u cv ` H y cv ` = x cv R y cv br /\ anbi1d H v cv ` H x cv ` = H u cv ` H y cv ` = x cv R y cv br anass syl5bb y A birexdv y A x cv v cv = H u cv ` H y cv ` = x cv R y cv br /\ r19.42v syl6bb birexdva x cv v cv R y cv breq1 H u cv ` H y cv ` = anbi2d y A birexdv A ceqsrexv H A B :-1-1-> adantl bitrd H A B u cv y cv f1fveq u cv y cv eqcom syl6bb anassrs v cv R y cv br anbi1d birexdva y cv u cv v cv R breq2 A ceqsrexv H A B :-1-1-> adantl bitrd sylan9bb anandis H v cv fvex H u cv fvex z cv H v cv ` H x cv ` eqeq1 w cv H y cv ` = anbi1d x cv R y cv br anbi1d y A birexdv x A birexdv w cv H u cv ` H y cv ` eqeq1 H v cv ` H x cv ` = anbi2d x cv R y cv br anbi1d y A birexdv x A birexdv opelopab syl5bb sylan9bbr an1rs H v cv ` S H u cv ` df-br syl5rbb exp32 r19.21adv r19.21aiv H A B f1of1 sylan jca H R S A B v u df-iso sylibr))
+thm (f1owe ((w z) (R z) (R w) (x y) (x z) (w x) (S x) (y z) (w y) (S y) (S z) (S w) (A z) (A w) (B z) (B w) (F x) (F y) (F z) (F w)) ((f1owe.1 (= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y))))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A))) (x cv z cv F fveq2 S F y cv ` breq1d y cv w cv F fveq2 F z cv ` S breq2d f1owe.1 A A brabg rgen2 F R S A B z w df-iso F R S A B isowe sylbir mpan2 biimprd))
+thm (f1oweALT ((w z) (v z) (u z) (f z) (R z) (v w) (u w) (f w) (R w) (u v) (f v) (R v) (f u) (R u) (R f) (x y) (x z) (w x) (v x) (u x) (f x) (S x) (y z) (w y) (v y) (u y) (f y) (S y) (S z) (S w) (S v) (S u) (S f) (A z) (A w) (A v) (A u) (A f) (B z) (B w) (B v) (B u) (B f) (F x) (F y) (F z) (F w) (F v) (F u) (F f)) ((f1oweALT.1 (= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y))))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A))) (F A B df-f1o pm3.27bd F A B df-fo F ran B S freq2 biimprd F A df-fn z visset F funimaex w cv F z cv " F ran sseq1 w cv F z cv " {/} eqeq1 negbid anbi12d w cv F z cv " f f cv S u cv br -. raleq u rexeqd imbi12d V cla4gv F z cv w cv funfvima2 F w cv ` F z cv " n0i syl6 w 19.23adv z cv w n0 syl5ib imp F z cv imassrn jctil syl7 syl S F ran w u f df-fr syl5ib com23 exp3a anabsi5 imp3a F z cv fores F z cv |` v cv ` f cv S F z cv |` w cv ` breq1 negbid z cv F z cv " cbvfo w z cv birexdv F z cv |` w cv ` u cv f cv S breq2 negbid f F z cv " biraldv z cv F z cv " cbvexfo bitrd v cv z cv F fvres w cv z cv F fvres S breqan12rd v visset w visset x cv v cv F fveq2 S F y cv ` breq1d y cv w cv F fveq2 F v cv ` S breq2d f1oweALT.1 brab syl6rbbr negbid biraldva birexa syl5bb syl sylibrd exp4b com34 com23 imp4a z 19.21adv R F dom z w v df-fr syl6ibr F dom A R freq2 biimpd sylan9 sylbi sylan9r sylbi syl F A B df-f1o w visset v visset x cv w cv F fveq2 S F y cv ` breq1d y cv v cv F fveq2 F w cv ` S breq2d f1oweALT.1 brab F A B :-1-1-> w cv A e. v cv A e. /\ /\ a1i F A B w cv v cv f1fveq bicomd v visset w visset x cv v cv F fveq2 S F y cv ` breq1d y cv w cv F fveq2 F v cv ` S breq2d f1oweALT.1 brab F A B :-1-1-> w cv A e. v cv A e. /\ /\ a1i 3orbi123d bi2raldva F w cv ` u cv S F v cv ` breq1 F w cv ` u cv F v cv ` eqeq1 F w cv ` u cv F v cv ` S breq2 3orbi123d v A biraldv A B cbvfo F v cv ` f cv u cv S breq2 F v cv ` f cv u cv eqeq2 F v cv ` f cv S u cv breq1 3orbi123d A B cbvfo u B biraldv bitrd sylan9bb sylbi biimprd anim12d S B u f dfwe2 R A w v dfwe2 3imtr4g))
+thm (canth ((x y) (A x) (A y) (F x) (F y)) ((canth.1 (e. A (V)))) (-. (:-onto-> F A (P~ A))) (F A A P~ forn F A A P~ fof x cv y cv = id x cv y cv F fveq2 eleq12d negbid A elrab baibr y cv F y cv ` e. y cv x A x cv F x cv ` e. -. {e.|} e. nbbn F y cv ` x A x cv F x cv ` e. -. {e.|} y cv eleq2 con3i sylbi syl rgen F A A P~ ffn F A x A x cv F x cv ` e. -. {e.|} y fvelrn biimpd syl con3d y A F y cv ` x A x cv F x cv ` e. -. {e.|} = ralnex syl5ib mpi x A x cv F x cv ` e. -. ssrab canth.1 x x cv F x cv ` e. -. rabex A elpw mpbir F ran A P~ x A x cv F x cv ` e. -. {e.|} eleq2 mpbiri con3i 3syl pm2.65i))
+thm (ncanth () () (:-onto-> (I) (V) (P~ (V))) (f1ovi pwv V P~ V I V f1oeq3 ax-mp mpbir I V V P~ f1ofo ax-mp))
+thm (iunon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((iunon.1 (e. A (V))) (iunon.2 (e. B (V)))) (-> (A.e. x A (e. B (On))) (e. (U_ x A B) (On))) (x A B On e. hbra1 y cv On e. x ax-17 x A B On e. ra4 B On y cv eleq1a syl6 r19.23ad y x A y cv B = E.e. abid syl5ib y 19.21aiv z y x A y cv B = E.e. hbab1 z cv On e. y ax-17 dfss2f sylibr iunon.1 y x B abrexex onuni syl iunon.2 x A y dfiun2 On eleq1i sylibr))
+thm (iinon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((iinon.1 (e. B (V)))) (-> (/\ (A.e. x A (e. B (On))) (-. (= A ({/})))) (e. (|^|_ x A B) (On))) (y x A y cv B = E.e. {|} oninton y x cv A e. y cv B = 19.42v iinon.1 y isseti mpbiran2 x biex x y x cv A e. y cv B = /\ excom bitr3 x A y cv B = df-rex y biex bitr4 A x n0 y x A y cv B = E.e. abn0 3bitr4 sylan2b x A B On e. hbra1 y cv On e. x ax-17 x A B On e. ra4 B On y cv eleq1a syl6 r19.23ad y x A y cv B = E.e. abid syl5ib y 19.21aiv z y x A y cv B = E.e. hbab1 z cv On e. y ax-17 dfss2f sylibr sylan iinon.1 x A y dfiin2 On eleq1i sylibr))
+thm (tfrlem1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (w x) (F x) (w y) (F y) (w z) (F z) (F w) (G x) (G y) (G z) (G w)) () (-> (e. A (On)) (-> (/\ (Fn F A) (Fn G A)) (-> (A.e. x A (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x)))))) (A.e. x A (= (` F (cv x)) (` G (cv x))))))) (A ssid y cv A A sseq1 y cv A x F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ raleq y cv A x F x cv ` G x cv ` = raleq imbi12d imbi12d F A Fn G A Fn /\ imbi2d y cv z cv A sseq1 F A Fn G A Fn /\ anbi2d y cv z cv x F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ raleq anbi12d y cv z cv x F x cv ` G x cv ` = raleq imbi12d y cv z cv onelsst z cv y cv A sstr2 F A Fn G A Fn /\ anim2d z cv y cv C_ x ax-17 x y cv F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ hbra1 z cv y cv x cv ssel x y cv F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ ra4 syl9 r19.21ad anim12d syl6 com23 r19.21adv y cv x cv onelsst x cv y cv A sstr2 F x cv ` B F x cv |` ` G x cv ` B G x cv |` ` eqeq12 F A G x cv w fvreseq biimpar B fveq2d syl5bir exp4c com4l syl9 syl6 imp4a com23 imp31 r19.20dva x y cv F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ F x cv ` G x cv ` = r19.20 syl6 x z cv F x cv ` G x cv ` = hbra1 w x cv F w cv ` G w cv ` = A.e. z ax-17 z cv x cv w F w cv ` G w cv ` = raleq x cv w cv F fveq2 x cv w cv G fveq2 eqeq12d z cv cbvralv syl5bb y cv cbvral syl5ib exp com23 imp4a imim12d F A Fn G A Fn /\ y cv A C_ /\ x y cv F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ A.e. /\ x y cv F x cv ` G x cv ` = A.e. pm2.43 syl6 z y cv F A Fn G A Fn /\ z cv A C_ /\ x z cv F x cv ` B F x cv |` ` = G x cv ` B G x cv |` ` = /\ A.e. /\ x z cv F x cv ` G x cv ` = A.e. r19.20 syl5 tfis2 exp4c vtoclga mpii))
+thm (tfrlem2 ((A w) (B w) (F w) (G w) (w x) (w y)) () (-> (/\ (Fn F A) (Fn G A)) (-> (/\ (e. (<,> (cv x) (cv y)) F) (e. (<,> (cv x) (cv z)) G)) (-> (e. A (On)) (-> (A. w (-> (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))))) (= (cv y) (cv z)))))) (A F G w B tfrlem1 com12 imp3a x cv y cv <,> F e. adantr x visset y visset F A fnop G A Fn adantlr w cv x cv F fveq2 w cv x cv G fveq2 eqeq12d A rcla4v syl syld imp x cv z cv <,> G e. adantlrr w A F w cv ` B F w cv |` ` = G w cv ` B G w cv |` ` = /\ df-ral A On e. anbi2i sylan2br y visset F x cv funfvopi imp z visset G x cv funfvopi imp anim12i an4s F A fnfun G A fnfun anim12i sylan F x cv ` y cv G x cv ` z cv eqeq12 syl A On e. w w cv A e. F w cv ` B F w cv |` ` = G w cv ` B G w cv |` ` = /\ -> A. /\ adantr mpbid A On e. w cv A e. F w cv ` B F w cv |` ` = G w cv ` B G w cv |` ` = /\ -> abai w bial w A On e. w cv A e. F w cv ` B F w cv |` ` = G w cv ` B G w cv |` ` = /\ -> 19.28v w A On e. A On e. w cv A e. F w cv ` B F w cv |` ` = G w cv ` B G w cv |` ` = /\ -> -> 19.28v 3bitr3r sylan2b exp43))
+thm (tfrlem3 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z)) ((tfrlem3.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))))) (= A ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y))))))))) (tfrlem3.1 g visset f cv g cv x cv fneq1 f cv g cv y cv fveq1 f cv g cv y cv reseq1 G fveq2d eqeq12d y x cv biraldv anbi12d x On birexdv elab x cv z cv g cv fneq2 x cv z cv y g cv y cv ` G g cv y cv |` ` = raleq anbi12d On cbvrexv bitr biabri eqtr))
+thm (tfrlem4 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (g z) (G z) (G f) (G g)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (e. (cv g) A) (Fun (cv g))) (tfrlem.1 g z tfrlem3 eqabi g cv z cv fnfun y z cv g cv y cv ` G g cv y cv |` ` = A.e. adantr z cv On e. a1i r19.23aiv sylbi))
+thm (tfrlem5 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (f z) (F z) (v w) (u w) (f w) (F w) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (g z) (h z) (G z) (g w) (h w) (G w) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (/\ (e. (cv g) A) (e. (cv h) A)) (-> (/\ (e. (<,> (cv x) (cv u)) (cv g)) (e. (<,> (cv x) (cv v)) (cv h))) (= (cv u) (cv v)))) (tfrlem.1 g z tfrlem3 eqabi tfrlem.1 h w tfrlem3 eqabi anbi12i z On w On g cv z cv Fn y z cv g cv y cv ` G g cv y cv |` ` = A.e. /\ h cv w cv Fn y w cv h cv y cv ` G h cv y cv |` ` = A.e. /\ reeanv bitr4 g cv z cv h cv w cv x u v 2elresin g cv z cv w cv i^i |` z cv w cv i^i h cv z cv w cv i^i |` x u v y G tfrlem2 g cv z cv w cv fnresin1 h cv w cv z cv fnresin2 syl2an sylbid com24 com3r imp32 z cv w cv onin y z cv g cv y cv ` G g cv y cv |` ` = w cv h cv y cv ` G h cv y cv |` ` = r19.26m y cv z cv e. g cv y cv ` G g cv y cv |` ` = y cv w cv e. h cv y cv ` G h cv y cv |` ` = prth z cv w cv i^i On e. y cv z cv w cv i^i e. pm3.27 y cv z cv w cv elin sylib syl5 z cv w cv i^i y cv onelsst impac y cv z cv w cv i^i g cv fvres y cv z cv w cv i^i g cv resabs1 G fveq2d eqeqan12rd y cv z cv w cv i^i h cv fvres y cv z cv w cv i^i h cv resabs1 G fveq2d eqeqan12rd anbi12d syl bicomd mpbidi exp3a y 19.20i sylbir g cv z cv Fn h cv w cv Fn /\ anim2i an4s syl2an exp r19.23aivv sylbi))
+thm (tfrlem6 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (F x) (F y) (F f) (G x) (G y) (G f) (G g)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Rel F) (tfrlem.2 A g reluni tfrlem.1 tfrlem.2 g tfrlem4 g cv funrel syl mprgbir F A U. releq mpbiri ax-mp))
+thm (tfrlem7 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (v x) (u x) (F x) (v y) (u y) (F y) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Fun F) (F x u v dffun4 tfrlem.1 tfrlem.2 tfrlem6 tfrlem.2 x cv u cv <,> eleq2i x cv u cv <,> A g eluni bitr tfrlem.2 x cv v cv <,> eleq2i x cv v cv <,> A h eluni bitr anbi12i g h x cv u cv <,> g cv e. g cv A e. /\ x cv v cv <,> h cv e. h cv A e. /\ eeanv bitr4 x cv u cv <,> g cv e. g cv A e. x cv v cv <,> h cv e. h cv A e. an4 x cv u cv <,> g cv e. x cv v cv <,> h cv e. /\ g cv A e. h cv A e. /\ ancom bitr tfrlem.1 tfrlem.2 g h u v tfrlem5 imp sylbi g h 19.23aivv sylbi v ax-gen x u gen2 mpbir2an))
+thm (tfrlem8 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (f z) (F z) (v w) (f w) (F w) (f v) (F v) (F f) (G x) (G y) (G z) (G w) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (Ord (dom F)) (F dom v w dftr2 w visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr w cv z cv <,> eleq2i w cv z cv <,> f x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ E.e. eluniab bitr z biex bitr x On w cv z cv <,> f cv e. f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ r19.42v w visset z visset f cv x cv fnop ancoms y x cv f cv y cv ` G f cv y cv |` ` = A.e. adantrr x cv On e. adantl x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ ra4e tfrlem.1 eqabi f cv A elssuni tfrlem.2 f cv sseq2i f cv F dmss sylbir syl sylbir syl f cv x cv fndm F dom sseq1d x cv On e. y x cv f cv y cv ` G f cv y cv |` ` = A.e. ad2antrl mpbid w cv z cv <,> f cv e. adantrl jca exp r19.22i sylbir z f 19.23aivv sylbi v cv w cv e. anim2i x On v cv w cv e. w cv x cv e. x cv F dom C_ /\ r19.42v sylibr x cv v cv w cv ontr1 x cv F dom v cv ssel sylan9r exp4b com4l imp4d r19.23aiv syl w ax-gen mpgbir w visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr w cv z cv <,> eleq2i w cv z cv <,> f x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ E.e. eluniab bitr x cv w cv onelon w visset z visset f cv x cv fnop sylan2 exp32 com12 y x cv f cv y cv ` G f cv y cv |` ` = A.e. adantr com13 r19.23adv imp f 19.23aiv sylbi z 19.23aiv sylbi ssriv ordon F dom On trssord mp3an))
+thm (tfrlem9 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (G z) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A)))) (-> (e. (cv y) (dom F)) (= (` F (cv y)) (` G (|` F (cv y))))) (y visset F z eldm2 tfrlem.2 tfrlem.1 unieqi eqtr y cv z cv <,> eleq2i y cv z cv <,> f x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ E.e. eluniab bitr y visset z visset f cv x cv fnop x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ ra4e tfrlem.1 eqabi f cv A elssuni tfrlem.2 syl6ssr sylbir syl y x cv f cv y cv ` G f cv y cv |` ` = ra4 com12 f cv x cv fndm y cv eleq2d tfrlem.1 tfrlem.2 tfrlem7 F f cv y cv funssfv f cv x cv Fn x cv On e. /\ adantrl F f cv y cv fun2ssres G fveq2d f cv x cv fndm On eleq1d f cv dom y cv onelsst syl6bir imp31 sylan2 eqeq12d mpanl1 biimprd exp com3l exp31 com34 com24 sylbird com3l syld com24 imp4d mpdi syl exp4d exp com4r pm2.43i com3l imp4a r19.23adv imp f 19.23aiv sylbi z 19.23aiv sylbi))
+thm (tfrlem10 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (Fn C (suc (dom F)))) (x cv F dom G F F dom |` ` opeq1 x cv G F F dom |` ` <,> F dom G F F dom |` ` <,> sneq x cv G F F dom |` ` <,> {} F dom G F F dom |` ` <,> {} funeq 3syl x visset G F F dom |` fvex funsn On vtoclg tfrlem.1 tfrlem.2 tfrlem7 F F dom G F F dom |` ` <,> {} funun tfrlem.3 C F F dom G F F dom |` ` <,> {} u. funeq ax-mp sylibr x cv F dom G F F dom |` ` opeq1 sneqd dmeqd x cv F dom sneq eqeq12d x cv G F F dom |` ` dmsnop On vtoclg x cv eleq2d x cv F dom F dom eleq1 negbid F dom eloni F dom ordeirr syl syl5bir com12 x cv F dom elsni syl5 sylbid con2d r19.21aiv F dom F dom G F F dom |` ` <,> {} dom x disj sylibr sylan2 exp mpan mpcom x cv F dom G F F dom |` ` opeq1 x cv G F F dom |` ` <,> F dom G F F dom |` ` <,> sneq x cv G F F dom |` ` <,> {} F dom G F F dom |` ` <,> {} dmeq 3syl x cv F dom sneq eqeq12d x cv G F F dom |` ` dmsnop On vtoclg F dom uneq2d tfrlem.3 dmeqi F F dom G F F dom |` ` <,> {} dmun eqtr F dom df-suc 3eqtr4g jca C F dom suc df-fn sylibr))
+thm (tfrlem11 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (-> (e. (cv y) (suc (dom F))) (= (` C (cv y)) (` G (|` C (cv y)))))) (F F dom G F F dom |` ` <,> {} ssun1 tfrlem.3 sseqtr4 C F y cv funssfv F dom On e. adantrl C F y cv fun2ssres G fveq2d F dom y cv onelsst imp sylan2 eqeq12d tfrlem.1 tfrlem.2 tfrlem9 syl5bir mpanl2 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C F dom suc fnfun syl sylan exp32 pm2.43i pm2.43d y cv G C y cv |` ` opex snid y cv F dom G C y cv |` ` opeq1 F dom On e. adantl F F dom G F F dom |` ` <,> {} ssun1 tfrlem.3 sseqtr4 C F y cv fun2ssres mpanl2 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C F dom suc fnfun syl y cv F dom eqimss syl2an y cv F dom F reseq2 F dom On e. adantl eqtrd C y cv |` F F dom |` G fveq2 G C y cv |` ` G F F dom |` ` F dom opeq2 3syl eqtrd sneqd y cv G C y cv |` ` <,> eleq2d mpbii y cv G C y cv |` ` <,> F dom G F F dom |` ` <,> {} F elun2 syl tfrlem.3 syl6eleqr G C y cv |` fvex C F dom suc y cv fnfvop tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 y visset F dom eqelsuc syl2an mpbird exp jaod y cv F dom elsuci syl5))
+thm (tfrlem12 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (-> (e. (dom F) (On)) (e. C A)) (x cv F dom suc C fneq2 x cv F dom suc y C y cv ` G C y cv |` ` = raleq anbi12d On rcla4ev F dom suceloni tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 tfrlem.1 tfrlem.2 tfrlem.3 tfrlem11 r19.21aiv jca sylanc F dom suc On C fnex F dom suceloni tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 sylc f cv C x cv fneq1 f cv C y cv fveq1 f cv C y cv reseq1 G fveq2d eqeq12d y x cv biraldv anbi12d x On birexdv tfrlem.1 V elab2g syl mpbird))
+thm (tfrlem13 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfrlem.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfrlem.2 (= F (U. A))) (tfrlem.3 (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F))))))))) (= (dom F) (On)) (tfrlem.1 tfrlem.2 tfrlem8 F dom ordeirr C A elssuni tfrlem.2 syl6ssr C F dmss C dom F dom F dom ssel 3syl tfrlem.1 tfrlem.2 tfrlem.3 tfrlem12 F dom On sucidg tfrlem.1 tfrlem.2 tfrlem.3 tfrlem10 C F dom suc fndm syl eleqtrrd sylc nsyl ax-mp tfrlem.1 tfrlem.2 tfrlem8 F dom ordeleqon mpbi F dom On e. F dom On = orel1 mp2))
+thm (tfr1 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (Fn F (On)) (F On df-fn tfr.1 tfr.2 tfrlem7 tfr.1 tfr.2 F F dom G F F dom |` ` <,> {} u. eqid tfrlem13 mpbir2an))
+thm (tfr2 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (y z)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (-> (e. (cv z) (On)) (= (` F (cv z)) (` G (|` F (cv z))))) (y cv z cv F fveq2 y cv z cv F reseq2 G fveq2d eqeq12d tfr.1 tfr.2 F F dom G F F dom |` ` <,> {} u. eqid tfrlem13 y cv eleq2i tfr.1 tfr.2 tfrlem9 sylbir vtoclga))
+thm (tfr3 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (B x) (B y)) ((tfr.1 (= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (tfr.2 (= F (U. A)))) (-> (/\ (Fn B (On)) (A.e. x (On) (= (` B (cv x)) (` G (|` B (cv x)))))) (= B F)) (B On Fn x ax-17 x On B x cv ` G B x cv |` ` = hbra1 hban B On Fn x ax-17 x On B x cv ` G B x cv |` ` = hbra1 hban B y cv ` F y cv ` = x ax-17 hbim x cv y cv B fveq2 x cv y cv F fveq2 eqeq12d B On Fn x On B x cv ` G B x cv |` ` = A.e. /\ imbi2d x On B x cv ` G B x cv |` ` = ra4 tfr.1 tfr.2 tfr1 B On F x cv y fvreseq mpanl2 B x cv |` F x cv |` G fveq2 syl6bir x cv onsst sylan2 ancoms imp x cv On e. B x cv ` G B x cv |` ` = -> x cv On e. /\ adantr tfr.1 tfr.2 x tfr2 x cv On e. B x cv ` G B x cv |` ` = -> jctr x cv On e. B x cv ` G B x cv |` ` = F x cv ` G F x cv |` ` = jcab sylibr B x cv ` G B x cv |` ` F x cv ` G F x cv |` ` eqeq12 syl6 imp x cv On e. B On Fn /\ y x cv B y cv ` F y cv ` = A.e. /\ adantl mpbird exp43 com4t exp4a pm2.43d syl com3l imp3a a2d y x cv B On Fn x On B x cv ` G B x cv |` ` = A.e. /\ B y cv ` F y cv ` = r19.21v syl5ib tfis2f com12 r19.21ai On eqid tfr.1 tfr.2 tfr1 B On F On x eqfnfv mpan2 biimpar mpanr1 syldan))
+thm (tz7.44lem1 ((x y) (A x) (A y) (G x) (H y)) ((tz7.44lem1.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x)))))))))) (Fun G) (x y x cv {/} = y cv A = /\ x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ x cv dom Lim y cv x cv ran U. = /\ \/\/ funopab H x cv x cv dom U. ` fvex x visset x cv V rnexg x cv ran V uniexg syl ax-mp nlim0 dm0 {/} dom {/} limeq ax-mp mtbir x cv {/} dmeq x cv dom {/} dom limeq syl biimpa mto y A moeq3 mpgbir tz7.44lem1.1 G x y x cv {/} = y cv A = /\ x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ x cv dom Lim y cv x cv ran U. = /\ \/\/ {<,>|} funeq ax-mp mpbir))
+thm (tz7.44-1 ((x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.4 (e. A (V)))) (= (` F ({/})) A) (0elon x cv {/} F fveq2 x cv {/} F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp F res0 G fveq2i tz7.44.1 tz7.44lem1 x cv {/} = y cv A = /\ x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ x cv dom Lim y cv x cv ran U. = /\ 3mix1 x y ssopab2i tz7.44.1 sseqtr4 0ex tz7.44.4 x cv {/} {/} eqeq1 y cv A = anbi1d y cv A A eqeq1 {/} {/} = anbi2d opelopab {/} eqid A eqid mpbir2an sselii tz7.44.4 G {/} funfvopi mp2 3eqtr))
+thm (tz7.44-2 ((x y) (B x) (B y) (F y) (H x) (x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.5 (e. B (On)))) (= (` F (suc B)) (` H (` F B))) (tz7.44.5 onsuc x cv B suc F fveq2 x cv B suc F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp tz7.44.1 tz7.44lem1 x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ x cv {/} = y cv A = /\ x cv dom Lim y cv x cv ran U. = /\ 3mix2 x y ssopab2i tz7.44.1 sseqtr4 tz7.44.5 onsuc tz7.44.2 F On fnfun ax-mp B suc On F resfunexg mp2 H F B ` fvex x cv F B suc |` {/} eqeq1 x cv F B suc |` dmeq x cv dom F B suc |` dom limeq syl orbi12d negbid x cv F B suc |` dmeq x cv dom F B suc |` dom unieq x cv dom U. F B suc |` dom U. x cv fveq2 3syl x cv F B suc |` F B suc |` dom U. fveq1 eqtrd H fveq2d y cv eqeq2d anbi12d y cv H F B ` ` H F B suc |` F B suc |` dom U. ` ` eqeq1 F B suc |` {/} = F B suc |` dom Lim \/ -. anbi2d opelopab B nsuceq0 tz7.44.2 F On fndm ax-mp B suc ineq2i F B suc dmres tz7.44.5 onsuc onss B suc On dfss mpbi 3eqtr4 {/} eqeq1i mtbir F B suc |` {/} dmeq dm0 syl6eq mto tz7.44.5 elisseti nlimsuc tz7.44.2 F On fndm ax-mp B suc ineq2i F B suc dmres tz7.44.5 onsuc onss B suc On dfss mpbi 3eqtr4 F B suc |` dom B suc limeq ax-mp mtbir pm3.2ni tz7.44.5 elisseti sucid B B suc F fvres ax-mp tz7.44.2 F On fndm ax-mp B suc ineq2i F B suc dmres tz7.44.5 onsuc onss B suc On dfss mpbi 3eqtr4 unieqi tz7.44.5 onunisuc eqtr2 F B suc |` fveq2i eqtr3 H fveq2i mpbir2an sselii H F B ` fvex G F B suc |` funfvopi mp2 eqtr))
+thm (tz7.44-3 ((x y) (B x) (B y) (F y) (H x) (x y) (A x) (A y) (F x) (G x) (H y)) ((tz7.44.1 (= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (tz7.44.2 (Fn F (On))) (tz7.44.3 (-> (e. (cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x)))))) (tz7.44.5 (e. B (On)))) (-> (Lim B) (= (` F B) (U. (" F B)))) (tz7.44.2 F On fndm ax-mp B ineq2i F B dmres tz7.44.5 onss B On dfss mpbi 3eqtr4 F B |` dom B limeq ax-mp biimpr F B df-ima unieqi jctir tz7.44.5 tz7.44.2 F On fnfun ax-mp B On F resfunexg mp2 tz7.44.2 F On fnfun ax-mp tz7.44.5 elisseti F funimaex ax-mp uniex x cv F B |` dmeq x cv dom F B |` dom limeq syl x cv F B |` rneq unieqd y cv eqeq2d anbi12d y cv F B " U. F B |` ran U. eqeq1 F B |` dom Lim anbi2d opelopab sylibr x cv dom Lim y cv x cv ran U. = /\ x cv {/} = y cv A = /\ x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ 3mix3 x y ssopab2i tz7.44.1 sseqtr4 F B |` F B " U. <,> sseli tz7.44.1 tz7.44lem1 tz7.44.2 F On fnfun ax-mp tz7.44.5 elisseti F funimaex ax-mp uniex G F B |` funfvopi ax-mp 3syl tz7.44.5 x cv B F fveq2 x cv B F reseq2 G fveq2d eqeq12d tz7.44.3 vtoclga ax-mp syl5eq))
+thm (dfrdg2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g)) () (= (rec F A) (U. ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` ({<,>|} g z (\/\/ (/\ (= (cv g) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv g) ({/})) (Lim (dom (cv g))))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))))) (|` (cv f) (cv y)))))))))) (F A f x y g z df-rdg z cv g cv dom Lim g cv ran U. F g cv g cv dom U. ` ` eqif g cv {/} = -. anbi2i g cv dom Lim z cv g cv ran U. = /\ g cv dom Lim -. z cv F g cv g cv dom U. ` ` = /\ orcom g cv {/} = -. anbi2i g cv {/} = -. g cv dom Lim -. z cv F g cv g cv dom U. ` ` = /\ g cv dom Lim z cv g cv ran U. = /\ andi g cv {/} = g cv dom Lim ioran z cv F g cv g cv dom U. ` ` = anbi1i g cv {/} = -. g cv dom Lim -. z cv F g cv g cv dom U. ` ` = anass bitr g cv {/} dmeq dm0 syl6eq nlim0 g cv dom {/} limeq mtbiri syl con2i pm4.71ri z cv g cv ran U. = anbi1i g cv {/} = -. g cv dom Lim z cv g cv ran U. = anass bitr orbi12i bitr4 3bitr g cv {/} = z cv A = /\ orbi2i z cv g cv {/} = A g cv dom Lim g cv ran U. F g cv g cv dom U. ` ` if eqif g cv {/} = z cv A = /\ g cv {/} = g cv dom Lim \/ -. z cv F g cv g cv dom U. ` ` = /\ g cv dom Lim z cv g cv ran U. = /\ 3orass 3bitr4 g z biopabi f cv y cv |` fveq1i f cv y cv ` eqeq2i y x cv biral f cv x cv Fn anbi2i x On birex f biabi unieqi eqtr))
+thm (rdgeq1 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (G x) (G y) (G z) (G f) (G g) (A x) (A y) (A z) (A f) (A g)) () (-> (= F G) (= (rec F A) (rec G A))) (F G g cv g cv dom U. ` fveq1 g cv dom Lim g cv ran U. ifeq2d g cv {/} = A ifeq2d z cv eqeq2d g z biopabdv f cv y cv |` fveq1d f cv y cv ` eqeq2d y x cv biraldv f cv x cv Fn anbi2d x On birexdv f biabdv unieqd F A f x y g z df-rdg G A f x y g z df-rdg 3eqtr4g))
+thm (rdgeq2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g) (B x) (B y) (B z) (B f) (B g)) () (-> (= A B) (= (rec F A) (rec F B))) (A B g cv {/} = g cv dom Lim g cv ran U. F g cv g cv dom U. ` ` if ifeq1 z cv eqeq2d g z biopabdv f cv y cv |` fveq1d f cv y cv ` eqeq2d y x cv biraldv f cv x cv Fn anbi2d x On birexdv f biabdv unieqd F A f x y g z df-rdg F B f x y g z df-rdg 3eqtr4g))
+thm (hbrdg ((y z) (w y) (v y) (f y) (g y) (F y) (w z) (v z) (f z) (g z) (F z) (v w) (f w) (g w) (F w) (f v) (g v) (F v) (f g) (F f) (F g) (A y) (A z) (A w) (A v) (A f) (A g) (x y) (x z) (w x) (v x) (f x) (g x)) ((hbrdg.1 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbrdg.2 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (rec F A)) (A. x (e. (cv y) (rec F A)))) (w cv On e. x ax-17 f cv w cv Fn x ax-17 v cv w cv e. x ax-17 y cv f cv v cv ` e. x ax-17 y cv z cv e. x ax-17 g cv {/} = x ax-17 hbrdg.2 g cv dom Lim x ax-17 y cv g cv ran U. e. x ax-17 hbrdg.1 y cv g cv g cv dom U. ` e. x ax-17 hbfv hbif hbif hbeq y g z hbopab y cv f cv v cv |` e. x ax-17 hbfv hbeq hbral hban hbrex y f hbab hbuni F A f w v g z df-rdg y cv eleq2i F A f w v g z df-rdg y cv eleq2i x bial 3imtr4))
+thm (rdglem1 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z) (w y) (G y) (w z) (g w) (G w)) () (= ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))) ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. w (cv z) (= (` (cv g) (cv w)) (` G (|` (cv g) (cv w))))))))) (f x On f cv x cv Fn y x cv f cv y cv ` G f cv y cv |` ` = A.e. /\ E.e. {|} eqid g z tfrlem3 y cv w cv g cv fveq2 y cv w cv g cv reseq2 G fveq2d eqeq12d z cv cbvralv g cv z cv Fn anbi2i z On birex g biabi eqtr))
+thm (rdglem2 ((x y) (x z) (w x) (y z) (w y) (w z) (A x) (A z) (A w) (H x) (H z) (H w)) () (= ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))) ({<,>|} z y (\/\/ (/\ (= (cv z) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv z) ({/})) (Lim (dom (cv z))))) (= (cv y) (` H (` (cv z) (U. (dom (cv z))))))) (/\ (Lim (dom (cv z))) (= (cv y) (U. (ran (cv z)))))))) (x cv z cv y cv opeq1 w cv eqeq2d x cv z cv {/} eqeq1 y cv A = anbi1d x cv z cv {/} eqeq1 x cv z cv dmeq x cv dom z cv dom limeq syl orbi12d negbid x cv z cv dmeq x cv dom z cv dom unieq x cv dom U. z cv dom U. x cv fveq2 3syl x cv z cv z cv dom U. fveq1 eqtrd H fveq2d y cv eqeq2d anbi12d x cv z cv dmeq x cv dom z cv dom limeq syl x cv z cv rneq unieqd y cv eqeq2d anbi12d 3orbi123d anbi12d y biexdv cbvexv w biabi x y x cv {/} = y cv A = /\ x cv {/} = x cv dom Lim \/ -. y cv H x cv x cv dom U. ` ` = /\ x cv dom Lim y cv x cv ran U. = /\ \/\/ w df-opab z y z cv {/} = y cv A = /\ z cv {/} = z cv dom Lim \/ -. y cv H z cv z cv dom U. ` ` = /\ z cv dom Lim y cv z cv ran U. = /\ \/\/ w df-opab 3eqtr4))
+thm (rdgfnon ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (Fn (rec F A) (On)) (w u v g z z cv g cv {/} = A g cv dom Lim g cv ran U. F g cv g cv dom U. ` ` if if = {<,>|} f x y rdglem1 F A w u v g z df-rdg tfr1))
+thm (rdgval ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (-> (e. (cv g) (On)) (= (` (rec F A) (cv g)) (` ({<,>|} w z (\/\/ (/\ (= (cv w) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv w) ({/})) (Lim (dom (cv w))))) (= (cv z) (` F (` (cv w) (U. (dom (cv w))))))) (/\ (Lim (dom (cv w))) (= (cv z) (U. (ran (cv w))))))) (|` (rec F A) (cv g))))) (w u v g z g cv {/} = z cv A = /\ g cv {/} = g cv dom Lim \/ -. z cv F g cv g cv dom U. ` ` = /\ g cv dom Lim z cv g cv ran U. = /\ \/\/ {<,>|} f x y rdglem1 F A w u v g z dfrdg2 g tfr2 g z A F w rdglem2 F A rec g cv |` fveq1i syl6eq))
+thm (rdg0 ((g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.1 (e. A (V)))) (= (` (rec F A) ({/})) A) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.1 tz7.44-1))
+thm (rdgsuc ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.2 (e. B (On)))) (= (` (rec F A) (suc B)) (` F (` (rec F A) B))) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.2 tz7.44-2))
+thm (rdglim ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((rdg.2 (e. B (On)))) (-> (Lim B) (= (` (rec F A) B) (U. (" (rec F A) B)))) (w z A F g rdglem2 F A rdgfnon g F A w z rdgval rdg.2 tz7.44-3))
+thm (rdg0t ((A x) (F x)) () (-> (e. A C) (= (` (rec F A) ({/})) A)) (x cv A F rdgeq2 {/} fveq1d x cv A = id eqeq12d x visset F rdg0 C vtoclg))
+thm (rdgsuct () () (-> (e. B (On)) (= (` (rec F A) (suc B)) (` F (` (rec F A) B)))) (B B On e. B {/} if suceq F A rec fveq2d B B On e. B {/} if F A rec fveq2 F fveq2d eqeq12d 0elon B elimel F A rdgsuc dedth))
+thm (rdgsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((rdgsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (rdgsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (rdgsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (rdgsucopab.4 (= F (rec ({<,>|} x y (= (cv y) C)) A))) (rdgsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (/\ (e. B (On)) (e. D R)) (= (` F (suc B)) D)) (B x y y cv C = {<,>|} A rdgsuct rdgsucopab.4 B suc fveq1i syl5eq x y y cv C = {<,>|} A rec B fvex z x y y cv C = hbopab1 rdgsucopab.1 hbrdg rdgsucopab.2 hbfv rdgsucopab.3 rdgsucopab.4 B fveq1i x cv eqeq2i rdgsucopab.5 sylbir V R y fvopabgf mpan sylan9eq))
+thm (rdgsucopabn ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((rdgsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (rdgsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (rdgsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (rdgsucopab.4 (= F (rec ({<,>|} x y (= (cv y) C)) A))) (rdgsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (-. (e. D (V))) (= (` F (suc B)) ({/}))) (B x y y cv C = {<,>|} A rdgsuct rdgsucopab.4 B suc fveq1i syl5eq z x y y cv C = hbopab1 rdgsucopab.1 hbrdg rdgsucopab.2 hbfv rdgsucopab.3 rdgsucopab.4 B fveq1i x cv eqeq2i rdgsucopab.5 sylbir y fvopabnf sylan9eq exp B sucelon rdgsucopab.4 dmeqi x y y cv C = {<,>|} A rdgfnon x y y cv C = {<,>|} A rec On fndm ax-mp eqtr B suc eleq2i bitr4 negbii B suc F ndmfv sylbi D V e. -. a1d pm2.61i))
+thm (rdglimt () () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. (" (rec F A) B)))) (B B On e. B {/} if limeq B B On e. B {/} if F A rec fveq2 B B On e. B {/} if F A rec imaeq2 unieqd eqeq12d imbi12d 0elon B elimel F A rdglim dedth imp B C limelon sylan anabss3))
+thm (rdglim2 ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. ({|} y (E.e. x B (= (cv y) (` (rec F A) (cv x)))))))) (B C F A rdglimt B limord B x cv ordelord exp x visset elon syl6ibr syl F A rdgfnon y visset F A rec On x cv fnfvop mpan y cv F A rec x cv ` eqcom syl5bb syl6 pm5.32d x biexdv x B y cv F A rec x cv ` = df-rex syl5rbb y biabdv F A rec B y x dfima3 syl5eq unieqd B C e. adantl eqtrd))
+thm (rdglim2a ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U_ x B (` (rec F A) (cv x))))) (B C F A y x rdglim2 F A rec x cv fvex x B y dfiun2 syl6eqr))
+thm (frfnom () () (Fn (|` (rec F A) (om)) (om)) (F A rec om |` om df-fn F A rdgfnon F A rec On fnfun ax-mp F A rec om funres ax-mp F A rdgfnon F A rec On fndm ax-mp om ineq2i F A rec om dmres omsson om On dfss mpbi 3eqtr4 mpbir2an))
+thm (fr0t () () (-> (e. A B) (= (` (|` (rec F A) (om)) ({/})) A)) (A B F rdg0t peano1 {/} om F A rec fvres ax-mp syl5eq))
+thm (frsuct () () (-> (e. B (om)) (= (` (|` (rec F A) (om)) (suc B)) (` F (` (|` (rec F A) (om)) B)))) (B nnont B F A rdgsuct syl B peano2b B suc om F A rec fvres sylbi B om F A rec fvres F fveq2d 3eqtr4d))
+thm (frsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((frsucopab.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (frsucopab.2 (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (frsucopab.3 (-> (e. (cv z) D) (A. x (e. (cv z) D)))) (frsucopab.4 (= F (|` (rec ({<,>|} x y (= (cv y) C)) A) (om)))) (frsucopab.5 (-> (= (cv x) (` F B)) (= C D)))) (-> (/\ (e. B (om)) (e. D R)) (= (` F (suc B)) D)) (B x y y cv C = {<,>|} A frsuct frsucopab.4 B suc fveq1i syl5eq x y y cv C = {<,>|} A rec om |` B fvex z x y y cv C = hbopab1 frsucopab.1 hbrdg z cv om e. x ax-17 hbres frsucopab.2 hbfv frsucopab.3 frsucopab.4 B fveq1i x cv eqeq2i frsucopab.5 sylbir V R y fvopabgf mpan sylan9eq))
+thm (tz7.48lem ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w)) ((tz7.48.1 (Fn F (On)))) (-> (/\ (C_ A (On)) (A.e. x A (A.e. y (cv x) (-. (= (` F (cv x)) (` F (cv y))))))) (Fun (`' (|` F A)))) (x cv A F fvres y cv A F fvres eqeqan12d A On C_ x cv y cv e. F y cv ` F x cv ` = -. -> y cv x cv e. F x cv ` F y cv ` = -. -> /\ ad2antrl A On x cv ssel A On y cv ssel anim12d x cv y cv e. F y cv ` F x cv ` = -. y cv x cv e. F x cv ` F y cv ` = -. pm3.48 F x cv ` F y cv ` = -. oridm F x cv ` F y cv ` eqcom negbii F x cv ` F y cv ` = -. orbi1i bitr3 syl6ibr con2d x cv y cv ordtri3 biimprd x cv eloni y cv eloni syl2an syl9r syl6 imp32 sylbid exp32 a2d y 19.20dv x 19.20dv x A y A x cv y cv e. F y cv ` F x cv ` = -. -> y cv x cv e. F x cv ` F y cv ` = -. -> /\ r2al x A y A F A |` x cv ` F A |` y cv ` = x cv y cv = -> r2al 3imtr4g x cv A e. y cv A e. pm3.26 y cv x cv e. anim1i F x cv ` F y cv ` = -. imim1i exp3a x y 19.20i2 x A y x cv F x cv ` F y cv ` = -. r2al x A y A y cv x cv e. F x cv ` F y cv ` = -. -> r2al 3imtr4 y cv w cv x cv eleq1 y cv w cv F fveq2 F x cv ` eqeq2d negbid imbi12d A cbvralv x A biral x cv z cv w cv eleq2 x cv z cv F fveq2 F w cv ` eqeq1d negbid imbi12d w A biraldv A cbvralv w cv x cv z cv eleq1 w cv x cv F fveq2 F z cv ` eqeq2d negbid imbi12d A cbvralv z A biral z cv y cv x cv eleq2 z cv y cv F fveq2 F x cv ` eqeq1d negbid imbi12d x A biraldv A cbvralv bitr 3bitr y A x x cv y cv e. F y cv ` F x cv ` = -. -> ralcom2 sylbi ancri y A x cv y cv e. F y cv ` F x cv ` = -. -> y cv x cv e. F x cv ` F y cv ` = -. -> r19.26 x A biral x A y A x cv y cv e. F y cv ` F x cv ` = -. -> A.e. y A y cv x cv e. F x cv ` F y cv ` = -. -> A.e. r19.26 bitr sylibr syl syl5 imdistani F A |` A V x y f1fv F A |` A V df-f1 bitr3 pm3.27bd F A |` A fnf sylanb tz7.48.1 F On A fnssres mpan sylan syl))
+thm (tz7.48-1 ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (C_ (ran F) A)) (x On F x cv ` A F x cv " \ e. hbra1 y cv A e. x ax-17 x On F x cv ` A F x cv " \ e. ra4 F x cv ` y cv A eleq1 F x cv ` A F x cv " eldifi syl5bi com12 x cv On e. imim2i imp3a syl 19.23ad y visset F x elrn x visset y cv F opeldm tz7.48.1 F On fndm ax-mp x cv eleq2i sylib ancri tz7.48.1 y visset F On x cv fnfvop mpan pm5.32i sylibr x 19.22i sylbi syl5 ssrdv))
+thm (tz7.48-2 ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (Fun (`' F))) (x cv y cv onelon ancoms F x cv dmres y cv eleq2i y cv x cv F dom elin bitr tz7.48.1 F On fnfun ax-mp F x cv funres ax-mp F x cv |` y cv fvrn mpan sylbir y cv x cv F fvres F x cv |` ran eleq1d F x cv df-ima F y cv ` eleq2i syl6rbbr y cv F dom e. adantr mpbird F y cv ` F x cv " F x cv ` eleq1a F x cv ` A F x cv " eldifn nsyli syl tz7.48.1 F On fndm ax-mp y cv eleq2i sylan2br syldan exp imp3a com12 r19.21aiv exp r19.20i On ssid tz7.48.1 On x y tz7.48lem mpan syl tz7.48.1 F On fnrel ax-mp tz7.48.1 F On fndm ax-mp On ssid eqsstr F On relssres mp2an F On |` F cnveq ax-mp F On |` `' F `' funeq ax-mp sylib))
+thm (tz7.48-3 ((F x) (A x)) ((tz7.48.1 (Fn F (On)))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (-. (e. A (V)))) (onprc tz7.48.1 F On fndm ax-mp V eleq1i mtbir tz7.48.1 x A tz7.48-2 F `' V funrnex com12 F df-rn V eleq1i F dfdm4 V eleq1i 3imtr4g syl mtoi A V F ran ssexg tz7.48.1 x A tz7.48-1 syl5com mtod))
+thm (tz7.49 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((tz7.48.1 (Fn F (On))) (tz7.49.2 (e. A (V)))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (/\/\ (A.e. y (cv x) (-. (= (\ A (" F (cv y))) ({/})))) (= (" F (cv x)) A) (Fun (`' (|` F (cv x))))))) (tz7.49.2 tz7.48.1 x A tz7.48-3 mt2 x On A F x cv " \ {/} = -. F x cv ` A F x cv " \ e. r19.20 mtoi x On A F x cv " \ {/} = dfrex2 sylibr x cv y cv F imaeq2 A difeq2d {/} eqeq1d onminex syl x On A F x cv " \ {/} = -. F x cv ` A F x cv " \ e. -> hbra1 x On A F x cv " \ {/} = -. F x cv ` A F x cv " \ e. -> A.e. y x cv A F y cv " \ {/} = -. A.e. pm3.27 x cv On e. A F x cv " \ {/} = ad2antll x On A F x cv " \ {/} = -. F x cv ` A F x cv " \ e. -> A.e. y ax-17 y x cv A F y cv " \ {/} = -. hbra1 hban x cv On e. z cv A e. -> y ax-17 y x cv A F y cv " \ {/} = -. ra4 x cv On e. adantld x cv y cv onelon x cv y cv F imaeq2 A difeq2d {/} eqeq1d negbid x cv y cv F fveq2 x cv y cv F imaeq2 A difeq2d eleq12d imbi12d On rcla4v com23 syl sylcom com3r imp exp3a com23 F y cv ` z cv A eleq1 F y cv ` A F y cv " eldifi syl5bi com12 syl8 com34 r19.23ad tz7.48.1 F On fnfun ax-mp F z cv x cv y fvelima mpan syl5 com23 imp ssrdv A F x cv " ssdif0 biimpr anim12i F x cv " A eqss sylibr y x cv A F y cv " \ {/} = -. hbra1 x On A F x cv " \ {/} = -. F x cv ` A F x cv " \ e. -> A.e. y ax-17 hban x cv On C_ y ax-17 x cv On y cv ssel y cv onsst tz7.48.1 F On fndm ax-mp syl6ssr tz7.48.1 F On fnfun ax-mp F y cv z cv funfvima2 mpan syl syl6 y x cv A F y cv " \ {/} = -. ra4 com12 x cv On C_ a1i x cv On y cv ssel x cv y cv F imaeq2 A difeq2d {/} eqeq1d negbid x cv y cv F fveq2 x cv y cv F imaeq2 A difeq2d eleq12d imbi12d On rcla4v com23 syl6 syldd imp4a F z cv ` F y cv " F y cv ` eleq1a con3d F y cv ` A F y cv " eldifn syl5com syl8 com34 syldd com4r imp4a z 19.21adv 19.21ad y x cv z y cv F y cv ` F z cv ` = -. r2al syl6ibr ancld tz7.48.1 x cv y z tz7.48lem syl6 x cv onsst syl5 ancoms imp A F x cv " \ {/} = adantr 3jca exp41 com23 com34 imp4a r19.22d mpd))
+thm (tz7.49c ((x y) (F x) (F y) (A x) (A y)) ((tz7.48.1 (Fn F (On))) (tz7.49.2 (e. A (V)))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (:-1-1-onto-> (|` F (cv x)) (cv x) A))) (tz7.48.1 tz7.49.2 x y tz7.49 x cv onsst tz7.48.1 F On x cv fnssres mpan syl F x cv df-ima A eqeq1i biimp anim12i F x cv |` `' Fun anim1i F x cv |` x cv A f1o2 F x cv |` x cv Fn F x cv |` `' Fun F x cv |` ran A = df-3an F x cv |` x cv Fn F x cv |` `' Fun F x cv |` ran A = an23 3bitr sylibr exp31 imp3a y x cv A F y cv " \ {/} = -. A.e. F x cv " A = F x cv |` `' Fun 3simpc syl5 r19.22i syl))
+thm (abianfplem ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((abianfp.1 (e. A (V))) (abianfp.2 (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x))))) (-> (e. (cv v) (On)) (-> (= (` F (cv x)) (cv x)) (= (` G (cv v)) (cv x)))) (v cv {/} G fveq2 x cv eqeq1d v cv y cv G fveq2 x cv eqeq1d v cv y cv suc G fveq2 x cv eqeq1d abianfp.2 {/} fveq1i x visset z w w cv F z cv ` = {<,>|} rdg0 eqtr F x cv ` x cv = a1i F G y cv ` fvex u cv x cv e. z ax-17 u cv y cv e. z ax-17 u cv F e. z ax-17 u z w w cv F z cv ` = hbopab1 u cv x cv e. z ax-17 hbrdg abianfp.2 u cv eleq2i abianfp.2 u cv eleq2i z bial 3imtr4 u cv y cv e. z ax-17 hbfv hbfv abianfp.2 z cv G y cv ` F fveq2 V rdgsucopab mpan2 G y cv ` x cv F fveq2 F x cv ` x cv = id sylan9eqr sylan9eq exp32 v visset v cv V z w w cv F z cv ` = {<,>|} x cv y rdglim2a mpan abianfp.2 v cv fveq1i abianfp.2 y cv fveq1i y cv v cv e. a1i iuneq2i 3eqtr4g y v cv G y cv ` x cv = A.e. adantr y v cv G y cv ` x cv iuneq2 v cv df-lim v cv Ord v cv {/} = -. v cv v cv U. = 3simp2 sylbi v cv y x cv iunconst syl sylan9eqr eqtrd exp F x cv ` x cv = a1d tfinds2))
+thm (abianfp ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((abianfp.1 (e. A (V))) (abianfp.2 (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x))))) (<-> (E.e. x A (= (` F (cv x)) (cv x))) (E.e. x A (A.e. v (On) (/\ (e. (` G (cv v)) A) (-> (-. (= (` F (` G (cv v))) (` G (cv v)))) (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u)))))))))) (abianfp.1 abianfp.2 v abianfplem imp A eleq1d biimprd abianfp.1 abianfp.2 v abianfplem G v cv ` x cv F fveq2 G v cv ` x cv = id eqeq12d biimprcd sylcom imp F G v cv ` ` G v cv ` = negb syl u v cv G v cv ` G u cv ` = -. A.e. pm2.21d jctird exp com13 r19.21adv r19.22i onprc v On G v cv ` A e. F G v cv ` ` G v cv ` = -. u v cv G v cv ` G u cv ` = -. A.e. -> r19.26 y cv G v cv ` F fveq2 y cv G v cv ` = id eqeq12d negbid A rcla4v com12 u v cv G v cv ` G u cv ` = -. A.e. imim1d v On r19.20sdv v On G v cv ` A e. u v cv G v cv ` G u cv ` = -. A.e. r19.20 syl6 imp com12 z w w cv F z cv ` = {<,>|} x cv rdgfnon abianfp.2 G z w w cv F z cv ` = {<,>|} x cv rec On fneq1 ax-mp mpbir G On A v ffnfv biimpr mpan On ssid z w w cv F z cv ` = {<,>|} x cv rdgfnon abianfp.2 G z w w cv F z cv ` = {<,>|} x cv rec On fneq1 ax-mp mpbir On v u tz7.48lem mpan z w w cv F z cv ` = {<,>|} x cv rdgfnon abianfp.2 G z w w cv F z cv ` = {<,>|} x cv rec On fneq1 ax-mp mpbir G On fnresdm ax-mp G On |` G cnveq ax-mp G On |` `' G `' funeq ax-mp sylib anim12i G On A df-f1 sylibr abianfp.1 A V G On f1dmex ax-mp syl exp syld exp3a com23 imp sylbi mtoi x cv A e. a1i r19.23aiv x cv y cv F fveq2 x cv y cv = id eqeq12d A cbvrexv y A F y cv ` y cv = dfrex2 bitr2 sylib impbi))
+thm (opreq () () (-> (= F G) (= (opr A F B) (opr A G B))) (F G A B <,> fveq1 A F B df-opr A G B df-opr 3eqtr4g))
+thm (opreq1 () () (-> (= A B) (= (opr A F C) (opr B F C))) (A B C opeq1 F fveq2d A F C df-opr B F C df-opr 3eqtr4g))
+thm (opreq2 () () (-> (= A B) (= (opr C F A) (opr C F B))) (A B C opeq2 F fveq2d C F A df-opr C F B df-opr 3eqtr4g))
+thm (opreq12 () () (-> (/\ (= A B) (= C D)) (= (opr A F C) (opr B F D))) (A B F C opreq1 C D B F opreq2 sylan9eq))
+thm (opreq1i () ((opreq1i.1 (= A B))) (= (opr A F C) (opr B F C)) (opreq1i.1 A B F C opreq1 ax-mp))
+thm (opreq2i () ((opreq1i.1 (= A B))) (= (opr C F A) (opr C F B)) (opreq1i.1 A B C F opreq2 ax-mp))
+thm (opreq12i () ((opreq1i.1 (= A B)) (opreq12i.2 (= C D))) (= (opr A F C) (opr B F D)) (opreq1i.1 F C opreq1i opreq12i.2 B F opreq2i eqtr))
+thm (opreq1d () ((opreq1d.1 (-> ph (= A B)))) (-> ph (= (opr A F C) (opr B F C))) (opreq1d.1 A B F C opreq1 syl))
+thm (opreq2d () ((opreq1d.1 (-> ph (= A B)))) (-> ph (= (opr C F A) (opr C F B))) (opreq1d.1 A B C F opreq2 syl))
+thm (opreq12d () ((opreq1d.1 (-> ph (= A B))) (opreq12d.2 (-> ph (= C D)))) (-> ph (= (opr A F C) (opr B F D))) (opreq1d.1 F C opreq1d opreq12d.2 B F opreq2d eqtrd))
+thm (opreqan12d () ((opreq1d.1 (-> ph (= A B))) (opreqan12i.2 (-> ps (= C D)))) (-> (/\ ph ps) (= (opr A F C) (opr B F D))) (A B C D F opreq12 opreq1d.1 opreqan12i.2 syl2an))
+thm (opreqan12rd () ((opreq1d.1 (-> ph (= A B))) (opreqan12i.2 (-> ps (= C D)))) (-> (/\ ps ph) (= (opr A F C) (opr B F D))) (opreq1d.1 opreqan12i.2 F opreqan12d ancoms))
+thm (hbopr ((F y) (A y) (B y) (x y)) ((hbopr.1 (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (hbopr.2 (-> (e. (cv y) F) (A. x (e. (cv y) F)))) (hbopr.3 (-> (e. (cv y) B) (A. x (e. (cv y) B))))) (-> (e. (cv y) (opr A F B)) (A. x (e. (cv y) (opr A F B)))) (hbopr.2 hbopr.1 hbopr.3 hbop hbfv A F B df-opr y cv eleq2i A F B df-opr y cv eleq2i x bial 3imtr4))
+thm (oprex () () (e. (opr A F B) (V)) (A F B df-opr F A B <,> fvex eqeltr))
+thm (oprprc1 () ((oprprc1.1 (Rel (dom F)))) (-> (-. (e. A (V))) (= (opr A F B) ({/}))) (A F dom B df-br oprprc1.1 A B brrelexi sylbir con3i A B <,> F ndmfv syl A F B df-opr syl5eq))
+thm (oprprc2 () () (-> (-. (e. B (V))) (= (opr A F B) (opr A F A))) (B A opprc2 F fveq2d A F B df-opr A F A df-opr 3eqtr4g))
+thm (dfoprab2 ((x z) (w x) (v x) (w z) (v z) (v w) (y z) (w y) (v y) (ph w) (ph v)) () (= ({<<,>,>|} x y z ph) ({<,>|} w z (E. x (E. y (/\ (= (cv w) (<,> (cv x) (cv y))) ph))))) (z w x y v cv w cv z cv <,> = w cv x cv y cv <,> = ph /\ /\ E. E. excom z w x y v cv w cv z cv <,> = w cv x cv y cv <,> = ph /\ /\ exrot4 w v cv x cv y cv <,> z cv <,> = ph /\ w cv x cv y cv <,> = 19.42v w cv x cv y cv <,> z cv opeq1 v cv eqeq2d pm5.32ri ph anbi1i v cv w cv z cv <,> = w cv x cv y cv <,> = ph anass v cv x cv y cv <,> z cv <,> = w cv x cv y cv <,> = ph an23 3bitr3 w biex x cv y cv opex w isseti v cv x cv y cv <,> z cv <,> = ph /\ biantru 3bitr4 x y z bi3ex bitr x y v cv w cv z cv <,> = w cv x cv y cv <,> = ph /\ 19.42vv w z bi2ex 3bitr3 v biabi x y z ph v df-oprab w z x y w cv x cv y cv <,> = ph /\ E. E. v df-opab 3eqtr4))
+thm (reloprab ((x z) (w x) (w z) (y z) (w y) (ph w)) () (Rel ({<<,>,>|} x y z ph)) (w z x y w cv x cv y cv <,> = ph /\ E. E. relopab x y z ph w dfoprab2 x y z ph {<<,>,>|} w z x y w cv x cv y cv <,> = ph /\ E. E. {<,>|} releq ax-mp mpbir))
+thm (hboprab1 ((x y) (x z) (v x) (y z) (v y) (v z) (w x) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. x (e. (cv w) ({<<,>,>|} x y z ph)))) (x y z v cv x cv y cv <,> z cv <,> = ph /\ E. E. hbe1 w v hbab x y z ph v df-oprab w cv eleq2i x y z ph v df-oprab w cv eleq2i x bial 3imtr4))
+thm (hboprab2 ((x y) (x z) (v x) (y z) (v y) (v z) (w y) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. y (e. (cv w) ({<<,>,>|} x y z ph)))) (y z v cv x cv y cv <,> z cv <,> = ph /\ E. hbe1 x hbex w v hbab x y z ph v df-oprab w cv eleq2i x y z ph v df-oprab w cv eleq2i y bial 3imtr4))
+thm (bioprabd ((x z) (w x) (w z) (y z) (w y) (ph w) (ps w) (ch w)) ((bioprabd.1 (-> ph (A. x ph))) (bioprabd.2 (-> ph (A. y ph))) (bioprabd.3 (-> ph (A. z ph))) (bioprabd.4 (-> ph (<-> ps ch)))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))) (ph w ax-17 bioprabd.3 bioprabd.1 bioprabd.2 bioprabd.4 w cv x cv y cv <,> = anbi2d biexd biexd biopabd x y z ps w dfoprab2 x y z ch w dfoprab2 3eqtr4g))
+thm (bioprabdv ((x z) (ph x) (ph z) (y z) (ph y)) ((bioprabdv.1 (-> ph (<-> ps ch)))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))) (ph x ax-17 ph y ax-17 ph z ax-17 bioprabdv.1 bioprabd))
+thm (bioprabi ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((bioprab.1 (<-> ph ps))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)) (w cv eqid bioprab.1 w cv w cv = a1i x y z bioprabdv ax-mp))
+thm (cbvoprab12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u) (ps u)) ((cbvoprab12.1 (-> ph (A. w ph))) (cbvoprab12.2 (-> ph (A. v ph))) (cbvoprab12.3 (-> ps (A. x ps))) (cbvoprab12.4 (-> ps (A. y ps))) (cbvoprab12.5 (-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)) (u cv x cv y cv <,> z cv <,> = w ax-17 cbvoprab12.1 hban z hbex u cv x cv y cv <,> z cv <,> = v ax-17 cbvoprab12.2 hban z hbex u cv w cv v cv <,> z cv <,> = x ax-17 cbvoprab12.3 hban z hbex u cv w cv v cv <,> z cv <,> = y ax-17 cbvoprab12.4 hban z hbex x cv w cv y cv v cv opeq12 x cv y cv <,> w cv v cv <,> z cv opeq1 syl u cv eqeq2d cbvoprab12.5 anbi12d z biexdv cbvex2 u biabi x y z ph u df-oprab w v z ps u df-oprab 3eqtr4))
+thm (cbvoprab12v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps x) (ps y)) ((cbvoprab12v.1 (-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)) (ph w ax-17 ph v ax-17 ps x ax-17 ps y ax-17 cbvoprab12v.1 z cbvoprab12))
+thm (cbvoprab3v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps z) (ps v)) ((cbvoprab3v.1 (-> (= (cv z) (cv w)) (<-> ph ps)))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y w ps)) (cbvoprab3v.1 v cv x cv y cv <,> = anbi2d x y bi2exdv v cbvopab2v x y z ph v dfoprab2 x y w ps v dfoprab2 3eqtr4))
+thm (dmoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (dom ({<<,>,>|} x y z ph)) ({<,>|} x y (E. z ph))) (x y z ph w dfoprab2 dmeqi w z x y w cv x cv y cv <,> = ph /\ E. E. dmopab z x y w cv x cv y cv <,> = ph /\ exrot3 z w cv x cv y cv <,> = ph 19.42v x y bi2ex bitr w biabi x y z ph E. w df-opab eqtr4 3eqtr))
+thm (dmoprabss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (dom ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph))) (X. A B)) (x y z x cv A e. y cv B e. /\ ph /\ dmoprab z x cv A e. y cv B e. /\ ph 19.42v x y biopabi x y A B z ph E. opabssxp eqsstr eqsstr))
+thm (rnoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (ran ({<<,>,>|} x y z ph)) ({|} z (E. x (E. y ph)))) (x y z ph w dfoprab2 rneqi w z x y w cv x cv y cv <,> = ph /\ E. E. rnopab w x y w cv x cv y cv <,> = ph /\ exrot3 w w cv x cv y cv <,> = ph 19.41v x cv y cv opex w isseti mpbiran x y bi2ex bitr z biabi 3eqtr))
+thm (reldmoprab ((x y) (x z) (y z)) () (Rel (dom ({<<,>,>|} x y z ph))) (x y z ph E. relopab x y z ph dmoprab x y z ph {<<,>,>|} dom x y z ph E. {<,>|} releq ax-mp mpbir))
+thm (eloprabg ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D w) (D x) (D y) (D z) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (ph w) (ps x) (ch x) (ch y) (th x) (th y) (th z) (th w)) ((eloprabg.1 (-> (= (cv x) A) (<-> ph ps))) (eloprabg.2 (-> (= (cv y) B) (<-> ps ch))) (eloprabg.3 (-> (= (cv z) C) (<-> ch th)))) (-> (/\/\ (e. A D) (e. B R) (e. C S)) (<-> (e. (<,> (<,> A B) C) ({<<,>,>|} x y z ph)) th)) (A B <,> C opex w cv A B <,> C <,> x cv y cv <,> z cv <,> eqeq1 A B <,> C <,> x cv y cv <,> z cv <,> eqcom syl6bb x visset y visset z visset B R C S A otthg A D e. 3adant1 sylan9bbr ph anbi1d eloprabg.1 eloprabg.2 eloprabg.3 syl3an9b pm5.32i syl6bb x y z bi3exdv w cv A B <,> C <,> x y z ph {<<,>,>|} eleq1 x y z ph w df-oprab w cv eleq2i w x y z w cv x cv y cv <,> z cv <,> = ph /\ E. E. E. abid bitr2 syl5bb A D e. B R e. C S e. /\/\ adantl A D x elex B R y elex C S z elex 3anim123i x y z x cv A = y cv B = z cv C = eeeanv sylibr th biantrurd x y z x cv A = y cv B = z cv C = /\/\ th 19.41vvv syl6rbbr w cv A B <,> C <,> = adantr 3bitr3d expcom vtocle))
+thm (ssoprab2i ((ph w) (ps w) (x y) (x z) (w x) (y z) (w y) (w z)) ((ssoprab2i.1 (-> ph ps))) (C_ ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)) (ssoprab2i.1 w cv x cv y cv <,> = anim2i x y 19.22i2 w z ssopab2i x y z ph w dfoprab2 x y z ps w dfoprab2 3sstr4))
+thm (funoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) ((funoprab.1 (E* z ph))) (Fun ({<<,>,>|} x y z ph)) (w z x y w cv x cv y cv <,> = ph /\ E. E. funopab funoprab.1 x y w cv mosubop mpgbir x y z ph w dfoprab2 x y z ph {<<,>,>|} w z x y w cv x cv y cv <,> = ph /\ E. E. {<,>|} funeq ax-mp mpbir))
+thm (fnoprab ((x y) (x z) (y z) (ph z)) ((fnoprab.1 (-> ph (E! z ps)))) (Fn ({<<,>,>|} x y z (/\ ph ps)) ({<,>|} x y ph)) (x y z ph ps /\ {<<,>,>|} x y ph {<,>|} df-fn fnoprab.1 z ps eumo syl z ph ps moanimv mpbir x y funoprab x y z ph ps /\ dmoprab ph ps pm3.26 z 19.23aiv fnoprab.1 z ps euex syl ancli z ph ps 19.42v sylibr impbi x y biopabi eqtr mpbir2an))
+thm (fnoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((fnoprab2.1 (e. C (V))) (fnoprab2.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (Fn F (X. A B)) (fnoprab2.1 z eueq1 x cv A e. y cv B e. /\ a1i x y fnoprab fnoprab2.2 F x y z x cv A e. y cv B e. /\ z cv C = /\ {<<,>,>|} A B X. fneq1 ax-mp A B x y df-xp A B X. x y x cv A e. y cv B e. /\ {<,>|} x y z x cv A e. y cv B e. /\ z cv C = /\ {<<,>,>|} fneq2 ax-mp bitr mpbir))
+thm (dmoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((fnoprab2.1 (e. C (V))) (fnoprab2.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (= (dom F) (X. A B)) (fnoprab2.1 fnoprab2.2 fnoprab2 F A B X. fndm ax-mp))
+thm (ffnoprval ((x y) (w x) (A x) (w y) (A y) (A w) (B x) (B y) (B w) (C x) (C y) (C w) (F x) (F y) (F w)) () (<-> (:--> F (X. A B) C) (/\ (Fn F (X. A B)) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))) (F A B X. C w ffnfv w cv x cv y cv <,> F fveq2 x cv F y cv df-opr syl6eqr C eleq1d A B ralxp F A B X. Fn anbi2i bitr))
+thm (eqfnoprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z) (D z) (F x) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (Fn F (X. A B)) (Fn G (X. C D))) (<-> (= F G) (/\ (= (X. A B) (X. C D)) (A.e. x A (A.e. y B (= (opr (cv x) F (cv y)) (opr (cv x) G (cv y)))))))) (F A B X. G C D X. z eqfnfv z cv x cv y cv <,> F fveq2 z cv x cv y cv <,> G fveq2 eqeq12d x cv F y cv df-opr x cv G y cv df-opr eqeq12i syl6bbr A B ralxp A B X. C D X. = anbi2i syl6bb))
+thm (fnoprval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) () (<-> (Fn F (X. A B)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y))))))) (F A B X. w z fnopabfv w cv A B x y elxp z cv F w cv ` = anbi1i x y w cv x cv y cv <,> = x cv A e. y cv B e. /\ /\ z cv F w cv ` = 19.41vv w cv x cv y cv <,> = x cv A e. y cv B e. /\ z cv F w cv ` = anass w cv x cv y cv <,> F fveq2 x cv F y cv df-opr syl6eqr z cv eqeq2d x cv A e. y cv B e. /\ anbi2d pm5.32i bitr x y bi2ex bitr3 bitr w z biopabi x y z x cv A e. y cv B e. /\ z cv x cv F y cv opr = /\ w dfoprab2 eqtr4 F eqeq2i bitr))
+thm (foprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (F x) (F y) (F z)) () (<-> (:--> F (X. A B) C) (/\ (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))))) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))) (F A B C x y ffnoprval F A B x y z fnoprval x A y B x cv F y cv opr C e. A.e. A.e. anbi1i bitr))
+thm (oprabex ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((oprabex.1 (e. A (V))) (oprabex.2 (e. B (V))) (oprabex.3 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (E* z ph))) (oprabex.4 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph))))) (e. F (V)) (oprabex.4 oprabex.1 oprabex.2 xpex x y z A B ph dmoprabss ssexi oprabex.3 z x cv A e. y cv B e. /\ ph moanimv mpbir x y funoprab x y z x cv A e. y cv B e. /\ ph /\ {<<,>,>|} V funex mp2 eqeltr))
+thm (oprabex2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((oprabex2.1 (e. A (V))) (oprabex2.2 (e. B (V))) (oprabex2.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (e. F (V)) (oprabex2.1 oprabex2.2 z C moeq x cv A e. y cv B e. /\ a1i oprabex2.3 oprabex))
+thm (oprabex3 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z)) ((oprabex3.1 (e. H (V))) (oprabex3.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R))))))))))) (e. F (V)) (oprabex3.1 oprabex3.1 xpex oprabex3.1 oprabex3.1 xpex z R moeq u f y cv mosubop w v x cv mosubop x cv w cv v cv <,> = y cv u cv f cv <,> = z cv R = anass u f bi2ex u f x cv w cv v cv <,> = y cv u cv f cv <,> = z cv R = /\ 19.42vv bitr w v bi2ex z bimo mpbir x cv H H X. e. y cv H H X. e. /\ a1i oprabex3.2 oprabex))
+thm (oprabval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabval.1 (e. C (V))) (oprabval.2 (-> (= (cv x) A) (<-> ph ps))) (oprabval.3 (-> (= (cv y) B) (<-> ps ch))) (oprabval.4 (-> (= (cv z) C) (<-> ch th))) (oprabval.5 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E! z ph))) (oprabval.6 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\ (e. A R) (e. B S)) (<-> (= (opr A F B) C) th)) (x cv A R eleq1 y cv S e. anbi1d y cv B S eleq1 A R e. anbi2d R S opelopabg biimprd pm2.43i oprabval.5 x y fnoprab oprabval.1 x y z x cv R e. y cv S e. /\ ph /\ {<<,>,>|} x y x cv R e. y cv S e. /\ {<,>|} A B <,> fnfvop mpan syl oprabval.1 x cv A R eleq1 y cv S e. anbi1d oprabval.2 anbi12d y cv B S eleq1 A R e. anbi2d oprabval.3 anbi12d oprabval.4 A R e. B S e. /\ anbi2d R S V eloprabg mp3an3 bitrd A F B df-opr oprabval.6 A B <,> fveq1i eqtr C eqeq1i syl5bb A R e. B S e. /\ th ibar bitr4d))
+thm (oprabvalig ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (D x) (D y) (D z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabvalig.1 (-> (= (cv x) A) (<-> ph ps))) (oprabvalig.2 (-> (= (cv y) B) (<-> ps ch))) (oprabvalig.3 (-> (= (cv z) C) (<-> ch th))) (oprabvalig.4 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph))) (oprabvalig.5 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\/\ (e. A R) (e. B S) (e. C D)) (-> th (= (opr A F B) C))) (x cv A R eleq1 y cv S e. anbi1d oprabvalig.1 anbi12d y cv B S eleq1 A R e. anbi2d oprabvalig.2 anbi12d oprabvalig.3 A R e. B S e. /\ anbi2d R S D eloprabg biimpar exp32 com12 C D e. 3adant3 pm2.43i oprabvalig.4 z x cv R e. y cv S e. /\ ph moanimv mpbir x y funoprab C D x y z x cv R e. y cv S e. /\ ph /\ {<<,>,>|} A B <,> funopfvg mpan2 A R e. B S e. 3ad2ant3 syld A F B df-opr oprabvalig.5 A B <,> fveq1i eqtr C eqeq1i syl6ibr))
+thm (oprabvali ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((oprabvali.1 (e. C (V))) (oprabvali.2 (-> (= (cv x) A) (<-> ph ps))) (oprabvali.3 (-> (= (cv y) B) (<-> ps ch))) (oprabvali.4 (-> (= (cv z) C) (<-> ch th))) (oprabvali.5 (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph))) (oprabvali.6 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph))))) (-> (/\ (e. A R) (e. B S)) (-> th (= (opr A F B) C))) (oprabvali.1 oprabvali.2 oprabvali.3 oprabvali.4 oprabvali.5 oprabvali.6 V oprabvalig mp3an3))
+thm (oprabval2g ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (H x) (H y) (H z) (G x) (R z) (S x) (S y) (S z)) ((oprabval2g.1 (-> (= (cv x) A) (= R G))) (oprabval2g.2 (-> (= (cv y) B) (= G S))) (oprabval2g.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S)) (S eqid oprabval2g.1 z cv eqeq2d oprabval2g.2 z cv eqeq2d z cv S S eqeq1 z R moeq x cv C e. y cv D e. /\ a1i oprabval2g.3 H oprabvalig mpi))
+thm (oprabval2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (R z) (S x) (S y) (S z)) ((oprabval2.1 (e. S (V))) (oprabval2.2 (-> (= (cv x) A) (= R G))) (oprabval2.3 (-> (= (cv y) B) (= G S))) (oprabval2.4 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R)))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)) (oprabval2.1 oprabval2.2 oprabval2.3 oprabval2.4 V oprabval2g mp3an3))
+thm (oprabval5 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (G x) (R z) (S x) (S y) (S z)) ((oprabval5.1 (e. S (V))) (oprabval5.2 (-> (= (cv x) A) (= R G))) (oprabval5.3 (-> (= (cv y) B) (= G S))) (oprabval5.4 (= F ({<<,>,>|} x y z (= (cv z) R))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)) (oprabval5.1 oprabval5.2 oprabval5.3 oprabval5.4 x visset y visset pm3.2i z cv R = biantrur x y z bioprabi eqtr oprabval2 A C elisset B D elisset syl2an))
+thm (oprabval3 ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z) (S x) (S y) (S z) (S w) (S v) (S u) (S f)) ((oprabval3.1 (e. S (V))) (oprabval3.2 (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= R S))) (oprabval3.3 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R))))))))))) (-> (/\ (/\ (e. A H) (e. B H)) (/\ (e. C H) (e. D H))) (= (opr (<,> A B) F (<,> C D)) S)) (oprabval3.1 x cv A B <,> w cv v cv <,> eqeq1 y cv u cv f cv <,> = anbi1d z cv R = anbi1d w v u f bi4exdv y cv C D <,> u cv f cv <,> eqeq1 A B <,> w cv v cv <,> = anbi2d z cv R = anbi1d w v u f bi4exdv z cv S R eqeq1 A B <,> w cv v cv <,> = C D <,> u cv f cv <,> = /\ anbi2d w v u f bi4exdv z R moeq u f y cv mosubop w v x cv mosubop x cv w cv v cv <,> = y cv u cv f cv <,> = z cv R = anass u f bi2ex u f x cv w cv v cv <,> = y cv u cv f cv <,> = z cv R = /\ 19.42vv bitr w v bi2ex z bimo mpbir x cv H H X. e. y cv H H X. e. /\ a1i oprabval3.3 oprabvali A H B H opelxpi C H D H opelxpi anim12i S eqid oprabval3.2 S eqeq2d H H copsex4g mpbiri sylc))
+thm (oprabval4g ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C z) (C w) (C v) (C u) (C f) (D w) (D v)) ((oprabval4g.1 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. C D)) (= (opr (cv x) F (cv y)) C)) (x w u v cv y u cv C e. [/] {|} f sbab eqcomd eqcoms y v C u sbab eqcomd eqcoms oprabval4g.1 w cv A e. v cv B e. /\ x ax-17 w x f cv u v cv y u cv C e. [/] {|} e. hbs1 z f hbab hbeleq hban w cv A e. v cv B e. /\ y ax-17 v y u cv C e. hbs1 f u hbab w x hbsb z f hbab hbeleq hban x cv A e. y cv B e. /\ z cv C = /\ w ax-17 x cv A e. y cv B e. /\ z cv C = /\ v ax-17 w cv x cv A eleq1 v cv y cv B eleq1 bi2anan9 x w u v cv y u cv C e. [/] {|} f sbab eqcomd eqcoms y v C u sbab eqcomd eqcoms sylan9eq z cv eqeq2d anbi12d z cbvoprab12 eqtr4 V oprabval2g C D elisset syl3an3))
+thm (elrnoprab ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C z) (D x) (D y) (D w) (D v) (D u) (F w) (F v) (F u)) ((elrnoprab.1 (e. C (V))) (elrnoprab.2 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C)))))) (<-> (e. D (ran F)) (E.e. x A (E.e. y B (= D C)))) (elrnoprab.1 elrnoprab.2 fnoprab2 F A B X. D w fvelrn w cv v cv u cv <,> F fveq2 D eqeq1d A B cbvop u cv B e. x ax-17 w x y z x cv A e. y cv B e. /\ z cv C = /\ hboprab1 elrnoprab.2 w cv eleq2i elrnoprab.2 w cv eleq2i x bial 3imtr4 w cv v cv u cv <,> e. x ax-17 hbfv w cv D e. x ax-17 hbeq hbrex u B F x cv u cv <,> ` D = E.e. v ax-17 v cv x cv u cv opeq1 F fveq2d D eqeq1d u B birexdv A cbvrex w x y z x cv A e. y cv B e. /\ z cv C = /\ hboprab2 elrnoprab.2 w cv eleq2i elrnoprab.2 w cv eleq2i y bial 3imtr4 w cv x cv u cv <,> e. y ax-17 hbfv w cv D e. y ax-17 hbeq F x cv y cv <,> ` D = u ax-17 u cv y cv x cv opeq2 F fveq2d D eqeq1d B cbvrex x A birex 3bitr x cv F y cv df-opr D eqeq1i x A y B bi2rex bitr4 elrnoprab.1 elrnoprab.2 V oprabval4g mp3an3 D eqeq1d C D eqcom syl6bb bi2rexa bitr syl6bb ax-mp))
+thm (oprvalex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w) (B x) (B y) (B z) (B w)) ((oprvalex.1 (e. A (V))) (oprvalex.2 (e. B (V)))) (e. ({|} z (E.e. x A (E.e. y B (= (cv z) (opr (cv x) F (cv y)))))) (V)) (w cv x cv y cv <,> F fveq2 x cv F y cv df-opr syl6eqr z cv eqeq2d A B cbvop z biabi oprvalex.1 oprvalex.2 xpex z w F w cv ` abrexex eqeltrr))
+thm (oprssdm ((x y) (S x) (S y) (F x) (F y)) ((oprssdm.1 (-. (e. ({/}) S))) (oprssdm.2 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)))) (C_ (X. S S) (dom F)) (S S relxp y visset x cv S S opelxp x cv y cv <,> F ndmfv x cv F y cv df-opr {/} eqeq1i oprssdm.1 x cv F y cv opr {/} S eleq1 mtbiri oprssdm.2 nsyl sylbir syl a3i sylbi relssi))
+thm (ndmoprg () ((ndmoprg.1 (= (dom F) (X. S S)))) (-> (/\ (e. B C) (-. (/\ (e. A S) (e. B S)))) (= (opr A F B) ({/}))) (B C A S S opelxpg ndmoprg.1 A B <,> eleq2i syl5bb negbid A B <,> F ndmfv A F B df-opr syl5eq syl6bir imp))
+thm (ndmoprcl ((A x) (B x) (F x) (S x)) ((ndmoprcl.1 (= (dom F) (X. S S))) (ndmoprcl.2 (-> (/\ (e. A S) (e. (cv x) S)) (e. (opr A F (cv x)) S))) (ndmoprcl.3 (e. ({/}) S))) (e. (opr A F B) S) (B A F oprprc2 S eleq1d ndmoprcl.3 ndmoprcl.1 A V A ndmoprg S eleq1d mpbiri exp x cv A A F opreq2 S eleq1d A S e. imbi2d ndmoprcl.2 expcom vtoclga imp pm2.61d2 syl5bir com12 ndmoprcl.3 ndmoprcl.1 B V A ndmoprg S eleq1d mpbiri exp x cv B A F opreq2 S eleq1d A S e. imbi2d ndmoprcl.2 expcom vtoclga impcom pm2.61d2 pm2.61d2 ndmoprcl.3 S S relxp ndmoprcl.1 F dom S S X. releq ax-mp mpbir A B oprprc1 S eleq1d mpbiri pm2.61i))
+thm (ndmopr () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S)))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) ({/}))) (ndmopr.1 ndmopr.2 B V A ndmoprg mpan))
+thm (ndmoprrcl () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmoprrcl.3 (-. (e. ({/}) S)))) (-> (e. (opr A F B) S) (/\ (e. A S) (e. B S))) (ndmoprrcl.3 ndmopr.1 ndmopr.2 A ndmopr S eleq1d mtbiri a3i))
+thm (ndmoprcom () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.3 (e. A (V)))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) (opr B F A))) (ndmopr.1 ndmopr.2 A ndmopr A S e. B S e. ancom negbii ndmopr.3 ndmopr.2 B ndmopr sylbi eqtr4d))
+thm (ndmoprass () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.4 (e. C (V))) (ndmopr.5 (-. (e. ({/}) S)))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) (ndmopr.1 ndmopr.2 ndmopr.5 A ndmoprrcl C S e. anim1i A S e. B S e. C S e. df-3an sylibr con3i ndmopr.4 ndmopr.2 A F B opr ndmopr syl ndmopr.4 ndmopr.2 ndmopr.5 B ndmoprrcl A S e. anim2i A S e. B S e. C S e. 3anass sylibr con3i B F C oprex ndmopr.2 A ndmopr syl eqtr4d))
+thm (ndmoprdistr () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmopr.4 (e. C (V))) (ndmopr.5 (-. (e. ({/}) S))) (ndmopr.6 (= (dom G) (X. S S)))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))) (ndmopr.4 ndmopr.2 ndmopr.5 B ndmoprrcl A S e. anim2i A S e. B S e. C S e. 3anass sylibr con3i B F C oprex ndmopr.6 A ndmopr syl ndmopr.1 ndmopr.6 ndmopr.5 A ndmoprrcl ndmopr.4 ndmopr.6 ndmopr.5 A ndmoprrcl anim12i A S e. B S e. C S e. 3anass A S e. B S e. C S e. anandi bitr sylibr con3i A G C oprex ndmopr.2 A G B opr ndmopr syl eqtr4d))
+thm (ndmord () ((ndmopr.1 (e. B (V))) (ndmopr.2 (= (dom F) (X. S S))) (ndmord.3 (e. A (V))) (ndmord.4 (C_ R (X. S S))) (ndmord.5 (-. (e. ({/}) S))) (ndmord.6 (-> (/\/\ (e. A S) (e. B S) (e. C S)) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))) (ndmord.6 3exp imp ndmopr.1 ndmord.4 A brel C F B oprex ndmord.4 C F A opr brel ndmord.3 ndmopr.2 ndmord.5 C ndmoprrcl pm3.27d ndmopr.1 ndmopr.2 ndmord.5 C ndmoprrcl pm3.27d anim12i syl pm5.21ni C S e. a1d pm2.61i))
+thm (ndmordi () ((ndmordi.3 (e. A (V))) (ndmordi.2 (= (dom F) (X. S S))) (ndmordi.4 (C_ R (X. S S))) (ndmordi.5 (-. (e. ({/}) S))) (ndmordi.6 (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))) (-> (br (opr C F A) R (opr C F B)) (br A R B)) (C F B oprex ndmordi.4 C F A opr brel pm3.26d ndmordi.3 ndmordi.2 ndmordi.5 C ndmoprrcl pm3.26d syl ndmordi.6 biimprd mpcom))
+thm (caoprcl ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y)) ((caoprcl.1 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)))) (-> (/\ (e. A S) (e. B S)) (e. (opr A F B) S)) (x cv A F y cv opreq1 S eleq1d y cv B A F opreq2 S eleq1d caoprcl.1 vtocl2ga))
+thm (caoprcom ((x y) (F x) (F y) (A x) (A y) (B x) (B y)) ((caoprcom.1 (e. A (V))) (caoprcom.2 (e. B (V))) (caoprcom.3 (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (= (opr A F B) (opr B F A)) (caoprcom.1 caoprcom.2 x cv A F y cv opreq1 x cv A y cv F opreq2 eqeq12d y cv B A F opreq2 y cv B F A opreq1 eqeq12d sylan9bb caoprcom.3 vtocl2))
+thm (caoprass ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caoprass.1 (e. A (V))) (caoprass.2 (e. B (V))) (caoprass.3 (e. C (V))) (caoprass.4 (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr A F (opr B F C))) (caoprass.1 caoprass.2 caoprass.3 x cv A F y cv opreq1 F z cv opreq1d x cv A F y cv F z cv opr opreq1 eqeq12d y cv B A F opreq2 F z cv opreq1d y cv B F z cv opreq1 A F opreq2d eqeq12d z cv C A F B opr F opreq2 z cv C B F opreq2 A F opreq2d eqeq12d syl3an9b caoprass.4 vtocl3))
+thm (caoprcan ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caoprcan.1 (e. C (V))) (caoprcan.2 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))))) (-> (/\ (e. A S) (e. B S)) (-> (= (opr A F B) (opr A F C)) (= B C))) (x cv A F y cv opreq1 x cv A F C opreq1 eqeq12d y cv C = imbi1d y cv B A F opreq2 A F C opr eqeq1d y cv B C eqeq1 imbi12d caoprcan.1 z cv C x cv F opreq2 x cv F y cv opr eqeq2d z cv C y cv eqeq2 imbi12d x cv S e. y cv S e. /\ imbi2d caoprcan.2 vtocl vtocl2ga))
+thm (caoprord ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))) (z cv C F A opreq1 z cv C F B opreq1 R breq12d A R B br bibi2d caoprord.1 caoprord.2 x cv A R y cv breq1 x cv A z cv F opreq2 R z cv F y cv opr breq1d bibi12d y cv B A R breq2 y cv B z cv F opreq2 z cv F A opr R breq2d bibi12d sylan9bb z cv S e. imbi2d caoprord.3 vtocl2 vtoclga))
+thm (caoprord2 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (caoprord2.3 (e. C (V))) (caoprord2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (-> (e. C S) (<-> (br A R B) (br (opr A F C) R (opr B F C)))) (caoprord.1 caoprord.2 caoprord.3 C caoprord caoprord2.3 caoprord.1 caoprord2.com caoprcom caoprord2.3 caoprord.2 caoprord2.com caoprcom R breq12i syl6bb))
+thm (caoprord3 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (R x) (R y) (R z)) ((caoprord.1 (e. A (V))) (caoprord.2 (e. B (V))) (caoprord.3 (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (caoprord2.3 (e. C (V))) (caoprord2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprord3.4 (e. D (V)))) (-> (/\ (/\ (e. B S) (e. C S)) (= (opr A F B) (opr C F D))) (<-> (br A R C) (br D R B))) (caoprord.1 caoprord2.3 caoprord.3 caoprord.2 caoprord2.com caoprord2 C S e. adantr A F B opr C F D opr R C F B opr breq1 sylan9bb caoprord3.4 caoprord.2 caoprord.3 C caoprord B S e. A F B opr C F D opr = ad2antlr bitr4d))
+thm (caoprdistr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((caoprdistr.1 (e. A (V))) (caoprdistr.2 (e. B (V))) (caoprdistr.3 (e. C (V))) (caoprdistr.4 (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C))) (caoprdistr.1 caoprdistr.2 caoprdistr.3 x cv A G y cv F z cv opr opreq1 x cv A G y cv opreq1 x cv A G z cv opreq1 F opreq12d eqeq12d y cv B F z cv opreq1 A G opreq2d y cv B A G opreq2 F A G z cv opr opreq1d eqeq12d z cv C B F opreq2 A G opreq2d z cv C A G opreq2 A G B opr F opreq2d eqeq12d syl3an9b caoprdistr.4 vtocl3))
+thm (caopr32 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr (opr A F C) F B)) (caopr.2 caopr.3 caopr.com caoprcom A F opreq2i caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.1 caopr.3 caopr.2 caopr.ass caoprass 3eqtr4))
+thm (caopr12 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr A F (opr B F C)) (opr B F (opr A F C))) (caopr.1 caopr.2 caopr.com caoprcom F C opreq1i caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.2 caopr.1 caopr.3 caopr.ass caoprass 3eqtr3))
+thm (caopr31 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr A F B) F C) (opr (opr C F B) F A)) (caopr.1 caopr.3 caopr.2 caopr.ass caoprass caopr.1 caopr.3 caopr.2 caopr.com caopr.ass caopr12 eqtr caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr32 caopr.3 caopr.1 caopr.2 caopr.com caopr.ass caopr32 caopr.3 caopr.1 caopr.2 caopr.ass caoprass eqtr3 3eqtr4))
+thm (caopr13 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr A F (opr B F C)) (opr C F (opr B F A))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr31 caopr.1 caopr.2 caopr.3 caopr.ass caoprass caopr.3 caopr.2 caopr.1 caopr.ass caoprass 3eqtr3))
+thm (caopr4 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr B F D))) (caopr.2 caopr.3 caopr.4 caopr.com caopr.ass caopr12 A F opreq2i caopr.1 caopr.2 C F D oprex caopr.ass caoprass caopr.1 caopr.3 B F D oprex caopr.ass caoprass 3eqtr4))
+thm (caopr411 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr C F B) F (opr A F D))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr31 F D opreq1i A F B oprex caopr.3 caopr.4 caopr.ass caoprass C F B oprex caopr.1 caopr.4 caopr.ass caoprass 3eqtr3))
+thm (caopr42 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((caopr.1 (e. A (V))) (caopr.2 (e. B (V))) (caopr.3 (e. C (V))) (caopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caopr.4 (e. D (V)))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr D F B))) (caopr.1 caopr.2 caopr.3 caopr.com caopr.ass caopr.4 caopr4 caopr.2 caopr.4 caopr.com caoprcom A F C opr F opreq2i eqtr))
+thm (caoprdistrr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))))) (= (opr (opr A F B) G C) (opr (opr A G C) F (opr B G C))) (caoprd.3 caoprd.1 caoprd.2 caoprd.distr caoprdistr caoprd.3 A F B oprex caoprd.com caoprcom caoprd.3 caoprd.1 caoprd.com caoprcom caoprd.3 caoprd.2 caoprd.com caoprcom F opreq12i 3eqtr3))
+thm (caoprdilem ((x y) (x z) (H x) (y z) (H y) (H z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (caoprdl.4 (e. D (V))) (caoprdl.5 (e. H (V))) (caoprdl.ass (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z)))))) (= (opr (opr (opr A G C) F (opr B G D)) G H) (opr (opr A G (opr C G H)) F (opr B G (opr D G H)))) (A G C oprex B G D oprex caoprdl.5 caoprd.com caoprd.distr caoprdistrr caoprd.1 caoprd.3 caoprdl.5 caoprdl.ass caoprass caoprd.2 caoprdl.4 caoprdl.5 caoprdl.ass caoprass F opreq12i eqtr))
+thm (caoprlem2 ((x y) (x z) (H x) (y z) (H y) (H z) (R x) (R y) (R z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z) (R x) (R y) (R z)) ((caoprd.1 (e. A (V))) (caoprd.2 (e. B (V))) (caoprd.3 (e. C (V))) (caoprd.com (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (caoprd.distr (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (caoprdl.4 (e. D (V))) (caoprdl.5 (e. H (V))) (caoprdl.ass (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z))))) (caoprdl2.6 (e. R (V))) (caoprdl2.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprdl2.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))))) (= (opr (opr (opr (opr A G C) F (opr B G D)) G H) F (opr (opr (opr A G D) F (opr B G C)) G R)) (opr (opr A G (opr (opr C G H) F (opr D G R))) F (opr B G (opr (opr C G R) F (opr D G H))))) (A G C G H opr oprex B G D G H opr oprex A G D G R opr oprex caoprdl2.com caoprdl2.ass B G C G R opr oprex caopr42 caoprd.1 caoprd.2 caoprd.3 caoprd.com caoprd.distr caoprdl.4 caoprdl.5 caoprdl.ass caoprdilem caoprd.1 caoprd.2 caoprdl.4 caoprd.com caoprd.distr caoprd.3 caoprdl2.6 caoprdl.ass caoprdilem F opreq12i caoprd.1 C G H oprex D G R oprex caoprd.distr caoprdistr caoprd.2 C G R oprex D G H oprex caoprd.distr caoprdistr F opreq12i 3eqtr4))
+thm (caoprmo ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (S w) (S v) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z)) ((caoprmo.1 (e. A (V))) (caoprmo.2 (e. B S)) (caoprmo.dom (= (dom F) (X. S S))) (caoprmo.3 (-. (e. ({/}) S))) (caoprmo.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (caoprmo.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (caoprmo.id (-> (e. (cv x) S) (= (opr (cv x) F B) (cv x))))) (E* w (= (opr A F (cv w)) B)) (w cv v cv S eleq1 w cv v cv A F opreq2 B eqeq1d anbi12d mo4 A F v cv opr B w cv F opreq2 x cv w cv F B opreq1 x cv w cv = id eqeq12d caoprmo.id vtoclga sylan9eqr caoprmo.1 w visset v visset caoprmo.ass caoprass caoprmo.1 w visset v visset caoprmo.com caoprmo.ass caopr12 eqtr syl5eq v cv S e. adantrl A F w cv opr B = adantlr A F w cv opr B F v cv opreq1 x cv v cv F B opreq1 x cv v cv = id eqeq12d caoprmo.id vtoclga caoprmo.2 elisseti v visset caoprmo.com caoprcom syl5eq sylan9eq A F v cv opr B = adantrr w cv S e. adantll eqtr3d v ax-gen mpgbir w A F w cv opr B = w cv S e. A F w cv opr B = /\ immo mpi caoprmo.2 A F w cv opr B S eleq1 mpbiri w visset caoprmo.dom caoprmo.3 A ndmoprrcl pm3.27d syl ancri mpg))
+thm (1stval ((x y) (A x) (A y)) () (= (` (1st) A) (U. (dom ({} A)))) (A snex A {} V dmexg ax-mp uniex x cv A sneq dmeqd unieqd V V y fvopabg x y df-1st A fveq1i syl5eq mpan2 A 1st fvprc A snprc biimp dmeqd dm0 syl6eq unieqd uni0 syl6eq eqtr4d pm2.61i))
+thm (2ndval ((x y) (A x) (A y)) () (= (` (2nd) A) (U. (ran ({} A)))) (A snex A {} V rnexg ax-mp uniex x cv A sneq rneqd unieqd V V y fvopabg x y df-2nd A fveq1i syl5eq mpan2 A 2nd fvprc A snprc biimp rneqd rn0 syl6eq unieqd uni0 syl6eq eqtr4d pm2.61i))
+thm (op1st () ((op1st.1 (e. A (V)))) (= (` (1st) (<,> A B)) A) (A B <,> 1stval op1st.1 B op1sta eqtr))
+thm (op2nd () ((op1st.1 (e. A (V))) (op2n.2 (e. B (V)))) (= (` (2nd) (<,> A B)) B) (A B <,> 2ndval op1st.1 op2n.2 op2nda eqtr))
+thm (op1stg ((A x) (B x)) () (-> (e. A C) (= (` (1st) (<,> A B)) A)) (x cv A B opeq1 1st fveq2d x cv A = id eqeq12d x visset B op1st C vtoclg))
+thm (op2ndg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (= (` (2nd) (<,> A B)) B)) (x cv A y cv opeq1 2nd fveq2d y cv eqeq1d y cv B A opeq2 2nd fveq2d y cv B = id eqeq12d x visset y visset op2nd C D vtocl2g))
+thm (elxp6 () () (<-> (e. A (X. B C)) (/\ (= A (<,> (` (1st) A) (` (2nd) A))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C)))) (A B C elxp4 A 1stval A 2ndval 1st A ` A {} dom U. 2nd A ` A {} ran U. opeq12 mp2an A eqeq2i A 1stval B eleq1i A 2ndval C eleq1i anbi12i anbi12i bitr4))
+thm (fo1st ((x y)) () (:-onto-> (1st) (V) (V)) (x y y cv x cv {} dom U. = {<,>|} V V df-fo x cv snex x cv {} V dmexg ax-mp uniex x visset y cv x cv {} dom U. = biantrur x y biopabi fnopab2 y visset y cv op1sta eqcomi y cv y cv opex x cv y cv y cv <,> sneq dmeqd unieqd y cv eqeq2d cla4ev ax-mp y equid 2th y biabi x y y cv x cv {} dom U. = rnopab y df-v 3eqtr4 mpbir2an x y df-1st 1st x y y cv x cv {} dom U. = {<,>|} V V foeq1 ax-mp mpbir))
+thm (fo2nd ((x y)) () (:-onto-> (2nd) (V) (V)) (x y y cv x cv {} ran U. = {<,>|} V V df-fo x cv snex x cv {} V rnexg ax-mp uniex x visset y cv x cv {} ran U. = biantrur x y biopabi fnopab2 y visset y visset op2nda eqcomi y cv y cv opex x cv y cv y cv <,> sneq rneqd unieqd y cv eqeq2d cla4ev ax-mp y equid 2th y biabi x y y cv x cv {} ran U. = rnopab y df-v 3eqtr4 mpbir2an x y df-2nd 2nd x y y cv x cv {} ran U. = {<,>|} V V foeq1 ax-mp mpbir))
+thm (f1stres ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (:--> (|` (1st) (X. A B)) (X. A B) A) (y visset z cv op1sta A eleq1i biimpr z cv B e. adantr rgen2a x cv y cv z cv <,> sneq dmeqd unieqd A eleq1d A B ralxp mpbir x y df-1st 1st x y y cv x cv {} dom U. = {<,>|} A B X. reseq1 ax-mp x y y cv x cv {} dom U. = A B X. resopab x y x cv A B X. e. y cv x cv {} dom U. = /\ {<,>|} eqid 3eqtr A fopab2 mpbi))
+thm (1st2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv x))) A) (` (1st) A)) (w cv x y z z cv x cv = {<<,>,>|} v cv df-opr w visset v visset w visset x cv w cv = id w cv eqid y cv v cv = a1i x y z z cv x cv = {<<,>,>|} eqid V V oprabval5 mp2an eqtr3 w visset v cv op1st eqtr4 w cv v cv <,> A x y z z cv x cv = {<<,>,>|} fveq2 w cv v cv <,> A 1st fveq2 eqeq12d mpbii w v 19.23aivv x visset y visset pm3.2i z x a9e 2th x y biopabi V V x y df-xp x y z z cv x cv = dmoprab 3eqtr4r A eleq2i A w v elvv A w cv v cv <,> eqcom w v bi2ex 3bitr negbii A x y z z cv x cv = {<<,>,>|} ndmfv sylbir A {} dom w n0 w visset A {} v eldm2 w cv v cv opex A elsnc v biex bitr w biex bitr biimp con1i unieqd uni0 syl6eq A 1stval syl5eq eqtr4d pm2.61i))
+thm (df1st2 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (u x) (u y) (u z) (u w) (u v)) () (= ({<<,>,>|} x y z (= (cv z) (cv x))) (|` (1st) (X. (V) (V)))) (x visset x visset y visset pm3.2i z cv x cv = biantrur x y z bioprabi fnoprab2 fo1st 1st V V fof ax-mp 1st V V ffn ax-mp V V X. ssv 1st V V V X. fnssres mp2an x y z z cv x cv = {<<,>,>|} V V X. 1st V V X. |` V V X. u eqfnfv mp2an V V X. eqid x y z w cv v cv <,> 1st2val v visset w cv V V opelxp w visset v visset mpbir2an w cv v cv <,> V V X. 1st fvres ax-mp eqtr4 w cv V e. v cv V e. /\ a1i rgen2 u cv w cv v cv <,> x y z z cv x cv = {<<,>,>|} fveq2 u cv w cv v cv <,> 1st V V X. |` fveq2 eqeq12d V V ralxp mpbir mpbir2an))
+thm (1ord () () (e. (1o) (On)) (df-1o 0elon onsuc eqeltr))
+thm (2ord () () (e. (2o) (On)) (1ord 1o sucelon df-2o On eleq1i bitr4 mpbi))
+thm (df1o2 () () (= (1o) ({} ({/}))) (df-1o suc0 eqtr))
+thm (df2o2 () () (= (2o) ({,} ({/}) ({} ({/})))) (df-2o 1o df-suc df1o2 df1o2 sneqi uneq12i {/} {/} {} df-pr eqtr4 3eqtr))
+thm (0ne1oOLD () () (-. (= ({/}) (1o))) (0nep0 df1o2 {/} eqeq2i mtbir))
+thm (1ne0 () () (-. (= (1o) ({/}))) (0nep0 df1o2 {/} eqeq1i {/} {} {/} eqcom bitr mtbir))
+thm (ordgt0ge1 () () (-> (Ord A) (<-> (e. ({/}) A) (C_ (1o) A))) (0elon {/} On A ordelsuc mpan df-1o A sseq1i syl6bbr))
+thm (ordge1n0 () () (-> (Ord A) (<-> (C_ (1o) A) (-. (= A ({/}))))) (A ordgt0ge1 A ord0eln0 bitr3d))
+thm (el1o () () (<-> (e. A (1o)) (= A ({/}))) (df1o2 A eleq2i 0ex A elsnc2 bitr))
+thm (0lt1o () () (e. ({/}) (1o)) ({/} eqid {/} el1o mpbir))
+thm (fnoa ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (+o) (X. (On) (On))) (w v v cv w cv suc = {<,>|} x cv rec y cv fvex x y z w v df-oadd fnoprab2))
+thm (fnom ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (.o) (X. (On) (On))) (w v v cv w cv +o x cv opr = {<,>|} {/} rec y cv fvex x y z w v df-omul fnoprab2))
+thm (oav ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (` (rec ({<,>|} x y (= (cv y) (suc (cv x)))) A) B))) (x y y cv x cv suc = {<,>|} A rec B fvex w cv A x y y cv x cv suc = {<,>|} rdgeq2 v cv fveq1d v cv B x y y cv x cv suc = {<,>|} A rec fveq2 w v z x y df-oadd oprabval2))
+thm (omv ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (.o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) B))) (x y y cv x cv +o A opr = {<,>|} {/} rec B fvex w cv A x cv +o opreq2 y cv eqeq2d x y biopabdv x y y cv x cv +o w cv opr = {<,>|} x y y cv x cv +o A opr = {<,>|} {/} rdgeq1 x y y cv x cv +o w cv opr = {<,>|} {/} rec x y y cv x cv +o A opr = {<,>|} {/} rec v cv fveq1 3syl v cv B x y y cv x cv +o A opr = {<,>|} {/} rec fveq2 w v z x y df-omul oprabval2))
+thm (oe0lem () ((oe0lem.1 (-> (/\ ph (= A ({/}))) ps)) (oe0lem.2 (-> (/\ (/\ (e. A (On)) ph) (e. ({/}) A)) ps))) (-> (/\ (e. A (On)) ph) ps) (oe0lem.1 exp A On e. adantl A on0eln0 ph adantr oe0lem.2 exp sylbird pm2.61d))
+thm (oev ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) B) (if (= A ({/})) (\ (1o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B)))) (1ord 1o On B difexg ax-mp x y y cv x cv .o A opr = {<,>|} 1o rec B fvex A {/} = ifex w cv A {/} eqeq1 1o v cv \ x y y cv x cv .o w cv opr = {<,>|} 1o rec v cv ` ifbid w cv A x cv .o opreq2 y cv eqeq2d x y biopabdv x y y cv x cv .o w cv opr = {<,>|} x y y cv x cv .o A opr = {<,>|} 1o rdgeq1 x y y cv x cv .o w cv opr = {<,>|} 1o rec x y y cv x cv .o A opr = {<,>|} 1o rec v cv fveq1 3syl A {/} = 1o v cv \ ifeq2d eqtrd v cv B 1o difeq2 A {/} = x y y cv x cv .o A opr = {<,>|} 1o rec v cv ` ifeq1d v cv B x y y cv x cv .o A opr = {<,>|} 1o rec fveq2 A {/} = 1o B \ ifeq2d eqtrd w v z x y df-oexp oprabval2))
+thm (oevn0 ((x y) (A x) (A y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) A)) (= (opr A (^o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B))) (A on0eln0 B On e. adantr A B x y oev A {/} = 1o B \ x y y cv x cv .o A opr = {<,>|} 1o rec B ` iffalse sylan9eq exp sylbid imp))
+thm (oa0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (+o) ({/})) A)) (0elon A {/} x y oav mpan2 A On x y y cv x cv suc = {<,>|} rdg0t eqtrd))
+thm (om0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (.o) ({/})) ({/}))) (0elon A {/} x y omv mpan2 0ex x y y cv x cv +o A opr = {<,>|} rdg0 syl6eq))
+thm (om0x ((x y) (A x) (A y)) () (= (opr A (.o) ({/})) ({/})) (A {/} x y omv 0ex x y y cv x cv +o A opr = {<,>|} rdg0 syl6eq 0ex fnom .o On On X. fndm ax-mp A ndmopr pm2.61i))
+thm (oe0m ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr ({/}) (^o) A) (\ (1o) A))) (0elon {/} A x y oev mpan {/} eqid {/} {/} = 1o A \ x y y cv x cv .o {/} opr = {<,>|} 1o rec A ` iftrue ax-mp syl6eq))
+thm (oe0m0 () () (= (opr ({/}) (^o) ({/})) (1o)) (0elon {/} oe0m 1o dif0 syl6eq ax-mp))
+thm (oe0m1 () () (-> (e. A (On)) (<-> (e. ({/}) A) (= (opr ({/}) (^o) A) ({/})))) (A eloni A ordgt0ge1 syl A oe0m {/} eqeq1d 1o A ssdif0 syl6rbbr bitrd))
+thm (oe0m1OLD () () (-> (/\ (e. A (On)) (e. ({/}) A)) (= (opr ({/}) (^o) A) ({/}))) (A oe0m {/} A snssi df1o2 syl5ss 1o A ssdif0 sylib sylan9eq))
+thm (oe0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (^o) ({/})) (1o))) (A {/} ^o {/} opreq1 oe0m0 syl6eq A On e. adantl 0elon A {/} x y oevn0 mpanl2 1ord elisseti x y y cv x cv .o A opr = {<,>|} rdg0 syl6eq A On e. adantll oe0lem anidms))
+thm (oev2 ((x y) (A x) (A y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) B) (i^i (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B) (u. (\ (V) (|^| A)) (|^| B))))) (A {/} B {/} ^o opreq12 oe0m0 syl6eq B {/} x y y cv x cv .o A opr = {<,>|} 1o rec fveq2 1ord elisseti x y y cv x cv .o A opr = {<,>|} rdg0 syl6eq B {/} inteq int0 syl6eq ineq12d 1o inv A {/} = a1i sylan9eqr eqtr4d A {/} ^o B opreq1 B oe0m1OLD sylan9eqr an1rs B int0el x y y cv x cv .o A opr = {<,>|} 1o rec B ` ineq2d x y y cv x cv .o A opr = {<,>|} 1o rec B ` in0 syl6eq B On e. A {/} = /\ adantl eqtr4d oe0lem A {/} inteq int0 syl6eq V difeq2d V difid syl6eq B |^| uneq2d B |^| V A |^| \ uncom B |^| un0 3eqtr3g B On e. adantl x y y cv x cv .o A opr = {<,>|} 1o rec B ` ineq2d eqtr4d A B x y oevn0 A int0el V difeq2d V dif0 syl6eq B |^| uneq2d B |^| V A |^| \ uncom B |^| unv 3eqtr3g A On e. B On e. /\ adantl x y y cv x cv .o A opr = {<,>|} 1o rec B ` ineq2d x y y cv x cv .o A opr = {<,>|} 1o rec B ` inv syl6req eqtrd oe0lem))
+thm (oasuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) (suc B)) (suc (opr A (+o) B)))) (B x y y cv x cv suc = {<,>|} A rdgsuct A On e. adantl A B suc x y oav B suceloni sylan2 A B x y oav x y y cv x cv suc = {<,>|} fveq2d A +o B oprex A +o B oprex sucex x cv A +o B opr suceq y fvopab syl5eqr 3eqtr4d))
+thm (oa1suc () () (-> (e. A (On)) (= (opr A (+o) (1o)) (suc A))) (0elon A {/} oasuc mpan2 df-1o A +o opreq2i syl5eq A oa0 A +o {/} opr A suceq syl eqtrd))
+thm (omsuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (.o) (suc B)) (opr (opr A (.o) B) (+o) A))) (B x y y cv x cv +o A opr = {<,>|} {/} rdgsuct A On e. adantl A B suc x y omv B suceloni sylan2 A B x y omv x y y cv x cv +o A opr = {<,>|} fveq2d A .o B oprex A .o B opr +o A oprex x cv A .o B opr +o A opreq1 y fvopab syl5eqr 3eqtr4d))
+thm (oesuc ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) (suc B)) (opr (opr A (^o) B) (.o) A))) (A {/} ^o B suc opreq1 B eloni B 0elsuc syl B suc oe0m1OLD B suceloni sylan mpdan sylan9eqr A {/} ^o B opreq1 A {/} = id .o opreq12d oe0m0 1ord eqeltr B {/} {/} ^o opreq2 On eleq1d mpbiri B On e. adantl 0elon B oe0m1OLD On eleq1d mpbiri B On e. adantll oe0lem anidms {/} ^o B opr om0 syl sylan9eqr eqtr4d B x y y cv x cv .o A opr = {<,>|} 1o rdgsuct A On e. {/} A e. ad2antlr A B suc x y oevn0 B suceloni sylanl2 A B x y oevn0 x y y cv x cv .o A opr = {<,>|} fveq2d A ^o B oprex A ^o B opr .o A oprex x cv A ^o B opr .o A opreq1 y fvopab syl5eqr 3eqtr4d oe0lem))
+thm (oalim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (+o) B) (U_ x B (opr A (+o) (cv x))))) (B On y z z cv y cv suc = {<,>|} A x rdglim2a A On e. adantl A B y z oav A x cv y z oav B x cv onelon sylan2 anassrs exp iuneq2dv eqeq12d B Lim adantrr mpbird B C limelon B C e. B Lim pm3.27 jca sylan2))
+thm (omlim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (.o) B) (U_ x B (opr A (.o) (cv x))))) (B On y z z cv y cv +o A opr = {<,>|} {/} x rdglim2a A On e. adantl A B y z omv A x cv y z omv B x cv onelon sylan2 anassrs exp iuneq2dv eqeq12d B Lim adantrr mpbird B C limelon B C e. B Lim pm3.27 jca sylan2))
+thm (oelim ((x y) (x z) (A x) (y z) (A y) (A z) (B x)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (e. ({/}) A)) (= (opr A (^o) B) (U_ x B (opr A (^o) (cv x))))) (B On y z z cv y cv .o A opr = {<,>|} 1o x rdglim2a A On e. {/} A e. ad2antlr A B y z oevn0 A x cv y z oevn0 B x cv onelon sylanl2 exp42 com34 imp31 iuneq2dv eqeq12d B Lim adantlrr mpbird B C limelon B C e. B Lim pm3.27 jca sylanl2))
+thm (oacl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (+o) B) (On))) (x cv {/} A +o opreq2 On eleq1d x cv y cv A +o opreq2 On eleq1d x cv y cv suc A +o opreq2 On eleq1d x cv B A +o opreq2 On eleq1d A oa0 On eleq1d ibir A y cv oasuc On eleq1d A +o y cv opr suceloni syl5bir expcom x visset A x cv V y oalim mpanr1 On eleq1d x visset A +o y cv oprex y iunon syl5bir expcom tfinds3 impcom))
+thm (omcl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (.o) B) (On))) (x cv {/} A .o opreq2 On eleq1d x cv y cv A .o opreq2 On eleq1d x cv y cv suc A .o opreq2 On eleq1d x cv B A .o opreq2 On eleq1d 0elon A om0 On eleq1d mpbiri A y cv omsuc On eleq1d A .o y cv opr A oacl syl5bir exp4b com24 pm2.43i com3r x visset A x cv V y omlim mpanr1 ancoms On eleq1d x visset A .o y cv oprex y iunon syl5bir exp tfinds3 impcom))
+thm (oecl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (e. (opr A (^o) B) (On))) (A {/} ^o B opreq1 On eleq1d oe0m0 1ord eqeltr B {/} {/} ^o opreq2 On eleq1d mpbiri B On e. adantl 0elon B oe0m1OLD On eleq1d mpbiri B On e. adantll oe0lem anidms syl5bir impcom x cv {/} A ^o opreq2 On eleq1d x cv y cv A ^o opreq2 On eleq1d x cv y cv suc A ^o opreq2 On eleq1d x cv B A ^o opreq2 On eleq1d 1ord A oe0 On eleq1d mpbiri {/} A e. adantr A y cv oesuc On eleq1d A ^o y cv opr A omcl syl5bir exp4b com24 pm2.43i com3r {/} A e. adantrd x visset A x cv V y oelim mpanlr1 anasss an1s On eleq1d x visset A ^o y cv oprex y iunon syl5bir exp tfinds3 exp3a com12 imp31 oe0lem))
+thm (oa0r ((x y) (A x)) () (-> (e. A (On)) (= (opr ({/}) (+o) A) A)) (x cv {/} {/} +o opreq2 x cv {/} = id eqeq12d x cv y cv {/} +o opreq2 x cv y cv = id eqeq12d x cv y cv suc {/} +o opreq2 x cv y cv suc = id eqeq12d x cv A {/} +o opreq2 x cv A = id eqeq12d 0elon {/} oa0 ax-mp 0elon {/} y cv oasuc mpan {/} +o y cv opr y cv suceq sylan9eq exp x visset 0elon {/} x cv V y oalim mpan mpan x cv limuni eqeq12d y x cv {/} +o y cv opr y cv iuneq2 x cv y uniiun syl6eqr syl5bir tfinds))
+thm (om0r ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr ({/}) (.o) A) ({/}))) (x cv {/} {/} .o opreq2 {/} eqeq1d x cv y cv {/} .o opreq2 {/} eqeq1d x cv y cv suc {/} .o opreq2 {/} eqeq1d x cv A {/} .o opreq2 {/} eqeq1d {/} om0x 0elon {/} y cv omsuc mpan 0elon {/} oa0 ax-mp eqcomi y cv On e. a1i eqeq12d {/} .o y cv opr {/} +o {/} opreq1 syl5bir x visset 0elon {/} x cv V y omlim mpan mpan {/} eqeq1d y x cv {/} .o y cv opr {/} iuneq2 y x cv iun0 syl6eq syl5bir tfinds))
+thm (o1p1e2 () () (= (opr (1o) (+o) (1o)) (2o)) (1ord 1o oa1suc ax-mp df-2o eqtr4))
+thm (om1 () () (-> (e. A (On)) (= (opr A (.o) (1o)) A)) (0elon A {/} omsuc mpan2 df-1o A .o opreq2i syl5eq A om0 +o A opreq1d A oa0r 3eqtrd))
+thm (om1r ((x y) (A x)) () (-> (e. A (On)) (= (opr (1o) (.o) A) A)) (x cv {/} 1o .o opreq2 x cv {/} = id eqeq12d x cv y cv 1o .o opreq2 x cv y cv = id eqeq12d x cv y cv suc 1o .o opreq2 x cv y cv suc = id eqeq12d x cv A 1o .o opreq2 x cv A = id eqeq12d 1o om0x 1ord 1o y cv omsuc mpan 1o .o y cv opr y cv +o 1o opreq1 sylan9eq y cv oa1suc 1o .o y cv opr y cv = adantr eqtrd exp x visset 1ord 1o x cv V y omlim mpan mpan x cv limuni eqeq12d y x cv 1o .o y cv opr y cv iuneq2 x cv y uniiun syl6eqr syl5bir tfinds))
+thm (oe1 () () (-> (e. A (On)) (= (opr A (^o) (1o)) A)) (0elon A {/} oesuc mpan2 df-1o A ^o opreq2i syl5eq A oe0 .o A opreq1d A om1r 3eqtrd))
+thm (oe1m ((x y) (A x)) () (-> (e. A (On)) (= (opr (1o) (^o) A) (1o))) (x cv {/} 1o ^o opreq2 1o eqeq1d x cv y cv 1o ^o opreq2 1o eqeq1d x cv y cv suc 1o ^o opreq2 1o eqeq1d x cv A 1o ^o opreq2 1o eqeq1d 1ord 1o oe0 ax-mp 1ord 1o y cv oesuc mpan 1o ^o y cv opr 1o .o 1o opreq1 1ord 1o om1 ax-mp syl6eq sylan9eq exp x visset 1ord 0lt1o 1o x cv V y oelim mpan2 mpan mpan 1o eqeq1d x cv 0ellim {/} x cv n0i x cv y 1o iunconst 3syl y x cv 1o ^o y cv opr U_ eqeq2d bitr4d y x cv 1o ^o y cv opr 1o iuneq2 syl5bir tfinds))
+thm (oaordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B A onelon C On e. adantll B eloni B A ordsucss syl C On e. A On e. ad2antlr x cv A suc C +o opreq2 C +o A suc opr sseq2d C On e. imbi2d x cv y cv C +o opreq2 C +o A suc opr sseq2d C On e. imbi2d x cv y cv suc C +o opreq2 C +o A suc opr sseq2d C On e. imbi2d x cv B C +o opreq2 C +o A suc opr sseq2d C On e. imbi2d C +o A suc opr ssid C On e. a1i A suc On e. a1i C y cv oasuc ancoms C +o A suc opr sseq2d C +o y cv opr sssucid C +o A suc opr C +o y cv opr C +o y cv opr suc sstr2 mpi syl5bir exp A suc On e. A suc y cv C_ ad2antll a2d A sucelon A On x cv sucssel sylbir x cv A limsuc biimpd sylan9r imp y cv A suc C +o opreq2 x cv ssiun2s syl C On e. adantr x visset C x cv V y oalim mpanr1 ancoms A suc On e. adantlr A suc x cv C_ adantlr sseqtr4d exp y x cv A suc y cv C_ C On e. C +o A suc opr C +o y cv opr C_ -> -> A.e. a1d tfindsg exp31 A sucelon syl5ib com4r imp31 syld C A oasuc C +o B opr sseq1d C +o A oprex C +o A opr V C +o B opr sucssel ax-mp syl6bi B On e. adantlr syld imp an1rs mpdan exp ancoms))
+thm (oaord () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B C A oaordi A On e. 3adant1 A B C +o opreq2 A On e. B On e. C On e. /\/\ a1i A C B oaordi B On e. 3adant2 orim12d con3d A On e. B On e. C On e. df-3an A On e. B On e. /\ C On e. ancom C On e. A On e. B On e. anandi 3bitr C A oacl C +o A opr eloni syl C B oacl C +o B opr eloni syl anim12i sylbi C +o A opr C +o B opr ordtri2 syl A On e. B On e. C On e. 3simpa A eloni B eloni anim12i A B ordtri2 3syl 3imtr4d impbid))
+thm (oacan () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (= (opr A (+o) B) (opr A (+o) C)) (= B C))) (B C A oaord 3comr C B A oaord 3com13 orbi12d negbid bicomd A +o B opr A +o C opr ordtri3 A B oacl A +o B opr eloni syl A C oacl A +o C opr eloni syl syl2an 3impdi B C ordtri3 B eloni C eloni syl2an A On e. 3adant1 3bitr4d))
+thm (oaword () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (C_ A B) (C_ (opr C (+o) A) (opr C (+o) B)))) (A B C oaord C A B oacan 3coml bicomd orbi12d A B onsseleq C On e. 3adant3 C A oacl C B oacl anim12i anandis ancoms 3impa C +o A opr C +o B opr onsseleq syl 3bitr4d))
+thm (oawordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr A (+o) C) (opr B (+o) C)))) (x cv {/} A +o opreq2 x cv {/} B +o opreq2 sseq12d x cv y cv A +o opreq2 x cv y cv B +o opreq2 sseq12d x cv y cv suc A +o opreq2 x cv y cv suc B +o opreq2 sseq12d x cv C A +o opreq2 x cv C B +o opreq2 sseq12d A oa0 B On e. adantr B oa0 A On e. adantl sseq12d biimpar A +o y cv opr B +o y cv opr ordsucsssuc A y cv oacl A +o y cv opr eloni syl B y cv oacl B +o y cv opr eloni syl syl2an anandirs A y cv oasuc B On e. adantlr B y cv oasuc A On e. adantll sseq12d bitr4d biimpd expcom A B C_ adantrd x visset A x cv V y oalim B On e. adantlr B x cv V y oalim A On e. adantll sseq12d y x cv A +o y cv opr B +o y cv opr ss2iun syl5bir mpanr1 expcom A B C_ adantrd tfinds3 exp4c com3l 3imp))
+thm (oaord1 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. ({/}) B) (e. A (opr A (+o) B)))) (0elon {/} B A oaord mp3an1 A oa0 B On e. adantl A +o B opr eleq1d bitrd ancoms))
+thm (oaword1 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr A (+o) B))) (A oa0 B On e. adantr B 0ss 0elon {/} B A oaword 3com13 mp3an3 mpbii eqsstr3d))
+thm (oaword2 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr B (+o) A))) (B 0ss 0elon {/} B A oawordri mp3an1 A oa0r B On e. adantl B +o A opr sseq1d sylibd mpi ancoms))
+thm (oawordeulem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (S x) (S z)) ((oawordeulem.1 (e. A (On))) (oawordeulem.2 (e. B (On))) (oawordeulem.3 (= S ({e.|} y (On) (C_ B (opr A (+o) (cv y))))))) (-> (C_ A B) (E!e. x (On) (= (opr A (+o) (cv x)) B))) (oawordeulem.3 y On B A +o y cv opr C_ ssrab eqsstr oawordeulem.3 B eleq2i y cv B A +o opreq2 B sseq2d On elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an S |^| z onzsl mpbi S |^| {/} A +o opreq2 oawordeulem.1 A oa0 ax-mp syl6eq B sseq1d biimprd S |^| z cv suc A +o opreq2 oawordeulem.1 A z cv oasuc mpan sylan9eqr z visset sucid S |^| z cv suc z cv eleq2 mpbiri oawordeulem.3 y On B A +o y cv opr C_ ssrab eqsstr oawordeulem.3 B eleq2i y cv B A +o opreq2 B sseq2d On elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an z cv onel y cv z cv A +o opreq2 B sseq2d onnminsb oawordeulem.3 inteqi z cv eleq2i syl5ib oawordeulem.1 A z cv oacl mpan oawordeulem.2 B A +o z cv opr ontri1 mpan syl bicon2d sylibrd mpcom oawordeulem.2 onord B A +o z cv opr ordsucss ax-mp 3syl z cv On e. adantl eqsstrd exp r19.23aiv A B C_ a1d z S |^| A +o z cv opr B iunss oawordeulem.3 y On B A +o y cv opr C_ ssrab eqsstr oawordeulem.3 B eleq2i y cv B A +o opreq2 B sseq2d On elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an z cv onel y cv z cv A +o opreq2 B sseq2d onnminsb oawordeulem.3 inteqi z cv eleq2i syl5ib oawordeulem.1 A z cv oacl mpan oawordeulem.2 B A +o z cv opr ontri1 mpan syl bicon2d sylibrd mpcom oawordeulem.2 A +o z cv opr onelss syl mprgbir oawordeulem.1 A S |^| V z oalim mpan B sseq1d mpbiri A B C_ a1d 3jaoi ax-mp oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an y cv B A +o opreq2 B sseq2d On rcla4ev mp2an z cv B e. y ax-17 z cv A e. y ax-17 z cv +o e. y ax-17 z y On B A +o y cv opr C_ hbrab1 hbint hbopr hbss y cv y On B A +o y cv opr C_ {e.|} |^| A +o opreq2 B sseq2d onminsb ax-mp oawordeulem.3 inteqi A +o opreq2i sseqtr4 jctir A +o S |^| opr B eqss sylibr oawordeulem.3 y On B A +o y cv opr C_ ssrab eqsstr oawordeulem.3 B eleq2i y cv B A +o opreq2 B sseq2d On elrab bitr oawordeulem.2 oawordeulem.2 oawordeulem.1 B A oaword2 mp2an mpbir2an B S n0i ax-mp S oninton mp2an jctil x cv S |^| A +o opreq2 B eqeq1d On rcla4ev syl oawordeulem.1 A x cv y cv oacan mp3an1 A +o x cv opr B A +o y cv opr eqtr3t syl5bi rgen2 jctir x cv y cv A +o opreq2 B eqeq1d On reu4 sylibr))
+thm (oawordeu ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (E!e. x (On) (= (opr A (+o) (cv x)) B))) (A A On e. A {/} if B sseq1 A A On e. A {/} if +o x cv opreq1 B eqeq1d x On bireudv imbi12d B B On e. B {/} if A On e. A {/} if sseq2 B B On e. B {/} if A On e. A {/} if +o x cv opr eqeq2 x On bireudv imbi12d 0elon A elimel 0elon B elimel y On B On e. B {/} if A On e. A {/} if +o y cv opr C_ {e.|} eqid x oawordeulem dedth2h imp))
+thm (oawordexr ((A x) (B x)) () (-> (/\ (e. A (On)) (E.e. x (On) (= (opr A (+o) (cv x)) B))) (C_ A B)) (A x cv oaword1 A +o x cv opr B A sseq2 biimpcd syl exp r19.23adv imp))
+thm (oawordex ((A x) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (E.e. x (On) (= (opr A (+o) (cv x)) B)))) (A B x oawordeu exp x On A +o x cv opr B = reurex syl6 A x B oawordexr exp B On e. adantr impbid))
+thm (oaordex ((A x) (B x)) () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (E.e. x (On) (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B))))) (B A onelsst A On e. adantl A B x oawordex sylibd A x cv oaord1 A +o x cv opr B A eleq2 sylan9bb biimprcd exp4c com12 imp4b x cv On e. A +o x cv opr B = pm3.27 A On e. A B e. /\ a1i jcad exp3a r19.22dv exp B On e. adantr mpdd A x cv oaord1 A +o x cv opr B A eleq2 sylan9bb biimpd exp31 com34 imp4a r19.23adv B On e. adantr impbid))
+thm (oa00 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (opr A (+o) B) ({/})) (/\ (= A ({/})) (= B ({/}))))) (A on0eln0 B On e. adantr A B oaword1 {/} sseld sylbird {/} A +o B opr n0i syl6 a3d B on0eln0 A On e. adantl 0elon {/} B A oaord mp3an1 ancoms bitr3d A +o {/} opr A +o B opr n0i syl6bi a3d jcad A {/} B {/} +o opreq12 0elon {/} oa0 ax-mp syl6eq A On e. B On e. /\ a1i impbid))
+thm (oalimcl ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (Lim (opr A (+o) B))) (A B oacl A +o B opr eloni syl B C limelon sylan2 B 0ellim {/} B n0i syl A On e. B C e. ad2antrr A B oa00 A {/} = B {/} = pm3.27 syl6bi con3d B C limelon sylan2 mpd y visset sucid A +o B opr y cv suc x B A +o x cv opr U_ eqeq1 A B C x oalim syl5bi imp y cv eleq2d mpbii y cv x B A +o x cv opr eliun sylib B x cv onelon B C limelon sylan x cv B onnbtwn x cv B e. B x cv suc e. imnan sylibr com12 B C e. B Lim /\ adantl mpd A On e. y cv A +o x cv opr e. ad2antrl A x cv oacl A +o x cv opr eloni A +o x cv opr y cv ordsucelsuc 3syl A x cv oasuc y cv suc eleq2d bitr4d B x cv onelon B C limelon sylan sylan2 A +o B opr y cv suc A +o x cv suc opr eleq1 bicomd sylan9bbr B x cv suc A oaord 3expa B C limelon x cv B e. adantr B x cv onelon B C limelon sylan x cv sucelon sylib jca sylan ancoms A +o B opr y cv suc = adantl bitr4d biimpd exp32 com4l imp32 mtod exp44 imp r19.23adv A +o B opr y cv suc = adantl mpd expcom pm2.01d y cv On e. adantr nrexdv jca A +o B opr {/} = y On A +o B opr y cv suc = E.e. ioran sylibr jca A +o B opr y dflim3 sylibr))
+thm (oaass ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr (opr A (+o) B) (+o) C) (opr A (+o) (opr B (+o) C)))) (x cv {/} A +o B opr +o opreq2 x cv {/} B +o opreq2 A +o opreq2d eqeq12d x cv y cv A +o B opr +o opreq2 x cv y cv B +o opreq2 A +o opreq2d eqeq12d x cv y cv suc A +o B opr +o opreq2 x cv y cv suc B +o opreq2 A +o opreq2d eqeq12d x cv C A +o B opr +o opreq2 x cv C B +o opreq2 A +o opreq2d eqeq12d A B oacl A +o B opr oa0 syl B oa0 A +o opreq2d A On e. adantl eqtr4d A +o B opr y cv oasuc A B oacl sylan B y cv oasuc A +o opreq2d A On e. adantl A B +o y cv opr oasuc B y cv oacl sylan2 eqtrd anassrs eqeq12d A +o B opr +o y cv opr A +o B +o y cv opr opr suceq syl5bir expcom y x cv A +o B opr +o y cv opr A +o B +o y cv opr opr iuneq2 x cv Lim A On e. B On e. /\ /\ adantl x visset A +o B opr x cv V y oalim mpanr1 A B oacl sylan ancoms y x cv A +o B opr +o y cv opr A +o B +o y cv opr opr = A.e. adantr B +o x cv oprex A B +o x cv opr V z oalim mpanr1 x visset B x cv V oalimcl mpanr1 ancoms sylan2 x visset x cv V limelon mpan B x cv oacl ancoms B +o x cv opr z cv onelon exp syl A On e. adantld x cv Lim adantl z cv B ordtri2or ancoms B eloni z cv eloni syl2an z cv B +o x cv opr e. adantrl x cv On e. adantll x cv Lim adantl y cv {/} B +o opreq2 z cv sseq2d x cv rcla4ev x cv 0ellim B z cv onelsst B oa0 z cv sseq2d sylibrd imp syl2an exp32 imp x cv On e. adantrl z cv B +o x cv opr e. z cv On e. /\ adantrr B z cv y oawordex z cv B +o x cv opr e. adantrl x cv On e. adantll y cv x cv B oaord 3expb B +o y cv opr z cv B +o x cv opr eleq1 sylan9bb an1rs biimpar B +o y cv opr z cv eqimss2 y cv On e. adantl x cv On e. B On e. /\ z cv B +o x cv opr e. ad2antll jca anasss expcom r19.22dv2 z cv On e. adantrr sylbid x cv Lim adantl jaod mpd exp45 imp A On e. adantld imp32 z cv B +o y cv opr A oaword A On e. z cv B +o x cv opr e. /\ z cv On e. pm3.27 x cv Lim x cv On e. B On e. /\ /\ y cv x cv e. ad2antlr B y cv oacl x cv y cv onelon sylan2 exp32 com12 imp31 x cv Lim adantll A On e. z cv B +o x cv opr e. /\ z cv On e. /\ adantlr A On e. z cv B +o x cv opr e. pm3.26 z cv On e. adantr x cv Lim x cv On e. B On e. /\ /\ y cv x cv e. ad2antlr syl3anc birexdva mpbid exp32 mpdd exp32 mpd exp4a imp31 r19.21aiv z B +o x cv opr y x cv A +o z cv opr A +o B +o y cv opr opr iunss2 syl ancoms x cv B y cv oaordi w cv A +o B +o y cv opr opr e. anim1d z cv B +o y cv opr A +o opreq2 w cv eleq2d B +o x cv opr rcla4ev syl6 exp3a r19.23adv w cv y x cv A +o B +o y cv opr opr eliun w cv z B +o x cv opr A +o z cv opr eliun 3imtr4g ssrdv x visset x cv V limelon mpan sylan A On e. adantl eqssd eqtrd an1s y x cv A +o B opr +o y cv opr A +o B +o y cv opr opr = A.e. adantr 3eqtr4d exp31 tfinds3 com12 exp 3imp))
+thm (oarec ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (u. A ({|} x (E.e. y B (= (cv x) (opr A (+o) (cv y)))))))) (z cv {/} A +o opreq2 z cv {/} y x cv A +o y cv opr = rexeq x biabdv A uneq2d eqeq12d z cv w cv A +o opreq2 z cv w cv y x cv A +o y cv opr = rexeq x biabdv A uneq2d eqeq12d z cv w cv suc A +o opreq2 z cv w cv suc y x cv A +o y cv opr = rexeq x biabdv A uneq2d eqeq12d z cv B A +o opreq2 z cv B y x cv A +o y cv opr = rexeq x biabdv A uneq2d eqeq12d A oa0 y x cv A +o y cv opr = rex0 x nex x y {/} x cv A +o y cv opr = E.e. abn0 bicon1i mpbi A uneq2i A un0 eqtr2 syl6eq A w cv oasuc A +o w cv opr x df-sn A +o w cv opr A x y w cv x cv A +o y cv opr = E.e. {|} u. A +o w cv opr {} x x cv A +o w cv opr = {|} uneq12 mpan2 A +o w cv opr df-suc y visset w cv elsuc x cv A +o y cv opr = anbi1i y cv w cv e. y cv w cv = x cv A +o y cv opr = andir bitr y biex y y cv w cv e. x cv A +o y cv opr = /\ y cv w cv = x cv A +o y cv opr = /\ 19.43 bitr y w cv suc x cv A +o y cv opr = df-rex y w cv x cv A +o y cv opr = df-rex w visset y cv w cv A +o opreq2 x cv eqeq2d ceqsexv bicomi orbi12i 3bitr4 x biabi x y w cv x cv A +o y cv opr = E.e. x cv A +o w cv opr = unab eqtr4 A uneq2i A x y w cv x cv A +o y cv opr = E.e. {|} x x cv A +o w cv opr = {|} unass eqtr4 3eqtr4g sylan9eq exp31 com12 z visset A z cv V w oalim mpanr1 ancoms w z cv A +o w cv opr A x y w cv x cv A +o y cv opr = E.e. {|} u. = A.e. adantr w z cv A +o w cv opr A x y w cv x cv A +o y cv opr = E.e. {|} u. iuneq2 z cv Lim A On e. /\ adantl z cv 0ellim {/} z cv n0i z cv w A iunconst 3syl z cv limord z cv y cv w cv ordtr1 syl exp3a com23 imp x cv A +o y cv opr = anim1d r19.22dv2 exp r19.23adv z cv Lim y ax-17 w cv z cv e. y ax-17 y w cv x cv A +o y cv opr = hbre1 hbrex z cv y cv limsuc biimpd y cv z cv sucidg z cv Lim a1i jcad x cv A +o y cv opr = anim1d w cv y cv suc y cv eleq2 x cv A +o y cv opr = anbi1d z cv rcla4ev y w cv x cv A +o y cv opr = ra4e w z cv r19.22si syl anassrs syl6 exp3a r19.23ad impbid x biabdv w z cv x y w cv x cv A +o y cv opr = E.e. iunab syl5eq uneq12d w z cv A x y w cv x cv A +o y cv opr = E.e. {|} iunun syl5eq A On e. w z cv A +o w cv opr A x y w cv x cv A +o y cv opr = E.e. {|} u. = A.e. ad2antll 3eqtrd exp31 tfinds3 impcom))
+thm (omordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. B (On)) (e. C (On))) (e. ({/}) C)) (-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B)))) (B A onelon exp x cv {/} A eleq2 x cv {/} C .o opreq2 C .o A opr eleq2d imbi12d x cv y cv A eleq2 x cv y cv C .o opreq2 C .o A opr eleq2d imbi12d x cv y cv suc A eleq2 x cv y cv suc C .o opreq2 C .o A opr eleq2d imbi12d x cv B A eleq2 x cv B C .o opreq2 C .o A opr eleq2d imbi12d A noel C .o A opr C .o {/} opr e. pm2.21i A On e. C On e. /\ {/} C e. /\ a1i C .o y cv opr C oaword1 C .o A opr sseld A y cv e. imim2d imp {/} C e. adantrl A y cv C .o opreq2 C .o y cv opr +o C opr eleq1d C .o y cv opr C oaord1 biimpa syl5bir com12 A y cv e. C .o A opr C .o y cv opr e. -> adantrr jaod C y cv omcl C On e. y cv On e. pm3.26 jca sylan A y cv elsuci syl5 C y cv omsuc C .o A opr eleq2d {/} C e. A y cv e. C .o A opr C .o y cv opr e. -> /\ adantr sylibrd exp43 com12 A On e. adantld imp3a x cv A limsuc biimpa y cv A suc C .o opreq2 x cv ssiun2s syl C On e. adantll x visset C x cv V y omlim mpanr1 A x cv e. adantr sseqtr4d C On e. x cv Lim /\ id {/} C e. adantrr A On e. adantlr sylan C .o A opr C oaord1 C A omcl sylan anabss1 biimpa C A omsuc {/} C e. adantr eleqtrrd x cv Lim adantrl A x cv e. adantr sseldd exp exp43 com13 imp4c y x cv A y cv e. C .o A opr C .o y cv opr e. -> A.e. a1dd tfinds3 com23 exp4a exp4a mpdd com34 com24 imp31))
+thm (omord2 () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (<-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B)))) (B C A omordi A On e. 3adantl1 A B C .o opreq2 A On e. B On e. C On e. /\/\ {/} C e. /\ a1i A C B omordi B On e. 3adantl2 orim12d con3d C .o A opr C .o B opr ordtri2 C .o A opr eloni C .o B opr eloni syl2an C A omcl C B omcl syl2an anandis ancoms 3impa {/} C e. adantr A B ordtri2 A eloni B eloni syl2an C On e. 3adant3 {/} C e. adantr 3imtr4d impbid))
+thm (omord () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (A B C omord2 exp pm5.32rd C .o A opr C .o B opr e. {/} C e. pm3.26 A On e. B On e. C On e. /\/\ a1i C {/} .o B opreq1 {/} eqeq1d B om0r syl5bir com12 C .o A opr C .o B opr n0i nsyli C On e. adantr C on0eln0 B On e. adantl sylibrd A On e. 3adant1 ancld impbid bitrd))
+thm (omcan () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) A)) (<-> (= (opr A (.o) B) (opr A (.o) C)) (= B C))) (C A B omordi exp ancoms B On e. 3adant2 imp B A C omordi exp ancoms C On e. 3adant3 imp orim12d con3d A .o B opr A .o C opr ordtri3 A B omcl A .o B opr eloni syl A C omcl A .o C opr eloni syl syl2an 3impdi {/} A e. adantr B C ordtri3 B eloni C eloni syl2an A On e. 3adant1 {/} A e. adantr 3imtr4d B C A .o opreq2 A On e. B On e. C On e. /\/\ {/} A e. /\ a1i impbid))
+thm (omword () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (<-> (C_ A B) (C_ (opr C (.o) A) (opr C (.o) B)))) (A B C omord2 C A B omcan C On e. A On e. B On e. 3anrot sylanbr bicomd orbi12d A B onsseleq C On e. 3adant3 {/} C e. adantr C A omcl C B omcl anim12i anandis ancoms 3impa C .o A opr C .o B opr onsseleq syl {/} C e. adantr 3bitr4d))
+thm (omwordi () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr C (.o) A) (opr C (.o) B)))) (A B C omword biimpd exp C eloni C ord0eln0 bicon2d syl A On e. B On e. 3ad2ant3 C {/} .o A opreq1 C {/} .o B opreq1 sseq12d {/} ssid A om0r B On e. adantr B om0r A On e. adantl sseq12d mpbiri syl5bir com12 C On e. 3adant3 sylbird A B C_ a1dd pm2.61d))
+thm (omwordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr A (.o) C) (opr B (.o) C)))) (x cv {/} A .o opreq2 x cv {/} B .o opreq2 sseq12d x cv y cv A .o opreq2 x cv y cv B .o opreq2 sseq12d x cv y cv suc A .o opreq2 x cv y cv suc B .o opreq2 sseq12d x cv C A .o opreq2 x cv C B .o opreq2 sseq12d B .o {/} opr 0ss A om0 B .o {/} opr sseq1d mpbiri B On e. A B C_ ad2antll A .o y cv opr B .o y cv opr A oawordri A y cv omcl B On e. 3adant2 B y cv omcl A On e. 3adant1 A On e. B On e. y cv On e. 3simp1 syl3anc imp A B C_ adantrl A B B .o y cv opr oaword A On e. B On e. y cv On e. 3simp1 A On e. B On e. y cv On e. 3simp2 B y cv omcl A On e. 3adant1 syl3anc biimpa A .o y cv opr B .o y cv opr C_ adantrr sstrd A y cv omsuc B On e. 3adant2 A B C_ A .o y cv opr B .o y cv opr C_ /\ adantr B y cv omsuc A On e. 3adant1 A B C_ A .o y cv opr B .o y cv opr C_ /\ adantr 3sstr4d exp32 3exp com3r imp4c x visset A x cv V y omlim B On e. x cv V e. x cv Lim /\ /\ adantr B x cv V y omlim A On e. x cv V e. x cv Lim /\ /\ adantl sseq12d y x cv A .o y cv opr B .o y cv opr ss2iun syl5bir anandirs mpanr1 expcom A B C_ adantrd tfinds3 exp3a imp 3impb 3coml))
+thm (omword1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) B)) (C_ A (opr A (.o) B))) (B eloni B ordgt0ge1 syl A On e. adantl 1ord 1o B A omwordi mp3an1 ancoms A om1 B On e. adantr A .o B opr sseq1d sylibd sylbid imp))
+thm (omword2 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) B)) (C_ A (opr B (.o) A))) (A om1r B On e. {/} B e. ad2antll B ordgt0ge1 biimpa B eloni sylan A On e. adantll 1ord 1o B A omwordri mp3an1 ancoms {/} B e. adantr mpd eqsstr3d))
+thm (om00 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (opr A (.o) B) ({/})) (\/ (= A ({/})) (= B ({/}))))) (A eloni A ordge1n0 syl biimprd B On e. adantr B on0eln0 A On e. adantl A B omword1 exp sylbird anim12d 1o A A .o B opr sstr syl6 A {/} = B {/} = ioran syl5ib A B omcl A .o B opr eloni A .o B opr ordge1n0 3syl sylibd a3d A {/} .o B opreq1 B om0r sylan9eqr exp A On e. adantl B {/} A .o opreq2 A om0 sylan9eqr exp B On e. adantr jaod impbid))
+thm (om00el () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. ({/}) (opr A (.o) B)) (/\ (e. ({/}) A) (e. ({/}) B)))) (A B om00 negbid A B omcl A .o B opr on0eln0 syl A on0eln0 B on0eln0 bi2anan9 A {/} = B {/} = ioran syl6bbr 3bitr4d))
+thm (omordlim ((A x) (B x) (C x)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B D) (Lim B))) (e. C (opr A (.o) B))) (E.e. x B (e. C (opr A (.o) (cv x))))) (A B D x omlim C eleq2d C x B A .o x cv opr eliun syl6bb biimpa))
+thm (omlimcl ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (e. ({/}) A)) (Lim (opr A (.o) B))) (A B omcl A .o B opr eloni syl B C limelon sylan2 {/} A e. adantr {/} A n0i B 0ellim {/} B n0i syl anim12i ancoms B C e. adantll A On e. adantll A B om00 negbid A {/} = B {/} = ioran syl6bb B C limelon sylan2 {/} A e. adantr mpbird y visset sucid A .o B opr y cv suc x B A .o x cv opr U_ eqeq1 biimpac A B C x omlim sylan y cv eleq2d mpbii y cv x B A .o x cv opr eliun sylib {/} A e. adantlr B x cv onelon B C limelon sylan x cv B onnbtwn x cv B e. B x cv suc e. imnan sylibr com12 B C e. B Lim /\ adantl mpd A On e. adantll {/} A e. adantlr y cv A .o x cv opr e. adantr A x cv omcl A .o x cv opr eloni A .o x cv opr y cv ordsucelsuc syl A .o x cv opr oa1suc y cv suc eleq2d bitr4d syl {/} A e. adantr A eloni A ordgt0ge1 syl x cv On e. adantr A x cv omcl 1ord 1o A A .o x cv opr oaword mp3an1 syldan bitrd biimpa A x cv omsuc {/} A e. adantr sseqtr4d y cv suc sseld sylbid A .o B opr y cv suc A .o x cv suc opr eleq1 biimprd syl9 com23 B On e. adantlrl B x cv suc A omord B x cv suc e. {/} A e. pm3.26 syl6bir x cv sucelon syl3an2b 3comr 3expb {/} A e. adantr syl6d B On e. x cv B e. pm3.26 B x cv onelon jca B C limelon sylan A On e. anim2i anassrs sylan an1rs imp mtod exp31 r19.23adv A .o B opr y cv suc = adantr mpd exp pm2.01d y cv On e. adantr nrexdv jca A .o B opr {/} = y On A .o B opr y cv suc = E.e. ioran sylibr jca A .o B opr y dflim3 sylibr))
+thm (odi ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr A (.o) (opr B (+o) C)) (opr (opr A (.o) B) (+o) (opr A (.o) C)))) (x cv {/} B +o opreq2 A .o opreq2d x cv {/} A .o opreq2 A .o B opr +o opreq2d eqeq12d x cv y cv B +o opreq2 A .o opreq2d x cv y cv A .o opreq2 A .o B opr +o opreq2d eqeq12d x cv y cv suc B +o opreq2 A .o opreq2d x cv y cv suc A .o opreq2 A .o B opr +o opreq2d eqeq12d x cv C B +o opreq2 A .o opreq2d x cv C A .o opreq2 A .o B opr +o opreq2d eqeq12d A B omcl A .o B opr oa0 syl A om0 B On e. adantr A .o B opr +o opreq2d B oa0 A On e. adantl A .o opreq2d 3eqtr4rd B y cv oasuc A On e. 3adant1 A .o opreq2d A B +o y cv opr omsuc B y cv oacl sylan2 3impb eqtrd A y cv omsuc B On e. 3adant2 A .o B opr +o opreq2d A .o B opr A .o y cv opr A oaass A B omcl syl3an1 A y cv omcl syl3an2 3exp exp4b pm2.43a com4r pm2.43i 3imp eqtr4d eqeq12d A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr +o A opreq1 syl5bir 3exp com3r imp3a A {/} .o B +o x cv opr opreq1 A {/} .o B opreq1 A {/} .o x cv opreq1 +o opreq12d eqeq12d B x cv oacl B +o x cv opr om0r syl B om0r x cv om0r +o opreqan12d 0elon {/} oa0 ax-mp syl6req eqtrd x visset x cv V limelon mpan sylan2 ancoms syl5bir exp3a com3r imp y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. a1dd B z cv ontri1 B z cv v oawordex bitr3d x cv On e. B On e. pm3.27 z cv B +o x cv opr e. adantr B +o x cv opr z cv onelon B x cv oacl ancoms sylan sylanc v cv x cv B oaord 3expb B +o v cv opr z cv B +o x cv opr eleq1 sylan9bb B +o v cv opr z cv = v cv x cv e. iba v cv On e. x cv On e. B On e. /\ /\ adantl bitr3d an1rs biimpcd exp4c com4r imp r19.22dv sylbid orrd x visset x cv V limelon mpan B On e. anim1i sylan A On e. adantlrl {/} A e. y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. /\ adantlr A x cv om00el biimprd x cv 0ellim sylan2i x visset x cv V limelon mpan sylan2 exp4b com4r pm2.43a imp31 z cv B e. a1d B On e. adantlrr B A z cv omordi A On e. B On e. ancom sylanb A B omcl A .o B opr A .o z cv opr onelsst A .o B opr oa0 A .o z cv opr sseq2d sylibrd syl {/} A e. adantr syld x cv Lim adantll jcad w cv {/} A .o B opr +o opreq2 A .o z cv opr sseq2d A .o x cv opr rcla4ev syl6 y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantrr x cv A v cv omordi x visset x cv V limelon mpan A On e. anim1i sylan B +o v cv opr z cv = adantrd y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantrr y cv v cv B +o opreq2 A .o opreq2d y cv v cv A .o opreq2 A .o B opr +o opreq2d eqeq12d x cv rcla4v com12 A .o B +o v cv opr opr A .o B opr +o A .o v cv opr opr A .o z cv opr eqeq1 B +o v cv opr z cv A .o opreq2 syl5bi A .o B opr +o A .o v cv opr opr A .o z cv opr eqimss2 syl6 v cv x cv e. imim2i imp3a syl x cv Lim A On e. /\ {/} A e. ad2antrr jcad w cv A .o v cv opr A .o B opr +o opreq2 A .o z cv opr sseq2d A .o x cv opr rcla4ev syl6 v cv On e. a1d r19.23adv B On e. adantlrr jaod z cv B +o x cv opr e. adantr mpd exp r19.21aiv z B +o x cv opr w A .o x cv opr A .o z cv opr A .o B opr +o w cv opr iunss2 syl x visset A x cv V w cv v omordlim exp mpanr1 ancoms imp B On e. adantlrr y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantlr x cv B v cv oaordi x visset x cv V limelon mpan sylan imp A On e. adantlrl w cv A .o v cv opr e. a1d y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantlr A .o v cv opr A .o B opr w cv oaordi A v cv omcl ancoms B On e. adantrr A B omcl v cv On e. adantl sylanc x cv v cv ordelon x cv limord sylan sylan an1rs y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantlr y cv v cv B +o opreq2 A .o opreq2d y cv v cv A .o opreq2 A .o B opr +o opreq2d eqeq12d x cv rcla4v impcom A .o B opr +o w cv opr eleq2d x cv Lim A On e. B On e. /\ /\ adantll sylibrd A B +o v cv opr omcl B v cv oacl ancoms sylan2 an1s x cv v cv ordelon x cv limord sylan sylan an1rs A .o B +o v cv opr opr A .o B opr +o w cv opr onelsst syl y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantlr syld jcad z cv B +o v cv opr A .o opreq2 A .o B opr +o w cv opr sseq2d B +o x cv opr rcla4ev syl6 exp r19.23adv w cv A .o x cv opr e. adantr mpd exp r19.21aiv w A .o x cv opr z B +o x cv opr A .o B opr +o w cv opr A .o z cv opr iunss2 syl {/} A e. adantrl eqssd x visset B x cv V oalimcl mpanr1 ancoms A On e. anim2i an1s B +o x cv oprex A B +o x cv opr V z omlim mpanr1 syl {/} A e. y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. /\ adantr A .o B opr A .o x cv opr V w oalim A B omcl x cv Lim {/} A e. ad2antlr A x cv V omlimcl x visset x cv Lim jctl A On e. anim2i ancoms sylan B On e. adantlrr A .o x cv oprex jctil sylanc y x cv A .o B +o y cv opr opr A .o B opr +o A .o y cv opr opr = A.e. adantrr 3eqtr4d exp43 com3l imp oe0lem com12 tfinds3 exp3a com3l 3imp))
+thm (omass ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (= (opr (opr A (.o) B) (.o) C) (opr A (.o) (opr B (.o) C)))) (x cv {/} A .o B opr .o opreq2 x cv {/} B .o opreq2 A .o opreq2d eqeq12d x cv y cv A .o B opr .o opreq2 x cv y cv B .o opreq2 A .o opreq2d eqeq12d x cv y cv suc A .o B opr .o opreq2 x cv y cv suc B .o opreq2 A .o opreq2d eqeq12d x cv C A .o B opr .o opreq2 x cv C B .o opreq2 A .o opreq2d eqeq12d A B omcl A .o B opr om0 syl B om0 A .o opreq2d A om0 sylan9eqr eqtr4d A .o B opr y cv omsuc A B omcl sylan 3impa B y cv omsuc A On e. 3adant1 A .o opreq2d A B .o y cv opr B odi B y cv omcl syl3an2 3exp exp3a com34 pm2.43d 3imp eqtrd eqeq12d A .o B opr .o y cv opr A .o B .o y cv opr opr +o A .o B opr opreq1 syl5bir 3exp com3r imp3a x visset A .o B opr x cv V y omlim mpanr1 A B omcl ancoms sylan an1rs {/} B e. y x cv A .o B opr .o y cv opr A .o B .o y cv opr opr = A.e. ad2antll y x cv A .o B opr .o y cv opr A .o B .o y cv opr opr iuneq2 x cv B y cv omordi x visset x cv V limelon mpan B On e. anim1i ancoms sylan A .o B .o y cv opr opr ssid z cv B .o y cv opr A .o opreq2 A .o B .o y cv opr opr sseq2d B .o x cv opr rcla4ev mpan2 syl6 r19.21aiv y x cv z B .o x cv opr A .o B .o y cv opr opr A .o z cv opr iunss2 syl A On e. adantlr B .o x cv opr z cv onelon B x cv omcl x visset x cv V limelon mpan sylan2 sylan A On e. adantlr x visset B x cv V z cv y omordlim exp mpanr1 z cv On e. A On e. ad2antlr B .o y cv opr z cv onelsst z cv On e. A On e. 3ad2ant2 z cv B .o y cv opr A omwordi syld 3exp B y cv omcl x cv y cv onelon x visset x cv V limelon mpan sylan sylan2 syl5 exp4d imp32 com23 imp r19.22dv syld exp31 imp4c mpcom exp r19.21aiv z B .o x cv opr y x cv A .o z cv opr A .o B .o y cv opr opr iunss2 syl {/} B e. adantr eqssd B .o x cv oprex A B .o x cv opr V z omlim mpanr1 x visset B x cv V omlimcl mpanlr1 sylan2 ancoms an1rs eqtr4d sylan9eqr eqtrd exp31 B eloni B ord0eln0 bicon2d syl x cv Lim A On e. ad2antll B {/} A .o opreq2 A om0 sylan9eqr .o x cv opreq1d x cv om0r sylan9eqr anassrs B {/} .o x cv opreq1 x cv om0r sylan9eqr A .o opreq2d A om0 sylan9eq an1rs eqtr4d exp x visset x cv V limelon mpan sylan B On e. adantll sylbird y x cv A .o B opr .o y cv opr A .o B .o y cv opr opr = A.e. a1dd pm2.61d exp31 com3l imp3a tfinds3 exp3a com3l 3imp))
+thm (oen0 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) A)) (e. ({/}) (opr A (^o) B))) (x cv {/} A ^o opreq2 {/} eleq2d x cv y cv A ^o opreq2 {/} eleq2d x cv y cv suc A ^o opreq2 {/} eleq2d x cv B A ^o opreq2 {/} eleq2d 0lt1o A oe0 {/} eleq2d mpbiri {/} A e. adantr A A ^o y cv opr {/} omordi A ^o y cv opr om0 A ^o y cv opr .o A opr eleq1d A On e. {/} A ^o y cv opr e. ad2antlr sylibd A On e. y cv On e. pm3.26 A y cv oecl jca sylan A y cv oesuc {/} eleq2d {/} A ^o y cv opr e. adantr sylibrd exp31 com12 com34 imp3a x cv 0ellim A oe0 A ^o {/} opr 1o eqimss2 syl anim12i y cv {/} A ^o opreq2 1o sseq2d x cv rcla4ev y x cv 1o A ^o y cv opr ssiun 3syl {/} A e. adantrr x visset A x cv V y oelim mpanlr1 anasss an1s sseqtr4d A x cv oecl ancoms x visset x cv V limelon mpan sylan A ^o x cv opr eloni A ^o x cv opr ordgt0ge1 3syl {/} A e. adantrr mpbird exp y x cv {/} A ^o y cv opr e. A.e. a1dd tfinds3 exp3a com12 imp31))
+thm (oeordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. B (On)) (e. C (On))) (e. (1o) C)) (-> (e. A B) (e. (opr C (^o) A) (opr C (^o) B)))) (x cv A suc C ^o opreq2 C ^o A opr eleq2d C On e. 1o C e. /\ imbi2d x cv y cv C ^o opreq2 C ^o A opr eleq2d C On e. 1o C e. /\ imbi2d x cv y cv suc C ^o opreq2 C ^o A opr eleq2d C On e. 1o C e. /\ imbi2d x cv B C ^o opreq2 C ^o A opr eleq2d C On e. 1o C e. /\ imbi2d 0lt1o C {/} 1o ontr1 mpani A On e. adantr C A oen0 exp syld C On e. A On e. pm3.26 C A oecl jca jctild C C ^o A opr 1o omordi syli C A oecl C ^o A opr om1 syl C ^o A opr .o C opr eleq1d C A oesuc C ^o A opr eleq2d bitr4d sylibd expcom imp3a 0lt1o C {/} 1o ontr1 mpani y cv On e. adantr C y cv oen0 exp syld C On e. y cv On e. pm3.26 C y cv oecl jca jctild C C ^o y cv opr 1o omordi syli C y cv oecl C ^o y cv opr om1 syl C ^o y cv opr .o C opr eleq1d C y cv oesuc C ^o y cv opr eleq2d bitr4d sylibd C y cv suc oecl y cv sucelon sylan2b C ^o y cv suc opr C ^o A opr C ^o y cv opr ontr1 syl exp3a com23 syld expcom imp3a A y cv e. adantr a2d x cv A limsuc biimpa A suc x cv elisset A sucexb A V sucidg sylbir syl y cv A suc A eleq2 y cv A suc C ^o opreq2 C ^o A opr eleq2d imbi12d x cv rcla4v mpid anc2li y cv A suc C ^o opreq2 C ^o A opr eleq2d x cv rcla4ev C ^o A opr y x cv C ^o y cv opr eliun sylibr syl6 syl C On e. 1o C e. /\ adantr 0lt1o C {/} 1o ontr1 mpani x cv Lim adantr x visset C x cv V y oelim mpanlr1 exp syld imp anasss an1s A x cv e. adantlr C ^o A opr eleq2d sylibrd exp a2d A y cv e. C On e. 1o C e. /\ C ^o A opr C ^o y cv opr e. bi2.04 y x cv biral y x cv C On e. 1o C e. /\ A y cv e. C ^o A opr C ^o y cv opr e. -> r19.21v bitr syl5ib tfindsg2 exp4b com34 com24 imp31))
+thm (oeord () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) C)) (<-> (e. A B) (e. (opr C (^o) A) (opr C (^o) B)))) (B C A oeordi A On e. 3adantl1 A B C ^o opreq2 A On e. B On e. C On e. /\/\ 1o C e. /\ a1i A C B oeordi B On e. 3adantl2 orim12d con3d C ^o A opr C ^o B opr ordtri2 C ^o A opr eloni C ^o B opr eloni syl2an C A oecl C B oecl syl2an anandis ancoms 3impa 1o C e. adantr A B ordtri2 A eloni B eloni syl2an C On e. 3adant3 1o C e. adantr 3imtr4d impbid))
+thm (oecan () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) A)) (<-> (= (opr A (^o) B) (opr A (^o) C)) (= B C))) (C A B oeordi exp ancoms B On e. 3adant2 imp B A C oeordi exp ancoms C On e. 3adant3 imp orim12d con3d A ^o B opr A ^o C opr ordtri3 A B oecl A ^o B opr eloni syl A C oecl A ^o C opr eloni syl syl2an 3impdi 1o A e. adantr B C ordtri3 B eloni C eloni syl2an A On e. 3adant1 1o A e. adantr 3imtr4d B C A ^o opreq2 A On e. B On e. C On e. /\/\ 1o A e. /\ a1i impbid))
+thm (oeword () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. (1o) C)) (<-> (C_ A B) (C_ (opr C (^o) A) (opr C (^o) B)))) (A B C oeord C A B oecan C On e. A On e. B On e. 3anrot sylanbr bicomd orbi12d A B onsseleq C On e. 3adant3 1o C e. adantr C A oecl C B oecl anim12i anandis ancoms 3impa C ^o A opr C ^o B opr onsseleq syl 1o C e. adantr 3bitr4d))
+thm (oewordi () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (-> (C_ A B) (C_ (opr C (^o) A) (opr C (^o) B)))) (C eloni C ordgt0ge1 syl 1ord 1o C onsseleq mpan bitrd A On e. B On e. 3ad2ant3 A B C oeword biimpd exp 1o C ^o A opreq1 1o C ^o B opreq1 sseq12d A oe1m B On e. adantr B oe1m A On e. adantl eqtr4d 1o ^o A opr 1o ^o B opr eqimss syl syl5bi com12 C On e. 3adant3 A B C_ a1dd jaod sylbid imp))
+thm (oewordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (C_ (opr A (^o) C) (opr B (^o) C)))) (x cv {/} A ^o opreq2 x cv {/} B ^o opreq2 sseq12d x cv y cv A ^o opreq2 x cv y cv B ^o opreq2 sseq12d x cv y cv suc A ^o opreq2 x cv y cv suc B ^o opreq2 sseq12d x cv C A ^o opreq2 x cv C B ^o opreq2 sseq12d B A onelon A oe0 syl B oe0 A B e. adantr eqtr4d A ^o {/} opr B ^o {/} opr eqimss syl A ^o y cv opr B ^o y cv opr A omwordri A y cv oecl B On e. 3adant2 B y cv oecl A On e. 3adant1 A On e. B On e. y cv On e. 3simp1 syl3anc imp A B C_ adantrl A B B ^o y cv opr omwordi A On e. B On e. y cv On e. 3simp1 A On e. B On e. y cv On e. 3simp2 B y cv oecl A On e. 3adant1 syl3anc imp A ^o y cv opr B ^o y cv opr C_ adantrr sstrd A y cv oesuc B On e. 3adant2 A B C_ A ^o y cv opr B ^o y cv opr C_ /\ adantr B y cv oesuc A On e. 3adant1 A B C_ A ^o y cv opr B ^o y cv opr C_ /\ adantr 3sstr4d exp32 3exp com3r imp4c B A onelon B On e. A B e. pm3.26 jca B A onelsst imp jca syl5 A {/} ^o x cv opreq1 B ^o x cv opr sseq1d x cv oe0m1OLD x visset x cv V limelon mpan x cv 0ellim sylanc B ^o x cv opr 0ss x cv Lim a1i eqsstrd syl5bir B On e. A B e. /\ adantl y x cv A ^o y cv opr B ^o y cv opr C_ A.e. a1dd x visset A x cv V y oelim mpanlr1 an1rs B On e. A B e. /\ adantllr x visset B x cv V y oelim mpanlr1 B On e. A B e. pm3.26 x cv Lim anim1i B on0eln0 A B n0i syl5bir imp x cv Lim adantr sylanc {/} A e. adantlr A On e. adantlll sseq12d y x cv A ^o y cv opr B ^o y cv opr ss2iun syl5bir exp oe0lem com12 B A onelon ancri syl5 tfinds3 exp3a impcom))
+thm (oeworde ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. (1o) A)) (C_ B (opr A (^o) B))) (x cv {/} = id x cv {/} A ^o opreq2 sseq12d x cv y cv = id x cv y cv A ^o opreq2 sseq12d x cv y cv suc = id x cv y cv suc A ^o opreq2 sseq12d x cv B = id x cv B A ^o opreq2 sseq12d A ^o {/} opr 0ss A On e. 1o A e. /\ a1i y cv A ^o y cv opr ordsucsssuc y cv eloni A On e. adantr A y cv oecl ancoms A ^o y cv opr eloni syl sylanc 1o A e. adantrr y cv suc A ^o y cv opr suc A ^o y cv suc opr sstr2 y visset sucid y cv suc A y cv oeordi mpi anasss A y cv suc oecl A ^o y cv suc opr eloni syl ancoms A ^o y cv suc opr A ^o y cv opr ordsucss syl 1o A e. adantrr mpd y cv sucelon sylanb syl5com sylbid exp x cv limuni x cv y uniiun syl6eq A On e. {/} A e. /\ adantr x visset A x cv V y oelim mpanlr1 anasss an1s sseq12d y x cv y cv A ^o y cv opr ss2iun syl5bir exp A on0eln0 1o A n0i syl5bir imdistani syl5 tfinds3 exp3a com12 imp31))
+thm (oeordsuc () () (-> (/\ (e. B (On)) (e. C (On))) (-> (e. A B) (e. (opr A (^o) (suc C)) (opr B (^o) (suc C))))) (B A onelon exp C On e. adantr B C A oewordri A On e. 3adant1 A ^o C opr B ^o C opr A omwordri A C oecl B On e. 3adant2 B C oecl A On e. 3adant1 A On e. B On e. C On e. 3simp1 syl3anc syld A C oesuc B On e. 3adant2 B ^o C opr .o A opr sseq1d sylibrd B on0eln0 A B n0i syl5bir C On e. adantr B C oen0 exp syld B B ^o C opr A omordi B On e. C On e. pm3.26 B C oecl jca sylan exp com23 mpdd A On e. 3adant1 B C oesuc A On e. 3adant1 B ^o C opr .o A opr eleq2d sylibrd jcad 3expa A ^o C suc opr B ^o C suc opr B ^o C opr .o A opr ontr2 A C suc oecl B C suc oecl syl2an anandirs C sucelon sylan2b syld exp31 com4l imp mpdd))
+thm (oelim2 ((x y) (A x) (A y) (B x) (B y) (C y)) () (-> (/\ (e. A (On)) (/\ (e. B C) (Lim B))) (= (opr A (^o) B) (U_ x (\ B (1o)) (opr A (^o) (cv x))))) (A {/} ^o B opreq1 A {/} ^o x cv opreq1 x cv B 1o \ e. a1d iuneq2dv eqeq12d B oe0m1OLD B C limelon B 0ellim B C e. adantl sylanc B x cv ordelon B limord sylan x cv on0eln0 x cv el1o negbii syl6bbr x cv oe0m1OLD exp sylbird syl exp imp3a x cv B 1o eldif syl5ib iuneq2dv B 0ellim B {/} limsuc mpbid df-1o B eleq1i sylibr 1ord oneirr jctir 1o B 1o eldif sylibr 1o B 1o \ n0i B 1o \ x {/} iunconst 3syl eqtrd B C e. adantl eqtr4d syl5bir impcom A B C y oelim B y cv limsuc biimpa y cv nsuceq0 y cv suc el1o mtbir jctir y cv suc B 1o eldif sylibr exp A On e. {/} A e. ad2antlr y cv sssucid y cv y cv suc A oewordi y cv On e. y cv suc On e. A On e. /\/\ id 3expa ancoms B y cv ordelon B limord sylan y cv suceloni ancli syl sylan2 anassrs sylan an1rs mpi exp jcad x cv y cv suc A ^o opreq2 A ^o y cv opr sseq2d B 1o \ rcla4ev syl6 r19.21aiv y B x B 1o \ A ^o y cv opr A ^o x cv opr iunss2 syl B 1o difss B 1o \ B x A ^o x cv opr iunss1 ax-mp x cv y cv A ^o opreq2 B cbviunv sseqtr A On e. B Lim /\ {/} A e. /\ a1i eqssd B C e. adantlrl eqtrd oe0lem))
+thm (nna0 () () (-> (e. A (om)) (= (opr A (+o) ({/})) A)) (A nnont A oa0 syl))
+thm (nnm0 () () (-> (e. A (om)) (= (opr A (.o) ({/})) ({/}))) (A nnont A om0 syl))
+thm (nnasuc () () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (+o) (suc B)) (suc (opr A (+o) B)))) (A B oasuc A nnont B nnont syl2an))
+thm (nnmsuc () () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (.o) (suc B)) (opr (opr A (.o) B) (+o) A))) (A B omsuc A nnont B nnont syl2an))
+thm (nna0r () () (-> (e. A (om)) (= (opr ({/}) (+o) A) A)) (A nnont A oa0r syl))
+thm (nnm0r () () (-> (e. A (om)) (= (opr ({/}) (.o) A) ({/}))) (A nnont A om0r syl))
+thm (nnacl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (e. (opr A (+o) B) (om))) (x cv {/} A +o opreq2 om eleq1d A om e. imbi2d x cv y cv A +o opreq2 om eleq1d A om e. imbi2d x cv y cv suc A +o opreq2 om eleq1d A om e. imbi2d x cv B A +o opreq2 om eleq1d A om e. imbi2d A nna0 om eleq1d biimprd pm2.43i A y cv nnasuc om eleq1d A +o y cv opr peano2 syl5bir expcom a2d finds impcom))
+thm (nnmcl ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (e. (opr A (.o) B) (om))) (x cv {/} A .o opreq2 om eleq1d A om e. imbi2d x cv y cv A .o opreq2 om eleq1d A om e. imbi2d x cv y cv suc A .o opreq2 om eleq1d A om e. imbi2d x cv B A .o opreq2 om eleq1d A om e. imbi2d peano1 A nnm0 om eleq1d mpbiri A y cv nnmsuc om eleq1d A .o y cv opr A nnacl syl5bir exp4b com24 pm2.43i com3r a2d finds impcom))
+thm (nnarcl () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. (opr A (+o) B) (om)) (/\ (e. A (om)) (e. B (om))))) (A B oaword1 A eloni ordom A om A +o B opr ordtr2 mpan2 syl exp3a B On e. adantr mpd B A oaword2 ancoms B eloni ordom B om A +o B opr ordtr2 mpan2 syl exp3a A On e. adantl mpd jcad A B nnacl A On e. B On e. /\ a1i impbid))
+thm (nnacom ((A x) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (+o) B) (opr B (+o) A))) (x cv {/} +o B opreq1 x cv {/} B +o opreq2 eqeq12d B om e. imbi2d x cv y cv +o B opreq1 x cv y cv B +o opreq2 eqeq12d B om e. imbi2d x cv y cv suc +o B opreq1 x cv y cv suc B +o opreq2 eqeq12d B om e. imbi2d x cv A +o B opreq1 x cv A B +o opreq2 eqeq12d B om e. imbi2d B nna0r B nna0 eqtr4d x cv {/} y cv suc +o opreq2 x cv {/} y cv +o opreq2 y cv +o x cv opr y cv +o {/} opr suceq syl eqeq12d y cv om e. imbi2d x cv z cv y cv suc +o opreq2 x cv z cv y cv +o opreq2 y cv +o x cv opr y cv +o z cv opr suceq syl eqeq12d y cv om e. imbi2d x cv z cv suc y cv suc +o opreq2 x cv z cv suc y cv +o opreq2 y cv +o x cv opr y cv +o z cv suc opr suceq syl eqeq12d y cv om e. imbi2d x cv B y cv suc +o opreq2 x cv B y cv +o opreq2 y cv +o x cv opr y cv +o B opr suceq syl eqeq12d y cv om e. imbi2d y cv peano2b y cv suc nna0 sylbi y cv nna0 y cv +o {/} opr y cv suceq syl eqtr4d y cv suc z cv oasuc y cv nnont y cv suceloni syl z cv nnont syl2an y cv nnont z cv nnont anim12i y cv z cv oasuc y cv +o z cv suc opr y cv +o z cv opr suc suceq 3syl eqeq12d y cv suc +o z cv opr y cv +o z cv opr suc suceq syl5bir expcom a2d finds imp B y cv nnasuc eqeq12d y cv +o B opr B +o y cv opr suceq syl5bir expcom a2d finds imp))
+thm (nnaordi () () (-> (/\ (e. B (om)) (e. C (om))) (-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (B C A oaordi B nnont C nnont syl2an))
+thm (nnaord () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B)))) (A B C oaord A nnont B nnont C nnont syl3an))
+thm (nnaordr () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (e. A B) (e. (opr A (+o) C) (opr B (+o) C)))) (A B C nnaord C A nnacom ancoms B om e. 3adant2 C B nnacom ancoms A om e. 3adant1 eleq12d bitrd))
+thm (nnaass () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr (opr A (+o) B) (+o) C) (opr A (+o) (opr B (+o) C)))) (A B C oaass A nnont B nnont C nnont syl3an))
+thm (nndi () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr A (.o) (opr B (+o) C)) (opr (opr A (.o) B) (+o) (opr A (.o) C)))) (A B C odi A nnont B nnont C nnont syl3an))
+thm (nnmass () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (= (opr (opr A (.o) B) (.o) C) (opr A (.o) (opr B (.o) C)))) (A B C omass A nnont B nnont C nnont syl3an))
+thm (nnmsucr ((x y) (A x) (A y) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr (suc A) (.o) B) (opr (opr A (.o) B) (+o) B))) (x cv {/} A suc .o opreq2 x cv {/} A .o opreq2 x cv {/} = id +o opreq12d eqeq12d A om e. imbi2d x cv y cv A suc .o opreq2 x cv y cv A .o opreq2 x cv y cv = id +o opreq12d eqeq12d A om e. imbi2d x cv y cv suc A suc .o opreq2 x cv y cv suc A .o opreq2 x cv y cv suc = id +o opreq12d eqeq12d A om e. imbi2d x cv B A suc .o opreq2 x cv B A .o opreq2 x cv B = id +o opreq12d eqeq12d A om e. imbi2d A peano2b A suc nnm0 sylbi A nnm0 +o {/} opreq1d peano1 {/} nna0 ax-mp syl6eq eqtr4d A suc y cv nnmsuc A peano2b sylanb A y cv nnmsuc +o y cv suc opreq1d A y cv nnacom A +o y cv opr y cv +o A opr suceq syl A y cv nnasuc y cv A nnasuc ancoms 3eqtr4d A .o y cv opr +o opreq2d A .o y cv opr A y cv suc nnaass y cv peano2b syl3an3b A y cv nnmcl syl3an1 3expb anidms A .o y cv opr y cv A suc nnaass A peano2b syl3an3b A y cv nnmcl syl3an1 3expb an42s anidms 3eqtr4d eqtrd eqeq12d A suc .o y cv opr A .o y cv opr +o y cv opr +o A suc opreq1 syl5bir expcom a2d finds impcom))
+thm (nnmcom ((A x) (x y) (B x) (B y)) () (-> (/\ (e. A (om)) (e. B (om))) (= (opr A (.o) B) (opr B (.o) A))) (x cv {/} .o B opreq1 x cv {/} B .o opreq2 eqeq12d B om e. imbi2d x cv y cv .o B opreq1 x cv y cv B .o opreq2 eqeq12d B om e. imbi2d x cv y cv suc .o B opreq1 x cv y cv suc B .o opreq2 eqeq12d B om e. imbi2d x cv A .o B opreq1 x cv A B .o opreq2 eqeq12d B om e. imbi2d B nnm0r B nnm0 eqtr4d y cv B nnmsucr B y cv nnmsuc ancoms eqeq12d y cv .o B opr B .o y cv opr +o B opreq1 syl5bir exp a2d finds imp))
+thm (nnacan () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (= (opr A (+o) B) (opr A (+o) C)) (= B C))) (A B C oacan A nnont B nnont C nnont syl3an))
+thm (nnaword () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (C_ A B) (C_ (opr C (+o) A) (opr C (+o) B)))) (A B C oaword A nnont B nnont C nnont syl3an))
+thm (nnaword1 () () (-> (/\ (e. A (om)) (e. B (om))) (C_ A (opr A (+o) B))) (A B oaword1 A nnont B nnont syl2an))
+thm (nnaword2 () () (-> (/\ (e. A (om)) (e. B (om))) (C_ A (opr B (+o) A))) (A B nnaword1 A B nnacom sseqtrd))
+thm (nnmordi () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (B C A omordi exp B nnont C nnont syl2an com23 imp3a A om e. 3adant1))
+thm (nnmord () () (-> (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (<-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B)))) (A B C omord A nnont B nnont C nnont syl3an))
+thm (nnmcan () () (-> (/\ (/\/\ (e. A (om)) (e. B (om)) (e. C (om))) (e. ({/}) A)) (<-> (= (opr A (.o) B) (opr A (.o) C)) (= B C))) (A B C omcan A nnont B nnont C nnont 3anim123i sylan))
+thm (nnaordex ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (E.e. x (om) (/\ (e. ({/}) (cv x)) (= (opr A (+o) (cv x)) B))))) (A B x oaordex B nnont sylan2 A +o x cv opr B om eleq1 bicomd A x cv nnarcl sylan9bbr A om e. x cv om e. pm3.27 syl6bi exp31 com23 {/} x cv e. adantld com24 imp4b x cv On e. {/} x cv e. A +o x cv opr B = /\ pm3.27 A On e. B om e. /\ a1i jcad x cv nnont {/} x cv e. A +o x cv opr B = /\ anim1i A On e. B om e. /\ a1i impbid birexdv2 bitrd A nnont sylan))
+thm (nnawordex ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (C_ A B) (E.e. x (om) (= (opr A (+o) (cv x)) B)))) (A B x oawordex B nnont sylan2 A +o x cv opr B om eleq1 bicomd A x cv nnarcl sylan9bbr A om e. x cv om e. pm3.27 syl6bi exp31 com23 com24 imp4b x cv On e. A +o x cv opr B = pm3.27 A On e. B om e. /\ a1i jcad x cv nnont A +o x cv opr B = anim1i A On e. B om e. /\ a1i impbid birexdv2 bitrd A nnont sylan))
+thm (oaabslem ((A x)) () (-> (/\ (e. (om) (On)) (e. A (om))) (= (opr A (+o) (om)) (om))) (A om On x oalim A nnont limom om On e. jctr syl2an A x cv nnacl ordom om A +o x cv opr ordelss mpan syl exp r19.21aiv x om A +o x cv opr om iunss sylibr om On e. adantr eqsstrd ancoms om A oaword2 A nnont sylan2 eqssd))
+thm (oaabs ((x y) (A x) (A y) (B x)) () (-> (/\ (/\ (e. A (om)) (e. B (On))) (C_ (om) B)) (= (opr A (+o) B) B)) (B On om ssexg ordom om V elong mpbiri syl6 imp x cv om A +o opreq2 x cv om = id eqeq12d A om e. imbi2d x cv y cv A +o opreq2 x cv y cv = id eqeq12d A om e. imbi2d x cv y cv suc A +o opreq2 x cv y cv suc = id eqeq12d A om e. imbi2d x cv B A +o opreq2 x cv B = id eqeq12d A om e. imbi2d A oaabslem exp A y cv oasuc A nnont sylan A +o y cv opr y cv suceq sylan9eq exp31 com12 om On e. om y cv C_ ad2antll a2d A om On y oalim A nnont limom om On e. jctr syl2an A oaabslem ancoms limom om limuni ax-mp syl6eq eqtr3d x cv Lim om x cv C_ /\ y x cv om y cv C_ A +o y cv opr y cv = -> A.e. /\ adantl x cv y cv ordelon x cv limord sylan y cv eloni ordom om y cv ordtri1 mpan 3syl exp pm5.32d y cv x cv om eldif syl6bbr A +o y cv opr y cv = imbi1d y cv x cv e. om y cv C_ A +o y cv opr y cv = impexp syl5bbr biraldv2 y x cv om \ A +o y cv opr y cv iuneq2 x cv om \ y uniiun syl6eqr syl6bi imp om x cv C_ adantlr A om e. om On e. /\ adantr uneq12d A x cv V y oalim A nnont x visset x cv Lim jctl syl2an ancoms om x cv ssequn1 biimp om x cv undif2 syl5eq om x cv om \ u. x cv y A +o y cv opr iuneq1 syl y om x cv om \ A +o y cv opr iunxun syl5reqr sylan9eq an1rs om On e. adantrr y x cv om y cv C_ A +o y cv opr y cv = -> A.e. adantlr x cv limuni om x cv ssequn1 biimp om x cv undif2 syl5eq unieqd om x cv om \ uniun syl5reqr sylan9eq y x cv om y cv C_ A +o y cv opr y cv = -> A.e. A om e. om On e. /\ ad2antll 3eqtr4d exp43 com23 a2d om y cv C_ A om e. A +o y cv opr y cv = bi2.04 y x cv biral y x cv A om e. om y cv C_ A +o y cv opr y cv = -> r19.21v bitr syl5ib com4r impcom an1rs tfindsg an1rs mpdan exp com3r imp31))
+thm (1onn () () (e. (1o) (om)) (df-1o peano1 {/} peano2 ax-mp eqeltr))
+thm (2onn () () (e. (2o) (om)) (df-2o 1onn 1o peano2 ax-mp eqeltr))
+thm (omsmolem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F y) (F z) (F w)) () (-> (e. (cv y) (om)) (-> (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) (-> (e. (cv z) (cv y)) (e. (` F (cv z)) (` F (cv y)))))) (y cv {/} z cv eleq2 y cv {/} F fveq2 F z cv ` eleq2d imbi12d y cv w cv z cv eleq2 y cv w cv F fveq2 F z cv ` eleq2d imbi12d y cv w cv suc z cv eleq2 y cv w cv suc F fveq2 F z cv ` eleq2d imbi12d z cv noel F z cv ` F {/} ` e. pm2.21i A On C_ F om A :--> /\ x om F x cv ` F x cv suc ` e. A.e. /\ a1i x cv w cv F fveq2 x cv w cv suceq F fveq2d eleq12d om rcla4v impcom A On C_ F om A :--> /\ adantll A On F w cv suc ` ssel F om A w cv suc ffvrn w cv peano2b sylan2b syl5 F w cv suc ` F z cv ` F w cv ` ontr1 exp3a com23 syl6 exp3a imp31 x om F x cv ` F x cv suc ` e. A.e. adantlr mpd z cv w cv e. imim2d imp z cv w cv F fveq2 F w cv suc ` eleq1d x cv w cv F fveq2 x cv w cv suceq F fveq2d eleq12d om rcla4v impcom syl5bir com12 A On C_ F om A :--> /\ adantll z cv w cv e. F z cv ` F w cv ` e. -> adantr jaod z visset w cv elsuc syl5ib exp31 com12 finds2))
+thm (omsmo ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) () (-> (/\ (/\ (C_ A (On)) (:--> F (om) A)) (A.e. x (om) (e. (` F (cv x)) (` F (suc (cv x)))))) (:-1-1-> F (om) A)) (A On C_ F om A :--> pm3.27 x om F x cv ` F x cv suc ` e. A.e. adantr z A F x y omsmolem y cv om e. adantl imp y A F x z omsmolem z cv om e. adantr imp orim12d ancoms con3d A On F y cv ` ssel F om A y cv ffvrn syl5 exp3a imp F y cv ` eloni syl6 A On F z cv ` ssel F om A z cv ffvrn syl5 exp3a imp F z cv ` eloni syl6 anim12d imp F y cv ` F z cv ` ordtri3 syl x om F x cv ` F x cv suc ` e. A.e. adantlr y cv z cv ordtri3 y cv nnord z cv nnord syl2an A On C_ F om A :--> /\ x om F x cv ` F x cv suc ` e. A.e. /\ adantl 3imtr4d exp32 r19.21adv r19.21aiv jca F om A y z f1fv sylibr))
+thm (er2 ((x y) (x z) (R x) (y z) (R y) (R z)) () (<-> (Er R) (A. x (A. y (A. z (/\ (-> (br (cv x) R (cv y)) (br (cv y) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))))))) (R df-er R x y cnvsym R x y z cotr anbi12i R `' R R R o. unss z x cv R y cv br y cv R x cv br -> x cv R y cv br y cv R z cv br /\ x cv R z cv br -> 19.28v y bial y x cv R y cv br y cv R x cv br -> z x cv R y cv br y cv R z cv br /\ x cv R z cv br -> A. 19.26 bitr x bial x y x cv R y cv br y cv R x cv br -> A. y z x cv R y cv br y cv R z cv br /\ x cv R z cv br -> A. A. 19.26 bitr2 3bitr3 bitr))
+thm (ec2 ((A y) (x y) (A x) (R y) (R x)) ((ec2.1 (e. A (V)))) (= ([] A R) ({|} y (br A R (cv y)))) (R A {} y x dfima3 A R df-ec ec2.1 x cv A y cv opeq1 R eleq1d ceqsexv x A elsn x cv y cv <,> R e. anbi1i x biex A R y cv df-br 3bitr4r y biabi 3eqtr4))
+thm (ecexg () () (-> (e. R B) (e. ([] A R) (V))) (R B A {} imaexg A R df-ec V eleq1i sylibr))
+thm (ereq () () (-> (= R S) (<-> (Er R) (Er S))) (R S cnveq R S R coeq1 R S S coeq2 eqtrd uneq12d R sseq1d R S S `' S S o. u. sseq2 bitrd R df-er S df-er 3bitr4g))
+thm (ster ((x y) (x z) (R x) (y z) (R y) (R z)) ((ster.1 (-> (br (cv x) R (cv y)) (br (cv y) R (cv x)))) (ster.2 (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z))))) (Er R) (R x y z er2 ster.1 ster.2 pm3.2i y z gen2 mpgbir))
+thm (ider ((x y) (x z) (y z)) () (Er (I)) (x y equcomi x visset y visset ideq y visset x visset ideq 3imtr4 x cv y cv z cv eqtrt x visset y visset ideq y visset z visset ideq anbi12i x visset z visset ideq 3imtr4 ster))
+thm (ersym ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) ((ersym.1 (e. A (V))) (ersym.2 (e. B (V))) (ersym.3 (Er R))) (-> (br A R B) (br B R A)) (ersym.1 ersym.2 x cv A y cv B R breq12 y cv B x cv A R breq12 ancoms imbi12d ersym.3 R x y z er2 mpbi a4i a4i a4i pm3.26i vtocl2))
+thm (ersymb () ((ersymb.1 (e. A (V))) (ersymb.2 (e. B (V))) (ersymb.3 (Er R))) (<-> (br A R B) (br B R A)) (ersymb.1 ersymb.2 ersymb.3 ersym ersymb.2 ersymb.1 ersymb.3 ersym impbi))
+thm (ertr ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((ertr.1 (e. A (V))) (ertr.2 (e. B (V))) (ertr.3 (e. C (V))) (ertr.4 (Er R))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (ertr.1 ertr.2 ertr.3 x cv A R y cv breq1 y cv R z cv br anbi1d x cv A R z cv breq1 imbi12d y cv B A R breq2 y cv B R z cv breq1 anbi12d A R z cv br imbi1d z cv C B R breq2 A R B br anbi2d z cv C A R breq2 imbi12d syl3an9b ertr.4 R x y z er2 mpbi a4i a4i a4i pm3.27i vtocl3))
+thm (erref ((x y) (A x) (A y) (R x) (R y)) ((erref.1 (Er R))) (-> (e. A (u. (dom R) (ran R))) (br A R A)) (x cv A R x cv breq1 x cv A A R breq2 bitrd x cv R dom R ran elun x visset R y eldm x visset y visset x visset erref.1 ertr x visset y visset erref.1 ersymb sylan2b anidms y 19.23aiv sylbi x visset R y elrn2 x visset y visset x visset erref.1 ertr y visset x visset erref.1 ersymb sylanb anidms y 19.23aiv sylbi jaoi sylbi vtoclga))
+thm (erdmrn ((x y) (R x) (R y)) ((erdmrn.1 (Er R))) (= (dom R) (ran R)) (x visset y visset erdmrn.1 ersymb y biex x visset R y eldm x visset R y elrn2 3bitr4 eqri))
+thm (eceq1 () () (-> (= A B) (= ([] C A) ([] C B))) (A B C {} imaeq1 C A df-ec C B df-ec 3eqtr4g))
+thm (eceq2 () () (-> (= A B) (= ([] A C) ([] B C))) (A B sneq A {} B {} C imaeq2 syl A C df-ec B C df-ec 3eqtr4g))
+thm (elec ((A x) (B x) (R x)) ((elec.1 (e. A (V))) (elec.2 (e. B (V)))) (<-> (e. A ([] B R)) (br B R A)) (elec.1 x cv A B R breq2 elec.2 R x ec2 elab2))
+thm (ecdmn0 ((R x) (A x)) ((ecdmn0.1 (e. A (V)))) (<-> (e. A (dom R)) (-. (= ([] A R) ({/})))) (x visset ecdmn0.1 R elec x biex A R [] x n0 ecdmn0.1 R x eldm 3bitr4r))
+thm (erthi ((A x) (B x) (R x)) ((erthi.1 (e. A (V))) (erthi.2 (e. B (V))) (erthi.3 (Er R))) (-> (br A R B) (= ([] A R) ([] B R))) (erthi.1 erthi.2 erthi.3 ersymb erthi.2 erthi.1 x visset erthi.3 ertr exp sylbi erthi.1 erthi.2 x visset erthi.3 ertr exp impbid x visset erthi.1 R elec x visset erthi.2 R elec 3bitr4g eqrd))
+thm (erth ((A x) (B x) (R x)) ((erth.1 (e. B (V))) (erth.2 (Er R))) (-> (e. A (u. (dom R) (ran R))) (<-> (= ([] A R) ([] B R)) (br A R B))) (x cv A R eceq2 B R [] eqeq1d x cv A R B breq1 bibi12d x cv R [] B R [] x cv eleq2 x visset x visset R elec x visset erth.1 R elec 3bitr3g erth.1 x visset erth.2 ersym syl6bi erth.2 x cv erref syl5com x visset erth.1 erth.2 erthi x cv R dom R ran u. e. a1i impbid vtoclga))
+thm (erthdm () ((erthdm.1 (e. B (V))) (erthdm.2 (Er R))) (-> (e. A (dom R)) (<-> (= ([] A R) ([] B R)) (br A R B))) (A R dom R ran elun1 erthdm.1 erthdm.2 A erth syl))
+thm (erthdmr () ((erthdmr.1 (e. A (V))) (erthdmr.2 (e. B (V))) (erthdmr.3 (Er R))) (-> (e. B (dom R)) (<-> (= ([] A R) ([] B R)) (br A R B))) (erthdmr.1 erthdmr.3 B erthdm A R [] B R [] eqcom erthdmr.1 erthdmr.2 erthdmr.3 ersymb 3bitr4g))
+thm (ereldm () ((ereldm.1 (e. A (V))) (ereldm.2 (e. B (V))) (ereldm.3 (Er R)) (ereldm.4 (= (dom R) D))) (-> (= ([] A R) ([] B R)) (<-> (e. A D) (e. B D))) (ereldm.2 ereldm.3 A erthdm biimpcd ereldm.1 ereldm.2 ereldm.3 ersymb ereldm.2 R A breldm sylbi syl6 ereldm.1 ereldm.2 ereldm.3 erthdmr biimpcd ereldm.1 R B breldm syl6 impbid ereldm.4 A eleq2i ereldm.4 B eleq2i 3bitr3g))
+thm (erdisj ((A x) (B x) (R x)) ((erdisj.1 (e. A (V))) (erdisj.2 (e. B (V))) (erdisj.3 (Er R))) (\/ (= ([] A R) ([] B R)) (= (i^i ([] A R) ([] B R)) ({/}))) (x visset erdisj.1 R elec erdisj.1 x visset erdisj.2 erdisj.3 ertr exp erdisj.1 erdisj.2 erdisj.3 erthi syl6 x visset erdisj.2 R elec erdisj.2 x visset erdisj.3 ersymb bitr syl5ib sylbi con3d com12 x 19.21aiv A R [] B R [] x disj1 sylibr orri))
+thm (ecidsn () () (= ([] A (I)) ({} A)) (A I df-ec A {} imai eqtr))
+thm (qseq1 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (/. A C) (/. B C))) (A B x y cv x cv C [] = rexeq y biabdv A C y x df-qs B C y x df-qs 3eqtr4g))
+thm (qseq2 ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (= A B) (= (/. C A) (/. C B))) (A B x cv eceq1 y cv eqeq2d x C birexdv y biabdv C A y x df-qs C B y x df-qs 3eqtr4g))
+thm (elqs ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) ((elqs.1 (e. B (V)))) (<-> (e. B (/. A R)) (E. x (/\ (e. (cv x) A) (= B ([] (cv x) R))))) (elqs.1 y cv B x cv R [] eqeq1 x A birexdv A R y x df-qs elab2 x A B x cv R [] = df-rex bitr))
+thm (elqsi ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) () (-> (e. B (/. A R)) (E. x (/\ (e. (cv x) A) (= B ([] (cv x) R))))) (y cv B x cv R [] eqeq1 x cv A e. anbi2d x biexdv y visset A R x elqs biimp vtoclga))
+thm (ecelqsi ((x y) (A x) (A y) (B x) (B y) (R x) (R y)) ((ecelqsi.1 (e. R (V)))) (-> (e. B A) (e. ([] B R) (/. A R))) (y cv B R eceq2 A R /. eleq1d x y a9e y cv R [] eqid x cv y cv A eleq1 x cv y cv R eceq2 y cv R [] eqeq2d anbi12d biimprcd mpan2 x 19.22dv mpi ecelqsi.1 R V y cv ecexg ax-mp A R x elqs sylibr vtoclga))
+thm (ecopqsi () ((ecopqsi.1 (e. R (V))) (ecopqsi.2 (= S (/. (X. A A) R)))) (-> (/\ (e. B A) (e. C A)) (e. ([] (<,> B C) R) S)) (B A C A opelxpi ecopqsi.1 B C <,> A A X. ecelqsi ecopqsi.2 syl6eleqr syl))
+thm (qsex ((x y) (A x) (A y) (R x) (R y)) ((qsex.1 (e. A (V)))) (e. (/. A R) (V)) (x A y cv x cv R [] = df-rex y biabi A R y x df-qs x y x cv A e. y cv x cv R [] = /\ rnopab 3eqtr4 qsex.1 x y x cv R [] funopabex2 x y x cv A e. y cv x cv R [] = /\ {<,>|} V rnexg ax-mp eqeltr))
+thm (snec ((x y) (A x) (A y) (R x) (R y)) ((snec.1 (e. A (V)))) (= ({} ([] A R)) (/. ({} A) R)) (x A {} y cv x cv R [] = df-rex x A elsn y cv x cv R [] = anbi1i x biex snec.1 x cv A R eceq2 y cv eqeq2d ceqsexv 3bitr bicomi y biabi A R [] y df-sn A {} R y x df-qs 3eqtr4))
+thm (ecqs () ((ecqs.1 (e. A (V))) (ecqs.2 (e. R (V)))) (= ([] A R) (U. (/. ({} A) R))) (ecqs.2 R V A ecexg ax-mp unisn ecqs.1 R snec unieqi eqtr3))
+thm (0nelqs ((R x) (A x)) ((0nelqs.1 (Er R)) (0nelqs.2 (= (dom R) A))) (-. (e. ({/}) (/. A R))) (x visset R ecdmn0 0nelqs.2 x cv eleq2i x cv R [] {/} eqcom negbii 3bitr3 biimp x cv A e. {/} x cv R [] = imnan mpbi x nex {/} A R x elqsi mto))
+thm (ecelqsdm () ((ecelqsdm.1 (e. B (V))) (ecelqsdm.2 (Er R)) (ecelqsdm.3 (= (dom R) A))) (-> (e. ([] B R) (/. A R)) (e. B A)) (ecelqsdm.2 ecelqsdm.3 0nelqs ecelqsdm.1 R ecdmn0 ecelqsdm.3 B eleq2i bitr3 bicon1i B R [] {/} A R /. eleq1 sylbi mtbiri a3i))
+thm (ecid ((A y)) ((ecid.1 (e. A (V)))) (= ([] A (`' (E))) A) (ecid.1 E `' y ec2 ecid.1 y visset E brcnv y visset ecid.1 epelc bitr y biabi y A abid2 3eqtr))
+thm (qsid ((x y) (A x) (A y)) ((qsid.1 (e. A (V)))) (= (/. A (`' (E))) A) (A E `' y x df-qs x visset ecid y cv eqeq2i y cv x cv eqcom bitr x A birex y cv A x risset bitr4 y biabi y A abid2 3eqtr))
+thm (ectocl ((A x) (B x) (R x) (ps x)) ((ectocl.1 (= S (/. B R))) (ectocl.2 (-> (= ([] (cv x) R) A) (<-> ph ps))) (ectocl.3 (-> (e. (cv x) B) ph))) (-> (e. A S) ps) (ectocl.1 A eleq2i A B R x elqsi ectocl.2 eqcoms ectocl.3 syl5bi impcom x 19.23aiv syl sylbi))
+thm (ecoptocl ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (ps x) (ps y) (ps z)) ((ecoptocl.1 (= S (/. (X. B C) R))) (ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (ecoptocl.3 (-> (/\ (e. (cv x) B) (e. (cv y) C)) ph))) (-> (e. A S) ps) (ecoptocl.1 A eleq2i A B C X. R z elqsi B C X. eqid x cv y cv <,> z cv R eceq2 A eqeq2d ps imbi1d ecoptocl.2 eqcoms ecoptocl.3 syl5bi com12 optocl imp z 19.23aiv syl sylbi))
+thm (2ecoptocl ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (S z) (S w) (R x) (R y) (R z) (R w) (ps x) (ps y) (ch z) (ch w)) ((2ecoptocl.1 (= S (/. (X. C D) R))) (2ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (2ecoptocl.3 (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch))) (2ecoptocl.4 (-> (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (/\ (e. (cv z) C) (e. (cv w) D))) ph))) (-> (/\ (e. A S) (e. B S)) ch) (2ecoptocl.1 2ecoptocl.3 A S e. imbi2d 2ecoptocl.1 2ecoptocl.2 z cv C e. w cv D e. /\ imbi2d 2ecoptocl.4 exp ecoptocl com12 ecoptocl impcom))
+thm (3ecoptocl ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B z) (B w) (B v) (B u) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S z) (S w) (S v) (S u) (R x) (R y) (R z) (R w) (R v) (R u) (ps x) (ps y) (ch z) (ch w) (th v) (th u)) ((3ecoptocl.1 (= S (/. (X. D D) R))) (3ecoptocl.2 (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps))) (3ecoptocl.3 (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch))) (3ecoptocl.4 (-> (= ([] (<,> (cv v) (cv u)) R) C) (<-> ch th))) (3ecoptocl.5 (-> (/\/\ (/\ (e. (cv x) D) (e. (cv y) D)) (/\ (e. (cv z) D) (e. (cv w) D)) (/\ (e. (cv v) D) (e. (cv u) D))) ph))) (-> (/\/\ (e. A S) (e. B S) (e. C S)) th) (3ecoptocl.1 3ecoptocl.3 A S e. imbi2d 3ecoptocl.4 A S e. imbi2d 3ecoptocl.1 3ecoptocl.2 z cv D e. w cv D e. /\ v cv D e. u cv D e. /\ /\ imbi2d 3ecoptocl.5 3exp imp3a ecoptocl com12 2ecoptocl com12 exp3a 3imp))
+thm (brecop ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S x) (S y) (S z) (S w) (S v) (S u) (H x) (H y) (G z) (G w) (G v) (G u) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u)) ((brecop.1 (e. S (V))) (brecop.2 (Er S)) (brecop.3 (= (dom S) (X. G G))) (brecop.4 (= H (/. (X. G G) S))) (brecop.5 (= R ({<,>|} x y (/\ (/\ (e. (cv x) H) (e. (cv y) H)) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) ([] (<,> (cv z) (cv w)) S)) (= (cv y) ([] (<,> (cv v) (cv u)) S))) ph))))))))) (brecop.6 (-> (/\ (/\ (/\ (e. (cv z) G) (e. (cv w) G)) (/\ (e. A G) (e. B G))) (/\ (/\ (e. (cv v) G) (e. (cv u) G)) (/\ (e. C G) (e. D G)))) (-> (/\ (= ([] (<,> (cv z) (cv w)) S) ([] (<,> A B) S)) (= ([] (<,> (cv v) (cv u)) S) ([] (<,> C D) S))) (<-> ph ps))))) (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) ps)) (x cv A B <,> S [] H eleq1 y cv H e. anbi1d x cv A B <,> S [] z cv w cv <,> S [] eqeq1 y cv v cv u cv <,> S [] = anbi1d ph anbi1d z w v u bi4exdv anbi12d y cv C D <,> S [] H eleq1 A B <,> S [] H e. anbi2d y cv C D <,> S [] v cv u cv <,> S [] eqeq1 A B <,> S [] z cv w cv <,> S [] = anbi2d ph anbi1d z w v u bi4exdv anbi12d H H opelopabg A B <,> S [] H e. C D <,> S [] H e. /\ z w v u A B <,> S [] z cv w cv <,> S [] = C D <,> S [] v cv u cv <,> S [] = /\ ph /\ E. E. E. E. ibar bitr4d A B <,> S [] R C D <,> S [] df-br brecop.5 A B <,> S [] C D <,> S [] <,> eleq2i bitr syl5bb brecop.1 brecop.4 A B ecopqsi brecop.1 brecop.4 C D ecopqsi syl2an z cv A w cv B opeq12 z cv w cv <,> A B <,> S eceq2 syl v cv C u cv D opeq12 v cv u cv <,> C D <,> S eceq2 syl anim12i z cv w cv opex A B opex brecop.2 brecop.3 ereldm w visset z cv G G opelxp syl5bbr A G B G opelxpi syl5bir v cv u cv opex C D opex brecop.2 brecop.3 ereldm u visset v cv G G opelxp syl5bbr C G D G opelxpi syl5bir im2anan9 brecop.6 an4s exp com13 mpdd pm5.74d G G cgsex4g A B <,> S [] z cv w cv <,> S [] eqcom C D <,> S [] v cv u cv <,> S [] eqcom anbi12i A G e. B G e. /\ C G e. D G e. /\ /\ a1i A G e. B G e. /\ C G e. D G e. /\ /\ ph biimt anbi12d z w v u bi4exdv A G e. B G e. /\ C G e. D G e. /\ /\ ps biimt 3bitr4d bitrd))
+var (class Q)
+thm (brecop2 () ((brecop2.1 (e. S (V))) (brecop2.2 (e. B (V))) (brecop2.3 (e. C (V))) (brecop2.4 (e. D (V))) (brecop2.5 (Er S)) (brecop2.6 (= (dom S) (X. G G))) (brecop2.7 (= H (/. (X. G G) S))) (brecop2.8 (C_ R (X. H H))) (brecop2.9 (C_ Q (X. G G))) (brecop2.10 (-. (e. ({/}) G))) (brecop2.11 (= (dom F) (X. G G))) (brecop2.12 (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C)))))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C))) (brecop2.1 S V C D <,> ecexg ax-mp brecop2.8 A B <,> S [] brel brecop2.7 A B <,> S [] eleq2i A B opex brecop2.5 brecop2.6 ecelqsdm sylbi brecop2.2 A G G opelxp sylib brecop2.7 C D <,> S [] eleq2i C D opex brecop2.5 brecop2.6 ecelqsdm sylbi brecop2.4 C G G opelxp sylib anim12i syl B F C oprex brecop2.9 A F D opr brel brecop2.4 brecop2.11 brecop2.10 A ndmoprrcl brecop2.3 brecop2.11 brecop2.10 B ndmoprrcl anim12i syl A G e. D G e. B G e. C G e. an42 sylib brecop2.12 pm5.21nii))
+var (set s)
+var (set r)
+thm (ecopopreq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (S x) (S y) (S z) (S w) (S v) (S u)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C)))) (z cv A u cv D F opreq12 w cv B v cv C F opreq12 eqeqan12d an42s ecopopr.1 opbrop))
+thm (ecopoprdm ((f g) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (F y) (w z) (v z) (u z) (f z) (g z) (F z) (v w) (u w) (f w) (g w) (F w) (u v) (f v) (g v) (F v) (f u) (g u) (F u) (F f) (F g) (R f) (R g) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))))) (= (dom R) (X. S S)) (ecopopr.1 x y S S X. S S X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv F u cv opr w cv F v cv opr = /\ E. E. E. E. opabssxp eqsstr R S S X. S S X. X. dmss ax-mp S S X. dmxpid sseqtr S S relxp g visset f cv S S opelxp f visset g visset ecopopr.com caoprcom ecopopr.1 f cv g cv f cv g cv ecopopreq anidms mpbiri f cv g cv <,> R f cv g cv <,> df-br sylib f cv g cv opex f cv g cv <,> R opeldm syl sylbi relssi eqssi))
+thm (ecopoprsym ((f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B f) (B g) (B h) (B t) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (F w) (u v) (f v) (g v) (h v) (t v) (F v) (f u) (g u) (h u) (t u) (F u) (F f) (F g) (F h) (F t) (R f) (R g) (R h) (R t) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.2 (e. B (V)))) (-> (br A R B) (br B R A)) (ecopopr.2 ecopopr.1 x y S S X. S S X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv F u cv opr w cv F v cv opr = /\ E. E. E. E. opabssxp eqsstr A brel S S X. eqid f cv g cv <,> A R h cv t cv <,> breq1 f cv g cv <,> A h cv t cv <,> R breq2 bibi12d h cv t cv <,> B A R breq2 h cv t cv <,> B R A breq1 bibi12d ecopopr.1 f cv g cv h cv t cv ecopopreq f visset t visset ecopopr.com caoprcom g visset h visset ecopopr.com caoprcom eqeq12i t cv F f cv opr h cv F g cv opr eqcom bitr syl6bb ecopopr.1 h cv t cv f cv g cv ecopopreq ancoms bitr4d 2optocl syl ibi))
+thm (ecopoprtrn ((f g) (f h) (f t) (f s) (f r) (A f) (g h) (g t) (g s) (g r) (A g) (h t) (h s) (h r) (A h) (s t) (r t) (A t) (r s) (A s) (A r) (B f) (B g) (B h) (B t) (B s) (B r) (C f) (C g) (C h) (C t) (C s) (C r) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (F w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (F v) (f u) (g u) (h u) (t u) (s u) (r u) (F u) (F f) (F g) (F h) (F t) (F s) (F r) (R f) (R g) (R h) (R t) (R s) (R r) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t) (S s) (S r)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.cl (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (ecopopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (ecopopr.can (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (ecopopr.3 (e. B (V))) (ecopopr.4 (e. C (V)))) (-> (/\ (br A R B) (br B R C)) (br A R C)) (ecopopr.3 ecopopr.1 x y S S X. S S X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv F u cv opr w cv F v cv opr = /\ E. E. E. E. opabssxp eqsstr A brel pm3.26d ecopopr.4 ecopopr.1 x y S S X. S S X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv F u cv opr w cv F v cv opr = /\ E. E. E. E. opabssxp eqsstr B brel anim12i A S S X. e. B S S X. e. C S S X. e. 3anass sylibr S S X. eqid f cv g cv <,> A R h cv t cv <,> breq1 h cv t cv <,> R s cv r cv <,> br anbi1d f cv g cv <,> A R s cv r cv <,> breq1 imbi12d h cv t cv <,> B A R breq2 h cv t cv <,> B R s cv r cv <,> breq1 anbi12d A R s cv r cv <,> br imbi1d s cv r cv <,> C B R breq2 A R B br anbi2d s cv r cv <,> C A R breq2 imbi12d ecopopr.1 f cv g cv h cv t cv ecopopreq s cv S e. r cv S e. /\ 3adant3 ecopopr.1 h cv t cv s cv r cv ecopopreq f cv S e. g cv S e. /\ 3adant1 anbi12d f cv F t cv opr g cv F h cv opr h cv F r cv opr t cv F s cv opr F opreq12 h visset t visset f visset ecopopr.com ecopopr.ass r visset caopr411 g visset t visset h visset ecopopr.com ecopopr.ass s visset caopr411 g visset t visset h visset ecopopr.com ecopopr.ass s visset caopr4 eqtr3 3eqtr4g syl6bi g cv F s cv oprex ecopopr.can h cv F t cv opr f cv F r cv opr caoprcan ecopopr.cl h cv t cv caoprcl ecopopr.cl f cv r cv caoprcl syl2an 3impb 3com12 s cv S e. r cv S e. pm3.27 syl3an3 f cv S e. g cv S e. pm3.26 syl3an1 syld ecopopr.1 f cv g cv s cv r cv ecopopreq h cv S e. t cv S e. /\ 3adant2 sylibrd 3optocl mpcom))
+thm (ecopoprer ((f g) (f h) (g h) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (F z) (v w) (u w) (f w) (g w) (h w) (F w) (u v) (f v) (g v) (h v) (F v) (f u) (g u) (h u) (F u) (F f) (F g) (F h) (R f) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h)) ((ecopopr.1 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (ecopopr.com (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (ecopopr.cl (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (ecopopr.ass (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (ecopopr.can (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))))) (Er R) (ecopopr.1 ecopopr.com g visset f cv ecopoprsym ecopopr.1 ecopopr.com ecopopr.cl ecopopr.ass ecopopr.can g visset h visset f cv ecopoprtrn ster))
+thm (eceqopreq ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((eceqopreq.2 (e. B (V))) (eceqopreq.3 (e. C (V))) (eceqopreq.4 (e. D (V))) (eceqopreq.5 (Er R)) (eceqopreq.6 (= (dom R) (X. S S))) (eceqopreq.7 (= (dom F) (X. S S))) (eceqopreq.8 (-. (e. ({/}) S))) (eceqopreq.9 (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (eceqopreq.10 (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C)))))) (-> (/\ (e. A S) (e. C S)) (<-> (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)))) (A S e. B S e. /\ C S e. D S e. /\ pm3.26 A S B S opelxpi eceqopreq.6 syl6eleqr C D opex eceqopreq.5 A B <,> erthdm 3syl eceqopreq.10 bitrd exp43 3imp A B <,> R [] C D <,> R [] {/} eqeq1 biimprcd con3d eceqopreq.6 C D <,> eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 A B <,> eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 syl5ibr com12 C S e. 3adant3 A F D opr B F C opr S eleq1 eceqopreq.9 B C caoprcl syl5bir eceqopreq.4 eceqopreq.7 eceqopreq.8 A ndmoprrcl pm3.27d syl6com con3d A S e. 3adant1 jcad A B <,> R [] C D <,> R [] = A F D opr B F C opr = pm5.21 syl6 pm2.61d 3exp com23 imp A B <,> R [] C D <,> R [] {/} eqeq1 biimpcd con3d eceqopreq.6 A B <,> eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 C D <,> eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 syl5ibr com12 A S e. 3adant1 A F D opr B F C opr S eleq1 eceqopreq.9 A D caoprcl syl5bi eceqopreq.3 eceqopreq.7 eceqopreq.8 B ndmoprrcl pm3.26d syl6com con3d C S e. 3adant2 jcad A B <,> R [] C D <,> R [] = A F D opr B F C opr = pm5.21 syl6 3exp imp C D <,> R [] {/} A B <,> R [] eqeq2 eceqopreq.6 C D <,> eleq2i C D opex R ecdmn0 eceqopreq.4 C S S opelxp 3bitr3 pm3.27bd nsyl4 eceqopreq.6 A B <,> eleq2i A B opex R ecdmn0 eceqopreq.2 A S S opelxp 3bitr3 pm3.27bd con1i syl5bir B S e. C S e. pm3.26 con3i eceqopreq.3 eceqopreq.7 B ndmopr B F C opr {/} A F D opr eqeq2 A S e. D S e. pm3.27 con3i eceqopreq.4 eceqopreq.7 A ndmopr syl syl5bir 3syl com12 jcad A B <,> R [] C D <,> R [] = A F D opr B F C opr = pm5.1 syl6 pm2.61d1 pm2.61d))
+thm (th3qlem1 ((x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (A x) (A y) (A z) (A w) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((th3qlem1.1 (Er R)) (th3qlem1.2 (= (dom R) S)) (th3qlem1.3 (-> (/\ (/\ (e. (cv y) S) (e. (cv w) S)) (/\ (e. (cv z) S) (e. (cv v) S))) (-> (/\ (br (cv y) R (cv w)) (br (cv z) R (cv v))) (br (opr (cv y) F (cv z)) R (opr (cv w) F (cv v))))))) (-> (/\ (e. A (/. S R)) (e. B (/. S R))) (E* x (E. y (E. z (/\ (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R))))))) (A y cv R [] = B z cv R [] = A w cv R [] = B v cv R [] = an4 A S R /. e. A S R /. e. /\ B S R /. e. B S R /. e. /\ /\ anbi2i A S R /. e. A S R /. e. /\ B S R /. e. B S R /. e. /\ A y cv R [] = A w cv R [] = /\ B z cv R [] = B v cv R [] = /\ an4 A S R /. e. A S R /. e. A y cv R [] = A w cv R [] = an4 A y cv R [] S R /. eleq1 pm5.32ri A w cv R [] S R /. eleq1 pm5.32ri anbi12i bitr B S R /. e. B S R /. e. B z cv R [] = B v cv R [] = an4 B z cv R [] S R /. eleq1 pm5.32ri B v cv R [] S R /. eleq1 pm5.32ri anbi12i bitr anbi12i 3bitr y cv S e. w cv S e. pm3.26 th3qlem1.2 syl6eleqr w visset th3qlem1.1 y cv erthdm syl z cv S e. v cv S e. pm3.26 th3qlem1.2 syl6eleqr v visset th3qlem1.1 z cv erthdm syl bi2anan9 th3qlem1.3 sylbid x cv y cv F z cv opr R [] u cv w cv F v cv opr R [] eqeq12 y cv F z cv oprex w cv F v cv oprex th3qlem1.1 erthi syl5bir syl9 com23 imp y visset th3qlem1.1 th3qlem1.2 ecelqsdm w visset th3qlem1.1 th3qlem1.2 ecelqsdm anim12i z visset th3qlem1.1 th3qlem1.2 ecelqsdm v visset th3qlem1.1 th3qlem1.2 ecelqsdm anim12i anim12i A y cv R [] w cv R [] eqtr2t B z cv R [] v cv R [] eqtr2t anim12i syl2an an4s y cv R [] S R /. e. A y cv R [] = w cv R [] S R /. e. A w cv R [] = an4 z cv R [] S R /. e. B z cv R [] = v cv R [] S R /. e. B v cv R [] = an4 syl2anb sylbi A S R /. e. anidm B S R /. e. anidm anbi12i sylanbr exp imp3a A y cv R [] = B z cv R [] = /\ x cv y cv F z cv opr R [] = A w cv R [] = B v cv R [] = /\ u cv w cv F v cv opr R [] = an4 syl5ib w v 19.23advv y z 19.23advv y z w v A y cv R [] = B z cv R [] = /\ x cv y cv F z cv opr R [] = /\ A w cv R [] = B v cv R [] = /\ u cv w cv F v cv opr R [] = /\ ee4anv syl5ibr x u 19.21aivv x cv u cv y cv F z cv opr R [] eqeq1 A y cv R [] = B z cv R [] = /\ anbi2d y z bi2exdv y cv w cv R eceq2 A eqeq2d z cv v cv R eceq2 B eqeq2d bi2anan9 y cv w cv z cv v cv F opreq12 y cv F z cv opr w cv F v cv opr R eceq2 syl u cv eqeq2d anbi12d cbvex2v syl6bb mo4 sylibr))
+thm (th3qlem2 ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))))) (-> (/\ (e. A (/. (X. S S) R)) (e. B (/. (X. S S) R))) (E* z (E. w (E. v (E. u (E. t (/\ (/\ (= A ([] (<,> (cv w) (cv v)) R)) (= B ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))) (th3q.2 th3q.3 S S X. eqid w cv v cv <,> s cv R u cv t cv <,> breq1 x cv R y cv br anbi1d w cv v cv <,> s cv F x cv opreq1 R u cv t cv <,> F y cv opr breq1d imbi12d x cv S S X. e. y cv S S X. e. /\ imbi2d u cv t cv <,> f cv s cv R breq2 x cv R y cv br anbi1d u cv t cv <,> f cv F y cv opreq1 s cv F x cv opr R breq2d imbi12d x cv S S X. e. y cv S S X. e. /\ imbi2d S S X. eqid s cv f cv <,> x cv R g cv h cv <,> breq1 w cv v cv <,> R u cv t cv <,> br anbi2d s cv f cv <,> x cv w cv v cv <,> F opreq2 R u cv t cv <,> F g cv h cv <,> opr breq1d imbi12d w cv S e. v cv S e. /\ u cv S e. t cv S e. /\ /\ imbi2d g cv h cv <,> y cv x cv R breq2 w cv v cv <,> R u cv t cv <,> br anbi2d g cv h cv <,> y cv u cv t cv <,> F opreq2 w cv v cv <,> F x cv opr R breq2d imbi12d w cv S e. v cv S e. /\ u cv S e. t cv S e. /\ /\ imbi2d th3q.4 expcom 2optocl com12 2optocl imp A B z th3qlem1 z w v u t A w cv v cv <,> R [] = B u cv t cv <,> R [] = /\ z cv w cv v cv <,> F u cv t cv <,> opr R [] = /\ E. E. E. E. s x A s cv R [] = B x cv R [] = /\ z cv s cv F x cv opr R [] = /\ E. E. immo w cv v cv opex u cv t cv opex s cv w cv v cv <,> R eceq2 A eqeq2d x cv u cv t cv <,> R eceq2 B eqeq2d bi2anan9 s cv w cv v cv <,> x cv u cv t cv <,> F opreq12 s cv F x cv opr w cv v cv <,> F u cv t cv <,> opr R eceq2 syl z cv eqeq2d anbi12d cla4e2v u t 19.23aivv w v 19.23aivv mpg syl))
+thm (th3qcor ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (th3q.5 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)))))))))))) (Fun G) (th3q.1 th3q.2 th3q.3 th3q.4 x cv y cv z th3qlem2 z x cv S S X. R /. e. y cv S S X. R /. e. /\ w v u t x cv w cv v cv <,> R [] = y cv u cv t cv <,> R [] = /\ z cv w cv v cv <,> F u cv t cv <,> opr R [] = /\ E. E. E. E. moanimv mpbir x y funoprab th3q.5 G x y z x cv S S X. R /. e. y cv S S X. R /. e. /\ w v u t x cv w cv v cv <,> R [] = y cv u cv t cv <,> R [] = /\ z cv w cv v cv <,> F u cv t cv <,> opr R [] = /\ E. E. E. E. /\ {<<,>,>|} funeq ax-mp mpbir))
+thm (th3q ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((th3q.1 (e. R (V))) (th3q.2 (Er R)) (th3q.3 (= (dom R) (X. S S))) (th3q.4 (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (th3q.5 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) G ([] (<,> C D) R)) ([] (opr (<,> A B) F (<,> C D)) R))) (th3q.1 R V A B <,> F C D <,> opr ecexg ax-mp x cv A B <,> R [] w cv v cv <,> R [] eqeq1 y cv u cv t cv <,> R [] = anbi1d z cv w cv v cv <,> F u cv t cv <,> opr R [] = anbi1d w v u t bi4exdv y cv C D <,> R [] u cv t cv <,> R [] eqeq1 A B <,> R [] w cv v cv <,> R [] = anbi2d z cv w cv v cv <,> F u cv t cv <,> opr R [] = anbi1d w v u t bi4exdv z cv A B <,> F C D <,> opr R [] w cv v cv <,> F u cv t cv <,> opr R [] eqeq1 A B <,> R [] w cv v cv <,> R [] = C D <,> R [] u cv t cv <,> R [] = /\ anbi2d w v u t bi4exdv th3q.1 th3q.2 th3q.3 th3q.4 x cv y cv z th3qlem2 th3q.5 oprabvali A S B S opelxpi th3q.1 A B <,> S S X. ecelqsi syl C S D S opelxpi th3q.1 C D <,> S S X. ecelqsi syl anim12i A B <,> R [] eqid C D <,> R [] eqid pm3.2i A B <,> F C D <,> opr R [] eqid w cv A v cv B opeq12 w cv v cv <,> A B <,> R eceq2 A B <,> R [] eqeq2d C D <,> R [] C D <,> R [] = anbi1d w cv v cv <,> A B <,> F C D <,> opreq1 w cv v cv <,> F C D <,> opr A B <,> F C D <,> opr R eceq2 syl A B <,> F C D <,> opr R [] eqeq2d anbi12d syl S S cla4e2gv u cv C t cv D opeq12 u cv t cv <,> C D <,> R eceq2 C D <,> R [] eqeq2d A B <,> R [] w cv v cv <,> R [] = anbi2d u cv t cv <,> C D <,> w cv v cv <,> F opreq2 w cv v cv <,> F u cv t cv <,> opr w cv v cv <,> F C D <,> opr R eceq2 syl A B <,> F C D <,> opr R [] eqeq2d anbi12d syl S S cla4e2gv w v 19.22dvv sylan9 mp2ani sylc))
+var (set a)
+var (set b)
+var (set c)
+var (set d)
+var (class L)
+thm (oprec ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d) (F x) (F y) (F z) (F t) (F s) (F g) (F h) (F a) (F b) (F c) (F d) (G x) (G y) (G z) (G t) (G s) (G g) (G h) (G a) (G b) (G c) (G d) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (J x) (J y) (J z) (K x) (K y) (K z) (K w) (K v) (K u) (K f) (L x) (L y) (L z) (L w) (L v) (L u) (L f) (Q x) (Q y) (Q z) (Q a) (Q b) (Q c) (Q d) (R x) (R y) (R z) (R t) (R s) (R g) (R h) (R a) (R b) (R c) (R d) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (S a) (S b) (S c) (S d) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u) (ch z) (ch w) (ch v) (ch u)) ((oprec.1 (e. H (V))) (oprec.2 (e. K (V))) (oprec.3 (e. L (V))) (oprec.4 (e. R (V))) (oprec.5 (Er R)) (oprec.6 (= (dom R) (X. S S))) (oprec.7 (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) ph))))))))) (oprec.8 (-> (/\ (/\ (= (cv z) (cv a)) (= (cv w) (cv b))) (/\ (= (cv v) (cv c)) (= (cv u) (cv d)))) (<-> ph ps))) (oprec.9 (-> (/\ (/\ (= (cv z) (cv g)) (= (cv w) (cv h))) (/\ (= (cv v) (cv t)) (= (cv u) (cv s)))) (<-> ph ch))) (oprec.10 (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) J)))))))))) (oprec.11 (-> (/\ (/\ (= (cv w) (cv a)) (= (cv v) (cv b))) (/\ (= (cv u) (cv g)) (= (cv f) (cv h)))) (= J K))) (oprec.12 (-> (/\ (/\ (= (cv w) (cv c)) (= (cv v) (cv d))) (/\ (= (cv u) (cv t)) (= (cv f) (cv s)))) (= J L))) (oprec.13 (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= J H))) (oprec.14 (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) Q) (e. (cv y) Q)) (E. a (E. b (E. c (E. d (/\ (/\ (= (cv x) ([] (<,> (cv a) (cv b)) R)) (= (cv y) ([] (<,> (cv c) (cv d)) R))) (= (cv z) ([] (opr (<,> (cv a) (cv b)) G (<,> (cv c) (cv d))) R))))))))))) (oprec.15 (= Q (/. (X. S S) R))) (oprec.16 (-> (/\ (/\ (/\ (e. (cv a) S) (e. (cv b) S)) (/\ (e. (cv c) S) (e. (cv d) S))) (/\ (/\ (e. (cv g) S) (e. (cv h) S)) (/\ (e. (cv t) S) (e. (cv s) S)))) (-> (/\ ps ch) (br K R L))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) F ([] (<,> C D) R)) ([] H R))) (oprec.4 oprec.5 oprec.6 oprec.16 oprec.8 oprec.7 opbrop oprec.9 oprec.7 opbrop bi2anan9 oprec.2 oprec.11 oprec.10 oprabval3 oprec.3 oprec.12 oprec.10 oprabval3 R breqan12d an4s 3imtr4d oprec.14 oprec.15 x cv eleq2i oprec.15 y cv eleq2i anbi12i a b c d x cv a cv b cv <,> R [] = y cv c cv d cv <,> R [] = /\ z cv a cv b cv <,> G c cv d cv <,> opr R [] = /\ E. E. E. E. anbi1i x y z bioprabi eqtr A B C D th3q oprec.1 oprec.13 oprec.10 oprabval3 A B <,> G C D <,> opr H R eceq2 syl eqtrd))
+thm (ecoprcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (C z) (C w)) ((ecoprcom.1 (= C (/. (X. S S) R))) (ecoprcom.2 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> D G) R)))) (ecoprcom.3 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv x) S) (e. (cv y) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv x) (cv y)) R)) ([] (<,> H J) R)))) (ecoprcom.4 (= D H)) (ecoprcom.5 (= G J))) (-> (/\ (e. A C) (e. B C)) (= (opr A F B) (opr B F A))) (ecoprcom.1 x cv y cv <,> R [] A F z cv w cv <,> R [] opreq1 x cv y cv <,> R [] A z cv w cv <,> R [] F opreq2 eqeq12d z cv w cv <,> R [] B A F opreq2 z cv w cv <,> R [] B F A opreq1 eqeq12d ecoprcom.2 ecoprcom.4 ecoprcom.5 D H G J opeq12 D G <,> H J <,> R eceq2 syl mp2an syl6eq ecoprcom.3 ancoms eqtr4d 2ecoptocl))
+thm (ecoprass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (F x) (F y) (F z) (F w) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (D z) (D w) (D v) (D u)) ((ecoprass.1 (= D (/. (X. S S) R))) (ecoprass.2 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> G H) R)))) (ecoprass.3 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> N Q) R)))) (ecoprass.4 (-> (/\ (/\ (e. G S) (e. H S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> G H) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> J K) R)))) (ecoprass.5 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. N S) (e. Q S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> N Q) R)) ([] (<,> L M) R)))) (ecoprass.6 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (/\ (e. G S) (e. H S)))) (ecoprass.7 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. N S) (e. Q S)))) (ecoprass.8 (= J L)) (ecoprass.9 (= K M))) (-> (/\/\ (e. A D) (e. B D) (e. C D)) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) (ecoprass.1 x cv y cv <,> R [] A F z cv w cv <,> R [] opreq1 F v cv u cv <,> R [] opreq1d x cv y cv <,> R [] A F z cv w cv <,> R [] F v cv u cv <,> R [] opr opreq1 eqeq12d z cv w cv <,> R [] B A F opreq2 F v cv u cv <,> R [] opreq1d z cv w cv <,> R [] B F v cv u cv <,> R [] opreq1 A F opreq2d eqeq12d v cv u cv <,> R [] C A F B opr F opreq2 v cv u cv <,> R [] C B F opreq2 A F opreq2d eqeq12d ecoprass.2 F v cv u cv <,> R [] opreq1d v cv S e. u cv S e. /\ adantr ecoprass.4 ecoprass.6 sylan eqtrd 3impa ecoprass.8 ecoprass.9 J L K M opeq12 J K <,> L M <,> R eceq2 syl mp2an syl6eq ecoprass.3 x cv y cv <,> R [] F opreq2d x cv S e. y cv S e. /\ adantl ecoprass.5 ecoprass.7 sylan2 eqtrd 3impb eqtr4d 3ecoptocl))
+var (class W)
+var (class X)
+var (class Y)
+var (class Z)
+thm (ecoprdi ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (F x) (F y) (F z) (F w) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (G x) (G y) (G z) (G w) (G v) (G u) (D z) (D w) (D v) (D u)) ((ecoprdist.1 (= D (/. (X. S S) R))) (ecoprdist.2 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> M N) R)))) (ecoprdist.3 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. M S) (e. N S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> M N) R)) ([] (<,> H J) R)))) (ecoprdist.4 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> (cv z) (cv w)) R)) ([] (<,> W X) R)))) (ecoprdist.5 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv x) (cv y)) R) G ([] (<,> (cv v) (cv u)) R)) ([] (<,> Y Z) R)))) (ecoprdist.6 (-> (/\ (/\ (e. W S) (e. X S)) (/\ (e. Y S) (e. Z S))) (= (opr ([] (<,> W X) R) F ([] (<,> Y Z) R)) ([] (<,> K L) R)))) (ecoprdist.7 (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. M S) (e. N S)))) (ecoprdist.8 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (/\ (e. W S) (e. X S)))) (ecoprdist.9 (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (/\ (e. Y S) (e. Z S)))) (ecoprdist.10 (= H K)) (ecoprdist.11 (= J L))) (-> (/\/\ (e. A D) (e. B D) (e. C D)) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))) (ecoprdist.1 x cv y cv <,> R [] A G z cv w cv <,> R [] F v cv u cv <,> R [] opr opreq1 x cv y cv <,> R [] A G z cv w cv <,> R [] opreq1 x cv y cv <,> R [] A G v cv u cv <,> R [] opreq1 F opreq12d eqeq12d z cv w cv <,> R [] B F v cv u cv <,> R [] opreq1 A G opreq2d z cv w cv <,> R [] B A G opreq2 F A G v cv u cv <,> R [] opr opreq1d eqeq12d v cv u cv <,> R [] C B F opreq2 A G opreq2d v cv u cv <,> R [] C A G opreq2 A G B opr F opreq2d eqeq12d ecoprdist.2 x cv y cv <,> R [] G opreq2d x cv S e. y cv S e. /\ adantl ecoprdist.3 ecoprdist.7 sylan2 eqtrd 3impb ecoprdist.10 ecoprdist.11 H K J L opeq12 H J <,> K L <,> R eceq2 syl mp2an syl6eq ecoprdist.4 ecoprdist.5 F opreqan12d ecoprdist.6 ecoprdist.8 ecoprdist.9 syl2an eqtrd 3impdi eqtr4d 3ecoptocl))
+thm (mapprc ((A f) (B f)) () (-> (-. (e. A (V))) (= ({|} f (:--> (cv f) A B)) ({/}))) (f f cv A B :--> abn0 f visset f cv V dmexg ax-mp f cv A B fdm V eleq1d mpbii f 19.23aiv sylbi con1i))
+thm (mapex ((A f) (B f)) () (-> (/\ (e. A C) (e. B D)) (e. ({|} f (:--> (cv f) A B)) (V))) (f cv A B fssxp f ss2abi A B X. f df-pw sseqtr4 A C B D xpexg A B X. V pwexg A B X. P~ V f f cv A B :--> {|} ssexg 3syl mpi))
+thm (fnmap ((f x) (f y) (f z) (x y) (x z) (y z)) () (Fn (^m) (X. (V) (V))) (y visset x visset y cv V x cv V f mapex mp2an x y z f df-map x visset y visset pm3.2i z cv f f cv y cv x cv :--> {|} = biantrur x y z bioprabi eqtr fnoprab2))
+thm (mapvalg ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f) (B x) (B y) (B z) (B f)) () (-> (/\ (e. A C) (e. B D)) (= (opr A (^m) B) ({|} f (:--> (cv f) B A)))) (B D A C f mapex ancoms x cv A f cv y cv feq3 f biabdv y cv B f cv A feq2 f biabdv x y z f df-map x visset y visset pm3.2i z cv f f cv y cv x cv :--> {|} = biantrur x y z bioprabi eqtr V oprabval2g 3exp imp A C elisset B D elisset syl2an mpd))
+thm (mapval ((A f) (B f)) ((mapval.1 (e. A (V))) (mapval.2 (e. B (V)))) (= (opr A (^m) B) ({|} f (:--> (cv f) B A))) (mapval.1 mapval.2 A V B V f mapvalg mp2an))
+thm (elmap ((A f) (B f) (C f)) ((elmap.1 (e. A (V))) (elmap.2 (e. B (V)))) (<-> (e. C (opr A (^m) B)) (:--> C B A)) (elmap.1 elmap.2 f mapval C eleq2i elmap.2 B V C A fex ax-mp f cv C B A feq1 elab3g bitr))
+thm (map0e ((A f)) ((map0e.1 (e. A (V)))) (= (opr A (^m) ({/})) (1o)) (f cv fn0 f cv ran A C_ anbi1i f cv {/} A df-f A 0ss f cv {/} rneq rn0 syl6eq A sseq1d mpbiri pm4.71i 3bitr4 f biabi map0e.1 0ex f mapval {/} f df-sn 3eqtr4 df1o2 eqtr4))
+thm (map0b ((A f)) ((map0e.1 (e. A (V)))) (-> (-. (= A ({/}))) (= (opr ({/}) (^m) A) ({/}))) (f f cv A {/} :--> abn0 f cv A {/} fdm f cv A {/} frn f cv ran ss0 syl f cv dm0rn0 sylibr eqtr3d f 19.23aiv sylbi con1i 0ex map0e.1 f mapval syl5eq))
+thm (map0 ((f x) (A f) (A x) (B f) (B x)) ((map0.1 (e. A (V))) (map0.2 (e. B (V)))) (<-> (= (opr A (^m) B) ({/})) (/\ (= A ({/})) (-. (= B ({/}))))) (map0.1 map0.2 f mapval {/} eqeq1i x cv A snssi x visset B fconst B x cv {} X. B x cv {} A fss mpan map0.2 x cv snex xpex f cv B x cv {} X. B A feq1 cla4ev 3syl x 19.23aiv A x n0 f f cv B A :--> abn0 3imtr4 a3i sylbi 0nep0 map0.1 map0e {/} eqeq1i df1o2 {/} eqeq1i {/} {} {/} eqcom 3bitr mtbir B {/} A ^m opreq2 {/} eqeq1d mtbiri con2i jca A {/} ^m B opreq1 map0.2 map0b sylan9eq impbi))
+thm (mapsn ((f y) (A f) (A y) (B f) (B y)) ((map0.1 (e. A (V))) (map0.2 (e. B (V)))) (= (opr A (^m) ({} B)) ({|} f (E.e. y A (= (cv f) ({} (<,> B (cv y))))))) (map0.1 B snex f cv elmap map0.2 snid f cv B {} B y fneu f cv B {} A ffn sylan mpan2 f cv B {} A frel f cv B y relimasn syl f cv B {} A fdm f cv dom B {} f cv imaeq2 syl f cv imadmrn syl5reqr eqtr3d y cv {} eqeq1d y biexdv y B f cv y cv br eusn syl5bb mpbid f cv B {} A frn y cv sseld y visset snid f cv ran y cv {} y cv eleq2 mpbiri syl5 f cv ran y cv {} f cv B {} feq3 f cv B {} A ffn f cv B {} fnforn sylib f cv B {} f cv ran fof syl syl5bi com12 map0.2 y visset f cv fsn syl6ib jcad y 19.22dv mpd y A f cv B y cv <,> {} = df-rex sylibr f cv B {} y cv {} A fss map0.2 y visset f1osn f cv B y cv <,> {} B {} y cv {} f1oeq1 mpbiri f cv B {} y cv {} f1of syl y cv A snssi syl2an expcom r19.23aiv impbi bitr biabri))
+thm (mapss ((A f) (B f) (C f)) ((mapss.1 (e. B (V))) (mapss.2 (e. C (V)))) (-> (C_ A B) (C_ (opr A (^m) C) (opr B (^m) C))) (f cv C A B fss expcom f 19.21aiv f f cv C A :--> f cv C B :--> ss2ab sylibr mapss.1 A ssex mapss.2 A V C V f mapvalg mpan2 syl mapss.1 mapss.2 f mapval A B C_ a1i 3sstr4d))
+thm (relen ((x y) (f x) (f y)) () (Rel (~~)) (x y f f cv x cv y cv :-1-1-onto-> E. relopab x y f df-en ~~ x y f f cv x cv y cv :-1-1-onto-> E. {<,>|} releq ax-mp mpbir))
+thm (reldom ((x y) (f x) (f y)) () (Rel (~<_)) (x y f f cv x cv y cv :-1-1-> E. relopab x y f df-dom ~<_ x y f f cv x cv y cv :-1-1-> E. {<,>|} releq ax-mp mpbir))
+thm (relsdom () () (Rel (~<)) (reldom ~<_ ~~ reldif df-sdom ~< ~<_ ~~ \ releq ax-mp sylibr ax-mp))
+thm (breng ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B x) (B y) (C y)) () (-> (e. B C) (<-> (br A (~~) B) (E. f (:-1-1-onto-> (cv f) A B)))) (x cv A f cv y cv f1oeq2 f biexdv y cv B f cv A f1oeq3 f biexdv x y f df-en V C brabg exp relen A B brrelexi f cv A B f1ofn f visset f cv V dmexg ax-mp f cv A fndm V eleq1d mpbii syl f 19.23aiv pm5.21ni B C e. a1d pm2.61i))
+thm (brdomg ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B x) (B y) (C y)) () (-> (e. B C) (<-> (br A (~<_) B) (E. f (:-1-1-> (cv f) A B)))) (x cv A f cv y cv f1eq2 f biexdv y cv B f cv A f1eq3 f biexdv x y f df-dom V C brabg exp reldom A B brrelexi f cv A B f1f f visset f cv V dmexg ax-mp f cv A B fdm V eleq1d mpbii syl f 19.23aiv pm5.21ni B C e. a1d pm2.61i))
+thm (bren ((A f) (B f)) ((bren.1 (e. B (V)))) (<-> (br A (~~) B) (E. f (:-1-1-onto-> (cv f) A B))) (bren.1 B V A f breng ax-mp))
+thm (brdom ((A f) (B f)) ((bren.1 (e. B (V)))) (<-> (br A (~<_) B) (E. f (:-1-1-> (cv f) A B))) (bren.1 B V A f brdomg ax-mp))
+thm (domen ((f x) (A f) (A x) (B f) (B x)) ((bren.1 (e. B (V)))) (<-> (br A (~<_) B) (E. x (/\ (br A (~~) (cv x)) (C_ (cv x) B)))) (f visset A B x f11o f biex f x f cv A x cv :-1-1-onto-> x cv B C_ /\ excom bitr bren.1 A f brdom x visset A f bren x cv B C_ anbi1i f f cv A x cv :-1-1-onto-> x cv B C_ 19.41v bitr4 x biex 3bitr4))
+thm (domeng ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (<-> (br A (~<_) B) (E. x (/\ (br A (~~) (cv x)) (C_ (cv x) B))))) (y cv B A ~<_ breq2 y cv B x cv sseq2 A ~~ x cv br anbi2d x biexdv bibi12d y visset A x domen C vtoclg))
+thm (brsdom () () (<-> (br A (~<) B) (/\ (br A (~<_) B) (-. (br A (~~) B)))) (df-sdom A B <,> eleq2i A ~< B df-br A ~<_ B df-br A ~~ B df-br negbii anbi12i A B <,> ~<_ ~~ eldif bitr4 3bitr4))
+thm (enssdom ((x y) (f x) (f y)) () (C_ (~~) (~<_)) (relen f cv x cv y cv f1of1 f 19.22i x y f f cv x cv y cv :-1-1-onto-> E. opabid x y f f cv x cv y cv :-1-1-> E. opabid 3imtr4 x y f df-en x cv y cv <,> eleq2i x y f df-dom x cv y cv <,> eleq2i 3imtr4 relssi))
+thm (dfdom2 () () (= (~<_) (u. (~<) (~~))) (df-sdom ~~ uneq2i ~~ ~< uncom enssdom ~~ ~<_ ssundif mpbi 3eqtr3r))
+thm (endom () () (-> (br A (~~) B) (br A (~<_) B)) (enssdom A ~~ B br a1i A B ssbrd pm2.43i))
+thm (sdomdom () () (-> (br A (~<) B) (br A (~<_) B)) (A B brsdom pm3.26bd))
+thm (sdomnen () () (-> (br A (~<) B) (-. (br A (~~) B))) (A B brsdom pm3.27bd))
+thm (brdom2 () () (<-> (br A (~<_) B) (\/ (br A (~<) B) (br A (~~) B))) (dfdom2 A B <,> eleq2i A ~<_ B df-br A ~< B df-br A ~~ B df-br orbi12i A B <,> ~< ~~ elun bitr4 3bitr4))
+thm (bren2 () () (<-> (br A (~~) B) (/\ (br A (~<_) B) (-. (br A (~<) B)))) (A B endom A B sdomnen con2i jca A B brdom2 biimp orcanai impbi))
+thm (enrefg ((A f)) () (-> (e. A B) (br A (~~) A)) (funi A B I resfunexg mpi A f1oi f cv I A |` A A f1oeq1 V cla4egv mpi syl A B A f breng mpbird))
+thm (enref () ((enref.1 (e. A (V)))) (br A (~~) A) (enref.1 A V enrefg ax-mp))
+thm (eqeng () () (-> (e. A C) (-> (= A B) (br A (~~) B))) (A B A ~~ breq2 A C enrefg syl5bi com12))
+thm (domrefg () () (-> (e. A B) (br A (~<_) A)) (A B enrefg A A endom syl))
+thm (f1oeng ((A f) (B f) (F f)) () (-> (e. A C) (-> (:-1-1-onto-> F A B) (br A (~~) B))) (A C F fnex F A B f1ofn syl5 f cv F A B f1oeq1 V cla4egv syli B A ~~ brprc A C enrefg syl5bir f f cv A B :-1-1-onto-> E. a1d com3r B V A f breng biimprd pm2.61d2 syld))
+thm (f1domg ((A f) (B f) (F f)) () (-> (e. A C) (-> (:-1-1-> F A B) (br A (~<_) B))) (A C F fnex F A B f1f F A B ffn syl syl5 f cv F A B f1eq1 V cla4egv syli B A ~<_ brprc A C domrefg syl5bir f f cv A B :-1-1-> E. a1d com3r B V A f brdomg biimprd pm2.61d2 syld))
+thm (f1dom2g () () (-> (e. B C) (-> (:-1-1-> F A B) (br A (~<_) B))) (B C F A f1dmex A V F B f1domg syli))
+thm (f1oen () ((f1oen.1 (e. A (V)))) (-> (:-1-1-onto-> F A B) (br A (~~) B)) (f1oen.1 A V F B f1oeng ax-mp))
+thm (f1dom () ((f1oen.1 (e. A (V)))) (-> (:-1-1-> F A B) (br A (~<_) B)) (f1oen.1 A V F B f1domg ax-mp))
+thm (en2d ((x y) (A x) (A y) (B x) (B y) (C y) (D x) (ph x) (ph y)) ((en2d.1 (-> ph (e. A (V)))) (en2d.2 (-> ph (-> (e. (cv x) A) (e. C (V))))) (en2d.3 (-> ph (-> (e. (cv y) B) (e. D (V))))) (en2d.4 (-> ph (<-> (/\ (e. (cv x) A) (= (cv y) C)) (/\ (e. (cv y) B) (= (cv x) D)))))) (-> ph (br A (~~) B)) (A V x y y cv B e. x cv D = /\ {<,>|} B f1oeng en2d.1 en2d.2 C y eueq syl6ib r19.21aiv x y x cv A e. y cv C = /\ {<,>|} eqid fnopabg sylib en2d.4 x y biopabdv x y x cv A e. y cv C = /\ {<,>|} x y y cv B e. x cv D = /\ {<,>|} A fneq1 syl mpbid en2d.3 D x eueq syl6ib r19.21aiv x y y cv B e. x cv D = /\ cnvopab fnopabg sylib jca x y y cv B e. x cv D = /\ {<,>|} A B f1o4 sylibr sylc))
+thm (en3d ((x y) (A x) (A y) (B x) (B y) (C y) (D x) (ph x) (ph y)) ((en3d.1 (-> ph (e. A (V)))) (en3d.2 (-> ph (-> (e. (cv x) A) (e. C B)))) (en3d.3 (-> ph (-> (e. (cv y) B) (e. D A)))) (en3d.4 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> (= (cv x) D) (= (cv y) C)))))) (-> ph (br A (~~) B)) (en3d.1 en3d.2 C B elisset syl6 en3d.3 D A elisset syl6 en3d.2 C B y cv eleq1a syl6 imp32 en3d.4 imp biimpar exp42 com34 imp32 jcai exp en3d.3 D A x cv eleq1a syl6 imp32 en3d.4 imp biimpa exp42 com23 com34 imp32 jcai exp impbid en2d))
+thm (en2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((en2.1 (e. A (V))) (en2.2 (-> (e. (cv x) A) (e. C (V)))) (en2.3 (-> (e. (cv y) B) (e. D (V)))) (en2.4 (<-> (/\ (e. (cv x) A) (= (cv y) C)) (/\ (e. (cv y) B) (= (cv x) D))))) (br A (~~) B) (A eqid en2.1 A A = a1i en2.2 A A = a1i en2.3 A A = a1i en2.4 A A = a1i en2d ax-mp))
+thm (en3 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((en3.1 (e. A (V))) (en3.2 (-> (e. (cv x) A) (e. C B))) (en3.3 (-> (e. (cv y) B) (e. D A))) (en3.4 (-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> (= (cv x) D) (= (cv y) C))))) (br A (~~) B) (A eqid en3.1 A A = a1i en3.2 A A = a1i en3.3 A A = a1i en3.4 A A = a1i en3d ax-mp))
+thm (dom2d ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C y) (C z) (D x) (D z) (ph x) (ph y)) ((dom2d.1 (-> ph (-> (e. (cv x) A) (e. C B)))) (dom2d.2 (-> ph (-> (/\ (e. (cv x) A) (e. (cv y) A)) (<-> (= C D) (= (cv x) (cv y))))))) (-> ph (-> (e. A R) (br A (~<_) B))) (A R x y x cv A e. y cv C = /\ {<,>|} B f1domg dom2d.1 r19.21aiv x y x cv A e. y cv C = /\ {<,>|} eqid B fopab2 sylib dom2d.1 imp x A C B y fvopab2 ph adantll mpdan y cv A e. adantrr ph y cv A e. /\ x ax-17 z x y x cv A e. y cv C = /\ hbopab1 z cv y cv e. x ax-17 hbfv z cv D e. x ax-17 hbeq hbim x cv y cv A eleq1 ph anbi2d x y x cv A e. y cv C = /\ {<,>|} x cv ` C = imbi1d x cv y cv A eleq1 y cv A e. anbi1d y cv A e. anidm syl6bb ph anbi2d x cv y cv x y x cv A e. y cv C = /\ {<,>|} fveq2 ph x cv A e. y cv A e. /\ /\ adantr dom2d.2 imp biimparc eqeq12d exp sylbird pm5.74d bitrd dom2d.1 imp x A C B y fvopab2 ph adantll mpdan chvar x cv A e. adantrl eqeq12d dom2d.2 imp biimpd sylbid exp r19.21aivv jca z x y x cv A e. y cv C = /\ hbopab1 z x y x cv A e. y cv C = /\ hbopab2 A B f1fvf sylibr syl5com))
+thm (dom2 ((x y) (A x) (A y) (B x) (B y) (C y) (D x)) ((dom2.1 (-> (e. (cv x) A) (e. C B))) (dom2.2 (-> (/\ (e. (cv x) A) (e. (cv y) A)) (<-> (= C D) (= (cv x) (cv y)))))) (-> (e. A R) (br A (~<_) B)) (A eqid dom2.1 A A = a1i dom2.2 A A = a1i R dom2d ax-mp))
+thm (idssen ((x y)) () (C_ (I) (~~)) (reli x visset y visset ideq x cv f1oi x cv y cv I x cv |` x cv f1oeq3 mpbii x visset I x cv |` y cv f1oen syl sylbi x cv I y cv df-br x cv ~~ y cv df-br 3imtr3 relssi))
+thm (dmen () () (= (dom (~~)) (V)) (idssen I ~~ dmss dmi syl5ssr ax-mp ~~ dom vss mpbi))
+thm (ssdomg () () (-> (e. A C) (-> (C_ A B) (br A (~<_) B))) (A C I A |` B f1domg A f1oi I A |` A A f1o3 mpbi pm3.26i I A |` A A fof ax-mp I A |` A A B fss mpan funi cnvi I `' I funeq ax-mp mpbir I A funres11 ax-mp jctir I A |` A B df-f1 sylibr syl5))
+thm (ssdom2g () () (-> (e. B C) (-> (C_ A B) (br A (~<_) B))) (B C A ssexg A V B ssdomg syli))
+thm (ener ((f g) (f x) (f y) (f z) (g x) (g y) (g z) (x y) (x z) (y z)) () (Er (~~)) (y visset x cv f bren f cv x cv y cv f1ocnv y visset f cv `' x cv f1oen syl f 19.23aiv sylbi g f g cv x cv y cv :-1-1-onto-> f cv y cv z cv :-1-1-onto-> eeanv f cv y cv z cv g cv x cv f1oco ancoms x visset f cv g cv o. z cv f1oen syl g f 19.23aivv sylbir y visset x cv g bren z visset y cv f bren syl2anb ster))
+thm (sdomirr () () (-. (br A (~<) A)) (A A sdomnen A V enrefg nsyl3 relsdom A A brrelexi con3i pm2.61i))
+thm (sdomex () () (-> (br A (~<) B) (/\ (e. A (V)) (e. B (V)))) (relsdom A B brrelexi A sdomirr B A ~< brprc mtbiri a3i jca))
+thm (ensymg ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (-> (br A (~~) B) (br B (~~) A))) (relen A B brrelexi x cv A ~~ y cv breq1 x cv A y cv ~~ breq2 imbi12d y cv B A ~~ breq2 y cv B ~~ A breq1 imbi12d x visset y visset ener ersym V C vtocl2g exp syl pm2.43b))
+thm (ensym () ((ensym.1 (e. B (V)))) (-> (br A (~~) B) (br B (~~) A)) (ensym.1 B V A ensymg ax-mp))
+thm (ensymi () ((ensym.1 (e. B (V))) (ensymi.2 (br A (~~) B))) (br B (~~) A) (ensymi.2 ensym.1 A ensym ax-mp))
+thm (entrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (br A (~~) B) (br B (~~) C)) (br A (~~) C)) (relen x visset y visset z visset ener ertr x visset enref A B C vtoclrbr))
+thm (domtr ((x y) (x z) (f x) (g x) (A x) (y z) (f y) (g y) (A y) (f z) (g z) (A z) (f g) (A f) (A g) (B x) (B y) (B z) (B f) (B g) (C x) (C y) (C z) (C f) (C g)) () (-> (/\ (br A (~<_) B) (br B (~<_) C)) (br A (~<_) C)) (reldom g f g cv x cv y cv :-1-1-> f cv y cv z cv :-1-1-> eeanv f cv y cv z cv g cv x cv f1co ancoms x visset f cv g cv o. z cv f1dom syl g f 19.23aivv sylbir y visset x cv g brdom z visset y cv f brdom syl2anb x visset x cv V domrefg ax-mp A B C vtoclrbr))
+thm (entr () ((entr.1 (br A (~~) B)) (entr.2 (br B (~~) C))) (br A (~~) C) (entr.1 entr.2 A B C entrt mp2an))
+thm (entr2 () ((entr2.1 (e. C (V))) (entr2.2 (br A (~~) B)) (entr2.3 (br B (~~) C))) (br C (~~) A) (entr2.1 entr2.2 entr2.3 entr ensymi))
+thm (entr3 () ((entr3.1 (e. B (V))) (entr3.2 (br A (~~) B)) (entr3.3 (br A (~~) C))) (br B (~~) C) (entr3.1 entr3.2 ensymi entr3.3 entr))
+thm (entr4 () ((entr4.1 (e. B (V))) (entr4.2 (br A (~~) B)) (entr4.3 (br C (~~) B))) (br A (~~) C) (entr4.2 entr4.1 entr4.3 ensymi entr))
+thm (endomtr () () (-> (/\ (br A (~~) B) (br B (~<_) C)) (br A (~<_) C)) (A B C domtr A B endom sylan))
+thm (domentr () () (-> (/\ (br A (~<_) B) (br B (~~) C)) (br A (~<_) C)) (A B C domtr B C endom sylan2))
+thm (f1imaen () ((f1imaen.1 (e. C (V)))) (-> (/\ (:-1-1-> F A B) (C_ C A)) (br (" F C) (~~) C)) (F A B C f1ores f1imaen.1 F C |` F C " f1oen F C |` C F C " f1ofo f1imaen.1 C V F C |` F C " fornex ax-mp F C " V C ensymg 3syl mpd syl))
+thm (en0 ((A f)) () (<-> (br A (~~) ({/})) (= A ({/}))) (0ex A f bren f cv A {/} f1ocnv f cv `' A f1o00 pm3.27bd syl f 19.23aiv sylbi 0ex enref A {/} ~~ {/} breq1 mpbiri impbi))
+thm (ensn1 ((A f)) ((ensn1.1 (e. A (V)))) (br ({} A) (~~) (1o)) (ensn1.1 0ex f1osn A {/} <,> snex f cv A {/} <,> {} A {} {/} {} f1oeq1 cla4ev ax-mp p0ex A {} f bren mpbir df1o2 breqtrr))
+thm (ensn1g ((A x) (B x)) () (-> (e. A B) (br ({} A) (~~) (1o))) (x cv A sneq ~~ 1o breq1d x visset ensn1 B vtoclg))
+thm (en1 ((f x) (A x) (A f)) () (<-> (br A (~~) (1o)) (E. x (= A ({} (cv x))))) (df1o2 A ~~ breq2i p0ex A f bren bitr f cv A {/} {} f1ocnv f cv `' {/} {} A f1ofo f cv `' {/} {} A forn syl f cv `' {/} {} A f1of 0ex f cv `' A fsn2 pm3.27bd syl rneqd 0ex f cv `' {/} fvex rnsnop syl6eq eqtr3d f cv `' {/} fvex x cv f cv `' {/} ` sneq A eqeq2d cla4ev 3syl f 19.23aiv sylbi x visset ensn1 A x cv {} ~~ 1o breq1 mpbiri x 19.23aiv impbi))
+thm (2dom ((x y) (f x) (A x) (f y) (A y) (A f)) ((2dom.1 (e. A (V)))) (-> (br (2o) (~<_) A) (E.e. x A (E.e. y A (-. (= (cv x) (cv y)))))) (df2o2 ~<_ A breq1i 2dom.1 {/} {/} {} {,} f brdom bitr x cv f cv {/} ` y cv eqeq1 negbid y cv f cv {/} {} ` f cv {/} ` eqeq2 negbid A A rcla42ev f cv {/} {/} {} {,} A f1f 0ex {/} {} pri1 f cv {/} {/} {} {,} A {/} ffvrn mpan2 p0ex {/} pri2 f cv {/} {/} {} {,} A {/} {} ffvrn mpan2 jca syl 0nep0 0ex {/} {} pri1 p0ex {/} pri2 pm3.2i f cv {/} {/} {} {,} A {/} {/} {} f1fveq mpan2 mtbiri sylanc f 19.23aiv sylbi))
+thm (fundmen ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) ((fundmen.1 (e. F (V)))) (-> (Fun F) (br (dom F) (~~) F)) (fundmen.1 F V dmexg ax-mp F Fun a1i F x cv funopfv exp F funrel F y cv elreldm exp syl F V V X. y cv ssel2 F funrel F df-rel sylib sylan y cv z w elvv sylib x cv y cv |^| |^| z cv eqeq1 y cv z cv w cv <,> inteq inteqd z visset w cv op1stb syl6eq syl5bir x cv z cv w cv opeq1 syl6 imp x cv w cv <,> z cv w cv <,> y cv eqeq2 biimprcd x cv y cv |^| |^| = adantl mpd ancoms F Fun y cv F e. /\ adantl x cv y cv |^| |^| z cv eqeq1 y cv z cv w cv <,> inteq inteqd z visset w cv op1stb syl6eq syl5bir x cv z cv w cv opeq1 syl6 imp x cv w cv <,> z cv w cv <,> y cv eqeq2 biimprcd x cv y cv |^| |^| = adantl mpd F eleq1d F Fun adantl w visset F x cv funfvopi x cv y cv |^| |^| = y cv z cv w cv <,> = /\ adantr sylbid exp32 com24 imp43 F x cv ` w cv x cv opeq2 syl eqtr4d exp32 z w 19.23advv mpd x cv F dom e. adantrl y cv x cv F x cv ` <,> inteq inteqd x visset F x cv ` op1stb syl6req F Fun x cv F dom e. y cv F e. /\ /\ a1i impbid exp en3d))
+thm (mapsnen ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((mapsnen.1 (e. A (V))) (mapsnen.2 (e. B (V)))) (br (opr A (^m) ({} B)) (~~) A) (A ^m B {} oprex z cv B fvex z cv A ^m B {} opr e. a1i B w cv <,> snex w cv A e. a1i mapsnen.1 mapsnen.2 z y mapsn eqabi w cv z cv B ` = anbi1i y A z cv B y cv <,> {} = w cv z cv B ` = r19.41v bitr4 y A z cv B y cv <,> {} = w cv z cv B ` = /\ df-rex bitr z cv B y cv <,> {} B fveq1 mapsnen.2 y visset fvsn syl6eq w cv eqeq2d w y equcom syl6bb pm5.32i y cv A e. anbi2i y cv A e. z cv B y cv <,> {} = y cv w cv = anass bitr4 y cv A e. z cv B y cv <,> {} = /\ y cv w cv = ancom bitr y biex w visset y cv w cv A eleq1 y cv w cv B opeq2 sneqd z cv eqeq2d anbi12d ceqsexv 3bitr en2))
+thm (map1 ((x y) (A x) (A y)) ((map1.1 (e. A (V)))) (br (opr (1o) (^m) A) (~~) (1o)) (1o ^m A oprex 0ex x cv 1o ^m A opr e. a1i map1.1 p0ex xpex y cv 1o e. a1i y cv 1o e. x cv A {/} {} X. = ancom df1o2 ^m A opreq1i x cv eleq2i p0ex map1.1 x cv elmap bitr 0ex x cv A fconst2 bitr2 df1o2 y cv eleq2i y {/} elsn bitr anbi12i bitr2 en2))
+thm (en2sn ((A x) (B x)) () (-> (/\ (e. A C) (e. B D)) (br ({} A) (~~) ({} B))) (A {} 1o B {} entrt x cv A sneq ~~ 1o breq1d x visset ensn1 C vtoclg x cv B sneq ~~ 1o breq1d x visset ensn1 D vtoclg 1ord 1o On B {} ensymg ax-mp syl syl2an))
+thm (snfi ((A x)) () (E.e. x (om) (br ({} A) (~~) (cv x))) (x cv A sneq ~~ 1o breq1d x visset ensn1 V vtoclg 1onn x cv 1o A {} ~~ breq2 om rcla4ev mpan syl A snprc A {} en0 peano1 x cv {/} A {} ~~ breq2 om rcla4ev mpan sylbir sylbi pm2.61i))
+thm (unen ((f g) (f h) (A f) (g h) (A g) (A h) (B f) (B g) (B h) (C f) (C g) (C h) (D f) (D g) (D h)) () (-> (/\ (/\ (br A (~~) B) (br C (~~) D)) (/\ (= (i^i A C) ({/})) (= (i^i B D) ({/})))) (br (u. A C) (~~) (u. B D))) (B D unexb B V A f breng D V C g breng bi2anan9 sylbir B D u. V A C u. h breng f cv A B g cv C D f1oun f visset g visset unex h cv f cv g cv u. A C u. B D u. f1oeq1 cla4ev syl syl5bir exp3a f g 19.23advv f g f cv A B :-1-1-onto-> g cv C D :-1-1-onto-> eeanv syl5ibr sylbid imp3a B D u. A C u. ~~ brprc relen A B brrelexi relen C D brrelexi anim12i A C unexb sylib A C u. V enrefg syl syl5bir A C i^i {/} = B D i^i {/} = /\ adantrd pm2.61i))
+thm (xpsnen ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((xpsnen.1 (e. A (V))) (xpsnen.2 (e. B (V)))) (br (X. A ({} B)) (~~) A) (xpsnen.1 B snex xpex y cv A B {} x z elxp x visset y cv x cv z cv <,> inteq inteqd x visset z cv op1stb syl6eq V eleq1d mpbiri x cv A e. z cv B {} e. /\ adantr x z 19.23aivv sylbi x cv B opex x cv A e. a1i x visset x cv y cv |^| |^| V eleq1 mpbii x cv y cv |^| |^| B opeq1 y cv eqeq2d x cv y cv |^| |^| A eleq1 anbi12d V ceqsexgv y cv A B {} x z elxp y cv x cv z cv <,> = x cv A e. /\ z cv B {} e. ancom y cv x cv z cv <,> = x cv A e. z cv B {} e. anass z B elsn y cv x cv z cv <,> = x cv A e. /\ anbi1i 3bitr3 z biex xpsnen.2 z cv B x cv opeq2 y cv eqeq2d x cv A e. anbi1d ceqsexv y cv x cv B <,> inteq inteqd x visset B op1stb syl6req pm4.71ri x cv A e. anbi1i x cv y cv |^| |^| = y cv x cv B <,> = x cv A e. anass bitr 3bitr x biex bitr syl5bb syl pm5.32ri y cv x cv B <,> inteq inteqd x visset B op1stb syl6req x cv A e. adantr pm4.71i x cv y cv |^| |^| B opeq1 y cv eqeq2d x cv y cv |^| |^| A eleq1 anbi12d pm5.32ri bitr2 y cv x cv B <,> = x cv A e. ancom 3bitr en2))
+thm (xpsneng ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (br (X. A ({} B)) (~~) A)) (x cv A y cv {} xpeq1 x cv A = id ~~ breq12d y cv B sneq y cv {} B {} A xpeq2 syl ~~ A breq1d x visset y visset xpsnen C D vtocl2g))
+thm (endisj ((x y) (A x) (A y) (B x) (B y)) ((endisj.1 (e. A (V))) (endisj.2 (e. B (V)))) (E. x (E. y (/\ (/\ (br (cv x) (~~) A) (br (cv y) (~~) B)) (= (i^i (cv x) (cv y)) ({/}))))) (endisj.1 0ex xpsnen endisj.2 p0ex xpsnen pm3.2i 0nep0 {/} {/} {} A B xpsndisj ax-mp endisj.1 p0ex xpex endisj.2 {/} {} snex xpex x cv A {/} {} X. ~~ A breq1 y cv B {/} {} {} X. ~~ B breq1 bi2anan9 x cv A {/} {} X. y cv B {/} {} {} X. ineq12 {/} eqeq1d anbi12d cla4e2v mp2an))
+thm (undom ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((undom.1 (e. B (V))) (undom.2 (e. C (V))) (undom.3 (e. D (V)))) (-> (/\ (/\ (br A (~<_) B) (br C (~<_) D)) (= (i^i B D) ({/}))) (br (u. A C) (~<_) (u. B D))) (A C u. x cv y cv u. B D u. endomtr A x cv C A \ y cv unen A C undif2 syl5eqbrr B D i^i {/} x cv y cv i^i sseq2 x cv y cv i^i ss0b syl6bb x cv B y cv D ss2in syl5bi imp A C difdisj jctil sylan2 anassrs an1rs x cv B y cv D unss12 undom.1 undom.3 unex B D u. V x cv y cv u. ssdom2g ax-mp syl A ~~ x cv br C A \ ~~ y cv br /\ B D i^i {/} = ad2antlr sylanc exp an4s exp x 19.23aiv y 19.23adv imp undom.1 A x domen undom.3 C A \ y domen syl2anb undom.2 C A difss C V C A \ ssdom2g mp2 C A \ C D domtr mpan sylan2 imp))
+thm (xpcomen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((xpcomen.1 (e. A (V))) (xpcomen.2 (e. B (V)))) (br (X. A B) (~~) (X. B A)) (xpcomen.1 xpcomen.2 xpex x cv snex cnvex uniex x cv A B X. e. a1i y cv snex cnvex uniex y cv B A X. e. a1i x cv z cv w cv <,> sneq x cv {} z cv w cv <,> {} cnveq syl z visset w visset cnvsn syl6eq unieqd w cv z cv opex unisn syl6req y cv w cv z cv <,> sneq y cv {} w cv z cv <,> {} cnveq syl w visset z visset cnvsn syl6eq unieqd z cv w cv opex unisn syl6req eq2tr z cv A e. w cv B e. ancom anbi12i x cv z cv w cv <,> = z cv A e. w cv B e. /\ y cv x cv {} `' U. = an23 y cv w cv z cv <,> = w cv B e. z cv A e. /\ x cv y cv {} `' U. = an23 3bitr4 z w bi2ex z w x cv z cv w cv <,> = z cv A e. w cv B e. /\ /\ y cv x cv {} `' U. = 19.41vv z w y cv w cv z cv <,> = w cv B e. z cv A e. /\ /\ x cv y cv {} `' U. = 19.41vv 3bitr3 x cv A B z w elxp y cv x cv {} `' U. = anbi1i y cv B A w z elxp w z y cv w cv z cv <,> = w cv B e. z cv A e. /\ /\ excom bitr x cv y cv {} `' U. = anbi1i 3bitr4 en2))
+thm (xpassen ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((xpassen.1 (e. A (V))) (xpassen.2 (e. B (V))) (xpassen.3 (e. C (V)))) (br (X. (X. A B) C) (~~) (X. A (X. B C))) (xpassen.1 xpassen.2 xpex xpassen.3 xpex x cv {} dom U. {} dom U. x cv {} dom U. {} ran U. x cv {} ran U. <,> opex x cv A B X. C X. e. a1i y cv {} dom U. y cv {} ran U. {} dom U. <,> y cv {} ran U. {} ran U. opex y cv A B C X. X. e. a1i z cv x cv {} dom U. {} dom U. w cv v cv <,> x cv {} dom U. {} ran U. x cv {} ran U. <,> opeq12 x cv z cv w cv <,> v cv <,> sneq dmeqd unieqd sneqd dmeqd unieqd z cv w cv opex v cv op1sta sneqi dmeqi unieqi z visset w cv op1sta eqtr syl6req w cv x cv {} dom U. {} ran U. v cv x cv {} ran U. opeq12 x cv z cv w cv <,> v cv <,> sneq dmeqd unieqd sneqd rneqd unieqd z cv w cv opex v cv op1sta sneqi rneqi unieqi z visset w visset op2nda eqtr syl6req x cv z cv w cv <,> v cv <,> sneq rneqd unieqd z cv w cv opex v visset op2nda syl6req sylanc sylanc z cv w cv <,> y cv {} dom U. y cv {} ran U. {} dom U. <,> v cv y cv {} ran U. {} ran U. opeq12 z cv y cv {} dom U. w cv y cv {} ran U. {} dom U. opeq12 y cv z cv w cv v cv <,> <,> sneq dmeqd unieqd z visset w cv v cv <,> op1sta syl6req y cv z cv w cv v cv <,> <,> sneq rneqd unieqd sneqd dmeqd unieqd z visset w cv v cv opex op2nda sneqi dmeqi unieqi w visset v cv op1sta eqtr syl6req sylanc y cv z cv w cv v cv <,> <,> sneq rneqd unieqd sneqd rneqd unieqd z visset w cv v cv opex op2nda sneqi rneqi unieqi w visset v visset op2nda eqtr syl6req sylanc eq2tr z cv A e. w cv B e. v cv C e. anass anbi12i x cv z cv w cv <,> v cv <,> = z cv A e. w cv B e. /\ v cv C e. /\ y cv x cv {} dom U. {} dom U. x cv {} dom U. {} ran U. x cv {} ran U. <,> <,> = an23 y cv z cv w cv v cv <,> <,> = z cv A e. w cv B e. v cv C e. /\ /\ x cv y cv {} dom U. y cv {} ran U. {} dom U. <,> y cv {} ran U. {} ran U. <,> = an23 3bitr4 v biex v x cv z cv w cv <,> v cv <,> = z cv A e. w cv B e. /\ v cv C e. /\ /\ y cv x cv {} dom U. {} dom U. x cv {} dom U. {} ran U. x cv {} ran U. <,> <,> = 19.41v v y cv z cv w cv v cv <,> <,> = z cv A e. w cv B e. v cv C e. /\ /\ /\ x cv y cv {} dom U. y cv {} ran U. {} dom U. <,> y cv {} ran U. {} ran U. <,> = 19.41v 3bitr3 z w bi2ex z w v x cv z cv w cv <,> v cv <,> = z cv A e. w cv B e. /\ v cv C e. /\ /\ E. y cv x cv {} dom U. {} dom U. x cv {} dom U. {} ran U. x cv {} ran U. <,> <,> = 19.41vv z w v y cv z cv w cv v cv <,> <,> = z cv A e. w cv B e. v cv C e. /\ /\ /\ E. x cv y cv {} dom U. y cv {} ran U. {} dom U. <,> y cv {} ran U. {} ran U. <,> = 19.41vv 3bitr3 x cv A B X. C u v elxp u v x cv u cv v cv <,> = u cv A B X. e. v cv C e. /\ /\ excom u cv A B z w elxp x cv u cv v cv <,> = v cv C e. /\ anbi1i x cv u cv v cv <,> = u cv A B X. e. v cv C e. an12 z w u cv z cv w cv <,> = z cv A e. w cv B e. /\ /\ x cv u cv v cv <,> = v cv C e. /\ 19.41vv 3bitr4 v u bi2ex v u z w u cv z cv w cv <,> = z cv A e. w cv B e. /\ /\ x cv u cv v cv <,> = v cv C e. /\ /\ exrot4 u cv z cv w cv <,> = z cv A e. w cv B e. /\ x cv u cv v cv <,> = v cv C e. /\ anass u biex z cv w cv opex u cv z cv w cv <,> v cv opeq1 x cv eqeq2d v cv C e. anbi1d z cv A e. w cv B e. /\ anbi2d ceqsexv z cv A e. w cv B e. /\ x cv z cv w cv <,> v cv <,> = v cv C e. an12 3bitr z w v bi3ex 3bitr 3bitr y cv x cv {} dom U. {} dom U. x cv {} dom U. {} ran U. x cv {} ran U. <,> <,> = anbi1i y cv A B C X. z u elxp u cv B C w v elxp y cv z cv u cv <,> = z cv A e. /\ anbi2i y cv z cv u cv <,> = z cv A e. u cv B C X. e. anass w v y cv z cv u cv <,> = z cv A e. /\ u cv w cv v cv <,> = w cv B e. v cv C e. /\ /\ 19.42vv y cv z cv u cv <,> = z cv A e. /\ u cv w cv v cv <,> = w cv B e. v cv C e. /\ an12 y cv z cv u cv <,> = z cv A e. w cv B e. v cv C e. /\ anass u cv w cv v cv <,> = anbi2i bitr w v bi2ex bitr3 3bitr3 u biex u w v u cv w cv v cv <,> = y cv z cv u cv <,> = z cv A e. w cv B e. v cv C e. /\ /\ /\ /\ exrot3 w cv v cv opex u cv w cv v cv <,> z cv opeq2 y cv eqeq2d z cv A e. w cv B e. v cv C e. /\ /\ anbi1d ceqsexv w v bi2ex 3bitr z biex bitr x cv y cv {} dom U. y cv {} ran U. {} dom U. <,> y cv {} ran U. {} ran U. <,> = anbi1i 3bitr4 en2))
+thm (xpdom2 ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) ((xpdom2.1 (e. A (V))) (xpdom2.2 (e. B (V))) (xpdom2.3 (e. C (V)))) (-> (br A (~<_) B) (br (X. C A) (~<_) (X. C B))) (xpdom2.2 A f brdom xpdom2.3 xpdom2.1 xpex f cv A B f1f f cv A B x cv {} ran U. ffvrn exp syl x cv {} dom U. C e. anim2d x cv x cv {} dom U. x cv {} ran U. <,> = adantld x cv C A elxp4 f cv x cv {} ran U. fvex x cv {} dom U. C B opelxp 3imtr4g x cv z cv w cv <,> sneq dmeqd unieqd z visset w cv op1sta syl6eq x cv z cv w cv <,> sneq rneqd unieqd z visset w visset op2nda syl6eq f cv fveq2d jca x cv snex x cv {} V dmexg ax-mp uniex f cv x cv {} ran U. fvex f cv w cv fvex z cv opth sylibr y cv v cv u cv <,> sneq dmeqd unieqd v visset u cv op1sta syl6eq y cv v cv u cv <,> sneq rneqd unieqd v visset u visset op2nda syl6eq f cv fveq2d jca y cv snex y cv {} V dmexg ax-mp uniex f cv y cv {} ran U. fvex f cv u cv fvex v cv opth sylibr eqeqan12d z cv C e. w cv A e. /\ v cv C e. u cv A e. /\ /\ adantl f cv A B :-1-1-> adantr f cv A B w cv u cv f1fveq ancoms z cv v cv = anbi2d z visset f cv w cv fvex f cv u cv fvex v cv opth syl5bb exp v cv C e. adantrl z cv C e. adantll imp x cv z cv w cv <,> = y cv v cv u cv <,> = /\ adantlr bitrd x cv z cv w cv <,> y cv v cv u cv <,> eqeq12 z visset w visset u visset v cv opth syl6bb z cv C e. w cv A e. /\ v cv C e. u cv A e. /\ /\ adantl f cv A B :-1-1-> adantr bitr4d exp exp43 com23 r19.23aivv r19.23advv imp x cv C A z w elxp2 y cv C A v u elxp2 syl2anb com12 V dom2d mpi f 19.23aiv sylbi))
+thm (xpdom1 () ((xpdom2.1 (e. A (V))) (xpdom2.2 (e. B (V))) (xpdom2.3 (e. C (V)))) (-> (br A (~<_) B) (br (X. A C) (~<_) (X. B C))) (xpdom2.1 xpdom2.2 xpdom2.3 xpdom2 xpdom2.1 xpdom2.3 xpcomen A C X. C A X. C B X. endomtr mpan xpdom2.3 xpdom2.2 xpcomen A C X. C B X. B C X. domentr mpan2 3syl))
+thm (xpdom3 ((A x) (B x)) ((xpdom3.1 (e. A (V)))) (-> (-. (= B ({/}))) (br A (~<_) (X. A B))) (B x n0 x visset B snss A ssid A A x cv {} B ssxp mpan xpdom3.1 x cv snex xpex A x cv {} X. V A B X. ssdomg ax-mp xpdom3.1 xpdom3.1 x visset xpsnen ensymi A A x cv {} X. A B X. endomtr mpan 3syl sylbi x 19.23aiv sylbi))
+thm (pw2en ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u)) ((pw2en.1 (e. A (V)))) (br (P~ A) (~~) (opr (2o) (^m) A)) (pw2en.1 pwex pw2en.1 z w v v cv {/} = z cv x cv e. /\ {|} funopabex2 x cv A P~ e. a1i y visset cnvex y cv `' V {/} {} {} imaexg ax-mp y cv 2o ^m A opr e. a1i x cv A sseqin2 biimp y cv `' {/} {} {} " eqeq1d y cv z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} u cv {/} {} <,> eleq2 p0ex u visset y cv `' elimasn p0ex u visset y cv opelcnv bitr syl5bb v v cv {/} = v cv {/} = u cv x cv e. /\ eq2ab {/} v df-sn v v cv {/} = u cv x cv e. /\ {|} eqeq1i u cv x cv e. v cv {/} = iba v 19.21aiv 0ex v cv {/} {/} eqeq1 v cv {/} {/} eqeq1 u cv x cv e. anbi1d bibi12d cla4v {/} eqid u cv x cv e. a1bi {/} {/} = u cv x cv e. pm4.71 bitr sylibr impbi 3bitr4r u cv A e. anbi2i u cv A x cv elin u visset p0ex z cv u cv A eleq1 z cv u cv x cv eleq1 v cv {/} = anbi2d v biabdv w cv eqeq2d anbi12d w cv {/} {} v v cv {/} = u cv x cv e. /\ {|} eqeq1 u cv A e. anbi2d opelopab 3bitr4 syl6rbbr eqrd syl5bi com12 x cv y cv `' {/} {} {} " A sseq1 y cv `' {/} {} {} imassrn {/} v df-sn p0ex eqeltrr v cv {/} = z cv x cv e. pm3.26 v ss2abi ssexi z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} eqid fnopab2 y cv z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} A fneq1 mpbiri y cv A fndm syl y cv dfdm4 syl5eqr y cv `' {/} {} {} " sseq2d mpbii syl5bir com12 impbid x visset A elpw syl5bb pm5.32ri x cv y cv `' {/} {} {} " = y cv z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} = ancom z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} eqid p0ex {/} pri2 df2o2 eleqtrr z cv x cv e. v cv {/} = iba v {/} elsn syl5bb biabrdv 2o eleq1d mpbii 0ex {/} {} pri1 df2o2 eleqtrr v v cv {/} = z cv x cv e. /\ abn0 v cv {/} = z cv x cv e. pm3.27 v 19.23aiv sylbi con1i 2o eleq1d mpbiri pm2.61i z cv A e. a1i fopab y cv z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} A 2o feq1 mpbiri x cv y cv `' {/} {} {} " = a1i x cv y cv `' {/} {} {} " z cv eleq2 p0ex z visset y cv `' elimasn p0ex z visset y cv opelcnv bitr syl6bb p0ex y cv A z cv fnfvop y cv A 2o ffn sylan bicomd sylan9bb biimpa z cv x cv e. v cv {/} = iba v {/} elsn syl5bb biabrdv x cv y cv `' {/} {} {} " = y cv A 2o :--> z cv A e. /\ /\ adantl eqtrd y cv A 2o z cv ffvrn df2o2 y cv z cv ` eleq2i y cv z cv fvex {/} {/} {} elpr bitr sylib ord x cv y cv `' {/} {} {} " = adantl x cv y cv `' {/} {} {} " z cv eleq2 p0ex z visset y cv `' elimasn p0ex z visset y cv opelcnv bitr syl6bb p0ex y cv A z cv fnfvop y cv A 2o ffn sylan bicomd sylan9bb sylibrd con1d imp v v cv {/} = z cv x cv e. /\ abn0 v cv {/} = z cv x cv e. pm3.27 v 19.23aiv sylbi con1i x cv y cv `' {/} {} {} " = y cv A 2o :--> z cv A e. /\ /\ adantl eqtr4d pm2.61an2 {/} v df-sn p0ex eqeltrr v cv {/} = z cv x cv e. pm3.26 v ss2abi ssexi z A v v cv {/} = z cv x cv e. /\ {|} V w fvopab2 mpan2 x cv y cv `' {/} {} {} " = y cv A 2o :--> ad2antrr eqtr4d exp32 imp r19.21aiv y cv A 2o ffn {/} v df-sn p0ex eqeltrr v cv {/} = z cv x cv e. pm3.26 v ss2abi ssexi z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} eqid fnopab2 u cv y cv e. z ax-17 u z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ hbopab1 A A cleqfvf mpan2 A eqid z A y cv z cv ` z w z cv A e. w cv v v cv {/} = z cv x cv e. /\ {|} = /\ {<,>|} z cv ` = A.e. biantrur syl6bbr syl x cv y cv `' {/} {} {} " = adantl mpbird exp impbid 2ord elisseti pw2en.1 y cv elmap syl6bbr pm5.32ri 3bitr en2))
+thm (sbthlem1 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (C_ (U. D) (\ A (" (cv g) (\ B (" (cv f) (U. D)))))) (D A g cv B f cv D U. " \ " \ x unissb sbthlem.2 eqabi x cv A g cv B f cv x cv " \ " ssconb biimprd exp A x cv difss g cv B f cv x cv " \ " A x cv \ A sstr2 mpi syl5 pm2.43d imp sylbi x cv D elssuni x cv D U. f cv imass2 f cv x cv " f cv D U. " B sscon 3syl B f cv D U. " \ B f cv x cv " \ g cv imass2 g cv B f cv D U. " \ " g cv B f cv x cv " \ " A sscon 3syl sstrd mprgbir))
+thm (sbthlem2 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (C_ (ran (cv g)) A) (C_ (\ A (" (cv g) (\ B (" (cv f) (U. D))))) (U. D))) (sbthlem.1 sbthlem.2 sbthlem1 D U. A g cv B f cv D U. " \ " \ f cv imass2 ax-mp f cv D U. " f cv A g cv B f cv D U. " \ " \ " B sscon ax-mp B f cv A g cv B f cv D U. " \ " \ " \ B f cv D U. " \ g cv imass2 ax-mp g cv B f cv A g cv B f cv D U. " \ " \ " \ " g cv B f cv D U. " \ " A sscon ax-mp g cv B f cv A g cv B f cv D U. " \ " \ " \ imassrn g cv B f cv A g cv B f cv D U. " \ " \ " \ " g cv ran A sstr2 ax-mp A g cv B f cv D U. " \ " difss g cv B f cv A g cv B f cv D U. " \ " \ " \ " A A g cv B f cv D U. " \ " \ ssconb mpan2 syl mpbiri A g cv B f cv D U. " \ " difss jctil sbthlem.1 A g cv B f cv D U. " \ " difss ssexi x cv A g cv B f cv D U. " \ " \ A sseq1 x cv A g cv B f cv D U. " \ " \ A difeq2 g cv B f cv x cv " \ " sseq2d x cv A g cv B f cv D U. " \ " \ f cv imaeq2 B difeq2d B f cv x cv " \ B f cv A g cv B f cv D U. " \ " \ " \ g cv imaeq2 g cv B f cv x cv " \ " g cv B f cv A g cv B f cv D U. " \ " \ " \ " A A g cv B f cv D U. " \ " \ \ sseq1 3syl bitrd anbi12d elab sylibr sbthlem.2 syl6eleqr A g cv B f cv D U. " \ " \ D elssuni syl))
+thm (sbthlem3 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (C_ (ran (cv g)) A) (= (" (cv g) (\ B (" (cv f) (U. D)))) (\ A (U. D)))) (sbthlem.1 sbthlem.2 sbthlem2 sbthlem.1 sbthlem.2 sbthlem1 jctil D U. A g cv B f cv D U. " \ " \ eqss sylibr A difeq2d g cv B f cv D U. " \ imassrn g cv B f cv D U. " \ " g cv ran A sstr2 ax-mp g cv B f cv D U. " \ " A dfss4 sylib eqtr2d))
+thm (sbthlem4 ((A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))))) (-> (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g)))) (= (" (`' (cv g)) (\ A (U. D))) (\ B (" (cv f) (U. D))))) (B f cv D U. " difss g cv dom B B f cv D U. " \ sseq2 mpbiri B f cv D U. " \ g cv ssdmres sylib g cv B f cv D U. " \ |` dfdm4 syl5reqr g cv B f cv D U. " \ funcnvres sbthlem.1 sbthlem.2 sbthlem3 g cv B f cv D U. " \ " A D U. \ g cv `' reseq2 syl sylan9eqr rneqd sylan9eq anassrs g cv `' A D U. \ df-ima syl6reqr))
+thm (sbthlem5 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (= (dom (cv f)) A) (C_ (ran (cv g)) A)) (= (dom H) A)) (sbthlem.1 sbthlem.2 sbthlem1 A g cv B f cv D U. " \ " difss sstri f cv dom A D U. sseq2 mpbiri D U. f cv dom dfss sylib A D U. \ uneq1d g cv B f cv D U. " \ imassrn sbthlem.1 sbthlem.2 sbthlem3 g cv ran sseq1d mpbii A D U. \ g cv ran dfss sylib D U. f cv dom i^i uneq2d sylan9eq sbthlem.3 dmeqi f cv D U. |` g cv `' A D U. \ |` dmun f cv D U. dmres g cv `' A D U. \ dmres g cv df-rn eqcomi A D U. \ ineq2i eqtr uneq12i 3eqtr syl6reqr sbthlem.1 sbthlem.2 sbthlem1 A g cv B f cv D U. " \ " difss sstri D U. A ssundif mpbi syl6eq))
+thm (sbthlem6 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (C_ (ran (cv f)) B) (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (= (ran H) B)) (sbthlem.1 sbthlem.2 sbthlem4 g cv `' A D U. \ df-ima syl5reqr f cv D U. " uneq2d f cv D U. |` g cv `' A D U. \ |` rnun sbthlem.3 rneqi f cv D U. df-ima g cv `' A D U. \ |` ran uneq1i 3eqtr4 syl6reqr f cv D U. imassrn f cv D U. " f cv ran B sstr2 ax-mp f cv D U. " B ssundif sylib sylan9eqr))
+thm (sbthlem7 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (Fun (cv f)) (Fun (`' (cv g)))) (Fun H)) (f cv D U. dmres D U. f cv dom inss1 eqsstr f cv D U. |` dom D U. g cv `' A D U. \ |` dom ssrin ax-mp g cv `' A D U. \ dmres A D U. \ g cv `' dom inss1 eqsstr g cv `' A D U. \ |` dom A D U. \ D U. sslin ax-mp sstri D U. A difdisj sseqtr f cv D U. |` dom g cv `' A D U. \ |` dom i^i ss0 ax-mp f cv D U. |` g cv `' A D U. \ |` funun mpan2 f cv D U. funres g cv `' A D U. \ funres syl2an sbthlem.3 H f cv D U. |` g cv `' A D U. \ |` u. funeq ax-mp sylibr))
+thm (sbthlem8 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (Fun (`' (cv f))) (/\ (/\ (/\ (Fun (cv g)) (= (dom (cv g)) B)) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (Fun (`' H))) (f cv D U. |` `' g cv `' A D U. \ |` `' funun f cv D U. funres11 g cv funcnvcnv g cv `' A D U. \ funres11 syl g cv dom B = adantr g cv ran A C_ g cv `' Fun ad2antll anim12i sbthlem.1 sbthlem.2 sbthlem4 g cv `' A D U. \ df-ima g cv `' A D U. \ |` df-rn eqtr syl5eqr f cv D U. df-ima f cv D U. |` df-rn eqtr2 jctil f cv D U. |` `' dom f cv D U. " g cv `' A D U. \ |` `' dom B f cv D U. " \ ineq12 syl f cv D U. " B difdisj syl6eq g cv Fun adantlll f cv `' Fun adantl sylanc sbthlem.3 H f cv D U. |` g cv `' A D U. \ |` u. cnveq ax-mp f cv D U. |` g cv `' A D U. \ |` cnvun eqtr H `' f cv D U. |` `' g cv `' A D U. \ |` `' u. funeq ax-mp sylibr))
+thm (sbthlem9 ((H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))))) (-> (/\ (:-1-1-> (cv f) A B) (:-1-1-> (cv g) B A)) (:-1-1-onto-> H A B)) (sbthlem.1 sbthlem.2 sbthlem.3 sbthlem7 sbthlem.1 sbthlem.2 sbthlem.3 sbthlem5 g cv Fun g cv dom B = /\ adantrl anim12i an42s f cv ran B C_ adantlr f cv `' Fun adantlr sbthlem.1 sbthlem.2 sbthlem.3 sbthlem8 f cv Fun f cv dom A = /\ f cv ran B C_ /\ adantll sbthlem.1 sbthlem.2 sbthlem.3 sbthlem6 H df-rn syl5eqr g cv Fun g cv dom B = pm3.27 g cv ran A C_ anim1i g cv `' Fun anim1i sylan2 f cv Fun f cv dom A = /\ adantll f cv `' Fun adantlr jca jca f cv A B df-f1 f cv A B df-f f cv A df-fn f cv ran B C_ anbi1i bitr f cv `' Fun anbi1i bitr g cv B A df-f1 g cv B A df-f g cv B df-fn g cv ran A C_ anbi1i bitr g cv `' Fun anbi1i bitr anbi12i H A B f1o4 H A df-fn H `' B df-fn anbi12i bitr 3imtr4))
+thm (sbthlem10 ((f g) (A f) (A g) (B f) (B g) (H x) (A x) (B x) (D x) (f x) (g x)) ((sbthlem.1 (e. A (V))) (sbthlem.2 (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (sbthlem.3 (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (sbthlem.4 (e. B (V)))) (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B)) (sbthlem.4 A f brdom sbthlem.1 B g brdom anbi12i f g f cv A B :-1-1-> g cv B A :-1-1-> eeanv bitr4 sbthlem.1 sbthlem.2 sbthlem.3 sbthlem9 sbthlem.1 H B f1oen syl f g 19.23aivv sylbi))
+thm (sbth ((x y) (x z) (w x) (f x) (g x) (A x) (y z) (w y) (f y) (g y) (A y) (w z) (f z) (g z) (A z) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B f) (B g)) () (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B)) (z cv A ~<_ w cv breq1 z cv A w cv ~<_ breq2 anbi12d z cv A ~~ w cv breq1 imbi12d w cv B A ~<_ breq2 w cv B ~<_ A breq1 anbi12d w cv B A ~~ breq2 imbi12d z visset y cv x cv z cv sseq1 y cv x cv f cv imaeq2 w cv difeq2d w cv f cv y cv " \ w cv f cv x cv " \ g cv imaeq2 g cv w cv f cv y cv " \ " g cv w cv f cv x cv " \ " z cv y cv \ sseq1 3syl y cv x cv z cv difeq2 g cv w cv f cv x cv " \ " sseq2d bitrd anbi12d cbvabv f cv y y cv z cv C_ g cv w cv f cv y cv " \ " z cv y cv \ C_ /\ {|} U. |` g cv `' z cv y y cv z cv C_ g cv w cv f cv y cv " \ " z cv y cv \ C_ /\ {|} U. \ |` u. eqid w visset sbthlem10 V V vtocl2g reldom A B brrelexi reldom B A brrelexi syl2an pm2.43i))
+thm (sbthbg () () (-> (e. B C) (<-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B))) (A B sbth B C e. a1i A B endom B C e. a1i B C A ensymg B A endom syl6 jcad impbid))
+thm (sbthcl ((x y)) () (= (~~) (i^i (~<_) (`' (~<_)))) (relen reldom ~<_ ~<_ `' relin ax-mp y visset y cv V x cv sbthbg ax-mp x cv ~<_ y cv df-br y cv ~<_ x cv df-br x visset y visset ~<_ opelcnv bitr4 anbi12i x cv y cv <,> ~<_ ~<_ `' elin bitr4 x cv ~~ y cv df-br 3bitr3r cleqreli))
+thm (dfsdom2 () () (= (~<) (\ (~<_) (`' (~<_)))) (df-sdom sbthcl ~<_ difeq2i ~<_ ~<_ `' difin 3eqtr))
+thm (brsdom2 () ((brsdom2.1 (e. A (V))) (brsdom2.2 (e. B (V)))) (<-> (br A (~<) B) (/\ (br A (~<_) B) (-. (br B (~<_) A)))) (dfsdom2 A B <,> eleq2i A ~< B df-br A ~<_ B df-br B ~<_ A df-br brsdom2.1 brsdom2.2 ~<_ opelcnv bitr4 negbii anbi12i A B <,> ~<_ ~<_ `' eldif bitr4 3bitr4))
+thm (sdomnsym () () (-> (br A (~<) B) (-. (br B (~<) A))) (A B sdomnen A B sbth A B sdomdom B A sdomdom syl2an exp mtod))
+thm (domnsym () () (-> (br A (~<_) B) (-. (br B (~<) A))) (A B brdom2 A B sdomnsym B V A ensymg con3d relsdom B A brrelexi B A sdomnen sylc con2i jaoi sylbi))
+thm (0dom () () (br ({/}) (~<_) A) (0ex A 0ss {/} V A ssdomg mp2))
+thm (dom0 () () (<-> (br A (~<_) ({/})) (= A ({/}))) (A 0dom A ~<_ {/} br biantru 0ex {/} V A sbthbg ax-mp A en0 3bitr))
+thm (0sdom () ((0sdom.1 (e. A (V)))) (<-> (br ({/}) (~<) A) (-. (= A ({/})))) ({/} A brsdom A 0dom mpbiran 0sdom.1 {/} ensym 0ex A ensym impbi A en0 bitr negbii bitr))
+thm (sdom0 () () (-. (br A (~<) ({/}))) (A 0dom {/} A domnsym ax-mp))
+thm (sdomdomtr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~<_) C)) (br A (~<) C))) (A B sdomnen C D e. B ~<_ C br ad2antrl A B C domtr A B sdomdom sylan A C brdom2 A ~< C br A ~~ C br df-or bitr sylib C D e. adantl A B C domtr A B sdomdom sylan A C brdom2 A ~< C br A ~~ C br df-or bitr sylib C D A ensymg C A endom syl6 sylan9r A B sdomdom C D e. B ~<_ C br ad2antrl jctird C A B domtr syl6 A ~< B br B ~<_ C br pm3.27 C D e. adantl jctird C B sbth syl6 jcad A C B entrt syl6 mt3d exp))
+thm (sdomentr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~~) C)) (br A (~<) C))) (C D A B sdomdomtr B C endom sylan2i))
+thm (ensdomtr () () (-> (/\ (br A (~~) B) (br B (~<) C)) (br A (~<) C)) (A B C endomtr exp B V e. adantl B V A ensymg imp B A C entrt exp syl con3d anim12d B C brsdom A C brsdom 3imtr4g exp imp3a relsdom B C brrelexi con3i A ~< C br pm2.21d A ~~ B br adantld pm2.61i))
+thm (sdomtr () () (-> (/\ (br A (~<) B) (br B (~<) C)) (br A (~<) C)) (B C sdomex pm3.27d C V A B sdomdomtr B C sdomdom sylan2i syl anabsi7))
+thm (domsdomtr () () (-> (/\ (br A (~<_) B) (br B (~<) C)) (br A (~<) C)) (A B brdom2 A B C sdomtr exp relsdom B C brrelexi A B C endomtr exp B V e. adantl B V A ensymg B A C entrt exp syl6 imp con3d anim12d B C brsdom A C brsdom 3imtr4g exp syl pm2.43b jaoi sylbi imp))
+thm (enen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~~) C) (br B (~~) C))) (B D A ensymg imp B A C entrt exp syl A B C entrt B D e. adantll exp impbid))
+thm (enen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~~) A) (br C (~~) B))) (C A B entrt expcom B D e. adantl C B A entrt exp B D A ensymg imp syl5com impbid))
+thm (domen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<_) C) (br B (~<_) C))) (B D A ensymg imp B A C endomtr exp syl A B C endomtr exp B D e. adantl impbid))
+thm (domen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<_) A) (br C (~<_) B))) (C A B domentr expcom B D e. adantl C B A domentr exp B D A ensymg imp syl5com impbid))
+thm (sdomen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<) C) (br B (~<) C))) (B D A ensymg imp B A C ensdomtr exp syl A B C ensdomtr B D e. adantll exp impbid))
+thm (sdomen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<) A) (br C (~<) B))) (B D C A sdomentr exp3a com23 imp A V e. adantll B D A ensymg A V C B sdomentr exp3a com23 syl9r imp31 impbid exp31 imp3a relen A B brrelexi con3i C ~< A br C ~< B br <-> pm2.21d B D e. adantld pm2.61i))
+thm (canth2 ((x y) (f x) (A x) (f y) (A y) (A f)) ((canth2.1 (e. A (V)))) (br A (~<) (P~ A)) (A A P~ brsdom canth2.1 x visset A snelpw biimp x visset y cv sneqr x cv y cv sneq impbi x cv A e. y cv A e. /\ a1i V dom2 ax-mp canth2.1 f cv canth f cv A A P~ df-f1o pm3.27bd mto f nex canth2.1 pwex A f bren mtbir mpbir2an))
+thm (canth2g ((A x)) () (-> (e. A B) (br A (~<) (P~ A))) (x cv A pweq x cv A x cv P~ A P~ ~< breq12 mpdan x visset canth2 B vtoclg))
+thm (xpen () ((xpen.1 (e. A (V))) (xpen.2 (e. B (V))) (xpen.3 (e. C (V))) (xpen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (X. A C) (~~) (X. B D))) (A C X. B C X. B D X. entrt xpen.1 xpen.2 xpen.3 xpdom2 xpen.2 xpen.1 xpen.3 xpdom2 anim12i xpen.2 B V A sbthbg ax-mp xpen.3 xpen.2 xpex C B X. V C A X. sbthbg ax-mp 3imtr3 xpen.2 xpen.3 xpex xpen.3 xpen.2 xpcomen B C X. V C B X. C A X. enen2 mp2an xpen.1 xpen.3 xpex xpen.3 xpen.1 xpcomen A C X. V C A X. B C X. enen1 mp2an bitr sylib xpen.3 xpen.4 xpen.2 xpdom2 xpen.4 xpen.3 xpen.2 xpdom2 anim12i xpen.4 D V C sbthbg ax-mp xpen.2 xpen.4 xpex B D X. V B C X. sbthbg ax-mp 3imtr3 syl2an))
+thm (mapenlem1 ((f g) (f x) (f y) (f z) (f v) (A f) (g x) (g y) (g z) (g v) (A g) (x y) (x z) (v x) (A x) (y z) (v y) (A y) (v z) (A z) (A v) (B f) (B g) (B x) (B y) (B z) (B v) (C f) (C g) (C x) (C y) (C z) (C v) (D f) (D g) (D x) (D y) (D z) (D v) (H z) (H v)) ((mapenlem.1 (e. A (V))) (mapenlem.2 (e. B (V))) (mapenlem.3 (e. C (V))) (mapenlem.4 (e. D (V))) (mapenlem.5 (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g))))))))) (-> (/\ (/\ (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:--> (cv z) C A)) (e. (cv v) C)) (= (` (` H (cv z)) (` (cv g) (cv v))) (` (cv f) (` (cv z) (cv v))))) (mapenlem.1 mapenlem.3 z cv elmap x cv z cv f cv coeq2 g cv `' coeq1d mapenlem.5 f visset z visset coex g visset cnvex coex fvopab4 sylbir g cv v cv ` fveq1d f cv A B :-1-1-onto-> g cv C D :-1-1-onto-> /\ v cv C e. ad2antlr g cv C D f1ococnv1 f cv z cv o. coeq2d f cv z cv o. C B fcoi1 sylan9eqr f cv A B z cv C fco f cv A B f1of sylan sylan an1rs f cv z cv o. g cv `' g cv coass syl5eq v cv fveq1d v cv C e. adantr f cv z cv o. g cv `' o. g cv C D v cv fvco3 exp f cv z cv o. g cv `' funco f cv z cv funco f cv A B f1ofun z cv C A ffun syl2an g cv C D f1o3 pm3.27bd syl2an g cv C D f1of syl2an exp31 pm2.43d exp com23 imp41 f cv z cv C A v cv fvco3 anasss f cv A B f1ofun sylan g cv C D :-1-1-onto-> adantlr anassrs 3eqtr3d eqtrd))
+thm (mapenlem2 ((f g) (f x) (f y) (f z) (f w) (f v) (A f) (g x) (g y) (g z) (g w) (g v) (A g) (x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B f) (B g) (B x) (B y) (B z) (B w) (B v) (C f) (C g) (C x) (C y) (C z) (C w) (C v) (D f) (D g) (D x) (D y) (D z) (D w) (D v) (H z) (H w) (H v)) ((mapenlem.1 (e. A (V))) (mapenlem.2 (e. B (V))) (mapenlem.3 (e. C (V))) (mapenlem.4 (e. D (V))) (mapenlem.5 (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g))))))))) (-> (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:-1-1-onto-> H (opr A (^m) C) (opr B (^m) D))) (f cv x cv o. C B g cv `' D fco f cv A B x cv C fco f cv A B f1of sylan g cv C D f1ocnv g cv `' D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 x cv elmap mapenlem.2 mapenlem.4 f cv x cv o. g cv `' o. elmap 3imtr4g r19.21aiv mapenlem.5 B ^m D opr fopab2 sylib H z cv ` H w cv ` g cv v cv ` fveq1 z cv C A :--> w cv C A :--> /\ adantl f cv A B :-1-1-onto-> g cv C D :-1-1-onto-> /\ v cv C e. ad2antlr mapenlem.1 mapenlem.2 mapenlem.3 mapenlem.4 mapenlem.5 z v mapenlem1 H z cv ` H w cv ` = adantrl exp43 w cv C A :--> adantrd imp42 mapenlem.1 mapenlem.2 mapenlem.3 mapenlem.4 mapenlem.5 w v mapenlem1 H z cv ` H w cv ` = adantrl exp43 z cv C A :--> adantld imp42 3eqtr3d f cv A B z cv v cv ` w cv v cv ` f1fveq f cv A B f1of1 z cv C A v cv ffvrn w cv C A :--> adantlr w cv C A v cv ffvrn z cv C A :--> adantll jca H z cv ` H w cv ` = adantlr syl2an g cv C D :-1-1-onto-> adantlr anassrs mpbid exp r19.21aiv z cv C w cv C v eqfnfv C eqid v C z cv v cv ` w cv v cv ` = A.e. biantrur syl6bbr z cv C A ffn w cv C A ffn syl2an f cv A B :-1-1-onto-> g cv C D :-1-1-onto-> /\ H z cv ` H w cv ` = ad2antrl mpbird exp32 mapenlem.1 mapenlem.3 z cv elmap mapenlem.1 mapenlem.3 w cv elmap anbi12i syl5ib exp3a r19.21adv r19.21aiv jca H A ^m C opr B ^m D opr z w f1fv sylibr f cv x cv o. C B g cv `' D fco f cv A B x cv C fco f cv A B f1of sylan g cv C D f1ocnv g cv `' D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 x cv elmap mapenlem.2 mapenlem.4 f cv x cv o. g cv `' o. elmap 3imtr4g r19.21aiv mapenlem.5 B ^m D opr fopab2 sylib H A ^m C opr B ^m D opr frn syl f cv `' z cv o. D A g cv C fco f cv `' B A z cv D fco f cv A B f1ocnv f cv `' B A f1of syl sylan g cv C D f1of syl2an an1rs mapenlem.1 mapenlem.3 f cv `' z cv o. g cv o. elmap sylibr f cv `' z cv o. D A g cv C fco f cv `' B A z cv D fco f cv A B f1ocnv f cv `' B A f1of syl sylan g cv C D f1of syl2an an1rs mapenlem.1 mapenlem.3 f cv `' z cv o. g cv o. elmap sylibr x cv f cv `' z cv o. g cv o. f cv coeq2 g cv `' coeq1d mapenlem.5 f visset f visset cnvex z visset coex g visset coex coex g visset cnvex coex fvopab4 syl f cv A B f1ococnv2 z cv coeq1d z cv D B fcoi2 sylan9eq g cv g cv `' o. coeq1d g cv C D :-1-1-onto-> adantlr g cv C D f1ococnv2 z cv coeq2d z cv D B fcoi1 sylan9eq f cv A B :-1-1-onto-> adantll eqtrd f cv f cv `' z cv o. g cv o. g cv `' coass f cv `' z cv o. g cv g cv `' coass f cv coeq2i f cv f cv `' z cv coass g cv g cv `' o. coeq1i f cv f cv `' z cv o. g cv g cv `' o. coass eqtr2 3eqtr syl5eq eqtrd jca exp mapenlem.2 mapenlem.4 z cv elmap syl5ib w cv f cv `' z cv o. g cv o. H fveq2 z cv eqeq1d A ^m C opr rcla4ev syl6 f cv x cv o. C B g cv `' D fco f cv A B x cv C fco f cv A B f1of sylan g cv C D f1ocnv g cv `' D C f1of syl syl2an exp31 com23 imp mapenlem.1 mapenlem.3 x cv elmap mapenlem.2 mapenlem.4 f cv x cv o. g cv `' o. elmap 3imtr4g r19.21aiv mapenlem.5 B ^m D opr fopab2 sylib H A ^m C opr B ^m D opr ffn H A ^m C opr z cv w fvelrn 3syl sylibrd ssrdv eqssd jca H A ^m C opr B ^m D opr f1o5 sylibr))
+thm (mapen ((f g) (f x) (f y) (A f) (g x) (g y) (A g) (x y) (A x) (A y) (B f) (B g) (B x) (B y) (C f) (C g) (C x) (C y) (D f) (D g) (D x) (D y)) ((mapen.1 (e. A (V))) (mapen.2 (e. B (V))) (mapen.3 (e. C (V))) (mapen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (opr A (^m) C) (~~) (opr B (^m) D))) (mapen.1 mapen.2 mapen.3 mapen.4 x y x cv A ^m C opr e. y cv f cv x cv o. g cv `' o. = /\ {<,>|} eqid mapenlem2 A ^m C oprex x y x cv A ^m C opr e. y cv f cv x cv o. g cv `' o. = /\ {<,>|} B ^m D opr f1oen syl exp f 19.23aiv g 19.23adv imp mapen.2 A f bren mapen.4 C g bren syl2anb))
+thm (mapdom1 ((A x) (B x) (C x)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr A (^m) C) (~<_) (opr B (^m) C))) (mapdom1.2 A x domen A ^m C opr x cv ^m C opr B ^m C opr endomtr mapdom1.3 enref mapdom1.1 x visset mapdom1.3 mapdom1.3 mapen mpan2 mapdom1.2 mapdom1.3 x cv mapss x cv ^m C oprex x cv ^m C opr V B ^m C opr ssdomg ax-mp syl syl2an x 19.23aiv sylbi))
+thm (mapdom2lem ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w) (C x) (C z) (C w)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (e. (cv x) (opr C (^m) (cv z))) (= (i^i (cv x) (X. (\ B (cv z)) ({} (cv w)))) ({/}))) (mapdom1.3 z visset x cv elmap biimp x cv z cv C fdm syl w visset B z cv \ fconst B z cv \ w cv {} X. B z cv \ w cv {} fdm ax-mp x cv C ^m z cv opr e. a1i ineq12d z cv B difdisj syl6eq x cv B z cv \ w cv {} X. dmin x cv dom B z cv \ w cv {} X. dom i^i {/} x cv B z cv \ w cv {} X. i^i dom sseq2 mpbii x cv B z cv \ w cv {} X. i^i dom ss0 syl syl B z cv \ w cv {} relxp B z cv \ w cv {} X. x cv relin ax-mp B z cv \ w cv {} X. x cv incom B z cv \ w cv {} X. x cv i^i x cv B z cv \ w cv {} X. i^i releq ax-mp mpbi x cv B z cv \ w cv {} X. i^i reldm0 ax-mp sylibr))
+thm (mapdom2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((mapdom1.1 (e. A (V))) (mapdom1.2 (e. B (V))) (mapdom1.3 (e. C (V)))) (-> (/\ (br A (~<_) B) (-. (/\ (= A ({/})) (= C ({/}))))) (br (opr C (^m) A) (~<_) (opr C (^m) B))) (C {/} ^m A opreq1 mapdom1.1 map0b sylan9eqr C ^m B opr 0dom syl6eqbr A ~<_ B br a1i mapdom1.2 A z domen C ^m A opr C ^m z cv opr C ^m B opr endomtr mapdom1.3 enref mapdom1.3 mapdom1.3 mapdom1.1 z visset mapen mpan C ^m z cv oprex z cv B ssundif z cv B z cv \ u. B x cv B z cv \ w cv {} X. u. C w cv {} u. feq2 sylbi w cv C snssi w cv {} C ssequn2 sylib C w cv {} u. C x cv B z cv \ w cv {} X. u. B feq3 syl sylan9bb w visset B z cv \ fconst z cv B difdisj x cv z cv C B z cv \ w cv {} X. B z cv \ w cv {} fun mpan2 mpan2 syl5bi mapdom1.3 z visset x cv elmap mapdom1.3 mapdom1.2 x cv B z cv \ w cv {} X. u. elmap 3imtr4g mapdom1.1 mapdom1.2 mapdom1.3 x z w mapdom2lem mapdom1.1 mapdom1.2 mapdom1.3 y z w mapdom2lem eqcomd sylan9eq x cv B z cv \ w cv {} X. u. y cv B z cv \ w cv {} X. u. = biantrud x cv B z cv \ w cv {} X. y cv unineq syl6bb z cv B C_ w cv C e. /\ a1i V dom2d mpi syl2an anassrs exp z 19.23aiv sylbi w 19.23adv C w n0 syl5ib jaod imp C {/} = exmid A {/} = -. C {/} = -. \/ biantru A {/} = C {/} = ianor A {/} = -. C {/} = C {/} = -. ordir 3bitr4 sylan2b))
+thm (mapxpen ((x y) (x z) (w x) (v x) (f x) (g x) (A x) (y z) (w y) (v y) (f y) (g y) (A y) (w z) (v z) (f z) (g z) (A z) (v w) (f w) (g w) (A w) (f v) (g v) (A v) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (C x) (C y) (C z) (C w) (C v) (C f) (C g)) ((mapxpen.1 (e. A (V))) (mapxpen.2 (e. B (V))) (mapxpen.3 (e. C (V)))) (br (opr (opr A (^m) B) (^m) C) (~~) (opr A (^m) (X. B C))) (A ^m B opr ^m C oprex mapxpen.2 mapxpen.3 z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} eqid oprabex2 x cv A ^m B opr ^m C opr e. a1i mapxpen.3 w f z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} funopabex2 y cv A ^m B C X. opr e. a1i x cv w cv ` z cv fvex z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} eqid fnoprab2 y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} B C X. fneq1 mpbiri x cv A ^m B opr ^m C opr e. adantl x cv A ^m B opr ^m C opr e. z ax-17 y z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ hboprab1 hbeleq hban x cv A ^m B opr ^m C opr e. w ax-17 y z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ hboprab2 hbeleq hban z cv B e. w ax-17 x cv w cv ` B A z cv ffvrn mapxpen.1 mapxpen.2 x cv w cv ` elmap sylanb x cv C A ^m B opr w cv ffvrn A ^m B oprex mapxpen.3 x cv elmap sylanb sylan an1rs anasss y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = adantlr y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} z cv w cv opreq x cv w cv ` z cv fvex z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} eqid V oprabval4g mp3an3 sylan9eq A eleq1d x cv A ^m B opr ^m C opr e. adantll mpbird exp32 r19.21ad r19.21ai jca mapxpen.1 mapxpen.2 mapxpen.3 xpex y cv elmap y cv B C A z w ffnoprval bitr sylibr A ^m B oprex mapxpen.3 x cv elmap x cv C A ^m B opr w f fopabfv pm3.26bd sylbi y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = adantr x cv A ^m B opr ^m C opr e. w ax-17 y z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ hboprab2 hbeleq hban x cv A ^m B opr ^m C opr e. y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = /\ f ax-17 x cv C A ^m B opr w cv ffvrn A ^m B oprex mapxpen.3 x cv elmap sylanb y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = adantlr y z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ hboprab1 hbeleq w cv C e. z ax-17 hban y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = w cv C e. /\ g ax-17 y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} z cv w cv opreq x cv w cv ` z cv fvex z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} eqid V oprabval4g mp3an3 sylan9eq anassrs an1rs g cv eqeq2d exp pm5.32d biopabd x cv w cv ` eqeq2d mapxpen.1 mapxpen.2 x cv w cv ` elmap x cv w cv ` B A z g fopabfv pm3.26bd sylbi syl5bir x cv A ^m B opr ^m C opr e. adantll mpd f cv eqeq2d exp pm5.32d biopabd eqtrd jca w cv C e. z ax-17 f z g z cv B e. g cv z cv y cv w cv opr = /\ hbopab1 hbeleq hban x w f hbopab hbeleq x w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ hbopab1 hbeleq x cv w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} = v ax-17 x cv w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} w cv fveq1 mapxpen.2 z g z cv y cv w cv opr funopabex2 w C z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} V f fvopab2 mpan2 sylan9eq z cv fveq1d z cv y cv w cv oprex z B z cv y cv w cv opr V g fvopab2 mpan2 sylan9eq an1rs anasss v cv eqeq2d exp pm5.32d bioprabd y cv eqeq2d y cv B C z w v fnoprval syl6bbr z B w C z cv y cv w cv opr A e. A.e. A.e. anbi1d mapxpen.1 mapxpen.2 mapxpen.3 xpex y cv elmap y cv B C A z w ffnoprval bitr syl6bbr biimparc y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = z B w C z cv y cv w cv opr A e. A.e. A.e. ancom sylib mapxpen.2 z g z cv y cv w cv opr funopabex2 w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} eqid fnopab2 x cv w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} C fneq1 mpbiri w C x cv w cv ` A ^m B opr e. A.e. biantrurd A ^m B oprex mapxpen.3 x cv elmap x cv C A ^m B opr w ffnfv bitr2 syl6bb x w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ hbopab1 hbeleq w cv C e. z ax-17 f z g z cv B e. g cv z cv y cv w cv opr = /\ hbopab1 hbeleq hban x w f hbopab hbeleq w cv C e. z ax-17 hban x cv w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} w cv fveq1 mapxpen.2 z g z cv y cv w cv opr funopabex2 w C z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} V f fvopab2 mpan2 sylan9eq z cv fveq1d z cv y cv w cv oprex z B z cv y cv w cv opr V g fvopab2 mpan2 sylan9eq A eleq1d biralda x cv w cv ` B Fn anbi2d mapxpen.1 mapxpen.2 x cv w cv ` elmap x cv w cv ` B A z ffnfv bitr syl5bb x cv w f w cv C e. f cv z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} = /\ {<,>|} w cv fveq1 mapxpen.2 z g z cv y cv w cv opr funopabex2 w C z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} V f fvopab2 mpan2 sylan9eq z cv y cv w cv oprex z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} eqid fnopab2 x cv w cv ` z g z cv B e. g cv z cv y cv w cv opr = /\ {<,>|} B fneq1 mpbiri syl z B z cv y cv w cv opr A e. A.e. biantrurd bitr4d biralda bitr3d w C z B z cv y cv w cv opr A e. ralcom syl6bb y cv A ^m B C X. opr e. adantl y cv z w v z cv B e. w cv C e. /\ v cv x cv w cv ` z cv ` = /\ {<<,>,>|} = anbi1d mpbird impbi en2))
+thm (xpmapenlem1 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (/\ (-> (= (cv y) (<,> D R)) (A. z (= (cv y) (<,> D R)))) (-> (= (cv y) (<,> D R)) (A. w (= (cv y) (<,> D R))))) (y z w z cv C e. w cv x cv z cv ` {} dom U. = /\ hbopab1 xpmapenlem.4 y cv eleq2i xpmapenlem.4 y cv eleq2i z bial 3imtr4 y z w z cv C e. w cv x cv z cv ` {} ran U. = /\ hbopab1 xpmapenlem.5 y cv eleq2i xpmapenlem.5 y cv eleq2i z bial 3imtr4 hbop hbeleq y z w z cv C e. w cv x cv z cv ` {} dom U. = /\ hbopab2 xpmapenlem.4 y cv eleq2i xpmapenlem.4 y cv eleq2i w bial 3imtr4 y z w z cv C e. w cv x cv z cv ` {} ran U. = /\ hbopab2 xpmapenlem.5 y cv eleq2i xpmapenlem.5 y cv eleq2i w bial 3imtr4 hbop hbeleq pm3.2i))
+thm (xpmapenlem2 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (= (cv y) (<,> D R)) (e. (cv z) C)) (/\ (= (` (U. (dom ({} (cv y)))) (cv z)) (U. (dom ({} (` (cv x) (cv z)))))) (= (` (U. (ran ({} (cv y)))) (cv z)) (U. (ran ({} (` (cv x) (cv z)))))))) (y cv D R <,> sneq xpmapenlem.4 D z w z cv C e. w cv x cv z cv ` {} dom U. = /\ {<,>|} R opeq1 ax-mp sneqi syl6eq dmeqd unieqd xpmapen.3 z w x cv z cv ` {} dom U. funopabex2 R op1sta syl6eq z cv fveq1d x cv z cv ` snex x cv z cv ` {} V dmexg ax-mp uniex z C x cv z cv ` {} dom U. V w fvopab2 mpan2 sylan9eq y cv D R <,> sneq xpmapenlem.5 R z w z cv C e. w cv x cv z cv ` {} ran U. = /\ {<,>|} D opeq2 ax-mp sneqi syl6eq rneqd unieqd xpmapenlem.4 xpmapen.3 z w x cv z cv ` {} dom U. funopabex2 eqeltr xpmapen.3 z w x cv z cv ` {} ran U. funopabex2 op2nda syl6eq z cv fveq1d x cv z cv ` snex x cv z cv ` {} V rnexg ax-mp uniex z C x cv z cv ` {} ran U. V w fvopab2 mpan2 sylan9eq jca))
+thm (xpmapenlem3 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R))) (= (cv x) S)) (x cv C A B X. ffn x cv C z w fnopabfv sylib y cv D R <,> = adantr x cv C A B X. :--> z ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i hban x cv C A B X. :--> w ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i hban x cv C A B X. z cv ffvrn x cv z cv ` A B elxp4 pm3.26bd syl y cv D R <,> = adantlr xpmapen.1 xpmapen.1 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 y cv {} dom U. z cv ` x cv z cv ` {} dom U. y cv {} ran U. z cv ` x cv z cv ` {} ran U. opeq12 syl x cv C A B X. :--> adantll eqtr4d w cv eqeq2d exp pm5.32d biopabd xpmapenlem.6 syl6eqr eqtrd))
+thm (xpmapenlem4 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (-> (/\ (/\ (= (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y)))))) (/\ (:--> (U. (dom ({} (cv y)))) C A) (:--> (U. (ran ({} (cv y)))) C B))) (= (cv x) S)) (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R)))) (xpmapenlem.6 x cv S z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} eqtrt x cv z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} C A B X. feq1 syl mpan2 y cv {} ran U. z cv fvex y cv {} dom U. z cv ` A B opelxp z C biral z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} eqid A B X. fopab2 z C y cv {} dom U. z cv ` A e. y cv {} ran U. z cv ` B e. r19.26 3bitr3r syl6rbbr biimpac y cv {} dom U. C A z cv ffvrn exp r19.21aiv y cv {} ran U. C B z cv ffvrn exp r19.21aiv anim12i sylan y cv y cv {} dom U. y cv {} ran U. <,> = adantll y cv y cv {} dom U. y cv {} ran U. <,> D R <,> eqeq1 y cv {} dom U. C A z w fopabfv pm3.26bd x z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ hbopab1 xpmapenlem.6 x cv eleq2i xpmapenlem.6 x cv eleq2i z bial 3imtr4 hbeleq x z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ hbopab2 xpmapenlem.6 x cv eleq2i xpmapenlem.6 x cv eleq2i w bial 3imtr4 hbeleq xpmapenlem.6 x cv S z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} eqtrt mpan2 z cv fveq1d y cv {} dom U. z cv ` y cv {} ran U. z cv ` opex z C y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> V w fvopab2 mpan2 sylan9eq sneqd dmeqd unieqd y cv {} dom U. z cv fvex y cv {} ran U. z cv ` op1sta syl6eq w cv eqeq2d exp pm5.32d biopabd xpmapenlem.4 syl5req sylan9eq y cv {} ran U. C B :--> adantlr y cv {} ran U. C B z w fopabfv pm3.26bd x z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ hbopab1 xpmapenlem.6 x cv eleq2i xpmapenlem.6 x cv eleq2i z bial 3imtr4 hbeleq x z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ hbopab2 xpmapenlem.6 x cv eleq2i xpmapenlem.6 x cv eleq2i w bial 3imtr4 hbeleq xpmapenlem.6 x cv S z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} eqtrt mpan2 z cv fveq1d y cv {} dom U. z cv ` y cv {} ran U. z cv ` opex z C y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> V w fvopab2 mpan2 sylan9eq sneqd rneqd unieqd y cv {} dom U. z cv fvex y cv {} ran U. z cv fvex op2nda syl6eq w cv eqeq2d exp pm5.32d biopabd xpmapenlem.5 syl5req sylan9eq y cv {} dom U. C A :--> adantll jca y cv snex y cv {} V dmexg ax-mp uniex y cv snex y cv {} V rnexg ax-mp uniex xpmapenlem.5 xpmapen.3 z w x cv z cv ` {} ran U. funopabex2 eqeltr D opth sylibr syl5bir exp3a imp31 jca))
+thm (xpmapenlem5 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V))) (xpmapenlem.4 (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z)))))))))) (xpmapenlem.5 (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z)))))))))) (xpmapenlem.6 (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z))))))))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C))) (A B X. ^m C oprex D R opex x cv A B X. ^m C opr e. a1i xpmapenlem.6 xpmapen.3 z w y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> funopabex2 eqeltr y cv A ^m C opr B ^m C opr X. e. a1i xpmapenlem.5 xpmapen.3 z w x cv z cv ` {} ran U. funopabex2 eqeltr D V V opelxp xpmapenlem.4 xpmapen.3 z w x cv z cv ` {} dom U. funopabex2 eqeltr xpmapenlem.5 xpmapen.3 z w x cv z cv ` {} ran U. funopabex2 eqeltr mpbir2an y cv D R <,> V V X. eleq1 mpbiri x cv C A B X. :--> adantl y cv V V elxp4 pm3.26bd syl y cv D R <,> sneq dmeqd unieqd xpmapenlem.4 xpmapen.3 z w x cv z cv ` {} dom U. funopabex2 eqeltr R op1sta syl6eq xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 pm3.26d w cv eqeq2d exp pm5.32d biopabd xpmapenlem.4 syl6eqr eqtr4d y cv D R <,> sneq rneqd unieqd xpmapenlem.4 xpmapen.3 z w x cv z cv ` {} dom U. funopabex2 eqeltr xpmapenlem.5 xpmapen.3 z w x cv z cv ` {} ran U. funopabex2 eqeltr op2nda syl6eq xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.27i xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem2 pm3.27d w cv eqeq2d exp pm5.32d biopabd xpmapenlem.5 syl6eqr eqtr4d jca x cv C A B X. :--> adantl x cv C A B X. z cv ffvrn exp r19.21aiv y cv D R <,> = adantr x cv C A B X. :--> z ax-17 xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem1 pm3.26i hban xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem3 z cv fveq1d y cv {} dom U. z cv ` y cv {} ran U. z cv ` opex z C y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> V w fvopab2 mpan2 xpmapenlem.6 z cv fveq1i syl5eq sylan9eq A B X. eleq1d y cv {} ran U. z cv fvex y cv {} dom U. z cv ` A B opelxp syl6bb biralda z C y cv {} dom U. z cv ` A e. y cv {} ran U. z cv ` B e. r19.26 syl6bb mpbid jca y cv {} dom U. z w z cv C e. w cv y cv {} dom U. z cv ` = /\ {<,>|} = y cv {} ran U. z w z cv C e. w cv y cv {} ran U. z cv ` = /\ {<,>|} = z C y cv {} dom U. z cv ` A e. A.e. z C y cv {} ran U. z cv ` B e. A.e. an4 y cv {} dom U. C A z w fopabfv y cv {} ran U. C B z w fopabfv anbi12i bitr4 sylib jca xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem3 jca xpmapen.1 xpmapen.2 xpmapen.3 xpmapenlem.4 xpmapenlem.5 xpmapenlem.6 xpmapenlem4 impbi xpmapen.1 xpmapen.2 xpex xpmapen.3 x cv elmap y cv D R <,> = anbi1i y cv A ^m C opr B ^m C opr elxp4 xpmapen.1 xpmapen.3 y cv {} dom U. elmap xpmapen.2 xpmapen.3 y cv {} ran U. elmap anbi12i y cv y cv {} dom U. y cv {} ran U. <,> = anbi2i bitr x cv S = anbi1i 3bitr4 en2))
+thm (xpmapen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((xpmapen.1 (e. A (V))) (xpmapen.2 (e. B (V))) (xpmapen.3 (e. C (V)))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C))) (xpmapen.1 xpmapen.2 xpmapen.3 z w z cv C e. w cv x cv z cv ` {} dom U. = /\ {<,>|} eqid z w z cv C e. w cv x cv z cv ` {} ran U. = /\ {<,>|} eqid z w z cv C e. w cv y cv {} dom U. z cv ` y cv {} ran U. z cv ` <,> = /\ {<,>|} eqid xpmapenlem5))
+thm (mapunen ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((mapunen.1 (e. A (V))) (mapunen.2 (e. B (V))) (mapunen.3 (e. C (V)))) (-> (= (i^i A B) ({/})) (br (opr C (^m) (u. A B)) (~~) (X. (opr C (^m) A) (opr C (^m) B)))) (C ^m A B u. oprex A B i^i {/} = a1i A B ssun1 x cv A B u. C A fssres mpan2 mapunen.3 mapunen.1 x cv A |` elmap sylibr B A ssun2 x cv A B u. C B fssres mpan2 mapunen.3 mapunen.2 x cv B |` elmap sylibr jca mapunen.3 mapunen.1 mapunen.2 unex x cv elmap x visset x cv V B resexg ax-mp x cv A |` C ^m A opr C ^m B opr opelxp 3imtr4 A B i^i {/} = a1i y cv |^| |^| A C y cv {} ran U. B C fun mapunen.3 mapunen.1 mapunen.2 unex y cv |^| |^| y cv {} ran U. u. elmap C unidm C C u. C y cv |^| |^| y cv {} ran U. u. A B u. feq3 ax-mp bitr4 sylibr y cv C ^m A opr C ^m B opr elxp5 pm3.27bd mapunen.3 mapunen.1 y cv |^| |^| elmap mapunen.3 mapunen.2 y cv {} ran U. elmap anbi12i sylib sylan expcom y cv C ^m A opr C ^m B opr elxp5 x cv A |` y cv |^| |^| x cv B |` y cv {} ran U. opeq12 x cv y cv |^| |^| y cv {} ran U. u. A reseq1 y cv |^| |^| y cv {} ran U. A resundir syl6eq y cv |^| |^| A fnresdm y cv {} ran U. A |` uneq1d y cv {} ran U. B A fnresdisj biimpa A B incom {/} eqeq1i sylan2b y cv |^| |^| uneq2d y cv |^| |^| un0 syl6eq sylan9eq anassrs sylan9eqr x cv y cv |^| |^| y cv {} ran U. u. B reseq1 y cv |^| |^| y cv {} ran U. B resundir syl6eq y cv {} ran U. B fnresdm y cv |^| |^| B |` uneq2d y cv |^| |^| A B fnresdisj biimpa y cv {} ran U. uneq1d {/} y cv {} ran U. uncom y cv {} ran U. un0 eqtr syl6eq sylan9eqr an1rs sylan9eqr sylanc y cv eqeq2d biimparc exp44 imp mapunen.3 mapunen.1 y cv |^| |^| elmap y cv |^| |^| A C ffn sylbi mapunen.3 mapunen.2 y cv {} ran U. elmap y cv {} ran U. B C ffn sylbi anim12i sylan2 sylbi impcom x cv C ^m A B u. opr e. adantrl mapunen.3 mapunen.1 mapunen.2 unex x cv elmap x cv A B u. C ffn sylbi x cv A B u. fnresdm syl eqcomd y cv x cv A |` x cv B |` <,> inteq inteqd x visset x cv V A resexg ax-mp x cv B |` op1stb syl6eq y cv x cv A |` x cv B |` <,> sneq rneqd unieqd x visset x cv V A resexg ax-mp x visset x cv V B resexg ax-mp op2nda syl6eq uneq12d x cv A B resundi syl6reqr sylan9eq exp A B i^i {/} = y cv C ^m A opr C ^m B opr X. e. ad2antrl impbid exp en3d))
+thm (pwen () ((pwen.1 (e. A (V))) (pwen.2 (e. B (V)))) (-> (br A (~~) B) (br (P~ A) (~~) (P~ B))) (2ord 2o On enrefg ax-mp 2ord elisseti 2ord elisseti pwen.1 pwen.2 mapen mpan 2o ^m A oprex pwen.1 pw2en 2o ^m A opr V A P~ B P~ enen1 mp2an 2o ^m B oprex pwen.2 pw2en 2o ^m B opr V B P~ 2o ^m A opr enen2 mp2an bitr2 sylib))
+thm (ssenen ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f) (B x) (B y) (B z) (B f) (C x) (C y) (C z) (C f)) ((ssenen.1 (e. A (V))) (ssenen.2 (e. B (V))) (ssenen.3 (e. C (V)))) (-> (br A (~~) B) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) C))) (~~) ({|} x (/\ (C_ (cv x) B) (br (cv x) (~~) C))))) (ssenen.2 A f bren ssenen.1 pwex x x cv ~~ C br {|} inex1 f cv A B :-1-1-onto-> a1i f cv y cv " y cv C entrt y visset f cv A B f1imaen f cv A B f1of1 sylan sylan exp31 imp3a f cv A B f1ofo f cv y cv imassrn f cv A B forn f cv y cv " sseq2d mpbii syl jctild y cv A P~ x x cv ~~ C br {|} elin y visset A elpw y visset x cv y cv ~~ C breq1 elab anbi12i bitr f cv y cv " B P~ x x cv ~~ C br {|} elin f visset f cv V y cv imaexg ax-mp B elpw f visset f cv V y cv imaexg ax-mp x cv f cv y cv " ~~ C breq1 elab anbi12i bitr 3imtr4g f cv A B f1ocnv f cv `' z cv " z cv C entrt z visset f cv `' B A f1imaen f cv `' B A f1of1 sylan sylan exp31 imp3a f cv `' B A f1ofo f cv `' z cv imassrn f cv `' B A forn f cv `' z cv " sseq2d mpbii syl jctild syl z cv B P~ x x cv ~~ C br {|} elin z visset B elpw z visset x cv z cv ~~ C breq1 elab anbi12i bitr f cv `' z cv " A P~ x x cv ~~ C br {|} elin f visset cnvex f cv `' V z cv imaexg ax-mp A elpw f visset cnvex f cv `' V z cv imaexg ax-mp x cv f cv `' z cv " ~~ C breq1 elab anbi12i bitr 3imtr4g y cv f cv `' z cv " f cv imaeq2 f cv A B f1orel f cv dfrel2 sylib f cv `' `' f cv f cv `' z cv " imaeq1 syl z cv B C_ adantr f cv `' B A z cv f1imacnv f cv A B f1ocnv f cv `' B A f1of1 syl sylan eqtr3d sylan9eqr eqcomd exp z cv B P~ x x cv ~~ C br {|} elin z cv B P~ e. z cv x x cv ~~ C br {|} e. pm3.26 z visset B elpw sylib sylbi sylan2 y cv A P~ x x cv ~~ C br {|} i^i e. adantrl z cv f cv y cv " f cv `' imaeq2 f cv A B y cv f1imacnv f cv A B f1of1 sylan sylan9eqr eqcomd exp y cv A P~ x x cv ~~ C br {|} elin y cv A P~ e. y cv x x cv ~~ C br {|} e. pm3.26 y visset A elpw sylib sylbi sylan2 z cv B P~ x x cv ~~ C br {|} i^i e. adantrr impbid exp en3d f 19.23aiv sylbi A x df-pw x x cv ~~ C br {|} ineq1i x x cv A C_ x cv ~~ C br inab eqtr B x df-pw x x cv ~~ C br {|} ineq1i x x cv B C_ x cv ~~ C br inab eqtr 3brtr3g))
+thm (limenpsi ((x y) (A x) (A y)) ((limenpsi.1 (Lim A))) (-> (e. A B) (br A (~~) (\ A ({} ({/}))))) (A A {/} {} \ sbth limenpsi.1 A x cv limsuc ax-mp biimp x cv nsuceq0 x visset sucex {/} elsnc mtbir jctir x cv suc A {/} {} eldif sylibr x cv y cv suc11 limenpsi.1 A limord ax-mp A x cv ordelon mpan limenpsi.1 A limord ax-mp A y cv ordelon mpan syl2an B dom2 A {/} {} difss A B A {/} {} \ ssdom2g mpi sylanc))
+thm (limensuci () ((limensuci.1 (Lim A))) (-> (e. A B) (br A (~~) (suc A))) (A {/} {} \ {/} {} incom {/} {} A difdisj eqtr limensuci.1 A limord ax-mp A ordeirr ax-mp A A disjsn mpbir pm3.2i A {/} {} \ A {/} {} A {} unen mpan2 A {/} {} \ V A ensymg A B {/} {} difexg limensuci.1 B limenpsi sylc 0ex {/} V A B en2sn mpan sylanc limensuci.1 A 0ellim ax-mp 0ex A snss mpbi {/} {} A ssundif mpbi {/} {} A {/} {} \ uncom eqtr3 A df-suc 3brtr4g))
+thm (limensuc () () (-> (/\ (e. A B) (Lim A)) (br A (~~) (suc A))) (A A Lim A On if B eleq1 A A Lim A On if = id A A Lim A On if suceq ~~ breq12d imbi12d A A Lim A On if limeq On A Lim A On if limeq limon elimhyp B limensuci dedth impcom))
+thm (phplem1 () () (-> (e. B A) (= (u. ({} B) (\ A ({} B))) A)) (B A snssi B {} A ssundif sylib))
+thm (phplem2 () () (-> (/\ (e. A (om)) (e. B A)) (= (u. ({} A) (\ A ({} B))) (\ (suc A) ({} B)))) (A B nordeq A B disjsn2 syl A nnord sylan A {} B {} A undif4 A df-suc A A {} uncom eqtr B {} difeq1i syl6eqr syl))
+thm (phplem3 () ((phplem3.1 (e. A (V))) (phplem3.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B A)) (br A (~~) (\ (suc A) ({} B)))) (A B {} difss A B {} \ A A {} ssrin ax-mp A nnord A orddisj syl A B {} \ A {} i^i sseq2d mpbii A B {} \ A {} i^i ss0 syl A {} A B {} \ incom syl5eq B {} A difdisj jctil phplem3.2 phplem3.1 f1osn B snex B A <,> {} A {} f1oen ax-mp phplem3.1 A B {} difss ssexi enref pm3.2i jctil B {} A {} A B {} \ A B {} \ unen syl B A e. adantr B A phplem1 A om e. adantl A B phplem2 3brtr3d))
+thm (phplem4 () ((phplem3.1 (e. A (V))) (phplem3.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B (suc A))) (br A (~~) (\ (suc A) ({} B)))) (phplem3.1 phplem3.2 phplem3 A nnord A orddif syl A B sneq A suc difeq2d eqcoms sylan9eq phplem3.1 enref syl5breq jaodan B A elsuci sylan2))
+thm (phplem5 ((A f) (B f)) ((phplem3.1 (e. A (V))) (phplem3.2 (e. B (V)))) (-> (/\ (e. A (om)) (e. B (om))) (-> (br (suc A) (~~) (suc B)) (br A (~~) B))) (A B suc f cv A ` {} \ B entrt f cv A suc B suc f1of1 A sssucid jctir f cv A suc B suc A f1ores phplem3.1 f cv A |` f cv A " f1oen 3syl A om e. adantl A nnord A orddif A A suc A {} \ f cv imaeq2 3syl f cv A suc B suc f1ofn phplem3.1 sucid f cv A suc A fnsnfv mpan2 f cv A ` {} f cv A {} " f cv A suc " difeq2 3syl f cv imadmrn eqcomi f cv A suc B suc :-1-1-onto-> a1i f cv A suc B suc f1ofo f cv A suc B suc forn syl f cv A suc B suc f1ofn f cv A suc fndm f cv dom A suc f cv imaeq2 3syl 3eqtr3d f cv A ` {} difeq1d f cv A suc B suc f1o3 pm3.27bd f cv A suc A {} imadif syl 3eqtr4rd sylan9eq breqtrd phplem3.2 f cv A fvex phplem4 f cv A suc B suc f1ofn phplem3.1 sucid f cv A suc A fnfvrn mpan2 syl f cv A suc B suc f1ofo f cv A suc B suc forn f cv A ` eleq2d syl mpbid sylan2 phplem3.2 phplem3.2 sucex B suc f cv A ` {} difss ssexi ener ersym syl syl2an anandirs exp f 19.23adv phplem3.2 sucex A suc f bren syl5ib))
+thm (nneneq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~~) B) (= A B))) (z cv B A ~~ breq2 z cv B A eqeq2 imbi12d om rcla4v x cv {/} ~~ z cv breq1 x cv {/} z cv eqeq1 imbi12d z om biraldv x cv y cv ~~ z cv breq1 x cv y cv z cv eqeq1 imbi12d z om biraldv x cv y cv suc ~~ z cv breq1 x cv y cv suc z cv eqeq1 imbi12d z om biraldv x cv A ~~ z cv breq1 x cv A z cv eqeq1 imbi12d z om biraldv 0ex z visset ener ersym z cv en0 z cv {/} eqcom bitr sylib z cv om e. a1i rgen y cv suc en0 w cv {/} y cv suc ~~ breq2 w cv {/} y cv suc eqeq2 bibi12d mpbiri biimpd y cv om e. z om y cv ~~ z cv br y cv z cv = -> A.e. /\ a1i y cv om e. z ax-17 z om y cv ~~ z cv br y cv z cv = -> hbra1 hban y cv suc ~~ w cv br y cv suc w cv = -> z ax-17 y visset z visset phplem5 y cv z cv = imim1d exp a2d z om y cv ~~ z cv br y cv z cv = -> ra4 syl5 imp y cv z cv suceq syl8 w cv z cv suc y cv suc ~~ breq2 w cv z cv suc y cv suc eqeq2 imbi12d biimprcd syl6 r19.23ad jaod exp w cv z nn0suc syl7 r19.21adv w cv z cv y cv suc ~~ breq2 w cv z cv y cv suc eqeq2 imbi12d om cbvralv syl6ib finds syl5 impcom A om B eqeng B om e. adantr impbid))
+thm (php ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (om)) (C: B A)) (-. (br A (~~) B))) (A x nn0suc orcanai B 0ss {/} B A sspsstr mpan A 0pss sylib sylan2 A x cv suc B psseq2 A x cv suc ~~ B breq1 negbid imbi12d B x cv suc y pssnel B x cv suc y cv {} \ x cv domentr B y cv disjsn B y cv {} disj3 bitr3 B B y cv {} \ x cv suc y cv {} \ sseq1 sylbi B x cv suc y cv {} ssdif syl5bir x visset sucex x cv suc y cv {} difss ssexi x cv suc y cv {} \ V B ssdom2g ax-mp syl6 B x cv suc pssss syl5 imp x visset y visset phplem4 x visset sucex x cv suc y cv {} difss ssexi x cv ensym syl syl2an exp43 com4r imp y 19.23aiv mpcom x cv suc B x cv endomtr x visset sucex x cv sssucid x cv suc V x cv ssdom2g mp2 jctir x cv suc x cv sbth syl expcom x cv peano2b x cv suc nnord sylbi x visset sucid x cv suc x cv nordeq mpan2 syl x cv suc x cv nneneq x cv peano2b sylanb anidms mtbird nsyli syli com12 syl5bir com12 r19.23aiv syl exp pm2.43d imp))
+thm (php2 ((A x) (B x)) () (-> (/\ (e. A (om)) (C: B A)) (br B (~<) A)) (x cv A om eleq1 x cv A B psseq2 anbi12d x cv A B ~< breq2 imbi12d B x cv pssss x visset x cv V B ssdom2g ax-mp syl x cv om e. adantl x cv B php x visset B ensym nsyl jca B x cv brsdom sylibr om vtoclg anabsi5))
+thm (php3 ((A x) (x y) (x z) (f x) (A y) (A z) (A f) (y z) (f y) (f z) (B y) (B z) (B f)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C: B A)) (br B (~<) A)) (A V B ssdom2g imp relen A z cv brrelexi B A pssss syl2an z cv om e. adantll z cv f cv B " php B A pssss B A f cv imass2 syl f cv A z cv :-1-1-onto-> adantl f cv A B \ y cv funfvima2 f cv A z cv f1ofun A B difss f cv A z cv f1ofn f cv A fndm f cv dom A A B \ sseq2 3syl mpbiri sylanc f cv A z cv f1o3 pm3.27bd f cv A B imadif syl f cv y cv ` eleq2d sylibd f cv y cv ` f cv A " f cv B " \ n0i syl6 y cv A B eldif syl5ibr y 19.23adv imp B A y pssnel sylan2 f cv A " f cv B " ssdif0 negbii sylibr jca f cv B " f cv A " dfpss3 sylibr f cv imadmrn f cv A z cv :-1-1-onto-> a1i f cv A z cv f1ofn f cv A fndm f cv dom A f cv imaeq2 3syl f cv A z cv f1ofo f cv A z cv forn syl 3eqtr3d f cv B " psseq2d B A C: adantr mpbid sylan2 f cv A z cv B f1ores f cv A z cv f1of1 B A pssss syl2an f visset f cv V B resexg ax-mp y cv f cv B |` B f cv B " f1oeq1 cla4ev f visset f cv V B imaexg ax-mp B y bren sylibr z cv B f cv B " entrt expcom 3syl z cv om e. adantl mtod exp32 f 19.23adv z visset A f bren syl5ib imp31 B A z cv entrt exp z visset B ensym syl6com z cv om e. B A C: ad2antlr mtod jca B A brsdom sylibr exp31 r19.23aiv imp x cv z cv A ~~ breq2 om cbvrexv sylanb))
+thm (php4 () () (-> (e. A (om)) (br A (~<) (suc A))) (A om sucidg A nnord A ordsuc biimp ancli A A suc ordelpss 3syl mpbid A suc A php2 A peano2b sylanb mpdan))
+thm (php5 () () (-> (e. A (om)) (-. (br A (~~) (suc A)))) (A php4 A A suc sdomnen syl))
+thm (onomeneq () () (-> (/\ (e. A (On)) (e. B (om))) (<-> (br A (~~) B) (= A B))) (A B nneneq biimpa B php5 A ~~ B br adantr B om A B suc enen1 mtbird A On e. adantll A B B suc endomtr B sssucid B om B suc ssdomg mpi sylan2 ancoms om A C_ a1d A On e. adantll om A B ssel com12 A On e. adantr B om A ordelsuc A eloni sylan2 sylibd A On B suc ssdom2g B om e. adantl syld ancoms A ~~ B br adantr jcad A B suc sbth syl6 mtod A eloni ordom om A ordtri1 mpan syl bicon2d B om e. A ~~ B br ad2antll mpbird A On e. B om e. pm3.27 A ~~ B br adantr jca A On e. B om e. /\ A ~~ B br pm3.27 sylanc exp A On B eqeng B om e. adantr impbid))
+thm (onfin ((A x)) () (-> (e. A (On)) (<-> (E.e. x (om) (br A (~~) (cv x))) (e. A (om)))) (A x cv onomeneq x cv om A eleq1a A On e. adantl sylbid exp r19.23adv A om enrefg x cv A A ~~ breq2 om rcla4ev mpdan A On e. a1i impbid))
+thm (nndomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<_) B) (C_ A B))) (A B php2 exp A B domnsym nsyli B om e. adantr A B ordtri1 B A ordelpss ancoms negbid bitrd A nnord B nnord syl2an sylibrd A om B ssdomg B om e. adantr impbid))
+thm (nnsdomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (C: A B))) (A B nndomo A B nneneq negbid anbi12d A B brsdom A B dfpss2 3bitr4g))
+thm (omsucdom () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (br (suc A) (~<_) B))) (A B ordelpss A nnord sylan A om B ordelsuc bitr3d B nnord sylan2 A B nnsdomo A suc B nndomo A peano2b sylanb 3bitr4d))
+thm (sucdomi () () (-> (/\ (e. A (om)) (e. B C)) (-> (br (suc A) (~<_) B) (br A (~<) B))) (B C A A suc sdomdomtr exp3a impcom A php4 sylan))
+thm (0sdom1dom ((A x)) ((0sdom1dom.1 (e. A (V)))) (<-> (br ({/}) (~<) A) (br (1o) (~<_) A)) (0sdom1dom.1 0sdom A x n0 bitr x cv A snssi 0sdom1dom.1 A V x cv {} ssdom2g ax-mp 1ord elisseti x visset ensn1 ensymi 1o x cv {} A endomtr mpan 3syl x 19.23aiv sylbi df-1o ~<_ A breq1i peano1 0sdom1dom.1 {/} A V sucdomi mp2an sylbi impbi))
+thm (finsucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (E.e. x (om) (br B (~~) (cv x)))) (<-> (br A (~<) B) (br (suc A) (~<_) B))) (A x cv omsucdom B ~~ x cv br adantr x cv om B A sdomen2 A om e. adantll x cv om B A suc domen2 A om e. adantll 3bitr4d exp31 r19.23adv imp))
+thm (pssinf ((B x)) () (-> (/\ (C: A B) (br A (~~) B)) (-. (E.e. x (om) (br B (~~) (cv x))))) (x B A php3 exp A B sdomnen syl6com con2d imp))
+thm (ominf ((x y)) () (-. (E.e. x (om) (br (om) (~~) (cv x)))) (x cv y cv om ~~ breq2 om cbvrexv y cv om x pssinf y cv nnord ordom jctir y cv om ordelssne syl ibi y cv om dfpss2 sylibr y visset om ensym syl2an exp r19.23aiv sylbi x om om ~~ x cv br E.e. pm2.01 ax-mp))
+thm (omsdomnn ((A x)) () (-> (e. A (om)) (/\ (br A (~<_) (om)) (-. (br (om) (~~) A)))) (ordom om A ordelss mpan A om om ssdomg mpd x cv A om ~~ breq2 negbid x ominf x om om ~~ x cv br ralnex mpbir vtoclri jca))
+thm (isfinite1 ((A x)) () (-> (E.e. x (om) (br A (~~) (cv x))) (/\ (br A (~<_) (om)) (-. (br (om) (~~) A)))) (x cv omsdomnn A ~~ x cv br anim2i A ~~ x cv br x cv ~<_ om br om ~~ x cv br -. anandi sylib ancoms A x cv om endomtr om A x cv entrt expcom con3d imp anim12i syl exp r19.23aiv))
+thm (infsdomnn () ((infsdomnn.1 (e. A (V)))) (-> (/\ (br (om) (~<_) A) (e. B (om))) (br B (~<) A)) (reldom om A brrelexi om V B ensymg con3d B ~<_ om br anim2d B omsdomnn syl5 B om brsdom syl6ibr infsdomnn.1 A V B om sdomdomtr ax-mp expcom syl9 mpcom imp))
+thm (infn0 () ((infsdomnn.1 (e. A (V)))) (-> (br (om) (~<_) A) (-. (= A ({/})))) (peano1 infsdomnn.1 {/} infsdomnn mpan2 infsdomnn.1 0sdom sylib))
+thm (pssnn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (om)) (C: B A)) (E.e. x A (br B (~~) (cv x)))) (A om B ssexg B A pssss syl5 w cv B A psseq1 w cv B ~~ x cv breq1 x A birexdv imbi12d V cla4gv z cv {/} w cv psseq2 z cv {/} x w cv ~~ x cv br rexeq imbi12d w bialdv z cv y cv w cv psseq2 z cv y cv x w cv ~~ x cv br rexeq imbi12d w bialdv z cv y cv suc w cv psseq2 z cv y cv suc x w cv ~~ x cv br rexeq imbi12d w bialdv z cv A w cv psseq2 z cv A x w cv ~~ x cv br rexeq imbi12d w bialdv w cv npss0 x {/} w cv ~~ x cv br E.e. pm2.21i w ax-gen y cv om e. w ax-17 w w cv y cv C: x y cv w cv ~~ x cv br E.e. -> hba1 z cv y cv w cv eleq1 biimpcd con3d w cv y cv suc C: adantl w cv y cv suc pssss z cv sseld z cv y cv elsuci ord con1d syl6 imp syld exp com23 imp ssrdv w cv y cv = -. anim1i w cv y cv dfpss2 sylibr x cv y cv elelsuc w cv ~~ x cv br anim1i r19.22i2 imim12i exp4c w a4s y cv om e. adantl com4t y cv nnord y cv orddif syl w cv y cv {} \ sseq2d w cv y cv suc y cv {} ssdif syl5bir w cv y cv suc pssss syl5 w cv y cv {} \ y cv z cv eleq2 z cv w cv y cv {} eldifi syl6bir y cv w cv e. z cv y cv suc e. /\ adantl z cv y cv elsuci ord y cv w cv z cv eleq1a sylan9r w cv y cv {} \ y cv = adantr pm2.61d exp con3d exp imp3a z 19.23adv w cv y cv suc z pssnel syl5 im2anan9r w cv y cv suc C: anidm syl5ibr w cv y cv {} \ y cv dfpss2 syl6ibr w cv z cv y cv psseq1 w cv z cv ~~ x cv breq1 x y cv birexdv imbi12d cbvalv w visset w cv y cv {} difss ssexi z cv w cv y cv {} \ y cv psseq1 z cv w cv y cv {} \ ~~ x cv breq1 x y cv birexdv imbi12d cla4v sylbi sylan9 y cv nnord y cv x cv ordsucelsuc biimpd syl y cv w cv e. adantl w cv y cv {} \ ~~ x cv br adantrd y cv w cv snssi y cv {} w cv ssundif sylib y cv {} w cv y cv {} \ uncom syl5reqr x cv df-suc y cv w cv e. a1i ~~ breq12d w cv y cv {} \ x cv y cv {} x cv {} unen y visset x visset f1osn y cv snex y cv x cv <,> {} x cv {} f1oen ax-mp w cv y cv {} \ ~~ x cv br jctr x cv nnord x cv orddisj syl y cv {} w cv y cv {} \ incom y cv {} w cv difdisj eqtr3 jctil syl2an syl5bir x cv y cv elnn sylan2i exp4d com24 imp4b jcad z cv x cv suc w cv ~~ breq2 y cv suc rcla4ev syl6 x 19.23adv x y cv w cv y cv {} \ ~~ x cv br df-rex x cv z cv w cv ~~ breq2 y cv suc cbvrexv 3imtr4g w w cv y cv C: x y cv w cv ~~ x cv br E.e. -> A. adantr syld exp31 imp3a w visset y cv eqelsuc w visset enref jctir x cv w cv w cv ~~ breq2 y cv suc rcla4ev syl w cv y cv suc C: a1d y cv om e. w w cv y cv C: x y cv w cv ~~ x cv br E.e. -> A. /\ a1d pm2.61ii exp 19.21ad finds syl5 syl6 pm2.43a pm2.43d imp))
+thm (ssnn ((A x) (B x)) () (-> (/\ (e. A (om)) (C_ B A)) (E.e. x (om) (br B (~~) (cv x)))) (A B x pssnn x cv A elnn expcom B ~~ x cv br anim1d x 19.22dv x A B ~~ x cv br df-rex x om B ~~ x cv br df-rex 3imtr4g B A C: adantr mpd B A om eleq1 biimparc B om enrefg ancli x cv B B ~~ breq2 om rcla4ev 3syl jaodan B A sspss sylan2b))
+thm (ssfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C_ B A)) (E.e. x (om) (br B (~~) (cv x)))) (x cv om A z breng x cv z cv B " y ssnn z cv A x cv df-f1o pm3.27bd z cv B imassrn z cv A x cv forn z cv B " sseq2d mpbii syl sylan2 B A C_ adantrr B z cv B " y cv entrt z visset z cv V B resexg ax-mp x cv z cv B |` B z cv B " f1oeq1 cla4ev z visset z cv V B imaexg ax-mp B x bren sylibr sylan z cv A x cv B f1ores z cv A x cv f1of1 sylan sylan exp y om r19.22sdv x cv om e. adantl mpd exp32 z 19.23adv sylbid r19.23aiv imp y cv x cv B ~~ breq2 om cbvrexv sylib))
+thm (unblem1 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ B (om)) (A.e. x (om) (E.e. y B (e. (cv x) (cv y))))) (e. A B)) (e. (|^| (\ B (suc A))) B)) (omsson B om On sstr mpan2 B A suc difss B A suc \ B On sstr mpan syl x om y B x cv y cv e. E.e. A.e. A B e. ad2antll y cv B e. A suc y cv e. pm3.26 B om C_ a1i B om y cv ssel y cv nnord y cv A suc ordn2lp y cv A suc e. A suc y cv e. imnan sylibr con2d syl syl6 imp3a jcad y cv B A suc eldif y cv B A suc \ n0i sylbir syl6 exp3a r19.23adv x cv A suc y cv eleq1 y B birexdv om rcla4v impcom syl5 B om A ssel A peano2b syl6ib sylan2d exp3a imp31 jca B A suc \ onint B A suc \ |^| B A suc eldifi 3syl))
+thm (unblem2 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (F z) (F w) (F v) (F u)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) A))) (z cv {/} F fveq2 A eleq1d z cv u cv F fveq2 A eleq1d z cv u cv suc F fveq2 A eleq1d A onint omsson A om On sstr mpan2 peano1 w cv {/} v cv eleq1 v A birexdv om rcla4v ax-mp v A {/} v cv e. df-rex sylib v cv A e. {/} v cv e. pm3.26 v 19.22i syl A v n0 sylibr syl2an A |^| A x y y cv A x cv suc \ |^| = {<,>|} fr0t unblem.2 {/} fveq1i syl5req A eleq1d ibi syl w cv A |^| e. x ax-17 w cv u cv e. x ax-17 w cv A e. x ax-17 unblem.1 w cv u cv e. x ax-17 hbfv hbsuc hbdif hbint unblem.2 x cv F u cv ` suceq A difeq2d inteqd A frsucopab eqcomd A eleq1d exp ibd A w v F u cv ` unblem1 syl5 exp3a finds2 com12))
+thm (unblem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) (` F (suc (cv z)))))) (unblem.1 unblem.2 v z unblem2 imp omsson A om On sstr mpan2 A On F z cv ` ssel anc2li syl w om v A w cv v cv e. E.e. A.e. z cv om e. ad2antll mpd A F z cv ` onmindif syl unblem.1 unblem.2 v z unblem2 A w v F z cv ` unblem1 exp syld w cv A |^| e. x ax-17 w cv z cv e. x ax-17 w cv A e. x ax-17 unblem.1 w cv z cv e. x ax-17 hbfv hbsuc hbdif hbint unblem.2 x cv F z cv ` suceq A difeq2d inteqd A frsucopab exp sylcom imp eleqtrrd exp))
+thm (unblem4 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((unblem.1 (-> (e. (cv w) F) (A. x (e. (cv w) F)))) (unblem.2 (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om))))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (:-1-1-> F (om) A)) (A F z omsmo omsson A om On sstr mpan2 w om v A w cv v cv e. E.e. A.e. adantr unblem.1 unblem.2 v z unblem2 r19.21aiv x y y cv A x cv suc \ |^| = {<,>|} A |^| frfnom unblem.2 F x y y cv A x cv suc \ |^| = {<,>|} A |^| rec om |` om fneq1 ax-mp mpbir F om A z ffnfv biimpr mpan syl jca unblem.1 unblem.2 v z unblem3 r19.21aiv sylanc))
+thm (unbnn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((unbnn.1 (e. A (V)))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om))) (A om sbth unbnn.1 A V om ssdomg ax-mp x om y A x cv y cv e. E.e. A.e. adantr x z w w cv A z cv suc \ |^| = hbopab1 x cv A |^| e. z ax-17 hbrdg x cv om e. z ax-17 hbres z w w cv A z cv suc \ |^| = {<,>|} A |^| rec om |` eqid y unblem4 unbnn.1 A V z w w cv A z cv suc \ |^| = {<,>|} A |^| rec om |` om f1dom2g ax-mp syl sylanc))
+thm (unbnn2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((unbnn.1 (e. A (V)))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (C_ (cv x) (cv y))))) (br A (~~) (om))) (unbnn.1 z y unbnn x cv z cv suc y cv sseq1 y A birexdv om rcla4v z visset z cv V y cv sucssel ax-mp y A r19.22si syl6com z cv peano2 syl5 r19.21aiv sylan2))
+thm (isfinite2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x)))) (A om sdomex pm3.27d om V A y domeng A om sdomdom syl5bi y visset z w unbnn exp y cv om sdomnen nsyli y cv A om ensdomtr y visset A ensym sylan syl5 w cv z cv ordtri1 y cv On w cv ssel2 w visset elon sylib sylan an1rs biraldva y cv z cv w unissb w y cv z cv w cv e. ralnex bicomi 3bitr4g y cv z cv ordunisssuc bitr3d omsson y cv om On sstr mpan2 z cv nnord syl2an z cv suc y cv x ssnn z cv peano2b sylanb exp y cv om C_ adantl sylbid exp r19.23adv z om w y cv z cv w cv e. E.e. rexnal syl5ibr syld imp A y cv x cv entrt exp x om r19.22sdv y cv om C_ A ~< om br ad2antrl mpd exp32 com13 imp3a y 19.23adv sylcom mpcom))
+thm (unfilem1 ((x y) (A x) (A y) (B x) (B y)) ((unfilem1.1 (e. A (om))) (unfilem1.2 (e. B (om))) (unfilem1.3 (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x)))))))) (= (ran F) (\ (opr A (+o) B) A)) (x y x cv B e. y cv A +o x cv opr = /\ rnopab unfilem1.3 rneqi y cv A +o B opr A eldif unfilem1.1 unfilem1.2 A B nnacl mp2an y cv A +o B opr elnn mpan2 unfilem1.1 A y cv ordtri1 A nnord y cv nnord syl2an A y cv x nnawordex bitr3d mpan x om A +o x cv opr y cv = df-rex syl6bb syl unfilem1.2 unfilem1.1 x cv B A nnaord mp3an23 A +o x cv opr y cv A +o B opr eleq1 sylan9bb biimprcd A +o x cv opr y cv eqcom biimp x cv om e. adantl y cv A +o B opr e. a1i jcad x 19.22dv sylbid imp y cv A +o x cv opr A +o B opr eleq1 y cv A +o x cv opr A eleq1 negbid anbi12d biimparc unfilem1.2 x cv B elnn mpan2 unfilem1.2 unfilem1.1 x cv B A nnaord mp3an23 syl ibi unfilem1.2 x cv B elnn mpan2 unfilem1.1 A x cv nnaword1 A x cv nnacl A +o x cv opr nnord unfilem1.1 A nnord ax-mp A A +o x cv opr ordtri1 mpan 3syl mpbid mpan syl jca sylan x 19.23aiv impbi bitr biabri 3eqtr4))
+thm (unfilem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z) (F w)) ((unfilem1.1 (e. A (om))) (unfilem1.2 (e. B (om))) (unfilem1.3 (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x)))))))) (:-1-1-onto-> F B (\ (opr A (+o) B) A)) (F B A +o B opr A \ df-f1o F B A +o B opr A \ z w f1fv F B A +o B opr A \ df-fo A +o x cv oprex unfilem1.3 fnopab2 unfilem1.1 unfilem1.2 unfilem1.3 unfilem1 mpbir2an F B A +o B opr A \ fof ax-mp x cv z cv A +o opreq2 unfilem1.3 A +o z cv oprex fvopab4 x cv w cv A +o opreq2 unfilem1.3 A +o w cv oprex fvopab4 eqeqan12d unfilem1.1 A z cv w cv nnacan mp3an1 unfilem1.2 z cv B elnn mpan2 unfilem1.2 w cv B elnn mpan2 syl2an bitrd biimpd rgen2 mpbir2an F B A +o B opr A \ df-fo A +o x cv oprex unfilem1.3 fnopab2 unfilem1.1 unfilem1.2 unfilem1.3 unfilem1 mpbir2an mpbir2an))
+thm (unfilem3 ((x y) (f x) (A x) (f y) (A y) (A f) (B x) (B y) (B f)) () (-> (/\ (e. A (om)) (e. B (om))) (br B (~~) (\ (opr A (+o) B) A))) (f cv x y x cv B e. y cv A +o x cv opr = /\ {<,>|} B A +o B opr A \ f1oeq1 V cla4egv A A om e. A {/} if +o x cv opreq1 y cv eqeq2d x cv B e. anbi2d x y biopabdv x y x cv B e. y cv A +o x cv opr = /\ {<,>|} x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B A +o B opr A \ f1oeq1 syl A A om e. A {/} if +o B opreq1 A difeq1d A A om e. A {/} if A om e. A {/} if +o B opr difeq2 eqtrd A +o B opr A \ A om e. A {/} if +o B opr A om e. A {/} if \ x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B f1oeq3 syl bitrd B B om e. B {/} if x cv eleq2 y cv A om e. A {/} if +o x cv opr = anbi1d x y biopabdv x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B A om e. A {/} if +o B opr A om e. A {/} if \ f1oeq1 syl B B om e. B {/} if x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} A om e. A {/} if +o B opr A om e. A {/} if \ f1oeq2 B B om e. B {/} if A om e. A {/} if +o opreq2 A om e. A {/} if difeq1d A om e. A {/} if +o B opr A om e. A {/} if \ A om e. A {/} if +o B om e. B {/} if opr A om e. A {/} if \ x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B om e. B {/} if f1oeq3 syl 3bitrd peano1 A elimel peano1 B elimel x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} eqid unfilem2 dedth2h x y x cv B e. y cv A +o x cv opr = /\ {<,>|} B A +o B opr A \ f1ofn syl B om x y x cv B e. y cv A +o x cv opr = /\ {<,>|} fnex A om e. adantl mpd A A om e. A {/} if +o x cv opreq1 y cv eqeq2d x cv B e. anbi2d x y biopabdv x y x cv B e. y cv A +o x cv opr = /\ {<,>|} x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B A +o B opr A \ f1oeq1 syl A A om e. A {/} if +o B opreq1 A difeq1d A A om e. A {/} if A om e. A {/} if +o B opr difeq2 eqtrd A +o B opr A \ A om e. A {/} if +o B opr A om e. A {/} if \ x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B f1oeq3 syl bitrd B B om e. B {/} if x cv eleq2 y cv A om e. A {/} if +o x cv opr = anbi1d x y biopabdv x y x cv B e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B A om e. A {/} if +o B opr A om e. A {/} if \ f1oeq1 syl B B om e. B {/} if x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} A om e. A {/} if +o B opr A om e. A {/} if \ f1oeq2 B B om e. B {/} if A om e. A {/} if +o opreq2 A om e. A {/} if difeq1d A om e. A {/} if +o B opr A om e. A {/} if \ A om e. A {/} if +o B om e. B {/} if opr A om e. A {/} if \ x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} B om e. B {/} if f1oeq3 syl 3bitrd peano1 A elimel peano1 B elimel x y x cv B om e. B {/} if e. y cv A om e. A {/} if +o x cv opr = /\ {<,>|} eqid unfilem2 dedth2h sylc A +o B oprex A +o B opr V A difexg ax-mp B f bren sylibr))
+thm (unfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (E.e. x (om) (br B (~~) (cv x)))) (E.e. x (om) (br (u. A B) (~~) (cv x)))) (x om y om A ~~ x cv br B A \ ~~ y cv br reeanv A B undif2 x cv om e. y cv om e. /\ a1i x cv y cv nnaword1 x cv x cv +o y cv opr ssundif sylib ~~ breq12d A B difdisj x cv x cv +o y cv opr difdisj pm3.2i A x cv B A \ x cv +o y cv opr x cv \ unen mpan2 syl5bi x cv y cv unfilem3 B A \ y cv x cv +o y cv opr x cv \ entrt expcom syl sylan2d x cv y cv nnacl z cv x cv +o y cv opr A B u. ~~ breq2 om rcla4ev exp syl syld r19.23aivv sylbir x cv y cv B ~~ breq2 om cbvrexv B A difss y B B A \ ssfi mpan2 sylbi sylan2 z cv x cv A B u. ~~ breq2 om cbvrexv sylib))
+thm (unfi2 ((A x) (B x)) () (-> (/\ (br A (~<) (om)) (br B (~<) (om))) (br (u. A B) (~<) (om))) (x A B unfi A x isfinite2 B x isfinite2 syl2an x A B u. isfinite1 syl A om sdomex pm3.27d om V A B u. ensymg syl con3d B ~< om br adantr A B u. ~<_ om br anim2d mpd A B u. om brsdom sylibr))
+thm (infcntss ((A x)) ((infcntss.1 (e. A (V)))) (-> (br (om) (~<_) A) (E. x (/\ (C_ (cv x) A) (br (cv x) (~~) (om))))) (infcntss.1 om x domen x visset om ensym x cv A C_ anim2i ancoms x 19.22i sylbi))
+thm (prfi ((A x) (B x)) () (E.e. x (om) (br ({,} A B) (~~) (cv x))) (x A snfi x B snfi x A {} B {} unfi mp2an A B df-pr ~~ x cv breq1i x om birex mpbir))
+thm (fiint ((x y) (x z) (w x) (v x) (f x) (A x) (y z) (w y) (v y) (f y) (A y) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f)) () (<-> (A.e. x A (A.e. y A (e. (i^i (cv x) (cv y)) A))) (A. x (-> (/\ (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) A)))) (x cv z cv y cv ineq1 A eleq1d y A biraldv A cbvralv y cv w cv z cv ineq2 A eleq1d A cbvralv z A biral y cv {/} ~~ x cv breq1 x cv A C_ x cv {/} = -. /\ anbi2d x cv |^| A e. imbi1d x bialdv y cv v cv ~~ x cv breq1 x cv A C_ x cv {/} = -. /\ anbi2d x cv |^| A e. imbi1d x bialdv y cv v cv suc ~~ x cv breq1 x cv A C_ x cv {/} = -. /\ anbi2d x cv |^| A e. imbi1d x bialdv x visset {/} ensym x cv en0 sylib x cv {/} = -. anim1i ancoms x cv A C_ adantll x cv {/} = pm3.24 x cv |^| A e. pm2.21i syl x ax-gen z A w A z cv w cv i^i A e. A.e. A.e. a1i z A w A z cv w cv i^i A e. A.e. A.e. x ax-17 x x cv A C_ x cv {/} = -. /\ v cv ~~ x cv br /\ x cv |^| A e. -> hba1 x cv A f cv v cv ` ssel f cv v cv suc x cv df-f1o pm3.27bd f cv v cv suc x cv fof v visset sucid f cv v cv suc x cv v cv ffvrn mpan2 3syl syl5 imp x x cv A C_ x cv {/} = -. /\ v cv ~~ x cv br /\ x cv |^| A e. -> A. z A w A z cv w cv i^i A e. A.e. A.e. /\ adantr f cv v cv imassrn f cv v cv suc x cv f1o2 f cv v cv suc Fn f cv `' Fun f cv ran x cv = 3simp3 sylbi f cv v cv " sseq2d mpbii f cv v cv " x cv A sstr2 syl f cv v cv " {/} = -. anim1d f cv v cv suc x cv f1of1 v cv sssucid f cv v cv suc x cv v cv f1ores mpan2 v visset f cv v cv |` f cv v cv " f1oen 3syl jctird f visset f cv V v cv imaexg ax-mp x cv f cv v cv " A sseq1 x cv f cv v cv " {/} eqeq1 negbid anbi12d x cv f cv v cv " v cv ~~ breq2 anbi12d x cv f cv v cv " inteq A eleq1d imbi12d cla4v sylan9 z cv f cv v cv " |^| w cv ineq1 A eleq1d w cv f cv v cv ` f cv v cv " |^| ineq2 A eleq1d A A rcla42v exp syl6 com4r exp4a exp3a com14 imp43 f cv v cv " {/} inteq int0 syl6eq f cv v cv ` ineq1d f cv v cv ` ssv f cv v cv ` V sseqin2 mpbi syl6eq A eleq1d biimprd pm2.61d2 mpd f cv v cv suc x cv f1ofn v visset sucid f cv v cv suc v cv fnsnfv mpan2 syl f cv v cv " uneq2d v cv df-suc v cv suc v cv v cv {} u. f cv imaeq2 ax-mp f cv v cv v cv {} imaun eqtr2 syl6eq f cv v cv suc x cv df-f1o pm3.27bd f cv v cv suc x cv foima syl eqtrd inteqd f cv v cv fvex f cv v cv " intunsn syl5eqr A eleq1d x cv A C_ x x cv A C_ x cv {/} = -. /\ v cv ~~ x cv br /\ x cv |^| A e. -> A. z A w A z cv w cv i^i A e. A.e. A.e. /\ ad2antlr mpbid exp43 f 19.23adv x visset v cv suc f bren syl5ib imp x cv {/} = -. adantlr com13 19.21ad v cv om e. a1i finds2 x x cv A C_ x cv {/} = -. /\ y cv ~~ x cv br /\ x cv |^| A e. -> ax-4 syl6 exp4a com24 y visset x cv ensym syl5 r19.23aiv com13 imp3a x 19.21aiv z w zfpair x cv z cv w cv {,} A sseq1 x cv z cv w cv {,} {/} eqeq1 negbid anbi12d x cv z cv w cv {,} ~~ y cv breq1 y om birexdv anbi12d x cv z cv w cv {,} inteq A eleq1d imbi12d cla4v z visset w cv prnz z cv w cv {,} A C_ biantru y z cv w cv prfi z cv w cv {,} A C_ z cv w cv {,} {/} = -. /\ biantru bitr2 z visset w visset A prss bitr4 z visset w visset intpr A eleq1i 3imtr3g exp3a r19.21adv r19.21aiv impbi 3bitr))
+thm (zfregcl ((x y) (A x) (x z) (A y) (y z) (A z)) ((zfregcl.1 (e. A (V)))) (-> (E. x (e. (cv x) A)) (E.e. x A (A.e. y (cv x) (-. (e. (cv y) A))))) (zfregcl.1 z cv A x cv eleq2 x biexdv z cv A y cv eleq2 negbid y x cv biraldv x rexeqd imbi12d x z cv y x cv y cv z cv e. -. A.e. hbre1 x z y axreg y x cv y cv z cv e. -. df-ral x z cv birex x z cv y y cv x cv e. y cv z cv e. -. -> A. df-rex bitr2 sylib 19.23ai vtocl))
+thm (zfreg ((A x) (x y) (A y)) ((zfreg.1 (e. A (V)))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/})))) (zfreg.1 x y zfregcl A x n0 x cv A y disj x A birex 3imtr4))
+thm (zfreg2 ((A x) (x y) (A y)) ((zfreg2.1 (e. A (V)))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i A (cv x)) ({/})))) (zfreg2.1 x y zfregcl A x n0 A x cv incom {/} eqeq1i x cv A y disj bitr x A birex 3imtr4))
+thm (eirrv ((x y) (x z) (y z)) () (-. (e. (cv x) (cv x))) (y cv x cv x cv {} eleq1 x visset snid a4w1 x cv snex y z zfregcl ax-mp x y x ax-14 equcoms com12 y x cv elsn syl5ib x visset snid z cv x cv x cv {} eleq1 negbid y cv rcla4v com12 mt2i nsyli con2d r19.21aiv y x cv {} z y cv z cv x cv {} e. -. A.e. ralnex sylib mt2))
+thm (eirr ((A x)) () (-. (e. A A)) (x cv A x cv eleq2 x cv A A eleq1 bitrd negbid x eirrv A vtoclg A A e. pm2.01 ax-mp))
+thm (sucprcreg ((A x)) () (<-> (-. (e. A (V))) (= (suc A) A)) (A sucprc A eirr A A e. x ax-17 x cv A A eleq1 V ceqsalg mtbiri A ssid A df-suc A eqeq1i A A {} u. A A sseq1 sylbi mpbiri x cv sseld x cv A A {} elun syl5ibr x cv A {} e. x cv A e. olc syl5 x A elsn syl5ibr x 19.21aiv nsyl3 impbi))
+thm (zfregfr ((x y) (A x) (A y)) () (Fr (E) A) (A x y dfepfr x visset y zfreg2 x cv A C_ adantl mpgbir))
+thm (en2lp ((A x) (A y) (x y) (B x) (B y)) () (-. (/\ (e. A B) (e. B A))) (x cv A y cv eleq1 x cv A y cv eleq2 anbi12d negbid y cv B A eleq2 y cv B A eleq1 anbi12d negbid V zfregfr x visset y visset pm3.2i V x y efrn2lp mp2an V V vtocl2g A B elisset B A elisset anim12i con3i pm2.61i))
+thm (preleq () ((preleq.1 (e. A (V))) (preleq.2 (e. B (V))) (preleq.3 (e. C (V))) (preleq.4 (e. D (V)))) (-> (/\ (/\ (e. A B) (e. C D)) (= ({,} A B) ({,} C D))) (/\ (= A C) (= B D))) (preleq.1 preleq.2 preleq.3 preleq.4 preq12b biimp ord D C en2lp A D B C eleq12 C D e. anbi1d mtbiri syl6 a3d impcom))
+thm (opthreg () ((preleq.1 (e. A (V))) (preleq.2 (e. B (V))) (preleq.3 (e. C (V))) (preleq.4 (e. D (V)))) (-> (= ({,} A ({,} A B)) ({,} C ({,} C D))) (/\ (= A C) (= B D))) (preleq.1 B pri1 preleq.3 D pri1 preleq.1 A B prex preleq.3 C D prex preleq mpanl12 A C B preq1 C D {,} eqeq1d preleq.2 preleq.4 C prer2 syl6bi imdistani syl))
+thm (suc11reg () () (<-> (= (suc A) (suc B)) (= A B)) (A B en2lp A B e. B A e. ianor mpbi A suc B suc A eleq2 A V sucidg syl5bi com12 A V B elsucg sylibd imp ord exp com23 A suc B suc B eleq2 B V sucidg syl5bir com12 B V A elsucg sylibd imp ord B A eqcom syl6ib exp com23 jaao mpi A suc V B suc nelneq A sucexb B sucexb negbii syl2anb A B = pm2.21d B suc V A suc nelneq B sucexb A sucexb negbii syl2anb ancoms A B = pm2.21d A suc B suc eqcom syl5ib A sucprc B sucprc eqeqan12d biimpd 4cases A B suceq impbi))
+thm (inf0 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((inf0.1 (e. (om) (V)))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (y visset y cv V w v v cv w cv suc = {<,>|} fr0t ax-mp w v v cv w cv suc = {<,>|} y cv frfnom peano1 w v v cv w cv suc = {<,>|} y cv rec om |` om {/} fnfvrn mp2an eqeltrr w v v cv w cv suc = {<,>|} y cv frfnom w v v cv w cv suc = {<,>|} y cv rec om |` om u cv f fvelrn ax-mp w v v cv w cv suc = {<,>|} y cv rec om |` f cv ` u cv w v v cv w cv suc = {<,>|} y cv rec om |` f cv suc ` eleq1 w v v cv w cv suc = {<,>|} y cv rec om |` f cv fvex sucid w v v cv w cv suc = {<,>|} y cv rec om |` f cv fvex sucex z cv y cv e. w ax-17 z cv f cv e. w ax-17 z w v v cv w cv suc = hbopab1 z cv y cv e. w ax-17 hbrdg z cv om e. w ax-17 hbres z cv f cv e. w ax-17 hbfv hbsuc w v v cv w cv suc = {<,>|} y cv rec om |` eqid w cv w v v cv w cv suc = {<,>|} y cv rec om |` f cv ` suceq V frsucopab mpan2 w v v cv w cv suc = {<,>|} y cv rec om |` f cv ` eleq2d mpbiri syl5bi f cv peano2b w v v cv w cv suc = {<,>|} y cv frfnom w v v cv w cv suc = {<,>|} y cv rec om |` om f cv suc fnfvrn mpan sylbi w v v cv w cv suc = {<,>|} y cv rec om |` f cv ` u cv = a1i jcad w v v cv w cv suc = {<,>|} y cv rec om |` f cv suc fvex z cv w v v cv w cv suc = {<,>|} y cv rec om |` f cv suc ` u cv eleq2 z cv w v v cv w cv suc = {<,>|} y cv rec om |` f cv suc ` w v v cv w cv suc = {<,>|} y cv rec om |` ran eleq1 anbi12d cla4ev syl6com r19.23aiv sylbi u ax-gen w v v cv w cv suc = {<,>|} y cv frfnom w v v cv w cv suc = {<,>|} y cv rec om |` om fndm ax-mp inf0.1 eqeltr w v v cv w cv suc = {<,>|} y cv frfnom w v v cv w cv suc = {<,>|} y cv rec om |` om fnfun ax-mp w v v cv w cv suc = {<,>|} y cv rec om |` V funrnex mp2 x cv w v v cv w cv suc = {<,>|} y cv rec om |` ran y cv eleq2 x cv w v v cv w cv suc = {<,>|} y cv rec om |` ran u cv eleq2 x cv w v v cv w cv suc = {<,>|} y cv rec om |` ran z cv eleq2 u cv z cv e. anbi2d z biexdv imbi12d u bialdv anbi12d cla4ev mp2an y cv u cv x cv eleq1 y cv u cv z cv eleq1 z cv x cv e. anbi1d z biexdv imbi12d cbvalv y cv x cv e. anbi2i x biex mpbir))
+thm (inf1 ((x y) (x z) (y z)) ((inf1.1 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))) (inf1.1 y cv x cv n0i y y cv x cv e. z y cv z cv e. z cv x cv e. /\ E. -> A. anim1i x 19.22i ax-mp))
+thm (inf2 ((x y) (x z) (y z)) ((inf1.1 (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x))))) (inf1.1 inf1 x cv x cv U. y dfss2 y cv x cv z eluni y cv x cv e. imbi2i y bial bitr x cv {/} = -. anbi2i x biex mpbir))
+thm (inf3lema ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (<-> (e. A (` G B)) (/\ (e. A (cv x)) (C_ (i^i A (cv x)) B))) (inf3lem.1 z cv u cv = id y cv v cv w cv x cv i^i sseq2 w x cv birabsdv w cv f cv x cv ineq1 v cv sseq1d x cv cbvrabv syl6eq eqeqan12rd cbvopabv eqtr B fveq1i inf3lem.4 x visset f f cv x cv i^i B C_ rabex v cv B f cv x cv i^i sseq2 f x cv birabsdv u fvopab eqtr A eleq2i f cv A x cv ineq1 B sseq1d x cv elrab bitr))
+thm (inf3lemb ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (= (` F ({/})) ({/})) (inf3lem.2 {/} fveq1i 0ex {/} V G fr0t ax-mp eqtr))
+thm (inf3lemc ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (= (` F (suc A)) (` G (` F A)))) (A G {/} frsuct inf3lem.2 A suc fveq1i inf3lem.2 A fveq1i G fveq2i 3eqtr4g))
+thm (inf3lemd ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (C_ (` F A) (cv x))) (A v nnsuc A v cv suc F fveq2 x cv sseq1d inf3lem.1 inf3lem.2 v visset inf3lem.4 inf3lemc u cv eleq2d inf3lem.1 inf3lem.2 u visset F v cv fvex inf3lema pm3.26bd syl6bi ssrdv syl5bir com12 r19.23aiv syl exp x cv 0ss A {/} F fveq2 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemb syl6eq x cv sseq1d mpbiri pm2.61d2))
+thm (inf3lem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (e. A (om)) (C_ (` F A) (` F (suc A)))) (v cv {/} F fveq2 v cv {/} suceq F fveq2d sseq12d v cv u cv F fveq2 v cv u cv suceq F fveq2d sseq12d v cv u cv suc F fveq2 v cv u cv suc suceq F fveq2d sseq12d v cv A F fveq2 v cv A suceq F fveq2d sseq12d inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.3 inf3lemb F {/} suc ` 0ss eqsstr inf3lem.1 inf3lem.2 u visset u visset inf3lemc v cv eleq2d inf3lem.1 inf3lem.2 v visset F u cv fvex inf3lema syl6bb u cv peano2b inf3lem.1 inf3lem.2 u visset sucex u visset inf3lemc sylbi v cv eleq2d inf3lem.1 inf3lem.2 v visset F u cv suc fvex inf3lema syl6bb imbi12d v cv x cv i^i F u cv ` F u cv suc ` sstr2 com12 v cv x cv e. anim2d syl5bir imp ssrdv exp finds))
+thm (inf3lem2 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (cv x))))) (v cv {/} F fveq2 x cv eqeq1d negbid x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv u cv F fveq2 x cv eqeq1d negbid x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv u cv suc F fveq2 x cv eqeq1d negbid x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv A F fveq2 x cv eqeq1d negbid x cv {/} = -. x cv x cv U. C_ /\ imbi2d inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemb x cv eqeq1i {/} x cv eqcom bitr biimp con3i x cv x cv U. C_ adantr x cv x cv U. v cv ssel v cv x cv f eluni syl6ib inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lemc f cv eleq2d inf3lem.1 inf3lem.2 f visset F u cv fvex inf3lema pm3.27bd v cv sseld v cv f cv x cv elin syl5ibr syl6bi F u cv suc ` x cv f cv eleq2 biimparc syl5 com23 exp4a exp3a com34 imp3a f 19.23adv sylan9r pm2.43d F u cv suc ` x cv = v cv F u cv ` e. con3 syl6 imp3a v 19.23adv F u cv ` x cv dfpss2 F u cv ` x cv v pssnel sylbir syl5 inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lemd sylani exp4b pm2.43a x cv {/} = -. adantld a2d finds com12))
+thm (inf3lem3 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G v) (F v) (A v) (B v)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (` F (suc A)))))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem2 com12 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemd jctild F A ` x cv pssdifn0 syl6 inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lemc v cv eleq2d v cv x cv F A ` eldifi v cv x cv F A ` inssdif0 biimpr anim12i inf3lem.1 inf3lem.2 v visset F A fvex inf3lema sylibr syl5bir v cv x cv F A ` eldifn v cv x cv F A ` \ i^i {/} = adantr A om e. a1i jcad F A ` F A suc ` v cv eleq2 biimprd v cv F A suc ` e. v cv F A ` e. iman sylib con2i syl6 exp3a r19.23adv x visset x cv F A ` difss ssexi v zfreg syl5 syld com12))
+thm (inf3lem4 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (C: (` F A) (` F (suc A))))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem1 x cv {/} = -. x cv x cv U. C_ /\ a1i inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem3 jcad F A ` F A suc ` dfpss2 syl6ibr))
+thm (inf3lem5 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (/\ (e. A (om)) (e. B A)) (C: (` F B) (` F A)))) (B A elnn ancoms A nnord A B ordsucss syl B om e. adantr v cv B suc F fveq2 F B ` psseq2d x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv u cv F fveq2 F B ` psseq2d x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv u cv suc F fveq2 F B ` psseq2d x cv {/} = -. x cv x cv U. C_ /\ imbi2d v cv A F fveq2 F B ` psseq2d x cv {/} = -. x cv x cv U. C_ /\ imbi2d B peano2b inf3lem.1 inf3lem.2 inf3lem.4 inf3lem.4 inf3lem4 com12 sylbir inf3lem.1 inf3lem.2 u visset inf3lem.4 inf3lem4 F B ` F u cv ` F u cv suc ` psstr expcom syl6com a2d B suc om e. B suc u cv C_ ad2antll findsg exp B peano2b sylan2b syld exp com23 imp mpd com12))
+thm (inf3lem6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (:-1-1-> F (om) (P~ (cv x)))) (inf3lem.1 inf3lem.2 u visset v visset inf3lem5 F v cv ` F u cv ` dfpss2 pm3.27bd syl6 exp3a imp v cv om e. adantrl inf3lem.1 inf3lem.2 v visset u visset inf3lem5 F u cv ` F v cv ` dfpss2 pm3.27bd F u cv ` F v cv ` eqcom negbii sylib syl6 exp3a imp u cv om e. adantrr jaod con2d v cv u cv ordtri3 v cv nnord u cv nnord syl2an x cv {/} = -. x cv x cv U. C_ /\ adantl sylibrd exp v u 19.21aivv v om u om F v cv ` F u cv ` = v cv u cv = -> r2al sylibr inf3lem.2 G {/} frfnom F G {/} rec om |` om fneq1 mpbiri ax-mp F om u cv v fvelrn F v cv ` u cv x cv P~ eleq1 inf3lem.1 inf3lem.2 v visset inf3lem.4 inf3lemd F v cv fvex x cv elpw sylibr syl5bi com12 r19.23aiv syl6bi ssrdv ancli ax-mp F om x cv P~ df-f mpbir jctil F om x cv P~ v u f1fv sylibr))
+thm (inf3lem7 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3lem.1 (= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y))))))) (inf3lem.2 (= F (|` (rec G ({/})) (om)))) (inf3lem.3 (e. A (V))) (inf3lem.4 (e. B (V)))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (e. (om) (V))) (inf3lem.1 inf3lem.2 inf3lem.3 inf3lem.4 inf3lem6 F om x cv P~ f1f F om x cv P~ fdm syl F dfdm4 syl5eqr F `' V funrnex F om x cv P~ f1f F om x cv P~ frn x visset pwex F ran ssex 3syl F df-rn V eleq1i sylib F om x cv P~ df-f1 pm3.27bd sylc eqeltrrd syl))
+thm (inf3 ((x y) (x z) (w x) (y z) (w y) (w z)) ((inf3.1 (E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x))))))) (e. (om) (V)) (inf3.1 y z z cv w x cv w cv x cv i^i y cv C_ {e.|} = {<,>|} eqid y z z cv w x cv w cv x cv i^i y cv C_ {e.|} = {<,>|} {/} rec om |` eqid x visset x visset inf3lem7 x 19.23aiv ax-mp))
+thm (infeq5 ((x y) (x z) (w x) (y z) (w y) (w z)) () (<-> (E. x (C: (cv x) (U. (cv x)))) (e. (om) (V))) (x cv x cv U. dfpss2 pm3.27bd x cv {/} unieq uni0 syl6req x cv {/} x cv U. eqtrt mpdan nsyl x cv x cv U. pssss jca x 19.22i y z z cv w x cv w cv x cv i^i y cv C_ {e.|} = {<,>|} eqid y z z cv w x cv w cv x cv i^i y cv C_ {e.|} = {<,>|} {/} rec om |` eqid x visset x visset inf3lem7 x 19.23aiv syl om V {/} {} difexg 0ex snid om {/} {} disj4 om {/} {} disj3 bitr3 peano1 om om {/} {} \ {/} eleq2 mpbii {/} om {/} {} eldif sylib pm3.27d sylbi a3i ax-mp om unidif0 limom om limuni ax-mp eqtr4 om {/} {} \ psseq2i mpbir x cv om {/} {} \ x cv U. psseq1 x cv om {/} {} \ unieq om {/} {} \ psseq2d bitrd V cla4egv mpi syl impbi))
+thm (inf4 ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (E. y (/\ (e. (cv y) (cv x)) (A. z (-. (e. (cv z) (cv y)))))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv z) (cv x)) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y))))))))))) (peano1 y cv peano2 y ax-gen x y z axinf inf2 inf3 x cv om {/} eleq2 x cv om y cv eleq2 x cv om y cv suc eleq2 imbi12d y bialdv anbi12d cla4ev mp2an x cv y z 0el y x cv z z cv y cv e. -. A. df-rex bitr y cv x cv z w sucel z x cv w w cv z cv e. w cv y cv e. w cv y cv = \/ <-> A. df-rex bitr y cv x cv e. imbi2i y bial anbi12i x biex mpbi))
+thm (zfinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x))))) (x y z w inf4 x cv y z 0el y x cv z z cv y cv e. -. A. df-rex bitr y cv x cv z w sucel z x cv w w cv z cv e. w cv y cv e. w cv y cv = \/ <-> A. df-rex bitr y x cv biral y x cv z z cv x cv e. w w cv z cv e. w cv y cv e. w cv y cv = \/ <-> A. /\ E. df-ral bitr anbi12i x biex mpbir))
+thm (omex ((x y)) () (e. (om) (V)) (x y zfinf x cv y peano5 y cv x cv e. y cv suc x cv e. -> y cv om e. ax-1 r19.20i2 sylan2 x 19.22i ax-mp x visset om ssex x 19.23aiv ax-mp))
+thm (inf5 () () (E. x (C: (cv x) (U. (cv x)))) (omex x infeq5 mpbir))
+thm (omelon () () (e. (om) (On)) (omex onprc om On V eleq1 mtbiri mt2 omon ori mt3))
+thm (dfom3 ((x y) (x z) (y z)) () (= (om) (|^| ({|} x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x))))))) (0ex x {/} x cv e. y x cv y cv suc x cv e. A.e. /\ elintab {/} x cv e. y x cv y cv suc x cv e. A.e. pm3.26 mpgbir y cv z cv suceq x cv eleq1d x cv rcla4v com12 {/} x cv e. adantl a2i x 19.20i z visset x {/} x cv e. y x cv y cv suc x cv e. A.e. /\ elintab z visset sucex x {/} x cv e. y x cv y cv suc x cv e. A.e. /\ elintab 3imtr4 z cv om e. a1i rgen x {/} x cv e. y x cv y cv suc x cv e. A.e. /\ {|} |^| z peano5 mp2an peano1 y cv peano2 rgen z visset x {/} x cv e. y x cv y cv suc x cv e. A.e. /\ elintab omex x cv om {/} eleq2 x cv om y cv suc eleq2 y raleqd anbi12d x cv om z cv eleq2 imbi12d cla4v sylbi mp2ani ssriv eqssi))
+thm (elom3 ((A x)) () (<-> (e. A (om)) (A. x (-> (Lim (cv x)) (e. A (cv x))))) (A om x elomg ibi pm3.27d limom omex x cv om limeq x cv om A eleq2 imbi12d cla4v mpi impbi))
+thm (dfom4 ((x y)) () (= (om) ({|} x (A. y (-> (Lim (cv y)) (e. (cv x) (cv y)))))) (x cv y elom3 biabri))
+thm (oancom () () (=/= (opr (1o) (+o) (om)) (opr (om) (+o) (1o))) (omelon 1onn 1o oaabslem mp2an omex sucid omelon om oa1suc ax-mp eleqtrr eqeltr 1ord omelon 1o om oacl mp2an omelon 1ord om 1o oacl mp2an 1o +o om opr om +o 1o opr onelpsst mp2an pm3.27bd ax-mp 1o +o om opr om +o 1o opr df-ne mpbir))
+thm (isfinite ((A x)) () (<-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x)))) (A x isfinite2 x A isfinite1 omex A ensym con3i A ~<_ om br anim2i A om brsdom sylibr syl impbi))
+thm (nnsdom () () (-> (e. A (om)) (br A (~<) (om))) (A omsdomnn omex A ensym con3i A ~<_ om br anim2i syl A om brsdom sylibr))
+thm (omenps () () (br (om) (~~) (\ (om) ({} ({/})))) (omex limom V limenpsi ax-mp))
+thm (omensuc () () (br (om) (~~) (suc (om))) (omex limom V limensuci ax-mp))
+thm (infensuc ((x y) (A x) (A y)) () (-> (/\ (e. A (On)) (C_ (om) A)) (br A (~~) (suc A))) (omelon x cv om = id x cv om suceq ~~ breq12d x cv y cv = id x cv y cv suceq ~~ breq12d x cv y cv suc = id x cv y cv suc suceq ~~ breq12d x cv A = id x cv A suceq ~~ breq12d omensuc om On e. a1i y visset y visset sucex y cv V y cv suc V en2sn mp2an y cv y cv suc y cv {} y cv suc {} unen y cv df-suc y cv suc df-suc 3brtr4g exp y cv eloni y cv ordeirr syl y cv y cv disjsn sylibr y cv suc eloni y cv suc ordeirr syl y cv sucelon y cv suc y cv suc disjsn 3imtr4 jca syl5 mpan2 com12 om On e. om y cv C_ ad2antll x visset x cv V limensuc mpan om On e. om x cv C_ ad2antll y x cv om y cv C_ y cv ~~ y cv suc br -> A.e. a1d tfindsg mpanl2))
+thm (unbnnt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om))) (omex A ssex z cv A om sseq1 z cv A y x cv y cv e. rexeq x om biraldv anbi12d z cv A ~~ om breq1 imbi12d z visset x y unbnn V vtoclg exp3a mpcom imp))
+thm (trcl ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A y) (A z) (A v) (F y) (F v) (F u)) ((trcl.1 (e. A (V))) (trcl.2 (= F (|` (rec ({<,>|} z w (= (cv w) (u. (cv z) (U. (cv z))))) A) (om)))) (trcl.3 (= C (U_ y (om) (` F (cv y)))))) (/\/\ (C_ A C) (Tr C) (A. x (-> (/\ (C_ A (cv x)) (Tr (cv x))) (C_ C (cv x))))) (peano1 trcl.2 {/} fveq1i trcl.1 A V z w w cv z cv z cv U. u. = {<,>|} fr0t ax-mp eqtr2 A F {/} ` eqimss ax-mp y cv {/} F fveq2 A sseq2d om rcla4ev mp2an y om A F y cv ` ssiun ax-mp trcl.3 sseqtr4 y om F y cv ` U_ v u dftr2 u cv y om F y cv ` eliun v cv u cv e. anbi2i y om v cv u cv e. u cv F y cv ` e. r19.42v bitr4 F y cv ` U. F y cv ` ssun2 F y cv fvex F y cv fvex uniex unex v cv A e. z ax-17 v cv y cv e. z ax-17 v z w w cv z cv z cv U. u. = hbopab1 v cv A e. z ax-17 hbrdg v cv om e. z ax-17 hbres trcl.2 v cv eleq2i trcl.2 v cv eleq2i z bial 3imtr4 v cv y cv e. z ax-17 hbfv v z w w cv z cv z cv U. u. = hbopab1 v cv A e. z ax-17 hbrdg v cv om e. z ax-17 hbres trcl.2 v cv eleq2i trcl.2 v cv eleq2i z bial 3imtr4 v cv y cv e. z ax-17 hbfv hbuni hbun trcl.2 z cv F y cv ` unieq z cv F y cv ` z cv U. F y cv ` U. uneq12 mpdan V frsucopab mpan2 F y cv ` U. sseq2d mpbiri v cv sseld v cv u cv F y cv ` elunii syl5 r19.22i sylbi y cv peano2 u cv y cv suc F fveq2 v cv eleq2d om rcla4ev exp syl r19.23aiv y cv u cv F fveq2 v cv eleq2d om cbvrexv sylibr v cv y om F y cv ` eliun sylibr syl u ax-gen mpgbir trcl.3 C y om F y cv ` U_ treq ax-mp mpbir v cv {/} F fveq2 x cv sseq1d v cv y cv F fveq2 x cv sseq1d v cv y cv suc F fveq2 x cv sseq1d trcl.2 {/} fveq1i trcl.1 A V z w w cv z cv z cv U. u. = {<,>|} fr0t ax-mp eqtr x cv sseq1i biimpr x cv Tr adantr F y cv ` x cv uniss F y cv ` U. x cv U. x cv sstr2 x cv df-tr syl5ib syl anc2li F y cv ` x cv F y cv ` U. unss syl6ib F y cv fvex F y cv fvex uniex unex v cv A e. z ax-17 v cv y cv e. z ax-17 v z w w cv z cv z cv U. u. = hbopab1 v cv A e. z ax-17 hbrdg v cv om e. z ax-17 hbres trcl.2 v cv eleq2i trcl.2 v cv eleq2i z bial 3imtr4 v cv y cv e. z ax-17 hbfv v z w w cv z cv z cv U. u. = hbopab1 v cv A e. z ax-17 hbrdg v cv om e. z ax-17 hbres trcl.2 v cv eleq2i trcl.2 v cv eleq2i z bial 3imtr4 v cv y cv e. z ax-17 hbfv hbuni hbun trcl.2 z cv F y cv ` unieq z cv F y cv ` z cv U. F y cv ` U. uneq12 mpdan V frsucopab mpan2 x cv sseq1d biimprd syl9r com23 A x cv C_ adantld finds2 com12 r19.21aiv trcl.3 y cv v cv F fveq2 om cbviunv eqtr x cv sseq1i v om F v cv ` x cv iunss bitr sylibr x ax-gen 3pm3.2i))
+thm (tz9.1 ((A x) (A y) (A z) (A w) (A v) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((tz9.1.1 (e. A (V)))) (E. x (/\/\ (C_ A (cv x)) (Tr (cv x)) (A. y (-> (/\ (C_ A (cv y)) (Tr (cv y))) (C_ (cv x) (cv y)))))) (tz9.1.1 w v v cv w cv w cv U. u. = {<,>|} A rec om |` eqid z om w v v cv w cv w cv U. u. = {<,>|} A rec om |` z cv ` U_ eqid y trcl omex w v v cv w cv w cv U. u. = {<,>|} A rec om |` z cv fvex z iunex x cv z om w v v cv w cv w cv U. u. = {<,>|} A rec om |` z cv ` U_ A sseq2 x cv z om w v v cv w cv w cv U. u. = {<,>|} A rec om |` z cv ` U_ treq x cv z om w v v cv w cv w cv U. u. = {<,>|} A rec om |` z cv ` U_ y cv sseq1 A y cv C_ y cv Tr /\ imbi2d y bialdv 3anbi123d cla4ev ax-mp))
+thm (zfregs ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/})))) (A z n0 z cv snex y w tz9.1 y cv w cv x cv trel y cv A x cv inass A x cv incom y cv ineq2i eqtr w cv eleq2i w cv y cv x cv A i^i elin bitr2 w cv y cv A i^i x cv i^i n0i sylbi exp syl6 exp3a com34 imp3a x cv A inss1 w cv sseli ancri syl5 w 19.23adv x cv A i^i w n0 syl5ib com23 imp a3d x cv A e. anim2d exp imp3a x cv y cv A elin y cv A i^i x cv i^i {/} = anbi1i x cv y cv e. x cv A e. y cv A i^i x cv i^i {/} = anass bitr syl5ib r19.22dv2 y visset A inex1 x zfreg2 syl5 z cv A snssi z cv {} y cv C_ anim2i z cv {} y cv A ssin z visset y cv A i^i snss bitr4 sylib z cv y cv A i^i n0i syl syl5 exp3a impcom w z cv {} w cv C_ w cv Tr /\ y cv w cv C_ -> A. 3adant3 y 19.23aiv ax-mp z 19.23aiv sylbi))
+thm (setind ((x y) (A x) (A y)) () (-> (A. x (-> (C_ (cv x) A) (e. (cv x) A))) (= A (V))) (x cv y cv A sseq1 x cv y cv A eleq1 imbi12d a4b1 y cv A ssindif0 syl5ibr y cv V A eldifn nsyli imp nrexdv V A \ y zfregs nsyl2 A vdif0 sylibr))
+thm (setind2 ((A x)) () (-> (C_ (P~ A) A) (= A (V))) (A P~ A x dfss2 x visset A elpw x cv A e. imbi1i x bial bitr x A setind sylbi))
+thm (r1fnon ((x y)) () (Fn (R1) (On)) (x y y cv x cv P~ = {<,>|} {/} rdgfnon x y df-r1 R1 x y y cv x cv P~ = {<,>|} {/} rec On fneq1 ax-mp mpbir))
+thm (r10 ((x y)) () (= (` (R1) ({/})) ({/})) (x y df-r1 {/} fveq1i 0ex x y y cv x cv P~ = {<,>|} rdg0 eqtr))
+thm (r1suc ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (R1) (suc A)) (P~ (` (R1) A)))) (R1 A fvex pwex z cv {/} e. x ax-17 z cv A e. x ax-17 z cv R1 A ` P~ e. x ax-17 x y df-r1 x cv R1 A ` pweq V rdgsucopab mpan2))
+thm (r1lim ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A B) (Lim A)) (= (` (R1) A) (U_ x A (` (R1) (cv x))))) (A B y z z cv y cv P~ = {<,>|} {/} x rdglim2a y z df-r1 A fveq1i y z df-r1 x cv fveq1i x cv A e. a1i iuneq2i 3eqtr4g))
+thm (r1tr ((x y) (x z) (A x) (y z) (A y) (A z)) () (Tr (` (R1) A)) (x cv {/} R1 fveq2 R1 x cv ` R1 {/} ` treq syl x cv y cv R1 fveq2 R1 x cv ` R1 y cv ` treq syl x cv y cv suc R1 fveq2 R1 x cv ` R1 y cv suc ` treq syl x cv A R1 fveq2 R1 x cv ` R1 A ` treq syl tr0 r10 R1 {/} ` {/} treq ax-mp mpbir y cv r1suc x cv eleq2d x visset R1 y cv ` elpw syl6bb R1 y cv ` Tr adantr x cv R1 y cv ` z cv ssel R1 y cv ` dftr4 R1 y cv ` R1 y cv ` P~ z cv ssel sylbi y cv r1suc z cv eleq2d biimprd sylan9r sylan9r ssrdv exp sylbid r19.21aiv R1 y cv suc ` x dftr3 sylibr exp x visset x cv V y r1lim mpan z cv eleq2d z cv y x cv R1 y cv ` eliun biimp syl6bi y x cv R1 y cv ` Tr hbra1 y x cv R1 y cv ` Tr ra4 R1 y cv ` z cv trss syl6 r19.22d sylan9 x visset x cv V y r1lim mpan z cv sseq2d y x cv z cv R1 y cv ` ssiun syl5bir y x cv R1 y cv ` Tr A.e. adantr syld r19.21aiv R1 x cv ` z dftr3 sylibr exp tfinds tr0 r1fnon R1 On fndm ax-mp A eleq2i negbii A R1 ndmfv sylbir R1 A ` {/} treq syl mpbiri pm2.61i))
+thm (r1ord ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (e. (` (R1) A) (` (R1) B)))) (x cv A suc A eleq2 x cv A suc R1 fveq2 R1 A ` eleq2d imbi12d x cv y cv A eleq2 x cv y cv R1 fveq2 R1 A ` eleq2d imbi12d x cv y cv suc A eleq2 x cv y cv suc R1 fveq2 R1 A ` eleq2d imbi12d x cv B A eleq2 x cv B R1 fveq2 R1 A ` eleq2d imbi12d A suc A onelon R1 A fvex pwid A r1suc R1 A ` eleq2d mpbiri syl exp R1 y cv fvex pwid y cv r1suc R1 y cv ` eleq2d mpbiri y cv suc r1tr R1 y cv suc ` R1 y cv ` trss ax-mp syl R1 A ` sseld A y cv e. imim2d A suc On elisset A sucexb sylibr A V y cv sucssel syl imp syl7 A y cv suc e. a1d com24 exp3a imp31 y cv A suc R1 fveq2 R1 A ` eleq2d x cv rcla4ev x cv A limsuc biimpa x cv A onelon x cv limord x visset elon sylibr sylan R1 A fvex pwid A r1suc R1 A ` eleq2d mpbiri syl sylanc R1 A ` y x cv R1 y cv ` eliun sylibr x visset x cv V y r1lim mpan R1 A ` eleq2d A x cv e. adantr mpbird exp A suc On e. A suc x cv C_ ad2antll y x cv A suc y cv C_ A y cv e. R1 A ` R1 y cv ` e. -> -> A.e. a1d tfindsg B On e. A B e. pm3.26 B A onelon A suceloni syl jca B eloni B A ordsucss syl imp sylanc exp pm2.43d))
+thm (r1ord2 () () (-> (e. B (On)) (-> (e. A B) (C_ (` (R1) A) (` (R1) B)))) (B A r1ord B r1tr R1 B ` R1 A ` trss ax-mp syl6))
+thm (r1ord3 () () (-> (/\ (e. A (On)) (e. B (On))) (-> (C_ A B) (C_ (` (R1) A) (` (R1) B)))) (A B onsseleq B A r1ord2 A On e. adantl A B R1 fveq2 R1 A ` R1 B ` eqimss syl A On e. B On e. /\ a1i jaod sylbid))
+thm (r1val1 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ (` (R1) (cv x)))))) (A x onzsl x A R1 x cv ` P~ U_ 0ss A {/} R1 fveq2 r10 syl6eq x A R1 x cv ` P~ U_ sseq1d mpbiri y cv R1 A ` e. x ax-17 y x A R1 x cv ` P~ hbiu1 hbss A x cv suc R1 fveq2 x cv r1suc sylan9eqr x visset sucid A x cv suc x cv eleq2 mpbiri x A R1 x cv ` P~ ssiun2 syl x cv On e. adantl eqsstrd exp r19.23ai A V x r1lim A x cv ordelon A limord sylan x cv sucelon x visset sucid x cv suc x cv r1ord2 mpi sylbi x cv r1suc sseqtrd syl exp r19.21aiv x A R1 x cv ` R1 x cv ` P~ ss2iun syl A V e. adantl eqsstrd 3jaoi sylbi A x cv onelon x cv r1suc syl x cv suc A r1ord3 A x cv onelon x cv sucelon sylib A On e. x cv A e. pm3.26 jca A eloni A x cv ordsucss syl imp sylc eqsstr3d exp r19.21aiv x A R1 x cv ` P~ R1 A ` iunss sylibr eqssd))
+thm (tz9.12lem1 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (C_ (" F A) (On)) (y visset F A x elima3 tz9.12lem.2 x cv y cv <,> eleq2i x visset y visset z cv x cv R1 v cv ` eleq1 v On birabsdv inteqd w cv eqeq2d w cv y cv v On x cv R1 v cv ` e. {e.|} |^| eqeq1 opelopab bitr y cv v On x cv R1 v cv ` e. {e.|} |^| = y 19.8a v On x cv R1 v cv ` e. {e.|} |^| y isset sylibr v On x cv R1 v cv ` e. {e.|} intex y cv v On x cv R1 v cv ` e. {e.|} |^| On eleq1 v On x cv R1 v cv ` e. ssrab v On x cv R1 v cv ` e. {e.|} oninton mpan syl5bir com12 sylbir mpcom sylbi x cv A e. adantl x 19.23aiv sylbi ssriv))
+thm (tz9.12lem2 ((w z) (v z) (A z) (v w) (A w) (A v)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (e. (suc (U. (" F A))) (On)) (tz9.12lem.1 tz9.12lem.2 tz9.12lem1 z w v On z cv R1 v cv ` e. {e.|} |^| funopabeq tz9.12lem.2 F z w w cv v On z cv R1 v cv ` e. {e.|} |^| = {<,>|} funeq ax-mp mpbir tz9.12lem.1 F funimaex ax-mp onuni ax-mp onsuc))
+thm (tz9.12lem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((tz9.12lem.1 (e. A (V))) (tz9.12lem.2 (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v)))))))))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (e. A (` (R1) (suc (suc (U. (" F A))))))) (v cv y cv R1 fveq2 x cv eleq2d On rcla4ev v On x cv R1 v cv ` e. rabn0 sylibr tz9.12lem.2 x cv y cv <,> eleq2i x visset y visset z cv x cv R1 v cv ` eleq1 v On birabsdv inteqd w cv eqeq2d w cv y cv v On x cv R1 v cv ` e. {e.|} |^| eqeq1 opelopab bitr y biex x visset F y eldm2 v On x cv R1 v cv ` e. {e.|} |^| y isset 3bitr4 v On x cv R1 v cv ` e. {e.|} intex bitr4 sylibr z w v On z cv R1 v cv ` e. {e.|} |^| funopabeq tz9.12lem.2 F z w w cv v On z cv R1 v cv ` e. {e.|} |^| = {<,>|} funeq ax-mp mpbir F x cv A funfvima mpan syl tz9.12lem.1 tz9.12lem.2 tz9.12lem1 F A " onsucuni ax-mp F x cv ` sseli tz9.12lem.1 tz9.12lem.2 tz9.12lem2 F A " U. suc F x cv ` r1ord2 ax-mp syl syl6 imp v cv y cv R1 fveq2 x cv eleq2d On rcla4ev v On x cv R1 v cv ` e. rabn0 sylibr v On x cv R1 v cv ` e. {e.|} intex x visset z cv x cv R1 v cv ` eleq1 v On birabsdv inteqd V V w fvopabg mpan tz9.12lem.2 x cv fveq1i syl5eq sylbi v On x cv R1 v cv ` e. ssrab v On x cv R1 v cv ` e. {e.|} onint mpan eqeltrd w v On z cv R1 v cv ` e. hbrab1 hbint hbeleq y z w hbopab tz9.12lem.2 y cv eleq2i tz9.12lem.2 y cv eleq2i v bial 3imtr4 y cv x cv e. v ax-17 hbfv y cv On e. v ax-17 y cv x cv e. v ax-17 y cv R1 e. v ax-17 w v On z cv R1 v cv ` e. hbrab1 hbint hbeleq y z w hbopab tz9.12lem.2 y cv eleq2i tz9.12lem.2 y cv eleq2i v bial 3imtr4 y cv x cv e. v ax-17 hbfv hbfv hbel v cv F x cv ` R1 fveq2 x cv eleq2d elrabf pm3.27bd 3syl x cv A e. adantr sseldd exp31 com3r r19.23adv r19.20i tz9.12lem.1 tz9.12lem.2 tz9.12lem2 F A " U. suc r1suc ax-mp A eleq2i tz9.12lem.1 R1 F A " U. suc ` elpw A R1 F A " U. suc ` x dfss3 3bitr sylibr))
+thm (tz9.12 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) ((tz9.12.1 (e. A (V)))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (E.e. y (On) (e. A (` (R1) (cv y))))) (tz9.12.1 z w w cv v On z cv R1 v cv ` e. {e.|} |^| = {<,>|} eqid x y tz9.12lem3 tz9.12.1 z w w cv v On z cv R1 v cv ` e. {e.|} |^| = {<,>|} eqid tz9.12lem2 onsuc y cv z w w cv v On z cv R1 v cv ` e. {e.|} |^| = {<,>|} A " U. suc suc R1 fveq2 A eleq2d On rcla4ev mpan syl))
+thm (tz9.13 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((tz9.13.1 (e. A (V)))) (E.e. x (On) (e. A (` (R1) (cv x)))) (tz9.13.1 z y x On y cv R1 x cv ` e. E.e. {|} setind z cv y x On y cv R1 x cv ` e. E.e. {|} w cv ssel w visset y cv w cv R1 x cv ` eleq1 x On birexdv elab syl6ib r19.21aiv z visset w x tz9.12 syl z visset y cv z cv R1 x cv ` eleq1 x On birexdv elab sylibr mpg eleqtrr tz9.13.1 y cv A R1 x cv ` eleq1 x On birexdv elab mpbi))
+thm (tz9.13g ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (cv x))))) (y cv A = x ax-17 y cv A R1 x cv ` eleq1 On birexd y visset x tz9.13 B vtoclg))
+thm (rankwflem ((A x)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (suc (cv x)))))) (A B x tz9.13g x cv suceloni x visset sucid x cv suc x cv r1ord2 mpi syl A sseld r19.22i syl))
+thm (jech9.3 ((x y)) () (= (U_ x (On) (` (R1) (cv x))) (V)) (y visset x tz9.13 y cv x On R1 x cv ` eliun mpbir y visset 2th eqri))
+thm (unir1 () () (= (U. (" (R1) (On))) (V)) (r1fnon R1 On fndm ax-mp R1 dom On R1 imaeq2 ax-mp R1 imadmrn eqtr3 unieqi r1fnon R1 On x fniunfv ax-mp eqtr4 x jech9.3 eqtr))
+thm (rankval ((x y) (x z) (A x) (y z) (A y) (A z)) ((rankval.1 (e. A (V)))) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x))))))) (y z x df-rank A fveq1i rankval.1 rankval.1 A V x rankwflem ax-mp x On A R1 x cv suc ` e. rabn0 mpbir x On A R1 x cv suc ` e. {e.|} intex mpbi y cv A R1 x cv suc ` eleq1 x On birabsdv inteqd z fvopab eqtr))
+thm (rankvalg ((x y) (A x) (A y)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))))) (y cv A rank fveq2 y cv A R1 x cv suc ` eleq1 x On birabsdv inteqd eqeq12d y visset x rankval B vtoclg))
+thm (rankval2 ((A x)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (C_ A (` (R1) (cv x))))))) (A B x rankvalg x cv r1suc A eleq2d R1 x cv fvex R1 x cv ` V A elpw2g ax-mp syl6bb birabi inteqi syl6eq))
+thm (rankon ((A x)) () (e. (` (rank) A) (On)) (A V x rankvalg A V x rankwflem x On A R1 x cv suc ` e. rabn0 sylibr x On A R1 x cv suc ` e. ssrab x On A R1 x cv suc ` e. {e.|} oninton mpan syl eqeltrd 0elon A rank fvprc On eleq1d mpbiri pm2.61i))
+thm (rankid ((x y) (A x) (A y)) ((rankid.1 (e. A (V)))) (e. A (` (R1) (suc (` (rank) A)))) (rankid.1 A V x rankwflem ax-mp y cv A e. x ax-17 y cv R1 e. x ax-17 y x On A R1 x cv suc ` e. hbrab1 hbint hbsuc hbfv hbel x cv x On A R1 x cv suc ` e. {e.|} |^| suceq R1 fveq2d A eleq2d onminsb ax-mp rankid.1 x rankval rank A ` x On A R1 x cv suc ` e. {e.|} |^| suceq ax-mp R1 fveq2i eleqtrr))
+thm (rankr1lem () ((rankr1lem.1 (e. A (V)))) (-> (e. B (On)) (-> (-. (e. A (` (R1) B))) (C_ B (` (rank) A)))) (A rankon onsuc rank A ` suc B ontri1 mpan rankr1lem.1 rankid A rankon onsuc rank A ` suc B r1ord3 mpan R1 rank A ` suc ` R1 B ` A ssel syl6 mpii sylbird con1d A rankon B rank A ` onsssuc mpan2 sylibrd))
+thm (rankr1 ((x y) (A x) (A y) (B x) (B y)) ((rankr1.1 (e. A (V)))) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B))))) (A rankon B rank A ` On eleq1 mpbiri B eloni B x ordzsl A noel B {/} R1 fveq2 r10 syl6eq A eleq2d mtbiri B rank A ` = a1d x visset sucid B x cv suc x cv eleq2 mpbiri x cv On e. adantl B x cv ontri1 bicon2d x cv suc On B eleq1a imp x cv suceloni sylan x cv On e. B x cv suc = pm3.26 sylanc mpbid B rank A ` = adantr rankr1.1 y rankval B eqeq2i biimp x cv sseq1d y cv x cv suceq R1 fveq2d A eleq2d onintss imp syl5bir B x cv suc R1 fveq2 A eleq2d biimpa sylan2i exp4d com3l imp31 mtod exp31 r19.23aiv B rank A ` x cv eleq2 A rankon x cv onel syl6bi anc2li x cv suceloni x visset sucid x cv suc x cv r1ord2 mpi syl A sseld y cv x cv suceq R1 fveq2d A eleq2d onintss rankr1.1 y rankval x cv sseq1i syl6ibr syld B rank A ` x cv sseq1 biimprd sylan9r B x cv ontri1 A rankon B rank A ` On eleq1 mpbiri sylan sylibd syl6 com3l B rank A ` = x cv B e. con2 syl6 pm2.43a r19.23aiv con2i B Lim adantr B V x r1lim rank A fvex B rank A ` V eleq1 mpbiri sylan A eleq2d A x B R1 x cv ` eliun syl6bb mtbird expcom 3jaoi sylbi syl mpcom rankr1.1 y rankval y On A R1 y cv suc ` e. ssrab rankr1.1 A V y rankwflem ax-mp y On A R1 y cv suc ` e. rabn0 mpbir y On A R1 y cv suc ` e. {e.|} onint mp2an eqeltr B rank A ` y On A R1 y cv suc ` e. {e.|} eleq1 mpbiri y cv B suceq R1 fveq2d A eleq2d On elrab pm3.27bd syl jca rankr1.1 B rankr1lem com12 A noel B suc R1 ndmfv A eleq2d mtbiri a3i r1fnon R1 On fndm ax-mp B suc eleq2i B sucelon bitr4 sylib syl5 imp A noel B suc R1 ndmfv A eleq2d mtbiri a3i r1fnon R1 On fndm ax-mp B suc eleq2i B sucelon bitr4 sylib y cv B suceq R1 fveq2d A eleq2d onintss mpcom rankr1.1 y rankval syl5ss A R1 B ` e. -. adantl eqssd impbi))
+thm (rankr1g ((A x) (B x)) () (-> (e. A C) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B)))))) (x cv A rank fveq2 B eqeq2d x cv A R1 B ` eleq1 negbid x cv A R1 B suc ` eleq1 anbi12d bibi12d x visset B rankr1 C vtoclg))
+thm (ssrankr1 () ((ssrankr1.1 (e. A (V)))) (-> (e. B (On)) (<-> (C_ B (` (rank) A)) (-. (e. A (` (R1) B))))) (rank A ` eqid ssrankr1.1 rank A ` rankr1 mpbi pm3.26i A rankon B rank A ` r1ord3 mpan2 imp A sseld mtoi exp ssrankr1.1 B rankr1lem impbid))
+thm (rankr1a () ((rankr1a.1 (e. A (V)))) (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (` (rank) A) B))) (rankr1a.1 B ssrankr1 A rankon B rank A ` ontri1 mpan2 bitr3d bicon4d))
+thm (r1val2 ((A x)) () (-> (e. A (On)) (= (` (R1) A) ({|} x (e. (` (rank) (cv x)) A)))) (x visset A ssrankr1 A eloni x cv rankon onord A rank x cv ` ordtri1 mpan2 syl bitr3d bicon4d biabrdv))
+thm (r1val3 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ ({|} y (e. (` (rank) (cv y)) (cv x))))))) (A x r1val1 A x cv onelon x cv y r1val2 R1 x cv ` y rank y cv ` x cv e. {|} pweq 3syl exp r19.21aiv x A R1 x cv ` P~ y rank y cv ` x cv e. {|} P~ iuneq2 syl eqtrd))
+thm (rankel () ((rankel.1 (e. B (V)))) (-> (e. A B) (e. (` (rank) A) (` (rank) B))) (rank A ` eqid A B rank A ` rankr1g mpbii pm3.26d A rankon rank A ` r1suc ax-mp B eleq2i rankel.1 R1 rank A ` ` elpw bitr B R1 rank A ` ` A ssel sylbi com12 mtod B rankon A rankon rank B ` rank A ` ontri1 mp2an B rankon onord A rankon onord rank B ` rank A ` ordsucsssuc mp2an rankel.1 rankid B rankon onsuc A rankon onsuc rank B ` suc rank A ` suc r1ord3 mp2an B sseld mpi sylbi sylbir nsyl2))
+thm (rankval3 ((x y) (A x) (A y)) ((rankval3.1 (e. A (V)))) (= (` (rank) A) (|^| ({e.|} x (On) (A.e. y A (e. (` (rank) (cv y)) (cv x)))))) (rankval3.1 x rankval y visset rankid x cv eloni x cv rank y cv ` ordsucss syl y cv rankon onsuc rank y cv ` suc x cv r1ord3 mpan syld R1 rank y cv ` suc ` R1 x cv ` y cv ssel syl6 mpii y A r19.20sdv x cv r1suc A eleq2d rankval3.1 R1 x cv ` elpw A R1 x cv ` y dfss3 bitr syl6bb sylibrd ss2rabi x On y A rank y cv ` x cv e. A.e. {e.|} x On A R1 x cv suc ` e. {e.|} intss ax-mp eqsstr A rankon rankval3.1 y cv rankel rgen x cv rank A ` rank y cv ` eleq2 y A biraldv onintss mp2 eqssi))
+thm (bndrank ((x y) (A x) (A y)) () (-> (E.e. x (On) (A.e. y A (C_ (` (rank) (cv y)) (cv x)))) (e. A (V))) (y visset rankid x cv eloni y cv rankon onord rank y cv ` x cv ordsucsssuc mpan syl x cv suceloni y cv rankon onsuc rank y cv ` suc x cv suc r1ord3 mpan syl sylbid R1 rank y cv ` suc ` R1 x cv suc ` y cv ssel syl6 mpii y A r19.20sdv A R1 x cv suc ` y dfss3 R1 x cv suc fvex A ssex sylbir syl6 r19.23aiv))
+thm (unbndrank ((x y) (A x) (A y)) () (-> (-. (e. A (V))) (A.e. x (On) (E.e. y A (e. (cv x) (` (rank) (cv y)))))) (y cv rankon rank y cv ` x cv ontri1 mpan y A biraldv y A x cv rank y cv ` e. ralnex syl6bb birexa x On y A x cv rank y cv ` e. E.e. rexnal bitr x y A bndrank sylbir con1i))
+thm (rankpw () ((rankpw.1 (e. A (V)))) (= (` (rank) (P~ A)) (suc (` (rank) A))) (rankpw.1 pwex rank A ` suc rankr1 rank A ` eqid rankpw.1 rank A ` rankr1 pm3.26bd ax-mp rankpw.1 pwid A P~ R1 rank A ` ` A ssel mpi mto rankpw.1 pwex R1 rank A ` ` elpw mtbir A rankon rank A ` r1suc ax-mp A P~ eleq2i mtbir rankpw.1 rankid A rankon rank A ` r1suc ax-mp eleqtr rankpw.1 R1 rank A ` ` elpw mpbi A R1 rank A ` ` sspwb mpbi A rankon rank A ` r1suc ax-mp sseqtr4 rankpw.1 pwex R1 rank A ` suc ` elpw mpbir A rankon onsuc rank A ` suc r1suc ax-mp eleqtrr mpbir2an eqcomi))
+thm (r1pw ((A x) (B x)) () (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) (suc B))))) (x cv A R1 B ` eleq1 x cv A pweq R1 B suc ` eleq1d bibi12d B On e. imbi2d x visset B rankr1a B eloni B rank x cv ` ordsucelsuc syl bitrd x visset rankpw B suc eleq1i syl6bbr B suceloni x visset pwex B suc rankr1a syl bitr4d V vtoclg A R1 B ` elisset A P~ R1 B suc ` elisset A pwexb sylibr pm5.21ni B On e. a1d pm2.61i))
+thm (r1pwcl ((A x) (A y) (x y) (B x) (B y)) () (-> (Lim B) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) B)))) (B V x r1lim A eleq2d A x B R1 x cv ` eliun syl6bb B x cv onelon B V limelon sylan x cv A r1pw syl birexdva B x cv limsuc A P~ R1 x cv suc ` e. anbi1d x visset sucex y cv x cv suc B eleq1 y cv x cv suc R1 fveq2 A P~ eleq2d anbi12d cla4ev syl6bi x 19.23adv x B A P~ R1 x cv suc ` e. df-rex y B A P~ R1 y cv ` e. df-rex 3imtr4g x cv y cv R1 fveq2 A P~ eleq2d B cbvrexv syl6ibr B V e. adantl B x cv onelon B V limelon sylan exp x cv sssucid x cv x cv suc r1ord3 mpi x cv sucelon sylan2b anidms A P~ sseld syl6 r19.22dv impbid 3bitrd B V x r1lim A P~ eleq2d A P~ x B R1 x cv ` eliun syl6bb bitr4d exp A R1 B ` n0i B R1 fvprc nsyl2 A P~ R1 B ` n0i B R1 fvprc nsyl2 pm5.21ni B Lim a1d pm2.61i))
+thm (rankss () ((rankss.1 (e. B (V)))) (-> (C_ A B) (C_ (` (rank) A) (` (rank) B))) (rankss.1 pwex A rankel rankss.1 B V A elpw2g ax-mp rankss.1 rankpw rank A ` eleq2i A rankon B rankon rank A ` rank B ` onsssuc mp2an bitr4 3imtr3))
+thm (ranksn ((x y) (A x) (A y)) ((ranksn.1 (e. A (V)))) (= (` (rank) ({} A)) (suc (` (rank) A))) (y A {} rank y cv ` x cv e. df-ral y A elsn rank y cv ` x cv e. imbi1i y bial ranksn.1 y cv A rank fveq2 x cv eleq1d ceqsalv 3bitr x cv On e. a1i birabi inteqi A snex x y rankval3 A rankon rank A ` x onsucmin ax-mp 3eqtr4))
+thm (rankuni ((x y) (x z) (A x) (y z) (A y) (A z)) ((ranksn.1 (e. A (V)))) (= (` (rank) (U. A)) (U_ x A (` (rank) (cv x)))) (ranksn.1 uniex z y rankval3 z cv x A rank x cv ` U_ rank y cv ` eleq2 y A U. biraldv On elrab ranksn.1 rank x cv fvex x iunon x cv rankon x cv A e. a1i mprg y cv A x eluni2 z cv rank y cv ` e. x ax-17 z x A rank x cv ` hbiu1 hbel x A rank x cv ` ssiun2 rank y cv ` sseld x visset y cv rankel syl5 r19.23ai sylbi rgen mpbir2an x A rank x cv ` U_ z On y A U. rank y cv ` z cv e. A.e. {e.|} intss1 ax-mp eqsstr x A rank x cv ` rank A U. ` iunss x cv A elssuni ranksn.1 uniex x cv rankss syl mprgbir eqssi))
+thm (rankuniss ((A x)) ((ranksn.1 (e. A (V)))) (C_ (` (rank) (U. A)) (` (rank) A)) (ranksn.1 x rankuni x A rank x cv ` rank A ` iunss ranksn.1 x cv rankel A rankon rank x cv ` onelss syl mprgbir eqsstr))
+thm (rankun ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((rankun.1 (e. A (V))) (rankun.2 (e. B (V)))) (= (` (rank) (u. A B)) (u. (` (rank) A) (` (rank) B))) (rankun.1 rankun.2 unex y z rankval3 x cv eleq2i x visset y On z A B u. rank z cv ` y cv e. A.e. elintrab bitr z cv A B elun rankun.1 z cv rankel rank z cv ` rank A ` rank B ` elun1 syl rankun.2 z cv rankel rank z cv ` rank B ` rank A ` elun2 syl jaoi sylbi rgen A rankon B rankon onun y cv rank A ` rank B ` u. rank z cv ` eleq2 z A B u. biraldv y cv rank A ` rank B ` u. x cv eleq2 imbi12d On rcla4v ax-mp mpi sylbi ssriv A B ssun1 rankun.1 rankun.2 unex A rankss ax-mp B A ssun2 rankun.1 rankun.2 unex B rankss ax-mp unssi eqssi))
+thm (rankpr () ((rankun.1 (e. A (V))) (rankun.2 (e. B (V)))) (= (` (rank) ({,} A B)) (suc (u. (` (rank) A) (` (rank) B)))) (A B df-pr rank fveq2i A snex B snex rankun rankun.1 ranksn rankun.2 ranksn uneq12i 3eqtr A rankon onord B rankon onord rank A ` rank B ` ordsucun mp2an eqtr4))
+thm (r1rankid () () (-> (e. A B) (C_ A (` (R1) (` (rank) A)))) (rank A ` eqid A B rank A ` rankr1g mpbii pm3.27d A rankon rank A ` r1suc ax-mp A eleq2i sylib A B R1 rank A ` ` elpwg mpbid))
+thm (rankonid ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (On)) (= (` (rank) A) A)) (x cv y cv rank fveq2 x cv y cv = id eqeq12d x cv A rank fveq2 x cv A = id eqeq12d rank y cv ` y cv z cv eleq1 y x cv r19.20si y x cv rank y cv ` z cv e. y cv z cv e. r19.15 syl x cv z cv y dfss3 syl6bbr z On birabsdv inteqd x visset z y rankval3 syl5eq x cv On z intmin eqcomd sylan9eqr exp tfis3 A rankon rank A ` A On eleq1 mpbii impbi))
+thm (rankr1id ((x y) (A x) (A y)) () (<-> (e. A (On)) (= (` (rank) (` (R1) A)) A)) (x cv A R1 fveq2 rank fveq2d x cv A = id eqeq12d x cv y cv r1ord3 exp r19.21aiv y On x cv y cv C_ R1 x cv ` R1 y cv ` C_ ss2rab sylibr y On x cv y cv C_ {e.|} y On R1 x cv ` R1 y cv ` C_ {e.|} intss syl x cv On y intmin sseqtr4d R1 x cv fvex R1 x cv ` V y rankval2 ax-mp syl5ss x cv rankonid x visset x cv V r1rankid ax-mp rank x cv ` x cv R1 fveq2 x cv sseq2d mpbii R1 x cv fvex x cv rankss syl rank x cv ` x cv rank R1 x cv ` ` sseq1 mpbid sylbi eqssd vtoclga R1 A ` rankon rank R1 A ` ` A On eleq1 mpbii impbi))
+thm (ranklon ((x y) (A x) (A y)) ((ranklon.1 (e. A (V)))) (= (` (rank) A) (U_ x A (suc (` (rank) (cv x))))) (y cv A e. x ax-17 y cv R1 e. x ax-17 y x A rank x cv ` suc hbiu1 hbfv dfss2f x visset rankid x A rank x cv ` suc ssiun2 x cv rankon onsuc ranklon.1 rank x cv fvex sucex x iunon x cv rankon onsuc x cv A e. a1i mprg rank x cv ` suc x A rank x cv ` suc U_ r1ord3 mp2an syl x cv sseld mpi mpgbir R1 x A rank x cv ` suc U_ fvex A rankss ax-mp ranklon.1 rank x cv fvex sucex x iunon x cv rankon onsuc x cv A e. a1i mprg x A rank x cv ` suc U_ y cv r1ord3 mpan ss2rabi y On x A rank x cv ` suc U_ y cv C_ {e.|} y On R1 x A rank x cv ` suc U_ ` R1 y cv ` C_ {e.|} intss ax-mp R1 x A rank x cv ` suc U_ fvex R1 x A rank x cv ` suc U_ ` V y rankval2 ax-mp ranklon.1 rank x cv fvex sucex x iunon x cv rankon onsuc x cv A e. a1i mprg x A rank x cv ` suc U_ On y intmin ax-mp 3sstr4 sstri x A rank x cv ` suc rank A ` iunss ranklon.1 x cv rankel x cv rankon A rankon onsucss sylib mprgbir eqssi))
+thm (scottex ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) (V)) (0ex A {/} V eleq1 mpbiri A V x y A rank x cv ` rank y cv ` C_ A.e. rabexg syl A y n0 y A rank x cv ` rank y cv ` C_ hbra1 z cv A e. y ax-17 x hbrab z cv V e. y ax-17 hbel y A rank x cv ` rank y cv ` C_ ra4 com12 x cv A e. a1d r19.21aiv x A y A rank x cv ` rank y cv ` C_ A.e. rank x cv ` rank y cv ` C_ ss2rab sylibr y cv rankon x cv w cv rank fveq2 rank y cv ` sseq1d A elrab pm3.27bd rgen z cv rank y cv ` rank w cv ` sseq2 w x A rank x cv ` rank y cv ` C_ {e.|} biraldv On rcla4ev mp2an z w x A rank x cv ` rank y cv ` C_ {e.|} bndrank ax-mp x A y A rank x cv ` rank y cv ` C_ A.e. {e.|} ssex syl 19.23ai sylbi pm2.61i))
+thm (scott0 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (= A ({/})) (= ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) ({/}))) (A {/} x y A rank x cv ` rank y cv ` C_ A.e. rabeq x y A rank x cv ` rank y cv ` C_ A.e. rab0 syl6eq A x n0 x A rank x cv ` rank x cv ` = hbre1 rank x cv ` eqid x A rank x cv ` rank x cv ` = ra4e mpan2 19.23ai sylbi rank x cv fvex y cv rank x cv ` rank x cv ` eqeq1 x cv A e. anbi2d cla4ev x 19.22i y x x cv A e. y cv rank x cv ` = /\ excom sylibr x A rank x cv ` rank x cv ` = df-rex x A y cv rank x cv ` = df-rex y biex 3imtr4 syl y x A y cv rank x cv ` = E.e. abn0 sylibr z y x A y cv rank x cv ` = E.e. hbab1 z cv On e. y ax-17 dfss2f y x A y cv rank x cv ` = E.e. abid x cv rankon y cv rank x cv ` On eleq1 mpbiri x cv A e. a1i r19.23aiv sylbi mpgbir y x A y cv rank x cv ` = E.e. {|} onint mpan rank x cv fvex x A y dfiin2 y x A y cv rank x cv ` = E.e. {|} eleq1i sylibr syl y x A rank x cv ` hbii1 hbeleq y cv x A rank x cv ` |^|_ rank x cv ` eqeq1 A birexd y x A y cv rank x cv ` = E.e. {|} elabg ibi x A rank x cv ` |^|_ rank x cv ` rank y cv ` sseq1 rank y cv ` ssid x cv y cv rank fveq2 rank y cv ` sseq1d A rcla4ev mpan2 x A rank x cv ` rank y cv ` iinss syl syl5bi r19.21aiv x A r19.22si 3syl x A y A rank x cv ` rank y cv ` C_ A.e. rabn0 sylibr a3i impbi))
+thm (scottexs ((x y) (x z) (y z) (ph y) (ph z)) () (e. ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)) (y cv x ph {|} e. z ax-17 y x ph hbab1 y x ph hbab1 rank z cv ` rank y cv ` C_ x ax-17 hbral y x ph {|} rank x cv ` rank y cv ` C_ A.e. z ax-17 z cv x cv rank fveq2 rank y cv ` sseq1d y x ph {|} biraldv cbvrab x x ph {|} y x ph {|} rank x cv ` rank y cv ` C_ A.e. df-rab x ph abid y x ph {|} rank x cv ` rank y cv ` C_ df-ral y x ph df-clab rank x cv ` rank y cv ` C_ imbi1i y bial bitr anbi12i x biabi 3eqtr z x ph {|} y scottex eqeltrr))
+thm (scott0s ((x y) (x z) (y z) (ph y) (ph z)) () (<-> (E. x ph) (-. (= ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) ({/})))) (x ph abn0 x ph {|} z y scott0 y cv x ph {|} e. z ax-17 y x ph hbab1 y x ph hbab1 rank z cv ` rank y cv ` C_ x ax-17 hbral y x ph {|} rank x cv ` rank y cv ` C_ A.e. z ax-17 z cv x cv rank fveq2 rank y cv ` sseq1d y x ph {|} biraldv cbvrab x x ph {|} y x ph {|} rank x cv ` rank y cv ` C_ A.e. df-rab x ph abid y x ph {|} rank x cv ` rank y cv ` C_ df-ral y x ph df-clab rank x cv ` rank y cv ` C_ imbi1i y bial bitr anbi12i x biabi 3eqtr {/} eqeq1i bitr negbii bitr3))
+thm (cplem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w) (D w)) ((cplem1.1 (= C ({e.|} y B (A.e. z B (C_ (` (rank) (cv y)) (` (rank) (cv z))))))) (cplem1.2 (= D (U_ x A C)))) (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B D) ({/}))))) (cplem1.1 y B z B rank y cv ` rank z cv ` C_ A.e. ssrab eqsstr w cv sseli x cv A e. a1i x A C ssiun2 cplem1.2 syl6ssr w cv sseld jcad w cv B D inelcm syl6 w 19.23adv B y z scott0 cplem1.1 {/} eqeq1i bitr4 negbii C w n0 bitr syl5ib rgen))
+thm (cplem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((cplem2.1 (e. A (V)))) (E. y (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B (cv y)) ({/})))))) (z B w B rank z cv ` rank w cv ` C_ A.e. {e.|} eqid x A z B w B rank z cv ` rank w cv ` C_ A.e. {e.|} U_ eqid cplem1 cplem2.1 z B w scottex x iunex y x A z B w B rank z cv ` rank w cv ` C_ A.e. {e.|} hbiu1 hbeleq y cv x A z B w B rank z cv ` rank w cv ` C_ A.e. {e.|} U_ B ineq2 {/} eqeq1d negbid B {/} = -. imbi2d A birald cla4ev ax-mp))
+thm (cp ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (E. w (A.e. x (cv z) (-> (E. y ph) (E.e. y (cv w) ph)))) (z visset w x y ph {|} cplem2 y ph abn0 y cv y ph {|} w cv elin y ph abid y cv w cv e. anbi1i ph y cv w cv e. ancom 3bitr y biex z y ph hbab1 z cv w cv e. y ax-17 hbin n0f y w cv ph df-rex 3bitr4 imbi12i x z cv biral w biex mpbi))
+thm (bnd ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (-> (A.e. x (cv z) (E. y ph)) (E. w (A.e. x (cv z) (E.e. y (cv w) ph)))) (w x z y ph cp x z cv y ph E. y w cv ph E.e. r19.20 w 19.22i ax-mp w x z cv y ph E. A.e. x z cv y w cv ph E.e. A.e. 19.37v mpbi))
+thm (bnd2 ((ph z) (ph w) (ph v) (w z) (v z) (v w) (x z) (w x) (v x) (A x) (A z) (A w) (A v) (x y) (B x) (y z) (w y) (v y) (B y) (B z) (B w) (B v)) ((bnd2.1 (e. A (V)))) (-> (A.e. x A (E.e. y B ph)) (E. z (/\ (C_ (cv z) B) (A.e. x A (E.e. y (cv z) ph))))) (y B ph df-rex x A biral bnd2.1 v cv A x y y cv B e. ph /\ E. raleq v cv A x y w cv y cv B e. ph /\ E.e. raleq w biexdv imbi12d x v y y cv B e. ph /\ w bnd vtocl sylbi w visset B inex1 w cv B inss2 z cv w cv B i^i B sseq1 mpbiri x A y z cv ph E.e. A.e. biantrurd z cv w cv B i^i y ph rexeq y cv w cv B elin ph anbi1i y cv w cv e. y cv B e. ph anass bitr birex2 syl6bb x A biraldv bitr3d cla4ev w 19.23aiv syl))
+thm (kardex ((x y) (x z) (A x) (y z) (A y) (A z)) ((kardex.1 (e. A (V)))) (e. ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)) (x z z cv ~~ A br {|} y z z cv ~~ A br {|} rank x cv ` rank y cv ` C_ A.e. df-rab x visset z cv x cv ~~ A breq1 elab y z z cv ~~ A br {|} rank x cv ` rank y cv ` C_ df-ral y visset z cv y cv ~~ A breq1 elab rank x cv ` rank y cv ` C_ imbi1i y bial bitr anbi12i x biabi eqtr x z z cv ~~ A br {|} y scottex eqeltrr))
+thm (karden ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C z) (D z)) ((karden.1 (e. A (V))) (karden.2 (e. B (V))) (karden.3 (= C ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (karden.4 (= D ({|} x (/\ (br (cv x) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))))) (<-> (= C D) (br A (~~) B)) (karden.1 enref karden.1 w cv A ~~ A breq1 cla4ev ax-mp w w cv ~~ A br abn0 mpbir w w cv ~~ A br {|} z y scott0 mtbi z w w cv ~~ A br {|} y w w cv ~~ A br {|} rank z cv ` rank y cv ` C_ A.e. rabn0 mpbi z cv ~~ A br y y cv ~~ A br rank z cv ` rank y cv ` C_ -> A. pm3.26 C D = a1i karden.3 karden.4 eqeq12i x x cv ~~ A br y y cv ~~ A br rank x cv ` rank y cv ` C_ -> A. /\ x cv ~~ B br y y cv ~~ B br rank x cv ` rank y cv ` C_ -> A. /\ eq2ab bitr x cv z cv ~~ A breq1 x cv z cv rank fveq2 rank y cv ` sseq1d y cv ~~ A br imbi2d y bialdv anbi12d x cv z cv ~~ B breq1 x cv z cv rank fveq2 rank y cv ` sseq1d y cv ~~ B br imbi2d y bialdv anbi12d bibi12d a4b1 sylbi z cv ~~ B br y y cv ~~ B br rank z cv ` rank y cv ` C_ -> A. pm3.26 syl6bi jcad A z cv B entrt karden.1 z cv ensym sylan syl6 z visset w cv z cv ~~ A breq1 elab y w w cv ~~ A br {|} rank z cv ` rank y cv ` C_ df-ral y visset w cv y cv ~~ A breq1 elab rank z cv ` rank y cv ` C_ imbi1i y bial bitr anbi12i syl5ib exp3a r19.23adv mpi karden.2 B V A x cv enen2 mpan karden.2 B V A y cv enen2 mpan rank x cv ` rank y cv ` C_ imbi1d y bialdv anbi12d x biabdv karden.3 karden.4 3eqtr4g impbi))
+thm (aceq1 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (y z) (w y) (v y) (u y) (t y) (h y) (w z) (v z) (u z) (t z) (h z) (v w) (u w) (t w) (h w) (u v) (t v) (h v) (t u) (h u) (h t)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))))))) (w cv t cv = v h cv u y cv h cv u cv e. v cv u cv e. /\ E.e. E!e. pm4.2i h cv cbvralv v cv z cv u cv eleq1 h cv u cv e. anbi2d u y cv birexdv h cv cbvreuv z h cv u y cv h cv u cv e. z cv u cv e. /\ E.e. E!e. pm4.2 bitr t h cv biral bitr h x cv biral z cv h cv u cv eleq1 v cv u cv e. anbi1d u y cv birexdv v reueqd w raleqd x cv cbvralv w cv h cv u cv eleq1 z cv u cv e. anbi1d u y cv birexdv z reueqd t raleqd x cv cbvralv 3bitr4 y biex z w cv x cv e. z cv w cv e. x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. -> 19.21v z cv w cv e. w cv x cv e. x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. impexp z cv w cv e. w cv x cv e. x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. bi2.04 bitr z bial z z cv w cv e. u y cv w cv u cv e. z cv u cv e. /\ E.e. /\ x df-eu z w cv u y cv w cv u cv e. z cv u cv e. /\ E.e. df-reu x z cv w cv e. x cv y cv e. w cv x cv e. z cv x cv e. /\ /\ 19.42v z cv w cv e. x cv y cv e. w cv x cv e. z cv x cv e. an42 z cv w cv e. x cv y cv e. w cv x cv e. z cv x cv e. /\ anass bitr3 x biex u y cv w cv u cv e. z cv u cv e. /\ df-rex u cv x cv y cv eleq1 u cv x cv w cv eleq2 u cv x cv z cv eleq2 anbi12d anbi12d cbvexv bitr z cv w cv e. anbi2i 3bitr4 z cv x cv = bibi1i z bial x biex 3bitr4 t cv w cv e. imbi2i t bial t w cv z w cv u y cv w cv u cv e. z cv u cv e. /\ E.e. E!e. df-ral z cv w cv e. x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. -> t ax-17 t cv w cv e. z ax-17 z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> hba1 x hbex hbim z cv t cv w cv eleq1 x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. imbi1d cbval 3bitr4 w cv x cv e. imbi2i 3bitr4 w bial z w z cv w cv e. w cv x cv e. /\ x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. -> alcom w x cv t w cv z w cv u y cv w cv u cv e. z cv u cv e. /\ E.e. E!e. A.e. df-ral 3bitr4r y biex bitr))
+thm (aceq0 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v)))))))))) (y z x w v u aceq1 v x u equequ2 t u cv w cv e. w cv t cv e. /\ u cv t cv e. t cv y cv e. /\ /\ E. bibi2d t x w elequ2 u cv w cv e. anbi2d t x u elequ2 t x y elequ1 anbi12d anbi12d cbvexv u cv x cv = bibi1i syl6bb u bialdv u z w elequ1 w cv x cv e. anbi1d u z x elequ1 x cv y cv e. anbi1d anbi12d x biexdv u z x equequ1 bibi12d cbvalv syl6bb cbvexv z cv w cv e. w cv x cv e. /\ imbi2i z w bi2al y biex bitr4))
+thm (aceq2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) (w cv t cv = v z cv u y cv z cv u cv e. v cv u cv e. /\ E.e. E!e. pm4.2i z cv cbvralv t z cv v z cv u y cv z cv u cv e. v cv u cv e. /\ E.e. E!e. df-ral t t cv z cv e. v z cv u y cv z cv u cv e. v cv u cv e. /\ E.e. E!e. 19.23v bitr z cv t n0 v cv u cv z cv eleq2 v cv u cv w cv eleq2 anbi12d y cv cbvrexv w z cv bireu w cv v cv u cv eleq1 z cv u cv e. anbi2d u y cv birexdv z cv cbvreuv bitr imbi12i bicomi bitr bitr z x cv biral y biex))
+thm (aceq3lem ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (f x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (f y) (w z) (v z) (u z) (t z) (h z) (g z) (f z) (v w) (u w) (t w) (h w) (g w) (f w) (u v) (t v) (h v) (g v) (f v) (t u) (h u) (g u) (f u) (h t) (g t) (f t) (g h) (f h) (f g) (F t) (F h) (F g)) ((aceq3lem.1 (= F ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u)))))))))) (-> (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))) (E. f (/\ (C_ (cv f) (cv y)) (Fn (cv f) (dom (cv y)))))) (y visset y cv V rnexg ax-mp pwex x cv y cv ran P~ z z cv {/} = -. f cv z cv ` z cv e. -> raleq f biexdv cla4v w v w cv y cv dom e. v cv f cv u w cv y cv u cv br {|} ` = /\ relopab aceq3lem.1 F w v w cv y cv dom e. v cv f cv u w cv y cv u cv br {|} ` = /\ {<,>|} releq ax-mp mpbir z y cv ran P~ z cv {/} = -. f cv z cv ` z cv e. -> A.e. a1i aceq3lem.1 t cv h cv <,> eleq2i t visset h visset w cv t cv y cv dom eleq1 w cv t cv y cv u cv breq1 u biabdv f cv fveq2d v cv eqeq2d anbi12d v cv h cv f cv u t cv y cv u cv br {|} ` eqeq1 t cv y cv dom e. anbi2d opelopab bitr pm3.26bd t cv y cv u cv br t 19.8a u ss2abi y cv u t dfrn2 sseqtr4 y visset y cv V rnexg ax-mp y cv ran V u t cv y cv u cv br {|} elpw2g ax-mp mpbir z cv u t cv y cv u cv br {|} {/} eqeq1 negbid z cv u t cv y cv u cv br {|} f cv fveq2 z cv u t cv y cv u cv br {|} = id eleq12d imbi12d y cv ran P~ rcla4v ax-mp t visset y cv u eldm u t cv y cv u cv br abn0 bitr4 syl5ib com12 w cv t cv y cv u cv breq1 u biabdv f cv fveq2d aceq3lem.1 f cv u t cv y cv u cv br {|} fvex fvopab4 u t cv y cv u cv br {|} eleq1d F t cv fvex h cv F t cv ` t cv y cv breq2 u cv h cv t cv y cv breq2 cbvabv elab2 t cv y cv F t cv ` df-br bitr2 syl5bb sylibrd syl f cv u w cv y cv u cv br {|} fvex aceq3lem.1 fnopab2 F y cv dom fnfun ax-mp h visset F t cv funfvopi ax-mp F t cv ` h cv t cv opeq2 y cv eleq1d syl sylibd com12 relssdv f cv u w cv y cv u cv br {|} fvex aceq3lem.1 fnopab2 jctir y visset y cv V dmexg ax-mp f cv u w cv y cv u cv br {|} fvex aceq3lem.1 fnopab2 y cv dom V F fnex mp2 g cv F y cv sseq1 g cv F y cv dom fneq1 anbi12d cla4ev syl f 19.23aiv syl g cv f cv y cv sseq1 g cv f cv y cv dom fneq1 anbi12d cbvexv sylib))
+thm (aceq3 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))))))) (x cv y cv f cv sseq2 x cv y cv dmeq x cv dom y cv dom f cv fneq2 syl anbi12d f biexdv cbvalv x visset x visset uniex xpex w cv x cv e. v cv w cv e. pm3.26 v cv w cv x cv elunii ancoms jca w v ssopab2i x cv x cv U. w v df-xp sseqtr4 ssexi y cv w v w cv x cv e. v cv w cv e. /\ {<,>|} f cv sseq2 y cv w v w cv x cv e. v cv w cv e. /\ {<,>|} dmeq y cv dom w v w cv x cv e. v cv w cv e. /\ {<,>|} dom f cv fneq2 syl anbi12d f biexdv cla4v f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} dom fndm f cv dom w v w cv x cv e. v cv w cv e. /\ {<,>|} dom z cv eleq2 w v w cv x cv e. v cv w cv e. /\ dmopab z cv eleq2i v z cv x cv e. v cv z cv e. 19.42v z visset w cv z cv x cv eleq1 w cv z cv v cv eleq2 anbi12d v biexdv elab z cv v n0 z cv x cv e. anbi2i 3bitr4 bitr2 syl6rbbr syl f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} C_ adantl f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} z cv funfvima3 ancoms f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} dom fnfun sylan2 sylbid imp z cv x cv e. u cv z cv e. ibar biabrdv z visset z cv V w v w cv x cv e. v cv w cv e. /\ {<,>|} u imasng ax-mp z visset u visset w cv z cv x cv eleq1 w cv z cv v cv eleq2 anbi12d v cv u cv z cv eleq1 z cv x cv e. anbi2d w v w cv x cv e. v cv w cv e. /\ {<,>|} eqid brab u biabi eqtr syl6reqr f cv z cv ` eleq2d f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} C_ f cv w v w cv x cv e. v cv w cv e. /\ {<,>|} dom Fn /\ z cv {/} = -. ad2antrl mpbid exp32 r19.21aiv f 19.22i syl x 19.21aiv w v w cv y cv dom e. v cv f cv u w cv y cv u cv br {|} ` = /\ {<,>|} eqid x z aceq3lem y 19.21aiv impbi bitr))
+thm (aceq4 ((f x) (f z) (f y) (f w) (f v) (x z) (x y) (w x) (v x) (y z) (w z) (v z) (w y) (v y) (v w)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (/\ (Fn (cv f) (cv x)) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))))) (x f z aceq3 f cv y cv z cv fveq1 z cv eleq1d z cv {/} = -. imbi2d z x cv biraldv cbvexv w cv z cv y cv fveq2 w v w cv x cv e. v cv y cv w cv ` = /\ {<,>|} eqid y cv z cv fvex fvopab4 z cv eleq1d z cv {/} = -. imbi2d birala w v w cv x cv e. v cv y cv w cv ` = /\ {<,>|} x cv Fn anbi2i y cv w cv fvex w v w cv x cv e. v cv y cv w cv ` = /\ {<,>|} eqid fnopab2 mpbiran x visset w v y cv w cv ` funopabex2 f cv w v w cv x cv e. v cv y cv w cv ` = /\ {<,>|} x cv fneq1 f cv w v w cv x cv e. v cv y cv w cv ` = /\ {<,>|} z cv fveq1 z cv eleq1d z cv {/} = -. imbi2d z x cv biraldv anbi12d cla4ev sylbir y 19.23aiv sylbi f cv x cv Fn z x cv z cv {/} = -. f cv z cv ` z cv e. -> A.e. pm3.27 f 19.22i impbi x bial bitr))
+thm (aceq5lem1 ((v w) (v y) (g v) (t v) (w y) (g w) (t w) (g y) (t y) (g t)) () (<-> (E! v (e. (cv v) (i^i (X. ({} (cv w)) (cv w)) (cv y)))) (E! g (/\ (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y))))) (v cv w cv {} w cv X. y cv elin v cv w cv {} w cv t g elxp t g v cv t cv g cv <,> = t cv w cv {} e. g cv w cv e. /\ /\ excom bitr v cv y cv e. anbi1i g t v cv t cv g cv <,> = t cv w cv {} e. g cv w cv e. /\ /\ v cv y cv e. 19.41vv v cv t cv g cv <,> = t cv w cv {} e. g cv w cv e. /\ v cv y cv e. an23 v cv t cv g cv <,> y cv eleq1 pm5.32i t w cv elsn g cv w cv e. anbi1i anbi12i v cv t cv g cv <,> = t cv g cv <,> y cv e. t cv w cv = g cv w cv e. an4 v cv t cv g cv <,> = t cv w cv = ancom t cv g cv <,> y cv e. g cv w cv e. ancom anbi12i t cv w cv = v cv t cv g cv <,> = g cv w cv e. t cv g cv <,> y cv e. /\ anass 3bitr 3bitr t biex w visset t cv w cv g cv opeq1 v cv eqeq2d t cv w cv g cv opeq1 y cv eleq1d g cv w cv e. anbi2d anbi12d ceqsexv bitr g biex bitr3 3bitr v bieu v g w cv g cv w cv e. w cv g cv <,> y cv e. /\ euop2 bitr))
+thm (aceq5lem2 ((u w) (t w) (h w) (g w) (t u) (h u) (g u) (h t) (g t) (g h) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))))) (<-> (e. (<,> (cv w) (cv g)) (U. A)) (/\ (e. (cv w) (cv h)) (e. (cv g) (cv w)))) (aceq5lem.1 unieqi w cv g cv <,> eleq2i w cv g cv <,> u u cv {/} = -. t h cv u cv t cv {} t cv X. = E.e. /\ eluniab t h cv w cv g cv <,> u cv e. u cv {/} = -. /\ u cv t cv {} t cv X. = r19.42v w cv g cv <,> u cv e. u cv {/} = -. t h cv u cv t cv {} t cv X. = E.e. anass bitr2 u biex t h cv u w cv g cv <,> u cv e. u cv {/} = -. /\ u cv t cv {} t cv X. = /\ rexcom4 t h cv u w cv g cv <,> u cv e. u cv {/} = -. /\ u cv t cv {} t cv X. = /\ E. df-rex bitr3 3bitr w cv g cv <,> u cv e. u cv {/} = -. /\ u cv t cv {} t cv X. = ancom w cv g cv <,> u cv n0i pm4.71i u cv t cv {} t cv X. = anbi2i bitr4 u biex t cv snex t visset xpex u cv t cv {} t cv X. w cv g cv <,> eleq2 ceqsexv bitr t cv h cv e. anbi2i g visset w cv t cv {} t cv opelxp w t cv elsn w cv t cv eqcom bitr g cv t cv e. anbi1i bitr t cv h cv e. anbi2i t cv h cv e. t cv w cv = g cv t cv e. an12 3bitr t biex w visset t cv w cv h cv eleq1 t cv w cv g cv eleq2 anbi12d ceqsexv bitr 3bitr))
+thm (aceq5lem3 ((u w) (t w) (h w) (t u) (h u) (h t) (A w)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))))) (<-> (e. (X. ({} (cv w)) (cv w)) A) (/\ (-. (= (cv w) ({/}))) (e. (cv w) (cv h)))) (w cv snex w visset xpex u cv w cv {} w cv X. {/} eqeq1 negbid u cv w cv {} w cv X. t cv {} t cv X. eqeq1 t h cv birexdv anbi12d elab aceq5lem.1 w cv {} w cv X. eleq2i w cv {/} w cv {} xpeq2 w cv {} xp0 syl6eq w cv {} w cv X. {/} rneq w visset snnz w cv {} w cv rnxp ax-mp rn0 3eqtr3g impbi negbii w cv {} w cv X. t cv {} t cv X. rneq w visset snnz w cv {} w cv rnxp ax-mp t visset snnz t cv {} t cv rnxp ax-mp 3eqtr3g w cv t cv sneq w cv {} t cv {} w cv xpeq1 syl w cv t cv t cv {} xpeq2 eqtrd impbi w cv t cv eqcom bitr t cv h cv e. anbi2i t cv h cv e. t cv w cv = ancom bitr t biex w visset t cv w cv h cv eleq1 ceqsexv bitr2 t h cv w cv {} w cv X. t cv {} t cv X. = df-rex bitr4 anbi12i 3bitr4))
+thm (aceq5lem4 ((x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B g) (A x) (A y) (A z) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (aceq5lem.2 (= B (i^i (U. A) (cv y)))) (aceq5lem.3 (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))) (-> ph (E. y (A.e. z A (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (aceq5lem.1 z cv eleq2i z visset u cv z cv {/} eqeq1 negbid u cv z cv t cv {} t cv X. eqeq1 t h cv birexdv anbi12d elab pm3.26bd sylbi rgen z cv t cv {} t cv X. x cv eleq2 x cv t cv {} t cv u v elxp u v x cv u cv v cv <,> = u cv t cv {} e. v cv t cv e. /\ /\ excom bitr syl6bb w cv g cv {} g cv X. x cv eleq2 x cv g cv {} g cv u y elxp u y x cv u cv y cv <,> = u cv g cv {} e. y cv g cv e. /\ /\ excom bitr syl6bb bi2anan9 v y u x cv u cv v cv <,> = u cv t cv {} e. v cv t cv e. /\ /\ E. u x cv u cv y cv <,> = u cv g cv {} e. y cv g cv e. /\ /\ E. eeanv syl6bbr u cv t cv v cv opeq1 x cv eqeq2d biimpac u t cv elsn sylan2b v cv t cv e. adantrr u 19.23aiv u cv g cv y cv opeq1 x cv eqeq2d biimpac u g cv elsn sylan2b y cv g cv e. adantrr u 19.23aiv anim12i x cv t cv v cv <,> g cv y cv <,> eqtr2t t visset v visset y visset g cv opth pm3.26bd 3syl v y 19.23aivv syl6bi t cv g cv sneq t cv {} g cv {} t cv xpeq1 syl t cv g cv g cv {} xpeq2 eqtrd syl6 z cv t cv {} t cv X. w cv g cv {} g cv X. eqeq12 sylibrd exp t cv h cv e. a1i r19.23aiv g cv h cv e. a1d r19.23adv imp z visset u cv z cv {/} eqeq1 negbid u cv z cv t cv {} t cv X. eqeq1 t h cv birexdv anbi12d aceq5lem.1 elab2 pm3.27bd w visset u cv w cv {/} eqeq1 negbid u cv w cv t cv {} t cv X. eqeq1 t h cv birexdv anbi12d aceq5lem.1 elab2 pm3.27bd t cv g cv sneq t cv {} g cv {} t cv xpeq1 syl t cv g cv g cv {} xpeq2 eqtrd w cv eqeq2d h cv cbvrexv sylib syl2an x cv z cv e. x cv w cv e. df-an syl5ibr con1d x 19.21adv z cv w cv x disj1 syl6ibr rgen2 aceq5lem.3 aceq5lem.1 h visset u t t cv {} t cv X. abrexex u cv {/} = -. t h cv u cv t cv {} t cv X. = E.e. pm3.27 u ss2abi ssexi eqeltr x cv A z z cv {/} = -. raleq x cv A w z cv w cv = -. z cv w cv i^i {/} = -> raleq z raleqd anbi12d x cv A z v v cv z cv y cv i^i e. E! raleq y biexdv imbi12d cla4v sylbi mp2ani))
+thm (aceq5lem5 ((f x) (f z) (f y) (f w) (f v) (f u) (f t) (f h) (f g) (x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B f) (B g) (A x) (A y) (A z) (A w) (A g)) ((aceq5lem.1 (= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (aceq5lem.2 (= B (i^i (U. A) (cv y)))) (aceq5lem.3 (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))) (-> ph (E. f (A.e. w (cv h) (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w)))))) (aceq5lem.1 aceq5lem.2 aceq5lem.3 aceq5lem4 w cv {/} = -. w cv h cv e. pm3.27 z A v v cv z cv y cv i^i e. E! A.e. a1i z cv w cv {} w cv X. y cv ineq1 v cv eleq2d v bieudv A rcla4v com12 aceq5lem.1 w aceq5lem3 v w y g aceq5lem1 3imtr3g jcad aceq5lem.2 w cv g cv <,> eleq2i w cv g cv <,> A U. y cv elin aceq5lem.1 w g aceq5lem2 w cv g cv <,> y cv e. anbi1i w cv h cv e. g cv w cv e. w cv g cv <,> y cv e. anass bitr 3bitr g bieu g w cv h cv e. g cv w cv e. w cv g cv <,> y cv e. /\ euanv bitr2 syl6ib g w cv g cv <,> B e. euex g w cv g cv <,> B e. hbeu1 B w cv ` w cv e. g ax-17 hbim aceq5lem.2 w cv g cv <,> eleq2i w cv g cv <,> A U. y cv elin aceq5lem.1 w g aceq5lem2 w cv g cv <,> y cv e. anbi1i w cv h cv e. g cv w cv e. w cv g cv <,> y cv e. anass bitr 3bitr pm3.27bd pm3.26d w visset g B tz6.12 w cv eleq1d biimparc exp32 mpcom 19.23ai mpcom syl6 exp3a com23 r19.21aiv aceq5lem.2 y visset A U. inex2 eqeltr f cv B w cv fveq1 w cv eleq1d w cv {/} = -. imbi2d w h cv biraldv cla4ev syl y 19.23aiv syl))
+thm (aceq5 ((f x) (f z) (f y) (f w) (f v) (f h) (f u) (f t) (x z) (x y) (w x) (v x) (h x) (u x) (t x) (y z) (w z) (v z) (h z) (u z) (t z) (w y) (v y) (h y) (u y) (t y) (v w) (h w) (u w) (t w) (h v) (u v) (t v) (h u) (h t) (t u)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) (x f w aceq4 w cv t cv f cv fveq2 w cv t cv = id eleq12d w cv t cv z cv eqeq2 negbid w cv t cv z cv ineq2 {/} eqeq1d imbi12d anbi12d x cv rcla4v v cv f cv t cv ` t cv eleq1 negbid z cv rcla4v z cv t cv v disj syl5ib z cv t cv = -. imim2d f cv t cv ` t cv e. z cv t cv = pm4.1 syl6ibr com13 imp31 f cv t cv ` v cv z cv eleq1 biimpar sylan2 f cv t cv ` v cv f cv z cv ` eqeq2 f cv z cv ` v cv eqcom syl6bb z cv t cv f cv fveq2 syl5bi f cv t cv ` t cv e. z cv t cv = -. z cv t cv i^i {/} = -> /\ v cv z cv e. ad2antrl mpd exp32 syl6com com14 r19.23adv f cv x cv v cv t fvelrn biimpac syl5 exp3a com4t imp4b v cv z cv f cv ran elin syl5ib w x cv f cv w cv ` w cv e. z cv w cv = -. z cv w cv i^i {/} = -> r19.26 sylan2br anassrs z cv x cv e. adantlr v cv f cv z cv ` z cv f cv ran i^i eleq1 w cv z cv f cv fveq2 w cv z cv = id eleq12d x cv rcla4v f cv x cv z cv fnfvrn expcom anim12d f cv z cv ` z cv f cv ran elin syl6ibr exp3a com13 imp31 syl5bir com12 w x cv z cv w cv = -. z cv w cv i^i {/} = -> A.e. adantr impbid exp v 19.21adv f cv z cv fvex h cv f cv z cv ` v cv eqeq2 v cv z cv f cv ran i^i e. bibi2d v bialdv cla4ev v v cv z cv f cv ran i^i e. h df-eu sylibr syl6 r19.20dva exp z cv w cv {/} eqeq1 negbid x cv cbvralv w x cv w cv {/} = -. f cv w cv ` w cv e. -> A.e. anbi2i w x cv w cv {/} = -. f cv w cv ` w cv e. -> w cv {/} = -. r19.26 bitr4 w cv {/} = -. f cv w cv ` w cv e. pm3.35 ancoms w x cv r19.20si sylbi syl5 exp3a imp4b f visset f cv V rnexg ax-mp y cv f cv ran z cv ineq2 v cv eleq2d v bieudv z x cv biraldv cla4ev syl6 f 19.23aiv x 19.20i sylbi u u cv {/} = -. t h cv u cv t cv {} t cv X. = E.e. /\ {|} eqid u u cv {/} = -. t h cv u cv t cv {} t cv X. = E.e. /\ {|} U. y cv i^i eqid x z x cv z cv {/} = -. A.e. z x cv w x cv z cv w cv = -. z cv w cv i^i {/} = -> A.e. A.e. /\ y z x cv v v cv z cv y cv i^i e. E! A.e. E. -> A. pm4.2 f aceq5lem5 h 19.21aiv x f w aceq3 x cv h cv w w cv {/} = -. f cv w cv ` w cv e. -> raleq f biexdv cbvalv bitr2 sylib impbi))
+thm (aceq6a ((x z) (f x) (x y) (w x) (v x) (u x) (g x) (f z) (y z) (w z) (v z) (u z) (g z) (f y) (f w) (f v) (f u) (f g) (w y) (v y) (u y) (g y) (v w) (u w) (g w) (u v) (g v) (g u)) () (-> (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x))))))) (u cv z cv w cv eleq2 u cv z cv v cv eleq1 w cv v cv e. anbi1d v y cv birexdv anbi12d w biabdv w u cv v y cv u cv v cv e. w cv v cv e. /\ E.e. df-rab w z cv v y cv z cv v cv e. w cv v cv e. /\ E.e. df-rab 3eqtr4g unieqd u g u cv x cv e. g cv w u cv v y cv u cv v cv e. w cv v cv e. /\ E.e. {e.|} U. = /\ {<,>|} eqid z visset w v y cv z cv v cv e. w cv v cv e. /\ E.e. rabex uniex fvopab4 z cv eleq1d w z cv v y cv z cv v cv e. w cv v cv e. /\ E.e. reucl syl5bir z cv {/} = -. imim2d r19.20i x visset u g w u cv v y cv u cv v cv e. w cv v cv e. /\ E.e. {e.|} U. funopabex2 f cv u g u cv x cv e. g cv w u cv v y cv u cv v cv e. w cv v cv e. /\ E.e. {e.|} U. = /\ {<,>|} z cv fveq1 z cv eleq1d z cv {/} = -. imbi2d z x cv biraldv cla4ev syl y 19.23aiv x 19.20i x f z aceq3 sylibr))
+thm (aceq6b ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (y z) (w y) (v y) (u y) (f y) (g y) (w z) (v z) (u z) (f z) (g z) (v w) (u w) (f w) (g w) (u v) (f v) (g v) (f u) (g u) (f g)) () (-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))))))))) (x f z aceq3 z x cv z cv {/} = -. f cv z cv ` z cv e. -> hbra1 z x cv z cv {/} = -. f cv z cv ` z cv e. -> ra4 f cv z cv fvex w cv f cv z cv ` z cv eleq1 w cv f cv z cv ` v cv eleq1 z cv v cv e. anbi2d v g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} birexdv anbi12d cla4ev z cv eqid u cv z cv {/} eqeq1 negbid u cv z cv z cv eqeq1 anbi12d x cv rcla4ev mpanr2 u cv z cv f cv fveq2 f cv u cv ` f cv z cv ` u cv preq1 syl u cv z cv f cv z cv ` preq2 eqtr2d u cv {/} = -. anim2i u x cv r19.22si syl f cv z cv ` z cv prex g cv f cv z cv ` z cv {,} f cv u cv ` u cv {,} eqeq1 u cv {/} = -. anbi2d u x cv birexdv elab sylibr z visset f cv z cv ` pri2 f cv z cv fvex z cv pri1 pm3.2i jctir v cv f cv z cv ` z cv {,} z cv eleq2 v cv f cv z cv ` z cv {,} f cv z cv ` eleq2 anbi12d g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} rcla4ev syl sylan2 exp syl8 imp3a pm2.43d v visset g cv v cv f cv u cv ` u cv {,} eqeq1 u cv {/} = -. anbi2d u x cv birexdv elab z cv u cv {/} eqeq1 negbid z cv u cv f cv fveq2 z cv eleq1d z cv u cv f cv u cv ` eleq2 bitrd imbi12d x cv rcla4v com12 w visset z visset f cv u cv fvex u visset prel12 v cv f cv u cv ` u cv {,} w cv eleq2 v cv f cv u cv ` u cv {,} z cv eleq2 anbi12d w cv v cv e. z cv v cv e. ancom syl5rbbr sylan9bbr w eirrv w cv z cv w cv eleq2 mtbii con2i sylan2 f cv u cv ` u cv e. adantrr exp pm5.32d w visset z visset f cv u cv fvex u visset preleq syl6bir z cv u cv f cv fveq2 w cv eqeq2d biimparc syl6 exp4c com13 syl8 com4r imp imp4a com3l r19.23aiv sylbi exp3a com13 imp4b v 19.23adv v g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} z cv v cv e. w cv v cv e. /\ df-rex syl5ib exp imp3a w 19.21aiv w w cv z cv e. v g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} z cv v cv e. w cv v cv e. /\ E.e. /\ f cv z cv ` mo2icl syl jctird w z cv v g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} z cv v cv e. w cv v cv e. /\ E.e. df-reu w w cv z cv e. v g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} z cv v cv e. w cv v cv e. /\ E.e. /\ eu5 bitr syl6ibr exp3a r19.21ai x visset g u f cv u cv ` u cv {,} abrexex u cv {/} = -. g cv f cv u cv ` u cv {,} = pm3.27 u x cv r19.22si g ss2abi ssexi y cv g u x cv u cv {/} = -. g cv f cv u cv ` u cv {,} = /\ E.e. {|} v z cv v cv e. w cv v cv e. /\ rexeq w z cv bireudv z cv {/} = -. imbi2d z x cv biraldv cla4ev syl f 19.23aiv x 19.20i sylbi))
+thm (aceq7 ((x z) (f x) (x y) (w x) (v x) (u x) (f z) (y z) (w z) (v z) (u z) (f y) (f w) (f v) (f u) (w y) (v y) (u y) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))))) (x f y z w v aceq6b x y z w v f aceq6a impbi y z x w v u aceq2 x bial bitr4))
+thm (ac2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (y z w x v u t ax-ac y z x w v u t aceq0 mpbir))
+thm (ac3 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))))))) (y z x w v u ac2 y z x w v u aceq2 mpbi))
+thm (ac7 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x))))) (x f y z w v u aceq7 y z x w v u ac2 mpgbir a4i))
+thm (ac7g ((f x) (R f) (R x)) () (-> (e. R A) (E. f (/\ (C_ (cv f) R) (Fn (cv f) (dom R))))) (x cv R f cv sseq2 x cv R dmeq x cv dom R dom f cv fneq2 syl anbi12d f biexdv f x ac7 A vtoclg))
+thm (ac4 ((x z) (f x) (f z)) () (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))))) (x f z aceq3 f x ac7 mpgbi a4i))
+thm (ac4c ((x y) (f x) (A x) (f y) (A y) (A f)) ((ac4c.1 (e. A (V)))) (E. f (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))))) (ac4c.1 y cv A x x cv {/} = -. f cv x cv ` x cv e. -> raleq f biexdv f x y ac4 vtocl))
+thm (ac5 ((f x) (f y) (A f) (x y) (A x) (A y)) ((ac5.1 (e. A (V)))) (E. f (/\ (Fn (cv f) A) (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x)))))) (ac5.1 y cv A f cv fneq2 y cv A x x cv {/} = -. f cv x cv ` x cv e. -> raleq anbi12d f biexdv y f x aceq3 f x y ac4 mpgbir y f x aceq4 mpbi a4i vtocl))
+thm (ac5b ((f x) (A x) (A f)) ((ac5b.1 (e. A (V)))) (-> (A.e. x A (-. (= (cv x) ({/})))) (E. f (/\ (:--> (cv f) A (U. A)) (A.e. x A (e. (` (cv f) (cv x)) (cv x)))))) (ac5b.1 f x ac5 f x A x cv {/} = -. A.e. f cv A Fn x A x cv {/} = -. f cv x cv ` x cv e. -> A.e. /\ 19.42v f cv A x chfnrn exp anc2li f cv A A U. df-f syl6ibr impac x A x cv {/} = -. x cv {/} = -. f cv x cv ` x cv e. -> r19.26 x cv {/} = -. f cv x cv ` x cv e. pm3.35 x A r19.20si sylbir sylan2 an1s f 19.22i sylbir mpan2))
+thm (ac6lem ((f z) (g z) (C z) (f g) (C f) (C g) (H z) (H f) (H g) (f g) (f x) (f y) (f z) (A f) (g x) (g y) (g z) (A g) (x y) (x z) (A x) (y z) (A y) (A z) (B f) (B g) (B x) (B y) (B z) (ph z) (f ph)) ((ac6.1 (e. A (V))) (ac6.2 (e. B (V))) (ac6lem.4 (= C ({e.|} y B ph))) (ac6lem.5 (= H ({<,>|} x z (/\ (e. (cv x) A) (= (cv z) C)))))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A (e. (` (cv f) (cv x)) C))))) (ac6lem.4 z cv eqeq2i biimp {/} eqeq1d negbid y B ph rabn0 syl6bb biimprcd x A r19.20si x A z cv C = z cv {/} = -. r19.23v sylib z x x cv A e. z cv C = /\ E. abid ac6lem.5 rneqi x z x cv A e. z cv C = /\ rnopab eqtr z cv eleq2i x A z cv C = df-rex 3bitr4 syl5ib r19.21aiv ac6.1 ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 A V H fnex mp2 H V rnexg ax-mp z g ac5b ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 H A fnfrn mpbi g cv H ran B H A fco mpan2 z H ran g cv z cv ` z cv e. A.e. adantr g cv Fun x ax-17 z x z x cv A e. z cv C = /\ hbopab1 ac6lem.5 z cv eleq2i ac6lem.5 z cv eleq2i x bial 3imtr4 hbrn g cv z cv ` z cv e. x ax-17 hbral hban ac6lem.4 ac6.2 y ph rabex eqeltr z isseti x cv A e. z cv C = /\ x 19.8a ac6lem.5 rneqi x z x cv A e. z cv C = /\ rnopab eqtr eqabi sylibr expcom z cv C H ran eleq1 sylibd z 19.23aiv ax-mp z cv C g cv fveq2 z cv C = id eleq12d H ran rcla4v syl impcom g cv Fun adantll ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 H A fnfun ax-mp g cv H x cv fvco ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 dmopab2 x cv eleq2i sylan2br mpanl2 ac6lem.4 ac6.2 y ph rabex eqeltr x A C V z fvopab2 mpan2 ac6lem.5 x cv fveq1i syl5eq g cv fveq2d g cv Fun adantl eqtrd C eleq1d z H ran g cv z cv ` z cv e. A.e. adantlr mpbird exp r19.21ai g cv H ran B ffun sylan jca H ran B z unissb z x x cv A e. z cv C = /\ E. abid ac6lem.5 rneqi x z x cv A e. z cv C = /\ rnopab eqtr z cv eleq2i x A z cv C = df-rex 3bitr4 ac6lem.4 y B ph ssrab eqsstr z cv C B sseq1 mpbiri x cv A e. a1i r19.23aiv sylbi mprgbir g cv H ran H ran U. B fss mpan2 sylan g visset ac6.1 ac6lem.4 ac6.2 y ph rabex eqeltr ac6lem.5 fnopab2 A V H fnex mp2 coex f cv g cv H o. A B feq1 f cv g cv e. x ax-17 f x z x cv A e. z cv C = /\ hbopab1 ac6lem.5 f cv eleq2i ac6lem.5 f cv eleq2i x bial 3imtr4 hbco hbeleq f cv g cv H o. x cv fveq1 C eleq1d A birald anbi12d cla4ev syl g 19.23aiv 3syl))
+thm (ac6 ((f x) (f y) (f z) (f w) (A f) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B f) (B x) (B y) (B z) (B w) (ph z) (ph w) (f ph) (ps y)) ((ac6.1 (e. A (V))) (ac6.2 (e. B (V))) (ac6.3 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps)))) (ac6.1 ac6.2 y B ph {e.|} eqid w cv z cv y B ph {e.|} eqeq1 x cv A e. anbi2d x cbvopab2v f ac6lem ac6.3 B elrab pm3.27bd x A r19.20si f cv A B :--> anim2i f 19.22i syl))
+thm (ac6s ((x y) (f x) (x z) (A x) (f y) (y z) (A y) (f z) (A f) (A z) (B x) (B y) (B f) (B z) (f ph) (ph z) (ps y) (ps z)) ((ac6s.1 (e. A (V))) (ac6s.2 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps)))) (ac6s.1 x y B ph z bnd2 ac6s.1 z visset ac6s.2 ac6 z cv B C_ anim2i z 19.22i f cv A z cv B fss expcom x A ps A.e. anim1d f 19.22dv imp z 19.23aiv syl syl))
+thm (ac6s2 ((x y) (f x) (A x) (f y) (A y) (A f) (f ph) (ps y)) ((ac6s.1 (e. A (V))) (ac6s.2 (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps)))) (-> (A.e. x A (E. y ph)) (E. f (A.e. x A ps))) (y ph rexv x A biral ac6s.1 ac6s.2 V ac6s f cv A V :--> x A ps A.e. pm3.27 f 19.22i syl sylbir))
+thm (ac8 ((x z) (x y) (w x) (v x) (f x) (y z) (w z) (v z) (f z) (w y) (v y) (f y) (v w) (f w) (f v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (x f z w y v aceq5 f x ac7 mpgbi a4i))
+thm (kmlem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph x) (ph y) (ph v) (ps x) (ps v)) () (-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) ph))) (E. y (A.e. z (cv x) ps)))) (A. x (-> (A.e. z (cv x) (A.e. w (cv x) ph)) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) ps)))))) (v visset u u cv {/} = -. rabex x cv u v cv u cv {/} = -. {e.|} z z cv {/} = -. raleq x cv u v cv u cv {/} = -. {e.|} w ph raleq z raleqd anbi12d x cv u v cv u cv {/} = -. {e.|} z ps raleq y biexdv imbi12d cla4v v 19.21aiv u v cv u cv {/} = -. ssrab z cv sseli u v cv u cv {/} = -. ssrab w cv sseli ph imim1i r19.20i2 imim12i r19.20i2 u cv z cv {/} eqeq1 negbid v cv elrab pm3.27bd rgen jctil u cv z cv {/} eqeq1 negbid v cv elrab biimpr ps imim1i exp3a r19.20i2 y 19.22i imim12i v 19.20i syl v cv x cv w ph raleq z raleqd v cv x cv z z cv {/} = -. ps -> raleq y biexdv imbi12d cbvalv sylib))
+thm (kmlem2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph y) (ph w) (ph u)) () (<-> (E. y (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) (y cv w cv z cv ineq2 v cv eleq2d v bieudv ph imbi2d z x cv biraldv cbvexv x visset uniex y cv x cv U. u cv eleq2 negbid u biexdv y u nalset y u u cv y cv e. alexn mpbir a4i vtocl z cv x cv elssuni u cv sseld con3d z cv u cv disjsn syl6ibr impcom z cv w cv i^i uneq2d z cv w cv i^i un0 syl6eq z cv w cv u cv {} indi syl5req v cv eleq2d v bieudv ph imbi2d biraldva u visset snid u cv w cv e. -. a1i orri u cv w cv u cv {} elun mpbir w cv u cv {} u. x cv elssuni u cv sseld mpi con3i z x cv ph v v cv z cv w cv u cv {} u. i^i e. E! -> A.e. biantrurd bitrd w visset u cv snex unex y cv w cv u cv {} u. x cv eleq1 negbid y cv w cv u cv {} u. z cv ineq2 v cv eleq2d v bieudv ph imbi2d z x cv biraldv anbi12d cla4ev syl6bi u 19.23aiv ax-mp w 19.23aiv sylbi y cv x cv e. -. z x cv ph v v cv z cv y cv i^i e. E! -> A.e. pm3.27 y 19.22i impbi))
+thm (kmlem3 ((x z) (w x) (v x) (w z) (v z) (v w)) () (<-> (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/}))) (E.e. v (cv z) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w)))))))) (z cv x cv z cv {} \ U. v dfdif2 v z cv dfnul3 v z cv v cv x cv z cv {} \ U. e. -. {e.|} uneq2i v z cv v cv x cv z cv {} \ U. e. -. {e.|} un0 v z cv v cv x cv z cv {} \ U. e. -. v cv z cv e. -. unrab 3eqtr3 v cv x cv z cv {} \ U. e. v cv z cv e. ianor w x cv z cv w cv = -. v cv z cv w cv i^i e. -. -> rexnal w x cv z cv w cv = -. v cv z cv w cv i^i e. -. -> -. df-rex w cv x cv z cv {} eldif w z cv elsn w cv z cv eqcom bitr negbii w cv x cv e. anbi2i bitr v cv w cv e. v cv z cv e. /\ anbi2i v cv w cv e. v cv z cv e. ancom w cv x cv e. z cv w cv = -. /\ anbi1i v cv z cv e. v cv w cv e. /\ w cv x cv e. z cv w cv = -. /\ ancom bitr w cv x cv e. z cv w cv = -. v cv z cv e. v cv w cv e. /\ anass 3bitr v cv w cv e. v cv z cv e. w cv x cv z cv {} \ e. an23 v cv z cv w cv elin z cv w cv = -. anbi2i z cv w cv = -. v cv z cv w cv i^i e. df-an bitr3 w cv x cv e. anbi2i 3bitr3r w biex w v cv w cv e. w cv x cv z cv {} \ e. /\ v cv z cv e. 19.41v 3bitr v cv x cv z cv {} \ w eluni v cv z cv e. anbi1i bitr4 bitr3 bicon1i bitr3 v cv z cv e. a1i birabi 3eqtr {/} eqeq1i negbii v z cv w x cv z cv w cv = -. v cv z cv w cv i^i e. -. -> A.e. rabn0 bitr))
+thm (kmlem4 ((v y) (x y) (v x) (y z) (v z) (w y) (v w)) () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (cv w)) ({/}))) (w visset v cv w cv x cv eleq1 v cv w cv z cv eqeq2 negbid anbi12d v cv w cv y cv eleq2 negbid imbi12d cla4v com12 y cv z cv x cv z cv {} \ U. eldif y cv z cv e. y cv x cv z cv {} \ U. e. -. pm3.27 y cv x cv z cv {} \ v eluni negbii v y cv v cv e. v cv x cv z cv {} \ e. /\ alnex y cv v cv e. v cv x cv z cv {} \ e. ianor v cv x cv z cv {} eldif v z cv elsn v cv z cv eqcom bitr negbii v cv x cv e. anbi2i bitr negbii v cv x cv e. z cv v cv = iman bitr4 y cv v cv e. imbi2i y cv v cv e. v cv x cv z cv {} \ e. -. imor y cv v cv e. z cv v cv = pm4.1 v cv x cv e. imbi2i y cv v cv e. v cv x cv e. z cv v cv = bi2.04 v cv x cv e. z cv v cv = -. y cv v cv e. -. impexp 3bitr4 3bitr3 bitr v bial bitr3 bitr sylib sylbi syl5 r19.21aiv z cv x cv z cv {} \ U. \ w cv y disj sylibr))
+thm (kmlem5 () () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) ({/}))) (w cv x cv w cv {} \ U. difss w cv x cv w cv {} \ U. \ w cv z cv x cv z cv {} \ U. \ sslin ax-mp w x z kmlem4 z cv x cv z cv {} \ U. \ w cv x cv w cv {} \ U. \ i^i sseq2d mpbii z cv x cv z cv {} \ U. \ w cv x cv w cv {} \ U. \ i^i ss0b sylib))
+thm (kmlem6 ((x z) (w x) (v x) (w z) (v z) (v w) (ph v) (A v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> ph (= A ({/})))))) (A.e. z (cv x) (E.e. v (cv z) (A.e. w (cv x) (-> ph (-. (e. (cv v) A))))))) (z x cv z cv {/} = -. w x cv ph A {/} = -> A.e. r19.26 v v cv z cv e. w x cv ph v cv A e. -. -> A.e. 19.29r v z cv w x cv ph v cv A e. -. -> A.e. df-rex sylibr z cv v n0 biimp v cv A n0i con2i ph imim2i w x cv r19.20si v 19.21aiv syl2an z x cv r19.20si sylbir))
+thm (kmlem7 ((x z) (w x) (v x) (w z) (v z) (v w)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))))) (z x w z cv w cv = -. z cv w cv i^i v kmlem6 z cv w cv = -. v cv z cv w cv i^i e. imnan w x cv biral w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ ralnex bitr v z cv birex v z cv w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ E.e. rexnal bitr z x cv biral z x cv v z cv w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ E.e. A.e. ralnex bitr sylib))
+thm (kmlem8 ((x z) (w x) (u x) (t x) (h x) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A z) (A w) (A h)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (A.e. z A (A.e. w A (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) (t x cv h x cv z cv t cv x cv t cv {} \ U. \ = w cv h cv x cv h cv {} \ U. \ = reeanv z cv t cv x cv t cv {} \ U. \ w cv h cv x cv h cv {} \ U. \ ineq12 {/} eqeq1d h x t kmlem5 syl5bir exp3a z cv t cv x cv t cv {} \ U. \ w cv h cv x cv h cv {} \ U. \ eqeq12 t cv h cv x cv t cv {} \ U. difeq1 t cv h cv sneq x cv difeq2d unieqd h cv difeq2d eqtrd syl5bir con3d syl5d com12 t cv x cv e. adantl r19.23aivv sylbir z visset u cv z cv t cv x cv t cv {} \ U. \ eqeq1 t x cv birexdv kmlem8.1 elab2 w visset u cv w cv t cv x cv t cv {} \ U. \ eqeq1 t x cv birexdv kmlem8.1 elab2 t cv h cv x cv t cv {} \ U. difeq1 t cv h cv sneq x cv difeq2d unieqd h cv difeq2d eqtrd w cv eqeq2d x cv cbvrexv bitr syl2anb rgen2))
+thm (kmlem9 ((h ph) (x y) (x z) (w x) (u x) (t x) (h x) (y z) (w y) (u y) (t y) (h y) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A y) (A z) (A w) (A h)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (A. h (-> (A.e. z (cv h) (A.e. w (cv h) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) (E. y (A.e. z (cv h) ph)))) (E. y (A.e. z A ph))) (kmlem8.1 z w kmlem8 kmlem8.1 x visset u t t cv x cv t cv {} \ U. \ abrexex eqeltr h cv A w z cv w cv = -. z cv w cv i^i {/} = -> raleq z raleqd h cv A z ph raleq y biexdv imbi12d cla4v mpi))
+thm (kmlem10 ((x z) (u x) (t x) (u z) (t z) (t u) (A z)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (e. (cv z) (cv x)) (= (i^i (cv z) (U. A)) (\ (cv z) (U. (\ (cv x) ({} (cv z))))))) (z cv x cv snssi z cv {} x cv ssequn1 sylib z cv {} x cv undif2 syl5req x cv z cv {} x cv z cv {} \ u. t z cv t cv x cv t cv {} \ U. \ i^i iuneq1 syl z x t kmlem4 z cv t cv x cv t cv {} \ U. \ incom syl5eq exp t cv x cv z cv {} eldifn t z cv elsn negbii sylib syl5 r19.21aiv t x cv z cv {} \ z cv t cv x cv t cv {} \ U. \ i^i {/} iuneq2 syl t x cv z cv {} \ iun0 syl6eq z cv z cv x cv z cv {} \ U. \ i^i uneq2d t z cv {} x cv z cv {} \ z cv t cv x cv t cv {} \ U. \ i^i iunxun z visset t cv z cv x cv t cv {} \ U. difeq1 t cv z cv sneq x cv difeq2d unieqd z cv difeq2d eqtrd z cv ineq2d iunxsn t x cv z cv {} \ z cv t cv x cv t cv {} \ U. \ i^i U_ uneq1i eqtr syl5eq eqtrd z cv z cv x cv z cv {} \ U. \ i^i un0 z cv x cv z cv {} \ U. difss z cv x cv z cv {} \ U. \ z cv sseqin2 mpbi eqtr syl6eq kmlem8.1 unieqi t visset t cv V x cv t cv {} \ U. difexg ax-mp t x cv u dfiun2 eqtr4 z cv ineq2i t x cv z cv t cv x cv t cv {} \ U. \ iunin2 eqtr4 syl5eq))
+thm (kmlem11 ((x y) (x z) (v x) (u x) (t x) (y z) (v y) (u y) (t y) (v z) (u z) (t z) (u v) (t v) (t u) (A y) (A z) (A v)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (-> (A.e. z (cv x) (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/})))) (-> (A.e. z A (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (i^i (cv y) (U. A))))))))) (t cv z cv x cv t cv {} \ U. difeq1 t cv z cv sneq x cv difeq2d unieqd z cv difeq2d eqtrd {/} eqeq1d negbid x cv cbvralv t cv z cv x cv t cv {} \ U. difeq1 t cv z cv sneq x cv difeq2d unieqd z cv difeq2d eqtrd y cv ineq1d v cv eleq2d v bieudv x cv cbvralv imbi12i kmlem8.1 z kmlem10 y cv ineq1d z cv y cv A U. in12 y cv z cv A U. i^i incom eqtr syl5req v cv eleq2d v bieudv v v cv z cv y cv A U. i^i i^i e. E! z cv {/} = -. ax-1 syl6bi r19.20i z x cv z cv x cv z cv {} \ U. \ {/} = -. A.e. imim2i sylbi com12 kmlem8.1 A u t x cv u cv t cv x cv t cv {} \ U. \ = E.e. {|} z z cv {/} = -. v v cv z cv y cv i^i e. E! -> raleq ax-mp z u t x cv u cv t cv x cv t cv {} \ U. \ = E.e. {|} z cv {/} = -. v v cv z cv y cv i^i e. E! -> df-ral z visset u cv z cv t cv x cv t cv {} \ U. \ eqeq1 t x cv birexdv elab z cv {/} = -. v v cv z cv y cv i^i e. E! -> imbi1i t x cv z cv t cv x cv t cv {} \ U. \ = z cv {/} = -. v v cv z cv y cv i^i e. E! -> r19.23v bitr4 z bial t x cv z z cv t cv x cv t cv {} \ U. \ = z cv {/} = -. v v cv z cv y cv i^i e. E! -> -> ralcom4 t visset t cv V x cv t cv {} \ U. difexg ax-mp z cv t cv x cv t cv {} \ U. \ {/} eqeq1 negbid z cv t cv x cv t cv {} \ U. \ y cv ineq1 v cv eleq2d v bieudv imbi12d ceqsalv t x cv biral bitr3 bitr 3bitr t x cv t cv x cv t cv {} \ U. \ {/} = -. v v cv t cv x cv t cv {} \ U. \ y cv i^i e. E! r19.20 sylbi syl5))
+thm (kmlem12 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (w z) (v z) (u z) (t z) (h z) (g z) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (A y) (A z) (A w) (A v) (A h) (A g)) ((kmlem8.1 (= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t))))))))))) (<-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (A. x (-> (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) (x z w z cv w cv = -. z cv w cv i^i {/} = -> y v v cv z cv y cv i^i e. E! kmlem1 x z x cv w x cv z cv w cv = -. z cv w cv i^i {/} = -> A.e. A.e. y z x cv z cv {/} = -. v v cv z cv y cv i^i e. E! -> A.e. E. -> A. pm4.2 x cv h cv w z cv w cv = -. z cv w cv i^i {/} = -> raleq z raleqd x cv h cv z z cv {/} = -. v v cv z cv y cv i^i e. E! -> raleq y biexdv imbi12d cbvalv kmlem8.1 h z w y z cv {/} = -. v v cv z cv y cv i^i e. E! -> kmlem9 y cv g cv z cv ineq2 v cv eleq2d v bieudv z cv {/} = -. imbi2d z A biraldv cbvexv kmlem8.1 z v g kmlem11 g visset A U. inex1 y cv g cv A U. i^i z cv ineq2 v cv eleq2d v bieudv z cv {/} = -. imbi2d z x cv biraldv cla4ev syl6 g 19.23adv com12 z x v w kmlem3 z cv w cv = -. v cv z cv w cv i^i e. imnan w x cv biral w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ ralnex bitr v z cv birex v z cv w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ E.e. rexnal 3bitr z x cv biral z x cv v z cv w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ E.e. A.e. ralnex bitr syl5ibr sylbi syl x 19.21aiv sylbi sylbi syl z x w v kmlem7 y z x cv z cv {/} = -. v v cv z cv y cv i^i e. E! -> A.e. E. imim1i z cv {/} = -. v v cv z cv y cv i^i e. E! biimt z x cv r19.20si z x cv v v cv z cv y cv i^i e. E! z cv {/} = -. v v cv z cv y cv i^i e. E! -> r19.15 syl y biexdv z x cv w x cv z cv w cv = -. z cv w cv i^i {/} = -> A.e. A.e. adantr pm5.74i sylibr x 19.20i impbi))
+thm (kmlem13 ((x y) (x z) (v x) (y z) (v y) (v z)) () (<-> (-> (-. (E.e. z (cv x) (A.e. v (cv z) ph))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (\/ (E.e. z (cv x) (A.e. v (cv z) ph)) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))) (z x cv v z cv ph A.e. ralnex v z cv ph -. df-rex v z cv ph rexnal bitr3 v cv z cv e. ph -. pm3.26 v 19.22i z cv v n0 sylibr sylbir z x cv r19.20si sylbir z cv {/} = -. v v cv z cv y cv i^i e. E! biimt z x cv r19.20si z x cv v v cv z cv y cv i^i e. E! z cv {/} = -. v v cv z cv y cv i^i e. E! -> r19.15 syl y cv x cv e. -. anbi2d y biexdv y z x z cv {/} = -. v kmlem2 syl6rbbr syl pm5.74i z x cv v z cv ph A.e. E.e. y y cv x cv e. -. z x cv v v cv z cv y cv i^i e. E! A.e. /\ E. pm4.64 bitr))
+thm (kmlem14 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (A. z (E. v (A. u (/\ (e. (cv y) (cv x)) ph)))))) (z cv y cv w cv eqeq1 negbid z cv y cv w cv ineq1 v cv eleq2d anbi12d w x cv birexdv v raleqd x cv cbvrexv y x cv v y cv w x cv y cv w cv = -. v cv y cv w cv i^i e. /\ E.e. A.e. df-rex v cv z cv y cv w cv i^i eleq1 y cv w cv = -. anbi2d w x cv birexdv y cv cbvralv z y cv w x cv y cv w cv = -. z cv y cv w cv i^i e. /\ E.e. df-ral bitr y cv x cv e. anbi2i z y cv x cv e. z cv y cv e. w x cv y cv w cv = -. z cv y cv w cv i^i e. /\ E.e. -> 19.28v w cv v cv y cv eqeq2 negbid w cv v cv y cv ineq2 z cv eleq2d anbi12d x cv cbvrexv v x cv y cv v cv = -. z cv y cv v cv i^i e. /\ df-rex bitr z cv y cv e. imbi2i v z cv y cv e. v cv x cv e. y cv v cv = -. z cv y cv v cv i^i e. /\ /\ 19.37v bitr4 y cv x cv e. anbi2i v y cv x cv e. z cv y cv e. v cv x cv e. y cv v cv = -. z cv y cv v cv i^i e. /\ /\ -> 19.42v kmlem14.1 z cv y cv v cv elin baibr v cv x cv e. y cv v cv = -. /\ anbi2d v cv x cv e. y cv v cv = -. z cv y cv v cv i^i e. anass syl6bb pm5.74i bitr y cv x cv e. anbi2i y cv x cv e. ph /\ u ax-17 19.3r bitr3 v biex bitr3 bitr z bial bitr3 bitr y biex 3bitr))
+thm (kmlem15 ((x y) (x z) (v x) (u x) (y z) (v y) (u y) (v z) (u z) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (/\ (-. (e. (cv y) (cv x))) ch) (A. z (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps))))) (kmlem14.3 v cv z cv y cv i^i e. u ax-17 v eu1 v cv z cv y cv elin u cv z cv y cv i^i e. v ax-17 v cv u cv z cv y cv i^i eleq1 sbie u cv z cv y cv elin bitr v cv u cv eqcom imbi12i u bial anbi12i u v cv z cv e. v cv y cv e. /\ u cv z cv e. u cv y cv e. /\ u cv v cv = -> 19.28v bitr4 v biex bitr z x cv biral z x cv v u v cv z cv e. v cv y cv e. /\ u cv z cv e. u cv y cv e. /\ u cv v cv = -> /\ A. E. df-ral kmlem14.2 u bial u z cv x cv e. v cv z cv e. v cv y cv e. /\ u cv z cv e. u cv y cv e. /\ u cv v cv = -> /\ 19.21v bitr v biex v z cv x cv e. u v cv z cv e. v cv y cv e. /\ u cv z cv e. u cv y cv e. /\ u cv v cv = -> /\ A. 19.37v bitr z bial bitr4 3bitr y cv x cv e. -. anbi2i z y cv x cv e. -. v u ps A. E. 19.28v u y cv x cv e. -. ps 19.28v v biex v y cv x cv e. -. u ps A. 19.42v bitr2 z bial bitr3 bitr))
+thm (kmlem16 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((kmlem14.1 (<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v)))))) (kmlem14.2 (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))) (kmlem14.3 (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (<-> (\/ (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (/\ (-. (e. (cv y) (cv x))) ch))) (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) ph) (/\ (-. (e. (cv y) (cv x))) ps))))))) (kmlem14.1 kmlem14.2 kmlem14.3 w kmlem14 kmlem14.1 kmlem14.2 kmlem14.3 kmlem15 y biex orbi12i y z v u y cv x cv e. ph /\ A. E. A. z v u y cv x cv e. -. ps /\ A. E. A. 19.43 y cv x cv e. pm3.24 y cv x cv e. ph pm3.26 u a4s z v 19.23aivv y cv x cv e. -. ps pm3.26 u a4s z v 19.23aivv anim12i mto z v u y cv x cv e. ph /\ A. E. v u y cv x cv e. -. ps /\ A. E. 19.33b ax-mp y cv x cv e. pm3.24 y cv x cv e. ph pm3.26 u 19.23aiv y cv x cv e. -. ps pm3.26 u 19.23aiv anim12i mto u y cv x cv e. ph /\ y cv x cv e. -. ps /\ 19.33b ax-mp v biex v u y cv x cv e. ph /\ A. u y cv x cv e. -. ps /\ A. 19.43 bitr2 z bial bitr3 y biex bitr3 bitr))
+thm (aceqkm ((x y) (x z) (w x) (v x) (u x) (f x) (t x) (h x) (y z) (w y) (v y) (u y) (f y) (t y) (h y) (w z) (v z) (u z) (f z) (t z) (h z) (v w) (u w) (f w) (t w) (h w) (u v) (f v) (t v) (h v) (f u) (t u) (h u) (f t) (f h) (h t)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))))))))) (x f z w y v aceq5 t h x cv t cv h cv x cv h cv {} \ U. \ = E.e. {|} eqid z w y v kmlem12 z x v w x cv z cv w cv = -. v cv z cv w cv i^i e. /\ E.e. y kmlem13 x bial bitr z cv y cv e. v cv x cv e. y cv v cv = -. /\ z cv v cv e. /\ -> pm4.2 z cv x cv e. v cv z cv e. v cv y cv e. /\ u cv z cv e. u cv y cv e. /\ u cv v cv = -> /\ -> pm4.2 z x cv v v cv z cv y cv i^i e. E! A.e. pm4.2 w kmlem16 x bial 3bitr))
+thm (ackm ((x y) (x z) (v x) (u x) (f x) (y z) (v y) (u y) (f y) (v z) (u z) (f z) (u v) (f v) (f u)) () (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))))))))) (x f y z v u aceqkm f x ac7 mpgbi))
+thm (numthlem ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (B x) (B y) (B f) (F x) (F y) (F f) (G x) (G y) (G f)) ((numthlem.1 (e. A (V))) (numthlem.2 (= B ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (numthlem.3 (= F (U. B))) (numthlem.4 (= G ({<,>|} f y (= (cv y) (` (cv g) (\ A (ran (cv f))))))))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))) (numthlem.1 pwex g y ac4c numthlem.2 numthlem.3 x tfr2 x visset numthlem.2 numthlem.3 tfrlem7 x cv V F resfunexg mp2 g cv A F x cv |` ran \ fvex f cv F x cv |` rneq f cv ran F x cv |` ran A difeq2 A f cv ran \ A F x cv |` ran \ g cv fveq2 3syl y fvopab numthlem.4 F x cv |` fveq1i F x cv df-ima A difeq2i g cv fveq2i 3eqtr4 syl6eq A F x cv " \ eleq1d A F x cv " difss numthlem.1 A F x cv " difss ssexi A elpw mpbir y cv A F x cv " \ {/} eqeq1 negbid y cv A F x cv " \ g cv fveq2 y cv A F x cv " \ = id eleq12d imbi12d A P~ rcla4v ax-mp imp syl5bir exp3a com12 r19.21aiv numthlem.2 numthlem.3 tfr1 numthlem.1 x tz7.49c x visset numthlem.2 numthlem.3 tfrlem7 x cv V F resfunexg mp2 f cv F x cv |` x cv A f1oeq1 cla4ev x On r19.22si 3syl g 19.23aiv ax-mp))
+thm (numth ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h)) ((numth.1 (e. A (V)))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))) (numth.1 g z w v u u cv h cv A v cv ran \ ` = {<,>|} f x y rdglem1 g z On g cv z cv Fn w z cv g cv w cv ` v u u cv h cv A v cv ran \ ` = {<,>|} g cv w cv |` ` = A.e. /\ E.e. {|} U. eqid u cv y cv = id v cv f cv rneq v cv ran f cv ran A difeq2 A v cv ran \ A f cv ran \ h cv fveq2 3syl eqeqan12rd cbvopabv numthlem))
+thm (numth2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (E.e. x (On) (br (cv x) (~~) A)) (y cv A x cv ~~ breq2 x On birexdv y visset x z numth y visset x cv z bren x On birex mpbir V vtoclg 0elon {/} On enrefg ax-mp A {/} ~~ brprc mpbiri 0elon x cv {/} ~~ A breq1 On rcla4ev mpan syl pm2.61i))
+thm (numthcor ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (br A (~<) (cv x)))) (y cv A ~< x cv breq1 x On birexdv x y cv P~ numth2 y visset pwex x cv ensym y visset canth2 x visset x cv V y cv y cv P~ sdomentr ax-mp mpan syl x On r19.22si ax-mp B vtoclg))
+thm (weth ((x y) (x z) (w x) (f x) (A x) (y z) (w y) (f y) (A y) (w z) (f z) (A z) (f w) (A w) (A f)) ((weth.1 (e. A (V)))) (E. x (We (cv x) A)) (weth.1 y f numth f cv y cv A f1ocnv z w f cv `' z cv ` E f cv `' w cv ` br {<,>|} eqid A y cv f1owe z w f cv `' z cv ` E f cv `' w cv ` br {<,>|} A weinxp weth.1 weth.1 xpex z w f cv `' z cv ` E f cv `' w cv ` br {<,>|} inex2 x cv z w f cv `' z cv ` E f cv `' w cv ` br {<,>|} A A X. i^i A weeq1 cla4ev sylbi syl6 y cv eloni y cv ordwe syl syl5 syl f 19.23aiv com12 r19.23aiv ax-mp))
+thm (zornlem1 ((w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D u) (D v) (D f) (D t) (R x) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (e. (` F (cv x)) D)) (zornlem.2 zornlem.3 x tfr2 zornlem.6 F x cv |` fveq1i x visset zornlem.2 zornlem.3 tfrlem7 x cv V F resfunexg mp2 zornlem.1 zornlem.5 z A g F x cv " g cv R z cv br A.e. ssrab eqsstr ssexi v u D u cv w cv v cv br -. A.e. rabex uniex f cv F x cv |` rneq F x cv df-ima syl6eqr g cv eleq2d g cv R z cv br imbi1d biraldv2 z A birabsdv zornlem.4 zornlem.5 3eqtr4g v cv eleq2d f cv F x cv |` rneq F x cv df-ima syl6eqr g cv eleq2d g cv R z cv br imbi1d biraldv2 z A birabsdv zornlem.4 zornlem.5 3eqtr4g u cv eleq2d u cv w cv v cv br -. imbi1d biraldv2 anbi12d v biabdv v C u C u cv w cv v cv br -. A.e. df-rab v D u D u cv w cv v cv br -. A.e. df-rab 3eqtr4g unieqd t fvopab eqtr syl6eq D eleq1d zornlem.5 z A g F x cv " g cv R z cv br A.e. ssrab eqsstr zornlem.1 zornlem.5 z A g F x cv " g cv R z cv br A.e. ssrab eqsstr ssexi w cv A v u wereu mpanr1 v D u D u cv w cv v cv br -. A.e. reucl syl syl5bir imp))
+thm (zornlem2 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (-> (e. (cv y) (cv x)) (br (` F (cv y)) R (` F (cv x))))) (x cv onsst zornlem.2 zornlem.3 tfr1 F On fndm ax-mp syl6ssr zornlem.2 zornlem.3 tfrlem7 F x cv y cv funfvima2 mpan syl w cv A We D {/} = -. /\ adantr zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 F x cv ` eleq2i z cv F x cv ` g cv R breq2 g F x cv " biraldv A elrab bitr pm3.27bd g cv F y cv ` R F x cv ` breq1 F x cv " rcla4v com12 3syl syld))
+thm (zornlem3 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (Po R A) (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/})))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))))) (zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 y zornlem2 R A Po adantl F x cv ` F y cv ` R F x cv ` breq1 biimprcd R A F x cv ` poirr nsyli com12 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 z A g F x cv " g cv R z cv br A.e. ssrab eqsstr F x cv ` sseli syl sylan2 syld))
+thm (zornlem4 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))))) (-> (/\ (Po R A) (We (cv w) A)) (E.e. x (On) (= D ({/})))) (F ran V e. pm3.24 w cv A We x ax-17 x x cv On e. D {/} = -. -> hba1 hban y cv A e. x ax-17 F x cv ` y cv A eleq1 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem1 zornlem.5 z A g F x cv " g cv R z cv br A.e. ssrab eqsstr F x cv ` sseli syl syl5bi com12 exp32 com12 a2d x a4sd imp r19.23ad zornlem.2 zornlem.3 tfr1 F On y cv x fvelrn ax-mp syl5ib ssrdv zornlem.1 F ran ssex syl exp R A Po adantl zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 y zornlem3 exp45 com23 imp a2d imp4a y 19.21adv x 19.20dv x On y x cv F x cv ` F y cv ` = -. r2al syl6ibr On ssid zornlem.2 zornlem.3 tfr1 On x y tz7.48lem mpan zornlem.2 zornlem.3 tfrlem6 zornlem.2 zornlem.3 tfr1 F On fndm ax-mp On ssid eqsstr F On relssres mp2an F On |` F cnveq ax-mp F On |` `' F `' funeq ax-mp sylib syl6 onprc F `' V funrnex com12 F df-rn V eleq1i zornlem.2 zornlem.3 tfr1 F On fndm ax-mp F dfdm4 eqtr3 V eleq1i 3imtr4g mtoi syl6 jcad x x cv On e. D {/} = alinexa syl5ibr mt3i x On D {/} = df-rex sylibr))
+thm (zornlem5 ((u x) (v x) (f x) (t x) (s x) (H x) (u v) (f u) (t u) (s u) (H u) (f v) (t v) (s v) (H v) (f t) (f s) (H f) (s t) (H t) (H s) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (s x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (s y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (s w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h s) (A h) (t z) (f t) (g t) (t u) (t v) (s t) (A t) (f z) (g z) (u z) (v z) (s z) (A z) (f g) (f u) (f v) (f s) (A f) (g u) (g v) (g s) (A g) (u v) (s u) (A u) (s v) (A v) (A s) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F s) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (C_ (" F (cv x)) A)) (w cv A We x cv On e. /\ y ax-17 y x cv H {/} = -. hbra1 hban s cv A e. y ax-17 F y cv ` s cv A eleq1 zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.7 zornlem.6 zornlem1 zornlem.7 z A g F y cv " g cv R z cv br A.e. ssrab eqsstr F y cv ` sseli syl syl5bi x cv y cv onelon sylani com12 exp43 com3r imp a2d y a4sd y x cv H {/} = -. df-ral syl5ib imp r19.23ad zornlem.2 zornlem.3 tfrlem7 F s cv x cv y fvelima mpan syl5 ssrdv))
+thm (zornlem6 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (Po R A) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (Or R (" F (cv x))))) (zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 F x cv " A R poss syl com12 x cv a cv onelon x cv b cv onelon anim12i anandis exp zornlem.1 zornlem.2 zornlem.3 zornlem.4 z A g F b cv " g cv R z cv br A.e. {e.|} eqid zornlem.6 a zornlem2 a cv On e. adantll F a cv ` r cv F b cv ` s cv R breq12 biimpcd syl6 com23 z A g F a cv " g cv R z cv br A.e. {e.|} {/} = -. adantrrl imp F a cv ` r cv F b cv ` s cv eqeq12 a cv b cv F fveq2 syl5bi a cv On e. b cv On e. /\ w cv A We z A g F a cv " g cv R z cv br A.e. {e.|} {/} = -. z A g F b cv " g cv R z cv br A.e. {e.|} {/} = -. /\ /\ /\ adantl zornlem.1 zornlem.2 zornlem.3 zornlem.4 z A g F a cv " g cv R z cv br A.e. {e.|} eqid zornlem.6 b zornlem2 b cv On e. adantlr F b cv ` s cv F a cv ` r cv R breq12 ancoms biimpcd syl6 com23 z A g F b cv " g cv R z cv br A.e. {e.|} {/} = -. adantrrr imp 3orim123d a cv b cv ordtri3or a cv eloni b cv eloni syl2an syl5 exp31 com4r pm2.43i syl6 exp4a com3r imp a2d zornlem.7 {/} eqeq1i negbii y x cv biral y cv a cv F imaeq2 F y cv " F a cv " g g cv R z cv br raleq syl z A birabsdv {/} eqeq1d negbid x cv rcla4v com12 y cv b cv F imaeq2 F y cv " F b cv " g g cv R z cv br raleq syl z A birabsdv {/} eqeq1d negbid x cv rcla4v com12 anim12d sylbi syl5 imp4b a b 19.23advv zornlem.2 zornlem.3 tfr1 F On fnfun ax-mp F r cv x cv a fvelima a x cv F a cv ` r cv = df-rex sylib exp F s cv x cv b fvelima b x cv F b cv ` s cv = df-rex sylib exp anim12d ax-mp a cv x cv e. b cv x cv e. F a cv ` r cv = F b cv ` s cv = an4 a b bi2ex a b a cv x cv e. F a cv ` r cv = /\ b cv x cv e. F b cv ` s cv = /\ eeanv bitr sylibr syl5 r s 19.21aivv r F x cv " s F x cv " r cv R s cv br r cv s cv = s cv R r cv br \/\/ r2al sylibr R A Po a1i jcad R F x cv " r s df-so syl6ibr))
+thm (zornlem7 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((zornlem.1 (e. A (V))) (zornlem.2 (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t)))))))))) (zornlem.3 (= F (U. B))) (zornlem.4 (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z)))))) (zornlem.5 (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z)))))) (zornlem.6 (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (zornlem.7 (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z))))))) (-> (/\ (Po R A) (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a)))))))) (E.e. a A (A.e. b A (-. (br (cv a) R (cv b)))))) (zornlem.1 w weth zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem4 x cv y cv F imaeq2 F x cv " F y cv " g g cv R z cv br raleq syl z A birabsdv zornlem.5 zornlem.7 3eqtr4g {/} eqeq1d onminex zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 R A Po a1i zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem6 jcad zornlem.2 zornlem.3 tfrlem7 x visset F funimaex ax-mp s cv F x cv " A sseq1 s cv F x cv " R soeq2 anbi12d s cv F x cv " r r cv R a cv br r cv a cv = \/ raleq a A birexdv imbi12d cla4v sylan9 D {/} = adantld imp b cv noel zornlem.1 zornlem.2 zornlem.3 zornlem.4 zornlem.5 zornlem.6 zornlem.7 zornlem5 r cv sseld R A r cv a cv b cv potr r cv A e. a cv A e. b cv A e. 3anass sylan2br exp3a com23 imp r cv a cv R b cv breq1 biimprcd R A Po r cv A e. a cv A e. b cv A e. /\ /\ /\ adantl jaod exp42 sylan9r com24 com23 imp31 a2d r 19.20dv r F x cv " r cv R a cv br r cv a cv = \/ df-ral r F x cv " r cv R b cv br df-ral 3imtr4g zornlem.5 {/} eqeq1i z A g F x cv " g cv R z cv br A.e. {e.|} {/} b cv eleq2 sylbi z cv b cv g cv R breq2 g F x cv " biraldv A elrab syl5bbr biimpd exp3a imp r cv g cv R b cv breq1 F x cv " cbvralv syl5ib sylan9r exp32 com34 imp31 mtoi exp42 exp4a com34 exp com4r pm2.43a imp3a com4l imp3a b 19.21adv b A a cv R b cv br -. df-ral syl6ibr exp3a imdistand a 19.22dv a A r F x cv " r cv R a cv br r cv a cv = \/ A.e. df-rex a A b A a cv R b cv br -. A.e. df-rex 3imtr4g exp32 com12 s s cv A C_ R s cv Or /\ a A r s cv r cv R a cv br r cv a cv = \/ A.e. E.e. -> A. adantr imp32 mpd exp45 com23 exp3a imp imp4a com3l r19.23aiv 3syl s s cv A C_ R s cv Or /\ a A r s cv r cv R a cv br r cv a cv = \/ A.e. E.e. -> A. adantlr pm2.43i exp w 19.23adv mpi))
+thm (zorn2lem ((x y) (x z) (y z) (w x) (w y)) () (<-> (br (cv z) ({<,>|} x y (C: (cv x) (cv y))) (cv w)) (C: (cv z) (cv w))) (z visset w visset x cv z cv y cv psseq1 y cv w cv z cv psseq2 x y x cv y cv C: {<,>|} eqid brab))
+var (set q)
+thm (zorn ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (q x) (a x) (b x) (c x) (d x) (j x) (k x) (m x) (n x) (R x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (q y) (a y) (b y) (c y) (d y) (j y) (k y) (m y) (n y) (R y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (q z) (a z) (b z) (c z) (d z) (j z) (k z) (m z) (n z) (R z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (q w) (a w) (b w) (c w) (d w) (j w) (k w) (m w) (n w) (R w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (q v) (a v) (b v) (c v) (d v) (j v) (k v) (m v) (n v) (R v) (f u) (g u) (h u) (t u) (s u) (r u) (q u) (a u) (b u) (c u) (d u) (j u) (k u) (m u) (n u) (R u) (f g) (f h) (f t) (f s) (f r) (f q) (a f) (b f) (c f) (d f) (f j) (f k) (f m) (f n) (R f) (g h) (g t) (g s) (g r) (g q) (a g) (b g) (c g) (d g) (g j) (g k) (g m) (g n) (R g) (h t) (h s) (h r) (h q) (a h) (b h) (c h) (d h) (h j) (h k) (h m) (h n) (R h) (s t) (r t) (q t) (a t) (b t) (c t) (d t) (j t) (k t) (m t) (n t) (R t) (r s) (q s) (a s) (b s) (c s) (d s) (j s) (k s) (m s) (n s) (R s) (q r) (a r) (b r) (c r) (d r) (j r) (k r) (m r) (n r) (R r) (a q) (b q) (c q) (d q) (j q) (k q) (m q) (n q) (R q) (a b) (a c) (a d) (a j) (a k) (a m) (a n) (R a) (b c) (b d) (b j) (b k) (b m) (b n) (R b) (c d) (c j) (c k) (c m) (c n) (R c) (d j) (d k) (d m) (d n) (R d) (j k) (j m) (j n) (R j) (k m) (k n) (R k) (m n) (R m) (R n) (A x) (A y) (A z) (A w) (A v) (A u) (A f) (A g) (A h) (A t) (A s) (A r) (A q) (A a) (A b) (A c) (A d) (A j) (A k) (A m) (A n)) ((zorn.1 (e. A (V)))) (-> (/\ (Po R A) (A. w (-> (/\ (C_ (cv w) A) (Or R (cv w))) (E.e. x A (A.e. z (cv w) (\/ (br (cv z) R (cv x)) (= (cv z) (cv x)))))))) (E.e. x A (A.e. y A (-. (br (cv x) R (cv y)))))) (zorn.1 a b c h k k cv m v A q h cv ran q cv R v cv br A.e. {e.|} k v A q h cv ran q cv R v cv br A.e. {e.|} k cv q cv m cv br -. A.e. {e.|} U. = {<,>|} d f g rdglem1 a b On a cv b cv Fn c b cv a cv c cv ` h k k cv m v A q h cv ran q cv R v cv br A.e. {e.|} k v A q h cv ran q cv R v cv br A.e. {e.|} k cv q cv m cv br -. A.e. {e.|} U. = {<,>|} a cv c cv |` ` = A.e. /\ E.e. {|} U. eqid v cv r cv s cv R breq2 s d cv ran biraldv q cv s cv R v cv breq1 d cv ran cbvralv syl5bb A cbvrabv r A s a b On a cv b cv Fn c b cv a cv c cv ` h k k cv m v A q h cv ran q cv R v cv br A.e. {e.|} k v A q h cv ran q cv R v cv br A.e. {e.|} k cv q cv m cv br -. A.e. {e.|} U. = {<,>|} a cv c cv |` ` = A.e. /\ E.e. {|} U. t cv " s cv R r cv br A.e. {e.|} eqid k cv g cv = id h cv d cv rneq h cv ran d cv ran q q cv R v cv br raleq syl v A birabsdv n cv eleq2d h cv d cv rneq h cv ran d cv ran q q cv R v cv br raleq syl v A birabsdv v A q h cv ran q cv R v cv br A.e. {e.|} v A q d cv ran q cv R v cv br A.e. {e.|} j j cv q cv n cv br -. raleq k cv j cv q cv n cv breq1 negbid v A q h cv ran q cv R v cv br A.e. {e.|} cbvralv syl5bb syl anbi12d n biabdv m cv n cv v A q h cv ran q cv R v cv br A.e. {e.|} eleq1 m cv n cv k cv q cv breq2 negbid k v A q h cv ran q cv R v cv br A.e. {e.|} biraldv anbi12d cbvabv syl5eq m v A q h cv ran q cv R v cv br A.e. {e.|} k v A q h cv ran q cv R v cv br A.e. {e.|} k cv q cv m cv br -. A.e. df-rab n v A q d cv ran q cv R v cv br A.e. {e.|} j v A q d cv ran q cv R v cv br A.e. {e.|} j cv q cv n cv br -. A.e. df-rab 3eqtr4g unieqd eqeqan12rd cbvopabv r A s a b On a cv b cv Fn c b cv a cv c cv ` h k k cv m v A q h cv ran q cv R v cv br A.e. {e.|} k v A q h cv ran q cv R v cv br A.e. {e.|} k cv q cv m cv br -. A.e. {e.|} U. = {<,>|} a cv c cv |` ` = A.e. /\ E.e. {|} U. u cv " s cv R r cv br A.e. {e.|} eqid w x z y zornlem7))
+thm (zorn2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A x) (A y) (A z) (A w) (A v) (A u)) ((zorn.1 (e. A (V)))) (-> (A. z (-> (/\ (C_ (cv z) A) (A.e. x (cv z) (A.e. y (cv z) (\/ (C_ (cv x) (cv y)) (C_ (cv y) (cv x)))))) (e. (U. (cv z)) A))) (E.e. x A (A.e. y A (-. (C: (cv x) (cv y)))))) (w v w cv v cv C: {<,>|} z cv x y df-so pm3.27bd x w v y zorn2lem x cv y cv = pm4.2 y w v x zorn2lem 3orbi123i x cv y cv sspsstri bitr4 x z cv y z cv bi2ral sylib z cv A C_ anim2i z cv U. A x risset x cv z cv U. eqimss2 z cv x cv u unissb sylib u w v x zorn2lem u cv x cv = orbi1i u cv x cv sspss bitr4 u z cv biral sylibr x A r19.22si sylbi imim12i z 19.20i u cv pssirr u w v u zorn2lem mtbir u cv y cv x cv psstr u w v x zorn2lem sylibr u w v y zorn2lem y w v x zorn2lem syl2anb pm3.2i u cv A e. y cv A e. x cv A e. /\/\ a1i rgen3 w v w cv v cv C: {<,>|} A u y x df-po mpbir zorn.1 w v w cv v cv C: {<,>|} z x u y zorn mpan syl x w v y zorn2lem negbii y A biral x A birex sylib))
+thm (fodom ((F f) (A f) (B f)) ((fodom.1 (e. A (V)))) (-> (:-onto-> F A B) (br B (~<_) A)) (F A B fof fodom.1 A V F B fex ax-mp syl F V cnvexg F `' V f ac7g 3syl F A B forn F df-rn syl5eqr F `' dom B f cv fneq2 syl B f cv ran A domtr f cv B fnfrn biimp F A B :-onto-> f cv F `' C_ ad2antlr f cv `' F funss impcom F A B fof F A B ffun syl f cv F `' cnvss F cnvcnvss f cv `' F `' `' F sstr mpan2 syl syl2an f cv B Fn adantlr jca f cv B f cv ran df-f1 sylibr f visset f cv V rnexg ax-mp f cv ran V f cv B f1dom2g ax-mp syl f cv F `' rnss F A B :-onto-> adantl F A B fof F A B fdm syl F dfdm4 syl5eqr f cv F `' C_ adantr sseqtrd f visset f cv V rnexg ax-mp f cv ran V A ssdomg ax-mp syl f cv B Fn adantlr sylanc exp31 sylbid com23 imp3a f 19.23adv mpd))
+thm (fodomg ((A x) (B x) (F x)) () (-> (e. A C) (-> (:-onto-> F A B) (br B (~<_) A))) (x cv A F B foeq2 x cv A B ~<_ breq2 imbi12d x visset F B fodom C vtoclg))
+thm (fodomb ((f g) (f z) (A f) (g z) (A g) (A z) (B f) (B g) (B z)) ((fodomb.1 (e. A (V))) (fodomb.2 (e. B (V)))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A))) (f cv A B fof f cv A B fdm syl {/} eqeq1d f cv A B forn {/} eqeq1d f cv dm0rn0 syl5bb bitr3d negbid biimpcd fodomb.2 0sdom syl6ibr fodomb.1 f cv B fodom A {/} = -. a1i jcad f 19.23adv imp fodomb.1 A V {/} B sdomdomtr ax-mp fodomb.1 0sdom sylib g cv B A df-f1 pm3.27bd z visset A g cv ran \ fconst A g cv ran \ z cv {} X. A g cv ran \ z cv {} ffun ax-mp jctir g cv df-rn eqcomi z visset snnz z cv {} A g cv ran \ dmxp ax-mp ineq12i g cv ran A difdisj eqtr g cv `' A g cv ran \ z cv {} X. funun mpan2 syl z cv B e. adantl g cv B A f1f g cv B A frn syl g cv ran A ssundif sylib g cv `' A g cv ran \ z cv {} X. dmun g cv df-rn A g cv ran \ z cv {} X. dom uneq1i z visset snnz z cv {} A g cv ran \ dmxp ax-mp g cv ran uneq2i eqtr3 eqtr syl5eq z cv B e. adantl jca g cv `' A g cv ran \ z cv {} X. u. A df-fn sylibr g cv B A f1f g cv B A fdm syl g cv dfdm4 syl5eqr A g cv ran \ z cv {} X. ran uneq1d B 0ss A g cv ran \ {/} z cv {} xpeq1 z cv {} xp0r syl6eq rneqd rn0 syl6eq B sseq1d mpbiri z cv B e. a1d A g cv ran \ z cv {} rnxp z cv B e. adantr z cv B snssi A g cv ran \ {/} = -. adantl eqsstrd exp pm2.61i A g cv ran \ z cv {} X. ran B ssequn2 sylib sylan9eqr g cv `' A g cv ran \ z cv {} X. rnun syl5eq jca g cv `' A g cv ran \ z cv {} X. u. A B df-fo sylibr g visset cnvex fodomb.1 A V g cv ran difexg ax-mp z cv snex xpex unex f cv g cv `' A g cv ran \ z cv {} X. u. A B foeq1 cla4ev syl exp g 19.23adv z 19.23aiv imp fodomb.2 0sdom B z n0 bitr fodomb.1 B g brdom syl2anb jca impbi))
+thm (imadomg () () (-> (e. A B) (-> (Fun F) (br (" F A) (~<_) A))) (F A funres F A |` funforn sylib A B F resfunexg F A |` V dmexg syl6 F A |` dom V F A |` F A |` ran fodomg F A df-ima ~<_ F A |` dom breq1i syl6ibr syl6 mpdi F A " F A |` dom A domtr F A dmres A F dom inss1 eqsstr A B F A |` dom ssdom2g mpi sylan2 expcom syld))
+thm (fnrndomg () () (-> (e. A B) (-> (Fn F A) (br (ran F) (~<_) A))) (A B F F ran fodomg F A fnforn syl5ib))
+thm (htalem ((x y) (A x) (A y) (R x) (R y)) ((htalem.1 (e. A (V))) (htalem.2 (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x))))))))) (-> (/\ (We R A) (-. (= A ({/})))) (e. B A)) (A ssid htalem.1 R A x y wereu mpanr1 x A y A y cv R x cv br -. A.e. reucl syl htalem.2 A eleq1i sylibr))
+thm (hta ((v z) (A z) (A v) (x y) (R x) (x z) (v x) (R y) (y z) (v y) (R z) (R v) (ph y) (ph z) (ph v)) ((hta.1 (= A ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (hta.2 (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x))))))))) (-> (We R A) (-> ph ([/] B x ph))) (hta.1 A x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} R weeq2 ax-mp x ph y scottexs hta.2 hta.1 ph y ax-17 y y cv x ph [/] rank x cv ` rank y cv ` C_ -> hba1 hban z x hbab hta.1 z cv eleq2i hta.1 z cv eleq2i y bial 3imtr4 ph y ax-17 y y cv x ph [/] rank x cv ` rank y cv ` C_ -> hba1 hban z x hbab y cv R x cv br -. raleqf ax-mp x cv A e. a1i birabi hta.1 z x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 hta.1 z cv eleq2i hta.1 z cv eleq2i x bial 3imtr4 z x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 y x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} y cv R x cv br -. A.e. rabeqf ax-mp v x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 v cv x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} e. z ax-17 y x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} y cv R x cv br -. A.e. z ax-17 y x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 y cv R z cv br -. x ax-17 hbral x cv z cv y cv R breq2 negbid y x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} biraldv cbvrab ph y ax-17 y y cv x ph [/] rank x cv ` rank y cv ` C_ -> hba1 hban z x hbab z cv x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} e. v ax-17 y cv R z cv br -. v ax-17 v cv R z cv br -. y ax-17 y cv v cv R z cv breq1 negbid cbvralf z cv x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ {|} e. a1i birabi eqtr 3eqtr unieqi eqtr htalem exp sylbi ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. pm3.26 x ss2abi B sseli hta.2 hta.1 x ph y scottexs eqeltr z cv v cv e. x ax-17 z x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 hta.1 z cv eleq2i hta.1 z cv eleq2i x bial 3imtr4 hbel v cv A e. z ax-17 y A y cv R x cv br -. A.e. z ax-17 z cv y cv e. x ax-17 z x ph y y cv x ph [/] rank x cv ` rank y cv ` C_ -> A. /\ hbab1 hta.1 z cv eleq2i hta.1 z cv eleq2i x bial 3imtr4 hbel y cv R z cv br -. x ax-17 hbral x cv z cv y cv R breq2 negbid y A biraldv cbvrab z A y A y cv R z cv br -. A.e. ssrab eqsstr ssexi uniex eqeltr x ph elabs sylib syl6 ph x 19.8a x ph y scott0s sylib syl5))
+thm (oncardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A))))) (A On enrefg x cv A ~~ A breq1 On rcla4ev mpdan x On x cv ~~ A br rabn0 sylibr x On x cv ~~ A br ssrab x On x cv ~~ A br {e.|} oninton mpan syl y cv A x cv ~~ breq2 x On birabsdv inteqd On On z fvopabg mpdan y z x df-card A fveq1i syl5eq))
+thm (oncardon ((A x)) () (-> (e. A (On)) (e. (` (card) A) (On))) (A x oncardval card A fvex A x oncardval V eleq1d mpbii x On x cv ~~ A br {e.|} intex sylibr x On x cv ~~ A br ssrab x On x cv ~~ A br {e.|} oninton mpan syl eqeltrd))
+thm (oncardid ((A x)) () (-> (e. A (On)) (br (` (card) A) (~~) A)) (A x oncardval card A fvex A x oncardval V eleq1d mpbii x On x cv ~~ A br {e.|} intex sylibr x On x cv ~~ A br ssrab x On x cv ~~ A br {e.|} onint mpan syl eqeltrd x cv card A ` ~~ A breq1 On elrab pm3.27bd syl))
+thm (cardonle ((A x)) () (-> (e. A (On)) (C_ (` (card) A) A)) (A x oncardval A On enrefg x cv A ~~ A breq1 On elrab A x On x cv ~~ A br {e.|} intss1 sylbir mpdan eqsstrd))
+thm (card0 () () (= (` (card) ({/})) ({/})) (0elon {/} cardonle ax-mp card {/} ` ss0b mpbi))
+thm (cardnn () () (-> (e. A (om)) (= (` (card) A) A)) (card A ` A sdomnen A nnont A oncardid syl nsyl3 A nnont A oncardon card A ` A ordelpss card A ` eloni A eloni syl2an mpancom syl A nnont A cardonle A oncardon card A ` A onsseleq mpancom mpbid syl card A ` A elnn expcom A om card A ` eleq1a jaod mpd card A ` A nnsdomo mpancom bitr4d mtbird A nnont A cardonle A oncardon card A ` A onsseleq mpancom mpbid syl ord mpd))
+thm (cardom () () (= (` (card) (om)) (om)) (omelon om oncardid ax-mp card om ` nnsdom card om ` om sdomnen syl mt2 omelon om cardonle ax-mp omelon om oncardon ax-mp omelon onssel mpbi ori ax-mp))
+thm (cardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A)))) (x A numth2 x On x cv ~~ A br intexrab mpbi y cv A x cv ~~ breq2 x On birabsdv inteqd V V z fvopabg mpan2 y z x df-card A fveq1i syl5eq A card fvprc x visset enref A x cv ~~ brprc mpbiri x cv On e. biantrud x biabdv x On x cv ~~ A br df-rab syl6reqr x On abid2 syl6eq inteqd inton syl6eq eqtr4d pm2.61i))
+thm (cardon ((A x)) () (e. (` (card) A) (On)) (A x cardval x On x cv ~~ A br ssrab A x cardval card A fvex eqeltrr x On x cv ~~ A br {e.|} intex mpbir x On x cv ~~ A br {e.|} oninton mp2an eqeltr))
+thm (cardid ((A x)) () (br (` (card) A) (~~) A) (A x cardval x On x cv ~~ A br ssrab A x cardval card A fvex eqeltrr x On x cv ~~ A br {e.|} intex mpbir x On x cv ~~ A br {e.|} onint mp2an eqeltr x cv card A ` ~~ A breq1 On elrab pm3.27bd ax-mp))
+thm (oncard ((x y) (A x) (A y)) () (<-> (E. x (= A (` (card) (cv x)))) (= A (` (card) A))) (x cv cardid A card x cv ` ~~ x cv breq1 mpbiri A cardid card A ` A x cv entrt mpan A cardon y cv card A ` ~~ x cv breq1 onintss ax-mp x cv y cardval syl5ss 3syl A card x cv ` card A ` sseq1 mpbird x cv cardon A card x cv ` On eleq1 mpbiri A cardonle syl eqssd x 19.23aiv card A fvex A card A ` V eleq1 mpbiri x cv A card fveq2 A eqeq2d V cla4egv mpcom impbi))
+thm (cardne ((A x) (B x)) () (-> (e. A (` (card) B)) (-. (br A (~~) B))) (B cardon A onel x cv A ~~ B breq1 onintss B x cardval A sseq1i syl6ibr B cardon card B ` A ontri1 mpan sylibd con2d mpcom))
+thm (carden () () (-> (/\ (e. A C) (e. B D)) (<-> (= (` (card) A) (` (card) B)) (br A (~~) B))) (card A ` card B ` A ~~ breq2 B cardid A card B ` B entrt mpan2 syl6bi A cardid A C card A ` ensymg mpi syl5com B D e. adantr B D A ensymg B cardid card B ` B A entrt mpan card B ` A cardne con2i A cardon B cardon card A ` card B ` ontri1 mp2an sylibr syl syl6 A cardid card A ` A B entrt mpan card A ` B cardne con2i B cardon A cardon card B ` card A ` ontri1 mp2an sylibr syl B D e. a1i jcad card A ` card B ` eqss syl6ibr A C e. adantl impbid))
+thm (cardeq0 () () (-> (e. A B) (<-> (= (` (card) A) ({/})) (= A ({/})))) (0ex A B {/} V carden mpan2 card0 card A ` eqeq2i A en0 3bitr3g))
+thm (cardsn () () (-> (e. A B) (= (` (card) ({} A)) (1o))) (A B ensn1g A snex 1onn A {} V 1o om carden mp2an sylibr 1onn 1o cardnn ax-mp syl6eq))
+thm (carddomi () () (-> (e. A C) (-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))) (A card A ` card B ` endomtr A cardid A C card A ` ensymg mpi A cardon card A ` On card B ` ssdomg ax-mp syl2an B cardid A card B ` B domentr mpan2 syl exp))
+thm (carddom () () (-> (/\ (e. A C) (e. B D)) (<-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))) (A C B carddomi B D e. adantr B D A carddomi A cardon card B ` onelss syl5 B A domnsym syl6 con2d A cardon B cardon card A ` card B ` ontri1 mp2an syl6ibr A C e. adantl A C B D carden card A ` card B ` eqimss syl6bir jaod A B brdom2 syl5ib impbid))
+thm (cardsdom () () (-> (/\ (e. A C) (e. B D)) (<-> (e. (` (card) A) (` (card) B)) (br A (~<) B))) (A C B D carddom A C B D carden negbid anbi12d A cardon B cardon card A ` card B ` onelpsst mp2an A B brsdom 3bitr4g))
+thm (domtri () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (~<_) B) (-. (br B (~<) A)))) (A C B D carddom B D A C cardsdom ancoms negbid A cardon B cardon card A ` card B ` ontri1 mp2an syl5bb bitr3d))
+thm (entri () () (-> (/\ (e. A C) (e. B D)) (\/\/ (br A (~<) B) (br A (~~) B) (br B (~<) A))) (A C B D domtri biimprd A B brdom2 syl6ib con1d orrd A ~< B br A ~~ B br B ~< A br df-3or sylibr))
+thm (entri2 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<) A))) (A C B D entri A B brdom2 B ~< A br orbi1i A ~< B br A ~~ B br B ~< A br df-3or bitr4 sylibr))
+thm (entri3 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<_) A))) (A C B D entri2 B A sdomdom A ~<_ B br orim2i syl))
+thm (sucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B C)) (<-> (br A (~<) B) (br (suc A) (~<_) B))) (omex om V B C entri2 mpan A om e. adantl B C A om sdomdomtr exp3a imp A nnsdom sylan2 ancoms A peano2b A suc nnsdom sylbi A suc om sdomdom A suc om B domtr exp 3syl B C e. adantr jcad A ~< B br A suc ~<_ B br pm5.1 syl6 A x B finsucdom exp B x isfinite2 syl5 B C e. adantr jaod mpd))
+thm (unxpdomlem ((x y) (x z) (w x) (v x) (u x) (t x) (f x) (A x) (y z) (w y) (v y) (u y) (t y) (f y) (A y) (w z) (v z) (u z) (t z) (f z) (A z) (v w) (u w) (t w) (f w) (A w) (u v) (t v) (f v) (A v) (t u) (f u) (A u) (f t) (A t) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B f)) ((unxpdomlem.1 (e. A (V))) (unxpdomlem.2 (e. B (V)))) (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B))) (x cv f cv w cv eqeq1 y cv v cv = w cv y cv if y cv v cv = v cv x cv if ifbid x cv f cv y cv v cv = v cv ifeq2 f cv w cv = y cv v cv = w cv y cv if ifeq2d eqtrd f cv eqeq1d y cv v cv = w cv y cv if v cv v cv = w cv v cv if y cv v cv = v cv f cv if v cv v cv = v cv f cv if f cv w cv = ifeq12 y cv v cv v cv eqeq1 w cv y cv ifbid y cv v cv v cv v cv = w cv ifeq2 eqtrd y cv v cv v cv eqeq1 v cv f cv ifbid sylanc f cv eqeq1d A B rcla42ev f cv w cv = v cv v cv = w cv v cv if v cv v cv = v cv f cv if iftrue v cv eqid v cv v cv = w cv v cv iftrue ax-mp syl6eq f cv w cv = id eqtr4d sylan2 exp31 com3l t cv B e. t cv v cv = -. ad2antlr x cv f cv w cv eqeq1 y cv v cv = w cv y cv if y cv v cv = v cv x cv if ifbid x cv f cv y cv v cv = v cv ifeq2 f cv w cv = y cv v cv = w cv y cv if ifeq2d eqtrd f cv eqeq1d y cv v cv = w cv y cv if t cv v cv = w cv t cv if y cv v cv = v cv f cv if t cv v cv = v cv f cv if f cv w cv = ifeq12 y cv t cv v cv eqeq1 w cv y cv ifbid y cv t cv t cv v cv = w cv ifeq2 eqtrd y cv t cv v cv eqeq1 v cv f cv ifbid sylanc f cv eqeq1d A B rcla42ev f cv w cv = t cv v cv = w cv t cv if t cv v cv = v cv f cv if iffalse t cv v cv = v cv f cv iffalse sylan9eqr sylan2 exp43 imp3a com3l v cv B e. adantlr pm2.61d u cv A e. w cv A e. /\ u cv w cv = -. /\ adantl x cv u cv w cv eqeq1 y cv v cv = w cv y cv if y cv v cv = v cv x cv if ifbid x cv u cv y cv v cv = v cv ifeq2 u cv w cv = y cv v cv = w cv y cv if ifeq2d eqtrd f cv eqeq1d y cv v cv = w cv y cv if f cv v cv = w cv f cv if y cv v cv = v cv u cv if f cv v cv = v cv u cv if u cv w cv = ifeq12 y cv f cv v cv eqeq1 w cv y cv ifbid y cv f cv f cv v cv = w cv ifeq2 eqtrd y cv f cv v cv eqeq1 v cv u cv ifbid sylanc f cv eqeq1d A B rcla42ev u cv w cv = f cv v cv = w cv f cv if f cv v cv = v cv u cv if iffalse f cv v cv = v cv u cv iftrue f v equcomi eqtrd sylan9eqr sylan2 exp43 com24 imp w cv A e. adantlr x cv w cv w cv eqeq1 y cv v cv = w cv y cv if y cv v cv = v cv x cv if ifbid x cv w cv y cv v cv = v cv ifeq2 w cv w cv = y cv v cv = w cv y cv if ifeq2d eqtrd f cv eqeq1d y cv v cv = w cv y cv if f cv v cv = w cv f cv if y cv v cv = v cv w cv if f cv v cv = v cv w cv if w cv w cv = ifeq12 y cv f cv v cv eqeq1 w cv y cv ifbid y cv f cv f cv v cv = w cv ifeq2 eqtrd y cv f cv v cv eqeq1 v cv w cv ifbid sylanc f cv eqeq1d A B rcla42ev f cv v cv = w cv f cv iffalse w cv eqid w cv w cv = f cv v cv = w cv f cv if f cv v cv = v cv w cv if iftrue ax-mp syl5eq sylan2 exp31 com23 u cv A e. u cv w cv = -. ad2antlr pm2.61d t cv B e. v cv B e. /\ t cv v cv = -. /\ adantr jaod f cv A B elun syl5ib x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if f cv eqcom x A y B bi2rex w visset y visset y cv v cv = ifex v visset x visset y cv v cv = ifex x cv w cv = ifex x y z x cv A e. y cv B e. /\ z cv x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if = /\ {<<,>,>|} eqid f cv elrnoprab bitr4 syl6ib ssrdv unxpdomlem.1 unxpdomlem.2 unex A B u. V x y z x cv A e. y cv B e. /\ z cv x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if = /\ {<<,>,>|} ran ssdomg ax-mp unxpdomlem.1 unxpdomlem.2 xpex w visset y visset y cv v cv = ifex v visset x visset y cv v cv = ifex x cv w cv = ifex x y z x cv A e. y cv B e. /\ z cv x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if = /\ {<<,>,>|} eqid fnoprab2 A B X. V x y z x cv A e. y cv B e. /\ z cv x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if = /\ {<<,>,>|} fnrndomg mp2 A B u. x y z x cv A e. y cv B e. /\ z cv x cv w cv = y cv v cv = w cv y cv if y cv v cv = v cv x cv if if = /\ {<<,>,>|} ran A B X. domtr mpan2 3syl exp43 r19.23aivv r19.23advv imp 1onn unxpdomlem.1 1o A V sucdom mp2an df-2o ~<_ A breq1i bitr4 unxpdomlem.1 u w 2dom sylbi 1onn unxpdomlem.2 1o B V sucdom mp2an df-2o ~<_ B breq1i bitr4 unxpdomlem.2 t v 2dom sylbi syl2an))
+thm (unxpdom ((x y) (A x) (A y) (B y)) () (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B))) (1o A sdomex pm3.27d 1o B sdomex pm3.27d anim12i x cv A 1o ~< breq2 1o ~< y cv br anbi1d x cv A y cv uneq1 x cv A y cv xpeq1 ~<_ breq12d imbi12d y cv B 1o ~< breq2 1o ~< A br anbi2d y cv B A uneq2 y cv B A xpeq2 ~<_ breq12d imbi12d x visset y visset unxpdomlem V V vtocl2g mpcom))
+thm (unxpdom2 () ((unxpdom2.1 (e. A (V))) (unxpdom2.2 (e. B (V)))) (-> (/\ (br (1o) (~<) A) (br B (~<_) A)) (br (u. A B) (~<_) (X. A A))) (A B u. A {/} {} X. A 1o {} X. u. A A X. domtr unxpdom2.1 unxpdom2.1 1onn elisseti xpsnen ensymi B A A 1o {} X. domentr mpan2 unxpdom2.1 unxpdom2.1 0ex xpsnen ensymi A A {/} {} X. endom ax-mp 0ne1oOLD {/} 1o A A xpsndisj ax-mp unxpdom2.1 p0ex xpex unxpdom2.2 unxpdom2.1 1o snex xpex A undom mpan2 mpan syl A {/} {} X. A 1o {} X. unxpdom unxpdom2.1 unxpdom2.1 0ex xpsnen ensymi unxpdom2.1 p0ex xpex A {/} {} X. V 1o A sdomentr ax-mp mpan2 unxpdom2.1 unxpdom2.1 1onn elisseti xpsnen ensymi unxpdom2.1 1o snex xpex A 1o {} X. V 1o A sdomentr ax-mp mpan2 sylanc unxpdom2.1 0ex xpsnen unxpdom2.1 1onn elisseti xpsnen unxpdom2.1 p0ex xpex unxpdom2.1 unxpdom2.1 1o snex xpex unxpdom2.1 xpen mp2an A {/} {} X. A 1o {} X. u. A {/} {} X. A 1o {} X. X. A A X. domentr mpan2 syl syl2an ancoms))
+thm (sucxpdom ((A x)) () (-> (br (1o) (~<) A) (br (suc A) (~<_) (X. A A))) (1o A sdomex pm3.27d x cv A 1o ~< breq2 x cv A suceq x cv A x cv xpeq1 x cv A A xpeq2 eqtrd ~<_ breq12d imbi12d x visset x visset 1ord elisseti xpsnen x cv V x cv 1o {} X. x cv {} sdomen2 mp2an 1ord elisseti x visset ensn1 1o V x cv {} x cv sdomen1 mp2an bitr x cv {} x cv 1o {} X. sdomdom sylbir x visset x cv V domrefg ax-mp x visset x visset 0ex xpsnen x cv V x cv {/} {} X. x cv domen2 mp2an mpbir 0ne1oOLD {/} 1o x cv x cv xpsndisj ax-mp x visset p0ex xpex x cv snex x visset 1o snex xpex x cv undom mpan2 mpan syl x cv {/} {} X. x cv 1o {} X. unxpdom x visset x visset 0ex xpsnen x cv V x cv {/} {} X. 1o sdomen2 mp2an x visset x visset 1ord elisseti xpsnen x cv V x cv 1o {} X. 1o sdomen2 mp2an sylancbr jca x cv x cv {} u. x cv {/} {} X. x cv 1o {} X. u. x cv {/} {} X. x cv 1o {} X. X. domtr x visset 0ex xpsnen x visset 1ord elisseti xpsnen x visset p0ex xpex x visset x visset 1o snex xpex x visset xpen mp2an x cv x cv {} u. x cv {/} {} X. x cv 1o {} X. X. x cv x cv X. domentr mpan2 3syl x cv df-suc syl5eqbr V vtoclg mpcom))
+thm (sdomel () () (-> (/\ (e. A (On)) (e. B (On))) (-> (br A (~<) B) (e. A B))) (B On A ssdomg A On e. adantr B A ontri1 B On A On domtri 3imtr3d a3d ancoms))
+thm (sdomsdomcard () () (<-> (br A (~<) B) (br A (~<) (` (card) B))) (A B sdomex pm3.27d A sdom0 B card fvprc A ~< breq2d mtbiri a3i card B fvex card B ` V A B sdomentr ax-mp exp B cardid B V card B ` ensymg mpi syl5com B cardid B V A card B ` sdomentr mpan2i impbid pm5.21nii))
+thm (cardidm () () (= (` (card) (` (card) A)) (` (card) A)) (A cardid card A fvex card A ` V A V carden mpan mpbiri A card fvprc card fveq2d card0 syl6eq A card fvprc eqtr4d pm2.61i))
+thm (canth3 () () (-> (e. A B) (e. (` (card) A) (` (card) (P~ A)))) (A B canth2g A B pwexg A B A P~ V cardsdom mpdan mpbird))
+thm (cardlim ((A x)) () (<-> (C_ (om) (` (card) A)) (Lim (` (card) A))) (card A ` x cv suc om sseq2 biimpd x cv infensuc exp limom om x cv limsssuc ax-mp syl5ibr sylan9r card A ` x cv suc x cv ~~ breq2 x cv On e. adantl sylibrd exp com3r imp x visset sucid card A ` x cv suc x cv eleq2 mpbiri A cardidm syl6eleqr x cv card A ` cardne syl om card A ` C_ x cv On e. /\ a1i pm2.65d nrexdv peano1 om card A ` {/} ssel mpi {/} card A ` n0i A cardon onord card A ` x ordzsl mpbi card A ` {/} = x On card A ` x cv suc = E.e. card A ` Lim 3orass mpbi ori 3syl ord mpd card A ` limomss impbi))
+thm (cardsdomel () () (-> (e. A (On)) (<-> (br A (~<) B) (e. A (` (card) B)))) (A On card B ` ssdom2g B cardon card B ` A ontri1 mpan B cardon card B ` On A On domtri mpan 3imtr3d a3d B cardon A onelss B cardon card B ` On A ssdom2g ax-mp syl B cardidm A eleq2i A card B ` cardne sylbir jca A card B ` brsdom sylibr A On e. a1i impbid A B sdomsdomcard syl5bb))
+thm (iscard ((A x)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x A (br (cv x) (~<) A)))) (A cardon card A ` A On eleq1 mpbii pm4.71ri A cardonle card A ` A eqss baibr syl A x cv onelon x cv A cardsdomel syl biraldva A card A ` x dfss3 syl6rbbr bitr3d pm5.32i bitr))
+thm (iscard2 ((x y) (A x) (A y)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x (On) (-> (br A (~~) (cv x)) (C_ A (cv x)))))) (A cardon card A ` A On eleq1 mpbii pm4.71ri A cardonle A card A ` C_ biantrurd card A ` A eqss syl6rbbr A On x cv ensymg x visset A ensym A On e. a1i impbid x cv On e. anbi2d y cv x cv ~~ A breq1 On elrab syl5bb A x cv C_ imbi1d x cv On e. A ~~ x cv br A x cv C_ impexp syl6bb biraldv2 A y On y cv ~~ A br {e.|} x ssint syl5bb A y cardval A sseq2i syl5bb bitrd pm5.32i bitr))
+thm (cardval2 ((A x)) () (= (` (card) A) ({e.|} x (On) (br (cv x) (~<) A))) (x card A ` abid2 A cardon x cv onel pm4.71ri x cv A cardsdomel pm5.32i bitr4 x biabi x On x cv ~< A br df-rab eqtr4 eqtr3))
+thm (ondomon ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A B) (e. ({e.|} x (On) (br (cv x) (~<_) A)) (On))) (y cv z cv A domtr y cv On e. anim2i anassrs z cv y cv onelon z cv y cv onelsst imp y visset y cv V z cv ssdomg ax-mp syl jca sylan exp31 com12 imp3a x cv z cv ~<_ A breq1 On elrab x cv y cv ~<_ A breq1 On elrab 3imtr4g imp y z gen2 x On x cv ~<_ A br {e.|} y z dftr2 mpbir x On x cv ~<_ A br ssrab ordon x On x cv ~<_ A br {e.|} On trssord mp3an A B elisset x cv A A P~ domsdomtr A V canth2g sylan2 expcom x cv On e. a1d r19.21aiv syl x On x cv ~<_ A br x cv ~< A P~ br ss2rab sylibr A P~ x cardval2 card A P~ fvex eqeltrr x On x cv ~<_ A br {e.|} ssex x On x cv ~<_ A br {e.|} V elong 3syl mpbiri))
+thm (ondomcard ((x y) (A x) (A y)) () (-> (e. A B) (= (` (card) ({e.|} x (On) (br (cv x) (~<_) A))) ({e.|} x (On) (br (cv x) (~<_) A)))) (A B elisset A V x ondomon y cv A x On x cv ~<_ A br {e.|} domsdomtr x cv y cv ~<_ A breq1 On elrab pm3.27bd A V x ondomon x On x cv ~<_ A br {e.|} eloni x On x cv ~<_ A br {e.|} ordeirr syl y x On x cv ~<_ A br hbrab1 y cv On e. x ax-17 y x On x cv ~<_ A br hbrab1 y cv ~<_ e. x ax-17 y cv A e. x ax-17 hbbr x cv x On x cv ~<_ A br {e.|} ~<_ A breq1 elrabf biimpr exp mtod syl A V x ondomon x On x cv ~<_ A br {e.|} On A V domtri bicon2d mpancom mpbird syl2an y cv x On x cv ~<_ A br {e.|} sdomnen syl expcom con2d y visset x On x cv ~<_ A br {e.|} ensym syl5 y cv On e. adantr x On x cv ~<_ A br {e.|} y cv ontri1 A V x ondomon sylan sylibrd exp r19.21aiv jca x On x cv ~<_ A br {e.|} y iscard2 sylibr syl))
+thm (carduni ((x y) (A x) (A y) (B y)) () (-> (e. A B) (-> (A.e. x A (= (` (card) (cv x)) (cv x))) (= (` (card) (U. A)) (U. A)))) (A B onunit x cv y cv card fveq2 x cv y cv = id eqeq12d A rcla4v y cv cardon card y cv ` y cv On eleq1 mpbii syl6com ssrdv syl5 imp A U. cardonle syl A U. cardon oneirr card A U. ` A y eluni x cv y cv card fveq2 x cv y cv = id eqeq12d A rcla4v A B uniexg y visset y cv V A U. V carddom mpan bicomd syl card y cv ` y cv card A U. ` sseq1 sylan9bb y cv A elssuni y visset y cv V A U. ssdomg ax-mp syl syl5bi y cv card A U. ` card A U. ` ssel syl6 exp com13 syld com4r imp y 19.23aiv sylbi com13 imp mtoi jca A B onunit x cv y cv card fveq2 x cv y cv = id eqeq12d A rcla4v y cv cardon card y cv ` y cv On eleq1 mpbii syl6com ssrdv syl5 imp A U. eloni A U. cardon onord card A U. ` A U. ordtri4 mpan 3syl mpbird exp))
+thm (cardiun ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (-> (A.e. x A (= (` (card) B) B)) (= (` (card) (U_ x A B)) (U_ x A B)))) (A C z x card B ` abrexexg y visset z cv y cv card B ` eqeq1 x A birexdv elab y cv card B ` card fveq2 B cardidm syl6eq y cv card B ` = id eqtr4d x cv A e. a1i r19.23aiv sylbi rgen z x A z cv card B ` = E.e. {|} V y carduni mpi syl card B fvex x A z dfiun2 card fveq2i card B fvex x A z dfiun2 3eqtr4g x A card B ` B = A.e. adantr x A card B ` B iuneq2 A C e. adantl card fveq2d x A card B ` B iuneq2 A C e. adantl 3eqtr3d exp))
+thm (cardmin ((x y) (A x) (A y) (B y)) () (-> (e. A B) (= (` (card) (|^| ({e.|} x (On) (br A (~<) (cv x))))) (|^| ({e.|} x (On) (br A (~<) (cv x)))))) (A B x numthcor x A ~< x cv br onintrab2 sylib A B x numthcor x A ~< x cv br onintrab2 sylib x On A ~< x cv br {e.|} |^| y cv onelon exp syl x cv y cv A ~< breq2 On elrab x On A ~< x cv br ssrab x On A ~< x cv br {e.|} y cv onnmin mpan sylbir exp con2d syli y visset y cv V A B domtri mpan sylibrd A B x numthcor y cv A e. x ax-17 y cv ~< e. x ax-17 y x On A ~< x cv br hbrab1 hbint hbbr x cv x On A ~< x cv br {e.|} |^| A ~< breq2 onminsb syl jctird y cv A x On A ~< x cv br {e.|} |^| domsdomtr syl6 r19.21aiv jca x On A ~< x cv br {e.|} |^| y iscard sylibr))
+thm (cardprc ((x y)) () (-. (e. ({|} x (= (` (card) (cv x)) (cv x))) (V))) (x card x cv ` x cv = {|} U. V canth3 card x card x cv ` x cv = {|} U. P~ fvex x card x cv ` x cv = {|} U. P~ cardidm card x card x cv ` x cv = {|} U. P~ fvex y cv card e. x ax-17 y x card x cv ` x cv = hbab1 hbuni hbpw hbfv y cv card e. x ax-17 y cv card e. x ax-17 y x card x cv ` x cv = hbab1 hbuni hbpw hbfv hbfv y cv card e. x ax-17 y x card x cv ` x cv = hbab1 hbuni hbpw hbfv hbeq x cv card x card x cv ` x cv = {|} U. P~ ` card fveq2 x cv card x card x cv ` x cv = {|} U. P~ ` = id eqeq12d V elabgf ax-mp mpbir card x card x cv ` x cv = {|} U. P~ ` x card x cv ` x cv = {|} elssuni ax-mp card x card x cv ` x cv = {|} U. P~ ` V x card x cv ` x cv = {|} U. ssdomg mp2 card x card x cv ` x cv = {|} U. P~ fvex card x card x cv ` x cv = {|} U. P~ ` V x card x cv ` x cv = {|} U. V carddom mpan mpbiri x card x cv ` x cv = {|} U. P~ cardidm syl5ssr x card x cv ` x cv = {|} U. P~ cardon x card x cv ` x cv = {|} U. cardon card x card x cv ` x cv = {|} U. P~ ` card x card x cv ` x cv = {|} U. ` ontri1 mp2an sylib pm2.65i x card x cv ` x cv = {|} V uniexg mto))
+thm (alephfnon ((x y) (x z) (y z)) () (Fn (aleph) (On)) (x y y cv z On x cv ~< z cv br {e.|} |^| = {<,>|} om rdgfnon x y z df-aleph aleph x y y cv z On x cv ~< z cv br {e.|} |^| = {<,>|} om rec On fneq1 ax-mp mpbir))
+thm (aleph0 ((x y) (x z) (y z)) () (= (` (aleph) ({/})) (om)) (x y z df-aleph {/} fveq1i omex x y y cv z On x cv ~< z cv br {e.|} |^| = {<,>|} rdg0 eqtr))
+thm (alephlim ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A B) (Lim A)) (= (` (aleph) A) (U_ x A (` (aleph) (cv x))))) (A B y z z cv w On y cv ~< w cv br {e.|} |^| = {<,>|} om x rdglim2a y z w df-aleph A fveq1i y z w df-aleph x cv fveq1i x cv A e. a1i iuneq2i 3eqtr4g))
+thm (alephon ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. (` (aleph) A) (On)) (x cv {/} aleph fveq2 On eleq1d x cv y cv aleph fveq2 On eleq1d x cv y cv suc aleph fveq2 On eleq1d x cv A aleph fveq2 On eleq1d aleph0 omelon eqeltr w cv om e. z ax-17 w cv y cv e. z ax-17 w cv x On aleph y cv ` ~< x cv br {e.|} |^| e. z ax-17 z y x df-aleph z cv aleph y cv ` ~< x cv breq1 x On birabsdv inteqd V rdgsucopab On eleq1d x aleph y cv ` ~< x cv br onintrab syl6rbbr exp ibd 0elon w cv om e. z ax-17 w cv y cv e. z ax-17 w cv x On aleph y cv ` ~< x cv br {e.|} |^| e. z ax-17 z y x df-aleph z cv aleph y cv ` ~< x cv breq1 x On birabsdv inteqd rdgsucopabn On eleq1d mpbiri pm2.61d1 aleph y cv ` On e. a1d x visset x cv V y alephlim mpan On eleq1d x visset aleph y cv fvex y iunon syl5bir tfinds 0elon alephfnon aleph On fndm ax-mp A eleq2i negbii A aleph ndmfv sylbir On eleq1d mpbiri pm2.61i))
+thm (alephsuc ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (On)) (= (` (aleph) (suc A)) (|^| ({e.|} x (On) (br (` (aleph) A) (~<) (cv x)))))) (aleph A fvex aleph A ` V x numthcor ax-mp x On aleph A ` ~< x cv br intexrab mpbi w cv om e. y ax-17 w cv A e. y ax-17 w cv x On aleph A ` ~< x cv br {e.|} |^| e. y ax-17 y z x df-aleph y cv aleph A ` ~< x cv breq1 x On birabsdv inteqd V rdgsucopab mpan2))
+thm (alephcard ((x y) (A x) (A y)) () (= (` (card) (` (aleph) A)) (` (aleph) A)) (x cv {/} aleph fveq2 card fveq2d x cv {/} aleph fveq2 eqeq12d x cv y cv aleph fveq2 card fveq2d x cv y cv aleph fveq2 eqeq12d x cv y cv suc aleph fveq2 card fveq2d x cv y cv suc aleph fveq2 eqeq12d x cv A aleph fveq2 card fveq2d x cv A aleph fveq2 eqeq12d cardom aleph0 card fveq2i aleph0 3eqtr4 aleph y cv fvex aleph y cv ` V x cardmin ax-mp y cv On e. a1i y cv x alephsuc card fveq2d y cv x alephsuc 3eqtr4d card aleph y cv ` ` aleph y cv ` = a1d x visset x cv V y aleph y cv ` cardiun ax-mp x cv Lim adantl x visset x cv V y alephlim mpan y x cv card aleph y cv ` ` aleph y cv ` = A.e. adantr card fveq2d x visset x cv V y alephlim mpan y x cv card aleph y cv ` ` aleph y cv ` = A.e. adantr 3eqtr4d exp tfinds card0 alephfnon aleph On fndm ax-mp A eleq2i negbii A aleph ndmfv sylbir card fveq2d alephfnon aleph On fndm ax-mp A eleq2i negbii A aleph ndmfv sylbir eqeq12d mpbiri pm2.61i))
+thm (alephnbtwn ((A x) (B x)) () (-> (= (` (card) B) B) (-. (/\ (e. (` (aleph) A) B) (e. B (` (aleph) (suc A)))))) (card B ` B aleph A ` eleq2 A alephon aleph A ` B cardsdomel ax-mp syl5bb A On e. adantl A x alephsuc B eleq2d biimpd x cv B aleph A ` ~< breq2 onnminsb sylan9 con2d B cardon card B ` B On eleq1 mpbii sylan2 sylbird aleph A ` B e. B aleph A suc ` e. imnan sylib exp B aleph A suc ` n0i alephfnon aleph On fndm ax-mp A suc eleq2i negbii A suc aleph ndmfv sylbir nsyl2 A sucelon sylibr aleph A ` B e. adantl con3i card B ` B = a1d pm2.61i))
+thm (alephnbtwn2 () () (-. (/\ (br (` (aleph) A) (~<) B) (br B (~<) (` (aleph) (suc A))))) (B cardidm card B ` A alephnbtwn ax-mp A alephon aleph A ` B cardsdomel ax-mp B V e. a1i A suc alephon B V aleph A suc ` On cardsdom mpan2 A suc alephcard card B ` eleq2i syl5rbbr anbi12d mtbiri relsdom B aleph A suc ` brrelexi aleph A ` ~< B br adantl con3i pm2.61i))
+thm (alephsucpw () () (br (` (aleph) (suc A)) (~<_) (P~ (` (aleph) A))) (aleph A fvex canth2 A aleph A ` P~ alephnbtwn2 aleph A ` ~< aleph A ` P~ br aleph A ` P~ ~< aleph A suc ` br imnan mpbir ax-mp aleph A suc fvex aleph A fvex pwex aleph A suc ` V aleph A ` P~ V domtri mp2an mpbir))
+thm (aleph1 () () (br (` (aleph) (1o)) (~<_) (opr (2o) (^m) (` (aleph) ({/})))) (df-1o aleph fveq2i {/} alephsucpw 2o ^m aleph {/} ` oprex aleph {/} fvex pw2en 2o ^m aleph {/} ` opr V aleph {/} ` P~ aleph {/} suc ` domen2 mp2an mpbi eqbrtr))
+thm (alephordlem1 ((x y) (A x) (A y)) () (-> (e. A (On)) (br (` (aleph) A) (~<) (` (aleph) (suc A)))) (A x alephsuc A alephon aleph A ` On x numthcor ax-mp y cv aleph A ` e. x ax-17 y cv ~< e. x ax-17 y x On aleph A ` ~< x cv br hbrab1 hbint hbbr x cv x On aleph A ` ~< x cv br {e.|} |^| aleph A ` ~< breq2 onminsb ax-mp syl5breqr))
+thm (alephordlem2 ((A x) (B x)) () (-> (/\ (e. B (V)) (Lim B)) (-> (e. A B) (br (` (aleph) A) (~<_) (` (aleph) B)))) (B V x alephlim aleph A ` sseq2d x cv A aleph fveq2 B ssiun2s syl5bir A alephon aleph A ` On aleph B ` ssdomg ax-mp syl6))
+thm (alephordi ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B)))) (x cv {/} A eleq2 x cv {/} aleph fveq2 aleph A ` ~< breq2d imbi12d x cv y cv A eleq2 x cv y cv aleph fveq2 aleph A ` ~< breq2d imbi12d x cv y cv suc A eleq2 x cv y cv suc aleph fveq2 aleph A ` ~< breq2d imbi12d x cv B A eleq2 x cv B aleph fveq2 aleph A ` ~< breq2d imbi12d A noel aleph A ` ~< aleph {/} ` br pm2.21i aleph A ` aleph y cv ` aleph y cv suc ` sdomtr y cv alephordlem1 sylan2 expcom A y cv e. imim2d com23 A y cv aleph fveq2 ~< aleph y cv suc ` breq1d y cv alephordlem1 syl5bir A y cv e. aleph A ` ~< aleph y cv ` br -> a1d com3r jaod y visset A elsuc2 syl5ib com23 x visset x cv V y alephlim mpan aleph A ` sseq2d y cv A aleph fveq2 x cv ssiun2s syl5bir A alephon aleph A ` On aleph x cv ` ssdomg ax-mp syl6 x cv A limsuc x visset x cv A suc alephordlem2 mpan sylbid imp aleph A suc ` aleph x cv ` domnsym syl aleph x cv fvex aleph A ` ensym aleph x cv ` aleph A ` aleph A suc ` ensdomtr exp syl A alephordlem1 syl5 x cv A onelon x visset x cv V limelon mpan sylan syl5com mtod exp jcad aleph A ` aleph x cv ` brsdom syl6ibr y x cv A y cv e. aleph A ` ~< aleph y cv ` br -> A.e. a1d tfinds))
+thm (alephord () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B)))) (B A alephordi A On e. adantl A B alephordi con3d A alephon B alephon aleph A ` On aleph B ` On domtri mp2an syl5ib B On e. adantr A B ontri1 sylibrd A B aleph fveq2 A alephon aleph A ` On aleph B ` eqeng ax-mp syl con3i A On e. B On e. /\ a1i anim12d A B onelpsst sylibrd aleph A ` aleph B ` brsdom syl5ib impbid))
+thm (alephord2 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))) (A B alephord A alephon aleph A ` aleph B ` cardsdomel ax-mp B alephcard aleph A ` eleq2i bitr syl6bb))
+thm (alephord2i () () (-> (e. B (On)) (-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))) (B A onelon A B alephord2 biimpd exp imp3a mpcom exp))
+thm (alephord3 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (C_ A B) (C_ (` (aleph) A) (` (aleph) B)))) (B A alephord2 ancoms negbid A B ontri1 A alephon B alephon aleph A ` aleph B ` ontri1 mp2an A On e. B On e. /\ a1i 3bitr4d))
+thm (aleph11 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (= (` (aleph) A) (` (aleph) B)) (= A B))) (A B alephord3 B A alephord3 ancoms anbi12d A B eqss aleph A ` aleph B ` eqss 3bitr4g bicomd))
+thm (alephsucdom () () (-> (e. B (On)) (<-> (br A (~<_) (` (aleph) B)) (br A (~<) (` (aleph) (suc B))))) (A aleph B ` aleph B suc ` domsdomtr exp B alephordlem1 syl5com A V e. adantl aleph B fvex A V aleph B ` V domtri mpan2 B A alephnbtwn2 aleph B ` ~< A br A ~< aleph B suc ` br imnan mpbir con2i syl5bir B On e. adantr impbid exp reldom A aleph B ` brrelexi relsdom A aleph B suc ` brrelexi pm5.21ni B On e. a1d pm2.61i))
+thm (alephsuc2 ((A x)) () (-> (e. A (On)) (= (` (aleph) (suc A)) ({e.|} x (On) (br (cv x) (~<_) (` (aleph) A))))) (A x cv alephsucdom x On birabsdv A suc alephcard aleph A suc ` x cardval2 eqtr3 syl6reqr))
+thm (alephgeom () () (<-> (e. A (On)) (C_ (om) (` (aleph) A))) (A 0ss 0elon {/} A alephord3 mpan mpbii aleph0 syl5ssr peano1 ordom ord0 om {/} ordtri1 mp2an bicon2i mpbi A aleph ndmfv om sseq2d mtbiri a3i alephfnon aleph On fndm ax-mp A eleq2i sylib impbi))
+thm (alephislim () () (<-> (e. A (On)) (Lim (` (aleph) A))) (A alephgeom aleph A ` cardlim A alephcard om sseq2i A alephcard card aleph A ` ` aleph A ` limeq ax-mp 3bitr3 bitr))
+thm (alephle ((x y) (A x) (A y)) () (-> (e. A (On)) (C_ A (` (aleph) A))) (x cv y cv = id x cv y cv aleph fveq2 sseq12d x cv A = id x cv A aleph fveq2 sseq12d x cv y cv alephord2i imp x cv y cv onelon x cv alephon y cv aleph x cv ` aleph y cv ` ontr2 mpan2 syl mpan2d r19.20dva x cv alephon x cv aleph x cv ` ontri1 mpan2 x cv alephon oneirr y cv aleph x cv ` aleph x cv ` eleq1 x cv rcla4v com12 mtoi syl5bir syld tfis3))
+thm (cardaleph ((x y) (A x) (A y)) () (-> (/\ (C_ (om) A) (= (` (card) A) A)) (= A (` (aleph) (|^| ({e.|} x (On) (C_ A (` (aleph) (cv x)))))))) (A cardon card A ` A On eleq1 mpbii A alephle x cv A aleph fveq2 A sseq2d On rcla4ev mpdan x A aleph x cv ` C_ onintrab2 sylib x On A aleph x cv ` C_ {e.|} |^| eloni x On A aleph x cv ` C_ {e.|} |^| y ordzsl x On A aleph x cv ` C_ {e.|} |^| {/} = y On x On A aleph x cv ` C_ {e.|} |^| y cv suc = E.e. x On A aleph x cv ` C_ {e.|} |^| Lim 3orass bitr sylib 3syl om A C_ adantl A cardon card A ` A On eleq1 mpbii A alephle x cv A aleph fveq2 A sseq2d On rcla4ev mpdan y cv A e. x ax-17 y cv aleph e. x ax-17 y x On A aleph x cv ` C_ hbrab1 hbint hbfv hbss x cv x On A aleph x cv ` C_ {e.|} |^| aleph fveq2 A sseq2d onminsb 3syl x On A aleph x cv ` C_ {e.|} |^| {/} = a1i x On A aleph x cv ` C_ {e.|} |^| {/} aleph fveq2 aleph0 syl6eq A sseq1d biimprd anim12d A aleph x On A aleph x cv ` C_ {e.|} |^| ` eqss syl6ibr com12 ancoms x cv y cv aleph fveq2 A sseq2d onnminsb y visset sucid x On A aleph x cv ` C_ {e.|} |^| y cv suc y cv eleq2 mpbiri syl5 imp card A ` A = adantl x On A aleph x cv ` C_ {e.|} |^| y cv suc aleph fveq2 y cv x alephsuc sylan9eqr A eleq2d biimpd A cardon card A ` A On eleq1 mpbii x cv A aleph y cv ` ~< breq2 onnminsb aleph y cv fvex A On aleph y cv ` V domtri mpan2 sylibrd aleph y cv fvex A On aleph y cv ` V carddom mpan2 sylibrd syl card A ` A card aleph y cv ` ` sseq1 y cv alephcard A sseq2i syl6bb sylibd sylan9r mtod exp32 r19.23adv A cardon card A ` A On eleq1 mpbii x On A aleph x cv ` C_ {e.|} |^| y cv onelon A alephle x cv A aleph fveq2 A sseq2d On rcla4ev mpdan x A aleph x cv ` C_ onintrab2 sylib sylan x cv y cv aleph fveq2 A sseq2d onnminsb A On e. adantld mpcom y cv alephon A onelss nsyl nrexdv x On A aleph x cv ` C_ {e.|} |^| Lim adantr x On A aleph x cv ` C_ {e.|} |^| On y alephlim A alephle x cv A aleph fveq2 A sseq2d On rcla4ev mpdan x A aleph x cv ` C_ onintrab2 sylib sylan A eleq2d A y x On A aleph x cv ` C_ {e.|} |^| aleph y cv ` eliun syl6bb mtbird exp syl jaod A cardon card A ` A On eleq1 mpbii A alephle x cv A aleph fveq2 A sseq2d On rcla4ev mpdan y cv A e. x ax-17 y cv aleph e. x ax-17 y x On A aleph x cv ` C_ hbrab1 hbint hbfv hbss x cv x On A aleph x cv ` C_ {e.|} |^| aleph fveq2 A sseq2d onminsb syl x On A aleph x cv ` C_ {e.|} |^| alephon A aleph x On A aleph x cv ` C_ {e.|} |^| ` onsseleq mpan2 mpbid ord syl syld om A C_ adantl jaod mpd))
+thm (cardalephex ((x y) (A x) (A y)) () (-> (C_ (om) A) (<-> (= (` (card) A) A) (E.e. x (On) (= A (` (aleph) (cv x)))))) (x cv y On A aleph y cv ` C_ {e.|} |^| aleph fveq2 A eqeq2d On rcla4ev om A C_ card A ` A = pm3.26 A y cardaleph om sseq2d y On A aleph y cv ` C_ {e.|} |^| alephgeom syl6bbr mpbid A y cardaleph sylanc exp x cv alephcard A aleph x cv ` card fveq2 A aleph x cv ` = id eqeq12d mpbiri x cv On e. a1i r19.23aiv om A C_ a1i impbid))
+thm (isinfcard ((A x)) () (<-> (/\ (C_ (om) A) (= (` (card) A) A)) (e. A (ran (aleph)))) (aleph x cv ` A eqcom x On birex alephfnon aleph On A x fvelrn ax-mp A x cardalephex pm5.32i A aleph x cv ` om sseq2 x cv alephgeom biimp syl5bir com12 r19.23aiv pm4.71ri bitr4 3bitr4r))
+thm (iscard3 () () (<-> (= (` (card) A) A) (e. A (u. (om) (ran (aleph))))) (A cardon card A ` A On eleq1 mpbii A eloni ordom A om ordtri2or mpan2 3syl ord A isinfcard biimp expcom syld orrd A cardnn A isinfcard bicomi pm3.27bd jaoi impbi A om aleph ran elun bitr4))
+thm (cardnum () () (= ({|} x (= (` (card) (cv x)) (cv x))) (u. (om) (ran (aleph)))) (x cv iscard3 bicomi biabri eqcomi))
+thm (carduniima ((F x) (A x)) () (-> (e. A B) (-> (:--> F A (u. (om) (ran (aleph)))) (e. (U. (" F A)) (u. (om) (ran (aleph)))))) (A B F funimaexg F A om aleph ran u. ffun syl5 F A " V x carduni F A om aleph ran u. ffn F A fnima syl F A om aleph ran u. frn eqsstrd x cv sseld x cv iscard3 syl6ibr r19.21aiv syl5 syli F A " U. iscard3 syl6ib))
+thm (cardinfima ((F x) (A x)) () (-> (e. A B) (-> (/\ (:--> F A (u. (om) (ran (aleph)))) (E.e. x A (e. (` F (cv x)) (ran (aleph))))) (e. (U. (" F A)) (ran (aleph))))) (A B elisset F x cv ` isinfcard bicomi pm3.26bd F A x cv fnfvrn exp F A fnima F x cv ` eleq2d sylibrd F x cv ` F A " elssuni syl6 imp F A om aleph ran u. ffn sylan sylan9ssr anasss A V e. a1i A V F carduniima F A " U. iscard3 syl6ibr x cv A e. F x cv ` aleph ran e. /\ adantrd jcad F A " U. isinfcard syl6ib exp4d imp r19.23adv exp imp3a syl))
+thm (alephiso ((x y) (x z) (y z)) () (Isom (aleph) (E) (E) (On) ({|} x (/\ (C_ (om) (cv x)) (= (` (card) (cv x)) (cv x))))) (aleph E E On x om x cv C_ card x cv ` x cv = /\ {|} y z df-iso aleph On x om x cv C_ card x cv ` x cv = /\ {|} df-f1o aleph On x om x cv C_ card x cv ` x cv = /\ {|} y z f1fv aleph On x om x cv C_ card x cv ` x cv = /\ {|} df-fo alephfnon x cv isinfcard bicomi biabri mpbir2an aleph On x om x cv C_ card x cv ` x cv = /\ {|} fof ax-mp y cv z cv aleph11 biimpd rgen2 mpbir2an aleph On x om x cv C_ card x cv ` x cv = /\ {|} df-fo alephfnon x cv isinfcard bicomi biabri mpbir2an mpbir2an y cv z cv alephord2 y z epel aleph y cv fvex aleph z cv fvex epelc 3bitr4g rgen2 mpbir2an))
+thm (alephprc () () (-. (e. (ran (aleph)) (V))) (x cardprc x cardnum V eleq1i mtbi omex om aleph ran unexb biimp mpan mto))
+thm (alephsson () () (C_ (ran (aleph)) (On)) (x cv isinfcard x cv cardon card x cv ` x cv On eleq1 mpbii om x cv C_ adantl sylbir ssriv))
+thm (unialeph () () (= (U. (ran (aleph))) (On)) (alephprc aleph ran uniexb mtbi aleph ran U. On elisset mto alephsson aleph ran ssorduni ax-mp aleph ran U. ordeleqon mpbi ori ax-mp))
+thm (alephfplem1 ((x y)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (e. (` H ({/})) (ran (aleph))) (alephfplem.1 {/} fveq1i aleph {/} fvex aleph {/} ` V x y y cv aleph x cv ` = {<,>|} fr0t ax-mp eqtr alephfnon 0elon aleph On {/} fnfvrn mp2an eqeltr))
+thm (alephfplem2 ((x y) (x z) (w x) (y z) (w y) (w z) (H z) (H w)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (-> (e. (cv w) (om)) (= (` H (suc (cv w))) (` (aleph) (` H (cv w))))) (aleph H w cv ` fvex z cv aleph {/} ` e. x ax-17 z cv w cv e. x ax-17 z cv aleph e. x ax-17 z x y y cv aleph x cv ` = hbopab1 z cv aleph {/} ` e. x ax-17 hbrdg z cv om e. x ax-17 hbres alephfplem.1 z cv eleq2i alephfplem.1 z cv eleq2i x bial 3imtr4 z cv w cv e. x ax-17 hbfv hbfv alephfplem.1 x cv H w cv ` aleph fveq2 V frsucopab mpan2))
+thm (alephfplem3 ((x y) (w x) (v x) (w y) (v y) (v w) (H w) (H v)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (-> (e. (cv v) (om)) (e. (` H (cv v)) (ran (aleph)))) (y equid v cv {/} H fveq2 aleph ran eleq1d v cv w cv H fveq2 aleph ran eleq1d v cv w cv suc H fveq2 aleph ran eleq1d alephfplem.1 alephfplem1 y cv y cv = a1i alephfplem.1 w alephfplem2 aleph ran eleq1d alephsson H w cv ` sseli alephfnon aleph On H w cv ` fnfvrn mpan syl syl5bir y cv y cv = a1d finds2 mpi))
+thm (alephfplem4 ((x y) (x z) (y z) (H z)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (e. (U. (" H (om))) (ran (aleph))) (H om aleph ran z ffnfv x y y cv aleph x cv ` = {<,>|} aleph {/} ` frfnom alephfplem.1 H x y y cv aleph x cv ` = {<,>|} aleph {/} ` rec om |` om fneq1 ax-mp mpbir alephfplem.1 z alephfplem3 rgen mpbir2an aleph ran om ssun2 H om aleph ran om aleph ran u. fss mp2an peano1 alephfplem.1 alephfplem1 z cv {/} H fveq2 aleph ran eleq1d om rcla4ev mp2an omex om V H z cardinfima ax-mp mp2an))
+thm (alephfp ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (H z) (H w) (H v)) ((alephfplem.1 (= H (|` (rec ({<,>|} x y (= (cv y) (` (aleph) (cv x)))) (` (aleph) ({/}))) (om))))) (= (` (aleph) (U. (" H (om)))) (U. (" H (om)))) (alephfplem.1 alephfplem4 H om " U. isinfcard H om " U. z cardalephex biimpa sylbir ax-mp z cv alephle aleph z cv ` eirr z cv H om " w eluni x y y cv aleph x cv ` = {<,>|} aleph {/} ` frfnom alephfplem.1 H x y y cv aleph x cv ` = {<,>|} aleph {/} ` rec om |` om fneq1 ax-mp mpbir H om fnima ax-mp w cv eleq2i x y y cv aleph x cv ` = {<,>|} aleph {/} ` frfnom alephfplem.1 H x y y cv aleph x cv ` = {<,>|} aleph {/} ` rec om |` om fneq1 ax-mp mpbir H om w cv v fvelrn ax-mp bitr z cv w cv e. anbi2i v om z cv w cv e. H v cv ` w cv = r19.42v bitr4 H v cv ` w cv z cv eleq2 biimparc v om r19.22si sylbi w 19.23aiv sylbi alephfplem.1 v alephfplem3 alephsson H v cv ` sseli H v cv ` z cv alephord2i 3syl alephfplem.1 v alephfplem2 v cv peano2 x y y cv aleph x cv ` = {<,>|} aleph {/} ` frfnom alephfplem.1 H x y y cv aleph x cv ` = {<,>|} aleph {/} ` rec om |` om fneq1 ax-mp mpbir H om v cv suc fnfvrn mpan x y y cv aleph x cv ` = {<,>|} aleph {/} ` frfnom alephfplem.1 H x y y cv aleph x cv ` = {<,>|} aleph {/} ` rec om |` om fneq1 ax-mp mpbir H om fnima ax-mp syl6eleqr syl eqeltrrd aleph H v cv ` ` H om " elssuni syl aleph z cv ` sseld syld r19.23aiv syl H om " U. aleph z cv ` z cv eleq2 H om " U. aleph z cv ` aleph z cv ` eleq2 imbi12d mpbii mtoi anim12i z cv eloni z cv alephon onord z cv aleph z cv ` ordtri4 mpan2 syl H om " U. aleph z cv ` = adantr mpbird H om " U. aleph z cv ` z cv eqeq2 z cv On e. adantl mpbird eqcomd aleph fveq2d H om " U. aleph z cv ` aleph H om " U. ` eqeq2 z cv On e. adantl mpbird exp r19.23aiv ax-mp))
+thm (alephfp2 ((x y) (x z) (y z)) () (E.e. x (On) (= (` (aleph) (cv x)) (cv x))) (alephsson y z z cv aleph y cv ` = {<,>|} aleph {/} ` rec om |` eqid alephfplem4 sselii y z z cv aleph y cv ` = {<,>|} aleph {/} ` rec om |` eqid alephfp x cv y z z cv aleph y cv ` = {<,>|} aleph {/} ` rec om |` om " U. aleph fveq2 x cv y z z cv aleph y cv ` = {<,>|} aleph {/} ` rec om |` om " U. = id eqeq12d On rcla4ev mp2an))
+thm (alephval2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A (On)) (e. ({/}) A)) (= (` (aleph) A) (|^| ({e.|} x (On) (A.e. y A (br (` (aleph) (cv y)) (~<) (cv x))))))) (x On y A aleph y cv ` ~< x cv br A.e. ssrab x On y A aleph y cv ` ~< x cv br A.e. {e.|} aleph A ` z oneqmini ax-mp A y cv alephordi r19.21aiv A alephon jctil x cv aleph A ` aleph y cv ` ~< breq2 y A biraldv On elrab sylibr {/} A e. adantr omex z visset om V z cv V entri3 mp2an x cv A alephord ancoms card z cv ` aleph x cv ` ~< aleph A ` breq1 z visset z cv cardid z cv V card z cv ` aleph A ` sdomen1 mp2an syl5rbbr sylan9bb y cv x cv aleph fveq2 ~< z cv breq1d A rcla4v aleph x cv ` sdomirr card z cv ` aleph x cv ` aleph x cv ` ~< breq2 z visset z cv cardid z cv V card z cv ` aleph x cv ` sdomen2 mp2an syl5bbr mtbiri nsyli com12 A On e. x cv On e. /\ adantl sylbird exp31 r19.23adv z cv cardidm card z cv ` x cardalephex mpbii syl5 omex z visset om V z cv V carddom mp2an cardom card z cv ` sseq1i bitr3 syl5ib {/} A e. adantr {/} A n0i A y aleph y cv ` ~< z cv br -. r19.2z z cv om aleph y cv ` domtr y cv alephgeom omex om V aleph y cv ` ssdomg ax-mp sylbi sylan2 z cv aleph y cv ` domnsym syl A y cv onelon sylan2 exp32 imp r19.21aiv syl5 syl y A aleph y cv ` ~< z cv br rexnal syl6ib com12 exp imp3a z cv ~< aleph A ` br a1d com3r jaod mpi A alephon z cv onel z cv aleph A ` cardsdomel A alephcard z cv eleq2i syl6rbb syl ibi syl5 x cv z cv aleph y cv ` ~< breq2 y A biraldv On elrab pm3.27bd con3i syl6 r19.21aiv sylanc))
+thm (alephval3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (aleph) A) (|^| ({|} x (/\/\ (= (` (card) (cv x)) (cv x)) (C_ (om) (cv x)) (A.e. y A (-. (= (cv x) (` (aleph) (cv y)))))))))) (x cv cardon card x cv ` x cv On eleq1 mpbii om x cv C_ y A x cv aleph y cv ` = -. A.e. 3ad2ant1 x ss2abi x On abid2 sseqtr x card x cv ` x cv = om x cv C_ y A x cv aleph y cv ` = -. A.e. /\/\ {|} aleph A ` z oneqmini ax-mp A alephcard A On e. a1i A alephgeom biimp A y cv alephord2i y cv alephon oneirr aleph A ` aleph y cv ` aleph y cv ` eleq2 mtbiri con2i syl6 r19.21aiv 3jca aleph A fvex x cv aleph A ` card fveq2 x cv aleph A ` = id eqeq12d x cv aleph A ` om sseq2 x cv aleph A ` aleph y cv ` eqeq1 negbid y A biraldv 3anbi123d elab sylibr z cv aleph y cv ` aleph A ` eleq1 y cv A alephord2 bicomd sylan9bbr biimpcd y cv On e. A On e. /\ z cv aleph y cv ` = pm3.27 z cv aleph A ` e. a1i jcad exp4c com3r imp4b y 19.22dv y On z cv aleph y cv ` = df-rex y A z cv aleph y cv ` = df-rex 3imtr4g z cv y cardalephex biimpac syl5 imp y A z cv aleph y cv ` = dfrex2 sylib A On e. z cv aleph A ` e. /\ card z cv ` z cv = om z cv C_ /\ y A z cv aleph y cv ` = -. A.e. nan mpbir exp z visset x cv z cv card fveq2 x cv z cv = id eqeq12d x cv z cv om sseq2 x cv z cv aleph y cv ` eqeq1 negbid y A biraldv 3anbi123d elab card z cv ` z cv = om z cv C_ y A z cv aleph y cv ` = -. A.e. df-3an bitr negbii syl6ibr r19.21aiv sylanc))
+thm (cflem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A B) (E. x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))))) (A ssid z cv ssid w cv z cv z cv sseq2 A rcla4ev mpan2 rgen y cv A A sseq1 y cv A w z cv w cv C_ rexeq z A biraldv anbi12d B cla4egv mp2ani x x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ 19.41v card y cv fvex x isseti mpbiran y biex y x x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ excom bitr3 sylib))
+thm (cfval ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u)) () (-> (e. A (On)) (= (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (A.e. z A (E.e. w (cv y) (C_ (cv z) (cv w))))))))))) (A On x y z w cflem x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. intexab sylib v cv A y cv sseq2 v cv A z w y cv z cv w cv C_ E.e. raleq anbi12d x cv card y cv ` = anbi2d y biexdv x biabdv inteqd v u x y z w df-cf V fvopab4g mpdan))
+thm (cffnon ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Fn (cf) (On)) (v visset v cv V x y z w cflem ax-mp x y x cv card y cv ` = y cv v cv C_ z v cv w y cv z cv w cv C_ E.e. A.e. /\ /\ E. intexab mpbi v u x y z w df-cf fnopab2))
+thm (cfub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (C_ (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (C_ A (U. (cv y))))))))) (A x y z w cfval y cv A w cv ssel A w cv onelon exp sylan9r w cv z cv onelsst syl6 imdistand ancomsd w 19.22dv z cv y cv w eluni w y cv z cv w cv C_ df-rex 3imtr4g z A r19.20sdv A y cv U. z dfss3 syl5ib exp imdistand x cv card y cv ` = anim2d y 19.22dv x 19.21aiv x y x cv card y cv ` = y cv A C_ A y cv U. C_ /\ /\ E. y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. ss2ab sylibr x y x cv card y cv ` = y cv A C_ A y cv U. C_ /\ /\ E. {|} x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} intss syl eqsstrd x y x cv card y cv ` = y cv A C_ A y cv U. C_ /\ /\ E. {|} |^| 0ss cffnon cf On fndm ax-mp A eleq2i negbii A cf ndmfv sylbir x y x cv card y cv ` = y cv A C_ A y cv U. C_ /\ /\ E. {|} |^| sseq1d mpbiri pm2.61i))
+thm (cflim ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\ (e. A B) (Lim A)) (= (` (cf) A) (|^| ({|} x (E. y (/\ (= (cv x) (` (card) (cv y))) (/\ (C_ (cv y) A) (= A (U. (cv y)))))))))) (A v cv limsuc biimpd z cv v cv suc w cv sseq1 w y cv birexdv A rcla4v v visset v cv V w cv sucssel ax-mp w y cv r19.22si v cv y cv w eluni2 sylibr syl6com syl9 r19.21adv A y cv U. v dfss3 syl6ibr y cv A C_ adantr A limuni y cv U. sseq2d y cv A uniss syl5bir imp jctird A y cv U. eqss syl6ibr exp imdistand x cv card y cv ` = anim2d y 19.22dv x 19.21aiv x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. y x cv card y cv ` = y cv A C_ A y cv U. = /\ /\ E. ss2ab sylibr x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} x y x cv card y cv ` = y cv A C_ A y cv U. = /\ /\ E. {|} intss syl A V e. adantl A V limelon A x y z w cfval syl sseqtr4d A x y cfub A y cv U. eqimss y cv A C_ anim2i x cv card y cv ` = anim2i y 19.22i x ss2abi x y x cv card y cv ` = y cv A C_ A y cv U. = /\ /\ E. {|} x y x cv card y cv ` = y cv A C_ A y cv U. C_ /\ /\ E. {|} intss ax-mp sstri jctil cf A ` x y x cv card y cv ` = y cv A C_ A y cv U. = /\ /\ E. {|} |^| eqss sylibr A B elisset sylan))
+thm (cf0 ((x y)) () (= (` (cf) ({/})) ({/})) ({/} x y cfub y cv U. 0ss y cv {/} C_ biantru y cv ss0b bitr3 x cv card y cv ` = anbi2i x cv card y cv ` = y cv {/} = ancom bitr y biex 0ex y cv {/} card fveq2 x cv eqeq2d ceqsexv card0 x cv eqeq2i 3bitr x biabi {/} x df-sn eqtr4 inteqi 0ex intsn eqtr sseqtr cf {/} ` ss0b mpbi))
+thm (cardcf ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` (card) (` (cf) A)) (` (cf) A)) (A x y z w cfval cf A fvex A x y z w cfval V eleq1d mpbii x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} intex sylibr v cv cardon v visset x cv v cv card y cv ` eqeq1 y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab v cv card y cv ` card fveq2 y cv cardidm syl6eq v cv card y cv ` card v cv ` eqeq2 mpbird y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ adantr y 19.23aiv sylbi On eleq1d mpbii ssriv x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} onint mpan syl eqeltrd v cv cf A ` card fveq2 v cv cf A ` = id eqeq12d v visset x cv v cv card y cv ` eqeq1 y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab v cv card y cv ` card fveq2 y cv cardidm syl6eq v cv card y cv ` card v cv ` eqeq2 mpbird y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ adantr y 19.23aiv sylbi vtoclga syl cffnon cf On fndm ax-mp A eleq2i negbii bicomi A cf ndmfv sylbi card0 cf A ` {/} card fveq2 cf A ` {/} = id eqeq12d mpbiri syl pm2.61i))
+thm (cflecard ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (C_ (` (cf) A) (` (card) A)) (A x y z w cfval A ssid z cv ssid w cv z cv z cv sseq2 A rcla4ev mpan2 rgen pm3.2i y cv A card fveq2 x cv eqeq2d y cv A A sseq1 y cv A w z cv w cv C_ rexeq z A biraldv anbi12d anbi12d On cla4egv mpan2i x 19.21aiv x x cv card A ` = y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. ss2ab sylibr card A ` x df-sn syl5ss card A ` {} x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} intss syl card A fvex intsn syl6ss eqsstrd card A ` 0ss cffnon cf On fndm ax-mp A eleq2i negbii A cf ndmfv sylbir card A ` sseq1d mpbiri pm2.61i))
+thm (cfle () () (C_ (` (cf) A) A) (A cardonle A cflecard cf A ` card A ` A sstr2 ax-mp syl A 0ss cffnon cf On fndm ax-mp A eleq2i negbii A cf ndmfv sylbir A sseq1d mpbiri pm2.61i))
+thm (cfeq0 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (<-> (= (` (cf) A) ({/})) (= A ({/})))) (A x y z w cfval {/} eqeq1d v cv cardon v visset x cv v cv card y cv ` eqeq1 y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab v cv card y cv ` card fveq2 y cv cardidm syl6eq v cv card y cv ` card v cv ` eqeq2 mpbird y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ adantr y 19.23aiv sylbi On eleq1d mpbii ssriv x y x cv card y cv ` = y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} onint0 ax-mp 0ex x cv {/} card y cv ` eqeq1 y cv A C_ z A w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab y cv {/} A sseq1 y cv {/} w z cv w cv C_ rexeq z A biraldv anbi12d biimpa {/} card y cv ` eqcom y visset y cv V cardeq0 ax-mp bitr sylanb w z cv w cv C_ rex0 z cv A e. a1i rgen A z w {/} z cv w cv C_ E.e. -. r19.2z mpi z A w {/} z cv w cv C_ E.e. rexnal sylib a3i {/} A C_ adantl syl y 19.23aiv sylbi sylbi syl6bi A {/} cf fveq2 cf0 syl6eq A On e. a1i impbid))
+thm (cfsuc ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (= (` (cf) (suc A)) (1o))) (A sucelon A suc x y z w cfval sylbi A snex y cv A {} card fveq2 1o eqeq2d y cv A {} A suc sseq1 y cv A {} w z cv w cv C_ rexeq z A suc biraldv anbi12d anbi12d cla4ev A On cardsn eqcomd A z cv onelsst z cv A eqimss A On e. a1i jaod z cv A elsuci syl5 A On snidg jctild w cv A z cv sseq2 A {} rcla4ev syl6 r19.21aiv A {} A ssun2 A df-suc sseqtr4 jctil sylanc 1ord elisseti x cv 1o card y cv ` eqeq1 y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab sylibr df1o2 v cv eleq2i v {/} elsn bitr {/} card y cv ` eqcom y visset y cv V cardeq0 ax-mp bitr w z cv w cv C_ rex0 z cv A suc e. a1i nrex A nsuceq0 A suc z w {/} z cv w cv C_ E.e. r19.2z ax-mp mto y cv {/} w z cv w cv C_ rexeq z A suc biraldv mtbiri sylbi y cv A suc C_ intnand {/} card y cv ` = y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ imnan mpbi y nex 0ex x cv {/} card y cv ` eqeq1 y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ anbi1d y biexdv elab mtbir v cv {/} x y x cv card y cv ` = y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} eleq1 mtbiri sylbi rgen jctir y cv cardon x cv card y cv ` On eleq1 mpbiri y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ adantr y 19.23aiv x ss2abi x On abid2 sseqtr x y x cv card y cv ` = y cv A suc C_ z A suc w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} 1o v oneqmini ax-mp syl eqtr4d))
+thm (cfom ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (` (cf) (om)) (om)) (om cflecard cardom sseqtr omex intsn x cv card y cv ` om eqtrt y visset z w unbnn2 y visset omex y cv V om V carden mp2an sylibr cardom syl6eq sylan2 y 19.23aiv x ss2abi om x df-sn sseqtr4 x y x cv card y cv ` = y cv om C_ z om w y cv z cv w cv C_ E.e. A.e. /\ /\ E. {|} om {} intss ax-mp omelon om x y z w cfval ax-mp sseqtr4 eqsstr3 eqssi))
+thm (cdavalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A C) (e. B D)) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. B ({} (1o)))))) (p0ex A V {/} {} V xpexg mpan2 1o snex B V 1o {} V xpexg mpan2 anim12i A {/} {} X. B 1o {} X. unexb sylib x cv A {/} {} xpeq1 y cv 1o {} X. uneq1d y cv B 1o {} xpeq1 A {/} {} X. uneq2d x y z df-cda x visset y visset pm3.2i z cv x cv {/} {} X. y cv 1o {} X. u. = biantrur x y z bioprabi eqtr V oprabval2g 3expa mpdan A C elisset B D elisset syl2an))
+thm (cdaval () ((cdaval.1 (e. A (V))) (cdaval.2 (e. B (V)))) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. B ({} (1o))))) (cdaval.1 cdaval.2 A V B V cdavalt mp2an))
+thm (uncdadom () ((cdaval.1 (e. A (V))) (cdaval.2 (e. B (V)))) (br (u. A B) (~<_) (opr A (+c) B)) (cdaval.1 cdaval.1 0ex xpsnen ensymi A A {/} {} X. endom ax-mp cdaval.2 cdaval.2 1ord elisseti xpsnen ensymi B B 1o {} X. endom ax-mp pm3.2i 0ne1oOLD {/} 1o A B xpsndisj ax-mp cdaval.1 p0ex xpex cdaval.2 cdaval.2 1o snex xpex A undom mp2an cdaval.1 cdaval.2 cdaval breqtrr))
+thm (cdaen () ((cdaen.1 (e. A (V))) (cdaen.2 (e. B (V))) (cdaen.3 (e. C (V))) (cdaen.4 (e. D (V)))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (opr A (+c) C) (~~) (opr B (+c) D))) (0ne1oOLD {/} 1o A C xpsndisj ax-mp 0ne1oOLD {/} 1o B D xpsndisj ax-mp pm3.2i A {/} {} X. B {/} {} X. C 1o {} X. D 1o {} X. unen mpan2 cdaen.1 cdaen.1 0ex xpsnen A V A {/} {} X. B {/} {} X. enen1 mp2an cdaen.2 cdaen.2 0ex xpsnen B V B {/} {} X. A enen2 mp2an bitr cdaen.3 cdaen.3 1ord elisseti xpsnen C V C 1o {} X. D 1o {} X. enen1 mp2an cdaen.4 cdaen.4 1ord elisseti xpsnen D V D 1o {} X. C enen2 mp2an bitr syl2anbr cdaen.1 cdaen.3 cdaval cdaen.2 cdaen.4 cdaval 3brtr4g))
+thm (cda0en () ((cda0en.1 (e. A (V)))) (br (opr A (+c) ({/})) (~~) A) (cda0en.1 0ex cdaval 1o {} xp0r A {/} {} X. uneq2i A {/} {} X. un0 3eqtr cda0en.1 0ex xpsnen eqbrtr))
+thm (cda1en () ((cda0en.1 (e. A (V)))) (br (opr A (+c) (1o)) (~~) (suc (` (card) A))) (cda0en.1 cda0en.1 0ex xpsnen A cardid entr4 1ord elisseti 1ord elisseti 1ord elisseti xpsnen card A fvex ensn1 entr4 pm3.2i 0ne1oOLD {/} 1o A 1o xpsndisj ax-mp A cardon onord card A ` orddisj ax-mp pm3.2i A {/} {} X. card A ` 1o 1o {} X. card A ` {} unen mp2an cda0en.1 1ord elisseti cdaval card A ` df-suc 3brtr4))
+thm (xp1en () ((cda0en.1 (e. A (V)))) (br (X. A (1o)) (~~) A) (df1o2 1o {/} {} A xpeq2 ax-mp cda0en.1 0ex xpsnen eqbrtr))
+thm (xp2cda () ((cda0en.1 (e. A (V)))) (= (X. A (2o)) (opr A (+c) A)) (A {/} {} 1o {} xpundi {/} {/} {} df-pr df2o2 df1o2 sneqi {/} {} uneq2i 3eqtr4 2o {/} {} 1o {} u. A xpeq2 ax-mp cda0en.1 cda0en.1 cdaval 3eqtr4))
+thm (cdacomen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V)))) (br (opr A (+c) B) (~~) (opr B (+c) A)) (0ne1oOLD {/} 1o A B xpsndisj ax-mp 0ne1oOLD {/} 1o eqcom mtbi 1o {/} A B xpsndisj ax-mp cdacomen.1 cdacomen.1 0ex xpsnen cdacomen.1 1ord elisseti xpsnen entr4 cdacomen.2 cdacomen.2 1ord elisseti xpsnen cdacomen.2 0ex xpsnen entr4 A {/} {} X. A 1o {} X. B 1o {} X. B {/} {} X. unen mpanl12 mp2an A 1o {} X. B {/} {} X. uncom breqtr cdacomen.1 cdacomen.2 cdaval cdacomen.2 cdacomen.1 cdaval 3brtr4))
+thm (cdaassen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (opr (opr A (+c) B) (+c) C) (~~) (opr A (+c) (opr B (+c) C))) (cdacomen.1 p0ex xpex cdacomen.2 1o snex xpex unex p0ex xpex A {/} {} X. B 1o {} X. {/} {} xpundir A {/} {} X. B 1o {} X. u. {/} {} X. V A {/} {} X. {/} {} X. B 1o {} X. {/} {} X. u. eqeng mp2 cdaassen.3 1o snex xpex cdaassen.3 1o snex xpex 1ord elisseti xpsnen ensymi pm3.2i 0ne1oOLD {/} 1o A {/} {} X. B 1o {} X. u. C xpsndisj ax-mp 0ne1oOLD {/} 1o A {/} {} X. C 1o {} X. xpsndisj ax-mp 0ne1oOLD {/} 1o B 1o {} X. C 1o {} X. xpsndisj ax-mp pm3.2i A {/} {} X. {/} {} X. C 1o {} X. 1o {} X. B 1o {} X. {/} {} X. undisj1 mpbi pm3.2i A {/} {} X. B 1o {} X. u. {/} {} X. A {/} {} X. {/} {} X. B 1o {} X. {/} {} X. u. C 1o {} X. C 1o {} X. 1o {} X. unen mp2an cdacomen.1 p0ex xpex p0ex xpex cdacomen.2 p0ex xpex 1o snex xpex cdaassen.3 1o snex xpex 1o snex xpex unex unex cdacomen.1 p0ex xpex p0ex xpex enref cdacomen.2 1o snex p0ex xpassen cdacomen.2 p0ex 1o snex xpex xpex cdacomen.2 enref 1o snex p0ex xpcomen cdacomen.2 cdacomen.2 1o snex p0ex xpex p0ex 1o snex xpex xpen mp2an cdacomen.2 p0ex 1o snex xpassen entr4 entr pm3.2i 0ne1oOLD {/} 1o A B xpsndisj ax-mp A {/} {} X. B 1o {} X. i^i {/} {/} {} xpeq1 ax-mp A {/} {} X. B 1o {} X. {/} {} xpindir {/} {} xp0r 3eqtr3 0ne1oOLD {/} 1o A {/} {} X. B {/} {} X. xpsndisj ax-mp pm3.2i A {/} {} X. {/} {} X. A {/} {} X. {/} {} X. B 1o {} X. {/} {} X. B {/} {} X. 1o {} X. unen mp2an cdaassen.3 1o snex xpex 1o snex xpex enref pm3.2i 0ne1oOLD {/} 1o A {/} {} X. C 1o {} X. xpsndisj ax-mp 0ne1oOLD {/} 1o B 1o {} X. C 1o {} X. xpsndisj ax-mp pm3.2i A {/} {} X. {/} {} X. C 1o {} X. 1o {} X. B 1o {} X. {/} {} X. undisj1 mpbi 0ne1oOLD {/} 1o A {/} {} X. C 1o {} X. xpsndisj ax-mp 0ne1oOLD {/} 1o B C xpsndisj ax-mp B {/} {} X. C 1o {} X. i^i {/} 1o {} xpeq1 ax-mp B {/} {} X. C 1o {} X. 1o {} xpindir 1o {} xp0r 3eqtr3 pm3.2i A {/} {} X. {/} {} X. C 1o {} X. 1o {} X. B {/} {} X. 1o {} X. undisj1 mpbi pm3.2i A {/} {} X. {/} {} X. B 1o {} X. {/} {} X. u. A {/} {} X. {/} {} X. B {/} {} X. 1o {} X. u. C 1o {} X. 1o {} X. C 1o {} X. 1o {} X. unen mp2an A {/} {} X. {/} {} X. B {/} {} X. 1o {} X. C 1o {} X. 1o {} X. unass breqtr cdacomen.1 p0ex xpex cdacomen.1 p0ex xpex 0ex xpsnen ensymi cdacomen.2 p0ex xpex cdaassen.3 1o snex xpex unex 1o snex xpex B {/} {} X. C 1o {} X. 1o {} xpundir B {/} {} X. C 1o {} X. u. 1o {} X. V B {/} {} X. 1o {} X. C 1o {} X. 1o {} X. u. eqeng mp2 pm3.2i 0ne1oOLD {/} 1o A B {/} {} X. C 1o {} X. u. xpsndisj ax-mp 0ne1oOLD {/} 1o A {/} {} X. B {/} {} X. xpsndisj ax-mp 0ne1oOLD {/} 1o A {/} {} X. C 1o {} X. xpsndisj ax-mp pm3.2i A {/} {} X. {/} {} X. B {/} {} X. 1o {} X. C 1o {} X. 1o {} X. undisj2 mpbi pm3.2i A {/} {} X. A {/} {} X. {/} {} X. B {/} {} X. C 1o {} X. u. 1o {} X. B {/} {} X. 1o {} X. C 1o {} X. 1o {} X. u. unen mp2an entr4 entr cdacomen.1 cdacomen.2 cdaval +c C opreq1i cdacomen.1 p0ex xpex cdacomen.2 1o snex xpex unex cdaassen.3 cdaval eqtr cdacomen.2 cdaassen.3 cdaval A +c opreq2i cdacomen.1 cdacomen.2 p0ex xpex cdaassen.3 1o snex xpex unex cdaval eqtr 3brtr4))
+thm (xpcdaen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (X. A (opr B (+c) C)) (~~) (opr (X. A B) (+c) (X. A C))) (cdacomen.1 cdacomen.2 p0ex xpex cdaassen.3 1o snex xpex unex xpex cdacomen.1 cdacomen.2 p0ex xpassen cdacomen.1 cdaassen.3 1o snex xpassen pm3.2i 0ne1oOLD {/} 1o A B X. A C X. xpsndisj ax-mp 0ne1oOLD {/} 1o B C xpsndisj ax-mp B {/} {} X. C 1o {} X. i^i {/} A xpeq2 ax-mp A B {/} {} X. C 1o {} X. xpindi A xp0 3eqtr3 pm3.2i A B X. {/} {} X. A B {/} {} X. X. A C X. 1o {} X. A C 1o {} X. X. unen mp2an A B {/} {} X. C 1o {} X. xpundi breqtrr ensymi cdacomen.2 cdaassen.3 cdaval B +c C opr B {/} {} X. C 1o {} X. u. A xpeq2 ax-mp cdacomen.1 cdacomen.2 xpex cdacomen.1 cdaassen.3 xpex cdaval 3brtr4))
+thm (mapcdaen () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (br (opr A (^m) (opr B (+c) C)) (~~) (X. (opr A (^m) B) (opr A (^m) C))) (cdacomen.2 cdaassen.3 cdaval A ^m opreq2i 0ne1oOLD {/} 1o B C xpsndisj ax-mp cdacomen.2 p0ex xpex cdaassen.3 1o snex xpex cdacomen.1 mapunen ax-mp cdacomen.1 enref cdacomen.2 0ex xpsnen cdacomen.1 cdacomen.1 cdacomen.2 p0ex xpex cdacomen.2 mapen mp2an cdacomen.1 enref cdaassen.3 1ord elisseti xpsnen cdacomen.1 cdacomen.1 cdaassen.3 1o snex xpex cdaassen.3 mapen mp2an A ^m B {/} {} X. oprex A ^m B oprex A ^m C 1o {} X. oprex A ^m C oprex xpen mp2an entr eqbrtr))
+thm (cdadom1 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr A (+c) C) (~<_) (opr B (+c) C))) (cdacomen.1 cdacomen.1 0ex xpsnen A V A {/} {} X. B {/} {} X. domen1 mp2an cdacomen.2 cdacomen.2 0ex xpsnen B V B {/} {} X. A domen2 mp2an bitr cdaassen.3 1o snex xpex C 1o {} X. V domrefg ax-mp 0ne1oOLD {/} 1o B C xpsndisj ax-mp cdacomen.2 p0ex xpex cdaassen.3 1o snex xpex cdaassen.3 1o snex xpex A {/} {} X. undom mpan2 mpan2 sylbir cdacomen.1 cdaassen.3 cdaval cdacomen.2 cdaassen.3 cdaval 3brtr4g))
+thm (cdadom2 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V))) (cdaassen.3 (e. C (V)))) (-> (br A (~<_) B) (br (opr C (+c) A) (~<_) (opr C (+c) B))) (cdacomen.1 cdacomen.2 cdaassen.3 cdadom1 cdaassen.3 cdacomen.1 cdacomen C +c A opr A +c C opr B +c C opr endomtr mpan cdacomen.2 cdaassen.3 cdacomen C +c A opr B +c C opr C +c B opr domentr mpan2 3syl))
+thm (cdadom3 () ((cdacomen.1 (e. A (V))) (cdacomen.2 (e. B (V)))) (br A (~<_) (opr A (+c) B)) (cdacomen.1 A B ssun1 A V A B u. ssdomg mp2 cdacomen.1 cdacomen.2 uncdadom A A B u. A +c B opr domtr mp2an))
+thm (cdafi () () (-> (/\ (br A (~<) (om)) (br B (~<) (om))) (br (opr A (+c) B) (~<) (om))) (A V B V cdavalt A om sdomex pm3.26d B om sdomex pm3.26d syl2an A {/} {} X. B 1o {} X. unfi2 A V A {/} {} X. om sdomen1 A om sdomex pm3.26d A om sdomex pm3.26d 0elon A V {/} On xpsneng mpan2 syl sylanc ibir B V B 1o {} X. om sdomen1 B om sdomex pm3.26d B om sdomex pm3.26d 1ord B V 1o On xpsneng mpan2 syl sylanc ibir syl2an eqbrtrd))
+thm (cdainf () ((cdainf.1 (e. A (V)))) (<-> (br (om) (~<_) A) (br (om) (~<_) (opr A (+c) A))) (cdainf.1 cdainf.1 cdadom3 om A A +c A opr domtr mpan2 A A cdafi anidms con3i omex A +c A oprex om V A +c A opr V domtri mp2an omex cdainf.1 om V A V domtri mp2an 3imtr4 impbi))
+thm (nd1 () () (-> (A. x (= (cv x) (cv y))) (-. (A. x (e. (cv y) (cv z))))) (z eirrv y y cv z cv e. z stdpc4 z eirrv y z cv z cv e. A. pm2.21i y z z elequ1 sbie sylib mto x y y cv z cv e. ax-10 mtoi))
+thm (nd2 () () (-> (A. x (= (cv x) (cv y))) (-. (A. x (e. (cv z) (cv y))))) (z eirrv y z cv y cv e. z stdpc4 z eirrv y z cv z cv e. A. pm2.21i y z z elequ2 sbie sylib mto x y z cv y cv e. ax-10 mtoi))
+thm (nd3 () () (-> (A. x (= (cv x) (cv y))) (-. (A. z (e. (cv x) (cv y))))) (x x cv y cv = ax-4 x eirrv x y x elequ2 mtbii z x cv y cv e. ax-4 con3i 3syl))
+thm (nd4 () () (-> (A. x (= (cv x) (cv y))) (-. (A. z (e. (cv y) (cv x))))) (y x z nd3 alequcoms))
+thm (nd5 ((x z)) () (-> (-. (A. y (= (cv y) (cv x)))) (-> (= (cv z) (cv y)) (A. x (= (cv z) (cv y))))) (x y z dveeq2 nalequcoms))
+thm (axextnd ((w x) (w y) (w z)) () (E. x (-> (<-> (e. (cv x) (cv y)) (e. (cv x) (cv z))) (= (cv y) (cv z)))) (x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y w dveel2 x x cv z cv = A. -. adantr x z w dveel2 x x cv y cv = A. -. adantl hbbid w x y elequ1 w x z elequ1 bibi12d x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i cbvald w y z zfext2 syl6bir y cv z cv = x 19.8a syl6 exp x z a9e x x cv y cv = hba1 x y z ax-8 x a4s 19.22d mpi x x cv y cv e. x cv z cv e. <-> A. a1d x y a9e x x cv z cv = hba1 x z y ax-8 z y equcomi syl6 x a4s 19.22d mpi x x cv y cv e. x cv z cv e. <-> A. a1d pm2.61ii 19.35ri))
+thm (axrepndlem1 ((x z) (w x) (w z) (x y) (w y) (ph w)) () (-> (-. (A. y (= (cv y) (cv z)))) (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph)))))))) (x y w w cv z ph [/] axrep y z x hbnae y z y hbnae y z z hbnae y z z hbnae ph w ax-17 z hbsb3 y y cv z cv = A. -. a1i y z w nd5 hbimd w z ph sbequ12r w z y equequ1 imbi12d y y cv z cv = A. -. a1i cbvald biexd y z z hbnae y z z hbnae w cv x cv e. z ax-17 y y cv z cv = A. -. a1i y z x hbnae z y x dveel2 nalequcoms ph w ax-17 z hbsb3 y hbal y y cv z cv = A. -. a1i hband hbexd hbbid w z x elequ1 y y cv z cv = A. -. adantl y z w dveeq2 imp y w cv z cv = hba1 w z ph sbequ12r y a4s biald syl x cv y cv e. anbi2d x biexdv bibi12d exp cbvald imbi12d biexd mpbii))
+thm (axrepndlem2 ((w x) (w y) (w z) (ph w)) () (-> (/\ (/\ (-. (A. x (= (cv x) (cv y)))) (-. (A. x (= (cv x) (cv z))))) (-. (A. y (= (cv y) (cv z))))) (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph)))))))) (x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban ph w ax-17 x hbsb3 x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i x z y ax-12 impcom hbimd hbald hbexd x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 x x cv y cv = A. -. adantl x x cv y cv = A. -. x x cv z cv = A. -. /\ w ax-17 x y w dveel2 x x cv z cv = A. -. adantr x y y hbnae x z y hbnae hban ph w ax-17 x hbsb3 x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i hbald hband hbexd hbbid hbald hbimd x y w nd5 x x cv z cv = A. -. adantr imdistani x y y hbnae x z y hbnae hban y w cv x cv = hba1 hban x z w nd5 imdistani x z z hbnae z w cv x cv = hba1 hban w x ph sbequ12r z cv y cv = imbi1d z a4s x x cv z cv = A. -. adantl biald syl x x cv y cv = A. -. x x cv z cv = A. -. pm3.27 y w cv x cv = ax-4 syl2an biexd syl x z w nd5 x x cv y cv = A. -. adantl imdistani x y z hbnae x z z hbnae hban z w cv x cv = hba1 hban w x z elequ2 x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl x y x hbnae x z x hbnae hban x y w dveel2 x x cv z cv = A. -. adantr x y y hbnae x z y hbnae hban ph w ax-17 x hbsb3 x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i hbald hband w x y elequ1 x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl y x w ph sbal2 nalequcoms bicomd w x y ph A. sbequ12r sylan9bb x x cv z cv = A. -. adantlr anbi12d exp cbvexd w cv x cv = adantr bibi12d z w cv x cv = ax-4 sylan2 biald syl imbi12d exp cbvexd y z w w cv x ph [/] axrepndlem1 syl5bi imp))
+thm (axrepnd () () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (A. y (e. (cv z) (cv x))) (E. x (/\ (A. z (e. (cv x) (cv y))) (A. y ph))))))) (x y z ph axrepndlem2 x y x hbnae x z x hbnae hban y z x hbnae hban x y z hbnae x z z hbnae hban y z z hbnae hban y z x ax15 com12 nalequcoms imp x x cv z cv = A. -. adantlr y z cv x cv e. ax-4 x x cv y cv = A. -. x x cv z cv = A. -. /\ y y cv z cv = A. -. /\ a1i impbid x y x hbnae x z x hbnae hban y z x hbnae hban z x y ax15 imp z x alequcom con3i z y alequcom con3i syl2an x x cv y cv = A. -. adantll z x cv y cv e. ax-4 x x cv y cv = A. -. x x cv z cv = A. -. /\ y y cv z cv = A. -. /\ a1i impbid y ph A. anbi1d biexd bibi12d biald y z ph z cv y cv = -> A. E. imbi2d biexd mpbid exp31 x y z hbae y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. pm5.21 y x z nd2 alequcoms x y x hbae x y z nd3 y ph A. intnanrd nexd sylanc 19.21ai y z ph z cv y cv = -> A. E. a1d y z ph z cv y cv = -> A. E. z y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. <-> A. -> x 19.8a syl x z z hbae y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. pm5.21 x z y nd4 x z x hbae z x y nd1 alequcoms y ph A. intnanrd nexd sylanc 19.21ai y z ph z cv y cv = -> A. E. a1d y z ph z cv y cv = -> A. E. z y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. <-> A. -> x 19.8a syl y z z hbae y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. pm5.21 y z x nd1 y z x hbae z y x nd2 alequcoms y ph A. intnanrd nexd sylanc 19.21ai y z ph z cv y cv = -> A. E. a1d y z ph z cv y cv = -> A. E. z y z cv x cv e. A. x z x cv y cv e. A. y ph A. /\ E. <-> A. -> x 19.8a syl pm2.61iii))
+thm (axunndlem1 ((x y) (w x) (w y) (x z) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (y z x hbae y cv x cv en2lp y z x elequ2 y cv x cv e. anbi2d mtbii y a4s nexd y cv x cv e. pm2.21d a5i y x y cv x cv e. x cv z cv e. /\ E. y cv x cv e. -> A. x 19.8a syl x w z axun y z x hbnae y z y hbnae y z y hbnae y z x hbnae w cv x cv e. y ax-17 y y cv z cv = A. -. a1i y z x dveel2 hband hbexd w cv x cv e. y ax-17 y y cv z cv = A. -. a1i hbimd w y x elequ1 x cv z cv e. anbi1d x biexdv w y x elequ1 imbi12d y y cv z cv = A. -. a1i cbvald biexd mpbii pm2.61i))
+thm (axunnd ((w x) (w y) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))) (w y z axunndlem1 x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x x cv y cv = A. -. x x cv z cv = A. -. /\ w ax-17 x y w dveel1 x x cv z cv = A. -. adantr x z w dveel2 x x cv y cv = A. -. adantl hband hbexd x y w dveel1 x x cv z cv = A. -. adantr hbimd hbald x y w nd5 x x cv z cv = A. -. adantr imdistani x y y hbnae x z y hbnae hban y w cv x cv = hba1 hban x y x hbnae x z x hbnae hban x y w dveel1 x x cv z cv = A. -. adantr x z w dveel2 x x cv y cv = A. -. adantl hband w x y elequ2 w x z elequ1 anbi12d x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i cbvexd y w cv x cv = A. adantr w x y elequ2 y a4s x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl imbi12d biald syl exp cbvexd mpbii exp x y y hbae x y x hbae y eirrv x y y elequ2 y cv x cv e. x cv z cv e. pm3.26 syl5bi mtoi x a4s nexd y cv x cv e. pm2.21d 19.21ai y x y cv x cv e. x cv z cv e. /\ E. y cv x cv e. -> A. x 19.8a syl x z y hbae x z x hbae z eirrv x z z elequ1 y cv x cv e. x cv z cv e. pm3.27 syl5bi mtoi x a4s nexd y cv x cv e. pm2.21d 19.21ai y x y cv x cv e. x cv z cv e. /\ E. y cv x cv e. -> A. x 19.8a syl pm2.61ii))
+thm (axpowndlem1 () () (-> (A. x (= (cv x) (cv y))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))) (x cv y cv = x y x z x cv y cv e. E. y x cv z cv e. A. -> A. y cv x cv e. -> A. E. pm2.24 x a4s))
+thm (axpowndlem2 ((w x) (y z) (w z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (-. (A. x (= (cv x) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))))) (w y z axpow w cv y cv e. z 19.8a y w cv z cv e. ax-4 imim12i w 19.20i y cv w cv e. imim1i y 19.20i w 19.22i ax-mp w cv y cv = -. a1i w ax-gen x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y x hbnae x y w dveeq2 hbnd x x cv z cv = A. -. adantr x x cv y cv = A. -. x x cv z cv = A. -. /\ w ax-17 x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x y w hbnae x z w hbnae hban x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 x x cv z cv = A. -. adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 x x cv y cv = A. -. adantl hbald hbimd hbald x y w dveel1 x x cv z cv = A. -. adantr hbimd hbald hbexd hbimd w x y equequ1 negbid x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl x y x hbnae x z x hbnae hban x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x x cv y cv = A. -. x x cv z cv = A. -. /\ w ax-17 x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 x x cv z cv = A. -. adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 x x cv y cv = A. -. adantl hbald hbimd hbald x y w dveel1 x x cv z cv = A. -. adantr hbimd hbald x y w nd5 x x cv z cv = A. -. adantr imdistani x y y hbnae x z y hbnae hban y w cv x cv = hba1 hban x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y z hbnae x z z hbnae hban x y w dveel2 x x cv z cv = A. -. adantr hbexd x y y hbnae x z y hbnae hban x z w dveel2 x x cv y cv = A. -. adantl hbald hbimd x z w nd5 imdistani z w cv x cv = hba1 w x y elequ1 z a4s biexd x x cv z cv = A. -. adantl syl x x cv y cv = A. -. adantll x y w nd5 imdistani y w cv x cv = hba1 w x z elequ1 y a4s biald x x cv y cv = A. -. adantl syl x x cv z cv = A. -. adantlr imbi12d exp cbvald y w cv x cv = A. adantr w x y elequ2 y a4s x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl imbi12d biald syl exp cbvexd w cv x cv = adantr imbi12d exp cbvald mpbii 19.21bi exp))
+thm (axpowndlem3 ((w x) (y z) (w y) (w z)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))) (x y z axpowndlem2 x y z axpowndlem1 x z x hbae x z y hbae x z y nd3 y x cv z cv e. A. z x cv y cv e. E. mtt syl z x x cv y cv e. -. ax-10 alequcoms z x cv y cv e. alnex x x cv y cv e. alnex 3imtr3g sylbird x a4sd y cv x cv e. imim1d 19.20d 19.22d p0ex x cv {/} {} w cv eleq2 w cv {/} = imbi2d w bialdv cla4ev 0ex snid w cv {/} {/} {} eleq1 mpbiri mpg w cv x n0 bicon1i w cv x cv e. imbi1i w bial x biex mpbir x y x hbnae x y y hbnae x y y hbnae x y y hbnae x y x hbnae y x w dveel1 nalequcoms hbexd hbnd y x w dveel2 nalequcoms hbimd x y w dveeq2 imdistani x w cv y cv = hba1 w y x elequ2 x a4s biexd x x cv y cv = A. -. adantl syl negbid w y x elequ1 x x cv y cv = A. -. adantl imbi12d exp cbvald biexd mpbii syl5 imp x cv y cv = -. a1d exp x y z axpowndlem1 pm2.61d2 pm2.61ii))
+thm (axpowndlem4 ((w x) (w y) (w z)) () (-> (-. (A. y (= (cv y) (cv x)))) (-> (-. (A. y (= (cv y) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))))) (x w z axpowndlem3 w ax-gen y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x y hbnae y x w dveeq1 hbnd y y cv z cv = A. -. adantr y x x hbnae y z x hbnae hban y y cv x cv = A. -. y y cv z cv = A. -. /\ w ax-17 y x y hbnae y z y hbnae hban y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x z hbnae y x w dveel1 hbexd y y cv z cv = A. -. adantr y y cv x cv = A. -. y y cv z cv = A. -. /\ w ax-17 y x z ax15 imp hbald hbimd hbald y x w dveel2 y y cv z cv = A. -. adantr hbimd hbald hbexd hbimd w y x equequ2 negbid y y cv x cv = A. -. y y cv z cv = A. -. /\ adantl y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x x hbnae y z x hbnae hban y x y hbnae y z y hbnae hban y x z hbnae y x w dveel1 hbexd y y cv z cv = A. -. adantr y y cv x cv = A. -. y y cv z cv = A. -. /\ w ax-17 y x z ax15 imp hbald hbimd hbald y x w dveel2 y y cv z cv = A. -. adantr hbimd y x w nd5 y y cv z cv = A. -. adantr imdistani y x x hbnae y z x hbnae hban x w cv y cv = hba1 hban y z w nd5 imdistani z w cv y cv = hba1 w y x elequ2 z a4s biexd y y cv z cv = A. -. adantl syl x w cv y cv = ax-4 sylan2 y y cv x cv = A. -. adantll y x y hbnae y z y hbnae hban y x z ax15 imp w cv y cv = x cv z cv e. pm4.2i y y cv x cv = A. -. y y cv z cv = A. -. /\ a1i cbvald x w cv y cv = A. adantr imbi12d biald syl w y x elequ1 y y cv x cv = A. -. y y cv z cv = A. -. /\ adantl imbi12d exp cbvald biexd w cv y cv = adantr imbi12d exp cbvald mpbii 19.21bi exp))
+thm (axpownd ((w x) (w y)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))) (y x z axpowndlem4 x y z axpowndlem1 alequcoms x y z axpowndlem1 y y cv z cv = A. a1d x y y hbnae y z y hbae hban w x el x y y hbnae y x w dveel1 nalequcoms w y x elequ2 x x cv y cv = A. -. a1i cbvexd mpbii y x cv y cv e. E. x 19.8a syl x y x cv y cv e. E. df-ex sylib y y cv z cv = A. adantr y z x hbae y y cv z cv = A. x cv y cv e. -. pm4.2i dral1 y x cv y cv e. alnex z x cv y cv e. alnex 3bitr3g y z x nd2 y x cv z cv e. A. z x cv y cv e. E. mtt syl bitrd biald x x cv y cv = A. -. adantl mtbid y cv x cv e. pm2.21d 19.21ai y x z x cv y cv e. E. y x cv z cv e. A. -> A. y cv x cv e. -> A. x 19.8a syl x cv y cv = -. a1d exp pm2.61i pm2.61ii))
+thm (axregndlem1 () () (-> (A. x (= (cv x) (cv z))) (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))))) (x z x hbae x z z hbae x eirrv x z x elequ1 mtbii x a4s z cv y cv e. -. pm2.21d 19.21ai x cv y cv e. anim2i expcom 19.22d x cv y cv e. x 19.8a syl5))
+thm (axregndlem2 ((w x) (y z) (w z) (w y)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (w y z axreg w ax-gen x y x hbnae x z x hbnae hban x y x hbnae x z x hbnae hban x y w dveel2 x x cv z cv = A. -. adantr x x cv y cv = A. -. x x cv z cv = A. -. /\ w ax-17 x y w dveel2 x x cv z cv = A. -. adantr x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 x x cv y cv = A. -. adantl x y x hbnae x z x hbnae hban x z y ax15 impcom hbnd hbimd hbald hband hbexd hbimd w x y elequ1 x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl x y x hbnae x z x hbnae hban x y w dveel2 x x cv z cv = A. -. adantr x y z hbnae x z z hbnae hban x y x hbnae x z x hbnae hban x z w dveel1 x x cv y cv = A. -. adantl x y x hbnae x z x hbnae hban x z y ax15 impcom hbnd hbimd hbald hband w x y elequ1 x x cv z cv = A. -. adantl x z w nd5 imdistani x z z hbnae z w cv x cv = hba1 hban w x z elequ2 z cv y cv e. -. imbi1d z a4s x x cv z cv = A. -. adantl biald syl anbi12d exp x x cv y cv = A. -. adantl cbvexd w cv x cv = adantr imbi12d exp cbvald mpbii 19.21bi exp x eirrv x y x elequ2 mtbii x a4s x x cv y cv e. z z cv x cv e. z cv y cv e. -. -> A. /\ E. pm2.21d x z y axregndlem1 pm2.61ii))
+thm (axregnd ((w x) (w y) (w z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))) (z x x hbnae z y x hbnae hban z x z hbnae z y z hbnae hban z x z hbnae z y z hbnae hban z x w dveel2 z z cv y cv = A. -. adantr z x z hbnae z y z hbnae hban z y w dveel2 z z cv x cv = A. -. adantl hbnd hbimd w z x elequ1 w z y elequ1 negbid imbi12d z z cv x cv = A. -. z z cv y cv = A. -. /\ a1i cbvald x cv y cv e. anbi2d biexd x y w axregndlem2 syl5bi exp x z y axregndlem1 alequcoms z y x hbae z eirrv z y z elequ2 mtbii z a4s z cv x cv e. a1d a5i x cv y cv e. anim2i expcom 19.22d x cv y cv e. x 19.8a syl5 pm2.61ii))
+thm (axinfndlem1 ((w x) (y z) (w z) (w y)) () (-> (A. x (e. (cv y) (cv z))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (w y z axinf w y cv z cv e. A. a1i x y x hbnae x z x hbnae hban x y z ax15 imp w cv x cv = y cv z cv e. pm4.2i x x cv y cv = A. -. x x cv z cv = A. -. /\ a1i cbvald x y x hbnae x z x hbnae hban x y w dveel1 x x cv z cv = A. -. adantr x y y hbnae x z y hbnae hban x y x hbnae x z x hbnae hban x y w dveel1 x x cv z cv = A. -. adantr x y z hbnae x z z hbnae hban x y z ax15 imp x z w dveel1 x x cv y cv = A. -. adantl hband hbexd hbimd hbald hband w x y elequ2 x x cv y cv = A. -. x x cv z cv = A. -. /\ adantl x y w nd5 x x cv z cv = A. -. adantr imdistani x z y hbnae y w cv x cv = hba1 hban w x y elequ2 x x cv z cv = A. -. adantl x z w nd5 imdistani z w cv x cv = hba1 w x z elequ2 y cv z cv e. anbi2d z a4s biexd x x cv z cv = A. -. adantl syl imbi12d y w cv x cv = ax-4 sylan2 biald x x cv y cv = A. -. adantll syl anbi12d exp cbvexd imbi12d mpbii exp x y z nd1 x y cv x cv e. y y cv x cv e. z y cv z cv e. z cv x cv e. /\ E. -> A. /\ E. pm2.21d x z y nd2 x y cv x cv e. y y cv x cv e. z y cv z cv e. z cv x cv e. /\ E. -> A. /\ E. pm2.21d pm2.61ii))
+thm (axinfnd ((w x) (w y) (w z)) () (E. x (-> (e. (cv y) (cv z)) (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (x w z axinfndlem1 w ax-gen y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y z x hbnae y z w dveel2 hbald y y cv x cv = A. -. adantl y x x hbnae y z x hbnae hban y x w dveel2 y y cv z cv = A. -. adantr y y cv x cv = A. -. y y cv z cv = A. -. /\ w ax-17 y x y hbnae y z y hbnae hban y x w dveel2 y y cv z cv = A. -. adantr y x z hbnae y z z hbnae hban y z w dveel2 y y cv x cv = A. -. adantl y z x ax15 impcom hband hbexd hbimd hbald hband hbexd hbimd y x w nd5 y y cv z cv = A. -. adantr imdistani x w cv y cv = hba1 w y z elequ1 x a4s biald y y cv x cv = A. -. y y cv z cv = A. -. /\ adantl y x x hbnae y z x hbnae hban x w cv y cv = hba1 hban w y x elequ1 x a4s y x y hbnae y z y hbnae hban y x y hbnae y z y hbnae hban y x w dveel2 y y cv z cv = A. -. adantr y x z hbnae y z z hbnae hban y z w dveel2 y y cv x cv = A. -. adantl y z x ax15 impcom hband hbexd hbimd w y x elequ1 y y cv z cv = A. -. adantl y z w nd5 imdistani z w cv y cv = hba1 w y z elequ1 z cv x cv e. anbi1d z a4s biexd y y cv z cv = A. -. adantl syl imbi12d y y cv x cv = A. -. adantll exp cbvald bi2anan9r biexd imbi12d syl exp cbvald mpbii 19.21bi exp x y z nd1 alequcoms x y cv x cv e. y y cv x cv e. z y cv z cv e. z cv x cv e. /\ E. -> A. /\ E. pm2.21d y z x nd3 x y cv x cv e. y y cv x cv e. z y cv z cv e. z cv x cv e. /\ E. -> A. /\ E. pm2.21d pm2.61ii 19.35ri))
+thm (axacndlem1 () () (-> (A. x (= (cv x) (cv y))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (x y y hbae x y z hbae x y z nd1 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d y cv z cv e. z cv w cv e. pm3.26 x 19.20i syl5 19.21ai 19.21ai y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl))
+thm (axacndlem2 () () (-> (A. x (= (cv x) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (x z y hbae x z z hbae x z w nd1 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d y cv z cv e. z cv w cv e. pm3.27 x 19.20i syl5 19.21ai 19.21ai y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl))
+thm (axacndlem3 () () (-> (A. y (= (cv y) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) (y z z hbae y z x nd3 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d y cv z cv e. z cv w cv e. pm3.26 x 19.20i syl5 19.21ai a5i y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl))
+thm (axacndlem4 ((v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (v y z w axac v y cv z cv e. z cv w cv e. /\ ax-4 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv v cv e. /\ /\ E. y cv w cv = <-> A. E. imim1i y z 19.20i2 v 19.22i ax-mp x z x hbnae x y x hbnae x w x hbnae hban hban x z y hbnae x y y hbnae x w y hbnae hban hban x z z hbnae x y z hbnae x w z hbnae hban hban x z x hbnae x y x hbnae x w x hbnae hban hban x x cv z cv = A. -. x x cv y cv = A. -. x x cv w cv = A. -. /\ /\ v ax-17 x y z ax15 impcom x x cv w cv = A. -. adantrr x z w ax15 imp x x cv y cv = A. -. adantrl hband hbald x z w hbnae x y w hbnae x w w hbnae hban hban x z y hbnae x y y hbnae x w y hbnae hban hban x z x hbnae x y x hbnae x w x hbnae hban hban x z w hbnae x y w hbnae x w w hbnae hban hban x y z ax15 impcom x x cv w cv = A. -. adantrr x z w ax15 imp x x cv y cv = A. -. adantrl hband x y w ax15 imp x w v dveel1 x x cv y cv = A. -. adantl hband x x cv z cv = A. -. adantl hband hbexd x y w ax-12 imp x x cv z cv = A. -. adantl hbbid hbald hbexd hbimd hbald hbald x z v nd5 x y z hbnae x y v nd5 hbald sylan9 x x cv w cv = A. -. adantrr x z y hbnae x y y hbnae x w y hbnae hban hban y z v cv x cv = A. hba1 hban x z z hbnae x y z hbnae x w z hbnae hban hban z v cv x cv = hba1 y hbal hban x z x hbnae x y x hbnae x w x hbnae hban hban x y z ax15 impcom x x cv w cv = A. -. adantrr x z w ax15 imp x x cv y cv = A. -. adantrl hband v cv x cv = y cv z cv e. z cv w cv e. /\ pm4.2i x x cv z cv = A. -. x x cv y cv = A. -. x x cv w cv = A. -. /\ /\ a1i cbvald y z v cv x cv = A. A. adantr x w y hbnae x w v nd5 19.20d z v cv x cv = ax-4 y 19.20i syl5 x x cv y cv = A. -. adantl imdistani w v cv x cv = hba1 y hbal y w v cv x cv = A. hba1 w v cv x cv = hba1 v x w elequ2 y cv w cv e. anbi2d y cv z cv e. z cv w cv e. /\ anbi2d w a4s biexd y cv w cv = bibi1d y a4s biald biexd x x cv y cv = A. -. x x cv w cv = A. -. /\ adantl syl x x cv z cv = A. -. adantll imbi12d biald biald exp syld cbvexd mpbii exp32 x z y w axacndlem2 x y z w axacndlem1 x w y hbae x w z hbae x w z nd2 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d y cv z cv e. z cv w cv e. pm3.27 x 19.20i syl5 19.21ai 19.21ai y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl pm2.61iii))
+thm (axacndlem5 ((v x) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x v z w axacndlem4 y z x hbnae y x x hbnae y w x hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z z hbnae y x z hbnae y w z hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z x hbnae y x x hbnae y w x hbnae hban hban y z v dveel2 y y cv x cv = A. -. y y cv w cv = A. -. /\ adantr y z w ax15 imp y y cv x cv = A. -. adantrl hband hbald y z w hbnae y x w hbnae y w w hbnae hban hban y y cv z cv = A. -. y y cv x cv = A. -. y y cv w cv = A. -. /\ /\ v ax-17 y z y hbnae y x y hbnae y w y hbnae hban hban y z w hbnae y x w hbnae y w w hbnae hban hban y z v dveel2 y y cv x cv = A. -. y y cv w cv = A. -. /\ adantr y z w ax15 imp y y cv x cv = A. -. adantrl hband y w v dveel2 y y cv z cv = A. -. y y cv x cv = A. -. ad2antrr y w x ax15 impcom y y cv z cv = A. -. adantl hband hband hbexd y w v dveeq2 y y cv z cv = A. -. y y cv x cv = A. -. ad2antrr hbbid hbald hbexd hbimd hbald y z v nd5 y y cv x cv = A. -. y y cv w cv = A. -. /\ adantr imdistani y z z hbnae y x z hbnae y w z hbnae hban hban z v cv y cv = hba1 hban y x v nd5 imdistani x v cv y cv = hba1 v y z elequ1 z cv w cv e. anbi1d x a4s biald y y cv x cv = A. -. adantl syl z v cv y cv = ax-4 sylan2 y y cv w cv = A. -. adantlr y y cv z cv = A. -. adantll y z w hbnae y x w hbnae y w w hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z y hbnae y x y hbnae y w y hbnae hban hban y z w hbnae y x w hbnae y w w hbnae hban hban y z v dveel2 y y cv x cv = A. -. y y cv w cv = A. -. /\ adantr y z w ax15 imp y y cv x cv = A. -. adantrl hband y w v dveel2 y y cv z cv = A. -. y y cv x cv = A. -. ad2antrr y w x ax15 impcom y y cv z cv = A. -. adantl hband hband hbexd y w v dveeq2 y y cv z cv = A. -. y y cv x cv = A. -. ad2antrr hbbid y w v nd5 imdistani w v cv y cv = hba1 v y z elequ1 z cv w cv e. anbi1d v y w elequ1 w cv x cv e. anbi1d anbi12d w a4s biexd y y cv w cv = A. -. adantl syl y y cv x cv = A. -. adantll y y cv z cv = A. -. adantll v y w equequ1 y y cv z cv = A. -. y y cv x cv = A. -. y y cv w cv = A. -. /\ /\ adantl bibi12d exp cbvald biexd z v cv y cv = A. adantr imbi12d biald syl exp cbvald biexd mpbii exp32 y z x w axacndlem3 x y z w axacndlem1 alequcoms y w z hbae y cv z cv en2lp y w z elequ2 y cv z cv e. anbi2d mtbii y a4s w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d x a4sd 19.21ai a5i y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl pm2.61iii))
+thm (axacnd ((v x) (v y) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))) (x y v w axacndlem5 z x x hbnae z y x hbnae z w x hbnae hban hban z x y hbnae z y y hbnae z w y hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x x hbnae z y x hbnae z w x hbnae hban hban z y v dveel1 z z cv x cv = A. -. z z cv w cv = A. -. ad2antrl z w v dveel2 z z cv x cv = A. -. z z cv y cv = A. -. ad2antrr hband hbald z x w hbnae z y w hbnae z w w hbnae hban hban z x y hbnae z y y hbnae z w y hbnae hban hban z x z hbnae z y z hbnae z w z hbnae hban hban z x w hbnae z y w hbnae z w w hbnae hban hban z y v dveel1 z z cv x cv = A. -. z z cv w cv = A. -. ad2antrl z w v dveel2 z z cv x cv = A. -. z z cv y cv = A. -. ad2antrr hband z y w ax15 imp z z cv x cv = A. -. adantl z w x ax15 impcom z z cv y cv = A. -. adantrl hband hband hbexd z y w ax-12 imp z z cv x cv = A. -. adantl hbbid hbald hbexd hbimd z x v nd5 imdistani x v cv z cv = hba1 v z y elequ2 v z w elequ1 anbi12d x a4s biald z z cv x cv = A. -. adantl syl z z cv y cv = A. -. z z cv w cv = A. -. /\ adantlr z y v nd5 z w y hbnae z w v nd5 hbald sylan9 imdistani w y v cv z cv = A. hba1 y v cv z cv = hba1 w hbal w y v cv z cv = A. hba1 v z y elequ2 v z w elequ1 anbi12d y a4s w a4s y cv w cv e. w cv x cv e. /\ anbi1d biexd y cv w cv = bibi1d biald biexd z z cv y cv = A. -. z z cv w cv = A. -. /\ adantl syl z z cv x cv = A. -. adantll imbi12d exp cbvald biald biexd mpbii exp32 x z y w axacndlem2 alequcoms y z x w axacndlem3 alequcoms z w y hbae z w x nd3 w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. pm2.21d y cv z cv e. z cv w cv e. pm3.27 x 19.20i syl5 a5i 19.21ai y z x y cv z cv e. z cv w cv e. /\ A. w y w y cv z cv e. z cv w cv e. /\ y cv w cv e. w cv x cv e. /\ /\ E. y cv w cv = <-> A. E. -> A. A. x 19.8a syl pm2.61iii))
+thm (zfcndext ((x y) (x z) (y z)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))) (z x y axextnd 19.35i x cv y cv = z 19.9rv sylibr))
+thm (zfcndrep ((x y) (x z) (w x) (y z) (w y) (w z)) () (-> (A. w (E. y (A. z (-> (A. y ph) (= (cv z) (cv y)))))) (E. y (A. z (<-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv w) (cv x)) (A. y ph))))))) (y z y ph A. z cv y cv = -> A. hbe1 z cv w cv e. y ax-17 w cv x cv e. y ax-17 y y ph A. hba1 hban w hbex hbbi z hbal hbim w hbex x visset y x w elequ2 y y ph A. A. anbi1d w biexdv z cv w cv e. bibi2d z bialdv y z y ph A. z cv y cv = -> A. E. imbi2d w biexdv w y z y ph A. axrepnd z cv w cv e. y ax-17 19.3r w cv y cv e. z ax-17 19.3r y y ph A. A. anbi1i w biex bibi12i z bial y z y ph A. z cv y cv = -> A. E. imbi2i w biex mpbir vtoclf 19.35i z cv y cv e. w ax-17 w w cv x cv e. y ph A. /\ hbe1 hbbi z hbal z cv w cv e. y ax-17 w cv x cv e. y ax-17 y y ph A. hba1 hban w hbex hbbi z hbal y w z elequ2 y ph hba1 19.3r w cv x cv e. anbi2i w biex y cv w cv = a1i bibi12d z bialdv cbvex sylibr))
+thm (zfcndun ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))) (y z x axunnd w y z elequ2 w y x elequ1 anbi12d cbvexv z cv y cv e. imbi1i z bial y biex mpbir))
+thm (zfcndpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))) (y z dtru y y cv z cv = exnal mpbir y z y x y cv z cv e. E. z y cv x cv e. A. -> A. z cv y cv e. -> A. hbe1 y z x axpownd 19.23ai ax-mp y cv z cv e. x 19.9rv y cv x cv e. z ax-17 19.3r imbi12i y bial z cv y cv e. imbi1i z bial y biex mpbir w y z elequ1 w y x elequ1 imbi12d cbvalv z cv y cv e. imbi1i z bial y biex mpbir))
+thm (zfcndreg ((x y) (x z) (y z)) () (-> (E. y (e. (cv y) (cv x))) (E. y (/\ (e. (cv y) (cv x)) (A. z (-> (e. (cv z) (cv y)) (-. (e. (cv z) (cv x)))))))) (y y cv x cv e. z z cv y cv e. z cv x cv e. -. -> A. /\ hbe1 y x z axregnd 19.23ai))
+thm (zfcndinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv y)))))))) (w x el x cv y cv e. w ax-17 x cv y cv e. w ax-17 w x cv w cv e. w cv y cv e. /\ hbe1 hbim x hbal hban y hbex x cv w cv e. y ax-17 y x w axinfnd 19.35i syl 19.23ai ax-mp z x y elequ1 z x w elequ1 w cv y cv e. anbi1d w biexdv imbi12d cbvalv x cv y cv e. anbi2i y biex mpbir))
+thm (zfcndac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v))))))))) (y z w x axacnd z cv w cv e. w cv x cv e. /\ y ax-17 19.3r x z x z cv w cv e. w cv x cv e. /\ z cv x cv e. x cv y cv e. /\ /\ E. z cv x cv = <-> A. E. imbi1i z w bi2al y biex mpbir v x u equequ2 t u cv w cv e. w cv t cv e. /\ u cv t cv e. t cv y cv e. /\ /\ E. bibi2d t x w elequ2 u cv w cv e. anbi2d t x u elequ2 t x y elequ1 anbi12d anbi12d cbvexv u cv x cv = bibi1i syl6bb u bialdv u z w elequ1 w cv x cv e. anbi1d u z x elequ1 x cv y cv e. anbi1d anbi12d x biexdv u z x equequ1 bibi12d cbvalv syl6bb cbvexv z cv w cv e. w cv x cv e. /\ imbi2i z w bi2al y biex mpbir))
+thm (elni () () (<-> (e. A (N.)) (/\ (e. A (om)) (-. (= A ({/}))))) (df-ni A eleq2i A om {/} {} eldif 0ex A elsnc2 negbii A om e. anbi2i 3bitr))
+thm (elni2 () () (<-> (e. A (N.)) (/\ (e. A (om)) (e. ({/}) A))) (A elni A nnord A ord0eln0 syl pm5.32i bitr4))
+thm (pinn () () (-> (e. A (N.)) (e. A (om))) (df-ni om {/} {} difss eqsstr A sseli))
+thm (pion () () (-> (e. A (N.)) (e. A (On))) (A pinn A nnont syl))
+thm (piord () () (-> (e. A (N.)) (Ord A)) (A pinn A nnord syl))
+thm (niex () () (e. (N.) (V)) (omex df-ni om {/} {} difss eqsstr ssexi))
+thm (0npi () () (-. (e. ({/}) (N.))) ({/} eqid {/} elni pm3.27bd mt2))
+thm (1pi () () (e. (1o) (N.)) (1o elni 1onn {/} nsuceq0 df-1o {/} eqeq1i mtbir mpbir2an))
+thm (addpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (+N) B) (opr A (+o) B))) (A N. B N. opelxpi A B <,> N. N. X. +o fvres A +N B df-opr df-pli A B <,> fveq1i eqtr A +o B df-opr 3eqtr4g syl))
+thm (mulpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (.N) B) (opr A (.o) B))) (A N. B N. opelxpi A B <,> N. N. X. .o fvres A .N B df-opr df-mi A B <,> fveq1i eqtr A .o B df-opr 3eqtr4g syl))
+thm (mulidpi () () (-> (e. A (N.)) (= (opr A (.N) (1o)) A)) (1pi A 1o mulpiord mpan2 A pinn A nnont A om1 3syl eqtrd))
+thm (ltpiord ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (N.)) (e. B (N.))) (<-> (br A (<N) B) (e. A B))) (x cv A <N y cv breq1 x cv A y cv eleq1 bibi12d y cv B A <N breq2 y cv B A eleq2 bibi12d y visset x cv N. N. opelxp x cv y cv <,> N. N. X. e. x cv y cv <,> E e. iba x cv E y cv df-br x y epel bitr3 syl5bbr x cv <N y cv df-br df-lti x cv y cv <,> eleq2i x cv y cv <,> E N. N. X. elin 3bitr syl6rbbr sylbir vtocl2ga))
+thm (ltsopi ((x y) (x z) (y z)) () (Or (<N) (N.)) (x cv y cv ordtri2 x cv piord y cv piord syl2an x cv y cv ltpiord y cv x cv ltpiord ancoms x cv y cv = orbi2d negbid 3bitr4d z cv N. e. 3adant3 x cv N. e. y cv N. e. z cv N. e. 3simp3 z cv pion z cv x cv y cv ontr1 3syl x cv y cv ltpiord z cv N. e. 3adant3 y cv z cv ltpiord x cv N. e. 3adant1 anbi12d x cv z cv ltpiord y cv N. e. 3adant2 3imtr4d jca so))
+thm (ltrelpi () () (C_ (<N) (X. (N.) (N.))) (df-lti E N. N. X. inss2 eqsstr))
+thm (dmaddpi () () (= (dom (+N)) (X. (N.) (N.))) (+o N. N. X. dmres fnoa +o On On X. fndm ax-mp N. N. X. ineq2i eqtr df-pli dmeqi df-ni om {/} {} difss eqsstr omsson sstri N. On C_ anidm mpbir N. On N. On ssxp ax-mp N. N. X. On On X. dfss mpbi 3eqtr4))
+thm (dmmulpi () () (= (dom (.N)) (X. (N.) (N.))) (.o N. N. X. dmres fnom .o On On X. fndm ax-mp N. N. X. ineq2i eqtr df-mi dmeqi df-ni om {/} {} difss eqsstr omsson sstri N. On C_ anidm mpbir N. On N. On ssxp ax-mp N. N. X. On On X. dfss mpbi 3eqtr4))
+thm (addclpi () () (-> (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (+N) B) (N.))) (A B addpiord A B nnacl B pinn sylan2 B A {/} nnaordi A +o {/} opr A +o B opr n0i syl6 expcom imp32 B elni2 sylan2b jca A +o B opr elni sylibr A pinn sylan eqeltrd))
+thm (mulclpi () () (-> (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (.N) B) (N.))) (A B mulpiord A B nnmcl A pinn B pinn syl2an peano1 {/} B A nnmordi mp3an1 imp an4s B elni2 A elni2 syl2anb ancoms A .o {/} opr A .o B opr n0i syl jca A .o B opr elni sylibr eqeltrd))
+thm (addcompi () ((addcompi.1 (e. A (V))) (addcompi.2 (e. B (V)))) (= (opr A (+N) B) (opr B (+N) A)) (A B nnacom A pinn B pinn syl2an A B addpiord B A addpiord ancoms 3eqtr4d addcompi.2 dmaddpi addcompi.1 ndmoprcom pm2.61i))
+thm (addasspi () ((addasspi.1 (e. B (V))) (addasspi.2 (e. C (V)))) (= (opr (opr A (+N) B) (+N) C) (opr A (+N) (opr B (+N) C))) (A B C nnaass A pinn B pinn C pinn syl3an A +N B opr C addpiord A B addclpi sylan A B addpiord +o C opreq1d C N. e. adantr eqtrd 3impa A B +N C opr addpiord B C addclpi sylan2 B C addpiord A +o opreq2d A N. e. adantl eqtrd 3impb 3eqtr4d addasspi.1 dmaddpi addasspi.2 0npi A ndmoprass pm2.61i))
+thm (mulcompi () ((mulcompi.1 (e. A (V))) (mulcompi.2 (e. B (V)))) (= (opr A (.N) B) (opr B (.N) A)) (A B nnmcom A pinn B pinn syl2an A B mulpiord B A mulpiord ancoms 3eqtr4d mulcompi.2 dmmulpi mulcompi.1 ndmoprcom pm2.61i))
+thm (mulasspi () ((mulasspi.1 (e. B (V))) (mulasspi.2 (e. C (V)))) (= (opr (opr A (.N) B) (.N) C) (opr A (.N) (opr B (.N) C))) (A B C nnmass A pinn B pinn C pinn syl3an A .N B opr C mulpiord A B mulclpi sylan A B mulpiord .o C opreq1d C N. e. adantr eqtrd 3impa A B .N C opr mulpiord B C mulclpi sylan2 B C mulpiord A .o opreq2d A N. e. adantl eqtrd 3impb 3eqtr4d mulasspi.1 dmmulpi mulasspi.2 0npi A ndmoprass pm2.61i))
+thm (distrpi () ((distrpi.1 (e. B (V))) (distrpi.2 (e. C (V)))) (= (opr A (.N) (opr B (+N) C)) (opr (opr A (.N) B) (+N) (opr A (.N) C))) (A B C nndi A pinn B pinn C pinn syl3an A B +N C opr mulpiord B C addclpi sylan2 B C addpiord A .o opreq2d A N. e. adantl eqtrd 3impb A .N B opr A .N C opr addpiord A B mulclpi A C mulclpi syl2an A B mulpiord A C mulpiord +o opreqan12d eqtrd 3impdi 3eqtr4d distrpi.1 dmaddpi distrpi.2 0npi dmmulpi A ndmoprdistr pm2.61i))
+thm (mulcanpi () ((mulcanpi.1 (e. C (V)))) (-> (/\ (e. A (N.)) (e. B (N.))) (-> (= (opr A (.N) B) (opr A (.N) C)) (= B C))) (A B mulpiord C N. e. adantr A C mulpiord B N. e. adantlr eqeq12d A B C nnmcan biimpd A elni2 pm3.27bd sylan2 exp A pinn B pinn C pinn syl3an 3exp com4r pm2.43i imp31 sylbid A .N B opr A .N C opr N. eleq1 A B mulclpi syl5bi imp mulcanpi.1 dmmulpi 0npi A ndmoprrcl A N. e. C N. e. pm3.27 3syl sylan2 exp32 imp4b pm2.43i exp))
+thm (addnidpi () ((addnidpi.1 (e. B (V)))) (-> (e. A (N.)) (-. (= (opr A (+N) B) A))) (B A {/} nnaordi A nna0 A +o B opr eleq1d A +o B opr A A eleq2 negbid A nnord A ordeirr syl syl5bir com12 con2d sylbid B om e. adantl syld expcom imp32 B elni2 sylan2b A pinn sylan A B addpiord A eqeq1d mtbird A N. e. a1d addnidpi.1 dmaddpi A ndmopr A eqeq1d 0npi {/} A N. eleq1 mtbii syl6bi con2d pm2.61i))
+thm (ltexpi ((A x) (B x)) () (-> (/\ (e. A (N.)) (e. B (N.))) (<-> (br A (<N) B) (E. x (/\ (e. (cv x) (N.)) (= (opr A (+N) (cv x)) B))))) (A B x nnaordex x om {/} x cv e. A +o x cv opr B = /\ df-rex syl6bb A pinn B pinn syl2an A B ltpiord A x cv addpiord B eqeq1d exp pm5.32d x cv elni2 A +o x cv opr B = anbi1i x cv om e. {/} x cv e. A +o x cv opr B = anass bitr syl6bb x biexdv B N. e. adantr 3bitr4d))
+thm (ltapi () ((ltapi.1 (e. A (V))) (ltapi.2 (e. B (V)))) (-> (e. C (N.)) (<-> (br A (<N) B) (br (opr C (+N) A) (<N) (opr C (+N) B)))) (ltapi.2 dmaddpi ltapi.1 ltrelpi 0npi A B C nnaord A pinn B pinn C pinn syl3an 3expa A B ltpiord C N. e. adantr C +N A opr C +N B opr ltpiord C A addclpi C B addclpi syl2an C A addpiord C N. e. B N. e. /\ adantr C B addpiord C N. e. A N. e. /\ adantl eleq12d bitrd anandis ancoms 3bitr4d 3impa ndmord))
+thm (ltmpi () ((ltmpi.1 (e. A (V))) (ltmpi.2 (e. B (V)))) (-> (e. C (N.)) (<-> (br A (<N) B) (br (opr C (.N) A) (<N) (opr C (.N) B)))) (ltmpi.2 dmmulpi ltmpi.1 ltrelpi 0npi {/} C e. A B e. iba A B C nnmord sylan9bbr 3exp1 imp4b C elni2 syl5ib A pinn B pinn syl2an imp A B ltpiord C N. e. adantr C .N A opr C .N B opr ltpiord C A mulclpi C B mulclpi syl2an C A mulpiord C N. e. B N. e. /\ adantr C B mulpiord C N. e. A N. e. /\ adantl eleq12d bitrd anandis ancoms 3bitr4d 3impa ndmord))
+thm (1lt2pi () () (br (1o) (<N) (opr (1o) (+N) (1o))) (1onn 1o nna0 ax-mp 0ex sucid df-1o eleqtrr peano1 1onn 1onn {/} 1o 1o nnaord mp3an mpbi eqeltrr 1pi 1pi 1o 1o addpiord mp2an eleqtrr 1pi 1pi 1pi 1o 1o addclpi mp2an 1o 1o +N 1o opr ltpiord mp2an mpbir))
+thm (nlt1pi () () (-. (br A (<N) (1o))) (A elni pm3.27bd A noel 1pi A 1o ltpiord mpan2 A N. {/} elsucg df-1o A eleq2i syl5bb bitrd biimpa ord mpi exp mtod 1pi elisseti ltrelpi A brel pm3.26d con3i pm2.61i))
+thm (indpi ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((indpi.1 (-> (= (cv x) (1o)) (<-> ph ps))) (indpi.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (indpi.3 (-> (= (cv x) (opr (cv y) (+N) (1o))) (<-> ph th))) (indpi.4 (-> (= (cv x) A) (<-> ph ta))) (indpi.5 ps) (indpi.6 (-> (e. (cv y) (N.)) (-> ch th)))) (-> (e. A (N.)) ta) (A nlt1pi 1pi ltsopi <N N. A 1o sotric mpan mpan2 mtbii A 1o = 1o <N A br \/ nega syl 1pi elisseti A x eqvinc indpi.4 indpi.5 indpi.1 mpbiri gencl eqcoms A N. e. a1i 1onn x cv 1o N. eleq1 x cv 1o 1o <N breq2 anbi12d indpi.1 imbi12d x cv y cv N. eleq1 x cv y cv 1o <N breq2 anbi12d indpi.2 imbi12d x cv y cv suc om eleq1 y cv peano2b syl6bbr x cv pinn syl5bi 1o <N x cv br adantrd x cv y cv suc 1o eleq2 1o y cv elsuci 1o y cv n0i 0ex sucid df-1o eleqtrr 1o y cv {/} eleq2 mpbii {/} y cv n0i syl jaoi syl syl6bi 1pi 1o x cv ltpiord mpan biimpa syl5 jcad y cv elni syl6ibr x cv y cv suc 1o <N breq2 x cv N. e. 1o <N x cv br pm3.27 syl5bi jcad 1pi y cv 1o addpiord mpan2 y cv pion y cv oa1suc syl eqtrd x cv eqeq2d biimparc N. eleq1d 1pi y cv 1o addclpi mpan2 syl5bir exp pm2.43d x cv y cv suc 1o <N breq2 biimprd anim12d impbid ph imbi1d 1pi y cv 1o addpiord mpan2 y cv pion y cv oa1suc syl eqtrd x cv eqeq2d indpi.3 syl6bir 1o <N y cv suc br adantr com12 pm5.74d bitrd x cv A N. eleq1 x cv A 1o <N breq2 anbi12d indpi.4 imbi12d indpi.5 1o N. e. 1o <N 1o br /\ a1i 1o om e. a1i y cv N. e. 1o y cv e. pm3.2 1pi 1o y cv ltpiord mpan pm5.32i syl6ibr ch imim1d 1pi elisseti y cv x eqvinc indpi.2 indpi.5 indpi.1 mpbiri gencl 1o y cv e. ch 1o y cv = jao mpi 1o y cv elsuci syl5 y visset sucex ltrelpi 1o brel 1o y cv suc ltpiord syl ibi syl5 syl6com imp3a y cv elni2 indpi.6 sylbir df-1o y cv sseq1i 0ex {/} V y cv sucssel ax-mp sylbi sylan2 syl9r 1o om e. adantlr findsg mpanl2 A pinn A elni2 A nnord A {/} ordsucss syl df-1o A sseq1i syl6ibr imp sylbi sylanc exp3a pm2.43i jaod mpd))
+thm (enqbreq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (<-> (br (<,> A B) (~Q) (<,> C D)) (= (opr A (.N) D) (opr B (.N) C)))) (x y z w v u df-enq A B C D ecopopreq))
+thm (dmenq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (~Q)) (X. (N.) (N.))) (x y z w v u df-enq x visset y visset mulcompi ecopoprdm))
+thm (enqer ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Er (~Q)) (x y z w v u df-enq x visset y visset mulcompi x cv y cv mulclpi y visset z visset x cv mulasspi z visset x cv y cv mulcanpi ecopoprer))
+thm (enqeceq () () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (<-> (= ([] (<,> A B) (~Q)) ([] (<,> C D) (~Q))) (= (opr A (.N) D) (opr B (.N) C)))) (A N. e. B N. e. /\ C N. e. D N. e. /\ pm3.26 A N. B N. opelxpi dmenq syl6eleqr C D opex enqer A B <,> erthdm 3syl A B C D enqbreq bitrd))
+thm (enqex ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (e. (~Q) (V)) (niex niex xpex niex niex xpex xpex x y z w v u df-enq x y N. N. X. N. N. X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv .N u cv opr w cv .N v cv opr = /\ E. E. E. E. opabssxp eqsstr ssexi))
+thm (nqex () () (e. (Q.) (V)) (df-nq niex niex xpex ~Q qsex eqeltr))
+thm (0npq () () (-. (e. ({/}) (Q.))) (enqer dmenq 0nelqs df-nq {/} eleq2i mtbir))
+thm (ltrelpq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ (<Q) (X. (Q.) (Q.))) (x y z w v u df-ltq x y Q. Q. z w v u x cv z cv w cv <,> ~Q [] = y cv v cv u cv <,> ~Q [] = /\ z cv .N u cv opr <N w cv .N v cv opr br /\ E. E. E. E. opabssxp eqsstr))
+thm (addcmpblnq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnq.1 (e. A (V))) (cmpblnq.2 (e. B (V))) (cmpblnq.3 (e. C (V))) (cmpblnq.4 (e. D (V))) (cmpblnq.5 (e. F (V))) (cmpblnq.6 (e. G (V))) (cmpblnq.7 (e. R (V))) (cmpblnq.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr (opr A (.N) G) (+N) (opr B (.N) F)) (opr B (.N) G)) (~Q) (<,> (opr (opr C (.N) S) (+N) (opr D (.N) R)) (opr D (.N) S))))) (A .N G opr B .N F opr addclpi A G mulclpi B F mulclpi syl2an an42s B G mulclpi A N. e. F N. e. /\ adantl an4s jca C .N S opr D .N R opr addclpi C S mulclpi D R mulclpi syl2an an42s D S mulclpi C N. e. R N. e. /\ adantl an4s jca anim12i an4s A .N G opr +N B .N F opr opr B .N G opr C .N S opr +N D .N R opr opr D .N S opr enqbreq syl A .N D opr B .N C opr .N G .N S opr opreq1 F .N S opr G .N R opr B .N D opr .N opreq2 +N opreqan12d A .N G oprex B .N F oprex D .N S oprex x visset y visset mulcompi y visset z visset x cv distrpi caoprdistrr cmpblnq.1 cmpblnq.6 cmpblnq.4 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.5 cmpblnq.4 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.8 caopr4 +N opreq12i eqtr C .N S oprex D .N R oprex B .N G opr distrpi cmpblnq.2 cmpblnq.6 cmpblnq.3 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.6 cmpblnq.4 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.7 caopr4 +N opreq12i eqtr 3eqtr4g syl5bir))
+thm (mulcmpblnq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnq.1 (e. A (V))) (cmpblnq.2 (e. B (V))) (cmpblnq.3 (e. C (V))) (cmpblnq.4 (e. D (V))) (cmpblnq.5 (e. F (V))) (cmpblnq.6 (e. G (V))) (cmpblnq.7 (e. R (V))) (cmpblnq.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr A (.N) F) (opr B (.N) G)) (~Q) (<,> (opr C (.N) R) (opr D (.N) S))))) (A F mulclpi B G mulclpi anim12i an4s C R mulclpi D S mulclpi anim12i an4s anim12i an4s A .N F opr B .N G opr C .N R opr D .N S opr enqbreq syl cmpblnq.1 cmpblnq.5 cmpblnq.4 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.8 caopr4 cmpblnq.2 cmpblnq.6 cmpblnq.3 x visset y visset mulcompi y visset z visset x cv mulasspi cmpblnq.7 caopr4 eqeq12i syl6bb A .N D opr B .N C opr F .N S opr G .N R opr .N opreq12 syl5bir))
+thm (addpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (+Q) ([] (<,> C D) (~Q))) ([] (<,> (opr (opr A (.N) D) (+N) (opr B (.N) C)) (opr B (.N) D)) (~Q)))) (A .N D opr +N B .N C opr opr B .N D opr opex a cv .N h cv opr +N b cv .N g cv opr opr b cv .N h cv opr opex c cv .N s cv opr +N d cv .N t cv opr opr d cv .N s cv opr opex enqex enqer dmenq x y z w v u df-enq z cv a cv u cv d cv .N opreq12 w cv b cv v cv c cv .N opreq12 eqeqan12d an42s z cv g cv u cv s cv .N opreq12 w cv h cv v cv t cv .N opreq12 eqeqan12d an42s x y z w v u f df-plpq w cv .N f cv opr +N v cv .N u cv opr opr a cv .N h cv opr +N b cv .N g cv opr opr v cv .N f cv opr b cv .N h cv opr opeq12 w cv a cv f cv h cv .N opreq12 v cv b cv u cv g cv .N opreq12 +N opreqan12d an42s v cv b cv f cv h cv .N opreq12 w cv a cv = u cv g cv = /\ adantl an4s sylanc w cv .N f cv opr +N v cv .N u cv opr opr c cv .N s cv opr +N d cv .N t cv opr opr v cv .N f cv opr d cv .N s cv opr opeq12 w cv c cv f cv s cv .N opreq12 v cv d cv u cv t cv .N opreq12 +N opreqan12d an42s v cv d cv f cv s cv .N opreq12 w cv c cv = u cv t cv = /\ adantl an4s sylanc w cv .N f cv opr +N v cv .N u cv opr opr A .N D opr +N B .N C opr opr v cv .N f cv opr B .N D opr opeq12 w cv A f cv D .N opreq12 v cv B u cv C .N opreq12 +N opreqan12d an42s v cv B f cv D .N opreq12 w cv A = u cv C = /\ adantl an4s sylanc x y z a b c d df-plq df-nq a visset b visset c visset d visset g visset h visset t visset s visset addcmpblnq oprec))
+thm (mulpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (.Q) ([] (<,> C D) (~Q))) ([] (<,> (opr A (.N) C) (opr B (.N) D)) (~Q)))) (A .N C opr B .N D opr opex a cv .N g cv opr b cv .N h cv opr opex c cv .N t cv opr d cv .N s cv opr opex enqex enqer dmenq x y z w v u df-enq z cv a cv u cv d cv .N opreq12 w cv b cv v cv c cv .N opreq12 eqeqan12d an42s z cv g cv u cv s cv .N opreq12 w cv h cv v cv t cv .N opreq12 eqeqan12d an42s x y z w v u f df-mpq w cv .N u cv opr a cv .N g cv opr v cv .N f cv opr b cv .N h cv opr opeq12 w cv a cv u cv g cv .N opreq12 v cv b cv f cv h cv .N opreq12 syl2an an4s w cv .N u cv opr c cv .N t cv opr v cv .N f cv opr d cv .N s cv opr opeq12 w cv c cv u cv t cv .N opreq12 v cv d cv f cv s cv .N opreq12 syl2an an4s w cv .N u cv opr A .N C opr v cv .N f cv opr B .N D opr opeq12 w cv A u cv C .N opreq12 v cv B f cv D .N opreq12 syl2an an4s x y z a b c d df-mq df-nq a visset b visset c visset d visset g visset h visset t visset s visset mulcmpblnq oprec))
+thm (ordpipq ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((ordpipq.1 (e. A (V))) (ordpipq.2 (e. B (V))) (ordpipq.3 (e. C (V))) (ordpipq.4 (e. D (V)))) (<-> (br ([] (<,> A B) (~Q)) (<Q) ([] (<,> C D) (~Q))) (br (opr A (.N) D) (<N) (opr B (.N) C))) (enqex ordpipq.2 ordpipq.3 ordpipq.4 enqer dmenq df-nq ltrelpq ltrelpi 0npi dmmulpi enqex enqer dmenq df-nq x y z w v u df-ltq z cv w cv A B enqeceq v cv u cv C D enqeceq v cv .N D opr u cv .N C opr eqcom syl6bb bi2anan9 z cv .N B opr w cv .N A opr u cv .N C opr v cv .N D opr .N opreq12 z visset u visset ordpipq.2 x visset y visset mulcompi y visset f visset x cv mulasspi ordpipq.3 caopr4 w visset v visset ordpipq.1 x visset y visset mulcompi y visset f visset x cv mulasspi ordpipq.4 caopr4 3eqtr4g syl6bi B C mulclpi D N. e. adantrr A N. e. adantll w cv v cv mulclpi u cv N. e. adantrr z cv N. e. adantll anim12i ancoms an4s z cv .N u cv oprex B .N C oprex x visset y visset f cv ltmpi w cv .N v cv oprex x visset y visset mulcompi A .N D oprex caoprord3 exp syl syld brecop brecop2))
+thm (1q () () (e. (1Q) (Q.)) (1pi 1pi 1o N. 1o N. opelxpi mp2an enqex 1o 1o <,> N. N. X. ecelqsi ax-mp df-1q df-nq eleq12i mpbir))
+thm (addclpq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (+Q) B) (Q.))) (df-nq x cv y cv <,> ~Q [] A +Q z cv w cv <,> ~Q [] opreq1 N. N. X. ~Q /. eleq1d z cv w cv <,> ~Q [] B A +Q opreq2 N. N. X. ~Q /. eleq1d x cv y cv z cv w cv addpipq x cv .N w cv opr y cv .N z cv opr addclpi x cv w cv mulclpi y cv z cv mulclpi syl2an an42s y cv w cv mulclpi x cv N. e. z cv N. e. /\ adantl an4s jca x cv .N w cv opr +N y cv .N z cv opr opr N. y cv .N w cv opr N. opelxpi enqex x cv .N w cv opr +N y cv .N z cv opr opr y cv .N w cv opr <,> N. N. X. ecelqsi 3syl eqeltrd 2ecoptocl df-nq syl6eleqr))
+thm (dmaddpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+Q)) (X. (Q.) (Q.))) (x y z w v u f df-plq dmeqi x y z Q. Q. w v u f x cv w cv v cv <,> ~Q [] = y cv u cv f cv <,> ~Q [] = /\ z cv w cv v cv <,> +pQ u cv f cv <,> opr ~Q [] = /\ E. E. E. E. dmoprabss eqsstr 0npq x cv y cv addclpq oprssdm eqssi))
+thm (mulclpq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (.Q) B) (Q.))) (df-nq x cv y cv <,> ~Q [] A .Q z cv w cv <,> ~Q [] opreq1 N. N. X. ~Q /. eleq1d z cv w cv <,> ~Q [] B A .Q opreq2 N. N. X. ~Q /. eleq1d x cv y cv z cv w cv mulpipq x cv z cv mulclpi y cv w cv mulclpi anim12i an4s x cv .N z cv opr N. y cv .N w cv opr N. opelxpi enqex x cv .N z cv opr y cv .N w cv opr <,> N. N. X. ecelqsi 3syl eqeltrd 2ecoptocl df-nq syl6eleqr))
+thm (dmmulpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.Q)) (X. (Q.) (Q.))) (x y z w v u f df-mq dmeqi x y z Q. Q. w v u f x cv w cv v cv <,> ~Q [] = y cv u cv f cv <,> ~Q [] = /\ z cv w cv v cv <,> .pQ u cv f cv <,> opr ~Q [] = /\ E. E. E. E. dmoprabss eqsstr 0npq x cv y cv mulclpq oprssdm eqssi))
+thm (addcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((addcompq.1 (e. A (V))) (addcompq.2 (e. B (V)))) (= (opr A (+Q) B) (opr B (+Q) A)) (df-nq x cv y cv z cv w cv addpipq z cv w cv x cv y cv addpipq x visset w visset mulcompi y visset z visset mulcompi +N opreq12i w cv .N x cv oprex z cv .N y cv oprex addcompi eqtr y visset w visset mulcompi A B ecoprcom addcompq.2 dmaddpq addcompq.1 ndmoprcom pm2.61i))
+thm (addasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((addasspq.1 (e. B (V))) (addasspq.2 (e. C (V)))) (= (opr (opr A (+Q) B) (+Q) C) (opr A (+Q) (opr B (+Q) C))) (df-nq x cv y cv z cv w cv addpipq z cv w cv v cv u cv addpipq x cv .N w cv opr +N y cv .N z cv opr opr y cv .N w cv opr v cv u cv addpipq x cv y cv z cv .N u cv opr +N w cv .N v cv opr opr w cv .N u cv opr addpipq x cv .N w cv opr y cv .N z cv opr addclpi x cv w cv mulclpi y cv z cv mulclpi syl2an an42s y cv w cv mulclpi x cv N. e. z cv N. e. /\ adantl an4s jca z cv .N u cv opr w cv .N v cv opr addclpi z cv u cv mulclpi w cv v cv mulclpi syl2an an42s w cv u cv mulclpi z cv N. e. v cv N. e. /\ adantl an4s jca y cv .N z cv .N u cv opr oprex y cv .N w cv .N v cv opr oprex x cv .N w cv .N u cv opr opr addasspi x visset y visset w visset f visset g visset mulcompi g visset h visset f cv distrpi z visset u visset g visset h visset f cv mulasspi caoprdilem w visset v visset y cv mulasspi +N opreq12i z cv .N u cv oprex w cv .N v cv oprex y cv distrpi x cv .N w cv .N u cv opr opr +N opreq2i 3eqtr4 w visset u visset y cv mulasspi A B C ecoprass addasspq.1 dmaddpq addasspq.2 0npq A ndmoprass pm2.61i))
+thm (mulcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((mulcompq.1 (e. A (V))) (mulcompq.2 (e. B (V)))) (= (opr A (.Q) B) (opr B (.Q) A)) (df-nq x cv y cv z cv w cv mulpipq z cv w cv x cv y cv mulpipq x visset z visset mulcompi y visset w visset mulcompi A B ecoprcom mulcompq.2 dmmulpq mulcompq.1 ndmoprcom pm2.61i))
+thm (mulasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((mulasspq.1 (e. B (V))) (mulasspq.2 (e. C (V)))) (= (opr (opr A (.Q) B) (.Q) C) (opr A (.Q) (opr B (.Q) C))) (df-nq x cv y cv z cv w cv mulpipq z cv w cv v cv u cv mulpipq x cv .N z cv opr y cv .N w cv opr v cv u cv mulpipq x cv y cv z cv .N v cv opr w cv .N u cv opr mulpipq x cv z cv mulclpi y cv w cv mulclpi anim12i an4s z cv v cv mulclpi w cv u cv mulclpi anim12i an4s z visset v visset x cv mulasspi w visset u visset y cv mulasspi A B C ecoprass mulasspq.1 dmmulpq mulasspq.2 0npq A ndmoprass pm2.61i))
+thm (distrpqlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((distrpqlem.1 (e. A (V))) (distrpqlem.2 (e. B (V))) (distrpqlem.3 (e. C (V)))) (-> (/\/\ (e. A (N.)) (e. B (N.)) (e. C (N.))) (= ([] (<,> (opr A (.N) B) (opr A (.N) C)) (~Q)) ([] (<,> B C) (~Q)))) (distrpqlem.1 distrpqlem.2 distrpqlem.3 x visset y visset mulcompi y visset z visset x cv mulasspi caopr32 A B mulclpi A C mulclpi anim12i A N. e. A N. e. /\ B N. e. C N. e. /\ pm3.27 an4s jca 3impdi A .N B opr A .N C opr B C enqbreq syl mpbiri A .N B opr A .N C opr opex B C opex enqer erthi syl))
+thm (distrpq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((distrpq.1 (e. B (V))) (distrpq.2 (e. C (V)))) (= (opr A (.Q) (opr B (+Q) C)) (opr (opr A (.Q) B) (+Q) (opr A (.Q) C))) (df-nq z cv w cv v cv u cv addpipq x cv y cv z cv .N u cv opr +N w cv .N v cv opr opr w cv .N u cv opr mulpipq x cv z cv .N u cv opr +N w cv .N v cv opr opr mulclpi y cv N. e. w cv .N u cv opr N. e. pm3.26 y cv w cv .N u cv opr mulclpi jca anim12i x cv .N z cv .N u cv opr +N w cv .N v cv opr opr opr N. e. y cv N. e. y cv .N w cv .N u cv opr opr N. e. an12 y cv N. e. x cv .N z cv .N u cv opr +N w cv .N v cv opr opr opr N. e. y cv .N w cv .N u cv opr opr N. e. 3anass bitr4 sylib an4s y visset x cv .N z cv .N u cv opr +N w cv .N v cv opr opr oprex y cv .N w cv .N u cv opr oprex distrpqlem syl eqtr4d x cv y cv z cv w cv mulpipq x cv y cv v cv u cv mulpipq x cv .N z cv opr y cv .N w cv opr x cv .N v cv opr y cv .N u cv opr addpipq z cv .N u cv opr w cv .N v cv opr addclpi z cv u cv mulclpi w cv v cv mulclpi syl2an an42s w cv u cv mulclpi z cv N. e. v cv N. e. /\ adantl an4s jca x cv z cv mulclpi y cv w cv mulclpi anim12i an4s x cv v cv mulclpi y cv u cv mulclpi anim12i an4s z cv .N u cv oprex w cv .N v cv oprex y cv .N x cv opr distrpi x visset z cv .N u cv opr +N w cv .N v cv opr oprex y cv mulasspi x visset y visset mulcompi .N z cv .N u cv opr opreq1i x visset y visset z visset f visset g visset mulcompi g visset h visset f cv mulasspi u visset caopr4 eqtr3 y visset x visset w visset f visset g visset mulcompi g visset h visset f cv mulasspi v visset caopr4 +N opreq12i 3eqtr3 y visset w cv .N u cv oprex y cv mulasspi y visset y visset w visset f visset g visset mulcompi g visset h visset f cv mulasspi u visset caopr4 eqtr3 A B C ecoprdi distrpq.1 dmaddpq distrpq.2 0npq dmmulpq A ndmoprdistr pm2.61i))
+thm (1qec () ((1qec.1 (e. A (V)))) (-> (e. A (N.)) (= (1Q) ([] (<,> A A) (~Q)))) (1pi 1pi 1qec.1 1pi elisseti 1pi elisseti distrpqlem mp3an23 A mulidpi A mulidpi jca A .N 1o opr A A .N 1o opr A opeq12 A .N 1o opr A .N 1o opr <,> A A <,> ~Q eceq2 3syl eqtr3d df-1q syl5eq))
+thm (mulidpq ((x y) (A x) (A y)) () (-> (e. A (Q.)) (= (opr A (.Q) (1Q)) A)) (df-nq x cv y cv <,> ~Q [] A .Q 1Q opreq1 x cv y cv <,> ~Q [] A = id eqeq12d 1pi 1pi pm3.2i x cv y cv 1o 1o mulpipq mpan2 df-1q x cv y cv <,> ~Q [] .Q opreq2i 1pi elisseti x visset mulcompi 1pi elisseti y visset mulcompi 1o .N x cv opr x cv .N 1o opr 1o .N y cv opr y cv .N 1o opr opeq12 mp2an 1o .N x cv opr 1o .N y cv opr <,> x cv .N 1o opr y cv .N 1o opr <,> ~Q eceq2 ax-mp 3eqtr4g 1pi 1pi elisseti x visset y visset distrpqlem mp3an1 eqtrd ecoptocl))
+thm (recmulpq ((x y) (A x) (A y) (B x) (B y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((recmulpq.1 (e. B (V)))) (-> (e. A (Q.)) (<-> (= (` (*Q) A) B) (= (opr A (.Q) B) (1Q)))) (recmulpq.1 x cv A .Q y cv opreq1 1Q eqeq1d y cv B A .Q opreq2 1Q eqeq1d df-nq z cv w cv <,> ~Q [] x cv .Q y cv opreq1 1Q eqeq1d y biexdv z cv w cv w cv z cv mulpipq an42s anidms z cv w cv mulclpi z cv .N w cv oprex 1qec z visset w visset mulcompi z cv .N w cv opr w cv .N z cv opr z cv .N w cv opr opeq2 ax-mp z cv .N w cv opr z cv .N w cv opr <,> z cv .N w cv opr w cv .N z cv opr <,> ~Q eceq2 ax-mp syl6eq syl eqtr4d enqex ~Q V w cv z cv <,> ecexg ax-mp y cv w cv z cv <,> ~Q [] z cv w cv <,> ~Q [] .Q opreq2 1Q eqeq1d cla4ev syl ecoptocl y x cv .Q y cv opr 1Q = eu5 x visset 1q dmmulpq 0npq z visset w visset mulcompq w visset v visset z cv mulasspq z cv mulidpq y caoprmo mpbiran2 sylibr x y df-rq fvopab3))
+thm (recidpq () () (-> (e. A (Q.)) (= (opr A (.Q) (` (*Q) A)) (1Q))) (*Q A ` eqid *Q A fvex A recmulpq mpbii))
+thm (recclpq () () (-> (e. A (Q.)) (e. (` (*Q) A) (Q.))) (1q A recidpq Q. eleq1d mpbiri *Q A fvex dmmulpq 0npq A ndmoprrcl pm3.27d syl))
+thm (recrecpq () ((recrecpq.1 (e. A (V)))) (-> (e. A (Q.)) (= (` (*Q) (` (*Q) A)) A)) (A recidpq *Q A fvex recrecpq.1 mulcompq syl5eq A recclpq recrecpq.1 *Q A ` recmulpq syl mpbird))
+thm (dmrecpq ((x y)) () (= (dom (*Q)) (Q.)) (x y df-rq dmeqi x cv recidpq *Q x cv fvex y cv *Q x cv ` x cv .Q opreq2 1Q eqeq1d cla4ev syl rgen x Q. y x cv .Q y cv opr 1Q = dmopab3 mpbi eqtr))
+thm (ltsopq ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or (<Q) (Q.)) (df-nq x cv y cv <,> ~Q [] f cv <Q z cv w cv <,> ~Q [] breq1 x cv y cv <,> ~Q [] f cv z cv w cv <,> ~Q [] eqeq1 x cv y cv <,> ~Q [] f cv z cv w cv <,> ~Q [] <Q breq2 orbi12d negbid bibi12d x cv y cv <,> ~Q [] f cv <Q z cv w cv <,> ~Q [] breq1 z cv w cv <,> ~Q [] <Q v cv u cv <,> ~Q [] br anbi1d x cv y cv <,> ~Q [] f cv <Q v cv u cv <,> ~Q [] breq1 imbi12d anbi12d z cv w cv <,> ~Q [] g cv f cv <Q breq2 z cv w cv <,> ~Q [] g cv f cv eqeq2 z cv w cv <,> ~Q [] g cv <Q f cv breq1 orbi12d negbid bibi12d z cv w cv <,> ~Q [] g cv f cv <Q breq2 z cv w cv <,> ~Q [] g cv <Q v cv u cv <,> ~Q [] breq1 anbi12d f cv <Q v cv u cv <,> ~Q [] br imbi1d anbi12d v cv u cv <,> ~Q [] h cv g cv <Q breq2 f cv <Q g cv br anbi2d v cv u cv <,> ~Q [] h cv f cv <Q breq2 imbi12d f cv <Q g cv br f cv g cv = g cv <Q f cv br \/ -. <-> anbi2d ltsopi <N N. x cv .N w cv opr y cv .N z cv opr sotric mpan x cv w cv mulclpi y cv z cv mulclpi syl2an an42s x cv y cv z cv w cv enqeceq z visset w visset x visset y visset ordpipq z visset y visset mulcompi w visset x visset mulcompi <N breq12i bitr x cv N. e. y cv N. e. /\ z cv N. e. w cv N. e. /\ /\ a1i orbi12d negbid bitr4d x visset y visset z visset w visset ordpipq syl5bb v cv N. e. u cv N. e. /\ 3adant3 x cv .N w cv oprex y cv .N z cv oprex f visset g visset h cv ltmpi u visset f visset g visset mulcompi caoprord2 w visset u visset x cv mulasspi z visset u visset y cv mulasspi <N breq12i syl6bb x visset y visset z visset w visset ordpipq syl5bb z cv .N u cv oprex w cv .N v cv oprex y cv ltmpi z visset w visset v visset u visset ordpipq syl5bb bi2anan9r x cv .N w cv .N u cv opr oprex ltsopi ltrelpi y cv .N z cv .N u cv opr oprex y cv .N w cv .N v cv opr oprex sotri x cv .N u cv oprex dmmulpi ltrelpi 0npi x cv .N u cv oprex y cv .N v cv oprex w cv ltmpi ndmordi x visset w visset u visset f visset g visset mulcompi g visset h visset f cv mulasspi caopr12 y visset w visset v visset f visset g visset mulcompi g visset h visset f cv mulasspi caopr12 <N breq12i x visset y visset v visset u visset ordpipq 3imtr4 syl syl6bi v cv N. e. adantrl x cv N. e. adantll z cv N. e. w cv N. e. /\ 3adant2 jca 3ecoptocl so))
+thm (ltapq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((ltapq.1 (e. A (V))) (ltapq.2 (e. B (V)))) (-> (e. C (Q.)) (<-> (br A (<Q) B) (br (opr C (+Q) A) (<Q) (opr C (+Q) B)))) (ltapq.2 dmaddpq ltapq.1 ltrelpq 0npq df-nq x cv y cv <,> ~Q [] A <Q z cv w cv <,> ~Q [] breq1 x cv y cv <,> ~Q [] A v cv u cv <,> ~Q [] +Q opreq2 <Q v cv u cv <,> ~Q [] +Q z cv w cv <,> ~Q [] opr breq1d bibi12d z cv w cv <,> ~Q [] B A <Q breq2 z cv w cv <,> ~Q [] B v cv u cv <,> ~Q [] +Q opreq2 v cv u cv <,> ~Q [] +Q A opr <Q breq2d bibi12d v cv u cv <,> ~Q [] C +Q A opreq1 v cv u cv <,> ~Q [] C +Q B opreq1 <Q breq12d A <Q B br bibi2d u cv u cv mulclpi x cv .N w cv oprex y cv .N z cv oprex u cv .N u cv opr ltmpi syl anidms v cv N. e. y cv N. e. /\ w cv N. e. ad2antrl v cv y cv mulclpi u cv w cv mulclpi anim12i v cv .N y cv opr u cv .N w cv opr mulclpi u cv .N x cv opr .N u cv .N w cv opr oprex u cv .N y cv opr .N u cv .N z cv opr oprex v cv .N y cv opr .N u cv .N w cv opr opr ltapi 3syl u visset u visset x visset f visset g visset mulcompi g visset h visset f cv mulasspi w visset caopr4 u visset u visset y visset f visset g visset mulcompi g visset h visset f cv mulasspi z visset caopr4 <N breq12i v cv .N y cv oprex u cv .N x cv oprex u cv .N w cv oprex f visset g visset mulcompi g visset h visset f cv distrpi caoprdistrr v cv .N w cv oprex u cv .N z cv oprex u cv .N y cv opr distrpi u visset y visset v visset f visset g visset mulcompi g visset h visset f cv mulasspi w visset caopr411 +N u cv .N y cv opr .N u cv .N z cv opr opr opreq1i eqtr <N breq12i 3bitr4g bitrd x visset y visset z visset w visset ordpipq v cv .N y cv opr +N u cv .N x cv opr oprex u cv .N y cv oprex v cv .N w cv opr +N u cv .N z cv opr oprex u cv .N w cv oprex ordpipq 3bitr4g an4s x cv N. e. y cv N. e. pm3.27 z cv N. e. w cv N. e. pm3.27 anim12i sylan2 v cv u cv x cv y cv addpipq z cv N. e. w cv N. e. /\ adantrr v cv u cv z cv w cv addpipq x cv N. e. y cv N. e. /\ adantrl <Q breq12d bitr4d 3impb 3coml 3ecoptocl ndmord))
+thm (ltmpq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((ltapq.1 (e. A (V))) (ltapq.2 (e. B (V)))) (-> (e. C (Q.)) (<-> (br A (<Q) B) (br (opr C (.Q) A) (<Q) (opr C (.Q) B)))) (ltapq.2 dmmulpq ltapq.1 ltrelpq 0npq df-nq x cv y cv <,> ~Q [] A <Q z cv w cv <,> ~Q [] breq1 x cv y cv <,> ~Q [] A v cv u cv <,> ~Q [] .Q opreq2 <Q v cv u cv <,> ~Q [] .Q z cv w cv <,> ~Q [] opr breq1d bibi12d z cv w cv <,> ~Q [] B A <Q breq2 z cv w cv <,> ~Q [] B v cv u cv <,> ~Q [] .Q opreq2 v cv u cv <,> ~Q [] .Q A opr <Q breq2d bibi12d v cv u cv <,> ~Q [] C .Q A opreq1 v cv u cv <,> ~Q [] C .Q B opreq1 <Q breq12d A <Q B br bibi2d v cv u cv mulclpi x cv .N w cv oprex y cv .N z cv oprex v cv .N u cv opr ltmpi v visset x visset u visset f visset g visset mulcompi g visset h visset f cv mulasspi w visset caopr4 u visset y visset v visset f visset g visset mulcompi g visset h visset f cv mulasspi z visset caopr4 u visset v visset mulcompi .N y cv .N z cv opr opreq1i eqtr <N breq12i syl6bbr syl x visset y visset z visset w visset ordpipq v cv .N x cv oprex u cv .N y cv oprex v cv .N z cv oprex u cv .N w cv oprex ordpipq 3bitr4g x cv N. e. y cv N. e. /\ z cv N. e. w cv N. e. /\ /\ adantr v cv u cv x cv y cv mulpipq z cv N. e. w cv N. e. /\ adantrr v cv u cv z cv w cv mulpipq x cv N. e. y cv N. e. /\ adantrl <Q breq12d bitr4d 3impb 3coml 3ecoptocl ndmord))
+thm (1lt2pq () () (br (1Q) (<Q) (opr (1Q) (+Q) (1Q))) (1lt2pi 1pi 1o mulidpi ax-mp 1pi 1o mulidpi ax-mp 1pi 1o mulidpi ax-mp +N opreq12i 3brtr4 1pi 1o .N 1o oprex 1o .N 1o opr +N 1o .N 1o opr oprex 1o ltmpi ax-mp mpbi 1pi elisseti 1pi elisseti 1o .N 1o opr +N 1o .N 1o opr oprex 1o .N 1o oprex ordpipq mpbir df-1q df-1q df-1q +Q opreq12i 1pi 1pi pm3.2i 1pi 1pi pm3.2i 1o 1o 1o 1o addpipq mp2an eqtr 3brtr4))
+thm (ltaddpq () ((ltaddpq.1 (e. A (V))) (ltaddpq.2 (e. B (V)))) (-> (/\ (e. A (Q.)) (e. B (Q.))) (br A (<Q) (opr A (+Q) B))) (ltaddpq.2 B +Q B oprex A ltapq 1lt2pq 1q elisseti 1Q +Q 1Q oprex B ltmpq mpbii B mulidpq B mulidpq B mulidpq +Q opreq12d 1q elisseti 1q elisseti B distrpq syl5eq 3brtr3d syl5bi ltaddpq.2 ltaddpq.1 addcompq ltaddpq.2 A +Q B oprex addcompq ltaddpq.2 ltaddpq.2 A addasspq eqtr <Q breq12i syl6ibr imp ltaddpq.1 A +Q B oprex B ltapq A Q. e. adantl mpbird))
+thm (ltexpq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((ltexpq.1 (e. A (V)))) (-> (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A (<Q) B) (E. x (= (opr A (+Q) (cv x)) B)))) (df-nq y cv z cv <,> ~Q [] A <Q w cv v cv <,> ~Q [] breq1 y cv z cv <,> ~Q [] A +Q x cv opreq1 w cv v cv <,> ~Q [] eqeq1d x biexdv imbi12d w cv v cv <,> ~Q [] B A <Q breq2 w cv v cv <,> ~Q [] B A +Q x cv opr eqeq2 x biexdv imbi12d y cv v cv mulclpi z cv w cv mulclpi anim12i an42s y cv .N v cv opr z cv .N w cv opr u ltexpi syl y cv N. e. z cv N. e. /\ w cv N. e. v cv N. e. /\ pm3.26 u cv N. e. adantr y cv N. e. z cv N. e. /\ w cv N. e. v cv N. e. /\ /\ u cv N. e. pm3.27 y cv N. e. z cv N. e. pm3.27 w cv N. e. v cv N. e. pm3.27 anim12i u cv N. e. adantr z cv v cv mulclpi syl jca jca y cv .N v cv opr +N u cv opr z cv .N w cv opr = adantrr y cv z cv u cv z cv .N v cv opr addpipq syl y cv N. e. z cv N. e. pm3.27 w cv N. e. v cv N. e. /\ u cv N. e. ad2antll y cv .N v cv opr u cv addclpi y cv v cv mulclpi y cv N. e. z cv N. e. pm3.26 w cv N. e. v cv N. e. pm3.27 syl2an sylan y cv N. e. z cv N. e. pm3.27 w cv N. e. v cv N. e. pm3.27 anim12i u cv N. e. adantr z cv v cv mulclpi syl 3jca y cv .N v cv opr +N u cv opr z cv .N w cv opr = adantrr z visset y cv .N v cv opr +N u cv oprex z cv .N v cv oprex distrpqlem y visset z visset v visset x visset w visset mulcompi w visset u visset x cv mulasspi caopr12 +N z cv .N u cv opr opreq1i y cv .N v cv oprex u visset z cv distrpi eqtr4 y cv .N z cv .N v cv opr opr +N z cv .N u cv opr opr z cv .N y cv .N v cv opr +N u cv opr opr z cv .N z cv .N v cv opr opr opeq1 ax-mp y cv .N z cv .N v cv opr opr +N z cv .N u cv opr opr z cv .N z cv .N v cv opr opr <,> z cv .N y cv .N v cv opr +N u cv opr opr z cv .N z cv .N v cv opr opr <,> ~Q eceq2 ax-mp syl5eq syl z cv N. e. w cv N. e. v cv N. e. 3anass biimpr y cv N. e. adantll y cv .N v cv opr +N u cv opr z cv .N w cv opr = anim1i u cv N. e. adantrl y cv .N v cv opr +N u cv opr z cv .N w cv opr z cv .N v cv opr opeq1 y cv .N v cv opr +N u cv opr z cv .N v cv opr <,> z cv .N w cv opr z cv .N v cv opr <,> ~Q eceq2 syl z visset w visset v visset distrpqlem sylan9eqr syl 3eqtrd enqex ~Q V u cv z cv .N v cv opr <,> ecexg ax-mp x cv u cv z cv .N v cv opr <,> ~Q [] y cv z cv <,> ~Q [] +Q opreq2 w cv v cv <,> ~Q [] eqeq1d cla4ev syl exp u 19.23adv sylbid y visset z visset w visset v visset ordpipq syl5ib 2ecoptocl A +Q x cv opr B Q. eleq1 x visset dmaddpq 0npq A ndmoprrcl pm3.27d syl6bir ltexpq.1 x visset ltaddpq exp syl9 imp3a A +Q x cv opr B A <Q breq2 sylibd com12 x 19.23adv impbid))
+thm (ltexpq2 ((A x) (B x)) ((ltexpq2.1 (e. A (V)))) (-> (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A (<Q) B) (E. x (/\ (e. (cv x) (Q.)) (= (opr A (+Q) (cv x)) B))))) (ltexpq2.1 B x ltexpq x cv Q. e. A +Q x cv opr B = pm3.27 A Q. e. B Q. e. /\ a1i A +Q x cv opr B Q. eleq1 x visset dmaddpq 0npq A ndmoprrcl pm3.27d syl6bir com12 ancrd A Q. e. adantl impbid x biexdv bitr4d))
+thm (halfpq ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (Q.)) (E. x (= (opr (cv x) (+Q) (cv x)) A))) (df-nq y cv z cv <,> ~Q [] A x cv +Q x cv opr eqeq2 x biexdv y cv z cv +N z cv opr y cv z cv +N z cv opr addpipq y visset y visset z cv +N z cv opr distrpi y visset z cv +N z cv oprex mulcompi +N z cv +N z cv opr .N y cv opr opreq1i eqtr4 z cv +N z cv opr .N y cv +N y cv opr opr y cv .N z cv +N z cv opr opr +N z cv +N z cv opr .N y cv opr opr z cv +N z cv opr .N z cv +N z cv opr opr opeq1 ax-mp z cv +N z cv opr .N y cv +N y cv opr opr z cv +N z cv opr .N z cv +N z cv opr opr <,> y cv .N z cv +N z cv opr opr +N z cv +N z cv opr .N y cv opr opr z cv +N z cv opr .N z cv +N z cv opr opr <,> ~Q eceq2 ax-mp syl6eqr z cv z cv addclpi anidms y cv N. e. anim2i z cv z cv addclpi anidms y cv N. e. anim2i sylanc z cv +N z cv oprex y cv +N y cv oprex z cv +N z cv oprex distrpqlem z cv z cv addclpi anidms y cv N. e. adantl y cv y cv addclpi anidms z cv N. e. adantr z cv z cv addclpi anidms y cv N. e. adantl syl3anc y cv mulidpi y cv mulidpi +N opreq12d y visset 1o +N 1o oprex mulcompi 1pi elisseti 1pi elisseti y cv distrpi eqtr3 syl5eq z cv mulidpi z cv mulidpi +N opreq12d z visset 1o +N 1o oprex mulcompi 1pi elisseti 1pi elisseti z cv distrpi eqtr3 syl5eq anim12i 1o +N 1o opr .N y cv opr y cv +N y cv opr 1o +N 1o opr .N z cv opr z cv +N z cv opr opeq12 1o +N 1o opr .N y cv opr 1o +N 1o opr .N z cv opr <,> y cv +N y cv opr z cv +N z cv opr <,> ~Q eceq2 3syl 1pi 1pi 1o 1o addclpi mp2an 1o +N 1o oprex y visset z visset distrpqlem mp3an1 eqtr3d 3eqtrd enqex ~Q V y cv z cv +N z cv opr <,> ecexg ax-mp x cv y cv z cv +N z cv opr <,> ~Q [] x cv y cv z cv +N z cv opr <,> ~Q [] +Q opreq12 anidms y cv z cv <,> ~Q [] eqeq1d cla4ev syl ecoptocl))
+thm (nsmallpq ((A x)) () (-> (e. A (Q.)) (E. x (br (cv x) (<Q) A))) (A x halfpq x cv +Q x cv opr A Q. eleq1 x visset dmaddpq 0npq x cv ndmoprrcl syl6bir com12 x cv +Q x cv opr A x cv <Q breq2 x visset x visset ltaddpq syl5bi sylcom x 19.22dv mpd))
+thm (ltbtwnpq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((ltbtwnpq.1 (e. A (V))) (ltbtwnpq.2 (e. B (V)))) (-> (br A (<Q) B) (E. x (/\ (br A (<Q) (cv x)) (br (cv x) (<Q) B)))) (ltbtwnpq.2 ltrelpq A brel ltbtwnpq.1 B y ltexpq A +Q y cv opr B Q. eleq1 y visset dmaddpq 0npq A ndmoprrcl syl6bir y cv z halfpq A Q. e. adantl z cv +Q z cv opr y cv Q. eleq1 z visset dmaddpq 0npq z cv ndmoprrcl pm3.26d syl6bir A Q. e. anim2d z cv +Q z cv opr y cv A +Q opreq2 B eqeq1d A +Q z cv +Q z cv opr opr B A +Q z cv opr <Q breq2 A +Q z cv oprex z visset ltaddpq z visset z visset A addasspq syl6breq syl5bi syl6bir A z cv addclpq A Q. e. z cv Q. e. pm3.27 jca syl7 ltbtwnpq.1 z visset ltaddpq A Q. e. z cv Q. e. /\ A <Q A +Q z cv opr br A +Q z cv opr <Q B br pm3.43i ax-mp syl6 A +Q z cv oprex x cv A +Q z cv opr A <Q breq2 x cv A +Q z cv opr <Q B breq1 anbi12d cla4ev syl8 com23 syld z 19.23aiv mpcom com12 syld com12 A Q. e. adantl y 19.23adv sylbid mpcom))
+thm (ltrpq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((ltrpq.1 (e. A (V))) (ltrpq.2 (e. B (V)))) (-> (br A (<Q) B) (br (` (*Q) B) (<Q) (` (*Q) A))) (ltrpq.2 ltrelpq A brel 1q elisseti ltsopq ltrelpq soirri A recidpq B recidpq <Q breqan12rd mtbiri A <Q B br adantll A recclpq ltrpq.1 ltrpq.2 y visset z visset x cv ltmpq *Q A fvex y visset z visset mulcompq caoprord2 syl B Q. e. anbi1d biimpac B .Q *Q B ` opr B .Q *Q A ` opr A .Q *Q A ` opr <Q breq2 biimprcd *Q B ` *Q A ` B .Q opreq2 syl5 B Q. e. adantr A .Q *Q A ` oprex ltsopq ltrelpq B .Q *Q A ` oprex B .Q *Q B ` oprex sotri *Q A fvex *Q B fvex B ltmpq biimpa sylan2 exp32 imp jaod syl mtod exp31 imp3a com12 ltsopq <Q Q. *Q B ` *Q A ` sotric mpan B recclpq A recclpq syl2an sylibrd ancoms mpcom))
+thm (npex ((x y) (x z) (y z)) () (e. (P.) (V)) (nqex pwex x cv Q. pssss {/} x cv C: y x cv z z cv <Q y cv br z cv x cv e. -> A. z x cv y cv <Q z cv br E.e. /\ A.e. ad2antlr x ss2abi x y z df-np Q. x df-pw 3sstr4 ssexi))
+thm (elnp ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (P.)) (/\ (/\ (C: ({/}) A) (C: A (Q.))) (A.e. x A (/\ (A. y (-> (br (cv y) (<Q) (cv x)) (e. (cv y) A))) (E.e. y A (br (cv x) (<Q) (cv y))))))) (A P. elisset A Q. pssss nqex A ssex syl {/} A C: x A y y cv <Q x cv br y cv A e. -> A. y A x cv <Q y cv br E.e. /\ A.e. ad2antlr z cv A {/} psseq2 z cv A Q. psseq1 anbi12d z cv A y cv eleq2 y cv <Q x cv br imbi2d y bialdv z cv A y x cv <Q y cv br rexeq anbi12d x raleqd anbi12d z x y df-np V elab2g pm5.21nii))
+thm (prn0 ((x y) (A x) (A y)) () (-> (e. A (P.)) (-. (= A ({/})))) (A x y elnp {/} A C: A Q. C: /\ x A y y cv <Q x cv br y cv A e. -> A. y A x cv <Q y cv br E.e. /\ A.e. pm3.26 pm3.26d sylbi A 0pss sylib))
+thm (prpssnq ((x y) (A x) (A y)) () (-> (e. A (P.)) (C: A (Q.))) (A x y elnp {/} A C: A Q. C: /\ x A y y cv <Q x cv br y cv A e. -> A. y A x cv <Q y cv br E.e. /\ A.e. pm3.26 pm3.27d sylbi))
+thm (elprpq () () (-> (/\ (e. A (P.)) (e. B A)) (e. B (Q.))) (A prpssnq pssssd B sseld imp))
+thm (0npr () () (-. (e. ({/}) (P.))) ({/} eqid {/} prn0 mt2))
+thm (prcdpq ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. A (P.)) (e. B A)) (-> (br C (<Q) B) (e. C A))) (x cv B A eleq1 A P. e. anbi2d x cv B y cv <Q breq2 anbi12d y cv A e. imbi1d y cv C <Q B breq1 A P. e. B A e. /\ anbi2d y cv C A eleq1 imbi12d A x y elnp pm3.27bd r19.21bi pm3.26d 19.21bi imp A V vtocl2g ltrelpq Q. Q. relxp <Q Q. Q. X. ssrel mp2 C B brrelexi sylan2 A P. e. adantll pm2.43i exp))
+thm (prub () () (-> (/\ (/\ (e. A (P.)) (e. B A)) (e. C (Q.))) (-> (-. (e. C A)) (br B (<Q) C))) (B C A eleq1 biimpcd A P. e. adantl A B C prcdpq jaod con3d C Q. e. adantr ltsopq <Q Q. B C sotric mpan A B elprpq sylan sylibrd))
+thm (prnmax ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (P.)) (e. B A)) (E. x (/\ (e. (cv x) A) (br B (<Q) (cv x))))) (y cv B A eleq1 A P. e. anbi2d y cv B <Q x cv breq1 x A birexdv imbi12d A y x elnp pm3.27bd r19.21bi pm3.27d A vtoclg anabsi7 x A B <Q x cv br df-rex sylib))
+thm (prnmadd ((x y) (A x) (A y) (B x) (B y)) ((prnmadd.1 (e. B (V)))) (-> (/\ (e. A (P.)) (e. B A)) (E. x (e. (opr B (+Q) (cv x)) A))) (A B y prnmax y visset ltrelpq B brel prnmadd.1 y cv x ltexpq biimpcd mpd y cv B +Q x cv opr eqcom x biex sylibr y cv A e. anim1i x y cv B +Q x cv opr = y cv A e. 19.41v sylibr ancoms y 19.22i y x y cv B +Q x cv opr = y cv A e. /\ excomim 3syl B +Q x cv opr A y df-clel x biex sylibr))
+thm (ltrelpr ((x y)) () (C_ (<P) (X. (P.) (P.))) (x y df-ltp x y P. P. x cv y cv C: opabssxp eqsstr))
+thm (genpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A F B) ({|} f (E. g (E. h (/\ (/\ (e. (cv g) A) (e. (cv h) B)) (= (cv f) (opr (cv g) G (cv h))))))))) (f cv A F g cv opreq1 f cv A y z g cv x cv y cv G z cv opr = E.e. rexeq x biabdv eqeq12d g cv B A F opreq2 g cv B z x cv y cv G z cv opr = rexeq y A birexdv x biabdv eqeq12d f visset g visset x y z G oprvalex w cv f cv y z v cv x cv y cv G z cv opr = E.e. rexeq x biabdv v cv g cv z x cv y cv G z cv opr = rexeq y f cv birexdv x biabdv genp.1 oprabval2 vtocl2ga x cv f cv g cv G h cv opr eqeq1 g cv A e. h cv B e. /\ anbi2d g h bi2exdv y A z B x cv y cv G z cv opr = r2ex y cv g cv A eleq1 z cv h cv B eleq1 bi2anan9 y cv g cv z cv h cv G opreq12 x cv eqeq2d anbi12d cbvex2v bitr syl5bb cbvabv syl6eq))
+thm (genpelv ((f g) (f h) (C f) (g h) (C g) (C h) (F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpelv.2 (e. C (V)))) (-> (/\ (e. A (P.)) (e. B (P.))) (<-> (e. C (opr A F B)) (E. f (E. g (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= C (opr (cv f) G (cv g)))))))) (genp.1 A B h f g genpv C eleq2d genpelv.2 h cv C f cv G g cv opr eqeq1 f cv A e. g cv B e. /\ anbi2d f g bi2exdv elab syl6bb))
+thm (genpprecl ((f g) (f h) (C f) (g h) (C g) (C h) (D f) (D g) (D h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (/\ (e. C A) (e. D B)) (e. (opr C G D) (opr A F B)))) (C G D opr eqid genp.1 A B f g h genpv C G D opr eleq2d C G D oprex f cv C G D opr g cv G h cv opr eqeq1 g cv A e. h cv B e. /\ anbi2d g h bi2exdv elab syl6bb g cv C A eleq1 h cv D B eleq1 bi2anan9 g cv C h cv D G opreq12 C G D opr eqeq2d anbi12d A B cla4e2gv anabsi5 syl5bir mpan2i))
+thm (genpdm ((x y) (x z) (y z) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (= (dom F) (X. (P.) (P.))) (w v u w cv P. e. v cv P. e. /\ u cv x y w cv z v cv x cv y cv G z cv opr = E.e. E.e. {|} = /\ dmoprab genp.1 dmeqi P. P. w v df-xp u w cv P. e. v cv P. e. /\ u cv x y w cv z v cv x cv y cv G z cv opr = E.e. E.e. {|} = 19.42v w visset v visset x y z G oprvalex u isseti mpbiran2 w v biopabi eqtr4 3eqtr4))
+thm (genpn0 ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (C: ({/}) (opr A F B))) (A prn0 A g n0 sylib B prn0 B h n0 sylib anim12i f g h g cv A e. h cv B e. /\ f cv g cv G h cv opr = /\ exrot3 f g cv A e. h cv B e. /\ f cv g cv G h cv opr = 19.42v g cv G h cv oprex f isseti mpbiran2 g h bi2ex g h g cv A e. h cv B e. eeanv 3bitr sylibr genp.1 A B f g h genpv eqabd f biexdv mpbird A F B opr 0pss A F B opr f n0 bitr sylibr))
+thm (genpss ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpss.2 (-> (/\ (e. (cv g) (Q.)) (e. (cv h) (Q.))) (e. (opr (cv g) G (cv h)) (Q.))))) (-> (/\ (e. A (P.)) (e. B (P.))) (C_ (opr A F B) (Q.))) (genp.1 A B f g h genpv eqabd A prpssnq pssssd g cv sseld B prpssnq pssssd h cv sseld im2anan9 genpss.2 syl6 g cv G h cv opr Q. f cv eleq1a syl6 imp3a g h 19.23advv sylbid ssrdv))
+thm (genpnnp ((v w) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpnnp.2 (-> (/\ (e. (cv w) (Q.)) (e. (cv v) (Q.))) (e. (opr (cv w) G (cv v)) (Q.)))) (genpnnp.3 (-> (e. (cv z) (Q.)) (<-> (br (cv x) (<Q) (cv y)) (br (opr (cv z) G (cv x)) (<Q) (opr (cv z) G (cv y)))))) (genpnnp.4 (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-. (= (opr A F B) (Q.)))) (A prpssnq A Q. w pssnel syl B prpssnq B Q. v pssnel syl anim12i w v w cv Q. e. w cv A e. -. /\ v cv Q. e. v cv B e. -. /\ eeanv sylibr A g cv w cv prub B h cv v cv prub im2anan9 ltsopq <Q Q. g cv w cv so2nr mpan h cv Q. e. v cv Q. e. /\ w cv G v cv opr g cv G h cv opr = ad2antll w visset v visset genpnnp.3 g visset genpnnp.4 h visset caoprord3 g cv <Q w cv br anbi2d h cv Q. e. v cv Q. e. pm3.27 g cv Q. e. w cv Q. e. pm3.26 anim12i ancoms sylan mtbid exp con2d A g cv elprpq w cv Q. e. anim1i B h cv elprpq v cv Q. e. anim1i syl2an syld an4s exp an4s exp com24 imp32 g cv A e. h cv B e. /\ w cv G v cv opr g cv G h cv opr = imnan sylib h nexdv g nexdv exp genp.1 A B f g h genpv w cv G v cv opr eleq2d w cv G v cv oprex f cv w cv G v cv opr g cv G h cv opr eqeq1 g cv A e. h cv B e. /\ anbi2d g h bi2exdv elab syl6bb negbid sylibrd com12 ancoms an4s genpnnp.2 A F B opr Q. w cv G v cv opr eleq2 biimprcd con3d syl w cv A e. -. adantlr v cv B e. -. adantrr syld com12 w v 19.23advv mpd))
+thm (genpcd ((F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpcd.2 (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) (<Q) (opr (cv g) G (cv h))) (e. (cv x) (opr A F B)))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (-> (br (cv x) (<Q) (cv f)) (e. (cv x) (opr A F B))))) (genp.1 f visset A B g h genpelv x cv Q. e. adantr f cv g cv G h cv opr x cv <Q breq2 biimpd genpcd.2 sylan9r exp31 an4s exp com23 imp4b g h 19.23advv sylbid exp f visset ltrelpq x cv brel pm3.26d syl5 com34 pm2.43d com23))
+thm (genpnmax ((x y) (h x) (F x) (h y) (F y) (F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpnmax.2 (-> (e. (cv v) (Q.)) (<-> (br (cv z) (<Q) (cv w)) (br (opr (cv v) G (cv z)) (<Q) (opr (cv v) G (cv w)))))) (genpnmax.3 (= (opr (cv z) G (cv w)) (opr (cv w) G (cv z))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (E. x (/\ (e. (cv x) (opr A F B)) (br (cv f) (<Q) (cv x)))))) (genp.1 A B f g h genpv eqabd f cv g cv G h cv opr <Q x cv breq1 x cv A F B opr e. anbi2d x biexdv A g cv y prnmax B P. e. h cv B e. /\ adantr genp.1 A B y cv h cv genpprecl exp4b com34 imp32 B h cv elprpq g visset y visset genpnmax.2 h visset genpnmax.3 caoprord2 biimpd syl A P. e. adantl anim12d y cv G h cv oprex x cv y cv G h cv opr A F B opr eleq1 x cv y cv G h cv opr g cv G h cv opr <Q breq2 anbi12d cla4ev syl6 g cv A e. adantlr y 19.23adv mpd an4s syl5bir exp3a com3l imp3a g h 19.23advv sylbid))
+thm (genpcl ((v w) (A w) (A v) (B w) (B v) (x y) (w x) (v x) (h x) (F x) (w y) (v y) (h y) (F y) (h w) (F w) (h v) (F v) (F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpcl.2 (-> (/\ (e. (cv x) (Q.)) (e. (cv y) (Q.))) (e. (opr (cv x) G (cv y)) (Q.)))) (genpcl.3 (-> (e. (cv h) (Q.)) (<-> (br (cv f) (<Q) (cv g)) (br (opr (cv h) G (cv f)) (<Q) (opr (cv h) G (cv g)))))) (genpcl.4 (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x)))) (genpcl.5 (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) (<Q) (opr (cv g) G (cv h))) (e. (cv x) (opr A F B)))))) (-> (/\ (e. A (P.)) (e. B (P.))) (e. (opr A F B) (P.))) (genp.1 A B genpn0 genp.1 genpcl.2 g cv h cv caoprcl A B genpss genp.1 genpcl.2 w cv v cv caoprcl x visset y visset genpcl.3 z cv caoprord genpcl.4 A B genpnnp jca A F B opr Q. dfpss2 sylibr jca genp.1 genpcl.5 f genpcd x 19.21adv genp.1 z visset w visset genpcl.3 v cv caoprord z visset w visset genpcl.4 caoprcom A B f genpnmax x A F B opr f cv <Q x cv br df-rex syl6ibr jcad r19.21aiv jca A F B opr f x elnp sylibr))
+thm (genpass ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (v w) (u w) (f w) (g w) (h w) (t w) (u v) (f v) (g v) (h v) (t v) (f u) (g u) (h u) (t u) (f g) (f h) (f t) (g h) (g t) (h t) (A x) (A f) (A g) (A h) (A t) (B x) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C f) (C g) (C h) (C t) (F x) (F y) (F z) (F h) (F t) (G t) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((genp.1 (= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (genpass.2 (e. B (V))) (genpass.3 (e. C (V))) (genpass.4 (= (dom F) (X. (P.) (P.)))) (genpass.5 (-> (/\ (e. (cv f) (P.)) (e. (cv g) (P.))) (e. (opr (cv f) F (cv g)) (P.)))) (genpass.6 (= (opr (opr (cv f) G (cv g)) G (cv h)) (opr (cv f) G (opr (cv g) G (cv h)))))) (= (opr (opr A F B) F C) (opr A F (opr B F C))) (genp.1 x visset A F B opr C t h genpelv genpass.5 A B caoprcl sylan 3impa genp.1 t visset A B f g genpelv C P. e. 3adant3 h cv C e. x cv t cv G h cv opr = /\ anbi1d t cv A F B opr e. h cv C e. x cv t cv G h cv opr = anass f g f cv A e. g cv B e. /\ t cv f cv G g cv opr = /\ h cv C e. x cv t cv G h cv opr = /\ 19.41vv 3bitr4g t h bi2exdv bitrd t h f g f cv A e. g cv B e. /\ t cv f cv G g cv opr = /\ h cv C e. x cv t cv G h cv opr = /\ /\ exrot4 t h f cv A e. g cv B e. /\ t cv f cv G g cv opr = /\ h cv C e. x cv t cv G h cv opr = /\ /\ excom f cv A e. g cv B e. /\ t cv f cv G g cv opr = h cv C e. x cv t cv G h cv opr = an4 f cv A e. g cv B e. h cv C e. anass t cv f cv G g cv opr G h cv opreq1 genpass.6 syl6eq x cv eqeq2d pm5.32i anbi12i f cv A e. g cv B e. h cv C e. /\ /\ t cv f cv G g cv opr = x cv f cv G g cv G h cv opr opr = an12 3bitr t biex t t cv f cv G g cv opr = f cv A e. g cv B e. h cv C e. /\ /\ x cv f cv G g cv G h cv opr opr = /\ 19.41v f cv G g cv oprex t isseti mpbiran bitr h biex bitr f g bi2ex bitr syl6bb genp.1 x visset A B F C opr f t genpelv genpass.5 B C caoprcl sylan2 3impb genp.1 t visset B C g h genpelv A P. e. 3adant1 f cv A e. x cv f cv G t cv opr = /\ anbi1d f cv A e. t cv B F C opr e. x cv f cv G t cv opr = anass f cv A e. t cv B F C opr e. x cv f cv G t cv opr = an12 bitr g h g cv B e. h cv C e. /\ t cv g cv G h cv opr = /\ f cv A e. x cv f cv G t cv opr = /\ 19.41vv 3bitr4g f t bi2exdv t g h g cv B e. h cv C e. /\ t cv g cv G h cv opr = /\ f cv A e. x cv f cv G t cv opr = /\ /\ exrot3 g cv B e. h cv C e. /\ t cv g cv G h cv opr = f cv A e. x cv f cv G t cv opr = an4 g cv B e. h cv C e. /\ f cv A e. ancom t cv g cv G h cv opr f cv G opreq2 x cv eqeq2d pm5.32i anbi12i f cv A e. g cv B e. h cv C e. /\ /\ t cv g cv G h cv opr = x cv f cv G g cv G h cv opr opr = an12 3bitr t biex t t cv g cv G h cv opr = f cv A e. g cv B e. h cv C e. /\ /\ x cv f cv G g cv G h cv opr opr = /\ 19.41v g cv G h cv oprex t isseti mpbiran bitr g h bi2ex bitr f biex syl6bb bitrd bitr4d eqrd genpass.2 genpass.4 genpass.3 0npr A ndmoprass pm2.61i))
+thm (plpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (+P.) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (+Q) (cv z))))))))) (w v u f g h df-plp A B x y z genpv))
+thm (mpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (.P) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (.Q) (cv z))))))))) (w v u f g h df-mp A B x y z genpv))
+thm (dmplp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (+P.)) (X. (P.) (P.))) (x y z w v u df-plp genpdm))
+thm (dmmp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (.P)) (X. (P.) (P.))) (x y z w v u df-mp genpdm))
+thm (1pr ((x y)) () (e. (1P) (P.)) (1P x y elnp 1lt2pq 1q elisseti 1Q +Q 1Q oprex ltrpq *Q 1Q fvex 1q elisseti mulcompq 1q 1Q recclpq ax-mp *Q 1Q ` mulidpq ax-mp 1q 1Q recidpq ax-mp 3eqtr3 syl6breq ax-mp *Q 1Q +Q 1Q opr fvex x cv *Q 1Q +Q 1Q opr ` <Q 1Q breq1 x df-1p elab2 mpbir *Q 1Q +Q 1Q opr ` 1P n0i ax-mp 1P 0pss mpbir 1P Q. dfpss2 x df-1p eqabi 1q elisseti ltrelpq x cv brel pm3.26d sylbi ssriv 1q elisseti ltsopq ltrelpq soirri 1q elisseti x cv 1Q <Q 1Q breq1 x df-1p elab2 mtbir 1q 1P Q. 1Q eleq2 mpbiri mto mpbir2an pm3.2i y visset ltsopq ltrelpq x visset 1q elisseti sotri exp x df-1p eqabi y df-1p eqabi 3imtr4g com12 y 19.21aiv x visset 1q elisseti y ltbtwnpq x df-1p eqabi y df-1p eqabi x cv <Q y cv br anbi1i y cv <Q 1Q br x cv <Q y cv br ancom bitr y biex 3imtr4 y 1P x cv <Q y cv br df-rex sylibr jca rgen mpbir2an))
+thm (addclprlem1 ((x y) (x z) (w x) (y z) (w y) (w z) (g y) (g z) (g w) (h y) (h z) (h w)) () (-> (/\ (/\ (e. A (P.)) (e. (cv g) A)) (e. (cv x) (Q.))) (-> (br (cv x) (<Q) (opr (cv g) (+Q) (cv h))) (e. (opr (opr (cv x) (.Q) (` (*Q) (opr (cv g) (+Q) (cv h)))) (.Q) (cv g)) A))) (*Q g cv +Q h cv opr fvex *Q x cv fvex x cv ltmpq x cv .Q *Q g cv +Q h cv opr ` oprex x cv .Q *Q x cv ` oprex y visset z visset w cv ltmpq g visset y visset z visset mulcompq caoprord2 sylan9bbr x visset g cv +Q h cv oprex ltrpq syl5bi x cv recidpq .Q g cv opreq1d g cv mulidpq 1q elisseti g visset mulcompq syl5eq sylan9eqr x cv .Q *Q g cv +Q h cv opr ` opr .Q g cv opr <Q breq2d sylibd A g cv elprpq sylan A g cv x cv .Q *Q g cv +Q h cv opr ` opr .Q g cv opr prcdpq x cv Q. e. adantr syld))
+thm (addclprlem2 ((x y) (x z) (w x) (v x) (u x) (g x) (h x) (y z) (w y) (v y) (u y) (g y) (h y) (w z) (v z) (u z) (g z) (h z) (v w) (u w) (g w) (h w) (u v) (g v) (h v) (g u) (h u) (g h) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) (<Q) (opr (cv g) (+Q) (cv h))) (e. (cv x) (opr A (+P.) B)))) (A g x h addclprlem1 B P. e. h cv B e. /\ adantlr B h x g addclprlem1 g visset h visset addcompq x cv <Q breq2i g visset h visset addcompq *Q fveq2i x cv .Q opreq2i .Q h cv opreq1i B eleq1i 3imtr4g A P. e. g cv A e. /\ adantll jcad A P. e. g cv A e. /\ B P. e. h cv B e. /\ /\ x cv Q. e. pm3.26 A P. e. g cv A e. pm3.26 B P. e. h cv B e. pm3.26 anim12i w v u x y z df-plp A B x cv .Q *Q g cv +Q h cv opr ` opr .Q g cv opr x cv .Q *Q g cv +Q h cv opr ` opr .Q h cv opr genpprecl 3syl syld A g cv elprpq B h cv elprpq anim12i g cv h cv addclpq g cv +Q h cv opr recidpq 3syl *Q g cv +Q h cv opr fvex g cv +Q h cv oprex mulcompq syl5eq x cv .Q opreq2d x cv mulidpq sylan9eq g visset h visset x cv .Q *Q g cv +Q h cv opr ` opr distrpq *Q g cv +Q h cv opr fvex g cv +Q h cv oprex x cv mulasspq eqtr3 syl5eq A +P. B opr eleq1d sylibd))
+thm (addclpr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h) (A x) (A y) (A z) (A w) (A v) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h)) () (-> (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (+P.) B) (P.))) (w v u x y z df-plp x cv y cv addclpq f visset g visset h cv ltapq x visset y visset addcompq A g B h x addclprlem2 genpcl))
+thm (mulclprlem ((x y) (x z) (w x) (v x) (u x) (g x) (h x) (y z) (w y) (v y) (u y) (g y) (h y) (w z) (v z) (u z) (g z) (h z) (v w) (u w) (g w) (h w) (u v) (g v) (h v) (g u) (h u) (g h) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) (<Q) (opr (cv g) (.Q) (cv h))) (e. (cv x) (opr A (.P) B)))) (h cv recclpq g cv Q. e. adantl x visset g cv .Q h cv oprex y visset z visset w cv ltmpq *Q h cv fvex y visset z visset mulcompq caoprord2 syl h cv recidpq g cv .Q opreq2d h visset *Q h cv fvex g cv mulasspq syl5eq g cv mulidpq sylan9eqr x cv .Q *Q h cv ` opr <Q breq2d bitrd A g cv elprpq B h cv elprpq syl2an A g cv x cv .Q *Q h cv ` opr prcdpq B P. e. h cv B e. /\ adantr sylbid w v u x y z df-mp A B x cv .Q *Q h cv ` opr h cv genpprecl exp4b com34 imp32 g cv A e. adantlr syld x cv Q. e. adantr h cv recidpq *Q h cv fvex h visset mulcompq syl5eq x cv .Q opreq2d *Q h cv fvex h visset x cv mulasspq syl5eq x cv mulidpq sylan9eq A .P B opr eleq1d B h cv elprpq A P. e. g cv A e. /\ adantl sylan sylibd))
+thm (mulclpr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h) (A x) (A y) (A z) (A w) (A v) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h)) () (-> (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (.P) B) (P.))) (w v u x y z df-mp x cv y cv mulclpq f visset g visset h cv ltmpq x visset y visset mulcompq A g B h x mulclprlem genpcl))
+thm (addcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((addcompr.1 (e. A (V))) (addcompr.2 (e. B (V)))) (= (opr A (+P.) B) (opr B (+P.) A)) (A B x y z plpv B A x z y plpv z cv B e. y cv A e. ancom z visset y visset addcompq x cv eqeq2i anbi12i z y bi2ex z y y cv A e. z cv B e. /\ x cv y cv +Q z cv opr = /\ excom bitr x biabi syl6eq ancoms eqtr4d addcompr.2 dmplp addcompr.1 ndmoprcom pm2.61i))
+thm (addasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((addasspr.1 (e. B (V))) (addasspr.2 (e. C (V)))) (= (opr (opr A (+P.) B) (+P.) C) (opr A (+P.) (opr B (+P.) C))) (w v u x y z df-plp addasspr.1 addasspr.2 dmplp f cv g cv addclpr g visset h visset f cv addasspq A genpass))
+thm (mulcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((mulcompr.1 (e. A (V))) (mulcompr.2 (e. B (V)))) (= (opr A (.P) B) (opr B (.P) A)) (A B x y z mpv B A x z y mpv z cv B e. y cv A e. ancom z visset y visset mulcompq x cv eqeq2i anbi12i z y bi2ex z y y cv A e. z cv B e. /\ x cv y cv .Q z cv opr = /\ excom bitr x biabi syl6eq ancoms eqtr4d mulcompr.2 dmmp mulcompr.1 ndmoprcom pm2.61i))
+thm (mulasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((mulasspr.1 (e. B (V))) (mulasspr.2 (e. C (V)))) (= (opr (opr A (.P) B) (.P) C) (opr A (.P) (opr B (.P) C))) (w v u x y z df-mp mulasspr.1 mulasspr.2 dmmp f cv g cv mulclpr g visset h visset f cv mulasspq A genpass))
+thm (distrlem1pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr A (.P) (opr B (+P.) C)) (opr (opr A (.P) B) (+P.) (opr A (.P) C)))) (y z u f g h df-mp w visset A B +P. C opr x v genpelv B C addclpr sylan2 x w u f g h df-plp v visset B C y z genpelv w cv x cv .Q v cv opr = anbi1d x cv A e. anbi2d x cv A e. v cv B +P. C opr e. w cv x cv .Q v cv opr = anass y z x cv A e. y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = /\ w cv x cv .Q v cv opr = /\ 19.42vv y z y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = /\ w cv x cv .Q v cv opr = 19.41vv x cv A e. anbi2i bitr 3bitr4g A P. e. adantl x v bi2exdv bitrd x v y z x cv A e. y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = /\ w cv x cv .Q v cv opr = /\ /\ exrot4 y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = w cv x cv .Q v cv opr = anass v biex v y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = w cv x cv .Q v cv opr = /\ 19.42v y cv +Q z cv oprex v cv y cv +Q z cv opr x cv .Q opreq2 w cv eqeq2d ceqsexv y cv B e. z cv C e. /\ anbi2i 3bitr x cv A e. anbi2i v x cv A e. y cv B e. z cv C e. /\ v cv y cv +Q z cv opr = /\ w cv x cv .Q v cv opr = /\ 19.42v x cv A e. y cv B e. z cv C e. /\ w cv x cv .Q y cv +Q z cv opr opr = anass 3bitr4 y z x bi3ex bitr syl6bb w v u f g h df-mp A B x cv y cv genpprecl w v u f g h df-mp A C x cv z cv genpprecl im2anan9 w v u f g h df-plp A .P B opr A .P C opr x cv .Q y cv opr x cv .Q z cv opr genpprecl A B mulclpr A C mulclpr syl2an syld x cv A e. y cv B e. z cv C e. anandi y visset z visset x cv distrpq A .P B opr +P. A .P C opr opr eleq1i 3imtr4g anandis x cv .Q y cv +Q z cv opr opr A .P B opr +P. A .P C opr opr w cv eleq1a syl6 imp3a z x 19.23advv y 19.23adv sylbid ssrdv))
+thm (distrlem2pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (-> (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv x) (.Q) (cv z))) (opr A (.P) (opr B (+P.) C))))) (w v u f g h df-plp B C y cv z cv genpprecl x cv A e. anim2d A P. e. adantl w v u f g h df-mp A B +P. C opr x cv y cv +Q z cv opr genpprecl B C addclpr sylan2 syld y visset z visset x cv distrpq A .P B +P. C opr opr eleq1i syl6ib))
+thm (distrlem3pr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C)))) (/\ (e. (cv x) (Q.)) (/\ (e. (cv y) (Q.)) (e. (cv z) (Q.))))) (A P. e. B P. e. C P. e. x cv A e. y cv B e. z cv C e. an6 A x cv elprpq B y cv elprpq C z cv elprpq 3anim123i sylbi A P. e. B P. e. C P. e. 3anass x cv A e. y cv B e. z cv C e. 3anass anbi12i x cv Q. e. y cv Q. e. z cv Q. e. 3anass 3imtr3))
+thm (distrlem4pr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C)))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) (opr A (.P) (opr B (+P.) C)))) (A B C f y z distrlem3pr x visset f visset w visset v visset u cv ltmpq y visset w visset v visset mulcompq caoprord2 f cv z cv mulclpq x cv .Q y cv oprex f cv .Q y cv oprex w visset v visset u cv ltapq f cv .Q z cv oprex w visset v visset addcompq caoprord2 syl sylan9bb an1s syl A B C f y z distrlem2pr A B +P. C opr mulclpr B C addclpr sylan2 A .P B +P. C opr opr f cv .Q y cv opr +Q f cv .Q z cv opr opr x cv .Q y cv opr +Q f cv .Q z cv opr opr prcdpq exp syl syld imp sylbid x cv A e. adantrll A B C x y z distrlem3pr f visset x visset w visset v visset u cv ltmpq z visset w visset v visset mulcompq caoprord2 x cv y cv mulclpq f cv .Q z cv oprex x cv .Q z cv oprex x cv .Q y cv opr ltapq syl sylan9bbr anasss syl A B C x y z distrlem2pr A B +P. C opr mulclpr B C addclpr sylan2 A .P B +P. C opr opr x cv .Q y cv opr +Q x cv .Q z cv opr opr x cv .Q y cv opr +Q f cv .Q z cv opr opr prcdpq exp syl syld imp sylbid f cv A e. adantrlr A x cv elprpq A f cv elprpq anim12i anandis ltsopq <Q Q. x cv f cv sotrieq mpan syl B P. e. C P. e. /\ adantlr y cv B e. z cv C e. /\ adantrr x cv f cv .Q z cv opreq1 x cv .Q y cv opr +Q opreq2d A .P B +P. C opr opr eleq1d A B C x y z distrlem2pr x cv A e. f cv A e. pm3.26 sylani imp syl5bi com12 sylbird ecase3d))
+thm (distrlem5pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr (opr A (.P) B) (+P.) (opr A (.P) C)) (opr A (.P) (opr B (+P.) C)))) (x y z f g h df-plp w visset A .P B opr A .P C opr v u genpelv A B mulclpr A C mulclpr syl2an w v u f g h df-mp v visset A B x y genpelv w v u x g h df-mp u visset A C f z genpelv bi2anan9 x y f z x cv A e. y cv B e. /\ v cv x cv .Q y cv opr = /\ f cv A e. z cv C e. /\ u cv f cv .Q z cv opr = /\ ee4anv syl6bbr w cv v cv +Q u cv opr = anbi1d f z x cv A e. y cv B e. /\ v cv x cv .Q y cv opr = /\ f cv A e. z cv C e. /\ u cv f cv .Q z cv opr = /\ /\ w cv v cv +Q u cv opr = 19.41vv x y bi2ex x y f z x cv A e. y cv B e. /\ v cv x cv .Q y cv opr = /\ f cv A e. z cv C e. /\ u cv f cv .Q z cv opr = /\ /\ E. E. w cv v cv +Q u cv opr = 19.41vv bitr syl6bbr v u bi2exdv v u x y f z x cv A e. y cv B e. /\ v cv x cv .Q y cv opr = /\ f cv A e. z cv C e. /\ u cv f cv .Q z cv opr = /\ /\ w cv v cv +Q u cv opr = /\ E. E. exrot4 v u f z x cv A e. y cv B e. /\ v cv x cv .Q y cv opr = /\ f cv A e. z cv C e. /\ u cv f cv .Q z cv opr = /\ /\ w cv v cv +Q u cv opr = /\ exrot4 v u x cv A e. f cv A e. /\ y cv B e. z cv C e. /\ /\ v cv x cv .Q y cv opr = u cv f cv .Q z cv opr = /\ w cv v cv +Q u cv opr = /\ 19.42vv x cv A e. f cv A e. /\ y cv B e. z cv C e. /\ /\ v cv x cv .Q y cv opr = u cv f cv .Q z cv opr = /\ w cv v cv +Q u cv opr = anass x cv A e. f cv A e. y cv B e. z cv C e. an4 v cv x cv .Q y cv opr = u cv f cv .Q z cv opr = /\ anbi1i x cv A e. y cv B e. /\ f cv A e. z cv C e. /\ v cv x cv .Q y cv opr = u cv f cv .Q z cv opr = an4 bitr w cv v cv +Q u cv opr = anbi1i bitr3 v u bi2ex x cv .Q y cv oprex f cv .Q z cv oprex v cv x cv .Q y cv opr = u cv f cv .Q z cv opr = /\ id v cv x cv .Q y cv opr u cv f cv .Q z cv opr +Q opreq12 w cv eqeq2d V V cgsex2g mp2an x cv A e. f cv A e. /\ y cv B e. z cv C e. /\ /\ anbi2i 3bitr3 f z bi2ex bitr x y bi2ex bitr syl6bb bitrd anandis w cv x cv .Q y cv opr +Q f cv .Q z cv opr opr A .P B +P. C opr opr eleq1 A B C x f y z distrlem4pr syl5bir com12 exp imp3a f z 19.23advv x y 19.23advv sylbid ssrdv))
+thm (distrpr () ((distrpr.1 (e. B (V))) (distrpr.2 (e. C (V)))) (= (opr A (.P) (opr B (+P.) C)) (opr (opr A (.P) B) (+P.) (opr A (.P) C))) (A B C distrlem1pr A B C distrlem5pr eqssd 3impb distrpr.1 dmplp distrpr.2 0npr dmmp A ndmoprdistr pm2.61i))
+thm (1idpr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g)) () (-> (e. A (P.)) (= (opr A (.P) (1P)) A)) (x cv f cv .Q g cv opr <Q f cv breq1 g visset 1q elisseti f cv ltmpq f cv mulidpq f cv .Q g cv opr <Q breq2d bitrd g df-1p eqabi syl5rbb sylan9bbr A f cv elprpq sylan exp31 imp3a pm5.32d f cv A e. x cv <Q f cv br x cv f cv .Q g cv opr = an23 f cv A e. g cv 1P e. x cv f cv .Q g cv opr = an23 3bitr4g g biexdv g f cv A e. x cv <Q f cv br /\ x cv f cv .Q g cv opr = 19.42v syl5rbbr f biexdv 1pr w v u x y z df-mp x visset A 1P f g genpelv mpan2 A x cv f prnmax f visset ltrelpq x cv brel f cv recidpq x cv .Q opreq2d f visset x visset *Q f cv fvex y visset z visset mulcompq z visset w visset y cv mulasspq caopr12 syl5eq x cv mulidpq sylan9eqr eqcomd x cv .Q *Q f cv ` oprex g cv x cv .Q *Q f cv ` opr f cv .Q opreq2 x cv eqeq2d cla4ev 3syl f cv A e. adantl ancli f 19.22i syl exp A f cv x cv prcdpq exp imp3a g x cv f cv .Q g cv opr = E. adantrd f 19.23adv impbid 3bitr4d eqrd))
+thm (ltprord ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (P.)) (e. B (P.))) (<-> (br A (<P) B) (C: A B))) (x cv A P. eleq1 y cv P. e. anbi1d x cv A y cv psseq1 anbi12d y cv B P. eleq1 A P. e. anbi2d y cv B A psseq2 anbi12d x y df-ltp P. P. brabg A P. e. B P. e. /\ A B C: ibar bitr4d))
+thm (psslinpr ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (P.)) (e. B (P.))) (\/\/ (C: A B) (= A B) (C: B A))) (B y cv x cv prub A x cv elprpq sylan2 A x cv y cv prcdpq B P. e. y cv B e. /\ adantl syld exp43 com3r imp imp4a com23 y 19.21adv x 19.23adv A B sspss negbii A B x nss bitr3 B A sspss B A y dfss2 bitr3 3imtr4g orrd A B C: A B = B A C: df-3or A B C: A B = B A C: or23 A B C: B A C: A B = orordir B A eqcom B A C: orbi2i A B C: A B = \/ orbi2i bitr4 3bitr sylibr))
+thm (ltsopr ((x y) (x z) (y z)) () (Or (<P) (P.)) (x cv pssirr x cv x cv ltprord mtbiri anidms x cv y cv z cv psstr x cv y cv ltprord z cv P. e. 3adant3 y cv z cv ltprord x cv P. e. 3adant1 anbi12d x cv z cv ltprord y cv P. e. 3adant2 imbi12d mpbiri x cv y cv psslinpr x cv y cv ltprord x cv P. e. y cv P. e. /\ x cv y cv = pm4.2i y cv x cv ltprord ancoms 3orbi123d mpbird itlso))
+thm (prlem934a ((x y) (w x) (C x) (w y) (C y) (C w) (x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (f x) (B x) (f y) (B y) (f z) (B z) (f w) (B w) (f v) (B v) (f u) (B u) (B f)) ((prlem934a.1 (e. B (V)))) (-> (e. C (N.)) (-> (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) (e. (opr (cv y) (+Q) (opr ([] (<,> C (1o)) (~Q)) (.Q) B)) A))) (w cv 1o 1o opeq1 w cv 1o <,> 1o 1o <,> ~Q eceq2 syl .Q B opreq1d y cv +Q opreq2d A eleq1d B Q. e. x x cv A e. x cv +Q B opr A e. -> A. /\ y cv A e. /\ imbi2d w cv z cv 1o opeq1 w cv 1o <,> z cv 1o <,> ~Q eceq2 syl .Q B opreq1d y cv +Q opreq2d A eleq1d B Q. e. x x cv A e. x cv +Q B opr A e. -> A. /\ y cv A e. /\ imbi2d w cv z cv +N 1o opr 1o opeq1 w cv 1o <,> z cv +N 1o opr 1o <,> ~Q eceq2 syl .Q B opreq1d y cv +Q opreq2d A eleq1d B Q. e. x x cv A e. x cv +Q B opr A e. -> A. /\ y cv A e. /\ imbi2d w cv C 1o opeq1 w cv 1o <,> C 1o <,> ~Q eceq2 syl .Q B opreq1d y cv +Q opreq2d A eleq1d B Q. e. x x cv A e. x cv +Q B opr A e. -> A. /\ y cv A e. /\ imbi2d x cv y cv A eleq1 x cv y cv +Q B opreq1 A eleq1d imbi12d a4b1 B mulidpq prlem934a.1 1q elisseti mulcompq syl5eqr df-1q .Q B opreq1i syl5eqr y cv +Q opreq2d A eleq1d biimprd sylan9r imp y cv +Q z cv 1o <,> ~Q [] .Q B opr oprex x cv y cv +Q z cv 1o <,> ~Q [] .Q B opr opr A eleq1 x cv y cv +Q z cv 1o <,> ~Q [] .Q B opr opr +Q B opreq1 A eleq1d imbi12d cla4v B mulidpq prlem934a.1 1q elisseti mulcompq syl5reqr z cv 1o <,> ~Q [] .Q B opr +Q opreq2d enqex ~Q V z cv 1o <,> ecexg ax-mp 1q elisseti prlem934a.1 v visset u visset mulcompq u visset f visset v cv distrpq caoprdistrr syl6eqr 1pi 1pi 1pi pm3.2i z cv 1o 1o 1o addpipq mpan2 mpan2 df-1q z cv 1o <,> ~Q [] +Q opreq2i syl5eq z cv mulidpi 1pi 1o mulidpi ax-mp z cv N. e. a1i +N opreq12d 1pi 1o mulidpi ax-mp jctir z cv .N 1o opr +N 1o .N 1o opr opr z cv +N 1o opr 1o .N 1o opr 1o opeq12 z cv .N 1o opr +N 1o .N 1o opr opr 1o .N 1o opr <,> z cv +N 1o opr 1o <,> ~Q eceq2 3syl eqtrd .Q B opreq1d sylan9eqr y cv +Q opreq2d z cv 1o <,> ~Q [] .Q B oprex prlem934a.1 y cv addasspq syl5eq A eleq1d biimpd sylan9r exp31 imp3a y cv A e. adantrd a2d indpi))
+thm (prlem934b ((w z) (v z) (u z) (v w) (u w) (u v) (f z) (f w) (f v) (f u) (g z) (h z) (g w) (h w) (g v) (h v) (g u) (h u) (f g) (f h) (g h)) () (-> (/\ (/\ (e. (cv u) (N.)) (e. (cv w) (N.))) (/\ (e. (cv v) (N.)) (e. (cv z) (N.)))) (\/ (= (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (cv v) (cv u)) (~Q))) (br ([] (<,> (cv v) (cv u)) (~Q)) (<Q) (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q)))))) (u cv z cv mulclpi u cv .N z cv opr nlt1pi 1pi ltsopi <N N. u cv .N z cv opr 1o sotric mpan mpan2 bicon2d mpbiri syl v cv N. e. adantl 1pi v cv .N z cv opr 1o v cv u cv enqeceq ancoms mpanr2 v cv z cv mulclpi sylan2 an4s anabsan u cv .N z cv opr 1o .N v cv opreq1 v visset z visset u visset f visset g visset mulcompi g visset h visset f cv mulasspi caopr31 syl5eq syl5bir 1pi elisseti u cv .N z cv oprex v cv ltmpi v visset u visset v cv .N z cv oprex 1pi elisseti ordpipq u visset v visset z visset f visset g visset mulcompi g visset h visset f cv mulasspi caopr12 v cv .N 1o opr <N breq2i bitr syl6bbr biimpd u cv N. e. z cv N. e. /\ adantr orim12d mpd an1s w cv N. e. adantlr w cv N. e. w cv N. e. v cv N. e. z cv N. e. an42 1pi w cv .N v cv opr 1o z cv w cv mulpipq mpanl2 w cv v cv mulclpi sylan sylbi anabsan 1pi w visset v cv .N z cv oprex 1pi elisseti distrpqlem mp3an3 v cv z cv mulclpi sylan2 v visset z visset w cv mulasspi 1pi elisseti w visset mulcompi w cv .N v cv opr .N z cv opr w cv .N v cv .N z cv opr opr 1o .N w cv opr w cv .N 1o opr opeq12 mp2an w cv .N v cv opr .N z cv opr 1o .N w cv opr <,> w cv .N v cv .N z cv opr opr w cv .N 1o opr <,> ~Q eceq2 ax-mp syl5eq eqtrd u cv N. e. adantll w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr v cv .N z cv opr 1o <,> ~Q [] v cv u cv <,> ~Q [] eqeq1 w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr v cv .N z cv opr 1o <,> ~Q [] v cv u cv <,> ~Q [] <Q breq2 orbi12d syl mpbird))
+thm (prlem934 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (P.)) (e. B (Q.))) (E. x (/\ (e. (cv x) A) (-. (e. (opr (cv x) (+Q) B) A))))) (A prpssnq A Q. dfpss3 sylib pm3.27d B Q. e. adantl df-nq v cv u cv <,> ~Q [] y cv A eleq1 x x cv A e. x cv +Q z cv w cv <,> ~Q [] opr A e. -> A. imbi2d A P. e. imbi2d z cv w cv <,> ~Q [] B x cv +Q opreq2 A eleq1d x cv A e. imbi2d x bialdv y cv A e. imbi1d A P. e. imbi2d u cv N. e. w cv N. e. pm3.27 u cv N. e. w cv N. e. pm3.27 jca v cv N. e. z cv N. e. /\ anim1i w cv N. e. w cv N. e. v cv N. e. z cv N. e. an42 sylib w cv v cv mulclpi enqex ~Q V z cv w cv <,> ecexg ax-mp w cv .N v cv opr x A y prlem934a exp4c enqex df-nq z cv w cv ecopqsi syl5 syl imp syl com23 A P. e. adantld A y cv +Q w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr opr v cv u cv <,> ~Q [] prcdpq w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] oprex y visset ltaddpq u cv N. e. w cv N. e. pm3.27 u cv N. e. w cv N. e. pm3.27 jca v cv N. e. z cv N. e. /\ anim1i w cv N. e. w cv N. e. v cv N. e. z cv N. e. an42 sylib w cv v cv mulclpi 1pi enqex df-nq w cv .N v cv opr 1o ecopqsi mpan2 syl enqex df-nq z cv w cv ecopqsi anim12i w cv .N v cv opr 1o <,> ~Q [] z cv w cv <,> ~Q [] mulclpq 3syl sylan u w v z prlem934b w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr v cv u cv <,> ~Q [] <Q w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr +Q y cv opr breq1 biimpd enqex ~Q V v cv u cv <,> ecexg ax-mp ltsopq ltrelpq w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] oprex w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] opr +Q y cv oprex sotri exp jaoi syl y cv Q. e. adantr mpd w cv .N v cv opr 1o <,> ~Q [] .Q z cv w cv <,> ~Q [] oprex y visset addcompq syl6breq syl5 exp4b com24 A y cv elprpq syl5 anabsi5 com12 syldd y 19.23adv A prn0 A y n0 sylib ancli y A P. e. y cv A e. 19.42v sylibr syl5 ancoms an4s 2ecoptocl exp com4l imp y 19.21adv Q. A y dfss2 syl6ibr mtod x x cv A e. x cv +Q B opr A e. exanali sylibr ancoms))
+thm (ltaddpr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (br A (<P) (opr A (+P.) B))) (B prn0 B y n0 sylib A P. e. adantl A +P. B opr x cv +Q y cv opr elprpq y visset dmaddpq 0npq x cv ndmoprrcl x visset y visset ltaddpq 3syl A +P. B opr x cv +Q y cv opr x cv prcdpq mpd A B addclpr x cv A e. y cv B e. /\ adantr w v u x y z df-plp A B x cv y cv genpprecl imp sylanc exp32 com23 x 19.21adv A A +P. B opr x dfss2 syl6ibr w v u x y z df-plp A B x cv y cv genpprecl A A +P. B opr x cv +Q y cv opr eleq2 biimprcd con3d syl6 exp3a com34 imp3a x 19.23adv A y cv x prlem934 syl5 B y cv elprpq sylan2i exp4d imp3a pm2.43i pm2.43d jcad A A +P. B opr dfpss2 syl6ibr y 19.23adv mpd A B addclpr A A +P. B opr ltprord syldan mpbird))
+thm (ltaddpr2 () ((ltaddpr2.1 (e. B (V)))) (-> (e. C (P.)) (-> (= (opr A (+P.) B) C) (br A (<P) C))) (A +P. B opr C P. eleq1 ltaddpr2.1 dmplp 0npr A ndmoprrcl syl6bir A +P. B opr C A <P breq2 A B ltaddpr syl5bi syld com12))
+thm (ltexprlem1 ((x y) (A x) (A y) (B x) (B y) (C x)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (-> (C: A B) (-. (= C ({/}))))) (y visset B x prnmadd y cv A e. -. anim2i x y cv A e. -. y cv +Q x cv opr B e. 19.42v sylibr exp32 com3l imp3a y 19.22dv A B y pssnel syl5 ltexprlem.1 eqabi x biex C x n0 y x y cv A e. -. y cv +Q x cv opr B e. /\ excom 3bitr4 syl6ibr))
+thm (ltexprlem2 ((x y) (A x) (A y) (B x) (B y) (C x)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (C: C (Q.))) (C B Q. sspsstr B y cv +Q x cv opr elprpq x visset dmaddpq 0npq y cv ndmoprrcl syl B y cv +Q x cv opr x cv prcdpq x visset y visset ltaddpq ancoms x visset y visset addcompq syl6breq syl5 mpd exp y cv A e. -. adantld y 19.23adv ltexprlem.1 eqabi syl5ib ssrdv B prpssnq sylanc))
+thm (ltexprlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C z)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (-> (e. (cv x) C) (A. z (-> (br (cv z) (<Q) (cv x)) (e. (cv z) C))))) (B y cv +Q x cv opr elprpq x visset dmaddpq 0npq y cv ndmoprrcl pm3.26d z visset x visset y cv ltapq 3syl B y cv +Q x cv opr y cv +Q z cv opr prcdpq sylbid exp com23 imp y cv A e. -. anim2d y 19.22dv ltexprlem.1 eqabi z visset x cv z cv y cv +Q opreq2 B eleq1d y cv A e. -. anbi2d y biexdv ltexprlem.1 elab2 3imtr4g exp com23 z 19.21adv))
+thm (ltexprlem4 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C z) (C w)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (e. B (P.)) (-> (e. (cv x) C) (E. z (/\ (e. (cv z) C) (br (cv x) (<Q) (cv z)))))) (B y cv +Q x cv opr w prnmax w visset ltrelpq y cv +Q x cv opr brel pm3.26d x visset dmaddpq 0npq y cv ndmoprrcl syl y visset ltsopq ltrelpq y cv +Q x cv oprex w visset sotri y visset x visset ltaddpq sylan mpancom w visset ltrelpq y cv brel y visset w cv z ltexpq biimpd mpcom syl w cv y cv +Q z cv opr eqcom z biex sylibr ancri w cv B e. anim2i z w cv y cv +Q z cv opr = E. w cv B e. y cv +Q x cv opr <Q w cv br an12 sylibr z w cv y cv +Q z cv opr = w cv B e. y cv +Q x cv opr <Q w cv br /\ 19.41v sylibr w 19.22i z w w cv y cv +Q z cv opr = w cv B e. y cv +Q x cv opr <Q w cv br /\ /\ excom sylibr y cv +Q z cv oprex w cv y cv +Q z cv opr B eleq1 w cv y cv +Q z cv opr y cv +Q x cv opr <Q breq2 anbi12d ceqsexv x visset dmaddpq ltrelpq 0npq x visset z visset y cv ltapq ndmordi y cv +Q z cv opr B e. anim2i sylbi z 19.22i 3syl y cv A e. -. anim2i an1s z y cv A e. -. y cv +Q z cv opr B e. x cv <Q z cv br /\ 19.42v sylibr exp y 19.22dv ltexprlem.1 eqabi z visset x cv z cv y cv +Q opreq2 B eleq1d y cv A e. -. anbi2d y biexdv ltexprlem.1 elab2 x cv <Q z cv br anbi1i y cv A e. -. y cv +Q z cv opr B e. x cv <Q z cv br anass y biex y y cv A e. -. y cv +Q z cv opr B e. /\ x cv <Q z cv br 19.41v bitr3 bitr4 z biex y z y cv A e. -. y cv +Q z cv opr B e. x cv <Q z cv br /\ /\ excom bitr4 3imtr4g))
+thm (ltexprlem5 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C z)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (/\ (e. B (P.)) (C: A B)) (e. C (P.))) (ltexprlem.1 ltexprlem1 C 0pss syl6ibr imp ltexprlem.1 ltexprlem2 A B C: adantr jca ltexprlem.1 z ltexprlem3 ltexprlem.1 z ltexprlem4 z C x cv <Q z cv br df-rex syl6ibr jcad r19.21aiv A B C: adantr jca C x z elnp sylibr))
+thm (ltexprlem6 ((x y) (x z) (w x) (v x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (f y) (g y) (h y) (A y) (w z) (v z) (f z) (g z) (h z) (A z) (v w) (f w) (g w) (h w) (A w) (f v) (g v) (h v) (A v) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (C x) (C z) (C w) (C v) (C f) (C g) (C h) (u x) (u y) (u z) (u w) (u v) (f u) (g u) (h u)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ (opr A (+P.) C) B)) (z v u f g h df-plp z visset A C w x genpelv ltexprlem.1 ltexprlem5 sylan2 z cv w cv +Q x cv opr B eleq1 biimparc A w cv y cv prub B y cv +Q x cv opr elprpq x visset dmaddpq 0npq y cv ndmoprrcl pm3.26d syl sylan2 B y cv +Q x cv opr elprpq x visset dmaddpq 0npq y cv ndmoprrcl pm3.27d w visset y visset z visset v visset u cv ltapq x visset z visset v visset addcompq caoprord2 3syl B y cv +Q x cv opr w cv +Q x cv opr prcdpq sylbid A P. e. w cv A e. /\ adantl syld exp32 com34 imp4b y 19.23adv ltexprlem.1 eqabi syl5ib exp31 com23 imp43 sylan exp31 imp3a w x 19.23advv A B C: adantrr sylbid ssrdv anassrs))
+thm (ltexprlem7 ((x y) (x z) (w x) (v x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (f y) (g y) (h y) (A y) (w z) (v z) (f z) (g z) (h z) (A z) (v w) (f w) (g w) (h w) (A w) (f v) (g v) (h v) (A v) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (C x) (C z) (C w) (C v) (C f) (C g) (C h) (u x) (u y) (u z) (u w) (u v) (f u) (g u) (h u)) ((ltexprlem.1 (= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B))))))) (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ B (opr A (+P.) C))) (A C ltaddpr A C addclpr A A +P. C opr ltprord syldan mpbid pssssd w cv sseld w cv B e. a1d B P. e. a1d com4r exp3a w visset B v prnmadd B w cv +Q v cv opr elprpq v visset dmaddpq 0npq w cv ndmoprrcl syl A v cv z prlem934 A z cv w cv prub z visset w cv x ltexpq A z cv elprpq sylan sylibd exp C P. e. adantlr z cv +Q v cv opr A e. -. adantrr z v u f g h df-plp A C z cv x cv genpprecl z cv +Q v cv oprex y cv z cv +Q v cv opr A eleq1 negbid y cv z cv +Q v cv opr +Q x cv opreq1 B eleq1d anbi12d cla4ev ltexprlem.1 eqabi sylibr z cv +Q x cv opr w cv +Q v cv opreq1 z visset v visset x visset f visset g visset addcompq g visset h visset f cv addasspq caopr32 syl5eq B eleq1d biimpar sylan2 sylan2i exp4d imp42 z cv +Q x cv opr w cv A +P. C opr eleq1 A P. e. C P. e. /\ z cv A e. z cv +Q v cv opr A e. -. /\ /\ w cv +Q v cv opr B e. ad2antrl mpbid exp32 x 19.23adv syl6d exp31 com23 z 19.23adv v cv Q. e. adantr mpd com24 exp com14 imp4b com4r com12 B P. e. adantld mpcom exp v 19.23adv w cv B e. adantr mpd exp com4t exp pm2.61i ltexprlem.1 ltexprlem5 syl5 exp3a com34 pm2.43d imp31 ssrdv))
+thm (ltexpri ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((ltexpri.1 (e. B (V)))) (-> (br A (<P) B) (E. x (/\ (e. (cv x) (P.)) (= (opr A (+P.) (cv x)) B)))) (ltexpri.1 ltrelpr A brel A B ltprord x cv y w w cv A e. -. w cv +Q y cv opr B e. /\ E. {|} A +P. opreq2 B eqeq1d P. cla4egv y cv z cv w cv +Q opreq2 B eleq1d w cv A e. -. anbi2d w biexdv cbvabv ltexprlem5 A P. e. adantll y cv z cv w cv +Q opreq2 B eleq1d w cv A e. -. anbi2d w biexdv cbvabv ltexprlem6 y cv z cv w cv +Q opreq2 B eleq1d w cv A e. -. anbi2d w biexdv cbvabv ltexprlem7 eqssd sylc exp sylbid A +P. x cv opr B P. eleq1 x visset dmplp 0npr A ndmoprrcl pm3.27d syl6bir com12 A P. e. adantl ancrd x 19.22dv syld mpcom))
+thm (ltaprlem ((A x) (B x) (C x)) ((ltaprlem.1 (e. A (V))) (ltaprlem.2 (e. B (V)))) (-> (e. C (P.)) (-> (br A (<P) B) (br (opr C (+P.) A) (<P) (opr C (+P.) B)))) (ltaprlem.2 A x ltexpri A +P. x cv opr B C +P. opreq2 ltaprlem.1 x visset C addasspr syl5eq C +P. A opr <P breq2d C +P. A opr x cv ltaddpr syl5bi exp3a C A addclpr syl5 com3r imp x 19.23aiv syl ltaprlem.2 ltrelpr A brel pm3.26d sylan2i exp3a pm2.43b))
+thm (ltapr () ((ltapr.1 (e. A (V))) (ltapr.2 (e. B (V)))) (-> (e. C (P.)) (<-> (br A (<P) B) (br (opr C (+P.) A) (<P) (opr C (+P.) B)))) (ltapr.2 dmplp ltapr.1 ltrelpr 0npr ltapr.1 ltapr.2 C ltaprlem B P. e. A P. e. /\ adantr ltapr.2 ltapr.1 C ltaprlem B P. e. A P. e. /\ adantr ltsopr <P P. B A sotric mpan C P. e. adantl C B addclpr C A addclpr anim12i anandis ltsopr <P P. C +P. B opr C +P. A opr sotric mpan syl 3imtr3d a3d C +P. A opr <P C +P. B opr br C +P. B opr C +P. A opr = olc syl5 B A = A <P B br df-or syl6ib com23 C +P. A oprex ltsopr ltrelpr soirri B A C +P. opreq2 C +P. A opr <P breq2d mtbiri A <P B br pm2.21d pm2.61d2 impbid 3impb 3com13 ndmord))
+thm (addcanpr () ((addcanpr.1 (e. B (V))) (addcanpr.2 (e. C (V)))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (= (opr A (+P.) B) (opr A (+P.) C)) (= B C))) (A +P. B opr A +P. C opr P. eleq1 addcanpr.2 dmplp 0npr A ndmoprrcl syl6bi A B addclpr syl5com addcanpr.1 addcanpr.2 A ltapr addcanpr.2 addcanpr.1 A ltapr orbi12d negbid B P. e. A P. e. C P. e. /\ ad2antll A P. e. B P. e. C P. e. anandi ltsopr <P P. B C sotrieq mpan A P. e. adantl sylbir ltsopr <P P. A +P. B opr A +P. C opr sotrieq mpan A B addclpr A C addclpr syl2an 3bitr4d biimprd exp syld pm2.43d))
+thm (prlem936a ((w x) (v x) (u x) (v w) (u w) (u v) (w y) (v y) (u y) (w z) (v z) (u z)) () (-> (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> (br (opr (cv y) (+Q) (cv z)) (<Q) (opr (cv x) (+Q) (cv z))) (br (opr (cv x) (+Q) (cv z)) (<Q) (opr (opr (cv x) (.Q) (` (*Q) (cv y))) (.Q) (opr (cv y) (+Q) (cv z)))))) (y cv recclpq z cv Q. e. anim2i z cv *Q y cv ` mulclpq y visset x visset z cv .Q *Q y cv ` opr ltmpq 3syl y visset x visset z cv ltapq z visset y visset addcompq z visset x visset addcompq <Q breq12i syl6bb y cv Q. e. adantr y cv recidpq z cv .Q opreq2d z cv mulidpq sylan9eqr *Q y cv fvex y visset z cv mulasspq y visset *Q y cv fvex mulcompq z cv .Q opreq2i eqtr4 syl5eq <Q z cv .Q *Q y cv ` opr .Q x cv opr breq1d 3bitr3d z visset z cv .Q *Q y cv ` opr .Q x cv oprex x cv ltapq sylan9bbr y cv recidpq *Q y cv fvex y visset mulcompq syl5eq x cv .Q opreq2d *Q y cv fvex y visset x cv mulasspq syl5eq x cv mulidpq sylan9eqr x visset *Q y cv fvex z visset w visset v visset mulcompq v visset u visset w cv mulasspq caopr31 x cv Q. e. y cv Q. e. /\ a1i +Q opreq12d y visset z visset x cv .Q *Q y cv ` opr distrpq syl5eq z cv Q. e. adantrl x cv +Q z cv opr <Q breq2d bitr4d))
+thm (prlem936b () ((prlem936b.1 (-> (/\ (e. (opr (cv y) (.Q) B) A) ph) (e. (opr (cv y) (+Q) (cv z)) A))) (prlem936b.2 (-> (/\ (/\ (e. A (P.)) (e. (opr (cv y) (+Q) (cv z)) A)) (/\ (e. (cv x) (Q.)) (e. (cv z) (Q.)))) (-> ps ch))) (prlem936b.3 (-> (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> ch th))) (prlem936b.4 (-> (/\ (/\ (/\ (br (1Q) (<Q) B) (e. (cv x) (Q.))) (e. (cv y) (Q.))) ph) (<-> th ta))) (prlem936b.5 (-> (/\ (e. A (P.)) ta) (-> ps (-. (e. (opr (cv x) (.Q) B) A)))))) (-> (/\ (/\ (e. A (P.)) (e. (cv z) (Q.))) (/\ (/\ ph (e. (cv y) (Q.))) (/\ (br (1Q) (<Q) B) (e. (opr (cv y) (.Q) B) A)))) (-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) A) (-. (e. (opr (cv x) (.Q) B) A))))) (prlem936b.2 exp prlem936b.1 sylan2 com12 1Q <Q B br adantll y cv Q. e. adantrr exp4d com24 imp41 prlem936b.3 1Q <Q B br adantll ph adantr prlem936b.4 z cv Q. e. adantlrl bitrd y cv .Q B opr A e. A P. e. ad2antll sylibd exp imp32 prlem936b.5 expcom imp3a 1Q <Q B br x cv Q. e. /\ z cv Q. e. y cv Q. e. /\ /\ ph /\ y cv .Q B opr A e. /\ adantld mpcom exp43 com3r exp4c A x cv elprpq sylan2i exp4d pm2.43a imp3a com24 exp4a com23 imp43 exp3a com34 exp imp45 imdistand))
+thm (prlem936 ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (B x) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v)) ((prlem936.1 (e. B (V)))) (-> (/\ (e. A (P.)) (br (1Q) (<Q) B)) (E. x (/\ (e. (cv x) A) (-. (e. (opr (cv x) (.Q) B) A))))) (A prn0 A y n0 sylib A z cv x prlem934 y cv +Q z cv opr y cv .Q B opr A eleq1 biimparc A y cv +Q z cv opr x cv +Q z cv opr prub x cv z cv addclpq sylan2 x z y prlem936a y cv +Q z cv opr y cv .Q B opr x cv .Q *Q y cv ` opr .Q opreq2 y visset *Q y cv fvex prlem936.1 z visset w visset mulcompq w visset v visset z cv mulasspq x visset caopr42 B .Q x cv oprex y cv .Q *Q y cv ` oprex mulcompq x cv .Q *Q y cv ` oprex y cv .Q B oprex mulcompq 3eqtr4 syl6eqr y cv recidpq B .Q x cv opr .Q opreq2d prlem936.1 ltrelpq 1Q brel pm3.27d x cv Q. e. anim1i B x cv mulclpq B .Q x cv opr mulidpq x visset prlem936.1 mulcompq syl6eqr 3syl sylan9eqr sylan9eqr x cv +Q z cv opr <Q breq2d A x cv .Q B opr x cv +Q z cv opr prcdpq exp com23 imp con3d prlem936b x 19.22dv exp com23 mpd exp4d exp4b imp3a z 19.23adv 1q elisseti prlem936.1 y cv ltmpq y cv mulidpq <Q y cv .Q B opr breq1d bitrd y cv .Q B oprex ltrelpq y cv brel y visset y cv .Q B opr z ltexpq2 biimpcd mpd syl6bi imp syl5 imp4a pm2.43d exp3a A P. e. y cv A e. pm3.26 A y cv elprpq sylc com23 y visset x cv y cv A eleq1 x cv y cv .Q B opreq1 A eleq1d negbid anbi12d cla4ev 1Q <Q B br a1d exp A P. e. adantl pm2.61d exp y 19.23adv mpd imp))
+thm (reclem1pr ((x y) (A x) (A y) (B x)) ((reclempr.1 (= B ({|} x (E. y (/\ (br (cv x) (<Q) (cv y)) (-. (e. (` (*Q) (cv y)) A)))))))) (-> (e. A (P.)) (C: ({/}) B)) (A prpssnq A Q. x pssnel x cv recclpq dmrecpq 0npq x cv ndmfvrcl impbi *Q *Q x cv ` ` A e. -. anbi1i x visset recrecpq A eleq1d negbid pm5.32i bitr3 *Q x cv fvex y cv *Q x cv ` Q. eleq1 y cv *Q x cv ` *Q fveq2 A eleq1d negbid anbi12d cla4ev sylbir x 19.23aiv 3syl y cv x nsmallpq *Q y cv ` A e. -. anim1i x x cv <Q y cv br *Q y cv ` A e. -. 19.41v sylibr y 19.22i syl x y x cv <Q y cv br *Q y cv ` A e. -. /\ excom sylibr reclempr.1 eqabi x biex sylibr B 0pss B x n0 bitr sylibr))
+thm (reclem2pr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z)) ((reclempr.1 (= B ({|} x (E. y (/\ (br (cv x) (<Q) (cv y)) (-. (e. (` (*Q) (cv y)) A)))))))) (-> (e. A (P.)) (e. B (P.))) (reclempr.1 reclem1pr A prn0 A z cv elprpq z visset recrecpq A eleq1d A P. e. anbi2d syl *Q z cv fvex x cv *Q z cv ` *Q fveq2 A eleq1d A P. e. anbi2d cla4ev syl6bir pm2.43i A *Q x cv ` elprpq dmrecpq 0npq x cv ndmfvrcl syl A *Q x cv ` *Q y cv ` prcdpq x visset y visset ltrpq syl5 y 19.21aiv reclempr.1 eqabi y x cv <Q y cv br *Q y cv ` A e. exanali bitr bicon2i sylib jca x 19.22i syl exp z 19.23adv A z n0 Q. B x nss 3imtr4g mpd reclempr.1 eqabi y visset ltrelpq x cv brel pm3.26d *Q y cv ` A e. -. adantr y 19.23aiv sylbi ssriv jctil B Q. dfpss3 sylibr jca z visset ltsopq ltrelpq x visset y visset sotri exp *Q y cv ` A e. -. anim1d y 19.22dv reclempr.1 eqabi z visset x cv z cv <Q y cv breq1 *Q y cv ` A e. -. anbi1d y biexdv reclempr.1 elab2 3imtr4g com12 z 19.21aiv x visset y visset z ltbtwnpq *Q y cv ` A e. -. anim1i z x cv <Q z cv br z cv <Q y cv br /\ *Q y cv ` A e. -. 19.41v sylibr y 19.22i reclempr.1 eqabi y z cv <Q y cv br *Q y cv ` A e. -. /\ x cv <Q z cv br 19.41v x cv <Q z cv br z cv <Q y cv br *Q y cv ` A e. -. anass x cv <Q z cv br z cv <Q y cv br *Q y cv ` A e. -. /\ ancom bitr y biex z visset x cv z cv <Q y cv breq1 *Q y cv ` A e. -. anbi1d y biexdv reclempr.1 elab2 x cv <Q z cv br anbi1i 3bitr4 z biex z y x cv <Q z cv br z cv <Q y cv br /\ *Q y cv ` A e. -. /\ excom bitr3 3imtr4 z B x cv <Q z cv br df-rex sylibr jca rgen jctir B x z elnp sylibr))
+thm (reclem3pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g) (B x) (B z) (B w) (B v) (B u) (B f) (B g)) ((reclempr.1 (= B ({|} x (E. y (/\ (br (cv x) (<Q) (cv y)) (-. (e. (` (*Q) (cv y)) A)))))))) (-> (e. A (P.)) (C_ (1P) (opr A (.P) B))) (*Q w cv fvex A v prlem936 *Q z cv ` .Q w cv oprex x isseti *Q z cv fvex *Q v cv fvex x visset y visset u cv ltmpq w visset x visset y visset mulcompq caoprord2 v visset z visset ltrpq syl5bi v cv Q. e. adantl v cv recidpq v visset *Q v cv fvex mulcompq syl5eqr w cv recidpq .Q opreqan12d *Q v cv fvex v visset w visset x visset y visset mulcompq y visset u visset x cv mulasspq *Q w cv fvex caopr4 1q 1Q mulidpq ax-mp 3eqtr3g *Q v cv ` w cv mulclpq v cv recclpq sylan v cv .Q *Q w cv ` oprex *Q v cv ` .Q w cv opr recmulpq syl mpbird A eleq1d negbid biimprd anim12d x cv *Q z cv ` .Q w cv opr <Q *Q v cv ` .Q w cv opr breq1 *Q *Q v cv ` .Q w cv opr ` A e. -. anbi1d biimprcd syl6 *Q v cv ` .Q w cv oprex y cv *Q v cv ` .Q w cv opr x cv <Q breq2 y cv *Q v cv ` .Q w cv opr *Q fveq2 A eleq1d negbid anbi12d cla4ev reclempr.1 eqabi sylibr syl8 z visset ltrelpq v cv brel pm3.26d sylan exp4b pm2.43a imp42 z cv A e. anim2i x cv *Q z cv ` .Q w cv opr z cv .Q opreq2 z cv recidpq .Q w cv opreq1d *Q z cv fvex w visset z cv mulasspq syl5eqr w cv mulidpq w visset 1q elisseti mulcompq syl5eqr sylan9eq z visset ltrelpq v cv brel pm3.27d w cv Q. e. v cv .Q *Q w cv ` opr A e. -. pm3.26 syl2an sylan9eqr eqcomd z cv A e. adantl jca exp44 imp31 x 19.22dv mpi expcom z 19.22dv A v cv z prnmax syl5 exp4b com14 imp4b v 19.23aiv syl w visset 1q elisseti ltrpq *Q 1Q fvex 1q elisseti mulcompq 1q 1Q recclpq ax-mp *Q 1Q ` mulidpq ax-mp 1q 1Q recidpq ax-mp 3eqtr3 syl5eqbrr sylan2 1q elisseti ltrelpq w cv brel pm3.26d sylan2i pm2.43i exp reclempr.1 reclem2pr y w v u f g df-mp w visset A B z x genpelv mpdan sylibrd w df-1p eqabi syl5ib ssrdv))
+thm (reclem4pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g) (B x) (B z) (B w) (B v) (B u) (B f) (B g)) ((reclempr.1 (= B ({|} x (E. y (/\ (br (cv x) (<Q) (cv y)) (-. (e. (` (*Q) (cv y)) A)))))))) (-> (e. A (P.)) (= (opr A (.P) B) (1P))) (reclempr.1 reclem2pr y w v u f g df-mp w visset A B z x genpelv mpdan A z cv elprpq x visset y visset z cv ltmpq syl biimpd y cv Q. e. adantr A z cv *Q y cv ` prub y cv recclpq sylan2 z visset *Q y cv fvex y cv ltmpq y visset z visset mulcompq y cv Q. e. a1i y cv recidpq <Q breq12d bitrd A P. e. z cv A e. /\ adantl sylibd anim12d z cv .Q x cv oprex ltsopq ltrelpq z cv .Q y cv oprex 1q elisseti sotri syl6 exp4b y visset ltrelpq x cv brel pm3.27d syl5 pm2.43d imp3a y 19.23adv reclempr.1 eqabi syl5ib w cv z cv .Q x cv opr <Q 1Q breq1 biimprcd syl6 exp imp4c z x 19.23advv sylbid w df-1p eqabi syl6ibr ssrdv reclempr.1 reclem3pr eqssd))
+thm (recexpr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (P.)) (E. x (/\ (e. (cv x) (P.)) (= (opr A (.P) (cv x)) (1P))))) (x cv z y z cv <Q y cv br *Q y cv ` A e. -. /\ E. {|} A .P opreq2 1P eqeq1d P. cla4egv z cv w cv <Q y cv breq1 *Q y cv ` A e. -. anbi1d y biexdv cbvabv reclem2pr z cv w cv <Q y cv breq1 *Q y cv ` A e. -. anbi1d y biexdv cbvabv reclem4pr sylc 1pr A .P x cv opr 1P P. eleq1 mpbiri x visset dmmp 0npr A ndmoprrcl syl pm3.27d ancri x 19.22i syl))
+thm (suplem1pr ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) (<P) (cv x)))))))) (e. (U. A) (P.))) (A P. z cv ssel z cv prn0 z cv x n0 sylib syl6 ancrd x x cv z cv e. z cv A e. 19.41v syl6ibr z 19.22dv A z n0 A U. 0pss A U. x n0 bitr x cv A z eluni x biex x z x cv z cv e. z cv A e. /\ excom 3bitr 3imtr4g x cv prpssnq A P. C_ a1i A P. y cv ssel y cv A e. y cv <P x cv br -> imim1d y cv A e. y cv <P x cv br pm2.43 x visset ltrelpr y cv brel y cv x cv ltprord y cv x cv pssss syl6bi mpcom syl6 syl6 y 19.20dv A x cv y unissb y A y cv x cv C_ df-ral bitr syl6ibr anim12d A U. x cv Q. sspsstr ancoms syl6 x 19.23adv anim12d A P. z cv ssel z cv x cv y cv prcdpq exp syl6 com24 imp31 ancrd y cv z cv A elunii syl6 exp imp3a z 19.23adv x cv A z eluni syl5ib exp com23 y 19.21adv A P. z cv ssel z cv x cv y prnmax exp syl6 com23 imp ancrd y y cv z cv e. x cv <Q y cv br /\ z cv A e. 19.41v y cv z cv A elunii x cv <Q y cv br anim1i an1rs y 19.22i sylbir syl6 exp imp3a z 19.23adv x cv A z eluni y A U. x cv <Q y cv br df-rex 3imtr4g jcad r19.21aiv jctird A U. x y elnp syl6ibr exp3a imp31))
+thm (suplem2pr ((y z) (A y) (A z)) () (-> (C_ A (P.)) (/\ (-> (e. (cv y) A) (-. (br (U. A) (<P) (cv y)))) (-> (br (cv y) (<P) (U. A)) (E. z (/\ (e. (cv z) (P.)) (/\ (e. (cv z) A) (br (cv y) (<P) (cv z)))))))) (npex A ssex A V uniexg ltrelpr A U. V y cv brelg y cv P. e. A U. P. e. pm3.26 syl6 3syl ltsopr <P P. y cv z cv sotric mpan bicon2d ancoms z cv y cv ltprord y cv z cv = orbi2d z cv y cv sspss z cv y cv eqcom z cv y cv C: orbi2i z cv y cv C: y cv z cv = orcom 3bitr syl6bbr bitr3d A P. z cv ssel2 sylan an1rs exp pm5.74d z bialdv z z cv A e. y cv <P z cv br alinexa A y cv z unissb z A z cv y cv C_ df-ral bitr2 3bitr3g A U. y cv ssnpss y visset A U. ssex ltrelpr A U. V y cv brelg y cv A U. ltprord biimpd syli syl mtod syl6bi a3d exp syld pm2.43d z visset ltrelpr y cv brel pm3.27d z cv A e. adantl ancri z 19.22i syl6 y cv A elssuni y cv A U. ssnpss syl y visset ltrelpr A U. brel A U. y cv ltprord biimpd mpcom nsyl jctil))
+thm (suppr ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) (<P) (cv x)))))))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<P) (cv y)))) (-> (br (cv y) (<P) (cv x)) (E. z (/\ (e. (cv z) (P.)) (/\ (e. (cv z) A) (br (cv y) (<P) (cv z)))))))))))) (x cv A U. P. eleq1 x cv A U. <P y cv breq1 negbid y cv A e. imbi2d x cv A U. y cv <P breq2 z z cv P. e. z cv A e. y cv <P z cv br /\ /\ E. imbi1d anbi12d y cv P. e. imbi2d y bialdv anbi12d P. cla4egv A x y suplem1pr A x y suplem1pr A y z suplem2pr y cv P. e. a1d y 19.21aiv A {/} = -. x x cv P. e. y y cv P. e. y cv A e. y cv <P x cv br -> -> A. /\ E. ad2antll jca sylc))
+thm (enrbreq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (br (<,> A B) (~R) (<,> C D)) (= (opr A (+P.) D) (opr B (+P.) C)))) (x y z w v u df-enr A B C D ecopopreq))
+thm (dmenr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (~R)) (X. (P.) (P.))) (x y z w v u df-enr x visset y visset addcompr ecopoprdm))
+thm (enrer ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Er (~R)) (x y z w v u df-enr x visset y visset addcompr x cv y cv addclpr y visset z visset x cv addasspr y visset z visset x cv addcanpr ecopoprer))
+thm (enreceq () () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (= ([] (<,> A B) (~R)) ([] (<,> C D) (~R))) (= (opr A (+P.) D) (opr B (+P.) C)))) (A P. e. B P. e. /\ C P. e. D P. e. /\ pm3.26 A P. B P. opelxpi dmenr syl6eleqr C D opex enrer A B <,> erthdm 3syl A B C D enrbreq bitrd))
+thm (enrex ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (e. (~R) (V)) (npex npex xpex npex npex xpex xpex x y z w v u df-enr x y P. P. X. P. P. X. z w v u x cv z cv w cv <,> = y cv v cv u cv <,> = /\ z cv +P. u cv opr w cv +P. v cv opr = /\ E. E. E. E. opabssxp eqsstr ssexi))
+thm (srex () () (e. (R.) (V)) (df-nr npex npex xpex ~R qsex eqeltr))
+thm (ltrelsr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ (<R) (X. (R.) (R.))) (x y z w v u df-ltr x y R. R. z w v u x cv z cv w cv <,> ~R [] = y cv v cv u cv <,> ~R [] = /\ z cv +P. u cv opr <P w cv +P. v cv opr br /\ E. E. E. E. opabssxp eqsstr))
+thm (addcmpblnr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnr.1 (e. A (V))) (cmpblnr.2 (e. B (V))) (cmpblnr.3 (e. C (V))) (cmpblnr.4 (e. D (V))) (cmpblnr.5 (e. F (V))) (cmpblnr.6 (e. G (V))) (cmpblnr.7 (e. R (V))) (cmpblnr.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr A (+P.) F) (opr B (+P.) G)) (~R) (<,> (opr C (+P.) R) (opr D (+P.) S))))) (A F addclpr B G addclpr anim12i an4s C R addclpr D S addclpr anim12i an4s anim12i an4s A +P. F opr B +P. G opr C +P. R opr D +P. S opr enrbreq syl cmpblnr.1 cmpblnr.5 cmpblnr.4 x visset y visset addcompr y visset z visset x cv addasspr cmpblnr.8 caopr4 cmpblnr.2 cmpblnr.6 cmpblnr.3 x visset y visset addcompr y visset z visset x cv addasspr cmpblnr.7 caopr4 eqeq12i syl6bb A +P. D opr B +P. C opr F +P. S opr G +P. R opr +P. opreq12 syl5bir))
+thm (mulcmpblnrlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((cmpblnr.1 (e. A (V))) (cmpblnr.2 (e. B (V))) (cmpblnr.3 (e. C (V))) (cmpblnr.4 (e. D (V))) (cmpblnr.5 (e. F (V))) (cmpblnr.6 (e. G (V))) (cmpblnr.7 (e. R (V))) (cmpblnr.8 (e. S (V)))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (= (opr (opr D (.P) F) (+P.) (opr (opr (opr A (.P) F) (+P.) (opr B (.P) G)) (+P.) (opr (opr C (.P) S) (+P.) (opr D (.P) R)))) (opr (opr D (.P) F) (+P.) (opr (opr (opr A (.P) G) (+P.) (opr B (.P) F)) (+P.) (opr (opr C (.P) R) (+P.) (opr D (.P) S)))))) (A +P. D opr B +P. C opr .P F opreq1 cmpblnr.1 cmpblnr.4 cmpblnr.5 x visset y visset mulcompr y visset z visset x cv distrpr caoprdistrr cmpblnr.2 cmpblnr.3 cmpblnr.5 x visset y visset mulcompr y visset z visset x cv distrpr caoprdistrr 3eqtr3g +P. C .P S opr opreq1d F +P. S opr G +P. R opr C .P opreq2 cmpblnr.5 cmpblnr.8 C distrpr cmpblnr.6 cmpblnr.7 C distrpr 3eqtr3g B .P F opr +P. opreq2d C .P F oprex C .P S oprex B .P F opr addasspr syl5eq sylan9eq A .P F oprex D .P F oprex C .P S oprex x visset y visset addcompr y visset z visset x cv addasspr caopr32 B .P F oprex C .P G oprex C .P R oprex x visset y visset addcompr y visset z visset x cv addasspr caopr12 3eqtr3g B .P G opr +P. D .P R opr opr +P. opreq2d F +P. S opr G +P. R opr D .P opreq2 cmpblnr.5 cmpblnr.8 D distrpr cmpblnr.6 cmpblnr.7 D distrpr 3eqtr3g A .P G opr +P. opreq2d D .P G oprex D .P R oprex A .P G opr addasspr syl6eqr A +P. D opr B +P. C opr .P G opreq1 cmpblnr.1 cmpblnr.4 cmpblnr.6 x visset y visset mulcompr y visset z visset x cv distrpr caoprdistrr cmpblnr.2 cmpblnr.3 cmpblnr.6 x visset y visset mulcompr y visset z visset x cv distrpr caoprdistrr 3eqtr3g +P. D .P R opr opreq1d sylan9eqr A .P G oprex D .P F oprex D .P S oprex x visset y visset addcompr y visset z visset x cv addasspr caopr12 B .P G oprex C .P G oprex D .P R oprex x visset y visset addcompr y visset z visset x cv addasspr caopr32 3eqtr3g +P. B .P F opr +P. C .P R opr opr opreq1d C .P G oprex B .P F opr +P. C .P R opr oprex B .P G opr +P. D .P R opr opr addasspr syl6eq eqtr4d B .P G opr +P. D .P R opr oprex A .P F opr +P. C .P S opr oprex D .P F oprex x visset y visset addcompr y visset z visset x cv addasspr caopr13 A .P G opr +P. D .P S opr oprex B .P F opr +P. C .P R opr oprex D .P F opr addasspr 3eqtr3g A .P F oprex C .P S oprex B .P G oprex x visset y visset addcompr y visset z visset x cv addasspr D .P R oprex caopr4 D .P F opr +P. opreq2i A .P G oprex D .P S oprex B .P F oprex x visset y visset addcompr y visset z visset x cv addasspr C .P R oprex caopr42 D .P F opr +P. opreq2i 3eqtr3g))
+thm (mulcmpblnr () ((cmpblnr.1 (e. A (V))) (cmpblnr.2 (e. B (V))) (cmpblnr.3 (e. C (V))) (cmpblnr.4 (e. D (V))) (cmpblnr.5 (e. F (V))) (cmpblnr.6 (e. G (V))) (cmpblnr.7 (e. R (V))) (cmpblnr.8 (e. S (V)))) (-> (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr (opr A (.P) F) (+P.) (opr B (.P) G)) (opr (opr A (.P) G) (+P.) (opr B (.P) F))) (~R) (<,> (opr (opr C (.P) R) (+P.) (opr D (.P) S)) (opr (opr C (.P) S) (+P.) (opr D (.P) R)))))) (D F mulclpr C P. e. D P. e. pm3.27 F P. e. G P. e. pm3.26 syl2an A .P F opr +P. B .P G opr opr C .P S opr +P. D .P R opr opr addclpr A .P F opr +P. B .P G opr opr +P. C .P S opr +P. D .P R opr opr oprex A .P G opr +P. B .P F opr opr +P. C .P R opr +P. D .P S opr opr oprex D .P F opr addcanpr cmpblnr.1 cmpblnr.2 cmpblnr.3 cmpblnr.4 cmpblnr.5 cmpblnr.6 cmpblnr.7 cmpblnr.8 mulcmpblnrlem syl5 expcom imp3a syl C .P R opr +P. D .P S opr opr P. e. adantrl A .P G opr +P. B .P F opr opr P. e. adantlr A .P F opr +P. B .P G opr opr A .P G opr +P. B .P F opr opr C .P R opr +P. D .P S opr opr C .P S opr +P. D .P R opr opr enrbreq sylibrd A P. e. B P. e. F P. e. G P. e. rnlem A .P F opr B .P G opr addclpr A F mulclpr B G mulclpr syl2an A .P G opr B .P F opr addclpr A G mulclpr B F mulclpr syl2an anim12i sylbi C P. e. D P. e. R P. e. S P. e. rnlem C .P R opr D .P S opr addclpr C R mulclpr D S mulclpr syl2an C .P S opr D .P R opr addclpr C S mulclpr D R mulclpr syl2an anim12i sylbi syl2an an42s an4s exp4b mpdi imp an42s))
+thm (addsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (+R) ([] (<,> C D) (~R))) ([] (<,> (opr A (+P.) C) (opr B (+P.) D)) (~R)))) (A +P. C opr B +P. D opr opex a cv +P. g cv opr b cv +P. h cv opr opex c cv +P. t cv opr d cv +P. s cv opr opex enrex enrer dmenr x y z w v u df-enr z cv a cv u cv d cv +P. opreq12 w cv b cv v cv c cv +P. opreq12 eqeqan12d an42s z cv g cv u cv s cv +P. opreq12 w cv h cv v cv t cv +P. opreq12 eqeqan12d an42s x y z w v u f df-plpr w cv +P. u cv opr a cv +P. g cv opr v cv +P. f cv opr b cv +P. h cv opr opeq12 w cv a cv u cv g cv +P. opreq12 v cv b cv f cv h cv +P. opreq12 syl2an an4s w cv +P. u cv opr c cv +P. t cv opr v cv +P. f cv opr d cv +P. s cv opr opeq12 w cv c cv u cv t cv +P. opreq12 v cv d cv f cv s cv +P. opreq12 syl2an an4s w cv +P. u cv opr A +P. C opr v cv +P. f cv opr B +P. D opr opeq12 w cv A u cv C +P. opreq12 v cv B f cv D +P. opreq12 syl2an an4s x y z a b c d df-plr df-nr a visset b visset c visset d visset g visset h visset t visset s visset addcmpblnr oprec))
+thm (mulsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (.R) ([] (<,> C D) (~R))) ([] (<,> (opr (opr A (.P) C) (+P.) (opr B (.P) D)) (opr (opr A (.P) D) (+P.) (opr B (.P) C))) (~R)))) (A .P C opr +P. B .P D opr opr A .P D opr +P. B .P C opr opr opex a cv .P g cv opr +P. b cv .P h cv opr opr a cv .P h cv opr +P. b cv .P g cv opr opr opex c cv .P t cv opr +P. d cv .P s cv opr opr c cv .P s cv opr +P. d cv .P t cv opr opr opex enrex enrer dmenr x y z w v u df-enr z cv a cv u cv d cv +P. opreq12 w cv b cv v cv c cv +P. opreq12 eqeqan12d an42s z cv g cv u cv s cv +P. opreq12 w cv h cv v cv t cv +P. opreq12 eqeqan12d an42s x y z w v u f df-mpr w cv .P u cv opr +P. v cv .P f cv opr opr a cv .P g cv opr +P. b cv .P h cv opr opr w cv .P f cv opr +P. v cv .P u cv opr opr a cv .P h cv opr +P. b cv .P g cv opr opr opeq12 w cv a cv u cv g cv .P opreq12 v cv b cv f cv h cv .P opreq12 +P. opreqan12d an4s w cv a cv f cv h cv .P opreq12 v cv b cv u cv g cv .P opreq12 +P. opreqan12d an42s sylanc w cv .P u cv opr +P. v cv .P f cv opr opr c cv .P t cv opr +P. d cv .P s cv opr opr w cv .P f cv opr +P. v cv .P u cv opr opr c cv .P s cv opr +P. d cv .P t cv opr opr opeq12 w cv c cv u cv t cv .P opreq12 v cv d cv f cv s cv .P opreq12 +P. opreqan12d an4s w cv c cv f cv s cv .P opreq12 v cv d cv u cv t cv .P opreq12 +P. opreqan12d an42s sylanc w cv .P u cv opr +P. v cv .P f cv opr opr A .P C opr +P. B .P D opr opr w cv .P f cv opr +P. v cv .P u cv opr opr A .P D opr +P. B .P C opr opr opeq12 w cv A u cv C .P opreq12 v cv B f cv D .P opreq12 +P. opreqan12d an4s w cv A f cv D .P opreq12 v cv B u cv C .P opreq12 +P. opreqan12d an42s sylanc x y z a b c d df-mr df-nr a visset b visset c visset d visset g visset h visset t visset s visset mulcmpblnr oprec))
+thm (ltsrpr ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((ltsrpr.1 (e. A (V))) (ltsrpr.2 (e. B (V))) (ltsrpr.3 (e. C (V))) (ltsrpr.4 (e. D (V)))) (<-> (br ([] (<,> A B) (~R)) (<R) ([] (<,> C D) (~R))) (br (opr A (+P.) D) (<P) (opr B (+P.) C))) (enrex ltsrpr.2 ltsrpr.3 ltsrpr.4 enrer dmenr df-nr ltrelsr ltrelpr 0npr dmplp enrex enrer dmenr df-nr x y z w v u df-ltr z cv w cv A B enreceq v cv u cv C D enreceq v cv +P. D opr u cv +P. C opr eqcom syl6bb bi2anan9 z cv +P. B opr w cv +P. A opr u cv +P. C opr v cv +P. D opr +P. opreq12 z visset u visset ltsrpr.2 x visset y visset addcompr y visset f visset x cv addasspr ltsrpr.3 caopr4 w visset v visset ltsrpr.1 x visset y visset addcompr y visset f visset x cv addasspr ltsrpr.4 caopr4 3eqtr4g syl6bi B C addclpr D P. e. adantrr A P. e. adantll w cv v cv addclpr u cv P. e. adantrr z cv P. e. adantll anim12i ancoms an4s z cv +P. u cv oprex B +P. C oprex x visset y visset f cv ltapr w cv +P. v cv oprex x visset y visset addcompr A +P. D oprex caoprord3 exp syl syld brecop brecop2))
+thm (gt0srpr () ((gt0srpr.1 (e. A (V))) (gt0srpr.2 (e. B (V)))) (<-> (br (0R) (<R) ([] (<,> A B) (~R))) (br B (<P) A)) (1pr elisseti 1pr elisseti gt0srpr.1 gt0srpr.2 ltsrpr df-0r <R A B <,> ~R [] breq1i 1pr gt0srpr.2 gt0srpr.1 1P ltapr ax-mp 3bitr4))
+thm (0nsr () () (-. (e. ({/}) (R.))) (enrer dmenr 0nelqs df-nr {/} eleq2i mtbir))
+thm (0r () () (e. (0R) (R.)) (1pr 1pr 1P P. 1P P. opelxpi mp2an enrex 1P 1P <,> P. P. X. ecelqsi ax-mp df-0r df-nr eleq12i mpbir))
+thm (1r () () (e. (1R) (R.)) (1pr 1pr 1P 1P addclpr mp2an 1pr 1P +P. 1P opr P. 1P P. opelxpi mp2an enrex 1P +P. 1P opr 1P <,> P. P. X. ecelqsi ax-mp df-1r df-nr eleq12i mpbir))
+thm (m1r () () (e. (-1R) (R.)) (1pr 1pr 1pr 1P 1P addclpr mp2an 1P P. 1P +P. 1P opr P. opelxpi mp2an enrex 1P 1P +P. 1P opr <,> P. P. X. ecelqsi ax-mp df-m1r df-nr eleq12i mpbir))
+thm (addclsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (+R) B) (R.))) (df-nr x cv y cv <,> ~R [] A +R z cv w cv <,> ~R [] opreq1 P. P. X. ~R /. eleq1d z cv w cv <,> ~R [] B A +R opreq2 P. P. X. ~R /. eleq1d x cv y cv z cv w cv addsrpr x cv z cv addclpr y cv w cv addclpr anim12i an4s x cv +P. z cv opr P. y cv +P. w cv opr P. opelxpi enrex x cv +P. z cv opr y cv +P. w cv opr <,> P. P. X. ecelqsi 3syl eqeltrd 2ecoptocl df-nr syl6eleqr))
+thm (mulclsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (.R) B) (R.))) (df-nr x cv y cv <,> ~R [] A .R z cv w cv <,> ~R [] opreq1 P. P. X. ~R /. eleq1d z cv w cv <,> ~R [] B A .R opreq2 P. P. X. ~R /. eleq1d x cv y cv z cv w cv mulsrpr x cv .P z cv opr y cv .P w cv opr addclpr x cv z cv mulclpr y cv w cv mulclpr syl2an an4s x cv .P w cv opr y cv .P z cv opr addclpr x cv w cv mulclpr y cv z cv mulclpr syl2an an42s jca x cv .P z cv opr +P. y cv .P w cv opr opr P. x cv .P w cv opr +P. y cv .P z cv opr opr P. opelxpi enrex x cv .P z cv opr +P. y cv .P w cv opr opr x cv .P w cv opr +P. y cv .P z cv opr opr <,> P. P. X. ecelqsi 3syl eqeltrd 2ecoptocl df-nr syl6eleqr))
+thm (dmaddsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+R)) (X. (R.) (R.))) (x y z w v u f df-plr dmeqi x y z R. R. w v u f x cv w cv v cv <,> ~R [] = y cv u cv f cv <,> ~R [] = /\ z cv w cv v cv <,> +pR u cv f cv <,> opr ~R [] = /\ E. E. E. E. dmoprabss eqsstr 0nsr x cv y cv addclsr oprssdm eqssi))
+thm (dmmulsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.R)) (X. (R.) (R.))) (x y z w v u f df-mr dmeqi x y z R. R. w v u f x cv w cv v cv <,> ~R [] = y cv u cv f cv <,> ~R [] = /\ z cv w cv v cv <,> .pR u cv f cv <,> opr ~R [] = /\ E. E. E. E. dmoprabss eqsstr 0nsr x cv y cv mulclsr oprssdm eqssi))
+thm (addcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((addcomsr.1 (e. A (V))) (addcomsr.2 (e. B (V)))) (= (opr A (+R) B) (opr B (+R) A)) (df-nr x cv y cv z cv w cv addsrpr z cv w cv x cv y cv addsrpr x visset z visset addcompr y visset w visset addcompr A B ecoprcom addcomsr.2 dmaddsr addcomsr.1 ndmoprcom pm2.61i))
+thm (addasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((addasssr.1 (e. B (V))) (addasssr.2 (e. C (V)))) (= (opr (opr A (+R) B) (+R) C) (opr A (+R) (opr B (+R) C))) (df-nr x cv y cv z cv w cv addsrpr z cv w cv v cv u cv addsrpr x cv +P. z cv opr y cv +P. w cv opr v cv u cv addsrpr x cv y cv z cv +P. v cv opr w cv +P. u cv opr addsrpr x cv z cv addclpr y cv w cv addclpr anim12i an4s z cv v cv addclpr w cv u cv addclpr anim12i an4s z visset v visset x cv addasspr w visset u visset y cv addasspr A B C ecoprass addasssr.1 dmaddsr addasssr.2 0nsr A ndmoprass pm2.61i))
+thm (mulcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((mulcomsr.1 (e. A (V))) (mulcomsr.2 (e. B (V)))) (= (opr A (.R) B) (opr B (.R) A)) (df-nr x cv y cv z cv w cv mulsrpr z cv w cv x cv y cv mulsrpr x visset z visset mulcompr y visset w visset mulcompr +P. opreq12i x visset w visset mulcompr y visset z visset mulcompr +P. opreq12i w cv .P x cv oprex z cv .P y cv oprex addcompr eqtr A B ecoprcom mulcomsr.2 dmmulsr mulcomsr.1 ndmoprcom pm2.61i))
+thm (mulasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((mulasssr.1 (e. B (V))) (mulasssr.2 (e. C (V)))) (= (opr (opr A (.R) B) (.R) C) (opr A (.R) (opr B (.R) C))) (df-nr x cv y cv z cv w cv mulsrpr z cv w cv v cv u cv mulsrpr x cv .P z cv opr +P. y cv .P w cv opr opr x cv .P w cv opr +P. y cv .P z cv opr opr v cv u cv mulsrpr x cv y cv z cv .P v cv opr +P. w cv .P u cv opr opr z cv .P u cv opr +P. w cv .P v cv opr opr mulsrpr x cv .P z cv opr y cv .P w cv opr addclpr x cv z cv mulclpr y cv w cv mulclpr syl2an an4s x cv .P w cv opr y cv .P z cv opr addclpr x cv w cv mulclpr y cv z cv mulclpr syl2an an42s jca z cv .P v cv opr w cv .P u cv opr addclpr z cv v cv mulclpr w cv u cv mulclpr syl2an an4s z cv .P u cv opr w cv .P v cv opr addclpr z cv u cv mulclpr w cv v cv mulclpr syl2an an42s jca x visset y visset z visset f visset g visset mulcompr g visset h visset f cv distrpr w visset v visset g visset h visset f cv mulasspr u visset f visset g visset addcompr g visset h visset f cv addasspr caoprlem2 x visset y visset z visset f visset g visset mulcompr g visset h visset f cv distrpr w visset u visset g visset h visset f cv mulasspr v visset f visset g visset addcompr g visset h visset f cv addasspr caoprlem2 A B C ecoprass mulasssr.1 dmmulsr mulasssr.2 0nsr A ndmoprass pm2.61i))
+thm (distrsr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((distrsr.1 (e. B (V))) (distrsr.2 (e. C (V)))) (= (opr A (.R) (opr B (+R) C)) (opr (opr A (.R) B) (+R) (opr A (.R) C))) (df-nr z cv w cv v cv u cv addsrpr x cv y cv z cv +P. v cv opr w cv +P. u cv opr mulsrpr x cv y cv z cv w cv mulsrpr x cv y cv v cv u cv mulsrpr x cv .P z cv opr +P. y cv .P w cv opr opr x cv .P w cv opr +P. y cv .P z cv opr opr x cv .P v cv opr +P. y cv .P u cv opr opr x cv .P u cv opr +P. y cv .P v cv opr opr addsrpr z cv v cv addclpr w cv u cv addclpr anim12i an4s x cv .P z cv opr y cv .P w cv opr addclpr x cv z cv mulclpr y cv w cv mulclpr syl2an an4s x cv .P w cv opr y cv .P z cv opr addclpr x cv w cv mulclpr y cv z cv mulclpr syl2an an42s jca x cv .P v cv opr y cv .P u cv opr addclpr x cv v cv mulclpr y cv u cv mulclpr syl2an an4s x cv .P u cv opr y cv .P v cv opr addclpr x cv u cv mulclpr y cv v cv mulclpr syl2an an42s jca z visset v visset x cv distrpr w visset u visset y cv distrpr +P. opreq12i x cv .P z cv oprex x cv .P v cv oprex y cv .P w cv oprex f visset g visset addcompr g visset h visset f cv addasspr y cv .P u cv oprex caopr4 eqtr w visset u visset x cv distrpr z visset v visset y cv distrpr +P. opreq12i x cv .P w cv oprex x cv .P u cv oprex y cv .P z cv oprex f visset g visset addcompr g visset h visset f cv addasspr y cv .P v cv oprex caopr4 eqtr A B C ecoprdi distrsr.1 dmaddsr distrsr.2 0nsr dmmulsr A ndmoprdistr pm2.61i))
+thm (m1p1sr () () (= (opr (-1R) (+R) (1R)) (0R)) (1pr 1pr 1pr 1P 1P addclpr mp2an pm3.2i 1pr 1pr 1P 1P addclpr mp2an 1pr pm3.2i 1P 1P +P. 1P opr 1P +P. 1P opr 1P addsrpr mp2an 1pr elisseti 1pr elisseti 1P addasspr 1P +P. opreq2i 1pr 1pr pm3.2i 1pr 1pr 1pr 1P 1P addclpr mp2an 1P 1P +P. 1P opr addclpr mp2an 1pr 1pr 1P 1P addclpr mp2an 1pr 1P +P. 1P opr 1P addclpr mp2an pm3.2i 1P 1P 1P +P. 1P +P. 1P opr opr 1P +P. 1P opr +P. 1P opr enreceq mp2an mpbir eqtr4 df-m1r df-1r +R opreq12i df-0r 3eqtr4))
+thm (m1m1sr () () (= (opr (-1R) (.R) (-1R)) (1R)) (1pr 1pr 1pr 1P 1P addclpr mp2an pm3.2i 1pr 1pr 1pr 1P 1P addclpr mp2an pm3.2i 1P 1P +P. 1P opr 1P 1P +P. 1P opr mulsrpr mp2an 1pr elisseti 1P .P 1P +P. 1P opr opr +P. 1P +P. 1P opr .P 1P opr oprex 1P addasspr 1pr 1P 1idpr ax-mp 1pr elisseti 1pr elisseti 1P +P. 1P opr distrpr 1pr elisseti 1P +P. 1P oprex mulcompr +P. 1P +P. 1P opr .P 1P opr opreq1i eqtr4 +P. opreq12i 1P +P. opreq2i eqtr4 1pr 1pr 1P 1P addclpr mp2an 1pr pm3.2i 1pr 1pr 1P 1P mulclpr mp2an 1pr 1pr 1P 1P addclpr mp2an 1pr 1pr 1P 1P addclpr mp2an 1P +P. 1P opr 1P +P. 1P opr mulclpr mp2an 1P .P 1P opr 1P +P. 1P opr .P 1P +P. 1P opr opr addclpr mp2an 1pr 1pr 1pr 1P 1P addclpr mp2an 1P 1P +P. 1P opr mulclpr mp2an 1pr 1pr 1P 1P addclpr mp2an 1pr 1P +P. 1P opr 1P mulclpr mp2an 1P .P 1P +P. 1P opr opr 1P +P. 1P opr .P 1P opr addclpr mp2an pm3.2i 1P +P. 1P opr 1P 1P .P 1P opr +P. 1P +P. 1P opr .P 1P +P. 1P opr opr opr 1P .P 1P +P. 1P opr opr +P. 1P +P. 1P opr .P 1P opr opr enreceq mp2an mpbir eqtr4 df-m1r df-m1r .R opreq12i df-1r 3eqtr4))
+thm (ltsosr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or (<R) (R.)) (df-nr x cv y cv <,> ~R [] f cv <R z cv w cv <,> ~R [] breq1 x cv y cv <,> ~R [] f cv z cv w cv <,> ~R [] eqeq1 x cv y cv <,> ~R [] f cv z cv w cv <,> ~R [] <R breq2 orbi12d negbid bibi12d x cv y cv <,> ~R [] f cv <R z cv w cv <,> ~R [] breq1 z cv w cv <,> ~R [] <R v cv u cv <,> ~R [] br anbi1d x cv y cv <,> ~R [] f cv <R v cv u cv <,> ~R [] breq1 imbi12d anbi12d z cv w cv <,> ~R [] g cv f cv <R breq2 z cv w cv <,> ~R [] g cv f cv eqeq2 z cv w cv <,> ~R [] g cv <R f cv breq1 orbi12d negbid bibi12d z cv w cv <,> ~R [] g cv f cv <R breq2 z cv w cv <,> ~R [] g cv <R v cv u cv <,> ~R [] breq1 anbi12d f cv <R v cv u cv <,> ~R [] br imbi1d anbi12d v cv u cv <,> ~R [] h cv g cv <R breq2 f cv <R g cv br anbi2d v cv u cv <,> ~R [] h cv f cv <R breq2 imbi12d f cv <R g cv br f cv g cv = g cv <R f cv br \/ -. <-> anbi2d ltsopr <P P. x cv +P. w cv opr y cv +P. z cv opr sotric mpan x cv w cv addclpr y cv z cv addclpr syl2an an42s x cv y cv z cv w cv enreceq z visset w visset x visset y visset ltsrpr z visset y visset addcompr w visset x visset addcompr <P breq12i bitr x cv P. e. y cv P. e. /\ z cv P. e. w cv P. e. /\ /\ a1i orbi12d negbid bitr4d x visset y visset z visset w visset ltsrpr syl5bb v cv P. e. u cv P. e. /\ 3adant3 x cv +P. w cv oprex y cv +P. z cv oprex f visset g visset h cv ltapr u visset f visset g visset addcompr caoprord2 w visset u visset x cv addasspr z visset u visset y cv addasspr <P breq12i syl6bb x visset y visset z visset w visset ltsrpr syl5bb z cv +P. u cv oprex w cv +P. v cv oprex y cv ltapr z visset w visset v visset u visset ltsrpr syl5bb bi2anan9r x cv +P. w cv +P. u cv opr oprex ltsopr ltrelpr y cv +P. z cv +P. u cv opr oprex y cv +P. w cv +P. v cv opr oprex sotri x cv +P. u cv oprex dmplp ltrelpr 0npr x cv +P. u cv oprex y cv +P. v cv oprex w cv ltapr ndmordi x visset w visset u visset f visset g visset addcompr g visset h visset f cv addasspr caopr12 y visset w visset v visset f visset g visset addcompr g visset h visset f cv addasspr caopr12 <P breq12i x visset y visset v visset u visset ltsrpr 3imtr4 syl syl6bi v cv P. e. adantrl x cv P. e. adantll z cv P. e. w cv P. e. /\ 3adant2 jca 3ecoptocl so))
+thm (0lt1sr () () (br (0R) (<R) (1R)) (1pr 1pr 1P 1P addclpr mp2an 1pr 1P +P. 1P opr 1P ltaddpr mp2an 1pr elisseti 1P +P. 1P oprex addcompr breqtrr 1pr elisseti 1pr elisseti 1P +P. 1P oprex 1pr elisseti ltsrpr mpbir df-0r df-1r 3brtr4))
+thm (1ne0sr () () (-. (= (1R) (0R))) (ltsosr 1r <R R. 1R sonr mp2an 0lt1sr 1R 0R <R 1R breq1 mpbiri mto))
+thm (0idsr ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A x) (A y)) () (-> (e. A (R.)) (= (opr A (+R) (0R)) A)) (df-nr x cv y cv <,> ~R [] A +R 0R opreq1 x cv y cv <,> ~R [] A = id eqeq12d 1pr 1pr pm3.2i x cv y cv 1P 1P addsrpr mpan2 1pr x cv 1P addclpr mpan2 1pr y cv 1P addclpr mpan2 anim12i x visset y visset 1pr elisseti z visset w visset addcompr w visset v visset z cv addasspr caopr12 x cv y cv x cv +P. 1P opr y cv +P. 1P opr enreceq mpbiri mpdan eqtr4d df-0r x cv y cv <,> ~R [] +R opreq2i syl5eq ecoptocl))
+thm (1idsr ((x y) (A x) (A y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (-> (e. A (R.)) (= (opr A (.R) (1R)) A)) (df-nr x cv y cv <,> ~R [] A .R 1R opreq1 x cv y cv <,> ~R [] A = id eqeq12d 1pr 1pr 1P 1P addclpr mp2an 1pr pm3.2i x cv y cv 1P +P. 1P opr 1P mulsrpr mpan2 x cv 1idpr +P. x cv .P 1P opr opreq1d 1pr elisseti 1pr elisseti x cv distrpr syl5req y cv 1idpr +P. y cv .P 1P opr opreq1d 1pr elisseti 1pr elisseti y cv distrpr syl5eq +P. opreqan12d x cv .P 1P oprex y cv .P 1P +P. 1P opr oprex x cv addasspr x cv .P 1P +P. 1P opr oprex y visset y cv .P 1P oprex z visset w visset addcompr w visset v visset z cv addasspr caopr12 3eqtr3g x cv .P 1P +P. 1P opr opr y cv .P 1P opr addclpr 1pr 1pr 1P 1P addclpr mp2an x cv 1P +P. 1P opr mulclpr mpan2 1pr y cv 1P mulclpr mpan2 syl2an x cv .P 1P opr y cv .P 1P +P. 1P opr opr addclpr 1pr x cv 1P mulclpr mpan2 1pr 1pr 1P 1P addclpr mp2an y cv 1P +P. 1P opr mulclpr mpan2 syl2an anim12i x cv y cv x cv .P 1P +P. 1P opr opr +P. y cv .P 1P opr opr x cv .P 1P opr +P. y cv .P 1P +P. 1P opr opr opr enreceq syldan anidms mpbird eqtr4d df-1r x cv y cv <,> ~R [] .R opreq2i syl5eq ecoptocl))
+thm (00sr ((x y) (A x) (A y)) () (-> (e. A (R.)) (= (opr A (.R) (0R)) (0R))) (df-nr x cv y cv <,> ~R [] A .R 0R opreq1 0R eqeq1d 1pr 1pr pm3.2i x cv y cv 1P 1P mulsrpr mpan2 1pr 1pr pm3.2i x cv .P 1P opr +P. y cv .P 1P opr opr +P. 1P opr eqid x cv .P 1P opr +P. y cv .P 1P opr opr x cv .P 1P opr +P. y cv .P 1P opr opr 1P 1P enreceq mpbiri x cv .P 1P opr y cv .P 1P opr addclpr 1pr x cv 1P mulclpr mpan2 1pr y cv 1P mulclpr mpan2 syl2an x cv .P 1P opr y cv .P 1P opr addclpr 1pr x cv 1P mulclpr mpan2 1pr y cv 1P mulclpr mpan2 syl2an anim12i sylan mpan2 anidms eqtrd df-0r x cv y cv <,> ~R [] .R opreq2i df-0r 3eqtr4g ecoptocl))
+thm (ltasr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (f y) (f z) (f w) (f v) (f u)) ((ltasr.1 (e. A (V))) (ltasr.2 (e. B (V)))) (-> (e. C (R.)) (<-> (br A (<R) B) (br (opr C (+R) A) (<R) (opr C (+R) B)))) (ltasr.2 dmaddsr ltasr.1 ltrelsr 0nsr df-nr v cv u cv <,> ~R [] C +R x cv y cv <,> ~R [] opreq1 v cv u cv <,> ~R [] C +R z cv w cv <,> ~R [] opreq1 <R breq12d x cv y cv <,> ~R [] <R z cv w cv <,> ~R [] br bibi2d x cv y cv <,> ~R [] A <R z cv w cv <,> ~R [] breq1 x cv y cv <,> ~R [] A C +R opreq2 <R C +R z cv w cv <,> ~R [] opr breq1d bibi12d z cv w cv <,> ~R [] B A <R breq2 z cv w cv <,> ~R [] B C +R opreq2 C +R A opr <R breq2d bibi12d v cv u cv addclpr x cv P. e. y cv P. e. /\ z cv P. e. w cv P. e. /\ 3ad2ant1 x cv +P. w cv oprex y cv +P. z cv oprex v cv +P. u cv opr ltapr x visset y visset z visset w visset ltsrpr v cv +P. x cv oprex u cv +P. y cv oprex v cv +P. z cv oprex u cv +P. w cv oprex ltsrpr v visset x visset u visset y visset z visset addcompr z visset f visset y cv addasspr w visset caopr4 u cv +P. y cv oprex v cv +P. z cv oprex addcompr v visset z visset u visset x visset w visset addcompr w visset f visset x cv addasspr y visset caopr42 eqtr <P breq12i bitr 3bitr4g syl v cv u cv x cv y cv addsrpr z cv P. e. w cv P. e. /\ 3adant3 v cv u cv z cv w cv addsrpr x cv P. e. y cv P. e. /\ 3adant2 <R breq12d bitr4d 3ecoptocl 3coml ndmord))
+thm (pn0sr () () (-> (e. A (R.)) (= (opr A (+R) (opr A (.R) (-1R))) (0R))) (A 1idsr +R A .R -1R opr opreq1d A 00sr m1r elisseti 1r elisseti A distrsr m1p1sr A .R opreq2i A .R -1R oprex A .R 1R oprex addcomsr 3eqtr3 syl5eqr eqtr3d))
+thm (negexsr ((A x)) () (-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr A (+R) (cv x)) (0R))))) (A .R -1R oprex x cv A .R -1R opr R. eleq1 x cv A .R -1R opr A +R opreq2 0R eqeq1d anbi12d cla4ev m1r A -1R mulclsr mpan2 A pn0sr sylanc))
+thm (recexsrlem ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (u x) (f x) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((recexsrlem.1 (e. A (V)))) (-> (br (0R) (<R) A) (E. x (/\ (e. (cv x) (R.)) (= (opr A (.R) (cv x)) (1R))))) (recexsrlem.1 ltrelsr 0R brel pm3.27d df-nr y cv z cv <,> ~R [] A 0R <R breq2 y cv z cv <,> ~R [] A .R x cv opreq1 1R eqeq1d x biexdv imbi12d 1pr v cv 1P addclpr mpan2 1pr jctir y cv P. e. z cv P. e. /\ anim2i w cv .P v cv opr 1P = z cv +P. w cv opr y cv = /\ adantr y cv z cv v cv +P. 1P opr 1P mulsrpr syl y cv .P v cv +P. 1P opr opr z cv .P 1P opr addclpr y cv v cv +P. 1P opr mulclpr 1pr v cv 1P addclpr mpan2 sylan2 1pr z cv 1P mulclpr mpan2 syl2an an1rs y cv .P 1P opr z cv .P v cv +P. 1P opr opr addclpr 1pr y cv 1P mulclpr mpan2 z cv v cv +P. 1P opr mulclpr 1pr v cv 1P addclpr mpan2 sylan2 syl2an anassrs jca 1pr 1pr 1P 1P addclpr mp2an 1pr pm3.2i jctir y cv .P v cv +P. 1P opr opr +P. z cv .P 1P opr opr y cv .P 1P opr +P. z cv .P v cv +P. 1P opr opr opr 1P +P. 1P opr 1P enreceq syl z cv +P. w cv opr y cv .P v cv opreq1 eqcomd w cv .P v cv opr 1P z cv .P v cv opr +P. opreq2 z visset w visset v visset u visset f visset mulcompr f visset x visset u cv distrpr caoprdistrr syl5eq sylan9eqr +P. y cv .P 1P opr +P. z cv .P 1P opr opr opreq1d z cv .P v cv oprex 1pr elisseti y cv .P 1P opr +P. z cv .P 1P opr oprex u visset f visset addcompr f visset x visset u cv addasspr caopr32 syl6eq +P. 1P opreq1d 1pr elisseti 1pr elisseti z cv .P v cv opr +P. y cv .P 1P opr +P. z cv .P 1P opr opr opr addasspr syl6eq v visset 1pr elisseti y cv distrpr +P. z cv .P 1P opr opreq1i y cv .P 1P oprex z cv .P 1P oprex y cv .P v cv opr addasspr eqtr +P. 1P opreq1i v visset 1pr elisseti z cv distrpr y cv .P 1P opr +P. opreq2i y cv .P 1P oprex z cv .P v cv oprex z cv .P 1P oprex u visset f visset addcompr f visset x visset u cv addasspr caopr12 eqtr +P. 1P +P. 1P opr opreq1i 3eqtr4g syl5bir imp eqtrd df-1r syl6eqr enrex ~R V v cv +P. 1P opr 1P <,> ecexg ax-mp x cv v cv +P. 1P opr 1P <,> ~R [] y cv z cv <,> ~R [] .R opreq2 1R eqeq1d cla4ev syl exp43 imp3a v 19.23adv w cv v recexpr syl5 imp3a w 19.23adv y visset z visset gt0srpr y visset z cv w ltexpri sylbi syl5 ecoptocl mpcom 1r A .R x cv opr 1R R. eleq1 mpbiri x visset dmmulsr 0nsr A ndmoprrcl syl pm3.27d ancri x 19.22i syl))
+thm (addgt0sr () ((addgt0sr.1 (e. A (V))) (addgt0sr.2 (e. B (V)))) (-> (/\ (br (0R) (<R) A) (br (0R) (<R) B)) (br (0R) (<R) (opr A (+R) B))) (addgt0sr.1 ltrelsr 0R brel pm3.27d 0r elisseti addgt0sr.2 A ltasr A 0idsr <R A +R B opr breq1d bitrd syl biimpa 0r elisseti ltsosr ltrelsr addgt0sr.1 A +R B oprex sotri syldan))
+thm (mulgt0sr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (v x) (u x) (f x) (g x) (h x) (v y) (u y) (f y) (g y) (h y) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((mulgt0sr.1 (e. A (V))) (mulgt0sr.2 (e. B (V)))) (-> (/\ (br (0R) (<R) A) (br (0R) (<R) B)) (br (0R) (<R) (opr A (.R) B))) (mulgt0sr.1 ltrelsr 0R brel pm3.27d mulgt0sr.2 ltrelsr 0R brel pm3.27d anim12i df-nr x cv y cv <,> ~R [] A 0R <R breq2 0R <R z cv w cv <,> ~R [] br anbi1d x cv y cv <,> ~R [] A .R z cv w cv <,> ~R [] opreq1 0R <R breq2d imbi12d z cv w cv <,> ~R [] B 0R <R breq2 0R <R A br anbi2d z cv w cv <,> ~R [] B A .R opreq2 0R <R breq2d imbi12d x cv .P z cv opr +P. y cv .P w cv opr oprex x cv .P w cv opr +P. y cv .P z cv opr opr +P. v cv .P u cv opr oprex v cv .P w cv opr addcanpr y cv +P. v cv opr x cv w cv +P. u cv opr z cv .P opreq12 +P. y cv .P w cv opr +P. v cv .P w cv opr opr opreq1d w cv +P. u cv opr z cv y cv .P opreq2 w visset u visset y cv distrpr syl5eqr +P. v cv .P w cv opr +P. v cv .P u cv opr opr opreq1d y visset v visset w visset f visset g visset mulcompr g visset h visset f cv distrpr caoprdistrr y visset v visset u visset f visset g visset mulcompr g visset h visset f cv distrpr caoprdistrr +P. opreq12i w visset u visset y cv +P. v cv opr distrpr y cv .P w cv oprex y cv .P u cv oprex v cv .P w cv oprex f visset g visset addcompr g visset h visset f cv addasspr v cv .P u cv oprex caopr4 3eqtr4 v cv .P w cv oprex y cv .P z cv oprex v cv .P u cv oprex f visset g visset addcompr g visset h visset f cv addasspr caopr12 3eqtr4g y cv +P. v cv opr x cv .P w cv opreq1 y visset v visset w visset f visset g visset mulcompr g visset h visset f cv distrpr caoprdistrr syl5eqr +P. opreqan12rd eqtr3d y cv .P w cv oprex v cv .P w cv oprex x cv .P z cv opr addasspr x cv .P z cv opr +P. y cv .P w cv opr oprex v cv .P w cv oprex addcompr eqtr3 x cv .P w cv oprex y cv .P z cv opr +P. v cv .P u cv opr oprex v cv .P w cv opr addasspr v cv .P w cv oprex y cv .P z cv opr +P. v cv .P u cv opr oprex x cv .P w cv oprex f visset g visset addcompr g visset h visset f cv addasspr caopr32 y cv .P z cv oprex v cv .P u cv oprex x cv .P w cv opr addasspr v cv .P w cv opr +P. opreq2i 3eqtr4 3eqtr3g syl5 v cv .P u cv oprex x cv .P z cv opr +P. y cv .P w cv opr opr x cv .P w cv opr +P. y cv .P z cv opr opr ltaddpr2 x cv .P z cv opr +P. y cv .P w cv opr opr x cv .P w cv opr +P. y cv .P z cv opr opr +P. v cv .P u cv opr opr eqcom syl5ib v cv .P w cv opr P. e. adantl syld v cv w cv mulclpr sylan u cv P. e. a1d exp31 com13 imp imp4c v cv P. e. y cv +P. v cv opr x cv = u cv P. e. w cv +P. u cv opr z cv = an4 syl5ib v u 19.23advv x visset y cv v ltexpri z visset w cv u ltexpri anim12i v u v cv P. e. y cv +P. v cv opr x cv = /\ u cv P. e. w cv +P. u cv opr z cv = /\ eeanv sylibr syl5 x cv .P z cv opr y cv .P w cv opr addclpr x cv z cv mulclpr y cv w cv mulclpr syl2an an4s z cv P. e. w cv P. e. pm3.27 x cv P. e. y cv P. e. /\ adantl sylanc x cv y cv z cv w cv mulsrpr 0R <R breq2d x cv .P z cv opr +P. y cv .P w cv opr oprex x cv .P w cv opr +P. y cv .P z cv opr oprex gt0srpr syl6bb sylibrd x visset y visset gt0srpr z visset w visset gt0srpr anbi12i syl5ib 2ecoptocl mpcom))
+thm (sqgt0sr ((x y) (x z) (A x) (y z) (A y) (A z)) ((sqgt0sr.1 (e. A (V)))) (-> (e. A (R.)) (-> (-. (= A (0R))) (br (0R) (<R) (opr A (.R) A)))) (0r ltsosr <R R. A 0R sotrieq mpan mpan2 bicon2d m1r A -1R mulclsr mpan2 sqgt0sr.1 0r elisseti A .R -1R opr ltasr syl A pn0sr A .R -1R oprex sqgt0sr.1 addcomsr syl5eq m1r A -1R mulclsr mpan2 A .R -1R opr 0idsr syl <R breq12d bitrd A .R -1R oprex A .R -1R oprex mulgt0sr anidms syl6bi A A mulclsr A .R A opr 1idsr syl anidms sqgt0sr.1 m1r elisseti sqgt0sr.1 x visset y visset mulcomsr y visset z visset x cv mulasssr m1r elisseti caopr4 m1m1sr A .R A opr .R opreq2i eqtr syl5eq 0R <R breq2d sylibd sqgt0sr.1 sqgt0sr.1 mulgt0sr anidms A R. e. a1i jaod sylbird))
+thm (recexsr ((x y) (A x) (A y)) ((recexsr.1 (e. A (V)))) (-> (e. A (R.)) (-> (-. (= A (0R))) (E. x (/\ (e. (cv x) (R.)) (= (opr A (.R) (cv x)) (1R)))))) (recexsr.1 sqgt0sr A .R y cv oprex x cv A .R y cv opr R. eleq1 x cv A .R y cv opr A .R opreq2 1R eqeq1d anbi12d cla4ev recexsr.1 y visset A mulasssr 1R eqeq1i sylan2b A y cv mulclsr sylan exp31 imp3a y 19.23adv A .R A oprex y recexsrlem syl5 syld))
+thm (ssgt0sr () ((ssgt0sr.1 (e. A (V))) (ssgt0sr.2 (e. B (V)))) (-> (/\ (e. A (R.)) (e. B (R.))) (-> (-. (/\ (= A (0R)) (= B (0R)))) (br (0R) (<R) (opr (opr A (.R) A) (+R) (opr B (.R) B))))) (ssgt0sr.1 sqgt0sr imp B 0R = adantrr B 0R B 0R .R opreq12 anidms 0r 0R 00sr ax-mp syl6eq A .R A opr +R opreq2d A A mulclsr anidms A .R A opr 0idsr syl sylan9eqr 0R <R breq2d A 0R = -. adantrl mpbird B R. e. adantlr ancoms exp31 ssgt0sr.1 sqgt0sr ssgt0sr.2 sqgt0sr im2anan9 A .R A oprex B .R B oprex addgt0sr syl6 exp3a com3l pm2.61d com12 ssgt0sr.2 sqgt0sr imp A 0R = adantrl A 0R A 0R .R opreq12 anidms 0r 0R 00sr ax-mp syl6eq +R B .R B opr opreq1d B B mulclsr anidms B .R B opr 0idsr 0r elisseti B .R B oprex addcomsr syl5eq syl sylan9eqr 0R <R breq2d B 0R = -. adantrr mpbird A R. e. adantll exp32 ssgt0sr.1 sqgt0sr ssgt0sr.2 sqgt0sr im2anan9 A .R A oprex B .R B oprex addgt0sr syl6 exp3a pm2.61d jaod A 0R = B 0R = ianor syl5ib))
+thm (mappsrpr () ((mappsrpr.1 (e. A (V)))) (<-> (br (0R) (<R) ([] (<,> (opr A (+P.) (1P)) (1P)) (~R))) (e. A (P.))) (df-0r <R A +P. 1P opr 1P <,> ~R [] breq1i 1pr elisseti 1pr elisseti A +P. 1P oprex 1pr elisseti ltsrpr mappsrpr.1 1pr elisseti addcompr 1P +P. opreq2i 1pr elisseti mappsrpr.1 1P addasspr eqtr4 1P +P. 1P opr <P breq2i 1P +P. 1P opr +P. A oprex ltrelpr 1P +P. 1P opr brel pm3.27d mappsrpr.1 dmplp 0npr 1P +P. 1P opr ndmoprrcl pm3.27d syl 1pr 1pr 1P 1P addclpr mp2an 1P +P. 1P opr A ltaddpr mpan impbi bitr 3bitr))
+thm (ltpsrpr () ((ltpsrpr.1 (e. A (V))) (ltpsrpr.2 (e. B (V)))) (<-> (br ([] (<,> (opr A (+P.) (1P)) (1P)) (~R)) (<R) ([] (<,> (opr B (+P.) (1P)) (1P)) (~R))) (br A (<P) B)) (ltpsrpr.1 1pr elisseti addcompr ltpsrpr.2 1pr elisseti addcompr <P breq12i A +P. 1P oprex 1pr elisseti addcompr <P 1P +P. B +P. 1P opr opr breq1i A +P. 1P oprex 1pr elisseti B +P. 1P oprex 1pr elisseti ltsrpr 1pr A +P. 1P oprex B +P. 1P oprex 1P ltapr ax-mp 3bitr4 1pr ltpsrpr.1 ltpsrpr.2 1P ltapr ax-mp 3bitr4))
+thm (map2psrpr ((x y) (x z) (A x) (y z) (A y) (A z)) ((map2psrpr.1 (e. A (V)))) (<-> (br (0R) (<R) A) (E. x (/\ (e. (cv x) (P.)) (= ([] (<,> (opr (cv x) (+P.) (1P)) (1P)) (~R)) A)))) (map2psrpr.1 ltrelsr 0R brel pm3.27d df-nr y cv z cv <,> ~R [] A 0R <R breq2 y cv z cv <,> ~R [] A x cv +P. 1P opr 1P <,> ~R [] eqeq2 x cv P. e. anbi2d x biexdv imbi12d x cv +P. 1P opr 1P y cv z cv enreceq 1pr x cv 1P addclpr mpan2 1pr jctir sylan 1pr elisseti z visset x cv addasspr x visset 1P +P. z cv oprex addcompr eqtr 1P +P. y cv opr eqeq1i syl6bb expcom pm5.32d x biexdv df-0r <R y cv z cv <,> ~R [] breq1i 1pr elisseti 1pr elisseti y visset z visset ltsrpr bitr 1P +P. y cv oprex 1P +P. z cv opr x ltexpri sylbi syl5bir ecoptocl mpcom x cv +P. 1P opr 1P <,> ~R [] A 0R <R breq2 x visset mappsrpr syl5bbr biimpac x 19.23aiv impbi))
+thm (suppsrlem ((w x) (A x) (A w) (B x) (B w)) ((suppsr.1 (= B ({|} w (e. ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) A))))) (-> (/\ (A. x (-> (e. (cv x) A) (br (0R) (<R) (cv x)))) (-. (= A ({/})))) (/\ (C_ B (P.)) (-. (= B ({/}))))) (enrex ~R V w cv +P. 1P opr 1P <,> ecexg ax-mp x cv w cv +P. 1P opr 1P <,> ~R [] A eleq1 x cv w cv +P. 1P opr 1P <,> ~R [] 0R <R breq2 imbi12d cla4v suppsr.1 eqabi syl5ib w visset mappsrpr syl6ib ssrdv A {/} = -. adantr x x cv A e. 0R <R x cv br -> hba1 B {/} = -. x ax-17 hbim x x cv A e. 0R <R x cv br -> ax-4 com12 w cv +P. 1P opr 1P <,> ~R [] x cv A eleq1 suppsr.1 eqabi syl5bb biimprcd w cv B n0i syl6 w cv P. e. adantld w 19.23adv x visset w map2psrpr syl5ib syld 19.23ai com12 A x n0 syl5ib imp jca))
+thm (suppsr ((w x) (v x) (u x) (A x) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (B x) (y z) (w y) (v y) (u y) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) ((suppsr.1 (= B ({|} w (e. ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) A))))) (-> (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) (<R) (cv x)))) (-. (= A ({/})))) (E. x (/\ (br (0R) (<R) (cv x)) (A. y (-> (br (0R) (<R) (cv y)) (-> (e. (cv y) A) (br (cv y) (<R) (cv x)))))))) (E. x (/\ (br (0R) (<R) (cv x)) (A. y (-> (br (0R) (<R) (cv y)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<R) (cv y)))) (-> (br (cv y) (<R) (cv x)) (E. z (/\ (br (0R) (<R) (cv z)) (/\ (e. (cv z) A) (br (cv y) (<R) (cv z)))))))))))) (B w v u suppr enrex ~R V w cv +P. 1P opr 1P <,> ecexg ax-mp w cv +P. 1P opr 1P <,> ~R [] x cv <R y cv breq1 negbid y cv A e. imbi2d w cv +P. 1P opr 1P <,> ~R [] x cv y cv <R breq2 z 0R <R z cv br z cv A e. y cv <R z cv br /\ /\ E. imbi1d anbi12d 0R <R y cv br imbi2d y bialdv enrex ~R V v cv +P. 1P opr 1P <,> ecexg ax-mp v cv +P. 1P opr 1P <,> ~R [] y cv A eleq1 v visset w cv v cv +P. 1P opreq1 w cv +P. 1P opr v cv +P. 1P opr 1P opeq1 w cv +P. 1P opr 1P <,> v cv +P. 1P opr 1P <,> ~R eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb v cv +P. 1P opr 1P <,> ~R [] y cv w cv +P. 1P opr 1P <,> ~R [] <R breq2 w visset v visset ltpsrpr syl5bbr negbid imbi12d v cv +P. 1P opr 1P <,> ~R [] y cv <R w cv +P. 1P opr 1P <,> ~R [] breq1 v visset w visset ltpsrpr syl5bbr v cv +P. 1P opr 1P <,> ~R [] y cv <R z cv breq1 z cv A e. anbi2d 0R <R z cv br anbi2d z biexdv enrex ~R V u cv +P. 1P opr 1P <,> ecexg ax-mp u cv +P. 1P opr 1P <,> ~R [] z cv A eleq1 u visset w cv u cv +P. 1P opreq1 w cv +P. 1P opr u cv +P. 1P opr 1P opeq1 w cv +P. 1P opr 1P <,> u cv +P. 1P opr 1P <,> ~R eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb u cv +P. 1P opr 1P <,> ~R [] z cv v cv +P. 1P opr 1P <,> ~R [] <R breq2 v visset u visset ltpsrpr syl5bbr anbi12d u cv +P. 1P opr 1P <,> ~R [] z cv 0R <R breq2 u visset mappsrpr syl5bbr z visset u map2psrpr gencbvex syl5bb imbi12d anbi12d v cv +P. 1P opr 1P <,> ~R [] y cv 0R <R breq2 v visset mappsrpr syl5bbr y visset v map2psrpr gencbval syl5bb w cv +P. 1P opr 1P <,> ~R [] x cv 0R <R breq2 w visset mappsrpr syl5bbr x visset w map2psrpr gencbvex sylib enrex ~R V w cv +P. 1P opr 1P <,> ecexg ax-mp w cv +P. 1P opr 1P <,> ~R [] x cv y cv <R breq2 y cv A e. imbi2d 0R <R y cv br imbi2d y bialdv enrex ~R V v cv +P. 1P opr 1P <,> ecexg ax-mp v cv +P. 1P opr 1P <,> ~R [] y cv A eleq1 v visset w cv v cv +P. 1P opreq1 w cv +P. 1P opr v cv +P. 1P opr 1P opeq1 w cv +P. 1P opr 1P <,> v cv +P. 1P opr 1P <,> ~R eceq2 3syl A eleq1d suppsr.1 elab2 syl5bb v cv +P. 1P opr 1P <,> ~R [] y cv <R w cv +P. 1P opr 1P <,> ~R [] breq1 v visset w visset ltpsrpr syl5bbr imbi12d v cv +P. 1P opr 1P <,> ~R [] y cv 0R <R breq2 v visset mappsrpr syl5bbr y visset v map2psrpr gencbval syl5bb w cv +P. 1P opr 1P <,> ~R [] x cv 0R <R breq2 w visset mappsrpr syl5bbr x visset w map2psrpr gencbvex sylan2br suppsr.1 x suppsrlem sylan))
+thm (suppsr2 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) (<R) (cv x)))) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) (<R) (cv x)))))))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<R) (cv y)))) (-> (br (cv y) (<R) (cv x)) (E. z (/\ (e. (cv z) (R.)) (/\ (e. (cv z) A) (br (cv y) (<R) (cv z)))))))))))) (x x cv A e. 0R <R x cv br -> hba1 A {/} = -. x ax-17 hban x cv z cv A eleq1 x cv z cv 0R <R breq2 imbi12d a4b1 y cv z cv R. eleq1 y cv z cv A eleq1 y cv z cv <R x cv breq1 imbi12d imbi12d a4b1 z visset ltrelsr 0R brel pm3.27d syl5 0R <R z cv br z cv A e. z cv <R x cv br anc2l syl 0r elisseti ltsosr ltrelsr z visset x visset sotri syl8 sylan9 pm2.43d z 19.23adv A z n0 syl5ib exp com23 imp y visset ltrelsr 0R brel pm3.27d y cv A e. y cv <R x cv br -> imim1i y 19.20i x x cv A e. 0R <R x cv br -> A. A {/} = -. /\ a1i jcad x cv R. e. adantld 19.22d imdistani v cv w cv +P. 1P opreq1 v cv +P. 1P opr w cv +P. 1P opr 1P opeq1 v cv +P. 1P opr 1P <,> w cv +P. 1P opr 1P <,> ~R eceq2 3syl A eleq1d cbvabv x y z suppsr syl x x cv A e. 0R <R x cv br -> hba1 A {/} = -. x ax-17 hban x visset ltrelsr 0R brel pm3.27d x x cv A e. 0R <R x cv br -> A. A {/} = -. /\ a1i x cv y cv A eleq1 x cv y cv 0R <R breq2 imbi12d a4b1 y cv A e. 0R <R y cv br y cv A e. x cv <R y cv br -. -> imim1 y cv A e. x cv <R y cv br -. pm2.43 y cv R. e. a1d syl6 syl A {/} = -. adantr x x cv A e. 0R <R x cv br -> A. A {/} = -. /\ y cv R. e. /\ 0R <R y cv br y cv <R x cv br z 0R <R z cv br z cv A e. y cv <R z cv br /\ /\ E. -> -> pm3.27 x cv z cv A eleq1 x cv z cv 0R <R breq2 imbi12d a4b1 ancld 0r ltsosr <R R. 0R y cv sotric mpan mpan bicon2d 0R y cv <R z cv breq1 biimpd y visset ltsosr ltrelsr 0r elisseti z visset sotri exp jaoi syl6bir impcom ancld z cv A e. anim2d z cv A e. 0R <R z cv br y cv <R z cv br an12 syl6ib sylan9 z 19.22dv A z n0 syl5ib exp32 com24 imp31 z 0R <R z cv br z cv A e. y cv <R z cv br /\ /\ E. y cv <R x cv br ax-1 syl6 0R <R y cv br y cv <R x cv br z 0R <R z cv br z cv A e. y cv <R z cv br /\ /\ E. -> -> adantr pm2.61d an1rs z visset ltrelsr 0R brel pm3.27d z cv A e. y cv <R z cv br /\ anim1i z 19.22i syl6 exp31 anim12d 0R <R y cv br y cv A e. x cv <R y cv br -. -> y cv <R x cv br z 0R <R z cv br z cv A e. y cv <R z cv br /\ /\ E. -> jcab y cv R. e. y cv A e. x cv <R y cv br -. -> y cv <R x cv br z z cv R. e. z cv A e. y cv <R z cv br /\ /\ E. -> jcab 3imtr4g y 19.20dv anim12d 19.22d x x cv R. e. y y cv R. e. y cv A e. y cv <R x cv br -> -> A. /\ E. adantr mpd))
+thm (suppsr3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((suppsr3.1 (= B ({|} y (/\ (e. (cv y) A) (br (0R) (<R) (cv y))))))) (-> (/\ (E. y (/\ (e. (cv y) A) (br (0R) (<R) (cv y)))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) (<R) (cv x)))))))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<R) (cv y)))) (-> (br (cv y) (<R) (cv x)) (E. z (/\ (e. (cv z) (R.)) (/\ (e. (cv z) A) (br (cv y) (<R) (cv z)))))))))))) (x visset y cv x cv A eleq1 y cv x cv 0R <R breq2 anbi12d suppsr3.1 elab2 pm3.27bd x ax-gen x B y z suppsr2 mpanl1 B y n0 suppsr3.1 eqabi y biex bitr biimpr suppsr3.1 eqabi pm3.26bd y cv <R x cv br imim1i y cv R. e. imim2i y 19.20i x cv R. e. anim2i x 19.22i syl2an y y cv A e. 0R <R y cv br /\ hbe1 x cv R. e. y ax-17 hban y y cv R. e. y cv B e. x cv <R y cv br -. -> y cv <R x cv br z z cv R. e. z cv B e. y cv <R z cv br /\ /\ E. -> /\ -> hba1 hban suppsr3.1 eqabi x cv <R y cv br -. imbi1i y cv A e. 0R <R y cv br x cv <R y cv br -. impexp bitr y cv A e. 0R <R y cv br x cv <R y cv br -. pm2.04 sylbi y cv R. e. imim2i com23 y a4s y y cv A e. 0R <R y cv br /\ E. x cv R. e. /\ adantl x cv R. e. y ax-17 y y cv R. e. y cv B e. x cv <R y cv br -. -> -> hba1 0R <R x cv br y ax-17 hbim hbim ltsosr <R R. x cv y cv sotric mpan bicon2d y visset ltrelsr 0R brel pm3.27d sylan2 x cv y cv 0R <R breq2 biimprcd 0r elisseti ltsosr ltrelsr y visset x visset sotri exp jaod x cv R. e. adantl sylbird exp y cv A e. adantld a2d suppsr3.1 eqabi x cv <R y cv br -. imbi1i syl5ib y cv R. e. imim2d y visset ltrelsr 0R brel pm3.27d y cv A e. 0R <R y cv br /\ 0R <R x cv br -> imim1i 0R <R y cv br y cv A e. anabs7 0R <R x cv br imbi1i 0R <R y cv br y cv A e. 0R <R y cv br /\ 0R <R x cv br impexp bitr3 sylibr syl6 y a4sd com3r 19.23ai imp31 0r elisseti ltsosr ltrelsr x visset y visset sotri exp con3d syl x cv <R y cv br -. y cv A e. ax-1 y cv R. e. a1d syl6 pm2.61d exp z visset y cv z cv A eleq1 y cv z cv 0R <R breq2 anbi12d suppsr3.1 elab2 pm3.26bd y cv <R z cv br anim1i z cv R. e. anim2i z 19.22i y cv <R x cv br imim2i y cv R. e. imim2i y a4s y y cv A e. 0R <R y cv br /\ E. x cv R. e. /\ a1i anim12d y cv R. e. y cv B e. x cv <R y cv br -. -> y cv <R x cv br z z cv R. e. z cv B e. y cv <R z cv br /\ /\ E. -> jcab y bial y y cv R. e. y cv B e. x cv <R y cv br -. -> -> y cv R. e. y cv <R x cv br z z cv R. e. z cv B e. y cv <R z cv br /\ /\ E. -> -> 19.26 bitr y cv R. e. y cv A e. x cv <R y cv br -. -> y cv <R x cv br z z cv R. e. z cv A e. y cv <R z cv br /\ /\ E. -> jcab 3imtr4g imp 19.21ai exp31 imdistand x 19.22dv x x cv R. e. y y cv R. e. y cv A e. y cv <R x cv br -> -> A. /\ E. adantr mpd))
+thm (supsrlem1 () ((supsrlem.1 (e. C (R.)))) (<-> (e. (opr C (+R) (opr A (+R) (-1R))) (R.)) (e. A (R.))) (A +R -1R oprex dmaddsr 0nsr C ndmoprrcl pm3.27d m1r elisseti dmaddsr 0nsr A ndmoprrcl pm3.26d syl m1r A -1R addclsr mpan2 supsrlem.1 C A +R -1R opr addclsr mpan syl impbi))
+thm (supsrlem2 ((A x) (C x)) ((supsrlem.1 (e. C (R.)))) (<-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr C (+R) (opr (cv x) (+R) (-1R))) A)))) (m1r A -1R mulclsr mpan2 supsrlem.1 jctir A .R -1R opr C addclsr syl m1r A .R -1R opr +R C opr -1R addclsr mpan2 A .R -1R opr +R C opr +R -1R opr x negexsr 3syl A pn0sr +R C +R x cv +R -1R opr opr opreq1d m1r x cv -1R addclsr mpan2 supsrlem.1 jctil C x cv +R -1R opr addclsr C +R x cv +R -1R opr opr 0idsr 3syl C +R x cv +R -1R opr oprex 0r elisseti addcomsr syl5eqr sylan9eq A .R -1R oprex C +R x cv +R -1R opr oprex A addasssr syl5eqr A 0idsr x cv R. e. adantr eqeq12d m1r elisseti x visset A .R -1R opr +R C opr addasssr m1r elisseti x visset addcomsr A .R -1R opr +R C opr +R opreq2i supsrlem.1 elisseti x cv +R -1R oprex A .R -1R opr addasssr 3eqtr 0R eqeq1i A .R -1R opr +R C +R x cv +R -1R opr opr opr 0R A +R opreq2 sylbi syl5bi exp imdistand x 19.22dv mpd C +R x cv +R -1R opr opr A R. eleq1 m1r x cv -1R addclsr mpan2 supsrlem.1 jctil C x cv +R -1R opr addclsr syl syl5bi impcom x 19.23aiv impbi))
+thm (supsrlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((supsrlem.1 (e. C (R.))) (supsrlem3.2 (e. A (V))) (supsrlem3.3 (e. B (V)))) (<-> (br (opr C (+R) (opr A (+R) (-1R))) (<R) (opr C (+R) (opr B (+R) (-1R)))) (br A (<R) B)) (m1r supsrlem3.2 supsrlem3.3 x visset y visset z cv ltasr m1r elisseti x visset y visset addcomsr caoprord2 ax-mp supsrlem.1 A +R -1R oprex B +R -1R oprex C ltasr ax-mp bitr2))
+thm (supsrlem4 ((D x) (w x) (A x) (A w) (B x) (B w) (C x) (C w)) ((supsrlem.1 (e. C (R.))) (supsrlem.2 (= B ({|} w (e. (opr C (+R) (opr (cv w) (+R) (-1R))) A)))) (supsrlem4.1 (e. D (V)))) (<-> (e. D B) (e. (opr C (+R) (opr D (+R) (-1R))) A)) (supsrlem4.1 x cv D +R -1R opreq1 C +R opreq2d A eleq1d supsrlem.2 w cv x cv +R -1R opreq1 C +R opreq2d A eleq1d cbvabv eqtr elab2))
+thm (supsrlem5 ((w y) (A y) (A w) (B y) (B w) (C y) (C w)) ((supsrlem.1 (e. C (R.))) (supsrlem.2 (= B ({|} w (e. (opr C (+R) (opr (cv w) (+R) (-1R))) A))))) (-> (e. C A) (E. y (/\ (e. (cv y) B) (br (0R) (<R) (cv y))))) (supsrlem.1 supsrlem.2 1r elisseti supsrlem4 0lt1sr 1R B e. biantru 1r elisseti m1r elisseti addcomsr m1p1sr eqtr C +R opreq2i supsrlem.1 C 0idsr ax-mp eqtr A eleq1i 3bitr3 1r elisseti y cv 1R B eleq1 y cv 1R 0R <R breq2 anbi12d cla4ev sylbir))
+thm (supsrlem6 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((supsrlem.1 (e. C (R.))) (supsrlem.2 (= B ({|} w (e. (opr C (+R) (opr (cv w) (+R) (-1R))) A))))) (-> (/\ (e. C A) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) (<R) (cv x)))))))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<R) (cv y)))) (-> (br (cv y) (<R) (cv x)) (E. z (/\ (e. (cv z) (R.)) (/\ (e. (cv z) A) (br (cv y) (<R) (cv z)))))))))))) (x cv v cv B eleq1 x cv v cv 0R <R breq2 anbi12d cbvabv w u suppsr3 C +R w cv +R -1R opr oprex C +R w cv +R -1R opr opr x cv <R y cv breq1 negbid y cv A e. imbi2d C +R w cv +R -1R opr opr x cv y cv <R breq2 z z cv R. e. z cv A e. y cv <R z cv br /\ /\ E. imbi1d anbi12d y cv R. e. imbi2d y bialdv C +R v cv +R -1R opr oprex C +R v cv +R -1R opr opr y cv A eleq1 supsrlem.1 supsrlem.2 v visset supsrlem4 syl5bb C +R v cv +R -1R opr opr y cv C +R w cv +R -1R opr opr <R breq2 supsrlem.1 w visset v visset supsrlem3 syl5bbr negbid imbi12d C +R v cv +R -1R opr opr y cv <R C +R w cv +R -1R opr opr breq1 supsrlem.1 v visset w visset supsrlem3 syl5bbr C +R v cv +R -1R opr opr y cv <R z cv breq1 z cv A e. anbi2d z cv R. e. anbi2d z biexdv C +R u cv +R -1R opr oprex C +R u cv +R -1R opr opr z cv A eleq1 supsrlem.1 supsrlem.2 u visset supsrlem4 syl5bb C +R u cv +R -1R opr opr z cv C +R v cv +R -1R opr opr <R breq2 supsrlem.1 v visset u visset supsrlem3 syl5bbr anbi12d C +R u cv +R -1R opr opr z cv R. eleq1 supsrlem.1 u cv supsrlem1 syl5bbr supsrlem.1 z cv u supsrlem2 gencbvex syl5bb imbi12d anbi12d C +R v cv +R -1R opr opr y cv R. eleq1 supsrlem.1 v cv supsrlem1 syl5bbr supsrlem.1 y cv v supsrlem2 gencbval syl5bb C +R w cv +R -1R opr opr x cv R. eleq1 supsrlem.1 w cv supsrlem1 syl5bbr supsrlem.1 x cv w supsrlem2 gencbvex sylib C +R w cv +R -1R opr oprex C +R w cv +R -1R opr opr x cv y cv <R breq2 y cv A e. imbi2d y cv R. e. imbi2d y bialdv C +R v cv +R -1R opr oprex C +R v cv +R -1R opr opr y cv A eleq1 supsrlem.1 supsrlem.2 v visset supsrlem4 syl5bb C +R v cv +R -1R opr opr y cv <R C +R w cv +R -1R opr opr breq1 supsrlem.1 v visset w visset supsrlem3 syl5bbr imbi12d C +R v cv +R -1R opr opr y cv R. eleq1 supsrlem.1 v cv supsrlem1 syl5bbr supsrlem.1 y cv v supsrlem2 gencbval syl5bb C +R w cv +R -1R opr opr x cv R. eleq1 supsrlem.1 w cv supsrlem1 syl5bbr supsrlem.1 x cv w supsrlem2 gencbvex sylan2br supsrlem.1 supsrlem.2 v supsrlem5 sylan))
+thm (supsr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u)) () (-> (/\ (/\ (C_ A (R.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) (<R) (cv x)))))))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<R) (cv y)))) (-> (br (cv y) (<R) (cv x)) (E. z (/\ (e. (cv z) (R.)) (/\ (e. (cv z) A) (br (cv y) (<R) (cv z)))))))))))) (A R. w cv ssel w cv w cv R. e. w cv 0R if A eleq1 x x cv R. e. y y cv R. e. y cv A e. y cv <R x cv br -> -> A. /\ E. x x cv R. e. y y cv R. e. y cv A e. x cv <R y cv br -. -> y cv <R x cv br z z cv R. e. z cv A e. y cv <R z cv br /\ /\ E. -> /\ -> A. /\ E. -> imbi1d 0r w cv elimel v cv u cv +R -1R opreq1 w cv R. e. w cv 0R if +R opreq2d A eleq1d cbvabv x y z supsrlem6 exp dedth syli w 19.23adv A w n0 syl5ib imp31))
+thm (opelcn () ((opelcn.1 (e. B (V)))) (<-> (e. (<,> A B) (CC)) (/\ (e. A (R.)) (e. B (R.)))) (df-c A B <,> eleq2i opelcn.1 A R. R. opelxp bitr))
+thm (opelreal () () (<-> (e. (<,> A (0R)) (RR)) (e. A (R.))) (df-r A 0R <,> eleq2i 0r elisseti A R. 0R {} opelxp 0r elisseti 0R elsnc A R. e. anbi2i 3bitr 0R eqid mpbiran2))
+thm (elreal ((x y) (A x) (A y)) () (<-> (e. A (RR)) (E. x (/\ (e. (cv x) (R.)) (= (<,> (cv x) (0R)) A)))) (df-r A eleq2i A R. 0R {} x y elxp A x cv y cv <,> = x cv R. e. y cv 0R {} e. /\ ancom x cv R. e. y cv 0R {} e. A x cv y cv <,> = anass y 0R elsn A x cv y cv <,> eqcom anbi12i y cv 0R x cv opeq2 A eqeq1d pm5.32i bitr x cv R. e. anbi2i x cv R. e. y cv 0R = x cv 0R <,> A = an12 3bitr bitr y biex y y cv 0R = x cv R. e. x cv 0R <,> A = /\ 19.41v bitr 0r elisseti y isseti mpbiran x biex 3bitr))
+thm (0ncn () () (-. (e. ({/}) (CC))) (R. R. 0nelxp df-c {/} eleq2i mtbir))
+thm (ltrelre ((x y) (x z) (w x) (y z) (w y) (w z)) () (C_ (<RR) (X. (RR) (RR))) (x y z w df-lt x y RR RR z w x cv z cv 0R <,> = y cv w cv 0R <,> = /\ z cv <R w cv br /\ E. E. opabssxp eqsstr))
+thm (addcnsr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (+) (<,> C D)) (<,> (opr A (+R) C) (opr B (+R) D)))) (A +R C opr B +R D opr opex w cv +R u cv opr A +R u cv opr v cv +R f cv opr B +R f cv opr opeq12 w cv A +R u cv opreq1 v cv B +R f cv opreq1 syl2an A +R u cv opr A +R C opr B +R f cv opr B +R D opr opeq12 u cv C A +R opreq2 f cv D B +R opreq2 syl2an sylan9eq x y z w v u f df-plus df-c x cv eleq2i df-c y cv eleq2i anbi12i w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv +R u cv opr v cv +R f cv opr <,> = /\ E. E. E. E. anbi1i x y z bioprabi eqtr oprabval3))
+thm (mulcnsr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (x.) (<,> C D)) (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D))))) (A .R C opr +R -1R .R B .R D opr opr opr B .R C opr +R A .R D opr opr opex w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr A .R u cv opr +R -1R .R B .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr B .R u cv opr +R A .R f cv opr opr opeq12 w cv A .R u cv opreq1 v cv B .R f cv opreq1 -1R .R opreq2d +R opreqan12d v cv B .R u cv opreq1 w cv A .R f cv opreq1 +R opreqan12rd sylanc A .R u cv opr +R -1R .R B .R f cv opr opr opr A .R C opr +R -1R .R B .R D opr opr opr B .R u cv opr +R A .R f cv opr opr B .R C opr +R A .R D opr opr opeq12 u cv C A .R opreq2 f cv D B .R opreq2 -1R .R opreq2d +R opreqan12d u cv C B .R opreq2 f cv D A .R opreq2 +R opreqan12d sylanc sylan9eq x y z w v u f df-mul df-c x cv eleq2i df-c y cv eleq2i anbi12i w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> = /\ E. E. E. E. anbi1i x y z bioprabi eqtr oprabval3))
+thm (eqresr () ((eqresr.1 (e. A (V)))) (<-> (= (<,> A (0R)) (<,> B (0R))) (= A B)) (eqresr.1 0r elisseti 0r elisseti B opth 0R eqid mpbiran2))
+thm (addresr () () (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (+) (<,> B (0R))) (<,> (opr A (+R) B) (0R)))) (0r 0r pm3.2i A 0R B 0R addcnsr an4s mpan2 0r 0R 0idsr ax-mp 0R +R 0R opr 0R A +R B opr opeq2 ax-mp syl6eq))
+thm (mulresr () ((mulresr.1 (e. B (V)))) (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (x.) (<,> B (0R))) (<,> (opr A (.R) B) (0R)))) (0r 0r pm3.2i A 0R B 0R mulcnsr an4s mpan2 A .R B opr +R -1R .R 0R .R 0R opr opr opr A .R B opr 0R .R B opr +R A .R 0R opr opr 0R opeq12 A B mulclsr A .R B opr 0idsr syl 0r 0R 00sr ax-mp -1R .R opreq2i m1r -1R 00sr ax-mp eqtr A .R B opr +R opreq2i syl5eq B 00sr 0r elisseti mulresr.1 mulcomsr syl5eq A 00sr +R opreqan12rd 0r 0R 0idsr ax-mp syl6eq sylanc eqtrd))
+thm (ltresr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((ltresr.1 (e. A (V))) (ltresr.2 (e. B (V)))) (<-> (br (<,> A (0R)) (<RR) (<,> B (0R))) (br A (<R) B)) (B 0R opex ltrelre A 0R <,> brel A opelreal B opelreal anbi12i sylib ltresr.2 ltrelsr A brel A 0R opex B 0R opex x cv A 0R <,> RR eleq1 y cv RR e. anbi1d x cv A 0R <,> z cv 0R <,> eqeq1 y cv w cv 0R <,> = anbi1d z cv <R w cv br anbi1d z w bi2exdv anbi12d y cv B 0R <,> RR eleq1 A 0R <,> RR e. anbi2d y cv B 0R <,> w cv 0R <,> eqeq1 A 0R <,> z cv 0R <,> = anbi2d z cv <R w cv br anbi1d z w bi2exdv anbi12d x y z w df-lt brab baib ltresr.1 z cv eqresr ltresr.2 w cv eqresr anbi12i ltresr.1 ltresr.2 w visset z cv opth bitr4 z cv <R w cv br anbi1i z w bi2ex syl6bb A opelreal B opelreal syl2anbr z cv A w cv B <R breq12 R. R. copsex2g bitrd pm5.21nii))
+thm (suprelem ((w x) (A x) (A w) (B x) (B w)) ((supre.1 (= B ({|} w (e. (<,> (cv w) (0R)) A))))) (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (/\ (C_ B (R.)) (-. (= B ({/}))))) (A RR w cv 0R <,> ssel supre.1 eqabi syl5ib w cv opelreal syl6ib ssrdv A {/} = -. adantr A RR x cv ssel com12 w cv 0R <,> x cv A eleq1 supre.1 eqabi syl5bb biimprcd w cv B n0i syl6 w cv R. e. adantld w 19.23adv x cv w elreal syl5ib syld x 19.23aiv com12 A x n0 syl5ib imp jca))
+thm (supre ((w x) (v x) (u x) (A x) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (B x) (y z) (w y) (v y) (u y) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) ((supre.1 (= B ({|} w (e. (<,> (cv w) (0R)) A))))) (-> (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (RR)) (A. y (-> (e. (cv y) (RR)) (-> (e. (cv y) A) (br (cv y) (<RR) (cv x)))))))) (E. x (/\ (e. (cv x) (RR)) (A. y (-> (e. (cv y) (RR)) (/\ (-> (e. (cv y) A) (-. (br (cv x) (<RR) (cv y)))) (-> (br (cv y) (<RR) (cv x)) (E. z (/\ (e. (cv z) (RR)) (/\ (e. (cv z) A) (br (cv y) (<RR) (cv z)))))))))))) (B w v u supsr w cv 0R opex w cv 0R <,> x cv <RR y cv breq1 negbid y cv A e. imbi2d w cv 0R <,> x cv y cv <RR breq2 z z cv RR e. z cv A e. y cv <RR z cv br /\ /\ E. imbi1d anbi12d y cv RR e. imbi2d y bialdv v cv 0R opex v cv 0R <,> y cv A eleq1 v visset w cv v cv 0R opeq1 A eleq1d supre.1 elab2 syl5bb v cv 0R <,> y cv w cv 0R <,> <RR breq2 w visset v visset ltresr syl5bbr negbid imbi12d v cv 0R <,> y cv <RR w cv 0R <,> breq1 v visset w visset ltresr syl5bbr v cv 0R <,> y cv <RR z cv breq1 z cv A e. anbi2d z cv RR e. anbi2d z biexdv u cv 0R opex u cv 0R <,> z cv A eleq1 u visset w cv u cv 0R opeq1 A eleq1d supre.1 elab2 syl5bb u cv 0R <,> z cv v cv 0R <,> <RR breq2 v visset u visset ltresr syl5bbr anbi12d u cv 0R <,> z cv RR eleq1 u cv opelreal syl5bbr z cv u elreal gencbvex syl5bb imbi12d anbi12d v cv 0R <,> y cv RR eleq1 v cv opelreal syl5bbr y cv v elreal gencbval syl5bb w cv 0R <,> x cv RR eleq1 w cv opelreal syl5bbr x cv w elreal gencbvex sylib w cv 0R opex w cv 0R <,> x cv y cv <RR breq2 y cv A e. imbi2d y cv RR e. imbi2d y bialdv v cv 0R opex v cv 0R <,> y cv A eleq1 v visset w cv v cv 0R opeq1 A eleq1d supre.1 elab2 syl5bb v cv 0R <,> y cv <RR w cv 0R <,> breq1 v visset w visset ltresr syl5bbr imbi12d v cv 0R <,> y cv RR eleq1 v cv opelreal syl5bbr y cv v elreal gencbval syl5bb w cv 0R <,> x cv RR eleq1 w cv opelreal syl5bbr x cv w elreal gencbvex sylan2br supre.1 suprelem sylan))
+thm (ltsor ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (/\ (Or (<RR) (RR)) (= (RR) (RR))) (x cv w elreal y cv v elreal z cv u elreal w cv 0R <,> x cv <RR v cv 0R <,> breq1 w cv 0R <,> x cv v cv 0R <,> eqeq1 w cv 0R <,> x cv v cv 0R <,> <RR breq2 orbi12d negbid bibi12d w cv 0R <,> x cv <RR v cv 0R <,> breq1 v cv 0R <,> <RR u cv 0R <,> br anbi1d w cv 0R <,> x cv <RR u cv 0R <,> breq1 imbi12d anbi12d v cv 0R <,> y cv x cv <RR breq2 v cv 0R <,> y cv x cv eqeq2 v cv 0R <,> y cv <RR x cv breq1 orbi12d negbid bibi12d v cv 0R <,> y cv x cv <RR breq2 v cv 0R <,> y cv <RR u cv 0R <,> breq1 anbi12d x cv <RR u cv 0R <,> br imbi1d anbi12d u cv 0R <,> z cv y cv <RR breq2 x cv <RR y cv br anbi2d u cv 0R <,> z cv x cv <RR breq2 imbi12d x cv <RR y cv br x cv y cv = y cv <RR x cv br \/ -. <-> anbi2d ltsosr <R R. w cv v cv sotric mpan w visset v visset ltresr w visset v cv eqresr v visset w visset ltresr orbi12i negbii 3bitr4g u cv R. e. 3adant3 w visset ltsosr ltrelsr v visset u visset sotri w visset v visset ltresr v visset u visset ltresr anbi12i w visset u visset ltresr 3imtr4 jctir 3gencl so RR eqid pm3.2i))
+thm (dfcnqs () () (= (CC) (/. (X. (R.) (R.)) (`' (E)))) (df-c srex srex xpex qsid eqtr4))
+thm (addcnsrec () () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (+) ([] (<,> C D) (`' (E)))) ([] (<,> (opr A (+R) C) (opr B (+R) D)) (`' (E))))) (A B C D addcnsr A B opex ecid C D opex ecid + opreq12i A +R C opr B +R D opr opex ecid 3eqtr4g))
+thm (mulcnsrec () () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (x.) ([] (<,> C D) (`' (E)))) ([] (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D))) (`' (E))))) (A B C D mulcnsr A B opex ecid C D opex ecid x. opreq12i A .R C opr +R -1R .R B .R D opr opr opr B .R C opr +R A .R D opr opr opex ecid 3eqtr4g))
+thm (axaddopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (+) (X. (CC) (CC)) (CC)) (+ CC CC CC x y ffnoprval + CC CC X. df-fn z w cv +R u cv opr v cv +R f cv opr <,> moeq u f y cv mosubop w v x cv mosubop x cv w cv v cv <,> = y cv u cv f cv <,> = z cv w cv +R u cv opr v cv +R f cv opr <,> = anass u f bi2ex u f x cv w cv v cv <,> = y cv u cv f cv <,> = z cv w cv +R u cv opr v cv +R f cv opr <,> = /\ 19.42vv bitr w v bi2ex z bimo mpbir x cv CC e. y cv CC e. /\ moani x y funoprab x y z w v u f df-plus + x y z x cv CC e. y cv CC e. /\ w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv +R u cv opr v cv +R f cv opr <,> = /\ E. E. E. E. /\ {<<,>,>|} funeq ax-mp mpbir x y z w v u f df-plus dmeqi x y z CC CC w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv +R u cv opr v cv +R f cv opr <,> = /\ E. E. E. E. dmoprabss eqsstr 0ncn df-c z cv w cv <,> x cv + v cv u cv <,> opreq1 R. R. X. eleq1d v cv u cv <,> y cv x cv + opreq2 R. R. X. eleq1d z cv w cv v cv u cv addcnsr z cv v cv addclsr w cv u cv addclsr anim12i an4s z cv +R v cv opr R. w cv +R u cv opr R. opelxpi syl eqeltrd 2optocl df-c syl6eleqr oprssdm eqssi mpbir2an df-c z cv w cv <,> x cv + v cv u cv <,> opreq1 R. R. X. eleq1d v cv u cv <,> y cv x cv + opreq2 R. R. X. eleq1d z cv w cv v cv u cv addcnsr z cv v cv addclsr w cv u cv addclsr anim12i an4s z cv +R v cv opr R. w cv +R u cv opr R. opelxpi syl eqeltrd 2optocl df-c syl6eleqr rgen2 mpbir2an))
+thm (axmulopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (x.) (X. (CC) (CC)) (CC)) (x. CC CC CC x y ffnoprval x. CC CC X. df-fn z w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> moeq u f y cv mosubop w v x cv mosubop x cv w cv v cv <,> = y cv u cv f cv <,> = z cv w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> = anass u f bi2ex u f x cv w cv v cv <,> = y cv u cv f cv <,> = z cv w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> = /\ 19.42vv bitr w v bi2ex z bimo mpbir x cv CC e. y cv CC e. /\ moani x y funoprab x y z w v u f df-mul x. x y z x cv CC e. y cv CC e. /\ w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> = /\ E. E. E. E. /\ {<<,>,>|} funeq ax-mp mpbir x y z w v u f df-mul dmeqi x y z CC CC w v u f x cv w cv v cv <,> = y cv u cv f cv <,> = /\ z cv w cv .R u cv opr +R -1R .R v cv .R f cv opr opr opr v cv .R u cv opr +R w cv .R f cv opr opr <,> = /\ E. E. E. E. dmoprabss eqsstr 0ncn df-c z cv w cv <,> x cv x. v cv u cv <,> opreq1 R. R. X. eleq1d v cv u cv <,> y cv x cv x. opreq2 R. R. X. eleq1d z cv w cv v cv u cv mulcnsr z cv .R v cv opr +R -1R .R w cv .R u cv opr opr opr R. w cv .R v cv opr +R z cv .R u cv opr opr R. opelxpi z cv .R v cv opr -1R .R w cv .R u cv opr opr addclsr z cv v cv mulclsr w cv u cv mulclsr m1r -1R w cv .R u cv opr mulclsr mpan syl syl2an an4s w cv .R v cv opr z cv .R u cv opr addclsr w cv v cv mulclsr z cv u cv mulclsr syl2an ancoms an42s sylanc eqeltrd 2optocl df-c syl6eleqr oprssdm eqssi mpbir2an df-c z cv w cv <,> x cv x. v cv u cv <,> opreq1 R. R. X. eleq1d v cv u cv <,> y cv x cv x. opreq2 R. R. X. eleq1d z cv w cv v cv u cv mulcnsr z cv .R v cv opr +R -1R .R w cv .R u cv opr opr opr R. w cv .R v cv opr +R z cv .R u cv opr opr R. opelxpi z cv .R v cv opr -1R .R w cv .R u cv opr opr addclsr z cv v cv mulclsr w cv u cv mulclsr m1r -1R w cv .R u cv opr mulclsr mpan syl syl2an an4s w cv .R v cv opr z cv .R u cv opr addclsr w cv v cv mulclsr z cv u cv mulclsr syl2an ancoms an42s sylanc eqeltrd 2optocl df-c syl6eleqr rgen2 mpbir2an))
+thm (axcnex () () (e. (CC) (V)) (df-c srex srex xpex eqeltr))
+thm (axresscn () () (C_ (RR) (CC)) (R. ssid 0r 0R R. snssi ax-mp R. R. 0R {} R. ssxp mp2an df-r df-c 3sstr4))
+thm (ax1re () () (e. (1) (RR)) (df-1 1r 1R opelreal mpbir eqeltr))
+thm (axicn () () (e. (i) (CC)) (df-i CC eleq1i 1r elisseti 0R opelcn bitr 0r 1r mpbir2an))
+thm (axaddcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (+) B) (CC))) (axaddopr + CC CC CC x y ffnoprval pm3.27bd ax-mp x cv A + y cv opreq1 CC eleq1d y cv B A + opreq2 CC eleq1d CC CC rcla42v mpi))
+thm (axaddrcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (+) B) (RR))) (A x elreal B y elreal x cv 0R <,> A + y cv 0R <,> opreq1 RR eleq1d y cv 0R <,> B A + opreq2 RR eleq1d x cv y cv addresr x cv y cv addclsr x cv +R y cv opr opelreal sylibr eqeltrd 2gencl))
+thm (axmulcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (x.) B) (CC))) (axmulopr x. CC CC CC x y ffnoprval pm3.27bd ax-mp x cv A x. y cv opreq1 CC eleq1d y cv B A x. opreq2 CC eleq1d CC CC rcla42v mpi))
+thm (axmulrcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (x.) B) (RR))) (A x elreal B y elreal x cv 0R <,> A x. y cv 0R <,> opreq1 RR eleq1d y cv 0R <,> B A x. opreq2 RR eleq1d y visset x cv mulresr x cv y cv mulclsr x cv .R y cv opr opelreal sylibr eqeltrd 2gencl))
+thm (axaddcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (+) B) (opr B (+) A))) (dfcnqs x cv y cv z cv w cv addcnsrec z cv w cv x cv y cv addcnsrec x visset z visset addcomsr y visset w visset addcomsr A B ecoprcom))
+thm (axmulcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (x.) B) (opr B (x.) A))) (dfcnqs x cv y cv z cv w cv mulcnsrec z cv w cv x cv y cv mulcnsrec x visset z visset mulcomsr y visset w visset mulcomsr -1R .R opreq2i +R opreq12i y visset z visset mulcomsr x visset w visset mulcomsr +R opreq12i z cv .R y cv oprex w cv .R x cv oprex addcomsr eqtr A B ecoprcom))
+thm (axaddass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) C) (opr A (+) (opr B (+) C)))) (dfcnqs x cv y cv z cv w cv addcnsrec z cv w cv v cv u cv addcnsrec x cv +R z cv opr y cv +R w cv opr v cv u cv addcnsrec x cv y cv z cv +R v cv opr w cv +R u cv opr addcnsrec x cv z cv addclsr y cv w cv addclsr anim12i an4s z cv v cv addclsr w cv u cv addclsr anim12i an4s z visset v visset x cv addasssr w visset u visset y cv addasssr A B C ecoprass))
+thm (axmulass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (x.) B) (x.) C) (opr A (x.) (opr B (x.) C)))) (dfcnqs x cv y cv z cv w cv mulcnsrec z cv w cv v cv u cv mulcnsrec x cv .R z cv opr +R -1R .R y cv .R w cv opr opr opr y cv .R z cv opr +R x cv .R w cv opr opr v cv u cv mulcnsrec x cv y cv z cv .R v cv opr +R -1R .R w cv .R u cv opr opr opr w cv .R v cv opr +R z cv .R u cv opr opr mulcnsrec x cv .R z cv opr -1R .R y cv .R w cv opr opr addclsr x cv z cv mulclsr y cv w cv mulclsr m1r -1R y cv .R w cv opr mulclsr mpan syl syl2an an4s y cv .R z cv opr x cv .R w cv opr addclsr y cv z cv mulclsr x cv w cv mulclsr syl2an ancoms an42s jca z cv .R v cv opr -1R .R w cv .R u cv opr opr addclsr z cv v cv mulclsr w cv u cv mulclsr m1r -1R w cv .R u cv opr mulclsr mpan syl syl2an an4s w cv .R v cv opr z cv .R u cv opr addclsr w cv v cv mulclsr z cv u cv mulclsr syl2an ancoms an42s jca x cv .R z cv .R v cv opr oprex x cv .R -1R .R w cv .R u cv opr opr oprex -1R .R y cv .R w cv .R v cv opr opr oprex f visset g visset addcomsr g visset h visset f cv addasssr -1R .R y cv .R z cv .R u cv opr opr oprex caopr42 z cv .R v cv oprex -1R .R w cv .R u cv opr oprex x cv distrsr w cv .R v cv oprex z cv .R u cv oprex y cv distrsr -1R .R opreq2i y cv .R w cv .R v cv opr oprex y cv .R z cv .R u cv opr oprex -1R distrsr eqtr +R opreq12i x visset m1r elisseti z visset f visset g visset mulcomsr g visset h visset f cv distrsr y cv .R w cv oprex v visset g visset h visset f cv mulasssr caoprdilem w visset v visset y cv mulasssr -1R .R opreq2i x cv .R z cv .R v cv opr opr +R opreq2i eqtr y visset x visset z visset f visset g visset mulcomsr g visset h visset f cv distrsr w visset u visset g visset h visset f cv mulasssr caoprdilem -1R .R opreq2i y cv .R z cv .R u cv opr oprex x cv .R w cv .R u cv opr oprex -1R distrsr m1r elisseti x visset w cv .R u cv oprex f visset g visset mulcomsr g visset h visset f cv mulasssr caopr12 -1R .R y cv .R z cv .R u cv opr opr opr +R opreq2i 3eqtr +R opreq12i 3eqtr4r y cv .R z cv .R v cv opr oprex y cv .R -1R .R w cv .R u cv opr opr oprex x cv .R w cv .R v cv opr oprex f visset g visset addcomsr g visset h visset f cv addasssr x cv .R z cv .R u cv opr oprex caopr42 z cv .R v cv oprex -1R .R w cv .R u cv opr oprex y cv distrsr w cv .R v cv oprex z cv .R u cv oprex x cv distrsr +R opreq12i y visset x visset z visset f visset g visset mulcomsr g visset h visset f cv distrsr w visset v visset g visset h visset f cv mulasssr caoprdilem x visset m1r elisseti z visset f visset g visset mulcomsr g visset h visset f cv distrsr y cv .R w cv oprex u visset g visset h visset f cv mulasssr caoprdilem w visset u visset y cv mulasssr -1R .R opreq2i m1r elisseti y visset w cv .R u cv oprex f visset g visset mulcomsr g visset h visset f cv mulasssr caopr12 eqtr x cv .R z cv .R u cv opr opr +R opreq2i eqtr +R opreq12i 3eqtr4r A B C ecoprass))
+thm (axdistr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (+) C)) (opr (opr A (x.) B) (+) (opr A (x.) C)))) (dfcnqs z cv w cv v cv u cv addcnsrec x cv y cv z cv +R v cv opr w cv +R u cv opr mulcnsrec x cv y cv z cv w cv mulcnsrec x cv y cv v cv u cv mulcnsrec x cv .R z cv opr +R -1R .R y cv .R w cv opr opr opr y cv .R z cv opr +R x cv .R w cv opr opr x cv .R v cv opr +R -1R .R y cv .R u cv opr opr opr y cv .R v cv opr +R x cv .R u cv opr opr addcnsrec z cv v cv addclsr w cv u cv addclsr anim12i an4s x cv .R z cv opr -1R .R y cv .R w cv opr opr addclsr x cv z cv mulclsr y cv w cv mulclsr m1r -1R y cv .R w cv opr mulclsr mpan syl syl2an an4s y cv .R z cv opr x cv .R w cv opr addclsr y cv z cv mulclsr x cv w cv mulclsr syl2an ancoms an42s jca x cv .R v cv opr -1R .R y cv .R u cv opr opr addclsr x cv v cv mulclsr y cv u cv mulclsr m1r -1R y cv .R u cv opr mulclsr mpan syl syl2an an4s y cv .R v cv opr x cv .R u cv opr addclsr y cv v cv mulclsr x cv u cv mulclsr syl2an ancoms an42s jca z visset v visset x cv distrsr w visset u visset y cv distrsr -1R .R opreq2i y cv .R w cv oprex y cv .R u cv oprex -1R distrsr eqtr +R opreq12i x cv .R z cv oprex x cv .R v cv oprex -1R .R y cv .R w cv opr oprex f visset g visset addcomsr g visset h visset f cv addasssr -1R .R y cv .R u cv opr oprex caopr4 eqtr z visset v visset y cv distrsr w visset u visset x cv distrsr +R opreq12i y cv .R z cv oprex y cv .R v cv oprex x cv .R w cv oprex f visset g visset addcomsr g visset h visset f cv addasssr x cv .R u cv oprex caopr4 eqtr A B C ecoprdi))
+thm (ax1ne0 () () (=/= (1) (0)) (1ne0sr 1r elisseti 0R eqresr mtbir df-1 df-0 eqeq12i mtbir 1 0 df-ne mpbir))
+thm (ax0id ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (opr A (+) (0)) A)) (df-c x cv y cv <,> A + 0 opreq1 x cv y cv <,> A = id eqeq12d 0r 0r pm3.2i x cv y cv 0R 0R addcnsr mpan2 x cv +R 0R opr x cv y cv +R 0R opr y cv opeq12 x cv 0idsr y cv 0idsr syl2an eqtrd df-0 x cv y cv <,> + opreq2i syl5eq optocl))
+thm (ax1id ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (opr A (x.) (1)) A)) (df-c x cv y cv <,> A x. 1 opreq1 x cv y cv <,> A = id eqeq12d 1r 0r pm3.2i x cv y cv 1R 0R mulcnsr mpan2 x cv .R 1R opr +R -1R .R y cv .R 0R opr opr opr x cv y cv .R 1R opr +R x cv .R 0R opr opr y cv opeq12 y cv 00sr -1R .R opreq2d m1r -1R 00sr ax-mp syl6eq x cv .R 1R opr +R opreq2d x cv 1idsr +R 0R opreq1d x cv 0idsr eqtrd sylan9eqr x cv 00sr y cv .R 1R opr +R opreq2d y cv 1idsr +R 0R opreq1d y cv 0idsr eqtrd sylan9eq sylanc eqtrd df-1 x cv y cv <,> x. opreq2i syl5eq optocl))
+thm (axnegex ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (CC)) (E.e. x (CC) (= (opr A (+) (cv x)) (0)))) (df-c y cv z cv <,> A + x cv opreq1 0 eqeq1d x cv CC e. anbi2d x biexdv y cv w negexsr z cv v negexsr anim12i w v w cv R. e. y cv +R w cv opr 0R = /\ v cv R. e. z cv +R v cv opr 0R = /\ eeanv sylibr w cv v cv opex x cv w cv v cv <,> CC eleq1 x cv w cv v cv <,> y cv z cv <,> + opreq2 0 eqeq1d anbi12d cla4ev w cv R. v cv R. opelxpi df-c syl6eleqr y cv R. e. z cv R. e. /\ y cv +R w cv opr 0R = z cv +R v cv opr 0R = /\ ad2antlr y cv z cv w cv v cv addcnsr y cv +R w cv opr 0R z cv +R v cv opr 0R opeq12 df-0 syl6eqr sylan9eq sylanc exp31 imp3a w cv R. e. y cv +R w cv opr 0R = v cv R. e. z cv +R v cv opr 0R = an4 syl5ib w v 19.23advv mpd optocl x CC A + x cv opr 0 = df-rex sylibr))
+thm (axrecex ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (-> (/\ (e. A (CC)) (=/= A (0))) (E.e. x (CC) (= (opr A (x.) (cv x)) (1)))) (df-c y cv z cv <,> A 0 neeq1 y cv z cv <,> A x. x cv opreq1 1 eqeq1d x CC birexdv imbi12d y visset z visset ssgt0sr y cv .R y cv opr +R z cv .R z cv opr oprex w recexsrlem syl6 y cv z cv <,> 0 df-ne df-0 y cv z cv <,> eqeq2i y visset z visset 0r elisseti 0R opth bitr negbii bitr syl5ib y cv .R w cv opr -1R .R z cv .R w cv opr opr opex x cv y cv .R w cv opr -1R .R z cv .R w cv opr opr <,> CC eleq1 x cv y cv .R w cv opr -1R .R z cv .R w cv opr opr <,> y cv z cv <,> x. opreq2 1 eqeq1d anbi12d cla4ev y cv w cv mulclsr z cv w cv mulclsr m1r jctil -1R z cv .R w cv opr mulclsr syl anim12i y cv R. e. z cv R. e. w cv R. e. anandir -1R .R z cv .R w cv opr oprex y cv .R w cv opr opelcn 3imtr4 y cv .R y cv opr +R z cv .R z cv opr opr .R w cv opr 1R = adantrr y cv R. e. z cv R. e. /\ w cv R. e. pm3.26 y cv w cv mulclsr z cv w cv mulclsr m1r jctil -1R z cv .R w cv opr mulclsr syl anim12i anandirs jca y cv .R y cv opr +R z cv .R z cv opr opr .R w cv opr 1R = adantrr y cv z cv y cv .R w cv opr -1R .R z cv .R w cv opr opr mulcnsr syl y cv .R y cv .R w cv opr opr +R -1R .R z cv .R -1R .R z cv .R w cv opr opr opr opr opr 1R z cv .R y cv .R w cv opr opr +R y cv .R -1R .R z cv .R w cv opr opr opr opr 0R opeq12 y cv .R y cv opr +R z cv .R z cv opr opr .R w cv opr 1R y cv .R y cv .R w cv opr opr +R -1R .R z cv .R -1R .R z cv .R w cv opr opr opr opr opr eqeq2 y visset w visset y cv mulasssr eqcomi z cv R. e. w cv R. e. /\ a1i z cv R. e. w cv R. e. pm3.26 z cv w cv mulclsr jca z cv z cv .R w cv opr mulclsr z cv .R z cv .R w cv opr opr 1idsr 3syl m1m1sr .R z cv .R z cv .R w cv opr opr opreq1i z visset m1r elisseti z cv .R w cv oprex v visset u visset mulcomsr u visset x visset v cv mulasssr caopr12 -1R .R opreq2i m1r elisseti z cv .R z cv .R w cv opr oprex -1R mulasssr eqtr4 z cv .R z cv .R w cv opr oprex 1r elisseti mulcomsr 3eqtr4 z visset w visset z cv mulasssr 3eqtr4g +R opreq12d y cv .R y cv oprex z cv .R z cv oprex w visset v visset u visset mulcomsr u visset x visset v cv distrsr caoprdistrr syl6eqr syl5bi exp3a com3l imp32 y cv R. e. adantll y cv z cv .R w cv opr mulclsr z cv w cv mulclsr sylan2 anassrs y cv .R y cv opr +R z cv .R z cv opr opr .R w cv opr 1R = adantrr y cv .R z cv .R w cv opr opr pn0sr z visset y visset w visset v visset u visset mulcomsr u visset x visset v cv mulasssr caopr12 y visset m1r elisseti z cv .R w cv oprex v visset u visset mulcomsr u visset x visset v cv mulasssr caopr12 m1r elisseti y cv .R z cv .R w cv opr oprex mulcomsr eqtr +R opreq12i syl5eq syl sylanc eqtrd df-1 syl6eqr sylanc exp w 19.23adv syld x CC y cv z cv <,> x. x cv opr 1 = df-rex syl6ibr optocl imp))
+thm (axrnegex ((x y) (A x) (A y) (x z) (y z)) () (-> (e. A (RR)) (E.e. x (RR) (= (opr A (+) (cv x)) (0)))) (A y elreal y cv 0R <,> A + x cv opreq1 0 eqeq1d x RR birexdv y cv z negexsr y cv z cv addresr 0 eqeq1d df-0 y cv +R z cv opr 0R <,> eqeq2i y cv +R z cv oprex 0R eqresr bitr syl6bb exp pm5.32d z cv 0R opex x cv z cv 0R <,> RR eleq1 x cv z cv 0R <,> y cv 0R <,> + opreq2 0 eqeq1d anbi12d cla4ev z cv opelreal sylanbr x RR y cv 0R <,> + x cv opr 0 = df-rex sylibr syl6bir z 19.23adv mpd gencl))
+thm (axrrecex ((x y) (A x) (A y) (x z) (y z)) () (-> (/\ (e. A (RR)) (=/= A (0))) (E.e. x (RR) (= (opr A (x.) (cv x)) (1)))) (A y elreal y cv 0R <,> A 0 neeq1 y cv 0R <,> A x. x cv opreq1 1 eqeq1d x RR birexdv imbi12d y visset z recexsr z visset y cv mulresr 1 eqeq1d df-1 y cv .R z cv opr 0R <,> eqeq2i y cv .R z cv oprex 1R eqresr bitr syl6bb exp pm5.32d z cv opelreal y cv 0R <,> x. z cv 0R <,> opr 1 = anbi1i syl5bb z cv 0R opex x cv z cv 0R <,> RR eleq1 x cv z cv 0R <,> y cv 0R <,> x. opreq2 1 eqeq1d anbi12d cla4ev syl6bir z 19.23adv syld df-0 y cv 0R <,> eqeq2i y visset 0R eqresr bitr negbii syl5ib y cv 0R <,> 0 df-ne x RR y cv 0R <,> x. x cv opr 1 = df-rex 3imtr4g gencl imp))
+thm (axi2m1 () () (= (opr (opr (i) (x.) (i)) (+) (1)) (0)) (0r 1r pm3.2i 0r 1r pm3.2i 0R 1R 0R 1R mulcnsr mp2an 0R .R 0R opr +R -1R .R 1R .R 1R opr opr oprex 1R .R 0R opr +R 0R .R 1R opr oprex 0r elisseti -1R opth 0r 0R 00sr ax-mp 1r 1R 1idsr ax-mp -1R .R opreq2i m1r -1R 1idsr ax-mp eqtr +R opreq12i 0r elisseti m1r elisseti addcomsr m1r -1R 0idsr ax-mp 3eqtr 1r 1R 00sr ax-mp 0r 0R 1idsr ax-mp +R opreq12i 0r 0R 0idsr ax-mp eqtr mpbir2an eqtr + 1R 0R <,> opreq1i m1r 1r -1R 1R addresr mp2an m1p1sr -1R +R 1R oprex 0R eqresr mpbir 3eqtr df-i df-i x. opreq12i df-1 + opreq12i df-0 3eqtr4))
+thm (pre-axlttri () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<RR) B) (-. (\/ (= A B) (br B (<RR) A))))) (ltsor pm3.26i <RR RR A B sotric mpan))
+thm (pre-axlttrn () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<RR) B) (br B (<RR) C)) (br A (<RR) C))) (ltsor pm3.26i <RR RR A B C sotr mpan))
+thm (pre-axltadd ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br A (<RR) B) (br (opr C (+) A) (<RR) (opr C (+) B)))) (A x elreal B y elreal C z elreal x cv 0R <,> A <RR y cv 0R <,> breq1 x cv 0R <,> A z cv 0R <,> + opreq2 <RR z cv 0R <,> + y cv 0R <,> opr breq1d bibi12d y cv 0R <,> B A <RR breq2 y cv 0R <,> B z cv 0R <,> + opreq2 z cv 0R <,> + A opr <RR breq2d bibi12d z cv 0R <,> C + A opreq1 z cv 0R <,> C + B opreq1 <RR breq12d A <RR B br bibi2d x visset y visset z cv ltasr x cv R. e. y cv R. e. /\ adantr x visset y visset ltresr z cv R. e. x cv R. e. y cv R. e. /\ /\ a1i z cv x cv addresr z cv y cv addresr <RR breqan12d anandis z cv +R x cv oprex z cv +R y cv oprex ltresr syl6bb 3bitr4d ancoms 3impa 3gencl biimpd))
+thm (pre-axmulgt0 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<RR) A) (br (0) (<RR) B)) (br (0) (<RR) (opr A (x.) B)))) (A x elreal B y elreal x cv 0R <,> A 0 <RR breq2 0 <RR y cv 0R <,> br anbi1d x cv 0R <,> A x. y cv 0R <,> opreq1 0 <RR breq2d imbi12d y cv 0R <,> B 0 <RR breq2 0 <RR A br anbi2d y cv 0R <,> B A x. opreq2 0 <RR breq2d imbi12d df-0 x cv R. e. y cv R. e. /\ a1i y visset x cv mulresr <RR breq12d 0r elisseti x cv .R y cv oprex ltresr syl6bb x visset y visset mulgt0sr df-0 <RR x cv 0R <,> breq1i 0r elisseti x visset ltresr bitr df-0 <RR y cv 0R <,> breq1i 0r elisseti y visset ltresr bitr syl2anb syl5bir 2gencl))
+thm (axcnre ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E.e. x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (df-c z cv w cv <,> A x cv + y cv x. i opr opr eqeq1 y RR birexdv x RR birexdv z cv 0R opex w cv 0R opex x cv z cv 0R <,> RR eleq1 y cv w cv 0R <,> RR eleq1 bi2anan9 x cv z cv 0R <,> + y cv x. i opr opreq1 y cv w cv 0R <,> x. i opreq1 z cv 0R <,> + opreq2d sylan9eq z cv w cv <,> eqeq2d anbi12d cla4e2v z cv opelreal w cv opelreal anbi12i biimpr 0r 0r 1r pm3.2i w cv 0R 0R 1R mulcnsr mpan2 mpan2 w cv .R 0R opr +R -1R .R 0R .R 1R opr opr opr 0R 0R .R 0R opr +R w cv .R 1R opr opr w cv opeq12 w cv 00sr +R -1R .R 0R .R 1R opr opr opreq1d 0r 0R 1idsr ax-mp -1R .R opreq2i m1r -1R 00sr ax-mp eqtr 0R +R opreq2i 0r 0R 0idsr ax-mp eqtr syl6eq w cv 1idsr 0R .R 0R opr +R opreq2d w cv 0idsr 0r 0R 00sr ax-mp +R w cv opreq1i 0r elisseti w visset addcomsr eqtr syl5eq eqtrd sylanc eqtrd df-i w cv 0R <,> x. opreq2i syl5eq z cv 0R <,> + opreq2d z cv R. e. adantl 0r 0r z cv 0R 0R w cv addcnsr mpanl2 mpanr1 z cv +R 0R opr z cv 0R +R w cv opr w cv opeq12 z cv 0idsr w cv 0idsr 0r elisseti w visset addcomsr syl5eq syl2an 3eqtrd eqcomd sylanc x RR y RR z cv w cv <,> x cv + y cv x. i opr opr = r2ex sylibr optocl))
+thm (pre-axsup ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (br (cv y) (<RR) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<RR) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<RR) (cv x)) (E.e. z A (br (cv y) (<RR) (cv z)))))))) (A RR y cv ssel y cv A e. x cv <RR y cv br -. -> imim1d y cv A e. x cv <RR y cv br -. pm2.43 syl6 y 19.20dv y A x cv <RR y cv br -. df-ral syl6ibr z cv RR e. z cv A e. y cv <RR z cv br /\ pm3.27 z 19.22i z A y cv <RR z cv br df-rex sylibr y cv <RR x cv br imim2i y cv RR e. imim2i y 19.20i y RR y cv <RR x cv br z A y cv <RR z cv br E.e. -> df-ral sylibr A RR C_ a1i anim12d y cv RR e. y cv A e. x cv <RR y cv br -. -> y cv <RR x cv br z z cv RR e. z cv A e. y cv <RR z cv br /\ /\ E. -> jcab y bial y y cv RR e. y cv A e. x cv <RR y cv br -. -> -> y cv RR e. y cv <RR x cv br z z cv RR e. z cv A e. y cv <RR z cv br /\ /\ E. -> -> 19.26 bitr syl5ib x cv RR e. anim2d x 19.22dv x RR y A x cv <RR y cv br -. A.e. y RR y cv <RR x cv br z A y cv <RR z cv br E.e. -> A.e. /\ df-rex syl6ibr A {/} = -. adantr v cv w cv 0R opeq1 A eleq1d cbvabv x y z supre syl5 anabsi5 x RR y A y cv <RR x cv br A.e. df-rex y A y cv <RR x cv br df-ral y cv A e. y cv <RR x cv br -> y cv RR e. ax-1 y 19.20i sylbi x cv RR e. anim2i x 19.22i sylbi sylan2 A {/} df-ne A RR C_ anbi2i sylanb 3impa))
+thm (reex () () (e. (RR) (V)) (axcnex axresscn ssexi))
+thm (recnt () () (-> (e. A (RR)) (e. A (CC))) (axresscn A sseli))
+thm (recn () ((recn.1 (e. A (RR)))) (e. A (CC)) (axresscn recn.1 sselii))
+thm (recnd () ((recnd.1 (-> ph (e. A (RR))))) (-> ph (e. A (CC))) (recnd.1 A recnt syl))
+thm (elimne0 () () (=/= (if (=/= A (0)) A (1)) (0)) (A A 0 =/= A 1 if 0 neeq1 1 A 0 =/= A 1 if 0 neeq1 ax1ne0 elimhyp))
+thm (addex () () (e. (+) (V)) (axcnex axcnex xpex axaddopr CC CC X. V + CC fex mp2))
+thm (mulex () () (e. (x.) (V)) (axcnex axcnex xpex axmulopr CC CC X. V x. CC fex mp2))
+thm (adddirt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (x.) C) (opr (opr A (x.) C) (+) (opr B (x.) C)))) (C A B axdistr 3coml A + B opr C axmulcom A B axaddcl sylan 3impa A C axmulcom B CC e. 3adant2 B C axmulcom A CC e. 3adant1 + opreq12d 3eqtr4d))
+thm (addcl () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (e. (opr A (+) B) (CC)) (axi.1 axi.2 A B axaddcl mp2an))
+thm (mulcl () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (e. (opr A (x.) B) (CC)) (axi.1 axi.2 A B axmulcl mp2an))
+thm (addcom () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (= (opr A (+) B) (opr B (+) A)) (axi.1 axi.2 A B axaddcom mp2an))
+thm (mulcom () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC)))) (= (opr A (x.) B) (opr B (x.) A)) (axi.1 axi.2 A B axmulcom mp2an))
+thm (addass () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (+) B) (+) C) (opr A (+) (opr B (+) C))) (axi.1 axi.2 axi.3 A B C axaddass mp3an))
+thm (mulass () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (x.) B) (x.) C) (opr A (x.) (opr B (x.) C))) (axi.1 axi.2 axi.3 A B C axmulass mp3an))
+thm (adddi () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr A (x.) (opr B (+) C)) (opr (opr A (x.) B) (+) (opr A (x.) C))) (axi.1 axi.2 axi.3 A B C axdistr mp3an))
+thm (adddir () ((axi.1 (e. A (CC))) (axi.2 (e. B (CC))) (axi.3 (e. C (CC)))) (= (opr (opr A (+) B) (x.) C) (opr (opr A (x.) C) (+) (opr B (x.) C))) (axi.1 axi.2 axi.3 A B C adddirt mp3an))
+thm (1cn () () (e. (1) (CC)) (ax1re recn))
+thm (0cn () () (e. (0) (CC)) (axi2m1 axicn axicn mulcl 1cn addcl eqeltrr))
+thm (addid1 () ((addid1.1 (e. A (CC)))) (= (opr A (+) (0)) A) (addid1.1 A ax0id ax-mp))
+thm (addid2 () ((addid1.1 (e. A (CC)))) (= (opr (0) (+) A) A) (0cn addid1.1 addcom addid1.1 addid1 eqtr))
+thm (mulid1 () ((addid1.1 (e. A (CC)))) (= (opr A (x.) (1)) A) (addid1.1 A ax1id ax-mp))
+thm (mulid2 () ((addid1.1 (e. A (CC)))) (= (opr (1) (x.) A) A) (1cn addid1.1 mulcom addid1.1 mulid1 eqtr))
+thm (negex ((A x)) ((addid1.1 (e. A (CC)))) (E.e. x (CC) (= (opr A (+) (cv x)) (0))) (addid1.1 A x axnegex ax-mp))
+thm (recex ((A x)) ((addid1.1 (e. A (CC))) (recex.2 (=/= A (0)))) (E.e. x (CC) (= (opr A (x.) (cv x)) (1))) (addid1.1 recex.2 A x axrecex mp2an))
+thm (readdcl () ((axri.1 (e. A (RR))) (axri.2 (e. B (RR)))) (e. (opr A (+) B) (RR)) (axri.1 axri.2 A B axaddrcl mp2an))
+thm (remulcl () ((axri.1 (e. A (RR))) (axri.2 (e. B (RR)))) (e. (opr A (x.) B) (RR)) (axri.1 axri.2 A B axmulrcl mp2an))
+thm (addcan ((A x) (B x) (C x)) ((addcan.1 (e. A (CC))) (addcan.2 (e. B (CC))) (addcan.3 (e. C (CC)))) (<-> (= (opr A (+) B) (opr A (+) C)) (= B C)) (addcan.1 x negex addcan.1 addcan.2 x cv A B axaddass mp3an3 addcan.3 x cv A C axaddass mp3an3 eqeq12d mpan2 A + B opr A + C opr x cv + opreq2 syl5bir A + x cv opr 0 = adantr addcan.1 A x cv axaddcom mpan 0 eqeq1d x cv + A opr 0 + B opreq1 addcan.2 addid2 syl6eq x cv + A opr 0 + C opreq1 addcan.3 addid2 syl6eq eqeq12d syl6bi imp sylibd exp r19.23aiv ax-mp B C A + opreq2 impbi))
+thm (addcan2 () ((addcan.1 (e. A (CC))) (addcan.2 (e. B (CC))) (addcan.3 (e. C (CC)))) (<-> (= (opr A (+) C) (opr B (+) C)) (= A B)) (addcan.1 addcan.3 addcom addcan.2 addcan.3 addcom eqeq12i addcan.3 addcan.1 addcan.2 addcan bitr))
+thm (addcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (+) B) (opr A (+) C)) (= B C))) (A A CC e. A 0 if + B opreq1 A A CC e. A 0 if + C opreq1 eqeq12d B C = bibi1d B B CC e. B 0 if A CC e. A 0 if + opreq2 A CC e. A 0 if + C opr eqeq1d B B CC e. B 0 if C eqeq1 bibi12d C C CC e. C 0 if A CC e. A 0 if + opreq2 A CC e. A 0 if + B CC e. B 0 if opr eqeq2d C C CC e. C 0 if B CC e. B 0 if eqeq2 bibi12d 0cn A elimel 0cn B elimel 0cn C elimel addcan dedth3h))
+thm (addcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (+) C) (opr B (+) C)) (= A B))) (C A axaddcom B CC e. 3adant3 C B axaddcom A CC e. 3adant2 eqeq12d C A B addcant bitr3d 3coml))
+thm (negeu ((x y) (A x) (A y) (B x) (B y)) ((negeu.1 (e. A (CC))) (negeu.2 (e. B (CC)))) (E!e. x (CC) (= (opr A (+) (cv x)) B)) (x cv y cv A + opreq2 B eqeq1d CC reu4 negeu.1 y negex negeu.2 y cv B axaddcl mpan2 y cv + B opr CC x risset sylib x cv y cv + B opr A + opreq2 negeu.1 negeu.2 A y cv B axaddass mp3an13 eqcomd sylan9eqr A + y cv opr 0 + B opreq1 negeu.2 addid2 syl6eq sylan9eqr exp32 impcom x cv CC e. a1d r19.21aiv exp x CC x cv y cv + B opr = A + x cv opr B = r19.22 syl6 com23 mpd r19.23aiv ax-mp negeu.1 A x cv y cv addcant A + x cv opr B A + y cv opr eqtr3t syl5bi mp3an1 rgen2 mpbir2an))
+thm (add12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (+) (opr B (+) C)) (opr B (+) (opr A (+) C)))) (A B axaddcom + C opreq1d C CC e. 3adant3 A B C axaddass B A C axaddass 3com12 3eqtr3d))
+thm (add23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) C) (opr (opr A (+) C) (+) B))) (B C axaddcom A + opreq2d A CC e. 3adant1 A B C axaddass A C B axaddass 3com23 3eqtr4d))
+thm (add4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr B (+) D)))) (A B C add23t + D opreq1d 3expa D CC e. adantrr A + B opr C D axaddass 3expb A B axaddcl sylan A + C opr B D axaddass 3expb A C axaddcl sylan an4s 3eqtr3d))
+thm (add42t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr D (+) B)))) (A B C D add4t B D axaddcom C CC e. adantrl A CC e. adantll A + C opr + opreq2d eqtrd))
+thm (add12 () ((add.1 (e. A (CC))) (add.2 (e. B (CC))) (add.3 (e. C (CC)))) (= (opr A (+) (opr B (+) C)) (opr B (+) (opr A (+) C))) (add.1 add.2 add.3 A B C add12t mp3an))
+thm (add23 () ((add.1 (e. A (CC))) (add.2 (e. B (CC))) (add.3 (e. C (CC)))) (= (opr (opr A (+) B) (+) C) (opr (opr A (+) C) (+) B)) (add.1 add.2 add.3 A B C add23t mp3an))
+thm (add4 () ((add4.1 (e. A (CC))) (add4.2 (e. B (CC))) (add4.3 (e. C (CC))) (add4.4 (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr B (+) D))) (add4.1 add4.2 pm3.2i add4.3 add4.4 pm3.2i A B C D add4t mp2an))
+thm (add42 () ((add4.1 (e. A (CC))) (add4.2 (e. B (CC))) (add4.3 (e. C (CC))) (add4.4 (e. D (CC)))) (= (opr (opr A (+) B) (+) (opr C (+) D)) (opr (opr A (+) C) (+) (opr D (+) B))) (add4.1 add4.2 add4.3 add4.4 add4 add4.2 add4.4 addcom A + C opr + opreq2i eqtr))
+thm (addid2t () () (-> (e. A (CC)) (= (opr (0) (+) A) A)) (0cn 0 A axaddcom mpan A ax0id eqtrd))
+thm (peano2cn () () (-> (e. A (CC)) (e. (opr A (+) (1)) (CC))) (1cn A 1 axaddcl mpan2))
+thm (peano2re () () (-> (e. A (RR)) (e. (opr A (+) (1)) (RR))) (ax1re A 1 axaddrcl mpan2))
+thm (subval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((subval.1 (e. A (CC))) (subval.2 (e. B (CC)))) (= (opr A (-) B) (U. ({e.|} x (CC) (= (opr B (+) (cv x)) A)))) (subval.1 subval.2 axcnex x B + x cv opr A = rabex uniex y cv A z cv + x cv opr eqeq2 x CC birabsdv unieqd z cv B + x cv opreq1 A eqeq1d x CC birabsdv unieqd y z w x df-sub oprabval2 mp2an))
+thm (negeq () () (-> (= A B) (= (-u A) (-u B))) (A B 0 - opreq2 A df-neg B df-neg 3eqtr4g))
+thm (negeqi () ((negeqi.1 (= A B))) (= (-u A) (-u B)) (negeqi.1 A B negeq ax-mp))
+thm (negeqd () ((negeqd.1 (-> ph (= A B)))) (-> ph (= (-u A) (-u B))) (negeqd.1 A B negeq syl))
+thm (hbneg ((A y) (x y)) ((hbneg.1 (-> (e. (cv y) A) (A. x (e. (cv y) A))))) (-> (e. (cv y) (-u A)) (A. x (e. (cv y) (-u A)))) (y cv 0 e. x ax-17 y cv - e. x ax-17 hbneg.1 hbopr A df-neg y cv eleq2i A df-neg y cv eleq2i x bial 3imtr4))
+thm (minusex () () (e. (-u A) (V)) (A df-neg 0 - A oprex eqeltr))
+thm (subcl ((A x) (B x)) ((subcl.1 (e. A (CC))) (subcl.2 (e. B (CC)))) (e. (opr A (-) B) (CC)) (subcl.1 subcl.2 x subval subcl.2 subcl.1 x negeu x CC B + x cv opr A = reucl ax-mp eqeltr))
+thm (subclt () () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (-) B) (CC))) (A A CC e. A 0 if - B opreq1 CC eleq1d B B CC e. B 0 if A CC e. A 0 if - opreq2 CC eleq1d 0cn A elimel 0cn B elimel subcl dedth2h))
+thm (negclt () () (-> (e. A (CC)) (e. (-u A) (CC))) (0cn 0 A subclt mpan A df-neg CC eleq1i sylibr))
+thm (negcl () ((negcl.1 (e. A (CC)))) (e. (-u A) (CC)) (negcl.1 A negclt ax-mp))
+thm (subadd ((A x) (B x) (C x)) ((subadd.1 (e. A (CC))) (subadd.2 (e. B (CC))) (subadd.3 (e. C (CC)))) (<-> (= (opr A (-) B) C) (= (opr B (+) C) A)) (subadd.3 x cv C A - B opr eqeq2 x cv C B + opreq2 A eqeq1d bibi12d subadd.2 subadd.1 x negeu x CC B + x cv opr A = reuuni1 mpan2 subadd.1 subadd.2 x subval x cv eqeq1i syl6rbbr vtoclga ax-mp))
+thm (subsub23 () ((subadd.1 (e. A (CC))) (subadd.2 (e. B (CC))) (subadd.3 (e. C (CC)))) (<-> (= (opr A (-) B) C) (= (opr A (-) C) B)) (subadd.2 subadd.3 addcom A eqeq1i subadd.1 subadd.2 subadd.3 subadd subadd.1 subadd.3 subadd.2 subadd 3bitr4))
+thm (subaddt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) B) C) (= (opr B (+) C) A))) (A A CC e. A 0 if - B opreq1 C eqeq1d A A CC e. A 0 if B + C opr eqeq2 bibi12d B B CC e. B 0 if A CC e. A 0 if - opreq2 C eqeq1d B B CC e. B 0 if + C opreq1 A CC e. A 0 if eqeq1d bibi12d C C CC e. C 0 if A CC e. A 0 if - B CC e. B 0 if opr eqeq2 C C CC e. C 0 if B CC e. B 0 if + opreq2 A CC e. A 0 if eqeq1d bibi12d 0cn A elimel 0cn B elimel 0cn C elimel subadd dedth3h))
+thm (pncan3t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (+) (opr B (-) A)) B)) (B - A opr eqid B A B - A opr subaddt A CC e. B CC e. pm3.27 A CC e. B CC e. pm3.26 B A subclt ancoms syl3anc mpbii))
+thm (pncan3 () ((subaddeq.1 (e. A (CC))) (subaddeq.2 (e. B (CC)))) (= (opr A (+) (opr B (-) A)) B) (subaddeq.1 subaddeq.2 A B pncan3t mp2an))
+thm (negidt () () (-> (e. A (CC)) (= (opr A (+) (-u A)) (0))) (0cn A 0 pncan3t mpan2 A df-neg A + opreq2i syl5eq))
+thm (negid () ((negid.1 (e. A (CC)))) (= (opr A (+) (-u A)) (0)) (negid.1 A negidt ax-mp))
+thm (subneg () ((subneg.1 (e. A (CC))) (subneg.2 (e. B (CC)))) (= (opr A (-) B) (opr A (+) (-u B))) (subneg.2 subneg.1 subneg.2 negcl addcl addcom subneg.1 subneg.2 negcl subneg.2 addass subneg.2 negcl subneg.2 addcom subneg.2 negid eqtr A + opreq2i subneg.1 addid1 3eqtr eqtr subneg.1 subneg.2 subneg.1 subneg.2 negcl addcl subadd mpbir))
+thm (subnegt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (-) B) (opr A (+) (-u B)))) (A A CC e. A 0 if - B opreq1 A A CC e. A 0 if + B -u opreq1 eqeq12d B B CC e. B 0 if A CC e. A 0 if - opreq2 B B CC e. B 0 if negeq A CC e. A 0 if + opreq2d eqeq12d 0cn A elimel 0cn B elimel subneg dedth2h))
+thm (addsubasst () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) C) (opr A (+) (opr B (-) C)))) (A B C -u axaddass C negclt syl3an3 A + B opr C subnegt A B axaddcl C CC e. 3adant3 A CC e. B CC e. C CC e. 3simp3 sylanc B C subnegt A CC e. 3adant1 A + opreq2d 3eqtr4d))
+thm (addsubt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) C) (opr (opr A (-) C) (+) B))) (A B axaddcom - C opreq1d C CC e. 3adant3 B A C addsubasst 3com12 B A - C opr axaddcom exp A C subclt syl5 exp3a com12 3imp 3eqtrd))
+thm (addsub12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (+) (opr B (-) C)) (opr B (+) (opr A (-) C)))) (A B axaddcom - C opreq1d C CC e. 3adant3 A B C addsubasst B A C addsubasst 3com12 3eqtr3d))
+thm (addsubass () ((addsubass.1 (e. A (CC))) (addsubass.2 (e. B (CC))) (addsubass.3 (e. C (CC)))) (= (opr (opr A (+) B) (-) C) (opr A (+) (opr B (-) C))) (addsubass.1 addsubass.2 addsubass.3 A B C addsubasst mp3an))
+thm (addsub () ((addsubass.1 (e. A (CC))) (addsubass.2 (e. B (CC))) (addsubass.3 (e. C (CC)))) (= (opr (opr A (+) B) (-) C) (opr (opr A (-) C) (+) B)) (addsubass.1 addsubass.2 addsubass.3 A B C addsubt mp3an))
+thm (negneg () ((negneg.1 (e. A (CC)))) (= (-u (-u A)) A) (negneg.1 negcl negid A + opreq2i negneg.1 negid + A -u -u opreq1i negneg.1 negneg.1 negcl negneg.1 negcl negcl addass negneg.1 negcl negcl addid2 3eqtr3 negneg.1 addid1 3eqtr3))
+thm (subid () ((negneg.1 (e. A (CC)))) (= (opr A (-) A) (0)) (negneg.1 negneg.1 subneg negneg.1 negid eqtr))
+thm (subid1 () ((negneg.1 (e. A (CC)))) (= (opr A (-) (0)) A) (negneg.1 addid2 negneg.1 0cn negneg.1 subadd mpbir))
+thm (negnegt () () (-> (e. A (CC)) (= (-u (-u A)) A)) (A A CC e. A 0 if negeq negeqd A A CC e. A 0 if = id eqeq12d 0cn A elimel negneg dedth))
+thm (subneg2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (-) (-u B)) (opr A (+) B))) (A B -u subnegt B negclt sylan2 B negnegt A + opreq2d A CC e. adantl eqtrd))
+thm (subidt () () (-> (e. A (CC)) (= (opr A (-) A) (0))) (A A CC e. A 0 if A A CC e. A 0 if - opreq12 anidms 0 eqeq1d 0cn A elimel subid dedth))
+thm (subid1t () () (-> (e. A (CC)) (= (opr A (-) (0)) A)) (A A CC e. A 0 if - 0 opreq1 A A CC e. A 0 if = id eqeq12d 0cn A elimel subid1 dedth))
+thm (pncant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (-) B) A)) (A B B addsubasst 3expb anabsan2 B subidt A + opreq2d A ax0id sylan9eqr eqtrd))
+thm (pncan2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (-) A) B)) (B A axaddcom - A opreq1d B A pncant eqtr3d ancoms))
+thm (npcant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr A (-) B) (+) B) A)) (A B B addsubt 3expb anabsan2 A B pncant eqtr3d))
+thm (npncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (+) (opr B (-) C)) (opr A (-) C))) (A - B opr B C addsubasst A B subclt C CC e. 3adant3 A CC e. B CC e. C CC e. 3simp2 A CC e. B CC e. C CC e. 3simp3 syl3anc A B npcant - C opreq1d C CC e. 3adant3 eqtr3d))
+thm (nppcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr (opr A (-) B) (+) C) (+) B) (opr A (+) C))) (A - B opr C B add23t A B subclt C CC e. 3adant3 A CC e. B CC e. C CC e. 3simp3 A CC e. B CC e. C CC e. 3simp2 syl3anc A B npcant + C opreq1d C CC e. 3adant3 eqtrd))
+thm (subeq0 () ((subeq0.1 (e. A (CC))) (subeq0.2 (e. B (CC)))) (<-> (= (opr A (-) B) (0)) (= A B)) (subeq0.1 subeq0.2 subneg 0 eqeq1i A + B -u opr 0 + B opreq1 sylbi subeq0.1 subeq0.2 negcl subeq0.2 add23 subeq0.1 subeq0.2 subeq0.2 negcl addass subeq0.2 negid A + opreq2i subeq0.1 addid1 eqtr 3eqtr subeq0.2 addid2 3eqtr3g A B - B opreq1 subeq0.2 subid syl6eq impbi))
+thm (neg11 () ((subeq0.1 (e. A (CC))) (subeq0.2 (e. B (CC)))) (<-> (= (-u A) (-u B)) (= A B)) (A df-neg B df-neg eqeq12i 0cn subeq0.1 0cn subeq0.2 subcl subadd subeq0.1 subeq0.2 subneg B df-neg A + opreq2i eqtr2 0 eqeq1i subeq0.1 subeq0.2 subeq0 bitr 3bitr))
+thm (negcon1 () ((subeq0.1 (e. A (CC))) (subeq0.2 (e. B (CC)))) (<-> (= (-u A) B) (= (-u B) A)) (subeq0.1 negneg B -u eqeq1i subeq0.1 negcl subeq0.2 neg11 A B -u eqcom 3bitr3))
+thm (negcon2 () ((subeq0.1 (e. A (CC))) (subeq0.2 (e. B (CC)))) (<-> (= A (-u B)) (= B (-u A))) (subeq0.2 subeq0.1 negcon1 A B -u eqcom B A -u eqcom 3bitr4))
+thm (neg11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (-u A) (-u B)) (= A B))) (A A CC e. A 0 if negeq B -u eqeq1d A A CC e. A 0 if B eqeq1 bibi12d B B CC e. B 0 if negeq A CC e. A 0 if -u eqeq2d B B CC e. B 0 if A CC e. A 0 if eqeq2 bibi12d 0cn A elimel 0cn B elimel neg11 dedth2h))
+thm (negcon1t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (-u A) B) (= (-u B) A))) (A -u B neg11t A negclt sylan A negnegt B CC e. adantr B -u eqeq1d bitr3d A B -u eqcom syl6bb))
+thm (negcon2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= A (-u B)) (= B (-u A)))) (A B negcon1t A B -u eqcom syl6rbbr A -u B eqcom syl6bb))
+thm (subcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) B) (opr A (-) C)) (= B C))) (A B subnegt C CC e. 3adant3 A C subnegt B CC e. 3adant2 eqeq12d A B -u C -u addcant C negclt syl3an3 B negclt syl3an2 B C neg11t A CC e. 3adant1 3bitrd))
+thm (subcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (-) C) (opr B (-) C)) (= A B))) (A C subnegt B CC e. 3adant2 B C subnegt A CC e. 3adant1 eqeq12d A B C -u addcan2t C negclt syl3an3 bitrd))
+thm (subcan () ((subcan.1 (e. A (CC))) (subcan.2 (e. B (CC))) (subcan.3 (e. C (CC)))) (<-> (= (opr A (-) B) (opr A (-) C)) (= B C)) (subcan.1 subcan.2 subcan.3 A B C subcant mp3an))
+thm (subcan2 () ((subcan.1 (e. A (CC))) (subcan.2 (e. B (CC))) (subcan.3 (e. C (CC)))) (<-> (= (opr A (-) C) (opr B (-) C)) (= A B)) (subcan.1 subcan.2 subcan.3 A B C subcan2t mp3an))
+thm (subeq0t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (opr A (-) B) (0)) (= A B))) (A A CC e. A 0 if - B opreq1 0 eqeq1d A A CC e. A 0 if B eqeq1 bibi12d B B CC e. B 0 if A CC e. A 0 if - opreq2 0 eqeq1d B B CC e. B 0 if A CC e. A 0 if eqeq2 bibi12d 0cn A elimel 0cn B elimel subeq0 dedth2h))
+thm (neg0 () () (= (-u (0)) (0)) (0 df-neg 0cn subid eqtr))
+thm (renegcl ((A x)) ((renegcl.1 (e. A (RR)))) (e. (-u A) (RR)) (renegcl.1 A x axrnegex ax-mp x RR A + x cv opr 0 = df-rex mpbi x cv recnt 0cn renegcl.1 recn 0 A x cv subaddt mp3an12 syl A df-neg x cv eqeq1i syl5bb x cv RR A -u eleq1a sylbird imp x 19.23aiv ax-mp))
+thm (renegclt () () (-> (e. A (RR)) (e. (-u A) (RR))) (A A RR e. A 1 if negeq RR eleq1d ax1re A elimel renegcl dedth))
+thm (resubclt () () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (-) B) (RR))) (A B subnegt A recnt B recnt syl2an A B -u axaddrcl B renegclt sylan2 eqeltrd))
+thm (resubcl () ((resubcl.1 (e. A (RR))) (resubcl.2 (e. B (RR)))) (e. (opr A (-) B) (RR)) (resubcl.1 resubcl.2 A B resubclt mp2an))
+thm (0re () () (e. (0) (RR)) (ax1re ax1re resubcl 1cn subid RR eleq1i mpbi))
+thm (mulid2t () () (-> (e. A (CC)) (= (opr (1) (x.) A) A)) (1cn 1 A axmulcom mpan A ax1id eqtrd))
+thm (mul12t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (x.) C)) (opr B (x.) (opr A (x.) C)))) (A B axmulcom x. C opreq1d C CC e. 3adant3 A B C axmulass B A C axmulass 3com12 3eqtr3d))
+thm (mul23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (x.) B) (x.) C) (opr (opr A (x.) C) (x.) B))) (B C axmulcom A x. opreq2d A CC e. 3adant1 A B C axmulass A C B axmulass 3com23 3eqtr4d))
+thm (mul4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (x.) B) (x.) (opr C (x.) D)) (opr (opr A (x.) C) (x.) (opr B (x.) D)))) (A B C mul23t x. D opreq1d 3expa D CC e. adantrr A x. B opr C D axmulass 3expb A B axmulcl sylan A x. C opr B D axmulass 3expb A C axmulcl sylan an4s 3eqtr3d))
+thm (muladd11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr (1) (+) A) (x.) (opr (1) (+) B)) (opr (opr (1) (+) A) (+) (opr B (+) (opr A (x.) B))))) (1cn 1 + A opr 1 B axdistr mp3an2 1cn 1 A axaddcl mpan sylan 1cn 1 A axaddcl mpan 1 + A opr ax1id syl B CC e. adantr 1cn 1 A B adddirt mp3an1 B mulid2t A CC e. adantl + A x. B opr opreq1d eqtrd + opreq12d eqtrd))
+thm (mul12 () ((mul.1 (e. A (CC))) (mul.2 (e. B (CC))) (mul.3 (e. C (CC)))) (= (opr A (x.) (opr B (x.) C)) (opr B (x.) (opr A (x.) C))) (mul.1 mul.2 mulcom x. C opreq1i mul.1 mul.2 mul.3 mulass mul.2 mul.1 mul.3 mulass 3eqtr3))
+thm (mul23 () ((mul.1 (e. A (CC))) (mul.2 (e. B (CC))) (mul.3 (e. C (CC)))) (= (opr (opr A (x.) B) (x.) C) (opr (opr A (x.) C) (x.) B)) (mul.1 mul.2 mul.3 A B C mul23t mp3an))
+thm (mul4 () ((mul4.1 (e. A (CC))) (mul4.2 (e. B (CC))) (mul4.3 (e. C (CC))) (mul4.4 (e. D (CC)))) (= (opr (opr A (x.) B) (x.) (opr C (x.) D)) (opr (opr A (x.) C) (x.) (opr B (x.) D))) (mul4.1 mul4.2 pm3.2i mul4.3 mul4.4 pm3.2i A B C D mul4t mp2an))
+thm (muladd () ((mul4.1 (e. A (CC))) (mul4.2 (e. B (CC))) (mul4.3 (e. C (CC))) (mul4.4 (e. D (CC)))) (= (opr (opr A (+) B) (x.) (opr C (+) D)) (opr (opr (opr A (x.) C) (+) (opr D (x.) B)) (+) (opr (opr A (x.) D) (+) (opr C (x.) B)))) (mul4.1 mul4.2 addcl mul4.3 mul4.4 adddi mul4.1 mul4.2 mul4.3 adddir mul4.1 mul4.2 mul4.4 adddir + opreq12i mul4.1 mul4.3 mulcl mul4.2 mul4.3 mulcl mul4.1 mul4.4 mulcl mul4.2 mul4.4 mulcl addcl add23 mul4.4 mul4.2 mulcom A x. C opr + A x. D opr opr + opreq2i mul4.1 mul4.3 mulcl mul4.1 mul4.4 mulcl mul4.4 mul4.2 mulcl add23 mul4.1 mul4.3 mulcl mul4.1 mul4.4 mulcl mul4.2 mul4.4 mulcl addass 3eqtr3r mul4.2 mul4.3 mulcom + opreq12i mul4.1 mul4.3 mulcl mul4.4 mul4.2 mulcl addcl mul4.1 mul4.4 mulcl mul4.3 mul4.2 mulcl addass 3eqtr 3eqtr))
+thm (subdi () ((subdi.1 (e. A (CC))) (subdi.2 (e. B (CC))) (subdi.3 (e. C (CC)))) (= (opr A (x.) (opr B (-) C)) (opr (opr A (x.) B) (-) (opr A (x.) C))) (subdi.1 subdi.3 subdi.2 subdi.3 subcl adddi subdi.3 subdi.2 pncan3 A x. opreq2i eqtr3 subdi.1 subdi.2 mulcl subdi.1 subdi.3 mulcl subdi.1 subdi.2 subdi.3 subcl mulcl subadd mpbir eqcomi))
+thm (subdir () ((subdi.1 (e. A (CC))) (subdi.2 (e. B (CC))) (subdi.3 (e. C (CC)))) (= (opr (opr A (-) B) (x.) C) (opr (opr A (x.) C) (-) (opr B (x.) C))) (subdi.3 subdi.1 subdi.2 subdi subdi.1 subdi.2 subcl subdi.3 mulcom subdi.1 subdi.3 mulcom subdi.2 subdi.3 mulcom - opreq12i 3eqtr4))
+thm (subdit () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (x.) (opr B (-) C)) (opr (opr A (x.) B) (-) (opr A (x.) C)))) (A A CC e. A 0 if x. B - C opr opreq1 A A CC e. A 0 if x. B opreq1 A A CC e. A 0 if x. C opreq1 - opreq12d eqeq12d B B CC e. B 0 if - C opreq1 A CC e. A 0 if x. opreq2d B B CC e. B 0 if A CC e. A 0 if x. opreq2 - A CC e. A 0 if x. C opr opreq1d eqeq12d C C CC e. C 0 if B CC e. B 0 if - opreq2 A CC e. A 0 if x. opreq2d C C CC e. C 0 if A CC e. A 0 if x. opreq2 A CC e. A 0 if x. B CC e. B 0 if opr - opreq2d eqeq12d 0cn A elimel 0cn B elimel 0cn C elimel subdi dedth3h))
+thm (mul01 () ((mulzer.1 (e. A (CC)))) (= (opr A (x.) (0)) (0)) (mulzer.1 0cn 0cn subdi 0cn subid A x. opreq2i mulzer.1 0cn mulcl subid 3eqtr3))
+thm (mul02 () ((mulzer.1 (e. A (CC)))) (= (opr (0) (x.) A) (0)) (0cn mulzer.1 mulcom mulzer.1 mul01 eqtr))
+thm (1p1times () ((mulzer.1 (e. A (CC)))) (= (opr (opr (1) (+) (1)) (x.) A) (opr A (+) A)) (1cn 1cn mulzer.1 adddir mulzer.1 mulid2 mulzer.1 mulid2 + opreq12i eqtr))
+thm (mul01t () () (-> (e. A (CC)) (= (opr A (x.) (0)) (0))) (A A CC e. A 0 if x. 0 opreq1 0 eqeq1d 0cn A elimel mul01 dedth))
+thm (mul02t () () (-> (e. A (CC)) (= (opr (0) (x.) A) (0))) (0cn 0 A axmulcom mpan A mul01t eqtrd))
+thm (mulneg1 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr (-u A) (x.) B) (-u (opr A (x.) B))) (mulneg.2 mul01 mulneg.2 mulneg.1 mulcom - opreq12i A df-neg x. B opreq1i 0cn mulneg.1 subcl mulneg.2 mulcom mulneg.2 0cn mulneg.1 subdi 3eqtr A x. B opr df-neg 3eqtr4))
+thm (mulneg2 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr A (x.) (-u B)) (-u (opr A (x.) B))) (mulneg.1 mulneg.2 negcl mulcom mulneg.2 mulneg.1 mulneg1 mulneg.2 mulneg.1 mulcom negeqi 3eqtr))
+thm (mul2neg () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (opr (-u A) (x.) (-u B)) (opr A (x.) B)) (mulneg.1 mulneg.2 negcl mulneg1 mulneg.1 mulneg.2 negcl mulcom mulneg.2 mulneg.1 mulneg1 eqtr negeqi mulneg.2 mulneg.1 mulcl negneg 3eqtr mulneg.2 mulneg.1 mulcom eqtr))
+thm (negdi () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (+) B)) (opr (-u A) (+) (-u B))) (mulneg.1 mulneg.2 addcl mulid2 negeqi 1cn negcl mulneg.1 mulneg.2 adddi 1cn mulneg.1 mulneg.2 addcl mulneg1 1cn mulneg.1 mulneg1 mulneg.1 mulid2 negeqi eqtr 1cn mulneg.2 mulneg1 mulneg.2 mulid2 negeqi eqtr + opreq12i 3eqtr3 eqtr3))
+thm (negdi2 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (-) B)) (opr (-u A) (+) B)) (mulneg.1 mulneg.2 subneg negeqi mulneg.1 mulneg.2 negcl negdi mulneg.2 negneg A -u + opreq2i 3eqtr))
+thm (negdi3 () ((mulneg.1 (e. A (CC))) (mulneg.2 (e. B (CC)))) (= (-u (opr A (-) B)) (opr B (-) A)) (mulneg.1 negcl mulneg.2 addcom mulneg.1 mulneg.2 negdi2 mulneg.2 mulneg.1 subneg 3eqtr4))
+thm (mulneg1t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) B) (-u (opr A (x.) B)))) (A A CC e. A 0 if negeq x. B opreq1d A A CC e. A 0 if x. B opreq1 negeqd eqeq12d B B CC e. B 0 if A CC e. A 0 if -u x. opreq2 B B CC e. B 0 if A CC e. A 0 if x. opreq2 negeqd eqeq12d 0cn A elimel 0cn B elimel mulneg1 dedth2h))
+thm (mulneg2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (x.) (-u B)) (-u (opr A (x.) B)))) (B A mulneg1t ancoms A B -u axmulcom B negclt sylan2 A B axmulcom negeqd 3eqtr4d))
+thm (mulneg12t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) B) (opr A (x.) (-u B)))) (A B mulneg1t A B mulneg2t eqtr4d))
+thm (mul2negt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (-u A) (x.) (-u B)) (opr A (x.) B))) (A A CC e. A 0 if negeq x. B -u opreq1d A A CC e. A 0 if x. B opreq1 eqeq12d B B CC e. B 0 if negeq A CC e. A 0 if -u x. opreq2d B B CC e. B 0 if A CC e. A 0 if x. opreq2 eqeq12d 0cn A elimel 0cn B elimel mul2neg dedth2h))
+thm (negdit () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (+) B)) (opr (-u A) (+) (-u B)))) (A A CC e. A 0 if + B opreq1 negeqd A A CC e. A 0 if negeq + B -u opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if + opreq2 negeqd B B CC e. B 0 if negeq A CC e. A 0 if -u + opreq2d eqeq12d 0cn A elimel 0cn B elimel negdi dedth2h))
+thm (negdi2t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (-) B)) (opr (-u A) (+) B))) (A B subnegt negeqd A B -u negdit B negclt sylan2 B negnegt A CC e. adantl A -u + opreq2d 3eqtrd))
+thm (negdi3t () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (-u (opr A (-) B)) (opr B (-) A))) (A -u B axaddcom A negclt sylan A B negdi2t B A subnegt ancoms 3eqtr4d))
+thm (subsub2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr A (+) (opr C (-) B)))) (A B - C opr subnegt B C subclt sylan2 3impb B C negdi3t A + opreq2d A CC e. 3adant1 eqtrd))
+thm (subsubt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr (opr A (-) B) (+) C))) (A B C subsub2t A C B addsubasst A C B addsubt eqtr3d 3com23 eqtrd))
+thm (subsub3t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr A (-) (opr B (-) C)) (opr (opr A (+) C) (-) B))) (A B C subsub2t A C B addsubasst 3com23 eqtr4d))
+thm (subsub4t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) C) (opr A (-) (opr B (+) C)))) (A - B opr C subnegt A B subclt sylan 3impa A B C -u subsubt C negclt syl3an3 eqtr4d B C subneg2t A CC e. 3adant1 A - opreq2d eqtrd))
+thm (sub23t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) C) (opr (opr A (-) C) (-) B))) (B C axaddcom A CC e. 3adant1 A - opreq2d A B C subsub4t A C B subsub4t 3com23 3eqtr4d))
+thm (nnncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) (opr B (-) C)) (-) C) (opr A (-) B))) (A B - C opr C subsub4t A CC e. B CC e. C CC e. 3simp1 B C subclt A CC e. 3adant1 A CC e. B CC e. C CC e. 3simp3 syl3anc B C npcant A - opreq2d A CC e. 3adant1 eqtrd))
+thm (nnncan1t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) B) (-) (opr A (-) C)) (opr C (-) B))) (A - B opr A - C opr subnegt A - B opr A - C opr -u axaddcom A - C opr negclt sylan2 eqtrd A B subclt C CC e. 3adant3 A C subclt B CC e. 3adant2 sylanc A C negdi3t B CC e. 3adant2 + A - B opr opreq1d C A B npncant 3coml 3eqtrd))
+thm (nnncan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) C) (-) (opr B (-) C)) (opr A (-) B))) (A B - C opr C sub23t A CC e. B CC e. C CC e. 3simp1 B C subclt A CC e. 3adant1 A CC e. B CC e. C CC e. 3simp3 syl3anc A B C nnncant eqtr3d))
+thm (nncant () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr A (-) (opr A (-) B)) B)) (0cn A 0 B nnncan1t mp3an2 A subid1t B CC e. adantr - A - B opr opreq1d B subid1t A CC e. adantl 3eqtr3d))
+thm (nppcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (-) (opr B (+) C)) (+) C) (opr A (-) B))) (A B + C opr C subsubt A CC e. B CC e. C CC e. 3simp1 B C axaddcl A CC e. 3adant1 A CC e. B CC e. C CC e. 3simp3 syl3anc B C pncant A CC e. 3adant1 A - opreq2d eqtr3d))
+thm (mulm1t () () (-> (e. A (CC)) (= (opr (-u (1)) (x.) A) (-u A))) (1cn 1 A mulneg1t mpan A mulid2t negeqd eqtrd))
+thm (mulm1 () ((mulm1.1 (e. A (CC)))) (= (opr (-u (1)) (x.) A) (-u A)) (mulm1.1 A mulm1t ax-mp))
+thm (sub4t () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (= (opr (opr A (+) B) (-) (opr C (+) D)) (opr (opr A (-) C) (+) (opr B (-) D)))) (C D negdit A CC e. B CC e. /\ adantl A + B opr + opreq2d A B C -u D -u add4t C negclt D negclt anim12i sylan2 eqtrd A + B opr C + D opr subnegt A B axaddcl C D axaddcl syl2an A C subnegt D CC e. adantrr B CC e. adantlr B D subnegt C CC e. adantrl A CC e. adantll + opreq12d 3eqtr4d))
+thm (sub4 () ((sub4.1 (e. A (CC))) (sub4.2 (e. B (CC))) (sub4.3 (e. C (CC))) (sub4.4 (e. D (CC)))) (= (opr (opr A (+) B) (-) (opr C (+) D)) (opr (opr A (-) C) (+) (opr B (-) D))) (sub4.1 sub4.2 pm3.2i sub4.3 sub4.4 pm3.2i A B C D sub4t mp2an))
+thm (pnpcant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) (opr A (+) C)) (opr B (-) C))) (A B A C sub4t anandis A subidt + B - C opr opreq1d B C subclt B - C opr addid2t syl sylan9eq eqtrd 3impb))
+thm (pnncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (-) (opr A (-) C)) (opr B (+) C))) (A C subnegt B CC e. 3adant2 A + B opr - opreq2d A B C -u pnpcant C negclt syl3an3 B C subneg2t A CC e. 3adant1 3eqtrd))
+thm (ppncant () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (opr (opr A (+) B) (+) (opr C (-) B)) (opr A (+) C))) (A B axaddcom C CC e. 3adant3 - B - C opr opreq1d A + B opr B C subsub2t A B axaddcl C CC e. 3adant3 A CC e. B CC e. C CC e. 3simp2 A CC e. B CC e. C CC e. 3simp3 syl3anc B A C pnncant 3com12 3eqtr3d))
+thm (mulcan ((A x) (B x) (C x)) ((mulcan.1 (e. A (CC))) (mulcan.2 (e. B (CC))) (mulcan.3 (e. C (CC))) (mulcan.4 (=/= A (0)))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C)) (mulcan.1 mulcan.4 x recex mulcan.1 mulcan.2 x cv A B axmulass mp3an3 mulcan.3 x cv A C axmulass mp3an3 eqeq12d mpan2 A x. B opr A x. C opr x cv x. opreq2 syl5bir A x. x cv opr 1 = adantr mulcan.1 A x cv axmulcom mpan 1 eqeq1d x cv x. A opr 1 x. B opreq1 mulcan.2 mulid2 syl6eq x cv x. A opr 1 x. C opreq1 mulcan.3 mulid2 syl6eq eqeq12d syl6bi imp sylibd exp r19.23aiv ax-mp B C A x. opreq2 impbi))
+thm (mulcant2 () ((mulcant2.1 (=/= A (0)))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C))) (A A CC e. A 1 if x. B opreq1 A A CC e. A 1 if x. C opreq1 eqeq12d B C = bibi1d B B CC e. B 1 if A CC e. A 1 if x. opreq2 A CC e. A 1 if x. C opr eqeq1d B B CC e. B 1 if C eqeq1 bibi12d C C CC e. C 1 if A CC e. A 1 if x. opreq2 A CC e. A 1 if x. B CC e. B 1 if opr eqeq2d C C CC e. C 1 if B CC e. B 1 if eqeq2 bibi12d 1cn A elimel 1cn B elimel 1cn C elimel A A CC e. A 1 if 0 neeq1 1 A CC e. A 1 if 0 neeq1 mulcant2.1 ax1ne0 keephyp mulcan dedth3h))
+thm (mulcant () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= A (0))) (<-> (= (opr A (x.) B) (opr A (x.) C)) (= B C))) (A A 0 =/= A 1 if CC eleq1 A A 0 =/= A 1 if = B CC e. pm4.2i A A 0 =/= A 1 if = C CC e. pm4.2i 3anbi123d A A 0 =/= A 1 if x. B opreq1 A A 0 =/= A 1 if x. C opreq1 eqeq12d B C = bibi1d imbi12d A elimne0 B C mulcant2 dedth impcom))
+thm (mulcan2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (<-> (= (opr A (x.) C) (opr B (x.) C)) (= A B))) (A C axmulcom B CC e. 3adant2 B C axmulcom A CC e. 3adant1 eqeq12d C 0 =/= adantr C A B mulcant exp 3coml imp bitrd))
+thm (mul0or () ((mul0or.1 (e. A (CC))) (mul0or.2 (e. B (CC)))) (<-> (= (opr A (x.) B) (0)) (\/ (= A (0)) (= B (0)))) (A 0 df-ne mul0or.1 mul0or.2 0cn 3pm3.2i A B 0 mulcant mpan mul0or.1 mul01 A x. B opr eqeq2i syl5bbr biimpd sylbir com12 orrd A 0 x. B opreq1 mul0or.2 mul02 syl6eq B 0 A x. opreq2 mul0or.1 mul01 syl6eq jaoi impbi))
+thm (sq0 () ((mul0or.1 (e. A (CC)))) (<-> (= (opr A (x.) A) (0)) (= A (0))) (mul0or.1 mul0or.1 mul0or A 0 = oridm bitr))
+thm (mul0ort () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (opr A (x.) B) (0)) (\/ (= A (0)) (= B (0))))) (A A CC e. A 0 if x. B opreq1 0 eqeq1d A A CC e. A 0 if 0 eqeq1 B 0 = orbi1d bibi12d B B CC e. B 0 if A CC e. A 0 if x. opreq2 0 eqeq1d B B CC e. B 0 if 0 eqeq1 A CC e. A 0 if 0 = orbi2d bibi12d 0cn A elimel 0cn B elimel mul0or dedth2h))
+thm (muln0bt () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (/\ (=/= A (0)) (=/= B (0))) (=/= (opr A (x.) B) (0)))) (A B mul0ort negbid A 0 = B 0 = ioran syl6rbb A 0 df-ne B 0 df-ne anbi12i A x. B opr 0 df-ne 3bitr4g))
+thm (muln0 () ((muln0.1 (e. A (CC))) (muln0.2 (e. B (CC))) (muln0.3 (=/= A (0))) (muln0.4 (=/= B (0)))) (=/= (opr A (x.) B) (0)) (muln0.1 muln0.2 muln0.3 muln0.4 pm3.2i A B muln0bt mpbii mp2an))
+thm (receu ((x y) (A x) (A y) (B x) (B y)) ((receu.1 (e. A (CC))) (receu.2 (e. B (CC))) (receu.3 (=/= A (0)))) (E!e. x (CC) (= (opr A (x.) (cv x)) B)) (x cv y cv A x. opreq2 B eqeq1d CC reu4 receu.1 receu.3 y recex receu.2 y cv B axmulcl mpan2 y cv x. B opr CC x risset sylib x cv y cv x. B opr A x. opreq2 receu.1 receu.2 A y cv B axmulass mp3an13 eqcomd sylan9eqr A x. y cv opr 1 x. B opreq1 receu.2 mulid2 syl6eq sylan9eqr exp32 impcom x cv CC e. a1d r19.21aiv exp x CC x cv y cv x. B opr = A x. x cv opr B = r19.22 syl6 com23 mpd r19.23aiv ax-mp receu.1 receu.3 x cv y cv mulcant2 A x. x cv opr B A x. y cv opr eqtr3t syl5bi mp3an1 rgen2 mpbir2an))
+thm (divval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((divval.1 (e. A (CC))) (divval.2 (e. B (CC))) (divval.3 (=/= B (0)))) (= (opr A (/) B) (U. ({e.|} x (CC) (= (opr B (x.) (cv x)) A)))) (divval.1 B CC 0 {} eldif divval.2 divval.3 B 0 df-ne mpbi divval.2 elisseti 0 elsnc mtbir mpbir2an axcnex x B x. x cv opr A = rabex uniex y cv A z cv x. x cv opr eqeq2 x CC birabsdv unieqd z cv B x. x cv opreq1 A eqeq1d x CC birabsdv unieqd y z w x df-div oprabval2 mp2an))
+thm (divmul ((A x) (B x) (C x)) ((divmul.1 (e. A (CC))) (divmul.2 (e. B (CC))) (divmul.3 (e. C (CC))) (divmul.4 (=/= B (0)))) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A)) (divmul.3 x cv C A / B opr eqeq2 x cv C B x. opreq2 A eqeq1d bibi12d divmul.2 divmul.1 divmul.4 x receu x CC B x. x cv opr A = reuuni1 mpan2 divmul.1 divmul.2 divmul.4 x divval x cv eqeq1i syl6rbbr vtoclga ax-mp))
+thm (divmulz () ((divmulz.1 (e. A (CC))) (divmulz.2 (e. B (CC))) (divmulz.3 (e. C (CC)))) (-> (=/= B (0)) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A))) (B B 0 =/= B 1 if A / opreq2 C eqeq1d B B 0 =/= B 1 if x. C opreq1 A eqeq1d bibi12d divmulz.1 divmulz.2 1cn B 0 =/= keepel divmulz.3 B elimne0 divmul dedth))
+thm (divmult () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (<-> (= (opr A (/) B) C) (= (opr B (x.) C) A))) (A A CC e. A 0 if / B opreq1 C eqeq1d A A CC e. A 0 if B x. C opr eqeq2 bibi12d B 0 =/= imbi2d B B CC e. B 0 if 0 neeq1 B B CC e. B 0 if A CC e. A 0 if / opreq2 C eqeq1d B B CC e. B 0 if x. C opreq1 A CC e. A 0 if eqeq1d bibi12d imbi12d C C CC e. C 0 if A CC e. A 0 if / B CC e. B 0 if opr eqeq2 C C CC e. C 0 if B CC e. B 0 if x. opreq2 A CC e. A 0 if eqeq1d bibi12d B CC e. B 0 if 0 =/= imbi2d 0cn A elimel 0cn B elimel 0cn C elimel divmulz dedth3h imp))
+thm (divcl ((A x) (B x)) ((divcl.1 (e. A (CC))) (divcl.2 (e. B (CC))) (divcl.3 (=/= B (0)))) (e. (opr A (/) B) (CC)) (divcl.1 divcl.2 divcl.3 x divval divcl.2 divcl.1 divcl.3 x receu x CC B x. x cv opr A = reucl ax-mp eqeltr))
+thm (divclz () ((divclz.1 (e. A (CC))) (divclz.2 (e. B (CC)))) (-> (=/= B (0)) (e. (opr A (/) B) (CC))) (B B 0 =/= B 1 if A / opreq2 CC eleq1d divclz.1 divclz.2 1cn B 0 =/= keepel B elimne0 divcl dedth))
+thm (divclt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (e. (opr A (/) B) (CC))) (A A CC e. A 0 if / B opreq1 CC eleq1d B 0 =/= imbi2d B B CC e. B 0 if 0 neeq1 B B CC e. B 0 if A CC e. A 0 if / opreq2 CC eleq1d imbi12d 0cn A elimel 0cn B elimel divclz dedth2h exp 3imp))
+thm (reccl () ((reccl.1 (e. A (CC))) (reccl.2 (=/= A (0)))) (e. (opr (1) (/) A) (CC)) (1cn reccl.1 reccl.2 divcl))
+thm (recclz () ((recclz.1 (e. A (CC)))) (-> (=/= A (0)) (e. (opr (1) (/) A) (CC))) (1cn recclz.1 divclz))
+thm (recclt () () (-> (/\ (e. A (CC)) (=/= A (0))) (e. (opr (1) (/) A) (CC))) (1cn 1 A divclt mp3an1))
+thm (divcan2 () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC))) (divcan.3 (=/= A (0)))) (= (opr A (x.) (opr B (/) A)) B) (B / A opr eqid divcan.2 divcan.1 divcan.2 divcan.1 divcan.3 divcl divcan.3 divmul mpbi))
+thm (divcan1 () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC))) (divcan.3 (=/= A (0)))) (= (opr (opr B (/) A) (x.) A) B) (divcan.2 divcan.1 divcan.3 divcl divcan.1 mulcom divcan.1 divcan.2 divcan.3 divcan2 eqtr))
+thm (divcan1z () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr B (/) A) (x.) A) B)) (A A 0 =/= A 1 if B / opreq2 A A 0 =/= A 1 if = id x. opreq12d B eqeq1d divcan.1 1cn A 0 =/= keepel divcan.2 A elimne0 divcan1 dedth))
+thm (divcan2z () ((divcan.1 (e. A (CC))) (divcan.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr A (x.) (opr B (/) A)) B)) (A A 0 =/= A 1 if = id A A 0 =/= A 1 if B / opreq2 x. opreq12d B eqeq1d divcan.1 1cn A 0 =/= keepel divcan.2 A elimne0 divcan2 dedth))
+thm (divcan1t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr B (/) A) (x.) A) B)) (A A CC e. A 0 if 0 neeq1 A A CC e. A 0 if B / opreq2 A A CC e. A 0 if = id x. opreq12d B eqeq1d imbi12d B B CC e. B 0 if / A CC e. A 0 if opreq1 x. A CC e. A 0 if opreq1d B B CC e. B 0 if = id eqeq12d A CC e. A 0 if 0 =/= imbi2d 0cn A elimel 0cn B elimel divcan1z dedth2h exp 3imp))
+thm (divcan2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr A (x.) (opr B (/) A)) B)) (A A CC e. A 0 if 0 neeq1 A A CC e. A 0 if = id A A CC e. A 0 if B / opreq2 x. opreq12d B eqeq1d imbi12d B B CC e. B 0 if / A CC e. A 0 if opreq1 A CC e. A 0 if x. opreq2d B B CC e. B 0 if = id eqeq12d A CC e. A 0 if 0 =/= imbi2d 0cn A elimel 0cn B elimel divcan2z dedth2h exp 3imp))
+thm (divneq0bt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (<-> (=/= A (0)) (=/= (opr A (/) B) (0)))) (B mul01t A eqeq1d A 0 eqcom syl6rbbr A CC e. B 0 =/= 3ad2ant2 0cn A B 0 divmult mp3anl3 3impa bitr4d eqneqd))
+thm (divneq0 () ((divneq0.1 (e. A (CC))) (divneq0.2 (e. B (CC))) (divneq0.3 (=/= A (0))) (divneq0.4 (=/= B (0)))) (=/= (opr A (/) B) (0)) (divneq0.1 divneq0.2 divneq0.4 divneq0.3 A B divneq0bt mpbii mp3an))
+thm (recneq0z () ((recneq0z.1 (e. A (CC)))) (-> (=/= A (0)) (=/= (opr (1) (/) A) (0))) (A A 0 =/= A 1 if 1 / opreq2 0 neeq1d 1cn recneq0z.1 1cn A 0 =/= keepel ax1ne0 A elimne0 divneq0 dedth))
+thm (recid () ((recid.1 (e. A (CC))) (recid.2 (=/= A (0)))) (= (opr A (x.) (opr (1) (/) A)) (1)) (recid.1 1cn recid.2 divcan2))
+thm (recidz () ((recidz.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr A (x.) (opr (1) (/) A)) (1))) (recidz.1 1cn divcan2z))
+thm (recidt () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr A (x.) (opr (1) (/) A)) (1))) (A A CC e. A 0 if 0 neeq1 A A CC e. A 0 if = id A A CC e. A 0 if 1 / opreq2 x. opreq12d 1 eqeq1d imbi12d 0cn A elimel recidz dedth imp))
+thm (divrec () ((divrec.1 (e. A (CC))) (divrec.2 (e. B (CC))) (divrec.3 (=/= B (0)))) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B))) (divrec.2 divrec.1 divrec.2 divrec.3 reccl mulcl mulcom divrec.1 divrec.2 divrec.3 reccl divrec.2 mulass divrec.2 1cn divrec.3 divcan1 A x. opreq2i divrec.1 mulid1 eqtr 3eqtr divrec.1 divrec.2 divrec.1 divrec.2 divrec.3 reccl mulcl divrec.3 divmul mpbir))
+thm (divrecz () ((divrec.1 (e. A (CC))) (divrec.2 (e. B (CC)))) (-> (=/= B (0)) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B)))) (B B 0 =/= B 1 if A / opreq2 B B 0 =/= B 1 if 1 / opreq2 A x. opreq2d eqeq12d divrec.1 divrec.2 1cn B 0 =/= keepel B elimne0 divrec dedth))
+thm (divrect () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (opr A (/) B) (opr A (x.) (opr (1) (/) B)))) (A A CC e. A 0 if / B opreq1 A A CC e. A 0 if x. 1 / B opr opreq1 eqeq12d B 0 =/= imbi2d B B CC e. B 0 if 0 neeq1 B B CC e. B 0 if A CC e. A 0 if / opreq2 B B CC e. B 0 if 1 / opreq2 A CC e. A 0 if x. opreq2d eqeq12d imbi12d 0cn A elimel 0cn B elimel divrecz dedth2h exp 3imp))
+thm (divasst () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C)))) (A CC e. id B CC e. id C recclt 3anim123i 3exp exp4a 3imp1 A B 1 / C opr axmulass syl A x. B opr C divrect 3expa A B axmulcl C CC e. anim1i 3impa sylan B C divrect 3expa A CC e. 3adantl1 A x. opreq2d 3eqtr4d))
+thm (div23t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (x.) B) (/) C) (opr (opr A (/) C) (x.) B))) (A B axmulcom C CC e. 3adant3 C 0 =/= adantr / C opreq1d B A C divasst exp 3com12 imp A CC e. B CC e. C CC e. 3simp2 C 0 =/= adantr A C divclt 3expa B CC e. 3adantl2 jca B A / C opr axmulcom syl 3eqtrd))
+thm (div13t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= B (0))) (= (opr (opr A (/) B) (x.) C) (opr (opr C (/) B) (x.) A))) (A C axmulcom / B opreq1d B CC e. 3adant2 B 0 =/= adantr A C B div23t exp 3com23 imp C A B div23t exp 3coml imp 3eqtr3d))
+thm (div12t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr A (x.) (opr B (/) C)) (opr B (x.) (opr A (/) C)))) (A B / C opr axmulcom A CC e. B CC e. C CC e. 3simp1 C 0 =/= adantr B C divclt 3expa A CC e. 3adantl1 sylanc B C A div13t exp 3comr imp A / C opr B axmulcom A C divclt 3expa B CC e. 3adantl2 A CC e. B CC e. C CC e. 3simp2 C 0 =/= adantr sylanc 3eqtrd))
+thm (divassz () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC)))) (-> (=/= C (0)) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C)))) (divass.1 divass.2 divass.3 3pm3.2i A B C divasst mpan))
+thm (divass () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (x.) B) (/) C) (opr A (x.) (opr B (/) C))) (divass.4 divass.1 divass.2 divass.3 divassz ax-mp))
+thm (divdir () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C))) (divass.1 divass.2 divass.3 divass.4 reccl adddir divass.1 divass.2 addcl divass.3 divass.4 divrec divass.1 divass.3 divass.4 divrec divass.2 divass.3 divass.4 divrec + opreq12i 3eqtr4))
+thm (div23 () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC))) (divass.4 (=/= C (0)))) (= (opr (opr A (x.) B) (/) C) (opr (opr A (/) C) (x.) B)) (divass.1 divass.2 mulcom / C opreq1i divass.2 divass.1 divass.3 divass.4 divass divass.2 divass.1 divass.3 divass.4 divcl mulcom 3eqtr))
+thm (divdirz () ((divass.1 (e. A (CC))) (divass.2 (e. B (CC))) (divass.3 (e. C (CC)))) (-> (=/= C (0)) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C)))) (C C 0 =/= C 1 if A + B opr / opreq2 C C 0 =/= C 1 if A / opreq2 C C 0 =/= C 1 if B / opreq2 + opreq12d eqeq12d divass.1 divass.2 divass.3 1cn C 0 =/= keepel C elimne0 divdir dedth))
+thm (divdirt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (+) B) (/) C) (opr (opr A (/) C) (+) (opr B (/) C)))) (A A CC e. A 0 if + B opreq1 / C opreq1d A A CC e. A 0 if / C opreq1 + B / C opr opreq1d eqeq12d C 0 =/= imbi2d B B CC e. B 0 if A CC e. A 0 if + opreq2 / C opreq1d B B CC e. B 0 if / C opreq1 A CC e. A 0 if / C opr + opreq2d eqeq12d C 0 =/= imbi2d C C CC e. C 0 if 0 neeq1 C C CC e. C 0 if A CC e. A 0 if + B CC e. B 0 if opr / opreq2 C C CC e. C 0 if A CC e. A 0 if / opreq2 C C CC e. C 0 if B CC e. B 0 if / opreq2 + opreq12d eqeq12d imbi12d 0cn A elimel 0cn B elimel 0cn C elimel divdirz dedth3h imp))
+thm (divcan3 () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC))) (divcan3.3 (=/= A (0)))) (= (opr (opr A (x.) B) (/) A) B) (divcan3.1 divcan3.2 divcan3.1 divcan3.3 divass divcan3.1 divcan3.2 divcan3.3 divcan2 eqtr))
+thm (divcan4 () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC))) (divcan3.3 (=/= A (0)))) (= (opr (opr B (x.) A) (/) A) B) (divcan3.2 divcan3.1 mulcom / A opreq1i divcan3.1 divcan3.2 divcan3.3 divcan3 eqtr))
+thm (divcan3z () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr A (x.) B) (/) A) B)) (A A 0 =/= A 1 if x. B opreq1 A A 0 =/= A 1 if = id / opreq12d B eqeq1d divcan3.1 1cn A 0 =/= keepel divcan3.2 A elimne0 divcan3 dedth))
+thm (divcan4z () ((divcan3.1 (e. A (CC))) (divcan3.2 (e. B (CC)))) (-> (=/= A (0)) (= (opr (opr B (x.) A) (/) A) B)) (divcan3.1 divcan3.2 divcan3z divcan3.2 divcan3.1 mulcom / A opreq1i syl5eq))
+thm (divcan3t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr A (x.) B) (/) A) B)) (A A CC e. A 0 if 0 neeq1 A A CC e. A 0 if x. B opreq1 A A CC e. A 0 if = id / opreq12d B eqeq1d imbi12d B B CC e. B 0 if A CC e. A 0 if x. opreq2 / A CC e. A 0 if opreq1d B B CC e. B 0 if = id eqeq12d A CC e. A 0 if 0 =/= imbi2d 0cn A elimel 0cn B elimel divcan3z dedth2h exp 3imp))
+thm (divcan4t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= A (0))) (= (opr (opr B (x.) A) (/) A) B)) (A B axmulcom / A opreq1d A 0 =/= 3adant3 A B divcan3t eqtr3d))
+thm (div11 () ((div11.1 (e. A (CC))) (div11.2 (e. B (CC))) (div11.3 (e. C (CC))) (div11.4 (=/= C (0)))) (<-> (= (opr A (/) C) (opr B (/) C)) (= A B)) (div11.3 div11.1 div11.3 div11.4 divcl div11.2 div11.3 div11.4 divcl div11.4 mulcan div11.3 div11.1 div11.4 divcan2 div11.3 div11.2 div11.4 divcan2 eqeq12i bitr3))
+thm (dividt () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr A (/) A) (1))) (A A divrect 3expa A CC e. anidm sylanbr A recidt eqtrd))
+thm (div0t () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr (0) (/) A) (0))) (0cn 0 A divrect mp3an1 A recclt 1 / A opr mul02t syl eqtrd))
+thm (recrec () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr (1) (/) (opr (1) (/) A)) A) (recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divneq0 recid A x. opreq2i recrec.1 recrec.2 recid x. 1 / 1 / A opr opr opreq1i recrec.1 recrec.1 recrec.2 reccl recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divneq0 reccl mulass recrec.1 recrec.2 reccl 1cn recrec.1 ax1ne0 recrec.2 divneq0 reccl mulid2 3eqtr3 recrec.1 mulid1 3eqtr3))
+thm (divid () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr A (/) A) (1)) (recrec.1 recrec.1 recrec.2 divrec recrec.1 recrec.2 recid eqtr))
+thm (div0 () ((recrec.1 (e. A (CC))) (recrec.2 (=/= A (0)))) (= (opr (0) (/) A) (0)) (recrec.1 recrec.2 A div0t mp2an))
+thm (div1 () ((div1.1 (e. A (CC)))) (= (opr A (/) (1)) A) (div1.1 mulid2 div1.1 1cn div1.1 ax1ne0 divmul mpbir))
+thm (div1t () () (-> (e. A (CC)) (= (opr A (/) (1)) A)) (A A CC e. A 1 if / 1 opreq1 A A CC e. A 1 if = id eqeq12d 1cn A elimel div1 dedth))
+thm (divnegt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (-u (opr A (/) B)) (opr (-u A) (/) B))) (A 1 / B opr mulneg1t B recclt sylan2 3impb A -u B divrect A negclt syl3an1 A B divrect negeqd 3eqtr4rd))
+thm (divsubdirt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (=/= C (0))) (= (opr (opr A (-) B) (/) C) (opr (opr A (/) C) (-) (opr B (/) C)))) (A B -u C divdirt B negclt syl3anl2 A B subnegt C CC e. 3adant3 / C opreq1d C 0 =/= adantr A / C opr B / C opr subnegt A C divclt 3expa B CC e. 3adantl2 B C divclt 3expa A CC e. 3adantl1 sylanc B C divnegt 3expa A CC e. 3adantl1 A / C opr + opreq2d eqtrd 3eqtr4d))
+thm (recrect () () (-> (/\ (e. A (CC)) (=/= A (0))) (= (opr (1) (/) (opr (1) (/) A)) A)) (A A CC e. A 0 =/= /\ A 1 if 1 / opreq2 1 / opreq2d A A CC e. A 0 =/= /\ A 1 if = id eqeq12d A A CC e. A 0 =/= /\ A 1 if CC eleq1 A A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1 A CC e. A 0 =/= /\ A 1 if CC eleq1 1 A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.26i A A CC e. A 0 =/= /\ A 1 if CC eleq1 A A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1 A CC e. A 0 =/= /\ A 1 if CC eleq1 1 A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.27i recrec dedth))
+thm (rec11i () ((rec11.1 (e. A (CC))) (rec11.2 (e. B (CC))) (rec11i.3 (=/= A (0))) (rec11i.4 (=/= B (0)))) (<-> (= (opr (1) (/) A) (opr (1) (/) B)) (= A B)) (1 / A opr 1 / B opr A x. B opr x. opreq2 rec11.1 rec11.2 rec11.1 rec11i.3 reccl mul23 rec11.1 rec11i.3 recid x. B opreq1i rec11.2 mulid2 3eqtr rec11.1 rec11.2 rec11.2 rec11i.4 reccl mulass rec11.2 rec11i.4 recid A x. opreq2i rec11.1 mulid1 3eqtr eqeq12i B A eqcom bitr sylib A B 1 / opreq2 impbi))
+thm (rec11 () ((rec11.1 (e. A (CC))) (rec11.2 (e. B (CC)))) (-> (/\ (=/= A (0)) (=/= B (0))) (<-> (= (opr (1) (/) A) (opr (1) (/) B)) (= A B))) (A A 0 =/= A 1 if 1 / opreq2 1 / B opr eqeq1d A A 0 =/= A 1 if B eqeq1 bibi12d B B 0 =/= B 1 if 1 / opreq2 1 / A 0 =/= A 1 if opr eqeq2d B B 0 =/= B 1 if A 0 =/= A 1 if eqeq2 bibi12d rec11.1 1cn A 0 =/= keepel rec11.2 1cn B 0 =/= keepel A elimne0 B elimne0 rec11i dedth2h))
+thm (divmuldivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (x.) C) (/) (opr B (x.) D)))) (B x. D opr A / B opr x. C / D opr opr divcan3t B D axmulcl C CC e. adantrl A CC e. adantll B 0 =/= D 0 =/= /\ adantr A / B opr C / D opr axmulcl A B divclt 3expa C D divclt 3expa syl2an an4s B D muln0bt biimpd C CC e. adantrl A CC e. adantll imp syl3anc B A / B opr D C / D opr mul4t A CC e. B CC e. B 0 =/= 3simp2 A B divclt jca 3expa C CC e. D CC e. pm3.27 D 0 =/= adantr C D divclt 3expa jca syl2an B A divcan2t 3exp imp ancoms imp D C divcan2t 3com12 3expa x. opreqan12d eqtr3d an4s / B x. D opr opreq1d eqtr3d))
+thm (divmul13t () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr C (/) B) (x.) (opr A (/) D)))) (A C axmulcom D CC e. adantrr B CC e. adantlr / B x. D opr opreq1d B 0 =/= D 0 =/= /\ adantr A B C D divmuldivt C B A D divmuldivt A CC e. B CC e. ancom C CC e. D CC e. /\ anbi1i B CC e. A CC e. /\ C CC e. D CC e. /\ ancom C CC e. D CC e. B CC e. A CC e. an42 3bitr sylanb 3eqtr4d))
+thm (divmul24t () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (/) D) (x.) (opr C (/) B)))) (B D axmulcom C CC e. adantrl A CC e. adantll A x. C opr / opreq2d B 0 =/= D 0 =/= /\ adantr A B C D divmuldivt A D C B divmuldivt A CC e. B CC e. C CC e. D CC e. an42 A CC e. C CC e. D CC e. B CC e. an4 bitr B 0 =/= D 0 =/= ancom syl2anb 3eqtr4d))
+thm (divadddivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (=/= B (0)) (=/= D (0)))) (= (opr (opr A (/) B) (+) (opr C (/) D)) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (/) (opr B (x.) D)))) (B D muln0bt C CC e. adantrl A CC e. adantll A x. D opr B x. C opr B x. D opr divdirt exp A D axmulcl C CC e. adantrl B CC e. adantlr B C axmulcl D CC e. adantrr A CC e. adantll B D axmulcl C CC e. adantrl A CC e. adantll syl3anc sylbid imp D dividt B 0 =/= adantrl C CC e. adantll A CC e. B CC e. /\ adantll A / B opr x. opreq2d A B D D divmuldivt C CC e. D CC e. pm3.27 C CC e. D CC e. pm3.27 jca sylanl2 A B divclt 3expa A / B opr ax1id syl D 0 =/= adantrr C CC e. D CC e. /\ adantlr 3eqtr3d B dividt D 0 =/= adantrr A CC e. adantll C CC e. D CC e. /\ adantlr x. C / D opr opreq1d B B C D divmuldivt A CC e. B CC e. pm3.27 A CC e. B CC e. pm3.27 jca C CC e. D CC e. /\ anim1i sylan C D divclt 3expa C / D opr mulid2t syl B 0 =/= adantrl A CC e. B CC e. /\ adantll 3eqtr3d + opreq12d eqtr2d))
+thm (divdivdivt () () (-> (/\ (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (CC)))) (/\/\ (=/= B (0)) (=/= C (0)) (=/= D (0)))) (= (opr (opr A (/) B) (/) (opr C (/) D)) (opr (opr A (x.) D) (/) (opr B (x.) C)))) (D / C opr A / B opr axmulcom D C divclt 3com12 3expa D 0 =/= adantrr B 0 =/= adantrl A CC e. B CC e. /\ adantll A B divclt 3expa C 0 =/= D 0 =/= /\ adantrr C CC e. D CC e. /\ adantlr sylanc A B D C divmuldivt C CC e. D CC e. pm3.22 A CC e. B CC e. /\ anim2i C 0 =/= D 0 =/= pm3.26 B 0 =/= anim2i syl2an eqtrd C / D opr x. opreq2d C CC e. D CC e. pm3.22 ancli C 0 =/= D 0 =/= pm3.22 B 0 =/= adantl anim12i A CC e. B CC e. /\ adantll C D D C divmuldivt syl C D axmulcom A CC e. B CC e. /\ B 0 =/= C 0 =/= D 0 =/= /\ /\ ad2antlr / D x. C opr opreq1d D x. C opr dividt D C axmulcl ancoms A CC e. B CC e. /\ B 0 =/= C 0 =/= D 0 =/= /\ /\ ad2antlr D C muln0bt C 0 =/= D 0 =/= ancom syl5bb ancoms biimpa B 0 =/= adantrl A CC e. B CC e. /\ adantll sylanc 3eqtrd x. A / B opr opreq1d C / D opr D / C opr A / B opr axmulass C D divclt 3expa C 0 =/= adantrl B 0 =/= adantrl A CC e. B CC e. /\ adantll D C divclt 3com12 3expa D 0 =/= adantrr B 0 =/= adantrl A CC e. B CC e. /\ adantll A B divclt 3expa C 0 =/= D 0 =/= /\ adantrr C CC e. D CC e. /\ adantlr syl3anc A B divclt 3expa C 0 =/= D 0 =/= /\ adantrr C CC e. D CC e. /\ adantlr A / B opr mulid2t syl 3eqtr3d eqtr3d A / B opr C / D opr A x. D opr / B x. C opr opr divmult A B divclt 3expa C 0 =/= D 0 =/= /\ adantrr C CC e. D CC e. /\ adantlr C D divclt 3expa C 0 =/= adantrl B 0 =/= adantrl A CC e. B CC e. /\ adantll A x. D opr B x. C opr divclt A D axmulcl C CC e. adantrl B CC e. adantlr B 0 =/= C 0 =/= D 0 =/= /\ /\ adantr B C axmulcl D CC e. adantrr A CC e. adantll B 0 =/= C 0 =/= D 0 =/= /\ /\ adantr B C muln0bt biimpd C 0 =/= D 0 =/= pm3.26 sylan2i D CC e. adantrr A CC e. adantll imp syl3anc 3jca C D divneq0bt biimpd 3exp com34 imp43 B 0 =/= adantrl A CC e. B CC e. /\ adantll sylanc mpbird B 0 =/= C 0 =/= D 0 =/= 3anass sylan2b))
+thm (divmuldiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr A (x.) C) (/) (opr B (x.) D))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divmuldiv.6 pm3.2i A B C D divmuldivt mp2an))
+thm (divmul13 () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (x.) (opr C (/) D)) (opr (opr C (/) B) (x.) (opr A (/) D))) (divmuldiv.3 divmuldiv.1 mulcom / B x. D opr opreq1i divmuldiv.3 divmuldiv.2 divmuldiv.1 divmuldiv.4 divmuldiv.5 divmuldiv.6 divmuldiv divmuldiv.1 divmuldiv.2 divmuldiv.3 divmuldiv.4 divmuldiv.5 divmuldiv.6 divmuldiv 3eqtr4r))
+thm (divadddiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0)))) (= (opr (opr A (/) B) (+) (opr C (/) D)) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (/) (opr B (x.) D))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divmuldiv.6 pm3.2i A B C D divadddivt mp2an))
+thm (divdivdiv () ((divmuldiv.1 (e. A (CC))) (divmuldiv.2 (e. B (CC))) (divmuldiv.3 (e. C (CC))) (divmuldiv.4 (e. D (CC))) (divmuldiv.5 (=/= B (0))) (divmuldiv.6 (=/= D (0))) (divdivdiv.7 (=/= C (0)))) (= (opr (opr A (/) B) (/) (opr C (/) D)) (opr (opr A (x.) D) (/) (opr B (x.) C))) (divmuldiv.1 divmuldiv.2 pm3.2i divmuldiv.3 divmuldiv.4 pm3.2i pm3.2i divmuldiv.5 divdivdiv.7 divmuldiv.6 3pm3.2i A B C D divdivdivt mp2an))
+thm (recdivt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (=/= A (0)) (=/= B (0)))) (= (opr (1) (/) (opr A (/) B)) (opr B (/) A))) (1cn 1cn pm3.2i 1 1 A B divdivdivt ax1ne0 A 0 =/= B 0 =/= /\ jctl 1 0 =/= A 0 =/= B 0 =/= 3anass sylibr sylan2 mpanl1 1cn div1 / A / B opr opreq1i A CC e. B CC e. /\ A 0 =/= B 0 =/= /\ /\ a1i B mulid2t A mulid2t / opreqan12rd A 0 =/= B 0 =/= /\ adantr 3eqtr3d))
+thm (divdiv23t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (/\ (=/= B (0)) (=/= C (0)))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B))) (A 1 / B opr C div23t A CC e. B CC e. C CC e. 3simp1 B 0 =/= C 0 =/= /\ adantr B recclt A CC e. B CC e. C CC e. 3simp2 sylan C 0 =/= adantrr A CC e. B CC e. C CC e. 3simp3 B 0 =/= C 0 =/= /\ adantr 3jca B 0 =/= C 0 =/= pm3.27 A CC e. B CC e. C CC e. /\/\ adantl sylanc A B divrect 3expa C CC e. 3adantl3 C 0 =/= adantrr / C opreq1d A / C opr B divrect A C divclt 3expa B CC e. 3adantl2 B 0 =/= adantrl A CC e. B CC e. C CC e. 3simp2 B 0 =/= C 0 =/= /\ adantr B 0 =/= C 0 =/= pm3.26 A CC e. B CC e. C CC e. /\/\ adantl syl3anc 3eqtr4d))
+thm (divdiv23 () ((divdiv23.1 (e. A (CC))) (divdiv23.2 (e. B (CC))) (divdiv23.3 (e. C (CC))) (divdiv23.4 (=/= B (0))) (divdiv23.5 (=/= C (0)))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B)) (divdiv23.1 divdiv23.2 divdiv23.4 reccl divdiv23.3 divdiv23.5 div23 divdiv23.1 divdiv23.2 divdiv23.4 divrec / C opreq1i divdiv23.1 divdiv23.3 divdiv23.5 divcl divdiv23.2 divdiv23.4 divrec 3eqtr4))
+thm (divdiv23z () ((divdiv23.1 (e. A (CC))) (divdiv23.2 (e. B (CC))) (divdiv23.3 (e. C (CC)))) (-> (/\ (=/= B (0)) (=/= C (0))) (= (opr (opr A (/) B) (/) C) (opr (opr A (/) C) (/) B))) (divdiv23.1 divdiv23.2 divdiv23.3 3pm3.2i A B C divdiv23t mpan))
+thm (redivcl ((B x)) ((redivcl.1 (e. A (RR))) (redivcl.2 (e. B (RR))) (redivcl.3 (=/= B (0)))) (e. (opr A (/) B) (RR)) (redivcl.1 recn redivcl.2 recn redivcl.3 divrec redivcl.1 redivcl.2 redivcl.3 B x axrrecex mp2an x RR B x. x cv opr 1 = df-rex x cv recnt x cv x cv CC e. x cv 1 if 1 / B opr eqeq2 x cv x cv CC e. x cv 1 if B x. opreq2 1 eqeq1d bibi12d 1cn redivcl.2 recn 1cn x cv elimel redivcl.3 divmul dedth syl x cv RR 1 / B opr eleq1a sylbird imp x 19.23aiv sylbi ax-mp remulcl eqeltr))
+thm (redivclz () ((redivclz.1 (e. A (RR))) (redivclz.2 (e. B (RR)))) (-> (=/= B (0)) (e. (opr A (/) B) (RR))) (B B 0 =/= B 1 if A / opreq2 RR eleq1d redivclz.1 redivclz.2 ax1re B 0 =/= keepel B elimne0 redivcl dedth))
+thm (redivclt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (=/= B (0))) (e. (opr A (/) B) (RR))) (A A RR e. A 0 if / B opreq1 RR eleq1d B 0 =/= imbi2d B B RR e. B 0 if 0 neeq1 B B RR e. B 0 if A RR e. A 0 if / opreq2 RR eleq1d imbi12d 0re A elimel 0re B elimel redivclz dedth2h exp 3imp))
+thm (rereccl () ((rereccl.1 (e. A (RR))) (rereccl.2 (=/= A (0)))) (e. (opr (1) (/) A) (RR)) (ax1re rereccl.1 rereccl.2 redivcl))
+thm (rerecclz () ((rerecclz.1 (e. A (RR)))) (-> (=/= A (0)) (e. (opr (1) (/) A) (RR))) (ax1re rerecclz.1 redivclz))
+thm (rerecclt () () (-> (/\ (e. A (RR)) (=/= A (0))) (e. (opr (1) (/) A) (RR))) (ax1re 1 A redivclt mp3an1))
+thm (pnfxr () () (e. (+oo) (RR*)) (df-pnf axcnex eqeltr -oo pri1 +oo +oo -oo {,} e. +oo RR e. olc ax-mp +oo RR +oo -oo {,} elun mpbir df-xr eleqtrr))
+thm (mnfxr () () (e. (-oo) (RR*)) (df-mnf CC snex eqeltr +oo pri2 -oo +oo -oo {,} e. -oo RR e. olc ax-mp -oo RR +oo -oo {,} elun mpbir df-xr eleqtrr))
+thm (ltxrt () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<) B) (\/ (\/ (/\ (/\ (e. A (RR)) (e. B (RR))) (br A (<RR) B)) (/\ (= A (-oo)) (= B (+oo)))) (\/ (/\ (e. A (RR)) (= B (+oo))) (/\ (= A (-oo)) (e. B (RR))))))) (A <RR B df-br bicomi B RR* e. a1i B RR* A RR RR opelxpg anbi12d A B <,> <RR RR RR X. elin A RR e. B RR e. /\ A <RR B br ancom 3bitr4g A RR* e. adantl pnfxr A RR* B RR* +oo RR* -oo opthgg mp3an3 A B opex -oo +oo <,> elsnc syl5bb orbi12d A B <,> <RR RR RR X. i^i -oo +oo <,> {} elun syl5bb B RR* A RR +oo {} opelxpg B RR* +oo elsncg A RR e. anbi2d bitrd A RR* e. adantl B RR* A -oo {} RR opelxpg A RR* -oo elsncg B RR e. anbi1d sylan9bbr orbi12d A B <,> RR +oo {} X. -oo {} RR X. elun syl5bb orbi12d A < B df-br df-ltxr A B <,> eleq2i A B <,> <RR RR RR X. i^i -oo +oo <,> {} u. RR +oo {} X. -oo {} RR X. u. elun 3bitr syl5bb))
+thm (pnfnre () () (e/ (+oo) (RR)) (CC eirr df-pnf CC eleq1i mtbir axresscn +oo sseli mto +oo RR df-nel mpbir))
+thm (minfnre () () (e/ (-oo) (RR)) (axcnex snid CC CC {} en2lp CC CC {} e. CC {} CC e. imnan mpbir ax-mp df-mnf CC eleq1i mtbir axresscn -oo sseli mto -oo RR df-nel mpbir))
+thm (ressxr () () (C_ (RR) (RR*)) (RR +oo -oo {,} ssun1 df-xr sseqtr4))
+thm (rexrt () () (-> (e. A (RR)) (e. A (RR*))) (ressxr A sseli))
+thm (ltxrltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br A (<RR) B))) (A B ltxrt A rexrt B rexrt syl2an A RR e. B RR e. /\ A <RR B br ibar pnfnre +oo RR df-nel mpbi B +oo RR eleq1 mtbiri con2i A -oo = intnand A RR e. adantl pnfnre +oo RR df-nel mpbi B +oo RR eleq1 mtbiri con2i A RR e. intnand minfnre -oo RR df-nel mpbi A -oo RR eleq1 mtbiri con2i B RR e. intnanrd anim12i A RR e. B +oo = /\ A -oo = B RR e. /\ ioran sylibr ancoms jca A -oo = B +oo = /\ A RR e. B +oo = /\ A -oo = B RR e. /\ \/ ioran sylibr A -oo = B +oo = /\ A RR e. B +oo = /\ A -oo = B RR e. /\ \/ \/ A RR e. B RR e. /\ A <RR B br /\ biorf syl bitr2d A RR e. B RR e. /\ A <RR B br /\ A -oo = B +oo = /\ A RR e. B +oo = /\ A -oo = B RR e. /\ \/ orass A RR e. B RR e. /\ A <RR B br /\ A -oo = B +oo = /\ A RR e. B +oo = /\ A -oo = B RR e. /\ \/ \/ orcom bitr syl5bb bitrd))
+thm (xrlenltt () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<_) B) (-. (br B (<) A)))) (A RR* B RR* opelxpi df-le A B <,> eleq2i A B <,> RR* RR* X. < `' eldif bitr baib syl A <_ B df-br syl5bb A RR* B RR* < opelcnvg B < A df-br syl6rbbr negbid bitr4d))
+thm (axlttri () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (-. (\/ (= A B) (br B (<) A))))) (A B pre-axlttri A B ltxrltt B A ltxrltt ancoms A B = orbi2d negbid 3bitr4d))
+thm (axlttrn () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C))) (A B C pre-axlttrn A B ltxrltt C RR e. 3adant3 B C ltxrltt A RR e. 3adant1 anbi12d A C ltxrltt B RR e. 3adant2 3imtr4d))
+thm (axltadd () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B)))) (A B C pre-axltadd A B ltxrltt C RR e. 3adant3 C + A opr C + B opr ltxrltt C A axaddrcl C B axaddrcl syl2an 3impdi 3coml 3imtr4d))
+thm (axmulgt0 () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (x.) B)))) (A B pre-axmulgt0 0re 0 A ltxrltt mpan 0re 0 B ltxrltt mpan bi2anan9 A B axmulrcl 0re 0 A x. B opr ltxrltt mpan syl 3imtr4d))
+thm (axsup ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (br (cv y) (<) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A x y z pre-axsup 3expa exp y cv x cv ltxrltt A RR y cv ssel2 sylan an1rs biraldva birexdva A {/} =/= adantr x cv y cv ltxrltt ancoms A RR y cv ssel2 sylan an1rs negbid biraldva y cv x cv ltxrltt ancoms A RR C_ adantll y cv z cv ltxrltt ancoms A RR z cv ssel2 sylan an1rs birexdva x cv RR e. adantlr imbi12d biraldva anbi12d birexdva A {/} =/= adantr 3imtr4d imp 3impa))
+thm (lenltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (-. (br B (<) A)))) (A B xrlenltt A rexrt B rexrt syl2an))
+thm (ltnlet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (-. (br B (<_) A)))) (B A lenltt ancoms bicon2d))
+thm (ltso ((x y) (x z) (y z)) () (Or (<) (RR)) (x cv y cv axlttri z cv RR e. 3adant3 x cv y cv z cv axlttrn jca so))
+thm (lttri2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (-. (= A B)) (\/ (br A (<) B) (br B (<) A)))) (ltso < RR A B sotrieq mpan bicomd bicon1d))
+thm (lttri3t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (-. (br A (<) B)) (-. (br B (<) A))))) (ltso < RR A B sotrieq2 mpan))
+thm (ltnet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (-. (= A B)))) (A B lttri2t A < B br B < A br orc syl5bir))
+thm (letri3t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (br A (<_) B) (br B (<_) A)))) (A B lttri3t B < A br -. A < B br -. ancom syl6bbr A B lenltt B A lenltt ancoms anbi12d bitr4d))
+thm (leloet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (\/ (br A (<) B) (= A B)))) (A B lenltt B A axlttri ancoms bicon2d B A eqcom A < B br orbi1i A B = A < B br orcom bitr syl5rbbr bitrd))
+thm (eqleltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (= A B) (/\ (br A (<_) B) (-. (br A (<) B))))) (A B letri3t B A lenltt ancoms A <_ B br anbi2d bitrd))
+thm (ltlet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (br A (<_) B))) (A B leloet A < B br A B = orc syl5bir))
+thm (leltnet () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (<-> (br A (<) B) (-. (= A B)))) (A B lttri3t A < B br -. B < A br -. pm3.26 syl6bi A <_ B br adantr A B leloet biimpa ord impbid bicon2d 3impa))
+thm (leltnetOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<_) B) (<-> (br A (<) B) (-. (= A B))))) (A B lttri3t A < B br -. B < A br -. pm3.26 syl6bi A <_ B br adantr A B leloet biimpa ord impbid bicon2d exp))
+thm (ltlent () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (/\ (br A (<_) B) (-. (= A B))))) (A B ltlet A B ltnet jcad A B leloet A < B br A B = -. ax-1 A B = A < B br pm2.24 jaoi syl6bi imp3a impbid))
+thm (lelttrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<_) B) (br B (<) C)) (br A (<) C))) (A B leloet C RR e. 3adant3 A B C axlttrn exp3a A B < C breq1 biimprd A RR e. B RR e. C RR e. /\/\ a1i jaod sylbid imp3a))
+thm (ltletrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<) B) (br B (<_) C)) (br A (<) C))) (B C leloet A RR e. 3adant1 A B C axlttrn exp3a com23 B C A < breq2 biimpd A RR e. B RR e. C RR e. /\/\ a1i jaod sylbid com23 imp3a))
+thm (letrt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br A (<_) B) (br B (<_) C)) (br A (<_) C))) (B C leloet A RR e. 3adant1 A <_ B br adantr A B C lelttrt A C ltlet B RR e. 3adant2 syld exp3a imp B C A <_ breq2 biimpcd A RR e. B RR e. C RR e. /\/\ adantl jaod sylbid exp imp3a))
+thm (letrd () ((letrd.1 (-> ph (e. A (RR)))) (letrd.2 (-> ph (e. B (RR)))) (letrd.3 (-> ph (e. C (RR)))) (letrd.4 (-> ph (br A (<_) B))) (letrd.5 (-> ph (br B (<_) C)))) (-> ph (br A (<_) C)) (letrd.4 letrd.5 letrd.1 letrd.2 letrd.3 3jca A B C letrt syl mp2and))
+thm (ltnrt () () (-> (e. A (RR)) (-. (br A (<) A))) (ltso < RR A sonr mpan))
+thm (leidt () () (-> (e. A (RR)) (br A (<_) A)) (A eqid A < A br -. a1i orri A A leloet mpbiri anidms))
+thm (ltnsymt () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (br A (<) B) (-. (br B (<) A)))) (A B axlttri A B = B < A br pm2.46 syl6bi))
+thm (ltnsym2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (-. (/\ (br A (<) B) (br B (<) A)))) (A B ltnsymt A < B br B < A br imnan sylib))
+thm (elxr () () (<-> (e. A (RR*)) (\/\/ (e. A (RR)) (= A (+oo)) (= A (-oo)))) (df-xr A eleq2i A RR +oo -oo {,} elun pnfxr elisseti mnfxr elisseti A elpr2 A RR e. orbi2i A RR e. A +oo = A -oo = 3orass bitr4 3bitr))
+thm (pnfnemnf () () (=/= (+oo) (-oo)) (CC eirr axcnex snid CC CC {} CC eleq2 mpbiri mto df-pnf df-mnf eqeq12i mtbir +oo -oo df-ne mpbir))
+thm (renepnft () () (-> (e. A (RR)) (=/= A (+oo))) (pnfnre +oo RR df-nel mpbi A RR +oo nelneq mpan2 A +oo df-ne sylibr))
+thm (xrltnrt () () (-> (e. A (RR*)) (-. (br A (<) A))) (A elxr A ltnrt pnfnre +oo RR df-nel mpbi +oo RR e. intnan +oo <RR +oo br intnanr pnfnemnf +oo -oo df-ne mpbi +oo +oo = intnanr pm3.2ni pnfnre +oo RR df-nel mpbi +oo +oo = intnanr pnfnre +oo RR df-nel mpbi +oo -oo = intnan pm3.2ni pm3.2ni pnfxr pnfxr +oo +oo ltxrt mp2an mtbir A +oo A +oo < breq12 anidms mtbiri minfnre -oo RR df-nel mpbi -oo RR e. intnan -oo <RR -oo br intnanr pnfnemnf +oo -oo necom mpbi -oo +oo df-ne mpbi -oo -oo = intnan pm3.2ni minfnre -oo RR df-nel mpbi -oo +oo = intnanr minfnre -oo RR df-nel mpbi -oo -oo = intnan pm3.2ni pm3.2ni mnfxr mnfxr -oo -oo ltxrt mp2an mtbir A -oo A -oo < breq12 anidms mtbiri 3jaoi sylbi))
+thm (ltpnft () () (-> (e. A (RR)) (br A (<) (+oo))) (+oo eqid A RR e. jctr A RR e. +oo +oo = /\ A -oo = +oo RR e. /\ orc A RR e. +oo +oo = /\ A -oo = +oo RR e. /\ \/ A RR e. +oo RR e. /\ A <RR +oo br /\ A -oo = +oo +oo = /\ \/ olc 3syl A rexrt pnfxr A +oo ltxrt mpan2 syl mpbird))
+thm (mnfltt () () (-> (e. A (RR)) (br (-oo) (<) A)) (-oo eqid A RR e. jctl -oo -oo = A RR e. /\ -oo RR e. A +oo = /\ olc -oo RR e. A +oo = /\ -oo -oo = A RR e. /\ \/ -oo RR e. A RR e. /\ -oo <RR A br /\ -oo -oo = A +oo = /\ \/ olc 3syl A rexrt mnfxr -oo A ltxrt mpan syl mpbird))
+thm (mnfltpnf () () (br (-oo) (<) (+oo)) (-oo eqid +oo eqid -oo -oo = +oo +oo = /\ -oo RR e. +oo RR e. /\ -oo <RR +oo br /\ olc mp2an -oo RR e. +oo RR e. /\ -oo <RR +oo br /\ -oo -oo = +oo +oo = /\ \/ -oo RR e. +oo +oo = /\ -oo -oo = +oo RR e. /\ \/ orc ax-mp mnfxr pnfxr -oo +oo ltxrt mp2an mpbir))
+thm (pnfnltt () () (-> (e. A (RR*)) (-. (br (+oo) (<) A))) (pnfnre +oo RR df-nel mpbi A RR e. intnanr +oo <RR A br intnanr pnfnemnf +oo -oo df-ne mpbi A +oo = intnanr pm3.2ni pnfnre +oo RR df-nel mpbi A +oo = intnanr pnfnemnf +oo -oo df-ne mpbi A RR e. intnanr pm3.2ni pm3.2ni pnfxr +oo A ltxrt mpan mtbiri))
+thm (nltmnft () () (-> (e. A (RR*)) (-. (br A (<) (-oo)))) (minfnre -oo RR df-nel mpbi A RR e. intnan A <RR -oo br intnanr pnfnemnf +oo -oo necom mpbi -oo +oo df-ne mpbi A -oo = intnan pm3.2ni pnfnemnf +oo -oo necom mpbi -oo +oo df-ne mpbi A RR e. intnan minfnre -oo RR df-nel mpbi A -oo = intnan pm3.2ni pm3.2ni mnfxr A -oo ltxrt mpan2 mtbiri))
+thm (xrltnsymt () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (-> (br A (<) B) (-. (br B (<) A)))) (A B ltnsymt A rexrt A pnfnltt syl B +oo = adantr B +oo < A breq1 A RR e. adantl mtbird A < B br a1d A rexrt A nltmnft syl B -oo = adantr B -oo A < breq2 A RR e. adantl mtbird B < A br -. pm2.21d 3jaodan B pnfnltt A +oo = adantl A +oo < B breq1 B RR* e. adantr mtbird B < A br -. pm2.21d B elxr sylan2br B rexrt B nltmnft syl A -oo = adantl A -oo B < breq2 B RR e. adantr mtbird A < B br a1d mnfxr -oo pnfnltt ax-mp B +oo A -oo < breq12 mtbiri ancoms A < B br a1d mnfxr -oo xrltnrt ax-mp A -oo B -oo < breq12 mtbiri B < A br -. pm2.21d 3jaodan 3jaoian A elxr B elxr syl2anb))
+thm (xrlttrit () () (-> (/\ (e. A (RR*)) (e. B (RR*))) (<-> (br A (<) B) (-. (\/ (= A B) (br B (<) A))))) (A xrltnrt A B = adantr A B A < breq2 A RR* e. adantl mtbid exp B RR* e. adantr B A xrltnsymt ancoms jaod A B axlttri biimprd con1d A ltpnft B +oo = adantr B +oo A < breq2 A RR e. adantl mpbird A < B br A B = B < A br \/ pm2.24 syl A mnfltt B -oo = adantr B -oo < A breq1 A RR e. adantl mpbird B < A br A B = olc syl A < B br -. a1d 3jaodan B ltpnft A +oo = adantl A +oo B < breq2 B RR e. adantr mpbird B < A br A B = olc syl A < B br -. a1d A +oo B eqtr3t A B = B < A br orc syl A < B br -. a1d mnfltpnf B -oo A +oo < breq12 mpbiri ancoms B < A br A B = olc syl A < B br -. a1d 3jaodan B mnfltt A -oo = adantl A -oo < B breq1 B RR e. adantr mpbird A < B br A B = B < A br \/ pm2.24 syl mnfltpnf A -oo B +oo < breq12 mpbiri A < B br A B = B < A br \/ pm2.24 syl A -oo B eqtr3t A B = B < A br orc syl A < B br -. a1d 3jaodan 3jaoian A elxr B elxr syl2anb impbid bicon2d))
+thm (xrlttrt () () (-> (/\/\ (e. A (RR*)) (e. B (RR*)) (e. C (RR*))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C))) (A B C axlttrn 3expa an1rs C rexrt C pnfnltt syl B +oo = adantr B +oo < C breq1 C RR e. adantl mtbird A < C br pm2.21d A RR e. adantll A < B br adantld A rexrt A nltmnft syl B -oo = adantr B -oo A < breq2 A RR e. adantl mtbird A < C br pm2.21d C RR e. adantlr B < C br adantrd 3jaodan B elxr sylan2b an1rs A ltpnft C +oo = adantr C +oo A < breq2 A RR e. adantl mpbird B RR* e. adantlr A < B br B < C br /\ a1d B nltmnft C -oo = adantr C -oo B < breq2 B RR* e. adantl mtbird A < C br pm2.21d A < B br adantld A RR e. adantll 3jaodan anasss B pnfnltt A +oo = adantl A +oo < B breq1 B RR* e. adantr mtbird A < C br pm2.21d B < C br adantrd C RR e. C +oo = C -oo = \/\/ adantrr C mnfltt A -oo = adantl A -oo < C breq1 C RR e. adantr mpbird A < B br B < C br /\ a1d B RR* e. adantlr mnfltpnf A -oo C +oo < breq12 mpbiri A < B br B < C br /\ a1d B RR* e. adantlr B nltmnft C -oo = adantr C -oo B < breq2 B RR* e. adantl mtbird A < C br pm2.21d A < B br adantld A -oo = adantll 3jaodan anasss 3jaoian 3impb C elxr syl3an3b A elxr syl3an1b))
+thm (xrltso ((x y) (x z) (y z)) () (Or (<) (RR*)) (x cv y cv xrlttrit z cv RR* e. 3adant3 x cv y cv z cv xrlttrt jca so))
+thm (eqlet () () (-> (/\ (e. A (RR)) (= A B)) (br A (<_) B)) (A B A <_ breq2 biimpac A leidt sylan))
+thm (lttri2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (-. (= A B)) (\/ (br A (<) B) (br B (<) A))) (lt.1 lt.2 A B lttri2t mp2an))
+thm (lttri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (= A B) (/\ (-. (br A (<) B)) (-. (br B (<) A)))) (lt.1 lt.2 A B lttri3t mp2an))
+thm (letri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (= A B) (/\ (br A (<_) B) (br B (<_) A))) (lt.1 lt.2 A B letri3t mp2an))
+thm (leloe () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (\/ (br A (<) B) (= A B))) (lt.1 lt.2 A B leloet mp2an))
+thm (ltlen () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (/\ (br A (<_) B) (-. (= A B)))) (lt.1 lt.2 A B ltlent mp2an))
+thm (ltnsym () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (-. (br B (<) A))) (lt.1 lt.2 A B ltnsymt mp2an))
+thm (lenlt () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (-. (br B (<) A))) (lt.1 lt.2 A B lenltt mp2an))
+thm (ltnle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (-. (br B (<_) A))) (lt.2 lt.1 lenlt bicon2i))
+thm (ltle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (br A (<_) B)) (lt.1 lt.2 A B ltlet mp2an))
+thm (ltlei () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (ltlei.1 (br A (<) B))) (br A (<_) B) (ltlei.1 lt.1 lt.2 ltle ax-mp))
+thm (eqle () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (= A B) (br A (<_) B)) (lt.1 lt.2 letri3 pm3.26bd))
+thm (ltne () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (br A (<) B) (-. (= A B))) (lt.1 lt.2 A B ltnet mp2an))
+thm (letri () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (\/ (br A (<_) B) (br B (<_) A)) (lt.2 lt.1 ltnle lt.2 lt.1 ltle sylbir orri))
+thm (lttr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<) B) (br B (<) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C axlttrn mp3an))
+thm (lelttr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<_) B) (br B (<) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C lelttrt mp3an))
+thm (ltletr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<) B) (br B (<_) C)) (br A (<) C)) (lt.1 lt.2 lt.3 A B C ltletrt mp3an))
+thm (letr () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (-> (/\ (br A (<_) B) (br B (<_) C)) (br A (<_) C)) (lt.1 lt.2 lt.3 A B C letrt mp3an))
+thm (le2tri3 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (/\/\ (br A (<_) B) (br B (<_) C) (br C (<_) A)) (/\/\ (= A B) (= B C) (= C A))) (lt.1 lt.2 letri3 biimpr lt.2 lt.3 lt.1 letr sylan2 3impb lt.2 lt.3 letri3 biimpr lt.3 lt.1 lt.2 letr sylan2 3impb 3comr lt.1 lt.3 letri3 biimpr eqcomd lt.1 lt.2 lt.3 letr sylan 3impa 3jca lt.1 lt.2 eqle lt.2 lt.3 eqle lt.3 lt.1 eqle 3anim123i impbi))
+thm (ltadd2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B))) (lt.1 lt.2 lt.3 A B C axltadd mp3an lt.3 lt.1 readdcl lt.3 lt.2 readdcl lt.3 renegcl C + A opr C + B opr C -u axltadd mp3an lt.3 renegcl recn lt.3 recn addcom lt.3 recn negid eqtr + A opreq1i lt.3 renegcl recn lt.3 recn lt.1 recn addass lt.1 recn addid2 3eqtr3 lt.3 renegcl recn lt.3 recn addcom lt.3 recn negid eqtr + B opreq1i lt.3 renegcl recn lt.3 recn lt.2 recn addass lt.2 recn addid2 3eqtr3 3brtr3g impbi))
+thm (ltadd1 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<) B) (br (opr A (+) C) (<) (opr B (+) C))) (lt.1 lt.2 lt.3 ltadd2 lt.3 recn lt.1 recn addcom lt.3 recn lt.2 recn addcom < breq12i bitr))
+thm (leadd1 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<_) B) (br (opr A (+) C) (<_) (opr B (+) C))) (lt.2 lt.1 lt.3 ltadd1 negbii lt.1 lt.2 lenlt lt.1 lt.3 readdcl lt.2 lt.3 readdcl lenlt 3bitr4))
+thm (leadd2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br A (<_) B) (br (opr C (+) A) (<_) (opr C (+) B))) (lt.1 lt.2 lt.3 leadd1 lt.1 recn lt.3 recn addcom lt.2 recn lt.3 recn addcom <_ breq12i bitr))
+thm (ltsubadd () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr C (+) B))) (lt.1 recn lt.2 recn subneg < C breq1i lt.1 lt.2 renegcl readdcl lt.3 lt.2 ltadd1 lt.1 recn lt.2 renegcl recn lt.2 recn addass lt.2 renegcl recn lt.2 recn addcom lt.2 recn negid eqtr A + opreq2i lt.1 recn addid1 3eqtr < C + B opr breq1i 3bitr))
+thm (lesubadd () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr C (+) B))) (lt.1 lt.2 lt.3 ltsubadd lt.1 recn lt.2 recn lt.3 recn subadd lt.2 recn lt.3 recn addcom A eqeq1i C + B opr A eqcom bitr bitr orbi12i lt.1 lt.2 resubcl lt.3 leloe lt.1 lt.3 lt.2 readdcl leloe 3bitr4))
+thm (lt2add () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR))) (lt.4 (e. D (RR)))) (-> (/\ (br A (<) C) (br B (<) D)) (br (opr A (+) B) (<) (opr C (+) D))) (lt.1 lt.2 readdcl lt.3 lt.2 readdcl lt.3 lt.4 readdcl lttr lt.1 lt.3 lt.2 ltadd1 lt.2 lt.4 lt.3 ltadd2 syl2anb))
+thm (le2add () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (lt.3 (e. C (RR))) (lt.4 (e. D (RR)))) (-> (/\ (br A (<_) C) (br B (<_) D)) (br (opr A (+) B) (<_) (opr C (+) D))) (lt.1 lt.2 readdcl lt.3 lt.2 readdcl lt.3 lt.4 readdcl letr lt.1 lt.3 lt.2 leadd1 lt.2 lt.4 lt.3 leadd2 syl2anb))
+thm (addgt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (+) B))) (0re lt.1 lt.1 lt.2 readdcl lttr 0re lt.2 lt.1 ltadd2 lt.1 recn addid1 < A + B opr breq1i bitr sylan2b))
+thm (addge0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (+) B))) (lt.1 lt.2 addgt0 0re lt.1 lt.2 readdcl ltle syl 0 A + B opreq1 lt.2 recn addid2 syl5eqr 0 < breq2d biimpa 0re lt.1 lt.2 readdcl ltle syl 0 B A + opreq2 lt.1 recn addid1 syl5eqr 0 < breq2d biimpac 0re lt.1 lt.2 readdcl ltle syl 0 A 0 B + opreq12 0cn addid1 syl5eqr 0re lt.1 lt.2 readdcl eqle syl ccase 0re lt.1 leloe 0re lt.2 leloe syl2anb))
+thm (addgegt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) B)) (br (0) (<) (opr A (+) B))) (0re lt.1 leloe lt.1 lt.2 addgt0 exp 0 A + B opreq1 lt.2 recn addid2 syl5eqr 0 < breq2d biimpd jaoi sylbi imp))
+thm (addgt0i () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (addgt0i.3 (br (0) (<) A)) (addgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (+) B)) (addgt0i.3 addgt0i.4 lt.1 lt.2 addgt0 mp2an))
+thm (add20 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (+) B) (0)) (/\ (= A (0)) (= B (0))))) (lt.1 lt.2 readdcl 0re lttri3 pm3.27bd lt.1 lt.2 addgt0 nsyl3 A 0 = B 0 = /\ pm2.21d 0 A + B opreq1 lt.2 recn addid2 syl5eqr 0 eqeq1d biimprd 0 A eqcom biimp jctild 0 B A + opreq2 lt.1 recn addid1 syl5eqr 0 eqeq1d biimprd 0 B eqcom biimp jctird ccase2 0re lt.1 leloe 0re lt.2 leloe syl2anb A 0 + B opreq1 B 0 0 + opreq2 0cn addid1 syl6eq sylan9eq 0 <_ A br 0 <_ B br /\ a1i impbid))
+thm (ltneg () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) B) (br (-u B) (<) (-u A))) (lt.1 lt.2 lt.1 renegcl lt.2 renegcl readdcl ltadd1 lt.1 recn negid + B -u opreq1i lt.1 recn lt.1 renegcl recn lt.2 renegcl recn addass lt.2 renegcl recn addid2 3eqtr3 lt.2 recn lt.1 renegcl recn lt.2 renegcl recn add12 lt.2 recn negid A -u + opreq2i lt.1 renegcl recn addid1 3eqtr < breq12i bitr))
+thm (leneg () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<_) B) (br (-u B) (<_) (-u A))) (lt.1 lt.2 ltneg lt.2 recn lt.1 recn neg11 B A eqcom bitr2 orbi12i lt.1 lt.2 leloe lt.2 renegcl lt.1 renegcl leloe 3bitr4))
+thm (ltnegcon2 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (<-> (br A (<) (-u B)) (br B (<) (-u A))) (lt.1 lt.2 renegcl ltneg lt.2 recn negneg < A -u breq1i bitr))
+thm (mulgt0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (x.) B))) (lt.1 lt.2 A B axmulgt0 mp2an))
+thm (mulge0 () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (x.) B))) (lt.1 lt.2 mulgt0 0re lt.1 lt.2 remulcl ltle syl 0 A x. B opreq1 lt.2 recn mul02 syl5eqr 0re lt.1 lt.2 remulcl eqle syl 0 B A x. opreq2 lt.1 recn mul01 syl5eqr 0re lt.1 lt.2 remulcl eqle syl ccase2 0re lt.1 leloe 0re lt.2 leloe syl2anb))
+thm (mulgt0i () ((lt.1 (e. A (RR))) (lt.2 (e. B (RR))) (mulgt0i.3 (br (0) (<) A)) (mulgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (x.) B)) (mulgt0i.3 mulgt0i.4 lt.1 lt.2 mulgt0 mp2an))
+thm (ltnr () ((lt.1 (e. A (RR)))) (-. (br A (<) A)) (lt.1 A ltnrt ax-mp))
+thm (leid () ((lt.1 (e. A (RR)))) (br A (<_) A) (lt.1 A leidt ax-mp))
+thm (gt0ne0 () ((lt.1 (e. A (RR)))) (-> (br (0) (<) A) (=/= A (0))) (0re lt.1 ltne 0 A df-ne sylibr A 0 necom sylibr))
+thm (lesub0 () ((lt.1 (e. A (RR))) (lesub0.1 (e. B (RR)))) (<-> (/\ (br (0) (<_) A) (br B (<_) (opr B (-) A))) (= A (0))) (lesub0.1 recn addid1 eqcomi lesub0.1 recn lt.1 recn subneg <_ breq12i 0re lt.1 renegcl lesub0.1 leadd2 lt.1 0re leneg neg0 <_ A -u breq1i bitr2 bitr3 bitr 0 <_ A br anbi1i 0 <_ A br B <_ B - A opr br ancom lt.1 0re letri3 3bitr4))
+thm (sqgt0 () ((lt.1 (e. A (RR)))) (-> (-. (= A (0))) (br (0) (<) (opr A (x.) A))) (lt.1 0re lttri2 lt.1 0re ltneg neg0 < A -u breq1i bitr lt.1 renegcl lt.1 renegcl mulgt0 anidms lt.1 recn lt.1 recn mul2neg syl6breq sylbi lt.1 lt.1 mulgt0 anidms jaoi sylbi))
+thm (sqge0 () ((lt.1 (e. A (RR)))) (br (0) (<_) (opr A (x.) A)) (A 0 A x. opreq2 lt.1 recn mul01 syl6req 0 A x. A opr = 0 < A x. A opr br olc syl 0re lt.1 lt.1 remulcl leloe sylibr lt.1 sqgt0 0re lt.1 lt.1 remulcl ltle syl pm2.61i))
+thm (gt0ne0i () ((gt0ne0i.1 (e. A (RR))) (gt0ne0i.2 (br (0) (<) A))) (=/= A (0)) (gt0ne0i.2 gt0ne0i.1 gt0ne0 ax-mp))
+thm (gt0ne0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (=/= A (0))) (A A RR e. A 0 if 0 < breq2 A A RR e. A 0 if 0 neeq1 imbi12d 0re A elimel gt0ne0 dedth imp))
+thm (letrit () () (-> (/\ (e. A (RR)) (e. B (RR))) (\/ (br A (<_) B) (br B (<_) A))) (A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if B <_ breq2 orbi12d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if <_ A RR e. A 0 if breq1 orbi12d 0re A elimel 0re B elimel letri dedth2h))
+thm (lelttrit () () (-> (/\ (e. A (RR)) (e. B (RR))) (\/ (br A (<_) B) (br B (<) A))) (A B eqlet A <_ B br B < A br orc syl exp B RR e. adantr A B lttri2t A B ltlet B < A br orim1d sylbid pm2.61d))
+thm (ltadd1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) B) (br (opr A (+) C) (<) (opr B (+) C)))) (A A RR e. A 0 if < B breq1 A A RR e. A 0 if + C opreq1 < B + C opr breq1d bibi12d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if + C opreq1 A RR e. A 0 if + C opr < breq2d bibi12d C C RR e. C 0 if A RR e. A 0 if + opreq2 C C RR e. C 0 if B RR e. B 0 if + opreq2 < breq12d A RR e. A 0 if < B RR e. B 0 if br bibi2d 0re A elimel 0re B elimel 0re C elimel ltadd1 dedth3h))
+thm (ltadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) B) (br (opr C (+) A) (<) (opr C (+) B)))) (A B C ltadd1t A C axaddcom A recnt C recnt syl2an B RR e. 3adant2 B C axaddcom B recnt C recnt syl2an A RR e. 3adant1 < breq12d bitrd))
+thm (leadd1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr A (+) C) (<_) (opr B (+) C)))) (A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if + C opreq1 <_ B + C opr breq1d bibi12d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if + C opreq1 A RR e. A 0 if + C opr <_ breq2d bibi12d C C RR e. C 0 if A RR e. A 0 if + opreq2 C C RR e. C 0 if B RR e. B 0 if + opreq2 <_ breq12d A RR e. A 0 if <_ B RR e. B 0 if br bibi2d 0re A elimel 0re B elimel 0re C elimel leadd1 dedth3h))
+thm (leadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr C (+) A) (<_) (opr C (+) B)))) (A B C leadd1t A C axaddcom A recnt C recnt syl2an B RR e. 3adant2 B C axaddcom B recnt C recnt syl2an A RR e. 3adant1 <_ breq12d bitrd))
+thm (addge01t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br A (<_) (opr A (+) B)))) (0re 0 B A leadd2t mp3an1 ancoms A recnt A ax0id syl B RR e. adantr <_ A + B opr breq1d bitrd))
+thm (addge02t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br A (<_) (opr B (+) A)))) (A B addge01t A B axaddcom A recnt B recnt syl2an A <_ breq2d bitrd))
+thm (addge1tOLD () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<_) B)) (br A (<_) (opr A (+) B))) (A recnt A ax0id syl B RR e. 0 <_ B br 3ad2ant1 0re 0 B A leadd2t mp3an1 ancoms biimp3a eqbrtrrd))
+thm (addge2tOLD () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<_) B)) (br A (<_) (opr B (+) A))) (A B addge1tOLD A B axaddcom A recnt B recnt syl2an 0 <_ B br 3adant3 breqtrd))
+thm (subge0t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) (opr A (-) B)) (br B (<_) A))) (B A B -u leadd1t A RR e. B RR e. pm3.27 A RR e. B RR e. pm3.26 B renegclt A RR e. adantl syl3anc B negidt A CC e. adantl A B subnegt eqcomd <_ breq12d A recnt B recnt syl2an bitr2d))
+thm (subge0 () ((subge0.1 (e. A (RR))) (subge0.2 (e. B (RR)))) (<-> (br (0) (<_) (opr A (-) B)) (br B (<_) A)) (subge0.1 subge0.2 A B subge0t mp2an))
+thm (ltsubaddt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr C (+) B)))) (A A RR e. A 0 if - B opreq1 < C breq1d A A RR e. A 0 if < C + B opr breq1 bibi12d B B RR e. B 0 if A RR e. A 0 if - opreq2 < C breq1d B B RR e. B 0 if C + opreq2 A RR e. A 0 if < breq2d bibi12d C C RR e. C 0 if A RR e. A 0 if - B RR e. B 0 if opr < breq2 C C RR e. C 0 if + B RR e. B 0 if opreq1 A RR e. A 0 if < breq2d bibi12d 0re A elimel 0re B elimel 0re C elimel ltsubadd dedth3h))
+thm (ltsubadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr B (+) C)))) (A B C ltsubaddt B C axaddcom B recnt C recnt syl2an A RR e. 3adant1 A < breq2d bitr4d))
+thm (lesubaddt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr C (+) B)))) (A A RR e. A 0 if - B opreq1 <_ C breq1d A A RR e. A 0 if <_ C + B opr breq1 bibi12d B B RR e. B 0 if A RR e. A 0 if - opreq2 <_ C breq1d B B RR e. B 0 if C + opreq2 A RR e. A 0 if <_ breq2d bibi12d C C RR e. C 0 if A RR e. A 0 if - B RR e. B 0 if opr <_ breq2 C C RR e. C 0 if + B RR e. B 0 if opreq1 A RR e. A 0 if <_ breq2d bibi12d 0re A elimel 0re B elimel 0re C elimel lesubadd dedth3h))
+thm (lesubadd2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr B (+) C)))) (A B C lesubaddt B C axaddcom B recnt C recnt syl2an A RR e. 3adant1 A <_ breq2d bitr4d))
+thm (ltaddsubt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<) C) (br A (<) (opr C (-) B)))) (C B A lesubaddt 3com13 C - B opr A lenltt C B resubclt sylan 3impa 3com13 C A + B opr lenltt A B axaddrcl sylan2 3impb 3coml 3bitr3d bicomd bicon4d))
+thm (ltaddsub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<) C) (br B (<) (opr C (-) A)))) (A B axaddcom A recnt B recnt syl2an C RR e. 3adant3 < C breq1d B A C ltaddsubt 3com12 bitrd))
+thm (leaddsubt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<_) C) (br A (<_) (opr C (-) B)))) (C B A ltsubaddt 3com13 C - B opr A ltnlet C B resubclt sylan 3impa 3com13 C A + B opr ltnlet A B axaddrcl sylan2 3impb 3coml 3bitr3d bicomd bicon4d))
+thm (leaddsub2t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (+) B) (<_) C) (br B (<_) (opr C (-) A)))) (A B axaddcom A recnt B recnt syl2an C RR e. 3adant3 <_ C breq1d B A C leaddsubt 3com12 bitrd))
+thm (sublet () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<_) C) (br (opr A (-) C) (<_) B))) (A B C lesubaddt A C B lesubadd2t 3com23 bitr4d))
+thm (ltsubadd2 () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<) C) (br A (<) (opr B (+) C))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C ltsubadd2t mp3an))
+thm (lesubadd2 () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (-) B) (<_) C) (br A (<_) (opr B (+) C))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C lesubadd2t mp3an))
+thm (ltaddsub () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (<-> (br (opr A (+) B) (<) C) (br A (<) (opr C (-) B))) (ltsubadd.1 ltsubadd.2 ltsubadd.3 A B C ltaddsubt mp3an))
+thm (ltmullem () ((ltsubadd.1 (e. A (RR))) (ltsubadd.2 (e. B (RR))) (ltsubadd.3 (e. C (RR)))) (-> (br (0) (<) C) (-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (ltsubadd.2 ltsubadd.1 resubcl ltsubadd.3 mulgt0 expcom ltsubadd.1 recn addid2 < B breq1i 0re ltsubadd.1 ltsubadd.2 ltaddsub bitr3 ltsubadd.1 ltsubadd.3 remulcl recn addid2 < B x. C opr breq1i 0re ltsubadd.1 ltsubadd.3 remulcl ltsubadd.2 ltsubadd.3 remulcl ltaddsub ltsubadd.2 recn ltsubadd.1 recn ltsubadd.3 recn subdir 0 < breq2i bitr4 bitr3 3imtr4g))
+thm (ltsub23t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br (opr A (-) B) (<) C) (br (opr A (-) C) (<) B))) (A B C ltsubaddt A C B ltsubadd2t 3com23 bitr4d))
+thm (ltsub13t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<) (opr B (-) C)) (br C (<) (opr B (-) A)))) (A C B ltaddsubt A C B ltaddsub2t bitr3d 3com23))
+thm (lt2addt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (-> (/\ (br A (<) C) (br B (<) D)) (br (opr A (+) B) (<) (opr C (+) D)))) (A A RR e. A 0 if < C breq1 B < D br anbi1d A A RR e. A 0 if + B opreq1 < C + D opr breq1d imbi12d B B RR e. B 0 if < D breq1 A RR e. A 0 if < C br anbi2d B B RR e. B 0 if A RR e. A 0 if + opreq2 < C + D opr breq1d imbi12d C C RR e. C 0 if A RR e. A 0 if < breq2 B RR e. B 0 if < D br anbi1d C C RR e. C 0 if + D opreq1 A RR e. A 0 if + B RR e. B 0 if opr < breq2d imbi12d D D RR e. D 0 if B RR e. B 0 if < breq2 A RR e. A 0 if < C RR e. C 0 if br anbi2d D D RR e. D 0 if C RR e. C 0 if + opreq2 A RR e. A 0 if + B RR e. B 0 if opr < breq2d imbi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel lt2add dedth4h))
+thm (le2addt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (-> (/\ (br A (<_) C) (br B (<_) D)) (br (opr A (+) B) (<_) (opr C (+) D)))) (A A RR e. A 0 if <_ C breq1 B <_ D br anbi1d A A RR e. A 0 if + B opreq1 <_ C + D opr breq1d imbi12d B B RR e. B 0 if <_ D breq1 A RR e. A 0 if <_ C br anbi2d B B RR e. B 0 if A RR e. A 0 if + opreq2 <_ C + D opr breq1d imbi12d C C RR e. C 0 if A RR e. A 0 if <_ breq2 B RR e. B 0 if <_ D br anbi1d C C RR e. C 0 if + D opreq1 A RR e. A 0 if + B RR e. B 0 if opr <_ breq2d imbi12d D D RR e. D 0 if B RR e. B 0 if <_ breq2 A RR e. A 0 if <_ C RR e. C 0 if br anbi2d D D RR e. D 0 if C RR e. C 0 if + opreq2 A RR e. A 0 if + B RR e. B 0 if opr <_ breq2d imbi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel le2add dedth4h))
+thm (addgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (br (0) (<) (opr A (+) B))) (A A RR e. A 0 if 0 < breq2 0 < B br anbi1d A A RR e. A 0 if + B opreq1 0 < breq2d imbi12d B B RR e. B 0 if 0 < breq2 0 < A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if + opreq2 0 < breq2d imbi12d 0re A elimel 0re B elimel addgt0 dedth2h imp))
+thm (addge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (br (0) (<_) (opr A (+) B))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if + B opreq1 0 <_ breq2d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if + opreq2 0 <_ breq2d imbi12d 0re A elimel 0re B elimel addge0 dedth2h imp))
+thm (ltaddpost () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br B (<) (opr B (+) A)))) (0re 0 A B ltadd2t mp3an1 B recnt B ax0id syl A RR e. adantl < B + A opr breq1d bitrd))
+thm (ltaddpos2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br B (<) (opr A (+) B)))) (A B ltaddpost A B axaddcom A recnt B recnt syl2an B < breq2d bitr4d))
+thm (leaddnnegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) A) (br B (<_) (opr B (+) A)))) (0re 0 A B leadd2t mp3an1 B recnt B ax0id syl A RR e. adantl <_ B + A opr breq1d bitrd))
+thm (subge02t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br (opr A (-) B) (<_) A))) (A B addge01t A B A lesubaddt A RR e. B RR e. pm3.26 A RR e. B RR e. pm3.27 A RR e. B RR e. pm3.26 syl3anc bitr4d))
+thm (ltsubpost () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br (opr B (-) A) (<) B))) (A B ltaddpost B A B ltsubaddt A RR e. B RR e. pm3.27 A RR e. B RR e. pm3.26 A RR e. B RR e. pm3.27 syl3anc bitr4d))
+thm (posdift () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A)))) (B - A opr A ltaddpost B A resubclt ancoms A RR e. B RR e. pm3.26 sylanc A B pncan3t A recnt B recnt syl2an A < breq2d bitr2d))
+thm (nnegdift () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (br (0) (<_) (opr B (-) A)))) (A B posdift B A subeq0t B recnt A recnt syl2an ancoms A B eqcom syl6rbbr B - A opr 0 eqcom syl6bb orbi12d A B leloet B A resubclt ancoms 0re 0 B - A opr leloet mpan syl 3bitr4d))
+thm (ltnegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (-u B) (<) (-u A)))) (A A RR e. A 0 if < B breq1 A A RR e. A 0 if negeq B -u < breq2d bibi12d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if negeq < A RR e. A 0 if -u breq1d bibi12d 0re A elimel 0re B elimel ltneg dedth2h))
+thm (ltnegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<) B) (br (-u B) (<) A))) (A -u B ltnegt A renegclt sylan A recnt A negnegt syl B -u < breq2d B RR e. adantr bitrd))
+thm (lenegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (br (-u B) (<_) (-u A)))) (A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if negeq B -u <_ breq2d bibi12d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if negeq <_ A RR e. A 0 if -u breq1d bibi12d 0re A elimel 0re B elimel leneg dedth2h))
+thm (lenegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A))) (A -u B lenegt A renegclt sylan A recnt A negnegt syl B -u <_ breq2d B RR e. adantr bitrd))
+thm (lenegcon2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) (-u B)) (br B (<_) (-u A)))) (A B -u lenegt B renegclt sylan2 B recnt B negnegt syl A RR e. adantl <_ A -u breq1d bitrd))
+thm (lesub1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr A (-) C) (<_) (opr B (-) C)))) (A B C -u leadd1t C renegclt syl3an3 A C subnegt B CC e. 3adant2 B C subnegt A CC e. 3adant1 <_ breq12d A recnt B recnt C recnt syl3an bitr4d))
+thm (ltaddpos () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (0) (<) A) (br B (<) (opr B (+) A))) (ltaddpos.1 ltaddpos.2 A B ltaddpost mp2an))
+thm (posdif () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A))) (ltaddpos.1 ltaddpos.2 A B posdift mp2an))
+thm (ltnegcon1 () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (-u A) (<) B) (br (-u B) (<) A)) (ltaddpos.1 ltaddpos.2 A B ltnegcon1t mp2an))
+thm (lenegcon1 () ((ltaddpos.1 (e. A (RR))) (ltaddpos.2 (e. B (RR)))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A)) (ltaddpos.1 ltaddpos.2 A B lenegcon1t mp2an))
+thm (lt0neg1t () () (-> (e. A (RR)) (<-> (br A (<) (0)) (br (0) (<) (-u A)))) (0re A 0 ltnegt mpan2 neg0 < A -u breq1i syl6bb))
+thm (lt0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (-u A) (<) (0)))) (0re 0 A ltnegt mpan neg0 A -u < breq2i syl6bb))
+thm (le0neg1t () () (-> (e. A (RR)) (<-> (br A (<_) (0)) (br (0) (<_) (-u A)))) (0re A 0 lenegt mpan2 neg0 <_ A -u breq1i syl6bb))
+thm (le0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<_) A) (br (-u A) (<_) (0)))) (0re 0 A lenegt mpan neg0 A -u <_ breq2i syl6bb))
+thm (lesub0t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (/\ (br (0) (<_) A) (br B (<_) (opr B (-) A))) (= A (0)))) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if B - opreq2 B <_ breq2d anbi12d A A RR e. A 0 if 0 eqeq1 bibi12d B B RR e. B 0 if = id B B RR e. B 0 if - A RR e. A 0 if opreq1 <_ breq12d 0 <_ A RR e. A 0 if br anbi2d A RR e. A 0 if 0 = bibi1d 0re A elimel 0re B elimel lesub0 dedth2h))
+thm (mulge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (br (0) (<_) (opr A (x.) B))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if x. B opreq1 0 <_ breq2d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if x. opreq2 0 <_ breq2d imbi12d 0re A elimel 0re B elimel mulge0 dedth2h imp))
+thm (mulge0tOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (x.) B)))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if x. B opreq1 0 <_ breq2d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if x. opreq2 0 <_ breq2d imbi12d 0re A elimel 0re B elimel mulge0 dedth2h))
+thm (ltsqt () () (-> (e. A (RR)) (-> (-. (= A (0))) (br (0) (<) (opr A (x.) A)))) (A A RR e. A 0 if 0 eqeq1 negbid A A RR e. A 0 if = id A A RR e. A 0 if = id x. opreq12d 0 < breq2d imbi12d 0re A elimel sqgt0 dedth))
+thm (lt01 () () (br (0) (<) (1)) (ax1ne0 1 0 df-ne mpbi ax1re sqgt0 ax-mp 1cn mulid1 breqtr))
+thm (eqneg () ((eqneg.1 (e. A (CC)))) (<-> (= A (-u A)) (= A (0))) (ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i 1 + 1 opr 0 df-ne mpbi A A -u A + opreq2 eqneg.1 1p1times eqneg.1 negid eqcomi 3eqtr4g ax1re ax1re readdcl recn eqneg.1 mul0or sylib ord mpi 0 df-neg 0cn subid eqtr2 A 0 = id A 0 negeq eqeq12d mpbiri impbi))
+thm (eqnegt () () (-> (e. A (CC)) (<-> (= A (-u A)) (= A (0)))) (A A CC e. A 0 if = id A A CC e. A 0 if negeq eqeq12d A A CC e. A 0 if 0 eqeq1 bibi12d 0cn A elimel eqneg dedth))
+thm (negeq0t () () (-> (e. A (CC)) (<-> (= A (0)) (= (-u A) (0)))) (A negnegt A -u eqeq2d A -u A eqcom syl6rbb A eqnegt A negclt A -u eqnegt syl 3bitr3d))
+thm (negne0 () ((negne0.1 (e. A (CC)))) (<-> (=/= A (0)) (=/= (-u A) (0))) (negne0.1 A negeq0t ax-mp eqneqi))
+thm (negn0 () ((negne0.1 (e. A (CC))) (negn0.2 (=/= A (0)))) (=/= (-u A) (0)) (negn0.2 negne0.1 negne0 mpbi))
+thm (elimgt0 () () (br (0) (<) (if (br (0) (<) A) A (1))) (A 0 < A br A 1 if 0 < breq2 1 0 < A br A 1 if 0 < breq2 lt01 elimhyp))
+thm (elimge0 () () (br (0) (<_) (if (br (0) (<_) A) A (0))) (A 0 <_ A br A 0 if 0 <_ breq2 0 0 <_ A br A 0 if 0 <_ breq2 0re leid elimhyp))
+thm (ltplus1t () () (-> (e. A (RR)) (br A (<) (opr A (+) (1)))) (A recnt A ax0id syl lt01 0re ax1re 0 1 A ltadd2t mp3an12 mpbii eqbrtrrd))
+thm (ltplus1 () ((ltplus1.1 (e. A (RR)))) (br A (<) (opr A (+) (1))) (ltplus1.1 A ltplus1t ax-mp))
+thm (recgt0i () ((ltplus1.1 (e. A (RR))) (recgt0i.2 (br (0) (<) A))) (br (0) (<) (opr (1) (/) A)) (1cn ltplus1.1 recn ax1ne0 ltplus1.1 recgt0i.2 gt0ne0i divneq0 1 / A opr 0 necom mpbi 0 1 / A opr df-ne mpbi lt01 0re ax1re ltnsym ax-mp recgt0i.2 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl renegcl ltplus1.1 mulgt0 mpan2 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl recn ltplus1.1 recn mulneg1 ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl recn ltplus1.1 recn mulcom ltplus1.1 recn ltplus1.1 recgt0i.2 gt0ne0i recid eqtr negeqi eqtr syl6breq ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl 1 / A opr lt0neg1t ax-mp ax1re 1 lt0neg1t ax-mp 3imtr4 mto pm3.2ni 0re ltplus1.1 ltplus1.1 recgt0i.2 gt0ne0i rereccl 0 1 / A opr axlttri mp2an mpbir))
+thm (recgt0 () ((ltplus1.1 (e. A (RR)))) (-> (br (0) (<) A) (br (0) (<) (opr (1) (/) A))) (A 0 < A br A 1 if 1 / opreq2 0 < breq2d ltplus1.1 ax1re 0 < A br keepel A elimgt0 recgt0i dedth))
+thm (leplus1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (br A (<_) (opr B (+) (1)))) (B ltplus1t A RR e. adantl A B B + 1 opr lelttrt 3expb B peano2re ancli sylan2 mpan2d imp 3impa A B + 1 opr ltlet B peano2re sylan2 A <_ B br 3adant3 mpd))
+thm (plus1let () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (opr A (+) (1)) (<_) B)) (br A (<_) B)) (A A + 1 opr ltlet A peano2re ancli A ltplus1t sylc B RR e. adantr A A + 1 opr B letrt 3expa A peano2re ancli sylan mpand imp 3impa))
+thm (prodgt0lem () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR))) (prodgt0i.3 (br (0) (<) A))) (-> (br (0) (<) (opr A (x.) B)) (br (0) (<) B)) (prodgt0.1 prodgt0i.3 recgt0i 0re prodgt0.1 prodgt0.2 remulcl prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl ltmullem ax-mp prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl recn mul02 prodgt0.1 recn prodgt0.2 recn prodgt0.1 prodgt0.1 prodgt0i.3 gt0ne0i rereccl recn mul23 prodgt0.1 recn prodgt0.1 prodgt0i.3 gt0ne0i recid x. B opreq1i prodgt0.2 recn mulid2 3eqtr 3brtr3g))
+thm (prodgt0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B))) (br (0) (<) B)) (0re prodgt0.1 leloe A 0 < A br A 1 if x. B opreq1 0 < breq2d 0 < B br imbi1d prodgt0.1 ax1re 0 < A br keepel prodgt0.2 A elimgt0 prodgt0lem dedth 0re ltnr 0 A x. B opreq1 prodgt0.2 recn mul02 syl5eqr 0 < breq2d mtbii 0 < B br pm2.21d jaoi sylbi imp))
+thm (prodge0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B))) (br (0) (<_) B)) (prodgt0.1 prodgt0.2 renegcl mulgt0 exp prodgt0.2 B lt0neg1t ax-mp prodgt0.1 prodgt0.2 remulcl A x. B opr lt0neg1t ax-mp prodgt0.1 recn prodgt0.2 recn mulneg2 0 < breq2i bitr4 3imtr4g con3d 0re prodgt0.1 prodgt0.2 remulcl lenlt 0re prodgt0.2 lenlt 3imtr4g imp))
+thm (divgt0lem () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR))) (divgt0lem.3 (br (0) (<) B))) (-> (br (0) (<) A) (br (0) (<) (opr A (/) B))) (prodgt0.2 divgt0lem.3 recgt0i prodgt0.1 prodgt0.2 prodgt0.2 divgt0lem.3 gt0ne0i rereccl mulgt0 mpan2 prodgt0.1 recn prodgt0.2 recn prodgt0.2 divgt0lem.3 gt0ne0i divrec syl6breqr))
+thm (divgt0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (/) B))) (B 0 < B br B 1 if A / opreq2 0 < breq2d 0 < A br imbi2d prodgt0.1 prodgt0.2 ax1re 0 < B br keepel B elimgt0 divgt0lem dedth impcom))
+thm (divgt0i () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR))) (divgt0i.3 (br (0) (<) A)) (divgt0i.4 (br (0) (<) B))) (br (0) (<) (opr A (/) B)) (divgt0i.3 prodgt0.1 prodgt0.2 divgt0i.4 divgt0lem ax-mp))
+thm (divge0 () ((prodgt0.1 (e. A (RR))) (prodgt0.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<) B)) (br (0) (<_) (opr A (/) B))) (prodgt0.1 prodgt0.2 divgt0 expcom 0 A / B opreq1 0 eqeq1d prodgt0.2 gt0ne0 B B 0 =/= B 1 if 0 / opreq2 0 eqeq1d prodgt0.2 recn 1cn B 0 =/= keepel B elimne0 div0 dedth syl syl5bi com12 A / B opr 0 eqcom syl6ib orim12d 0re prodgt0.1 leloe syl5ib prodgt0.2 gt0ne0 prodgt0.1 prodgt0.2 redivclz syl 0re jctil 0 A / B opr leloet syl sylibrd impcom))
+thm (ltmul1i () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR))) (ltmul1i.4 (br (0) (<) C))) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C))) (ltmul1i.4 ltmul1.1 ltmul1.2 ltmul1.3 ltmullem ax-mp ltmul1.3 ltmul1i.4 recgt0i ltmul1.1 ltmul1.3 remulcl ltmul1.2 ltmul1.3 remulcl ltmul1.3 ltmul1.3 ltmul1i.4 gt0ne0i rereccl ltmullem ax-mp ltmul1.1 recn ltmul1.3 recn ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i reccl mulass ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i recid A x. opreq2i ltmul1.1 recn mulid1 3eqtr ltmul1.2 recn ltmul1.3 recn ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i reccl mulass ltmul1.3 recn ltmul1.3 ltmul1i.4 gt0ne0i recid B x. opreq2i ltmul1.2 recn mulid1 3eqtr 3brtr3g impbi))
+thm (ltmul1 () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (C 0 < C br C 1 if A x. opreq2 C 0 < C br C 1 if B x. opreq2 < breq12d A < B br bibi2d ltmul1.1 ltmul1.2 ltmul1.3 ax1re 0 < C br keepel C elimgt0 ltmul1i dedth))
+thm (ltdiv1i () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR))) (ltdivi.4 (br (0) (<) C))) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C))) (ltmul1.1 ltmul1.2 ltmul1.3 ltmul1.3 ltdivi.4 gt0ne0i rereccl ltmul1.3 ltdivi.4 recgt0i ltmul1i ltmul1.1 recn ltmul1.3 recn ltmul1.3 ltdivi.4 gt0ne0i divrec ltmul1.2 recn ltmul1.3 recn ltmul1.3 ltdivi.4 gt0ne0i divrec < breq12i bitr4))
+thm (ltdiv1 () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))) (C 0 < C br C 1 if A / opreq2 C 0 < C br C 1 if B / opreq2 < breq12d A < B br bibi2d ltmul1.1 ltmul1.2 ltmul1.3 ax1re 0 < C br keepel C elimgt0 ltdiv1i dedth))
+thm (ltmuldiv () ((ltmul1.1 (e. A (RR))) (ltmul1.2 (e. B (RR))) (ltmul1.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br (opr A (x.) C) (<) B) (br A (<) (opr B (/) C)))) (ltmul1.1 ltmul1.3 remulcl ltmul1.2 ltmul1.3 ltdiv1 ltmul1.3 gt0ne0 ltmul1.3 recn ltmul1.1 recn divcan4z syl < B / C opr breq1d bitrd))
+thm (prodgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B)))) (br (0) (<) B)) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if x. B opreq1 0 < breq2d anbi12d 0 < B br imbi1d B B RR e. B 0 if A RR e. A 0 if x. opreq2 0 < breq2d 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if 0 < breq2 imbi12d 0re A elimel 0re B elimel prodgt0 dedth2h imp))
+thm (prodgt02tOLD () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) (opr B (x.) A)))) (br (0) (<) B)) (A B prodgt0t exp B A axmulcom B recnt A recnt syl2an ancoms 0 < breq2d biimpd sylan2d imp))
+thm (prodgt02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) B) (br (0) (<) (opr A (x.) B)))) (br (0) (<) A)) (B A prodgt0t exp ancoms A B axmulcom A recnt B recnt syl2an 0 < breq2d biimpd sylan2d imp))
+thm (prodge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) B)) (A A RR e. A 0 if 0 < breq2 A A RR e. A 0 if x. B opreq1 0 <_ breq2d anbi12d 0 <_ B br imbi1d B B RR e. B 0 if A RR e. A 0 if x. opreq2 0 <_ breq2d 0 < A RR e. A 0 if br anbi2d B B RR e. B 0 if 0 <_ breq2 imbi12d 0re A elimel 0re B elimel prodge0 dedth2h imp))
+thm (prodge02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) B) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) A)) (B A prodge0t exp ancoms A B axmulcom A recnt B recnt syl2an 0 <_ breq2d biimpd sylan2d imp))
+thm (ltmul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))) (A A RR e. A 0 if < B breq1 A A RR e. A 0 if x. C opreq1 < B x. C opr breq1d bibi12d 0 < C br imbi2d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if x. C opreq1 A RR e. A 0 if x. C opr < breq2d bibi12d 0 < C br imbi2d C C RR e. C 0 if 0 < breq2 C C RR e. C 0 if A RR e. A 0 if x. opreq2 C C RR e. C 0 if B RR e. B 0 if x. opreq2 < breq12d A RR e. A 0 if < B RR e. B 0 if br bibi2d imbi12d 0re A elimel 0re B elimel 0re C elimel ltmul1 dedth3h imp))
+thm (ltmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))) (A B C ltmul1t A C axmulcom A recnt sylan B RR e. 3adant2 B C axmulcom B recnt sylan A RR e. 3adant1 < breq12d C recnt syl3an3 0 < C br adantr bitrd))
+thm (lemul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))) (A B C ltmul1t A B C mulcan2t A recnt B recnt C recnt 3anim123i 0 < C br adantr C gt0ne0t A RR e. adantll B RR e. 3adantl2 sylanc bicomd orbi12d A B leloet C RR e. 3adant3 0 < C br adantr A x. C opr B x. C opr leloet A C axmulrcl B RR e. 3adant2 B C axmulrcl A RR e. 3adant1 sylanc 0 < C br adantr 3bitr4d))
+thm (lemul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))) (A B C lemul1t A C axmulcom A recnt C recnt syl2an B RR e. 3adant2 B C axmulcom B recnt C recnt syl2an A RR e. 3adant1 <_ breq12d 0 < C br adantr bitrd))
+thm (ltmul2 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C ltmul2t exp mp3an))
+thm (lemul1 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C lemul1t exp mp3an))
+thm (lemul2 () ((ltmul2.1 (e. A (RR))) (ltmul2.2 (e. B (RR))) (ltmul2.3 (e. C (RR)))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))) (ltmul2.1 ltmul2.2 ltmul2.3 A B C lemul2t exp mp3an))
+thm (lemul1it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr A (x.) C) (<_) (opr B (x.) C))) (0re 0 C leloet mpan A RR e. B RR e. 3ad2ant3 A B C lemul1t biimpd exp 0 C A x. opreq2 0 C B x. opreq2 <_ breq12d 0re leid A mul01t B mul01t <_ breqan12d A recnt B recnt syl2an mpbiri syl5bi com12 C RR e. 3adant3 A <_ B br a1dd jaod sylbid imp32))
+thm (lemul2it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr C (x.) A) (<_) (opr C (x.) B))) (A B C lemul1it A C axmulcom A recnt C recnt syl2an B RR e. 3adant2 0 <_ C br A <_ B br /\ adantr B C axmulcom B recnt C recnt syl2an A RR e. 3adant1 0 <_ C br A <_ B br /\ adantr 3brtr3d))
+thm (mulgt1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (1) (<) A) (br (1) (<) B))) (br (1) (<) (opr A (x.) B))) (1 < A br 1 < B br pm3.26 A RR e. B RR e. /\ a1i lt01 0re ax1re 0 1 A axlttrn mp3an12 mpani B RR e. adantr ax1re 1 B A ltmul2t biimpd exp mp3an1 ancoms syld imp3a A recnt A ax1id syl B RR e. adantr < A x. B opr breq1d sylibd jcad A B axmulrcl ax1re 1 A A x. B opr axlttrn mp3an1 syldan syld imp))
+thm (recgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr (1) (/) A))) (A A RR e. A 0 if 0 < breq2 A A RR e. A 0 if 1 / opreq2 0 < breq2d imbi12d 0re A elimel recgt0 dedth imp))
+thm (divgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (br (0) (<) (opr A (/) B))) (A A RR e. A 0 if 0 < breq2 0 < B br anbi1d A A RR e. A 0 if / B opreq1 0 < breq2d imbi12d B B RR e. B 0 if 0 < breq2 0 < A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if / opreq2 0 < breq2d imbi12d 0re A elimel 0re B elimel divgt0 dedth2h imp))
+thm (divge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) B))) (br (0) (<_) (opr A (/) B))) (A A RR e. A 0 if 0 <_ breq2 0 < B br anbi1d A A RR e. A 0 if / B opreq1 0 <_ breq2d imbi12d B B RR e. B 0 if 0 < breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if / opreq2 0 <_ breq2d imbi12d 0re A elimel 0re B elimel divge0 dedth2h imp))
+thm (ltdiv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))) (A A RR e. A 0 if < B breq1 A A RR e. A 0 if / C opreq1 < B / C opr breq1d bibi12d 0 < C br imbi2d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if / C opreq1 A RR e. A 0 if / C opr < breq2d bibi12d 0 < C br imbi2d C C RR e. C 0 if 0 < breq2 C C RR e. C 0 if A RR e. A 0 if / opreq2 C C RR e. C 0 if B RR e. B 0 if / opreq2 < breq12d A RR e. A 0 if < B RR e. B 0 if br bibi2d imbi12d 0re A elimel 0re B elimel 0re C elimel ltdiv1 dedth3h imp))
+thm (lediv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (/) C) (<_) (opr B (/) C)))) (B A C ltdiv1t exp 3com12 imp negbid A B lenltt C RR e. 3adant3 0 < C br adantr C gt0ne0t B RR e. adantll A RR e. 3adantl1 A C redivclt 3expa B RR e. 3adantl2 B C redivclt 3expa A RR e. 3adantl1 jca syldan A / C opr B / C opr lenltt syl 3bitr4d))
+thm (gt0divt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) B)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) B)))) (0re 0 A B ltdiv1t mp3anl1 3impa B gt0ne0t B div0t B recnt sylan syldan < A / B opr breq1d A RR e. 3adant1 bitrd))
+thm (ltmuldivt () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (x.) B) (<) C) (br A (<) (opr C (/) B)))) (A A RR e. A 0 if x. B opreq1 < C breq1d A A RR e. A 0 if < C / B opr breq1 bibi12d 0 < B br imbi2d B B RR e. B 0 if 0 < breq2 B B RR e. B 0 if A RR e. A 0 if x. opreq2 < C breq1d B B RR e. B 0 if C / opreq2 A RR e. A 0 if < breq2d bibi12d imbi12d C C RR e. C 0 if A RR e. A 0 if x. B RR e. B 0 if opr < breq2 C C RR e. C 0 if / B RR e. B 0 if opreq1 A RR e. A 0 if < breq2d bibi12d 0 < B RR e. B 0 if br imbi2d 0re A elimel 0re C elimel 0re B elimel ltmuldiv dedth3h imp))
+thm (ltmuldiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) A)) (<-> (br (opr A (x.) B) (<) C) (br B (<) (opr C (/) A)))) (B A axmulcom B recnt A recnt syl2an C RR e. 3adant3 0 < A br adantr < C breq1d B A C ltmuldivt bitr3d exp 3com12 imp))
+thm (ltdivmult () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr B (x.) C)))) (A B x. C opr B ltdiv1t A RR e. B RR e. C RR e. 3simp1 B C axmulrcl A RR e. 3adant1 A RR e. B RR e. C RR e. 3simp2 3jca sylan B gt0ne0t exp C RR e. adantr B C divcan3t 3exp imp B recnt C recnt syl2an syld A RR e. 3adant1 imp A / B opr < breq2d bitr2d))
+thm (ltdivmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr C (x.) B)))) (A B C ltdivmult B C axmulcom B recnt C recnt syl2an A RR e. 3adant1 0 < B br adantr A < breq2d bitrd))
+thm (ltreci () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR))) (ltreci.3 (br (0) (<) A)) (ltreci.4 (br (0) (<) B))) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A))) (ltrec.1 recn ltrec.1 ltreci.3 gt0ne0i divid ltrec.2 recn ltrec.2 ltreci.4 gt0ne0i recid eqtr4 ltrec.2 recn ltrec.1 recn ltrec.1 ltreci.3 gt0ne0i divrec < breq12i ltrec.1 ltrec.2 ltrec.1 ltreci.3 ltdiv1i ltreci.4 ltrec.2 ltrec.2 ltreci.4 gt0ne0i rereccl ltrec.1 ltrec.1 ltreci.3 gt0ne0i rereccl ltrec.2 ltmul2 ax-mp 3bitr4))
+thm (ltrec () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))) (A 0 < A br A 1 if < B breq1 A 0 < A br A 1 if 1 / opreq2 1 / B opr < breq2d bibi12d B 0 < B br B 1 if 0 < A br A 1 if < breq2 B 0 < B br B 1 if 1 / opreq2 < 1 / 0 < A br A 1 if opr breq1d bibi12d ltrec.1 ax1re 0 < A br keepel ltrec.2 ax1re 0 < B br keepel A elimgt0 B elimgt0 ltreci dedth2h))
+thm (lerec () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))) (ltrec.1 ltrec.2 ltrec ltrec.1 recn ltrec.2 recn rec11 ltrec.1 gt0ne0 ltrec.2 gt0ne0 syl2an 1 / A opr 1 / B opr eqcom syl5rbbr orbi12d ltrec.2 gt0ne0 ltrec.2 rerecclz syl ltrec.1 gt0ne0 ltrec.1 rerecclz syl anim12i ancoms 1 / B opr 1 / A opr leloet syl bitr4d ltrec.1 ltrec.2 leloe syl5bb))
+thm (lt2sq () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))) (ltrec.1 ltrec.2 ltrec.1 ltmul2 ltrec.1 ltrec.2 ltrec.2 ltmul1 bi2anan9 A < B br anidm syl5bbr ltrec.1 ltrec.1 remulcl ltrec.1 ltrec.2 remulcl ltrec.2 ltrec.2 remulcl lttr syl6bi ltrec.2 ltrec.1 ltrec.2 lemul2 ltrec.2 ltrec.1 ltrec.1 lemul1 bi2anan9r B <_ A br anidm syl5bbr ltrec.2 ltrec.2 remulcl ltrec.2 ltrec.1 remulcl ltrec.1 ltrec.1 remulcl letr syl6bi ltrec.2 ltrec.1 lenlt ltrec.2 ltrec.2 remulcl ltrec.1 ltrec.1 remulcl lenlt 3imtr3g a3d impbid 0 A < B breq1 0 < B br adantr 0re ltrec.2 ltrec.2 ltmul2 0 A A x. opreq2 < B x. B opr breq1d ltrec.2 recn mul01 ltrec.1 recn mul01 eqtr4 < B x. B opr breq1i syl5bb sylan9bbr bitr3d 0 B <_ A breq1 0 < A br adantl 0re ltrec.1 ltrec.1 lemul2 0 B B x. opreq2 <_ A x. A opr breq1d ltrec.1 recn mul01 ltrec.2 recn mul01 eqtr4 <_ A x. A opr breq1i syl5bb sylan9bb bitr3d ltrec.2 ltrec.1 lenlt ltrec.2 ltrec.2 remulcl ltrec.1 ltrec.1 remulcl lenlt 3bitr3g bicon4d A < B br A x. A opr < B x. B opr br pm5.21 ltrec.2 ltnr 0 B < B breq1 bicomd 0 A < B breq1 sylan9bbr mtbii ltrec.2 ltrec.2 remulcl ltnr 0 B B x. opreq2 < B x. B opr breq1d bicomd 0 A A x. opreq2 < B x. B opr breq1d ltrec.2 recn mul01 ltrec.1 recn mul01 eqtr4 < B x. B opr breq1i syl5bb sylan9bbr mtbii sylanc ccase 0re ltrec.1 leloe 0re ltrec.2 leloe syl2anb))
+thm (le2sq () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B)))) (ltrec.2 ltrec.1 lt2sq negbid ltrec.1 ltrec.2 lenlt ltrec.1 ltrec.1 remulcl ltrec.2 ltrec.2 remulcl lenlt 3bitr4g ancoms))
+thm (sq11 () ((ltrec.1 (e. A (RR))) (ltrec.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (x.) A) (opr B (x.) B)) (= A B))) (ltrec.1 ltrec.2 lt2sq ltrec.2 ltrec.1 lt2sq ancoms orbi12d ltrec.1 ltrec.2 lttri2 ltrec.1 ltrec.1 remulcl ltrec.2 ltrec.2 remulcl lttri2 3bitr4g bicon4d bicomd))
+thm (ltrect () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))) (A A RR e. A 0 if 0 < breq2 0 < B br anbi1d A A RR e. A 0 if < B breq1 A A RR e. A 0 if 1 / opreq2 1 / B opr < breq2d bibi12d imbi12d B B RR e. B 0 if 0 < breq2 0 < A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if 1 / opreq2 < 1 / A RR e. A 0 if opr breq1d bibi12d imbi12d 0re A elimel 0re B elimel ltrec dedth2h imp))
+thm (lerect () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))) (A A RR e. A 0 if 0 < breq2 0 < B br anbi1d A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if 1 / opreq2 1 / B opr <_ breq2d bibi12d imbi12d B B RR e. B 0 if 0 < breq2 0 < A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if 1 / opreq2 <_ 1 / A RR e. A 0 if opr breq1d bibi12d imbi12d 0re A elimel 0re B elimel lerec dedth2h imp))
+thm (lt2sqt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if < B breq1 A A RR e. A 0 if A A RR e. A 0 if x. opreq12 anidms < B x. B opr breq1d bibi12d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if B B RR e. B 0 if x. opreq12 anidms A RR e. A 0 if x. A RR e. A 0 if opr < breq2d bibi12d imbi12d 0re A elimel 0re B elimel lt2sq dedth2h imp))
+thm (ltdiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\/\ (br (0) (<) A) (br (0) (<) B) (br (0) (<) C))) (<-> (br A (<) B) (br (opr C (/) B) (<) (opr C (/) A)))) (A B ltrect 0 < C br 3adantr3 C RR e. 3adantl3 A RR e. B RR e. C RR e. 0 < A br 0 < B br 0 < C br an6 1 / B opr 1 / A opr C ltmul2t A rerecclt syl3anl2 B rerecclt syl3anl1 C B divrect B recnt syl3an2 3expb A RR e. A 0 =/= /\ 3adant3 C A divrect A recnt syl3an2 3expb B RR e. B 0 =/= /\ 3adant2 < breq12d 3coml C recnt syl3an3 0 < C br adantr bitr4d exp 3com12 3exp imp4a 3imp B RR e. 0 < B br pm3.26 B gt0ne0t jca syl3an2 A RR e. 0 < A br pm3.26 A gt0ne0t jca syl3an1 sylbi bitrd))
+thm (ltrec1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br (opr (1) (/) A) (<) B) (br (opr (1) (/) B) (<) A))) (1 / A opr B ltrect A gt0ne0t A rerecclt syldan 0 < B br adantrr B RR e. adantlr A RR e. B RR e. pm3.27 0 < A br 0 < B br /\ adantr jca A recgt0t 0 < B br adantrr B RR e. adantlr 0 < A br 0 < B br pm3.27 A RR e. B RR e. /\ adantl jca sylanc A recrect A recnt 0 < A br adantr A gt0ne0t sylanc 0 < B br adantrr B RR e. adantlr 1 / B opr < breq2d bitrd))
+thm (lerec2t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) (opr (1) (/) B)) (br B (<_) (opr (1) (/) A)))) (A 1 / B opr lerect A RR e. B RR e. pm3.26 0 < A br 0 < B br /\ adantr B gt0ne0t B rerecclt syldan 0 < A br adantrl A RR e. adantll jca 0 < A br 0 < B br pm3.26 A RR e. B RR e. /\ adantl B recgt0t 0 < A br adantrl A RR e. adantll jca sylanc B recrect B recnt 0 < B br adantr B gt0ne0t sylanc 0 < A br adantrl A RR e. adantll <_ 1 / A opr breq1d bitrd))
+thm (ltdiv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))) (A / B opr C B ltmul1t B gt0ne0t A RR e. adantll A B redivclt 3expa syldan C RR e. 3adantl3 A RR e. B RR e. C RR e. 3simp3 0 < B br adantr A RR e. B RR e. C RR e. 3simp2 0 < B br adantr 3jca A RR e. B RR e. C RR e. /\/\ 0 < B br pm3.27 sylanc 0 < C br adantrr B gt0ne0t A RR e. adantll B A divcan1t 3com12 3expa A recnt B recnt anim12i sylan syldan C RR e. 3adantl3 0 < C br adantrr < C x. B opr breq1d A C x. B opr C ltdiv1t A RR e. B RR e. C RR e. 3simp1 C B axmulrcl ancoms A RR e. 3adant1 A RR e. B RR e. C RR e. 3simp3 3jca sylan C gt0ne0t B RR e. adantll C B divcan3t B recnt syl3an2 C recnt syl3an1 3com12 3expa syldan A RR e. 3adantl1 A / C opr < breq2d bitrd 0 < B br adantrl 3bitrd))
+thm (lediv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<_) C) (br (opr A (/) C) (<_) B))) (A / B opr C B lemul1t B gt0ne0t A RR e. adantll A B redivclt 3expa syldan C RR e. 3adantl3 A RR e. B RR e. C RR e. 3simp3 0 < B br adantr A RR e. B RR e. C RR e. 3simp2 0 < B br adantr 3jca A RR e. B RR e. C RR e. /\/\ 0 < B br pm3.27 sylanc 0 < C br adantrr B gt0ne0t A RR e. adantll B A divcan1t 3com12 3expa A recnt B recnt anim12i sylan syldan C RR e. 3adantl3 0 < C br adantrr <_ C x. B opr breq1d A C x. B opr C lediv1t A RR e. B RR e. C RR e. 3simp1 C B axmulrcl ancoms A RR e. 3adant1 A RR e. B RR e. C RR e. 3simp3 3jca sylan C gt0ne0t B RR e. adantll C B divcan3t B recnt syl3an2 C recnt syl3an1 3com12 3expa syldan A RR e. 3adantl1 A / C opr <_ breq2d bitrd 0 < B br adantrl 3bitrd))
+thm (ltdiv23 () ((ltdiv23.1 (e. A (RR))) (ltdiv23.2 (e. B (RR))) (ltdiv23.3 (e. C (RR)))) (-> (/\ (br (0) (<) B) (br (0) (<) C)) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))) (ltdiv23.1 ltdiv23.2 ltdiv23.3 3pm3.2i A B C ltdiv23t mpan))
+thm (ltdiv23i () ((ltdiv23.1 (e. A (RR))) (ltdiv23.2 (e. B (RR))) (ltdiv23.3 (e. C (RR))) (ltdiv23i.4 (br (0) (<) B)) (ltdiv23i.5 (br (0) (<) C))) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B)) (ltdiv23i.4 ltdiv23i.5 ltdiv23.1 ltdiv23.2 ltdiv23.3 ltdiv23 mp2an))
+thm (reclt1t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (<-> (br A (<) (1)) (br (1) (<) (opr (1) (/) A)))) (lt01 ax1re A 1 ltrect mpanl2 mpanr2 1cn div1 < 1 / A opr breq1i syl6bb))
+thm (recgt1t () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (/\ (br (0) (<) (opr (1) (/) A)) (br (opr (1) (/) A) (<) (1)))) (lt01 0re ax1re 0 1 A axlttrn mp3an12 mpani imdistani A recgt0t syl ax1re lt01 1 A ltrect mpanr1 mpanl1 1cn div1 1 / A opr < breq2i syl6bb biimpa lt01 0re ax1re 0 1 A axlttrn mp3an12 mpani imdistani sylan anabss3 jca))
+thm (le2sqt () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B))))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if A A RR e. A 0 if x. opreq12 anidms <_ B x. B opr breq1d bibi12d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if B B RR e. B 0 if x. opreq12 anidms A RR e. A 0 if x. A RR e. A 0 if opr <_ breq2d bibi12d imbi12d 0re A elimel 0re B elimel le2sq dedth2h))
+thm (halfpos () ((halfpos.1 (e. A (RR)))) (<-> (br (0) (<) A) (br (opr A (/) (opr (1) (+) (1))) (<) A)) (0re halfpos.1 halfpos.1 ltadd1 halfpos.1 recn addid2 halfpos.1 recn 1p1times eqcomi < breq12i halfpos.1 ax1re ax1re readdcl halfpos.1 remulcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ltdiv1i bitr 1cn 1cn addcl halfpos.1 recn ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divcan3 A / 1 + 1 opr opr < breq2i 3bitr))
+thm (ledivp1 () ((ledivp1.1 (e. A (RR))) (ledivp1.2 (e. B (RR))) (ledivp1.3 (e. C (RR)))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<_) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<_) B)) (ledivp1.1 ledivp1.3 remulcl ledivp1.1 ledivp1.3 ax1re readdcl remulcl ledivp1.2 letr ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.3 ltplus1 ltlei ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.1 C C + 1 opr A lemul2it exp mp3an mpan2 0 <_ C br A <_ B / C + 1 opr opr br 3ad2ant1 ledivp1.3 ltplus1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.2 ledivp1.3 ax1re readdcl redivclz syl ledivp1.3 ax1re readdcl ledivp1.1 A B / C + 1 opr opr C + 1 opr lemul1t mp3anl1 mpanl2 mpancom biimpd syl imp ledivp1.3 ltplus1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.3 ax1re readdcl recn ledivp1.2 recn divcan1z 3syl A <_ B / C + 1 opr opr br adantr breqtrd 0 <_ A br 3adant1 sylanc))
+thm (ltdivp1 () ((ledivp1.1 (e. A (RR))) (ledivp1.2 (e. B (RR))) (ledivp1.3 (e. C (RR)))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<) B)) (ledivp1.1 ledivp1.3 remulcl ledivp1.1 ledivp1.3 ax1re readdcl remulcl ledivp1.2 lelttr ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.3 ltplus1 ltlei ledivp1.3 ledivp1.3 ax1re readdcl ledivp1.1 C C + 1 opr A lemul2it exp mp3an mpan2 0 <_ C br A < B / C + 1 opr opr br 3ad2ant1 ledivp1.3 ltplus1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.2 ledivp1.3 ax1re readdcl redivclz syl ledivp1.3 ax1re readdcl ledivp1.1 A B / C + 1 opr opr C + 1 opr ltmul1t mp3anl1 mpanl2 mpancom biimpd syl imp ledivp1.3 ltplus1 0re ledivp1.3 ledivp1.3 ax1re readdcl lelttr mpan2 ledivp1.3 ax1re readdcl gt0ne0 ledivp1.3 ax1re readdcl recn ledivp1.2 recn divcan1z 3syl A < B / C + 1 opr opr br adantr breqtrd 0 <_ A br 3adant1 sylanc))
+thm (posex ((A x) (B x)) ((posex.1 (e. A (RR))) (posex.2 (e. B (RR))) (posex.3 (br (0) (<) A)) (posex.4 (br (0) (<) B))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (/\ (br (cv x) (<) A) (br (cv x) (<) B)))) (posex.1 posex.2 lttri2 biimp orri A B = A < B br B < A br or12 mpbi posex.3 posex.1 halfpos mpbi posex.1 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl posex.1 posex.2 lttr mpan posex.3 posex.1 halfpos mpbi jctil posex.1 ax1re ax1re readdcl posex.3 ax1re ax1re lt01 lt01 addgt0i divgt0i jctil posex.1 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl jctil x cv A / 1 + 1 opr opr 0 < breq2 x cv A / 1 + 1 opr opr < A breq1 x cv A / 1 + 1 opr opr < B breq1 anbi12d anbi12d RR rcla4ev syl posex.4 posex.2 halfpos mpbi A B B / 1 + 1 opr opr < breq2 mpbiri posex.4 posex.2 halfpos mpbi posex.2 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl posex.2 posex.1 lttr mpan jaoi posex.4 posex.2 halfpos mpbi jctir posex.2 ax1re ax1re readdcl posex.4 ax1re ax1re lt01 lt01 addgt0i divgt0i jctil posex.2 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl jctil x cv B / 1 + 1 opr opr 0 < breq2 x cv B / 1 + 1 opr opr < A breq1 x cv B / 1 + 1 opr opr < B breq1 anbi12d anbi12d RR rcla4ev syl jaoi ax-mp))
+thm (max1 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br A (<_) (if (br A (<_) B) B A))) (A <_ B br B A iffalse A <_ breq2d A leidt syl5bir com12 A <_ B br id A <_ B br B A iftrue breqtrrd pm2.61d2 B RR e. adantr))
+thm (max1ALT () () (-> (e. A (RR)) (br A (<_) (if (br A (<_) B) B A))) (A <_ B br B A iffalse A <_ breq2d A leidt syl5bir com12 A <_ B br id A <_ B br B A iftrue breqtrrd pm2.61d2))
+thm (max2 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br B (<_) (if (br A (<_) B) B A))) (A <_ B br B A iftrue B <_ breq2d B leidt syl5bir A RR e. adantld A B letrit ord com12 A <_ B br B A iffalse B <_ breq2d sylibrd pm2.61i))
+thm (peano5nn ((x y) (A x) (A y)) ((peano5nn.1 (e. A (V)))) (-> (/\ (e. (1) A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ (NN) A)) (peano5nn.1 y cv A 1 eleq2 y cv A x cv + 1 opr eleq2 x raleqd anbi12d elab A y 1 y cv e. x y cv x cv + 1 opr y cv e. A.e. /\ {|} intss1 sylbir y x df-n syl5ss))
+thm (nnssre () () (C_ (NN) (RR)) (ax1re x cv peano2re rgen reex x peano5nn mp2an))
+thm (nnsscn () () (C_ (NN) (CC)) (nnssre axresscn sstri))
+thm (nnret () () (-> (e. A (NN)) (e. A (RR))) (nnssre A sseli))
+thm (nncnt () () (-> (e. A (NN)) (e. A (CC))) (nnsscn A sseli))
+thm (nnre () ((nnre.1 (e. A (NN)))) (e. A (RR)) (nnre.1 A nnret ax-mp))
+thm (nncn () ((nnre.1 (e. A (NN)))) (e. A (CC)) (nnre.1 nnre recn))
+thm (nnex () () (e. (NN) (V)) (axcnex axresscn ssexi nnssre ssexi))
+thm (1nn ((x y)) () (e. (1) (NN)) (1cn elisseti x 1 x cv e. y x cv y cv + 1 opr x cv e. A.e. /\ elintab 1 x cv e. y x cv y cv + 1 opr x cv e. A.e. pm3.26 mpgbir x y df-n eleqtrr))
+thm (peano2nn ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (NN)) (e. (opr A (+) (1)) (NN))) (z cv A + 1 opreq1 NN eleq1d y cv z cv + 1 opreq1 x cv eleq1d x cv rcla4v com12 1 x cv e. adantl a2i x 19.20i z visset x 1 x cv e. y x cv y cv + 1 opr x cv e. A.e. /\ elintab z cv + 1 oprex x 1 x cv e. y x cv y cv + 1 opr x cv e. A.e. /\ elintab 3imtr4 x y df-n z cv eleq2i x y df-n z cv + 1 opr eleq2i 3imtr4 vtoclga))
+thm (dfnn2 ((x y) (x z) (y z)) () (= (NN) (|^| ({|} x (/\/\ (C_ (cv x) (RR)) (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))) (x cv z cv 1 eleq2 x cv z cv y cv + 1 opr eleq2 y raleqd anbi12d z y df-n x cv eqeq2i x cv NN 1 eleq2 x cv NN y cv + 1 opr eleq2 y raleqd anbi12d sylbir z y df-n nnssre eqsstr3 1nn y cv peano2nn rgen pm3.2i pm3.2i intabs x cv RR C_ 1 x cv e. y x cv y cv + 1 opr x cv e. A.e. 3anass x biabi inteqi x y df-n 3eqtr4r))
+thm (nnind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((nnind.1 (-> (= (cv x) (1)) (<-> ph ps))) (nnind.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nnind.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nnind.4 (-> (= (cv x) A) (<-> ph ta))) (nnind.5 ps) (nnind.6 (-> (e. (cv y) (NN)) (-> ch th)))) (-> (e. A (NN)) ta) (nnind.1 NN elrab 1nn nnind.5 mpbir2an x NN ph ssrab y cv sseli y cv peano2nn y cv NN e. a1d nnind.6 anim12d nnind.2 NN elrab nnind.3 NN elrab 3imtr4g mpcom rgen nnex x ph rabex y peano5nn mp2an A sseli nnind.4 NN elrab sylib pm3.27d))
+thm (nn1suc ((x y) (x z) (y z) (A x) (A y) (A z) (ps x) (ch x) (th x) (th z) (ph y) (ph z)) ((nn1suc.1 (-> (= (cv x) (1)) (<-> ph ps))) (nn1suc.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph ch))) (nn1suc.4 (-> (= (cv x) A) (<-> ph th))) (nn1suc.5 ps) (nn1suc.6 (-> (e. (cv y) (NN)) ch))) (-> (e. A (NN)) th) (z cv 1 x A NN e. ph -> dfsbcq z y x A NN e. ph -> sbequ z cv y cv + 1 opr x A NN e. ph -> dfsbcq z cv A x ph dfsbcq A NN x elex z cv A e. x ax-17 NN ph hbsbc A NN e. th -> x ax-17 hbbi x A ph sbceq1 nn1suc.4 bitr3d A NN e. imbi2d 19.23ai syl pm5.74rd pm2.43i sylan9bbr expcom pm5.74d A NN e. x ax-17 z ph sb19.21 syl5bb 1nn elisseti x isseti 1nn elisseti x ph hbsbcv nn1suc.5 nn1suc.1 mpbiri x 1 ph sbceq1 mpbid 19.23ai ax-mp A NN e. a1i 1nn elisseti A NN e. x ax-17 1 V ph sbc19.21g ax-mp mpbir nn1suc.6 y cv + 1 oprex x isseti ch x ax-17 y cv + 1 oprex x ph hbsbcv hbbi nn1suc.3 x y cv + 1 opr ph sbceq1 bitr3d 19.23ai ax-mp sylib A NN e. a1d y cv + 1 oprex A NN e. x ax-17 y cv + 1 opr V ph sbc19.21g ax-mp sylibr y cv x A NN e. ph -> [/] a1d nnind pm2.43i))
+thm (nnaddclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (+) B) (NN))) (x cv 1 A + opreq2 NN eleq1d A NN e. imbi2d x cv y cv A + opreq2 NN eleq1d A NN e. imbi2d x cv y cv + 1 opr A + opreq2 NN eleq1d A NN e. imbi2d x cv B A + opreq2 NN eleq1d A NN e. imbi2d A peano2nn 1cn A y cv 1 axaddass mp3an3 A nncnt y cv nncnt syl2an NN eleq1d A + y cv opr peano2nn syl5bi expcom a2d nnind impcom))
+thm (nnmulclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (x.) B) (NN))) (x cv 1 A x. opreq2 NN eleq1d A NN e. imbi2d x cv y cv A x. opreq2 NN eleq1d A NN e. imbi2d x cv y cv + 1 opr A x. opreq2 NN eleq1d A NN e. imbi2d x cv B A x. opreq2 NN eleq1d A NN e. imbi2d A nncnt A ax1id NN eleq1d biimprd mpcom 1cn A y cv 1 axdistr mp3an3 A ax1id A x. y cv opr + opreq2d y cv CC e. adantr eqtrd A nncnt y cv nncnt syl2an NN eleq1d A x. y cv opr A nnaddclt ancoms syl5bir exp4b pm2.43b a2d nnind impcom))
+thm (nn2get ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (E.e. x (NN) (/\ (br A (<_) (cv x)) (br B (<_) (cv x))))) (A B letrit A nnret B nnret syl2an B nnret B leidt A <_ B br biantrud biimpd syl anc2li x cv B A <_ breq2 x cv B B <_ breq2 anbi12d NN rcla4ev syl6 A NN e. adantl A nnret A leidt B <_ A br biantrurd biimpd syl anc2li x cv A A <_ breq2 x cv A B <_ breq2 anbi12d NN rcla4ev syl6 B NN e. adantr jaod mpd))
+thm (nnge1t ((x y) (A x) (A y)) () (-> (e. A (NN)) (br (1) (<_) A)) (x cv 1 1 <_ breq2 x cv y cv 1 <_ breq2 x cv y cv + 1 opr 1 <_ breq2 x cv A 1 <_ breq2 ax1re leid y cv nnret y cv recnt y cv ax0id syl 1 <_ breq2d lt01 0re ax1re 0 1 y cv axltadd mp3an12 mpi ax1re y cv + 0 opr y cv + 1 opr 1 axlttrn mp3an3 0re y cv 0 axaddrcl mpan2 y cv peano2re sylanc mpand con3d 0re y cv 0 axaddrcl mpan2 ax1re jctil 1 y cv + 0 opr lenltt syl y cv peano2re ax1re jctil 1 y cv + 1 opr lenltt syl 3imtr4d sylbird syl nnind))
+thm (nngt1ne1t () () (-> (e. A (NN)) (<-> (br (1) (<) A) (-. (= A (1))))) (ax1re 1 A leltnetOLD mpan A nnret A nnge1t sylc 1 A eqcom negbii syl6bb))
+thm (nnle1eq1t () () (-> (e. A (NN)) (<-> (br A (<_) (1)) (= A (1)))) (A nnge1t A <_ 1 br biantrud A nnret ax1re A 1 letri3t mpan2 syl bitr4d))
+thm (nngt0t () () (-> (e. A (NN)) (br (0) (<) A)) (lt01 0re ax1re 0 1 A ltletrt mp3an12 mpani A nnret A nnge1t sylc))
+thm (lt1nnn () () (-> (/\ (e. A (RR)) (br A (<) (1))) (-. (e. A (NN)))) (ax1re A 1 ltnlet mpan2 A nnge1t con3i syl6bi imp))
+thm (0nnn () () (-. (e. (0) (NN))) (0re lt01 0 lt1nnn mp2an))
+thm (nnne0t () () (-> (e. A (NN)) (=/= A (0))) (0nnn A 0 NN eleq1 mtbiri con2i A 0 df-ne sylibr))
+thm (nngt0 () ((nngt0.1 (e. A (NN)))) (br (0) (<) A) (nngt0.1 A nngt0t ax-mp))
+thm (nnne0 () ((nngt0.1 (e. A (NN)))) (=/= A (0)) (nngt0.1 nnre nngt0.1 nngt0 gt0ne0i))
+thm (nnrecgt0t () () (-> (e. A (NN)) (br (0) (<) (opr (1) (/) A))) (A nnge1t lt01 A nnret 0re ax1re 0 1 A ltletrt mp3an12 A recgt0t exp syld syl mpani mpd))
+thm (nnleltp1t ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<_) B) (br A (<) (opr B (+) (1))))) (A B leloet A nnret B nnret syl2an lt01 0re ax1re pm3.2i A 0 B 1 lt2addt an4s mpan2 mpan2i A RR e. B RR e. pm3.26 A recnt A ax0id 3syl < B + 1 opr breq1d sylibd A B < B + 1 opr breq1 B ltplus1t syl5bir com12 A RR e. adantl jaod A nnret B nnret syl2an z cv A < B + 1 opr breq1 z cv A < B breq1 z cv A B eqeq1 orbi12d imbi12d NN rcla4v x cv 1 + 1 opreq1 z cv < breq2d x cv 1 z cv < breq2 x cv 1 z cv eqeq2 orbi12d imbi12d z NN biraldv x cv y cv + 1 opreq1 z cv < breq2d x cv y cv z cv < breq2 x cv y cv z cv eqeq2 orbi12d imbi12d z NN biraldv x cv y cv + 1 opr + 1 opreq1 z cv < breq2d x cv y cv + 1 opr z cv < breq2 x cv y cv + 1 opr z cv eqeq2 orbi12d imbi12d z NN biraldv x cv B + 1 opreq1 z cv < breq2d x cv B z cv < breq2 x cv B z cv eqeq2 orbi12d imbi12d z NN biraldv x cv 1 < 1 + 1 opr breq1 x cv 1 < 1 breq1 x cv 1 1 eqeq1 orbi12d imbi12d x cv y cv + 1 opr < 1 + 1 opr breq1 x cv y cv + 1 opr < 1 breq1 x cv y cv + 1 opr 1 eqeq1 orbi12d imbi12d x cv z cv < 1 + 1 opr breq1 x cv z cv < 1 breq1 x cv z cv 1 eqeq1 orbi12d imbi12d 1 eqid 1 < 1 br -. a1i orri 1 < 1 + 1 opr br a1i y cv nnret ax1re ax1re y cv 1 1 ltadd1t mp3an23 syl y cv nnge1t y cv nnret ax1re 1 y cv lenltt mpan syl mpbid y cv + 1 opr < 1 br y cv + 1 opr 1 = \/ pm2.21d sylbird nn1suc rgen x cv 1 < y cv + 1 opr + 1 opr breq1 x cv 1 < y cv + 1 opr breq1 x cv 1 y cv + 1 opr eqeq1 orbi12d imbi12d y cv NN e. w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. /\ imbi2d x cv v cv + 1 opr < y cv + 1 opr + 1 opr breq1 x cv v cv + 1 opr < y cv + 1 opr breq1 x cv v cv + 1 opr y cv + 1 opr eqeq1 orbi12d imbi12d y cv NN e. w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. /\ imbi2d x cv z cv < y cv + 1 opr + 1 opr breq1 x cv z cv < y cv + 1 opr breq1 x cv z cv y cv + 1 opr eqeq1 orbi12d imbi12d y cv NN e. w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. /\ imbi2d y cv nngt0t y cv nnret 0re ax1re 0 y cv 1 ltadd1t mp3an13 syl mpbid 1cn addid2 syl5eqbrr 1 y cv + 1 opr = -. a1d con1d orrd 1 < y cv + 1 opr + 1 opr br a1d w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. adantr w cv v cv < y cv + 1 opr breq1 w cv v cv < y cv breq1 w cv v cv y cv eqeq1 orbi12d imbi12d NN rcla4va y cv NN e. adantlr ax1re v cv y cv + 1 opr 1 ltadd1t mp3an3 v cv nnret y cv peano2nn y cv + 1 opr nnret syl syl2an w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. adantr ax1re v cv y cv 1 ltadd1t v cv recnt y cv recnt 1 recnt 3anim123i 1 CC e. v cv CC e. y cv CC e. 3anrot sylibr 1 CC e. v cv CC e. y cv CC e. 3simpc 1 CC e. v cv CC e. y cv CC e. 3simp1 jca v cv CC e. y cv CC e. 1 CC e. anandir sylib v cv 1 axaddcom y cv 1 axaddcom eqeqan12d syl 1 v cv y cv addcant bitr2d syl orbi12d mp3an3 v cv nnret y cv nnret syl2an w NN w cv < y cv + 1 opr br w cv < y cv br w cv y cv = \/ -> A.e. adantr 3imtr3d exp31 imp3a nn1suc exp3a com3l r19.21adv z cv w cv < y cv + 1 opr breq1 z cv w cv < y cv breq1 z cv w cv y cv eqeq1 orbi12d imbi12d NN cbvralv syl5ib nnind syl5 imp impbid bitrd))
+thm (nnltp1let () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (br (opr A (+) (1)) (<_) B))) (ax1re A B 1 ltadd1t mp3an3 A nnret B nnret syl2an A + 1 opr B nnleltp1t A peano2nn sylan bitr4d))
+thm (nnsub ((x y) (A x) (A y) (B x) (B y)) ((nnsub.1 (e. A (NN))) (nnsub.2 (e. B (NN)))) (<-> (br A (<) B) (e. (opr B (-) A) (NN))) (nnsub.2 x cv 1 A < breq2 x cv 1 - A opreq1 NN eleq1d imbi12d x cv y cv A < breq2 x cv y cv - A opreq1 NN eleq1d imbi12d x cv y cv + 1 opr A < breq2 x cv y cv + 1 opr - A opreq1 NN eleq1d imbi12d x cv B A < breq2 x cv B - A opreq1 NN eleq1d imbi12d nnsub.1 A nnge1t ax-mp ax1re nnsub.1 nnre lenlt mpbi 1 - A opr NN e. pm2.21i y cv nnret nnsub.1 nnre jctil A y cv leloet syl nnsub.1 A y cv nnleltp1t mpan bitr3d y cv nncnt nnsub.1 nncn jctir 1cn y cv 1 A addsubt mp3an2 syl NN eleq1d y cv - A opr peano2nn syl5bir A < y cv br imim2d com23 nnsub.1 nncn 1cn nnsub.1 nncn addsub nnsub.1 nncn subid + 1 opreq1i 1cn addid2 3eqtr 1nn eqeltr A y cv + 1 opreq1 - A opreq1d NN eleq1d mpbii A < y cv br y cv - A opr NN e. -> a1d y cv NN e. a1i jaod sylbird com23 nnind ax-mp B - A opr nngt0t nnsub.1 nnre nnsub.2 nnre posdif sylibr impbi))
+thm (nnsubt () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (e. (opr B (-) A) (NN)))) (A A NN e. A 1 if < B breq1 A A NN e. A 1 if B - opreq2 NN eleq1d bibi12d B B NN e. B 1 if A NN e. A 1 if < breq2 B B NN e. B 1 if - A NN e. A 1 if opreq1 NN eleq1d bibi12d 1nn A elimel 1nn B elimel nnsub dedth2h))
+thm (nnaddm1clt () () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr (opr A (+) B) (-) (1)) (NN))) (ax1re ax1re ax1re readdcl 1 1 + 1 opr A + B opr ltletrt mp3an12 A B axaddrcl A nnret B nnret syl2an ax1re ax1re readdcl A NN e. B NN e. /\ a1i B nnret ax1re 1 B axaddrcl mpan syl A NN e. adantl A B axaddrcl A nnret B nnret syl2an B nnge1t B nnret ax1re ax1re 1 B 1 leadd2t mp3an13 syl mpbid A NN e. adantl A nnge1t B NN e. adantr ax1re 1 A B leadd1t mp3an1 A nnret B nnret syl2an mpbid letrd ax1re ltplus1 jctil sylc A B nnaddclt 1nn 1 A + B opr nnsubt mpan syl mpbid))
+thm (nndivt ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (E.e. x (NN) (= (opr A (x.) (cv x)) B)) (e. (opr B (/) A) (NN)))) (A nnne0t B NN e. adantr A x cv divcan3t 3com23 3expa x cv nncnt sylan2 B CC e. 3adantl2 eqcomd A x. x cv opr B / A opreq1 sylan9eq A CC e. B CC e. A 0 =/= /\/\ x cv NN e. /\ A x. x cv opr B = pm3.26 pm3.27d eqeltrrd exp31 r19.23adv A B divcan2t x cv B / A opr A x. opreq2 B eqeq1d NN rcla4ev expcom syl impbid 3exp imp A nncnt B nncnt syl2an mpd))
+thm (nndivtrt () () (-> (/\ (/\/\ (e. A (NN)) (e. B (NN)) (e. C (CC))) (/\ (e. (opr B (/) A) (NN)) (e. (opr C (/) B) (NN)))) (e. (opr C (/) A) (NN))) (B A C B divmul24t B nncnt A nncnt anim12i ancoms C CC e. 3adant3 B nncnt C CC e. anim2i ancoms A NN e. 3adant1 jca A nnne0t B nnne0t anim12i C CC e. 3adant3 sylanc B dividt B nncnt B nnne0t sylanc x. C / A opr opreq1d A NN e. C CC e. 3ad2ant2 C A divclt 3expb A nncnt A nnne0t jca sylan2 ancoms C / A opr mulid2t syl B NN e. 3adant2 3eqtrd NN eleq1d B / A opr C / B opr nnmulclt syl5bi imp))
+thm (2re () () (e. (2) (RR)) (df-2 ax1re ax1re readdcl eqeltr))
+thm (2cn () () (e. (2) (CC)) (2re recn))
+thm (3re () () (e. (3) (RR)) (df-3 2re ax1re readdcl eqeltr))
+thm (4re () () (e. (4) (RR)) (df-4 3re ax1re readdcl eqeltr))
+thm (5re () () (e. (5) (RR)) (df-5 4re ax1re readdcl eqeltr))
+thm (6re () () (e. (6) (RR)) (df-6 5re ax1re readdcl eqeltr))
+thm (7re () () (e. (7) (RR)) (df-7 6re ax1re readdcl eqeltr))
+thm (8re () () (e. (8) (RR)) (df-8 7re ax1re readdcl eqeltr))
+thm (9re () () (e. (9) (RR)) (df-9 8re ax1re readdcl eqeltr))
+thm (2pos () () (br (0) (<) (2)) (ax1re ax1re lt01 lt01 addgt0i df-2 breqtrr))
+thm (3pos () () (br (0) (<) (3)) (2re ax1re 2pos lt01 addgt0i df-3 breqtrr))
+thm (4pos () () (br (0) (<) (4)) (3re ax1re 3pos lt01 addgt0i df-4 breqtrr))
+thm (5pos () () (br (0) (<) (5)) (4re ax1re 4pos lt01 addgt0i df-5 breqtrr))
+thm (6pos () () (br (0) (<) (6)) (5re ax1re 5pos lt01 addgt0i df-6 breqtrr))
+thm (7pos () () (br (0) (<) (7)) (6re ax1re 6pos lt01 addgt0i df-7 breqtrr))
+thm (8pos () () (br (0) (<) (8)) (7re ax1re 7pos lt01 addgt0i df-8 breqtrr))
+thm (9pos () () (br (0) (<) (9)) (8re ax1re 8pos lt01 addgt0i df-9 breqtrr))
+thm (2nn () () (e. (2) (NN)) (df-2 1nn 1nn 1 1 nnaddclt mp2an eqeltr))
+thm (2p2e4 () () (= (opr (2) (+) (2)) (4)) (df-2 2 + opreq2i df-4 df-3 + 1 opreq1i 2cn 1cn 1cn addass 3eqtr eqtr4))
+thm (2times () ((2times.1 (e. A (CC)))) (= (opr (2) (x.) A) (opr A (+) A)) (df-2 x. A opreq1i 2times.1 1p1times eqtr))
+thm (2timest () () (-> (e. A (CC)) (= (opr (2) (x.) A) (opr A (+) A))) (A A CC e. A 0 if 2 x. opreq2 A A CC e. A 0 if = id A A CC e. A 0 if = id + opreq12d eqeq12d 0cn A elimel 2times dedth))
+thm (times2 () ((times2.1 (e. A (CC)))) (= (opr A (x.) (2)) (opr A (+) A)) (times2.1 2cn mulcom times2.1 2times eqtr))
+thm (3p2e5 () () (= (opr (3) (+) (2)) (5)) (df-4 + 1 opreq1i df-5 df-2 3 + opreq2i 3re recn 1cn 1cn addass eqtr4 3eqtr4r))
+thm (3p3e6 () () (= (opr (3) (+) (3)) (6)) (3p2e5 + 1 opreq1i df-3 3 + opreq2i 3re recn 2cn 1cn addass eqtr4 df-6 3eqtr4))
+thm (4p2e6 () () (= (opr (4) (+) (2)) (6)) (df-5 + 1 opreq1i df-6 df-2 4 + opreq2i 4re recn 1cn 1cn addass eqtr4 3eqtr4r))
+thm (4p3e7 () () (= (opr (4) (+) (3)) (7)) (4p2e6 + 1 opreq1i df-3 4 + opreq2i 4re recn 2cn 1cn addass eqtr4 df-7 3eqtr4))
+thm (4p4e8 () () (= (opr (4) (+) (4)) (8)) (4p3e7 + 1 opreq1i df-4 4 + opreq2i 4re recn 3re recn 1cn addass eqtr4 df-8 3eqtr4))
+thm (5p2e7 () () (= (opr (5) (+) (2)) (7)) (df-6 + 1 opreq1i df-7 df-2 5 + opreq2i 5re recn 1cn 1cn addass eqtr4 3eqtr4r))
+thm (5p3e8 () () (= (opr (5) (+) (3)) (8)) (5p2e7 + 1 opreq1i df-3 5 + opreq2i 5re recn 2cn 1cn addass eqtr4 df-8 3eqtr4))
+thm (5p4e9 () () (= (opr (5) (+) (4)) (9)) (5p3e8 + 1 opreq1i df-4 5 + opreq2i 5re recn 3re recn 1cn addass eqtr4 df-9 3eqtr4))
+thm (6p2e8 () () (= (opr (6) (+) (2)) (8)) (df-7 + 1 opreq1i df-8 df-2 6 + opreq2i 6re recn 1cn 1cn addass eqtr4 3eqtr4r))
+thm (6p3e9 () () (= (opr (6) (+) (3)) (9)) (6p2e8 + 1 opreq1i df-3 6 + opreq2i 6re recn 2cn 1cn addass eqtr4 df-9 3eqtr4))
+thm (7p2e9 () () (= (opr (7) (+) (2)) (9)) (df-8 + 1 opreq1i df-9 df-2 7 + opreq2i 7re recn 1cn 1cn addass eqtr4 3eqtr4r))
+thm (2t2e4 () () (= (opr (2) (x.) (2)) (4)) (2cn 2times 2p2e4 eqtr))
+thm (3t2e6 () () (= (opr (3) (x.) (2)) (6)) (3re recn times2 3p3e6 eqtr))
+thm (3t3e9 () () (= (opr (3) (x.) (3)) (9)) (df-3 3 x. opreq2i 3re recn 2cn 1cn adddi 3t2e6 3re recn mulid1 + opreq12i eqtr 6p3e9 3eqtr))
+thm (4t2e8 () () (= (opr (4) (x.) (2)) (8)) (4re recn times2 4p4e8 eqtr))
+thm (4d2e2 () () (= (opr (4) (/) (2)) (2)) (2t2e4 4re recn 2cn 2cn 2re 2pos gt0ne0i divmul mpbir))
+thm (1lt2 () () (br (1) (<) (2)) (ax1re ltplus1 df-2 breqtrr))
+thm (halfgt0 () () (br (0) (<) (opr (1) (/) (2))) (2re 2pos recgt0i))
+thm (halflt1 () () (br (opr (1) (/) (2)) (<) (1)) (1cn div1 1lt2 eqbrtr ax1re ax1re 2re lt01 2pos ltdiv23i mpbi))
+thm (halfpost () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (opr A (/) (2)) (<) A))) (A A RR e. A 0 if 0 < breq2 A A RR e. A 0 if / 1 + 1 opr opreq1 A A RR e. A 0 if = id < breq12d bibi12d 0re A elimel halfpos dedth df-2 A / opreq2i < A breq1i syl6bbr))
+thm (halfpos2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) (2))))) (2re 2pos A 2 gt0divt mp3an3 mpan2))
+thm (2halvest () () (-> (e. A (CC)) (= (opr (opr A (/) (2)) (+) (opr A (/) (2))) A)) (A 2timest / 2 opreq1d 2cn 2re 2pos gt0ne0i A A 2 divdirt mpan2 mp3an3 anidms eqtr2d 2cn 2re 2pos gt0ne0i 2 A divcan3t mp3an13 eqtrd))
+thm (nominpos ((x y)) () (-. (E.e. x (RR) (/\ (br (0) (<) (cv x)) (-. (E.e. y (RR) (/\ (br (0) (<) (cv y)) (br (cv y) (<) (cv x)))))))) (2pos 2re x cv 2 divgt0t exp mpan2 mpan2i x cv halfpost biimpd jcad 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 jctild y cv x cv / 2 opr 0 < breq2 y cv x cv / 2 opr < x cv breq1 anbi12d RR rcla4ev syl6 0 < x cv br y RR 0 < y cv br y cv < x cv br /\ E.e. iman sylib nrex))
+thm (lbreu ((w x) (w y) (S w) (x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (E!e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (y cv w cv x cv <_ breq2 S rcla4v y cv x cv w cv <_ breq2 S rcla4v im2anan9 ancoms S RR x cv ssel S RR w cv ssel anim12d impcom x cv w cv letri3t syl biimprd exp com23 syld com3r r19.21aivv x S y S x cv <_ y cv br A.e. E.e. anim2i ancoms x cv w cv <_ y cv breq1 y S biraldv S reu4 sylibr))
+thm (lbcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) S)) (S x y lbreu x S y S x cv <_ y cv br A.e. reucl2 x S y S x cv <_ y cv br A.e. ssrab x S y S x cv <_ y cv br A.e. {e.|} U. sseli 3syl))
+thm (lble ((w x) (w y) (S w) (x y) (S x) (S y) (A w) (A y)) () (-> (/\/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. A S)) (br (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (<_) A)) (y S x cv <_ y cv br hbra1 w cv S e. y ax-17 x hbrab hbuni w cv <_ e. y ax-17 w cv A e. y ax-17 hbbr y cv A x S y S x cv <_ y cv br A.e. {e.|} U. <_ breq2 S rcla4 imp S x y lbreu x cv w cv <_ y cv breq1 y S biraldv S cbvreuv sylib w cv x cv <_ y cv breq1 y S biraldv y S x cv <_ y cv br hbra1 w cv S e. y ax-17 x hbrab hbuni hbeleq w cv x S y S x cv <_ y cv br A.e. {e.|} U. <_ y cv breq1 S birald reuuni3 syl sylan2 3impb 3coml))
+thm (lbinfm ((w x) (w y) (w z) (v w) (S w) (x y) (x z) (v x) (S x) (y z) (v y) (S y) (v z) (S z) (S v)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (= (sup S (RR) (`' (<))) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))))) (S x y z cv lble 3expa x S y S x cv <_ y cv br A.e. {e.|} U. z cv lenltt S x y lbcl S RR x S y S x cv <_ y cv br A.e. {e.|} U. ssel2 syldan z cv S e. adantr S RR z cv ssel2 x S y S x cv <_ y cv br A.e. E.e. adantlr sylanc mpbid reex S ssex S V x y S x cv <_ y cv br A.e. rabexg syl x S y S x cv <_ y cv br A.e. {e.|} V uniexg z visset x S y S x cv <_ y cv br A.e. {e.|} U. V z cv V < brcnvg mpan2 3syl negbid x S y S x cv <_ y cv br A.e. E.e. z cv S e. ad2antll mpbird exp r19.21aiv S x y lbcl z cv < `' x S y S x cv <_ y cv br A.e. {e.|} U. br a1d ancrd w cv x S y S x cv <_ y cv br A.e. {e.|} U. z cv < `' breq2 S rcla4ev syl6 z cv RR e. a1d r19.21aiv jca v cv x S y S x cv <_ y cv br A.e. {e.|} U. < `' z cv breq1 negbid z S biraldv v cv x S y S x cv <_ y cv br A.e. {e.|} U. z cv < `' breq2 w S z cv < `' w cv br E.e. imbi1d z RR biraldv anbi12d RR reuuni2 S x y lbcl S RR x S y S x cv <_ y cv br A.e. {e.|} U. ssel x S y S x cv <_ y cv br A.e. E.e. adantr mpd v cv x S y S x cv <_ y cv br A.e. {e.|} U. < `' z cv breq1 negbid z S biraldv v cv x S y S x cv <_ y cv br A.e. {e.|} U. z cv < `' breq2 w S z cv < `' w cv br E.e. imbi1d z RR biraldv anbi12d RR rcla4ev S x y lbcl S RR x S y S x cv <_ y cv br A.e. {e.|} U. ssel x S y S x cv <_ y cv br A.e. E.e. adantr mpd S x y z cv lble 3expa x S y S x cv <_ y cv br A.e. {e.|} U. z cv lenltt S x y lbcl S RR x S y S x cv <_ y cv br A.e. {e.|} U. ssel2 syldan z cv S e. adantr S RR z cv ssel2 x S y S x cv <_ y cv br A.e. E.e. adantlr sylanc mpbid reex S ssex S V x y S x cv <_ y cv br A.e. rabexg syl x S y S x cv <_ y cv br A.e. {e.|} V uniexg z visset x S y S x cv <_ y cv br A.e. {e.|} U. V z cv V < brcnvg mpan2 3syl negbid x S y S x cv <_ y cv br A.e. E.e. z cv S e. ad2antll mpbird exp r19.21aiv S x y lbcl z cv < `' x S y S x cv <_ y cv br A.e. {e.|} U. br a1d ancrd w cv x S y S x cv <_ y cv br A.e. {e.|} U. z cv < `' breq2 S rcla4ev syl6 z cv RR e. a1d r19.21aiv jca sylanc ltso < RR cnvso mpbi v z S w supeu syl sylanc mpbid S RR < `' v z w df-sup syl5eq))
+thm (lbinfmcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (sup S (RR) (`' (<))) S)) (S x y lbinfm S x y lbcl eqeltrd))
+thm (sup2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (x cv peano2re y A y cv < x cv br y cv x cv = \/ A.e. adantr A RR C_ a1i A RR y cv ssel y cv x cv x cv + 1 opr axlttrn 3expb x cv peano2re ancli sylan2 x cv ltplus1t sylan2i exp4b com34 pm2.43d imp x cv ltplus1t y cv x cv < x cv + 1 opr breq1 biimprcd syl y cv RR e. adantl jaod exp syl6 com23 imp a2d y 19.20dv y A y cv < x cv br y cv x cv = \/ df-ral y A y cv < x cv + 1 opr br df-ral 3imtr4g exp imp3a jcad x cv + 1 oprex z cv x cv + 1 opr RR eleq1 z cv x cv + 1 opr y cv < breq2 y A biraldv anbi12d cla4ev syl6 x 19.23adv z cv x cv RR eleq1 z cv x cv y cv < breq2 y A biraldv anbi12d cbvexv syl6ib x RR y A y cv < x cv br y cv x cv = \/ A.e. df-rex x RR y A y cv < x cv br A.e. df-rex 3imtr4g A {/} =/= adantr imdistani A RR C_ A {/} =/= x RR y A y cv < x cv br y cv x cv = \/ A.e. E.e. df-3an A RR C_ A {/} =/= x RR y A y cv < x cv br A.e. E.e. df-3an 3imtr4 A x y z axsup syl))
+thm (sup3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))) (A {/} df-ne bicomi A RR C_ a1i A RR y cv ssel y cv x cv leloet expcom syl9 imp31 biraldva birexdva anbi12d pm5.32i A RR C_ A {/} = -. x RR y A y cv <_ x cv br A.e. E.e. 3anass A RR C_ A {/} =/= x RR y A y cv < x cv br y cv x cv = \/ A.e. E.e. 3anass 3bitr4 A x y z sup2 sylbi))
+thm (infm3lem ((x y)) () (-> (e. (cv x) (RR)) (E.e. y (RR) (= (cv x) (-u (cv y))))) (y cv x cv -u negeq x cv eqeq2d RR rcla4ev x cv renegclt x cv recnt x cv negnegt syl eqcomd sylanc))
+thm (infm3 ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))) (A RR v cv ssel pm4.71rd v biexdv v RR v cv A e. df-rex w cv renegclt v w infm3lem v cv w cv -u A eleq1 rexxfr bitr3 syl6bb A v n0 w RR w cv -u A e. rabn0 3bitr4g A RR y cv ssel pm4.71rd x cv <_ y cv br imbi1d y cv RR e. y cv A e. x cv <_ y cv br impexp syl6bb y bialdv y A x cv <_ y cv br df-ral v cv renegclt y v infm3lem y cv v cv -u A eleq1 y cv v cv -u x cv <_ breq2 imbi12d ralxfr y RR y cv A e. x cv <_ y cv br -> df-ral bitr3 3bitr4g x RR birexdv u cv renegclt x u infm3lem x cv u cv -u <_ v cv -u breq1 v cv -u A e. imbi2d v RR biraldv rexxfr v cv u cv lenegt ancoms v cv -u A e. imbi2d biraldva w cv v cv negeq A eleq1d RR elrab v cv <_ u cv br imbi1i v cv RR e. v cv -u A e. v cv <_ u cv br impexp bitr v bial v w RR w cv -u A e. {e.|} v cv <_ u cv br df-ral v RR v cv -u A e. v cv <_ u cv br -> df-ral 3bitr4r syl5bbr birexa bitr4 syl6bb anbi12d w RR w cv -u A e. ssrab w RR w cv -u A e. {e.|} u v t sup3 mp3an1 syl6bi A RR y cv ssel pm4.71rd y cv < x cv br -. imbi1d y cv RR e. y cv A e. y cv < x cv br -. impexp syl6bb y bialdv y A y cv < x cv br -. df-ral v cv renegclt y v infm3lem y cv v cv -u A eleq1 y cv v cv -u < x cv breq1 negbid imbi12d ralxfr y RR y cv A e. y cv < x cv br -. -> df-ral bitr3 3bitr4g A RR z cv ssel z cv < v cv -u br adantrd pm4.71rd z biexdv z A z cv < v cv -u br df-rex t cv renegclt z t infm3lem z cv t cv -u A eleq1 z cv t cv -u < v cv -u breq1 anbi12d rexxfr z RR z cv A e. z cv < v cv -u br /\ df-rex bitr3 3bitr4g x cv < v cv -u br imbi2d v RR biraldv v cv renegclt y v infm3lem y cv v cv -u x cv < breq2 y cv v cv -u z cv < breq2 z A birexdv imbi12d ralxfr syl5bb anbi12d x RR birexdv u cv renegclt x u infm3lem x cv u cv -u v cv -u < breq2 negbid v cv -u A e. imbi2d v RR biraldv x cv u cv -u < v cv -u breq1 t RR t cv -u A e. t cv -u < v cv -u br /\ E.e. imbi1d v RR biraldv anbi12d rexxfr u cv v cv ltnegt negbid v cv -u A e. imbi2d biraldva w cv v cv negeq A eleq1d RR elrab u cv < v cv br -. imbi1i v cv RR e. v cv -u A e. u cv < v cv br -. impexp bitr v bial v w RR w cv -u A e. {e.|} u cv < v cv br -. df-ral v RR v cv -u A e. u cv < v cv br -. -> df-ral 3bitr4r syl5bbr v cv u cv ltnegt ancoms v cv t cv ltnegt t cv -u A e. anbi2d birexdva w cv t cv negeq A eleq1d RR elrab v cv < t cv br anbi1i t cv RR e. t cv -u A e. v cv < t cv br anass bitr birex2 syl5bb u cv RR e. adantl imbi12d biraldva anbi12d birexa bitr4 syl6bb sylibrd exp3a 3imp))
+thm (suprcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. (sup A (RR) (<)) (RR))) (A x y z sup3 ltso x y A z supcl syl))
+thm (suprub ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. B A)) (br B (<_) (sup A (RR) (<)))) (A x y z sup3 ltso x y A z B supub syl imp B A RR < sup lenltt A RR C_ A {/} = -. x RR y A y cv <_ x cv br A.e. E.e. 3simp1 B sseld imp A x y suprcl B A e. adantr sylanc mpbird))
+thm (suprlub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (br B (<) (sup A (RR) (<))))) (E.e. z A (br B (<) (cv z)))) (A x y z sup3 ltso x y A z B suplub syl imp))
+thm (suprnub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z)))))) (-. (br B (<) (sup A (RR) (<))))) (A x y z sup3 ltso x y A z B supnub syl imp))
+thm (suprleub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B)))) (br (sup A (RR) (<)) (<_) B)) (z cv B lenltt A RR z cv ssel2 sylan an1rs biraldva exp pm5.32d A {/} = -. x RR y A y cv <_ x cv br A.e. E.e. 3ad2ant1 A x y B z suprnub A RR < sup B lenltt A x y suprcl B RR e. z A B < z cv br -. A.e. pm3.26 syl2an mpbird exp sylbid imp))
+thm (sup3i ((x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))) (sup3i.1 A x y z sup3 ax-mp))
+thm (suprcli ((x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (e. (sup A (RR) (<)) (RR)) (ltso sup3i.1 z sup3i supcli))
+thm (suprubi ((x y) (A x) (A y)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (e. B A) (br B (<_) (sup A (RR) (<)))) (sup3i.1 A x y B suprub mpan))
+thm (suprlubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (br B (<) (sup A (RR) (<)))) (E.e. z A (br B (<) (cv z)))) (ltso sup3i.1 z sup3i B suplubi))
+thm (suprnubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z))))) (-. (br B (<) (sup A (RR) (<))))) (ltso sup3i.1 z sup3i B supnubi))
+thm (suprleubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((sup3i.1 (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))))) (-> (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B))) (br (sup A (RR) (<)) (<_) B)) (sup3i.1 A x y B z suprleub mpan))
+thm (reuunineg ((x y) (x z) (y z) (ph y) (ph z) (ps x) (ps z)) ((reuunineg.1 (-> (= (cv x) (-u (cv y))) (<-> ph ps)))) (-> (E!e. x (RR) ph) (= (U. ({e.|} x (RR) ph)) (-u (U. ({e.|} y (RR) ps))))) (z y RR ps hbrab1 hbuni hbneg y cv renegclt y RR ps {e.|} U. renegclt reuunineg.1 y cv y RR ps {e.|} U. negeq x cv renegclt x cv y cv negcon2t x cv recnt y cv recnt syl2an reuhyp reuunixfr))
+thm (infmsup ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (= (sup A (RR) (`' (<))) (-u (sup ({e.|} z (RR) (e. (-u (cv z)) A)) (RR) (<))))) (A x y w infm3 3exp x cv v cv -u y cv < breq2 negbid y A biraldv x cv v cv -u < y cv breq1 w A w cv < y cv br E.e. imbi1d y RR biraldv anbi12d reuunineg A RR < `' x y w df-sup x visset y visset < brcnv negbii y A biral y visset x visset < brcnv y visset w visset < brcnv w A birex imbi12i y RR biral anbi12i x cv RR e. a1i birabi unieqi eqtr syl5eq v cv u cv ltnegt negbid u cv -u A e. imbi2d biraldva z cv u cv negeq A eleq1d RR elrab v cv < u cv br -. imbi1i u cv RR e. u cv -u A e. v cv < u cv br -. impexp bitr u bial u z RR z cv -u A e. {e.|} v cv < u cv br -. df-ral u RR u cv -u A e. v cv < u cv br -. -> df-ral 3bitr4r syl5bbr u cv v cv ltnegt ancoms u cv t cv ltnegt t cv -u A e. anbi2d birexdva z cv t cv negeq A eleq1d RR elrab u cv < t cv br anbi1i t cv RR e. t cv -u A e. u cv < t cv br anass bitr birex2 syl5bb v cv RR e. adantl imbi12d biraldva anbi12d A RR y cv ssel pm4.71rd y cv < v cv -u br -. imbi1d y cv RR e. y cv A e. y cv < v cv -u br -. impexp syl6rbb y bialdv u cv renegclt y u infm3lem y cv u cv -u A eleq1 y cv u cv -u < v cv -u breq1 negbid imbi12d ralxfr y RR y cv A e. y cv < v cv -u br -. -> df-ral bitr3 y A y cv < v cv -u br -. df-ral 3bitr4g A RR w cv ssel w cv < u cv -u br adantrd pm4.71rd w biexdv w A w cv < u cv -u br df-rex t cv renegclt w t infm3lem w cv t cv -u A eleq1 w cv t cv -u < u cv -u breq1 anbi12d rexxfr w RR w cv A e. w cv < u cv -u br /\ df-rex bitr3 3bitr4g v cv -u < u cv -u br imbi2d u RR biraldv u cv renegclt y u infm3lem y cv u cv -u v cv -u < breq2 y cv u cv -u w cv < breq2 w A birexdv imbi12d ralxfr syl5rbb anbi12d sylan9bbr exp birabdv unieqd z RR z cv -u A e. {e.|} RR < v u t df-sup syl5eq negeqd eqcomd sylan9eqr exp ltso < RR cnvso mpbi x y A w supeu x visset y visset < brcnv negbii y A biral y visset x visset < brcnv y visset w visset < brcnv w A birex imbi12i y RR biral anbi12i x RR birex x visset y visset < brcnv negbii y A biral y visset x visset < brcnv y visset w visset < brcnv w A birex imbi12i y RR biral anbi12i x RR bireu 3imtr3 syl5 syl6d 3imp))
+thm (infmrcl ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (e. (sup A (RR) (`' (<))) (RR))) (A x y v infmsup v RR v cv -u A e. ssrab v RR v cv -u A e. {e.|} z w suprcl mp3an1 A RR z cv ssel z cv renegclt syl6 A RR z cv ssel2 z cv recnt z cv negnegt 3syl A RR C_ z cv A e. pm3.27 eqeltrd exp jcad v cv z cv -u negeq A eleq1d RR elrab z cv -u v RR v cv -u A e. {e.|} n0i sylbir syl6 z 19.23adv imp A z n0 sylan2b v RR v cv -u A e. {e.|} {/} = -. id syl x RR y A x cv <_ y cv br A.e. E.e. 3adant3 y cv w cv -u x cv <_ breq2 A rcla4va w cv RR e. adantll x cv RR e. adantll x cv w cv lenegcon2t w cv -u A e. adantrr y A x cv <_ y cv br A.e. adantr mpbid exp31 v cv w cv negeq A eleq1d RR elrab syl5ib com23 r19.21adv x cv renegclt jctild z cv x cv -u w cv <_ breq2 w v RR v cv -u A e. {e.|} biraldv RR rcla4ev syl6 r19.23aiv A RR C_ A {/} = -. 3ad2ant3 sylanc v RR v cv -u A e. {e.|} RR < sup renegclt syl eqeltrd))
+thm (nnunb ((x y) (x z) (y z)) () (-. (E.e. x (RR) (A.e. y (NN) (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))) (y NN x cv < y cv br -. A.e. pm3.24 x cv - 1 oprex y cv x cv - 1 opr RR eleq1 y cv x cv - 1 opr < x cv breq1 y cv x cv - 1 opr < z cv breq1 z NN birexdv imbi12d imbi12d cla4v x cv ltplus1t ax1re x cv 1 x cv ltsubaddt mp3an2 anidms mpbird syl7 ax1re x cv 1 resubclt mpan2 syl5 pm2.43d z NN x cv - 1 opr < z cv br df-rex syl6ib com12 ax1re x cv 1 z cv ltsubaddt mp3an2 z cv nnret sylan2 exp pm5.32d z biexdv z cv + 1 oprex y cv z cv + 1 opr NN eleq1 y cv z cv + 1 opr x cv < breq2 anbi12d cla4ev z cv peano2nn sylan z 19.23aiv syl6bi syld y RR y cv < x cv br z NN y cv < z cv br E.e. -> df-ral y NN x cv < y cv br -. df-ral y y cv NN e. x cv < y cv br alinexa bitr2 bicon1i 3imtr4g y NN x cv < y cv br -. A.e. anim2d mtoi x cv RR e. y NN x cv < y cv br -. A.e. y RR y cv < x cv br z NN y cv < z cv br E.e. -> A.e. /\ imnan mpbi x nex x RR y NN x cv < y cv br -. A.e. y RR y cv < x cv br z NN y cv < z cv br E.e. -> A.e. /\ df-rex mtbir nnssre 1nn 1 NN n0i ax-mp NN {/} df-ne mpbir NN x y z sup2 mp3an12 mto))
+thm (arch ((x y) (A x) (A y)) () (-> (e. A (RR)) (E.e. x (NN) (br A (<) (cv x)))) (y cv A < x cv breq1 x NN birexdv y x nnunb y RR x NN x cv < y cv br x cv y cv = \/ A.e. ralnex mpbir y cv x cv axlttri x cv nnret sylan2 y cv x cv eqcom x cv < y cv br orbi1i x cv y cv = x cv < y cv br orcom bitr negbii syl6bb biimprd r19.22dva x NN x cv < y cv br x cv y cv = \/ rexnal syl5ibr r19.20i ax-mp vtoclri))
+thm (nnreclt ((A x)) () (-> (/\ (e. A (RR)) (br (0) (<) A)) (E.e. x (NN) (br (opr (1) (/) (cv x)) (<) A))) (1 A redivclt 3expa A RR e. 0 < A br pm3.26 ax1re jctil A gt0ne0t sylanc 1 / A opr x arch syl 1 / A opr x cv ltrect an4s 1 A redivclt 3expa A RR e. 0 < A br pm3.26 ax1re jctil A gt0ne0t sylanc A recgt0t jca x cv nnret x cv nngt0t jca syl2an A recrect A recnt 0 < A br adantr A gt0ne0t sylanc 1 / x cv opr < breq2d x cv NN e. adantr bitrd birexdva mpbid))
+thm (avril1 ((F x)) () (-. (/\ (br A (P~ (RR)) (` (i) (1))) (br F ({/}) (opr (0) (x.) (1))))) (x equid x dfnul2 eqabi bicon2i mpbi x cv F 0 <,> {/} eleq1 mtbii V vtocleg F 0 <,> {/} elisset con3i pm2.61i F {/} 0 x. 1 opr df-br 0cn mulid1 0 x. 1 opr 0 F opeq2 ax-mp {/} eleq1i bitr mtbir A R. 0R {} X. P~ y 0R 1R <,> 1 {} " y cv {} = {|} U. br intnan df-i 1 fveq1i 0R 1R <,> 1 y df-fv eqtr A RR P~ breq2i df-r RR R. 0R {} X. z cv sseq2 z biabdv RR z df-pw R. 0R {} X. z df-pw 3eqtr4g ax-mp A y 0R 1R <,> 1 {} " y cv {} = {|} U. breqi bitr F {/} 0 x. 1 opr br anbi1i negbii mpbir))
+thm (elnn0 () () (<-> (e. A (NN0)) (\/ (e. A (NN)) (= A (0)))) (df-n0 A eleq2i A NN 0 {} elun 0cn elisseti A elsnc2 A NN e. orbi2i 3bitr))
+thm (nnssnn0 () () (C_ (NN) (NN0)) (NN 0 {} ssun1 df-n0 sseqtr4))
+thm (nn0ssre () () (C_ (NN0) (RR)) (df-n0 nnssre 0re 0 RR snssi ax-mp unssi eqsstr))
+thm (nn0sscn () () (C_ (NN0) (CC)) (nn0ssre axresscn sstri))
+thm (nn0ex () () (e. (NN0) (V)) (axcnex nn0sscn ssexi))
+thm (nnnn0t () () (-> (e. A (NN)) (e. A (NN0))) (nnssnn0 A sseli))
+thm (nnnn0 () ((nnnn0.1 (e. N (NN)))) (e. N (NN0)) (nnnn0.1 N nnnn0t ax-mp))
+thm (nn0ret () () (-> (e. A (NN0)) (e. A (RR))) (nn0ssre A sseli))
+thm (nn0cnt () () (-> (e. A (NN0)) (e. A (CC))) (nn0sscn A sseli))
+thm (nn0re () ((nn0re.1 (e. A (NN0)))) (e. A (RR)) (nn0ssre nn0re.1 sselii))
+thm (nn0cn () ((nn0re.1 (e. A (NN0)))) (e. A (CC)) (nn0re.1 nn0re recn))
+thm (0nn0 () () (e. (0) (NN0)) (0 eqid 0 0 = 0 NN e. olc ax-mp 0 elnn0 mpbir))
+thm (1nn0 () () (e. (1) (NN0)) (nnssnn0 1nn sselii))
+thm (2nn0 () () (e. (2) (NN0)) (nnssnn0 2nn sselii))
+thm (lt0nnn0 () () (-> (/\ (e. A (RR)) (br A (<) (0))) (-. (e. A (NN0)))) (lt01 0re ax1re A 0 1 axlttrn mp3an23 mpan2i A lt1nnn exp syld con2d A 0 A < breq2 negbid A ltnrt syl5bi com12 jaod A elnn0 syl5ib con2d imp))
+thm (nn0ge0t () () (-> (e. N (NN0)) (br (0) (<_) N)) (N nn0ret N lt0nnn0 exp con2d mpcom 0re 0 N lenltt N nn0ret sylan2 mpan mpbird))
+thm (nn0ge0 () ((nn0ge0.1 (e. N (NN0)))) (br (0) (<_) N) (nn0ge0.1 N nn0ge0t ax-mp))
+thm (nn0addclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0))) (M N nnaddclt M + N opr nnnn0t syl M 0 + N opreq1 N addid2t sylan9eq N nncnt sylan2 N nnnn0t M 0 = adantl eqeltrd N 0 M + opreq2 M ax0id sylan9eqr M nncnt sylan M nnnn0t N 0 = adantr eqeltrd M 0 N 0 + opreq12 0cn addid1 syl6eq 0nn0 M 0 = N 0 = /\ a1i eqeltrd ccase M elnn0 N elnn0 syl2anb))
+thm (nn0addcl () ((nn0addcl.1 (e. M (NN0))) (nn0addcl.2 (e. N (NN0)))) (e. (opr M (+) N) (NN0)) (nn0addcl.1 nn0addcl.2 M N nn0addclt mp2an))
+thm (nn0mulcl () ((nn0addcl.1 (e. M (NN0))) (nn0addcl.2 (e. N (NN0)))) (e. (opr M (x.) N) (NN0)) (nn0addcl.1 M elnn0 mpbi nn0addcl.2 N elnn0 mpbi M N nnmulclt M x. N opr nnnn0t syl M 0 x. N opreq1 nn0addcl.2 nn0cn mul02 syl6eq 0nn0 M 0 = a1i eqeltrd N 0 M x. opreq2 nn0addcl.1 nn0cn mul01 syl6eq 0nn0 N 0 = a1i eqeltrd ccase2 mp2an))
+thm (nn0mulclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0))) (M M NN0 e. M 0 if x. N opreq1 NN0 eleq1d N N NN0 e. N 0 if M NN0 e. M 0 if x. opreq2 NN0 eleq1d 0nn0 M elimel 0nn0 N elimel nn0mulcl dedth2h))
+thm (peano2nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN0))) (1nn0 N 1 nn0addclt mpan2))
+thm (nnnn0addclt () () (-> (/\ (e. M (NN)) (e. N (NN0))) (e. (opr M (+) N) (NN))) (M N nnaddclt exp N 0 M + opreq2 NN eleq1d M nncnt M ax0id syl NN eleq1d ibir syl5bir com12 jaod N elnn0 syl5ib imp))
+thm (nn0p1nnt () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN))) (N nn0cnt 1cn N 1 axaddcom mpan2 syl 1nn 1 N nnnn0addclt mpan eqeltrd))
+thm (nn0ltp1let () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))) (M N nnltp1let M 0 < N breq1 N nngt0t N nnge1t 1cn addid2 syl5eqbr 2th pm5.74ri sylan9bb M 0 + 1 opreq1 <_ N breq1d N NN e. adantr bitr4d N 0 M < breq2 M < 0 br M + 1 opr <_ 0 br pm5.21 M nngt0t M nnret 0re M 0 ltnsymt mpan2 syl mt2d 0re 0 M M + 1 opr axlttrn mp3an1 M nnret M nnret M peano2re syl jca M nngt0t M nnret M ltplus1t syl jca sylc M nnret M peano2re 0re 0 M + 1 opr ltnlet mpan 3syl mpbid sylanc sylan9bbr N 0 M + 1 opr <_ breq2 M NN e. adantl bitr4d N 0 M < breq2 M 0 < 0 breq1 0re ltnr 0re ltplus1 0re 0re ax1re readdcl ltnle mpbi 2false syl6bb M 0 + 1 opreq1 <_ 0 breq1d bitr4d sylan9bbr N 0 M + 1 opr <_ breq2 M 0 = adantl bitr4d ccase M elnn0 N elnn0 syl2anb))
+thm (nn0leltp1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))) (ax1re M N 1 leadd1t mp3an3 M nn0ret N nn0ret syl2an M N + 1 opr nn0ltp1let N peano2nn0 sylan2 bitr4d))
+thm (nn0ltlem1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (M N nn0ltp1let ax1re M N - 1 opr 1 leadd1t mp3an3 M nn0ret N nn0ret ax1re N 1 resubclt mpan2 syl syl2an N nn0cnt 1cn N 1 npcant mpan2 syl M + 1 opr <_ breq2d M NN0 e. adantl bitr2d bitrd))
+thm (nn0addge1t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N))) (A N addge1tOLD 3expb N nn0ret N nn0ge0t jca sylan2))
+thm (nn0addge2t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A))) (A N addge2tOLD 3expb N nn0ret N nn0ge0t jca sylan2))
+thm (nn0addge1 () ((nn0addge1.1 (e. A (RR))) (nn0addge1.2 (e. N (NN0)))) (br A (<_) (opr A (+) N)) (nn0addge1.1 nn0addge1.2 A N nn0addge1t mp2an))
+thm (nn0addge2 () ((nn0addge1.1 (e. A (RR))) (nn0addge1.2 (e. N (NN0)))) (br A (<_) (opr N (+) A)) (nn0addge1.1 nn0addge1.2 A N nn0addge2t mp2an))
+thm (nn0le2x () ((nn0le2x.1 (e. N (NN0)))) (br N (<_) (opr (2) (x.) N)) (nn0le2x.1 nn0re nn0le2x.1 nn0addge1 nn0le2x.1 nn0cn 2times breqtrr))
+thm (nn0lele2x () ((nn0lele2x.1 (e. M (NN0))) (nn0lele2x.2 (e. N (NN0)))) (-> (br N (<_) M) (br N (<_) (opr (2) (x.) M))) (nn0lele2x.1 nn0le2x nn0lele2x.2 nn0re nn0lele2x.1 nn0re 2re nn0lele2x.1 nn0re remulcl letr mpan2))
+thm (elz ((N x)) () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/\/ (= N (0)) (e. N (NN)) (e. (-u N) (NN))))) (x df-z N eleq2i x cv N 0 eqeq1 x cv N NN eleq1 x cv N negeq NN eleq1d 3orbi123d RR elrab bitr))
+thm (nnnegz () () (-> (e. N (NN)) (e. (-u N) (ZZ))) (N nnret N renegclt syl N nncnt N negnegt NN eleq1d biimprd mpcom N -u -u NN e. N -u 0 = N -u NN e. \/ olc syl N -u 0 = N -u NN e. N -u -u NN e. df-3or sylibr jca N -u elz sylibr))
+thm (zret () () (-> (e. N (ZZ)) (e. N (RR))) (N elz pm3.26bd))
+thm (zcnt () () (-> (e. N (ZZ)) (e. N (CC))) (N zret recnd))
+thm (zre () ((zre.1 (e. A (ZZ)))) (e. A (RR)) (zre.1 A zret ax-mp))
+thm (zssre () () (C_ (ZZ) (RR)) (x cv zret ssriv))
+thm (zsscn () () (C_ (ZZ) (CC)) (x cv zcnt ssriv))
+thm (zex () () (e. (ZZ) (V)) (axcnex zsscn ssexi))
+thm (elnnz () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (0) (<) N))) (N nnret N NN e. N -u NN e. N 0 = \/ orc jca N nngt0t jca N RR e. 0 < N br /\ N NN e. idd N lt0neg2t N renegclt 0re N -u 0 ltnsymt mpan2 syl sylbid imp N -u nngt0t nsyl 0re 0 N ltnet mpan imp N 0 eqcom negbii sylibr jca N -u NN e. N 0 = ioran sylibr N NN e. pm2.21d jaod exp com23 imp31 impbi N elz N 0 = N NN e. N -u NN e. 3orrot N NN e. N -u NN e. N 0 = 3orass bitr N RR e. anbi2i bitr 0 < N br anbi1i bitr4))
+thm (0z () () (e. (0) (ZZ)) (0 elz 0re 0 eqid 0 0 = 0 NN e. 0 -u NN e. \/ orc ax-mp 0 0 = 0 NN e. 0 -u NN e. 3orass mpbir mpbir2an))
+thm (elnn0z () () (<-> (e. N (NN0)) (/\ (e. N (ZZ)) (br (0) (<_) N))) (N elnn0 N elnnz N 0 eqcom orbi12i N ZZ e. id 0z 0 N ZZ eleq1 mpbii jaoi N ZZ e. 0 N = orc impbi 0 < N br 0 N = \/ anbi1i N ZZ e. 0 < N br 0 N = ordir N zret 0re 0 N leloet mpan syl pm5.32i 3bitr4 3bitr))
+thm (elznn0nn () () (<-> (e. N (ZZ)) (\/ (e. N (NN0)) (/\ (e. N (RR)) (e. (-u N) (NN))))) (N elz N 0 = N NN e. N -u NN e. df-3or N RR e. anbi2i N RR e. N 0 = N NN e. \/ N -u NN e. andi N nn0ret pm4.71ri N elnn0 N NN e. N 0 = orcom bitr N RR e. anbi2i bitr N RR e. N -u NN e. /\ orbi1i bitr4 3bitr))
+thm (elznn0 () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN0)) (e. (-u N) (NN0))))) (N elz N elnn0 N RR e. a1i N recnt 0cn N 0 negcon1t mpan2 syl neg0 N eqeq1i 0 N eqcom bitr syl6bb N -u NN e. orbi2d N -u elnn0 syl5bb orbi12d N 0 = N NN e. N -u NN e. 3orass N 0 = N NN e. N -u NN e. \/ orcom N NN e. N -u NN e. N 0 = orordir bitr bitr2 syl6rbb pm5.32i bitr))
+thm (elznn () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN)) (e. (-u N) (NN0))))) (N elz N recnt N negeq0t syl N -u NN e. orbi2d N -u elnn0 syl6rbbr N NN e. orbi2d N 0 = N NN e. N -u NN e. 3orrot N NN e. N -u NN e. N 0 = 3orass bitr syl6rbbr pm5.32i bitr))
+thm (nnssz () () (C_ (NN) (ZZ)) (x cv elnnz pm3.26bd ssriv))
+thm (nn0ssz () () (C_ (NN0) (ZZ)) (df-n0 nnssz 0z 0z elisseti ZZ snss mpbi unssi eqsstr))
+thm (nnzt () () (-> (e. N (NN)) (e. N (ZZ))) (nnssz N sseli))
+thm (nn0zt () () (-> (e. N (NN0)) (e. N (ZZ))) (nn0ssz N sseli))
+thm (elnnz1 () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (1) (<_) N))) (N nnzt N nnge1t jca lt01 ax1re 1 lt0neg2t ax-mp mpbi ax1re 1 N lenegt mpan N renegclt ax1re renegcl 0re N -u 1 -u 0 lelttrt mp3an23 0re N -u 0 ltlet mpan2 syld syl exp3a sylbid imp mpi N renegclt 0re N -u 0 lenltt mpan2 syl 1 <_ N br adantr mpbid N zret sylan N -u nngt0t nsyl lt01 N 0 1 <_ breq2 ax1re 0re lenlt syl6bb bicon2d mpbii con2i N ZZ e. adantl jca N -u NN e. N 0 = ioran sylibr N elz N RR e. N 0 = N NN e. N -u NN e. \/\/ pm3.27 N -u NN e. N 0 = N NN e. 3orrot N -u NN e. N 0 = N NN e. df-3or bitr3 sylib sylbi ord 1 <_ N br adantr mpd impbi))
+thm (znnnlt1t () () (-> (e. N (ZZ)) (<-> (-. (e. N (NN))) (br N (<) (1)))) (N elnnz1 baib negbid N zret ax1re N 1 ltnlet mpan2 syl bitr4d))
+thm (nnz () () (= (NN) ({e.|} x (ZZ) (br (1) (<_) (cv x)))) (x cv elnnz1 biabri x ZZ 1 <_ x cv br df-rab eqtr4))
+thm (nn0z () () (= (NN0) ({e.|} x (ZZ) (br (0) (<_) (cv x)))) (x cv elnn0z biabri x ZZ 0 <_ x cv br df-rab eqtr4))
+thm (1z () () (e. (1) (ZZ)) (1nn 1 nnzt ax-mp))
+thm (2z () () (e. (2) (ZZ)) (2nn 2 nnzt ax-mp))
+thm (nn0subt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))) (M N nnsubt exp N 0 M < breq2 N 0 - M opreq1 NN eleq1d bibi12d M nnret M lt0neg1t syl M nnnegz M -u elnnz baib syl M df-neg NN eleq1i syl5rbbr bitrd syl5bir com12 jaod M 0 < N breq1 M 0 N - opreq2 NN eleq1d bibi12d N nnzt N zcnt N subid1t NN eleq1d syl N elnnz baib bitr2d syl syl5bir M 0 < N breq1 M 0 N - opreq2 NN eleq1d bibi12d 0re ltnr 0nnn 0cn subid NN eleq1i mtbir 2false N 0 0 < breq2 N 0 - 0 opreq1 NN eleq1d bibi12d mpbiri syl5bir jaod jaoi imp N M subeq0t M N eqcom syl6rbbr ancoms M nncnt 0cn M 0 CC eleq1 mpbiri jaoi N nncnt 0cn N 0 CC eleq1 mpbiri jaoi syl2an orbi12d M N leloet M nnret 0re M 0 RR eleq1 mpbiri jaoi N nnret 0re N 0 RR eleq1 mpbiri jaoi syl2an N - M opr elnn0 M NN e. M 0 = \/ N NN e. N 0 = \/ /\ a1i 3bitr4d M elnn0 N elnn0 syl2anb))
+thm (nn0sub2t () () (-> (/\/\ (e. M (NN0)) (e. N (NN0)) (br M (<_) N)) (e. (opr N (-) M) (NN0))) (M N nn0subt biimp3a))
+thm (znegclt () () (-> (e. N (ZZ)) (e. (-u N) (ZZ))) (N elz 0z N 0 negeq neg0 syl6eq ZZ eleq1d mpbiri N nnnegz nnssz N -u sseli 3jaoi N RR e. adantl sylbi))
+thm (nn0negz () () (-> (e. N (NN0)) (e. (-u N) (ZZ))) (N nn0zt N znegclt syl))
+thm (zaddclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (+) N) (ZZ))) (M N nn0addclt M + N opr NN0 e. M + N opr -u NN0 e. orc syl M RR e. N RR e. /\ a1i M N axaddrcl jctild M -u N letrit M renegclt sylan M -u NN0 e. N NN0 e. /\ adantr M -u N nn0subt N M subneg2t ancoms M N axaddcom eqtr4d M recnt N recnt syl2an NN0 eleq1d sylan9bbr N M -u nn0subt ancoms M -u N subnegt M negclt sylan M N negdit eqtr4d M recnt N recnt syl2an NN0 eleq1d sylan9bbr orbi12d mpbid exp M N axaddrcl jctild N -u M letrit N renegclt sylan ancoms M NN0 e. N -u NN0 e. /\ adantr N -u M nn0subt ancoms M N subneg2t M recnt N recnt syl2an NN0 eleq1d sylan9bbr M N -u nn0subt N -u M subnegt ancoms M -u N -u axaddcom M negclt sylan eqtr4d N negclt sylan2 M N negdit eqtr4d M recnt N recnt syl2an NN0 eleq1d sylan9bbr orbi12d mpbid exp M N axaddrcl jctild M N negdit M recnt N recnt syl2an NN0 eleq1d M -u N -u nn0addclt syl5bir M + N opr -u NN0 e. M + N opr NN0 e. olc syl6 M N axaddrcl jctild ccased M + N opr elznn0 syl6ibr imp an4s M elznn0 N elznn0 syl2anb))
+thm (peano2z () () (-> (e. N (ZZ)) (e. (opr N (+) (1)) (ZZ))) (1z N 1 zaddclt mpan2))
+thm (zsubclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (-) N) (ZZ))) (M N subnegt M zcnt N zcnt syl2an M N -u zaddclt N znegclt sylan2 eqeltrd))
+thm (peano2zr () () (-> (e. N (ZZ)) (e. (opr N (-) (1)) (ZZ))) (1z N 1 zsubclt mpan2))
+thm (zrevaddclt () () (-> (e. N (ZZ)) (<-> (/\ (e. M (CC)) (e. (opr M (+) N) (ZZ))) (e. M (ZZ)))) (M N pncant N zcnt sylan2 ancoms M + N opr ZZ e. adantr M + N opr N zsubclt ancoms M CC e. adantlr eqeltrrd exp M N zaddclt expcom M CC e. adantr impbid exp pm5.32d M zcnt pm4.71ri syl6bbr))
+thm (elnn0nn () () (<-> (e. N (NN0)) (/\ (e. N (CC)) (e. (opr N (+) (1)) (NN)))) (N peano2nn 1nn N 0 + 1 opreq1 1cn addid2 syl6eq NN eleq1d mpbiri jaoi N RR e. a1i N recnt 1z 1 N zrevaddclt ax-mp biimp exp syl 0re ax1re 0 N 1 leadd1t mp3an13 1cn addid2 <_ N + 1 opr breq1i syl6bb 0re 0 N leloet mpan bitr3d biimpd anim12d N ZZ e. 0 < N br 0 N = andi N elnnz N 0 eqcom 0z 0 N ZZ eleq1 mpbii pm4.71ri bitr orbi12i bitr4 syl6ib N + 1 opr elnnz1 syl5ib impbid N elnn0 syl5bb pm5.32i N nn0ret pm4.71ri 1z 1 N zrevaddclt ax-mp N zret sylbi N + 1 opr nnzt sylan2 N CC e. N + 1 opr NN e. pm3.27 jca N recnt N + 1 opr NN e. anim1i impbi 3bitr4))
+thm (elnnnn0 () () (<-> (e. N (NN)) (/\ (e. N (CC)) (e. (opr N (-) (1)) (NN0)))) (N nncnt pm4.71ri 1cn N 1 npcant mpan2 NN eleq1d 1cn N 1 subclt mpan2 N - 1 opr + 1 opr NN e. biantrurd bitr3d N - 1 opr elnn0nn syl6bbr pm5.32i bitr))
+thm (elnnnn0b () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (0) (<) N))) (N nnnn0t N nngt0t jca N elnnz biimpr N nn0zt sylan impbi))
+thm (znnsubt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (e. (opr N (-) M) (NN)))) (M N posdift M zret N zret syl2an N M zsubclt ancoms 0 < N - M opr br biantrurd bitrd N - M opr elnnz syl6bbr))
+thm (znn0subt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))) (M N nnegdift M zret N zret syl2an N M zsubclt ancoms 0 <_ N - M opr br biantrurd bitrd N - M opr elnn0z syl6bbr))
+thm (znn0sub2t () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. (opr N (-) M) (NN0))) (M N znn0subt biimp3a))
+thm (zmulclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (x.) N) (ZZ))) (M N nn0mulclt M x. N opr NN0 e. M x. N opr -u NN0 e. orc syl M RR e. N RR e. /\ a1i M N axmulrcl jctild M N mulneg1t M recnt N recnt syl2an NN0 eleq1d M -u N nn0mulclt syl5bi M x. N opr -u NN0 e. M x. N opr NN0 e. olc syl6 M N axmulrcl jctild M N mulneg2t M recnt N recnt syl2an NN0 eleq1d M N -u nn0mulclt syl5bi M x. N opr -u NN0 e. M x. N opr NN0 e. olc syl6 M N axmulrcl jctild M N mul2negt M recnt N recnt syl2an NN0 eleq1d M -u N -u nn0mulclt syl5bi M x. N opr NN0 e. M x. N opr -u NN0 e. orc syl6 M N axmulrcl jctild ccased M x. N opr elznn0 syl6ibr imp an4s M elznn0 N elznn0 syl2anb))
+thm (zltp1let () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))) (M N nn0ltp1let M RR e. N RR e. /\ a1i M recnt 0cn M 0 negcon1t mpan2 neg0 M eqeq1i 0 M eqcom bitr syl6bb syl M -u NN e. orbi2d M -u elnn0 syl5bb N elnn0 M RR e. a1i anbi12d N RR e. adantr M lt0neg1t M -u nngt0t syl5bir imp N nngt0t anim12i 0re M 0 N axlttrn mp3an2 N nnret sylan2 M -u NN e. adantlr mpd M peano2re M -u NN e. N NN e. ad2antll ax1re M RR e. M -u NN e. /\ N NN e. /\ a1i N nnret M RR e. M -u NN e. /\ adantl M lt0neg1t M -u nngt0t syl5bir imp 0re M 0 ltlet mpan2 M -u NN e. adantr mpd 0re ax1re M 0 1 leadd1t mp3an23 M -u NN e. adantr mpbid 1cn addid2 syl6breq N NN e. adantr N nnge1t M RR e. M -u NN e. /\ adantl letrd jca exp31 imp3a M < N br M + 1 opr <_ N br pm5.1 syl6 N RR e. adantr M 0 < N breq1 M 0 + 1 opreq1 1cn addid2 syl6eq <_ N breq1d bibi12d 0 < N br 1 <_ N br pm5.1 N nngt0t N nnge1t sylanc syl5bir imp M RR e. N RR e. /\ a1i N 0 M < breq2 N 0 M + 1 opr <_ breq2 bibi12d M -u nnnn0t 0nn0 0 M -u nn0ltlem1t mpan syl M RR e. adantl M lt0neg1t M -u NN e. adantr M peano2re M + 1 opr le0neg1t syl M recnt 1cn M 1 negdit M -u 1 subnegt M negclt sylan eqtr4d mpan2 syl 0 <_ breq2d bitrd M -u NN e. adantr 3bitr4d syl5bir com12 exp imp3a N RR e. adantr 0re ltnr lt01 0re ax1re ltnle mpbi 2false M 0 < N breq1 M 0 + 1 opreq1 1cn addid2 syl6eq <_ N breq1d bibi12d N 0 0 < breq2 N 0 1 <_ breq2 bibi12d sylan9bb mpbiri M RR e. N RR e. /\ a1i ccased sylbid M RR e. N RR e. pm3.27 M NN0 e. N -u NN0 e. /\ adantr 0re M RR e. N RR e. /\ M NN0 e. N -u NN0 e. /\ /\ a1i M RR e. N RR e. pm3.26 M NN0 e. N -u NN0 e. /\ adantr N -u nn0ge0t N RR e. adantl N le0neg1t N -u NN0 e. adantr mpbird M NN0 e. adantrl M RR e. adantll M nn0ge0t M RR e. N RR e. /\ N -u NN0 e. ad2antrl letrd N M lenltt ancoms M NN0 e. N -u NN0 e. /\ adantr mpbid M RR e. N RR e. pm3.27 M NN0 e. N -u NN0 e. /\ adantr 0re M RR e. N RR e. /\ M NN0 e. N -u NN0 e. /\ /\ a1i M RR e. N RR e. pm3.26 M NN0 e. N -u NN0 e. /\ adantr N -u nn0ge0t N RR e. adantl N le0neg1t N -u NN0 e. adantr mpbird M NN0 e. adantrl M RR e. adantll M nn0ge0t M RR e. N RR e. /\ N -u NN0 e. ad2antrl letrd M ltplus1t N RR e. M NN0 e. N -u NN0 e. /\ ad2antll N M M + 1 opr lelttrt 3expb M peano2re ancli sylan2 ancoms M NN0 e. N -u NN0 e. /\ adantr mp2and N M + 1 opr ltnlet M peano2re sylan2 ancoms M NN0 e. N -u NN0 e. /\ adantr mpbid jca exp M < N br M + 1 opr <_ N br pm5.21 syl6 N -u M -u nn0ltlem1t ancoms M RR e. N RR e. /\ adantl M N ltnegt M -u NN0 e. N -u NN0 e. /\ adantr M + 1 opr N lenegt M peano2re sylan M recnt 1cn M 1 negdit mpan2 M negclt 1cn M -u 1 subnegt mpan2 syl eqtr4d syl N -u <_ breq2d N RR e. adantr bitrd M -u NN0 e. N -u NN0 e. /\ adantr 3bitr4d exp ccased imp an4s M elznn0 N elznn0 syl2anb))
+thm (zleltp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))) (ax1re M N 1 leadd1t mp3an3 M zret N zret syl2an M N + 1 opr zltp1let N peano2z sylan2 bitr4d))
+thm (zlem1ltt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) (M - 1 opr N zltp1let M peano2zr sylan M zcnt 1cn M 1 npcant mpan2 syl N ZZ e. adantr <_ N breq1d bitr2d))
+thm (zltlem1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (M N - 1 opr zleltp1t N peano2zr sylan2 N zcnt 1cn N 1 npcant mpan2 syl M ZZ e. adantl M < breq2d bitr2d))
+thm (nn0lem1ltt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) (M N zlem1ltt M nn0zt N nn0zt syl2an))
+thm (nnlem1ltt () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))) (M N zlem1ltt M nnzt N nnzt syl2an))
+thm (nnltlem1t () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))) (M N zltlem1t M nnzt N nnzt syl2an))
+thm (z2get ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (E.e. k (ZZ) (/\ (br M (<_) (cv k)) (br N (<_) (cv k))))) (k cv M <_ N br N M if M <_ breq2 k cv M <_ N br N M if N <_ breq2 anbi12d ZZ rcla4ev N ZZ M M <_ N br ifcl ancoms M N max1 M N max2 jca M zret N zret syl2an sylanc))
+thm (recnzt () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (-. (e. (opr (1) (/) A) (ZZ)))) (A recgt1t pm3.27d A recgt1t pm3.26d 0z 0 1 / A opr zltp1let mpan 1cn addid2 <_ 1 / A opr breq1i syl6bb biimpcd syl lt01 0re ax1re 0 1 A axlttrn mp3an12 mpani imdistani A gt0ne0t syl A rerecclt syldan ax1re 1 1 / A opr lenltt mpan syl sylibd mt2d))
+thm (btwnnzt () () (-> (/\/\ (e. A (ZZ)) (br A (<) B) (br B (<) (opr A (+) (1)))) (-. (e. B (ZZ)))) (A B zltp1let A + 1 opr B lenltt A peano2z A + 1 opr zret syl B zret syl2an bitrd biimpd exp com23 imp con2d exp 3imp))
+thm (gtndivt () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br B (<) A)) (-. (e. (opr B (/) A) (ZZ)))) (0z 0 B / A opr btwnnzt mp3an1 B A divgt0t B nnret A RR e. B < A br 3ad2ant2 A RR e. B NN e. B < A br 3simp1 jca B nngt0t A RR e. B < A br 3ad2ant2 B nngt0t A RR e. adantl 0re 0 B A axlttrn mp3an1 B nnret sylan ancoms mpand imp 3impa jca sylanc A RR e. B NN e. B < A br 3simp3 ax1re B A 1 ltdivmul2t mp3anl3 B nnret A RR e. B < A br 3ad2ant2 A RR e. B NN e. B < A br 3simp1 jca B nngt0t A RR e. adantl 0re 0 B A axlttrn mp3an1 B nnret sylan ancoms mpand imp 3impa sylanc A recnt A mulid2t syl B < breq2d B NN e. B < A br 3ad2ant1 bitrd mpbird 1cn addid2 syl6breqr sylanc))
+thm (halfnz () () (-. (e. (opr (1) (/) (2)) (ZZ))) (2re 1lt2 2 recnzt mp2an))
+thm (primet ((A x)) () (-> (e. A (NN)) (<-> (A.e. x (NN) (-> (e. (opr A (/) (cv x)) (NN)) (\/ (= (cv x) (1)) (= (cv x) A)))) (A.e. x (NN) (-> (/\/\ (br (1) (<) (cv x)) (br (cv x) (<_) A) (e. (opr A (/) (cv x)) (NN))) (= (cv x) A))))) (x cv nngt1ne1t A NN e. adantl A / x cv opr NN e. anbi1d x cv A gtndivt 3expa exp x cv nnret sylan con2d x cv A lenltt x cv nnret A nnret syl2an sylibrd ancoms A / x cv opr nnzt syl5 pm4.71rd 1 < x cv br anbi2d 1 < x cv br x cv <_ A br A / x cv opr NN e. 3anass syl6bbr bitr3d x cv A = imbi1d x cv 1 = -. A / x cv opr NN e. x cv A = bi2.04 x cv 1 = -. A / x cv opr NN e. x cv A = impexp x cv 1 = x cv A = df-or A / x cv opr NN e. imbi2i 3bitr4r syl5bb biraldva))
+thm (sqznn () () (-> (/\ (e. A (ZZ)) (=/= A (0))) (e. (opr A (x.) A) (NN))) (A A zmulclt anidms A 0 =/= adantr A ltsqt A 0 df-ne syl5ib imp A zret sylan jca A x. A opr elnnz sylibr))
+thm (zneo () () (-> (/\ (e. A (ZZ)) (e. B (ZZ))) (-. (= (opr (2) (x.) A) (opr (opr (2) (x.) B) (+) (1))))) (halfnz B zcnt 2cn 2re 2pos gt0ne0i reccl B 1 / 2 opr axaddcom mpan2 - B opreq1d 2cn 2re 2pos gt0ne0i reccl 1 / 2 opr B pncant mpan eqtrd syl ZZ eleq1d mtbiri B + 1 / 2 opr opr B zsubclt expcom mtod A ZZ e. adantl 2cn 2re 2pos gt0ne0i 2 A divcan3t mp3an13 2cn 2 B axmulcl mpan 1cn 2cn 2re 2pos gt0ne0i 2 x. B opr 1 2 divdirt mpan2 mp3an23 syl 2cn 2re 2pos gt0ne0i 2 B divcan3t mp3an13 + 1 / 2 opr opreq1d eqtrd eqeqan12d 2 x. A opr 2 x. B opr + 1 opr / 2 opreq1 syl5bi A zcnt B zcnt syl2an imp ZZ eleq1d exp31 com3l A ZZ e. B + 1 / 2 opr opr ZZ e. ibib syl6ibr com3r imp mtod))
+thm (peano2uz2 ((A x) (B x)) () (-> (/\ (e. A (ZZ)) (e. B ({e.|} x (ZZ) (br A (<_) (cv x))))) (e. (opr B (+) (1)) ({e.|} x (ZZ) (br A (<_) (cv x))))) (B peano2z A ZZ e. A <_ B br ad2antrl B B + 1 opr ltlet B peano2re ancli B ltplus1t sylc A RR e. adantl A B B + 1 opr letrt 3expb B peano2re ancli sylan2 mpan2d A zret B zret syl2an exp imp32 jca x cv B A <_ breq2 ZZ elrab A ZZ e. anbi2i x cv B + 1 opr A <_ breq2 ZZ elrab 3imtr4))
+thm (dfuz ((x y) (x z) (w x) (v x) (u x) (m x) (N x) (y z) (w y) (v y) (u y) (m y) (N y) (w z) (v z) (u z) (m z) (N z) (v w) (u w) (m w) (N w) (u v) (m v) (N v) (m u) (N u) (N m)) ((dfuz.1 (e. N (ZZ)))) (= ({e.|} z (ZZ) (br N (<_) (cv z))) (|^| ({|} x (/\ (e. N (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))) (z cv m cv N <_ breq2 ZZ elrab pm3.26bd zex x cv ZZ N eleq2 x cv ZZ y cv + 1 opr eleq2 y raleqd anbi12d dfuz.1 y cv peano2z rgen pm3.2i V intmin3 ax-mp m cv sseli u cv m cv e. z ax-17 u cv m cv + 1 - N opr opr e. z ax-17 1z dfuz.1 1 N zsubclt mp2an z cv 1 - N opr zaddclt mpan2 v cv z cv + 1 - N opr opr 1 <_ breq2 z cv m cv + 1 - N opr opreq1 rabxfr z cv zret ax1re dfuz.1 zre resubcl dfuz.1 zre N z cv 1 - N opr leadd1t mp3an1 mpan2 syl dfuz.1 zre recn 1cn pncan3 <_ z cv + 1 - N opr opr breq1i syl6bb birabi m cv eleq2i syl6bbr 1 w cv e. v w cv v cv + 1 opr w cv e. A.e. pm3.26 v cv y cv + 1 - N opr opr + 1 opreq1 w cv eleq1d w cv rcla4v y cv recnt 1cn ax1re dfuz.1 zre resubcl recn y cv 1 - N opr 1 add23t mp3an2 mpan2 syl w cv eleq1d biimpd y cv peano2re jctild sylan9r com12 y 19.21aiv 1 w cv e. adantl jca m cv RR e. m cv + 1 - N opr opr w cv e. pm3.27 imim12i m cv ZZ e. a1i reex u u cv + 1 - N opr opr w cv e. rabex x cv u RR u cv + 1 - N opr opr w cv e. {e.|} N eleq2 u cv N + 1 - N opr opreq1 dfuz.1 zre recn 1cn pncan3 syl6eq w cv eleq1d RR elrab dfuz.1 zre mpbiran syl6bb x cv u RR u cv + 1 - N opr opr w cv e. {e.|} y cv eleq2 u cv y cv + 1 - N opr opreq1 w cv eleq1d RR elrab syl6bb x cv u RR u cv + 1 - N opr opr w cv e. {e.|} y cv + 1 opr eleq2 u cv y cv + 1 opr + 1 - N opr opreq1 w cv eleq1d RR elrab syl6bb imbi12d y bialdv y x cv y cv + 1 opr x cv e. df-ral syl5bb anbi12d x cv u RR u cv + 1 - N opr opr w cv e. {e.|} m cv eleq2 u cv m cv + 1 - N opr opreq1 w cv eleq1d RR elrab syl6bb imbi12d cla4v syl5 w 19.21adv N x cv e. y x cv y cv + 1 opr x cv e. A.e. pm3.26 y cv v cv - 1 - N opr opr + 1 opreq1 x cv eleq1d x cv rcla4v v cv recnt 1cn ax1re dfuz.1 zre resubcl recn v cv 1 1 - N opr addsubt mp3an3 mpan2 syl x cv eleq1d biimprd v cv peano2re jctild sylan9r com12 v 19.21aiv N x cv e. adantl jca m cv ZZ e. a1i m cv zcnt ax1re dfuz.1 zre resubcl recn m cv 1 - N opr pncant mpan2 syl x cv eleq1d biimpd m cv + 1 - N opr opr RR e. adantld imim12d reex u u cv - 1 - N opr opr x cv e. rabex w cv u RR u cv - 1 - N opr opr x cv e. {e.|} 1 eleq2 u cv 1 - 1 - N opr opreq1 1cn dfuz.1 zre recn 1 N nncant mp2an syl6eq x cv eleq1d RR elrab ax1re mpbiran syl6bb w cv u RR u cv - 1 - N opr opr x cv e. {e.|} v cv eleq2 u cv v cv - 1 - N opr opreq1 x cv eleq1d RR elrab syl6bb w cv u RR u cv - 1 - N opr opr x cv e. {e.|} v cv + 1 opr eleq2 u cv v cv + 1 opr - 1 - N opr opreq1 x cv eleq1d RR elrab syl6bb imbi12d v bialdv v w cv v cv + 1 opr w cv e. df-ral syl5bb anbi12d w cv u RR u cv - 1 - N opr opr x cv e. {e.|} m cv + 1 - N opr opr eleq2 u cv m cv + 1 - N opr opr - 1 - N opr opreq1 x cv eleq1d RR elrab syl6bb imbi12d cla4v syl5 x 19.21adv impbid m visset x N x cv e. y x cv y cv + 1 opr x cv e. A.e. /\ elintab m cv + 1 - N opr oprex w 1 w cv e. v w cv v cv + 1 opr w cv e. A.e. /\ elintab 3bitr4g w v df-n v nnz eqtr3 m cv + 1 - N opr opr eleq2i syl6rbb bitr3d pm5.21nii eqri))
+thm (peano5uz ((x y) (A x) (A y) (k x) (N x) (k y) (N y) (N k)) ((peano5uz.1 (e. A (V))) (peano5uz.2 (e. N (ZZ)))) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A)) (peano5uz.1 y cv A N eleq2 y cv A x cv + 1 opr eleq2 x raleqd anbi12d elab A y N y cv e. x y cv x cv + 1 opr y cv e. A.e. /\ {|} intss1 sylbir peano5uz.2 k y x dfuz syl5ss))
+thm (peano5uzt ((A x) (k x) (N x) (N k)) ((peano5uz.1 (e. A (V)))) (-> (e. N (ZZ)) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A))) (N N ZZ e. N 1 if A eleq1 x A x cv + 1 opr A e. A.e. anbi1d N N ZZ e. N 1 if <_ k cv breq1 k ZZ birabsdv A sseq1d imbi12d peano5uz.1 1z N elimel x k peano5uz dedth))
+thm (uzind ((j w) (N j) (N w) (j ps) (ch j) (j th) (j ta) (k w) (k ph) (ph w) (j k) (M j) (M k) (M w)) ((uzind.1 (-> (= (cv j) M) (<-> ph ps))) (uzind.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind.4 (-> (= (cv j) N) (<-> ph ta))) (uzind.5 (-> (e. M (ZZ)) ps)) (uzind.6 (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<_) (cv k))) (-> ch th)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) ta) (M zret M leidt syl uzind.5 jca ancli j cv M M <_ breq2 uzind.1 anbi12d ZZ elrab sylibr k cv peano2z M ZZ e. a1i M <_ k cv br ch /\ adantrd k cv ltplus1t M RR e. adantl M k cv k cv + 1 opr lelttrt 3expb k cv peano2re ancli sylan2 mpan2d M k cv + 1 opr ltlet k cv peano2re sylan2 syld M zret k cv zret syl2an ch adantrd exp4b imp4d uzind.6 3exp imp4d jcad jcad j cv k cv M <_ breq2 uzind.2 anbi12d ZZ elrab j cv k cv + 1 opr M <_ breq2 uzind.3 anbi12d ZZ elrab 3imtr4g r19.21aiv zex j M <_ j cv br ph /\ rabex M k w peano5uzt mp2and N sseld w cv N M <_ breq2 ZZ elrab j cv N M <_ breq2 uzind.4 anbi12d ZZ elrab 3imtr3g exp3a 3imp pm3.27d pm3.27d))
+thm (uzind2 ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((uzind2.1 (-> (= (cv j) (opr M (+) (1))) (<-> ph ps))) (uzind2.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind2.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind2.4 (-> (= (cv j) N) (<-> ph ta))) (uzind2.5 (-> (e. M (ZZ)) ps)) (uzind2.6 (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<) (cv k))) (-> ch th)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) ta) (M N zltp1let M peano2z uzind2.1 M ZZ e. imbi2d uzind2.2 M ZZ e. imbi2d uzind2.3 M ZZ e. imbi2d uzind2.4 M ZZ e. imbi2d uzind2.5 M + 1 opr ZZ e. a1i M k cv zltp1let uzind2.6 3exp imp sylbird exp com3l imp M + 1 opr ZZ e. 3adant1 a2d uzind 3exp syl com34 pm2.43a imp sylbid imp 3impa))
+thm (uzind3 ((j k) (N j) (N k) (j ps) (ch j) (j th) (j ta) (m ph) (j m) (M j) (k m) (M m) (M k)) ((uzind3.1 (-> (= (cv j) M) (<-> ph ps))) (uzind3.2 (-> (= (cv j) (cv m)) (<-> ph ch))) (uzind3.3 (-> (= (cv j) (opr (cv m) (+) (1))) (<-> ph th))) (uzind3.4 (-> (= (cv j) N) (<-> ph ta))) (uzind3.5 (-> (e. M (ZZ)) ps)) (uzind3.6 (-> (/\ (e. M (ZZ)) (e. (cv m) ({e.|} k (ZZ) (br M (<_) (cv k))))) (-> ch th)))) (-> (/\ (e. M (ZZ)) (e. N ({e.|} k (ZZ) (br M (<_) (cv k))))) ta) (uzind3.1 uzind3.2 uzind3.3 uzind3.4 uzind3.5 uzind3.6 k cv m cv M <_ breq2 ZZ elrab sylan2br 3impb uzind 3expb k cv N M <_ breq2 ZZ elrab sylan2b))
+thm (uzindOLD ((x z) (w x) (A x) (w z) (A z) (A w) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (w y) (ph y) (ph w) (x y) (B x) (y z) (B y) (B z) (B w)) ((uzindOLD.1 (-> (= (cv x) B) (<-> ph ps))) (uzindOLD.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (uzindOLD.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (uzindOLD.4 (-> (= (cv x) A) (<-> ph ta))) (uzindOLD.5 ps) (uzindOLD.6 (-> (/\ (/\ (e. (cv y) (ZZ)) (e. B (ZZ))) (br B (<_) (cv y))) (-> ch th)))) (-> (/\ (/\ (e. A (ZZ)) (e. B (ZZ))) (br B (<_) A)) ta) (B A nnegdift ancoms A B resubclt 0re ax1re 0 A - B opr 1 leadd1t mp3an13 syl 1cn addid2 <_ A - B opr + 1 opr breq1i syl6bb bitrd A zret B zret syl2an A B zsubclt A - B opr peano2z syl A - B opr + 1 opr elnnz1 z cv 1 ZZ eleq1 z cv - 1 opr + B oprex x isseti z cv 1 = B ZZ e. /\ x ax-17 z cv - 1 opr + B oprex x ph hbsbcv ps x ax-17 hbbi hbim x z cv - 1 opr + B opr ph sbceq1 z cv 1 = B ZZ e. /\ adantr x cv z cv - 1 opr + B opr B eqtrt z cv 1 - 1 opreq1 + B opreq1d B zcnt B addid2t syl 1cn subid + B opreq1i syl5eq sylan9eq sylan2 uzindOLD.1 syl bitr3d exp 19.23ai ax-mp exp A ZZ e. adantld pm5.74d imbi12d z cv w cv ZZ eleq1 z cv - 1 opr + B oprex x isseti w cv - 1 opr + B oprex y isseti x y x cv z cv - 1 opr + B opr = y cv w cv - 1 opr + B opr = eeanv z cv w cv = x ax-17 z cv - 1 opr + B oprex x ph hbsbcv w cv - 1 opr + B opr y ch [/] x ax-17 hbbi hbim z cv w cv = y ax-17 z cv - 1 opr + B opr x ph [/] y ax-17 w cv - 1 opr + B oprex y ch hbsbcv hbbi hbim x cv z cv - 1 opr + B opr y cv w cv - 1 opr + B opr eqeq12 z cv w cv - 1 opreq1 + B opreq1d syl5bir uzindOLD.2 syl6 x z cv - 1 opr + B opr ph sbceq1 y w cv - 1 opr + B opr ch sbceq1 bi2bian9 sylibd 19.23ai 19.23ai sylbir mp2an A ZZ e. B ZZ e. /\ imbi2d imbi12d z cv w cv + 1 opr ZZ eleq1 z cv - 1 opr + B opr - 1 opr + 1 oprex x isseti w cv + 1 opr - 1 opr + B opr - 1 oprex y isseti x y x cv z cv - 1 opr + B opr - 1 opr + 1 opr = y cv w cv + 1 opr - 1 opr + B opr - 1 opr = eeanv z cv w cv + 1 opr = x ax-17 z cv - 1 opr + B opr - 1 opr + 1 oprex x ph hbsbcv w cv + 1 opr - 1 opr + B opr - 1 opr y th [/] x ax-17 hbbi hbim z cv w cv + 1 opr = y ax-17 z cv - 1 opr + B opr - 1 opr + 1 opr x ph [/] y ax-17 w cv + 1 opr - 1 opr + B opr - 1 oprex y th hbsbcv hbbi hbim x cv z cv - 1 opr + B opr - 1 opr + 1 opr y cv + 1 opr w cv + 1 opr - 1 opr + B opr - 1 opr + 1 opr eqeq12 y cv w cv + 1 opr - 1 opr + B opr - 1 opr + 1 opreq1 sylan2 z cv w cv + 1 opr - 1 opreq1 + B opreq1d - 1 opreq1d + 1 opreq1d syl5bir uzindOLD.3 syl6 x z cv - 1 opr + B opr - 1 opr + 1 opr ph sbceq1 y w cv + 1 opr - 1 opr + B opr - 1 opr th sbceq1 bi2bian9 sylibd 19.23ai 19.23ai sylbir mp2an A ZZ e. B ZZ e. /\ imbi2d imbi12d z cv - 1 opr B axaddcl 1cn z cv 1 subclt mpan2 sylan 1cn z cv - 1 opr + B opr 1 npcant mpan2 syl z cv zcnt B zcnt syl2an exp A ZZ e. adantld z cv - 1 opr + B opr - 1 opr + 1 opr z cv - 1 opr + B opr x ph dfsbcq syl6 pm5.74d pm5.74i syl5bbr z cv A - B opr + 1 opr ZZ eleq1 z cv - 1 opr + B oprex x isseti z cv A - B opr + 1 opr = A ZZ e. B ZZ e. /\ /\ x ax-17 z cv - 1 opr + B oprex x ph hbsbcv ta x ax-17 hbbi hbim x z cv - 1 opr + B opr ph sbceq1 z cv A - B opr + 1 opr = A ZZ e. B ZZ e. /\ /\ adantr x cv z cv - 1 opr + B opr A - B opr + 1 opr - 1 opr + B opr eqtrt z cv A - B opr + 1 opr - 1 opreq1 + B opreq1d sylan2 A - B opr 1 B add23t A B subclt 1cn A CC e. B CC e. /\ a1i A CC e. B CC e. pm3.27 syl3anc A B npcant + 1 opreq1d eqtrd - 1 opreq1d A - B opr + 1 opr B 1 addsubt A B subclt A - B opr peano2cn syl A CC e. B CC e. pm3.27 1cn A CC e. B CC e. /\ a1i syl3anc 1cn A 1 pncant mpan2 B CC e. adantr 3eqtr3d A zcnt B zcnt syl2an sylan9eq anasss uzindOLD.4 syl bitr3d exp 19.23ai ax-mp exp pm5.74d imbi12d uzindOLD.5 A ZZ e. B ZZ e. /\ a1i 1 ZZ e. a1i w cv nnzt w cv + 1 opr ZZ e. a1d w cv - 1 opr + B oprex y isseti w cv NN e. B ZZ e. /\ y ax-17 w cv - 1 opr + B oprex y ch hbsbcv w cv - 1 opr + B oprex y th hbsbcv hbim hbim uzindOLD.6 y cv w cv - 1 opr + B opr ZZ eleq1 B ZZ e. anbi1d y cv w cv - 1 opr + B opr B <_ breq2 anbi12d w cv - 1 opr B zaddclt w cv peano2zr sylan w cv nnzt sylan w cv NN e. B ZZ e. pm3.27 jca B w cv - 1 opr + B opr B - 1 opr lesub1t w cv RR e. B RR e. pm3.27 w cv - 1 opr B axaddrcl ax1re w cv 1 resubclt mpan2 sylan ax1re B 1 resubclt mpan2 w cv RR e. adantl syl3anc 1cn B B 1 subsubt mp3an3 anidms B subidt + 1 opreq1d 1cn addid2 syl6eq eqtrd w cv CC e. adantl 1cn w cv - 1 opr + B opr B 1 subsubt mp3an3 w cv - 1 opr B axaddcl 1cn w cv 1 subclt mpan2 sylan w cv CC e. B CC e. pm3.27 sylanc w cv - 1 opr B pncant 1cn w cv 1 subclt mpan2 sylan + 1 opreq1d 1cn w cv 1 npcant mpan2 B CC e. adantr 3eqtrd <_ breq12d w cv recnt B recnt syl2an bitrd w cv zret B zret syl2an biimpar an1rs w cv elnnz1 sylanb jca syl5bir y w cv - 1 opr + B opr ch sbceq1 y w cv - 1 opr + B opr th sbceq1 imbi12d biimpd imim12d mpi 19.23ai ax-mp 1cn w cv + 1 opr - 1 opr B 1 addsubt mp3an3 w cv peano2cn 1cn w cv + 1 opr 1 subclt mpan2 syl sylan 1cn w cv 1 pncant mpan2 - 1 opreq1d + B opreq1d B CC e. adantr eqtrd w cv nncnt B zcnt syl2an w cv + 1 opr - 1 opr + B opr - 1 opr w cv - 1 opr + B opr y th dfsbcq syl sylibrd exp A ZZ e. adantld a2d imim12d nnind sylbir exp pm2.43a com23 mpcom sylbid imp))
+thm (uzind3OLD ((x z) (A x) (A z) (ps x) (ch x) (th x) (ta x) (ph y) (x y) (B x) (y z) (B y) (B z)) ((uzind3OLD.1 (-> (= (cv x) B) (<-> ph ps))) (uzind3OLD.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (uzind3OLD.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (uzind3OLD.4 (-> (= (cv x) A) (<-> ph ta))) (uzind3OLD.5 ps) (uzind3OLD.6 (-> (/\ (e. B (ZZ)) (e. (cv y) ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> ch th)))) (-> (/\ (e. B (ZZ)) (e. A ({e.|} z (ZZ) (br B (<_) (cv z))))) ta) (uzind3OLD.1 uzind3OLD.2 uzind3OLD.3 uzind3OLD.4 uzind3OLD.5 uzind3OLD.6 z cv y cv B <_ breq2 ZZ elrab sylan2br an1s anassrs uzindOLD anasss an1s z cv A B <_ breq2 ZZ elrab sylan2b))
+thm (uzind3iOLD ((k n) (N n) (N k) (n ps) (ch n) (n th) (n ta) (j ph) (j n) (M n) (j k) (M j) (M k)) ((uzind3iOLD.1 (e. M (ZZ))) (uzind3iOLD.2 (-> (= (cv n) M) (<-> ph ps))) (uzind3iOLD.3 (-> (= (cv n) (cv j)) (<-> ph ch))) (uzind3iOLD.4 (-> (= (cv n) (opr (cv j) (+) (1))) (<-> ph th))) (uzind3iOLD.5 (-> (= (cv n) N) (<-> ph ta))) (uzind3iOLD.6 ps) (uzind3iOLD.7 (-> (e. (cv j) ({e.|} k (ZZ) (br M (<_) (cv k)))) (-> ch th)))) (-> (e. N ({e.|} k (ZZ) (br M (<_) (cv k)))) ta) (uzind3iOLD.1 uzind3iOLD.2 uzind3iOLD.3 uzind3iOLD.4 uzind3iOLD.5 uzind3iOLD.6 uzind3iOLD.7 M ZZ e. adantl uzind3OLD mpan))
+thm (uzwoOLD ((x y) (w x) (v x) (u x) (A x) (w y) (v y) (u y) (A y) (v w) (u w) (A w) (u v) (A v) (A u) (y z) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (v cv B <_ y cv breq1 y A biraldv A z ZZ B <_ z cv br {e.|} C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv u cv <_ y cv breq1 y A biraldv A z ZZ B <_ z cv br {e.|} C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv u cv + 1 opr <_ y cv breq1 y A biraldv A z ZZ B <_ z cv br {e.|} C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv w cv <_ y cv breq1 y A biraldv A z ZZ B <_ z cv br {e.|} C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d A z ZZ B <_ z cv br {e.|} y cv ssel z cv y cv B <_ breq2 ZZ elrab pm3.27bd syl6 r19.21aiv x A y A x cv <_ y cv br A.e. E.e. -. adantr z ZZ B <_ z cv br ssrab u cv sseli x cv u cv <_ y cv breq1 y A biraldv A rcla4ev expcom con3d com12 u cv y cv letri3t u cv zret y cv zret syl2an y cv u cv zleltp1t y cv u cv + 1 opr ltnlet y cv zret u cv zret u cv peano2re syl syl2an bitrd ancoms u cv <_ y cv br anbi2d bitrd A ZZ y cv ssel2 sylan2 y cv A u cv eleq1a u cv ZZ e. A ZZ C_ ad2antrr sylbird exp3a u cv + 1 opr <_ y cv br u cv A e. con1 syl6 com23 exp32 com34 imp41 r19.20dva exp sylan9r pm2.43d exp31 imp3a syl z ZZ B <_ z cv br ssrab A z ZZ B <_ z cv br {e.|} ZZ sstr mpan2 sylani a2d B ZZ e. adantl uzind3OLD x cv w cv <_ y cv breq1 y A biraldv A rcla4ev expcom con3d com12 A z ZZ B <_ z cv br {e.|} C_ adantl sylcom exp A z ZZ B <_ z cv br {e.|} w cv ssel con3d com12 x A y A x cv <_ y cv br A.e. E.e. -. adantrd pm2.61d1 exp3a imp w 19.21adv A w eq0 syl6ibr con1d A {/} df-ne syl5ib imp anasss))
+thm (uzwo2OLD ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (B A z x y uzwoOLD z ZZ B <_ z cv br ssrab zssre sstri A z ZZ B <_ z cv br {e.|} RR sstr mpan2 y cv w cv x cv <_ breq2 A rcla4v y cv x cv w cv <_ breq2 A rcla4v im2anan9 ancoms A RR x cv ssel A RR w cv ssel anim12d impcom x cv w cv letri3t syl biimprd exp com23 syld com3r r19.21aivv syl B ZZ e. A {/} =/= ad2antrl jca x cv w cv <_ y cv breq1 y A biraldv A reu4 sylibr))
+thm (nnwo ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (1z 1 A z x y uzwoOLD mpan z nnz A sseq2i sylanb))
+thm (nnwoOLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (v cv 1 <_ y cv breq1 y A biraldv A NN C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv z cv <_ y cv breq1 y A biraldv A NN C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv z cv + 1 opr <_ y cv breq1 y A biraldv A NN C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d v cv w cv <_ y cv breq1 y A biraldv A NN C_ x A y A x cv <_ y cv br A.e. E.e. -. /\ imbi2d A NN y cv ssel y cv nnge1t syl6 r19.21aiv x A y A x cv <_ y cv br A.e. E.e. -. adantr x cv z cv <_ y cv breq1 y A biraldv A rcla4ev expcom con3d com12 z cv y cv letri3t z cv nnret y cv nnret syl2an y cv z cv nnleltp1t y cv z cv + 1 opr ltnlet z cv peano2re sylan2 y cv nnret z cv nnret syl2an bitrd ancoms z cv <_ y cv br anbi2d bitrd A NN y cv ssel2 sylan2 y cv A z cv eleq1a z cv NN e. A NN C_ ad2antrr sylbird exp3a z cv + 1 opr <_ y cv br z cv A e. con1 syl6 com23 exp32 com34 imp41 r19.20dva exp sylan9r pm2.43d exp31 imp3a a2d nnind x cv w cv <_ y cv breq1 y A biraldv A rcla4ev expcom con3d com12 A NN C_ adantl sylcom A NN w cv ssel con3d com12 x A y A x cv <_ y cv br A.e. E.e. -. adantrd pm2.61i exp w 19.21adv A w eq0 syl6ibr con1d A {/} df-ne syl5ib imp))
+thm (nnwof ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (A z)) ((nnwof.1 (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (nnwof.2 (-> (e. (cv z) A) (A. y (e. (cv z) A))))) (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (A w v nnwo z cv v cv e. y ax-17 nnwof.2 hbel w cv <_ v cv br y ax-17 hbim y cv A e. w cv <_ y cv br -> v ax-17 v cv y cv A eleq1 v cv y cv w cv <_ breq2 imbi12d cbval w cv A e. anbi2i w biex z cv w cv e. x ax-17 nnwof.1 hbel z cv y cv e. x ax-17 nnwof.1 hbel w cv <_ y cv br x ax-17 hbim y hbal hban x cv A e. y y cv A e. x cv <_ y cv br -> A. /\ w ax-17 w cv x cv A eleq1 w cv x cv <_ y cv breq1 y cv A e. imbi2d y bialdv anbi12d cbvex bitr w A v v cv A e. w cv <_ v cv br -> A. df-rex x A y y cv A e. x cv <_ y cv br -> A. df-rex 3bitr4 v A w cv <_ v cv br df-ral w A birex y A x cv <_ y cv br df-ral x A birex 3bitr4 sylib))
+thm (nnwos ((x y) (x z) (y z) (ph y) (ph z) (ps x)) ((nnwos.1 (-> (= (cv x) (cv y)) (<-> ph ps)))) (-> (E.e. x (NN) ph) (E.e. x (NN) (/\ ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y))))))) (z x NN ph hbrab1 z cv x NN ph {e.|} e. y ax-17 nnwof x NN ph {e.|} {/} df-ne x NN ph ssrab x NN ph {e.|} {/} =/= biantrur x NN ph rabn0 3bitr3 x x NN ph {e.|} y x NN ph {e.|} x cv <_ y cv br A.e. df-rex x NN ph rabid y x NN ph {e.|} x cv <_ y cv br df-ral nnwos.1 NN elrab x cv <_ y cv br imbi1i y cv NN e. ps x cv <_ y cv br impexp bitr y bial bitr anbi12i x biex y NN ps x cv <_ y cv br -> df-ral x cv NN e. ph /\ anbi2i x cv NN e. ph y NN ps x cv <_ y cv br -> A.e. anass bitr3 x biex x NN ph y NN ps x cv <_ y cv br -> A.e. /\ df-rex bitr4 3bitr 3imtr3))
+thm (indstr ((x y) (ph y) (ps x)) ((indstr.1 (-> (= (cv x) (cv y)) (<-> ph ps))) (indstr.2 (-> (e. (cv x) (NN)) (-> (A.e. y (NN) (-> (br (cv y) (<) (cv x)) ps)) ph)))) (-> (e. (cv x) (NN)) ph) (ph pm3.24 x cv y cv lenltt x cv nnret y cv nnret syl2an ps -. imbi2d y cv < x cv br ps pm4.1 syl6bbr biraldva indstr.2 sylbid ph -. anim2d ph -. ph ancom syl6ib mtoi rgen x NN ph -. y NN ps -. x cv <_ y cv br -> A.e. /\ ralnex mpbi indstr.1 negbid nnwos mto x NN ph dfral2 mpbir rspec))
+thm (nn0ind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((nn0ind.1 (-> (= (cv x) (0)) (<-> ph ps))) (nn0ind.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nn0ind.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nn0ind.4 (-> (= (cv x) A) (<-> ph ta))) (nn0ind.5 ps) (nn0ind.6 (-> (e. (cv y) (NN0)) (-> ch th)))) (-> (e. A (NN0)) ta) (A elnn0z 0z nn0ind.1 nn0ind.2 nn0ind.3 nn0ind.4 nn0ind.5 0 ZZ e. a1i y cv elnn0z nn0ind.6 sylbir 0 ZZ e. 3adant1 uzind mp3an1 sylbi))
+thm (nn0indALT ((x y) (x z) (y z) (A x) (A z) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (ph y) (ph z)) ((nn0indALT.1 (-> (= (cv x) (0)) (<-> ph ps))) (nn0indALT.2 (-> (= (cv x) (cv y)) (<-> ph ch))) (nn0indALT.3 (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th))) (nn0indALT.4 (-> (= (cv x) A) (<-> ph ta))) (nn0indALT.5 ps) (nn0indALT.6 (-> (e. (cv y) (NN0)) (-> ch th)))) (-> (e. A (NN0)) ta) (A elnn0 z cv 1 x ph dfsbcq nn0indALT.2 z vsbcint nn0indALT.3 z vsbcint nn0indALT.4 z vsbcint ax1re elisseti x ph hbsbcv ax1re elisseti 0nn0 elisseti nn0indALT.6 0nn0 0 NN0 y cv eleq1a ax-mp 0nn0 elisseti nn0indALT.5 nn0indALT.1 mpbiri y cv 0 x cv eqeq2 nn0indALT.2 syl6bir pm5.74d mpbii com12 vtocle sylc x cv 1 = adantr y cv 0 + 1 opreq1 1cn addid2 syl6eq x cv eqeq2d nn0indALT.3 syl6bir imp mpbird exp vtocle x 1 ph sbceq1 mpbid vtoclef y cv nnnn0t nn0indALT.6 syl nnind 0 A = x ax-17 ta x ax-17 hbim 0nn0 elisseti x cv 0 A eqeq1 nn0indALT.5 nn0indALT.1 bicomd nn0indALT.4 sylan9bb mpbii exp sylbird vtoclef eqcoms jaoi sylbi))
+thm (btwnz ((x z) (A x) (A z) (A y)) () (-> (e. A (RR)) (/\ (E.e. x (ZZ) (br (cv x) (<) A)) (E.e. y (ZZ) (br A (<) (cv y))))) (A renegclt A -u z arch syl A z cv ltnegcon1t exp z cv nnret syl5 pm5.32d x cv z cv -u < A breq1 ZZ rcla4ev z cv nnnegz sylan syl6bi exp3a r19.23adv mpd A y arch y cv nnzt A < y cv br anim1i r19.22i2 syl jca))
+thm (uzwo3lem1 ((v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((uzwo3lem.1 (= R ({e.|} z (ZZ) (br B (<_) (cv z)))))) (-> (e. B (RR)) (E!e. x R (A.e. y R (br (cv x) (<_) (cv y))))) (B v z btwnz pm3.26d v cv B z cv ltletrt v cv z cv ltlet B RR e. 3adant2 syld z cv zret syl3an3 v cv zret syl3an1 exp3a 3exp com34 imp32 r19.21aiv z ZZ B <_ z cv br v cv <_ z cv br ss2rab sylibr uzwo3lem.1 syl5ss B v z btwnz pm3.27d B z cv ltlet z cv zret sylan2 r19.22dva uzwo3lem.1 {/} eqeq1i negbii z ZZ B <_ z cv br rabn0 bitr2 R {/} df-ne bitr4 syl6ib mpd v cv ZZ e. v cv < B br ad2antrl jca exp v cv R z x y uzwo2OLD exp syld exp3a com12 r19.23adv mpd))
+thm (uzwo3lem2 ((x y) (x z) (w x) (v x) (t x) (y z) (w y) (v y) (t y) (w z) (v z) (t z) (v w) (t w) (t v) (A x) (A y) (B t) (S y) (S t) (R w) (v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((uzwo3lem.1 (= R ({e.|} z (ZZ) (br B (<_) (cv z))))) (uzwo3lem.2 (= S (U. ({e.|} w R (A.e. v R (br (cv w) (<_) (cv v)))))))) (-> (/\ (e. B (RR)) (/\ (C_ A R) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (A R t ZZ S <_ t cv br {e.|} sstr2 y cv t cv S <_ breq2 R rcla4v uzwo3lem.1 x y uzwo3lem1 x cv w cv <_ y cv breq1 y R biraldv y cv v cv w cv <_ breq2 R cbvralv syl6bb x cv w R v R w cv <_ v cv br A.e. {e.|} U. <_ y cv breq1 y R biraldv reuuni3 uzwo3lem.2 <_ y cv breq1i y R biral sylibr syl syl5com uzwo3lem.1 t cv eleq2i z cv t cv B <_ breq2 ZZ elrab bitr syl5ibr exp3a r19.21aiv t ZZ B <_ t cv br S <_ t cv br ss2rab sylibr uzwo3lem.1 z cv t cv B <_ breq2 ZZ cbvrabv eqtr syl5ss syl5com uzwo3lem.1 w v uzwo3lem1 w R v R w cv <_ v cv br A.e. reucl syl uzwo3lem.2 R eleq1i sylibr uzwo3lem.1 z cv t cv B <_ breq2 ZZ cbvrabv eqtr S eleq2i t cv S B <_ breq2 ZZ elrab bitr pm3.26bd S A t x y uzwo2OLD exp32 3syl syld imp32))
+thm (uzwo3 ((x y) (u x) (t x) (A x) (u y) (t y) (A y) (t u) (A u) (A t) (w z) (v z) (u z) (t z) (B z) (v w) (u w) (t w) (B w) (u v) (t v) (B v) (B u) (B t)) () (-> (/\ (e. B (RR)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))) (z ZZ B <_ z cv br {e.|} eqid w z ZZ B <_ z cv br {e.|} v z ZZ B <_ z cv br {e.|} w cv <_ v cv br A.e. {e.|} U. eqid A u t uzwo3lem2 t cv y cv u cv <_ breq2 A cbvralv u A bireu u cv x cv <_ y cv breq1 y A biraldv A cbvreuv bitr sylib))
+thm (zmin ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y))))))) (z ZZ A <_ z cv br {e.|} eqid x y uzwo3lem1 z cv x cv A <_ breq2 ZZ elrab z cv y cv A <_ breq2 ZZ elrab x cv <_ y cv br imbi1i y cv ZZ e. A <_ y cv br x cv <_ y cv br impexp bitr y bial y z ZZ A <_ z cv br {e.|} x cv <_ y cv br df-ral y ZZ A <_ y cv br x cv <_ y cv br -> df-ral 3bitr4 anbi12i x cv ZZ e. A <_ x cv br y ZZ A <_ y cv br x cv <_ y cv br -> A.e. anass bitr x bieu x z ZZ A <_ z cv br {e.|} y z ZZ A <_ z cv br {e.|} x cv <_ y cv br A.e. df-reu x ZZ A <_ x cv br y ZZ A <_ y cv br x cv <_ y cv br -> A.e. /\ df-reu 3bitr4r sylibr))
+thm (zmax ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))) (A renegclt A -u z w zmin syl x cv A lenegt x cv zret sylan ancoms w cv znegclt y cv w cv -u <_ A breq1 y cv w cv -u <_ x cv breq1 imbi12d ZZ rcla4v syl w cv A lenegcon1t x cv ZZ e. adantrr w cv x cv lenegcon1t x cv zret sylan2 A RR e. adantrl imbi12d w cv zret sylan biimpd exp com23 syld com13 r19.21adv y cv znegclt w cv y cv -u A -u <_ breq2 w cv y cv -u x cv -u <_ breq2 imbi12d ZZ rcla4v syl y cv A lenegt x cv ZZ e. adantrr y cv x cv lenegt x cv zret sylan2 A RR e. adantrl imbi12d y cv zret sylan biimprd exp com23 syld com13 r19.21adv impbid anbi12d bireudva x cv znegclt z cv znegclt z cv x cv negcon2t z cv zcnt x cv zcnt syl2an reuhyp z cv x cv -u A -u <_ breq2 z cv x cv -u <_ w cv breq1 A -u <_ w cv br imbi2d w ZZ biraldv anbi12d reuxfr syl6rbbr mpbid))
+thm (zbtwnre ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (br (cv x) (<) (opr A (+) (1)))))) (A x y zmin x cv - 1 opr A y cv ltletrt x cv zret ax1re x cv 1 resubclt mpan2 syl syl3an1 3expa y cv zret sylan2 x cv y cv zlem1ltt A RR e. adantlr sylibrd exp4b com23 r19.21adv x cv zret ax1re x cv 1 resubclt mpan2 x cv - 1 opr ltnrt 3syl x cv peano2zr x cv x cv - 1 opr zlem1ltt mpdan mtbird A RR e. adantr y ZZ A <_ y cv br x cv <_ y cv br -> A.e. adantr A x cv - 1 opr lenltt x cv zret ax1re x cv 1 resubclt mpan2 syl sylan2 ancoms y ZZ A <_ y cv br x cv <_ y cv br -> A.e. adantr x cv peano2zr y cv x cv - 1 opr A <_ breq2 y cv x cv - 1 opr x cv <_ breq2 imbi12d ZZ rcla4v syl imp A RR e. adantlr sylbird mt3d exp impbid ax1re x cv 1 A ltsubaddt mp3an2 x cv zret sylan bitr3d ancoms A <_ x cv br anbi2d bireudva mpbid))
+thm (rebtwnz ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))) (A renegclt A -u y zbtwnre syl x cv A lenegt ancoms A - 1 opr x cv ltnegt ax1re A 1 resubclt mpan2 sylan ax1re A 1 x cv ltsubaddt mp3an2 A recnt 1cn A 1 negdi2t mpan2 syl x cv RR e. adantr x cv -u < breq2d 3bitr3d anbi12d x cv zret sylan2 bicomd bireudva x cv znegclt y cv znegclt y cv x cv negcon2t y cv zcnt x cv zcnt syl2an reuhyp y cv x cv -u A -u <_ breq2 y cv x cv -u < A -u + 1 opr breq1 anbi12d reuxfr syl5bb mpbid))
+thm (flvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))))) (y cv A x cv <_ breq2 y cv A < x cv + 1 opr breq1 anbi12d x ZZ birabsdv unieqd y z x df-fl zex x x cv <_ A br A < x cv + 1 opr br /\ rabex uniex fvopab4))
+thm (flclt ((A z)) () (-> (e. A (RR)) (e. (` (floor) A) (ZZ))) (A z flvalt A z rebtwnz z ZZ z cv <_ A br A < z cv + 1 opr br /\ reucl syl eqeltrd))
+thm (flleltt ((x y) (A x) (A y)) () (-> (e. A (RR)) (/\ (br (` (floor) A) (<_) A) (br A (<) (opr (` (floor) A) (+) (1))))) (A x rebtwnz x cv y cv <_ A breq1 x cv y cv + 1 opreq1 A < breq2d anbi12d x cv y ZZ y cv <_ A br A < y cv + 1 opr br /\ {e.|} U. <_ A breq1 x cv y ZZ y cv <_ A br A < y cv + 1 opr br /\ {e.|} U. + 1 opreq1 A < breq2d anbi12d reuuni3 syl A y flvalt <_ A breq1d A y flvalt + 1 opreq1d A < breq2d anbi12d mpbird))
+thm (fllet () () (-> (e. A (RR)) (br (` (floor) A) (<_) A)) (A flleltt pm3.26d))
+thm (flltp1t () () (-> (e. A (RR)) (br A (<) (opr (` (floor) A) (+) (1)))) (A flleltt pm3.27d))
+thm (flget () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br B (<_) A) (br B (<_) (` (floor) A)))) (A flltp1t B ZZ e. adantl B A floor A ` + 1 opr lelttrt 3expb B zret A flclt floor A ` zret floor A ` peano2re 3syl ancli syl2an mpan2d B floor A ` zleltp1t A flclt sylan2 sylibrd A fllet B ZZ e. adantl B floor A ` A letrt 3expb B zret A flclt floor A ` zret syl ancri syl2an mpan2d impbid ancoms))
+thm (flltt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br A (<) B) (br (` (floor) A) (<) B))) (A B flget B A lenltt B zret sylan ancoms B floor A ` lenltt B zret A flclt floor A ` zret syl syl2an ancoms 3bitr3d bicon4d))
+thm (flidt () () (-> (e. A (ZZ)) (= (` (floor) A) A)) (A zret A fllet syl A zret A leidt syl A zret A A flget mpancom mpbid jca floor A ` A letri3t A zret A flclt floor A ` zret 3syl A zret sylanc mpbird))
+thm (flval2t ((x y) (A x) (A y)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))))) (A fllet A y cv flget biimpd exp r19.21aiv jca x cv floor A ` <_ A breq1 x cv floor A ` y cv <_ breq2 y cv <_ A br imbi2d y ZZ biraldv anbi12d ZZ reuuni2 A flclt A x y zmax sylanc mpbid eqcomd))
+thm (flval3t ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (= (` (floor) A) (sup ({e.|} x (ZZ) (br (cv x) (<_) A)) (RR) (<)))) (x ZZ x cv <_ A br {e.|} y z floor A ` suprub x ZZ x cv <_ A br ssrab zssre sstri A RR e. a1i A flclt A fllet jca x cv floor A ` <_ A breq1 ZZ elrab sylibr floor A ` x ZZ x cv <_ A br {e.|} n0i syl y cv floor A ` z cv <_ breq2 z x ZZ x cv <_ A br {e.|} biraldv RR rcla4ev A flclt floor A ` zret syl A z cv flget biimpd exp imp3a x cv z cv <_ A breq1 ZZ elrab syl5ib r19.21aiv sylanc 3jca A flclt A fllet jca x cv floor A ` <_ A breq1 ZZ elrab sylibr sylanc x ZZ x cv <_ A br {e.|} y z floor A ` w suprnub x ZZ x cv <_ A br ssrab zssre sstri A RR e. a1i A flclt A fllet jca x cv floor A ` <_ A breq1 ZZ elrab sylibr floor A ` x ZZ x cv <_ A br {e.|} n0i syl y cv floor A ` z cv <_ breq2 z x ZZ x cv <_ A br {e.|} biraldv RR rcla4ev A flclt floor A ` zret syl A z cv flget biimpd exp imp3a x cv z cv <_ A breq1 ZZ elrab syl5ib r19.21aiv sylanc 3jca A flclt floor A ` zret syl A w cv flget w cv floor A ` lenltt w cv zret A flclt floor A ` zret syl syl2an ancoms bitrd biimpd exp imp3a x cv w cv <_ A breq1 ZZ elrab syl5ib r19.21aiv jca sylanc jca floor A ` x ZZ x cv <_ A br {e.|} RR < sup eqleltt A flclt floor A ` zret syl x ZZ x cv <_ A br {e.|} y z suprcl x ZZ x cv <_ A br ssrab zssre sstri A RR e. a1i A flclt A fllet jca x cv floor A ` <_ A breq1 ZZ elrab sylibr floor A ` x ZZ x cv <_ A br {e.|} n0i syl y cv floor A ` z cv <_ breq2 z x ZZ x cv <_ A br {e.|} biraldv RR rcla4ev A flclt floor A ` zret syl A z cv flget biimpd exp imp3a x cv z cv <_ A breq1 ZZ elrab syl5ib r19.21aiv sylanc syl3anc sylanc mpbird))
+thm (flbit ((A x) (B x)) () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (= (` (floor) A) B) (/\ (br B (<_) A) (br A (<) (opr B (+) (1)))))) (A x flvalt B eqeq1d B ZZ e. adantr x cv B <_ A breq1 x cv B + 1 opreq1 A < breq2d anbi12d ZZ reuuni2 A x rebtwnz sylan2 ancoms bitr4d))
+thm (flge0nn0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (floor) A) (NN0))) (A flclt 0 <_ A br adantr 0z A 0 flget mpan2 biimpa jca floor A ` elnn0z sylibr))
+thm (flge1nnt () () (-> (/\ (e. A (RR)) (br (1) (<_) A)) (e. (` (floor) A) (NN))) (A flclt 1 <_ A br adantr 1z A 1 flget mpan2 biimpa jca floor A ` elnnz1 sylibr))
+thm (fladdzt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (= (` (floor) (opr A (+) B)) (opr (` (floor) A) (+) B))) (A fllet B ZZ e. adantr floor A ` A B leadd1t 3expa A flclt floor A ` zret syl ancri B zret syl2an mpbid A flltp1t B ZZ e. adantr A floor A ` + 1 opr B ltadd1t 3expa A flclt floor A ` peano2z floor A ` + 1 opr zret 3syl ancli B zret syl2an mpbid 1cn floor A ` 1 B add23t mp3an2 A flclt floor A ` zcnt syl B zcnt syl2an breqtrd jca A + B opr floor A ` + B opr flbit A B axaddrcl B zret sylan2 floor A ` B zaddclt A flclt sylan sylanc mpbird))
+thm (btwnzge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (ZZ))) (/\ (br B (<_) A) (br A (<) (opr B (+) (1))))) (<-> (br (0) (<_) A) (br (0) (<_) B))) (0z A 0 flget mpan2 B ZZ e. B <_ A br A < B + 1 opr br /\ ad2antll A B flbit biimpar 0 <_ breq2d bitrd))
+thm (elq ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (QQ)) (E.e. x (ZZ) (E.e. y (NN) (= A (opr (cv x) (/) (cv y)))))) (z x y df-q A eleq2i x cv / y cv oprex A x cv / y cv opr V eleq1 mpbiri y cv NN e. a1i r19.23aiv x cv ZZ e. a1i r19.23aiv z cv A x cv / y cv opr eqeq1 y NN birexdv x ZZ birexdv elab3g bitr))
+thm (znq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (/) B) (QQ))) (A / B opr eqid x cv A / y cv opreq1 A / B opr eqeq2d y cv B A / opreq2 A / B opr eqeq2d ZZ NN rcla42ev mpan2 A / B opr x y elq sylibr))
+thm (qret ((x y) (A x) (A y)) () (-> (e. A (QQ)) (e. A (RR))) (A x y elq A x cv / y cv opr RR eleq1 x cv y cv redivclt 3expb x cv zret y cv nnret y cv nnne0t jca syl2an syl5bir com12 r19.23aivv sylbi))
+thm (zqt ((x y) (A x) (A y)) () (-> (e. A (ZZ)) (e. A (QQ))) (x cv zret x cv recnt x cv div1t 3syl A eqeq2d x cv A eqcom syl6rbbr 1nn y cv 1 x cv / opreq2 A eqeq2d NN rcla4ev mpan syl6bi r19.22i A ZZ x risset A x y elq 3imtr4))
+thm (zssq () () (C_ (ZZ) (QQ)) (x cv zqt ssriv))
+thm (nn0ssq () () (C_ (NN0) (QQ)) (nn0ssz zssq sstri))
+thm (nnssq () () (C_ (NN) (QQ)) (nnssz zssq sstri))
+thm (qssre () () (C_ (QQ) (RR)) (x cv qret ssriv))
+thm (qsscn () () (C_ (QQ) (CC)) (qssre axresscn sstri))
+thm (nnqt () () (-> (e. A (NN)) (e. A (QQ))) (nnssq A sseli))
+thm (qcnt () () (-> (e. A (QQ)) (e. A (CC))) (qsscn A sseli))
+thm (qex () () (e. (QQ) (V)) (reex qssre ssexi))
+thm (qaddclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (+) B) (QQ))) (u cv x cv x. w cv opr + y cv x. z cv opr opr / v cv opreq1 A + B opr eqeq2d v cv y cv x. w cv opr x cv x. w cv opr + y cv x. z cv opr opr / opreq2 A + B opr eqeq2d ZZ NN rcla42ev A + B opr u v elq sylibr x cv x. w cv opr y cv x. z cv opr zaddclt x cv w cv zmulclt w cv nnzt sylan2 z cv ZZ e. adantrl y cv NN e. adantlr y cv z cv zmulclt y cv nnzt sylan w cv NN e. adantrr x cv ZZ e. adantll sylanc y cv w cv nnmulclt z cv ZZ e. adantrl x cv ZZ e. adantll jca A x cv / y cv opr = B z cv / w cv opr = /\ adantr A x cv / y cv opr B z cv / w cv opr + opreq12 x cv y cv z cv w cv divadddivt x cv zcnt y cv nncnt anim12i z cv zcnt w cv nncnt anim12i anim12i y cv nnne0t x cv ZZ e. adantl w cv nnne0t z cv ZZ e. adantl anim12i sylanc sylan9eqr sylanc an4s exp43 r19.23aivv r19.23advv imp A x y elq B z w elq syl2anb))
+thm (qnegclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (QQ)) (e. (-u A) (QQ))) (A x y elq x cv y cv divnegt 3expb x cv zcnt y cv nncnt y cv nnne0t jca syl2an A -u eqeq2d A x cv / y cv opr negeq syl5bi x cv znegclt y cv NN e. anim1i jctild z cv x cv -u / w cv opreq1 A -u eqeq2d w cv y cv x cv -u / opreq2 A -u eqeq2d ZZ NN rcla42ev A -u z w elq sylibr syl6 r19.23aivv sylbi))
+thm (qmulclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (x.) B) (QQ))) (v cv x cv x. z cv opr / u cv opreq1 A x. B opr eqeq2d u cv y cv x. w cv opr x cv x. z cv opr / opreq2 A x. B opr eqeq2d ZZ NN rcla42ev A x. B opr v u elq sylibr x cv z cv zmulclt y cv w cv nnmulclt anim12i an4s A x cv / y cv opr = B z cv / w cv opr = /\ adantr A x cv / y cv opr B z cv / w cv opr x. opreq12 x cv y cv z cv w cv divmuldivt x cv zcnt y cv nncnt anim12i z cv zcnt w cv nncnt anim12i anim12i y cv nnne0t x cv ZZ e. adantl w cv nnne0t z cv ZZ e. adantl anim12i sylanc sylan9eqr sylanc an4s exp43 r19.23aivv r19.23advv imp A x y elq B z w elq syl2anb))
+thm (qsubclt () () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (-) B) (QQ))) (A B subnegt A qcnt B qcnt syl2an A B -u qaddclt B qnegclt sylan2 eqeltrd))
+thm (qrecclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (QQ)) (=/= A (0))) (e. (opr (1) (/) A) (QQ))) (A x y elq A x cv / y cv opr 0 neeq1 x cv y cv divneq0bt 3expa x cv zcnt y cv nncnt anim12i sylan bicomd sylan9bbr x cv y cv zmulclt y cv nnzt sylan2 x cv 0 =/= adantr x cv sqznn y cv NN e. adantlr jca y cv 0 =/= adantlr A x cv / y cv opr = adantlr A x cv / y cv opr 1 / opreq2 x cv dividt y cv 0 =/= adantrr y cv CC e. adantlr / x cv / y cv opr opreq1d x cv x cv x cv y cv divdivdivt x cv CC e. y cv CC e. pm3.26 x cv CC e. y cv CC e. pm3.26 jca ancri x cv 0 =/= y cv 0 =/= pm3.26 x cv 0 =/= y cv 0 =/= pm3.26 x cv 0 =/= y cv 0 =/= pm3.27 3jca syl2an eqtr3d x cv zcnt y cv nncnt anim12i sylan anassrs an1rs sylan9eqr an1rs jca exp sylbid exp anasss y cv nnne0t ancli sylan2 z cv x cv x. y cv opr / w cv opreq1 1 / A opr eqeq2d w cv x cv x. x cv opr x cv x. y cv opr / opreq2 1 / A opr eqeq2d ZZ NN rcla42ev 1 / A opr z w elq sylibr syl8 r19.23aivv sylbi imp))
+thm (qdivclt () () (-> (/\ (/\ (e. A (QQ)) (e. B (QQ))) (=/= B (0))) (e. (opr A (/) B) (QQ))) (A B divrect 3expa A qcnt B qcnt anim12i sylan A 1 / B opr qmulclt B qrecclt sylan2 anassrs eqeltrd))
+thm (qrevaddclt () () (-> (e. B (QQ)) (<-> (/\ (e. A (CC)) (e. (opr A (+) B) (QQ))) (e. A (QQ)))) (A B pncant B qcnt sylan2 ancoms A + B opr QQ e. adantr A + B opr B qsubclt ancoms A CC e. adantlr eqeltrrd exp A B qaddclt expcom A CC e. adantr impbid exp pm5.32d A qcnt pm4.71ri syl6bbr))
+thm (nnrecqt () () (-> (e. A (NN)) (e. (opr (1) (/) A) (QQ))) (1z 1 A znq mpan))
+thm (qbtwnre ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (br A (<) B)) (E.e. x (QQ) (/\ (br A (<) (cv x)) (br (cv x) (<) B)))) (A B posdift B - A opr y nnreclt B A resubclt sylan exp ancoms sylbid y cv B axmulrcl y cv nnret sylan ax1re y cv x. B opr 1 resubclt mpan2 syl A RR e. adantrl y cv x. B opr - 1 opr z zbtwnre z ZZ y cv x. B opr - 1 opr <_ z cv br z cv < y cv x. B opr - 1 opr + 1 opr br /\ reurex 3syl y cv B A subdit y cv nncnt B recnt A recnt syl3an 3com23 3expb 1 < breq2d ax1re 1 y cv B - A opr ltdivmult mp3anl1 exp31 com23 y cv nnret y cv nngt0t sylc B A resubclt ancoms syl5 imp ax1re y cv x. A opr y cv x. B opr 1 ltsub13t mp3an3 y cv A axmulrcl y cv B axmulrcl syl2an anandis y cv nnret sylan 3bitr4d z cv RR e. adantr y cv x. B opr - 1 opr <_ z cv br anbi1d y cv x. A opr y cv x. B opr - 1 opr z cv ltletrt 3expa y cv A axmulrcl y cv nnret sylan B RR e. adantrr y cv B axmulrcl y cv nnret sylan ax1re y cv x. B opr 1 resubclt mpan2 syl A RR e. adantrl jca sylan sylbid y cv nngt0t y cv nnret y cv A z cv ltmuldiv2t 3exp1 syl com4r mpcom B RR e. adantrd imp31 sylibd y cv B axmulcl y cv nncnt B recnt syl2an 1cn y cv x. B opr 1 npcant mpan2 syl z cv RR e. adantr z cv < breq2d y cv nngt0t z cv RR e. B RR e. 3ad2ant2 z cv y cv B ltdivmult y cv nnret syl3anl2 mpdan 3coml 3expa bitr4d biimpd A RR e. adantlrl anim12d z cv zret sylan2 z cv y cv znq ancoms A RR e. B RR e. /\ adantlr jctild x cv z cv / y cv opr A < breq2 x cv z cv / y cv opr < B breq1 anbi12d QQ rcla4ev syl6 1 / y cv opr < B - A opr br y cv x. B opr - 1 opr <_ z cv br z cv < y cv x. B opr - 1 opr + 1 opr br anass syl5ibr exp4b com34 r19.23adv mpd expcom r19.23adv syld imp))
+thm (monoord ((y z) (A y) (A z) (B y) (x y) (x z) (F x) (F y) (F z)) ((monoord.1 (:--> F (NN) (RR))) (monoord.2 (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1))))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (` F B))) (y cv 1 A <_ breq2 y cv 1 F fveq2 F A ` <_ breq2d imbi12d A NN e. imbi2d y cv z cv A <_ breq2 y cv z cv F fveq2 F A ` <_ breq2d imbi12d A NN e. imbi2d y cv z cv + 1 opr A <_ breq2 y cv z cv + 1 opr F fveq2 F A ` <_ breq2d imbi12d A NN e. imbi2d y cv B A <_ breq2 y cv B F fveq2 F A ` <_ breq2d imbi12d A NN e. imbi2d A nnge1t A nnret ax1re A 1 letri3t mpan2 syl F A ` F 1 ` eqlet monoord.1 A ffvrni A 1 F fveq2 syl2an exp sylbird mpan2d A z cv + 1 opr leloet A nnret z cv peano2nn z cv + 1 opr nnret syl syl2an A <_ z cv br F A ` <_ F z cv ` br -> adantr A z cv nnleltp1t A <_ z cv br F A ` <_ F z cv ` br -> adantr monoord.1 A ffvrni z cv NN e. F A ` <_ F z cv ` br ad2antll monoord.1 z cv ffvrni A NN e. F A ` <_ F z cv ` br ad2antlr z cv peano2nn monoord.1 z cv + 1 opr ffvrni syl A NN e. F A ` <_ F z cv ` br ad2antlr A NN e. z cv NN e. /\ F A ` <_ F z cv ` br pm3.27 x cv z cv F fveq2 x cv z cv + 1 opreq1 F fveq2d <_ breq12d monoord.2 vtoclga A NN e. F A ` <_ F z cv ` br ad2antlr letrd exp A <_ z cv br imim2d imp sylbird F A ` F z cv + 1 opr ` eqlet monoord.1 A ffvrni A z cv + 1 opr F fveq2 syl2an exp z cv NN e. A <_ z cv br F A ` <_ F z cv ` br -> ad2antll jaod sylbid exp31 com12 a2d nnind com12 3imp))
+thm (om2uz0 ((x y) (C x) (C y)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (= (` G ({/})) C) (om2uz.2 {/} fveq1i om2uz.1 C ZZ x y y cv x cv + 1 opr = {<,>|} fr0t ax-mp eqtr))
+thm (om2uzsuc ((x y) (w x) (v x) (w y) (v y) (v w) (G w) (G v) (A w) (A v) (C x) (C y) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A (om)) (= (` G (suc A)) (opr (` G A) (+) (1)))) (w cv A suceq G fveq2d w cv A G fveq2 + 1 opreq1d eqeq12d G w cv ` + 1 oprex v cv C e. x ax-17 v cv w cv e. x ax-17 v x y y cv x cv + 1 opr = hbopab1 v cv C e. x ax-17 hbrdg v cv om e. x ax-17 hbres v cv w cv e. x ax-17 hbfv v cv + e. x ax-17 v cv 1 e. x ax-17 hbopr om2uz.2 w cv fveq1i + 1 opreq1i v cv eleq2i om2uz.2 w cv fveq1i + 1 opreq1i v cv eleq2i x bial 3imtr4 om2uz.2 x cv G w cv ` + 1 opreq1 V frsucopab mpan2 vtoclga))
+thm (om2uzuz ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A (om)) (e. (` G A) ({e.|} z (ZZ) (br C (<_) (cv z))))) (v cv {/} G fveq2 z ZZ C <_ z cv br {e.|} eleq1d v cv w cv G fveq2 z ZZ C <_ z cv br {e.|} eleq1d v cv w cv suc G fveq2 z ZZ C <_ z cv br {e.|} eleq1d v cv A G fveq2 z ZZ C <_ z cv br {e.|} eleq1d om2uz.1 om2uz.2 om2uz0 z cv C C <_ breq2 ZZ elrab om2uz.1 om2uz.1 zre leid mpbir2an eqeltr om2uz.1 om2uz.2 w cv om2uzsuc z ZZ C <_ z cv br {e.|} eleq1d om2uz.1 C G w cv ` z peano2uz2 mpan syl5bir finds))
+thm (om2uzlt ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (B z) (B w) (B v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (/\ (e. A (om)) (e. B (om))) (-> (e. A B) (br (` G A) (<) (` G B)))) (v cv {/} A eleq2 v cv {/} G fveq2 G A ` < breq2d imbi12d A om e. imbi2d v cv w cv A eleq2 v cv w cv G fveq2 G A ` < breq2d imbi12d A om e. imbi2d v cv w cv suc A eleq2 v cv w cv suc G fveq2 G A ` < breq2d imbi12d A om e. imbi2d v cv B A eleq2 v cv B G fveq2 G A ` < breq2d imbi12d A om e. imbi2d A noel G A ` < G {/} ` br pm2.21i A om e. a1i w cv om A elsuc2g bicomd A om e. adantl G A ` G w cv ` leloet om2uz.1 om2uz.2 A z om2uzuz z ZZ C <_ z cv br ssrab G A ` sseli G A ` zret 3syl om2uz.1 om2uz.2 w cv z om2uzuz z ZZ C <_ z cv br ssrab G w cv ` sseli G w cv ` zret 3syl syl2an G A ` G w cv ` zleltp1t z ZZ C <_ z cv br ssrab G A ` sseli z ZZ C <_ z cv br ssrab G w cv ` sseli syl2an om2uz.1 om2uz.2 A z om2uzuz om2uz.1 om2uz.2 w cv z om2uzuz syl2an om2uz.1 om2uz.2 w cv om2uzsuc G A ` < breq2d A om e. adantl bitr4d bitr3d imbi12d A w cv e. G A ` < G w cv ` br -> id A w cv G fveq2 A w cv e. G A ` < G w cv ` br -> a1i orim12d syl5bi expcom a2d finds impcom))
+thm (om2uzlt2 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (br (` G A) (<) (` G B)))) (om2uz.1 om2uz.2 A B om2uzlt om2uz.1 om2uz.2 B A om2uzlt B A G fveq2 B om e. A om e. /\ a1i orim12d ancoms B A onsseleq B A ontri1 bitr3d B nnont A nnont syl2an ancoms G B ` G A ` leloet G B ` G A ` lenltt bitr3d om2uz.1 om2uz.2 B z om2uzuz z ZZ C <_ z cv br ssrab zssre sstri G B ` sseli syl om2uz.1 om2uz.2 A z om2uzuz z ZZ C <_ z cv br ssrab zssre sstri G A ` sseli syl syl2an ancoms 3imtr3d a3d impbid))
+thm (om2uzran ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G z) (G w) (G v) (G u) (C x) (C y) (C z) (C w) (C v) (C u)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (= (ran G) ({e.|} z (ZZ) (br C (<_) (cv z)))) (x y y cv x cv + 1 opr = {<,>|} C frfnom om2uz.2 G x y y cv x cv + 1 opr = {<,>|} C rec om |` om fneq1 ax-mp mpbir G om u cv w fvelrn ax-mp G w cv ` u cv z ZZ C <_ z cv br {e.|} eleq1 om2uz.1 om2uz.2 w cv z om2uzuz syl5bi com12 r19.23aiv sylbi z cv u cv C <_ breq2 ZZ elrab om2uz.1 w cv C G ran eleq1 w cv v cv G ran eleq1 w cv v cv + 1 opr G ran eleq1 w cv u cv G ran eleq1 om2uz.1 om2uz.2 om2uz0 x y y cv x cv + 1 opr = {<,>|} C frfnom om2uz.2 G x y y cv x cv + 1 opr = {<,>|} C rec om |` om fneq1 ax-mp mpbir peano1 G om {/} fnfvrn mp2an eqeltrr C ZZ e. a1i x y y cv x cv + 1 opr = {<,>|} C frfnom om2uz.2 G x y y cv x cv + 1 opr = {<,>|} C rec om |` om fneq1 ax-mp mpbir G om v cv w fvelrn ax-mp om2uz.1 om2uz.2 w cv om2uzsuc G w cv ` v cv + 1 opreq1 sylan9eq w cv peano2 x y y cv x cv + 1 opr = {<,>|} C frfnom om2uz.2 G x y y cv x cv + 1 opr = {<,>|} C rec om |` om fneq1 ax-mp mpbir G om w cv suc fnfvrn mpan syl G w cv ` v cv = adantr eqeltrrd exp r19.23aiv sylbi C ZZ e. v cv ZZ e. C <_ v cv br /\/\ a1i uzind mp3an1 sylbi impbi eqri))
+thm (om2uzf1o ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (C x) (C y) (C z) (C w) (C v)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (:-1-1-onto-> G (om) ({e.|} z (ZZ) (br C (<_) (cv z)))) (G om z ZZ C <_ z cv br {e.|} f1o5 G om z ZZ C <_ z cv br {e.|} w v f1fv G om z ZZ C <_ z cv br {e.|} df-f x y y cv x cv + 1 opr = {<,>|} C frfnom om2uz.2 G x y y cv x cv + 1 opr = {<,>|} C rec om |` om fneq1 ax-mp mpbir om2uz.1 om2uz.2 z om2uzran G ran z ZZ C <_ z cv br {e.|} eqimss ax-mp mpbir2an G w cv ` G v cv ` lttri3t om2uz.1 om2uz.2 w cv z om2uzuz z ZZ C <_ z cv br ssrab G w cv ` sseli G w cv ` zret 3syl om2uz.1 om2uz.2 v cv z om2uzuz z ZZ C <_ z cv br ssrab G v cv ` sseli G v cv ` zret 3syl syl2an G w cv ` < G v cv ` br G v cv ` < G w cv ` br ioran syl6bbr w cv v cv ordtri3 w cv nnord v cv nnord syl2an bicon2d om2uz.1 om2uz.2 w cv v cv om2uzlt om2uz.1 om2uz.2 v cv w cv om2uzlt ancoms orim12d sylbird con1d sylbid rgen2 mpbir2an om2uz.1 om2uz.2 z om2uzran mpbir2an))
+thm (uzrdgval ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) B) (` (rec F A) (` (`' G) B)))) (om2uz.1 om2uz.2 z om2uzran G df-rn eqtr3 B eleq2i F A rdgfnon F A rec On fnfun ax-mp om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} f1ocnv ax-mp G `' z ZZ C <_ z cv br {e.|} om f1ofun ax-mp F A rec G `' B fvco mpanl12 sylbi))
+thm (uzrdgini ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) C) A)) (A B F rdg0t z cv C C <_ breq2 ZZ elrab om2uz.1 om2uz.1 zre leid mpbir2an om2uz.1 om2uz.2 C z F A uzrdgval ax-mp om2uz.1 om2uz.2 om2uz0 om2uz.1 om2uz.2 z om2uzf1o peano1 G om z ZZ C <_ z cv br {e.|} {/} C f1ocnvfv mp2an ax-mp F A rec fveq2i eqtr syl5eq))
+thm (uzrdgsuc ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) (opr B (+) (1))) (` F (` (o. (rec F A) (`' G)) B)))) (om2uz.1 C B z peano2uz2 mpan om2uz.1 om2uz.2 B + 1 opr z F A uzrdgval syl om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} f1ocnv ax-mp G `' z ZZ C <_ z cv br {e.|} om f1of ax-mp B ffvrni om2uz.1 om2uz.2 G `' B ` om2uzsuc syl om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} B f1ocnvfv2 mpan + 1 opreq1d eqtrd om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} f1ocnv ax-mp G `' z ZZ C <_ z cv br {e.|} om f1of ax-mp B ffvrni G `' B ` peano2 om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} G `' B ` suc B + 1 opr f1ocnvfv mpan 3syl mpd F A rec fveq2d om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} f1ocnv ax-mp G `' z ZZ C <_ z cv br {e.|} om f1of ax-mp B ffvrni G `' B ` nnont G `' B ` F A rdgsuct 3syl 3eqtrd om2uz.1 om2uz.2 B z F A uzrdgval F fveq2d eqtr4d))
+thm (uzrdginip1 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) (opr C (+) (1))) (` F A))) (om2uz.1 om2uz.2 A B F uzrdgini F fveq2d z cv C C <_ breq2 ZZ elrab om2uz.1 om2uz.1 zre leid mpbir2an om2uz.1 om2uz.2 C z F A uzrdgsuc ax-mp syl5eq))
+thm (uzrdgfnuz ((x y) (x z) (y z) (G z) (A z) (C x) (C y) (C z)) ((om2uz.1 (e. C (ZZ))) (om2uz.2 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om))))) (Fn (o. (rec F A) (`' G)) ({e.|} z (ZZ) (br C (<_) (cv z)))) (F A rdgfnon F A rec On fnf mpbi om2uz.1 om2uz.2 z om2uzf1o G om z ZZ C <_ z cv br {e.|} f1ocnv ax-mp G `' z ZZ C <_ z cv br {e.|} om f1of ax-mp omsson G `' z ZZ C <_ z cv br {e.|} om On fss mp2an F A rec On V G `' z ZZ C <_ z cv br {e.|} fco mp2an F A rec G `' o. z ZZ C <_ z cv br {e.|} fnf mpbir))
+thm (seq1lem1 ((x y) (v x) (v y) (w z) (v z) (u z) (t z) (f z) (v w) (u w) (t w) (f w) (u v) (t v) (f v) (t u) (f u) (f t) (A v) (B w) (B v) (B u) (B t) (B f) (u x) (t x) (f x) (C x) (C v) (C u) (C t) (C f) (G v) (G u) (G t) (G f)) ((seq1lem1.1 (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (1)) (om))))) (-> (e. A (NN)) (= (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) A)) A)) (v cv 1 z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. fveq2 1st fveq2d v cv 1 = id eqeq12d v cv u cv z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. fveq2 1st fveq2d v cv u cv = id eqeq12d v cv u cv + 1 opr z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. fveq2 1st fveq2d v cv u cv + 1 opr = id eqeq12d v cv A z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. fveq2 1st fveq2d v cv A = id eqeq12d 1 C opex 1z seq1lem1.1 1 C <,> V z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} uzrdgini ax-mp 1st fveq2i 1z elisseti C op1st eqtr v nnz u cv eleq2i 1z seq1lem1.1 u cv v z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> uzrdgsuc sylbi w cv 1st z cv ` + 1 opr B <,> = v ax-17 w cv 1st v cv ` + 1 opr e. z ax-17 v z t cv B e. hbs1 w t hbab hbop hbeleq 1st z cv ` + 1 opr 1st v cv ` + 1 opr B t v cv z t cv B e. [/] {|} opeq12 z cv v cv 1st fveq2 + 1 opreq1d z v B t sbab sylanc w cv eqeq2d w cbvopab1 z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` fveq1i f cv z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` e. v ax-17 f cv 1st z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` ` + 1 opr t z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` z t cv B e. [/] {|} <,> e. v ax-17 z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv fvex 1st z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` ` + 1 opr t z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` z t cv B e. [/] {|} opex 1st v cv ` + 1 opr 1st z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` ` + 1 opr t v cv z t cv B e. [/] {|} t z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` z t cv B e. [/] {|} opeq12 v cv z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` 1st fveq2 + 1 opreq1d v cv z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` z t cv B e. dfsbcq t biabdv sylanc w fvopabf eqtr syl6eq 1st fveq2d 1st z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` ` + 1 oprex t z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` z t cv B e. [/] {|} op1st syl6eq 1st z w w cv 1st z cv ` + 1 opr B <,> = {<,>|} 1 C <,> rec G `' o. u cv ` ` u cv + 1 opreq1 sylan9eq exp nnind))
+thm (seq1lem2 () () (-> (e. A (NN)) (e. (opr A (+) (1)) (\ (NN) ({} (1))))) (A + 1 opr nngt1ne1t 1nn elisseti A + 1 opr elsnc2 negbii syl6bbr pm5.32i A + 1 opr NN 1 {} eldif bitr4 biimp A peano2nn A nngt0t A nnret 0re ax1re 0 A 1 ltadd1t mp3an13 1cn addid2 < A + 1 opr breq1i syl6bb syl mpbid sylanc))
+thm (seq1rval ((A z) (S z) (S w) (w z) (F z) (F w)) ((seq1rval.1 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (seq1rval.2 (e. A (V)))) (= (` H A) (<,> (opr (` (1st) A) (+) (1)) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1)))))) (seq1rval.1 A fveq1i seq1rval.2 1st A ` + 1 opr 2nd A ` S F 1st A ` + 1 opr ` opr opex 1st z cv ` + 1 opr 1st A ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr 2nd A ` S F 1st A ` + 1 opr ` opr opeq12 z cv A 1st fveq2 + 1 opreq1d z cv A 2nd fveq2 z cv A 1st fveq2 + 1 opreq1d F fveq2d S opreq12d sylanc w fvopab eqtr))
+thm (seq1val ((x y) (f x) (g x) (h x) (G x) (f y) (g y) (h y) (G y) (f g) (f h) (G f) (g h) (G g) (G h) (H x) (H y) (H f) (H g) (H h) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (f z) (g z) (h z) (F z) (f w) (g w) (h w) (F w) (F f) (F g) (F h) (S x) (S y) (S z) (S w) (S f) (S g) (S h)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (= (opr S (seq1) F) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x))))))) (seq1val.1 seq1val.2 nnex x y 2nd H 1 F 1 ` <,> rec G `' o. x cv ` ` funopabex2 f cv S 2nd z cv ` g cv 1st z cv ` + 1 opr ` opreq 2nd z cv ` f cv g cv 1st z cv ` + 1 opr ` opr 2nd z cv ` S g cv 1st z cv ` + 1 opr ` opr 1st z cv ` + 1 opr opeq2 syl w cv eqeq2d z w biopabdv z w w cv 1st z cv ` + 1 opr 2nd z cv ` f cv g cv 1st z cv ` + 1 opr ` opr <,> = {<,>|} z w w cv 1st z cv ` + 1 opr 2nd z cv ` S g cv 1st z cv ` + 1 opr ` opr <,> = {<,>|} 1 g cv 1 ` <,> rdgeq1 syl z w w cv z cv + 1 opr = {<,>|} 1 rec om |` `' coeq1d seq1val.3 G z w w cv z cv + 1 opr = {<,>|} 1 rec om |` cnveq ax-mp z w w cv 1st z cv ` + 1 opr 2nd z cv ` S g cv 1st z cv ` + 1 opr ` opr <,> = {<,>|} 1 g cv 1 ` <,> rec coeq2i syl6eqr x cv fveq1d 2nd fveq2d y cv eqeq2d x cv NN e. anbi2d x y biopabdv g cv F 1st z cv ` + 1 opr fveq1 2nd z cv ` S opreq2d 2nd z cv ` S g cv 1st z cv ` + 1 opr ` opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr 1st z cv ` + 1 opr opeq2 syl w cv eqeq2d z w biopabdv seq1val.4 syl6eqr z w w cv 1st z cv ` + 1 opr 2nd z cv ` S g cv 1st z cv ` + 1 opr ` opr <,> = {<,>|} H 1 g cv 1 ` <,> rdgeq1 syl g cv F 1 fveq1 g cv 1 ` F 1 ` 1 opeq2 1 g cv 1 ` <,> 1 F 1 ` <,> H rdgeq2 3syl eqtrd G `' coeq1d x cv fveq1d 2nd fveq2d y cv eqeq2d x cv NN e. anbi2d x y biopabdv f g h x y z w df-seq1 V V oprabval5 mp2an))
+thm (seq1fnlem ((x y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (Fn (opr S (seq1) F) (NN)) (2nd H 1 F 1 ` <,> rec G `' o. x cv ` fvex seq1val.1 seq1val.2 seq1val.3 seq1val.4 x y seq1val fnopab2))
+thm (seq1val2 ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) A) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)))) (x cv A H 1 F 1 ` <,> rec G `' o. fveq2 2nd fveq2d seq1val.1 seq1val.2 seq1val.3 seq1val.4 x y seq1val 2nd H 1 F 1 ` <,> rec G `' o. A ` fvex fvopab4))
+thm (seq11lem ((w z) (F z) (F w) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (= (` (opr S (seq1) F) (1)) (` F (1))) (1nn seq1val.1 seq1val.2 seq1val.3 seq1val.4 1 seq1val2 ax-mp 1 F 1 ` opex 1z seq1val.3 1 F 1 ` <,> V H uzrdgini ax-mp 2nd fveq2i 1nn elisseti F 1 fvex op2nd 3eqtr))
+thm (seq1suclem ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((seq1val.1 (e. S (V))) (seq1val.2 (e. F (V))) (seq1val.3 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (seq1val.4 (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))) (x nnz A eleq2i 1z seq1val.3 A x H 1 F 1 ` <,> uzrdgsuc sylbi seq1val.4 1st x cv ` + 1 opr 1st z cv ` + 1 opr 2nd x cv ` S F 1st x cv ` + 1 opr ` opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr opeq12 x cv z cv 1st fveq2 + 1 opreq1d x cv z cv 2nd fveq2 x cv z cv 1st fveq2 + 1 opreq1d F fveq2d S opreq12d sylanc y cv eqeq2d y cv w cv 1st z cv ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr <,> eqeq1 sylan9bb cbvopabv eqtr4 H 1 F 1 ` <,> rec G `' o. A fvex seq1rval syl6eq 2nd fveq2d 1st H 1 F 1 ` <,> rec G `' o. A ` ` + 1 oprex 2nd H 1 F 1 ` <,> rec G `' o. A ` ` S F 1st H 1 F 1 ` <,> rec G `' o. A ` ` + 1 opr ` oprex op2nd syl6eq seq1val.3 A z w 2nd z cv ` S F 1st z cv ` + 1 opr ` opr F 1 ` seq1lem1 seq1val.4 H z w w cv 1st z cv ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr <,> = {<,>|} 1 F 1 ` <,> rdgeq1 ax-mp G `' coeq1i A fveq1i 1st fveq2i syl5eq + 1 opreq1d F fveq2d 2nd H 1 F 1 ` <,> rec G `' o. A ` ` S opreq2d eqtrd A peano2nn seq1val.1 seq1val.2 seq1val.3 seq1val.4 A + 1 opr seq1val2 syl seq1val.1 seq1val.2 seq1val.3 seq1val.4 A seq1val2 S F A + 1 opr ` opreq1d 3eqtr4d))
+thm (seq11 ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (= (` (opr S (seq1) F) (1)) (` F (1))) (seq111.1 seq111.2 z w w cv z cv + 1 opr = {<,>|} 1 rec om |` eqid z w w cv 1st z cv ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr <,> = {<,>|} eqid seq11lem))
+thm (seq1p1 ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))) (seq111.1 seq111.2 z w w cv z cv + 1 opr = {<,>|} 1 rec om |` eqid z w w cv 1st z cv ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr <,> = {<,>|} eqid A seq1suclem))
+thm (seq1m1 () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\ (e. N (NN)) (br (1) (<) N)) (= (` (opr S (seq1) F) N) (opr (` (opr S (seq1) F) (opr N (-) (1))) S (` F N)))) (1nn 1 N nnsubt mpan biimpa seq111.1 seq111.2 N - 1 opr seq1p1 syl N nncnt 1cn N 1 npcant mpan2 syl S seq1 F opr fveq2d 1 < N br adantr N nncnt 1cn N 1 npcant mpan2 syl F fveq2d S seq1 F opr N - 1 opr ` S opreq2d 1 < N br adantr 3eqtr3d))
+thm (seq1fn ((w z) (S z) (S w) (F z) (F w)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (Fn (opr S (seq1) F) (NN)) (seq111.1 seq111.2 z w w cv z cv + 1 opr = {<,>|} 1 rec om |` eqid z w w cv 1st z cv ` + 1 opr 2nd z cv ` S F 1st z cv ` + 1 opr ` opr <,> = {<,>|} eqid seq1fnlem))
+thm (seq1rn ((x y) (x z) (w x) (v x) (S x) (y z) (w y) (v y) (S y) (w z) (v z) (S z) (v w) (S w) (S v) (F x) (F y) (F z) (F w) (F v) (C x) (C y) (C z) (C w) (C v) (D x) (D y) (D z) (D w) (D v)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) (C_ (ran (opr S (seq1) F)) C)) (y cv 1 S seq1 F opr fveq2 C eleq1d F 1 ` C e. F NN 1 {} \ |` NN 1 {} \ D :--> S C D X. C :--> /\/\ imbi2d y cv z cv S seq1 F opr fveq2 C eleq1d F 1 ` C e. F NN 1 {} \ |` NN 1 {} \ D :--> S C D X. C :--> /\/\ imbi2d y cv z cv + 1 opr S seq1 F opr fveq2 C eleq1d F 1 ` C e. F NN 1 {} \ |` NN 1 {} \ D :--> S C D X. C :--> /\/\ imbi2d y cv x cv S seq1 F opr fveq2 C eleq1d F 1 ` C e. F NN 1 {} \ |` NN 1 {} \ D :--> S C D X. C :--> /\/\ imbi2d F 1 ` C e. F NN 1 {} \ |` NN 1 {} \ D :--> pm3.26 seq111.1 seq111.2 seq11 C eleq1i sylibr S C D X. C :--> 3adant3 F NN 1 {} \ |` NN 1 {} \ D z cv + 1 opr ffvrn z cv + 1 opr NN 1 {} \ F fvres D eleq1d F NN 1 {} \ |` NN 1 {} \ D :--> adantl mpbid z cv seq1lem2 sylan2 seq111.1 seq111.2 z cv seq1p1 C eleq1d w cv S seq1 F opr z cv ` S v cv opreq1 C eleq1d v cv F z cv + 1 opr ` S seq1 F opr z cv ` S opreq2 C eleq1d C D rcla42v ancoms S C D C w v ffnoprval pm3.27bd syl5 imp syl5bir exp4c F NN 1 {} \ |` NN 1 {} \ D :--> adantl mpd exp com4r impcom F 1 ` C e. 3adant1 com12 a2d nnind com12 r19.21aiv seq111.1 seq111.2 seq1fn S seq1 F opr NN x C fnfvrnss mpan syl))
+thm (seq1rn2 () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (C_ (ran (opr S (seq1) F)) C)) (seq111.1 seq111.2 C C seq1rn 3expa 1nn F NN C 1 ffvrn mpan2 NN 1 {} difss F NN C NN 1 {} \ fssres mpan2 jca sylan))
+thm (seq1cl () ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (-> (/\/\ (e. A (NN)) (:--> F (NN) C) (:--> S (X. C C) C)) (e. (` (opr S (seq1) F) A) C)) (S seq1 F opr ran C S seq1 F opr A ` ssel2 ancoms seq111.1 seq111.2 seq1fn S seq1 F opr NN A fnfvrn mpan seq111.1 seq111.2 C seq1rn2 syl2an 3impb))
+thm (seq1res ((x y) (x z) (S x) (y z) (S y) (S z) (F x) (F y) (F z)) ((seq111.1 (e. S (V))) (seq111.2 (e. F (V)))) (= (opr S (seq1) (|` F (NN))) (opr S (seq1) F)) (seq111.1 seq111.2 F V NN resexg ax-mp seq1fn seq111.1 seq111.2 seq1fn S seq1 F NN |` opr NN S seq1 F opr NN x eqfnfv mp2an NN eqid y cv 1 S seq1 F NN |` opr fveq2 y cv 1 S seq1 F opr fveq2 eqeq12d y cv z cv S seq1 F NN |` opr fveq2 y cv z cv S seq1 F opr fveq2 eqeq12d y cv z cv + 1 opr S seq1 F NN |` opr fveq2 y cv z cv + 1 opr S seq1 F opr fveq2 eqeq12d y cv x cv S seq1 F NN |` opr fveq2 y cv x cv S seq1 F opr fveq2 eqeq12d 1nn 1 NN F fvres ax-mp seq111.1 seq111.2 F V NN resexg ax-mp seq11 seq111.1 seq111.2 seq11 3eqtr4 S seq1 F NN |` opr z cv ` S seq1 F opr z cv ` = id z cv peano2nn z cv + 1 opr NN F fvres syl S opreqan12rd seq111.1 seq111.2 F V NN resexg ax-mp z cv seq1p1 S seq1 F NN |` opr z cv ` S seq1 F opr z cv ` = adantr seq111.1 seq111.2 z cv seq1p1 S seq1 F NN |` opr z cv ` S seq1 F opr z cv ` = adantr 3eqtr4d exp nnind rgen mpbir2an))
+thm (ser1ft ((F f)) () (-> (:--> F (NN) (CC)) (:--> (opr (+) (seq1) F) (NN) (CC))) (nnex NN V F CC fex ax-mp f cv F NN CC feq1 f cv F + seq1 opreq2 + seq1 f cv opr + seq1 F opr NN CC feq1 syl imbi12d axaddopr addex f visset CC seq1rn2 mpan2 addex f visset seq1fn jctil + seq1 f cv opr NN CC df-f sylibr V vtoclg mpcom))
+thm (ser1f () ((ser1f.1 (:--> F (NN) (CC)))) (:--> (opr (+) (seq1) F) (NN) (CC)) (ser1f.1 F ser1ft ax-mp))
+thm (ser1cl () ((ser1cl.1 (:--> F (NN) (CC)))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (CC))) (ser1cl.1 axaddopr addex nnex ser1cl.1 NN V F CC fex mp2 A CC seq1cl mp3an23))
+thm (ser1recl ((A x) (x y) (F x) (F y)) ((ser1recl.1 (:--> F (NN) (RR)))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (RR))) (x cv 1 + seq1 F opr fveq2 RR eleq1d x cv y cv + seq1 F opr fveq2 RR eleq1d x cv y cv + 1 opr + seq1 F opr fveq2 RR eleq1d x cv A + seq1 F opr fveq2 RR eleq1d addex nnex ser1recl.1 NN V F RR fex mp2 seq11 ser1recl.1 1nn F NN RR 1 ffvrn mp2an eqeltr + seq1 F opr y cv ` F y cv + 1 opr ` axaddrcl y cv peano2nn ser1recl.1 y cv + 1 opr ffvrni syl sylan2 addex nnex ser1recl.1 NN V F RR fex mp2 y cv seq1p1 RR eleq1d + seq1 F opr y cv ` RR e. adantl mpbird expcom nnind))
+thm (ser1re ((F x)) ((ser1recl.1 (:--> F (NN) (RR)))) (:--> (opr (+) (seq1) F) (NN) (RR)) (+ seq1 F opr NN RR x ffnfv addex nnex ser1recl.1 NN V F RR fex mp2 seq1fn ser1recl.1 x cv ser1recl rgen mpbir2an))
+thm (ser1cl2 ((x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1cl2.2 (A.e. x (NN) (e. A (CC))))) (-> (e. B (NN)) (e. (` (opr (+) (seq1) F) B) (CC))) (ser1cl2.2 ser1cl2.1 CC fopab2 mpbi B ser1cl))
+thm (ser1f2 ((x y) (x z) (y z) (A y) (F z)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1cl2.2 (A.e. x (NN) (e. A (CC))))) (:--> (opr (+) (seq1) F) (NN) (CC)) (+ seq1 F opr NN CC z ffnfv addex ser1cl2.1 nnex x y A funopabex2 eqeltr seq1fn ser1cl2.1 ser1cl2.2 z cv ser1cl2 rgen mpbir2an))
+thm (ser112 ((B y) (x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser112.2 (e. B (V))) (ser112.3 (-> (= (cv x) (1)) (= A B)))) (= (` (opr (+) (seq1) F) (1)) B) (addex ser1cl2.1 nnex x y A funopabex2 eqeltr seq11 1nn ser112.3 ser1cl2.1 ser112.2 fvopab4 ax-mp eqtr))
+thm (ser1suc2 ((B y) (x y) (C x) (C y) (x y) (A y) (B x)) ((ser1cl2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A))))) (ser1suc2.2 (e. C (V))) (ser1suc2.3 (-> (= (cv x) (opr B (+) (1))) (= A C)))) (-> (e. B (NN)) (= (` (opr (+) (seq1) F) (opr B (+) (1))) (opr (` (opr (+) (seq1) F) B) (+) C))) (addex ser1cl2.1 nnex x y A funopabex2 eqeltr B seq1p1 B peano2nn ser1suc2.3 ser1cl2.1 ser1suc2.2 fvopab4 syl + seq1 F opr B ` + opreq2d eqtrd))
+thm (ser1mono ((A y) (x y) (F x) (F y)) ((ser1mono.1 (:--> F (NN) (RR))) (ser1mono.2 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) F) A) (<_) (` (opr (+) (seq1) F) (opr A (+) (1))))) (ser1mono.1 axresscn F NN RR CC fss mp2an A ser1cl + seq1 F opr A ` ax0id syl eqcomd A peano2nn y cv A + 1 opr F fveq2 0 <_ breq2d x cv y cv F fveq2 0 <_ breq2d ser1mono.2 vtoclga vtoclga syl A peano2nn ser1mono.1 A + 1 opr ffvrni syl ser1mono.1 A ser1recl jca 0re 0 F A + 1 opr ` + seq1 F opr A ` leadd2t mp3an1 syl mpbid addex nnex ser1mono.1 NN V F RR fex mp2 A seq1p1 eqcomd breqtrd eqbrtrd))
+thm (ser1mulc ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z) (A x) (A y) (A z) (B y) (B z)) ((ser1mlc.1 (e. A (CC))) (ser1mlc.2 (:--> F (NN) (CC))) (ser1mlc.3 (e. G (V))) (ser1mlc.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x))))))) (-> (e. B (NN)) (= (` (opr (+) (seq1) G) B) (opr A (x.) (` (opr (+) (seq1) F) B)))) (y cv 1 + seq1 G opr fveq2 y cv 1 + seq1 F opr fveq2 A x. opreq2d eqeq12d y cv z cv + seq1 G opr fveq2 y cv z cv + seq1 F opr fveq2 A x. opreq2d eqeq12d y cv z cv + 1 opr + seq1 G opr fveq2 y cv z cv + 1 opr + seq1 F opr fveq2 A x. opreq2d eqeq12d y cv B + seq1 G opr fveq2 y cv B + seq1 F opr fveq2 A x. opreq2d eqeq12d 1nn x cv 1 G fveq2 x cv 1 F fveq2 A x. opreq2d eqeq12d ser1mlc.4 vtoclga ax-mp addex ser1mlc.3 seq11 addex nnex ser1mlc.2 NN V F CC fex mp2 seq11 A x. opreq2i 3eqtr4 addex ser1mlc.3 z cv seq1p1 + seq1 G opr z cv ` A x. + seq1 F opr z cv ` opr = adantr + seq1 G opr z cv ` A x. + seq1 F opr z cv ` opr + G z cv + 1 opr ` opreq1 z cv NN e. adantl ser1mlc.1 A + seq1 F opr z cv ` F z cv + 1 opr ` axdistr mp3an1 ser1mlc.2 ser1f z cv ffvrni z cv peano2nn ser1mlc.2 z cv + 1 opr ffvrni syl sylanc addex nnex ser1mlc.2 NN V F CC fex mp2 z cv seq1p1 A x. opreq2d z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 A x. opreq2d eqeq12d ser1mlc.4 vtoclga syl A x. + seq1 F opr z cv ` opr + opreq2d 3eqtr4rd + seq1 G opr z cv ` A x. + seq1 F opr z cv ` opr = adantr 3eqtrd exp nnind))
+thm (ser1add2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((ser1add2.1 (:--> F (NN) (CC))) (ser1add2.2 (:--> G (NN) (CC))) (ser1add2.3 (e. H (V))) (ser1add2.4 (-> (/\/\ (e. (cv k) (NN)) (e. N (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))) (N nnret N leidt syl j cv 1 <_ N breq1 N NN e. anbi2d j cv 1 + seq1 H opr fveq2 j cv 1 + seq1 F opr fveq2 j cv 1 + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv m cv <_ N breq1 N NN e. anbi2d j cv m cv + seq1 H opr fveq2 j cv m cv + seq1 F opr fveq2 j cv m cv + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv m cv + 1 opr <_ N breq1 N NN e. anbi2d j cv m cv + 1 opr + seq1 H opr fveq2 j cv m cv + 1 opr + seq1 F opr fveq2 j cv m cv + 1 opr + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv N <_ N breq1 N NN e. anbi2d j cv N + seq1 H opr fveq2 j cv N + seq1 F opr fveq2 j cv N + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d 1nn k cv 1 <_ N breq1 N NN e. anbi2d k cv 1 H fveq2 k cv 1 F fveq2 k cv 1 G fveq2 + opreq12d eqeq12d imbi12d ser1add2.4 3exp imp3a vtoclga ax-mp addex ser1add2.3 seq11 addex nnex ser1add2.1 NN V F CC fex mp2 seq11 addex nnex ser1add2.2 NN V G CC fex mp2 seq11 + opreq12i 3eqtr4g m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc N RR e. adantr m cv m cv + 1 opr N letrt m cv RR e. N RR e. pm3.26 m cv peano2re N RR e. adantr m cv RR e. N RR e. pm3.27 syl3anc mpand m cv nnret N nnret syl2an imp N NN e. adantllr m cv NN e. N NN e. /\ m cv + 1 opr <_ N br /\ + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = pm3.27 m cv peano2nn k cv m cv + 1 opr <_ N breq1 k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d imbi12d N NN e. imbi2d ser1add2.4 3exp vtoclga syl imp31 + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = adantr + opreq12d addex ser1add2.3 m cv seq1p1 m cv + 1 opr <_ N br + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = ad2antll N NN e. adantllr addex nnex ser1add2.1 NN V F CC fex mp2 m cv seq1p1 addex nnex ser1add2.2 NN V G CC fex mp2 m cv seq1p1 + opreq12d + seq1 F opr m cv ` F m cv + 1 opr ` + seq1 G opr m cv ` G m cv + 1 opr ` add4t ser1add2.1 m cv ser1cl m cv peano2nn ser1add2.1 m cv + 1 opr ffvrni syl jca ser1add2.2 m cv ser1cl m cv peano2nn ser1add2.2 m cv + 1 opr ffvrni syl jca sylanc eqtrd N NN e. + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = ad2antll m cv + 1 opr <_ N br adantlr 3eqtr4d exp N NN e. m cv <_ N br /\ imim2d com23 imp an4s anassrs mpdan exp41 pm2.43d imp3a com23 nnind mpan2d pm2.43i))
+thm (ser1add ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((ser1add.1 (:--> F (NN) (CC))) (ser1add.2 (:--> G (NN) (CC))) (ser1add.3 (e. H (V))) (ser1add.4 (-> (/\ (e. (cv k) (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))) (N nnret N leidt syl j cv 1 <_ N breq1 N NN e. anbi2d j cv 1 + seq1 H opr fveq2 j cv 1 + seq1 F opr fveq2 j cv 1 + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv m cv <_ N breq1 N NN e. anbi2d j cv m cv + seq1 H opr fveq2 j cv m cv + seq1 F opr fveq2 j cv m cv + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv m cv + 1 opr <_ N breq1 N NN e. anbi2d j cv m cv + 1 opr + seq1 H opr fveq2 j cv m cv + 1 opr + seq1 F opr fveq2 j cv m cv + 1 opr + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d j cv N <_ N breq1 N NN e. anbi2d j cv N + seq1 H opr fveq2 j cv N + seq1 F opr fveq2 j cv N + seq1 G opr fveq2 + opreq12d eqeq12d imbi12d 1nn k cv 1 <_ N breq1 k cv 1 H fveq2 k cv 1 F fveq2 k cv 1 G fveq2 + opreq12d eqeq12d imbi12d ser1add.4 exp vtoclga ax-mp addex ser1add.3 seq11 addex nnex ser1add.1 NN V F CC fex mp2 seq11 addex nnex ser1add.2 NN V G CC fex mp2 seq11 + opreq12i 3eqtr4g N NN e. adantl m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc N RR e. adantr m cv m cv + 1 opr N letrt m cv RR e. N RR e. pm3.26 m cv peano2re N RR e. adantr m cv RR e. N RR e. pm3.27 syl3anc mpand m cv nnret N nnret syl2an imp m cv NN e. m cv + 1 opr <_ N br /\ + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = pm3.27 k cv m cv + 1 opr <_ N breq1 k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d imbi12d ser1add.4 exp vtoclga imp m cv peano2nn sylan + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = adantr + opreq12d addex ser1add.3 m cv seq1p1 m cv + 1 opr <_ N br + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = ad2antll addex nnex ser1add.1 NN V F CC fex mp2 m cv seq1p1 addex nnex ser1add.2 NN V G CC fex mp2 m cv seq1p1 + opreq12d + seq1 F opr m cv ` F m cv + 1 opr ` + seq1 G opr m cv ` G m cv + 1 opr ` add4t ser1add.1 m cv ser1cl m cv peano2nn ser1add.1 m cv + 1 opr ffvrni syl jca ser1add.2 m cv ser1cl m cv peano2nn ser1add.2 m cv + 1 opr ffvrni syl jca sylanc eqtrd m cv + 1 opr <_ N br + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = ad2antll 3eqtr4d exp N NN e. m cv <_ N br /\ imim2d com23 imp an4s anassrs mpdan exp31 imp3a com23 nnind mpan2d pm2.43i))
+thm (ser1addOLD ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N m)) ((ser1addOLD.1 (:--> F (NN) (CC))) (ser1addOLD.2 (:--> G (NN) (CC))) (ser1addOLD.3 (e. H (V))) (ser1addOLD.4 (-> (e. (cv k) (NN)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))) (j cv 1 + seq1 H opr fveq2 j cv 1 + seq1 F opr fveq2 j cv 1 + seq1 G opr fveq2 + opreq12d eqeq12d j cv m cv + seq1 H opr fveq2 j cv m cv + seq1 F opr fveq2 j cv m cv + seq1 G opr fveq2 + opreq12d eqeq12d j cv m cv + 1 opr + seq1 H opr fveq2 j cv m cv + 1 opr + seq1 F opr fveq2 j cv m cv + 1 opr + seq1 G opr fveq2 + opreq12d eqeq12d j cv N + seq1 H opr fveq2 j cv N + seq1 F opr fveq2 j cv N + seq1 G opr fveq2 + opreq12d eqeq12d 1nn k cv 1 H fveq2 k cv 1 F fveq2 k cv 1 G fveq2 + opreq12d eqeq12d ser1addOLD.4 vtoclga ax-mp addex ser1addOLD.3 seq11 addex nnex ser1addOLD.1 NN V F CC fex mp2 seq11 addex nnex ser1addOLD.2 NN V G CC fex mp2 seq11 + opreq12i 3eqtr4 addex ser1addOLD.3 m cv seq1p1 + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = adantr + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr + H m cv + 1 opr ` opreq1 m cv NN e. adantl + seq1 F opr m cv ` F m cv + 1 opr ` + seq1 G opr m cv ` G m cv + 1 opr ` add4t ser1addOLD.1 m cv ser1cl m cv peano2nn ser1addOLD.1 m cv + 1 opr ffvrni syl jca ser1addOLD.2 m cv ser1cl m cv peano2nn ser1addOLD.2 m cv + 1 opr ffvrni syl jca sylanc addex nnex ser1addOLD.1 NN V F CC fex mp2 m cv seq1p1 addex nnex ser1addOLD.2 NN V G CC fex mp2 m cv seq1p1 + opreq12d m cv peano2nn k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d ser1addOLD.4 vtoclga syl + seq1 F opr m cv ` + + seq1 G opr m cv ` opr + opreq2d 3eqtr4rd + seq1 H opr m cv ` + seq1 F opr m cv ` + + seq1 G opr m cv ` opr = adantr 3eqtrd exp nnind))
+thm (ser1dir ((F k) (G k) (H k) (R k) (S k) (T k)) ((ser1dir.1 (:--> F (NN) (CC))) (ser1dir.2 (:--> G (NN) (CC))) (ser1dir.3 (:--> H (NN) (CC))) (ser1dir.4 (e. R (V))) (ser1dir.5 (Fn S (NN))) (ser1dir.6 (Fn T (NN))) (ser1dir.7 (-> (e. (cv k) (NN)) (= (` R (cv k)) (opr (opr (` F (cv k)) (+) (` G (cv k))) (x.) (` H (cv k)))))) (ser1dir.8 (-> (e. (cv k) (NN)) (= (` S (cv k)) (opr (` F (cv k)) (x.) (` H (cv k)))))) (ser1dir.9 (-> (e. (cv k) (NN)) (= (` T (cv k)) (opr (` G (cv k)) (x.) (` H (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) R) N) (opr (` (opr (+) (seq1) S) N) (+) (` (opr (+) (seq1) T) N)))) (S NN CC k ffnfv ser1dir.5 ser1dir.8 F k cv ` H k cv ` axmulcl ser1dir.1 k cv ffvrni ser1dir.3 k cv ffvrni sylanc eqeltrd rgen mpbir2an T NN CC k ffnfv ser1dir.6 ser1dir.9 G k cv ` H k cv ` axmulcl ser1dir.2 k cv ffvrni ser1dir.3 k cv ffvrni sylanc eqeltrd rgen mpbir2an ser1dir.4 F k cv ` G k cv ` H k cv ` adddirt ser1dir.1 k cv ffvrni ser1dir.2 k cv ffvrni ser1dir.3 k cv ffvrni syl3anc ser1dir.7 ser1dir.8 ser1dir.9 + opreq12d 3eqtr4d N ser1addOLD))
+thm (shftfval ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (F x) (F y) (F w) (F f) (F g) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (e. A B) (= (opr F (shift) A) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))))) (shftfval.1 axcnex x y F x cv - A opr ` funopabex2 f cv F x cv - w cv opr fveq1 y cv eqeq2d x cv CC e. anbi2d x y biopabdv w cv A = x cv CC e. pm4.2i w cv A x cv - opreq2 F fveq2d y cv eqeq2d anbi12d x y biopabdv f w g x y df-shft V B oprabval5 mpan))
+thm (shftfn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (e. A B) (Fn (opr F (shift) A) (CC))) (F x cv - A opr fvex x y x cv CC e. y cv F x cv - A opr ` = /\ {<,>|} eqid fnopab2 shftfval.1 A B x y shftfval F shift A opr x y x cv CC e. y cv F x cv - A opr ` = /\ {<,>|} CC fneq1 syl mpbiri))
+thm (shftres () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A C) (C_ B (CC))) (Fn (|` (opr F (shift) A) B) B)) (F shift A opr CC B fnssres shftfval.1 A C shftfn sylan))
+thm (shftresvalt () ((shftfval.1 (e. F (V)))) (-> (e. B C) (= (` (|` (opr F (shift) A) C) B) (` (opr F (shift) A) B))) (B C F shift A opr fvres))
+thm (shftvalt ((x y) (A x) (A y) (F x) (F y) (B x) (B y) (C x) (C y)) ((shftfval.1 (e. F (V)))) (-> (/\ (e. A C) (e. B (CC))) (= (` (opr F (shift) A) B) (` F (opr B (-) A)))) (shftfval.1 A C x y shftfval B fveq1d x cv B - A opreq1 F fveq2d x y x cv CC e. y cv F x cv - A opr ` = /\ {<,>|} eqid F B - A opr fvex fvopab4 sylan9eq))
+thm (shftval2t () ((shftfval.1 (e. F (V)))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (` (opr F (shift) (opr A (-) B)) (opr A (+) C)) (` F (opr B (+) C)))) (shftfval.1 A - B opr CC A + C opr shftvalt A B subclt C CC e. 3adant3 A C axaddcl B CC e. 3adant2 sylanc A C B pnncant 3com23 B C axaddcom A CC e. 3adant1 eqtr4d F fveq2d eqtrd))
+thm (shftval3t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (opr A (-) B)) A) (` F B))) (0cn shftfval.1 A B 0 shftval2t mp3an3 A ax0id B CC e. adantr F shift A - B opr opr fveq2d B ax0id A CC e. adantl F fveq2d 3eqtr3d))
+thm (shftval4t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (-u A)) B) (` F (opr A (+) B)))) (shftfval.1 A -u CC B shftvalt A negclt sylan B A subneg2t ancoms A B axaddcom eqtr4d F fveq2d eqtrd))
+thm (shftval5t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) A) (opr B (+) A)) (` F B))) (shftfval.1 A CC B + A opr shftvalt B CC e. A CC e. pm3.27 B A axaddcl sylanc B A pncant F fveq2d eqtrd ancoms))
+thm (shftf ((A x) (F x) (B x) (C x)) ((shftfval.1 (e. F (V)))) (-> (/\/\ (e. A D) (C_ B (CC)) (A.e. x B (e. (` F (opr (cv x) (-) A)) C))) (:--> (|` (opr F (shift) A) B) B C)) (shftfval.1 A V B shftres x B F x cv - A opr ` C e. A.e. 3adant3 shftfval.1 x cv B A shftresvalt A V e. B CC C_ /\ adantl B CC x cv ssel2 A V e. anim2i anassrs shftfval.1 A V x cv shftvalt syl eqtr2d C eleq1d biraldva biimp3a jca F shift A opr B |` B C x ffnfv sylibr A D elisset syl3an1))
+thm (2shft ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr F (shift) A) (shift) B) (opr F (shift) (opr A (+) B)))) (shftfval.1 A CC x cv - B opr shftvalt A CC e. B CC e. pm3.26 x cv CC e. adantl x cv B subclt A CC e. adantrl sylanc x cv A B sub23t x cv A B subsub4t eqtr3d 3expb F fveq2d eqtrd ancoms y cv eqeq2d exp pm5.32d x y biopabdv F shift A oprex B CC x y shftfval A CC e. adantl A B axaddcl shftfval.1 A + B opr CC x y shftfval syl 3eqtr4d))
+thm (shftcan2t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) (-u A)) (shift) A) B) (` F B))) (F shift A -u oprex A CC B shftvalt shftfval.1 A -u CC B - A opr shftvalt A negclt B CC e. adantl B A subclt sylanc B - A opr A subneg2t B A subclt B CC e. A CC e. pm3.27 sylanc B A npcant eqtrd F fveq2d eqtrd ancoms eqtrd))
+thm (shftcan1t () ((shftfval.1 (e. F (V)))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) A) (shift) (-u A)) B) (` F B))) (A negnegt F shift opreq2d shift A -u opreq1d B fveq1d B CC e. adantr shftfval.1 A -u B shftcan2t A negclt sylan eqtr3d))
+thm (shftidt () ((shftfval.1 (e. F (V)))) (-> (e. A (CC)) (= (` (opr F (shift) (0)) A) (` F A))) (0cn shftfval.1 0 A shftcan1t mpan 0cn 0cn negcl shftfval.1 0 0 -u 2shft mp2an 0cn negid F shift opreq2i eqtr A fveq1i syl5eqr))
+thm (seq1shftid ((j k) (j n) (S j) (k n) (S k) (S n) (F j) (F k) (F n)) ((seq1shftid.1 (e. S (V))) (seq1shftid.2 (e. F (V)))) (= (opr S (seq1) (opr F (shift) (0))) (opr S (seq1) F)) (NN eqid j cv 1 S seq1 F shift 0 opr opr fveq2 j cv 1 S seq1 F opr fveq2 eqeq12d j cv k cv S seq1 F shift 0 opr opr fveq2 j cv k cv S seq1 F opr fveq2 eqeq12d j cv k cv + 1 opr S seq1 F shift 0 opr opr fveq2 j cv k cv + 1 opr S seq1 F opr fveq2 eqeq12d j cv n cv S seq1 F shift 0 opr opr fveq2 j cv n cv S seq1 F opr fveq2 eqeq12d 1cn seq1shftid.2 1 shftidt ax-mp seq1shftid.1 F shift 0 oprex seq11 seq1shftid.1 seq1shftid.2 seq11 3eqtr4 S seq1 F shift 0 opr opr k cv ` S seq1 F opr k cv ` S F shift 0 opr k cv + 1 opr ` opreq1 k cv nncnt k cv peano2cn syl seq1shftid.2 k cv + 1 opr shftidt syl S seq1 F opr k cv ` S opreq2d sylan9eqr seq1shftid.1 F shift 0 oprex k cv seq1p1 S seq1 F shift 0 opr opr k cv ` S seq1 F opr k cv ` = adantr seq1shftid.1 seq1shftid.2 k cv seq1p1 S seq1 F shift 0 opr opr k cv ` S seq1 F opr k cv ` = adantr 3eqtr4d exp nnind rgen pm3.2i seq1shftid.1 F shift 0 oprex seq1fn seq1shftid.1 seq1shftid.2 seq1fn S seq1 F shift 0 opr opr NN S seq1 F opr NN n eqfnfv mp2an mpbir))
+thm (uzvalt ((j k) (j y) (N j) (k y) (N k) (N y)) () (-> (e. N (ZZ)) (= (` (ZZ>) N) ({e.|} k (ZZ) (br N (<_) (cv k))))) (j cv N <_ k cv breq1 k ZZ birabsdv j y k df-uz zex k N <_ k cv br rabex fvopab4))
+thm (eluz1t ((N k) (M k)) () (-> (e. M (ZZ)) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N)))) (M k uzvalt N eleq2d k cv N M <_ breq2 ZZ elrab syl6bb))
+thm (eluz2t ((j k) (j y) (N j) (k y) (N k) (N y) (M k)) () (<-> (e. N (` (ZZ>) M)) (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N))) (N ZZ> M ` n0i M ZZ> ndmfv nsyl2 zex k j cv <_ k cv br rabex j y k df-uz dmopab2 M eleq2i sylib M ZZ e. N ZZ e. M <_ N br 3simp1 M N eluz1t M ZZ e. N ZZ e. M <_ N br /\ ibar bitrd M ZZ e. N ZZ e. M <_ N br 3anass syl6bbr pm5.21nii))
+thm (eluz1 () ((eluz.1 (e. M (ZZ)))) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N))) (eluz.1 M N eluz1t ax-mp))
+thm (eluzelz () () (-> (e. N (` (ZZ>) M)) (e. N (ZZ))) (N M eluz2t M ZZ e. N ZZ e. M <_ N br 3simp2 sylbi))
+thm (eluzel2 () () (-> (e. N (` (ZZ>) M)) (e. M (ZZ))) (N M eluz2t M ZZ e. N ZZ e. M <_ N br 3simp1 sylbi))
+thm (eluzle () () (-> (e. N (` (ZZ>) M)) (br M (<_) N)) (N M eluz2t M ZZ e. N ZZ e. M <_ N br 3simp3 sylbi))
+thm (eluzt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. N (` (ZZ>) M)) (br M (<_) N))) (M N eluz1t N M eluzelz pm4.71ri syl5bbr N ZZ e. N ZZ> M ` e. M <_ N br pm5.32 sylibr imp))
+thm (uzidt () () (-> (e. M (ZZ)) (e. M (` (ZZ>) M))) (M zret M leidt syl ancli M M eluz1t mpbird))
+thm (uztrn () () (-> (/\ (e. M (` (ZZ>) K)) (e. K (` (ZZ>) N))) (e. M (` (ZZ>) N))) (K N eluzel2 M ZZ> K ` e. adantl M K eluzelz K ZZ> N ` e. adantr K N eluzle M ZZ> K ` e. adantl M K eluzle K ZZ> N ` e. adantr N K M letrt N zret K zret M zret syl3an K N eluzel2 M ZZ> K ` e. adantl K N eluzelz M ZZ> K ` e. adantl M K eluzelz K ZZ> N ` e. adantr syl3anc mp2and 3jca M N eluz2t sylibr))
+thm (uzssz ((j k) (j y) (M j) (k y) (M k) (M y)) () (C_ (` (ZZ>) M) (ZZ)) (M k uzvalt k ZZ M <_ k cv br ssrab M ZZ e. a1i eqsstrd j y k df-uz dmeqi j y ZZ y cv k ZZ j cv <_ k cv br {e.|} = dmopabss eqsstr M sseli con3i M ZZ> ndmfv syl ZZ 0ss M ZZ e. -. a1i eqsstrd pm2.61i))
+thm (uzss ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (C_ (` (ZZ>) N) (` (ZZ>) M))) (N M eluzle k cv ZZ e. adantr M N k cv letrt M zret N zret k cv zret syl3an 3expa N M eluzel2 N M eluzelz jca sylan mpand exp imdistand N M eluzelz N k cv eluz1t syl N M eluzel2 M k cv eluz1t syl 3imtr4d ssrdv))
+thm (uzssOLD ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (C_ (` (ZZ>) N) (` (ZZ>) M))) (M N k cv letrt M zret N zret k cv zret syl3an exp3a 3exp com34 3imp imdistand N k cv eluz1t M ZZ e. M <_ N br 3ad2ant2 M k cv eluz1t N ZZ e. M <_ N br 3ad2ant1 3imtr4d ssrdv))
+thm (eluzp1m1t () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. (opr N (-) (1)) (` (ZZ>) M))) (N peano2zr M ZZ e. M + 1 opr <_ N br ad2antrl ax1re M 1 N leaddsubt mp3an2 M zret N zret syl2an biimpa anasss jca exp M peano2z M + 1 opr N eluz1t syl M N - 1 opr eluz1t 3imtr4d imp))
+thm (eluzp1lt () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (br M (<) N)) (N M + 1 opr eluzle M ZZ e. adantl M N zltp1let N M + 1 opr eluzelz sylan2 mpbird))
+thm (eluzp1p1t () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) (opr M (+) (1))))) (M peano2z N ZZ e. M <_ N br 3ad2ant1 N peano2z M ZZ e. M <_ N br 3ad2ant2 ax1re M N 1 leadd1t mp3an3 M zret N zret syl2an biimp3a 3jca N M eluz2t N + 1 opr M + 1 opr eluz2t 3imtr4))
+thm (nnuz () () (= (NN) (` (ZZ>) (1))) (k nnz 1z 1 k uzvalt ax-mp eqtr4))
+thm (nn0uz () () (= (NN0) (` (ZZ>) (0))) (k nn0z 0z 0 k uzvalt ax-mp eqtr4))
+thm (raluz ((M x)) () (-> (e. M (ZZ)) (<-> (A.e. x (` (ZZ>) M) ph) (A.e. x (ZZ) (-> (br M (<_) (cv x)) ph)))) (M x cv eluz1t ph imbi1d x cv ZZ e. M <_ x cv br ph impexp syl6bb biraldv2))
+thm (raluz2 ((M x)) () (<-> (A.e. x (` (ZZ>) M) ph) (-> (e. M (ZZ)) (A.e. x (ZZ) (-> (br M (<_) (cv x)) ph)))) (x cv M eluz2t M ZZ e. x cv ZZ e. M <_ x cv br 3anass bitr ph imbi1i M ZZ e. x cv ZZ e. M <_ x cv br /\ ph impexp x cv ZZ e. M <_ x cv br ph impexp M ZZ e. imbi2i bitr M ZZ e. x cv ZZ e. M <_ x cv br ph -> bi2.04 3bitr biral2 x ZZ M ZZ e. M <_ x cv br ph -> r19.21v bitr))
+thm (rexuz ((M x)) () (-> (e. M (ZZ)) (<-> (E.e. x (` (ZZ>) M) ph) (E.e. x (ZZ) (/\ (br M (<_) (cv x)) ph)))) (M x cv eluz1t ph anbi1d x cv ZZ e. M <_ x cv br ph anass syl6bb birexdv2))
+thm (rexuz2 ((M x)) () (<-> (E.e. x (` (ZZ>) M) ph) (/\ (e. M (ZZ)) (E.e. x (ZZ) (/\ (br M (<_) (cv x)) ph)))) (x cv M eluz2t M ZZ e. x cv ZZ e. M <_ x cv br df-3an bitr ph anbi1i M ZZ e. x cv ZZ e. /\ M <_ x cv br ph anass M ZZ e. x cv ZZ e. M <_ x cv br ph /\ anass M ZZ e. x cv ZZ e. M <_ x cv br ph /\ an12 bitr 3bitr birex2 x ZZ M ZZ e. M <_ x cv br ph /\ r19.42v bitr))
+thm (peano2uz () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) M))) (M ZZ e. N ZZ e. M <_ N br 3simp1 N peano2z M ZZ e. M <_ N br 3ad2ant2 M N leplus1t N zret syl3an2 M zret syl3an1 3jca N M eluz2t N + 1 opr M eluz2t 3imtr4))
+thm (uzind4 ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((uzind4.1 (-> (= (cv j) M) (<-> ph ps))) (uzind4.2 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind4.3 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind4.4 (-> (= (cv j) N) (<-> ph ta))) (uzind4.5 (-> (e. M (ZZ)) ps)) (uzind4.6 (-> (e. (cv k) (` (ZZ>) M)) (-> ch th)))) (-> (e. N (` (ZZ>) M)) ta) (uzind4.1 uzind4.2 uzind4.3 uzind4.4 uzind4.5 k cv M eluz2t biimpr 3expb m cv k cv M <_ breq2 ZZ elrab sylan2b uzind4.6 syl uzind3 N M eluzel2 N M eluzelz N M eluzle jca m cv N M <_ breq2 ZZ elrab sylibr sylanc))
+thm (uzind4i ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((uzind4i.1 (e. M (ZZ))) (uzind4i.2 (-> (= (cv j) M) (<-> ph ps))) (uzind4i.3 (-> (= (cv j) (cv k)) (<-> ph ch))) (uzind4i.4 (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th))) (uzind4i.5 (-> (= (cv j) N) (<-> ph ta))) (uzind4i.6 ps) (uzind4i.7 (-> (e. (cv k) (` (ZZ>) M)) (-> ch th)))) (-> (e. N (` (ZZ>) M)) ta) (uzind4i.1 M m uzvalt ax-mp N eleq2i uzind4i.1 uzind4i.2 uzind4i.3 uzind4i.4 uzind4i.5 uzind4i.6 M ZZ e. a1i uzind4i.1 M m uzvalt ax-mp k cv eleq2i uzind4i.7 sylbir M ZZ e. adantl uzind3 mpan sylbi))
+thm (uzwo ((h t) (h n) (h m) (M h) (n t) (m t) (M t) (m n) (M n) (M m) (h j) (h k) (S h) (j t) (j k) (j m) (j n) (S j) (k t) (S t) (k m) (k n) (S k) (S m) (S n)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E.e. j S (A.e. k S (br (cv j) (<_) (cv k))))) (h cv M <_ t cv breq1 t S biraldv S ZZ> M ` C_ j S t S j cv <_ t cv br A.e. E.e. -. /\ imbi2d h cv m cv <_ t cv breq1 t S biraldv S ZZ> M ` C_ j S t S j cv <_ t cv br A.e. E.e. -. /\ imbi2d h cv m cv + 1 opr <_ t cv breq1 t S biraldv S ZZ> M ` C_ j S t S j cv <_ t cv br A.e. E.e. -. /\ imbi2d h cv n cv <_ t cv breq1 t S biraldv S ZZ> M ` C_ j S t S j cv <_ t cv br A.e. E.e. -. /\ imbi2d S ZZ> M ` t cv ssel t cv M eluzle syl6 r19.21aiv j S t S j cv <_ t cv br A.e. E.e. -. adantr M ZZ e. a1i m cv M eluzelz j cv m cv <_ t cv breq1 t S biraldv S rcla4ev expcom con3d com12 m cv t cv letri3t m cv zret t cv zret syl2an t cv m cv zleltp1t t cv m cv + 1 opr ltnlet t cv zret m cv zret m cv peano2re syl syl2an bitrd ancoms m cv <_ t cv br anbi2d bitrd S ZZ t cv ssel2 sylan2 t cv S m cv eleq1a m cv ZZ e. S ZZ C_ ad2antrr sylbird exp3a m cv + 1 opr <_ t cv br m cv S e. con1 syl6 com23 exp32 com34 imp41 r19.20dva exp sylan9r pm2.43d exp31 imp3a syl M uzssz S ZZ> M ` ZZ sstr mpan2 sylani a2d uzind4 j cv n cv <_ t cv breq1 t S biraldv S rcla4ev expcom con3d com12 S ZZ> M ` C_ adantl sylcom S ZZ> M ` n cv ssel con3d com12 j S t S j cv <_ t cv br A.e. E.e. -. adantrd pm2.61i exp n 19.21adv S n eq0 syl6ibr con1d imp t cv k cv j cv <_ breq2 S cbvralv j S birex sylib))
+thm (uzinfm ((j k) (M j) (M k)) ((uzinfm.1 (e. M (ZZ)))) (= (sup (` (ZZ>) M) (RR) (`' (<))) M) (M uzssz zssre sstri uzinfm.1 M uzidt ax-mp k cv M eluzle rgen j cv M <_ k cv breq1 k ZZ> M ` biraldv ZZ> M ` rcla4ev mp2an ZZ> M ` j k lbinfm mp2an k cv M eluzle rgen uzinfm.1 M uzidt ax-mp M uzssz zssre sstri uzinfm.1 M uzidt ax-mp k cv M eluzle rgen j cv M <_ k cv breq1 k ZZ> M ` biraldv ZZ> M ` rcla4ev mp2an ZZ> M ` j k lbreu mp2an j cv M <_ k cv breq1 k ZZ> M ` biraldv ZZ> M ` reuuni2 mp2an mpbi eqtr))
+thm (nninfm () () (= (sup (NN) (RR) (`' (<))) (1)) (nnuz NN ZZ> 1 ` RR < `' supeq1 ax-mp 1z uzinfm eqtr))
+thm (nn0infm () () (= (sup (NN0) (RR) (`' (<))) (0)) (nn0uz NN0 ZZ> 0 ` RR < `' supeq1 ax-mp 0z uzinfm eqtr))
+thm (seq0fval ((f g) (f h) (S f) (g h) (S g) (S h) (F f) (F g) (F h)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq0) F) (|` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0))) (seq0val.1 seq0val.2 S seq1 F shift 1 opr opr shift 1 -u oprex S seq1 F shift 1 opr opr shift 1 -u opr V NN0 resexg ax-mp f cv S seq1 g cv shift 1 opr opreq1 shift 1 -u opreq1d f cv seq1 g cv shift 1 opr opr shift 1 -u opr S seq1 g cv shift 1 opr opr shift 1 -u opr NN0 reseq1 syl g cv F shift 1 opreq1 S seq1 opreq2d shift 1 -u opreq1d S seq1 g cv shift 1 opr opr shift 1 -u opr S seq1 F shift 1 opr opr shift 1 -u opr NN0 reseq1 syl f g h df-seq0 V V oprabval5 mp2an))
+thm (seq0valt () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) N) (` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) N))) (N NN0 S seq1 F shift 1 opr opr shift 1 -u opr fvres seq0val.1 seq0val.2 seq0fval N fveq1i syl5eq))
+thm (seqzfval ((g h) (g k) (g x) (S g) (h k) (h x) (S h) (k x) (S k) (S x) (F g) (F h) (F k) (F x) (A k) (M g) (M h) (M k) (M x)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M A) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) ({e.|} k (ZZ) (br M (<_) (cv k)))))) (seq0val.1 M A S V op2ndg mpan2 M A S op1stg 1 - opreq2d F shift opreq2d seq1 opreq12d M A S op1stg - 1 opreq1d shift opreq12d 2nd M S <,> ` seq1 F shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr k ZZ 1st M S <,> ` <_ k cv br {e.|} reseq1 syl M A S op1stg <_ k cv breq1d k ZZ birabsdv k ZZ 1st M S <,> ` <_ k cv br {e.|} k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr reseq2 syl eqtrd M S opex seq0val.2 2nd M S <,> ` seq1 F shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr oprex 2nd M S <,> ` seq1 F shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr V k ZZ 1st M S <,> ` <_ k cv br {e.|} resexg ax-mp x cv M S <,> 2nd fveq2 x cv M S <,> 1st fveq2 1 - opreq2d g cv shift opreq2d seq1 opreq12d x cv M S <,> 1st fveq2 - 1 opreq1d shift opreq12d 2nd x cv ` seq1 g cv shift 1 - 1st x cv ` opr opr opr shift 1st x cv ` - 1 opr opr 2nd M S <,> ` seq1 g cv shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr k ZZ 1st x cv ` <_ k cv br {e.|} reseq1 syl x cv M S <,> 1st fveq2 <_ k cv breq1d k ZZ birabsdv k ZZ 1st x cv ` <_ k cv br {e.|} k ZZ 1st M S <,> ` <_ k cv br {e.|} 2nd M S <,> ` seq1 g cv shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr reseq2 syl eqtrd g cv F shift 1 - 1st M S <,> ` opr opreq1 2nd M S <,> ` seq1 opreq2d shift 1st M S <,> ` - 1 opr opreq1d 2nd M S <,> ` seq1 g cv shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr 2nd M S <,> ` seq1 F shift 1 - 1st M S <,> ` opr opr opr shift 1st M S <,> ` - 1 opr opr k ZZ 1st M S <,> ` <_ k cv br {e.|} reseq1 syl x g h k df-seqz V V oprabval5 mp2an syl5eq))
+thm (seqzfval2 ((S k) (F k) (M k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (` (ZZ>) M)))) (seq0val.1 seq0val.2 M ZZ k seqzfval M k uzvalt ZZ> M ` k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr reseq2 syl eqcomd eqtrd))
+thm (seqzfn () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (Fn (opr (<,> M S) (seq) F) (` (ZZ>) M))) (M - 1 oprex S seq1 F shift 1 - M opr opr oprex M - 1 opr V shftfn ax-mp M uzssz zsscn sstri S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr CC ZZ> M ` fnssres mp2an seq0val.1 seq0val.2 M seqzfval2 M S <,> seq F opr S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr ZZ> M ` |` ZZ> M ` fneq1 syl mpbiri))
+thm (seqzvalt ((S k) (F k) (A k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M A) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) N))) (seq0val.1 seq0val.2 M A k seqzfval N fveq1d N ZZ e. M <_ N br 3ad2ant1 M A e. N ZZ e. M <_ N br 3simpc k cv N M <_ breq2 ZZ elrab sylibr N k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres syl eqtrd))
+thm (seq1seqz ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq1) F) (opr (<,> (1) S) (seq) F)) (seq0val.1 F shift 0 oprex seq1fn 0cn nnsscn S seq1 F shift 0 opr oprex 0 CC NN shftres mp2an S seq1 F shift 0 opr opr NN S seq1 F shift 0 opr opr shift 0 opr NN |` NN k eqfnfv mp2an NN eqid k cv NN S seq1 F shift 0 opr opr shift 0 opr fvres k cv nncnt S seq1 F shift 0 opr oprex k cv shftidt syl eqtr2d rgen mpbir2an 1cn subid F shift opreq2i S seq1 opreq2i 1cn subid shift opreq12i S seq1 F shift 1 - 1 opr opr opr shift 1 - 1 opr opr S seq1 F shift 0 opr opr shift 0 opr NN reseq1 ax-mp eqtr4 seq0val.1 seq0val.2 seq1shftid k nnz NN k ZZ 1 <_ k cv br {e.|} S seq1 F shift 1 - 1 opr opr opr shift 1 - 1 opr opr reseq2 ax-mp 3eqtr3 1cn seq0val.1 seq0val.2 1 CC k seqzfval ax-mp eqtr4))
+thm (seq0seqz ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq0) F) (opr (<,> (0) S) (seq) F)) (seq0val.1 seq0val.2 seq0fval 1cn subid1 eqcomi F shift opreq2i S seq1 opreq2i 1 df-neg shift opreq12i S seq1 F shift 1 opr opr shift 1 -u opr S seq1 F shift 1 - 0 opr opr opr shift 0 - 1 opr opr NN0 reseq1 ax-mp k nn0z NN0 k ZZ 0 <_ k cv br {e.|} S seq1 F shift 1 - 0 opr opr opr shift 0 - 1 opr opr reseq2 ax-mp 3eqtr 0cn seq0val.1 seq0val.2 0 CC k seqzfval ax-mp eqtr4))
+thm (seq1seq02t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN)) (= (` (opr S (seq1) (opr F (shift) (1))) N) (` (opr (opr S (seq0) F) (shift) (1)) N))) (N elnnnn0 pm3.27bd seq0val.1 seq0val.2 N - 1 opr seq0valt syl N nncnt 1cn N 1 subclt mpan2 1cn S seq1 F shift 1 opr oprex 1 N - 1 opr shftval4t mpan 3syl N nncnt 1cn 1 N pncan3t mpan syl S seq1 F shift 1 opr opr fveq2d eqtrd eqtr2d N nncnt 1cn S seq0 F oprex 1 CC N shftvalt mpan syl eqtr4d))
+thm (seq1seq0t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN)) (= (` (opr S (seq1) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) N))) (seq0val.1 F shift 1 -u oprex N seq1seq02t 1cn negcl 1cn seq0val.2 1 -u 1 2shft mp2an 1cn negcl 1cn addcom 1cn negid eqtr F shift opreq2i eqtr S seq1 opreq2i seq0val.1 seq0val.2 seq1shftid eqtr2 N fveq1i syl5eq))
+thm (seq1seq0 ((S k) (F k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (opr S (seq1) F) (|` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) (NN))) (seq0val.1 seq0val.2 seq1fn 1cn nnsscn S seq0 F shift 1 -u opr oprex 1 CC NN shftres mp2an S seq1 F opr NN S seq0 F shift 1 -u opr opr shift 1 opr NN |` NN k eqfnfv mp2an NN eqid seq0val.1 seq0val.2 k cv seq1seq0t k cv NN S seq0 F shift 1 -u opr opr shift 1 opr fvres eqtr4d rgen mpbir2an))
+thm (seq0fn () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (Fn (opr S (seq0) F) (NN0)) (1cn negcl elisseti S seq1 F shift 1 opr oprex 1 -u V shftfn ax-mp nn0sscn S seq1 F shift 1 opr opr shift 1 -u opr CC NN0 fnssres mp2an seq0val.1 seq0val.2 seq0fval S seq0 F opr S seq1 F shift 1 opr opr shift 1 -u opr NN0 |` NN0 fneq1 ax-mp mpbir))
+thm (seqz1 ((S k) (F k) (M k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. M (ZZ)) (= (` (opr (<,> M S) (seq) F) M) (` F M))) (seq0val.1 seq0val.2 M ZZ k seqzfval M fveq1d M zret M leidt syl ancli k cv M M <_ breq2 ZZ elrab sylibr M k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres syl M zcnt M - 1 oprex S seq1 F shift 1 - M opr opr oprex M - 1 opr V M shftvalt mpan syl M zcnt 1cn M 1 nncant mpan2 syl S seq1 F shift 1 - M opr opr opr fveq2d seq0val.1 F shift 1 - M opr oprex seq11 syl6eq M zcnt 1cn seq0val.2 1 M shftval3t mpan syl 3eqtrd 3eqtrd))
+thm (seqzp1 ((S k) (F k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))) (N M eluzel2 seq0val.1 seq0val.2 M ZZ k seqzfval N + 1 opr fveq1d syl N M peano2uz N + 1 opr M eluzel2 M k uzvalt syl N + 1 opr eleq2d ibi N + 1 opr k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres 3syl S seq1 F shift 1 - M opr opr oprex M - 1 opr CC N + 1 opr shftvalt 1cn M 1 subclt mpan2 N peano2cn syl2an 1cn M 1 subclt mpan2 N CC e. anim2i ancoms 1cn N 1 M - 1 opr addsubt mp3an2 syl S seq1 F shift 1 - M opr opr opr fveq2d eqtrd N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc N M eluz2t 1cn N M 1 subsubt mp3an3 N zcnt M zcnt syl2an ancoms M <_ N br 3adant3 M N znn0sub2t N - M opr nn0p1nnt syl eqeltrd sylbi seq0val.1 F shift 1 - M opr oprex N - M - 1 opr opr seq1p1 syl seq0val.2 1 - M opr CC N - M - 1 opr opr + 1 opr shftvalt 1cn 1 M subclt mpan N CC e. adantr N M - 1 opr subclt 1cn M 1 subclt mpan2 sylan2 ancoms N - M - 1 opr opr peano2cn syl sylanc 1cn N - M - 1 opr opr + 1 opr 1 M subsub2t mp3an2 N M - 1 opr subclt 1cn M 1 subclt mpan2 sylan2 ancoms N - M - 1 opr opr peano2cn syl M CC e. N CC e. pm3.26 sylanc 1cn M 1 subclt mpan2 N CC e. anim2i ancoms 1cn N M - 1 opr 1 nppcant mp3an3 syl eqtrd F fveq2d eqtrd N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc S seq1 F shift 1 - M opr opr opr N - M - 1 opr opr ` S opreq2d 3eqtrd N M eluzel2 M k uzvalt N eleq2d syl ibi N k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres syl S seq1 F shift 1 - M opr opr oprex M - 1 opr ZZ N shftvalt N M eluzel2 M peano2zr syl N M eluzelz N zcnt syl sylanc eqtr2d N M eluzel2 seq0val.1 seq0val.2 M ZZ k seqzfval N fveq1d syl eqtr4d S F N + 1 opr ` opreq1d 3eqtrd eqtrd))
+thm (seqzp1OLD ((S k) (F k) (M k) (N k)) ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))) (seq0val.1 seq0val.2 M ZZ k seqzfval N + 1 opr fveq1d N ZZ e. M <_ N br 3ad2ant1 N peano2z M ZZ e. M <_ N br 3ad2ant2 M N leplus1t N zret syl3an2 M zret syl3an1 jca k cv N + 1 opr M <_ breq2 ZZ elrab sylibr N + 1 opr k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres syl S seq1 F shift 1 - M opr opr oprex M - 1 opr CC N + 1 opr shftvalt 1cn M 1 subclt mpan2 N peano2cn syl2an 1cn M 1 subclt mpan2 N CC e. anim2i ancoms 1cn N 1 M - 1 opr addsubt mp3an2 syl S seq1 F shift 1 - M opr opr opr fveq2d eqtrd M zcnt N zcnt syl2an M <_ N br 3adant3 1cn N M 1 subsubt mp3an3 N zcnt M zcnt syl2an ancoms M <_ N br 3adant3 M N znn0sub2t N - M opr nn0p1nnt syl eqeltrd seq0val.1 F shift 1 - M opr oprex N - M - 1 opr opr seq1p1 syl seq0val.2 1 - M opr CC N - M - 1 opr opr + 1 opr shftvalt 1cn 1 M subclt mpan N CC e. adantr N M - 1 opr subclt 1cn M 1 subclt mpan2 sylan2 ancoms N - M - 1 opr opr peano2cn syl sylanc 1cn N - M - 1 opr opr + 1 opr 1 M subsub2t mp3an2 N M - 1 opr subclt 1cn M 1 subclt mpan2 sylan2 ancoms N - M - 1 opr opr peano2cn syl M CC e. N CC e. pm3.26 sylanc 1cn M 1 subclt mpan2 N CC e. anim2i ancoms 1cn N M - 1 opr 1 nppcant mp3an3 syl eqtrd F fveq2d eqtrd M zcnt N zcnt syl2an S seq1 F shift 1 - M opr opr opr N - M - 1 opr opr ` S opreq2d M <_ N br 3adant3 3eqtrd k cv N M <_ breq2 ZZ elrab biimpr M ZZ e. 3adant1 N k ZZ M <_ k cv br {e.|} S seq1 F shift 1 - M opr opr opr shift M - 1 opr opr fvres syl M peano2zr N zcnt anim12i M <_ N br 3adant3 S seq1 F shift 1 - M opr opr oprex M - 1 opr ZZ N shftvalt syl eqtr2d seq0val.1 seq0val.2 M ZZ k seqzfval N fveq1d N ZZ e. M <_ N br 3ad2ant1 eqtr4d S F N + 1 opr ` opreq1d 3eqtrd eqtrd))
+thm (seqzm1 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (` (opr (<,> M S) (seq) F) N) (opr (` (opr (<,> M S) (seq) F) (opr N (-) (1))) S (` F N)))) (N - 1 opr M eluz2t seq0val.1 seq0val.2 N - 1 opr M seqzp1 sylbir M ZZ e. N ZZ e. M < N br 3simp1 N peano2zr M ZZ e. M < N br 3ad2ant2 M N zltlem1t biimp3a syl3anc N zcnt 1cn N 1 npcant mpan2 syl M S <,> seq F opr fveq2d M ZZ e. M < N br 3ad2ant2 N zcnt 1cn N 1 npcant mpan2 syl F fveq2d M S <,> seq F opr N - 1 opr ` S opreq2d M ZZ e. M < N br 3ad2ant2 3eqtr3d))
+thm (seq00 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (` (opr S (seq0) F) (0)) (` F (0))) (seq0val.1 seq0val.2 seq0fval 0 fveq1i 0nn0 0 NN0 S seq1 F shift 1 opr opr shift 1 -u opr fvres ax-mp eqtr 1cn negcl 0cn S seq1 F shift 1 opr oprex 1 -u CC 0 shftvalt mp2an 1 -u df-neg 1cn negneg eqtr3 S seq1 F shift 1 opr opr fveq2i seq0val.1 F shift 1 oprex seq11 1cn 1cn seq0val.2 1 CC 1 shftvalt mp2an 1cn subid F fveq2i eqtr 3eqtr 3eqtr))
+thm (seq0p1 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) (opr N (+) (1))) (opr (` (opr S (seq0) F) N) S (` F (opr N (+) (1)))))) (N peano2nn0 N + 1 opr NN0 S seq1 F shift 1 opr opr shift 1 -u opr fvres syl N nn0cnt N peano2cn 1cn negcl S seq1 F shift 1 opr oprex 1 -u CC N + 1 opr shftvalt mpan 3syl N nn0cnt 1cn 1cn negcl N 1 1 -u addsubt mp3an23 syl S seq1 F shift 1 opr opr fveq2d N nn0cnt 1cn N 1 subneg2t mpan2 syl N nn0p1nnt eqeltrd seq0val.1 F shift 1 oprex N - 1 -u opr seq1p1 syl N nn0cnt 1cn negcl N 1 -u subclt mpan2 syl N - 1 -u opr peano2cn 1cn seq0val.2 1 CC N - 1 -u opr + 1 opr shftvalt mpan 3syl N nn0cnt 1cn negcl N 1 -u subclt mpan2 1cn N - 1 -u opr 1 pncant mpan2 syl 1cn N 1 subneg2t mpan2 eqtrd syl F fveq2d eqtrd S seq1 F shift 1 opr opr N - 1 -u opr ` S opreq2d eqtrd 3eqtrd N NN0 S seq1 F shift 1 opr opr shift 1 -u opr fvres N nn0cnt 1cn negcl S seq1 F shift 1 opr oprex 1 -u CC N shftvalt mpan syl eqtrd seq0val.1 seq0val.2 seq0fval N fveq1i syl5req S F N + 1 opr ` opreq1d 3eqtrd seq0val.1 seq0val.2 seq0fval N + 1 opr fveq1i syl5eq))
+thm (seq01 () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (= (` (opr S (seq0) F) (1)) (opr (` F (0)) S (` F (1)))) (0nn0 seq0val.1 seq0val.2 0 seq0p1 ax-mp 1cn addid2 S seq0 F opr fveq2i seq0val.1 seq0val.2 seq00 1cn addid2 F fveq2i S opreq12i 3eqtr3))
+thm (seqzval2t () ((seq0val.1 (e. S (V))) (seq0val.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u M))) (shift) M) N))) (N zcnt M - 1 oprex S seq1 F shift 1 - M opr opr oprex M - 1 opr V N shftvalt mpan syl M ZZ e. M <_ N br 3ad2ant2 M zcnt M negclt 1cn seq0val.2 M -u 1 2shft mpan2 syl M negclt 1cn M -u 1 axaddcom mpan2 syl 1cn 1 M subnegt mpan eqtr4d F shift opreq2d eqtrd syl S seq1 opreq2d N - M - 1 opr opr fveq1d N ZZ e. M <_ N br 3ad2ant1 1cn N M 1 subsubt mp3an3 N zcnt M zcnt syl2an ancoms M <_ N br 3adant3 M N znn0sub2t N - M opr nn0p1nnt syl eqeltrd seq0val.1 F shift M -u oprex N - M - 1 opr opr seq1seq02t syl eqtr3d N M - 1 opr subclt 1cn M 1 subclt mpan2 sylan2 1cn S seq0 F shift M -u opr oprex 1 CC N - M - 1 opr opr shftvalt mpan syl 1cn N M 1 nnncant mp3an3 S seq0 F shift M -u opr opr fveq2d eqtrd ancoms M zcnt N zcnt syl2an M <_ N br 3adant3 3eqtrd seq0val.1 seq0val.2 M ZZ N seqzvalt S seq0 F shift M -u opr oprex M ZZ N shftvalt N zcnt sylan2 M <_ N br 3adant3 3eqtr4d))
+thm (seqzrn ((j k) (j n) (j v) (j w) (B j) (k n) (k v) (k w) (B k) (n v) (n w) (B n) (v w) (B v) (B w) (C j) (C k) (C n) (C v) (C w) (F j) (F k) (F n) (F v) (F w) (M j) (M k) (M n) (M v) (M w) (S j) (S k) (S n) (S v) (S w)) ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\ (/\ (e. M (ZZ)) (e. (` F M) C)) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) (C_ (ran (opr (<,> M S) (seq) F)) C)) (j cv M M S <,> seq F opr fveq2 C eleq1d F M ` C e. F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> S C B X. C :--> /\ /\ imbi2d j cv k cv M S <,> seq F opr fveq2 C eleq1d F M ` C e. F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> S C B X. C :--> /\ /\ imbi2d j cv k cv + 1 opr M S <,> seq F opr fveq2 C eleq1d F M ` C e. F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> S C B X. C :--> /\ /\ imbi2d j cv n cv M S <,> seq F opr fveq2 C eleq1d F M ` C e. F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> S C B X. C :--> /\ /\ imbi2d seqzrn.1 seqzrn.2 M seqz1 C eleq1d F M ` C e. F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> S C B X. C :--> /\ pm3.26 syl5bir F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B k cv + 1 opr ffvrn k cv + 1 opr ZZ> M + 1 opr ` F fvres B eleq1d F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> adantl mpbid k cv M eluzp1p1t sylan2 seqzrn.1 seqzrn.2 k cv M seqzp1 C eleq1d w cv M S <,> seq F opr k cv ` S v cv opreq1 C eleq1d v cv F k cv + 1 opr ` M S <,> seq F opr k cv ` S opreq2 C eleq1d C B rcla42v ancoms S C B C w v ffnoprval pm3.27bd syl5 imp syl5bir exp4c F ZZ> M + 1 opr ` |` ZZ> M + 1 opr ` B :--> adantl mpd exp com4r impcom F M ` C e. adantl com12 a2d uzind4 com12 r19.21aiv M ZZ e. a1i M S <,> seq F opr ZZ> M ` n C fnfvrnss seqzrn.1 seqzrn.2 M seqzfn sylan exp syld imp anassrs))
+thm (seqzrn2 () ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)) (seqzrn.1 seqzrn.2 M C C seqzrn exp43 imp4c F ZZ> M ` C M ffvrn M uzidt sylan2 F ZZ> M ` C ZZ> M + 1 opr ` fssres M M + 1 opr uzssOLD M ZZ e. id M peano2z M zret M M + 1 opr ltlet M peano2re ancli M ltplus1t sylc syl syl3anc sylan2 jca expcom syland imp 3impb))
+thm (seqzcl () ((seqzrn.1 (e. S (V))) (seqzrn.2 (e. F (V)))) (-> (/\/\ (e. N (` (ZZ>) M)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C)) (seqzrn.1 seqzrn.2 M C seqzrn2 N M eluzel2 syl3an1 N M eluzel2 seqzrn.1 seqzrn.2 M seqzfn syl M S <,> seq F opr ZZ> M ` N fnfvrn mpancom F ZZ> M ` C :--> S C C X. C :--> 3ad2ant1 sseldd))
+thm (seqzrnlemOLD () ((seqzrnlem.1 (e. S (V))) (seqzrnlem.2 (e. F (V))) (seqzrnlem.3 (e. M (ZZ))) (seqzrnlem.4 (= D (` (ZZ>) M)))) (-> (e. N D) (e. (opr N (+) (1)) (\ D ({} M)))) (seqzrnlem.4 N eleq2i seqzrnlem.3 N eluz1 bitr seqzrnlem.4 N + 1 opr eleq2i seqzrnlem.3 N + 1 opr eluz1 bitr N + 1 opr M {} e. -. anbi1i N + 1 opr D M {} eldif seqzrnlem.3 zre M N + 1 opr leltnet mp3an1 N + 1 opr zret sylan seqzrnlem.3 elisseti N + 1 opr elsnc2 N + 1 opr M eqcom bitr negbii syl6bbr pm5.32i 3bitr4r biimp N peano2z M <_ N br adantr seqzrnlem.3 zre M N leplus1t mp3an1 N zret sylan jca seqzrnlem.3 M N zleltp1t mpan biimpa sylanc sylbi))
+thm (seqzrnOLD ((j k) (j n) (j v) (j w) (B j) (k n) (k v) (k w) (B k) (n v) (n w) (B n) (v w) (B v) (B w) (C j) (C k) (C n) (C v) (C w) (D j) (D k) (D n) (F j) (F k) (F n) (F v) (F w) (M j) (M k) (M n) (M v) (M w) (S j) (S k) (S n) (S v) (S w)) ((seqzrnlem.1 (e. S (V))) (seqzrnlem.2 (e. F (V))) (seqzrnlem.3 (e. M (ZZ))) (seqzrnlem.4 (= D (` (ZZ>) M)))) (-> (/\/\ (e. (` F M) C) (:--> (|` F (\ D ({} M))) (\ D ({} M)) B) (:--> S (X. C B) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)) (seqzrnlem.3 n cv eluz1 seqzrnlem.3 j cv M M S <,> seq F opr fveq2 C eleq1d F M ` C e. F D M {} \ |` D M {} \ B :--> S C B X. C :--> /\/\ imbi2d j cv k cv M S <,> seq F opr fveq2 C eleq1d F M ` C e. F D M {} \ |` D M {} \ B :--> S C B X. C :--> /\/\ imbi2d j cv k cv + 1 opr M S <,> seq F opr fveq2 C eleq1d F M ` C e. F D M {} \ |` D M {} \ B :--> S C B X. C :--> /\/\ imbi2d j cv n cv M S <,> seq F opr fveq2 C eleq1d F M ` C e. F D M {} \ |` D M {} \ B :--> S C B X. C :--> /\/\ imbi2d F M ` C e. F D M {} \ |` D M {} \ B :--> pm3.26 seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M seqz1 ax-mp C eleq1i sylibr S C B X. C :--> 3adant3 M ZZ e. a1i F D M {} \ |` D M {} \ B k cv + 1 opr ffvrn k cv + 1 opr D M {} \ F fvres B eleq1d F D M {} \ |` D M {} \ B :--> adantl mpbid seqzrnlem.4 k cv eleq2i seqzrnlem.3 k cv eluz1 bitr seqzrnlem.1 seqzrnlem.1 seqzrnlem.3 seqzrnlem.4 k cv seqzrnlemOLD sylbir sylan2 seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M k cv seqzp1OLD mp3an1 C eleq1d w cv M S <,> seq F opr k cv ` S v cv opreq1 C eleq1d v cv F k cv + 1 opr ` M S <,> seq F opr k cv ` S opreq2 C eleq1d C B rcla42v ancoms S C B C w v ffnoprval pm3.27bd syl5 imp syl5bir exp4c F D M {} \ |` D M {} \ B :--> adantl mpd exp com4r impcom F M ` C e. 3adant1 com12 a2d M ZZ e. 3adant1 uzind mp3an1 sylbi com12 r19.21aiv seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M seqzfn ax-mp M S <,> seq F opr ZZ> M ` n C fnfvrnss mpan syl))
+thm (seqzrn2OLD () ((seqzrnlem.1 (e. S (V))) (seqzrnlem.2 (e. F (V))) (seqzrnlem.3 (e. M (ZZ))) (seqzrnlem.4 (= D (` (ZZ>) M)))) (-> (/\ (:--> F D C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)) (seqzrnlem.1 seqzrnlem.2 seqzrnlem.3 seqzrnlem.4 C C seqzrnOLD 3expa seqzrnlem.3 M uzidt ax-mp seqzrnlem.4 eleqtrr F D C M ffvrn mpan2 D M {} difss F D C D M {} \ fssres mpan2 jca sylan))
+thm (seqzclOLD () ((seqzrnlem.1 (e. S (V))) (seqzrnlem.2 (e. F (V))) (seqzrnlem.3 (e. M (ZZ))) (seqzrnlem.4 (= D (` (ZZ>) M)))) (-> (/\/\ (e. N D) (:--> F D C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C)) (M S <,> seq F opr ran C M S <,> seq F opr N ` ssel2 ancoms seqzrnlem.4 N eleq2i seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M seqzfn ax-mp M S <,> seq F opr ZZ> M ` N fnfvrn mpan sylbi seqzrnlem.1 seqzrnlem.2 seqzrnlem.3 seqzrnlem.4 C seqzrn2OLD syl2an 3impb))
+thm (seqzres ((j k) (j n) (k n) (D j) (D k) (D n) (F j) (F k) (F n) (M j) (M k) (M n) (S j) (S k) (S n)) ((seqzrnlem.1 (e. S (V))) (seqzrnlem.2 (e. F (V))) (seqzrnlem.3 (e. M (ZZ))) (seqzrnlem.4 (= D (` (ZZ>) M)))) (= (opr (<,> M S) (seq) (|` F D)) (opr (<,> M S) (seq) F)) (seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 F V D resexg ax-mp M seqzfn ax-mp seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M seqzfn ax-mp M S <,> seq F D |` opr ZZ> M ` M S <,> seq F opr ZZ> M ` n eqfnfv mp2an ZZ> M ` eqid seqzrnlem.3 j cv M M S <,> seq F D |` opr fveq2 j cv M M S <,> seq F opr fveq2 eqeq12d j cv k cv M S <,> seq F D |` opr fveq2 j cv k cv M S <,> seq F opr fveq2 eqeq12d j cv k cv + 1 opr M S <,> seq F D |` opr fveq2 j cv k cv + 1 opr M S <,> seq F opr fveq2 eqeq12d j cv n cv M S <,> seq F D |` opr fveq2 j cv n cv M S <,> seq F opr fveq2 eqeq12d seqzrnlem.3 M uzidt ax-mp seqzrnlem.4 eleqtrr M D F fvres ax-mp seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 F V D resexg ax-mp M seqz1 ax-mp seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M seqz1 ax-mp 3eqtr4 M S <,> seq F D |` opr k cv ` M S <,> seq F opr k cv ` = id k cv peano2z M <_ k cv br adantr seqzrnlem.3 zre M k cv leplus1t mp3an1 k cv zret sylan jca seqzrnlem.3 k cv + 1 opr eluz1 sylibr seqzrnlem.4 syl6eleqr k cv + 1 opr D F fvres syl S opreqan12rd seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 F V D resexg ax-mp M k cv seqzp1OLD mp3an1 M S <,> seq F D |` opr k cv ` M S <,> seq F opr k cv ` = adantr seqzrnlem.3 seqzrnlem.1 seqzrnlem.2 M k cv seqzp1OLD mp3an1 M S <,> seq F D |` opr k cv ` M S <,> seq F opr k cv ` = adantr 3eqtr4d seqzrnlem.3 k cv eluz1 sylanb exp uzind4i rgen mpbir2an))
+thm (serzcl1 () ((serzcl.1 (:--> F (` (ZZ>) M) (CC)))) (-> (e. N (` (ZZ>) M)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))) (serzcl.1 axaddopr addex ZZ> M fvex serzcl.1 ZZ> M ` V F CC fex mp2 N M CC seqzcl mp3an23))
+thm (serzcl () ((serzcli.1 (e. M (ZZ))) (serzcli.2 (:--> F (` (ZZ>) M) (CC)))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))) (serzcli.1 N eluz1 serzcli.2 axaddopr addex ZZ> M fvex serzcli.2 ZZ> M ` V F CC fex mp2 serzcli.1 ZZ> M ` eqid N CC seqzclOLD mp3an23 sylbir))
+thm (serzf ((M k) (F k)) ((serzcli.1 (e. M (ZZ))) (serzcli.2 (:--> F (` (ZZ>) M) (CC)))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (CC)) (M + <,> seq F opr ZZ> M ` CC k ffnfv serzcli.1 addex ZZ> M fvex serzcli.2 ZZ> M ` V F CC fex mp2 M seqzfn ax-mp serzcli.1 k cv eluz1 serzcli.1 serzcli.2 k cv serzcl sylbi rgen mpbir2an))
+thm (serzrecl ((j k) (F j) (F k) (M j) (M k) (N j)) ((serzrecli.1 (e. M (ZZ))) (serzrecli.2 (:--> F (` (ZZ>) M) (RR)))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (RR))) (serzrecli.1 j cv M M + <,> seq F opr fveq2 RR eleq1d j cv k cv M + <,> seq F opr fveq2 RR eleq1d j cv k cv + 1 opr M + <,> seq F opr fveq2 RR eleq1d j cv N M + <,> seq F opr fveq2 RR eleq1d serzrecli.1 addex ZZ> M fvex serzrecli.2 ZZ> M ` V F RR fex mp2 M seqz1 ax-mp serzrecli.1 M uzidt ax-mp serzrecli.2 M ffvrni ax-mp eqeltr M ZZ e. a1i M + <,> seq F opr k cv ` F k cv + 1 opr ` axaddrcl serzrecli.1 k cv eluz1 k cv M peano2uz sylbir serzrecli.2 k cv + 1 opr ffvrni syl sylan2 serzrecli.1 addex ZZ> M fvex serzrecli.2 ZZ> M ` V F RR fex mp2 M k cv seqzp1OLD mp3an1 RR eleq1d M + <,> seq F opr k cv ` RR e. adantl mpbird expcom M ZZ e. 3adant1 uzind mp3an1))
+thm (serzref ((F k) (M k)) ((serzrecli.1 (e. M (ZZ))) (serzrecli.2 (:--> F (` (ZZ>) M) (RR)))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (RR)) (M + <,> seq F opr ZZ> M ` RR k ffnfv serzrecli.1 addex ZZ> M fvex serzrecli.2 ZZ> M ` V F RR fex mp2 M seqzfn ax-mp serzrecli.1 k cv eluz1 serzrecli.1 serzrecli.2 k cv serzrecl sylbi rgen mpbir2an))
+thm (dfseq0 ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (seq0) ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g))))) (f cv seq1 g cv shift 1 opr opr shift 1 -u oprex f cv seq1 g cv shift 1 opr opr shift 1 -u opr V NN0 resexg ax-mp f visset g visset pm3.2i h cv f cv seq1 g cv shift 1 opr opr shift 1 -u opr NN0 |` = biantrur f g h bioprabi fnoprab2 f g h df-seq0 seq0 f g h h cv f cv seq1 g cv shift 1 opr opr shift 1 -u opr NN0 |` = {<<,>,>|} V V X. fneq1 ax-mp mpbir 0 f cv <,> seq g cv oprex f visset g visset pm3.2i h cv 0 f cv <,> seq g cv opr = biantrur f g h bioprabi fnoprab2 seq0 V V f g h h cv 0 f cv <,> seq g cv opr = {<<,>,>|} V V x y eqfnoprval mp2an V V X. eqid 0 x cv <,> seq y cv oprex f cv x cv 0 opeq2 seq g cv opreq1d g cv y cv 0 x cv <,> seq opreq2 f g h h cv 0 f cv <,> seq g cv opr = {<<,>,>|} eqid V V oprabval5 x visset y visset seq0seqz syl6reqr rgen2 mpbir2an))
+thm (serzmulc ((j k) (j m) (C j) (k m) (C k) (C m) (F j) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((serzmlc.1 (e. M (ZZ))) (serzmlc.2 (e. C (CC))) (serzmlc.3 (:--> F (` (ZZ>) M) (CC))) (serzmlc.4 (e. G (V))) (serzmlc.5 (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (= (` G (cv k)) (opr C (x.) (` F (cv k))))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M (+)) (seq) G) N) (opr C (x.) (` (opr (<,> M (+)) (seq) F) N)))) (serzmlc.1 j cv M M + <,> seq G opr fveq2 j cv M M + <,> seq F opr fveq2 C x. opreq2d eqeq12d j cv m cv M + <,> seq G opr fveq2 j cv m cv M + <,> seq F opr fveq2 C x. opreq2d eqeq12d j cv m cv + 1 opr M + <,> seq G opr fveq2 j cv m cv + 1 opr M + <,> seq F opr fveq2 C x. opreq2d eqeq12d j cv N M + <,> seq G opr fveq2 j cv N M + <,> seq F opr fveq2 C x. opreq2d eqeq12d serzmlc.1 serzmlc.1 zre leid k cv M M <_ breq2 k cv M G fveq2 k cv M F fveq2 C x. opreq2d eqeq12d imbi12d serzmlc.5 exp vtoclga mp2 serzmlc.1 addex serzmlc.4 M seqz1 ax-mp serzmlc.1 addex ZZ> M fvex serzmlc.3 ZZ> M ` V F CC fex mp2 M seqz1 ax-mp C x. opreq2i 3eqtr4 M ZZ e. a1i serzmlc.1 addex serzmlc.4 M m cv seqzp1OLD mp3an1 M + <,> seq G opr m cv ` C x. M + <,> seq F opr m cv ` opr = adantr M + <,> seq G opr m cv ` C x. M + <,> seq F opr m cv ` opr + G m cv + 1 opr ` opreq1 m cv ZZ e. M <_ m cv br /\ adantl serzmlc.2 C M + <,> seq F opr m cv ` F m cv + 1 opr ` axdistr mp3an1 serzmlc.1 serzmlc.3 m cv serzcl serzmlc.1 m cv eluz1 m cv M peano2uz sylbir serzmlc.3 m cv + 1 opr ffvrni syl sylanc serzmlc.1 addex ZZ> M fvex serzmlc.3 ZZ> M ` V F CC fex mp2 M m cv seqzp1OLD mp3an1 C x. opreq2d serzmlc.1 zre M m cv leplus1t mp3an1 m cv zret sylan k cv m cv + 1 opr M <_ breq2 k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 C x. opreq2d eqeq12d imbi12d serzmlc.5 exp vtoclga imp m cv peano2z sylan syldan C x. M + <,> seq F opr m cv ` opr + opreq2d 3eqtr4rd M + <,> seq G opr m cv ` C x. M + <,> seq F opr m cv ` opr = adantr 3eqtrd exp M ZZ e. 3adant1 uzind mp3an1))
+thm (ser0mulc ((j k) (F j) (F k) (G j) (G k) (C j) (C k)) ((ser0mlc.1 (e. C (CC))) (ser0mlc.2 (:--> F (NN0) (CC))) (ser0mlc.3 (e. G (V))) (ser0mlc.4 (-> (e. (cv k) (NN0)) (= (` G (cv k)) (opr C (x.) (` F (cv k))))))) (-> (e. N (NN0)) (= (` (opr (+) (seq0) G) N) (opr C (x.) (` (opr (+) (seq0) F) N)))) (N nn0p1nnt ser0mlc.1 1cn nnsscn j cv elnnnn0 pm3.27bd ser0mlc.2 j cv - 1 opr ffvrni syl rgen nn0ex ser0mlc.2 NN0 V F CC fex mp2 1 CC NN j CC shftf mp3an G shift 1 oprex j cv elnnnn0 pm3.27bd k cv j cv - 1 opr G fveq2 k cv j cv - 1 opr F fveq2 C x. opreq2d eqeq12d ser0mlc.4 vtoclga syl j cv nncnt 1cn ser0mlc.3 1 CC j cv shftvalt mpan syl j cv NN F shift 1 opr fvres j cv nncnt 1cn nn0ex ser0mlc.2 NN0 V F CC fex mp2 1 CC j cv shftvalt mpan syl eqtrd C x. opreq2d 3eqtr4d N + 1 opr ser1mulc syl N nn0p1nnt addex ser0mlc.3 N + 1 opr seq1seq02t syl N nn0cnt 1cn + seq0 G oprex 1 N shftval5t mpan syl eqtrd N nn0p1nnt addex nn0ex ser0mlc.2 NN0 V F CC fex mp2 N + 1 opr seq1seq02t syl N nn0cnt 1cn + seq0 F oprex 1 N shftval5t mpan syl eqtrd addex F shift 1 oprex seq1res N + 1 opr fveq1i syl5eq C x. opreq2d 3eqtr3d))
+thm (expnn0valt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CC)) (e. B (NN0))) (= (opr A (^) B) (if (= B (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) B)))) (1nn0 elisseti x. seq1 NN A {} X. opr B fvex B 0 = ifex x cv A sneq x cv {} A {} NN xpeq2 syl x. seq1 opreq2d y cv fveq1d y cv 0 = 1 ifeq2d y cv B x. seq1 NN A {} X. opr fveq2 y cv 0 = 1 ifeq2d y cv B 0 eqeq1 1 x. seq1 NN A {} X. opr B ` ifbid eqtrd x y z df-exp oprabval2))
+thm (expnn00t () () (-> (e. A (CC)) (= (opr A (^) (0)) (1))) (0nn0 A 0 expnn0valt mpan2 0 eqid 0 0 = 1 x. seq1 NN A {} X. opr 0 ` iftrue ax-mp syl6eq))
+thm (expvalt () () (-> (/\ (e. A (CC)) (e. B (NN))) (= (opr A (^) B) (` (opr (x.) (seq1) (X. (NN) ({} A))) B))) (A B expnn0valt B nnnn0t sylan2 B nnne0t B 0 df-ne sylib B 0 = 1 x. seq1 NN A {} X. opr B ` iffalse syl A CC e. adantl eqtrd))
+thm (expp1t () () (-> (/\ (e. A (CC)) (e. B (NN))) (= (opr A (^) (opr B (+) (1))) (opr (opr A (^) B) (x.) A))) (mulex nnex A snex xpex B seq1p1 A CC e. adantl NN A {} X. NN A B + 1 opr fvconst A CC NN fconstg B peano2nn syl2an x. seq1 NN A {} X. opr B ` x. opreq2d eqtrd A B + 1 opr expvalt B peano2nn sylan2 A B expvalt x. A opreq1d 3eqtr4d))
+thm (exp1t () () (-> (e. A (CC)) (= (opr A (^) (1)) A)) (1nn A 1 expvalt mpan2 A CC NN fconstg 1nn NN A {} X. NN A 1 fvconst mpan2 syl mulex nnex A snex xpex seq11 syl5eq eqtrd))
+thm (1expt ((x y) (A x) (A y)) () (-> (e. A (NN)) (= (opr (1) (^) A) (1))) (x cv 1 1 ^ opreq2 1 eqeq1d x cv y cv 1 ^ opreq2 1 eqeq1d x cv y cv + 1 opr 1 ^ opreq2 1 eqeq1d x cv A 1 ^ opreq2 1 eqeq1d 1cn 1 exp1t ax-mp 1cn 1 y cv expp1t mpan 1 eqeq1d 1 ^ y cv opr 1 x. 1 opreq1 1cn mulid1 syl6eq syl5bir nnind))
+thm (expcllem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (F x) (F y) (F z) (F w)) ((expcllem.1 (C_ F (CC))) (expcllem.2 (-> (/\ (e. (cv x) F) (e. (cv y) F)) (e. (opr (cv x) (x.) (cv y)) F)))) (-> (/\ (e. A F) (e. B (NN))) (e. (opr A (^) B) F)) (z cv 1 A ^ opreq2 F eleq1d A F e. imbi2d z cv w cv A ^ opreq2 F eleq1d A F e. imbi2d z cv w cv + 1 opr A ^ opreq2 F eleq1d A F e. imbi2d z cv B A ^ opreq2 F eleq1d A F e. imbi2d expcllem.1 A sseli A exp1t syl F eleq1d ibir x cv A ^ w cv opr x. y cv opreq1 F eleq1d y cv A A ^ w cv opr x. opreq2 F eleq1d expcllem.2 vtocl2ga ancoms w cv NN e. adantlr A w cv expp1t expcllem.1 A sseli sylan F eleq1d A ^ w cv opr F e. adantr mpbird exp31 com12 a2d nnind impcom))
+thm (nnexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (^) B) (NN))) (nnsscn x cv y cv nnmulclt A B expcllem))
+thm (nn0expclt ((x y) (A x) (A y)) () (-> (/\ (e. A (NN0)) (e. B (NN))) (e. (opr A (^) B) (NN0))) (nn0sscn x cv y cv nn0mulclt A B expcllem))
+thm (zexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (^) B) (ZZ))) (zsscn x cv y cv zmulclt A B expcllem))
+thm (qexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (QQ)) (e. B (NN))) (e. (opr A (^) B) (QQ))) (qsscn x cv y cv qmulclt A B expcllem))
+thm (reexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (RR)) (e. B (NN))) (e. (opr A (^) B) (RR))) (axresscn x cv y cv axmulrcl A B expcllem))
+thm (expclt ((x y) (A x) (A y)) () (-> (/\ (e. A (CC)) (e. B (NN))) (e. (opr A (^) B) (CC))) (CC ssid x cv y cv axmulcl A B expcllem))
+thm (expnn0p1t () () (-> (/\ (e. A (CC)) (e. B (NN0))) (= (opr A (^) (opr B (+) (1))) (opr (opr A (^) B) (x.) A))) (A B expp1t exp B 0 + 1 opreq1 A ^ opreq2d B 0 A ^ opreq2 x. A opreq1d eqeq12d A mulid2t A expnn00t x. A opreq1d A exp1t 1cn addid2 A ^ opreq2i syl5eq 3eqtr4rd syl5bir com12 jaod B elnn0 syl5ib imp))
+thm (expnn0clt () () (-> (/\ (e. A (CC)) (e. B (NN0))) (e. (opr A (^) B) (CC))) (A B expclt exp 1cn B 0 A ^ opreq2 A expnn00t sylan9eqr CC eleq1d mpbiri exp jaod B elnn0 syl5ib imp))
+thm (expeq0t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (NN))) (<-> (= (opr A (^) B) (0)) (= A (0)))) (x cv 1 A ^ opreq2 0 eqeq1d A 0 = bibi1d A CC e. imbi2d x cv y cv A ^ opreq2 0 eqeq1d A 0 = bibi1d A CC e. imbi2d x cv y cv + 1 opr A ^ opreq2 0 eqeq1d A 0 = bibi1d A CC e. imbi2d x cv B A ^ opreq2 0 eqeq1d A 0 = bibi1d A CC e. imbi2d A exp1t 0 eqeq1d A y cv expp1t 0 eqeq1d A ^ y cv opr A mul0ort A y cv expclt A CC e. y cv NN e. pm3.26 sylanc bitrd A ^ y cv opr 0 = A 0 = bi1 A ^ y cv opr 0 = A 0 = <-> A 0 = idd jaod A 0 = A ^ y cv opr 0 = olc jctir A ^ y cv opr 0 = A 0 = \/ A 0 = bi sylibr sylan9bb exp31 com12 a2d nnind impcom))
+thm (expne0t () () (-> (/\ (e. A (CC)) (e. B (NN))) (<-> (=/= A (0)) (=/= (opr A (^) B) (0)))) (A B expeq0t bicomd eqneqd))
+thm (expgt0t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (0) (<) A)) (br (0) (<) (opr A (^) B))) (x cv 1 A ^ opreq2 0 < breq2d A RR e. 0 < A br /\ imbi2d x cv y cv A ^ opreq2 0 < breq2d A RR e. 0 < A br /\ imbi2d x cv y cv + 1 opr A ^ opreq2 0 < breq2d A RR e. 0 < A br /\ imbi2d x cv B A ^ opreq2 0 < breq2d A RR e. 0 < A br /\ imbi2d A recnt A exp1t syl 0 < breq2d biimpar A ^ y cv opr A axmulgt0 A y cv reexpclt A RR e. y cv NN e. pm3.26 sylanc A y cv expp1t A recnt sylan 0 < breq2d sylibrd exp4b com12 com34 imp3a a2d nnind exp3a com12 3imp))
+thm (0expt () () (-> (e. A (NN)) (= (opr (0) (^) A) (0))) (0 eqid 0cn 0 A expeq0t mpan mpbiri))
+thm (expge0t () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (0) (<_) A)) (br (0) (<_) (opr A (^) B))) (A B expgt0t 3expa exp 0 A ^ B opreq1 0 eqeq2d B 0expt eqcomd syl5bi com12 A RR e. adantl orim12d 0re 0 A leloet mpan B NN e. adantr A B reexpclt 0re 0 A ^ B opr leloet mpan syl 3imtr4d imp 3impa))
+thm (expgt1t ((x y) (A x) (A y) (B x)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (1) (<) A)) (br (1) (<) (opr A (^) B))) (x cv 1 A ^ opreq2 1 < breq2d A RR e. 1 < A br /\ imbi2d x cv y cv A ^ opreq2 1 < breq2d A RR e. 1 < A br /\ imbi2d x cv y cv + 1 opr A ^ opreq2 1 < breq2d A RR e. 1 < A br /\ imbi2d x cv B A ^ opreq2 1 < breq2d A RR e. 1 < A br /\ imbi2d A recnt A exp1t syl 1 < breq2d biimpar A ^ y cv opr A mulgt1t A y cv reexpclt A RR e. y cv NN e. pm3.26 jca sylan exp A y cv expp1t A recnt sylan 1 < breq2d sylibrd exp4b com12 com34 imp3a a2d nnind exp3a com12 3imp))
+thm (expge1t () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (1) (<_) A)) (br (1) (<_) (opr A (^) B))) (A B expgt1t 3expa exp 1 A ^ B opreq1 1 eqeq2d B 1expt eqcomd syl5bi com12 A RR e. adantl orim12d ax1re 1 A leloet mpan B NN e. adantr A B reexpclt ax1re 1 A ^ B opr leloet mpan syl 3imtr4d imp 3impa))
+thm (mulexpt ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (NN))) (= (opr (opr A (x.) B) (^) C) (opr (opr A (^) C) (x.) (opr B (^) C)))) (x cv 1 A x. B opr ^ opreq2 x cv 1 A ^ opreq2 x cv 1 B ^ opreq2 x. opreq12d eqeq12d A CC e. B CC e. /\ imbi2d x cv y cv A x. B opr ^ opreq2 x cv y cv A ^ opreq2 x cv y cv B ^ opreq2 x. opreq12d eqeq12d A CC e. B CC e. /\ imbi2d x cv y cv + 1 opr A x. B opr ^ opreq2 x cv y cv + 1 opr A ^ opreq2 x cv y cv + 1 opr B ^ opreq2 x. opreq12d eqeq12d A CC e. B CC e. /\ imbi2d x cv C A x. B opr ^ opreq2 x cv C A ^ opreq2 x cv C B ^ opreq2 x. opreq12d eqeq12d A CC e. B CC e. /\ imbi2d A B axmulcl A x. B opr exp1t syl A exp1t B exp1t x. opreqan12d eqtr4d A x. B opr y cv expp1t A B axmulcl sylan A x. B opr ^ y cv opr A ^ y cv opr x. B ^ y cv opr opr = adantr A x. B opr ^ y cv opr A ^ y cv opr x. B ^ y cv opr opr x. A x. B opr opreq1 A CC e. B CC e. /\ y cv NN e. /\ adantl A ^ y cv opr B ^ y cv opr A B mul4t A y cv expclt B y cv expclt anim12i anandirs A CC e. B CC e. /\ y cv NN e. pm3.26 sylanc A y cv expp1t B CC e. adantlr B y cv expp1t A CC e. adantll x. opreq12d eqtr4d A x. B opr ^ y cv opr A ^ y cv opr x. B ^ y cv opr opr = adantr 3eqtrd exp31 com12 a2d nnind exp3a com3l 3imp))
+thm (recexpt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) B) (opr (1) (/) (opr A (^) B)))) (x cv 1 1 / A opr ^ opreq2 x cv 1 A ^ opreq2 1 / opreq2d eqeq12d A CC e. A 0 =/= /\ imbi2d x cv y cv 1 / A opr ^ opreq2 x cv y cv A ^ opreq2 1 / opreq2d eqeq12d A CC e. A 0 =/= /\ imbi2d x cv y cv + 1 opr 1 / A opr ^ opreq2 x cv y cv + 1 opr A ^ opreq2 1 / opreq2d eqeq12d A CC e. A 0 =/= /\ imbi2d x cv B 1 / A opr ^ opreq2 x cv B A ^ opreq2 1 / opreq2d eqeq12d A CC e. A 0 =/= /\ imbi2d A recclt 1 / A opr exp1t syl A exp1t 1 / opreq2d A 0 =/= adantr eqtr4d 1 / A opr ^ y cv opr 1 / A ^ y cv opr opr x. 1 / A opr opreq1 A CC e. y cv NN e. /\ A 0 =/= ad2antrr 1 / A opr y cv expp1t A recclt sylan an1rs 1 / A opr ^ y cv opr 1 / A ^ y cv opr opr = adantrr A y cv expp1t 1 / opreq2d A 0 =/= adantr 1 A ^ y cv opr 1 A divmuldivt A y cv expclt 1cn jctil A CC e. y cv NN e. pm3.26 1cn jctil jca A 0 =/= adantr A y cv expne0t biimpa A CC e. y cv NN e. /\ A 0 =/= pm3.27 jca sylanc 1cn mulid1 / A ^ y cv opr x. A opr opreq1i syl6req eqtrd 1 / A opr ^ y cv opr 1 / A ^ y cv opr opr = adantrr 3eqtr4d exp43 com12 imp3a a2d nnind exp3a com12 3imp))
+thm (expaddt ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (= (opr A (^) (opr B (+) C)) (opr (opr A (^) B) (x.) (opr A (^) C)))) (x cv 1 B + opreq2 A ^ opreq2d x cv 1 A ^ opreq2 A ^ B opr x. opreq2d eqeq12d A CC e. B NN e. /\ imbi2d x cv y cv B + opreq2 A ^ opreq2d x cv y cv A ^ opreq2 A ^ B opr x. opreq2d eqeq12d A CC e. B NN e. /\ imbi2d x cv y cv + 1 opr B + opreq2 A ^ opreq2d x cv y cv + 1 opr A ^ opreq2 A ^ B opr x. opreq2d eqeq12d A CC e. B NN e. /\ imbi2d x cv C B + opreq2 A ^ opreq2d x cv C A ^ opreq2 A ^ B opr x. opreq2d eqeq12d A CC e. B NN e. /\ imbi2d A B expp1t A exp1t B NN e. adantr A ^ B opr x. opreq2d eqtr4d 1cn B y cv 1 axaddass mp3an3 B nncnt y cv nncnt syl2an A CC e. adantll A ^ opreq2d A B + y cv opr expp1t A CC e. B NN e. pm3.26 y cv NN e. adantr B y cv nnaddclt A CC e. adantll sylanc eqtr3d A y cv expp1t B NN e. adantlr A ^ B opr x. opreq2d A ^ B opr A ^ y cv opr A axmulass A B expclt y cv NN e. adantr A y cv expclt B NN e. adantlr A CC e. B NN e. pm3.26 y cv NN e. adantr syl3anc eqtr4d eqeq12d A ^ B + y cv opr opr A ^ B opr x. A ^ y cv opr opr x. A opreq1 syl5bir expcom a2d nnind exp3a com3l 3imp))
+thm (expmult ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (= (opr A (^) (opr B (x.) C)) (opr (opr A (^) B) (^) C))) (x cv 1 B x. opreq2 A ^ opreq2d x cv 1 A ^ B opr ^ opreq2 eqeq12d A CC e. B NN e. /\ imbi2d x cv y cv B x. opreq2 A ^ opreq2d x cv y cv A ^ B opr ^ opreq2 eqeq12d A CC e. B NN e. /\ imbi2d x cv y cv + 1 opr B x. opreq2 A ^ opreq2d x cv y cv + 1 opr A ^ B opr ^ opreq2 eqeq12d A CC e. B NN e. /\ imbi2d x cv C B x. opreq2 A ^ opreq2d x cv C A ^ B opr ^ opreq2 eqeq12d A CC e. B NN e. /\ imbi2d B nncnt B ax1id syl A CC e. adantl A ^ opreq2d A B expclt A ^ B opr exp1t syl eqtr4d 1cn B y cv 1 axdistr mp3an3 B ax1id y cv CC e. adantr B x. y cv opr + opreq2d eqtrd B nncnt y cv nncnt syl2an A CC e. adantll A ^ opreq2d A B x. y cv opr B expaddt A CC e. B NN e. pm3.26 y cv NN e. adantr B y cv nnmulclt A CC e. adantll A CC e. B NN e. pm3.27 y cv NN e. adantr syl3anc eqtrd A ^ B opr y cv expp1t A B expclt sylan eqeq12d A ^ B x. y cv opr opr A ^ B opr ^ y cv opr x. A ^ B opr opreq1 syl5bir expcom a2d nnind exp3a com3l 3imp))
+thm (expsubt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (/\ (=/= A (0)) (br C (<) B))) (= (opr A (^) (opr B (-) C)) (opr (opr A (^) B) (/) (opr A (^) C)))) (A C B - C opr expaddt A CC e. B NN e. C NN e. 3simp1 C < B br adantr A CC e. B NN e. C NN e. 3simp3 C < B br adantr C B nnsubt ancoms biimpa A CC e. 3adantl1 syl3anc C B pncan3t C nncnt B nncnt syl2an ancoms A ^ opreq2d A CC e. 3adant1 C < B br adantr eqtr3d A 0 =/= adantrl A ^ B opr A ^ C opr A ^ B - C opr opr divmult A B expclt C NN e. 3adant3 A 0 =/= C < B br /\ adantr A C expclt B NN e. 3adant2 A 0 =/= C < B br /\ adantr A B - C opr expclt A CC e. B NN e. C NN e. 3simp1 C < B br adantr C B nnsubt ancoms biimpa A CC e. 3adantl1 sylanc A 0 =/= adantrl 3jca A C expne0t biimpa B NN e. 3adantl2 C < B br adantrr sylanc mpbird eqcomd))
+thm (divexpt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (NN))) (=/= B (0))) (= (opr (opr A (/) B) (^) C) (opr (opr A (^) C) (/) (opr B (^) C)))) (A B divrect 3expb ^ C opreq1d C NN e. adantr A 1 / B opr C mulexpt 3expa B recclt sylanl2 B C recexpt 3expa an1rs A CC e. adantll A ^ C opr x. opreq2d A ^ C opr B ^ C opr divrect A C expclt B CC e. B 0 =/= /\ adantlr B C expclt A CC e. adantll B 0 =/= adantlrr B C expne0t biimpa an1rs A CC e. adantll syl3anc eqtr4d 3eqtrd exp42 com34 3imp1))
+thm (expordit () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (/\ (br (1) (<) A) (br B (<) C))) (br (opr A (^) B) (<) (opr A (^) C))) (A C - B opr expgt1t A RR e. B NN e. C NN e. 3simp1 1 < A br B < C br /\ adantr B C nnsubt A RR e. 3adant1 biimpa 1 < A br adantrl 1 < A br B < C br pm3.26 A RR e. B NN e. C NN e. /\/\ adantl syl3anc 1 A ^ C - B opr opr A ^ B opr ltmul1t ax1re A RR e. B NN e. C NN e. /\/\ 1 < A br B < C br /\ /\ a1i A C - B opr reexpclt A RR e. B NN e. C NN e. 3simp1 B < C br adantr B C nnsubt A RR e. 3adant1 biimpa sylanc 1 < A br adantrl A B reexpclt C NN e. 3adant3 1 < A br B < C br /\ adantr 3jca A B expgt1t 3exp imp A B reexpclt lt01 0re ax1re 0 1 A ^ B opr axlttrn mp3an12 mpani syl syld C NN e. 3adant3 imp B < C br adantrr sylanc mpbid A B expclt A recnt sylan A ^ B opr mulid2t syl C NN e. 3adant3 1 < A br B < C br /\ adantr A RR e. B NN e. C NN e. 3simp1 recnd B < C br adantr B C nnsubt A RR e. 3adant1 biimpa A RR e. B NN e. C NN e. 3simp2 B < C br adantr 3jca 1 < A br adantrl A C - B opr B expaddt syl C B npcant C nncnt B nncnt syl2an ancoms A RR e. 3adant1 A ^ opreq2d 1 < A br B < C br /\ adantr eqtr3d 3brtr3d))
+thm (expcant () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (br (1) (<) A)) (<-> (= (opr A (^) B) (opr A (^) C)) (= B C))) (A B C expordit exp32 imp A C B expordit exp32 3com23 imp orim12d B C lttri2t B nnret C nnret syl2an A RR e. 3adant1 1 < A br adantr A ^ B opr A ^ C opr lttri2t A B reexpclt A C reexpclt syl2an 3impdi 1 < A br adantr 3imtr4d a3d B C A ^ opreq2 A RR e. B NN e. C NN e. /\/\ 1 < A br /\ a1i impbid))
+thm (expordt () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN)) (e. N (NN))) (br (1) (<) A)) (<-> (br M (<) N) (br (opr A (^) M) (<) (opr A (^) N)))) (A M N expordit exp32 imp A M N expcant biimprd A N M expordit exp32 3com23 imp orim12d con3d A ^ M opr A ^ N opr axlttri A M reexpclt A N reexpclt syl2an 3impdi 1 < A br adantr M N axlttri M nnret N nnret syl2an A RR e. 3adant1 1 < A br adantr 3imtr4d impbid))
+thm (explt1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (/\ (br (0) (<) A) (br A (<) (1)))) (<-> (br B (<) C) (br (opr A (^) C) (<) (opr A (^) B)))) (1 / A opr B C expordt A gt0ne0t A rerecclt syldan expcom 0 < A br B NN e. idd 0 < A br C NN e. idd 3anim123d impcom A < 1 br adantrr A reclt1t biimpa anasss B NN e. adantlr C NN e. 3adantl3 sylanc A ^ C opr A ^ B opr ltrect A C reexpclt 0 < A br adantr B NN e. 3adantl2 A B reexpclt 0 < A br adantr C NN e. 3adantl3 jca A C expgt0t 3expa B NN e. 3adantl2 A B expgt0t 3expa C NN e. 3adantl3 jca sylanc A B recexpt 3expa C NN e. 3adantl3 A C recexpt 3expa B NN e. 3adantl2 < breq12d A recnt B NN e. id C NN e. id 3anim123i 0 < A br adantr A gt0ne0t B NN e. adantlr C NN e. 3adantl3 sylanc bitr4d A < 1 br adantrr bitr4d))
+thm (sqvalt () () (-> (e. A (CC)) (= (opr A (^) (2)) (opr A (x.) A))) (1nn A 1 expp1t mpan2 df-2 A ^ opreq2i syl5eq A exp1t x. A opreq1d eqtrd))
+thm (sqclt () () (-> (e. A (CC)) (e. (opr A (^) (2)) (CC))) (A sqvalt A A axmulcl anidms eqeltrd))
+thm (sqval () ((sqval.1 (e. A (CC)))) (= (opr A (^) (2)) (opr A (x.) A)) (sqval.1 A sqvalt ax-mp))
+thm (sqcl () ((sqval.1 (e. A (CC)))) (e. (opr A (^) (2)) (CC)) (sqval.1 A sqclt ax-mp))
+thm (sqe0 () ((sqval.1 (e. A (CC)))) (<-> (= (opr A (^) (2)) (0)) (= A (0))) (sqval.1 sqval 0 eqeq1i sqval.1 sq0 bitr))
+thm (sqmul () ((sqval.1 (e. A (CC))) (sqmul.2 (e. B (CC)))) (= (opr (opr A (x.) B) (^) (2)) (opr (opr A (^) (2)) (x.) (opr B (^) (2)))) (sqval.1 sqmul.2 2nn A B 2 mulexpt mp3an))
+thm (sqdiv () ((sqval.1 (e. A (CC))) (sqmul.2 (e. B (CC))) (sqdiv.3 (=/= B (0)))) (= (opr (opr A (/) B) (^) (2)) (opr (opr A (^) (2)) (/) (opr B (^) (2)))) (sqval.1 sqmul.2 sqval.1 sqmul.2 sqdiv.3 sqdiv.3 divmuldiv sqval.1 sqmul.2 sqdiv.3 divcl sqval sqval.1 sqval sqmul.2 sqval / opreq12i 3eqtr4))
+thm (sqreci () ((sqval.1 (e. A (CC))) (sqreci.1 (=/= A (0)))) (= (opr (opr (1) (/) A) (^) (2)) (opr (1) (/) (opr A (^) (2)))) (1cn sqval.1 1cn sqval.1 sqreci.1 sqreci.1 divmuldiv 1cn mulid1 / A x. A opr opreq1i eqtr sqval.1 sqreci.1 reccl sqval sqval.1 sqval 1 / opreq2i 3eqtr4))
+thm (sqreclt () () (-> (e. A (RR)) (e. (opr A (^) (2)) (RR))) (2nn A 2 reexpclt mpan2))
+thm (sqrecl () ((sqrecl.1 (e. A (RR)))) (e. (opr A (^) (2)) (RR)) (sqrecl.1 A sqreclt ax-mp))
+thm (lt2sqe () ((sqrecl.1 (e. A (RR))) (lt2sqe.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2))))) (sqrecl.1 lt2sqe.2 lt2sq sqrecl.1 recn sqval lt2sqe.2 recn sqval < breq12i syl6bbr))
+thm (le2sqe () ((sqrecl.1 (e. A (RR))) (lt2sqe.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2))))) (sqrecl.1 lt2sqe.2 le2sq sqrecl.1 recn sqval lt2sqe.2 recn sqval <_ breq12i syl6bbr))
+thm (sqe11 () ((sqrecl.1 (e. A (RR))) (lt2sqe.2 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B))) (sqrecl.1 lt2sqe.2 sq11 sqrecl.1 recn sqval lt2sqe.2 recn sqval eqeq12i syl5bb))
+thm (sqegt0 () ((sqrecl.1 (e. A (RR)))) (-> (=/= A (0)) (br (0) (<) (opr A (^) (2)))) (sqrecl.1 sqgt0 A 0 df-ne sqrecl.1 recn sqval 0 < breq2i 3imtr4))
+thm (sqege0 () ((sqrecl.1 (e. A (RR)))) (br (0) (<_) (opr A (^) (2))) (sqrecl.1 sqge0 sqrecl.1 recn sqval breqtrr))
+thm (sqe11t () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B)))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if ^ 2 opreq1 B ^ 2 opr eqeq1d A A RR e. A 0 if B eqeq1 bibi12d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if ^ 2 opreq1 A RR e. A 0 if ^ 2 opr eqeq2d B B RR e. B 0 if A RR e. A 0 if eqeq2 bibi12d imbi12d 0re A elimel 0re B elimel sqe11 dedth2h))
+thm (lt2sqet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2)))))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if < B breq1 A A RR e. A 0 if ^ 2 opreq1 < B ^ 2 opr breq1d bibi12d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if ^ 2 opreq1 A RR e. A 0 if ^ 2 opr < breq2d bibi12d imbi12d 0re A elimel 0re B elimel lt2sqe dedth2h))
+thm (le2sqet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2)))))) (A B le2sqt imp A sqvalt B sqvalt <_ breqan12d A recnt B recnt syl2an 0 <_ A br 0 <_ B br /\ adantr bitr4d exp))
+thm (sqege0t () () (-> (e. A (RR)) (br (0) (<_) (opr A (^) (2)))) (A A RR e. A 0 if ^ 2 opreq1 0 <_ breq2d 0re A elimel sqege0 dedth))
+thm (sumsqne0 () ((sumsqne0.1 (e. A (RR))) (sumsqne0.2 (e. B (RR)))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr (opr A (^) (2)) (+) (opr B (^) (2))) (0))) (sumsqne0.1 sqegt0 sumsqne0.1 sqrecl sumsqne0.2 sqrecl sumsqne0.2 sqege0 A ^ 2 opr B ^ 2 opr addge1tOLD mp3an 0re sumsqne0.1 sqrecl sumsqne0.1 sqrecl sumsqne0.2 sqrecl readdcl ltletr mpan2 sumsqne0.1 sqrecl sumsqne0.2 sqrecl readdcl gt0ne0 3syl sumsqne0.2 sqegt0 sumsqne0.2 sqrecl sumsqne0.1 sqrecl sumsqne0.1 sqege0 B ^ 2 opr A ^ 2 opr addge2tOLD mp3an 0re sumsqne0.2 sqrecl sumsqne0.1 sqrecl sumsqne0.2 sqrecl readdcl ltletr mpan2 sumsqne0.1 sqrecl sumsqne0.2 sqrecl readdcl gt0ne0 3syl jaoi A 0 ^ 2 opreq1 B 0 ^ 2 opreq1 + opreqan12d 2nn 2 0expt ax-mp 2nn 2 0expt ax-mp + opreq12i 0cn addid1 eqtr syl6eq con3i A ^ 2 opr + B ^ 2 opr opr 0 df-ne A 0 df-ne B 0 df-ne orbi12i A 0 = B 0 = ianor bitr4 3imtr4 impbi))
+thm (sq1 () () (= (opr (1) (^) (2)) (1)) (1cn sqval 1cn mulid1 eqtr))
+thm (sq2 () () (= (opr (2) (^) (2)) (4)) (2cn sqval 2t2e4 eqtr))
+thm (cu2 () () (= (opr (2) (^) (3)) (8)) (df-3 2 ^ opreq2i 2cn 2nn 2 2 expp1t mp2an sq2 x. 2 opreq1i 4t2e8 eqtr 3eqtr))
+thm (binom2 () ((binom2.1 (e. A (CC))) (binom2.2 (e. B (CC)))) (= (opr (opr A (+) B) (^) (2)) (opr (opr (opr A (^) (2)) (+) (opr (2) (x.) (opr A (x.) B))) (+) (opr B (^) (2)))) (binom2.1 binom2.2 addcl binom2.1 binom2.2 adddi binom2.1 binom2.2 binom2.1 adddir binom2.2 binom2.1 mulcom A x. A opr + opreq2i eqtr binom2.1 binom2.2 binom2.2 adddir + opreq12i binom2.1 binom2.1 mulcl binom2.1 binom2.2 mulcl addcl binom2.1 binom2.2 mulcl binom2.2 binom2.2 mulcl addass binom2.1 binom2.1 mulcl binom2.1 binom2.2 mulcl binom2.1 binom2.2 mulcl addass + B x. B opr opreq1i eqtr3 3eqtr binom2.1 binom2.2 addcl sqval binom2.1 sqval binom2.1 binom2.2 mulcl 2times + opreq12i binom2.2 sqval + opreq12i 3eqtr4))
+thm (binom2a () ((binom2.1 (e. A (CC))) (binom2.2 (e. B (CC)))) (= (opr (opr A (+) B) (x.) (opr A (-) B)) (opr (opr A (^) (2)) (-) (opr B (^) (2)))) (binom2.1 binom2.2 binom2.1 binom2.2 subcl adddir binom2.1 binom2.1 binom2.2 subdi binom2.1 sqval - A x. B opr opreq1i eqtr4 binom2.2 binom2.1 binom2.2 subdi binom2.1 binom2.2 mulcom binom2.2 sqval - opreq12i eqtr4 + opreq12i binom2.1 sqcl binom2.1 binom2.2 mulcl subcl binom2.1 binom2.2 mulcl binom2.2 sqcl addsubass binom2.1 sqcl binom2.1 binom2.2 mulcl subcl binom2.1 binom2.2 mulcl addcom binom2.1 binom2.2 mulcl binom2.1 sqcl pncan3 eqtr - B ^ 2 opr opreq1i eqtr3 3eqtr))
+thm (bernneq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) B)) (<_) (opr (opr (1) (+) A) (^) B))) (x cv 1 A x. opreq2 1 + opreq2d x cv 1 1 + A opr ^ opreq2 <_ breq12d A RR e. 1 -u <_ A br /\ imbi2d x cv y cv A x. opreq2 1 + opreq2d x cv y cv 1 + A opr ^ opreq2 <_ breq12d A RR e. 1 -u <_ A br /\ imbi2d x cv y cv + 1 opr A x. opreq2 1 + opreq2d x cv y cv + 1 opr 1 + A opr ^ opreq2 <_ breq12d A RR e. 1 -u <_ A br /\ imbi2d x cv B A x. opreq2 1 + opreq2d x cv B 1 + A opr ^ opreq2 <_ breq12d A RR e. 1 -u <_ A br /\ imbi2d ax1re 1 A axaddrcl mpan 1 + A opr leidt syl A recnt A ax1id syl 1 + opreq2d A recnt 1cn 1 A axaddcl mpan 1 + A opr exp1t 3syl 3brtr4d 1 -u <_ A br adantr 1 + A x. y cv opr opr A axaddrcl A y cv axmulrcl y cv nnret sylan2 ax1re 1 A x. y cv opr axaddrcl mpan syl A RR e. y cv NN e. pm3.26 sylanc 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 1 + A x. y cv opr opr 1 + A opr axmulrcl A y cv axmulrcl y cv nnret sylan2 ax1re 1 A x. y cv opr axaddrcl mpan syl ax1re 1 A axaddrcl mpan y cv NN e. adantr sylanc 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 1 + A opr ^ y cv opr 1 + A opr axmulrcl 1 + A opr y cv reexpclt ax1re 1 A axaddrcl mpan sylan ax1re 1 A axaddrcl mpan y cv NN e. adantr sylanc 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr A x. A opr y cv mulge0tOLD A A axmulrcl anidms y cv nnret anim12i A sqege0t A recnt A sqvalt syl breqtrd 0re 0 y cv ltlet mpan y cv nnret y cv nngt0t sylc anim12i sylc A A y cv mul23t A recnt y cv NN e. adantr A recnt y cv NN e. adantr y cv nncnt A RR e. adantl syl3anc breqtrd A x. y cv opr x. A opr 1 + A x. y cv opr opr + A opr leaddnnegt A x. y cv opr A axmulrcl A y cv axmulrcl A RR e. y cv RR e. pm3.26 sylanc y cv nnret sylan2 1 + A x. y cv opr opr A axaddrcl A y cv axmulrcl y cv nnret sylan2 ax1re 1 A x. y cv opr axaddrcl mpan syl A RR e. y cv NN e. pm3.26 sylanc sylanc mpbid 1 + A x. y cv opr opr A A x. y cv opr x. A opr axaddass A y cv axmulcl 1cn 1 A x. y cv opr axaddcl mpan syl A CC e. y cv CC e. pm3.26 A x. y cv opr A axmulcl A y cv axmulcl A CC e. y cv CC e. pm3.26 sylanc syl3anc A x. y cv opr A muladd11t A y cv axmulcl A CC e. y cv CC e. pm3.26 sylanc eqtr4d A recnt y cv nncnt syl2an breqtrd 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 1 + A x. y cv opr opr 1 + A opr ^ y cv opr 1 + A opr lemul1it A y cv axmulrcl y cv nnret sylan2 ax1re 1 A x. y cv opr axaddrcl mpan syl 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 1 + A opr y cv reexpclt ax1re 1 A axaddrcl mpan sylan 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr ax1re 1 A axaddrcl mpan y cv NN e. 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ ad2antll 3jca ax1re renegcl ax1re 1 -u A 1 leadd2t mp3an13 1cn negid <_ 1 + A opr breq1i syl6bb biimpa y cv NN e. adantlr 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br adantrr 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br pm3.27 A RR e. y cv NN e. /\ adantl jca sylanc letrd 1cn A y cv 1 axdistr mp3an3 A ax1id y cv CC e. adantr A x. y cv opr + opreq2d eqtrd 1 + opreq2d 1cn 1 A x. y cv opr A axaddass mp3an1 A y cv axmulcl A CC e. y cv CC e. pm3.26 sylanc eqtr4d A recnt y cv nncnt syl2an 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 1 + A opr y cv expp1t A recnt 1cn 1 A axaddcl mpan syl sylan 1 -u <_ A br 1 + A x. y cv opr opr <_ 1 + A opr ^ y cv opr br /\ adantr 3brtr4d exp43 com12 imp3a a2d nnind exp3a com12 3imp))
+thm (discrlem1 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem1.4 (= D (-u (opr B (/) (opr (2) (x.) A))))) (discrlem1.5 (br (0) (<) A))) (<-> (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (4re recn discrlem.1 recn mulcl mul01 4re recn discrlem.1 recn mulcl discrlem.1 recn discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr recn sqcl mulcl discrlem.2 recn discrlem1.4 discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl negcl eqeltr mulcl addcl discrlem.3 recn adddi <_ breq12i 4re discrlem.1 4pos discrlem1.5 mulgt0i 0re discrlem.1 discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr sqrecl remulcl discrlem.2 discrlem1.4 discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl eqeltr remulcl readdcl discrlem.3 readdcl 4re discrlem.1 remulcl lemul2 ax-mp neg0 discrlem.2 sqrecl recn 4re recn discrlem.1 recn discrlem.3 recn mulcl mulcl negdi2 2re discrlem.1 remulcl recn 2cn 2re 2pos gt0ne0i divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mulass 2re discrlem.1 remulcl recn 2cn discrlem.2 recn 2re discrlem.1 remulcl recn 2re 2pos gt0ne0i 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divmul13 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divid B / 2 opr x. opreq2i discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl 1cn mulcom 3eqtr x. B / 2 x. A opr opr opreq1i 2cn discrlem.1 recn 2re 2pos gt0ne0i divcan3 discrlem1.4 ^ 2 opreq1i discrlem.2 2re discrlem.1 remulcl 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl renegcl recn sqval discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mul2neg eqtr eqtr2 x. opreq12i 3eqtr3r 2cn negneg x. B / 2 opr opreq1i 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulneg1 2cn discrlem.2 recn 2re 2pos gt0ne0i divcan2 3eqtr3r discrlem1.4 x. opreq12i 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mul2neg eqtr + opreq12i df-2 negeqi 1cn 1cn negdi eqtr 1 + opreq2i 1cn negid + 1 -u opreq1i 1cn 1cn negcl 1cn negcl addass 1cn negcl addid2 3eqtr3 eqtr x. B / 2 opr opreq1i 1cn 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl adddir discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulm1 3eqtr3 x. B / 2 x. A opr opr opreq1i 1cn discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl 2cn negcl discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl mulcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl adddir discrlem.2 recn 2cn 2re 2pos gt0ne0i divcl discrlem.2 recn 2re discrlem.1 remulcl recn 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcl mulneg1 discrlem.2 recn sqval 2t2e4 x. A opreq1i 2cn 2cn discrlem.1 recn mulass eqtr3 / opreq12i discrlem.2 recn 2cn discrlem.2 recn 2re discrlem.1 remulcl recn 2re 2pos gt0ne0i 2cn discrlem.1 recn 2re 2pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divmuldiv eqtr4 negeqi eqtr4 3eqtr3 eqtr 4 x. A opr x. opreq2i 4re discrlem.1 remulcl recn discrlem.2 sqrecl 4re discrlem.1 remulcl 4re recn discrlem.1 recn 4re 4pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 redivcl recn mulneg2 4re discrlem.1 remulcl recn discrlem.2 sqrecl recn 4re recn discrlem.1 recn 4re 4pos gt0ne0i discrlem.1 discrlem1.5 gt0ne0i muln0 divcan2 negeqi 3eqtr 4re recn discrlem.1 recn discrlem.3 recn mulass + opreq12i eqtr4 <_ breq12i 3bitr4 discrlem.2 sqrecl 4re discrlem.1 discrlem.3 remulcl remulcl resubcl 0re leneg bitr4))
+thm (discrlem2 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem1.4 (= D (-u (opr B (/) (opr (2) (x.) A))))) (discrlem2.5 (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C))))) (-> (br (0) (<) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (2pos 2re discrlem.1 mulgt0 mpan 2re discrlem.1 remulcl gt0ne0 syl discrlem.2 2re discrlem.1 remulcl redivclz B / 2 x. A opr opr renegclt 3syl discrlem1.4 RR eleq1i sylibr discrlem2.5 syl A 0 < A br A 1 if = id A 0 < A br A 1 if 2 x. opreq2 B / opreq2d negeqd discrlem1.4 syl5eq ^ 2 opreq1d x. opreq12d A 0 < A br A 1 if 2 x. opreq2 B / opreq2d negeqd discrlem1.4 syl5eq B x. opreq2d + opreq12d + C opreq1d 0 <_ breq2d A 0 < A br A 1 if x. C opreq1 4 x. opreq2d B ^ 2 opr - opreq2d <_ 0 breq1d bibi12d discrlem.1 ax1re 0 < A br keepel discrlem.2 discrlem.3 B / 2 x. 0 < A br A 1 if opr opr -u eqid A elimgt0 discrlem1 dedth mpbid))
+thm (discrlem3 () ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem3.4 (= D (opr (opr C (+) (1)) (/) (-u B)))) (discrlem3.5 (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C))))) (-> (= (0) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (discrlem.3 ltplus1 B 0 df-ne discrlem.2 recn negne0 bitr3 discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 RR eleq1i sylibr discrlem3.5 syl 0 A = adantr 0 A x. D ^ 2 opr opreq1 eqcomd discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 RR eleq1i sylibr recnd D sqclt D ^ 2 opr mul02t 3syl sylan9eqr + B x. D opr opreq1d discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 RR eleq1i sylibr discrlem.2 jctil B D axmulrcl syl recnd B x. D opr addid2t syl 0 A = adantr eqtrd + C opreq1d breqtrd discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 RR eleq1i sylibr discrlem.2 jctil B D axmulrcl 0re discrlem.3 0 B x. D opr C lesubadd2t mp3an13 3syl 0 A = adantr mpbird discrlem.3 ax1re readdcl discrlem.2 renegcl redivclz discrlem3.4 RR eleq1i sylibr discrlem.2 jctil B recnt D recnt anim12i B D mulneg1t 3syl eqcomd B x. D opr df-neg discrlem3.4 B -u x. opreq2i 3eqtr3g discrlem.2 recn negcl discrlem.3 ax1re readdcl recn divcan2z eqtrd <_ C breq1d 0 A = adantr mpbid discrlem.3 ax1re readdcl discrlem.3 lenlt sylib exp sylbi mt2i a3i x. B opreq1d discrlem.2 recn mul02 syl6eq discrlem.2 recn sqval syl5eq 0 A x. C opreq1 discrlem.3 recn mul02 syl5reqr 4 x. opreq2d 4re recn mul01 syl6eq - opreq12d 0cn subid syl6eq discrlem.2 sqrecl 4re discrlem.1 discrlem.3 remulcl remulcl resubcl 0re eqle syl))
+thm (discrlem ((A x) (B x) (C x)) ((discrlem.1 (e. A (RR))) (discrlem.2 (e. B (RR))) (discrlem.3 (e. C (RR))) (discrlem.4 (A.e. x (RR) (br (0) (<_) (opr (opr (opr A (x.) (opr (cv x) (^) (2))) (+) (opr B (x.) (cv x))) (+) C))))) (-> (br (0) (<_) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))) (0re discrlem.1 leloe discrlem.1 discrlem.2 discrlem.3 B / 2 x. A opr opr -u eqid x cv B / 2 x. A opr opr -u ^ 2 opreq1 A x. opreq2d x cv B / 2 x. A opr opr -u B x. opreq2 + opreq12d + C opreq1d 0 <_ breq2d discrlem.4 vtoclri discrlem2 discrlem.1 discrlem.2 discrlem.3 C + 1 opr / B -u opr eqid x cv C + 1 opr / B -u opr ^ 2 opreq1 A x. opreq2d x cv C + 1 opr / B -u opr B x. opreq2 + opreq12d + C opreq1d 0 <_ breq2d discrlem.4 vtoclri discrlem3 jaoi sylbi))
+thm (nnsqcl () ((nnsqcl.1 (e. A (NN)))) (e. (opr A (^) (2)) (NN)) (nnsqcl.1 nncn sqval nnsqcl.1 nnsqcl.1 A A nnmulclt mp2an eqeltr))
+thm (nnlesq () ((nnsqcl.1 (e. A (NN)))) (br A (<_) (opr A (^) (2))) (nnsqcl.1 nncn mulid1 nnsqcl.1 A nnge1t ax-mp nnsqcl.1 nngt0 ax1re nnsqcl.1 nnre nnsqcl.1 nnre lemul2 ax-mp mpbi nnsqcl.1 nncn sqval breqtrr eqbrtrr))
+thm (nneo ((x y) (A x) (A y)) ((nnsqcl.1 (e. A (NN)))) (<-> (e. (opr A (/) (2)) (NN)) (-. (e. (opr (opr A (+) (1)) (/) (2)) (NN)))) (ax1re ltplus1 df-2 breqtrr ax1re 2re nnsqcl.1 nnre ltadd2 mpbi 2cn nnsqcl.1 nnre ax1re readdcl recn 2re 2pos gt0ne0i divcan2 2cn nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divcl 1cn adddi 2cn nnsqcl.1 nncn 2re 2pos gt0ne0i divcan2 2cn mulid1 + opreq12i eqtr 3brtr4 2pos nnsqcl.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nnsqcl.1 nnre 2re 2re 2pos gt0ne0i redivcl ax1re readdcl 2re ltmul2 ax-mp mpbir nnsqcl.1 nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nnsqcl.1 nnre 2re 2re 2pos gt0ne0i redivcl ax1re readdcl ltnle mpbi nnsqcl.1 nnre ltplus1 nnsqcl.1 nnre nnsqcl.1 nnre ax1re readdcl 2re 2pos ltdiv1i mpbi A / 2 opr A + 1 opr / 2 opr nnltp1let mpbii mto A / 2 opr NN e. A + 1 opr / 2 opr NN e. imnan mpbir nnsqcl.1 x cv 1 + 1 opreq1 / 2 opreq1d NN eleq1d x cv 1 / 2 opreq1 NN eleq1d orbi12d x cv y cv + 1 opreq1 / 2 opreq1d NN eleq1d x cv y cv / 2 opreq1 NN eleq1d orbi12d x cv y cv + 1 opr + 1 opreq1 / 2 opreq1d NN eleq1d x cv y cv + 1 opr / 2 opreq1 NN eleq1d orbi12d x cv A + 1 opreq1 / 2 opreq1d NN eleq1d x cv A / 2 opreq1 NN eleq1d orbi12d df-2 / 2 opreq1i 2cn 2re 2pos gt0ne0i divid eqtr3 1nn eqeltr 1 / 2 opr NN e. pm2.21ni orri y cv nncnt 1cn 1cn y cv 1 1 axaddass mp3an23 df-2 y cv + opreq2i syl6eqr / 2 opreq1d y cv y cv CC e. y cv 2 if + 2 opreq1 / 2 opreq1d y cv y cv CC e. y cv 2 if / 2 opreq1 + 2 / 2 opr opreq1d eqeq12d 2cn y cv elimel 2cn 2cn 2re 2pos gt0ne0i divdir dedth 2cn 2re 2pos gt0ne0i divid y cv / 2 opr + opreq2i syl6eq eqtrd syl NN eleq1d y cv / 2 opr peano2nn syl5bir y cv + 1 opr / 2 opr NN e. orim2d y cv + 1 opr / 2 opr NN e. y cv + 1 opr + 1 opr / 2 opr NN e. orcom syl6ib nnind ax-mp ori impbi))
+thm (nnesq () ((nnsqcl.1 (e. A (NN)))) (<-> (e. (opr A (/) (2)) (NN)) (e. (opr (opr A (^) (2)) (/) (2)) (NN))) (A / 2 opr A / 2 opr nnmulclt anidms 2nn 2 A / 2 opr x. A / 2 opr opr nnmulclt mpan 2cn nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divcl nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divcl mulass nnsqcl.1 nncn nnsqcl.1 nncn 2cn 2re 2pos gt0ne0i divass nnsqcl.1 nncn sqval / 2 opreq1i 2cn nnsqcl.1 nncn 2re 2pos gt0ne0i divcan2 x. A / 2 opr opreq1i 3eqtr4r eqtr3 NN eleq1i sylib syl A + 1 opr / 2 opr A + 1 opr / 2 opr nnmulclt anidms 2nn 2 A + 1 opr / 2 opr x. A + 1 opr / 2 opr opr nnmulclt mpan 2cn nnsqcl.1 nncn 1cn addcl 2re 2pos gt0ne0i divcan2 x. A + 1 opr / 2 opr opreq1i 2cn nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divcl nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divcl mulass nnsqcl.1 nncn 1cn addcl nnsqcl.1 nncn 1cn addcl 2cn 2re 2pos gt0ne0i divass nnsqcl.1 nncn 1cn addcl sqval nnsqcl.1 nncn 1cn binom2 nnsqcl.1 nncn mulid1 2 x. opreq2i A ^ 2 opr + opreq2i sq1 + opreq12i nnsqcl.1 nnsqcl nncn 2cn nnsqcl.1 nncn mulcl 1cn add23 3eqtr eqtr3 / 2 opreq1i nnsqcl.1 nnsqcl nncn 1cn addcl 2cn nnsqcl.1 nncn mulcl 2cn 2re 2pos gt0ne0i divdir 2cn nnsqcl.1 nncn 2re 2pos gt0ne0i divcan3 A ^ 2 opr + 1 opr / 2 opr + opreq2i 3eqtr eqtr3 3eqtr3 NN eleq1i nnsqcl.1 nncn addid2 nnsqcl.1 nnsqcl nnre ax1re readdcl 2re nnsqcl.1 nnsqcl nnre ax1re nnsqcl.1 nnsqcl nngt0 lt01 addgt0i 2pos divgt0i 0re nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl nnsqcl.1 nnre ltadd1 mpbi eqbrtrr nnsqcl.1 A A ^ 2 opr + 1 opr / 2 opr + A opr nnsubt mpan mpbii nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl recn nnsqcl.1 nncn nnsqcl.1 nncn addsubass nnsqcl.1 nncn subid A ^ 2 opr + 1 opr / 2 opr + opreq2i nnsqcl.1 nnsqcl nnre ax1re readdcl 2re 2re 2pos gt0ne0i redivcl recn addid1 3eqtr NN eleq1i sylib sylbi 3syl con3i nnsqcl.1 nnsqcl nneo nnsqcl.1 nneo 3imtr4 impbi))
+thm (nn0le2sqet () ((nn0le2sqet.1 (e. A (NN0))) (nn0le2sqet.2 (e. B (NN0)))) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B))) (nn0le2sqet.1 nn0ge0 nn0le2sqet.2 nn0ge0 nn0le2sqet.1 nn0re nn0le2sqet.2 nn0re le2sqe mp2an nn0le2sqet.1 nn0cn sqval nn0le2sqet.2 nn0cn sqval <_ breq12i bitr))
+thm (nn0opthlem1 () ((nn0opthlem1.1 (e. A (NN0))) (nn0opthlem1.2 (e. C (NN0)))) (<-> (br A (<) C) (br (opr (opr A (x.) A) (+) (opr (2) (x.) A)) (<) (opr C (x.) C))) (nn0opthlem1.1 1nn0 nn0addcl nn0opthlem1.2 nn0le2sqet nn0opthlem1.1 nn0opthlem1.2 A C nn0ltp1let mp2an nn0opthlem1.1 nn0opthlem1.1 nn0mulcl 2nn0 nn0opthlem1.1 nn0mulcl nn0addcl nn0opthlem1.2 nn0opthlem1.2 nn0mulcl A x. A opr + 2 x. A opr opr C x. C opr nn0ltp1let mp2an nn0opthlem1.1 nn0cn 1cn binom2 nn0opthlem1.1 nn0cn 1cn addcl sqval nn0opthlem1.1 nn0cn sqval + 2 x. A x. 1 opr opr opreq1i 1cn sqval + opreq12i 3eqtr3 nn0opthlem1.1 nn0cn mulid1 2 x. opreq2i A x. A opr + opreq2i 1cn mulid1 + opreq12i eqtr <_ C x. C opr breq1i bitr4 3bitr4))
+thm (nn0opthlem2 () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (-> (/\ (br B (<_) A) (br D (<_) C)) (-> (br A (<) C) (-. (= (opr (opr A (x.) A) (+) B) (opr (opr C (x.) C) (+) D))))) (nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.1 nn0re nn0opth.1 nn0re remulcl 2re nn0opth.1 nn0re remulcl readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl lelttr nn0opth.1 nn0opth.2 nn0lele2x nn0opth.2 nn0re 2re nn0opth.1 nn0re remulcl nn0opth.1 nn0re nn0opth.1 nn0re remulcl leadd2 sylib nn0opth.1 nn0opth.3 nn0opthlem1 biimp syl2an nn0opth.3 nn0re nn0opth.3 nn0re remulcl recn addid1 nn0opth.4 nn0ge0 0re nn0opth.4 nn0re nn0opth.3 nn0re nn0opth.3 nn0re remulcl leadd2 mpbi eqbrtrr nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.4 nn0re readdcl ltletr mpan2 syl exp D <_ C br adantr nn0opth.1 nn0re nn0opth.1 nn0re remulcl nn0opth.2 nn0re readdcl nn0opth.3 nn0re nn0opth.3 nn0re remulcl nn0opth.4 nn0re readdcl ltne syl6))
+thm (nn0opth () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) (/\ (= A C) (= B D))) (nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr = id nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an x. opreq12d + D opreq1d eqtr4d nn0opth.1 nn0cn nn0opth.2 nn0cn addcl nn0opth.1 nn0cn nn0opth.2 nn0cn addcl mulcl nn0opth.2 nn0cn nn0opth.4 nn0cn addcan sylib C + opreq2d eqtr4d nn0opth.1 nn0cn nn0opth.3 nn0cn nn0opth.2 nn0cn addcan2 sylib A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr = id nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an nn0opth.2 nn0re nn0opth.1 nn0addge2 nn0opth.4 nn0re nn0opth.3 nn0addge2 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opthlem2 nn0opth.3 nn0opth.4 nn0addcl nn0opth.4 nn0opth.1 nn0opth.2 nn0addcl nn0opth.2 nn0opthlem2 ancoms A + B opr x. A + B opr opr + B opr C + D opr x. C + D opr opr + D opr eqcom negbii syl6ibr jaod nn0opth.1 nn0re nn0opth.2 nn0re readdcl nn0opth.3 nn0opth.4 nn0addcl nn0re lttri2 syl5ib a3d mp2an x. opreq12d + D opreq1d eqtr4d nn0opth.1 nn0cn nn0opth.2 nn0cn addcl nn0opth.1 nn0cn nn0opth.2 nn0cn addcl mulcl nn0opth.2 nn0cn nn0opth.4 nn0cn addcan sylib jca A C B D + opreq12 A C B D + opreq12 x. opreq12d A C = B D = pm3.27 + opreq12d impbi))
+thm (nn0opth2 () ((nn0opth.1 (e. A (NN0))) (nn0opth.2 (e. B (NN0))) (nn0opth.3 (e. C (NN0))) (nn0opth.4 (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D))) (nn0opth.1 nn0cn nn0opth.2 nn0cn addcl sqval + B opreq1i nn0opth.3 nn0cn nn0opth.4 nn0cn addcl sqval + D opreq1i eqeq12i nn0opth.1 nn0opth.2 nn0opth.3 nn0opth.4 nn0opth bitr))
+thm (nn0opth2t () () (-> (/\ (/\ (e. A (NN0)) (e. B (NN0))) (/\ (e. C (NN0)) (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D)))) (A A NN0 e. A 0 if + B opreq1 ^ 2 opreq1d + B opreq1d C + D opr ^ 2 opr + D opr eqeq1d A A NN0 e. A 0 if C eqeq1 B D = anbi1d bibi12d B B NN0 e. B 0 if A NN0 e. A 0 if + opreq2 ^ 2 opreq1d B B NN0 e. B 0 if = id + opreq12d C + D opr ^ 2 opr + D opr eqeq1d B B NN0 e. B 0 if D eqeq1 A NN0 e. A 0 if C = anbi2d bibi12d C C NN0 e. C 0 if + D opreq1 ^ 2 opreq1d + D opreq1d A NN0 e. A 0 if + B NN0 e. B 0 if opr ^ 2 opr + B NN0 e. B 0 if opr eqeq2d C C NN0 e. C 0 if A NN0 e. A 0 if eqeq2 B NN0 e. B 0 if D = anbi1d bibi12d D D NN0 e. D 0 if C NN0 e. C 0 if + opreq2 ^ 2 opreq1d D D NN0 e. D 0 if = id + opreq12d A NN0 e. A 0 if + B NN0 e. B 0 if opr ^ 2 opr + B NN0 e. B 0 if opr eqeq2d D D NN0 e. D 0 if B NN0 e. B 0 if eqeq2 A NN0 e. A 0 if C NN0 e. C 0 if = anbi2d bibi12d 0nn0 A elimel 0nn0 B elimel 0nn0 C elimel 0nn0 D elimel nn0opth2 dedth4h))
+thm (sqrval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (` (sqr) A) (sup ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))) (RR) (<)))) (w cv A 0 <_ breq2 RR elrab y cv A x cv x. x cv opr <_ breq2 0 <_ x cv br anbi2d x RR birabsdv x RR 0 <_ x cv br x cv x. x cv opr <_ y cv br /\ {e.|} x RR 0 <_ x cv br x cv x. x cv opr <_ A br /\ {e.|} RR < supeq1 syl y z x df-sqr w cv y cv 0 <_ breq2 RR elrab z cv x RR 0 <_ x cv br x cv x. x cv opr <_ y cv br /\ {e.|} RR < sup = anbi1i y z biopabi eqtr4 ltso x RR 0 <_ x cv br x cv x. x cv opr <_ A br /\ {e.|} supex fvopab4 sylbir))
+thm (sqr0 () () (= (` (sqr) (0)) (0)) (0re 0re leid 0 x sqrval mp2an x cv x cv axmulrcl anidms 0re jctir x cv x. x cv opr 0 lenltt syl x cv ltsqt con1d sylbid 0 <_ x cv br adantld imp 0re 0re leid 0cn mul01 0re 0re remulcl 0re eqle ax-mp pm3.2i pm3.2i x cv 0 RR eleq1 x cv 0 0 <_ breq2 x cv 0 x cv 0 x. opreq12 anidms <_ 0 breq1d anbi12d anbi12d mpbiri impbi x biabi x RR 0 <_ x cv br x cv x. x cv opr <_ 0 br /\ df-rab 0 x df-sn 3eqtr4 x RR 0 <_ x cv br x cv x. x cv opr <_ 0 br /\ {e.|} 0 {} RR < supeq1 ax-mp 0re ltso 0 supsn ax-mp 3eqtr))
+thm (sqrlem1 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br A (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A))) (sqrlem1.1 recn addid2 lt01 0re ax1re sqrlem1.1 ltadd1 mpbi eqbrtrr ax1re sqrlem1.1 readdcl recn mulid2 1cn addid1 sqrlem1.2 0re sqrlem1.1 ax1re ltadd2 mpbi eqbrtrr ax1re ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i ltmul1i mpbi eqbrtrr sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an))
+thm (sqrlem2 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (opr (opr A (/) (opr (1) (+) A)) (x.) (opr A (/) (opr (1) (+) A))) (<) A) (sqrlem1.1 sqrlem1.2 sqrlem1 sqrlem1.2 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.1 ltmul2 ax-mp mpbi sqrlem1.1 sqrlem1.1 remulcl sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl remulcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an ltdiv1i mpbi sqrlem1.1 recn ax1re sqrlem1.1 readdcl recn sqrlem1.1 recn ax1re sqrlem1.1 readdcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i divmuldiv sqrlem1.1 recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an gt0ne0i divass ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl recn ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem1 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl lttr mp2an gt0ne0i divid A x. opreq2i eqtr sqrlem1.1 recn mulid1 eqtr2 3brtr4))
+thm (sqrlem3 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (0) (<) (opr A (/) (opr (1) (+) A))) (sqrlem1.1 ax1re sqrlem1.1 readdcl sqrlem1.2 ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i divgt0i))
+thm (sqrlem4 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (<-> (e. D S) (/\ (e. D (RR)) (/\ (br (0) (<_) D) (br (opr D (x.) D) (<_) A)))) (sqrlem4.3 D eleq2i x cv D 0 <_ breq2 x cv D x cv D x. opreq12 anidms <_ A breq1d anbi12d RR elrab bitr))
+thm (sqrlem5 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (/\ (e. D (RR)) (/\ (br (0) (<) D) (br (opr D (x.) D) (<) A))) (e. D S)) (0re 0 D ltlet mpan sqrlem1.1 D x. D opr A ltlet D D axmulrcl anidms sylan mpan2 anim12d imdistani sqrlem1.1 sqrlem1.2 sqrlem4.3 D sqrlem4 sylibr))
+thm (sqrlem6 ((x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (/\/\ (C_ S (RR)) (=/= S ({/})) (E.e. x (RR) (A.e. y S (br (cv y) (<) (cv x))))) (sqrlem4.3 x RR 0 <_ x cv br x cv x. x cv opr <_ A br /\ ssrab eqsstr sqrlem1.1 sqrlem1.2 sqrlem4.3 0 sqrlem4 0re 0re leid 0cn mul01 0re sqrlem1.1 sqrlem1.2 ltlei eqbrtr pm3.2i mpbir2an 0 S n0i ax-mp S {/} df-ne mpbir ax1re sqrlem1.1 readdcl sqrlem1.1 sqrlem1.2 sqrlem4.3 y cv sqrlem4 0re 0 y cv leloet mpan y cv y cv RR e. y cv 1 if 0 < breq2 y cv y cv RR e. y cv 1 if y cv y cv RR e. y cv 1 if x. opreq12 anidms < 1 + A opr x. 1 + A opr opr breq1d y cv y cv RR e. y cv 1 if < 1 + A opr breq1 imbi12d imbi12d ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i ax1re y cv elimel ax1re sqrlem1.1 readdcl lt2sq 0re ax1re y cv elimel ltle 0re ax1re sqrlem1.1 readdcl ltle syl2an mpan2 biimprd dedth ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i 0 y cv < 1 + A opr breq1 mpbii y cv x. y cv opr < 1 + A opr x. 1 + A opr opr br a1d y cv RR e. a1i jaod sylbid y cv y cv axmulrcl anidms sqrlem1.1 y cv x. y cv opr A leloet sqrlem1.1 sqrlem1.2 sqrlem1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl remulcl y cv x. y cv opr A 1 + A opr x. 1 + A opr opr axlttrn mp3an3 mpan2i sqrlem1.1 sqrlem1.2 sqrlem1 y cv x. y cv opr A < 1 + A opr x. 1 + A opr opr breq1 mpbiri y cv x. y cv opr RR e. A RR e. /\ a1i jaod sylbid mpan2 syl syl5d imp32 sylbi rgen x cv 1 + A opr y cv < breq2 y S biraldv RR rcla4ev mp2an 3pm3.2i))
+thm (sqrlem7 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem7.4 (= B (sup S (RR) (<))))) (e. B (RR)) (sqrlem7.4 ltso sqrlem1.1 sqrlem1.2 sqrlem4.3 y sqrlem6 S x y z axsup ax-mp supcli eqeltr))
+thm (sqrlem8 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem4.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem7.4 (= B (sup S (RR) (<))))) (br (0) (<) B) (sqrlem1.1 sqrlem1.2 sqrlem3 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem1.1 sqrlem1.2 sqrlem3 sqrlem1.1 sqrlem1.2 sqrlem2 pm3.2i sqrlem1.1 sqrlem1.2 sqrlem4.3 A / 1 + A opr opr sqrlem5 mp2an ltso sqrlem1.1 sqrlem1.2 sqrlem4.3 y sqrlem6 S x y z axsup ax-mp A / 1 + A opr opr supubi ax-mp sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem7.4 sqrlem1.1 sqrlem1.2 sqrlem4.3 sqrlem7.4 sqrlem7 eqeltrr lenlt mpbir 0re sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl sqrlem7.4 sqrlem1.1 sqrlem1.2 sqrlem4.3 sqrlem7.4 sqrlem7 eqeltrr ltletr mp2an sqrlem7.4 breqtrr))
+thm (sqrlem9 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br (0) (<) C) (sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl ax1re ax1re readdcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl sqrlem9.5 sqrlem1.1 sqrlem9.3 sqrlem1.2 sqrlem9.5 divgt0i addgt0i ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem9.7 breqtrr))
+thm (sqrlem10 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br C (<) B) (sqrlem9.7 sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.5 ltdiv1i mpbi sqrlem9.3 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcan3 breqtr sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl sqrlem9.3 sqrlem9.3 ltadd2 mpbi sqrlem9.3 recn 1cn 1cn addcl mulcom sqrlem9.3 recn 1p1times eqtr breqtrr sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl sqrlem9.3 ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ltdiv1i mpbi 1cn 1cn addcl sqrlem9.3 recn ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divcan4 breqtr eqbrtr))
+thm (sqrlem11 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))))) (br A (<) (opr C (x.) C)) (sqrlem9.3 sqrlem9.3 remulcl recn 1cn 1cn addcl negcl sqrlem1.1 renegcl recn mul12 1cn 1cn addcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl recn mulneg1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl recn 1p1times negeqi B x. B opr x. A -u opr + B x. B opr x. A -u opr opr df-neg eqtr 3eqtr sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl readdcl sqrlem1.1 recn negid sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl ltadd1 mpbi eqbrtrr sqrlem1.1 recn negid sqrlem9.6 sqrlem1.1 sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl ltadd1 mpbi eqbrtrr mulgt0i sqrlem9.3 sqrlem9.3 remulcl recn sqrlem1.1 renegcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem1.1 renegcl recn muladd breqtr 0re sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem1.1 renegcl remulcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl readdcl ltsubadd mpbir eqbrtr sqrlem9.3 sqrlem9.3 remulcl ax1re ax1re readdcl renegcl sqrlem1.1 renegcl remulcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl readdcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i ltdiv1i mpbi sqrlem9.3 sqrlem9.3 remulcl recn 1cn 1cn addcl negcl sqrlem1.1 renegcl recn mulcl sqrlem9.3 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i sqrlem9.3 sqrlem9.5 gt0ne0i muln0 divcan3 1cn 1cn addcl sqrlem1.1 recn mul2neg eqtr sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl recn sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i divdir 3brtr3 ax1re ax1re readdcl sqrlem1.1 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i redivcl sqrlem1.1 renegcl sqrlem1.1 renegcl remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i redivcl readdcl ax1re ax1re readdcl sqrlem1.1 remulcl ltadd1 mpbi sqrlem1.1 recn 1cn 1cn addcl 1cn 1cn addcl mulcl mulcom 1cn 1cn addcl 1cn 1cn addcl sqrlem1.1 recn mulass 1cn 1cn addcl sqrlem1.1 recn mulcl 1p1times 3eqtr sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl muladd sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl recn sqrlem9.3 sqrlem9.3 remulcl sqrlem9.3 sqrlem9.3 sqrlem9.5 sqrlem9.5 mulgt0i gt0ne0i divcan4 eqcomi sqrlem1.1 recn sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i sqrlem9.3 sqrlem9.5 gt0ne0i divmuldiv sqrlem1.1 recn sqrlem1.1 recn mul2neg / B x. B opr opreq1i eqtr4 + opreq12i sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl mulcl 1p1times sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcan2 1 + 1 opr x. opreq2i eqtr3 + opreq12i eqtr 3brtr4 sqrlem1.1 ax1re ax1re readdcl ax1re ax1re readdcl remulcl remulcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl sqrlem9.3 sqrlem1.1 sqrlem9.3 sqrlem9.3 sqrlem9.5 gt0ne0i redivcl readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ax1re ax1re lt01 lt01 addgt0i mulgt0i ltdiv1i mpbi ax1re ax1re readdcl ax1re ax1re readdcl remulcl recn sqrlem1.1 recn ax1re ax1re readdcl ax1re ax1re readdcl remulcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i ax1re ax1re lt01 lt01 addgt0i mulgt0i gt0ne0i divcan4 sqrlem9.7 sqrlem9.7 x. opreq12i sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl addcl 1cn 1cn addcl sqrlem9.3 recn sqrlem1.1 recn sqrlem9.3 recn sqrlem9.3 sqrlem9.5 gt0ne0i divcl addcl 1cn 1cn addcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i divmuldiv eqtr2 3brtr3))
+thm (sqrlem12 ((A x) (S x) (D x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (e. D S) (br D (<) C)) (sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.27bd pm3.26d sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd 0re 0 D leloet mpan syl mpbid sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.27bd pm3.27d sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem11 sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd D D axmulrcl anidms sqrlem1.1 sqrlem9.4 sqrlem9.4 remulcl D x. D opr A C x. C opr lelttrt mp3an23 3syl mpan2i mpd 0 < D br adantr sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem9 sqrlem1.1 sqrlem1.2 sqrlem12.8 D sqrlem4 pm3.26bd D D RR e. D 0 if 0 < breq2 0 < C br anbi1d D D RR e. D 0 if < C breq1 D D RR e. D 0 if D D RR e. D 0 if x. opreq12 anidms < C x. C opr breq1d bibi12d imbi12d 0re D elimel sqrlem9.4 lt2sq 0re 0re D elimel ltle 0re sqrlem9.4 ltle syl2an dedth syl mpan2i imp mpbird exp sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem9 0 D < C breq1 mpbii D S e. a1i jaod mpd))
+thm (sqrlem13 ((x y) (A x) (A y) (x z) (S x) (y z) (S y) (S z) (C y) (C z) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (= B (sup S (RR) (<))) (-. (br C (<) B))) (sqrlem9.4 sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem12.8 z cv sqrlem12 sqrlem1.1 sqrlem1.2 sqrlem12.8 z cv sqrlem4 pm3.26bd sqrlem9.4 z cv C axlttri mpan2 syl mpbid z cv C = C < z cv br ioran sylib pm3.27d rgen ltso sqrlem1.1 sqrlem1.2 sqrlem12.8 y sqrlem6 S x y z axsup ax-mp C supnubi mp2an B S RR < sup C < breq2 mtbiri))
+thm (sqrlem14 ((A x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem9.3 (e. B (RR))) (sqrlem9.4 (e. C (RR))) (sqrlem9.5 (br (0) (<) B)) (sqrlem9.6 (br A (<) (opr B (x.) B))) (sqrlem9.7 (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (sqrlem12.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-. (= B (sup S (RR) (<)))) (sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem10 sqrlem1.1 sqrlem1.2 sqrlem9.3 sqrlem9.4 sqrlem9.5 sqrlem9.6 sqrlem9.7 sqrlem12.8 sqrlem13 mt2))
+thm (sqrlem15 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C))) (br B (<) (opr B (+) C)) (sqrlem15.3 recn addid1 sqrlem15.6 0re sqrlem15.5 sqrlem15.3 ltadd2 mpbi eqbrtrr))
+thm (sqrlem16 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (br (opr (opr B (+) C) (x.) (opr B (+) C)) (<) A)) (1cn 1cn addcl 1cn addcl sqrlem15.3 recn sqrlem15.5 recn mulass 1cn 1cn addcl 1cn addcl sqrlem15.3 recn mulcl sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl recn ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 divcan2 < breq12i ax1re ax1re readdcl ax1re readdcl sqrlem15.3 ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i sqrlem15.4 mulgt0i sqrlem15.5 sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 redivcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ltmul2 ax-mp ax1re ax1re readdcl ax1re readdcl sqrlem15.3 sqrlem15.5 remulcl remulcl sqrlem15.3 sqrlem15.3 remulcl sqrlem1.1 ltaddsub 3bitr4 sqrlem16.7 sqrlem15.6 sqrlem15.5 sqrlem15.3 sqrlem15.5 ltmul2 ax-mp mpbi sqrlem15.5 sqrlem15.5 remulcl sqrlem15.5 sqrlem15.3 remulcl sqrlem15.3 sqrlem15.3 remulcl ltadd2 mpbi sqrlem15.3 sqrlem15.3 remulcl sqrlem15.5 sqrlem15.5 remulcl readdcl sqrlem15.3 sqrlem15.3 remulcl sqrlem15.5 sqrlem15.3 remulcl readdcl sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl ltadd1 mpbi sqrlem15.3 recn sqrlem15.5 recn sqrlem15.3 recn sqrlem15.5 recn muladd sqrlem15.5 sqrlem15.3 remulcl sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl readdcl recn sqrlem15.3 sqrlem15.3 remulcl recn addcom 1cn 1cn addcl 1cn sqrlem15.3 sqrlem15.5 remulcl recn adddir sqrlem15.3 sqrlem15.5 remulcl recn 1p1times sqrlem15.3 sqrlem15.5 remulcl recn mulid2 sqrlem15.3 recn sqrlem15.5 recn mulcom eqtr + opreq12i sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl recn sqrlem15.5 sqrlem15.3 remulcl recn addcom 3eqtr + B x. B opr opreq1i sqrlem15.3 sqrlem15.3 remulcl recn sqrlem15.5 sqrlem15.3 remulcl recn sqrlem15.3 sqrlem15.5 remulcl sqrlem15.3 sqrlem15.5 remulcl readdcl recn addass 3eqtr4 3brtr4 sqrlem15.3 sqrlem15.5 readdcl sqrlem15.3 sqrlem15.5 readdcl remulcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 sqrlem15.5 remulcl remulcl sqrlem15.3 sqrlem15.3 remulcl readdcl sqrlem1.1 lttr mpan sylbi))
+thm (sqrlem17 ((A x) (B x) (C x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B)) (sqrlem17.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (e. (opr B (+) C) S)) (sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem16.7 sqrlem16 sqrlem15.3 sqrlem15.5 sqrlem15.4 sqrlem15.6 addgt0i jctil sqrlem15.3 sqrlem15.5 readdcl sqrlem1.1 sqrlem1.2 sqrlem17.8 B + C opr sqrlem5 mpan syl))
+thm (sqrlem18 ((x y) (A x) (A y) (x z) (B x) (y z) (B y) (B z) (C x) (S x) (S y) (S z) (x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem15.5 (e. C (RR))) (sqrlem15.6 (br (0) (<) C)) (sqrlem16.7 (br C (<) B)) (sqrlem17.8 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (-. (= B (sup S (RR) (<))))) (sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem15 B S RR < sup < B + C opr breq1 negbid sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem15.5 sqrlem15.6 sqrlem16.7 sqrlem17.8 sqrlem17 ltso sqrlem1.1 sqrlem1.2 sqrlem17.8 y sqrlem6 S x y z axsup ax-mp B + C opr supubi syl syl5bir com12 mt2i))
+thm (sqrlem19 () ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem19.5 (br (opr B (x.) B) (<) A))) (br (0) (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl sqrlem15.3 sqrlem15.3 remulcl recn addid2 sqrlem19.5 eqbrtr 0re sqrlem15.3 sqrlem15.3 remulcl sqrlem1.1 ltaddsub mpbi ax1re ax1re readdcl ax1re readdcl sqrlem15.3 ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i sqrlem15.4 mulgt0i divgt0i))
+thm (sqrlem20 ((w x) (A x) (A w) (B x) (B w) (S x) (S w) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem15.3 (e. B (RR))) (sqrlem15.4 (br (0) (<) B)) (sqrlem19.5 (br (opr B (x.) B) (<) A)) (sqrlem20.6 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))))) (-. (= B (sup S (RR) (<)))) (sqrlem15.3 sqrlem1.1 sqrlem15.3 sqrlem15.3 remulcl resubcl ax1re ax1re readdcl ax1re readdcl sqrlem15.3 remulcl ax1re ax1re readdcl ax1re readdcl recn sqrlem15.3 recn ax1re ax1re readdcl ax1re readdcl ax1re ax1re readdcl ax1re ax1re ax1re lt01 lt01 addgt0i lt01 addgt0i gt0ne0i sqrlem15.3 sqrlem15.4 gt0ne0i muln0 redivcl sqrlem15.4 sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 sqrlem19.5 sqrlem19 w posex w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < A - B x. B opr opr / 1 + 1 opr + 1 opr x. B opr opr breq1 B S RR < sup = -. imbi1d sqrlem1.1 sqrlem1.2 sqrlem15.3 sqrlem15.4 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp1i w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp2i w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 w cv w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if RR eleq1 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if 0 < breq2 B / 1 + 1 opr opr w cv RR e. 0 < w cv br w cv < B br /\/\ w cv B / 1 + 1 opr opr if < B breq1 3anbi123d sqrlem15.3 ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl sqrlem15.3 ax1re ax1re readdcl sqrlem15.4 ax1re ax1re lt01 lt01 addgt0i divgt0i sqrlem15.4 sqrlem15.3 halfpos mpbi 3pm3.2i elimhyp 3simp3i sqrlem20.6 sqrlem18 dedth 3exp imp4d r19.23aiv ax-mp))
+thm (sqrlem21 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (-. (br A (<) (opr B (x.) B))) (sqrlem21.4 B A < B x. B opr br B 1 + A opr if S RR < sup eqeq1 negbid sqrlem1.1 sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 ax1re sqrlem1.1 readdcl A < B x. B opr br keepel B A < B x. B opr br B 1 + A opr if = id B A < B x. B opr br B 1 + A opr if A / opreq2 + opreq12d / 1 + 1 opr opreq1d RR eleq1d 1 + A opr A < B x. B opr br B 1 + A opr if = id 1 + A opr A < B x. B opr br B 1 + A opr if A / opreq2 + opreq12d / 1 + 1 opr opreq1d RR eleq1d sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 gt0ne0i redivcl readdcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl ax1re sqrlem1.1 readdcl sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl readdcl ax1re ax1re readdcl ax1re ax1re readdcl ax1re ax1re lt01 lt01 addgt0i gt0ne0i redivcl keephyp B A < B x. B opr br B 1 + A opr if 0 < breq2 1 + A opr A < B x. B opr br B 1 + A opr if 0 < breq2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i keephyp B A < B x. B opr br B 1 + A opr if B A < B x. B opr br B 1 + A opr if x. opreq12 anidms A < breq2d 1 + A opr A < B x. B opr br B 1 + A opr if 1 + A opr A < B x. B opr br B 1 + A opr if x. opreq12 anidms A < breq2d sqrlem1.1 sqrlem1.2 sqrlem1 elimhyp A < B x. B opr br B 1 + A opr if + A / A < B x. B opr br B 1 + A opr if opr opr / 1 + 1 opr opr eqid sqrlem21.3 sqrlem14 dedth mt2))
+thm (sqrlem22 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (-. (br (opr B (x.) B) (<) A)) (sqrlem21.4 B B x. B opr < A br B A / 1 + A opr opr if S RR < sup eqeq1 negbid sqrlem1.1 sqrlem1.2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 readdcl ax1re sqrlem1.1 lt01 sqrlem1.2 addgt0i gt0ne0i redivcl B x. B opr < A br keepel B B x. B opr < A br B A / 1 + A opr opr if 0 < breq2 A / 1 + A opr opr B x. B opr < A br B A / 1 + A opr opr if 0 < breq2 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem8 sqrlem1.1 sqrlem1.2 sqrlem3 keephyp B B x. B opr < A br B A / 1 + A opr opr if B B x. B opr < A br B A / 1 + A opr opr if x. opreq12 anidms < A breq1d A / 1 + A opr opr B x. B opr < A br B A / 1 + A opr opr if A / 1 + A opr opr B x. B opr < A br B A / 1 + A opr opr if x. opreq12 anidms < A breq1d sqrlem1.1 sqrlem1.2 sqrlem2 elimhyp sqrlem21.3 sqrlem20 dedth mt2))
+thm (sqrlem23 ((A x) (B x) (S x) (A x)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A)) (sqrlem21.3 (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (sqrlem21.4 (= B (sup S (RR) (<))))) (= (opr B (x.) B) A) (sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem7 remulcl sqrlem1.1 lttri3 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem22 sqrlem1.1 sqrlem1.2 sqrlem21.3 sqrlem21.4 sqrlem21 mpbir2an))
+thm (sqrlem24 ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (e. (` (sqr) A) (RR)) (sqrlem1.1 sqrlem1.2 x cv y cv 0 <_ breq2 x cv y cv x cv y cv x. opreq12 anidms <_ A breq1d anbi12d RR cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem7))
+thm (sqrgt0i ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (br (0) (<) (` (sqr) A)) (sqrlem1.1 sqrlem1.2 x cv y cv 0 <_ breq2 x cv y cv x cv y cv x. opreq12 anidms <_ A breq1d anbi12d RR cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem8))
+thm (sqrlem26 ((x y) (A x) (A y)) ((sqrlem1.1 (e. A (RR))) (sqrlem1.2 (br (0) (<) A))) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A) (sqrlem1.1 sqrlem1.2 x cv y cv 0 <_ breq2 x cv y cv x cv y cv x. opreq12 anidms <_ A breq1d anbi12d RR cbvrabv sqrlem1.1 0re sqrlem1.1 sqrlem1.2 ltlei A x sqrval mp2an sqrlem23))
+thm (sqrth () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A)) (0re sqrth.1 leloe A 0 < A br A 1 if sqr fveq2 A 0 < A br A 1 if sqr fveq2 x. opreq12d A 0 < A br A 1 if = id eqeq12d sqrth.1 ax1re 0 < A br keepel A elimgt0 sqrlem26 dedth sqr0 sqr0 x. opreq12i 0cn mul01 eqtr 0 A sqr fveq2 0 A sqr fveq2 x. opreq12d 0 A = id eqeq12d mpbii jaoi sylbi))
+thm (sqrcl () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (e. (` (sqr) A) (RR))) (0re sqrth.1 leloe A 0 < A br A 1 if sqr fveq2 RR eleq1d sqrth.1 ax1re 0 < A br keepel A elimgt0 sqrlem24 dedth 0re 0 A sqr fveq2 sqr0 syl5eqr RR eleq1d mpbii jaoi sylbi))
+thm (sqrgt0 () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<) A) (br (0) (<) (` (sqr) A))) (A 0 < A br A 1 if sqr fveq2 0 < breq2d sqrth.1 ax1re 0 < A br keepel A elimgt0 sqrgt0i dedth))
+thm (sqrge0 () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (br (0) (<_) (` (sqr) A))) (0re sqrth.1 leloe 0re sqrth.1 ltle sqrth.1 sqrcl syl 0re 0 sqr A ` leloet mpan sqrth.1 sqrgt0 0 < sqr A ` br 0 sqr A ` = orc syl syl5bir mpcom 0 A sqr fveq2 sqr0 syl5eqr 0re leid syl5breq jaoi sylbi))
+thm (sqr11 () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) (` (sqr) B)) (= A B))) (sqr A ` sqr B ` sqr A ` sqr B ` x. opreq12 anidms 0 <_ A br 0 <_ B br /\ adantl sqrth.1 sqrth 0 <_ B br sqr A ` sqr B ` = ad2antll sqr11.1 sqrth 0 <_ A br sqr A ` sqr B ` = ad2antlr 3eqtr3d exp A B sqr fveq2 0 <_ A br 0 <_ B br /\ a1i impbid))
+thm (sqrmuli () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR))) (sqrmuli.1 (br (0) (<_) A)) (sqrmuli.2 (br (0) (<_) B))) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B))) (sqrmuli.1 sqrth.1 sqrth ax-mp sqrmuli.2 sqr11.1 sqrth ax-mp x. opreq12i sqrmuli.1 sqrth.1 sqrcl ax-mp recn sqrmuli.2 sqr11.1 sqrcl ax-mp recn sqrmuli.1 sqrth.1 sqrcl ax-mp recn sqrmuli.2 sqr11.1 sqrcl ax-mp recn mul4 sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrth ax-mp 3eqtr4r sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrge0 ax-mp sqrmuli.1 sqrth.1 sqrge0 ax-mp sqrmuli.2 sqr11.1 sqrge0 ax-mp sqrmuli.1 sqrth.1 sqrcl ax-mp sqrmuli.2 sqr11.1 sqrcl ax-mp mulge0 mp2an sqrmuli.1 sqrmuli.2 sqrth.1 sqr11.1 mulge0 mp2an sqrth.1 sqr11.1 remulcl sqrcl ax-mp sqrmuli.1 sqrth.1 sqrcl ax-mp sqrmuli.2 sqr11.1 sqrcl ax-mp remulcl sq11 mp2an mpbi))
+thm (sqrmul () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B)))) (A 0 <_ A br A 0 if x. B opreq1 sqr fveq2d A 0 <_ A br A 0 if sqr fveq2 x. sqr B ` opreq1d eqeq12d B 0 <_ B br B 0 if 0 <_ A br A 0 if x. opreq2 sqr fveq2d B 0 <_ B br B 0 if sqr fveq2 sqr 0 <_ A br A 0 if ` x. opreq2d eqeq12d sqrth.1 0re 0 <_ A br keepel sqr11.1 0re 0 <_ B br keepel A elimge0 B elimge0 sqrmuli dedth2h))
+thm (sqrsq () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) B) (= A (opr B (x.) B)))) (sqr11.1 sqr11.1 sqrmul anidms sqr11.1 sqrth eqtrd sqr A ` eqeq2d 0 <_ A br adantl sqr11.1 sqge0 sqrth.1 sqr11.1 sqr11.1 remulcl sqr11 mpan2 0 <_ B br adantr bitr3d))
+thm (sqrle () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B)))) (sqr A ` sqr B ` le2sqt sqrth.1 sqrcl sqr11.1 sqrcl syl2an sqrth.1 sqrge0 sqr11.1 sqrge0 syl2ani pm2.43i sqrth.1 sqrth sqr11.1 sqrth <_ breqan12d bitr2d))
+thm (sqrlt () ((sqrth.1 (e. A (RR))) (sqr11.1 (e. B (RR)))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (` (sqr) A) (<) (` (sqr) B)))) (sqr A ` sqr B ` lt2sqt sqrth.1 sqrcl sqr11.1 sqrcl anim12i sqrth.1 sqrge0 sqr11.1 sqrge0 anim12i sylanc sqrth.1 sqrth sqr11.1 sqrth < breqan12d bitr2d))
+thm (sqrsqid () ((sqrth.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (x.) A)) A)) (sqrth.1 sqge0 A x. A opr eqid sqrth.1 sqrth.1 remulcl sqrth.1 sqrsq mpbiri mpan))
+thm (sqrclt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (sqr) A) (RR))) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if sqr fveq2 RR eleq1d imbi12d 0re A elimel sqrcl dedth imp))
+thm (sqrgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (` (sqr) A))) (A A RR e. A 0 if 0 < breq2 A A RR e. A 0 if sqr fveq2 0 < breq2d imbi12d 0re A elimel sqrgt0 dedth imp))
+thm (sqrge0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (br (0) (<_) (` (sqr) A))) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if sqr fveq2 0 <_ breq2d imbi12d 0re A elimel sqrge0 dedth imp))
+thm (sqrlet () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B)))) (A A RR e. A 0 if 0 <_ breq2 0 <_ B br anbi1d A A RR e. A 0 if <_ B breq1 A A RR e. A 0 if sqr fveq2 <_ sqr B ` breq1d bibi12d imbi12d B B RR e. B 0 if 0 <_ breq2 0 <_ A RR e. A 0 if br anbi2d B B RR e. B 0 if A RR e. A 0 if <_ breq2 B B RR e. B 0 if sqr fveq2 sqr A RR e. A 0 if ` <_ breq2d bibi12d imbi12d 0re A elimel 0re B elimel sqrle dedth2h imp))
+thm (sqr00t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (<-> (= (` (sqr) A) (0)) (= A (0)))) (0re leid A A RR e. A 0 if 0 <_ breq2 0 <_ 0 br anbi1d A A RR e. A 0 if sqr fveq2 sqr 0 ` eqeq1d A A RR e. A 0 if 0 eqeq1 bibi12d imbi12d 0re A elimel 0re sqr11 dedth mpan2i imp sqr0 sqr A ` eqeq2i syl5bbr))
+thm (sqr1 () () (= (` (sqr) (1)) (1)) (1cn mulid1 sqr fveq2i 0re ax1re lt01 ltlei ax1re sqrsqid ax-mp eqtr3))
+thm (sqr4 () () (= (` (sqr) (4)) (2)) (2t2e4 eqcomi 4pos 2pos 4re 2re sqrsq 0re 4re ltle 0re 2re ltle syl2an mp2an mpbir))
+thm (sqr9 () () (= (` (sqr) (9)) (3)) (3t3e9 eqcomi 9pos 3pos 9re 3re sqrsq 0re 9re ltle 0re 3re ltle syl2an mp2an mpbir))
+thm (sqr2gt1lt2 () () (/\ (br (1) (<) (` (sqr) (2))) (br (` (sqr) (2)) (<) (2))) (sqr1 1lt2 0re ax1re lt01 ltlei 0re 2re 2pos ltlei ax1re 2re sqrlt mp2an mpbi eqbrtrr 2re 2pos 2re 2pos 2re 2pos sqrlem15 2p2e4 breqtr 0re 2re 2pos ltlei 0re 4re 4pos ltlei 2re 4re sqrlt mp2an mpbi sqr4 breqtr pm3.2i))
+thm (sqrsqe () ((sqrsqe.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (^) (2))) A)) (sqrsqe.1 sqrsqid sqrsqe.1 recn sqval sqr fveq2i syl5eq))
+thm (sqsqr () ((sqrsqe.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (^) (2)) A)) (sqrsqe.1 sqrcl recnd sqr A ` sqvalt syl sqrsqe.1 sqrth eqtrd))
+thm (sqsqrt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (opr (` (sqr) A) (^) (2)) A)) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if sqr fveq2 ^ 2 opreq1d A A RR e. A 0 if = id eqeq12d imbi12d 0re A elimel sqsqr dedth imp))
+thm (sqr2irrlem1 () ((sqr2irrlem1.1 (e. A (NN))) (sqr2irrlem1.2 (e. B (NN)))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2)))))) (sqr2irrlem1.2 nnre sqrecl recn mulid2 ax1re ltplus1 df-2 breqtrr ax1re 2re sqr2irrlem1.2 nnre sqrecl sqr2irrlem1.2 nnsqcl nngt0 ltmul1i mpbi eqbrtrr A ^ 2 opr 2 x. B ^ 2 opr opr B ^ 2 opr < breq2 mpbiri 0re sqr2irrlem1.2 nnre sqr2irrlem1.2 nngt0 ltlei 0re sqr2irrlem1.1 nnre sqr2irrlem1.1 nngt0 ltlei sqr2irrlem1.2 nnre sqr2irrlem1.1 nnre lt2sqe mp2an sylibr sqr2irrlem1.1 nnre sqrecl recn 2cn sqr2irrlem1.2 nnre sqrecl recn 2re 2pos gt0ne0i divmul sqr2irrlem1.2 nnsqcl A ^ 2 opr / 2 opr B ^ 2 opr NN eleq1 mpbiri sqr2irrlem1.1 nnesq sylibr sylbir eqcoms jca 2cn 2re sqr2irrlem1.1 nnre 2re 2re 2pos gt0ne0i redivcl sqrecl remulcl recn sqr2irrlem1.2 nnre sqrecl recn 2re 2pos gt0ne0i mulcan sqr2irrlem1.1 nncn 2cn 2re 2pos gt0ne0i sqdiv 2cn sqval A ^ 2 opr / opreq2i eqtr 2 x. 2 opr x. opreq2i 2cn 2cn sqr2irrlem1.1 nnre 2re 2re 2pos gt0ne0i redivcl sqrecl recn mulass 2cn 2cn mulcl sqr2irrlem1.1 nnre sqrecl recn 2cn 2cn 2re 2pos gt0ne0i 2re 2pos gt0ne0i muln0 divcan2 3eqtr3 2 x. B ^ 2 opr opr eqeq1i bitr3 biimpr eqcomd jca))
+thm (sqr2irrlem2 () () (-> (/\ (e. A (NN)) (e. B (NN))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2))))))) (A A NN e. A 2 if ^ 2 opreq1 2 x. B ^ 2 opr opr eqeq1d A A NN e. A 2 if B < breq2 A A NN e. A 2 if / 2 opreq1 NN eleq1d anbi12d A A NN e. A 2 if / 2 opreq1 ^ 2 opreq1d 2 x. opreq2d B ^ 2 opr eqeq2d anbi12d imbi12d B B NN e. B 2 if ^ 2 opreq1 2 x. opreq2d A NN e. A 2 if ^ 2 opr eqeq2d B B NN e. B 2 if < A NN e. A 2 if breq1 A NN e. A 2 if / 2 opr NN e. anbi1d B B NN e. B 2 if ^ 2 opreq1 2 x. A NN e. A 2 if / 2 opr ^ 2 opr opr eqeq1d anbi12d imbi12d 2nn A elimel 2nn B elimel sqr2irrlem1 dedth2h))
+thm (sqr2irrlem3 ((x y) (x z) (w x) (y z) (w y) (w z)) () (-. (E.e. x (NN) (E.e. y (NN) (= (opr (cv x) (^) (2)) (opr (2) (x.) (opr (cv y) (^) (2))))))) (x cv z cv ^ 2 opreq1 2 x. y cv ^ 2 opr opr eqeq1d negbid y NN biraldv y cv w cv ^ 2 opreq1 2 x. opreq2d z cv ^ 2 opr eqeq2d negbid NN cbvralv syl6bb z cv y cv < x cv breq1 z cv y cv ^ 2 opreq1 2 x. w cv ^ 2 opr opr eqeq1d negbid w NN biraldv imbi12d NN rcla4v impcom w cv x cv / 2 opr ^ 2 opreq1 2 x. opreq2d y cv ^ 2 opr eqeq2d negbid NN rcla4v com12 syl6 imp3a y cv < x cv br x cv / 2 opr NN e. /\ y cv ^ 2 opr 2 x. x cv / 2 opr ^ 2 opr opr = imnan sylib x cv NN e. adantll x cv y cv sqr2irrlem2 z NN z cv < x cv br w NN z cv ^ 2 opr 2 x. w cv ^ 2 opr opr = -. A.e. -> A.e. adantlr mtod exp31 r19.21adv indstr y NN x cv ^ 2 opr 2 x. y cv ^ 2 opr opr = ralnex sylib rgen x NN y NN x cv ^ 2 opr 2 x. y cv ^ 2 opr opr = E.e. ralnex mpbi))
+thm (sqr2irrlem4 () ((sqr2irrlem4.1 (e. A (NN))) (sqr2irrlem4.2 (e. B (NN)))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2))))) (sqr2irrlem4.1 nncn sqr2irrlem4.2 nncn sqr2irrlem4.2 nnne0 sqdiv eqcomi sqr2irrlem4.2 nnsqcl nncn 2cn sqr2irrlem4.2 nnsqcl nnne0 divcan4 eqeq12i sqr2irrlem4.1 nnre sqrecl recn 2re sqr2irrlem4.2 nnsqcl nnre remulcl recn sqr2irrlem4.2 nnsqcl nncn sqr2irrlem4.2 nnsqcl nnne0 div11 A / B opr ^ 2 opr 2 eqcom 3bitr3 0re 2re 2pos ltlei sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqege0 2re sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqrecl sqr11 mp2an bitr4 0re sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.1 nngt0 sqr2irrlem4.2 nngt0 divgt0i ltlei sqr2irrlem4.1 nnre sqr2irrlem4.2 nnre sqr2irrlem4.2 nnne0 redivcl sqrsqe ax-mp sqr 2 ` eqeq2i bitr2))
+thm (sqr2irrlem5 () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))))) (A A NN e. A 2 if / B opreq1 sqr 2 ` eqeq2d A A NN e. A 2 if ^ 2 opreq1 2 x. B ^ 2 opr opr eqeq1d bibi12d B B NN e. B 2 if A NN e. A 2 if / opreq2 sqr 2 ` eqeq2d B B NN e. B 2 if ^ 2 opreq1 2 x. opreq2d A NN e. A 2 if ^ 2 opr eqeq2d bibi12d 2nn A elimel 2nn B elimel sqr2irrlem4 dedth2h))
+thm (sqr2irr ((x y)) () (e/ (` (sqr) (2)) (QQ)) (x y sqr2irrlem3 x cv y cv sqr2irrlem5 bi2rexa mtbir y cv nngt0t x cv ZZ e. adantr 0re 0 y cv x cv ltmuldivt exp mp3an1 y cv nnret x cv zret syl2an mpd ancoms 2re 2pos sqrgt0i sqr 2 ` x cv / y cv opr 0 < breq2 mpbii syl5bir y cv nncnt y cv mul02t syl < x cv breq1d x cv ZZ e. adantl sylibd exp r19.23adv anc2li x cv elnnz syl6ibr impac r19.22i2 mto sqr 2 ` x y elq mtbir sqr 2 ` QQ df-nel mpbir))
+thm (sqr2re () () (e. (` (sqr) (2)) (RR)) (0re 2re 2pos ltlei 2re sqrcl ax-mp))
+thm (ine0 () () (=/= (i) (0)) (ax1ne0 1 0 df-ne mpbi i 0 i x. opreq2 axicn mul01 syl6req + 1 opreq1d 1cn addid2 axi2m1 3eqtr3g mto i 0 df-ne mpbir))
+thm (isqm1 () () (= (opr (i) (^) (2)) (-u (1))) (axicn sqval axicn axicn mulcl 1cn 1cn negcl addass axi2m1 + 1 -u opreq1i 1cn negid i x. i opr + opreq2i 3eqtr3r axicn axicn mulcl addid1 1cn negcl addid2 3eqtr3 eqtr))
+thm (itimesi () () (= (opr (i) (x.) (i)) (-u (1))) (axicn axicn mulcl 1cn 1cn negcl addass axi2m1 + 1 -u opreq1i 1cn negid i x. i opr + opreq2i 3eqtr3r axicn axicn mulcl addid1 1cn negcl addid2 3eqtr3))
+thm (irec () () (= (opr (1) (/) (i)) (-u (i))) (axicn axicn mulneg2 itimesi 1cn axicn axicn mulcl negcon2 mpbir eqtr4 1cn axicn axicn negcl ine0 divmul mpbir))
+thm (inelr () () (-. (e. (i) (RR))) (ine0 i 0 df-ne mpbi lt01 0re ax1re ltnsym ax-mp 0re itimesi ax1re renegcl eqeltr ax1re ltadd1 1cn addid2 axi2m1 < breq12i bitr mtbir i ltsqt mt3i mto))
+thm (crulem () ((cru.1 (e. A (RR))) (cru.2 (e. B (RR))) (cru.3 (e. C (RR))) (cru.4 (e. D (RR)))) (-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (= B D)) (inelr cru.3 cru.1 renegcl readdcl recn cru.2 recn cru.4 recn subcl axicn divmulz A + B x. i opr opr C + D x. i opr opr + A -u + D x. i opr -u opr opreq1 cru.1 recn cru.2 recn axicn mulcl cru.1 renegcl recn cru.4 recn axicn mulcl negcl add4 cru.1 recn negid + B x. i opr + D x. i opr -u opr opreq1i cru.2 recn axicn mulcl cru.4 recn axicn mulcl negcl addcl addid2 3eqtr cru.3 recn cru.4 recn axicn mulcl cru.1 renegcl recn cru.4 recn axicn mulcl negcl add4 cru.4 recn axicn mulcl negid C + A -u opr + opreq2i cru.3 cru.1 renegcl readdcl recn addid1 3eqtr 3eqtr3g cru.2 recn cru.4 recn axicn subdir cru.2 recn axicn mulcl cru.4 recn axicn mulcl subneg eqtr syl5eq syl5bir imp RR eleq1d cru.3 cru.1 renegcl readdcl cru.2 cru.4 resubcl redivclz syl5bi exp pm2.43b B - D opr 0 df-ne syl5ibr mt3i cru.2 recn cru.4 recn subeq0 sylib))
+thm (cru () ((cru.1 (e. A (RR))) (cru.2 (e. B (RR))) (cru.3 (e. C (RR))) (cru.4 (e. D (RR)))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D))) (cru.1 cru.2 cru.3 cru.4 crulem B D x. i opreq1 C + opreq2d A + B x. i opr opr eqeq2d cru.1 recn cru.2 recn axicn mulcl addcom cru.3 recn cru.2 recn axicn mulcl addcom eqeq12i cru.2 recn axicn mulcl cru.1 recn cru.3 recn addcan biimp sylbi syl6bir mpcom cru.1 cru.2 cru.3 cru.4 crulem jca A C = id B D x. i opreq1 + opreqan12d impbi))
+thm (crut () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D)))) (A A RR e. A 0 if + B x. i opr opreq1 C + D x. i opr opr eqeq1d A A RR e. A 0 if C eqeq1 B D = anbi1d bibi12d B B RR e. B 0 if x. i opreq1 A RR e. A 0 if + opreq2d C + D x. i opr opr eqeq1d B B RR e. B 0 if D eqeq1 A RR e. A 0 if C = anbi2d bibi12d C C RR e. C 0 if + D x. i opr opreq1 A RR e. A 0 if + B RR e. B 0 if x. i opr opr eqeq2d C C RR e. C 0 if A RR e. A 0 if eqeq2 B RR e. B 0 if D = anbi1d bibi12d D D RR e. D 0 if x. i opreq1 C RR e. C 0 if + opreq2d A RR e. A 0 if + B RR e. B 0 if x. i opr opr eqeq2d D D RR e. D 0 if B RR e. B 0 if eqeq2 A RR e. A 0 if C RR e. C 0 if = anbi2d bibi12d 0re A elimel 0re B elimel 0re C elimel 0re D elimel cru dedth4h))
+thm (crne0 () ((crne0.1 (e. A (RR))) (crne0.2 (e. B (RR)))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr A (+) (opr B (x.) (i))) (0))) (A 0 df-ne B 0 df-ne orbi12i A 0 = B 0 = ianor A + B x. i opr opr 0 df-ne A + B x. i opr opr 0 = id 0cn axicn mulcl addid2 axicn mul02 eqtr2 syl6eq crne0.1 crne0.2 0re 0re cru sylib A 0 = id B 0 x. i opreq1 + opreqan12d 0cn axicn mulcl addid2 axicn mul02 eqtr syl6eq impbi negbii bitr2 bitr3 bitr))
+thm (crmul () ((crmul.1 (e. A (CC))) (crmul.2 (e. B (CC))) (crmul.3 (e. C (CC))) (crmul.4 (e. D (CC)))) (= (opr (opr A (+) (opr B (x.) (i))) (x.) (opr C (+) (opr D (x.) (i)))) (opr (opr (opr A (x.) C) (-) (opr B (x.) D)) (+) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (x.) (i)))) (crmul.1 crmul.3 mulcl crmul.1 crmul.4 mulcl axicn mulcl crmul.2 crmul.4 mulcl negcl crmul.2 crmul.3 mulcl axicn mulcl add4 crmul.1 crmul.3 crmul.4 axicn mulcl adddi crmul.1 crmul.4 axicn mulass A x. C opr + opreq2i eqtr4 crmul.2 axicn mulcl crmul.3 crmul.4 axicn mulcl adddi crmul.2 axicn crmul.3 mul23 crmul.2 axicn crmul.4 axicn mul4 itimesi B x. D opr x. opreq2i crmul.2 crmul.4 mulcl 1cn negcl mulcom 1cn crmul.2 crmul.4 mulcl mulneg1 crmul.2 crmul.4 mulcl mulid2 negeqi 3eqtr 3eqtr + opreq12i crmul.2 crmul.3 mulcl axicn mulcl crmul.2 crmul.4 mulcl negcl addcom 3eqtr + opreq12i crmul.1 crmul.4 mulcl crmul.2 crmul.3 mulcl axicn adddir A x. C opr + B x. D opr -u opr + opreq2i 3eqtr4 crmul.1 crmul.2 axicn mulcl crmul.3 crmul.4 axicn mulcl addcl adddir crmul.1 crmul.3 mulcl crmul.2 crmul.4 mulcl subneg + A x. D opr + B x. C opr opr x. i opr opreq1i 3eqtr4))
+thm (crrecz () ((crrecz.1 (e. A (RR))) (crrecz.2 (e. B (RR)))) (-> (\/ (=/= A (0)) (=/= B (0))) (= (opr (1) (/) (opr A (+) (opr B (x.) (i)))) (opr (opr A (-) (opr B (x.) (i))) (/) (opr (opr A (^) (2)) (+) (opr B (^) (2)))))) (crrecz.1 crrecz.2 renegcl crne0 crrecz.2 recn negne0 A 0 =/= orbi2i crrecz.1 recn crrecz.2 recn axicn mulcl subneg crrecz.2 recn axicn mulneg1 A + opreq2i eqtr4 0 neeq1i 3bitr4 crrecz.1 recn crrecz.2 recn axicn mulcl subcl A - B x. i opr opr dividt mpan sylbi 1 / A + B x. i opr opr opr x. opreq2d crrecz.1 crrecz.2 crne0 crrecz.1 recn crrecz.2 recn axicn mulcl addcl recclz sylbi 1 / A + B x. i opr opr opr ax1id syl eqtr2d 1cn crrecz.1 recn crrecz.2 recn axicn mulcl addcl pm3.2i crrecz.1 recn crrecz.2 recn axicn mulcl subcl crrecz.1 recn crrecz.2 recn axicn mulcl subcl pm3.2i 1 A + B x. i opr opr A - B x. i opr opr A - B x. i opr opr divmuldivt mpanl12 crrecz.1 crrecz.2 crne0 crrecz.1 crrecz.2 renegcl crne0 crrecz.2 recn negne0 A 0 =/= orbi2i crrecz.1 recn crrecz.2 recn axicn mulcl subneg crrecz.2 recn axicn mulneg1 A + opreq2i eqtr4 0 neeq1i 3bitr4 sylancb eqtrd crrecz.1 recn crrecz.2 recn axicn mulcl subcl mulid2 crrecz.1 recn crrecz.2 recn axicn mulcl binom2a crrecz.1 recn sqcl crrecz.2 recn axicn mulcl sqcl subneg crrecz.2 recn axicn sqmul isqm1 B ^ 2 opr x. opreq2i crrecz.2 recn sqcl 1cn mulneg2 3eqtr negeqi crrecz.2 recn sqcl 1cn mulcl negneg crrecz.2 recn sqcl mulid1 3eqtr A ^ 2 opr + opreq2i 3eqtr / opreq12i syl6eq))
+thm (creur ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (A x y axcnre x cv y cv z cv w cv crut x cv z cv = y cv w cv = pm3.26 syl6bi A x cv + y cv x. i opr opr z cv + w cv x. i opr opr eqtr2t syl5 an4s exp imp3a y cv RR e. A x cv + y cv x. i opr opr = w cv RR e. A z cv + w cv x. i opr opr = an4 syl5ib y w 19.23advv y RR A x cv + y cv x. i opr opr = df-rex w RR A z cv + w cv x. i opr opr = df-rex anbi12i y w y cv RR e. A x cv + y cv x. i opr opr = /\ w cv RR e. A z cv + w cv x. i opr opr = /\ eeanv bitr4 syl5ib rgen2 jctir x cv z cv + y cv x. i opr opreq1 A eqeq2d y RR birexdv y cv w cv x. i opreq1 z cv + opreq2d A eqeq2d RR cbvrexv syl6bb RR reu4 sylibr))
+thm (creui ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))) (A x y axcnre x RR y RR A x cv + y cv x. i opr opr = rexcom sylib x cv y cv z cv w cv crut x cv z cv = y cv w cv = pm3.27 syl6bi A x cv + y cv x. i opr opr z cv + w cv x. i opr opr eqtr2t syl5 an4s expcom imp3a x cv RR e. A x cv + y cv x. i opr opr = z cv RR e. A z cv + w cv x. i opr opr = an4 syl5ib x z 19.23advv x RR A x cv + y cv x. i opr opr = df-rex z RR A z cv + w cv x. i opr opr = df-rex anbi12i x z x cv RR e. A x cv + y cv x. i opr opr = /\ z cv RR e. A z cv + w cv x. i opr opr = /\ eeanv bitr4 syl5ib rgen2 jctir y cv w cv x. i opreq1 x cv + opreq2d A eqeq2d x RR birexdv x cv z cv + w cv x. i opr opreq1 A eqeq2d RR cbvrexv syl6bb RR reu4 sylibr))
+thm (rimul () () (-> (/\ (e. A (RR)) (e. (opr A (x.) (i)) (RR))) (= A (0))) (A x. i opr recnt 0cn axicn mulcl 0 x. i opr A x. i opr axaddcom mpan axicn mul02 + A x. i opr opreq1i syl5eqr syl A RR e. adantl 0re 0re 0 A A x. i opr 0 crut mpanr2 mpanl1 mpbid pm3.27d))
+thm (nthruc () () (/\ (/\ (C: (NN) (ZZ)) (C: (ZZ) (QQ))) (/\ (C: (QQ) (RR)) (C: (RR) (CC)))) (nnssz 0z 0nnn pm3.2i NN ZZ 0 ssnelpss mp2 zssq nnssz 1nn sselii 2nn 1 2 znq mp2an halfnz pm3.2i ZZ QQ 1 / 2 opr ssnelpss mp2 pm3.2i qssre 2re 2pos sqrlem24 sqr2irr sqr 2 ` QQ df-nel mpbi pm3.2i QQ RR sqr 2 ` ssnelpss mp2 axresscn axicn inelr pm3.2i RR CC i ssnelpss mp2 pm3.2i pm3.2i))
+thm (nthruz () () (/\ (C: (NN) (NN0)) (C: (NN0) (ZZ))) (nnssnn0 0nn0 0nnn pm3.2i NN NN0 0 ssnelpss mp2 nn0ssz 1nn 1 nnnegz ax-mp ax1re renegcl neg0 lt01 eqbrtr ax1re 0re ltnegcon1 mpbir 1 -u lt0nnn0 mp2an pm3.2i NN0 ZZ 1 -u ssnelpss mp2 pm3.2i))
+thm (revalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Re) A) (U. ({e.|} x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))))) (z cv A x cv + y cv x. i opr opr eqeq1 y RR birexdv x RR birabsdv unieqd z w x y df-re axcnex axresscn ssexi x y RR A x cv + y cv x. i opr opr = E.e. rabex uniex fvopab4))
+thm (imvalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Im) A) (U. ({e.|} y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i))))))))) (z cv A x cv + y cv x. i opr opr eqeq1 x RR birexdv y RR birabsdv unieqd z w y x df-im axcnex axresscn ssexi y x RR A x cv + y cv x. i opr opr = E.e. rabex uniex fvopab4))
+thm (reclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Re) A) (RR))) (A x y revalt A x y creur x RR y RR A x cv + y cv x. i opr opr = E.e. reucl syl eqeltrd))
+thm (imclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Im) A) (RR))) (A y x imvalt A y x creui y RR x RR A x cv + y cv x. i opr opr = E.e. reucl syl eqeltrd))
+thm (replimt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i))))) (Im A ` eqid A imclt y cv Im A ` x. i opreq1 x cv + opreq2d A eqeq2d x RR birexdv y cv Im A ` Im A ` eqeq2 bibi12d A CC e. imbi2d y RR x RR A x cv + y cv x. i opr opr = E.e. reuuni1 A y x creui sylan2 A y x imvalt y cv eqeq1d y cv RR e. adantl bitr4d exp vtoclga mpcom mpbiri x RR A x cv + Im A ` x. i opr opr = df-rex sylib Re A ` eqid A reclt x cv Re A ` + y cv x. i opr opreq1 A eqeq2d y RR birexdv x cv Re A ` Re A ` eqeq2 bibi12d A CC e. imbi2d x RR y RR A x cv + y cv x. i opr opr = E.e. reuuni1 A x y creur sylan2 A x y revalt x cv eqeq1d x cv RR e. adantl bitr4d exp vtoclga mpcom mpbiri y RR A Re A ` + y cv x. i opr opr = df-rex sylib jca x cv RR e. y cv RR e. A x cv + Im A ` x. i opr opr = A Re A ` + y cv x. i opr opr = an4 x y bi2ex x y x cv RR e. A x cv + Im A ` x. i opr opr = /\ y cv RR e. A Re A ` + y cv x. i opr opr = /\ eeanv bitr sylibr A x cv + Im A ` x. i opr opr = A Re A ` + y cv x. i opr opr = pm3.26 A CC e. x cv RR e. y cv RR e. /\ ad2antrr A imclt x cv RR e. a1d ancld A reclt y cv RR e. a1d ancrd anim12d imp x cv Im A ` Re A ` y cv crut syl A x cv + Im A ` x. i opr opr Re A ` + y cv x. i opr opr eqtr2t syl5bi exp imp32 pm3.26d + Im A ` x. i opr opreq1d eqtrd exp x y 19.23advv mpd))
+thm (cjvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (*) A) (opr (` (Re) A) (-) (opr (` (Im) A) (x.) (i))))) (x cv A Re fveq2 x cv A Im fveq2 x. i opreq1d - opreq12d x y df-cj Re A ` - Im A ` x. i opr oprex fvopab4))
+thm (absvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (abs) A) (` (sqr) (opr A (x.) (` (*) A))))) (x cv A = id x cv A * fveq2 x. opreq12d sqr fveq2d x y df-abs sqr A x. * A ` opr fvex fvopab4))
+thm (cjclt () () (-> (e. A (CC)) (e. (` (*) A) (CC))) (A cjvalt Re A ` Im A ` x. i opr subclt A reclt recnd Im A ` i axmulcl A imclt recnd axicn A CC e. a1i sylanc sylanc eqeltrd))
+thm (recl () ((recl.1 (e. A (CC)))) (e. (` (Re) A) (RR)) (recl.1 A reclt ax-mp))
+thm (imcl () ((recl.1 (e. A (CC)))) (e. (` (Im) A) (RR)) (recl.1 A imclt ax-mp))
+thm (cjcl () ((recl.1 (e. A (CC)))) (e. (` (*) A) (CC)) (recl.1 A cjclt ax-mp))
+thm (replim () ((recl.1 (e. A (CC)))) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i)))) (recl.1 A replimt ax-mp))
+thm (crret () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A)) (A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr replimt syl eqcomd A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr reclt syl A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr imclt syl jca Re A + B x. i opr opr ` Im A + B x. i opr opr ` A B crut mpancom mpbid pm3.26d))
+thm (crimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B)) (A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr replimt syl eqcomd A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr reclt syl A B x. i opr axaddcl A recnt B recnt axicn B i axmulcl mpan2 syl syl2an A + B x. i opr opr imclt syl jca Re A + B x. i opr opr ` Im A + B x. i opr opr ` A B crut mpancom mpbid pm3.27d))
+thm (crre () ((crre.1 (e. A (RR))) (crre.2 (e. B (RR)))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A) (crre.1 crre.2 A B crret mp2an))
+thm (crim () ((crre.1 (e. A (RR))) (crre.2 (e. B (RR)))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B) (crre.1 crre.2 A B crimt mp2an))
+thm (imret () () (-> (e. A (CC)) (= (` (Im) A) (` (Re) (opr (-u (i)) (x.) A)))) (A replimt i -u x. opreq2d axicn negcl i -u Re A ` Im A ` x. i opr axdistr mp3an1 A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl sylanc A reclt recnd axicn negcl i -u Re A ` axmulcom mpan syl A reclt recnd axicn Re A ` i mulneg12t mpan2 syl eqtr4d A imclt recnd axicn negcl axicn i -u Im A ` i mul12t mp3an13 syl A imclt recnd Im A ` ax1id syl axicn axicn mulneg1 itimesi negeqi 1cn negneg 3eqtr Im A ` x. opreq2i syl5eq eqtrd + opreq12d Re A ` -u x. i opr Im A ` axaddcom A reclt Re A ` renegclt syl recnd axicn Re A ` -u i axmulcl mpan2 syl A imclt recnd sylanc eqtrd 3eqtrd Re fveq2d Im A ` Re A ` -u crret A imclt A reclt Re A ` renegclt syl sylanc eqtr2d))
+thm (reim0t () () (-> (e. A (RR)) (= (` (Im) A) (0))) (Im A ` rimul A recnt A imclt syl A recnt A replimt syl eqcomd A Re A ` Im A ` x. i opr subaddt A recnt A recnt A reclt syl Re A ` recnt syl A recnt A imclt syl Im A ` recnt syl axicn Im A ` i axmulcl mpan2 syl syl3anc mpbird A recnt A reclt syl A Re A ` resubclt mpdan eqeltrrd sylanc))
+thm (reim0bt () () (-> (e. A (CC)) (<-> (e. A (RR)) (= (` (Im) A) (0)))) (A reim0t A CC e. a1i A replimt Im A ` 0 = adantr Im A ` 0 x. i opreq1 axicn mul02 syl6eq Re A ` + opreq2d A reclt recnd Re A ` ax0id syl sylan9eqr eqtrd A reclt Im A ` 0 = adantr eqeltrd exp impbid))
+thm (cjcj () ((cjcj.1 (e. A (CC)))) (= (` (*) (` (*) A)) A) (cjcj.1 cjcl * A ` cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg cjcj.1 A cjvalt ax-mp cjcj.1 imcl recn axicn mulneg1 Re A ` + opreq2i 3eqtr4 Re fveq2i cjcj.1 recl cjcj.1 imcl renegcl crre eqtr cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg cjcj.1 A cjvalt ax-mp cjcj.1 imcl recn axicn mulneg1 Re A ` + opreq2i 3eqtr4 Im fveq2i x. i opreq1i cjcj.1 recl cjcj.1 imcl renegcl crim x. i opreq1i cjcj.1 imcl recn axicn mulneg1 3eqtr - opreq12i cjcj.1 imcl recn axicn mulcl negneg Re A ` + opreq2i cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negcl subneg cjcj.1 replim 3eqtr4 3eqtr))
+thm (reim0b () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (Im) A) (0))) (cjcj.1 A reim0bt ax-mp))
+thm (rereb () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (Re) A) A)) (cjcj.1 reim0b Im A ` 0 x. i opreq1 sylbi axicn mul02 syl6eq Re A ` + opreq2d cjcj.1 replim syl5req cjcj.1 recl recn addid1 syl5eqr cjcj.1 recl Re A ` A RR eleq1 mpbii impbi))
+thm (cjreb () ((cjcj.1 (e. A (CC)))) (<-> (e. A (RR)) (= (` (*) A) A)) (cjcj.1 reim0b cjcj.1 imcl recn eqneg axicn cjcj.1 imcl recn mulcom axicn cjcj.1 imcl recn negcl mulcom cjcj.1 imcl recn axicn mulneg1 eqtr eqeq12i axicn cjcj.1 imcl recn cjcj.1 imcl recn negcl ine0 mulcan cjcj.1 replim cjcj.1 A cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg eqtr eqeq12i cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl cjcj.1 imcl recn axicn mulcl negcl addcan bitr2 3bitr3 bitr3 A * A ` eqcom 3bitr))
+thm (recj () ((cjcj.1 (e. A (CC)))) (= (` (Re) (` (*) A)) (` (Re) A)) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg cjcj.1 A cjvalt ax-mp cjcj.1 imcl recn axicn mulneg1 Re A ` + opreq2i 3eqtr4 Re fveq2i cjcj.1 recl cjcj.1 imcl renegcl crre eqtr))
+thm (imcj () ((cjcj.1 (e. A (CC)))) (= (` (Im) (` (*) A)) (-u (` (Im) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg cjcj.1 A cjvalt ax-mp cjcj.1 imcl recn axicn mulneg1 Re A ` + opreq2i 3eqtr4 Im fveq2i cjcj.1 recl cjcj.1 imcl renegcl crim eqtr))
+thm (readd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (+) B)) (opr (` (Re) A) (+) (` (Re) B))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl readd.2 recl recn readd.2 imcl recn axicn mulcl add4 cjcj.1 replim readd.2 replim + opreq12i cjcj.1 imcl recn readd.2 imcl recn axicn adddir Re A ` + Re B ` opr + opreq2i 3eqtr4 Re fveq2i cjcj.1 recl readd.2 recl readdcl cjcj.1 imcl readd.2 imcl readdcl crre eqtr))
+thm (imadd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Im) (opr A (+) B)) (opr (` (Im) A) (+) (` (Im) B))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl readd.2 recl recn readd.2 imcl recn axicn mulcl add4 cjcj.1 replim readd.2 replim + opreq12i cjcj.1 imcl recn readd.2 imcl recn axicn adddir Re A ` + Re B ` opr + opreq2i 3eqtr4 Im fveq2i cjcj.1 recl readd.2 recl readdcl cjcj.1 imcl readd.2 imcl readdcl crim eqtr))
+thm (remul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (x.) B)) (opr (opr (` (Re) A) (x.) (` (Re) B)) (-) (opr (` (Im) A) (x.) (` (Im) B)))) (cjcj.1 replim readd.2 replim x. opreq12i cjcj.1 recl recn cjcj.1 imcl recn readd.2 recl recn readd.2 imcl recn crmul eqtr Re fveq2i cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl crre eqtr))
+thm (immul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Im) (opr A (x.) B)) (opr (opr (` (Re) A) (x.) (` (Im) B)) (+) (opr (` (Im) A) (x.) (` (Re) B)))) (cjcj.1 replim readd.2 replim x. opreq12i cjcj.1 recl recn cjcj.1 imcl recn readd.2 recl recn readd.2 imcl recn crmul eqtr Im fveq2i cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl crim eqtr))
+thm (cjadd () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (*) (opr A (+) B)) (opr (` (*) A) (+) (` (*) B))) (cjcj.1 readd.2 readd cjcj.1 readd.2 imadd x. i opreq1i cjcj.1 imcl recn readd.2 imcl recn axicn adddir eqtr - opreq12i cjcj.1 recl recn readd.2 recl recn cjcj.1 imcl recn axicn mulcl readd.2 imcl recn axicn mulcl sub4 eqtr cjcj.1 readd.2 addcl A + B opr cjvalt ax-mp cjcj.1 A cjvalt ax-mp readd.2 B cjvalt ax-mp + opreq12i 3eqtr4))
+thm (cjmul () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (*) (opr A (x.) B)) (opr (` (*) A) (x.) (` (*) B))) (cjcj.1 recl readd.2 recl remulcl cjcj.1 imcl readd.2 imcl remulcl resubcl recn cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl recn axicn mulcl subneg cjcj.1 imcl recn readd.2 imcl recn mul2neg eqcomi Re A ` x. Re B ` opr - opreq2i cjcj.1 recl readd.2 imcl remulcl cjcj.1 imcl readd.2 recl remulcl readdcl recn axicn mulneg1 cjcj.1 recl readd.2 imcl remulcl recn cjcj.1 imcl readd.2 recl remulcl recn negdi cjcj.1 recl recn readd.2 imcl recn mulneg2 cjcj.1 imcl recn readd.2 recl recn mulneg1 + opreq12i eqtr4 x. i opreq1i eqtr3 + opreq12i eqtr cjcj.1 readd.2 remul cjcj.1 readd.2 immul x. i opreq1i - opreq12i cjcj.1 recl recn cjcj.1 imcl recn negcl readd.2 recl recn readd.2 imcl recn negcl crmul 3eqtr4 cjcj.1 readd.2 mulcl A x. B opr cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg cjcj.1 A cjvalt ax-mp cjcj.1 imcl recn axicn mulneg1 Re A ` + opreq2i 3eqtr4 readd.2 recl recn readd.2 imcl recn axicn mulcl subneg readd.2 B cjvalt ax-mp readd.2 imcl recn axicn mulneg1 Re B ` + opreq2i 3eqtr4 x. opreq12i 3eqtr4))
+thm (ipcn () ((cjcj.1 (e. A (CC))) (readd.2 (e. B (CC)))) (= (` (Re) (opr A (x.) (` (*) B))) (opr (opr (` (Re) A) (x.) (` (Re) B)) (+) (opr (` (Im) A) (x.) (` (Im) B)))) (cjcj.1 readd.2 cjcl remul cjcj.1 recl readd.2 cjcl recl remulcl recn cjcj.1 imcl readd.2 cjcl imcl remulcl recn subneg readd.2 recj Re A ` x. opreq2i readd.2 imcj readd.2 cjcl imcl recn readd.2 imcl recn negcon2 mpbi Im A ` x. opreq2i cjcj.1 imcl recn readd.2 cjcl imcl recn mulneg2 eqtr2 + opreq12i 3eqtr))
+thm (cjmulrcl () ((cjcj.1 (e. A (CC)))) (e. (opr A (x.) (` (*) A)) (RR)) (cjcj.1 cjcj * A ` x. opreq2i cjcj.1 cjcj.1 cjcl cjmul cjcj.1 cjcj.1 cjcl mulcom 3eqtr4 cjcj.1 cjcj.1 cjcl mulcl cjreb mpbir))
+thm (cjmulval () ((cjcj.1 (e. A (CC)))) (= (opr A (x.) (` (*) A)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))) (cjcj.1 cjmulrcl cjcj.1 cjmulrcl recn rereb mpbi cjcj.1 cjcj.1 ipcn cjcj.1 recl recn sqval cjcj.1 imcl recn sqval + opreq12i eqtr4 eqtr3))
+thm (cjmulge0 () ((cjcj.1 (e. A (CC)))) (br (0) (<_) (opr A (x.) (` (*) A))) (cjcj.1 recl sqege0 cjcj.1 imcl sqege0 cjcj.1 recl sqrecl cjcj.1 imcl sqrecl addge0 mp2an cjcj.1 cjmulval breqtrr))
+thm (reneg () ((cjcj.1 (e. A (CC)))) (= (` (Re) (-u A)) (-u (` (Re) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negdi cjcj.1 replim negeqi cjcj.1 imcl recn axicn mulneg1 Re A ` -u + opreq2i 3eqtr4 Re fveq2i cjcj.1 recl renegcl cjcj.1 imcl renegcl crre eqtr))
+thm (negreb () ((cjcj.1 (e. A (CC)))) (<-> (e. (-u A) (RR)) (e. A (RR))) (cjcj.1 negcl rereb cjcj.1 reneg A -u eqeq1i cjcj.1 recl recn cjcj.1 neg11 3bitr cjcj.1 recl Re A ` A RR eleq1 mpbii sylbi A renegclt impbi))
+thm (imneg () ((cjcj.1 (e. A (CC)))) (= (` (Im) (-u A)) (-u (` (Im) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negdi cjcj.1 replim negeqi cjcj.1 imcl recn axicn mulneg1 Re A ` -u + opreq2i 3eqtr4 Im fveq2i cjcj.1 recl renegcl cjcj.1 imcl renegcl crim eqtr))
+thm (cjneg () ((cjcj.1 (e. A (CC)))) (= (` (*) (-u A)) (-u (` (*) A))) (cjcj.1 recl renegcl recn cjcj.1 imcl recn axicn mulcl negcl subneg cjcj.1 reneg cjcj.1 imneg x. i opreq1i cjcj.1 imcl recn axicn mulneg1 eqtr - opreq12i cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negcl negdi 3eqtr4 cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg negeqi eqtr4 cjcj.1 negcl A -u cjvalt ax-mp cjcj.1 A cjvalt ax-mp negeqi 3eqtr4))
+thm (addcj () ((cjcj.1 (e. A (CC)))) (= (opr A (+) (` (*) A)) (opr (2) (x.) (` (Re) A))) (cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl negcl add4 cjcj.1 imcl recn axicn mulcl negid Re A ` + Re A ` opr + opreq2i cjcj.1 recl recn cjcj.1 recl recn addcl addid1 3eqtr cjcj.1 replim cjcj.1 A cjvalt ax-mp cjcj.1 recl recn cjcj.1 imcl recn axicn mulcl subneg eqtr + opreq12i cjcj.1 recl recn 2times 3eqtr4))
+thm (cjrebt () () (-> (e. A (CC)) (<-> (e. A (RR)) (= (` (*) A) A))) (A A CC e. A 0 if RR eleq1 A A CC e. A 0 if * fveq2 A A CC e. A 0 if = id eqeq12d bibi12d 0cn A elimel cjreb dedth))
+thm (cjmulrclt () () (-> (e. A (CC)) (e. (opr A (x.) (` (*) A)) (RR))) (A A CC e. A 0 if = id A A CC e. A 0 if * fveq2 x. opreq12d RR eleq1d 0cn A elimel cjmulrcl dedth))
+thm (cjmulge0t () () (-> (e. A (CC)) (br (0) (<_) (opr A (x.) (` (*) A)))) (A A CC e. A 0 if = id A A CC e. A 0 if * fveq2 x. opreq12d 0 <_ breq2d 0cn A elimel cjmulge0 dedth))
+thm (renegt () () (-> (e. A (CC)) (= (` (Re) (-u A)) (-u (` (Re) A)))) (A A CC e. A 0 if negeq Re fveq2d A A CC e. A 0 if Re fveq2 negeqd eqeq12d 0cn A elimel reneg dedth))
+thm (readdt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (+) B)) (opr (` (Re) A) (+) (` (Re) B)))) (A A CC e. A 0 if + B opreq1 Re fveq2d A A CC e. A 0 if Re fveq2 + Re B ` opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if + opreq2 Re fveq2d B B CC e. B 0 if Re fveq2 Re A CC e. A 0 if ` + opreq2d eqeq12d 0cn A elimel 0cn B elimel readd dedth2h))
+thm (resubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (-) B)) (opr (` (Re) A) (-) (` (Re) B)))) (A B -u readdt B negclt sylan2 B renegt A CC e. adantl Re A ` + opreq2d eqtrd A B subnegt Re fveq2d Re A ` Re B ` subnegt A reclt recnd B reclt recnd syl2an 3eqtr4d))
+thm (imnegt () () (-> (e. A (CC)) (= (` (Im) (-u A)) (-u (` (Im) A)))) (A A CC e. A 0 if negeq Im fveq2d A A CC e. A 0 if Im fveq2 negeqd eqeq12d 0cn A elimel imneg dedth))
+thm (imaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (+) B)) (opr (` (Im) A) (+) (` (Im) B)))) (A A CC e. A 0 if + B opreq1 Im fveq2d A A CC e. A 0 if Im fveq2 + Im B ` opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if + opreq2 Im fveq2d B B CC e. B 0 if Im fveq2 Im A CC e. A 0 if ` + opreq2d eqeq12d 0cn A elimel 0cn B elimel imadd dedth2h))
+thm (imsubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (-) B)) (opr (` (Im) A) (-) (` (Im) B)))) (A B -u imaddt B negclt sylan2 B imnegt A CC e. adantl Im A ` + opreq2d eqtrd A B subnegt Im fveq2d Im A ` Im B ` subnegt A imclt recnd B imclt recnd syl2an 3eqtr4d))
+thm (cjret () () (-> (e. A (RR)) (= (` (*) A) A)) (A recnt A cjrebt biimpd mpcom))
+thm (cjcjt () () (-> (e. A (CC)) (= (` (*) (` (*) A)) A)) (A A CC e. A 0 if * fveq2 * fveq2d A A CC e. A 0 if = id eqeq12d 0cn A elimel cjcj dedth))
+thm (cjaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (+) B)) (opr (` (*) A) (+) (` (*) B)))) (A A CC e. A 0 if + B opreq1 * fveq2d A A CC e. A 0 if * fveq2 + * B ` opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if + opreq2 * fveq2d B B CC e. B 0 if * fveq2 * A CC e. A 0 if ` + opreq2d eqeq12d 0cn A elimel 0cn B elimel cjadd dedth2h))
+thm (cjmult () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (x.) B)) (opr (` (*) A) (x.) (` (*) B)))) (A A CC e. A 0 if x. B opreq1 * fveq2d A A CC e. A 0 if * fveq2 x. * B ` opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if x. opreq2 * fveq2d B B CC e. B 0 if * fveq2 * A CC e. A 0 if ` x. opreq2d eqeq12d 0cn A elimel 0cn B elimel cjmul dedth2h))
+thm (cjnegt () () (-> (e. A (CC)) (= (` (*) (-u A)) (-u (` (*) A)))) (A A CC e. A 0 if negeq * fveq2d A A CC e. A 0 if * fveq2 negeqd eqeq12d 0cn A elimel cjneg dedth))
+thm (cjsubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (*) (opr A (-) B)) (opr (` (*) A) (-) (` (*) B)))) (A B subnegt * fveq2d A B -u cjaddt B negclt sylan2 B cjnegt A CC e. adantl * A ` + opreq2d * A ` * B ` subnegt A cjclt B cjclt syl2an eqtr4d 3eqtrd))
+thm (cjexpt ((j k) (A j) (A k) (N j) (N k)) () (-> (/\ (e. A (CC)) (e. N (NN))) (= (` (*) (opr A (^) N)) (opr (` (*) A) (^) N))) (j cv 1 A ^ opreq2 * fveq2d j cv 1 * A ` ^ opreq2 eqeq12d A CC e. imbi2d j cv k cv A ^ opreq2 * fveq2d j cv k cv * A ` ^ opreq2 eqeq12d A CC e. imbi2d j cv k cv + 1 opr A ^ opreq2 * fveq2d j cv k cv + 1 opr * A ` ^ opreq2 eqeq12d A CC e. imbi2d j cv N A ^ opreq2 * fveq2d j cv N * A ` ^ opreq2 eqeq12d A CC e. imbi2d A exp1t * fveq2d A cjclt * A ` exp1t syl eqtr4d A k cv expp1t * fveq2d A ^ k cv opr A cjmult A k cv expclt A CC e. k cv NN e. pm3.26 sylanc eqtrd * A ^ k cv opr ` * A ` ^ k cv opr = adantr * A ^ k cv opr ` * A ` ^ k cv opr x. * A ` opreq1 * A ` k cv expp1t A cjclt sylan eqcomd sylan9eqr eqtrd exp31 com12 a2d nnind impcom))
+thm (recjt () () (-> (e. A (CC)) (= (` (Re) A) (opr (opr A (+) (` (*) A)) (/) (2)))) (2cn 2re 2pos gt0ne0i Re A ` Re A ` 2 divdirt mpan2 mp3an3 A reclt recnd A reclt recnd sylanc A reclt recnd Re A ` 2halvest syl eqtr2d A replimt A cjvalt + opreq12d Re A ` Im A ` x. i opr subnegt A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl sylanc Re A ` + Im A ` x. i opr opr + opreq2d A imclt recnd axicn Im A ` i axmulcl mpan2 Im A ` x. i opr negidt 3syl Re A ` + Re A ` opr + opreq2d Re A ` Re A ` Im A ` x. i opr Im A ` x. i opr -u add4t A reclt recnd A reclt recnd jca A imclt recnd axicn Im A ` i axmulcl mpan2 syl A imclt recnd axicn Im A ` i axmulcl mpan2 Im A ` x. i opr negclt 3syl jca sylanc A reclt recnd A reclt recnd jca Re A ` Re A ` axaddcl Re A ` + Re A ` opr ax0id 3syl 3eqtr3d 3eqtrd / 2 opreq1d eqtr4d))
+thm (imcjt () () (-> (e. A (CC)) (= (` (Im) A) (opr (opr A (-) (` (*) A)) (/) (opr (2) (x.) (i))))) (A imclt recnd 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 2 x. i opr Im A ` divcan4t mp3an13 syl Re A ` + Im A ` x. i opr opr Re A ` Im A ` x. i opr subsubt Re A ` Im A ` x. i opr axaddcl A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl sylanc A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl syl3anc A replimt A cjvalt - opreq12d A imclt recnd axicn Im A ` i axmulcl mpan2 Im A ` x. i opr 2timest 3syl A imclt recnd 2cn axicn Im A ` 2 i mul12t mp3an23 syl Re A ` Im A ` x. i opr pncan2t A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl sylanc + Im A ` x. i opr opreq1d 3eqtr4d 3eqtr4rd / 2 x. i opr opreq1d eqtr3d))
+thm (re0 () () (= (` (Re) (0)) (0)) (0re 0cn rereb mpbi))
+thm (im0 () () (= (` (Im) (0)) (0)) (0re 0cn reim0b mpbi))
+thm (cj0 () () (= (` (*) (0)) (0)) (0re 0cn cjreb mpbi))
+thm (cji () () (= (` (*) (i)) (-u (i))) (1cn axicn mulcl addid2 axicn mulid2 eqtr Re fveq2i 0re ax1re crre eqtr3 1cn axicn mulcl addid2 axicn mulid2 eqtr Im fveq2i 0re ax1re crim eqtr3 x. i opreq1i axicn mulid2 eqtr - opreq12i axicn i cjvalt ax-mp i df-neg 3eqtr4))
+thm (cjreimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (*) (opr A (+) (opr (i) (x.) B))) (opr A (-) (opr (i) (x.) B)))) (A cjret B recnt axicn i B cjmult mpan syl cji B RR e. a1i B cjret x. opreq12d B recnt axicn i B mulneg1t mpan syl 3eqtrd + opreqan12d A i x. B opr cjaddt A recnt B recnt axicn i B axmulcl mpan syl syl2an A i x. B opr subnegt A recnt B recnt axicn i B axmulcl mpan syl syl2an 3eqtr4d))
+thm (cjreim2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (*) (opr A (-) (opr (i) (x.) B))) (opr A (+) (opr (i) (x.) B)))) (A i x. B opr cjsubt A recnt B recnt axicn i B axmulcl mpan syl syl2an A cjret B recnt axicn i B cjmult mpan syl cji B RR e. a1i B cjret x. opreq12d B recnt axicn i B mulneg1t mpan syl 3eqtrd - opreqan12d A i x. B opr subneg2t A recnt B recnt axicn i B axmulcl mpan syl syl2an 3eqtrd))
+thm (cj11t () () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (= (` (*) A) (` (*) B)) (= A B))) (Im A ` Im B ` neg11t A imclt recnd B imclt recnd syl2an Re A ` Re B ` = anbi2d Re A ` Im A ` -u Re B ` Im B ` -u crut A reclt A imclt Im A ` renegclt syl jca B reclt B imclt Im B ` renegclt syl jca syl2an Re A ` Im A ` Re B ` Im B ` crut A reclt A imclt jca B reclt B imclt jca syl2an 3bitr4d Re A ` Im A ` x. i opr subnegt A reclt recnd A imclt recnd axicn Im A ` i axmulcl mpan2 syl sylanc A cjvalt A imclt recnd axicn jctir Im A ` i mulneg1t syl Re A ` + opreq2d 3eqtr4d Re B ` Im B ` x. i opr subnegt B reclt recnd B imclt recnd axicn Im B ` i axmulcl mpan2 syl sylanc B cjvalt B imclt recnd axicn jctir Im B ` i mulneg1t syl Re B ` + opreq2d 3eqtr4d eqeqan12d A replimt B replimt eqeqan12d 3bitr4d))
+thm (cjne0t () () (-> (e. A (CC)) (<-> (=/= A (0)) (=/= (` (*) A) (0)))) (0cn A 0 cj11t mpan2 cj0 * A ` eqeq2i syl5rbbr eqneqd))
+thm (absnegt () () (-> (e. A (CC)) (= (` (abs) (-u A)) (` (abs) A))) (A cjnegt A -u x. opreq2d A cjclt A * A ` mul2negt mpdan eqtrd sqr fveq2d A negclt A -u absvalt syl A absvalt 3eqtr4d))
+thm (absclt () () (-> (e. A (CC)) (e. (` (abs) A) (RR))) (A absvalt A x. * A ` opr sqrclt A cjmulrclt A cjmulge0t sylanc eqeltrd))
+thm (abscjt () () (-> (e. A (CC)) (= (` (abs) (` (*) A)) (` (abs) A))) (A cjclt A * A ` axmulcom mpdan A cjcjt * A ` x. opreq2d eqtr4d sqr fveq2d A absvalt A cjclt * A ` absvalt syl 3eqtr4rd))
+thm (absval2 () ((absval2.1 (e. A (CC)))) (= (` (abs) A) (` (sqr) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2))))) (absval2.1 A absvalt ax-mp absval2.1 cjmulval sqr fveq2i eqtr))
+thm (absvalsq () ((absval2.1 (e. A (CC)))) (= (opr (` (abs) A) (^) (2)) (opr A (x.) (` (*) A))) (absval2.1 A absvalt ax-mp ^ 2 opreq1i absval2.1 cjmulge0 absval2.1 cjmulrcl sqsqr ax-mp eqtr))
+thm (absvalsq2 () ((absval2.1 (e. A (CC)))) (= (opr (` (abs) A) (^) (2)) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))) (absval2.1 absvalsq absval2.1 cjmulval eqtr))
+thm (abs00 () ((absval2.1 (e. A (CC)))) (<-> (= (` (abs) A) (0)) (= A (0))) (absval2.1 absval2 0 eqeq1i absval2.1 recl sqrecl absval2.1 imcl sqrecl readdcl absval2.1 recl sqege0 absval2.1 imcl sqege0 absval2.1 recl sqrecl absval2.1 imcl sqrecl addge0 mp2an Re A ` ^ 2 opr + Im A ` ^ 2 opr opr sqr00t mp2an absval2.1 recl sqege0 absval2.1 imcl sqege0 absval2.1 recl sqrecl absval2.1 imcl sqrecl add20 mp2an absval2.1 recl recn sqe0 absval2.1 imcl recn sqe0 anbi12i Re A ` 0 + Im A ` x. i opr opreq1 Im A ` 0 x. i opreq1 0 + opreq2d sylan9eq 0cn axicn mulcl addid2 axicn mul02 eqtr syl6eq absval2.1 replim syl5eq A 0 Re fveq2 re0 syl6eq A 0 Im fveq2 im0 syl6eq jca impbi 3bitr 3bitr))
+thm (abscl () ((absval2.1 (e. A (CC)))) (e. (` (abs) A) (RR)) (absval2.1 A absclt ax-mp))
+thm (absge0 () ((absval2.1 (e. A (CC)))) (br (0) (<_) (` (abs) A)) (absval2.1 cjmulge0 absval2.1 cjmulrcl sqrge0 ax-mp absval2.1 A absvalt ax-mp breqtrr))
+thm (absgt0 () ((absval2.1 (e. A (CC)))) (<-> (=/= A (0)) (br (0) (<) (` (abs) A))) (A 0 df-ne absval2.1 abs00 abs A ` 0 eqcom bitr3 negbii absval2.1 absge0 0re absval2.1 abscl leloe mpbi 0 < abs A ` br 0 abs A ` = orcom mpbi ori 0re absval2.1 abscl ltne impbi 3bitr))
+thm (absneg () ((absval2.1 (e. A (CC)))) (= (` (abs) (-u A)) (` (abs) A)) (absval2.1 A absnegt ax-mp))
+thm (abscj () ((absval2.1 (e. A (CC)))) (= (` (abs) (` (*) A)) (` (abs) A)) (absval2.1 A abscjt ax-mp))
+thm (abssub () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (` (abs) (opr A (-) B)) (` (abs) (opr B (-) A))) (abssub.2 absval2.1 negdi3 abs fveq2i abssub.2 absval2.1 subcl absneg eqtr3))
+thm (absmul () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (` (abs) (opr A (x.) B)) (opr (` (abs) A) (x.) (` (abs) B))) (absval2.1 abssub.2 cjmul A x. B opr x. opreq2i absval2.1 abssub.2 absval2.1 cjcl abssub.2 cjcl mul4 eqtr sqr fveq2i absval2.1 cjmulrcl abssub.2 cjmulrcl absval2.1 cjmulge0 abssub.2 cjmulge0 sqrmuli eqtr absval2.1 abssub.2 mulcl A x. B opr absvalt ax-mp absval2.1 A absvalt ax-mp abssub.2 B absvalt ax-mp x. opreq12i 3eqtr4))
+thm (sqabsadd () ((absval2.1 (e. A (CC))) (abssub.2 (e. B (CC)))) (= (opr (` (abs) (opr A (+) B)) (^) (2)) (opr (opr (opr (` (abs) A) (^) (2)) (+) (opr (` (abs) B) (^) (2))) (+) (opr (2) (x.) (` (Re) (opr A (x.) (` (*) B)))))) (absval2.1 abssub.2 cjadd A + B opr x. opreq2i absval2.1 abssub.2 absval2.1 cjcl abssub.2 cjcl muladd eqtr absval2.1 abssub.2 addcl absvalsq absval2.1 absvalsq abssub.2 absvalsq abssub.2 abssub.2 cjcl mulcom eqtr + opreq12i absval2.1 abssub.2 cjcl mulcl addcj absval2.1 abssub.2 cjcl cjmul abssub.2 cjcj * A ` x. opreq2i eqtr A x. * B ` opr + opreq2i eqtr3 + opreq12i 3eqtr4))
+thm (absval2t () () (-> (e. A (CC)) (= (` (abs) A) (` (sqr) (opr (opr (` (Re) A) (^) (2)) (+) (opr (` (Im) A) (^) (2)))))) (A A CC e. A 0 if abs fveq2 A A CC e. A 0 if Re fveq2 ^ 2 opreq1d A A CC e. A 0 if Im fveq2 ^ 2 opreq1d + opreq12d sqr fveq2d eqeq12d 0cn A elimel absval2 dedth))
+thm (absge0t () () (-> (e. A (CC)) (br (0) (<_) (` (abs) A))) (A A CC e. A 0 if abs fveq2 0 <_ breq2d 0cn A elimel absge0 dedth))
+thm (absmult () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (abs) (opr A (x.) B)) (opr (` (abs) A) (x.) (` (abs) B)))) (A A CC e. A 0 if x. B opreq1 abs fveq2d A A CC e. A 0 if abs fveq2 x. abs B ` opreq1d eqeq12d B B CC e. B 0 if A CC e. A 0 if x. opreq2 abs fveq2d B B CC e. B 0 if abs fveq2 abs A CC e. A 0 if ` x. opreq2d eqeq12d 0cn A elimel 0cn B elimel absmul dedth2h))
+thm (absdivz () ((absdivz.1 (e. A (CC))) (absdivz.2 (e. B (CC)))) (-> (=/= B (0)) (= (` (abs) (opr A (/) B)) (opr (` (abs) A) (/) (` (abs) B)))) (absdivz.1 absdivz.2 divclz absdivz.2 B A / B opr absmult mpan syl absdivz.2 absdivz.1 divcan2z abs fveq2d eqtr3d abs A ` abs B ` abs A / B opr ` divmult absdivz.1 abscl recn B 0 =/= a1i absdivz.2 abscl recn B 0 =/= a1i absdivz.1 absdivz.2 divclz A / B opr absclt abs A / B opr ` recnt 3syl 3jca absdivz.2 abs00 eqneqi biimpr sylanc mpbird eqcomd))
+thm (absdivt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (=/= B (0))) (= (` (abs) (opr A (/) B)) (opr (` (abs) A) (/) (` (abs) B)))) (A A CC e. A 0 if / B opreq1 abs fveq2d A A CC e. A 0 if abs fveq2 / abs B ` opreq1d eqeq12d B 0 =/= imbi2d B B CC e. B 0 if 0 neeq1 B B CC e. B 0 if A CC e. A 0 if / opreq2 abs fveq2d B B CC e. B 0 if abs fveq2 abs A CC e. A 0 if ` / opreq2d eqeq12d imbi12d 0cn A elimel 0cn B elimel absdivz dedth2h exp 3imp))
+thm (absexpt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (NN))) (= (` (abs) (opr A (^) B)) (opr (` (abs) A) (^) B))) (x cv 1 A ^ opreq2 abs fveq2d x cv 1 abs A ` ^ opreq2 eqeq12d A CC e. imbi2d x cv y cv A ^ opreq2 abs fveq2d x cv y cv abs A ` ^ opreq2 eqeq12d A CC e. imbi2d x cv y cv + 1 opr A ^ opreq2 abs fveq2d x cv y cv + 1 opr abs A ` ^ opreq2 eqeq12d A CC e. imbi2d x cv B A ^ opreq2 abs fveq2d x cv B abs A ` ^ opreq2 eqeq12d A CC e. imbi2d A exp1t abs fveq2d A absclt abs A ` recnt abs A ` exp1t 3syl eqtr4d abs A ^ y cv opr ` abs A ` ^ y cv opr x. abs A ` opreq1 A CC e. y cv NN e. /\ adantl A y cv expp1t abs fveq2d A ^ y cv opr A absmult A y cv expclt A CC e. y cv NN e. pm3.26 sylanc eqtrd abs A ^ y cv opr ` abs A ` ^ y cv opr = adantr abs A ` y cv expp1t A absclt recnd sylan abs A ^ y cv opr ` abs A ` ^ y cv opr = adantr 3eqtr4d exp31 com12 a2d nnind impcom))
+thm (absid () ((absid.1 (e. A (RR)))) (-> (br (0) (<_) A) (= (` (abs) A) A)) (absid.1 sqrsqid absid.1 recn A absvalt ax-mp absid.1 absid.1 recn cjreb mpbi A x. opreq2i sqr fveq2i eqtr syl5eq))
+thm (absidt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (` (abs) A) A)) (A A RR e. A 0 if 0 <_ breq2 A A RR e. A 0 if abs fveq2 A A RR e. A 0 if = id eqeq12d imbi12d 0re A elimel absid dedth imp))
+thm (absnidt () () (-> (/\ (e. A (RR)) (br A (<_) (0))) (= (` (abs) A) (-u A))) (A le0neg1t A recnt A absnegt syl 0 <_ A -u br adantr A -u absidt A renegclt sylan eqtr3d exp sylbid imp))
+thm (leabst () () (-> (e. A (RR)) (br A (<_) (` (abs) A))) (0re 0 A letrit mpan A absidt A abs A ` eqlet abs A ` A eqcom sylan2b syldan exp A recnt A absge0t syl A recnt A absclt syl 0re A 0 abs A ` letrt mp3an2 mpdan mpan2d jaod mpd))
+thm (absort () () (-> (e. A (RR)) (\/ (= (` (abs) A) A) (= (` (abs) A) (-u A)))) (0re 0 A letrit mpan A absidt exp A absnidt exp orim12d mpd))
+thm (absret () () (-> (e. A (RR)) (= (` (abs) A) (` (sqr) (opr A (^) (2))))) (A recnt A absvalt syl A recnt A sqvalt syl A cjret A x. opreq2d eqtr4d sqr fveq2d eqtr4d))
+thm (absrelet () () (-> (e. A (CC)) (br (` (abs) (` (Re) A)) (<_) (` (abs) A))) (Re A ` ^ 2 opr Im A ` ^ 2 opr addge1tOLD A reclt Re A ` sqreclt syl A imclt Im A ` sqreclt syl A imclt Im A ` sqege0t syl syl3anc Re A ` ^ 2 opr Re A ` ^ 2 opr + Im A ` ^ 2 opr opr sqrlet A reclt Re A ` sqreclt syl Re A ` ^ 2 opr Im A ` ^ 2 opr axaddrcl A reclt Re A ` sqreclt syl A imclt Im A ` sqreclt syl sylanc jca A reclt Re A ` sqege0t syl Re A ` ^ 2 opr Im A ` ^ 2 opr addge0t A reclt Re A ` sqreclt syl A imclt Im A ` sqreclt syl jca A reclt Re A ` sqege0t syl A imclt Im A ` sqege0t syl jca sylanc jca sylanc mpbid A reclt Re A ` absret syl A absval2t 3brtr4d))
+thm (absimlet () () (-> (e. A (CC)) (br (` (abs) (` (Im) A)) (<_) (` (abs) A))) (Im A ` ^ 2 opr Re A ` ^ 2 opr addge2tOLD A imclt Im A ` sqreclt syl A reclt Re A ` sqreclt syl A reclt Re A ` sqege0t syl syl3anc Im A ` ^ 2 opr Re A ` ^ 2 opr + Im A ` ^ 2 opr opr sqrlet A imclt Im A ` sqreclt syl Re A ` ^ 2 opr Im A ` ^ 2 opr axaddrcl A reclt Re A ` sqreclt syl A imclt Im A ` sqreclt syl sylanc jca A imclt Im A ` sqege0t syl Re A ` ^ 2 opr Im A ` ^ 2 opr addge0t A reclt Re A ` sqreclt syl A imclt Im A ` sqreclt syl jca A reclt Re A ` sqege0t syl A imclt Im A ` sqege0t syl jca sylanc jca sylanc mpbid A imclt Im A ` absret syl A absval2t 3brtr4d))
+thm (absnid () ((absnid.1 (e. A (RR)))) (-> (br A (<_) (0)) (= (` (abs) A) (-u A))) (absnid.1 A absnidt mpan))
+thm (leabs () ((absnid.1 (e. A (RR)))) (br A (<_) (` (abs) A)) (0re absnid.1 letri absnid.1 absid absnid.1 absnid.1 recn abscl eqle eqcoms syl absnid.1 recn absge0 absnid.1 0re absnid.1 recn abscl letr mpan2 jaoi ax-mp))
+thm (absor () ((absnid.1 (e. A (RR)))) (\/ (= (` (abs) A) A) (= (` (abs) A) (-u A))) (absnid.1 A absort ax-mp))
+thm (absre () ((absnid.1 (e. A (RR)))) (= (` (abs) A) (` (sqr) (opr A (^) (2)))) (absnid.1 A absret ax-mp))
+thm (abslt () ((absnid.1 (e. A (RR))) (abslt.1 (e. B (RR)))) (<-> (br (` (abs) A) (<) B) (/\ (br (-u B) (<) A) (br A (<) B))) (absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl abslt.1 lelttr mpan absnid.1 leabs absnid.1 absnid.1 recn abscl abslt.1 lelttr mpan jca absnid.1 absor abs A ` A < B breq1 biimprcd abs A ` A -u < B breq1 biimprcd jaao mpi ancoms impbi absnid.1 abslt.1 ltnegcon1 A < B br anbi1i bitr))
+thm (absle () ((absnid.1 (e. A (RR))) (abslt.1 (e. B (RR)))) (<-> (br (` (abs) A) (<_) B) (/\ (br (-u B) (<_) A) (br A (<_) B))) (absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl abslt.1 letr mpan absnid.1 leabs absnid.1 absnid.1 recn abscl abslt.1 letr mpan jca absnid.1 absor abs A ` A <_ B breq1 biimprcd abs A ` A -u <_ B breq1 biimprcd jaao mpi ancoms impbi absnid.1 abslt.1 lenegcon1 A <_ B br anbi1i bitr))
+thm (absltOLD () ((absnid.1 (e. A (RR))) (absltOLD.1 (e. B (RR)))) (<-> (br (` (abs) A) (<) B) (/\ (br A (<) B) (br (-u A) (<) B))) (absnid.1 leabs absnid.1 absnid.1 recn abscl absltOLD.1 lelttr mpan absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl absltOLD.1 lelttr mpan jca absnid.1 absor abs A ` A < B breq1 biimprcd abs A ` A -u < B breq1 biimprcd jaao mpi impbi))
+thm (absleOLD () ((absnid.1 (e. A (RR))) (absltOLD.1 (e. B (RR)))) (<-> (br (` (abs) A) (<_) B) (/\ (br A (<_) B) (br (-u A) (<_) B))) (absnid.1 leabs absnid.1 absnid.1 recn abscl absltOLD.1 letr mpan absnid.1 renegcl leabs absnid.1 recn absneg breqtr absnid.1 renegcl absnid.1 recn abscl absltOLD.1 letr mpan jca absnid.1 absor abs A ` A <_ B breq1 biimprcd abs A ` A -u <_ B breq1 biimprcd jaao mpi impbi))
+thm (abs0 () () (= (` (abs) (0)) (0)) (0 eqid 0cn abs00 mpbir))
+thm (absi () () (= (` (abs) (i)) (1)) (axicn i absvalt ax-mp axicn axicn mulneg2 cji i x. opreq2i itimesi axicn axicn mulcl 1cn negcon2 mpbi 3eqtr4 sqr fveq2i sqr1 3eqtr))
+thm (nn0absclt () () (-> (e. A (ZZ)) (e. (` (abs) A) (NN0))) (A zret A absort syl abs A ` A ZZ eleq1 biimprd abs A ` A -u ZZ eleq1 A znegclt syl5bir jaoi mpcom A zcnt A absge0t syl jca abs A ` elnn0z sylibr))
+thm (absltt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<) B) (/\ (br (-u B) (<) A) (br A (<) B)))) (A A RR e. A 0 if abs fveq2 < B breq1d A A RR e. A 0 if B -u < breq2 A A RR e. A 0 if < B breq1 anbi12d bibi12d B B RR e. B 0 if abs A RR e. A 0 if ` < breq2 B B RR e. B 0 if negeq < A RR e. A 0 if breq1d B B RR e. B 0 if A RR e. A 0 if < breq2 anbi12d bibi12d 0re A elimel 0re B elimel abslt dedth2h))
+thm (abslttOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<) B) (/\ (br A (<) B) (br (-u A) (<) B)))) (A A RR e. A 0 if abs fveq2 < B breq1d A A RR e. A 0 if < B breq1 A A RR e. A 0 if negeq < B breq1d anbi12d bibi12d B B RR e. B 0 if abs A RR e. A 0 if ` < breq2 B B RR e. B 0 if A RR e. A 0 if < breq2 B B RR e. B 0 if A RR e. A 0 if -u < breq2 anbi12d bibi12d 0re A elimel 0re B elimel absltOLD dedth2h))
+thm (abslet () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (` (abs) A) (<_) B) (/\ (br (-u B) (<_) A) (br A (<_) B)))) (A A RR e. A 0 if abs fveq2 <_ B breq1d A A RR e. A 0 if B -u <_ breq2 A A RR e. A 0 if <_ B breq1 anbi12d bibi12d B B RR e. B 0 if abs A RR e. A 0 if ` <_ breq2 B B RR e. B 0 if negeq <_ A RR e. A 0 if breq1d B B RR e. B 0 if A RR e. A 0 if <_ breq2 anbi12d bibi12d 0re A elimel 0re B elimel absle dedth2h))
+thm (releabst () () (-> (e. A (CC)) (br (` (Re) A) (<_) (` (abs) A))) (A reclt A reclt Re A ` recnt Re A ` absclt 3syl A absclt A reclt Re A ` leabst syl A absrelet letrd))
+thm (recvalz () ((recvalz.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr (1) (/) A) (opr (` (*) A) (/) (opr (` (abs) A) (^) (2))))) (recvalz.1 cjcl recvalz.1 mulcom recvalz.1 absvalsq eqtr4 recvalz.1 A cjne0t ax-mp recvalz.1 abscl sqrecl recn recvalz.1 cjcl recvalz.1 divmulz sylbi mpbiri 1 / opreq2d recvalz.1 abscl sqrecl recn recvalz.1 cjcl abs A ` ^ 2 opr * A ` recdivt mpanl12 recvalz.1 abscl gt0ne0 recvalz.1 absgt0 recvalz.1 abscl recn sqe0 eqneqi 3imtr4 recvalz.1 A cjne0t ax-mp biimp sylanc eqtr3d))
+thm (releabs () ((releabs.1 (e. A (CC)))) (br (` (Re) A) (<_) (` (abs) A)) (releabs.1 A releabst ax-mp))
+thm (abstri () ((releabs.1 (e. A (CC))) (abstri.2 (e. B (CC)))) (br (` (abs) (opr A (+) B)) (<_) (opr (` (abs) A) (+) (` (abs) B))) (releabs.1 abstri.2 cjcl mulcl releabs releabs.1 abstri.2 cjcl absmul abstri.2 abscj abs A ` x. opreq2i eqtr breqtr 2pos releabs.1 abstri.2 cjcl mulcl recl releabs.1 abscl abstri.2 abscl remulcl 2re lemul2 ax-mp mpbi 2re releabs.1 abstri.2 cjcl mulcl recl remulcl 2re releabs.1 abscl abstri.2 abscl remulcl remulcl releabs.1 abscl sqrecl abstri.2 abscl sqrecl readdcl leadd2 mpbi releabs.1 abstri.2 sqabsadd releabs.1 abscl recn abstri.2 abscl recn binom2 releabs.1 abscl recn sqcl 2re releabs.1 abscl abstri.2 abscl remulcl remulcl recn abstri.2 abscl recn sqcl add23 eqtr 3brtr4 releabs.1 abstri.2 addcl absge0 releabs.1 absge0 abstri.2 absge0 releabs.1 abscl abstri.2 abscl addge0 mp2an releabs.1 abstri.2 addcl abscl releabs.1 abscl abstri.2 abscl readdcl le2sqe mp2an mpbir))
+thm (abs3dif () ((releabs.1 (e. A (CC))) (abstri.2 (e. B (CC))) (abs3dif.3 (e. C (CC)))) (br (` (abs) (opr A (-) B)) (<_) (opr (` (abs) (opr A (-) C)) (+) (` (abs) (opr C (-) B)))) (releabs.1 abstri.2 negcl abs3dif.3 abs3dif.3 negcl add4 releabs.1 abstri.2 negcl addcl addid1 abs3dif.3 negid A + B -u opr + opreq2i releabs.1 abstri.2 subneg 3eqtr4r releabs.1 abs3dif.3 negcl abs3dif.3 abstri.2 negcl add42 3eqtr4 releabs.1 abs3dif.3 subneg abs3dif.3 abstri.2 subneg + opreq12i eqtr4 abs fveq2i releabs.1 abs3dif.3 subcl abs3dif.3 abstri.2 subcl abstri eqbrtr))
+thm (abs3lem () ((releabs.1 (e. A (CC))) (abstri.2 (e. B (CC))) (abs3dif.3 (e. C (CC))) (abs3lem.4 (e. D (RR)))) (-> (/\ (br (` (abs) (opr A (-) C)) (<) (opr D (/) (2))) (br (` (abs) (opr C (-) B)) (<) (opr D (/) (2)))) (br (` (abs) (opr A (-) B)) (<) D)) (releabs.1 abs3dif.3 subcl abscl abs3dif.3 abstri.2 subcl abscl abs3lem.4 2re 2re 2pos gt0ne0i redivcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl lt2add releabs.1 abstri.2 abs3dif.3 abs3dif releabs.1 abstri.2 subcl abscl releabs.1 abs3dif.3 subcl abscl abs3dif.3 abstri.2 subcl abscl readdcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl abs3lem.4 2re 2re 2pos gt0ne0i redivcl readdcl lelttr mpan syl abs3lem.4 recn 2cn 2re 2pos gt0ne0i divcl 2times 2cn abs3lem.4 recn 2re 2pos gt0ne0i divcan2 eqtr3 syl6breq))
+thm (absidmt () () (-> (e. A (CC)) (= (` (abs) (` (abs) A)) (` (abs) A))) (abs A ` absidt A absclt A absge0t sylanc))
+thm (absgt0t () () (-> (e. A (CC)) (<-> (=/= A (0)) (br (0) (<) (` (abs) A)))) (A A CC e. A 0 if 0 neeq1 A A CC e. A 0 if abs fveq2 0 < breq2d bibi12d 0cn A elimel absgt0 dedth))
+thm (abssubt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (abs) (opr A (-) B)) (` (abs) (opr B (-) A)))) (A A CC e. A 0 if - B opreq1 abs fveq2d A A CC e. A 0 if B - opreq2 abs fveq2d eqeq12d B B CC e. B 0 if A CC e. A 0 if - opreq2 abs fveq2d B B CC e. B 0 if - A CC e. A 0 if opreq1 abs fveq2d eqeq12d 0cn A elimel 0cn B elimel abssub dedth2h))
+thm (abstrit () () (-> (/\ (e. A (CC)) (e. B (CC))) (br (` (abs) (opr A (+) B)) (<_) (opr (` (abs) A) (+) (` (abs) B)))) (A A CC e. A 0 if + B opreq1 abs fveq2d A A CC e. A 0 if abs fveq2 + abs B ` opreq1d <_ breq12d B B CC e. B 0 if A CC e. A 0 if + opreq2 abs fveq2d B B CC e. B 0 if abs fveq2 abs A CC e. A 0 if ` + opreq2d <_ breq12d 0cn A elimel 0cn B elimel abstri dedth2h))
+thm (abs1m ((A x)) ((abs1m.1 (e. A (CC)))) (E.e. x (CC) (/\ (= (` (abs) (cv x)) (1)) (= (` (abs) A) (opr (cv x) (x.) A)))) (abs1m.1 abs00 biimpr A 0 1 x. opreq2 0cn mulid2 syl6eq eqtr4d 0re ax1re lt01 ltlei ax1re absid ax-mp jctil 1cn jctil x cv 1 abs fveq2 1 eqeq1d x cv 1 x. A opreq1 abs A ` eqeq2d anbi12d CC rcla4ev syl A 0 df-ne abs1m.1 abs00 eqneqi abs1m.1 cjcl abs1m.1 abscl recn divclz abs1m.1 cjcl abs1m.1 abscl recn absdivz abs1m.1 abscl recn abs A ` dividt mpan abs1m.1 abscj abs1m.1 A absidmt ax-mp / opreq12i syl5eq eqtrd abs1m.1 absvalsq abs1m.1 abscl recn sqval abs1m.1 abs1m.1 cjcl mulcom 3eqtr3 abs1m.1 cjcl abs1m.1 mulcl abs1m.1 abscl recn abs1m.1 abscl recn divmulz mpbiri abs1m.1 cjcl abs1m.1 abs1m.1 abscl recn 3pm3.2i * A ` A abs A ` div23t mpan eqtr3d jca jca sylbir x cv * A ` / abs A ` opr abs fveq2 1 eqeq1d x cv * A ` / abs A ` opr x. A opreq1 abs A ` eqeq2d anbi12d CC rcla4ev syl sylbir pm2.61i))
+thm (recant ((A x) (B x)) () (-> (/\ (e. A (CC)) (e. B (CC))) (<-> (A.e. x (CC) (= (` (Re) (opr (cv x) (x.) A)) (` (Re) (opr (cv x) (x.) B)))) (= A B))) (A replimt A mulid2t eqcomd Re fveq2d A imret x. i opreq1d + opreq12d eqtrd B replimt B mulid2t eqcomd Re fveq2d B imret x. i opreq1d + opreq12d eqtrd eqeqan12d 1cn x cv 1 x. A opreq1 Re fveq2d x cv 1 x. B opreq1 Re fveq2d eqeq12d CC rcla4v ax-mp axicn negcl x cv i -u x. A opreq1 Re fveq2d x cv i -u x. B opreq1 Re fveq2d eqeq12d CC rcla4v ax-mp x. i opreq1d + opreq12d syl5bir A B x cv x. opreq2 Re fveq2d x cv CC e. a1d r19.21aiv A CC e. B CC e. /\ a1i impbid))
+thm (abs3lemt () () (-> (/\ (/\ (e. A (CC)) (e. B (CC))) (/\ (e. C (CC)) (e. D (RR)))) (-> (/\ (br (` (abs) (opr A (-) C)) (<) (opr D (/) (2))) (br (` (abs) (opr C (-) B)) (<) (opr D (/) (2)))) (br (` (abs) (opr A (-) B)) (<) D))) (A A CC e. A 0 if - C opreq1 abs fveq2d < D / 2 opr breq1d abs C - B opr ` < D / 2 opr br anbi1d A A CC e. A 0 if - B opreq1 abs fveq2d < D breq1d imbi12d B B CC e. B 0 if C - opreq2 abs fveq2d < D / 2 opr breq1d abs A CC e. A 0 if - C opr ` < D / 2 opr br anbi2d B B CC e. B 0 if A CC e. A 0 if - opreq2 abs fveq2d < D breq1d imbi12d C C CC e. C 0 if A CC e. A 0 if - opreq2 abs fveq2d < D / 2 opr breq1d C C CC e. C 0 if - B CC e. B 0 if opreq1 abs fveq2d < D / 2 opr breq1d anbi12d abs A CC e. A 0 if - B CC e. B 0 if opr ` < D br imbi1d D D RR e. D 0 if / 2 opreq1 abs A CC e. A 0 if - C CC e. C 0 if opr ` < breq2d D D RR e. D 0 if / 2 opreq1 abs C CC e. C 0 if - B CC e. B 0 if opr ` < breq2d anbi12d D D RR e. D 0 if abs A CC e. A 0 if - B CC e. B 0 if opr ` < breq2 imbi12d 0cn A elimel 0cn B elimel 0cn C elimel 0re D elimel abs3lem dedth4h))
+thm (abslem2i () ((abslem2.1 (e. A (CC))) (abslem2i.2 (=/= A (0)))) (= (opr (opr (` (*) (opr A (/) (` (abs) A))) (x.) A) (+) (opr (opr A (/) (` (abs) A)) (x.) (` (*) A))) (opr (2) (x.) (` (abs) A))) (abslem2.1 abscl recn 2times abslem2.1 abscl abslem2.1 abscl recn cjreb mpbi abslem2.1 abscl recn sqval abslem2.1 absvalsq eqtr3 / abs A ` opreq1i abslem2.1 abscl recn abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcan4 abslem2.1 abslem2.1 cjcl abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir div23 3eqtr3 * fveq2i eqtr3 abslem2.1 abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcl abslem2.1 cjcl cjmul abslem2.1 cjcj * A / abs A ` opr ` x. opreq2i 3eqtr abslem2.1 abscl recn sqval abslem2.1 absvalsq eqtr3 / abs A ` opreq1i abslem2.1 abscl recn abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir divcan4 abslem2.1 abslem2.1 cjcl abslem2.1 abscl recn abslem2i.2 abslem2.1 abs00 eqneqi mpbir div23 3eqtr3 + opreq12i eqtr2))
+thm (abslem2 () ((abslem2.1 (e. A (CC)))) (-> (=/= A (0)) (= (opr (opr (` (*) (opr A (/) (` (abs) A))) (x.) A) (+) (opr (opr A (/) (` (abs) A)) (x.) (` (*) A))) (opr (2) (x.) (` (abs) A)))) (A A 0 =/= A 1 if = id A A 0 =/= A 1 if abs fveq2 / opreq12d * fveq2d A A 0 =/= A 1 if = id x. opreq12d A A 0 =/= A 1 if = id A A 0 =/= A 1 if abs fveq2 / opreq12d A A 0 =/= A 1 if * fveq2 x. opreq12d + opreq12d A A 0 =/= A 1 if abs fveq2 2 x. opreq2d eqeq12d abslem2.1 1cn A 0 =/= keepel A elimne0 abslem2i dedth))
+thm (seq1bnd ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (F x) (w y) (F y) (w z) (F z) (F w)) ((seq1bnd.1 (:--> F (NN) (CC)))) (-> (e. A (NN)) (E.e. x (RR) (A.e. y (NN) (-> (br (cv y) (<_) A) (br (` (abs) (` F (cv y))) (<) (cv x)))))) (z cv 1 y cv <_ breq2 abs F y cv ` ` < x cv br imbi1d y NN biraldv x RR birexdv z cv w cv y cv <_ breq2 abs F y cv ` ` < x cv br imbi1d y NN biraldv x RR birexdv z cv w cv + 1 opr y cv <_ breq2 abs F y cv ` ` < x cv br imbi1d y NN biraldv x RR birexdv z cv A y cv <_ breq2 abs F y cv ` ` < x cv br imbi1d y NN biraldv x RR birexdv seq1bnd.1 1nn F NN CC 1 ffvrn mp2an abscl ax1re readdcl y cv nnle1eq1t y cv 1 F fveq2 abs fveq2d seq1bnd.1 1nn F NN CC 1 ffvrn mp2an abscl ltplus1 syl6eqbr syl6bi rgen x cv abs F 1 ` ` + 1 opr abs F y cv ` ` < breq2 y cv <_ 1 br imbi2d y NN biraldv RR rcla4ev mp2an y cv w cv + 1 opr leloet y cv nnret w cv peano2nn w cv + 1 opr nnret syl syl2an y cv w cv nnleltp1t y cv w cv + 1 opr = orbi1d bitr4d ancoms z cv RR e. adantlr y cv <_ w cv br abs F y cv ` ` < z cv br -> adantr z cv abs F w cv + 1 opr ` ` + 1 opr max1ALT w cv NN e. y cv NN e. ad2antlr abs F y cv ` ` z cv z cv <_ abs F w cv + 1 opr ` ` + 1 opr br abs F w cv + 1 opr ` ` + 1 opr z cv if ltletrt seq1bnd.1 y cv ffvrni F y cv ` absclt syl w cv NN e. z cv RR e. /\ adantl w cv NN e. z cv RR e. pm3.27 y cv NN e. adantr abs F w cv + 1 opr ` ` + 1 opr RR z cv z cv <_ abs F w cv + 1 opr ` ` + 1 opr br ifcl w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` peano2re 3syl sylan y cv NN e. adantr syl3anc mpan2d y cv <_ w cv br imim2d imp y cv w cv + 1 opr F fveq2 abs fveq2d < abs F w cv + 1 opr ` ` + 1 opr breq1d w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` ltplus1t 3syl syl5bir com12 z cv RR e. y cv NN e. ad2antll z cv abs F w cv + 1 opr ` ` + 1 opr max2 w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` peano2re 3syl sylan2 ancoms y cv NN e. adantr abs F y cv ` ` abs F w cv + 1 opr ` ` + 1 opr z cv <_ abs F w cv + 1 opr ` ` + 1 opr br abs F w cv + 1 opr ` ` + 1 opr z cv if ltletrt seq1bnd.1 y cv ffvrni F y cv ` absclt syl w cv NN e. z cv RR e. /\ adantl w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` peano2re 3syl z cv RR e. y cv NN e. ad2antll abs F w cv + 1 opr ` ` + 1 opr RR z cv z cv <_ abs F w cv + 1 opr ` ` + 1 opr br ifcl w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` peano2re 3syl sylan y cv NN e. adantr syl3anc mpan2d syld y cv <_ w cv br abs F y cv ` ` < z cv br -> adantr jaod sylbid exp r19.20dva abs F w cv + 1 opr ` ` + 1 opr RR z cv z cv <_ abs F w cv + 1 opr ` ` + 1 opr br ifcl w cv peano2nn seq1bnd.1 w cv + 1 opr ffvrni syl F w cv + 1 opr ` absclt abs F w cv + 1 opr ` ` peano2re 3syl sylan jctild x cv z cv <_ abs F w cv + 1 opr ` ` + 1 opr br abs F w cv + 1 opr ` ` + 1 opr z cv if abs F y cv ` ` < breq2 y cv <_ w cv + 1 opr br imbi2d y NN biraldv RR rcla4ev syl6 exp r19.23adv x cv z cv abs F y cv ` ` < breq2 y cv <_ w cv br imbi2d y NN biraldv RR cbvrexv syl5ib nnind))
+thm (seq1ublem ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (F y) (F z) (F w) (F v)) ((seq1ub.1 (:--> F (NN) (RR)))) (-> (e. B (NN)) (/\/\ (C_ (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (RR)) (-. (= (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) ({/}))) (E.e. y (RR) (A.e. z (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (br (cv z) (<_) (cv y)))))) (seq1ub.1 x NN x cv <_ B br ssrab F NN RR x NN x cv <_ B br {e.|} fssres mp2an F x NN x cv <_ B br {e.|} |` x NN x cv <_ B br {e.|} RR frn ax-mp B NN e. a1i B nnge1t 1nn jctil x cv 1 <_ B breq1 NN elrab sylibr seq1ub.1 F NN RR ffn ax-mp x NN x cv <_ B br ssrab F NN x NN x cv <_ B br {e.|} fnssres mp2an jctil F x NN x cv <_ B br {e.|} |` x NN x cv <_ B br {e.|} 1 fnfvrn syl F x NN x cv <_ B br {e.|} |` 1 ` F x NN x cv <_ B br {e.|} |` ran n0i syl seq1ub.1 axresscn F NN RR CC fss mp2an B y w seq1bnd w cv v cv <_ B breq1 w cv v cv F fveq2 abs fveq2d < y cv breq1d imbi12d NN rcla4v com12 imp32 y cv RR e. adantll F x NN x cv <_ B br {e.|} |` v cv ` z cv = adantr F v cv ` leabst y cv RR e. adantr F v cv ` abs F v cv ` ` y cv lelttrt 3expa F v cv ` recnt F v cv ` absclt syl ancli sylan mpand F v cv ` y cv ltlet syld seq1ub.1 v cv ffvrni sylan ancoms v cv <_ B br adantrr x cv v cv <_ B breq1 NN elrab v cv x NN x cv <_ B br {e.|} F fvres sylbir <_ y cv breq1d y cv RR e. adantl sylibrd F x NN x cv <_ B br {e.|} |` v cv ` z cv <_ y cv breq1 biimpd sylan9 w NN w cv <_ B br abs F w cv ` ` < y cv br -> A.e. adantllr mpd exp31 x cv v cv <_ B breq1 NN elrab syl5ib r19.23adv seq1ub.1 F NN RR ffn ax-mp x NN x cv <_ B br ssrab F NN x NN x cv <_ B br {e.|} fnssres mp2an F x NN x cv <_ B br {e.|} |` x NN x cv <_ B br {e.|} z cv v fvelrn ax-mp syl5ib exp r19.21adv r19.22i syl 3jca))
+thm (seq1ub ((A y) (x y) (x z) (B x) (y z) (B y) (B z) (F y) (F z)) ((seq1ub.1 (:--> F (NN) (RR)))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (sup (ran (|` F ({e.|} x (NN) (br (cv x) (<_) B)))) (RR) (<)))) (F x NN x cv <_ B br {e.|} |` ran y z F A ` suprub seq1ub.1 B x y z seq1ublem A NN e. A <_ B br 3ad2ant2 x cv y cv <_ B breq1 NN cbvrabv A eleq2i y cv A <_ B breq1 NN elrab bitr A x NN x cv <_ B br {e.|} F fvres sylbir x cv y cv <_ B breq1 NN cbvrabv A eleq2i y cv A <_ B breq1 NN elrab bitr biimpr seq1ub.1 F NN RR ffn ax-mp x NN x cv <_ B br ssrab F NN x NN x cv <_ B br {e.|} fnssres mp2an jctil F x NN x cv <_ B br {e.|} |` x NN x cv <_ B br {e.|} A fnfvrn syl eqeltrrd B NN e. 3adant2 sylanc))
+thm (cau2 ((x y) (x z) (y z) (R y) (R z)) ((cau2.1 (:--> F (NN) (CC))) (cau2.2 (-> (e. (cv x) (RR)) (-> (br (0) (<) (cv x)) (br (0) R (cv x)))))) (<-> (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x))))))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) R (cv x)))))))) (cau2.2 imp y cv z cv F fveq2 F z cv ` - opreq2d cau2.1 z cv ffvrni F z cv ` subidt syl sylan9eqr abs fveq2d abs0 syl6eq R x cv breq1d biimprcd exp3a syl imp y cv NN e. adantlr y cv < z cv br abs F z cv ` - F y cv ` opr ` R x cv br -> biantrud y cv < z cv br y cv z cv = abs F z cv ` - F y cv ` opr ` R x cv br jaob syl6bbr y cv z cv leloet y cv nnret z cv nnret syl2an x cv RR e. 0 < x cv br /\ adantll abs F z cv ` - F y cv ` opr ` R x cv br imbi1d bitr4d biraldva birexdva exp pm5.74d birala))
+thm (cvg1 ((y z) (w y) (w z) (ps y) (ps w)) () (<-> (A.e. x F (-> ph (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) ps))))) (A.e. x F (-> ph (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) ps)))))) (y cv z cv nnltp1let ps imbi1d biraldva biimpd y cv peano2nn jctild w cv y cv + 1 opr <_ z cv breq1 ps imbi1d z NN biraldv NN rcla4ev syl6 r19.23aiv w cv y cv <_ z cv breq1 ps imbi1d z NN biraldv NN cbvrexv sylib y cv z cv ltlet y cv nnret z cv nnret syl2an ps imim1d r19.20dva r19.22i impbi ph imbi2i x F biral))
+thm (cvg2 ((x y) (x z) (w x) (y z) (w y) (w z) (A x) (A w) (F x) (F z) (F w) (G x) (G w) (ph x) (ph w)) ((cvg2.1 (-> (/\ (e. (cv y) F) (e. (cv z) G)) (e. A (RR))))) (<-> (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y F (A.e. z G (-> ph (br A (<) (cv x))))))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y F (A.e. z G (-> ph (br A (<_) (cv x)))))))) (A x cv ltlet ancoms cvg2.1 sylan2 anassrs ph imim2d r19.20dva r19.22dva 0 < x cv br imim2d r19.20i x cv w cv / 2 opr 0 < breq2 x cv w cv / 2 opr A <_ breq2 ph imbi2d z G biraldv y F birexdv imbi12d RR rcla4v com12 w cv / 2 opr RR e. 0 < w cv / 2 opr br /\ y F z G ph A <_ w cv / 2 opr br -> A.e. E.e. biimt 2re 2re 2pos gt0ne0i w cv 2 redivclt mp3an23 0 < w cv br adantr w cv halfpos2t biimpa sylanc w cv / 2 opr RR e. 0 < w cv / 2 opr br y F z G ph A <_ w cv / 2 opr br -> A.e. E.e. impexp syl6bb w cv halfpost biimpa A RR e. adantr A w cv / 2 opr w cv lelttrt w cv RR e. 0 < w cv br /\ A RR e. pm3.27 2re 2re 2pos gt0ne0i w cv 2 redivclt mp3an23 0 < w cv br A RR e. ad2antll w cv RR e. 0 < w cv br pm3.26 A RR e. adantr syl3anc mpan2d cvg2.1 sylan2 anassrs ph imim2d r19.20dva r19.22dva sylbird com12 exp3a syl r19.21aiv w cv x cv 0 < breq2 w cv x cv A < breq2 ph imbi2d z G biraldv y F birexdv imbi12d RR cbvralv sylib impbi))
+thm (caubnd ((x y) (x z) (w x) (u x) (t x) (F x) (y z) (w y) (u y) (t y) (F y) (w z) (u z) (t z) (F z) (u w) (t w) (F w) (t u) (F u) (F t)) ((caubnd.1 (:--> F (NN) (CC))) (caubnd.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))))) (E.e. x (RR) (A.e. y (NN) (br (` (abs) (` F (cv y))) (<) (cv x)))) (lt01 ax1re caubnd.2 z cv 1 0 < breq2 z cv 1 abs F y cv ` - F w cv ` opr ` < breq2 w cv < y cv br imbi2d y NN biraldv w NN birexdv imbi12d RR rcla4v mp2 ax-mp ax1re abs F y cv ` - F w cv ` opr ` 1 abs F w cv ` ` ltadd1t mp3an2 F y cv ` F w cv ` subclt caubnd.1 y cv ffvrni caubnd.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absclt syl caubnd.1 w cv ffvrni F w cv ` absclt syl y cv NN e. adantl sylanc F y cv ` F w cv ` npcant caubnd.1 y cv ffvrni caubnd.1 w cv ffvrni syl2an abs fveq2d F y cv ` - F w cv ` opr F w cv ` abstrit F y cv ` F w cv ` subclt caubnd.1 y cv ffvrni caubnd.1 w cv ffvrni syl2an caubnd.1 w cv ffvrni y cv NN e. adantl sylanc eqbrtrrd abs F y cv ` ` abs F y cv ` - F w cv ` opr ` + abs F w cv ` ` opr 1 + abs F w cv ` ` opr lelttrt caubnd.1 y cv ffvrni F y cv ` absclt syl w cv NN e. adantr abs F y cv ` - F w cv ` opr ` abs F w cv ` ` axaddrcl F y cv ` F w cv ` subclt caubnd.1 y cv ffvrni caubnd.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absclt syl caubnd.1 w cv ffvrni F w cv ` absclt syl y cv NN e. adantl sylanc caubnd.1 w cv ffvrni F w cv ` absclt syl ax1re 1 abs F w cv ` ` axaddrcl mpan syl y cv NN e. adantl syl3anc mpand sylbid ancoms w cv < y cv br imim2d r19.20dva caubnd.1 w cv ffvrni F w cv ` absclt syl ax1re 1 abs F w cv ` ` axaddrcl mpan syl jctild u cv 1 + abs F w cv ` ` opr abs F y cv ` ` < breq2 w cv < y cv br imbi2d y NN biraldv RR rcla4ev syl6 r19.22i ax-mp caubnd.1 w cv t y seq1bnd y cv w cv lelttrit y cv nnret w cv nnret syl2an ancoms t cv RR e. adantlr u cv RR e. adantlr u cv t cv max2 ancoms y cv NN e. adantr abs F y cv ` ` t cv u cv <_ t cv br t cv u cv if ltletrt caubnd.1 y cv ffvrni F y cv ` absclt syl t cv RR e. u cv RR e. /\ adantl t cv RR e. u cv RR e. pm3.26 y cv NN e. adantr t cv RR u cv u cv <_ t cv br ifcl y cv NN e. adantr syl3anc mpan2d y cv <_ w cv br imim2d u cv t cv max1ALT t cv RR e. y cv NN e. ad2antlr abs F y cv ` ` u cv u cv <_ t cv br t cv u cv if ltletrt caubnd.1 y cv ffvrni F y cv ` absclt syl t cv RR e. u cv RR e. /\ adantl t cv RR e. u cv RR e. pm3.27 y cv NN e. adantr t cv RR u cv u cv <_ t cv br ifcl y cv NN e. adantr syl3anc mpan2d w cv < y cv br imim2d anim12d y cv <_ w cv br w cv < y cv br abs F y cv ` ` < u cv <_ t cv br t cv u cv if br jaob syl6ibr w cv NN e. adantlll mpid r19.20dva t cv RR u cv u cv <_ t cv br ifcl w cv NN e. adantll jctild x cv u cv <_ t cv br t cv u cv if abs F y cv ` ` < breq2 y NN biraldv RR rcla4ev syl6 y NN y cv <_ w cv br abs F y cv ` ` < t cv br -> w cv < y cv br abs F y cv ` ` < u cv br -> r19.26 syl5ibr exp4b com23 imp r19.23adv exp31 r19.23adv mpd r19.23aiv ax-mp))
+thm (caure ((y z) (w y) (w z) (x y) (w x) (F x) (G x)) ((caure.1 (:--> F (NN) (CC))) (caure.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caure.3 (Fn G (NN))) (caure.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))))) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (<) (cv z))))))) (caure.2 x cv y cv G fveq2 x cv y cv F fveq2 Re fveq2d eqeq12d caure.4 vtoclga x cv w cv G fveq2 x cv w cv F fveq2 Re fveq2d eqeq12d caure.4 vtoclga - opreqan12d F y cv ` F w cv ` resubt caure.1 y cv ffvrni caure.1 w cv ffvrni syl2an eqtr4d abs fveq2d F y cv ` F w cv ` subclt caure.1 y cv ffvrni caure.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absrelet syl eqbrtrd z cv RR e. 3adant3 abs G y cv ` - G w cv ` opr ` abs F y cv ` - F w cv ` opr ` z cv lelttrt G y cv ` G w cv ` resubclt G NN RR x ffnfv caure.3 caure.4 caure.1 x cv ffvrni F x cv ` reclt syl eqeltrd rgen mpbir2an y cv ffvrni G NN RR x ffnfv caure.3 caure.4 caure.1 x cv ffvrni F x cv ` reclt syl eqeltrd rgen mpbir2an w cv ffvrni syl2an recnd G y cv ` - G w cv ` opr absclt syl z cv RR e. 3adant3 F y cv ` F w cv ` subclt caure.1 y cv ffvrni caure.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absclt syl z cv RR e. 3adant3 y cv NN e. w cv NN e. z cv RR e. 3simp3 syl3anc mpand 3com13 3expa w cv < y cv br imim2d r19.20dva r19.22dva 0 < z cv br imim2d r19.20i ax-mp))
+thm (cauim ((y z) (w y) (w z) (x y) (w x) (F x) (G x)) ((cauim.1 (:--> F (NN) (CC))) (cauim.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (cauim.3 (Fn G (NN))) (cauim.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Im) (` F (cv x))))))) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` G (cv y)) (-) (` G (cv w)))) (<) (cv z))))))) (cauim.2 x cv y cv G fveq2 x cv y cv F fveq2 Im fveq2d eqeq12d cauim.4 vtoclga x cv w cv G fveq2 x cv w cv F fveq2 Im fveq2d eqeq12d cauim.4 vtoclga - opreqan12d F y cv ` F w cv ` imsubt cauim.1 y cv ffvrni cauim.1 w cv ffvrni syl2an eqtr4d abs fveq2d F y cv ` F w cv ` subclt cauim.1 y cv ffvrni cauim.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absimlet syl eqbrtrd z cv RR e. 3adant3 abs G y cv ` - G w cv ` opr ` abs F y cv ` - F w cv ` opr ` z cv lelttrt G y cv ` G w cv ` resubclt G NN RR x ffnfv cauim.3 cauim.4 cauim.1 x cv ffvrni F x cv ` imclt syl eqeltrd rgen mpbir2an y cv ffvrni G NN RR x ffnfv cauim.3 cauim.4 cauim.1 x cv ffvrni F x cv ` imclt syl eqeltrd rgen mpbir2an w cv ffvrni syl2an recnd G y cv ` - G w cv ` opr absclt syl z cv RR e. 3adant3 F y cv ` F w cv ` subclt cauim.1 y cv ffvrni cauim.1 w cv ffvrni syl2an F y cv ` - F w cv ` opr absclt syl z cv RR e. 3adant3 y cv NN e. w cv NN e. z cv RR e. 3simp3 syl3anc mpand 3com13 3expa w cv < y cv br imim2d r19.20dva r19.22dva 0 < z cv br imim2d r19.20i ax-mp))
+thm (ser1abs ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) ((ser1abs.1 (:--> F (NN) (CC))) (ser1abs.2 (Fn G (NN))) (ser1abs.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (abs) (` F (cv x))))))) (-> (e. A (NN)) (br (` (abs) (` (opr (+) (seq1) F) A)) (<_) (` (opr (+) (seq1) G) A))) (y cv 1 + seq1 F opr fveq2 abs fveq2d y cv 1 + seq1 G opr fveq2 <_ breq12d y cv z cv + seq1 F opr fveq2 abs fveq2d y cv z cv + seq1 G opr fveq2 <_ breq12d y cv z cv + 1 opr + seq1 F opr fveq2 abs fveq2d y cv z cv + 1 opr + seq1 G opr fveq2 <_ breq12d y cv A + seq1 F opr fveq2 abs fveq2d y cv A + seq1 G opr fveq2 <_ breq12d 1nn x cv 1 G fveq2 x cv 1 F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ax-mp addex nnex ser1abs.2 NN V G fnex mp2 seq11 addex nnex ser1abs.1 NN V F CC fex mp2 seq11 abs fveq2i 3eqtr4r 1nn ser1abs.1 1 ser1cl ax-mp abscl 1nn x cv 1 G fveq2 x cv 1 F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ax-mp addex nnex ser1abs.2 NN V G fnex mp2 seq11 addex nnex ser1abs.1 NN V F CC fex mp2 seq11 abs fveq2i 3eqtr4r 1nn ser1abs.1 1 ser1cl ax-mp abscl eqeltrr eqle ax-mp z cv peano2nn ser1abs.1 z cv + 1 opr ser1cl + seq1 F opr z cv + 1 opr ` absclt 3syl abs + seq1 F opr z cv ` ` <_ + seq1 G opr z cv ` br adantr abs + seq1 F opr z cv ` ` G z cv + 1 opr ` axaddrcl ser1abs.1 z cv ser1cl + seq1 F opr z cv ` absclt syl z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 z cv + 1 opr ffvrni F z cv + 1 opr ` absclt syl eqeltrd syl sylanc abs + seq1 F opr z cv ` ` <_ + seq1 G opr z cv ` br adantr z cv peano2nn y cv 1 + seq1 G opr fveq2 RR eleq1d y cv z cv + seq1 G opr fveq2 RR eleq1d y cv z cv + 1 opr + seq1 G opr fveq2 RR eleq1d y cv z cv + 1 opr + seq1 G opr fveq2 RR eleq1d 1nn x cv 1 G fveq2 x cv 1 F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ax-mp addex nnex ser1abs.2 NN V G fnex mp2 seq11 addex nnex ser1abs.1 NN V F CC fex mp2 seq11 abs fveq2i 3eqtr4r 1nn ser1abs.1 1 ser1cl ax-mp abscl eqeltrr + seq1 G opr z cv ` G z cv + 1 opr ` axaddrcl z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 z cv + 1 opr ffvrni F z cv + 1 opr ` absclt syl eqeltrd syl sylan2 expcom addex nnex ser1abs.2 NN V G fnex mp2 z cv seq1p1 RR eleq1d sylibrd nnind syl abs + seq1 F opr z cv ` ` <_ + seq1 G opr z cv ` br adantr + seq1 F opr z cv ` F z cv + 1 opr ` abstrit ser1abs.1 z cv ser1cl z cv peano2nn ser1abs.1 z cv + 1 opr ffvrni syl sylanc addex nnex ser1abs.1 NN V F CC fex mp2 z cv seq1p1 abs fveq2d z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga syl abs + seq1 F opr z cv ` ` + opreq2d 3brtr4d abs + seq1 F opr z cv ` ` <_ + seq1 G opr z cv ` br adantr abs + seq1 F opr z cv ` ` + seq1 G opr z cv ` G z cv + 1 opr ` leadd1t ser1abs.1 z cv ser1cl + seq1 F opr z cv ` absclt syl y cv 1 + seq1 G opr fveq2 RR eleq1d y cv z cv + seq1 G opr fveq2 RR eleq1d y cv z cv + 1 opr + seq1 G opr fveq2 RR eleq1d y cv z cv + seq1 G opr fveq2 RR eleq1d 1nn x cv 1 G fveq2 x cv 1 F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ax-mp addex nnex ser1abs.2 NN V G fnex mp2 seq11 addex nnex ser1abs.1 NN V F CC fex mp2 seq11 abs fveq2i 3eqtr4r 1nn ser1abs.1 1 ser1cl ax-mp abscl eqeltrr + seq1 G opr z cv ` G z cv + 1 opr ` axaddrcl z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 z cv + 1 opr ffvrni F z cv + 1 opr ` absclt syl eqeltrd syl sylan2 expcom addex nnex ser1abs.2 NN V G fnex mp2 z cv seq1p1 RR eleq1d sylibrd nnind z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 z cv + 1 opr ffvrni F z cv + 1 opr ` absclt syl eqeltrd syl syl3anc biimpa addex nnex ser1abs.2 NN V G fnex mp2 z cv seq1p1 abs + seq1 F opr z cv ` ` <_ + seq1 G opr z cv ` br adantr breqtrrd letrd exp nnind))
+thm (ser1absdiflem ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z) (A x) (A z) (B y) (B z)) ((ser1abs.1 (:--> F (NN) (CC))) (ser1abs.2 (Fn G (NN))) (ser1abs.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (abs) (` F (cv x))))))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (abs) (opr (` (opr (+) (seq1) F) (opr A (+) B)) (-) (` (opr (+) (seq1) F) A))) (<_) (opr (` (opr (+) (seq1) G) (opr A (+) B)) (-) (` (opr (+) (seq1) G) A)))) (y cv 1 A + opreq2 + seq1 F opr fveq2d - + seq1 F opr A ` opreq1d abs fveq2d y cv 1 A + opreq2 + seq1 G opr fveq2d - + seq1 G opr A ` opreq1d <_ breq12d A NN e. imbi2d y cv z cv A + opreq2 + seq1 F opr fveq2d - + seq1 F opr A ` opreq1d abs fveq2d y cv z cv A + opreq2 + seq1 G opr fveq2d - + seq1 G opr A ` opreq1d <_ breq12d A NN e. imbi2d y cv z cv + 1 opr A + opreq2 + seq1 F opr fveq2d - + seq1 F opr A ` opreq1d abs fveq2d y cv z cv + 1 opr A + opreq2 + seq1 G opr fveq2d - + seq1 G opr A ` opreq1d <_ breq12d A NN e. imbi2d y cv B A + opreq2 + seq1 F opr fveq2d - + seq1 F opr A ` opreq1d abs fveq2d y cv B A + opreq2 + seq1 G opr fveq2d - + seq1 G opr A ` opreq1d <_ breq12d A NN e. imbi2d A peano2nn ser1abs.1 A + 1 opr ffvrni syl F A + 1 opr ` absclt syl abs F A + 1 opr ` ` leidt syl addex nnex ser1abs.1 NN V F CC fex mp2 A seq1p1 - + seq1 F opr A ` opreq1d + seq1 F opr A ` F A + 1 opr ` pncan2t ser1abs.1 A ser1cl A peano2nn ser1abs.1 A + 1 opr ffvrni syl sylanc eqtrd abs fveq2d addex nnex ser1abs.2 NN V G fnex mp2 A seq1p1 - + seq1 G opr A ` opreq1d + seq1 G opr A ` G A + 1 opr ` pncan2t G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an axresscn G NN RR CC fss mp2an A ser1cl A peano2nn x cv A + 1 opr G fveq2 x cv A + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 A + 1 opr ffvrni F A + 1 opr ` absclt syl eqeltrd recnd syl sylanc A peano2nn x cv A + 1 opr G fveq2 x cv A + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga syl 3eqtrd 3brtr4d + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr F A + z cv opr + 1 opr ` abstrit + seq1 F opr A + z cv opr ` + seq1 F opr A ` subclt A z cv nnaddclt ser1abs.1 A + z cv opr ser1cl syl ser1abs.1 A ser1cl z cv NN e. adantr sylanc A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl sylanc abs + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr + F A + z cv opr + 1 opr ` opr ` abs + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr ` + abs F A + z cv opr + 1 opr ` ` opr + seq1 G opr A + z cv opr ` - + seq1 G opr A ` opr + abs F A + z cv opr + 1 opr ` ` opr letrt + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr F A + z cv opr + 1 opr ` axaddcl + seq1 F opr A + z cv opr ` + seq1 F opr A ` subclt A z cv nnaddclt ser1abs.1 A + z cv opr ser1cl syl ser1abs.1 A ser1cl z cv NN e. adantr sylanc A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl sylanc + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr + F A + z cv opr + 1 opr ` opr absclt syl abs + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr ` abs F A + z cv opr + 1 opr ` ` axaddrcl + seq1 F opr A + z cv opr ` + seq1 F opr A ` subclt A z cv nnaddclt ser1abs.1 A + z cv opr ser1cl syl ser1abs.1 A ser1cl z cv NN e. adantr sylanc + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr absclt syl A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl F A + z cv opr + 1 opr ` absclt syl sylanc + seq1 G opr A + z cv opr ` - + seq1 G opr A ` opr abs F A + z cv opr + 1 opr ` ` axaddrcl + seq1 G opr A + z cv opr ` + seq1 G opr A ` resubclt A z cv nnaddclt G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an A + z cv opr ser1recl syl G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an A ser1recl z cv NN e. adantr sylanc A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl F A + z cv opr + 1 opr ` absclt syl sylanc syl3anc mpand abs + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr ` + seq1 G opr A + z cv opr ` - + seq1 G opr A ` opr abs F A + z cv opr + 1 opr ` ` leadd1t + seq1 F opr A + z cv opr ` + seq1 F opr A ` subclt A z cv nnaddclt ser1abs.1 A + z cv opr ser1cl syl ser1abs.1 A ser1cl z cv NN e. adantr sylanc + seq1 F opr A + z cv opr ` - + seq1 F opr A ` opr absclt syl + seq1 G opr A + z cv opr ` + seq1 G opr A ` resubclt A z cv nnaddclt G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an A + z cv opr ser1recl syl G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an A ser1recl z cv NN e. adantr sylanc A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl F A + z cv opr + 1 opr ` absclt syl syl3anc 1cn A z cv 1 axaddass mp3an3 A nncnt z cv nncnt syl2an + seq1 F opr fveq2d A z cv nnaddclt addex nnex ser1abs.1 NN V F CC fex mp2 A + z cv opr seq1p1 syl eqtr3d - + seq1 F opr A ` opreq1d + seq1 F opr A + z cv opr ` F A + z cv opr + 1 opr ` + seq1 F opr A ` addsubt A z cv nnaddclt ser1abs.1 A + z cv opr ser1cl syl A z cv nnaddclt A + z cv opr peano2nn syl ser1abs.1 A + z cv opr + 1 opr ffvrni syl ser1abs.1 A ser1cl z cv NN e. adantr syl3anc eqtrd abs fveq2d 1cn A z cv 1 axaddass mp3an3 A nncnt z cv nncnt syl2an + seq1 G opr fveq2d A z cv nnaddclt addex nnex ser1abs.2 NN V G fnex mp2 A + z cv opr seq1p1 syl eqtr3d - + seq1 G opr A ` opreq1d + seq1 G opr A + z cv opr ` G A + z cv opr + 1 opr ` + seq1 G opr A ` addsubt A z cv nnaddclt G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an axresscn G NN RR CC fss mp2an A + z cv opr ser1cl syl A z cv nnaddclt A + z cv opr peano2nn syl x cv A + z cv opr + 1 opr G fveq2 x cv A + z cv opr + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga ser1abs.1 A + z cv opr + 1 opr ffvrni F A + z cv opr + 1 opr ` absclt syl eqeltrd recnd syl G NN RR x ffnfv ser1abs.2 ser1abs.3 ser1abs.1 x cv ffvrni F x cv ` absclt syl eqeltrd rgen mpbir2an axresscn G NN RR CC fss mp2an A ser1cl z cv NN e. adantr syl3anc A z cv nnaddclt A + z cv opr peano2nn syl x cv A + z cv opr + 1 opr G fveq2 x cv A + z cv opr + 1 opr F fveq2 abs fveq2d eqeq12d ser1abs.3 vtoclga syl + seq1 G opr A + z cv opr ` - + seq1 G opr A ` opr + opreq2d 3eqtrd <_ breq12d 3imtr4d expcom a2d nnind impcom))
+thm (ser1absdif ((F x) (G x) (A x)) ((ser1abs.1 (:--> F (NN) (CC))) (ser1abs.2 (Fn G (NN))) (ser1abs.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (abs) (` F (cv x))))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<) B)) (br (` (abs) (opr (` (opr (+) (seq1) F) B) (-) (` (opr (+) (seq1) F) A))) (<_) (opr (` (opr (+) (seq1) G) B) (-) (` (opr (+) (seq1) G) A)))) (A NN e. B NN e. A < B br 3simp1 A B nnsubt biimp3a jca ser1abs.1 ser1abs.2 ser1abs.3 A B - A opr ser1absdiflem syl A B pncan3t A nncnt B nncnt syl2an + seq1 F opr fveq2d - + seq1 F opr A ` opreq1d abs fveq2d A < B br 3adant3 A B pncan3t A nncnt B nncnt syl2an + seq1 G opr fveq2d - + seq1 G opr A ` opreq1d A < B br 3adant3 3brtr3d))
+thm (ser1re2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (N j) (N m)) ((ser1re2.1 (:--> F (NN) (CC))) (ser1re2.2 (e. G (V))) (ser1re2.3 (-> (e. (cv k) (NN)) (= (` G (cv k)) (` (Re) (` F (cv k))))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) G) N) (` (Re) (` (opr (+) (seq1) F) N)))) (j cv 1 + seq1 G opr fveq2 j cv 1 + seq1 F opr fveq2 Re fveq2d eqeq12d j cv m cv + seq1 G opr fveq2 j cv m cv + seq1 F opr fveq2 Re fveq2d eqeq12d j cv m cv + 1 opr + seq1 G opr fveq2 j cv m cv + 1 opr + seq1 F opr fveq2 Re fveq2d eqeq12d j cv N + seq1 G opr fveq2 j cv N + seq1 F opr fveq2 Re fveq2d eqeq12d 1nn k cv 1 G fveq2 k cv 1 F fveq2 Re fveq2d eqeq12d ser1re2.3 vtoclga ax-mp addex ser1re2.2 seq11 addex nnex ser1re2.1 NN V F CC fex mp2 seq11 Re fveq2i 3eqtr4 addex ser1re2.2 m cv seq1p1 + seq1 G opr m cv ` Re + seq1 F opr m cv ` ` = adantr + seq1 G opr m cv ` Re + seq1 F opr m cv ` ` + G m cv + 1 opr ` opreq1 m cv NN e. adantl + seq1 F opr m cv ` F m cv + 1 opr ` readdt ser1re2.1 ser1f m cv ffvrni m cv peano2nn ser1re2.1 m cv + 1 opr ffvrni syl sylanc addex nnex ser1re2.1 NN V F CC fex mp2 m cv seq1p1 Re fveq2d m cv peano2nn k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 Re fveq2d eqeq12d ser1re2.3 vtoclga syl Re + seq1 F opr m cv ` ` + opreq2d 3eqtr4rd + seq1 G opr m cv ` Re + seq1 F opr m cv ` ` = adantr 3eqtrd exp nnind))
+thm (ser1cj ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z) (A y) (A z)) ((ser1cj.1 (:--> F (NN) (CC))) (ser1cj.2 (e. G (V))) (ser1cj.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (*) (` F (cv x))))))) (-> (e. A (NN)) (= (` (opr (+) (seq1) G) A) (` (*) (` (opr (+) (seq1) F) A)))) (y cv 1 + seq1 G opr fveq2 y cv 1 + seq1 F opr fveq2 * fveq2d eqeq12d y cv z cv + seq1 G opr fveq2 y cv z cv + seq1 F opr fveq2 * fveq2d eqeq12d y cv z cv + 1 opr + seq1 G opr fveq2 y cv z cv + 1 opr + seq1 F opr fveq2 * fveq2d eqeq12d y cv A + seq1 G opr fveq2 y cv A + seq1 F opr fveq2 * fveq2d eqeq12d 1nn x cv 1 G fveq2 x cv 1 F fveq2 * fveq2d eqeq12d ser1cj.3 vtoclga ax-mp addex ser1cj.2 seq11 addex nnex ser1cj.1 NN V F CC fex mp2 seq11 * fveq2i 3eqtr4 addex ser1cj.2 z cv seq1p1 + seq1 G opr z cv ` * + seq1 F opr z cv ` ` + G z cv + 1 opr ` opreq1 sylan9eq + seq1 F opr z cv ` F z cv + 1 opr ` cjaddt ser1cj.1 z cv ser1cl z cv peano2nn ser1cj.1 z cv + 1 opr ffvrni syl sylanc addex nnex ser1cj.1 NN V F CC fex mp2 z cv seq1p1 * fveq2d z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 * fveq2d eqeq12d ser1cj.3 vtoclga syl * + seq1 F opr z cv ` ` + opreq2d 3eqtr4rd + seq1 G opr z cv ` * + seq1 F opr z cv ` ` = adantr eqtrd exp nnind))
+thm (facnnt () () (-> (e. A (NN)) (= (` (!) A) (` (opr (x.) (seq1) (|` (I) (NN))) A))) (df-fac ! x. seq1 I NN |` opr 0 1 <,> {} u. NN reseq1 ax-mp x. seq1 I NN |` opr 0 1 <,> {} NN resundir 0 {} NN incom 0nnn NN 0 disjsn mpbir eqtr 0nn0 elisseti 1nn elisseti f1osn 0 1 <,> {} 0 {} 1 {} f1ofn ax-mp 0 1 <,> {} 0 {} NN fnresdisj ax-mp mpbi x. seq1 I NN |` opr NN |` uneq2i eqtr x. seq1 I NN |` opr NN |` un0 3eqtr A fveq1i A NN e. a1i A NN ! fvres A NN x. seq1 I NN |` opr fvres 3eqtr3d))
+thm (fac0 () () (= (` (!) (0)) (1)) (df-fac ! x. seq1 I NN |` opr 0 1 <,> {} u. 0 {} reseq1 ax-mp x. seq1 I NN |` opr 0 1 <,> {} 0 {} resundir 0nnn NN 0 disjsn mpbir mulex nnex funi NN V I resfunexg mp2 seq1fn x. seq1 I NN |` opr NN 0 {} fnresdisj ax-mp mpbi 0nn0 elisseti 1 relsn 0 1 dmsnop 0 {} ssid eqsstr 0 1 <,> {} 0 {} relssres mp2an uneq12i {/} 0 1 <,> {} uncom 0 1 <,> {} un0 3eqtr 3eqtr 0 fveq1i 0nn0 elisseti snid 0 0 {} ! fvres ax-mp 0nn0 elisseti 1nn0 elisseti fvsn 3eqtr3))
+thm (fac1 () () (= (` (!) (1)) (1)) (1nn 1 facnnt ax-mp mulex nnex funi NN V I resfunexg mp2 seq11 1nn 1 NN I fvres ax-mp 1nn 1 NN fvi ax-mp eqtr eqtr eqtr))
+thm (facp1t () () (-> (e. A (NN0)) (= (` (!) (opr A (+) (1))) (opr (` (!) A) (x.) (opr A (+) (1))))) (A elnn0 A peano2nn A + 1 opr facnnt syl mulex nnex funi NN V I resfunexg mp2 A seq1p1 A peano2nn A + 1 opr NN I fvres syl A + 1 oprex A + 1 opr V fvi ax-mp syl6eq x. seq1 I NN |` opr A ` x. opreq2d A facnnt x. A + 1 opr opreq1d eqtr4d 3eqtrd A 0 + 1 opreq1 ! fveq2d 1cn addid2 ! fveq2i fac1 eqtr syl6eq A 0 ! fveq2 A 0 + 1 opreq1 x. opreq12d fac0 1cn addid2 x. opreq12i 1cn mulid1 eqtr syl6eq eqtr4d jaoi sylbi))
+thm (fac2 () () (= (` (!) (2)) (2)) (df-2 ! fveq2i 1nn0 1 facp1t ax-mp fac1 df-2 eqcomi x. opreq12i 2cn mulid2 eqtr 3eqtr))
+thm (fac3 () () (= (` (!) (3)) (6)) (df-3 ! fveq2i 2nn0 2 facp1t ax-mp fac2 df-3 eqcomi x. opreq12i 2cn 3re recn mulcom 3t2e6 3eqtr 3eqtr))
+thm (facnn2t () () (-> (e. A (NN)) (= (` (!) A) (opr (` (!) (opr A (-) (1))) (x.) A))) (A elnnnn0 A - 1 opr facp1t A CC e. adantl 1cn A 1 npcant mpan2 ! fveq2d A - 1 opr NN0 e. adantr 1cn A 1 npcant mpan2 ! A - 1 opr ` x. opreq2d A - 1 opr NN0 e. adantr 3eqtr3d sylbi))
+thm (facclt ((x y) (A x) (A y)) () (-> (e. A (NN0)) (e. (` (!) A) (NN))) (x cv 0 ! fveq2 NN eleq1d x cv y cv ! fveq2 NN eleq1d x cv y cv + 1 opr ! fveq2 NN eleq1d x cv A ! fveq2 NN eleq1d fac0 1nn eqeltr y cv facp1t ! y cv ` NN e. adantl ! y cv ` y cv + 1 opr nnmulclt y cv nn0p1nnt sylan2 eqeltrd expcom nn0ind))
+thm (faclbnd ((x y) (A x) (A y)) () (-> (e. A (NN)) (br (opr (opr (2) (^) A) (/) (2)) (<_) (` (!) A))) (x cv 1 2 ^ opreq2 / 2 opreq1d x cv 1 ! fveq2 <_ breq12d x cv y cv 2 ^ opreq2 / 2 opreq1d x cv y cv ! fveq2 <_ breq12d x cv y cv + 1 opr 2 ^ opreq2 / 2 opreq1d x cv y cv + 1 opr ! fveq2 <_ breq12d x cv A 2 ^ opreq2 / 2 opreq1d x cv A ! fveq2 <_ breq12d ax1re leid 2cn 2 exp1t ax-mp / 2 opreq1i 2cn 2re 2pos gt0ne0i divid eqtr fac1 3brtr4 2re 2pos 2 ^ y cv opr ! y cv ` x. 2 opr 2 lediv1t mpan2 mp3an3 2re 2 y cv reexpclt mpan y cv nnnn0t y cv facclt syl ! y cv ` nnret 2re ! y cv ` 2 axmulrcl mpan2 3syl sylanc y cv nnnn0t y cv facclt syl ! y cv ` nncnt 2cn 2re 2pos gt0ne0i 2 ! y cv ` divcan4t mp3an13 3syl 2 ^ y cv opr / 2 opr <_ breq2d bitrd y cv nnge1t y cv nnret ax1re ax1re 1 y cv 1 leadd1t mp3an13 syl mpbid df-2 syl5eqbr 2re 2 y cv + 1 opr ! y cv ` lemul2t mp3anl1 y cv peano2nn y cv + 1 opr nnret syl y cv nnnn0t y cv facclt ! y cv ` nnret 3syl jca y cv nnnn0t y cv facclt ! y cv ` nngt0t 3syl sylanc mpbid 2 ^ y cv opr ! y cv ` x. 2 opr ! y cv ` x. y cv + 1 opr opr letrt 2re 2 y cv reexpclt mpan y cv nnnn0t y cv facclt syl ! y cv ` nnret 2re ! y cv ` 2 axmulrcl mpan2 3syl ! y cv ` y cv + 1 opr axmulrcl y cv nnnn0t y cv facclt ! y cv ` nnret 3syl y cv peano2nn y cv + 1 opr nnret syl sylanc syl3anc mpan2d sylbird 2cn 2 y cv expp1t mpan / 2 opreq1d 2cn 2 y cv expclt mpan 2cn 2re 2pos gt0ne0i 2 2 ^ y cv opr divcan4t mp3an13 syl eqtrd y cv nnnn0t y cv facp1t syl <_ breq12d sylibrd nnind))
+thm (facnzt () () (-> (e. A (NN0)) (=/= (` (!) A) (0))) (A facclt ! A ` nnne0t syl))
+thm (facdivt ((A x) (x y) (B x) (B y)) () (-> (/\/\ (e. A (NN0)) (e. B (NN)) (br B (<_) A)) (e. (opr (` (!) A) (/) B) (NN))) (x cv 0 B <_ breq2 x cv 0 ! fveq2 / B opreq1d NN eleq1d imbi12d B NN e. imbi2d x cv y cv B <_ breq2 x cv y cv ! fveq2 / B opreq1d NN eleq1d imbi12d B NN e. imbi2d x cv y cv + 1 opr B <_ breq2 x cv y cv + 1 opr ! fveq2 / B opreq1d NN eleq1d imbi12d B NN e. imbi2d x cv A B <_ breq2 x cv A ! fveq2 / B opreq1d NN eleq1d imbi12d B NN e. imbi2d B nngt0t B nnret 0re 0 B ltnlet mpan syl mpbid ! 0 ` / B opr NN e. pm2.21d B y cv + 1 opr leloet B nnret y cv peano2nn0 y cv + 1 opr nn0ret syl syl2an B y cv nn0leltp1t B nnnn0t sylan ! y cv ` / B opr y cv + 1 opr nnmulclt y cv nn0p1nnt sylan2 expcom B NN e. adantl ! y cv ` y cv + 1 opr B div23t y cv facclt ! y cv ` nncnt syl B NN e. adantl y cv peano2nn0 y cv + 1 opr nn0ret syl recnd B NN e. adantl B nncnt y cv NN0 e. adantr 3jca B nnne0t y cv NN0 e. adantr sylanc NN eleq1d sylibrd B <_ y cv br imim2d com23 sylbird B y cv + 1 opr ! y cv ` x. opreq2 / B opreq1d NN eleq1d B ! y cv ` divcan4t B nncnt y cv NN0 e. adantr y cv facclt ! y cv ` nncnt syl B NN e. adantl B nnne0t y cv NN0 e. adantr syl3anc y cv facclt B NN e. adantl eqeltrd syl5bi com12 B <_ y cv br ! y cv ` / B opr NN e. -> a1dd jaod sylbid exp com34 com12 imp4d y cv facp1t / B opreq1d NN eleq1d sylibrd exp4d a2d nn0ind 3imp))
+thm (facndivt () () (-> (/\ (/\ (e. A (NN0)) (e. B (NN))) (/\ (br (1) (<) B) (br B (<_) A))) (-. (e. (opr (opr (` (!) A) (+) (1)) (/) B) (ZZ)))) (B recnzt B nnret sylan B <_ A br adantrr A NN0 e. adantll ! A ` + 1 opr / B opr ! A ` / B opr zsubclt exp A B facdivt 3expa ! A ` / B opr nnzt syl 1 < B br adantrl syl5com ! A ` + 1 opr ! A ` B divsubdirt A facclt ! A ` nncnt syl ! A ` peano2cn syl B NN e. 1 < B br B <_ A br /\ ad2antll A facclt ! A ` nncnt syl B NN e. 1 < B br B <_ A br /\ ad2antll B nncnt A NN0 e. 1 < B br B <_ A br /\ ad2antlr 3jca B nnne0t A NN0 e. 1 < B br B <_ A br /\ ad2antlr sylanc A facclt ! A ` nncnt syl 1cn ! A ` 1 pncan2t mpan2 syl / B opreq1d B NN e. 1 < B br B <_ A br /\ ad2antll eqtr3d ZZ eleq1d sylibd mtod))
+thm (bcvalt ((k n) (m n) (N n) (k m) (N k) (N m) (K n) (K k) (K m)) () (-> (/\ (e. N (NN0)) (e. K (ZZ))) (= (opr N (C.) K) (if (/\ (br (0) (<_) K) (br K (<_) N)) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))) (0)))) (! N ` / ! N - K opr ` x. ! K ` opr oprex 0nn0 elisseti 0 <_ K br K <_ N br /\ ifex n cv N k cv <_ breq2 0 <_ k cv br anbi2d ! n cv ` / ! n cv - k cv opr ` x. ! k cv ` opr opr 0 ifbid n cv N ! fveq2 n cv N - k cv opreq1 ! fveq2d x. ! k cv ` opreq1d / opreq12d 0 <_ k cv br k cv <_ N br /\ 0 ifeq1d eqtrd k cv K 0 <_ breq2 k cv K <_ N breq1 anbi12d ! N ` / ! N - k cv opr ` x. ! k cv ` opr opr 0 ifbid k cv K N - opreq2 ! fveq2d k cv K ! fveq2 x. opreq12d ! N ` / opreq2d 0 <_ K br K <_ N br /\ 0 ifeq1d eqtrd n k m df-bc oprabval2))
+thm (bcval3t () () (-> (/\ (/\ (e. N (NN0)) (e. K (ZZ))) (/\ (br (0) (<_) K) (br K (<_) N))) (= (opr N (C.) K) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))))) (N K bcvalt 0 <_ K br K <_ N br /\ ! N ` / ! N - K opr ` x. ! K ` opr opr 0 iftrue sylan9eq))
+thm (bcval2t () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (br K (<_) N)) (= (opr N (C.) K) (opr (` (!) N) (/) (opr (` (!) (opr N (-) K)) (x.) (` (!) K))))) (N K bcval3t K nn0zt N NN0 e. anim2i K <_ N br 3adant3 K nn0ge0t K <_ N br anim1i N NN0 e. 3adant1 sylanc))
+thm (bcval4t () () (-> (/\/\ (e. N (NN0)) (e. K (ZZ)) (\/ (br K (<) (0)) (br N (<) K))) (= (opr N (C.) K) (0))) (0re K 0 ltnlet mpan2 N RR e. adantl N K ltnlet orbi12d N nn0ret K zret syl2an biimpa N K bcvalt 0 <_ K br K <_ N br /\ ! N ` / ! N - K opr ` x. ! K ` opr opr 0 iffalse sylan9eq 0 <_ K br K <_ N br ianor sylan2br syldan 3impa))
+thm (bcval4tOLD () () (-> (/\ (/\ (e. N (NN0)) (e. K (ZZ))) (\/ (br K (<) (0)) (br N (<) K))) (= (opr N (C.) K) (0))) (0re K 0 ltnlet mpan2 N RR e. adantl N K ltnlet orbi12d N nn0ret K zret syl2an biimpa N K bcvalt 0 <_ K br K <_ N br /\ ! N ` / ! N - K opr ` x. ! K ` opr opr 0 iffalse sylan9eq 0 <_ K br K <_ N br ianor sylan2br syldan))
+thm (bccmplt () () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (br K (<_) N)) (= (opr N (C.) K) (opr N (C.) (opr N (-) K)))) (! N - K opr ` ! K ` axmulcom K N nn0subt ancoms biimp3a N - K opr facclt ! N - K opr ` nncnt 3syl K facclt ! K ` nncnt syl N NN0 e. K <_ N br 3ad2ant2 sylanc N K nncant N nn0cnt K nn0cnt syl2an ! fveq2d K <_ N br 3adant3 x. ! N - K opr ` opreq1d eqtr4d ! N ` / opreq2d N K bcval2t N N - K opr bcval2t N NN0 e. K NN0 e. K <_ N br 3simp1 K N nn0subt ancoms biimp3a N K nn0addge1t N K N lesubaddt N RR e. K RR e. pm3.26 N RR e. K RR e. pm3.27 N RR e. K RR e. pm3.26 syl3anc K nn0ret sylan2 mpbird N nn0ret sylan K <_ N br 3adant3 syl3anc 3eqtr4d))
+thm (bcn0t () () (-> (e. N (NN0)) (= (opr N (C.) (0)) (1))) (N nn0ge0t 0nn0 N 0 bcval2t mp3an2 mpdan N nn0cnt N subid1t syl ! fveq2d fac0 N NN0 e. a1i x. opreq12d N facclt ! N ` nncnt syl ! N ` ax1id syl eqtrd ! N ` / opreq2d ! N ` dividt N facclt ! N ` nncnt syl N facnzt sylanc 3eqtrd))
+thm (bcnnt () () (-> (e. N (NN0)) (= (opr N (C.) N) (1))) (N nn0ge0t 0nn0 N 0 bccmplt mp3an2 mpdan N nn0cnt N subid1t syl N C. opreq2d eqtr2d N bcn0t eqtrd))
+thm (bcnp11t () () (-> (e. N (NN0)) (= (opr (opr N (+) (1)) (C.) (1)) (opr N (+) (1)))) (1nn0 N + 1 opr 1 bcval2t mp3an2 N peano2nn0 ax1re 1 N nn0addge2t mpan sylanc N nn0cnt 1cn N 1 pncant mpan2 syl ! fveq2d fac1 N NN0 e. a1i x. opreq12d N facclt ! N ` nncnt syl ! N ` ax1id syl eqtrd ! N + 1 opr ` / opreq2d N facp1t eqcomd ! N + 1 opr ` ! N ` N + 1 opr divmult N peano2nn0 N + 1 opr facclt ! N + 1 opr ` nncnt syl syl N facclt ! N ` nncnt syl N peano2nn0 N + 1 opr nn0cnt syl 3jca N facnzt sylanc mpbird 3eqtrd))
+thm (bcpasc2 () ((bcpasc2.1 (e. N (NN))) (bcpasc2.2 (e. K (NN))) (bcpasc2.3 (br K (<_) N))) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K)) (1cn bcpasc2.2 nncn 1cn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr nn0p1nnt ax-mp nnne0 divadddiv bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulid2 bcpasc2.2 nncn mulid1 + opreq12i bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn bcpasc2.2 nncn add23 bcpasc2.1 nncn bcpasc2.2 nncn N K npcant mp2an + 1 opreq1i 3eqtr / K x. N - K opr + 1 opr opr opreq1i eqtr ! N ` / ! K - 1 opr ` x. ! N - K opr ` opr opr x. opreq2i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nnne0 divcl bcpasc2.2 nncn bcpasc2.2 nnne0 reccl bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr nn0p1nnt ax-mp nnne0 reccl adddi bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nncn bcpasc2.1 nnnn0 1nn0 nn0addcl nn0cn bcpasc2.2 nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulcl bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nnne0 bcpasc2.2 nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr nn0p1nnt ax-mp nnne0 muln0 divmuldiv bcpasc2.1 nnnn0 N facp1t ax-mp bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 1nn0 nn0addcl K N + 1 opr nn0subt mp2an mpbi N + 1 opr - K opr facclt ax-mp nncn bcpasc2.2 nnnn0 K facclt ax-mp nncn mulcom bcpasc2.2 K facnn2t ax-mp bcpasc2.1 nncn 1cn bcpasc2.2 nncn addsub ! fveq2i bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facp1t ax-mp eqtr x. opreq12i bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp nncn bcpasc2.2 nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mul4 3eqtr / opreq12i eqtr4 3eqtr3 bcpasc2.1 nnnn0 N facclt ax-mp nncn mulid1 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp nncn bcpasc2.2 nncn mul23 bcpasc2.2 K facnn2t ax-mp x. ! N - K opr ` opreq1i bcpasc2.2 nnnn0 K facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp nncn mulcom eqtr3 eqtr / opreq12i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nncn 1cn bcpasc2.2 nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nnne0 bcpasc2.2 nnne0 divmuldiv bcpasc2.1 nnnn0 bcpasc2.2 nnnn0 bcpasc2.3 N K bcval2t mp3an 3eqtr4r bcpasc2.1 nnnn0 N facclt ax-mp nncn mulid1 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp nncn bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp nncn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl mulass bcpasc2.1 nncn bcpasc2.2 nncn 1cn N K 1 subsubt mp3an ! fveq2i bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facp1t ax-mp eqtr2 ! K - 1 opr ` x. opreq2i bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp nncn bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnre ax1re bcpasc2.1 nnre lesubadd mpbir bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi bcpasc2.1 nnnn0 K - 1 opr N nn0subt mp2an mpbi N - K - 1 opr opr facclt ax-mp nncn mulcom 3eqtr / opreq12i bcpasc2.1 nnnn0 N facclt ax-mp nncn bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nncn 1cn bcpasc2.1 nncn bcpasc2.2 nncn subcl 1cn addcl bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi K - 1 opr facclt ax-mp bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr facclt ax-mp ! K - 1 opr ` ! N - K opr ` nnmulclt mp2an nnne0 bcpasc2.3 bcpasc2.2 nnnn0 bcpasc2.1 nnnn0 K N nn0subt mp2an mpbi N - K opr nn0p1nnt ax-mp nnne0 divmuldiv bcpasc2.1 nnnn0 bcpasc2.2 K nnge1t ax-mp 1nn0 bcpasc2.2 nnnn0 1 K nn0subt mp2an mpbi bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei bcpasc2.2 nnre ax1re bcpasc2.1 nnre lesubadd mpbir N K - 1 opr bcval2t mp3an 3eqtr4r + opreq12i bcpasc2.1 nnnn0 1nn0 nn0addcl bcpasc2.2 nnnn0 bcpasc2.2 nnre bcpasc2.1 nnnn0 1nn0 nn0addcl nn0re bcpasc2.3 bcpasc2.2 bcpasc2.1 K N nnleltp1t mp2an mpbi ltlei N + 1 opr K bcval2t mp3an 3eqtr4))
+thm (bcpasc2t () () (-> (/\/\ (e. N (NN)) (e. K (NN)) (br K (<_) N)) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K))) (N N NN e. K NN e. K <_ N br /\/\ N 1 if C. K opreq1 N N NN e. K NN e. K <_ N br /\/\ N 1 if C. K - 1 opr opreq1 + opreq12d N N NN e. K NN e. K <_ N br /\/\ N 1 if + 1 opreq1 C. K opreq1d eqeq12d K N NN e. K NN e. K <_ N br /\/\ K 1 if N NN e. K NN e. K <_ N br /\/\ N 1 if C. opreq2 K N NN e. K NN e. K <_ N br /\/\ K 1 if - 1 opreq1 N NN e. K NN e. K <_ N br /\/\ N 1 if C. opreq2d + opreq12d K N NN e. K NN e. K <_ N br /\/\ K 1 if N NN e. K NN e. K <_ N br /\/\ N 1 if + 1 opr C. opreq2 eqeq12d N N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 N N NN e. K NN e. K <_ N br /\/\ N 1 if = K NN e. pm4.2i N N NN e. K NN e. K <_ N br /\/\ N 1 if K <_ breq2 3anbi123d K N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i K N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 K N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ N 1 if = 1 NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ N 1 if 1 <_ breq2 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp1i N N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 N N NN e. K NN e. K <_ N br /\/\ N 1 if = K NN e. pm4.2i N N NN e. K NN e. K <_ N br /\/\ N 1 if K <_ breq2 3anbi123d K N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i K N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 K N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ N 1 if = 1 NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ N 1 if 1 <_ breq2 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp2i N N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 N N NN e. K NN e. K <_ N br /\/\ N 1 if = K NN e. pm4.2i N N NN e. K NN e. K <_ N br /\/\ N 1 if K <_ breq2 3anbi123d K N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i K N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 K N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ N 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ N 1 if = 1 NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ N 1 if 1 <_ breq2 3anbi123d 1 N NN e. K NN e. K <_ N br /\/\ K 1 if = N NN e. K NN e. K <_ N br /\/\ N 1 if NN e. pm4.2i 1 N NN e. K NN e. K <_ N br /\/\ K 1 if NN eleq1 1 N NN e. K NN e. K <_ N br /\/\ K 1 if <_ N NN e. K NN e. K <_ N br /\/\ N 1 if breq1 3anbi123d 1nn 1nn ax1re leid 3pm3.2i elimhyp2v 3simp3i bcpasc2 dedth2v))
+thm (bcpasc () ((bcpasc.1 (e. N (NN0))) (bcpasc.2 (e. K (ZZ)))) (= (opr (opr N (C.) K) (+) (opr N (C.) (opr K (-) (1)))) (opr (opr N (+) (1)) (C.) K)) (bcpasc.1 N elnn0 mpbi ori 1cn addid1 K 0 0 C. opreq2 0nn0 0 bcn0t ax-mp syl6eq K 0 - 1 opreq1 1 df-neg syl6eqr 0 C. opreq2d 0nn0 1z 1 znegclt ax-mp lt01 ax1re 1 lt0neg2t ax-mp mpbi 1 -u < 0 br 0 < 1 -u br orc ax-mp 0 1 -u bcval4t mp3an syl6eq + opreq12d K 0 1 C. opreq2 1nn0 1 bcn0t ax-mp syl6eq eqeq12d mpbiri bcpasc.2 zre 0re lttri2 0cn addid1 K < 0 br a1i 0nn0 bcpasc.2 0 K bcval4t mp3an12 orci bcpasc.2 zre leid bcpasc.2 bcpasc.2 K K zlem1ltt mp2an mpbi bcpasc.2 zre ax1re resubcl bcpasc.2 zre 0re lttr mpan K - 1 opr < 0 br 0 < K - 1 opr br orc 0nn0 bcpasc.2 1z K 1 zsubclt mp2an 0 K - 1 opr bcval4t mp3an12 3syl + opreq12d 1nn0 bcpasc.2 1 K bcval4t mp3an12 orci 3eqtr4d 0nn0 bcpasc.2 0 K bcval4t mp3an12 olci + 0 C. K - 1 opr opr opreq1d 0z bcpasc.2 0 K zltp1let mp2an 1cn addid2 <_ K breq1i ax1re bcpasc.2 zre leloe 3bitr 1nn0 bcpasc.2 1 K bcval4t mp3an12 olci ax1re bcpasc.2 zre posdif 0 < K - 1 opr br K - 1 opr < 0 br olc sylbi 0nn0 bcpasc.2 1z K 1 zsubclt mp2an 0 K - 1 opr bcval4t mp3an12 syl eqcomd 0 + opreq2d 0cn addid1 syl5eqr eqtr2d 1 K - 1 opreq1 1cn subid syl5eqr 0 C. opreq2d 0nn0 0 bcn0t ax-mp syl5eqr 0 + opreq2d 1cn addid2 syl5eqr 1 K 1 C. opreq2 1nn0 1 bcnnt ax-mp syl5eqr eqtr3d jaoi sylbi eqtrd jaoi sylbi pm2.61i N 0 C. K opreq1 N 0 C. K - 1 opr opreq1 + opreq12d N 0 + 1 opreq1 1cn addid2 syl6eq C. K opreq1d eqeq12d mpbiri syl bcpasc.2 K znnnlt1t ax-mp bcpasc.2 0z K 0 zleltp1t mp2an bcpasc.2 zre 0re leloe 1cn addid2 K < breq2i 3bitr3r bitr bcpasc.1 bcpasc.2 N K bcval4t mp3an12 orci bcpasc.2 zre leid bcpasc.2 bcpasc.2 K K zlem1ltt mp2an mpbi bcpasc.2 zre ax1re resubcl bcpasc.2 zre 0re lttr mpan K - 1 opr < 0 br N < K - 1 opr br orc bcpasc.1 bcpasc.2 1z K 1 zsubclt mp2an N K - 1 opr bcval4t mp3an12 3syl + opreq12d 0cn addid1 syl6eq bcpasc.1 1nn0 nn0addcl bcpasc.2 N + 1 opr K bcval4t mp3an12 orci eqtr4d K 0 N C. opreq2 bcpasc.1 N bcn0t ax-mp syl6eq K 0 - 1 opreq1 1 df-neg lt01 ax1re 1 lt0neg2t ax-mp mpbi eqbrtrr syl6eqbr K - 1 opr < 0 br N < K - 1 opr br orc bcpasc.1 bcpasc.2 1z K 1 zsubclt mp2an N K - 1 opr bcval4t mp3an12 3syl + opreq12d 1cn addid1 syl6eq K 0 N + 1 opr C. opreq2 bcpasc.1 1nn0 nn0addcl N + 1 opr bcn0t ax-mp syl6req eqtrd jaoi sylbi bcpasc.1 nn0re bcpasc.2 zre ltnle bcpasc.1 bcpasc.2 N K bcval4t mp3an12 olci + N C. K - 1 opr opr opreq1d bcpasc.1 N nn0zt ax-mp bcpasc.2 N K zltp1let mp2an bcpasc.1 nn0re ax1re readdcl bcpasc.2 zre leloe bitr bcpasc.1 nn0re ax1re bcpasc.2 zre ltaddsub N < K - 1 opr br K - 1 opr < 0 br olc sylbi bcpasc.1 bcpasc.2 1z K 1 zsubclt mp2an N K - 1 opr bcval4t mp3an12 syl 0 + opreq2d 0cn addid1 syl6eq bcpasc.1 1nn0 nn0addcl bcpasc.2 N + 1 opr K bcval4t mp3an12 olci eqtr4d N + 1 opr K - 1 opreq1 bcpasc.1 nn0cn 1cn N 1 pncant mp2an syl5eqr N C. opreq2d bcpasc.1 N bcnnt ax-mp syl5eqr 0 + opreq2d 1cn addid2 syl5eqr N + 1 opr K N + 1 opr C. opreq2 bcpasc.1 1nn0 nn0addcl N + 1 opr bcnnt ax-mp syl5eqr eqtr3d jaoi sylbi eqtrd sylbir N K bcpasc2t 3ecase))
+thm (bccl2t ((k n) (N k) (N n) (j n) (j k) (K k)) () (-> (/\/\ (e. N (NN0)) (e. K (NN0)) (br K (<_) N)) (e. (opr N (C.) K) (NN))) (k cv K <_ N breq1 k cv K N C. opreq2 NN eleq1d imbi12d NN rcla4v n cv 1 k cv <_ breq2 n cv 1 C. k cv opreq1 NN eleq1d imbi12d k NN biraldv n cv j cv k cv <_ breq2 n cv j cv C. k cv opreq1 NN eleq1d imbi12d k NN biraldv n cv j cv + 1 opr k cv <_ breq2 n cv j cv + 1 opr C. k cv opreq1 NN eleq1d imbi12d k NN biraldv n cv N k cv <_ breq2 n cv N C. k cv opreq1 NN eleq1d imbi12d k NN biraldv k cv nnle1eq1t 1nn0 1 bcnnt ax-mp 1nn eqeltr k cv 1 1 C. opreq2 NN eleq1d mpbiri syl6bi rgen j cv NN e. k ax-17 k NN k cv <_ j cv br j cv C. k cv opr NN e. -> hbra1 k cv j cv + 1 opr leloet k cv nnret j cv peano2nn j cv + 1 opr nnret syl syl2an k cv j cv nnleltp1t k cv j cv + 1 opr = orbi1d bitr4d k cv nnge1t k cv nnret ax1re 1 k cv leloet mpan syl mpbid j cv NN e. adantr j cv C. k cv opr j cv C. k cv - 1 opr opr nnaddclt k NN k cv <_ j cv br j cv C. k cv opr NN e. -> ra4 imp32 ancoms j cv NN e. adantllr 1 < k cv br adantlrl k cv nnzt k cv peano2zr syl 1nn 1 k cv nnltlem1t mpan biimpd k cv - 1 opr ZZ e. anim2d k cv - 1 opr elnnz1 syl6ibr mpand n cv k cv - 1 opr <_ j cv breq1 n cv k cv - 1 opr j cv C. opreq2 NN eleq1d imbi12d NN rcla4v k cv n cv <_ j cv breq1 k cv n cv j cv C. opreq2 NN eleq1d imbi12d NN cbvralv syl5ib com23 syl6 j cv NN e. adantr 1nn0 k cv 1 nn0addge1t mpan2 ax1re k cv 1 k cv lesubaddt mp3an2 anidms mpbird j cv RR e. adantr k cv - 1 opr k cv j cv letrt 3expa ax1re k cv 1 resubclt mpan2 ancri sylan mpand k cv nnret j cv nnret syl2an syl5d imp42 sylanc j cv k cv bcpasc2t 3com12 3expa NN eleq1d 1 < k cv br adantrl k NN k cv <_ j cv br j cv C. k cv opr NN e. -> A.e. adantr mpbid exp31 com12 1 k cv j cv + 1 opr C. opreq2 NN eleq1d j cv nnnn0t j cv bcnp11t syl j cv peano2nn eqeltrd syl5bi k cv NN e. adantld k NN k cv <_ j cv br j cv C. k cv opr NN e. -> A.e. a1dd k cv j cv + 1 opr C. k cv opreq1 NN eleq1d 1nn k cv nnnn0t k cv bcnnt syl NN eleq1d mpbiri syl5bi j cv NN e. adantrd k NN k cv <_ j cv br j cv C. k cv opr NN e. -> A.e. a1dd ccase2 com12 mpand sylbid exp com23 com4t r19.21ad nnind syl5com imp N 0 K <_ breq2 negbid K nngt0t K nnret 0re 0 K ltnlet mpan syl mpbid syl5bir imp N C. K opr NN e. pm2.21d 1nn N nnnn0t N bcn0t syl NN eleq1d mpbiri K 0 = adantr K 0 N C. opreq2 NN eleq1d N NN e. adantl mpbird K <_ N br a1d 0nn0 0 bcn0t ax-mp 1nn eqeltr N 0 K 0 C. opreq12 NN eleq1d mpbiri K <_ N br a1d ccase N elnn0 K elnn0 syl2anb imp 3impa))
+thm (bcclt () () (-> (/\ (e. N (NN0)) (e. K (ZZ))) (e. (opr N (C.) K) (NN0))) (N NN0 e. id K elnn0z biimpr K <_ N br id 3anim123i 3exp exp3a imp43 N K bccl2t N C. K opr nnnn0t 3syl N K bcvalt 0 <_ K br K <_ N br /\ ! N ` / ! N - K opr ` x. ! K ` opr opr 0 iffalse sylan9eq 0nn0 N NN0 e. K ZZ e. /\ 0 <_ K br K <_ N br /\ -. /\ a1i eqeltrd pm2.61an2))
+thm (fzvalt ((k m) (k n) (k z) (M k) (m n) (m z) (M m) (n z) (M n) (M z) (N k) (N m) (N n) (N z)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (= (opr M (...) N) ({e.|} k (ZZ) (/\ (br M (<_) (cv k)) (br (cv k) (<_) N))))) (zex k M <_ k cv br k cv <_ N br /\ rabex m cv M <_ k cv breq1 k cv <_ n cv br anbi1d k ZZ birabsdv n cv N k cv <_ breq2 M <_ k cv br anbi2d k ZZ birabsdv m n z k df-fz oprabval2))
+thm (elfz1t ((K j) (M j) (N j)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\/\ (e. K (ZZ)) (br M (<_) K) (br K (<_) N)))) (M N j fzvalt K eleq2d j cv K M <_ breq2 j cv K <_ N breq1 anbi12d ZZ elrab K ZZ e. M <_ K br K <_ N br 3anass bitr4 syl6bb))
+thm (elfzt () () (-> (/\/\ (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\ (br M (<_) K) (br K (<_) N)))) (M N K elfz1t K ZZ e. M <_ K br K <_ N br 3anass baib sylan9bb 3impa 3comr))
+thm (elfz2t ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))))) (M N K elfz1t M ZZ e. N ZZ e. /\ K ZZ e. M <_ K br K <_ N br /\ /\ ibar K ZZ e. M <_ K br K <_ N br 3anass syl5bb bitrd N A e. adantl zex k m cv <_ k cv br k cv <_ n cv br /\ rabex m n z k df-fz dmoprab2 N A M ndmoprg K eleq2d K noel M ZZ e. N ZZ e. /\ pm2.21i M ZZ e. N ZZ e. /\ K ZZ e. M <_ K br K <_ N br /\ /\ pm3.26 pm5.21ni N A e. adantl bitrd pm2.61an2 M ZZ e. N ZZ e. K ZZ e. df-3an M <_ K br K <_ N br /\ anbi1i M ZZ e. N ZZ e. /\ K ZZ e. M <_ K br K <_ N br /\ anass bitr2 syl6bb))
+thm (elfzlem ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. K (opr M (...) N)) (/\ (/\ (e. M (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N)))) (N V K M elfz2t M ZZ e. N ZZ e. K ZZ e. 3simpb M <_ K br K <_ N br /\ anim1i syl6bi N M ... oprprc2 K eleq2d N K <_ brprc M <_ K br anbi2d M ZZ e. K ZZ e. /\ anbi2d M V K M elfz2t M ZZ e. M ZZ e. K ZZ e. 3simpb M <_ K br K <_ M br pm3.26 anim12i syl6bi M V K M elfz2t K zret K leidt syl M ZZ e. M ZZ e. 3ad2ant3 M <_ K br K <_ M br /\ adantr syl6bi jcad M ZZ e. K ZZ e. /\ M <_ K br K <_ K br anass syl6ib ZZ ZZ relxp zex k m cv <_ k cv br k cv <_ n cv br /\ rabex m n z k df-fz dmoprab2 ... dom ZZ ZZ X. releq ax-mp mpbir M M oprprc1 M ... M opr {/} = M ZZ e. K ZZ e. /\ M <_ K br K <_ K br /\ /\ pm2.24 syl K M ... M opr n0i syl5 pm2.61i syl5bir sylbid pm2.61i))
+thm (elfz4t () () (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))) (e. K (opr M (...) N))) (M N K elfz1t biimprcd 3exp com4r exp 3imp imp32))
+thm (elfzuzb () () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))))) (N A K M elfz2t M ZZ e. K ZZ e. M <_ K br K ZZ e. N ZZ e. K <_ N br an6 K M eluz2t N K eluz2t anbi12i M ZZ e. N ZZ e. K ZZ e. df-3an M ZZ e. N ZZ e. K ZZ e. anandir N ZZ e. K ZZ e. ancom M ZZ e. K ZZ e. /\ anbi2i 3bitr M <_ K br K <_ N br /\ anbi1i M ZZ e. K ZZ e. /\ K ZZ e. N ZZ e. /\ M <_ K br K <_ N br /\ df-3an bitr4 3bitr4r syl6bb))
+thm (eluzfzt () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))) (e. K (opr M (...) N))) (N K eluzelz K ZZ> M ` e. adantl N ZZ K M elfzuzb biimpar mpancom))
+thm (elfzel2 () ((elfzel2.1 (e. N (V)))) (-> (e. K (opr M (...) N)) (e. N (ZZ))) (elfzel2.1 N V K M elfz2t ax-mp M ZZ e. N ZZ e. K ZZ e. 3simp2 M <_ K br K <_ N br /\ adantr sylbi))
+thm (elfzel1 () () (-> (e. K (opr M (...) N)) (e. M (ZZ))) (K M N elfzlem M ZZ e. K ZZ e. pm3.26 M <_ K br K <_ N br /\ adantr syl))
+thm (elfzelz () () (-> (e. K (opr M (...) N)) (e. K (ZZ))) (K M N elfzlem M ZZ e. K ZZ e. pm3.27 M <_ K br K <_ N br /\ adantr syl))
+thm (elfzle1 () () (-> (e. K (opr M (...) N)) (br M (<_) K)) (K M N elfzlem M <_ K br K <_ N br pm3.26 M ZZ e. K ZZ e. /\ adantl syl))
+thm (elfzle2 () () (-> (e. K (opr M (...) N)) (br K (<_) N)) (K M N elfzlem M <_ K br K <_ N br pm3.27 M ZZ e. K ZZ e. /\ adantl syl))
+thm (eluzfz1t () () (-> (e. N (` (ZZ>) M)) (e. M (opr M (...) N))) (N M eluz2t M N M elfz4t M ZZ e. N ZZ e. M <_ N br 3simp1 M ZZ e. N ZZ e. M <_ N br 3simp2 M ZZ e. N ZZ e. M <_ N br 3simp1 3jca M zret M leidt syl N ZZ e. M <_ N br 3ad2ant1 M ZZ e. N ZZ e. M <_ N br 3simp3 jca sylanc sylbi))
+thm (eluzfz2t () () (-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))) (M N N elfz4t N M eluzel2 N M eluzelz N M eluzelz 3jca N M eluzle N M eluzelz N zret N leidt 3syl jca sylanc))
+thm (eluzfz2tOLD () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. N (opr M (...) N))) (N M eluz2t N M eluzfz2t sylbir))
+thm (eluzfz2b () () (<-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))) (N M eluzfz2t N M N elfzel1 N M N elfzelz N M N elfzle1 3jca N M eluz2t sylibr impbi))
+thm (elfz3t () () (-> (e. N (ZZ)) (e. N (opr N (...) N))) (N uzidt N N eluzfz1t syl))
+thm (elfz1eqt () () (-> (e. K (opr N (...) N)) (= K N)) (K N N elfzlem N K letri3t N zret K zret syl2an biimpar eqcomd syl))
+thm (elfzle3 () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) N))) (br M (<_) N)) (M N K elfz1t M K N letrt M zret K zret N zret syl3an 3expa an1rs exp4b com4l 3imp com12 sylbid imp 3impa))
+thm (elfznn0t () () (-> (e. K (opr (0) (...) N)) (e. K (NN0))) (K 0 N elfzelz K 0 N elfzle1 jca K elnn0z sylibr))
+thm (fzaddelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (+) K) (opr (opr M (+) K) (...) (opr N (+) K))))) (J K zaddclt J ZZ e. a1d J ZZ e. K ZZ e. pm3.26 J + K opr ZZ e. a1d impbid M ZZ e. N ZZ e. /\ adantl M J K leadd1t M zret J zret K zret syl3an 3expb N ZZ e. adantlr J N K leadd1t J zret N zret K zret syl3an 3com12 3expb M ZZ e. adantll 3anbi123d M N J elfz1t J ZZ e. K ZZ e. /\ adantr M + K opr N + K opr J + K opr elfz1t M K zaddclt N K zaddclt syl2an anandirs J ZZ e. adantrl 3bitr4d))
+thm (fzsubelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (-) K) (opr (opr M (-) K) (...) (opr N (-) K))))) (M N J K -u fzaddelt K znegclt sylanr2 J K subnegt M CC e. N CC e. /\ adantl M K subnegt N K subnegt ... opreqan12d anandirs J CC e. adantrl eleq12d M zcnt N zcnt anim12i J zcnt K zcnt anim12i syl2an bitr4d))
+thm (fzss1t ((M k) (N k) (K k)) () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (C_ (opr K (...) N) (opr M (...) N))) (K M eluzle k cv ZZ e. adantr M K k cv letrt M zret K zret k cv zret syl3an 3expa K M eluzel2 K M eluzelz jca sylan mpand k cv <_ N br anim1d exp r19.21aiv k ZZ K <_ k cv br k cv <_ N br /\ M <_ k cv br k cv <_ N br /\ ss2rab sylibr N ZZ e. adantr K N k fzvalt K M eluzelz sylan M N k fzvalt K M eluzel2 sylan 3sstr4d))
+thm (fzss1tOLD ((M k) (N k) (K k)) () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (br M (<_) K))) (C_ (opr K (...) N) (opr M (...) N))) (M K k cv letrt M zret K zret k cv zret syl3an exp3a 3exp com34 imp42 k cv <_ N br anim1d exp N ZZ e. adantlr r19.21aiv k ZZ K <_ k cv br k cv <_ N br /\ M <_ k cv br k cv <_ N br /\ ss2rab sylibr K N k fzvalt ancoms M ZZ e. adantll M <_ K br adantrr M N k fzvalt K ZZ e. M <_ K br /\ adantr 3sstr4d))
+thm (fzss2tOLD ((M k) (N k) (K k)) () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (br K (<_) N))) (C_ (opr M (...) K) (opr M (...) N))) (k cv K N letrt k cv zret K zret N zret syl3an exp3a com23 3exp com23 com4l imp42 M <_ k cv br anim2d exp M ZZ e. adantll r19.21aiv k ZZ M <_ k cv br k cv <_ K br /\ M <_ k cv br k cv <_ N br /\ ss2rab sylibr M K k fzvalt N ZZ e. adantlr K <_ N br adantrr M N k fzvalt K ZZ e. K <_ N br /\ adantr 3sstr4d))
+thm (fzssp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr M (...) N) (opr M (...) (opr N (+) (1))))) (M N + 1 opr N fzss2tOLD N peano2z M ZZ e. anim2i M ZZ e. N ZZ e. pm3.27 N N leplus1t N zret N zret N zret N leidt syl syl3anc M ZZ e. adantl jca sylanc))
+thm (fzp1sst () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr (opr M (+) (1)) (...) N) (opr M (...) N))) (M peano2z N ZZ e. adantr M M leplus1t M zret M zret M zret M leidt syl syl3anc N ZZ e. adantr jca M N M + 1 opr fzss1tOLD mpdan))
+thm (elfzp1 () () (-> (e. N (` (ZZ>) M)) (<-> (e. K (opr M (...) (opr N (+) (1)))) (\/ (e. K (opr M (...) N)) (= K (opr N (+) (1)))))) (N M eluzel2 K M ... N + 1 opr opr e. K N + 1 opr = -. ad2antll N M eluzelz K M ... N + 1 opr opr e. K N + 1 opr = -. ad2antll K M N + 1 opr elfzelz N ZZ> M ` e. K N + 1 opr = -. ad2antlr 3jca K M N + 1 opr elfzle1 N ZZ> M ` e. K N + 1 opr = -. ad2antlr K M N + 1 opr elfzle2 N ZZ> M ` e. adantl K N + 1 opr = -. anim1i K N + 1 opr ltlent K M N + 1 opr elfzelz K zret syl N ZZ> M ` e. K N + 1 opr = -. ad2antlr N M eluzelz N zret N peano2re 3syl K M ... N + 1 opr opr e. K N + 1 opr = -. ad2antll sylanc mpbird K N zleltp1t K M N + 1 opr elfzelz N ZZ> M ` e. K N + 1 opr = -. ad2antlr N M eluzelz K M ... N + 1 opr opr e. K N + 1 opr = -. ad2antll sylanc mpbird jca jca exp N ZZ> M ` K M elfz2t K M ... N + 1 opr opr e. adantr sylibrd orrd K N + 1 opr = K M ... N opr e. orcom sylib exp M N fzssp1t N M eluzel2 N M eluzelz sylanc K sseld K N + 1 opr M ... N + 1 opr opr eleq1 N M peano2uz N + 1 opr M eluzfz2t syl syl5bir com12 jaod impbid))
+thm (fznn0t () () (-> (e. N (NN0)) (<-> (e. K (opr (0) (...) N)) (/\ (e. K (NN0)) (br K (<_) N)))) (N nn0zt 0z jctil 0 N K elfz1t syl K ZZ e. 0 <_ K br K <_ N br df-3an K elnn0z K <_ N br anbi1i bitr4 syl6bb))
+thm (fsequb ((j k) (j m) (j n) (j x) (j y) (F j) (k m) (k n) (k x) (k y) (F k) (m n) (m x) (m y) (F m) (n x) (n y) (F n) (x y) (F x) (F y) (M j) (M k) (M m) (M n) (M x) (M y) (N j) (N k) (N n) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (E.e. x (RR) (A.e. k (opr M (...) N) (br (` F (cv k)) (<) (cv x))))) (j cv M M ... opreq2 M ... j cv opr M ... M opr n F n cv ` RR e. raleq syl j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` < x cv br raleq syl x RR birexdv imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr n F n cv ` RR e. raleq syl j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` < x cv br raleq syl x RR birexdv imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr n F n cv ` RR e. raleq syl j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` < x cv br raleq syl x RR birexdv imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr n F n cv ` RR e. raleq syl j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` < x cv br raleq syl x RR birexdv imbi12d M elfz3t n cv M F fveq2 RR eleq1d M ... M opr rcla4v syl F M ` peano2re k cv M F fveq2 < F M ` + 1 opr breq1d F M ` ltplus1t syl5bir com12 k cv M elfz1eqt syl5 r19.21aiv jca M ZZ e. a1i x cv F M ` + 1 opr F k cv ` < breq2 k M ... M opr biraldv RR rcla4ev syl6 syld M m cv fzssp1t m cv M eluzel2 m cv M eluzelz sylanc n cv sseld F n cv ` RR e. imim1d r19.20dv2 x RR k M ... m cv opr F k cv ` < x cv br A.e. E.e. imim1d imp32 n cv m cv + 1 opr F fveq2 RR eleq1d M ... m cv + 1 opr opr rcla4va m cv M peano2uz m cv + 1 opr M eluzfz2t syl sylan n M ... m cv opr F n cv ` RR e. A.e. x RR k M ... m cv opr F k cv ` < x cv br A.e. E.e. -> adantrl m cv M k cv elfzp1 n M ... m cv + 1 opr opr F n cv ` RR e. A.e. adantr x cv RR e. F m cv + 1 opr ` RR e. /\ k cv M ... m cv opr e. F k cv ` < x cv br -> ad2antll F m cv + 1 opr ` + 1 opr x cv max2 F m cv + 1 opr ` peano2re sylan ancoms m cv ZZ> M ` e. n M ... m cv + 1 opr opr F n cv ` RR e. A.e. /\ k cv M ... m cv opr e. ad2antlr F k cv ` x cv F m cv + 1 opr ` + 1 opr <_ x cv br x cv F m cv + 1 opr ` + 1 opr if ltletrt n cv k cv F fveq2 RR eleq1d M ... m cv + 1 opr opr rcla4va M m cv fzssp1t m cv M eluzel2 m cv M eluzelz sylanc k cv sseld imp sylan an1rs x cv RR e. F m cv + 1 opr ` RR e. /\ adantlr x cv RR e. F m cv + 1 opr ` RR e. pm3.26 m cv ZZ> M ` e. n M ... m cv + 1 opr opr F n cv ` RR e. A.e. /\ k cv M ... m cv opr e. ad2antlr x cv RR F m cv + 1 opr ` + 1 opr F m cv + 1 opr ` + 1 opr <_ x cv br ifcl F m cv + 1 opr ` peano2re sylan2 m cv ZZ> M ` e. n M ... m cv + 1 opr opr F n cv ` RR e. A.e. /\ k cv M ... m cv opr e. ad2antlr syl3anc mpan2d exp a2d imp k cv m cv + 1 opr F fveq2 < F m cv + 1 opr ` + 1 opr <_ x cv br x cv F m cv + 1 opr ` + 1 opr if breq1d F m cv + 1 opr ` ltplus1t x cv RR e. adantl F m cv + 1 opr ` peano2re F m cv + 1 opr ` + 1 opr x cv max1ALT syl x cv RR e. adantl F m cv + 1 opr ` F m cv + 1 opr ` + 1 opr F m cv + 1 opr ` + 1 opr <_ x cv br x cv F m cv + 1 opr ` + 1 opr if ltletrt x cv RR e. F m cv + 1 opr ` RR e. pm3.27 F m cv + 1 opr ` peano2re x cv RR e. adantl x cv RR F m cv + 1 opr ` + 1 opr F m cv + 1 opr ` + 1 opr <_ x cv br ifcl F m cv + 1 opr ` peano2re sylan2 syl3anc mp2and syl5bir com12 m cv ZZ> M ` e. n M ... m cv + 1 opr opr F n cv ` RR e. A.e. /\ k cv M ... m cv opr e. F k cv ` < x cv br -> ad2antlr jaod sylbid exp r19.20dv2 x cv RR F m cv + 1 opr ` + 1 opr F m cv + 1 opr ` + 1 opr <_ x cv br ifcl F m cv + 1 opr ` peano2re sylan2 m cv ZZ> M ` e. n M ... m cv + 1 opr opr F n cv ` RR e. A.e. /\ adantl jctild y cv F m cv + 1 opr ` + 1 opr <_ x cv br x cv F m cv + 1 opr ` + 1 opr if F k cv ` < breq2 k M ... m cv + 1 opr opr biraldv RR rcla4ev syl6 exp32 com34 r19.23adv imp3a y cv x cv F k cv ` < breq2 k M ... m cv + 1 opr opr biraldv RR cbvrexv syl6ib n M ... m cv opr F n cv ` RR e. A.e. x RR k M ... m cv opr F k cv ` < x cv br A.e. E.e. -> adantrl mp2and exp32 uzind4 imp k cv n cv F fveq2 RR eleq1d M ... N opr cbvralv sylan2b))
+thm (fsumvalt ((f x) (x z) (M x) (f z) (M f) (M z) (N x) (N f) (N z) (F x) (F f) (F z)) () (-> (/\/\ (e. F (V)) (e. M A) (e. N B)) (= (opr (<,> M N) (sum) F) (if (br M (<_) N) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)) (0)))) (M N opex + seq0 F shift 1st M N <,> ` -u opr opr 2nd M N <,> ` - 1st M N <,> ` opr fvex 0z elisseti 1st M N <,> ` <_ 2nd M N <,> ` br ifex x cv M N <,> 1st fveq2 x cv M N <,> 2nd fveq2 <_ breq12d + seq0 f cv shift 1st x cv ` -u opr opr 2nd x cv ` - 1st x cv ` opr ` 0 ifbid x cv M N <,> 1st fveq2 negeqd f cv shift opreq2d + seq0 opreq2d 2nd x cv ` - 1st x cv ` opr fveq1d x cv M N <,> 2nd fveq2 x cv M N <,> 1st fveq2 - opreq12d + seq0 f cv shift 1st M N <,> ` -u opr opr fveq2d eqtrd 1st M N <,> ` <_ 2nd M N <,> ` br 0 ifeq1d eqtrd f cv F shift 1st M N <,> ` -u opreq1 + seq0 opreq2d 2nd M N <,> ` - 1st M N <,> ` opr fveq1d 1st M N <,> ` <_ 2nd M N <,> ` br 0 ifeq1d x f z df-fsum V V oprabval5 mpan M A N op1stg N B e. adantr M A N B op2ndg <_ breq12d + seq0 F shift 1st M N <,> ` -u opr opr 2nd M N <,> ` - 1st M N <,> ` opr ` 0 ifbid M A N op1stg N B e. adantr negeqd F shift opreq2d + seq0 opreq2d 2nd M N <,> ` - 1st M N <,> ` opr fveq1d M A N B op2ndg M A N op1stg N B e. adantr - opreq12d + seq0 F shift M -u opr opr fveq2d eqtrd M <_ N br 0 ifeq1d eqtrd sylan9eqr 3impa 3comr))
+thm (fsumalt () () (= (opr (<,> M N) (sum) F) (` (sum) (<,> (<,> M N) F))) (M N <,> sum F df-opr))
+thm (fsumval2t () ((fsumval2t.1 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M N) (sum) F) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)))) (N M eluz2t fsumval2t.1 F M ZZ N ZZ fsumvalt mp3an1 M <_ N br + seq0 F shift M -u opr opr N - M opr ` 0 iftrue sylan9eq 3impa sylbi))
+thm (fsumval2tOLD () ((fsumval2t.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M N) (sum) F) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)))) (fsumval2t.1 F M ZZ N ZZ fsumvalt mp3an1 M <_ N br + seq0 F shift M -u opr opr N - M opr ` 0 iftrue sylan9eq 3impa))
+thm (fsumval3t () ((fsumval2t.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br N (<) M)) (= (opr (<,> M N) (sum) F) (0))) (fsumval2t.1 F M ZZ N ZZ fsumvalt mp3an1 N < M br 3adant3 N M ltnlet ancoms M zret N zret syl2an biimp3a M <_ N br + seq0 F shift M -u opr opr N - M opr ` 0 iffalse syl eqtrd))
+thm (fsum1 () ((fsumval2t.1 (e. F (V)))) (-> (e. M (ZZ)) (= (opr (<,> M M) (sum) F) (` F M))) (M zret M leidt syl M ZZ e. adantr fsumval2t.1 M M fsumval2tOLD 3expa mpdan anidms M zcnt M subidt syl + seq0 F shift M -u opr opr fveq2d addex F shift M -u oprex seq00 syl6eq M zcnt 0cn fsumval2t.1 M 0 shftval4t mpan2 syl M zcnt M ax0id syl F fveq2d eqtrd 3eqtrd))
+thm (fsump1 () ((fsumval2t.1 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M (opr N (+) (1))) (sum) F) (opr (opr (<,> M N) (sum) F) (+) (` F (opr N (+) (1)))))) (N M eluz2t M N znn0sub2t sylbi addex F shift M -u oprex N - M opr seq0p1 syl N M peano2uz fsumval2t.1 N + 1 opr M fsumval2t syl 1cn N 1 M addsubt mp3an2 ancoms N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc + seq0 F shift M -u opr opr fveq2d eqtrd fsumval2t.1 N M fsumval2t N M subclt ancoms N - M opr peano2cn syl fsumval2t.1 M N - M opr + 1 opr shftval4t syldan N M subclt ancoms 1cn M N - M opr 1 axaddass mp3an3 syldan M N pncan3t + 1 opreq1d eqtr3d F fveq2d eqtr2d N M eluzel2 M zcnt syl N M eluzelz N zcnt syl sylanc + opreq12d 3eqtr4d))
+thm (fsump1OLD () ((fsumval2t.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M (opr N (+) (1))) (sum) F) (opr (opr (<,> M N) (sum) F) (+) (` F (opr N (+) (1)))))) (M N znn0sub2t addex F shift M -u oprex N - M opr seq0p1 syl fsumval2t.1 M N + 1 opr fsumval2tOLD M ZZ e. N ZZ e. M <_ N br 3simp1 N peano2z M ZZ e. M <_ N br 3ad2ant2 M N leplus1t N zret syl3an2 M zret syl3an1 syl3anc 1cn N 1 M addsubt mp3an2 ancoms M zcnt N zcnt syl2an M <_ N br 3adant3 + seq0 F shift M -u opr opr fveq2d eqtrd fsumval2t.1 M N fsumval2tOLD fsumval2t.1 M N - M opr + 1 opr shftval4t M CC e. N CC e. pm3.26 N M subclt ancoms N - M opr peano2cn syl sylanc 1cn M N - M opr 1 axaddass mp3an3 M CC e. N CC e. pm3.26 N M subclt ancoms sylanc M N pncan3t + 1 opreq1d eqtr3d F fveq2d eqtr2d M zcnt N zcnt syl2an M <_ N br 3adant3 + opreq12d 3eqtr4d))
+thm (fsumserz ((F j) (F k) (j k) (M j) (M k) (N j)) ((fsumval2t.1 (e. F (V)))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M N) (sum) F) (` (opr (<,> M (+)) (seq) F) N))) (j cv M M opeq2 sum F opreq1d j cv M M + <,> seq F opr fveq2 eqeq12d j cv k cv M opeq2 sum F opreq1d j cv k cv M + <,> seq F opr fveq2 eqeq12d j cv k cv + 1 opr M opeq2 sum F opreq1d j cv k cv + 1 opr M + <,> seq F opr fveq2 eqeq12d j cv N M opeq2 sum F opreq1d j cv N M + <,> seq F opr fveq2 eqeq12d fsumval2t.1 M fsum1 addex fsumval2t.1 M seqz1 eqtr4d M k cv <,> sum F opr M + <,> seq F opr k cv ` + F k cv + 1 opr ` opreq1 k cv ZZ> M ` e. adantl fsumval2t.1 k cv M fsump1 M k cv <,> sum F opr M + <,> seq F opr k cv ` = adantr addex fsumval2t.1 k cv M seqzp1 M k cv <,> sum F opr M + <,> seq F opr k cv ` = adantr 3eqtr4d exp uzind4))
+thm (fsumserzOLD ((F j) (F k) (j k) (M j) (M k) (N j)) ((fsumval2t.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M N) (sum) F) (` (opr (<,> M (+)) (seq) F) N))) (j cv M M opeq2 sum F opreq1d j cv M M + <,> seq F opr fveq2 eqeq12d j cv k cv M opeq2 sum F opreq1d j cv k cv M + <,> seq F opr fveq2 eqeq12d j cv k cv + 1 opr M opeq2 sum F opreq1d j cv k cv + 1 opr M + <,> seq F opr fveq2 eqeq12d j cv N M opeq2 sum F opreq1d j cv N M + <,> seq F opr fveq2 eqeq12d fsumval2t.1 M fsum1 addex fsumval2t.1 M seqz1 eqtr4d M k cv <,> sum F opr M + <,> seq F opr k cv ` + F k cv + 1 opr ` opreq1 M ZZ e. k cv ZZ e. M <_ k cv br /\/\ adantl fsumval2t.1 M k cv fsump1OLD M k cv <,> sum F opr M + <,> seq F opr k cv ` = adantr addex fsumval2t.1 M k cv seqzp1OLD M k cv <,> sum F opr M + <,> seq F opr k cv ` = adantr 3eqtr4d exp uzind))
+thm (fsumser0 () ((fsumval2t.1 (e. F (V)))) (-> (e. N (NN0)) (= (opr (<,> (0) N) (sum) F) (` (opr (+) (seq0) F) N))) (N elnn0z 0z fsumval2t.1 0 N fsumserzOLD mp3an1 sylbi addex fsumval2t.1 seq0seqz N fveq1i syl6eqr))
+thm (fsumm1 () ((fsumval2t.1 (e. F (V)))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (opr (<,> M N) (sum) F) (opr (opr (<,> M (opr N (-) (1))) (sum) F) (+) (` F N)))) (fsumval2t.1 M N - 1 opr fsump1OLD M ZZ e. N ZZ e. M < N br 3simp1 N peano2zr M ZZ e. M < N br 3ad2ant2 M N zltlem1t biimp3a syl3anc N zcnt 1cn N 1 npcant mpan2 syl N - 1 opr + 1 opr N M opeq2 syl sum F opreq1d M ZZ e. M < N br 3ad2ant2 N zcnt 1cn N 1 npcant mpan2 syl F fveq2d M N - 1 opr <,> sum F opr + opreq2d M ZZ e. M < N br 3ad2ant2 3eqtr3d))
+thm (fsumclt ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((fsumclt.1 (e. F (V)))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (opr (<,> M N) (sum) F) (CC))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` CC e. raleq syl j cv M M opeq2 sum F opreq1d CC eleq1d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. raleq syl j cv m cv M opeq2 sum F opreq1d CC eleq1d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d CC eleq1d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. raleq syl j cv N M opeq2 sum F opreq1d CC eleq1d imbi12d fsumclt.1 M fsum1 k M ... M opr F k cv ` CC e. A.e. adantr k cv M F fveq2 CC eleq1d M ... M opr rcla4va M elfz3t sylan eqeltrd exp M m cv fzssp1t m cv M eluzel2 m cv M eluzelz sylanc k cv sseld F k cv ` CC e. imim1d r19.20dv2 M m cv <,> sum F opr CC e. imim1d imp32 k cv m cv + 1 opr F fveq2 CC eleq1d M ... m cv + 1 opr opr rcla4va m cv M peano2uz m cv + 1 opr M eluzfz2t syl sylan k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr CC e. -> adantrl fsumclt.1 m cv M fsump1 CC eleq1d M m cv <,> sum F opr F m cv + 1 opr ` axaddcl syl5bir k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr CC e. -> k M ... m cv + 1 opr opr F k cv ` CC e. A.e. /\ adantr mp2and exp32 uzind4 imp))
+thm (fsumcltOLD ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((fsumclt.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (opr (<,> M N) (sum) F) (CC))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` CC e. raleq syl j cv M M opeq2 sum F opreq1d CC eleq1d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. raleq syl j cv m cv M opeq2 sum F opreq1d CC eleq1d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d CC eleq1d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. raleq syl j cv N M opeq2 sum F opreq1d CC eleq1d imbi12d fsumclt.1 M fsum1 k M ... M opr F k cv ` CC e. A.e. adantr M elfz3t k cv M F fveq2 CC eleq1d M ... M opr rcla4v syl imp eqeltrd exp M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. imim1d r19.20dv2 M m cv <,> sum F opr CC e. imim1d imp32 M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp 3jca M m cv + 1 opr eluzfz2tOLD syl k cv m cv + 1 opr F fveq2 CC eleq1d M ... m cv + 1 opr opr rcla4v syl imp k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr CC e. -> adantrl fsumclt.1 M m cv fsump1OLD CC eleq1d M m cv <,> sum F opr F m cv + 1 opr ` axaddcl syl5bir k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr CC e. -> k M ... m cv + 1 opr opr F k cv ` CC e. A.e. /\ adantr mp2and exp32 uzind imp))
+thm (fsumreclt ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((fsumclt.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (e. (opr (<,> M N) (sum) F) (RR))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` RR e. raleq syl j cv M M opeq2 sum F opreq1d RR eleq1d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` RR e. raleq syl j cv m cv M opeq2 sum F opreq1d RR eleq1d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` RR e. raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d RR eleq1d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` RR e. raleq syl j cv N M opeq2 sum F opreq1d RR eleq1d imbi12d fsumclt.1 M fsum1 k M ... M opr F k cv ` RR e. A.e. adantr M elfz3t k cv M F fveq2 RR eleq1d M ... M opr rcla4v syl imp eqeltrd exp M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` RR e. imim1d r19.20dv2 M m cv <,> sum F opr RR e. imim1d imp32 M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp 3jca M m cv + 1 opr eluzfz2tOLD syl k cv m cv + 1 opr F fveq2 RR eleq1d M ... m cv + 1 opr opr rcla4v syl imp k M ... m cv opr F k cv ` RR e. A.e. M m cv <,> sum F opr RR e. -> adantrl fsumclt.1 M m cv fsump1OLD RR eleq1d M m cv <,> sum F opr F m cv + 1 opr ` axaddrcl syl5bir k M ... m cv opr F k cv ` RR e. A.e. M m cv <,> sum F opr RR e. -> k M ... m cv + 1 opr opr F k cv ` RR e. A.e. /\ adantr mp2and exp32 uzind imp))
+thm (fsumnn0cl ((F k) (N k)) ((fsmnn0cl.1 (e. F (V))) (fsmnn0cl.2 (-> (/\/\ (e. (cv k) (NN0)) (e. N (NN0)) (br (cv k) (<_) N)) (e. (` F (cv k)) (CC))))) (-> (e. N (NN0)) (e. (opr (<,> (0) N) (sum) F) (CC))) (N elnn0z N elnn0z N k cv fznn0t fsmnn0cl.2 3com12 3expb exp sylbid r19.21aiv sylbir 0z fsmnn0cl.1 0 N k fsumcltOLD mp3anl1 mpdan sylbi))
+thm (fsum1p ((j k) (j m) (F j) (k m) (F k) (F m) (M j) (M k) (M m) (N j) (N k)) ((fsum1p.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (opr (<,> M N) (sum) F) (opr (` F M) (+) (opr (<,> (opr M (+) (1)) N) (sum) F)))) (j cv M + 1 opr M ... opreq2 M ... j cv opr M ... M + 1 opr opr k F k cv ` CC e. raleq syl j cv M + 1 opr M opeq2 sum F opreq1d j cv M + 1 opr M + 1 opr opeq2 sum F opreq1d F M ` + opreq2d eqeq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. raleq syl j cv m cv M opeq2 sum F opreq1d j cv m cv M + 1 opr opeq2 sum F opreq1d F M ` + opreq2d eqeq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d j cv m cv + 1 opr M + 1 opr opeq2 sum F opreq1d F M ` + opreq2d eqeq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. raleq syl j cv N M opeq2 sum F opreq1d j cv N M + 1 opr opeq2 sum F opreq1d F M ` + opreq2d eqeq12d imbi12d fsum1p.1 M M fsump1OLD M ZZ e. id M ZZ e. id M zret M leidt syl syl3anc fsum1p.1 M fsum1 M peano2z fsum1p.1 M + 1 opr fsum1 syl eqcomd + opreq12d eqtrd k M ... M + 1 opr opr F k cv ` CC e. A.e. a1d M m cv fzssp1t M < m cv br 3adant3 k cv sseld F k cv ` CC e. imim1d r19.20dv2 M m cv <,> sum F opr F M ` + M + 1 opr m cv <,> sum F opr opr = imim1d imp32 + F m cv + 1 opr ` opreq1d F M ` M + 1 opr m cv <,> sum F opr F m cv + 1 opr ` axaddass k cv M F fveq2 CC eleq1d M ... m cv + 1 opr opr rcla4va m cv + 1 opr M eluz2t m cv + 1 opr M eluzfz1t sylbir M ZZ e. m cv ZZ e. M < m cv br 3simp1 m cv peano2z M ZZ e. M < m cv br 3ad2ant2 m cv ltplus1t M RR e. adantl M m cv m cv + 1 opr axlttrn 3expb m cv peano2re ancli sylan2 mpan2d M m cv + 1 opr ltlet m cv peano2re sylan2 syld M zret m cv zret syl2an imp 3impa syl3anc sylan fsum1p.1 M + 1 opr m cv k fsumcltOLD M peano2z m cv ZZ e. M < m cv br 3ad2ant1 M ZZ e. m cv ZZ e. M < m cv br 3simp2 M m cv zltp1let biimp3a 3jca k M ... m cv + 1 opr opr F k cv ` CC e. A.e. adantr M m cv fzp1sst M m cv fzssp1t sstrd M < m cv br 3adant3 k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp sylanc k cv m cv + 1 opr F fveq2 CC eleq1d M ... m cv + 1 opr opr rcla4va m cv + 1 opr M eluz2t m cv + 1 opr M eluzfz2t sylbir M ZZ e. m cv ZZ e. M < m cv br 3simp1 m cv peano2z M ZZ e. M < m cv br 3ad2ant2 m cv ltplus1t M RR e. adantl M m cv m cv + 1 opr axlttrn 3expb m cv peano2re ancli sylan2 mpan2d M m cv + 1 opr ltlet m cv peano2re sylan2 syld M zret m cv zret syl2an imp 3impa syl3anc sylan syl3anc k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr F M ` + M + 1 opr m cv <,> sum F opr opr = -> adantrl eqtrd fsum1p.1 M m cv fsump1OLD M ZZ e. m cv ZZ e. M < m cv br 3simp1 M ZZ e. m cv ZZ e. M < m cv br 3simp2 M m cv ltlet M zret m cv zret syl2an exp 3imp syl3anc k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr F M ` + M + 1 opr m cv <,> sum F opr opr = -> k M ... m cv + 1 opr opr F k cv ` CC e. A.e. /\ adantr fsum1p.1 M + 1 opr m cv fsump1OLD M peano2z m cv ZZ e. M < m cv br 3ad2ant1 M ZZ e. m cv ZZ e. M < m cv br 3simp2 M m cv zltp1let biimp3a syl3anc F M ` + opreq2d k M ... m cv opr F k cv ` CC e. A.e. M m cv <,> sum F opr F M ` + M + 1 opr m cv <,> sum F opr opr = -> k M ... m cv + 1 opr opr F k cv ` CC e. A.e. /\ adantr 3eqtr4d exp32 uzind2 imp))
+thm (fsumsplit ((j k) (j m) (F j) (k m) (F k) (F m) (M j) (M k) (M m) (K j) (N j) (N k) (N m)) ((fsumsplit.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1))))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (opr (<,> M N) (sum) F) (opr (opr (<,> M K) (sum) F) (+) (opr (<,> (opr K (+) (1)) N) (sum) F)))) (M N - 1 opr K elfz1t N peano2zr sylan2 K N zltlem1t ancoms M ZZ e. adantll M <_ K br anbi2d exp pm5.32d K ZZ e. M <_ K br K <_ N - 1 opr br 3anass syl6bbr bitr4d j cv M < N breq1 N ZZ e. anbi2d k M ... N opr F k cv ` CC e. A.e. anbi1d j cv M M opeq2 sum F opreq1d j cv M + 1 opreq1 j cv + 1 opr M + 1 opr N opeq1 syl sum F opreq1d + opreq12d M N <,> sum F opr eqeq2d imbi12d j cv m cv < N breq1 N ZZ e. anbi2d k M ... N opr F k cv ` CC e. A.e. anbi1d j cv m cv M opeq2 sum F opreq1d j cv m cv + 1 opreq1 j cv + 1 opr m cv + 1 opr N opeq1 syl sum F opreq1d + opreq12d M N <,> sum F opr eqeq2d imbi12d j cv m cv + 1 opr < N breq1 N ZZ e. anbi2d k M ... N opr F k cv ` CC e. A.e. anbi1d j cv m cv + 1 opr M opeq2 sum F opreq1d j cv m cv + 1 opr + 1 opreq1 j cv + 1 opr m cv + 1 opr + 1 opr N opeq1 syl sum F opreq1d + opreq12d M N <,> sum F opr eqeq2d imbi12d j cv K < N breq1 N ZZ e. anbi2d k M ... N opr F k cv ` CC e. A.e. anbi1d j cv K M opeq2 sum F opreq1d j cv K + 1 opreq1 j cv + 1 opr K + 1 opr N opeq1 syl sum F opreq1d + opreq12d M N <,> sum F opr eqeq2d imbi12d fsumsplit.1 M N k fsum1p 3exp1 imp44 fsumsplit.1 M fsum1 + M + 1 opr N <,> sum F opr opreq1d N ZZ e. M < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr eqtr4d exp m cv ltplus1t N RR e. adantr m cv m cv + 1 opr N axlttrn 3expa m cv peano2re ancli sylan mpand m cv zret N zret syl2an exp M ZZ e. M <_ m cv br 3ad2ant2 imdistand k M ... N opr F k cv ` CC e. A.e. anim1d M N <,> sum F opr M m cv <,> sum F opr + m cv + 1 opr N <,> sum F opr opr = imim1d M N <,> sum F opr M m cv <,> sum F opr + m cv + 1 opr N <,> sum F opr opr = id fsumsplit.1 M m cv fsump1OLD N ZZ e. m cv + 1 opr < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr fsumsplit.1 m cv + 1 opr N k fsum1p m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr k M ... N opr F k cv ` CC e. A.e. adantrr N ZZ e. m cv + 1 opr < N br pm3.27 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ k M ... N opr F k cv ` CC e. A.e. ad2antrl 3jca M N m cv + 1 opr fzss1tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr jca m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ adantr M m cv leplus1t m cv zret syl3an2 M zret syl3an1 N ZZ e. m cv + 1 opr < N br /\ adantr jca sylanc k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp anasss sylanc eqcomd m cv + 1 opr N <,> sum F opr F m cv + 1 opr ` m cv + 1 opr + 1 opr N <,> sum F opr subaddt fsumsplit.1 m cv + 1 opr N k fsumcltOLD m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr k M ... N opr F k cv ` CC e. A.e. adantrr m cv + 1 opr N ltlet m cv zret m cv peano2re syl N zret syl2an imp anasss M <_ m cv br adantlr M ZZ e. 3adantl1 k M ... N opr F k cv ` CC e. A.e. adantrr 3jca M N m cv + 1 opr fzss1tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr jca m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ adantr M m cv leplus1t m cv zret syl3an2 M zret syl3an1 N ZZ e. m cv + 1 opr < N br /\ adantr jca sylanc k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp anasss sylanc k cv m cv + 1 opr F fveq2 CC eleq1d M ... N opr rcla4va M N m cv + 1 opr elfz4t M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ adantr 3jca M m cv leplus1t m cv zret syl3an2 M zret syl3an1 N ZZ e. m cv + 1 opr < N br /\ adantr m cv + 1 opr N ltlet m cv zret m cv peano2re syl N zret syl2an imp anasss M <_ m cv br adantlr M ZZ e. 3adantl1 jca sylanc sylan anasss fsumsplit.1 m cv + 1 opr + 1 opr N k fsumcltOLD m cv peano2z m cv + 1 opr peano2z syl M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr k M ... N opr F k cv ` CC e. A.e. adantrr m cv + 1 opr N zltp1let m cv peano2z sylan biimpa anasss M <_ m cv br adantlr M ZZ e. 3adantl1 k M ... N opr F k cv ` CC e. A.e. adantrr 3jca M N m cv + 1 opr + 1 opr fzss1tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. anim1i m cv peano2z m cv + 1 opr peano2z syl M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. adantr m cv m cv + 1 opr leplus1t m cv RR e. id m cv peano2re m cv m cv leplus1t m cv RR e. id m cv RR e. id m cv leidt syl3anc syl3anc M RR e. adantl M m cv m cv + 1 opr + 1 opr letrt 3expb m cv peano2re m cv + 1 opr peano2re syl ancli sylan2 mpan2d M zret m cv zret syl2an exp 3imp N ZZ e. adantr jca sylanc m cv + 1 opr < N br adantrr k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp anasss sylanc syl3anc mpbird eqcomd + opreq12d M m cv <,> sum F opr F m cv + 1 opr ` m cv + 1 opr N <,> sum F opr ppncant M N m cv fzss2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr jca M ZZ e. m cv ZZ e. M <_ m cv br 3simp2 N ZZ e. m cv + 1 opr < N br /\ adantr m cv ltplus1t N RR e. adantr m cv m cv + 1 opr N axlttrn 3expa m cv peano2re ancli sylan mpand m cv N ltlet syld m cv zret N zret syl2an imp anasss M <_ m cv br adantlr M ZZ e. 3adantl1 jca sylanc k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp anasss fsumsplit.1 M m cv k fsumcltOLD syldan k cv m cv + 1 opr F fveq2 CC eleq1d M ... N opr rcla4va M N m cv + 1 opr elfz4t M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ adantr 3jca M m cv leplus1t m cv zret syl3an2 M zret syl3an1 N ZZ e. m cv + 1 opr < N br /\ adantr m cv + 1 opr N ltlet m cv zret m cv peano2re syl N zret syl2an imp anasss M <_ m cv br adantlr M ZZ e. 3adantl1 jca sylanc sylan anasss fsumsplit.1 m cv + 1 opr N k fsumcltOLD m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ k M ... N opr F k cv ` CC e. A.e. /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr k M ... N opr F k cv ` CC e. A.e. adantrr m cv + 1 opr N ltlet m cv zret m cv peano2re syl N zret syl2an imp anasss M <_ m cv br adantlr M ZZ e. 3adantl1 k M ... N opr F k cv ` CC e. A.e. adantrr 3jca M N m cv + 1 opr fzss1tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 N ZZ e. m cv + 1 opr < N br /\ adantr M ZZ e. m cv ZZ e. M <_ m cv br /\/\ N ZZ e. pm3.27 m cv + 1 opr < N br adantrr jca m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 N ZZ e. m cv + 1 opr < N br /\ adantr M m cv leplus1t m cv zret syl3an2 M zret syl3an1 N ZZ e. m cv + 1 opr < N br /\ adantr jca sylanc k cv sseld F k cv ` CC e. imim1d r19.20dv2 imp anasss sylanc syl3anc eqtr2d sylan9eqr exp31 a2d syld uzind exp4c 3exp com23 com24 imp imp4d sylbid imp 3impa imp))
+thm (fsumadd ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (H j) (H k) (H m)) ((fsumadd.1 (e. F (V))) (fsumadd.2 (e. G (V))) (fsumadd.3 (e. H (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (= (opr (<,> M N) (sum) H) (opr (opr (<,> M N) (sum) F) (+) (opr (<,> M N) (sum) G)))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ raleq syl j cv M M opeq2 sum H opreq1d j cv M M opeq2 sum F opreq1d j cv M M opeq2 sum G opreq1d + opreq12d eqeq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ raleq syl j cv m cv M opeq2 sum H opreq1d j cv m cv M opeq2 sum F opreq1d j cv m cv M opeq2 sum G opreq1d + opreq12d eqeq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ raleq syl j cv m cv + 1 opr M opeq2 sum H opreq1d j cv m cv + 1 opr M opeq2 sum F opreq1d j cv m cv + 1 opr M opeq2 sum G opreq1d + opreq12d eqeq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ raleq syl j cv N M opeq2 sum H opreq1d j cv N M opeq2 sum F opreq1d j cv N M opeq2 sum G opreq1d + opreq12d eqeq12d imbi12d k cv M H fveq2 k cv M F fveq2 k cv M G fveq2 + opreq12d eqeq12d M ... M opr rcla4va M elfz3t sylan fsumadd.3 M fsum1 k M ... M opr H k cv ` F k cv ` + G k cv ` opr = A.e. adantr fsumadd.1 M fsum1 fsumadd.2 M fsum1 + opreq12d k M ... M opr H k cv ` F k cv ` + G k cv ` opr = A.e. adantr 3eqtr4d exp F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = 3simp3 k M ... M opr r19.20si syl5 M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ imim1d r19.20dv2 M m cv <,> sum H opr M m cv <,> sum F opr + M m cv <,> sum G opr opr = imim1d imp32 k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 CC eleq1d k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d 3anbi123d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan 3simp3d k M ... m cv opr F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ A.e. M m cv <,> sum H opr M m cv <,> sum F opr + M m cv <,> sum G opr opr = -> adantrl + opreq12d fsumadd.3 M m cv fsump1OLD k M ... m cv opr F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ A.e. M m cv <,> sum H opr M m cv <,> sum F opr + M m cv <,> sum G opr opr = -> k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ A.e. /\ adantr fsumadd.1 M m cv fsump1OLD fsumadd.2 M m cv fsump1OLD + opreq12d k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ A.e. adantr M m cv <,> sum F opr F m cv + 1 opr ` M m cv <,> sum G opr G m cv + 1 opr ` add4t M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = 3simp1 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsumadd.1 M m cv k fsumcltOLD syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 CC eleq1d k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d 3anbi123d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan 3simp1d jca M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = 3simp2 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsumadd.2 M m cv k fsumcltOLD syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 CC eleq1d k cv m cv + 1 opr H fveq2 k cv m cv + 1 opr F fveq2 k cv m cv + 1 opr G fveq2 + opreq12d eqeq12d 3anbi123d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan 3simp2d jca sylanc eqtrd k M ... m cv opr F k cv ` CC e. G k cv ` CC e. H k cv ` F k cv ` + G k cv ` opr = /\/\ A.e. M m cv <,> sum H opr M m cv <,> sum F opr + M m cv <,> sum G opr opr = -> adantrl 3eqtr4d exp32 uzind imp))
+thm (fsumeq ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((fsmeq.1 (e. F (V))) (fsmeq.2 (e. G (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (cv k))))) (= (opr (<,> M N) (sum) G) (opr (<,> M N) (sum) F))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k G k cv ` F k cv ` = raleq syl j cv M M opeq2 sum G opreq1d j cv M M opeq2 sum F opreq1d eqeq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k G k cv ` F k cv ` = raleq syl j cv m cv M opeq2 sum G opreq1d j cv m cv M opeq2 sum F opreq1d eqeq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k G k cv ` F k cv ` = raleq syl j cv m cv + 1 opr M opeq2 sum G opreq1d j cv m cv + 1 opr M opeq2 sum F opreq1d eqeq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k G k cv ` F k cv ` = raleq syl j cv N M opeq2 sum G opreq1d j cv N M opeq2 sum F opreq1d eqeq12d imbi12d k cv M G fveq2 k cv M F fveq2 eqeq12d M ... M opr rcla4va M elfz3t sylan fsmeq.2 M fsum1 k M ... M opr G k cv ` F k cv ` = A.e. adantr fsmeq.1 M fsum1 k M ... M opr G k cv ` F k cv ` = A.e. adantr 3eqtr4d exp M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 G k cv ` F k cv ` = imim1d r19.20dv2 M m cv <,> sum G opr M m cv <,> sum F opr = imim1d imp32 k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 eqeq12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan k M ... m cv opr G k cv ` F k cv ` = A.e. M m cv <,> sum G opr M m cv <,> sum F opr = -> adantrl + opreq12d fsmeq.2 M m cv fsump1OLD k M ... m cv opr G k cv ` F k cv ` = A.e. M m cv <,> sum G opr M m cv <,> sum F opr = -> k M ... m cv + 1 opr opr G k cv ` F k cv ` = A.e. /\ adantr fsmeq.1 M m cv fsump1OLD k M ... m cv opr G k cv ` F k cv ` = A.e. M m cv <,> sum G opr M m cv <,> sum F opr = -> k M ... m cv + 1 opr opr G k cv ` F k cv ` = A.e. /\ adantr 3eqtr4d exp32 uzind imp))
+thm (fsummulc ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (A j) (A k) (A m)) ((fsummulc.1 (e. F (V))) (fsummulc.2 (e. G (V))) (fsummulc.3 (e. A (CC)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (x.) (` F (cv k))))))) (= (opr (<,> M N) (sum) G) (opr A (x.) (opr (<,> M N) (sum) F)))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ raleq syl j cv M M opeq2 sum G opreq1d j cv M M opeq2 sum F opreq1d A x. opreq2d eqeq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ raleq syl j cv m cv M opeq2 sum G opreq1d j cv m cv M opeq2 sum F opreq1d A x. opreq2d eqeq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ raleq syl j cv m cv + 1 opr M opeq2 sum G opreq1d j cv m cv + 1 opr M opeq2 sum F opreq1d A x. opreq2d eqeq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ raleq syl j cv N M opeq2 sum G opreq1d j cv N M opeq2 sum F opreq1d A x. opreq2d eqeq12d imbi12d k cv M G fveq2 k cv M F fveq2 A x. opreq2d eqeq12d M ... M opr rcla4va M elfz3t sylan fsummulc.2 M fsum1 k M ... M opr G k cv ` A x. F k cv ` opr = A.e. adantr fsummulc.1 M fsum1 A x. opreq2d k M ... M opr G k cv ` A x. F k cv ` opr = A.e. adantr 3eqtr4d exp F k cv ` CC e. G k cv ` A x. F k cv ` opr = pm3.27 k M ... M opr r19.20si syl5 M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ imim1d r19.20dv2 M m cv <,> sum G opr A x. M m cv <,> sum F opr opr = imim1d imp32 k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 A x. opreq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan pm3.27d k M ... m cv opr F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ A.e. M m cv <,> sum G opr A x. M m cv <,> sum F opr opr = -> adantrl + opreq12d fsummulc.2 M m cv fsump1OLD k M ... m cv opr F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ A.e. M m cv <,> sum G opr A x. M m cv <,> sum F opr opr = -> k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ A.e. /\ adantr fsummulc.1 M m cv fsump1OLD A x. opreq2d k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ A.e. adantr fsummulc.3 A M m cv <,> sum F opr F m cv + 1 opr ` axdistr mp3an1 M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` A x. F k cv ` opr = pm3.26 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsummulc.1 M m cv k fsumcltOLD syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 A x. opreq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan pm3.26d sylanc eqtrd k M ... m cv opr F k cv ` CC e. G k cv ` A x. F k cv ` opr = /\ A.e. M m cv <,> sum G opr A x. M m cv <,> sum F opr opr = -> adantrl 3eqtr4d exp32 uzind imp))
+thm (fsumshft1 ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((fsumshft1.1 (e. F (V))) (fsumshft1.2 (e. G (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (opr (cv k) (+) (1)))))) (= (opr (<,> M N) (sum) G) (opr (<,> (opr M (+) (1)) (opr N (+) (1))) (sum) F))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k G k cv ` F k cv + 1 opr ` = raleq syl j cv M M opeq2 sum G opreq1d j cv M + 1 opreq1 j cv + 1 opr M + 1 opr M + 1 opr opeq2 syl sum F opreq1d eqeq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k G k cv ` F k cv + 1 opr ` = raleq syl j cv m cv M opeq2 sum G opreq1d j cv m cv + 1 opreq1 j cv + 1 opr m cv + 1 opr M + 1 opr opeq2 syl sum F opreq1d eqeq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k G k cv ` F k cv + 1 opr ` = raleq syl j cv m cv + 1 opr M opeq2 sum G opreq1d j cv m cv + 1 opr + 1 opreq1 j cv + 1 opr m cv + 1 opr + 1 opr M + 1 opr opeq2 syl sum F opreq1d eqeq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k G k cv ` F k cv + 1 opr ` = raleq syl j cv N M opeq2 sum G opreq1d j cv N + 1 opreq1 j cv + 1 opr N + 1 opr M + 1 opr opeq2 syl sum F opreq1d eqeq12d imbi12d k cv M G fveq2 k cv M + 1 opreq1 F fveq2d eqeq12d M ... M opr rcla4va M elfz3t sylan fsumshft1.2 M fsum1 k M ... M opr G k cv ` F k cv + 1 opr ` = A.e. adantr M peano2z fsumshft1.1 M + 1 opr fsum1 syl k M ... M opr G k cv ` F k cv + 1 opr ` = A.e. adantr 3eqtr4d exp M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 G k cv ` F k cv + 1 opr ` = imim1d r19.20dv2 M m cv <,> sum G opr M + 1 opr m cv + 1 opr <,> sum F opr = imim1d imp32 k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr + 1 opreq1 F fveq2d eqeq12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 m cv m cv + 1 opr ltlet m cv peano2re ancli m cv ltplus1t sylc M RR e. adantl M m cv m cv + 1 opr letrt M RR e. m cv RR e. pm3.26 M RR e. m cv RR e. pm3.27 m cv peano2re M RR e. adantl syl3anc mpan2d M zret m cv zret syl2an exp 3imp syl3anc sylan k M ... m cv opr G k cv ` F k cv + 1 opr ` = A.e. M m cv <,> sum G opr M + 1 opr m cv + 1 opr <,> sum F opr = -> adantrl + opreq12d fsumshft1.2 M m cv fsump1OLD k M ... m cv opr G k cv ` F k cv + 1 opr ` = A.e. M m cv <,> sum G opr M + 1 opr m cv + 1 opr <,> sum F opr = -> k M ... m cv + 1 opr opr G k cv ` F k cv + 1 opr ` = A.e. /\ adantr fsumshft1.1 M + 1 opr m cv + 1 opr fsump1OLD M peano2z m cv ZZ e. M <_ m cv br 3ad2ant1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 ax1re M m cv 1 leadd1t mp3an3 M zret m cv zret syl2an biimpd exp 3imp syl3anc k M ... m cv opr G k cv ` F k cv + 1 opr ` = A.e. M m cv <,> sum G opr M + 1 opr m cv + 1 opr <,> sum F opr = -> k M ... m cv + 1 opr opr G k cv ` F k cv + 1 opr ` = A.e. /\ adantr 3eqtr4d exp32 uzind imp))
+thm (fsumshftm1 ((j k) (M j) (M k) (N j) (N k) (F j) (F k) (G j) (G k)) ((fsumshftm1.1 (e. F (V))) (fsumshftm1.2 (e. G (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (opr (cv k) (-) (1)))))) (= (opr (<,> M N) (sum) G) (opr (<,> (opr M (-) (1)) (opr N (-) (1))) (sum) F))) (fsumshftm1.2 fsumshftm1.1 M - 1 opr N - 1 opr j fsumshft1 M peano2zr N ZZ e. M <_ N br 3ad2ant1 N peano2zr M ZZ e. M <_ N br 3ad2ant2 ax1re M N 1 lesub1t mp3an3 M zret N zret syl2an biimp3a 3jca k M ... N opr G k cv ` F k cv - 1 opr ` = A.e. adantr M - 1 opr N - 1 opr j cv elfz1t M peano2zr N peano2zr syl2an j cv ZZ e. M - 1 opr <_ j cv br j cv <_ N - 1 opr br 3simp1 syl6bi imp k M ... N opr G k cv ` F k cv - 1 opr ` = A.e. adantr k cv j cv + 1 opr G fveq2 k cv j cv + 1 opr - 1 opreq1 F fveq2d eqeq12d M ... N opr rcla4va 1z M N j cv + 1 opr 1 fzsubelt mpanr2 j cv peano2z sylan2 j cv zcnt 1cn j cv 1 pncant mpan2 syl M - 1 opr ... N - 1 opr opr eleq1d M ZZ e. N ZZ e. /\ adantl bitr2d biimpa an1rs sylan j cv zcnt 1cn j cv 1 pncant mpan2 syl F fveq2d M ZZ e. N ZZ e. /\ j cv M - 1 opr ... N - 1 opr opr e. /\ k M ... N opr G k cv ` F k cv - 1 opr ` = A.e. ad2antlr eqtr2d an1rs mpdan an1rs exp r19.21aiv M <_ N br 3adantl3 sylanc M - 1 opr + 1 opr M N - 1 opr + 1 opr N opeq12 M zcnt 1cn M 1 npcant mpan2 syl N zcnt 1cn N 1 npcant mpan2 syl syl2an sum G opreq1d M <_ N br 3adant3 k M ... N opr G k cv ` F k cv - 1 opr ` = A.e. adantr eqtr2d))
+thm (fsumabs ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((fsabs.1 (e. F (V))) (fsabs.2 (e. G (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (` (abs) (` F (cv k))))))) (br (` (abs) (opr (<,> M N) (sum) F)) (<_) (opr (<,> M N) (sum) G))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ raleq syl j cv M M opeq2 sum F opreq1d abs fveq2d j cv M M opeq2 sum G opreq1d <_ breq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ raleq syl j cv m cv M opeq2 sum F opreq1d abs fveq2d j cv m cv M opeq2 sum G opreq1d <_ breq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d abs fveq2d j cv m cv + 1 opr M opeq2 sum G opreq1d <_ breq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ raleq syl j cv N M opeq2 sum F opreq1d abs fveq2d j cv N M opeq2 sum G opreq1d <_ breq12d imbi12d M elfz3t k cv M F fveq2 CC eleq1d k cv M G fveq2 k cv M F fveq2 abs fveq2d eqeq12d anbi12d M ... M opr rcla4v syl F M ` absclt abs F M ` ` leidt syl G M ` abs F M ` ` = adantr F M ` CC e. G M ` abs F M ` ` = pm3.27 breqtrrd M ZZ e. adantl fsabs.1 M fsum1 abs fveq2d F M ` CC e. G M ` abs F M ` ` = /\ adantr fsabs.2 M fsum1 F M ` CC e. G M ` abs F M ` ` = /\ adantr 3brtr4d exp syld M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ imim1d r19.20dv2 abs M m cv <,> sum F opr ` <_ M m cv <,> sum G opr br imim1d M m cv <,> sum F opr F m cv + 1 opr ` abstrit M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` abs F k cv ` ` = pm3.26 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsabs.1 M m cv k fsumcltOLD syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 abs fveq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va pm3.26d M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan sylanc fsabs.1 M m cv fsump1OLD abs fveq2d k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ A.e. adantr k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 abs fveq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va pm3.27d M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan abs M m cv <,> sum F opr ` + opreq2d 3brtr4d abs M m cv <,> sum F opr ` <_ M m cv <,> sum G opr br adantr abs M m cv <,> sum F opr ` M m cv <,> sum G opr G m cv + 1 opr ` leadd1t M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` abs F k cv ` ` = pm3.26 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsabs.1 M m cv k fsumcltOLD syl M m cv <,> sum F opr absclt syl M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 abs F k cv ` ` RR G k cv ` eleq1a imp F k cv ` absclt sylan M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsabs.2 M m cv k fsumreclt syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 abs fveq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va abs F m cv + 1 opr ` ` RR G m cv + 1 opr ` eleq1a imp F m cv + 1 opr ` absclt sylan syl M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan syl3anc biimpa fsabs.2 M m cv fsump1OLD k M ... m cv + 1 opr opr F k cv ` CC e. G k cv ` abs F k cv ` ` = /\ A.e. abs M m cv <,> sum F opr ` <_ M m cv <,> sum G opr br ad2antll breqtrrd abs M m cv + 1 opr <,> sum F opr ` abs M m cv <,> sum F opr ` + G m cv + 1 opr ` opr M m cv + 1 opr <,> sum G opr letrt fsabs.1 M m cv + 1 opr k fsumcltOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 3jca F k cv ` CC e. G k cv ` abs F k cv ` ` = pm3.26 k M ... m cv + 1 opr opr r19.20si syl2an M m cv + 1 opr <,> sum F opr absclt syl abs M m cv <,> sum F opr ` G m cv + 1 opr ` axaddrcl M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` CC e. G k cv ` abs F k cv ` ` = pm3.26 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsabs.1 M m cv k fsumcltOLD syl M m cv <,> sum F opr absclt syl k cv m cv + 1 opr F fveq2 CC eleq1d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 abs fveq2d eqeq12d anbi12d M ... m cv + 1 opr opr rcla4va abs F m cv + 1 opr ` ` RR G m cv + 1 opr ` eleq1a imp F m cv + 1 opr ` absclt sylan syl M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan sylanc fsabs.2 M m cv + 1 opr k fsumreclt M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 3jca abs F k cv ` ` RR G k cv ` eleq1a imp F k cv ` absclt sylan k M ... m cv + 1 opr opr r19.20si syl2an syl3anc abs M m cv <,> sum F opr ` <_ M m cv <,> sum G opr br adantr mp2and exp31 a2d syld uzind imp))
+thm (fsumub ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (A j) (A k) (A m)) ((fsumub.1 (e. F (V))) (fsumub.2 (e. G (V))) (fsumub.3 (e. A (RR)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (RR)) (br (` F (cv k)) (<_) A)))) (br (opr (<,> M N) (sum) F) (<_) (opr (opr (opr N (-) M) (+) (1)) (x.) A))) (j cv M M ... opreq2 M ... j cv opr M ... M opr k F k cv ` RR e. F k cv ` <_ A br /\ raleq syl j cv M M opeq2 sum F opreq1d j cv M - M opreq1 + 1 opreq1d x. A opreq1d <_ breq12d imbi12d j cv m cv M ... opreq2 M ... j cv opr M ... m cv opr k F k cv ` RR e. F k cv ` <_ A br /\ raleq syl j cv m cv M opeq2 sum F opreq1d j cv m cv - M opreq1 + 1 opreq1d x. A opreq1d <_ breq12d imbi12d j cv m cv + 1 opr M ... opreq2 M ... j cv opr M ... m cv + 1 opr opr k F k cv ` RR e. F k cv ` <_ A br /\ raleq syl j cv m cv + 1 opr M opeq2 sum F opreq1d j cv m cv + 1 opr - M opreq1 + 1 opreq1d x. A opreq1d <_ breq12d imbi12d j cv N M ... opreq2 M ... j cv opr M ... N opr k F k cv ` RR e. F k cv ` <_ A br /\ raleq syl j cv N M opeq2 sum F opreq1d j cv N - M opreq1 + 1 opreq1d x. A opreq1d <_ breq12d imbi12d M elfz3t k cv M F fveq2 RR eleq1d k cv M F fveq2 <_ A breq1d anbi12d M ... M opr rcla4v syl fsumub.1 M fsum1 M zcnt M subidt syl + 1 opreq1d 1cn addid2 syl6eq x. A opreq1d fsumub.3 recn mulid2 syl6eq <_ breq12d biimpar F M ` RR e. adantrl exp syld M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` RR e. F k cv ` <_ A br /\ imim1d r19.20dv2 M m cv <,> sum F opr <_ m cv - M opr + 1 opr x. A opr br imim1d fsumub.1 M m cv fsump1OLD k M ... m cv + 1 opr opr F k cv ` RR e. F k cv ` <_ A br /\ A.e. M m cv <,> sum F opr <_ m cv - M opr + 1 opr x. A opr br ad2antll M m cv <,> sum F opr m cv - M opr + 1 opr x. A opr F m cv + 1 opr ` leadd1t M m cv fzssp1t k cv sseld M <_ m cv br 3adant3 F k cv ` RR e. F k cv ` <_ A br pm3.26 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ a1i imim12d r19.20dv2 imdistani fsumub.1 M m cv k fsumreclt syl m cv M resubclt m cv zret M zret syl2an ancoms m cv - M opr peano2re fsumub.3 m cv - M opr + 1 opr A axmulrcl mpan2 3syl M <_ m cv br 3adant3 k M ... m cv + 1 opr opr F k cv ` RR e. F k cv ` <_ A br /\ A.e. adantr k cv m cv + 1 opr F fveq2 RR eleq1d k cv m cv + 1 opr F fveq2 <_ A breq1d anbi12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan pm3.26d syl3anc biimpa eqbrtrd k cv m cv + 1 opr F fveq2 RR eleq1d k cv m cv + 1 opr F fveq2 <_ A breq1d anbi12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan F m cv + 1 opr ` A m cv - M opr + 1 opr x. A opr leadd2t M ZZ e. m cv ZZ e. M <_ m cv br /\/\ F m cv + 1 opr ` RR e. pm3.27 fsumub.3 M ZZ e. m cv ZZ e. M <_ m cv br /\/\ F m cv + 1 opr ` RR e. /\ a1i m cv M resubclt m cv zret M zret syl2an ancoms m cv - M opr peano2re fsumub.3 m cv - M opr + 1 opr A axmulrcl mpan2 3syl M <_ m cv br 3adant3 F m cv + 1 opr ` RR e. adantr syl3anc biimpa anasss fsumub.3 recn m cv + 1 opr - M opr 1 A adddirt mp3an3 m cv + 1 opr M resubclt m cv peano2re sylan ancoms recnd 1cn M RR e. m cv RR e. /\ a1i sylanc 1cn m cv 1 M addsubt mp3an2 m cv recnt M recnt syl2an ancoms x. A opreq1d fsumub.3 recn mulid2 M RR e. m cv RR e. /\ a1i + opreq12d eqtrd M zret m cv zret syl2an M <_ m cv br 3adant3 F m cv + 1 opr ` RR e. F m cv + 1 opr ` <_ A br /\ adantr breqtrrd syldan M m cv <,> sum F opr <_ m cv - M opr + 1 opr x. A opr br adantr M m cv + 1 opr <,> sum F opr m cv - M opr + 1 opr x. A opr + F m cv + 1 opr ` opr m cv + 1 opr - M opr + 1 opr x. A opr letrt fsumub.1 M m cv + 1 opr k fsumreclt M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 3jca F k cv ` RR e. F k cv ` <_ A br pm3.26 k M ... m cv + 1 opr opr r19.20si syl2an m cv - M opr + 1 opr x. A opr F m cv + 1 opr ` axaddrcl m cv M resubclt m cv zret M zret syl2an ancoms m cv - M opr peano2re fsumub.3 m cv - M opr + 1 opr A axmulrcl mpan2 3syl M <_ m cv br 3adant3 k M ... m cv + 1 opr opr F k cv ` RR e. F k cv ` <_ A br /\ A.e. adantr k cv m cv + 1 opr F fveq2 RR eleq1d k cv m cv + 1 opr F fveq2 <_ A breq1d anbi12d M ... m cv + 1 opr opr rcla4va M m cv + 1 opr eluzfz2tOLD M ZZ e. m cv ZZ e. M <_ m cv br 3simp1 m cv peano2z M ZZ e. M <_ m cv br 3ad2ant2 M m cv leplus1t m cv zret syl3an2 M zret syl3an1 syl3anc sylan pm3.26d sylanc m cv + 1 opr M resubclt m cv peano2re sylan ancoms M zret m cv zret syl2an m cv + 1 opr - M opr peano2re fsumub.3 m cv + 1 opr - M opr + 1 opr A axmulrcl mpan2 3syl M <_ m cv br 3adant3 k M ... m cv + 1 opr opr F k cv ` RR e. F k cv ` <_ A br /\ A.e. adantr syl3anc M m cv <,> sum F opr <_ m cv - M opr + 1 opr x. A opr br adantr mp2and exp31 a2d syld uzind imp))
+thm (serz0 ((j k) (M j) (M k) (N j)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M (+)) (seq) (X. (` (ZZ>) M) ({} (0)))) N) (0))) (j cv M M + <,> seq ZZ> M ` 0 {} X. opr fveq2 0 eqeq1d j cv k cv M + <,> seq ZZ> M ` 0 {} X. opr fveq2 0 eqeq1d j cv k cv + 1 opr M + <,> seq ZZ> M ` 0 {} X. opr fveq2 0 eqeq1d j cv N M + <,> seq ZZ> M ` 0 {} X. opr fveq2 0 eqeq1d addex ZZ> M fvex 0 snex xpex M seqz1 M uzidt 0cn elisseti M ZZ> M ` fvconst2 syl eqtrd addex ZZ> M fvex 0 snex xpex M k cv seqzp1OLD k cv M eluz2t k cv M peano2uz sylbir 0cn elisseti k cv + 1 opr ZZ> M ` fvconst2 syl M + <,> seq ZZ> M ` 0 {} X. opr k cv ` + opreq2d eqtrd M + <,> seq ZZ> M ` 0 {} X. opr k cv ` 0 + 0 opreq1 0cn addid1 syl6eq sylan9eq exp uzind))
+thm (serzcmp ((j k) (j m) (F j) (k m) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((serzcm.1 (e. M (ZZ))) (serzcm.2 (:--> F (` (ZZ>) M) (RR))) (serzcm.3 (:--> G (` (ZZ>) M) (RR))) (serzcm.4 (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (br (` G (cv k)) (<_) (` F (cv k)))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (br (` (opr (<,> M (+)) (seq) G) N) (<_) (` (opr (<,> M (+)) (seq) F) N))) (serzcm.1 j cv M M + <,> seq G opr fveq2 j cv M M + <,> seq F opr fveq2 <_ breq12d j cv m cv M + <,> seq G opr fveq2 j cv m cv M + <,> seq F opr fveq2 <_ breq12d j cv m cv + 1 opr M + <,> seq G opr fveq2 j cv m cv + 1 opr M + <,> seq F opr fveq2 <_ breq12d j cv N M + <,> seq G opr fveq2 j cv N M + <,> seq F opr fveq2 <_ breq12d serzcm.1 serzcm.1 zre leid serzcm.1 elisseti k cv M ZZ eleq1 k cv M M <_ breq2 anbi12d k cv M G fveq2 k cv M F fveq2 <_ breq12d imbi12d serzcm.4 vtocl mp2an serzcm.1 addex ZZ> M fvex serzcm.3 ZZ> M ` V G RR fex mp2 M seqz1 ax-mp serzcm.1 addex ZZ> M fvex serzcm.2 ZZ> M ` V F RR fex mp2 M seqz1 ax-mp 3brtr4 M ZZ e. a1i M + <,> seq G opr m cv ` G m cv + 1 opr ` axaddrcl serzcm.1 serzcm.3 m cv serzrecl serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.3 m cv + 1 opr ffvrni syl sylanc M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr M + <,> seq F opr m cv ` G m cv + 1 opr ` axaddrcl serzcm.1 serzcm.2 m cv serzrecl serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.3 m cv + 1 opr ffvrni syl sylanc M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr M + <,> seq F opr m cv ` F m cv + 1 opr ` axaddrcl serzcm.1 serzcm.2 m cv serzrecl serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.2 m cv + 1 opr ffvrni syl sylanc M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr M + <,> seq G opr m cv ` M + <,> seq F opr m cv ` G m cv + 1 opr ` leadd1t serzcm.1 serzcm.3 m cv serzrecl serzcm.1 serzcm.2 m cv serzrecl serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.3 m cv + 1 opr ffvrni syl syl3anc biimpa m cv + 1 oprex k cv m cv + 1 opr ZZ eleq1 k cv m cv + 1 opr M <_ breq2 anbi12d k cv m cv + 1 opr G fveq2 k cv m cv + 1 opr F fveq2 <_ breq12d imbi12d serzcm.4 vtocl m cv peano2z M <_ m cv br adantr serzcm.1 zre M m cv leplus1t mp3an1 m cv zret sylan sylanc G m cv + 1 opr ` F m cv + 1 opr ` M + <,> seq F opr m cv ` leadd2t serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.3 m cv + 1 opr ffvrni syl serzcm.1 m cv eluz1 m cv M peano2uz sylbir serzcm.2 m cv + 1 opr ffvrni syl serzcm.1 serzcm.2 m cv serzrecl syl3anc mpbid M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr letrd serzcm.1 addex ZZ> M fvex serzcm.3 ZZ> M ` V G RR fex mp2 M m cv seqzp1OLD mp3an1 M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr serzcm.1 addex ZZ> M fvex serzcm.2 ZZ> M ` V F RR fex mp2 M m cv seqzp1OLD mp3an1 M + <,> seq G opr m cv ` <_ M + <,> seq F opr m cv ` br adantr 3brtr4d exp M ZZ e. 3adant1 uzind mp3an1))
+thm (serzcmp0 ((F k) (M k)) ((serzcm0.1 (e. M (ZZ))) (serzcm0.2 (:--> F (` (ZZ>) M) (RR))) (serzcm0.3 (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (br (0) (<_) (` F (cv k)))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (br (0) (<_) (` (opr (<,> M (+)) (seq) F) N))) (serzcm0.1 M N serz0 mp3an1 serzcm0.1 serzcm0.2 0re elisseti ZZ> M ` fconst 0re 0 RR snssi ax-mp ZZ> M ` 0 {} X. ZZ> M ` 0 {} RR fss mp2an serzcm0.1 k cv eluz1 0re elisseti k cv ZZ> M ` fvconst2 sylbir serzcm0.3 eqbrtrd N serzcmp eqbrtrrd))
+thm (ser0split ((F k) (M k) (N k)) ((ser0split.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (NN0)) (e. N (NN0)) (br (0) (<) M)) (A.e. k (opr (0) (...) (opr M (+) N)) (e. (` F (cv k)) (CC)))) (= (` (opr (opr (+) (seq0) F) (shift) (-u M)) N) (opr (` (opr (+) (seq0) F) (opr M (-) (1))) (+) (` (opr (+) (seq0) (opr F (shift) (-u M))) N)))) (+ seq0 F oprex M N shftval4t M nn0cnt N nn0cnt syl2an M N nn0addclt ser0split.1 M + N opr fsumser0 syl eqtr4d 0 < M br 3adant3 k 0 ... M + N opr opr F k cv ` CC e. A.e. adantr 0z ser0split.1 0 M + N opr M - 1 opr k fsumsplit mp3anl1 M N nn0addclt M + N opr nn0zt syl 0 < M br 3adant3 0z 0 M + N opr - 1 opr M - 1 opr elfz4t mp3anl1 M N nn0addclt M + N opr nn0zt syl M + N opr peano2zr syl M nn0zt M peano2zr syl N NN0 e. adantr jca 0 < M br 3adant3 0nn0 0 M nn0ltlem1t mpan biimpa N NN0 e. 3adant2 M N nn0addge1t M nn0ret sylan ax1re M M + N opr 1 lesub1t mp3an3 M nn0ret N NN0 e. adantr M N nn0addclt M + N opr nn0zt syl M + N opr zret syl sylanc mpbid 0 < M br 3adant3 jca sylanc jca sylan M elnnnn0b M elnnnn0 bitr3 pm3.27bd ser0split.1 M - 1 opr fsumser0 syl N NN0 e. 3adant2 M nn0cnt 1cn M 1 npcant mpan2 syl M - 1 opr + 1 opr M M + N opr opeq1 syl sum F opreq1d N NN0 e. adantr ser0split.1 M M + N opr fsumval2tOLD M nn0zt N NN0 e. adantr M N nn0addclt M + N opr nn0zt syl M N nn0addge1t M nn0ret sylan syl3anc M N pncan2t M nn0cnt N nn0cnt syl2an + seq0 F shift M -u opr opr fveq2d 3eqtrd 0 < M br 3adant3 + opreq12d k 0 ... M + N opr opr F k cv ` CC e. A.e. adantr 3eqtrd))
+thm (serzsplit ((F k) (M k) (N k)) ((serzsplit.1 (e. F (V)))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1))))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` (opr (<,> M (+)) (seq) F) K) (+) (` (opr (<,> (opr K (+) (1)) (+)) (seq) F) N)))) (serzsplit.1 M N K k fsumsplit serzsplit.1 M N fsumserzOLD M ZZ e. N ZZ e. pm3.26 K ZZ e. M <_ K br K <_ N - 1 opr br /\/\ adantr M ZZ e. N ZZ e. pm3.27 K ZZ e. M <_ K br K <_ N - 1 opr br /\/\ adantr M K N - 1 opr letrt ax1re N 1 resubclt mpan2 syl3an3 M N - 1 opr leplus1t 3exp imp ax1re N 1 resubclt mpan2 sylan2 N recnt 1cn N 1 npcant mpan2 syl M <_ breq2d M RR e. adantl sylibd K RR e. 3adant2 syld M zret K zret N zret syl3an 3com23 3exp imp43 K ZZ e. M <_ K br K <_ N - 1 opr br 3anass sylan2b syl3anc M ZZ e. N ZZ e. K M ... N - 1 opr opr e. 3simpa M N - 1 opr K elfz1t N peano2zr sylan2 biimp3a sylanc k M ... N opr F k cv ` CC e. A.e. adantr M ZZ e. K ZZ e. M <_ K br 3anass biimpr N ZZ e. adantlr K <_ N - 1 opr br 3adantr3 serzsplit.1 M K fsumserzOLD syl serzsplit.1 K + 1 opr N fsumserzOLD K peano2z N ZZ e. K <_ N - 1 opr br ad2antrl N ZZ e. K ZZ e. K <_ N - 1 opr br /\ pm3.26 ax1re K 1 N leaddsubt mp3an2 K zret N zret syl2an ancoms biimpar anasss syl3anc M <_ K br 3adantr2 M ZZ e. adantll + opreq12d M ZZ e. N ZZ e. K M ... N - 1 opr opr e. 3simpa M N - 1 opr K elfz1t N peano2zr sylan2 biimp3a sylanc k M ... N opr F k cv ` CC e. A.e. adantr 3eqtr3d))
+thm (climrelOLD ((x y) (x z) (f x) (w x) (y z) (f y) (w y) (f z) (w z) (f w)) () (Rel (~~>OLD)) (f w f cv NN CC :--> w cv CC e. /\ x RR 0 < x cv br y NN z NN y cv <_ z cv br abs f cv z cv ` - w cv opr ` < x cv br -> A.e. E.e. -> A.e. /\ relopab f w x y z df-climOLD ~~>OLD f w f cv NN CC :--> w cv CC e. /\ x RR 0 < x cv br y NN z NN y cv <_ z cv br abs f cv z cv ` - w cv opr ` < x cv br -> A.e. E.e. -> A.e. /\ {<,>|} releq ax-mp mpbir))
+thm (climrel ((j k) (j x) (j y) (f j) (k x) (k y) (f k) (x y) (f x) (f y)) () (Rel (~~>)) (f y y cv CC e. x RR 0 < x cv br j ZZ k ZZ j cv <_ k cv br f cv k cv ` CC e. abs f cv k cv ` - y cv opr ` < x cv br /\ -> A.e. E.e. -> A.e. /\ relopab f y x j k df-clim ~~> f y y cv CC e. x RR 0 < x cv br j ZZ k ZZ j cv <_ k cv br f cv k cv ` CC e. abs f cv k cv ` - y cv opr ` < x cv br /\ -> A.e. E.e. -> A.e. /\ {<,>|} releq ax-mp mpbir))
+thm (clim ((j k) (j x) (j y) (f j) (F j) (k x) (k y) (f k) (F k) (x y) (f x) (F x) (f y) (F y) (F f) (A j) (A k) (A x) (A y) (A f)) () (-> (/\ (e. F C) (e. A D)) (<-> (br F (~~>) A) (/\ (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) (f cv F k cv fveq1 CC eleq1d f cv F k cv fveq1 - y cv opreq1d abs fveq2d < x cv breq1d anbi12d j cv <_ k cv br imbi2d k ZZ biraldv j ZZ birexdv 0 < x cv br imbi2d x RR biraldv y cv CC e. anbi2d y cv A CC eleq1 y cv A F k cv ` - opreq2 abs fveq2d < x cv breq1d F k cv ` CC e. anbi2d j cv <_ k cv br imbi2d k ZZ biraldv j ZZ birexdv 0 < x cv br imbi2d x RR biraldv anbi12d f y x j k df-clim C D brabg))
+thm (climuz ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) () (-> (/\ (e. F C) (e. A D)) (<-> (br F (~~>) A) (/\ (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (F C A D x j k clim j cv k F k cv ` CC e. abs F k cv ` - A opr ` < x cv br /\ raluz birexa 0 < x cv br imbi2i x RR biral A CC e. anbi2i syl6bbr))
+thm (climcl ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) () (-> (/\ (e. A C) (br F (~~>) A)) (e. A (CC))) (F V A C x j k clim biimpa pm3.26d exp climrel F A brrelexi con3i A CC e. pm2.21d A C e. adantr pm2.61an1 imp))
+thm (climcvgc1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x) (B j) (B k) (B x)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (x cv B 0 < breq2 x cv B abs F k cv ` - A opr ` < breq2 F k cv ` CC e. anbi2d j cv <_ k cv br imbi2d k ZZ biraldv j ZZ birexdv imbi12d RR rcla4v F V A C x j k clim biimpa pm3.27d syl5com exp31 climrel F A brrelexi con3i B RR e. 0 < B br j ZZ k ZZ j cv <_ k cv br F k cv ` CC e. abs F k cv ` - A opr ` < B br /\ -> A.e. E.e. -> -> pm2.21d A C e. a1d pm2.61i imp43))
+thm (climcvg1 ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) (A C F B j k climcvgc1 F k cv ` CC e. abs F k cv ` - A opr ` < B br pm3.27 j cv <_ k cv br imim2i k ZZ r19.20si j ZZ r19.22si syl))
+thm (climcvg1z ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) (A C F B j k climcvg1 j cv k abs F k cv ` - A opr ` < B br raluz birexa sylibr))
+thm (clim1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((clim.1 (e. F (V)))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) (clim.1 F V A CC x j k clim mpan A CC e. x RR 0 < x cv br j ZZ k ZZ j cv <_ k cv br F k cv ` CC e. abs F k cv ` - A opr ` < x cv br /\ -> A.e. E.e. -> A.e. ibar bitr4d))
+thm (clim1z ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((clim.1 (e. F (V)))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim.1 F V A CC x j k climuz mpan A CC e. x RR 0 < x cv br j ZZ k ZZ> j cv ` F k cv ` CC e. abs F k cv ` - A opr ` < x cv br /\ A.e. E.e. -> A.e. ibar bitr4d))
+thm (clim1a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x)) ((clim.1 (e. F (V)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim.1 A x n k clim1 m ZZ k ZZ m cv <_ k cv br F k cv ` CC e. -> A.e. E.e. adantr F k cv ` CC e. abs F k cv ` - A opr ` < x cv br pm3.27 n cv <_ k cv br imim2i k ZZ r19.20si n ZZ r19.22si n cv j cv <_ k cv breq1 abs F k cv ` - A opr ` < x cv br imbi1d k ZZ biraldv ZZ cbvrexv sylib m ZZ k ZZ m cv <_ k cv br F k cv ` CC e. -> A.e. E.e. a1i j cv ZZ m cv m cv <_ j cv br ifcl ancoms exp k ZZ m cv <_ k cv br F k cv ` CC e. -> j cv <_ k cv br abs F k cv ` - A opr ` < x cv br -> /\ A.e. adantr m cv j cv max1 k cv RR e. adantr m cv m cv <_ j cv br j cv m cv if k cv letrt 3expa m cv RR e. j cv RR e. pm3.26 j cv RR m cv m cv <_ j cv br ifcl ancoms jca sylan mpand m cv j cv max2 k cv RR e. adantr j cv m cv <_ j cv br j cv m cv if k cv letrt 3expa m cv RR e. j cv RR e. pm3.27 j cv RR m cv m cv <_ j cv br ifcl ancoms jca sylan mpand jcad m cv zret j cv zret anim12i k cv zret syl2an F k cv ` CC e. abs F k cv ` - A opr ` < x cv br /\ imim1d m cv <_ k cv br F k cv ` CC e. j cv <_ k cv br abs F k cv ` - A opr ` < x cv br prth syl5 r19.20dva exp com23 imp jcad n cv m cv <_ j cv br j cv m cv if <_ k cv breq1 F k cv ` CC e. abs F k cv ` - A opr ` < x cv br /\ imbi1d k ZZ biraldv ZZ rcla4ev syl6 k ZZ m cv <_ k cv br F k cv ` CC e. -> j cv <_ k cv br abs F k cv ` - A opr ` < x cv br -> r19.26 sylan2br exp32 imp com23 r19.23adv exp r19.23aiv impbid 0 < x cv br imbi2d x RR biraldv A CC e. adantl bitrd))
+thm (clim1az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x)) ((clim.1 (e. F (V)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (` (ZZ>) (cv m)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))) (m cv k F k cv ` CC e. raluz birexa A CC e. anbi2i biimp clim.1 A m k x j clim1a j cv k abs F k cv ` - A opr ` < x cv br raluz birexa 0 < x cv br imbi2i x RR biral bicomi syl6bb syl))
+thm (climcvg2 ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) ((climcvg2.1 (e. M (ZZ)))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (A C F B m k climcvg1 climcvg2.1 zre M m cv max2 mpan k cv RR e. adantr m cv M <_ m cv br m cv M if k cv letrt m cv RR e. k cv RR e. pm3.26 climcvg2.1 zre m cv RR M M <_ m cv br ifcl mpan2 k cv RR e. adantr m cv RR e. k cv RR e. pm3.27 syl3anc mpand m cv zret k cv zret syl2an abs F k cv ` - A opr ` < B br imim1d r19.20dva m cv zret climcvg2.1 zre M m cv max1 mpan syl jctild climcvg2.1 m cv ZZ M M <_ m cv br ifcl mpan2 jctild j cv M <_ m cv br m cv M if M <_ breq2 j cv M <_ m cv br m cv M if <_ k cv breq1 abs F k cv ` - A opr ` < B br imbi1d k ZZ biraldv anbi12d ZZ rcla4ev syl6 r19.23aiv syl))
+thm (climcvg2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))) (M m cv max2 k cv RR e. 3adant3 m cv M <_ m cv br m cv M if k cv letrt M RR e. m cv RR e. k cv RR e. 3simp2 m cv RR M M <_ m cv br ifcl ancoms k cv RR e. 3adant3 M RR e. m cv RR e. k cv RR e. 3simp3 syl3anc mpand M zret m cv zret k cv zret syl3an 3expa abs F k cv ` - A opr ` < B br imim1d r19.20dva M m cv max1 M zret m cv zret syl2an jctild m cv ZZ M M <_ m cv br ifcl ancoms jctild M j k ZZ> j cv ` abs F k cv ` - A opr ` < B br A.e. rexuz j cv M <_ m cv br m cv M if M <_ breq2 j cv M <_ m cv br m cv M if <_ k cv breq1 abs F k cv ` - A opr ` < B br imbi1d k ZZ biraldv anbi12d ZZ rcla4ev j cv k abs F k cv ` - A opr ` < B br raluz M <_ j cv br anbi2d birexa sylibr syl5bir m cv ZZ e. adantr syld exp r19.23adv A C F B m k climcvg1 syl5 imp))
+thm (climcvgc2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (M m cv max2 k cv RR e. 3adant3 m cv M <_ m cv br m cv M if k cv letrt M RR e. m cv RR e. k cv RR e. 3simp2 m cv RR M M <_ m cv br ifcl ancoms k cv RR e. 3adant3 M RR e. m cv RR e. k cv RR e. 3simp3 syl3anc mpand M zret m cv zret k cv zret syl3an 3expa F k cv ` CC e. abs F k cv ` - A opr ` < B br /\ imim1d r19.20dva M m cv max1 M zret m cv zret syl2an jctild m cv ZZ M M <_ m cv br ifcl ancoms jctild M j k ZZ j cv <_ k cv br F k cv ` CC e. abs F k cv ` - A opr ` < B br /\ -> A.e. rexuz j cv M <_ m cv br m cv M if M <_ breq2 j cv M <_ m cv br m cv M if <_ k cv breq1 F k cv ` CC e. abs F k cv ` - A opr ` < B br /\ imbi1d k ZZ biraldv anbi12d ZZ rcla4ev syl5bir m cv ZZ e. adantr syld exp r19.23adv A C F B m k climcvgc1 syl5 exp4c 3imp imp))
+thm (clim2az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((clim2x.1 (e. F (V)))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (` (ZZ>) (cv m)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))) (M A CC F x cv j k climcvg2z exp44 imp exp4a r19.21adv m ZZ k ZZ> m cv ` F k cv ` CC e. A.e. E.e. adantrr clim2x.1 A m k x j clim1az j cv M eluzelz k ZZ> j cv ` abs F k cv ` - A opr ` < x cv br A.e. anim1i r19.22i2 0 < x cv br imim2i x RR r19.20si syl5bir M ZZ e. adantl impbid))
+thm (clim3a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (N j) (N k) (N m) (N n) (N x)) ((clim3.1 (e. F (V))) (clim3.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim3.1 A m k x j clim1a j cv <_ k cv br abs F k cv ` - A opr ` < x cv br N <_ k cv br pm3.42 k ZZ r19.20si j ZZ r19.22si 0 < x cv br imim2i x RR r19.20si syl6bi clim3.1 A m k x n clim1a clim3.2 zre N j cv max1 mpan k cv RR e. adantr clim3.2 zre N N <_ j cv br j cv N if k cv letrt mp3an1 clim3.2 zre j cv RR N N <_ j cv br ifcl mpan2 sylan mpand clim3.2 zre N j cv max2 mpan k cv RR e. adantr j cv N <_ j cv br j cv N if k cv letrt j cv RR e. k cv RR e. pm3.26 clim3.2 zre j cv RR N N <_ j cv br ifcl mpan2 k cv RR e. adantr j cv RR e. k cv RR e. pm3.27 syl3anc mpand jcad j cv zret k cv zret syl2an abs F k cv ` - A opr ` < x cv br imim1d r19.20dva clim3.2 j cv ZZ N N <_ j cv br ifcl mpan2 jctild n cv N <_ j cv br j cv N if <_ k cv breq1 abs F k cv ` - A opr ` < x cv br imbi1d k ZZ biraldv ZZ rcla4ev syl6 r19.23aiv 0 < x cv br imim2i x RR r19.20si syl5bir impbid))
+thm (clim3 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((clim3.1 (e. F (V))) (clim3.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))) (clim3.1 clim3.2 A m k x j clim3a clim3.2 m cv N <_ k cv breq1 F k cv ` CC e. imbi1d k ZZ biraldv ZZ rcla4ev mpan sylan2))
+thm (climcvg4 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) ((climcvg4.1 (e. M (ZZ)))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (climcvg4.1 A C F B j k climcvg2 j cv <_ k cv br abs F k cv ` - A opr ` < B br N <_ k cv br pm3.42 k ZZ r19.20si M <_ j cv br anim2i j ZZ r19.22si syl))
+thm (climcvgc4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) (M A C F B j k climcvgc2z k cv N eluzelz j cv <_ k cv br F k cv ` CC e. abs F k cv ` - A opr ` < B br /\ -> imim1i r19.20i2 j ZZ> M ` r19.22si syl))
+thm (climOLD ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) ((climOLD.1 (e. A (V)))) (<-> (br F (~~>OLD) A) (/\ (/\ (:--> F (NN) (CC)) (e. A (CC))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x))))))))) (climrelOLD F A brrelexi nnex NN V F CC fex ax-mp A CC e. x RR 0 < x cv br y NN z NN y cv <_ z cv br abs F z cv ` - A opr ` < x cv br -> A.e. E.e. -> A.e. ad2antll climOLD.1 f cv F NN CC feq1 w cv CC e. anbi1d f cv F z cv fveq1 - w cv opreq1d abs fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d w cv A CC eleq1 F NN CC :--> anbi2d w cv A F z cv ` - opreq2 abs fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d f w x y z df-climOLD V V brabg mpan2 pm5.21nii))
+thm (climseqOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((climOLD.1 (e. A (V)))) (-> (br F (~~>OLD) A) (:--> F (NN) (CC))) (climOLD.1 F x y z climOLD pm3.26bd pm3.26d))
+thm (climclOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((climOLD.1 (e. A (V)))) (-> (br F (~~>OLD) A) (e. A (CC))) (climOLD.1 F x y z climOLD pm3.26bd pm3.27d))
+thm (climcvgOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((climOLD.1 (e. A (V)))) (-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x)))))))) (climOLD.1 F x y z climOLD pm3.27bd))
+thm (climcvg2OLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((climOLD.1 (e. A (V)))) (-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x)))))))) (climOLD.1 F x y z climcvgOLD x RR 0 < x cv br y z abs F z cv ` - A opr ` < x cv br cvg1 sylibr))
+thm (climfvclOLD () ((climOLD.1 (e. A (V)))) (-> (/\ (br F (~~>OLD) A) (e. B (NN))) (e. (` F B) (CC))) (F NN CC B ffvrn climOLD.1 F climseqOLD sylan))
+thm (climcltOLD ((F x) (A x)) () (-> (/\ (e. A B) (br F (~~>OLD) A)) (e. A (CC))) (x cv A F ~~>OLD breq2 x cv A CC eleq1 imbi12d x visset F climclOLD B vtoclg imp))
+thm (clim2OLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w)) () (-> (/\ (:--> F (NN) (CC)) (e. A (CC))) (<-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x))))))))) (w cv A F ~~>OLD breq2 w cv A F z cv ` - opreq2 abs fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv bibi12d F NN CC :--> imbi2d F NN CC :--> w cv CC e. /\ x RR 0 < x cv br y NN z NN y cv <_ z cv br abs F z cv ` - w cv opr ` < x cv br -> A.e. E.e. -> A.e. ibar bicomd w visset F x y z climOLD syl5bb expcom vtoclga impcom))
+thm (climconst ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (N j) (N k) (N x)) ((climcnst.1 (e. F (V))) (climcnst.2 (e. N (ZZ)))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (= (` F (cv k)) A)))) (br F (~~>) A)) (N <_ k cv br N <_ k cv br pm3.26 A CC e. 0 < x cv br /\ a1i F k cv ` A - A opreq1 A subidt sylan9eqr abs fveq2d abs0 syl6eq < x cv breq1d biimpar an1rs exp imim12d k ZZ r19.20sdv imp an1rs climcnst.2 jctil j cv N <_ k cv breq1 N <_ k cv br anbi2d abs F k cv ` - A opr ` < x cv br imbi1d k ZZ biraldv ZZ rcla4ev syl exp31 x cv RR e. a1dd r19.21adv imp A CC F k cv ` eleq1a N <_ k cv br imim2d k ZZ r19.20sdv imdistani climcnst.1 climcnst.2 A k x j clim3 syl mpbird))
+thm (climconst2 ((A k)) () (-> (e. A (CC)) (br (X. (ZZ) ({} A)) (~~>) A)) (A CC k cv ZZ fvconst2g exp 0 <_ k cv br a1dd r19.21aiv zex A snex xpex 0z A k climconst mpdan))
+thm (climconstOLD ((x y) (x z) (A x) (y z) (A y) (A z)) ((climconstOLD.1 (e. A (CC)))) (br (X. (NN) ({} A)) (~~>OLD) A) (climconstOLD.1 elisseti NN A {} X. x y z climOLD climconstOLD.1 elisseti NN fconst climconstOLD.1 A CC snssi ax-mp NN A {} X. NN A {} CC fss mp2an climconstOLD.1 pm3.2i climconstOLD.1 elisseti z cv NN fvconst2 - A opreq1d climconstOLD.1 subid syl6eq abs fveq2d abs0 syl6eq < x cv breq1d biimprd 1 <_ z cv br a1d com3r r19.21aiv 1nn jctil y cv 1 <_ z cv breq1 abs NN A {} X. z cv ` - A opr ` < x cv br imbi1d z NN biraldv NN rcla4ev syl x cv RR e. a1i rgen mpbir2an))
+thm (clim0 () () (br (X. (ZZ) ({} (0))) (~~>) (0)) (0cn 0 climconst2 ax-mp))
+thm (clim0OLD () () (br (X. (NN) ({} (0))) (~~>OLD) (0)) (0cn climconstOLD))
+thm (climunii ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (F j) (F k) (F m)) ((climuni.1 (e. A (V))) (climuni.2 (e. B (V))) (climunii.3 (/\ (br F (~~>) A) (br F (~~>) B)))) (= A B) (k cv m cv j z2get rgen2 climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl abscl 2re 2pos divgt0lem climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl abscl 2re 2re 2pos gt0ne0i redivcl climuni.1 climunii.3 pm3.26i A V F abs A - B opr ` / 2 opr k j climcvgc1 mpanl12 mpan climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl abscl 2re 2re 2pos gt0ne0i redivcl climuni.2 climunii.3 pm3.27i B V F abs A - B opr ` / 2 opr m j climcvgc1 mpanl12 mpan jca j ZZ k cv <_ j cv br F j cv ` CC e. abs F j cv ` - A opr ` < abs A - B opr ` / 2 opr br /\ -> m cv <_ j cv br F j cv ` CC e. abs F j cv ` - B opr ` < abs A - B opr ` / 2 opr br /\ -> r19.26 climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl abscl ltnr F j cv ` CC e. abs F j cv ` - A opr ` < abs A - B opr ` / 2 opr br abs F j cv ` - B opr ` < abs A - B opr ` / 2 opr br anandi climuni.1 climunii.3 pm3.26i A V F climcl mp2an F j cv ` A abssubt mpan2 < abs A - B opr ` / 2 opr breq1d abs F j cv ` - B opr ` < abs A - B opr ` / 2 opr br anbi1d climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl abscl climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an A B F j cv ` abs A - B opr ` abs3lemt mpanl12 mpan2 sylbid imp sylbir mto k cv <_ j cv br F j cv ` CC e. abs F j cv ` - A opr ` < abs A - B opr ` / 2 opr br /\ m cv <_ j cv br F j cv ` CC e. abs F j cv ` - B opr ` < abs A - B opr ` / 2 opr br /\ prth mtoi j ZZ r19.20si sylbir m ZZ r19.22si k ZZ r19.22si k ZZ m ZZ j ZZ k cv <_ j cv br F j cv ` CC e. abs F j cv ` - A opr ` < abs A - B opr ` / 2 opr br /\ -> A.e. j ZZ m cv <_ j cv br F j cv ` CC e. abs F j cv ` - B opr ` < abs A - B opr ` / 2 opr br /\ -> A.e. reeanv j ZZ k cv <_ j cv br m cv <_ j cv br /\ ralnex m ZZ birex m ZZ j ZZ k cv <_ j cv br m cv <_ j cv br /\ E.e. rexnal bitr k ZZ birex k ZZ m ZZ j ZZ k cv <_ j cv br m cv <_ j cv br /\ E.e. A.e. rexnal bitr 3imtr3 3syl mt2 A - B opr 0 df-ne climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subcl absgt0 bitr3 bicon1i mpbi climuni.1 climunii.3 pm3.26i A V F climcl mp2an climuni.2 climunii.3 pm3.27i B V F climcl mp2an subeq0 mpbi))
+thm (climuni () ((climuni.1 (e. A (V))) (climuni.2 (e. B (V)))) (-> (/\ (br F (~~>) A) (br F (~~>) B)) (= A B)) (A F ~~> A br F ~~> B br /\ A 0 if B eqeq1 B F ~~> A br F ~~> B br /\ B 0 if F ~~> A br F ~~> B br /\ A 0 if eqeq2 climuni.1 0cn elisseti F ~~> A br F ~~> B br /\ keepel climuni.2 0cn elisseti F ~~> A br F ~~> B br /\ keepel A F ~~> A br F ~~> B br /\ A 0 if F ~~> breq2 F ~~> B br anbi1d B F ~~> A br F ~~> B br /\ B 0 if F ~~> breq2 F ~~> F ~~> A br F ~~> B br /\ A 0 if br anbi2d F F ~~> A br F ~~> B br /\ F ZZ 0 {} X. if ~~> F ~~> A br F ~~> B br /\ A 0 if breq1 F F ~~> A br F ~~> B br /\ F ZZ 0 {} X. if ~~> F ~~> A br F ~~> B br /\ B 0 if breq1 anbi12d 0 F ~~> A br F ~~> B br /\ A 0 if ZZ 0 {} X. ~~> breq2 ZZ 0 {} X. ~~> 0 br anbi1d 0 F ~~> A br F ~~> B br /\ B 0 if ZZ 0 {} X. ~~> breq2 ZZ 0 {} X. ~~> F ~~> A br F ~~> B br /\ A 0 if br anbi2d ZZ 0 {} X. F ~~> A br F ~~> B br /\ F ZZ 0 {} X. if ~~> F ~~> A br F ~~> B br /\ A 0 if breq1 ZZ 0 {} X. F ~~> A br F ~~> B br /\ F ZZ 0 {} X. if ~~> F ~~> A br F ~~> B br /\ B 0 if breq1 anbi12d clim0 clim0 pm3.2i elimhyp3v climunii dedth2v))
+thm (climuniiOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((climuniOLD.1 (e. A (V))) (climuniOLD.2 (e. B (V))) (climuniiOLD.3 (/\ (br F (~~>OLD) A) (br F (~~>OLD) B)))) (= A B) (y cv w cv z nn2get rgen2 climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl abscl 2re 2pos divgt0lem climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl abscl 2re 2re 2pos gt0ne0i redivcl x cv abs A - B opr ` / 2 opr 0 < breq2 x cv abs A - B opr ` / 2 opr abs F z cv ` - A opr ` < breq2 y cv <_ z cv br imbi2d z NN biraldv y NN birexdv imbi12d climuniiOLD.3 pm3.26i climuniOLD.1 F x y z climcvgOLD ax-mp vtoclri ax-mp climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl abscl 2re 2re 2pos gt0ne0i redivcl x cv abs A - B opr ` / 2 opr 0 < breq2 x cv abs A - B opr ` / 2 opr abs F z cv ` - B opr ` < breq2 w cv <_ z cv br imbi2d z NN biraldv w NN birexdv imbi12d climuniiOLD.3 pm3.27i climuniOLD.2 F x w z climcvgOLD ax-mp vtoclri ax-mp jca z NN y cv <_ z cv br abs F z cv ` - A opr ` < abs A - B opr ` / 2 opr br -> w cv <_ z cv br abs F z cv ` - B opr ` < abs A - B opr ` / 2 opr br -> r19.26 y cv <_ z cv br w cv <_ z cv br /\ abs F z cv ` - A opr ` < abs A - B opr ` / 2 opr br abs F z cv ` - B opr ` < abs A - B opr ` / 2 opr br /\ con3 climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl abscl ltnr climuniiOLD.3 pm3.26i climuniOLD.1 F climseqOLD ax-mp z cv ffvrni climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp F z cv ` A abssubt mpan2 syl < abs A - B opr ` / 2 opr breq1d abs F z cv ` - B opr ` < abs A - B opr ` / 2 opr br anbi1d climuniiOLD.3 pm3.26i climuniOLD.1 F climseqOLD ax-mp z cv ffvrni climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl abscl climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp A B F z cv ` abs A - B opr ` abs3lemt mpanl12 mpan2 syl sylbid mtoi syl5com y cv <_ z cv br abs F z cv ` - A opr ` < abs A - B opr ` / 2 opr br w cv <_ z cv br abs F z cv ` - B opr ` < abs A - B opr ` / 2 opr br prth syl5 r19.20i sylbir w NN r19.22si y NN r19.22si y NN w NN z NN y cv <_ z cv br abs F z cv ` - A opr ` < abs A - B opr ` / 2 opr br -> A.e. z NN w cv <_ z cv br abs F z cv ` - B opr ` < abs A - B opr ` / 2 opr br -> A.e. reeanv z NN y cv <_ z cv br w cv <_ z cv br /\ ralnex w NN birex w NN z NN y cv <_ z cv br w cv <_ z cv br /\ E.e. rexnal bitr y NN birex y NN w NN z NN y cv <_ z cv br w cv <_ z cv br /\ E.e. A.e. rexnal bitr 3imtr3 3syl mt2 A - B opr 0 df-ne climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subcl absgt0 bitr3 bicon1i mpbi climuniiOLD.3 pm3.26i climuniOLD.1 F climclOLD ax-mp climuniiOLD.3 pm3.27i climuniOLD.2 F climclOLD ax-mp subeq0 mpbi))
+thm (climuniOLD () ((climuniOLD.1 (e. A (V))) (climuniOLD.2 (e. B (V)))) (-> (/\ (br F (~~>OLD) A) (br F (~~>OLD) B)) (= A B)) (A F ~~>OLD A br F ~~>OLD B br /\ A 0 if B eqeq1 B F ~~>OLD A br F ~~>OLD B br /\ B 0 if F ~~>OLD A br F ~~>OLD B br /\ A 0 if eqeq2 climuniOLD.1 0cn elisseti F ~~>OLD A br F ~~>OLD B br /\ keepel climuniOLD.2 0cn elisseti F ~~>OLD A br F ~~>OLD B br /\ keepel A F ~~>OLD A br F ~~>OLD B br /\ A 0 if F ~~>OLD breq2 F ~~>OLD B br anbi1d B F ~~>OLD A br F ~~>OLD B br /\ B 0 if F ~~>OLD breq2 F ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ A 0 if br anbi2d F F ~~>OLD A br F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ A 0 if breq1 F F ~~>OLD A br F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ B 0 if breq1 anbi12d 0 F ~~>OLD A br F ~~>OLD B br /\ A 0 if NN 0 {} X. ~~>OLD breq2 NN 0 {} X. ~~>OLD 0 br anbi1d 0 F ~~>OLD A br F ~~>OLD B br /\ B 0 if NN 0 {} X. ~~>OLD breq2 NN 0 {} X. ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ A 0 if br anbi2d NN 0 {} X. F ~~>OLD A br F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ A 0 if breq1 NN 0 {} X. F ~~>OLD A br F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br F ~~>OLD B br /\ B 0 if breq1 anbi12d clim0OLD clim0OLD pm3.2i elimhyp3v climuniiOLD dedth2v))
+thm (climmoOLD ((x y) (F x) (F y)) () (E* x (br F (~~>OLD) (cv x))) (x visset y visset F climuniOLD x y gen2 x cv y cv F ~~>OLD breq2 mo4 mpbir))
+thm (climshiftOLD ((y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (w x) (v x) (u x) (F x) (F y) (F w) (F v) (F u) (x z) (G x) (G y) (G z) (G w) (G v) (B x) (B y) (B z) (B w) (B v) (B u)) ((climshift.1 (e. A (V)))) (-> (/\/\ (:--> G (NN) (CC)) (e. B (NN0)) (A.e. x (NN) (-> (br B (<) (cv x)) (= (` G (cv x)) (` F (opr (cv x) (-) B)))))) (-> (br F (~~>OLD) A) (br G (~~>OLD) A))) (G NN CC :--> B NN0 e. x NN B < x cv br G x cv ` F x cv - B opr ` = -> A.e. /\ pm3.26 climshift.1 F climclOLD anim12i z cv v cv + B opr <_ w cv breq1 abs G w cv ` - A opr ` < y cv br imbi1d w NN biraldv NN rcla4ev v cv B nnnn0addclt x NN B < x cv br G x cv ` F x cv - B opr ` = -> A.e. adantrr u NN v cv <_ u cv br abs F u cv ` - A opr ` < y cv br -> A.e. adantr v cv B w cv leaddsubt v cv nnret B nn0ret w cv nnret syl3an 3expa u NN v cv <_ u cv br abs F u cv ` - A opr ` < y cv br -> A.e. adantlr v cv nnge1t B NN0 e. w cv NN e. ad2antll ax1re 1 v cv w cv - B opr letrt mp3an1 v cv nnret B NN0 e. w cv NN e. ad2antll w cv B zsubclt w cv nnzt B nn0zt syl2an ancoms v cv NN e. adantll w cv - B opr zret syl sylanc mpand w cv B zsubclt w cv nnzt B nn0zt syl2an ancoms v cv NN e. adantll jctild w cv - B opr elnnz1 syl6ibr u NN v cv <_ u cv br abs F u cv ` - A opr ` < y cv br -> A.e. adantlr u cv w cv - B opr v cv <_ breq2 u cv w cv - B opr F fveq2 - A opreq1d abs fveq2d < y cv breq1d imbi12d NN rcla4v com3l v cv NN e. B NN0 e. /\ w cv NN e. ad2antlr mpdd sylbid exp x NN B < x cv br G x cv ` F x cv - B opr ` = -> A.e. adantlrr imp31 B nn0cnt 1cn B 1 axaddcom mpan2 syl v cv NN e. adantl v cv nnge1t B NN0 e. adantr ax1re 1 v cv B leadd1t mp3an1 v cv nnret B nn0ret syl2an mpbid eqbrtrd w cv NN e. adantr B + 1 opr v cv + B opr w cv letrt B peano2nn0 B + 1 opr nn0ret syl v cv NN e. w cv NN e. ad2antlr v cv B nnnn0addclt v cv + B opr nnret syl w cv NN e. adantr w cv nnret v cv NN e. B NN0 e. /\ adantl syl3anc mpand B w cv nn0ltp1let w cv nnnn0t sylan2 v cv NN e. adantll sylibrd x NN B < x cv br G x cv ` F x cv - B opr ` = -> A.e. adantlrr x cv w cv B < breq2 x cv w cv G fveq2 x cv w cv - B opreq1 F fveq2d eqeq12d imbi12d NN rcla4v com12 v cv NN e. B NN0 e. ad2antrr imp syld imp - A opreq1d abs fveq2d < y cv breq1d u NN v cv <_ u cv br abs F u cv ` - A opr ` < y cv br -> A.e. adantllr mpbird exp31 r19.21aiv sylanc exp31 com12 r19.23adv 0 < y cv br imim2d y RR r19.20sdv climshift.1 F y v u climcvgOLD syl5 imp G NN CC :--> adantll jca climshift.1 G y z w climOLD sylibr exp42 3imp))
+thm (climreOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (G x) (G y) (G z) (G w) (A y) (A z) (A w)) ((climre.1 (e. A (V))) (climre.2 (br F (~~>OLD) A)) (climre.3 (Fn G (NN))) (climre.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))))) (br G (~~>OLD) (` (Re) A)) (Re A fvex G y z w climOLD G NN CC w ffnfv climre.3 x cv w cv G fveq2 x cv w cv F fveq2 Re fveq2d eqeq12d climre.4 vtoclga climre.2 climre.1 F w cv climfvclOLD mpan F w cv ` reclt syl eqeltrd recnd rgen mpbir2an climre.2 climre.1 F climclOLD ax-mp recl recn pm3.2i climre.2 climre.1 F y z w climcvgOLD ax-mp climre.2 climre.1 F w cv climfvclOLD mpan climre.2 climre.1 F climclOLD ax-mp F w cv ` A resubt mpan2 syl x cv w cv G fveq2 x cv w cv F fveq2 Re fveq2d eqeq12d climre.4 vtoclga - Re A ` opreq1d eqtr4d abs fveq2d climre.2 climre.1 F w cv climfvclOLD mpan climre.2 climre.1 F climclOLD ax-mp F w cv ` A subclt mpan2 syl F w cv ` - A opr absrelet syl eqbrtrrd y cv RR e. adantl abs G w cv ` - Re A ` opr ` abs F w cv ` - A opr ` y cv lelttrt x cv w cv G fveq2 x cv w cv F fveq2 Re fveq2d eqeq12d climre.4 vtoclga climre.2 climre.1 F w cv climfvclOLD mpan F w cv ` reclt syl eqeltrd climre.2 climre.1 F climclOLD ax-mp recl G w cv ` Re A ` resubclt mpan2 syl recnd G w cv ` - Re A ` opr absclt syl y cv RR e. adantl climre.2 climre.1 F w cv climfvclOLD mpan climre.2 climre.1 F climclOLD ax-mp F w cv ` A subclt mpan2 syl F w cv ` - A opr absclt syl y cv RR e. adantl y cv RR e. w cv NN e. pm3.26 syl3anc mpand z cv <_ w cv br imim2d r19.20dva z NN r19.22sdv 0 < y cv br imim2d r19.20i ax-mp mpbir2an))
+thm (climrecl ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((clmrecl.1 (e. A (V))) (clmrecl.2 (e. M (ZZ))) (clmrecl.3 (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (e. (` F (cv k)) (RR)))) (clmrecl.4 (br F (~~>) A))) (e. A (RR)) (0re clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn abscl 0 abs Im A ` ` eqleltt mp2an clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn absge0 j cv zret j cv leidt syl M <_ j cv br adantr k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d imbi12d clmrecl.3 chvarv F j cv ` reim0t syl - Im A ` opreq1d Im A ` df-neg syl6reqr k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d imbi12d clmrecl.3 chvarv recnd clmrecl.1 clmrecl.4 A V F climcl mp2an F j cv ` A imsubt mpan2 syl eqtr4d abs fveq2d clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn absneg syl5eqr k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d imbi12d clmrecl.3 chvarv recnd clmrecl.1 clmrecl.4 A V F climcl mp2an F j cv ` A subclt mpan2 F j cv ` - A opr absimlet 3syl eqbrtrd k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d imbi12d clmrecl.3 chvarv recnd clmrecl.1 clmrecl.4 A V F climcl mp2an F j cv ` A subclt mpan2 syl F j cv ` - A opr absclt clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn abscl abs Im A ` ` abs F j cv ` - A opr ` lenltt mpan 3syl mpbid jca m cv j cv j cv <_ breq2 m cv j cv F fveq2 - A opreq1d abs fveq2d < abs Im A ` ` breq1d negbid anbi12d ZZ rcla4ev syldan m ZZ j cv <_ m cv br abs F m cv ` - A opr ` < abs Im A ` ` br rexanali sylib exp M <_ j cv br m ZZ j cv <_ m cv br abs F m cv ` - A opr ` < abs Im A ` ` br -> A.e. imnan sylib nrex clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn abscl clmrecl.1 clmrecl.4 clmrecl.2 A V F abs Im A ` ` j m climcvg2 mpanl12 mpan mto mpbir2an eqcomi clmrecl.1 clmrecl.4 A V F climcl mp2an imcl recn abs00 mpbi clmrecl.1 clmrecl.4 A V F climcl mp2an reim0b mpbir))
+thm (climreclOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) ((climreclOLD.1 (e. A (V))) (climreclOLD.2 (:--> F (NN) (RR))) (climreclOLD.3 (br F (~~>OLD) A))) (e. A (RR)) (0re climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn abscl 0 abs Im A ` ` eqleltt mp2an climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn absge0 y cv nnret y cv leidt syl climreclOLD.2 y cv ffvrni F y cv ` reim0t syl - Im A ` opreq1d Im A ` df-neg syl6reqr climreclOLD.2 axresscn F NN RR CC fss mp2an y cv ffvrni climreclOLD.3 climreclOLD.1 F climclOLD ax-mp F y cv ` A imsubt mpan2 syl eqtr4d abs fveq2d climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn absneg syl5eqr climreclOLD.2 axresscn F NN RR CC fss mp2an y cv ffvrni climreclOLD.3 climreclOLD.1 F climclOLD ax-mp F y cv ` A subclt mpan2 F y cv ` - A opr absimlet 3syl eqbrtrd climreclOLD.2 axresscn F NN RR CC fss mp2an y cv ffvrni climreclOLD.3 climreclOLD.1 F climclOLD ax-mp F y cv ` A subclt mpan2 syl F y cv ` - A opr absclt climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn abscl abs Im A ` ` abs F y cv ` - A opr ` lenltt mpan 3syl mpbid jca y cv <_ y cv br abs F y cv ` - A opr ` < abs Im A ` ` br annim sylib z cv y cv y cv <_ breq2 z cv y cv F fveq2 - A opreq1d abs fveq2d < abs Im A ` ` breq1d imbi12d negbid NN rcla4ev mpdan z NN y cv <_ z cv br abs F z cv ` - A opr ` < abs Im A ` ` br -> rexnal sylib nrex climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn abscl climreclOLD.3 climreclOLD.1 F x y z climcvgOLD ax-mp x cv abs Im A ` ` 0 < breq2 x cv abs Im A ` ` abs F z cv ` - A opr ` < breq2 y cv <_ z cv br imbi2d z NN biraldv y NN birexdv imbi12d RR rcla4v mp2 mto mpbir2an eqcomi climreclOLD.3 climreclOLD.1 F climclOLD ax-mp imcl recn abs00 mpbi climreclOLD.3 climreclOLD.1 F climclOLD ax-mp reim0b mpbir))
+thm (climrecltOLD () ((climreclt.1 (e. A (V)))) (-> (/\ (:--> F (NN) (RR)) (br F (~~>OLD) A)) (e. A (RR))) (A F NN RR :--> F ~~>OLD A br /\ A 0 if RR eleq1 climreclt.1 0re elisseti F NN RR :--> F ~~>OLD A br /\ keepel F F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR feq1 F F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD A breq1 anbi12d A F NN RR :--> F ~~>OLD A br /\ A 0 if F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD breq2 F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR :--> anbi2d NN 0 {} X. F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR feq1 NN 0 {} X. F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD 0 breq1 anbi12d 0 F NN RR :--> F ~~>OLD A br /\ A 0 if F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD breq2 F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR :--> anbi2d 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an clim0OLD pm3.2i elimhyp2v pm3.26i F F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR feq1 F F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD A breq1 anbi12d A F NN RR :--> F ~~>OLD A br /\ A 0 if F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD breq2 F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR :--> anbi2d NN 0 {} X. F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR feq1 NN 0 {} X. F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD 0 breq1 anbi12d 0 F NN RR :--> F ~~>OLD A br /\ A 0 if F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if ~~>OLD breq2 F NN RR :--> F ~~>OLD A br /\ F NN 0 {} X. if NN RR :--> anbi2d 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an clim0OLD pm3.2i elimhyp2v pm3.27i climreclOLD dedth))
+thm (climge0 ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((clmge0.1 (e. A (V))) (clmge0.2 (e. M (ZZ))) (clmge0.3 (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) (clmge0.4 (br F (~~>) A))) (br (0) (<_) A) (clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl A lt0neg1t ax-mp clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl renegcl clmge0.1 clmge0.4 clmge0.2 A V F A -u j m climcvg2 mpanl12 mpan sylbi m cv j cv j cv <_ breq2 m cv j cv F fveq2 - A opreq1d abs fveq2d < A -u breq1d negbid anbi12d ZZ rcla4ev A < 0 br j cv ZZ e. pm3.27 M <_ j cv br adantr j cv zret j cv leidt syl A < 0 br M <_ j cv br ad2antlr k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.27d A < 0 br adantll k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl renegcl A -u F j cv ` addge02t mpan syl A < 0 br adantll mpbid k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d recnd clmge0.1 clmge0.4 A V F climcl mp2an F j cv ` A subnegt mpan2 syl abs fveq2d A < 0 br adantll F j cv ` + A -u opr absidt k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl renegcl F j cv ` A -u axaddrcl mpan2 syl A < 0 br adantll clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl renegcl F j cv ` A -u addge0t mpanl2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d A < 0 br adantll k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.27d A < 0 br adantll clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl 0re ltle clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl A le0neg1t ax-mp sylib j cv ZZ e. M <_ j cv br ad2antll jca sylanc sylanc eqtr2d breqtrd k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d recnd clmge0.1 clmge0.4 A V F climcl mp2an F j cv ` A subclt mpan2 syl F j cv ` - A opr absclt clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl renegcl A -u abs F j cv ` - A opr ` lenltt mpan 3syl A < 0 br adantll mpbid jca sylanc m ZZ j cv <_ m cv br abs F m cv ` - A opr ` < A -u br rexanali sylib exp M <_ j cv br m ZZ j cv <_ m cv br abs F m cv ` - A opr ` < A -u br -> A.e. imnan sylib nrexdv pm2.65i 0re clmge0.1 clmge0.2 k cv j cv ZZ eleq1 k cv j cv M <_ breq2 anbi12d k cv j cv F fveq2 RR eleq1d k cv j cv F fveq2 0 <_ breq2d anbi12d imbi12d clmge0.3 chvarv pm3.26d clmge0.4 climrecl lenlt mpbir))
+thm (climaddOLD ((x y) (v x) (f x) (g x) (t x) (u x) (F x) (v y) (f y) (g y) (t y) (u y) (F y) (f v) (g v) (t v) (u v) (F v) (f g) (f t) (f u) (F f) (g t) (g u) (F g) (t u) (F t) (F u) (G x) (G y) (G v) (G u) (G g) (G f) (G t) (h x) (A x) (h y) (A y) (h v) (A v) (h t) (A t) (f h) (g h) (h u) (A h) (A f) (A g) (A u) (B y) (B v) (B t) (B h) (B u) (B f) (B g) (H x) (H y) (H v) (H t) (H h) (H f) (H u)) ((climaddOLD.1 (e. A (V))) (climaddOLD.2 (e. B (V))) (climaddOLD.3 (br F (~~>OLD) A)) (climaddOLD.4 (br G (~~>OLD) B)) (climaddOLD.5 (Fn H (NN))) (climaddOLD.6 (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr (` F (cv x)) (+) (` G (cv x))))))) (br H (~~>OLD) (opr A (+) B)) (H NN CC x ffnfv climaddOLD.5 climaddOLD.6 F x cv ` G x cv ` axaddcl climaddOLD.3 climaddOLD.1 F climseqOLD ax-mp x cv ffvrni climaddOLD.4 climaddOLD.2 G climseqOLD ax-mp x cv ffvrni sylanc eqeltrd rgen mpbir2an climaddOLD.3 climaddOLD.1 F climclOLD ax-mp climaddOLD.4 climaddOLD.2 G climclOLD ax-mp addcl H A + B opr v h t clim2OLD mp2an y cv v cv / 2 opr 0 < breq2 y cv v cv / 2 opr abs F t cv ` - A opr ` < breq2 u cv <_ t cv br imbi2d t NN biraldv u NN birexdv imbi12d climaddOLD.3 climaddOLD.1 F y u t climcvgOLD ax-mp vtoclri y cv v cv / 2 opr 0 < breq2 y cv v cv / 2 opr abs G g cv ` - B opr ` < breq2 f cv <_ g cv br imbi2d g NN biraldv f NN birexdv imbi12d climaddOLD.4 climaddOLD.2 G y f g climcvgOLD ax-mp vtoclri jcad 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 0 < v cv br adantr 2re 2pos v cv 2 divgt0t mpanr2 mpanl2 sylc u cv NN e. f cv NN e. /\ t cv NN e. pm3.27 u cv f cv nn0addge1t u cv nnret f cv nnnn0t syl2an t cv NN e. adantr u cv u cv + f cv opr t cv letrt 3expa u cv nnret f cv NN e. adantr u cv f cv nnaddclt u cv + f cv opr nnret syl jca t cv nnret syl2an mpand g cv NN e. adantr f cv u cv nn0addge2t ancoms u cv nnnn0t f cv nnret syl2an g cv NN e. adantr f cv u cv + f cv opr g cv letrt 3expa f cv nnret u cv NN e. adantl u cv f cv nnaddclt u cv + f cv opr nnret syl jca g cv nnret syl2an mpand t cv NN e. adantlr anim12d u cv <_ t cv br abs F t cv ` - A opr ` < v cv / 2 opr br f cv <_ g cv br abs G g cv ` - B opr ` < v cv / 2 opr br prth syl9 exp4a r19.20dva g cv t cv u cv + f cv opr <_ breq2 g cv t cv G fveq2 - B opreq1d abs fveq2d < v cv / 2 opr breq1d abs F t cv ` - A opr ` < v cv / 2 opr br anbi2d imbi12d NN rcla4v com23 a2d g NN u cv + f cv opr <_ t cv br u cv + f cv opr <_ g cv br abs F t cv ` - A opr ` < v cv / 2 opr br abs G g cv ` - B opr ` < v cv / 2 opr br /\ -> r19.21v syl5ib com12 syl6 mpid v cv RR e. adantll abs F t cv ` - A opr ` abs G t cv ` - B opr ` v cv / 2 opr v cv / 2 opr lt2addt climaddOLD.3 climaddOLD.1 F t cv climfvclOLD mpan climaddOLD.3 climaddOLD.1 F climclOLD ax-mp F t cv ` A subclt mpan2 F t cv ` - A opr absclt 3syl climaddOLD.4 climaddOLD.2 G t cv climfvclOLD mpan climaddOLD.4 climaddOLD.2 G climclOLD ax-mp G t cv ` B subclt mpan2 G t cv ` - B opr absclt 3syl jca 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 jca syl2an 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 v cv / 2 opr recnt v cv / 2 opr 2timest 3syl v cv recnt 2cn 2re 2pos gt0ne0i 2 v cv divcan2t mp3an13 syl eqtr3d t cv NN e. adantl abs F t cv ` - A opr ` + abs G t cv ` - B opr ` opr < breq2d sylibd x cv t cv H fveq2 x cv t cv F fveq2 x cv t cv G fveq2 + opreq12d eqeq12d climaddOLD.6 vtoclga - A + B opr opreq1d climaddOLD.3 climaddOLD.1 F climclOLD ax-mp climaddOLD.4 climaddOLD.2 G climclOLD ax-mp pm3.2i F t cv ` G t cv ` A B sub4t mpan2 climaddOLD.3 climaddOLD.1 F t cv climfvclOLD mpan climaddOLD.4 climaddOLD.2 G t cv climfvclOLD mpan sylanc eqtrd abs fveq2d F t cv ` - A opr G t cv ` - B opr abstrit climaddOLD.3 climaddOLD.1 F t cv climfvclOLD mpan climaddOLD.3 climaddOLD.1 F climclOLD ax-mp F t cv ` A subclt mpan2 syl climaddOLD.4 climaddOLD.2 G t cv climfvclOLD mpan climaddOLD.4 climaddOLD.2 G climclOLD ax-mp G t cv ` B subclt mpan2 syl sylanc eqbrtrd v cv RR e. adantr abs H t cv ` - A + B opr opr ` abs F t cv ` - A opr ` + abs G t cv ` - B opr ` opr v cv lelttrt 3expa x cv t cv H fveq2 x cv t cv F fveq2 x cv t cv G fveq2 + opreq12d eqeq12d climaddOLD.6 vtoclga F t cv ` G t cv ` axaddcl climaddOLD.3 climaddOLD.1 F t cv climfvclOLD mpan climaddOLD.4 climaddOLD.2 G t cv climfvclOLD mpan sylanc eqeltrd climaddOLD.3 climaddOLD.1 F climclOLD ax-mp climaddOLD.4 climaddOLD.2 G climclOLD ax-mp addcl H t cv ` A + B opr subclt mpan2 H t cv ` - A + B opr opr absclt 3syl abs F t cv ` - A opr ` abs G t cv ` - B opr ` axaddrcl climaddOLD.3 climaddOLD.1 F t cv climfvclOLD mpan climaddOLD.3 climaddOLD.1 F climclOLD ax-mp F t cv ` A subclt mpan2 F t cv ` - A opr absclt 3syl climaddOLD.4 climaddOLD.2 G t cv climfvclOLD mpan climaddOLD.4 climaddOLD.2 G climclOLD ax-mp G t cv ` B subclt mpan2 G t cv ` - B opr absclt 3syl sylanc jca sylan mpand syld ancoms u cv NN e. f cv NN e. /\ adantlr syl6d 0 < v cv br adantllr r19.20dva h cv u cv + f cv opr <_ t cv breq1 abs H t cv ` - A + B opr opr ` < v cv br imbi1d t NN biraldv NN rcla4ev u cv f cv nnaddclt sylan exp v cv RR e. 0 < v cv br /\ adantl syld t NN g u cv <_ t cv br abs F t cv ` - A opr ` < v cv / 2 opr br -> f cv <_ g cv br abs G g cv ` - B opr ` < v cv / 2 opr br -> raaan syl5ibr exp r19.23advv u NN f NN t NN u cv <_ t cv br abs F t cv ` - A opr ` < v cv / 2 opr br -> A.e. g NN f cv <_ g cv br abs G g cv ` - B opr ` < v cv / 2 opr br -> A.e. reeanv syl5ibr mpd exp mprgbir))
+thm (climadd ((f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (F k) (F t) (F u) (G f) (G g) (G k) (G t) (G u) (H f) (h k) (H h) (k v) (H k) (H t) (H u) (H v) (N f) (N g) (N h) (N k) (N t) (N u) (N v)) ((climadd2.1 (e. F (V))) (climadd2.2 (e. G (V))) (climadd2.3 (e. H (V))) (climadd2.4 (e. A (CC))) (climadd2.5 (e. B (CC))) (climadd2.6 (br F (~~>) A)) (climadd2.7 (br G (~~>) B)) (climadd2.8 (e. N (ZZ))) (climadd2.9 (-> (/\ (e. (cv k) (ZZ)) (br N (<_) (cv k))) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))) (br H (~~>) (opr A (+) B)) (climadd2.4 climadd2.6 0z A CC F v cv / 2 opr u t N climcvg4 mpanl12 climadd2.5 climadd2.7 0z B CC G v cv / 2 opr f g N climcvg4 mpanl12 jca exp 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 0 < v cv br adantr 2re 2pos v cv 2 divgt0t mpanr2 mpanl2 sylc u cv NN0 e. f cv NN0 e. /\ t cv ZZ e. pm3.27 u cv f cv nn0addge1t u cv nn0ret sylan t cv ZZ e. adantr u cv u cv + f cv opr t cv letrt 3expa u cv nn0ret f cv NN0 e. adantr u cv f cv nn0addclt u cv + f cv opr nn0ret syl jca t cv zret syl2an mpand g cv ZZ e. adantr N <_ t cv br anim2d f cv u cv nn0addge2t ancoms f cv nn0ret sylan2 g cv ZZ e. adantr f cv u cv + f cv opr g cv letrt 3expa f cv nn0ret u cv NN0 e. adantl u cv f cv nn0addclt u cv + f cv opr nn0ret syl jca g cv zret syl2an mpand t cv ZZ e. adantlr N <_ g cv br anim2d anim12d N <_ t cv br u cv <_ t cv br /\ abs F t cv ` - A opr ` < v cv / 2 opr br N <_ g cv br f cv <_ g cv br /\ abs G g cv ` - B opr ` < v cv / 2 opr br prth syl9 exp4a r19.20dva g cv t cv N <_ breq2 g cv t cv u cv + f cv opr <_ breq2 anbi12d g cv t cv G fveq2 - B opreq1d abs fveq2d < v cv / 2 opr breq1d abs F t cv ` - A opr ` < v cv / 2 opr br anbi2d imbi12d ZZ rcla4v com23 a2d g ZZ N <_ t cv br u cv + f cv opr <_ t cv br /\ N <_ g cv br u cv + f cv opr <_ g cv br /\ abs F t cv ` - A opr ` < v cv / 2 opr br abs G g cv ` - B opr ` < v cv / 2 opr br /\ -> r19.21v syl5ib com12 syl6 mpid com23 v cv RR e. adantll abs F t cv ` - A opr ` abs G t cv ` - B opr ` v cv / 2 opr v cv / 2 opr lt2addt k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d climadd2.4 F t cv ` A subclt mpan2 syl F t cv ` - A opr absclt syl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d climadd2.5 G t cv ` B subclt mpan2 syl G t cv ` - B opr absclt syl jca 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 jca syl2an 2re 2re 2pos gt0ne0i v cv 2 redivclt mp3an23 v cv / 2 opr recnt v cv / 2 opr 2timest 3syl v cv recnt 2cn 2re 2pos gt0ne0i 2 v cv divcan2t mp3an13 syl eqtr3d t cv ZZ e. N <_ t cv br /\ adantl abs F t cv ` - A opr ` + abs G t cv ` - B opr ` opr < breq2d sylibd k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp3d - A + B opr opreq1d climadd2.4 climadd2.5 pm3.2i F t cv ` G t cv ` A B sub4t mpan2 k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d sylanc eqtrd abs fveq2d F t cv ` - A opr G t cv ` - B opr abstrit k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d climadd2.4 F t cv ` A subclt mpan2 syl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d climadd2.5 G t cv ` B subclt mpan2 syl sylanc eqbrtrd v cv RR e. adantr abs H t cv ` - A + B opr opr ` abs F t cv ` - A opr ` + abs G t cv ` - B opr ` opr v cv lelttrt 3expa k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp3d F t cv ` G t cv ` axaddcl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d sylanc eqeltrd climadd2.4 climadd2.5 addcl H t cv ` A + B opr subclt mpan2 H t cv ` - A + B opr opr absclt 3syl abs F t cv ` - A opr ` abs G t cv ` - B opr ` axaddrcl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d climadd2.4 F t cv ` A subclt mpan2 syl F t cv ` - A opr absclt syl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d climadd2.5 G t cv ` B subclt mpan2 syl G t cv ` - B opr absclt syl sylanc jca sylan mpand syld an1rs exp ancoms u cv + f cv opr <_ t cv br adantrd u cv NN0 e. f cv NN0 e. /\ adantlr syldd com23 0 < v cv br adantllr r19.20dva h cv u cv + f cv opr <_ t cv breq1 N <_ t cv br anbi2d abs H t cv ` - A + B opr opr ` < v cv br imbi1d t ZZ biraldv ZZ rcla4ev u cv f cv nn0addclt u cv + f cv opr nn0zt syl sylan exp v cv RR e. 0 < v cv br /\ adantl syld u cv elnn0z f cv elnn0z anbi12i u cv ZZ e. 0 <_ u cv br f cv ZZ e. 0 <_ f cv br an4 bitr sylan2br exp32 imp4b t ZZ g N <_ t cv br u cv <_ t cv br /\ abs F t cv ` - A opr ` < v cv / 2 opr br -> N <_ g cv br f cv <_ g cv br /\ abs G g cv ` - B opr ` < v cv / 2 opr br -> raaan 0 <_ u cv br 0 <_ f cv br /\ anbi2i 0 <_ u cv br 0 <_ f cv br t ZZ N <_ t cv br u cv <_ t cv br /\ abs F t cv ` - A opr ` < v cv / 2 opr br -> A.e. g ZZ N <_ g cv br f cv <_ g cv br /\ abs G g cv ` - B opr ` < v cv / 2 opr br -> A.e. an4 bitr syl5ibr exp r19.23advv u ZZ f ZZ 0 <_ u cv br t ZZ N <_ t cv br u cv <_ t cv br /\ abs F t cv ` - A opr ` < v cv / 2 opr br -> A.e. /\ 0 <_ f cv br g ZZ N <_ g cv br f cv <_ g cv br /\ abs G g cv ` - B opr ` < v cv / 2 opr br -> A.e. /\ reeanv syl5ibr mpd exp rgen climadd2.4 climadd2.5 addcl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp3d F t cv ` G t cv ` axaddcl k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp1d k cv t cv ZZ eleq1 k cv t cv N <_ breq2 anbi12d k cv t cv F fveq2 CC eleq1d k cv t cv G fveq2 CC eleq1d k cv t cv H fveq2 k cv t cv F fveq2 k cv t cv G fveq2 + opreq12d eqeq12d 3anbi123d imbi12d climadd2.9 chvarv 3simp2d sylanc eqeltrd exp rgen climadd2.3 climadd2.8 A + B opr t v h clim3 mp2an mpbir))
+thm (climaddc ((C k) (F k) (G k) (N k)) ((climaddc2.1 (e. F (V))) (climaddc2.2 (e. G (V))) (climaddc2.3 (e. A (CC))) (climaddc2.4 (e. C (CC))) (climaddc2.5 (br F (~~>) A)) (climaddc2.6 (e. N (ZZ))) (climaddc2.7 (-> (/\ (e. (cv k) (ZZ)) (br N (<_) (cv k))) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C)))))) (br G (~~>) (opr A (+) C)) (climaddc2.1 zex C snex xpex climaddc2.2 climaddc2.3 climaddc2.4 climaddc2.5 climaddc2.4 C climconst2 ax-mp climaddc2.6 climaddc2.7 pm3.26d climaddc2.4 C CC k cv ZZ fvconst2g mpan climaddc2.4 k cv ZZ e. a1i eqeltrd N <_ k cv br adantr climaddc2.7 pm3.27d climaddc2.4 C CC k cv ZZ fvconst2g mpan F k cv ` + opreq2d N <_ k cv br adantr eqtr4d 3jca climadd))
+thm (climaddtOLD ((F x) (G x) (A x) (B x) (H x)) ((climaddtOLD.1 (e. A (V))) (climaddtOLD.2 (e. B (V))) (climaddtOLD.5 (Fn H (NN))) (climaddtOLD.6 (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr (` F (cv x)) (+) (` G (cv x))))))) (-> (/\ (br F (~~>OLD) A) (br G (~~>OLD) B)) (br H (~~>OLD) (opr A (+) B))) (A F ~~>OLD A br G ~~>OLD B br /\ A 0 if + B opreq1 H ~~>OLD breq2d B F ~~>OLD A br G ~~>OLD B br /\ B 0 if F ~~>OLD A br G ~~>OLD B br /\ A 0 if + opreq2 H ~~>OLD breq2d H F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if + F ~~>OLD A br G ~~>OLD B br /\ B 0 if opr breq1 climaddtOLD.1 0cn elisseti F ~~>OLD A br G ~~>OLD B br /\ ifex climaddtOLD.2 0cn elisseti F ~~>OLD A br G ~~>OLD B br /\ ifex F F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD A breq1 G ~~>OLD B br anbi1d A F ~~>OLD A br G ~~>OLD B br /\ A 0 if F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 G ~~>OLD B br anbi1d G F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD B breq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d B F ~~>OLD A br G ~~>OLD B br /\ B 0 if F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD 0 breq1 NN 0 {} X. ~~>OLD 0 br anbi1d 0 F ~~>OLD A br G ~~>OLD B br /\ A 0 if F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 NN 0 {} X. ~~>OLD 0 br anbi1d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD 0 breq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d 0 F ~~>OLD A br G ~~>OLD B br /\ B 0 if F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d clim0OLD clim0OLD pm3.2i elimhyp4v pm3.26i F F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD A breq1 G ~~>OLD B br anbi1d A F ~~>OLD A br G ~~>OLD B br /\ A 0 if F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 G ~~>OLD B br anbi1d G F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD B breq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d B F ~~>OLD A br G ~~>OLD B br /\ B 0 if F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD 0 breq1 NN 0 {} X. ~~>OLD 0 br anbi1d 0 F ~~>OLD A br G ~~>OLD B br /\ A 0 if F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 NN 0 {} X. ~~>OLD 0 br anbi1d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD 0 breq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d 0 F ~~>OLD A br G ~~>OLD B br /\ B 0 if F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD F ~~>OLD A br G ~~>OLD B br /\ A 0 if br anbi2d clim0OLD clim0OLD pm3.2i elimhyp4v pm3.27i H F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if NN fneq1 NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if NN fneq1 climaddtOLD.5 0cn elisseti NN fconst NN 0 {} X. NN 0 {} ffn ax-mp keephyp H F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv fveq1 F x cv ` + G x cv ` opr eqeq1d x cv NN e. imbi2d F F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if x cv fveq1 + G x cv ` opreq1d F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d G F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if x cv fveq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if x cv ` + opreq2d F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv fveq1 NN 0 {} X. x cv ` + NN 0 {} X. x cv ` opr eqeq1d x cv NN e. imbi2d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if x cv fveq1 + NN 0 {} X. x cv ` opreq1d F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d NN 0 {} X. F ~~>OLD A br G ~~>OLD B br /\ G NN 0 {} X. if x cv fveq1 F ~~>OLD A br G ~~>OLD B br /\ F NN 0 {} X. if x cv ` + opreq2d F ~~>OLD A br G ~~>OLD B br /\ H NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d climaddtOLD.6 0cn elisseti x cv NN fvconst2 0cn elisseti x cv NN fvconst2 0cn elisseti x cv NN fvconst2 + opreq12d 0cn addid1 syl6req eqtrd keephyp3v climaddOLD dedth3v))
+thm (climaddcOLD ((A k) (F k) (G k)) ((climaddcOLD.1 (e. A (CC))) (climaddcOLD.2 (e. B (V))) (climaddcOLD.3 (br F (~~>OLD) B)) (climaddcOLD.4 (Fn G (NN))) (climaddcOLD.5 (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (+) (` F (cv k))))))) (br G (~~>OLD) (opr A (+) B)) (climaddcOLD.1 elisseti climaddcOLD.2 climaddcOLD.1 climconstOLD climaddcOLD.3 climaddcOLD.4 climaddcOLD.5 climaddcOLD.1 elisseti k cv NN fvconst2 + F k cv ` opreq1d eqtr4d climaddOLD))
+thm (climaddcxOLD ((j k) (j m) (j x) (A j) (k m) (k x) (A k) (m x) (A m) (A x) (B j) (B m) (B x) (F j) (F k) (F m) (F x) (G j) (G k) (G m) (G x)) ((climaddcOLD.1 (e. A (CC))) (climaddcOLD.2 (e. B (V))) (climaddcOLD.3 (br F (~~>OLD) B)) (climaddcOLD.4 (Fn G (NN))) (climaddc.5OLD (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (+) (` F (cv k))))))) (br G (~~>OLD) (opr A (+) B)) (climaddcOLD.3 climaddcOLD.2 F x j m climcvgOLD ax-mp k cv m cv NN eleq1 k cv m cv G fveq2 k cv m cv F fveq2 A + opreq2d eqeq12d imbi12d climaddc.5OLD chvarv - A + B opr opreq1d climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp m cv ffvrni climaddcOLD.1 climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp A F m cv ` B pnpcant mp3an13 syl eqtrd abs fveq2d < x cv breq1d j cv <_ m cv br imbi2d birala j NN birex 0 < x cv br imbi2i x RR biral mpbir G NN CC k ffnfv climaddcOLD.4 climaddc.5OLD climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp k cv ffvrni climaddcOLD.1 A F k cv ` axaddcl mpan syl eqeltrd rgen mpbir2an climaddcOLD.1 climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp addcl G A + B opr x j m clim2OLD mp2an mpbir))
+thm (climmulcOLD ((j k) (j m) (j y) (j z) (A j) (k m) (k y) (k z) (A k) (m y) (m z) (A m) (y z) (A y) (A z) (B j) (B m) (B y) (B z) (F j) (F k) (F m) (F y) (G j) (G k) (G m) (G z)) ((climaddcOLD.1 (e. A (CC))) (climaddcOLD.2 (e. B (V))) (climaddcOLD.3 (br F (~~>OLD) B)) (climaddcOLD.4 (Fn G (NN))) (climmulcOLD.5 (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (x.) (` F (cv k))))))) (br G (~~>OLD) (opr A (x.) B)) (k cv j cv NN eleq1 k cv j cv G fveq2 k cv j cv F fveq2 A x. opreq2d eqeq12d imbi12d climmulcOLD.5 chvarv A 0 = adantl A 0 x. F j cv ` opreq1 climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp j cv ffvrni F j cv ` mul02t syl sylan9eq eqtrd exp G j cv fvex 0 elsnc syl6ibr r19.21aiv G NN 0 {} j ffnfv G NN CC m ffnfv climaddcOLD.4 k cv m cv NN eleq1 k cv m cv G fveq2 k cv m cv F fveq2 A x. opreq2d eqeq12d imbi12d climmulcOLD.5 chvarv climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp m cv ffvrni climaddcOLD.1 A F m cv ` axmulcl mpan syl eqeltrd rgen mpbir2an G NN CC ffn ax-mp mpbiran sylibr 0cn elisseti G NN fconst2 sylib clim0OLD syl6eqbr A 0 x. B opreq1 climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp mul02 syl6eq breqtrrd A 0 df-ne y cv z cv / abs A ` opr 0 < breq2 y cv z cv / abs A ` opr abs F m cv ` - B opr ` < breq2 j cv <_ m cv br imbi2d m NN biraldv j NN birexdv imbi12d climaddcOLD.3 climaddcOLD.2 F y j m climcvgOLD ax-mp vtoclri z cv abs A ` redivclt 3expb climaddcOLD.1 abs00 eqneqi biimpr climaddcOLD.1 abscl jctil sylan2 0 < z cv br adantlr z cv abs A ` divgt0t z cv RR e. 0 < z cv br pm3.26 A 0 =/= adantr climaddcOLD.1 abscl jctir z cv RR e. 0 < z cv br pm3.27 climaddcOLD.1 absgt0 biimp anim12i sylanc sylc climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp m cv ffvrni climaddcOLD.1 climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp A F m cv ` B subdit mp3an13 syl k cv m cv NN eleq1 k cv m cv G fveq2 k cv m cv F fveq2 A x. opreq2d eqeq12d imbi12d climmulcOLD.5 chvarv - A x. B opr opreq1d eqtr4d abs fveq2d climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp m cv ffvrni climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp F m cv ` B subclt mpan2 climaddcOLD.1 A F m cv ` - B opr absmult mpan 3syl eqtr3d < z cv breq1d z cv RR e. A 0 =/= /\ adantl abs A ` abs F m cv ` - B opr ` z cv ltmuldiv2t climaddcOLD.1 abscl z cv RR e. A 0 =/= /\ m cv NN e. /\ a1i climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp m cv ffvrni climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp F m cv ` B subclt mpan2 F m cv ` - B opr absclt 3syl z cv RR e. A 0 =/= /\ adantl z cv RR e. A 0 =/= pm3.26 m cv NN e. adantr 3jca climaddcOLD.1 absgt0 biimp z cv RR e. m cv NN e. ad2antlr sylanc bitrd j cv <_ m cv br imbi2d biraldva j NN birexdv 0 < z cv br adantlr mpbird ancoms exp32 r19.21aiv G NN CC j ffnfv climaddcOLD.4 k cv j cv NN eleq1 k cv j cv G fveq2 k cv j cv F fveq2 A x. opreq2d eqeq12d imbi12d climmulcOLD.5 chvarv climaddcOLD.3 climaddcOLD.2 F climseqOLD ax-mp j cv ffvrni climaddcOLD.1 A F j cv ` axmulcl mpan syl eqeltrd rgen mpbir2an climaddcOLD.1 climaddcOLD.3 climaddcOLD.2 F climclOLD ax-mp mulcl G A x. B opr z j m clim2OLD mp2an sylibr sylbir pm2.61i))
+thm (climmulctxOLD ((F x) (G x) (A x) (B x)) ((climmulctOLD.1 (e. A (CC))) (climmulctOLD.2 (e. B (V))) (climmulctOLD.3 (Fn G (NN))) (climmulctOLD.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x))))))) (-> (br F (~~>OLD) B) (br G (~~>OLD) (opr A (x.) B))) (G F ~~>OLD B br G NN 0 {} X. if ~~>OLD A x. B opr breq1 B F ~~>OLD B br B 0 if A x. opreq2 F ~~>OLD B br G NN 0 {} X. if ~~>OLD breq2d climmulctOLD.1 climmulctOLD.2 0cn elisseti F ~~>OLD B br ifex F F ~~>OLD B br F NN 0 {} X. if ~~>OLD B breq1 B F ~~>OLD B br B 0 if F ~~>OLD B br F NN 0 {} X. if ~~>OLD breq2 NN 0 {} X. F ~~>OLD B br F NN 0 {} X. if ~~>OLD 0 breq1 0 F ~~>OLD B br B 0 if F ~~>OLD B br F NN 0 {} X. if ~~>OLD breq2 clim0OLD elimhyp2v G F ~~>OLD B br G NN 0 {} X. if NN fneq1 NN 0 {} X. F ~~>OLD B br G NN 0 {} X. if NN fneq1 climmulctOLD.3 0cn elisseti NN fconst NN 0 {} X. NN 0 {} ffn ax-mp keephyp G F ~~>OLD B br G NN 0 {} X. if x cv fveq1 A x. F x cv ` opr eqeq1d x cv NN e. imbi2d F F ~~>OLD B br F NN 0 {} X. if x cv fveq1 A x. opreq2d F ~~>OLD B br G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d NN 0 {} X. F ~~>OLD B br G NN 0 {} X. if x cv fveq1 A x. NN 0 {} X. x cv ` opr eqeq1d x cv NN e. imbi2d NN 0 {} X. F ~~>OLD B br F NN 0 {} X. if x cv fveq1 A x. opreq2d F ~~>OLD B br G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d climmulctOLD.4 0cn elisseti x cv NN fvconst2 0cn elisseti x cv NN fvconst2 A x. opreq2d climmulctOLD.1 mul01 syl6eq eqtr4d keephyp2v climmulcOLD dedth2v))
+thm (climmulctOLD ((F x) (G x) (A x) (B x)) ((climmulct.1 (e. B (V))) (climmulct.2 (Fn G (NN))) (climmulct.3 (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x))))))) (-> (/\ (e. A (CC)) (br F (~~>OLD) B)) (br G (~~>OLD) (opr A (x.) B))) (G A CC e. F ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD A x. B opr breq1 A A CC e. F ~~>OLD B br /\ A 0 if x. B opreq1 A CC e. F ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2d B A CC e. F ~~>OLD B br /\ B 0 if A CC e. F ~~>OLD B br /\ A 0 if x. opreq2 A CC e. F ~~>OLD B br /\ G NN 0 {} X. if ~~>OLD breq2d A A CC e. F ~~>OLD B br /\ A 0 if CC eleq1 F ~~>OLD B br anbi1d F A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD B breq1 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d B A CC e. F ~~>OLD B br /\ B 0 if A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0 A CC e. F ~~>OLD B br /\ A 0 if CC eleq1 NN 0 {} X. ~~>OLD 0 br anbi1d NN 0 {} X. A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD 0 breq1 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0 A CC e. F ~~>OLD B br /\ B 0 if A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0cn clim0OLD pm3.2i elimhyp3v pm3.26i climmulct.1 0cn elisseti A CC e. F ~~>OLD B br /\ keepel A A CC e. F ~~>OLD B br /\ A 0 if CC eleq1 F ~~>OLD B br anbi1d F A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD B breq1 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d B A CC e. F ~~>OLD B br /\ B 0 if A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0 A CC e. F ~~>OLD B br /\ A 0 if CC eleq1 NN 0 {} X. ~~>OLD 0 br anbi1d NN 0 {} X. A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD 0 breq1 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0 A CC e. F ~~>OLD B br /\ B 0 if A CC e. F ~~>OLD B br /\ F NN 0 {} X. if ~~>OLD breq2 A CC e. F ~~>OLD B br /\ A 0 if CC e. anbi2d 0cn clim0OLD pm3.2i elimhyp3v pm3.27i G A CC e. F ~~>OLD B br /\ G NN 0 {} X. if NN fneq1 NN 0 {} X. A CC e. F ~~>OLD B br /\ G NN 0 {} X. if NN fneq1 climmulct.2 0cn elisseti NN fconst NN 0 {} X. NN 0 {} ffn ax-mp keephyp G A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv fveq1 A x. F x cv ` opr eqeq1d x cv NN e. imbi2d A A CC e. F ~~>OLD B br /\ A 0 if x. F x cv ` opreq1 A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d F A CC e. F ~~>OLD B br /\ F NN 0 {} X. if x cv fveq1 A CC e. F ~~>OLD B br /\ A 0 if x. opreq2d A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d NN 0 {} X. A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv fveq1 0 x. NN 0 {} X. x cv ` opr eqeq1d x cv NN e. imbi2d 0 A CC e. F ~~>OLD B br /\ A 0 if x. NN 0 {} X. x cv ` opreq1 A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d NN 0 {} X. A CC e. F ~~>OLD B br /\ F NN 0 {} X. if x cv fveq1 A CC e. F ~~>OLD B br /\ A 0 if x. opreq2d A CC e. F ~~>OLD B br /\ G NN 0 {} X. if x cv ` eqeq2d x cv NN e. imbi2d climmulct.3 0cn elisseti x cv NN fvconst2 0cn elisseti x cv NN fvconst2 0 x. opreq2d 0cn mul01 syl6eq eqtr4d keephyp3v climmulcOLD dedth3v))
+thm (clim2serz ((j k) (F j) (F k) (M j) (M k) (N k)) ((clim2srz.1 (e. F (V))) (clim2srz.2 (e. A (V))) (clim2srz.3 (e. M (ZZ))) (clim2srz.4 (e. N (ZZ))) (clim2srz.5 (br M (<_) N)) (clim2srz.6 (br (opr (<,> M (+)) (seq) F) (~~>) A)) (clim2srz.7 (:--> F (` (ZZ>) M) (CC)))) (br (opr (<,> (opr N (+) (1)) (+)) (seq) F) (~~>) (opr A (-) (` (opr (<,> M (+)) (seq) F) N))) (M + <,> seq F oprex N + 1 opr + <,> seq F oprex clim2srz.2 clim2srz.6 A V M + <,> seq F opr climcl mp2an clim2srz.4 clim2srz.5 clim2srz.3 clim2srz.7 N serzcl mp2an negcl clim2srz.6 clim2srz.4 N peano2z ax-mp clim2srz.3 zre clim2srz.4 zre clim2srz.5 M N leplus1t mp3an k cv zret clim2srz.3 zre clim2srz.4 zre ax1re readdcl M N + 1 opr k cv letrt mp3an12 syl mpani imp clim2srz.3 clim2srz.7 k cv serzcl syldan clim2srz.3 clim2srz.1 M k cv N j serzsplit mp3anl1 k cv ZZ e. N + 1 opr <_ k cv br pm3.26 clim2srz.3 clim2srz.4 clim2srz.5 M k cv - 1 opr N elfz4t mpanr1 mp3anl3 mpanl1 k cv peano2zr N + 1 opr <_ k cv br adantr k cv zret clim2srz.4 zre ax1re N 1 k cv leaddsubt mp3an12 syl biimpa sylanc jca clim2srz.3 M k cv j cv elfz1t mpan clim2srz.3 j cv eluz1 clim2srz.7 j cv ffvrni sylbir j cv <_ k cv br 3adant3 syl6bi r19.21aiv N + 1 opr <_ k cv br adantr sylanc eqcomd M + <,> seq F opr k cv ` M + <,> seq F opr N ` N + 1 opr + <,> seq F opr k cv ` subaddt clim2srz.3 zre clim2srz.4 zre clim2srz.5 M N leplus1t mp3an k cv zret clim2srz.3 zre clim2srz.4 zre ax1re readdcl M N + 1 opr k cv letrt mp3an12 syl mpani imp clim2srz.3 clim2srz.7 k cv serzcl syldan clim2srz.4 clim2srz.5 clim2srz.3 clim2srz.7 N serzcl mp2an k cv ZZ e. N + 1 opr <_ k cv br /\ a1i clim2srz.4 N peano2z ax-mp clim2srz.7 clim2srz.3 clim2srz.4 N peano2z ax-mp clim2srz.3 zre clim2srz.4 zre clim2srz.5 M N leplus1t mp3an M N + 1 opr uzssOLD mp3an F ZZ> M ` CC ZZ> N + 1 opr ` fssres mp2an k cv serzcl addex clim2srz.1 clim2srz.4 N peano2z ax-mp ZZ> N + 1 opr ` eqid seqzres k cv fveq1i CC eleq1i sylib syl3anc mpbird clim2srz.3 zre clim2srz.4 zre clim2srz.5 M N leplus1t mp3an k cv zret clim2srz.3 zre clim2srz.4 zre ax1re readdcl M N + 1 opr k cv letrt mp3an12 syl mpani imp clim2srz.3 clim2srz.7 k cv serzcl syldan clim2srz.4 clim2srz.5 clim2srz.3 clim2srz.7 N serzcl mp2an jctir M + <,> seq F opr k cv ` M + <,> seq F opr N ` subnegt syl eqtr3d jca climaddc clim2srz.2 clim2srz.6 A V M + <,> seq F opr climcl mp2an clim2srz.4 clim2srz.5 clim2srz.3 clim2srz.7 N serzcl mp2an subneg breqtrr))
+thm (climcjOLD ((j x) (k x) (n x) (A x) (j k) (j n) (A j) (k n) (A k) (A n) (F x) (F j) (F k) (F n) (G x) (G j) (G k) (G n)) ((climcj.2 (e. A (V))) (climcj.3 (br F (~~>OLD) A)) (climcj.4 (Fn G (NN))) (climcj.5 (-> (e. (cv n) (NN)) (= (` G (cv n)) (` (*) (` F (cv n))))))) (br G (~~>OLD) (` (*) A)) (climcj.3 climcj.2 F x j k climcvgOLD ax-mp n cv k cv G fveq2 n cv k cv F fveq2 * fveq2d eqeq12d climcj.5 vtoclga - * A ` opreq1d climcj.3 climcj.2 F climseqOLD ax-mp k cv ffvrni climcj.3 climcj.2 F climclOLD ax-mp F k cv ` A cjsubt mpan2 syl eqtr4d abs fveq2d climcj.3 climcj.2 F climseqOLD ax-mp k cv ffvrni climcj.3 climcj.2 F climclOLD ax-mp F k cv ` A subclt mpan2 F k cv ` - A opr abscjt 3syl eqtrd < x cv breq1d j cv <_ k cv br imbi2d birala j NN birex 0 < x cv br imbi2i x RR biral mpbir G NN CC n ffnfv climcj.4 climcj.5 climcj.3 climcj.2 F climseqOLD ax-mp n cv ffvrni F n cv ` cjclt syl eqeltrd rgen mpbir2an climcj.3 climcj.2 F climclOLD ax-mp cjcl G * A ` x j k clim2OLD mp2an mpbir))
+thm (climubiOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A y) (A z) (A w) (B y) (B z) (B w)) ((climub.1 (e. A (V))) (climub.2 (:--> F (NN) (RR))) (climub.3 (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1)))))) (climub.4 (br F (~~>OLD) A)) (climubi.5 (e. B (NN)))) (br (` F B) (<_) A) (z cv B + y cv opr y cv <_ breq2 z cv B + y cv opr F fveq2 - A opreq1d abs fveq2d < F B ` - A opr breq1d imbi12d negbid NN rcla4ev climubi.5 B y cv nnaddclt mpan y cv nnret climubi.5 nnnn0 y cv B nn0addge2t mpan2 syl climub.2 climubi.5 F NN RR B ffvrn mp2an climub.1 climub.2 climub.4 climreclOLD resubcl F B ` - A opr F B + y cv opr ` - A opr abs F B + y cv opr ` - A opr ` letrt mp3an1 climubi.5 B y cv nnaddclt mpan climub.2 B + y cv opr ffvrni syl climub.1 climub.2 climub.4 climreclOLD F B + y cv opr ` A resubclt mpan2 syl climubi.5 B y cv nnaddclt mpan climub.2 B + y cv opr ffvrni syl climub.1 climub.2 climub.4 climreclOLD F B + y cv opr ` A resubclt mpan2 syl F B + y cv opr ` - A opr recnt syl F B + y cv opr ` - A opr absclt syl jca climubi.5 climub.2 climub.3 B B + y cv opr monoord mp3an1 climubi.5 B y cv nnaddclt mpan y cv nnnn0t climubi.5 nnre B y cv nn0addge1t mpan syl sylanc climubi.5 B y cv nnaddclt mpan climub.2 B + y cv opr ffvrni syl climub.2 climubi.5 F NN RR B ffvrn mp2an climub.1 climub.2 climub.4 climreclOLD F B ` F B + y cv opr ` A lesub1t mp3an13 syl mpbid climubi.5 B y cv nnaddclt mpan climub.2 B + y cv opr ffvrni syl climub.1 climub.2 climub.4 climreclOLD F B + y cv opr ` A resubclt mpan2 syl F B + y cv opr ` - A opr leabst syl jca sylc climubi.5 B y cv nnaddclt mpan climub.2 B + y cv opr ffvrni syl climub.1 climub.2 climub.4 climreclOLD F B + y cv opr ` A resubclt mpan2 syl F B + y cv opr ` - A opr recnt syl F B + y cv opr ` - A opr absclt syl climub.2 climubi.5 F NN RR B ffvrn mp2an climub.1 climub.2 climub.4 climreclOLD resubcl F B ` - A opr abs F B + y cv opr ` - A opr ` lenltt mpan syl mpbid jca y cv <_ B + y cv opr br abs F B + y cv opr ` - A opr ` < F B ` - A opr br annim sylib sylanc z NN y cv <_ z cv br abs F z cv ` - A opr ` < F B ` - A opr br -> rexnal sylib nrex climub.2 climubi.5 F NN RR B ffvrn mp2an climub.1 climub.2 climub.4 climreclOLD resubcl climub.4 climub.1 F w y z climcvgOLD ax-mp w cv F B ` - A opr 0 < breq2 w cv F B ` - A opr abs F z cv ` - A opr ` < breq2 y cv <_ z cv br imbi2d z NN biraldv y NN birexdv imbi12d RR rcla4v mp2 mto climub.1 climub.2 climub.4 climreclOLD climub.2 climubi.5 F NN RR B ffvrn mp2an posdif mtbir climub.2 climubi.5 F NN RR B ffvrn mp2an climub.1 climub.2 climub.4 climreclOLD lenlt mpbir))
+thm (climubOLD ((F x)) ((climub.1 (e. A (V))) (climub.2 (:--> F (NN) (RR))) (climub.3 (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1)))))) (climub.4 (br F (~~>OLD) A))) (-> (e. B (NN)) (br (` F B) (<_) A)) (B B NN e. B 1 if F fveq2 <_ A breq1d climub.1 climub.2 climub.3 climub.4 1nn B elimel climubiOLD dedth))
+thm (climsupOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) ((climsup.1 (:--> F (NN) (RR))) (climsup.2 (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1)))))) (climsup.3 (E.e. y (RR) (A.e. x (NN) (br (` F (cv x)) (<_) (cv y)))))) (br F (~~>OLD) (sup (ran F) (RR) (<))) (climsup.1 axresscn F NN RR CC fss mp2an climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli recn F F ran RR < sup z y x clim2OLD mp2an climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli z cv F ran RR < sup ltsubpost mpan2 climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i F ran RR < sup - z cv opr w suprlubi climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli F ran RR < sup z cv resubclt mpan sylan exp sylbid F ran RR < sup - z cv opr F y cv ` F x cv ` ltletrt climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli F ran RR < sup z cv resubclt mpan F ran RR < sup - z cv opr < w cv br adantr y cv NN e. F y cv ` w cv = /\ x cv NN e. y cv <_ x cv br /\ ad2antll climsup.1 y cv ffvrni F y cv ` w cv = adantr z cv RR e. F ran RR < sup - z cv opr < w cv br /\ x cv NN e. y cv <_ x cv br /\ ad2antlr climsup.1 x cv ffvrni z cv RR e. F ran RR < sup - z cv opr < w cv br /\ y cv NN e. F y cv ` w cv = /\ /\ y cv <_ x cv br ad2antrl 3jca F y cv ` w cv F ran RR < sup - z cv opr < breq2 biimparc y cv NN e. adantrl z cv RR e. adantll x cv NN e. y cv <_ x cv br /\ adantr climsup.1 climsup.2 y cv x cv monoord 3expb F y cv ` w cv = adantlr z cv RR e. F ran RR < sup - z cv opr < w cv br /\ adantll jca sylc climsup.1 x cv ffvrni F x cv ` recnt climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli recn F x cv ` F ran RR < sup abssubt mpan2 3syl F ran RR < sup - F x cv ` opr absidt climsup.1 x cv ffvrni climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli jctil F ran RR < sup F x cv ` resubclt syl climsup.1 F NN RR ffn ax-mp F NN x cv fnfvrn mpan climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i F x cv ` suprubi syl climsup.1 x cv ffvrni climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli F ran RR < sup F x cv ` subge0t mpan syl mpbird sylanc eqtrd < z cv breq1d z cv RR e. adantl climsup.1 F NN RR frn ax-mp 1nn 1 NN n0i ax-mp climsup.1 F NN RR fdm ax-mp {/} eqeq1i mtbir F dm0rn0 mtbi climsup.3 x NN F x cv ` <_ y cv br hbra1 z cv <_ y cv br x ax-17 x NN F x cv ` <_ y cv br ra4 F x cv ` z cv <_ y cv breq1 biimpcd syl6 r19.23ad climsup.1 F NN RR ffn ax-mp F NN z cv x fvelrn ax-mp syl5ib r19.21aiv y RR r19.22si ax-mp 3pm3.2i suprcli F ran RR < sup z cv F x cv ` ltsub23t mp3an1 climsup.1 x cv ffvrni sylan2 bitr4d y cv <_ x cv br adantrr F ran RR < sup - z cv opr < w cv br adantlr y cv NN e. F y cv ` w cv = /\ adantlr mpbird exp32 r19.21aiv exp32 r19.22dv climsup.1 F NN RR ffn ax-mp F NN w cv y fvelrn ax-mp syl5ib exp com23 r19.23adv syld mprgbir))
+thm (climcau ((x y) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (F z) (v w) (F w) (F v) (A y) (A z) (A w) (A v)) ((climcau.1 (e. A (V))) (climcau.2 (br F (~~>OLD) A))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) (<) (cv x))))))) (x cv halfpos2t 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 w cv x cv / 2 opr 0 < breq2 w cv x cv / 2 opr abs F v cv ` - A opr ` < breq2 y cv <_ v cv br imbi2d v NN biraldv y NN birexdv imbi12d climcau.2 climcau.1 F w y v climcvgOLD ax-mp vtoclri syl sylbid y cv z cv ltlet y cv nnret z cv nnret syl2an v NN y cv <_ v cv br abs F v cv ` - A opr ` < x cv / 2 opr br -> A.e. adantrl v cv z cv y cv <_ breq2 v cv z cv F fveq2 - A opreq1d abs fveq2d < x cv / 2 opr breq1d imbi12d NN rcla4v impcom y cv NN e. adantl syld imp climcau.2 climcau.1 F y cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp A F y cv ` abssubt mpan syl v NN y cv <_ v cv br abs F v cv ` - A opr ` < x cv / 2 opr br -> A.e. adantr y cv nnret y cv leidt syl v cv y cv y cv <_ breq2 v cv y cv F fveq2 - A opreq1d abs fveq2d < x cv / 2 opr breq1d imbi12d NN rcla4v mpid imp eqbrtrd z cv NN e. adantrr y cv < z cv br adantr jca x cv RR e. adantlll abs F z cv ` - A opr ` abs A - F y cv ` opr ` x cv / 2 opr x cv / 2 opr lt2addt climcau.2 climcau.1 F z cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp F z cv ` A subclt mpan2 syl F z cv ` - A opr absclt syl x cv RR e. y cv NN e. /\ adantl climcau.2 climcau.1 F y cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp A F y cv ` subclt mpan syl A - F y cv ` opr absclt syl x cv RR e. z cv NN e. ad2antlr jca 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 y cv NN e. z cv NN e. ad2antll 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 y cv NN e. z cv NN e. ad2antll jca sylanc v NN y cv <_ v cv br abs F v cv ` - A opr ` < x cv / 2 opr br -> A.e. adantrl y cv < z cv br adantr mpd x cv recnt x cv 2halvest syl y cv NN e. adantr v NN y cv <_ v cv br abs F v cv ` - A opr ` < x cv / 2 opr br -> A.e. z cv NN e. /\ y cv < z cv br ad2antll breqtrd exp climcau.2 climcau.1 F climclOLD ax-mp F z cv ` A F y cv ` npncant mp3an2 climcau.2 climcau.1 F z cv climfvclOLD mpan climcau.2 climcau.1 F y cv climfvclOLD mpan syl2an abs fveq2d F z cv ` - A opr A - F y cv ` opr abstrit climcau.2 climcau.1 F z cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp F z cv ` A subclt mpan2 syl climcau.2 climcau.1 F y cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp A F y cv ` subclt mpan syl syl2an eqbrtrrd x cv RR e. adantll abs F z cv ` - F y cv ` opr ` abs F z cv ` - A opr ` + abs A - F y cv ` opr ` opr x cv lelttrt F z cv ` F y cv ` subclt climcau.2 climcau.1 F z cv climfvclOLD mpan climcau.2 climcau.1 F y cv climfvclOLD mpan syl2an F z cv ` - F y cv ` opr absclt syl x cv RR e. adantll abs F z cv ` - A opr ` abs A - F y cv ` opr ` axaddrcl climcau.2 climcau.1 F z cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp F z cv ` A subclt mpan2 syl F z cv ` - A opr absclt syl climcau.2 climcau.1 F y cv climfvclOLD mpan climcau.2 climcau.1 F climclOLD ax-mp A F y cv ` subclt mpan syl A - F y cv ` opr absclt syl syl2an x cv RR e. adantll x cv RR e. z cv NN e. pm3.26 y cv NN e. adantr syl3anc mpand an1rs v NN y cv <_ v cv br abs F v cv ` - A opr ` < x cv / 2 opr br -> A.e. adantrl syld exp32 r19.21adv r19.22dva syld rgen))
+thm (caucvglem1 ((v y) (u y) (A y) (u v) (A v) (A u) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S z) (S w)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (<-> (e. A S) (/\ (e. A (RR)) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br A (<) (` F (cv y)))))))) (caucvg.3 A eleq2i u cv A < F y cv ` breq1 v cv <_ y cv br imbi2d y NN biraldv v NN birexdv RR elrab bitr))
+thm (caucvglem2 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. x (RR) (A.e. f S (br (cv f) (<_) (cv x))))) (caucvg.3 u RR v NN y NN v cv <_ y cv br u cv < F y cv ` br -> A.e. E.e. ssrab eqsstr caucvg.1 axresscn F NN RR CC fss mp2an caucvg.2 x caubnd F y cv ` x cv absltt x cv -u < F y cv ` br F y cv ` < x cv br pm3.26 syl6bi caucvg.1 y cv ffvrni sylan ancoms 1 <_ y cv br a1dd r19.20dva 1nn x cv RR e. a1i jctild v cv 1 <_ y cv breq1 x cv -u < F y cv ` br imbi1d y NN biraldv NN rcla4ev syl6 x cv renegclt jctild caucvg.1 caucvg.2 caucvg.3 x cv -u caucvglem1 syl6ibr x cv -u S n0i syl6 r19.23aiv ax-mp caucvg.1 axresscn F NN RR CC fss mp2an caucvg.2 x caubnd caucvg.1 caucvg.2 caucvg.3 f cv caucvglem1 v cv nnret v cv leidt syl y cv v cv v cv <_ breq2 y cv v cv F fveq2 f cv < breq2d imbi12d NN rcla4v mpid x cv RR e. adantr y cv v cv F fveq2 abs fveq2d < x cv breq1d NN rcla4v x cv RR e. adantr F v cv ` x cv absltt caucvg.1 v cv ffvrni sylan x cv -u < F v cv ` br F v cv ` < x cv br pm3.27 syl6bi syld anim12d f cv RR e. 3adant1 f cv F v cv ` x cv axlttrn caucvg.1 v cv ffvrni syl3an2 f cv x cv ltlet v cv NN e. 3adant2 syld syld exp3a 3exp com34 r19.23adv imp sylbi com3l r19.21adv r19.22i ax-mp 3pm3.2i))
+thm (caucvglem3 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (e. (sup S (RR) (<)) (RR)) (caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 suprcli))
+thm (caucvglem4 ((v y) (u y) (t y) (A y) (u v) (t v) (A v) (t u) (A u) (A t) (x y) (x z) (w x) (u x) (v x) (t x) (f x) (F x) (y z) (w y) (u y) (v y) (t y) (f y) (F y) (w z) (u z) (v z) (t z) (f z) (F z) (u w) (v w) (t w) (f w) (F w) (u v) (t u) (f u) (F u) (t v) (f v) (F v) (f t) (F t) (F f) (S x) (S z) (S w) (S t) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (e. A (RR)) (-> (br A (<) (sup S (RR) (<))) (e. A S))) (caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 A t suprlubi exp A t cv F y cv ` axlttrn caucvg.1 y cv ffvrni syl3an3 exp3a 3exp com3l com4t imp41 v cv <_ y cv br imim2d r19.20dva v NN r19.22sdv exp imp3a A RR e. A < t cv br pm3.26 jctild caucvg.1 caucvg.2 caucvg.3 t cv caucvglem1 caucvg.1 caucvg.2 caucvg.3 A caucvglem1 3imtr4g exp com23 r19.23adv syld))
+thm (caucvglem5 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (-u (opr (cv x) (/) (2))) (<_) (opr (sup S (RR) (<)) (-) (` F (cv w)))))) (w cv y cv nnltp1let biimprd x cv / 2 opr RR e. adantll F y cv ` - F w cv ` opr x cv / 2 opr absltt x cv / 2 opr -u < F y cv ` - F w cv ` opr br F y cv ` - F w cv ` opr < x cv / 2 opr br pm3.26 syl6bi F y cv ` F w cv ` resubclt sylan 3impa 3com13 x cv / 2 opr -u F w cv ` F y cv ` ltaddsubt x cv / 2 opr renegclt syl3an1 sylibrd caucvg.1 w cv ffvrni syl3an2 caucvg.1 y cv ffvrni syl3an3 3expa imim12d r19.20dva w cv peano2nn x cv / 2 opr RR e. adantl jctild v cv w cv + 1 opr <_ y cv breq1 x cv / 2 opr -u + F w cv ` opr < F y cv ` br imbi1d y NN biraldv NN rcla4ev syl6 x cv / 2 opr -u F w cv ` axaddrcl x cv / 2 opr renegclt caucvg.1 w cv ffvrni syl2an jctild caucvg.1 caucvg.2 caucvg.3 x cv / 2 opr -u + F w cv ` opr caucvglem1 caucvg.1 caucvg.2 caucvg.3 x f caucvglem2 x cv / 2 opr -u + F w cv ` opr suprubi sylbir syl6 caucvg.1 caucvg.2 caucvg.3 caucvglem3 x cv / 2 opr -u F w cv ` S RR < sup leaddsubt mp3an3 x cv / 2 opr renegclt caucvg.1 w cv ffvrni syl2an sylibd 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 sylan))
+thm (caucvglem6 ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S x) (S z) (S w)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (opr (sup S (RR) (<)) (-) (` F (cv w))) (<_) (opr (cv x) (/) (2))))) (y cv w cv + v cv opr w cv + v cv opr <_ breq2 NN rcla4ev w cv v cv nnaddclt w cv v cv nnaddclt w cv + v cv opr nnret syl w cv + v cv opr leidt syl sylanc x cv / 2 opr RR e. adantll x cv / 2 opr RR e. w cv NN e. v cv NN e. /\ /\ y ax-17 y NN w cv < y cv br abs F y cv ` - F w cv ` opr ` < x cv / 2 opr br -> hbra1 y NN v cv <_ y cv br x cv / 2 opr + F w cv ` opr < F y cv ` br -> hbra1 hbne hbim v cv nngt0t w cv NN e. adantl v cv w cv ltaddpost ancoms w cv nnret v cv nnret syl2an mpbid y cv RR e. adantl w cv w cv + v cv opr y cv ltletrt w cv nnret y cv RR e. v cv NN e. ad2antrl w cv v cv nnaddclt w cv + v cv opr nnret syl y cv RR e. adantl y cv RR e. w cv NN e. v cv NN e. /\ pm3.26 syl3anc mpand y cv nnret sylan x cv / 2 opr RR e. adantrl F y cv ` - F w cv ` opr x cv / 2 opr absltt x cv / 2 opr -u < F y cv ` - F w cv ` opr br F y cv ` - F w cv ` opr < x cv / 2 opr br pm3.27 syl6bi ancoms F y cv ` F w cv ` resubclt sylan2 an1s F y cv ` F w cv ` x cv / 2 opr ltsubaddt 3com23 3expb sylibd F y cv ` x cv / 2 opr + F w cv ` opr ltnsymt x cv / 2 opr F w cv ` axaddrcl sylan2 syld caucvg.1 y cv ffvrni sylan caucvg.1 w cv ffvrni sylanr2 v cv NN e. adantrrr imim12d exp com23 a2i imp4d com12 y NN w cv < y cv br abs F y cv ` - F w cv ` opr ` < x cv / 2 opr br -> ra4 syl5 w cv nngt0t v cv NN e. adantr w cv v cv ltaddpos2t w cv nnret v cv nnret syl2an mpbid y cv RR e. adantr v cv w cv + v cv opr y cv ltletrt v cv nnret w cv NN e. y cv RR e. ad2antlr w cv v cv nnaddclt w cv + v cv opr nnret syl y cv RR e. adantr w cv NN e. v cv NN e. /\ y cv RR e. pm3.27 syl3anc mpand v cv y cv ltlet v cv nnret sylan w cv NN e. adantll syld y cv nnret sylan2 ancoms x cv / 2 opr + F w cv ` opr < F y cv ` br imim1d exp com23 a2i imp4d com12 y NN v cv <_ y cv br x cv / 2 opr + F w cv ` opr < F y cv ` br -> ra4 syl5 x cv / 2 opr RR e. adantrll nsyld exp32 com12 r19.23ad anassrs mpd imp an1rs nrexdv exp x cv / 2 opr F w cv ` axaddrcl caucvg.1 w cv ffvrni sylan2 caucvg.1 caucvg.2 caucvg.3 x cv / 2 opr + F w cv ` opr caucvglem4 caucvg.1 caucvg.2 caucvg.3 x cv / 2 opr + F w cv ` opr caucvglem1 pm3.27bd syl6 con3d syl syld x cv / 2 opr F w cv ` axaddrcl caucvg.1 w cv ffvrni sylan2 caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup x cv / 2 opr + F w cv ` opr lenltt mpan syl sylibrd caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup F w cv ` x cv / 2 opr lesubaddt mp3an1 caucvg.1 w cv ffvrni sylan ancoms sylibrd 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 sylan))
+thm (caucvgOLD ((x y) (x z) (w x) (u x) (v x) (t x) (F x) (y z) (w y) (u y) (v y) (t y) (F y) (w z) (u z) (v z) (t z) (F z) (u w) (v w) (t w) (F w) (u v) (t u) (F u) (t v) (F v) (F t) (S x) (S z) (S w) (S t)) ((caucvg.1 (:--> F (NN) (RR))) (caucvg.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg.3 (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y)))))))))) (br F (~~>OLD) (sup S (RR) (<))) (caucvg.2 z cv x cv / 2 opr 0 < breq2 z cv x cv / 2 opr abs F y cv ` - F w cv ` opr ` < breq2 w cv < y cv br imbi2d y NN biraldv w NN birexdv imbi12d RR rcla4v com12 x cv / 2 opr RR e. 0 < x cv / 2 opr br /\ w NN y NN w cv < y cv br abs F y cv ` - F w cv ` opr ` < x cv / 2 opr br -> A.e. E.e. biimt x cv / 2 opr RR e. 0 < x cv / 2 opr br w NN y NN w cv < y cv br abs F y cv ` - F w cv ` opr ` < x cv / 2 opr br -> A.e. E.e. impexp syl6bb 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 0 < x cv br adantr x cv halfpos2t biimpa sylanc caucvg.1 caucvg.2 caucvg.3 x caucvglem5 caucvg.1 caucvg.2 caucvg.3 x caucvglem6 jcad S RR < sup - F w cv ` opr x cv / 2 opr abslet caucvg.1 w cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup F w cv ` resubclt mpan syl x cv RR e. adantl 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 w cv NN e. adantr sylanc sylibrd t cv NN e. adantr abs S RR < sup - F w cv ` opr ` x cv / 2 opr abs F t cv ` - F w cv ` opr ` leadd1t caucvg.1 w cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup F w cv ` resubclt mpan syl recnd S RR < sup - F w cv ` opr absclt syl x cv RR e. t cv NN e. ad2antlr 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 w cv NN e. t cv NN e. ad2antll F t cv ` F w cv ` resubclt caucvg.1 t cv ffvrni caucvg.1 w cv ffvrni syl2an ancoms recnd F t cv ` - F w cv ` opr absclt syl x cv RR e. adantll syl3anc sylibd caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn S RR < sup F w cv ` F t cv ` npncant mp3an1 caucvg.1 w cv ffvrni recnd caucvg.1 t cv ffvrni recnd syl2an abs fveq2d S RR < sup - F w cv ` opr F w cv ` - F t cv ` opr abstrit caucvg.1 w cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup F w cv ` resubclt mpan syl recnd t cv NN e. adantr F w cv ` F t cv ` resubclt caucvg.1 w cv ffvrni caucvg.1 t cv ffvrni syl2an recnd sylanc eqbrtrrd caucvg.1 t cv ffvrni recnd caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn S RR < sup F t cv ` abssubt mpan syl w cv NN e. adantl F w cv ` F t cv ` abssubt caucvg.1 w cv ffvrni recnd caucvg.1 t cv ffvrni recnd syl2an abs S RR < sup - F w cv ` opr ` + opreq2d 3brtr3d x cv RR e. adantll abs F t cv ` - S RR < sup opr ` abs S RR < sup - F w cv ` opr ` + abs F t cv ` - F w cv ` opr ` opr x cv / 2 opr + abs F t cv ` - F w cv ` opr ` opr letrt caucvg.1 t cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 F t cv ` S RR < sup resubclt mpan2 syl recnd F t cv ` - S RR < sup opr absclt syl x cv RR e. w cv NN e. /\ adantl abs S RR < sup - F w cv ` opr ` abs F t cv ` - F w cv ` opr ` axaddrcl caucvg.1 w cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 S RR < sup F w cv ` resubclt mpan syl recnd S RR < sup - F w cv ` opr absclt syl t cv NN e. adantr F t cv ` F w cv ` resubclt caucvg.1 t cv ffvrni caucvg.1 w cv ffvrni syl2an ancoms recnd F t cv ` - F w cv ` opr absclt syl sylanc x cv RR e. adantll x cv / 2 opr abs F t cv ` - F w cv ` opr ` axaddrcl 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 F t cv ` F w cv ` resubclt caucvg.1 t cv ffvrni caucvg.1 w cv ffvrni syl2an ancoms recnd F t cv ` - F w cv ` opr absclt syl syl2an anassrs syl3anc mpand syld w cv < t cv br adantrl y cv t cv w cv < breq2 y cv t cv F fveq2 - F w cv ` opreq1d abs fveq2d < x cv / 2 opr breq1d imbi12d NN rcla4v com3r imp x cv RR e. w cv NN e. /\ adantl abs F t cv ` - F w cv ` opr ` x cv / 2 opr x cv / 2 opr ltadd2t F t cv ` F w cv ` resubclt caucvg.1 t cv ffvrni caucvg.1 w cv ffvrni syl2an ancoms recnd F t cv ` - F w cv ` opr absclt syl x cv RR e. adantll 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 w cv NN e. t cv NN e. ad2antll 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 w cv NN e. t cv NN e. ad2antll syl3anc w cv < t cv br adantrl x cv recnt x cv 2halvest syl w cv NN e. w cv < t cv br t cv NN e. /\ ad2antll x cv / 2 opr + abs F t cv ` - F w cv ` opr ` opr < breq2d bitrd sylibd jcad abs F t cv ` - S RR < sup opr ` x cv / 2 opr + abs F t cv ` - F w cv ` opr ` opr x cv lelttrt caucvg.1 t cv ffvrni caucvg.1 caucvg.2 caucvg.3 caucvglem3 F t cv ` S RR < sup resubclt mpan2 syl recnd F t cv ` - S RR < sup opr absclt syl x cv RR e. w cv NN e. /\ adantl x cv / 2 opr abs F t cv ` - F w cv ` opr ` axaddrcl 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 F t cv ` F w cv ` resubclt caucvg.1 t cv ffvrni caucvg.1 w cv ffvrni syl2an ancoms recnd F t cv ` - F w cv ` opr absclt syl syl2an anassrs x cv RR e. w cv NN e. pm3.26 t cv NN e. adantr syl3anc w cv < t cv br adantrl syld exp32 com24 r19.21adv r19.22dva 0 < x cv br adantr sylbird com12 exp3a syl r19.21aiv ax-mp x RR 0 < x cv br w t abs F t cv ` - S RR < sup opr ` < x cv br cvg1 mpbi caucvg.1 axresscn F NN RR CC fss mp2an caucvg.1 caucvg.2 caucvg.3 caucvglem3 recn F S RR < sup x w t clim2OLD mp2an mpbir))
+thm (caucvg2OLD ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v)) ((caucvg2.1 (:--> F (NN) (RR))) (caucvg2.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))))) (E.e. x (RR) (br F (~~>OLD) (cv x))) (caucvg2.1 caucvg2.2 u RR v NN y NN v cv <_ y cv br u cv < F y cv ` br -> A.e. E.e. {e.|} eqid caucvglem3 caucvg2.1 caucvg2.2 u RR v NN y NN v cv <_ y cv br u cv < F y cv ` br -> A.e. E.e. {e.|} eqid caucvgOLD x cv u RR v NN y NN v cv <_ y cv br u cv < F y cv ` br -> A.e. E.e. {e.|} RR < sup F ~~>OLD breq2 RR rcla4ev mp2an))
+thm (caucvg3lemOLD ((u x) (v x) (F x) (u v) (F u) (F v) (x y) (x z) (w x) (G x) (y z) (w y) (u y) (v y) (G y) (w z) (u z) (v z) (G z) (u w) (v w) (G w) (G u) (G v) (t x) (H x) (t y) (H y) (t z) (H z) (t w) (H w) (H t) (R x) (t u) (R u) (t v) (R v) (R t)) ((caucvg3lem.1 (:--> F (NN) (CC))) (caucvg3lem.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg3lem.3 (Fn G (NN))) (caucvg3lem.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x)))))) (caucvg3lem.5 (Fn H (NN))) (caucvg3lem.6 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x)))))) (caucvg3lem.7 (Fn R (NN))) (caucvg3lem.8 (-> (e. (cv x) (NN)) (= (` R (cv x)) (opr (i) (x.) (` H (cv x))))))) (E.e. x (CC) (br F (~~>OLD) (cv x))) (G NN RR x ffnfv caucvg3lem.3 caucvg3lem.4 caucvg3lem.1 x cv ffvrni F x cv ` reclt syl eqeltrd rgen mpbir2an caucvg3lem.1 caucvg3lem.2 caucvg3lem.3 caucvg3lem.4 caure v caucvg2OLD H NN RR x ffnfv caucvg3lem.5 caucvg3lem.6 caucvg3lem.1 x cv ffvrni F x cv ` imclt syl eqeltrd rgen mpbir2an caucvg3lem.1 caucvg3lem.2 caucvg3lem.5 caucvg3lem.6 cauim t caucvg2OLD axicn t visset caucvg3lem.7 caucvg3lem.8 climmulctxOLD i x. t cv oprex R climclOLD u cv i x. t cv opr R ~~>OLD breq2 CC rcla4ev mpancom syl t cv RR e. a1i r19.23aiv ax-mp v visset u visset caucvg3lem.1 F NN CC ffn ax-mp caucvg3lem.1 x cv ffvrni F x cv ` replimt syl caucvg3lem.4 caucvg3lem.8 caucvg3lem.6 caucvg3lem.1 x cv ffvrni F x cv ` imclt syl eqeltrd recnd axicn i H x cv ` axmulcom mpan syl caucvg3lem.6 x. i opreq1d 3eqtrd + opreq12d eqtr4d climaddtOLD v cv + u cv oprex F climclOLD x cv v cv + u cv opr F ~~>OLD breq2 CC rcla4ev mpancom syl exp u cv CC e. a1d r19.23adv v cv RR e. a1i r19.23aiv mp2))
+thm (caucvg3OLD ((x y) (x z) (w x) (u x) (v x) (t x) (h x) (F x) (y z) (w y) (u y) (v y) (t y) (h y) (F y) (w z) (u z) (v z) (t z) (h z) (F z) (u w) (v w) (t w) (h w) (F w) (u v) (t u) (h u) (F u) (t v) (h v) (F v) (h t) (F t) (F h)) ((caucvg3.1 (:--> F (NN) (CC))) (caucvg3.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))))) (E.e. x (CC) (br F (~~>OLD) (cv x))) (caucvg3.1 caucvg3.2 Re F v cv ` fvex v u v cv NN e. u cv Re F v cv ` ` = /\ {<,>|} eqid fnopab2 v cv x cv F fveq2 Re fveq2d v u v cv NN e. u cv Re F v cv ` ` = /\ {<,>|} eqid Re F x cv ` fvex fvopab4 Im F v cv ` fvex v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} eqid fnopab2 v cv x cv F fveq2 Im fveq2d v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} eqid Im F x cv ` fvex fvopab4 i x. v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} t cv ` oprex t h t cv NN e. h cv i x. v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} t cv ` opr = /\ {<,>|} eqid fnopab2 t cv x cv v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} fveq2 i x. opreq2d t h t cv NN e. h cv i x. v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} t cv ` opr = /\ {<,>|} eqid i x. v u v cv NN e. u cv Im F v cv ` ` = /\ {<,>|} x cv ` oprex fvopab4 caucvg3lemOLD))
+thm (caucvg3aOLD ((v x) (u x) (F x) (u v) (F v) (F u) (x y) (x z) (w x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u) (D x) (D u) (D v) (R x) (R z) (R w) (S z) (S w)) ((caucvg3a.1 (:--> F (NN) (CC))) (caucvg3a.2 (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (caucvg3a.3 (Fn G (NN))) (caucvg3a.4 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x)))))) (caucvg3a.4a (= R ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` G (cv y))))))))) (caucvg3a.5 (Fn H (NN))) (caucvg3a.6 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x)))))) (caucvg3a.6a (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` H (cv y))))))))) (caucvg3a.7 (Fn D (NN))) (caucvg3a.8 (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` H (cv x))))))) (br F (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))) (ltso R supex i x. S RR < sup oprex G NN RR x ffnfv caucvg3a.3 caucvg3a.1 x cv ffvrni F x cv ` reclt syl caucvg3a.4 RR eleq1d mpbird rgen mpbir2an caucvg3a.1 caucvg3a.2 caucvg3a.3 caucvg3a.4 caure caucvg3a.4a caucvgOLD axicn ltso S supex H NN RR x ffnfv caucvg3a.5 caucvg3a.1 x cv ffvrni F x cv ` imclt syl caucvg3a.6 RR eleq1d mpbird rgen mpbir2an caucvg3a.1 caucvg3a.2 caucvg3a.5 caucvg3a.6 cauim caucvg3a.6a caucvgOLD caucvg3a.7 caucvg3a.8 climmulcOLD caucvg3a.1 F NN CC ffn ax-mp caucvg3a.1 x cv ffvrni F x cv ` replimt syl caucvg3a.4 caucvg3a.8 H NN RR x ffnfv caucvg3a.5 caucvg3a.1 x cv ffvrni F x cv ` imclt syl caucvg3a.6 RR eleq1d mpbird rgen mpbir2an axresscn H NN RR CC fss mp2an x cv ffvrni axicn i H x cv ` axmulcom mpan syl caucvg3a.6 x. i opreq1d eqtrd eqtrd + opreq12d eqcomd eqtrd climaddOLD))
+thm (ser1f0 ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A k) (A m) (A n)) ((ser1f0.1 (:--> F (NN) (CC))) (ser1f0.2 (e. A (CC))) (ser1f0.3 (br (opr (+) (seq1) F) (~~>) A))) (br F (~~>) (0)) (0cn k cv zret ax1re 2re 1lt2 ltlei ax1re 2re 1 2 k cv letrt mp3an12 mpani syl k cv elnnz1 ser1f0.1 k cv ffvrni sylbir exp syld rgen nnex ser1f0.1 NN V F CC fex mp2 2z 0 k x j clim3 mp2an ser1f0.2 ser1f0.3 A CC + seq1 F opr x cv / 2 opr m n climcvg1 mpanl12 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 0 < x cv br adantr x cv halfpos2t biimpa sylanc j cv m cv + 1 opr <_ k cv breq1 2 <_ k cv br anbi2d abs F k cv ` - 0 opr ` < x cv br imbi1d k ZZ biraldv ZZ rcla4ev m cv peano2z x cv RR e. n ZZ m cv <_ n cv br abs + seq1 F opr n cv ` - A opr ` < x cv / 2 opr br -> A.e. ad2antrl m cv k cv plus1let 3expa exp m cv zret k cv zret syl2an ancoms n ZZ m cv <_ n cv br abs + seq1 F opr n cv ` - A opr ` < x cv / 2 opr br -> A.e. adantrr n cv k cv m cv <_ breq2 n cv k cv + seq1 F opr fveq2 - A opreq1d abs fveq2d < x cv / 2 opr breq1d imbi12d ZZ rcla4va m cv ZZ e. adantrl syld ax1re m cv 1 k cv leaddsubt mp3an2 m cv zret k cv zret syl2an ancoms n ZZ m cv <_ n cv br abs + seq1 F opr n cv ` - A opr ` < x cv / 2 opr br -> A.e. adantrr k cv peano2zr n cv k cv - 1 opr m cv <_ breq2 n cv k cv - 1 opr + seq1 F opr fveq2 - A opreq1d abs fveq2d < x cv / 2 opr breq1d imbi12d ZZ rcla4v syl imp m cv ZZ e. adantrl sylbid jcad 2 <_ k cv br adantlr x cv RR e. adantrl abs + seq1 F opr k cv ` - A opr ` abs + seq1 F opr k cv - 1 opr ` - A opr ` x cv / 2 opr x cv / 2 opr lt2addt ser1f0.1 k cv ser1cl ser1f0.2 + seq1 F opr k cv ` A subclt mpan2 syl + seq1 F opr k cv ` - A opr absclt syl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.2 + seq1 F opr k cv - 1 opr ` A subclt mpan2 syl + seq1 F opr k cv - 1 opr ` - A opr absclt syl jca 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 2re 2re 2pos gt0ne0i x cv 2 redivclt mp3an23 jca syl2an 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.2 + seq1 F opr k cv - 1 opr ` A abssubt mpan2 syl abs + seq1 F opr k cv ` - A opr ` + opreq2d x cv recnt x cv 2halvest syl < breqan12d sylibd + seq1 F opr k cv ` - A opr A - + seq1 F opr k cv - 1 opr ` opr abstrit ser1f0.1 k cv ser1cl ser1f0.2 + seq1 F opr k cv ` A subclt mpan2 syl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.2 A + seq1 F opr k cv - 1 opr ` subclt mpan syl sylanc x cv RR e. adantr abs + seq1 F opr k cv ` - A opr + A - + seq1 F opr k cv - 1 opr ` opr opr ` abs + seq1 F opr k cv ` - A opr ` + abs A - + seq1 F opr k cv - 1 opr ` opr ` opr x cv lelttrt + seq1 F opr k cv ` - A opr A - + seq1 F opr k cv - 1 opr ` opr axaddcl ser1f0.1 k cv ser1cl ser1f0.2 + seq1 F opr k cv ` A subclt mpan2 syl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.2 A + seq1 F opr k cv - 1 opr ` subclt mpan syl sylanc + seq1 F opr k cv ` - A opr + A - + seq1 F opr k cv - 1 opr ` opr opr absclt syl x cv RR e. adantr abs + seq1 F opr k cv ` - A opr ` abs A - + seq1 F opr k cv - 1 opr ` opr ` axaddrcl ser1f0.1 k cv ser1cl ser1f0.2 + seq1 F opr k cv ` A subclt mpan2 syl + seq1 F opr k cv ` - A opr absclt syl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.2 A + seq1 F opr k cv - 1 opr ` subclt mpan syl A - + seq1 F opr k cv - 1 opr ` opr absclt syl sylanc x cv RR e. adantr k cv NN e. 1 < k cv br /\ x cv RR e. pm3.27 syl3anc mpand addex nnex ser1f0.1 NN V F CC fex mp2 k cv seq1m1 eqcomd + seq1 F opr k cv ` + seq1 F opr k cv - 1 opr ` F k cv ` subaddt ser1f0.1 k cv ser1cl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl ser1f0.1 k cv ffvrni 1 < k cv br adantr syl3anc mpbird ser1f0.2 + seq1 F opr k cv ` A + seq1 F opr k cv - 1 opr ` npncant mp3an2 ser1f0.1 k cv ser1cl 1 < k cv br adantr 1nn 1 k cv nnsubt mpan biimpa ser1f0.1 k cv - 1 opr ser1cl syl sylanc ser1f0.1 k cv ffvrni F k cv ` subid1t syl 1 < k cv br adantr 3eqtr4d abs fveq2d x equid x cv RR e. a1i < breqan12d sylibd syld 1z 1 k cv zltp1let mpan df-2 <_ k cv breq1i syl6bbr k cv zret ax1re 1 k cv ltlet mpan syl sylbird imdistani k cv elnnz1 sylibr 1z 1 k cv zltp1let mpan df-2 <_ k cv breq1i syl6bbr biimpar jca sylan m cv ZZ e. n ZZ m cv <_ n cv br abs + seq1 F opr n cv ` - A opr ` < x cv / 2 opr br -> A.e. /\ adantrr syld exp31 com3r imp4a r19.21aiv sylanc exp32 r19.23adv 0 < x cv br adantr mpd exp mprgbir))
+thm (ser1const ((j k) (A j) (A k) (N j)) ((ser1const.1 (e. A (CC)))) (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} A))) N) (opr N (x.) A))) (j cv 1 + seq1 NN A {} X. opr fveq2 j cv 1 x. A opreq1 eqeq12d j cv k cv + seq1 NN A {} X. opr fveq2 j cv k cv x. A opreq1 eqeq12d j cv k cv + 1 opr + seq1 NN A {} X. opr fveq2 j cv k cv + 1 opr x. A opreq1 eqeq12d j cv N + seq1 NN A {} X. opr fveq2 j cv N x. A opreq1 eqeq12d 1nn ser1const.1 elisseti 1 NN fvconst2 ax-mp addex nnex A snex xpex seq11 ser1const.1 mulid2 3eqtr4 addex nnex A snex xpex k cv seq1p1 k cv peano2nn ser1const.1 elisseti k cv + 1 opr NN fvconst2 syl + seq1 NN A {} X. opr k cv ` + opreq2d eqtrd + seq1 NN A {} X. opr k cv ` k cv x. A opr = adantr + seq1 NN A {} X. opr k cv ` k cv x. A opr + A opreq1 k cv nncnt 1cn ser1const.1 k cv 1 A adddirt mp3an23 syl ser1const.1 mulid2 k cv x. A opr + opreq2i syl6req sylan9eqr eqtrd exp nnind))
+thm (ser10 () () (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} (0)))) N) (0))) (0cn N ser1const N nncnt N mul01t syl eqtrd))
+thm (ser1clim0OLD ((x y) (x z) (y z)) () (br (opr (+) (seq1) (X. (NN) ({} (0)))) (~~>OLD) (0)) (0cn elisseti NN fconst 0cn 0 CC snssi ax-mp NN 0 {} X. NN 0 {} CC fss mp2an ser1f 0cn + seq1 NN 0 {} X. opr 0 x y z clim2OLD mp2an z cv ser10 - 0 opreq1d 0cn subid syl6eq abs fveq2d abs0 syl6eq < x cv breq1d biimprcd 1 <_ z cv br a1dd r19.21aiv 1nn jctil y cv 1 <_ z cv breq1 abs + seq1 NN 0 {} X. opr z cv ` - 0 opr ` < x cv br imbi1d z NN biraldv NN rcla4ev syl x cv RR e. a1i mprgbir))
+thm (ser1trunclem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z) (C y)) ((ser1trunc.1 (Fn F (NN))) (ser1trunc.2 (-> (e. (cv x) (NN)) (= (` F (cv x)) (if (br (cv x) (<_) A) B (0)))))) (-> (/\/\ (e. A (NN)) (e. B (CC)) (e. C (NN))) (= (` (opr (+) (seq1) F) C) (if (br C (<_) A) (opr C (x.) B) (opr A (x.) B)))) (y cv 1 + seq1 F opr fveq2 y cv 1 <_ A breq1 y cv x. B opr A x. B opr ifbid y cv 1 x. B opreq1 1 <_ A br A x. B opr ifeq1d eqtrd eqeq12d A NN e. B CC e. /\ imbi2d y cv z cv + seq1 F opr fveq2 y cv z cv <_ A breq1 y cv x. B opr A x. B opr ifbid y cv z cv x. B opreq1 z cv <_ A br A x. B opr ifeq1d eqtrd eqeq12d A NN e. B CC e. /\ imbi2d y cv z cv + 1 opr + seq1 F opr fveq2 y cv z cv + 1 opr <_ A breq1 y cv x. B opr A x. B opr ifbid y cv z cv + 1 opr x. B opreq1 z cv + 1 opr <_ A br A x. B opr ifeq1d eqtrd eqeq12d A NN e. B CC e. /\ imbi2d y cv C + seq1 F opr fveq2 y cv C <_ A breq1 y cv x. B opr A x. B opr ifbid y cv C x. B opreq1 C <_ A br A x. B opr ifeq1d eqtrd eqeq12d A NN e. B CC e. /\ imbi2d B mulid2t A nnge1t 1 <_ A br B 0 iftrue syl addex nnex ser1trunc.1 NN V F fnex mp2 seq11 1nn x cv 1 F fveq2 x cv 1 <_ A breq1 B 0 ifbid eqeq12d ser1trunc.2 vtoclga ax-mp eqtr2 syl5reqr sylan9eqr A nnge1t 1 <_ A br 1 x. B opr A x. B opr iftrue syl eqcomd B CC e. adantr eqtr3d addex nnex ser1trunc.1 NN V F fnex mp2 z cv seq1p1 A NN e. B CC e. /\ + seq1 F opr z cv ` z cv <_ A br z cv x. B opr A x. B opr if = ad2antll + seq1 F opr z cv ` z cv <_ A br z cv x. B opr A x. B opr if + F z cv + 1 opr ` opreq1 z cv NN e. A NN e. B CC e. /\ /\ adantl z cv <_ A br z cv x. B opr A x. B opr iftrue + F z cv + 1 opr ` opreq1d z cv NN e. A NN e. B CC e. /\ /\ adantl z cv peano2nn x cv z cv + 1 opr F fveq2 x cv z cv + 1 opr <_ A breq1 B 0 ifbid eqeq12d ser1trunc.2 vtoclga syl z cv + 1 opr <_ A br B 0 iftrue sylan9eq z cv x. B opr + opreq2d B CC e. adantlr z cv + 1 opr <_ A br z cv + 1 opr x. B opr A x. B opr iftrue 1cn z cv 1 B adddirt mp3an2 z cv nncnt sylan B mulid2t z cv NN e. adantl z cv x. B opr + opreq2d eqtrd sylan9eqr eqtr4d A NN e. adantlrl z cv <_ A br adantlr z cv <_ A br z cv + 1 opr <_ A br -. pm3.26 z cv NN e. A NN e. /\ adantl A z cv nnleltp1t A z cv + 1 opr ltnlet A nnret z cv nnret z cv peano2re syl syl2an bitr2d ancoms biimpa z cv <_ A br adantrl jca z cv A letri3t z cv nnret A nnret syl2an z cv <_ A br z cv + 1 opr <_ A br -. /\ adantr mpbird x. B opreq1d + 0 opreq1d B CC e. adantlrr A B axmulcl A nncnt sylan A x. B opr ax0id syl z cv NN e. z cv <_ A br z cv + 1 opr <_ A br -. /\ ad2antlr eqtrd anassrs z cv peano2nn x cv z cv + 1 opr F fveq2 x cv z cv + 1 opr <_ A breq1 B 0 ifbid eqeq12d ser1trunc.2 vtoclga syl z cv + 1 opr <_ A br B 0 iffalse sylan9eq z cv x. B opr + opreq2d z cv <_ A br adantlr A NN e. B CC e. /\ adantllr z cv + 1 opr <_ A br z cv + 1 opr x. B opr A x. B opr iffalse z cv NN e. A NN e. B CC e. /\ /\ z cv <_ A br /\ adantl 3eqtr4d pm2.61an2 eqtrd z cv <_ A br z cv x. B opr A x. B opr iffalse + F z cv + 1 opr ` opreq1d z cv NN e. A NN e. B CC e. /\ /\ adantl ax1re 0re ax1re lt01 ltlei z cv 1 addge1tOLD mp3an23 A RR e. adantr z cv z cv + 1 opr A letrt z cv RR e. A RR e. pm3.26 z cv peano2re A RR e. adantr z cv RR e. A RR e. pm3.27 syl3anc mpand z cv nnret A nnret syl2an con3d imp B CC e. adantlrr A B axmulcl A nncnt sylan A x. B opr ax0id syl z cv NN e. z cv + 1 opr <_ A br -. ad2antlr z cv peano2nn x cv z cv + 1 opr F fveq2 x cv z cv + 1 opr <_ A breq1 B 0 ifbid eqeq12d ser1trunc.2 vtoclga syl z cv + 1 opr <_ A br B 0 iffalse sylan9eq A NN e. B CC e. /\ adantlr A x. B opr + opreq2d z cv + 1 opr <_ A br z cv + 1 opr x. B opr A x. B opr iffalse z cv NN e. A NN e. B CC e. /\ /\ adantl 3eqtr4d syldan eqtrd pm2.61an2 + seq1 F opr z cv ` z cv <_ A br z cv x. B opr A x. B opr if = adantr 3eqtrd exp31 a2d nnind exp3a com3l 3imp))
+thm (ser1truncOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((ser1trunc.1 (Fn F (NN))) (ser1trunc.2 (-> (e. (cv x) (NN)) (= (` F (cv x)) (if (br (cv x) (<_) A) B (0)))))) (-> (/\ (e. A (NN)) (e. B (CC))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (x.) B))) (ser1trunc.2 CC eleq1d 0cn B CC 0 x cv <_ A br ifcl mpan2 syl5bir com12 r19.21aiv ser1trunc.1 jctil F NN CC x ffnfv sylibr F ser1ft syl A NN e. adantl A B axmulcl A nncnt sylan jca A NN e. 0 < y cv br ax-1 B CC e. adantr ser1trunc.1 ser1trunc.2 w cv ser1trunclem 3expa A < w cv br adantr A w cv ltnlet A nnret w cv nnret syl2an biimpa w cv <_ A br w cv x. B opr A x. B opr iffalse syl B CC e. adantllr eqtrd - A x. B opr opreq1d A B axmulcl A nncnt sylan A x. B opr subidt syl w cv NN e. A < w cv br ad2antll eqtrd abs fveq2d abs0 syl6eq 0 < y cv br adantlrl 0 < y cv br w cv NN e. pm3.26 A NN e. B CC e. /\ A < w cv br ad2antlr eqbrtrd exp42 r19.21adv jcad z cv A < w cv breq1 abs + seq1 F opr w cv ` - A x. B opr opr ` < y cv br imbi1d w NN biraldv NN rcla4ev syl6 y cv RR e. a1d r19.21aiv y RR 0 < y cv br z w abs + seq1 F opr w cv ` - A x. B opr opr ` < y cv br cvg1 sylib jca A x. B oprex + seq1 F opr y z w climOLD sylibr))
+thm (ser1trunc2OLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z)) ((ser1trunc2.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (if (br (cv x) (<_) A) B (0)))))))) (-> (/\ (e. A (NN)) (e. B (CC))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (x.) B))) (w cv B x cv <_ A br 0 ifeq1 y cv eqeq2d x cv NN e. anbi2d x y biopabdv + seq1 opreq2d w cv B A x. opreq2 ~~>OLD breq12d A NN e. imbi2d w visset 0cn elisseti x cv <_ A br ifex y eueq1 x cv NN e. a1i x y x cv NN e. y cv x cv <_ A br w cv 0 if = /\ {<,>|} eqid fnopab x cv z cv <_ A breq1 w cv 0 ifbid x y x cv NN e. y cv x cv <_ A br w cv 0 if = /\ {<,>|} eqid w visset 0cn elisseti z cv <_ A br ifex fvopab4 ser1truncOLD expcom vtoclga impcom ser1trunc2.1 + seq1 opreq2i syl5eqbr))
+thm (ser1cmp ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) ((ser1cmp.1 (:--> F (NN) (RR))) (ser1cmp.2 (:--> G (NN) (RR))) (ser1cmp.3 (-> (e. (cv x) (NN)) (br (` G (cv x)) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) G) A) (<_) (` (opr (+) (seq1) F) A))) (y cv 1 + seq1 G opr fveq2 y cv 1 + seq1 F opr fveq2 <_ breq12d y cv z cv + seq1 G opr fveq2 y cv z cv + seq1 F opr fveq2 <_ breq12d y cv z cv + 1 opr + seq1 G opr fveq2 y cv z cv + 1 opr + seq1 F opr fveq2 <_ breq12d y cv A + seq1 G opr fveq2 y cv A + seq1 F opr fveq2 <_ breq12d 1nn x cv 1 G fveq2 x cv 1 F fveq2 <_ breq12d ser1cmp.3 vtoclga ax-mp addex nnex ser1cmp.2 NN V G RR fex mp2 seq11 addex nnex ser1cmp.1 NN V F RR fex mp2 seq11 3brtr4 + seq1 G opr z cv ` G z cv + 1 opr ` axaddrcl ser1cmp.2 z cv ser1recl z cv peano2nn ser1cmp.2 z cv + 1 opr ffvrni syl sylanc + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr + seq1 F opr z cv ` G z cv + 1 opr ` axaddrcl ser1cmp.1 z cv ser1recl z cv peano2nn ser1cmp.2 z cv + 1 opr ffvrni syl sylanc + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr + seq1 F opr z cv ` F z cv + 1 opr ` axaddrcl ser1cmp.1 z cv ser1recl z cv peano2nn ser1cmp.1 z cv + 1 opr ffvrni syl sylanc + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr + seq1 G opr z cv ` + seq1 F opr z cv ` G z cv + 1 opr ` leadd1t ser1cmp.2 z cv ser1recl ser1cmp.1 z cv ser1recl z cv peano2nn ser1cmp.2 z cv + 1 opr ffvrni syl syl3anc biimpa z cv peano2nn x cv z cv + 1 opr G fveq2 x cv z cv + 1 opr F fveq2 <_ breq12d ser1cmp.3 vtoclga syl G z cv + 1 opr ` F z cv + 1 opr ` + seq1 F opr z cv ` leadd2t z cv peano2nn ser1cmp.2 z cv + 1 opr ffvrni syl z cv peano2nn ser1cmp.1 z cv + 1 opr ffvrni syl ser1cmp.1 z cv ser1recl syl3anc mpbid + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr letrd addex nnex ser1cmp.2 NN V G RR fex mp2 z cv seq1p1 + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr addex nnex ser1cmp.1 NN V F RR fex mp2 z cv seq1p1 + seq1 G opr z cv ` <_ + seq1 F opr z cv ` br adantr 3brtr4d exp nnind))
+thm (ser1cmp0 ((F x)) ((ser1cmp0.1 (:--> F (NN) (RR))) (ser1cmp0.2 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))))) (-> (e. A (NN)) (br (0) (<_) (` (opr (+) (seq1) F) A))) (A ser10 ser1cmp0.1 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an 0re elisseti x cv NN fvconst2 ser1cmp0.2 eqbrtrd A ser1cmp eqbrtrrd))
+thm (ser1cmp2lem ((A z) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F z) (G x) (G z) (S z)) ((ser1cmp2.1 (:--> F (NN) (RR))) (ser1cmp2.2 (:--> G (NN) (RR))) (ser1cmp2.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (ser1cmp2.4 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (ser1cmp2.5 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))))) (-> (/\ (e. A (NN)) (e. B (NN))) (-> (br A (<_) B) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A))))) (ser1cmp2.2 ser1re A B y seq1ub ser1cmp2.5 syl6breqr 3expa exp S + seq1 F opr A ` addge1tOLD ser1cmp2.2 ser1re B y x z seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran x z suprcl syl ser1cmp2.5 RR eleq1i sylibr A NN e. adantl ser1cmp2.1 A ser1recl B NN e. adantr ser1cmp2.1 ser1cmp2.3 A ser1cmp0 B NN e. adantr syl3anc + seq1 G opr A ` S S + + seq1 F opr A ` opr letrt ser1cmp2.2 A ser1recl B NN e. adantr ser1cmp2.2 ser1re B y x z seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran x z suprcl syl ser1cmp2.5 RR eleq1i sylibr A NN e. adantl S + seq1 F opr A ` axaddrcl ser1cmp2.2 ser1re B y x z seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran x z suprcl syl ser1cmp2.5 RR eleq1i sylibr ser1cmp2.1 A ser1recl syl2an ancoms syl3anc mpan2d syld))
+thm (ser1cmp2 ((A z) (x y) (w x) (x z) (B x) (w y) (y z) (B y) (w z) (B w) (B z) (F x) (F w) (F z) (G x) (G w) (G z) (S w) (S z)) ((ser1cmp2.1 (:--> F (NN) (RR))) (ser1cmp2.2 (:--> G (NN) (RR))) (ser1cmp2.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (ser1cmp2.4 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (ser1cmp2.5 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A)))) (z cv 1 + seq1 G opr fveq2 z cv 1 + seq1 F opr fveq2 S + opreq2d <_ breq12d B NN e. imbi2d z cv w cv + seq1 G opr fveq2 z cv w cv + seq1 F opr fveq2 S + opreq2d <_ breq12d B NN e. imbi2d z cv w cv + 1 opr + seq1 G opr fveq2 z cv w cv + 1 opr + seq1 F opr fveq2 S + opreq2d <_ breq12d B NN e. imbi2d z cv A + seq1 G opr fveq2 z cv A + seq1 F opr fveq2 S + opreq2d <_ breq12d B NN e. imbi2d B nnge1t 1nn ser1cmp2.1 ser1cmp2.2 ser1cmp2.3 ser1cmp2.4 ser1cmp2.5 1 ser1cmp2lem mpan mpd w cv + 1 opr B lelttrit w cv peano2nn w cv + 1 opr nnret syl B nnret syl2an ser1cmp2.1 ser1cmp2.2 ser1cmp2.3 ser1cmp2.4 ser1cmp2.5 w cv + 1 opr ser1cmp2lem w cv peano2nn sylan + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br a1dd + seq1 G opr w cv ` G w cv + 1 opr ` axaddrcl ser1cmp2.2 w cv ser1recl w cv peano2nn ser1cmp2.2 w cv + 1 opr ffvrni syl sylanc B NN e. adantr B < w cv + 1 opr br adantrr + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br adantr S + + seq1 F opr w cv ` opr G w cv + 1 opr ` axaddrcl S + seq1 F opr w cv ` axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran z w suprcl syl ser1cmp2.5 RR eleq1i sylibr ser1cmp2.1 w cv ser1recl syl2an ancoms w cv peano2nn ser1cmp2.2 w cv + 1 opr ffvrni syl B NN e. adantr sylanc B < w cv + 1 opr br adantrr + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br adantr S + + seq1 F opr w cv ` opr F w cv + 1 opr ` axaddrcl S + seq1 F opr w cv ` axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran z w suprcl syl ser1cmp2.5 RR eleq1i sylibr ser1cmp2.1 w cv ser1recl syl2an ancoms w cv peano2nn ser1cmp2.1 w cv + 1 opr ffvrni syl B NN e. adantr sylanc B < w cv + 1 opr br adantrr + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br adantr + seq1 G opr w cv ` S + + seq1 F opr w cv ` opr G w cv + 1 opr ` leadd1t ser1cmp2.2 w cv ser1recl B NN e. adantr S + seq1 F opr w cv ` axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran z w suprcl syl ser1cmp2.5 RR eleq1i sylibr ser1cmp2.1 w cv ser1recl syl2an ancoms w cv peano2nn ser1cmp2.2 w cv + 1 opr ffvrni syl B NN e. adantr syl3anc biimpa B < w cv + 1 opr br adantlrr x cv w cv + 1 opr B < breq2 x cv w cv + 1 opr G fveq2 x cv w cv + 1 opr F fveq2 <_ breq12d imbi12d ser1cmp2.4 exp vtoclga imp w cv peano2nn sylan B NN e. adantrl G w cv + 1 opr ` F w cv + 1 opr ` S + + seq1 F opr w cv ` opr leadd2t w cv peano2nn ser1cmp2.2 w cv + 1 opr ffvrni syl B NN e. adantr w cv peano2nn ser1cmp2.1 w cv + 1 opr ffvrni syl B NN e. adantr S + seq1 F opr w cv ` axaddrcl ser1cmp2.2 ser1re B y z w seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran z w suprcl syl ser1cmp2.5 RR eleq1i sylibr ser1cmp2.1 w cv ser1recl syl2an ancoms syl3anc B < w cv + 1 opr br adantrr mpbid + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br adantr letrd addex nnex ser1cmp2.2 NN V G RR fex mp2 w cv seq1p1 B NN e. B < w cv + 1 opr br /\ + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br ad2antll addex nnex ser1cmp2.1 NN V F RR fex mp2 w cv seq1p1 S + opreq2d B NN e. adantr S + seq1 F opr w cv ` F w cv + 1 opr ` axaddass ser1cmp2.2 ser1re B y z w seq1ublem + seq1 G opr y NN y cv <_ B br {e.|} |` ran z w suprcl syl ser1cmp2.5 RR eleq1i sylibr recnd w cv NN e. adantl ser1cmp2.1 w cv ser1recl recnd B NN e. adantr w cv peano2nn ser1cmp2.1 w cv + 1 opr ffvrni syl recnd B NN e. adantr syl3anc eqtr4d B < w cv + 1 opr br adantrr + seq1 G opr w cv ` <_ S + + seq1 F opr w cv ` opr br adantr 3brtr4d exp anassrs exp jaod mpd exp a2d nnind imp))
+thm (cvgcmp2lemOLD ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (S x) (S z) (S w) (H x) (H z)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2.8 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (cvgcmp2lem.9 (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (cvgcmp2lem.10 (Fn H (NN))) (cvgcmp2lem.11 (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr S (+) (` (opr (+) (seq1) F) (cv x))))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.3 ser1re cvgcmp2.3 cvgcmp2.5 x cv ser1mono cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr cvgcmp2.1 cvgcmp2.2 ser1re cvgcmp2.6 climreclOLD readdcl cvgcmp2.3 x cv ser1recl cvgcmp2lem.11 cvgcmp2.2 x cv ser1recl cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr S + seq1 F opr x cv ` axaddrcl mpan syl eqeltrd cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr cvgcmp2.1 cvgcmp2.2 ser1re cvgcmp2.6 climreclOLD readdcl x cv NN e. a1i cvgcmp2.7 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.8 cvgcmp2lem.9 x cv ser1cmp2 mpan2 cvgcmp2lem.11 breqtrrd S + A oprex H NN RR x ffnfv cvgcmp2lem.10 cvgcmp2lem.11 cvgcmp2.2 x cv ser1recl cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr S + seq1 F opr x cv ` axaddrcl mpan syl eqeltrd rgen mpbir2an cvgcmp2.2 cvgcmp2.4 z cv ser1mono cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr + seq1 F opr z cv ` + seq1 F opr z cv + 1 opr ` S leadd2t mp3an3 cvgcmp2.2 z cv ser1recl z cv peano2nn cvgcmp2.2 z cv + 1 opr ser1recl syl sylanc mpbid x cv z cv H fveq2 x cv z cv + seq1 F opr fveq2 S + opreq2d eqeq12d cvgcmp2lem.11 vtoclga z cv peano2nn x cv z cv + 1 opr H fveq2 x cv z cv + 1 opr + seq1 F opr fveq2 S + opreq2d eqeq12d cvgcmp2lem.11 vtoclga syl 3brtr4d cvgcmp2lem.9 cvgcmp2.7 cvgcmp2.3 ser1re B y z w seq1ublem ax-mp suprcli eqeltr recn cvgcmp2.1 cvgcmp2.6 cvgcmp2lem.10 cvgcmp2lem.11 climaddcxOLD x cv climubOLD letrd rgen z cv S + A opr + seq1 G opr x cv ` <_ breq2 x NN biraldv RR rcla4ev mp2an climsupOLD))
+thm (cvgcmp2OLD ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2.8 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.1 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.5 cvgcmp2.6 cvgcmp2.7 cvgcmp2.8 + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup eqid + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup + + seq1 F opr z cv ` oprex z w z cv NN e. w cv + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup + + seq1 F opr z cv ` opr = /\ {<,>|} eqid fnopab2 z cv x cv + seq1 F opr fveq2 + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup + opreq2d z w z cv NN e. w cv + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup + + seq1 F opr z cv ` opr = /\ {<,>|} eqid + seq1 G opr y NN y cv <_ B br {e.|} |` ran RR < sup + + seq1 F opr x cv ` oprex fvopab4 cvgcmp2lemOLD))
+thm (cvgcmp2clemOLD ((A x) (B x) (F x) (G x) (S x) (C x)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2c.9 (e. C (RR))) (cvgcmp2c.10 (br (0) (<_) C)) (cvgcmp2c.11 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x)))))) (cvgcmp2clem.12 (Fn S (NN))) (cvgcmp2clem.13 (-> (e. (cv x) (NN)) (= (` S (cv x)) (opr C (x.) (` F (cv x))))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (C x. A oprex S NN RR x ffnfv cvgcmp2clem.12 cvgcmp2clem.13 cvgcmp2.2 x cv ffvrni cvgcmp2c.9 C F x cv ` axmulrcl mpan syl eqeltrd rgen mpbir2an cvgcmp2.3 cvgcmp2c.9 C F x cv ` mulge0tOLD mpan cvgcmp2.2 x cv ffvrni cvgcmp2.4 cvgcmp2c.10 jctil sylc cvgcmp2clem.13 breqtrrd cvgcmp2.5 cvgcmp2c.9 recn cvgcmp2.1 cvgcmp2.6 addex nnex cvgcmp2clem.12 NN V S fnex mp2 seq1fn cvgcmp2c.9 recn cvgcmp2.2 axresscn F NN RR CC fss mp2an nnex cvgcmp2clem.12 NN V S fnex mp2 cvgcmp2clem.13 x cv ser1mulc climmulcOLD cvgcmp2.7 cvgcmp2c.11 cvgcmp2clem.13 B < x cv br adantr breqtrrd cvgcmp2OLD))
+thm (cvgcmp2cOLD ((x z) (A x) (A z) (w x) (B x) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (C x) (C z) (C w)) ((cvgcmp2.1 (e. A (V))) (cvgcmp2.2 (:--> F (NN) (RR))) (cvgcmp2.3 (:--> G (NN) (RR))) (cvgcmp2.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp2.5 (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x))))) (cvgcmp2.6 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp2.7 (e. B (NN))) (cvgcmp2c.9 (e. C (RR))) (cvgcmp2c.10 (br (0) (<_) C)) (cvgcmp2c.11 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x))))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp2.1 cvgcmp2.2 cvgcmp2.3 cvgcmp2.4 cvgcmp2.5 cvgcmp2.6 cvgcmp2.7 cvgcmp2c.9 cvgcmp2c.10 cvgcmp2c.11 C x. F z cv ` oprex z w z cv NN e. w cv C x. F z cv ` opr = /\ {<,>|} eqid fnopab2 z cv x cv F fveq2 C x. opreq2d z w z cv NN e. w cv C x. F z cv ` opr = /\ {<,>|} eqid C x. F x cv ` oprex fvopab4 cvgcmp2clemOLD))
+thm (cvgcmpOLD ((A x) (F x) (G x)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>OLD) A))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.3 x cv ffvrni cvgcmp.2 x cv ffvrni jca 0re 0 G x cv ` F x cv ` letrt mp3an1 syl mpd cvgcmp.4 pm3.26d cvgcmp.5 1nn cvgcmp.4 pm3.27d 1 < x cv br adantr cvgcmp2OLD))
+thm (cvgcmpALTOLD ((x y) (A x) (A y) (F x) (G x) (G y)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>OLD) A))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (cvgcmp.3 ser1re cvgcmp.3 cvgcmp.4 pm3.26d x cv ser1mono cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD cvgcmp.3 x cv ser1recl cvgcmp.2 x cv ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD x cv NN e. a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d x cv ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 cvgcmp.4 0re 0 G x cv ` F x cv ` letrt mp3an1 cvgcmp.3 x cv ffvrni cvgcmp.2 x cv ffvrni sylanc mpd x cv ser1mono cvgcmp.5 x cv climubOLD letrd rgen y cv A + seq1 G opr x cv ` <_ breq2 x NN biraldv RR rcla4ev mp2an climsupOLD))
+thm (cvgcmpub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (G x) (G y) (G z) (G w)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmpub.6 (e. B (V))) (cvgcmpub.7 (br (opr (+) (seq1) G) (~~>OLD) B))) (br B (<_) A) (cvgcmpub.6 cvgcmp.3 ser1re + seq1 G opr NN RR frn ax-mp 1nn 1 NN n0i ax-mp cvgcmp.3 ser1re + seq1 G opr NN RR fdm ax-mp {/} eqeq1i mtbir + seq1 G opr dm0rn0 mtbi cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD cvgcmp.3 ser1re + seq1 G opr NN RR ffn ax-mp + seq1 G opr NN z cv x fvelrn ax-mp + seq1 G opr x cv ` z cv <_ A breq1 cvgcmp.3 x cv ser1recl cvgcmp.2 x cv ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD x cv NN e. a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d x cv ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re 0 G x cv ` F x cv ` letrt mp3an1 cvgcmp.3 x cv ffvrni cvgcmp.2 x cv ffvrni jca cvgcmp.4 sylc x cv ser1mono cvgcmp.5 x cv climubOLD letrd syl5bi com12 r19.23aiv sylbi rgen y cv A z cv <_ breq2 z + seq1 G opr ran biraldv RR rcla4ev mp2an 3pm3.2i suprcli elisseti cvgcmpub.7 cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 cvgcmpOLD pm3.2i climuniiOLD cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD cvgcmp.3 ser1re + seq1 G opr NN RR ffn ax-mp + seq1 G opr NN w cv x fvelrn ax-mp + seq1 G opr x cv ` w cv <_ A breq1 cvgcmp.3 x cv ser1recl cvgcmp.2 x cv ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD x cv NN e. a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d x cv ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re 0 G x cv ` F x cv ` letrt mp3an1 cvgcmp.3 x cv ffvrni cvgcmp.2 x cv ffvrni jca cvgcmp.4 sylc x cv ser1mono cvgcmp.5 x cv climubOLD letrd syl5bi com12 r19.23aiv sylbi rgen cvgcmp.3 ser1re + seq1 G opr NN RR frn ax-mp 1nn 1 NN n0i ax-mp cvgcmp.3 ser1re + seq1 G opr NN RR fdm ax-mp {/} eqeq1i mtbir + seq1 G opr dm0rn0 mtbi cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD cvgcmp.3 ser1re + seq1 G opr NN RR ffn ax-mp + seq1 G opr NN z cv x fvelrn ax-mp + seq1 G opr x cv ` z cv <_ A breq1 cvgcmp.3 x cv ser1recl cvgcmp.2 x cv ser1recl cvgcmp.1 cvgcmp.2 ser1re cvgcmp.5 climreclOLD x cv NN e. a1i cvgcmp.2 cvgcmp.3 cvgcmp.4 pm3.27d x cv ser1cmp cvgcmp.1 cvgcmp.2 ser1re cvgcmp.2 0re 0 G x cv ` F x cv ` letrt mp3an1 cvgcmp.3 x cv ffvrni cvgcmp.2 x cv ffvrni jca cvgcmp.4 sylc x cv ser1mono cvgcmp.5 x cv climubOLD letrd syl5bi com12 r19.23aiv sylbi rgen y cv A z cv <_ breq2 z + seq1 G opr ran biraldv RR rcla4ev mp2an 3pm3.2i A w suprleubi mp2an eqbrtr))
+thm (cvgcmpubexOLD ((x y) (A x) (A y) (F x) (G x) (G y)) ((cvgcmp.1 (e. A (V))) (cvgcmp.2 (:--> F (NN) (RR))) (cvgcmp.3 (:--> G (NN) (RR))) (cvgcmp.4 (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))))) (cvgcmp.5 (br (opr (+) (seq1) F) (~~>OLD) A))) (E.e. y (RR) (/\ (br (cv y) (<_) A) (br (opr (+) (seq1) G) (~~>OLD) (cv y)))) (ltso + seq1 G opr ran supex cvgcmp.3 ser1re cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 cvgcmpOLD climreclOLD cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 ltso + seq1 G opr ran supex cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 cvgcmpOLD cvgcmpub cvgcmp.1 cvgcmp.2 cvgcmp.3 cvgcmp.4 cvgcmp.5 cvgcmpOLD pm3.2i y cv + seq1 G opr ran RR < sup <_ A breq1 y cv + seq1 G opr ran RR < sup + seq1 G opr ~~>OLD breq2 anbi12d RR rcla4ev mp2an))
+thm (cvgcmp3OLD ((A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((cvgcmp3.1 (:--> F (NN) (RR))) (cvgcmp3.2 (:--> T (NN) (CC))) (cvgcmp3.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3.6 (e. A (V))) (cvgcmp3.7 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp3.8 (Fn H (NN))) (cvgcmp3.9 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x)))))) (cvgcmp3.10 (Fn G (NN))) (cvgcmp3.11 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.12 (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v))))))))) (cvgcmp3.13 (Fn C (NN))) (cvgcmp3.14 (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.15 (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v))))))))) (cvgcmp3.16 (Fn D (NN))) (cvgcmp3.17 (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x)))))) (cvgcmp3.4 (e. B (NN))) (cvgcmp3.5 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (` F (cv x)))))) (br (opr (+) (seq1) T) (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))) (cvgcmp3.2 ser1f ltso + seq1 H opr ran supex cvgcmp3.6 cvgcmp3.1 H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an cvgcmp3.3 cvgcmp3.2 x cv ffvrni T x cv ` absge0t syl cvgcmp3.9 breqtrrd cvgcmp3.7 cvgcmp3.4 cvgcmp3.9 B < x cv br adantr cvgcmp3.5 eqbrtrd cvgcmp2OLD z w v climcau + seq1 T opr v cv ` + seq1 T opr w cv ` subclt cvgcmp3.2 v cv ser1cl cvgcmp3.2 w cv ser1cl syl2an + seq1 T opr v cv ` - + seq1 T opr w cv ` opr absclt syl w cv < v cv br adantr + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an w cv < v cv br adantr + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an recnd + seq1 H opr v cv ` - + seq1 H opr w cv ` opr absclt syl w cv < v cv br adantr cvgcmp3.2 cvgcmp3.8 cvgcmp3.9 w cv v cv ser1absdif 3com12 3expa + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an + seq1 H opr v cv ` - + seq1 H opr w cv ` opr leabst syl w cv < v cv br adantr letrd z cv RR e. adantlll abs + seq1 T opr v cv ` - + seq1 T opr w cv ` opr ` abs + seq1 H opr v cv ` - + seq1 H opr w cv ` opr ` z cv lelttrt + seq1 T opr v cv ` + seq1 T opr w cv ` subclt cvgcmp3.2 v cv ser1cl cvgcmp3.2 w cv ser1cl syl2an + seq1 T opr v cv ` - + seq1 T opr w cv ` opr absclt syl z cv RR e. adantll + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an recnd + seq1 H opr v cv ` - + seq1 H opr w cv ` opr absclt syl z cv RR e. adantll z cv RR e. v cv NN e. pm3.26 w cv NN e. adantr syl3anc w cv < v cv br adantr mpand exp an1rs a2d r19.20dva r19.22dva 0 < z cv br imim2d r19.20i ax-mp cvgcmp3.10 cvgcmp3.11 cvgcmp3.12 cvgcmp3.13 cvgcmp3.14 cvgcmp3.15 cvgcmp3.16 cvgcmp3.17 caucvg3aOLD))
+thm (cvgcmp3cOLD ((Q x) (A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((cvgcmp3.1 (:--> F (NN) (RR))) (cvgcmp3.2 (:--> T (NN) (CC))) (cvgcmp3.3 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3.6 (e. A (V))) (cvgcmp3.7 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp3.8 (Fn H (NN))) (cvgcmp3.9 (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x)))))) (cvgcmp3.10 (Fn G (NN))) (cvgcmp3.11 (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.12 (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v))))))))) (cvgcmp3.13 (Fn C (NN))) (cvgcmp3.14 (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x)))))) (cvgcmp3.15 (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v))))))))) (cvgcmp3.16 (Fn D (NN))) (cvgcmp3.17 (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x)))))) (cvgcmp3.4 (e. B (NN))) (cvgcmp3c.18 (e. Q (RR))) (cvgcmp3c.19 (br (0) (<_) Q)) (cvgcmp3c.5 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (opr Q (x.) (` F (cv x))))))) (br (opr (+) (seq1) T) (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))) (cvgcmp3.2 ser1f ltso + seq1 H opr ran supex cvgcmp3.6 cvgcmp3.1 H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an cvgcmp3.3 cvgcmp3.2 x cv ffvrni T x cv ` absge0t syl cvgcmp3.9 breqtrrd cvgcmp3.7 cvgcmp3.4 cvgcmp3c.18 cvgcmp3c.19 cvgcmp3.9 B < x cv br adantr cvgcmp3c.5 eqbrtrd cvgcmp2cOLD z w v climcau + seq1 T opr v cv ` + seq1 T opr w cv ` subclt cvgcmp3.2 v cv ser1cl cvgcmp3.2 w cv ser1cl syl2an + seq1 T opr v cv ` - + seq1 T opr w cv ` opr absclt syl w cv < v cv br adantr + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an w cv < v cv br adantr + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an recnd + seq1 H opr v cv ` - + seq1 H opr w cv ` opr absclt syl w cv < v cv br adantr cvgcmp3.2 cvgcmp3.8 cvgcmp3.9 w cv v cv ser1absdif 3com12 3expa + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an + seq1 H opr v cv ` - + seq1 H opr w cv ` opr leabst syl w cv < v cv br adantr letrd z cv RR e. adantlll abs + seq1 T opr v cv ` - + seq1 T opr w cv ` opr ` abs + seq1 H opr v cv ` - + seq1 H opr w cv ` opr ` z cv lelttrt + seq1 T opr v cv ` + seq1 T opr w cv ` subclt cvgcmp3.2 v cv ser1cl cvgcmp3.2 w cv ser1cl syl2an + seq1 T opr v cv ` - + seq1 T opr w cv ` opr absclt syl z cv RR e. adantll + seq1 H opr v cv ` + seq1 H opr w cv ` resubclt H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an v cv ser1recl H NN RR x ffnfv cvgcmp3.8 cvgcmp3.9 cvgcmp3.2 x cv ffvrni T x cv ` absclt syl eqeltrd rgen mpbir2an w cv ser1recl syl2an recnd + seq1 H opr v cv ` - + seq1 H opr w cv ` opr absclt syl z cv RR e. adantll z cv RR e. v cv NN e. pm3.26 w cv NN e. adantr syl3anc w cv < v cv br adantr mpand exp an1rs a2d r19.20dva r19.22dva 0 < z cv br imim2d r19.20i ax-mp cvgcmp3.10 cvgcmp3.11 cvgcmp3.12 cvgcmp3.13 cvgcmp3.14 cvgcmp3.15 cvgcmp3.16 cvgcmp3.17 caucvg3aOLD))
+thm (cvgcmp3ceOLD ((A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (G x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (G y) (w z) (v z) (u z) (t z) (h z) (g z) (G z) (v w) (u w) (t w) (h w) (g w) (G w) (u v) (t v) (h v) (g v) (G v) (t u) (h u) (g u) (G u) (h t) (g t) (G t) (g h) (G h) (G g) (C x)) ((cvgcmp3ce.1 (e. A (V))) (cvgcmp3ce.2 (e. B (NN))) (cvgcmp3ce.3 (:--> F (NN) (RR))) (cvgcmp3ce.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3ce.5 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp3ce.6 (e. C (RR))) (cvgcmp3ce.7 (br (0) (<_) C)) (cvgcmp3ce.8 (:--> G (NN) (CC))) (cvgcmp3ce.9 (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))) (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y))) (cvgcmp3ce.3 cvgcmp3ce.8 cvgcmp3ce.4 cvgcmp3ce.1 cvgcmp3ce.5 abs G u cv ` fvex u t u cv NN e. t cv abs G u cv ` ` = /\ {<,>|} eqid fnopab2 u cv x cv G fveq2 abs fveq2d u t u cv NN e. t cv abs G u cv ` ` = /\ {<,>|} eqid abs G x cv ` fvex fvopab4 Re + seq1 G opr u cv ` fvex u t u cv NN e. t cv Re + seq1 G opr u cv ` ` = /\ {<,>|} eqid fnopab2 u cv x cv + seq1 G opr fveq2 Re fveq2d u t u cv NN e. t cv Re + seq1 G opr u cv ` ` = /\ {<,>|} eqid Re + seq1 G opr x cv ` fvex fvopab4 z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Re + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} eqid Im + seq1 G opr u cv ` fvex u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} eqid fnopab2 u cv x cv + seq1 G opr fveq2 Im fveq2d u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} eqid Im + seq1 G opr x cv ` fvex fvopab4 z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} eqid i x. u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} h cv ` oprex h g h cv NN e. g cv i x. u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} h cv ` opr = /\ {<,>|} eqid fnopab2 h cv x cv u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} fveq2 i x. opreq2d h g h cv NN e. g cv i x. u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} h cv ` opr = /\ {<,>|} eqid i x. u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} x cv ` oprex fvopab4 cvgcmp3ce.2 cvgcmp3ce.6 cvgcmp3ce.7 cvgcmp3ce.9 cvgcmp3cOLD z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Re + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} RR < sup + i x. z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} RR < sup opr oprex y cv z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Re + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} RR < sup + i x. z RR w NN v NN w cv <_ v cv br z cv < u t u cv NN e. t cv Im + seq1 G opr u cv ` ` = /\ {<,>|} v cv ` br -> A.e. E.e. {e.|} RR < sup opr opr + seq1 G opr ~~>OLD breq2 cla4ev ax-mp))
+thm (cvgcmp3cetlem1OLD ((A z) (x z) (B x) (B z) (F x) (F z) (x y) (G x) (y z) (G y) (G z) (C x) (C z) (ph y) (ph z)) ((cvgcmp3ce.1 (e. A (V))) (cvgcmp3ce.2 (e. B (NN))) (cvgcmp3ce.3 (:--> F (NN) (RR))) (cvgcmp3ce.4 (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (cvgcmp3ce.5 (br (opr (+) (seq1) F) (~~>OLD) A)) (cvgcmp3cetlem1.6 (<-> ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))))) (-> ph (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))) (G ph G NN 0 {} X. if + seq1 opreq2 ~~>OLD y cv breq1d y biexdv cvgcmp3ce.1 cvgcmp3ce.2 cvgcmp3ce.3 x cv z cv F fveq2 0 <_ breq2d cvgcmp3ce.4 vtoclga cvgcmp3ce.5 G ph G NN 0 {} X. if NN CC feq1 G ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ C x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d C RR e. 0 <_ C br /\ anbi2d cvgcmp3cetlem1.6 x cv z cv B < breq2 x cv z cv G fveq2 abs fveq2d x cv z cv F fveq2 C x. opreq2d <_ breq12d imbi12d NN cbvralv G NN CC :--> anbi2i C RR e. 0 <_ C br /\ anbi2i bitr syl5bb C ph C 0 if RR eleq1 C ph C 0 if 0 <_ breq2 anbi12d C ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d NN 0 {} X. ph G NN 0 {} X. if NN CC feq1 NN 0 {} X. ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ 0 x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d 0 RR e. 0 <_ 0 br /\ anbi2d 0 ph C 0 if RR eleq1 0 ph C 0 if 0 <_ breq2 anbi12d 0 ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti NN fconst 0cn 0 CC snssi ax-mp NN 0 {} X. NN 0 {} CC fss mp2an 0cn elisseti z cv NN fvconst2 abs fveq2d abs0 syl6eq cvgcmp3ce.3 z cv ffvrni recnd F z cv ` mul02t syl 0re leid syl5breqr eqbrtrd B < z cv br a1d rgen pm3.2i pm3.2i elimhyp2v pm3.26i pm3.26i G ph G NN 0 {} X. if NN CC feq1 G ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ C x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d C RR e. 0 <_ C br /\ anbi2d cvgcmp3cetlem1.6 x cv z cv B < breq2 x cv z cv G fveq2 abs fveq2d x cv z cv F fveq2 C x. opreq2d <_ breq12d imbi12d NN cbvralv G NN CC :--> anbi2i C RR e. 0 <_ C br /\ anbi2i bitr syl5bb C ph C 0 if RR eleq1 C ph C 0 if 0 <_ breq2 anbi12d C ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d NN 0 {} X. ph G NN 0 {} X. if NN CC feq1 NN 0 {} X. ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ 0 x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d 0 RR e. 0 <_ 0 br /\ anbi2d 0 ph C 0 if RR eleq1 0 ph C 0 if 0 <_ breq2 anbi12d 0 ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti NN fconst 0cn 0 CC snssi ax-mp NN 0 {} X. NN 0 {} CC fss mp2an 0cn elisseti z cv NN fvconst2 abs fveq2d abs0 syl6eq cvgcmp3ce.3 z cv ffvrni recnd F z cv ` mul02t syl 0re leid syl5breqr eqbrtrd B < z cv br a1d rgen pm3.2i pm3.2i elimhyp2v pm3.26i pm3.27i G ph G NN 0 {} X. if NN CC feq1 G ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ C x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d C RR e. 0 <_ C br /\ anbi2d cvgcmp3cetlem1.6 x cv z cv B < breq2 x cv z cv G fveq2 abs fveq2d x cv z cv F fveq2 C x. opreq2d <_ breq12d imbi12d NN cbvralv G NN CC :--> anbi2i C RR e. 0 <_ C br /\ anbi2i bitr syl5bb C ph C 0 if RR eleq1 C ph C 0 if 0 <_ breq2 anbi12d C ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d NN 0 {} X. ph G NN 0 {} X. if NN CC feq1 NN 0 {} X. ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ 0 x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d 0 RR e. 0 <_ 0 br /\ anbi2d 0 ph C 0 if RR eleq1 0 ph C 0 if 0 <_ breq2 anbi12d 0 ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti NN fconst 0cn 0 CC snssi ax-mp NN 0 {} X. NN 0 {} CC fss mp2an 0cn elisseti z cv NN fvconst2 abs fveq2d abs0 syl6eq cvgcmp3ce.3 z cv ffvrni recnd F z cv ` mul02t syl 0re leid syl5breqr eqbrtrd B < z cv br a1d rgen pm3.2i pm3.2i elimhyp2v pm3.27i pm3.26i G ph G NN 0 {} X. if NN CC feq1 G ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ C x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d C RR e. 0 <_ C br /\ anbi2d cvgcmp3cetlem1.6 x cv z cv B < breq2 x cv z cv G fveq2 abs fveq2d x cv z cv F fveq2 C x. opreq2d <_ breq12d imbi12d NN cbvralv G NN CC :--> anbi2i C RR e. 0 <_ C br /\ anbi2i bitr syl5bb C ph C 0 if RR eleq1 C ph C 0 if 0 <_ breq2 anbi12d C ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d NN 0 {} X. ph G NN 0 {} X. if NN CC feq1 NN 0 {} X. ph G NN 0 {} X. if z cv fveq1 abs fveq2d <_ 0 x. F z cv ` opr breq1d B < z cv br imbi2d z NN biraldv anbi12d 0 RR e. 0 <_ 0 br /\ anbi2d 0 ph C 0 if RR eleq1 0 ph C 0 if 0 <_ breq2 anbi12d 0 ph C 0 if x. F z cv ` opreq1 abs ph G NN 0 {} X. if z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv ph G NN 0 {} X. if NN CC :--> anbi2d anbi12d 0re 0re leid pm3.2i 0cn elisseti NN fconst 0cn 0 CC snssi ax-mp NN 0 {} X. NN 0 {} CC fss mp2an 0cn elisseti z cv NN fvconst2 abs fveq2d abs0 syl6eq cvgcmp3ce.3 z cv ffvrni recnd F z cv ` mul02t syl 0re leid syl5breqr eqbrtrd B < z cv br a1d rgen pm3.2i pm3.2i elimhyp2v pm3.27i pm3.27i rspec imp y cvgcmp3ceOLD dedth))
+thm (cvgcmp3cetlem2OLD ((x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (G x) (G y) (G z) (C x) (C y) (C z) (ph y) (ph z)) ((cvgcmp3cetlem2.1 (e. A (V))) (cvgcmp3cetlem2.2 (e. B (NN))) (cvgcmp3cetlem2.3 (<-> ph (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A))))) (-> (/\ ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))) (F ph F NN 0 {} X. if z cv fveq1 C x. opreq2d abs G z cv ` ` <_ breq2d B < z cv br imbi2d z NN biraldv G NN CC :--> anbi2d C RR e. 0 <_ C br /\ anbi2d y + seq1 G opr ~~>OLD y cv br E. imbi1d cvgcmp3cetlem2.1 0re elisseti ph keepel cvgcmp3cetlem2.2 F ph F NN 0 {} X. if NN RR feq1 F ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d F ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD A breq1d anbi12d cvgcmp3cetlem2.3 x cv z cv F fveq2 0 <_ breq2d NN cbvralv F NN RR :--> anbi2i + seq1 F opr ~~>OLD A br anbi1i bitr syl5bb A ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d NN 0 {} X. ph F NN 0 {} X. if NN RR feq1 NN 0 {} X. ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d NN 0 {} X. ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD 0 breq1d anbi12d 0 ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an 0re elisseti z cv NN fvconst2 eqcomd 0re leid syl5breq rgen pm3.2i ser1clim0OLD pm3.2i elimhyp2v pm3.26i pm3.26i F ph F NN 0 {} X. if NN RR feq1 F ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d F ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD A breq1d anbi12d cvgcmp3cetlem2.3 x cv z cv F fveq2 0 <_ breq2d NN cbvralv F NN RR :--> anbi2i + seq1 F opr ~~>OLD A br anbi1i bitr syl5bb A ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d NN 0 {} X. ph F NN 0 {} X. if NN RR feq1 NN 0 {} X. ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d NN 0 {} X. ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD 0 breq1d anbi12d 0 ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an 0re elisseti z cv NN fvconst2 eqcomd 0re leid syl5breq rgen pm3.2i ser1clim0OLD pm3.2i elimhyp2v pm3.26i pm3.27i rspec F ph F NN 0 {} X. if NN RR feq1 F ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d F ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD A breq1d anbi12d cvgcmp3cetlem2.3 x cv z cv F fveq2 0 <_ breq2d NN cbvralv F NN RR :--> anbi2i + seq1 F opr ~~>OLD A br anbi1i bitr syl5bb A ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d NN 0 {} X. ph F NN 0 {} X. if NN RR feq1 NN 0 {} X. ph F NN 0 {} X. if z cv fveq1 0 <_ breq2d z NN biraldv anbi12d NN 0 {} X. ph F NN 0 {} X. if + seq1 opreq2 ~~>OLD 0 breq1d anbi12d 0 ph A 0 if + seq1 ph F NN 0 {} X. if opr ~~>OLD breq2 ph F NN 0 {} X. if NN RR :--> z NN 0 <_ ph F NN 0 {} X. if z cv ` br A.e. /\ anbi2d 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an 0re elisseti z cv NN fvconst2 eqcomd 0re leid syl5breq rgen pm3.2i ser1clim0OLD pm3.2i elimhyp2v pm3.27i C RR e. 0 <_ C br /\ G NN CC :--> z NN B < z cv br abs G z cv ` ` <_ C x. ph F NN 0 {} X. if z cv ` opr br -> A.e. /\ /\ pm4.2 y cvgcmp3cetlem1OLD dedth x cv z cv B < breq2 x cv z cv G fveq2 abs fveq2d x cv z cv F fveq2 C x. opreq2d <_ breq12d imbi12d NN cbvralv G NN CC :--> anbi2i C RR e. 0 <_ C br /\ anbi2i syl5ib imp))
+thm (cvgcmp3cetOLD ((A y) (x y) (B x) (B y) (F x) (F y) (G x) (G y) (C x) (C y)) ((cvgcmp3cet.6 (e. A (V)))) (-> (/\ (e. B (NN)) (/\ (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)) (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))) (B B NN e. B 1 if < x cv breq1 abs G x cv ` ` <_ C x. F x cv ` opr br imbi1d x NN biraldv G NN CC :--> anbi2d C RR e. 0 <_ C br /\ anbi2d F NN RR :--> x NN 0 <_ F x cv ` br A.e. /\ + seq1 F opr ~~>OLD A br /\ anbi2d y + seq1 G opr ~~>OLD y cv br E. imbi1d cvgcmp3cet.6 1nn B elimel F NN RR :--> x NN 0 <_ F x cv ` br A.e. /\ + seq1 F opr ~~>OLD A br /\ pm4.2 C G y cvgcmp3cetlem2OLD dedth imp))
+thm (expcnvlem1 ((ph x) (x y) (A x) (A y)) ((expcnvlem1.1 (e. A (RR))) (expcnvlem1.2 (-> (e. (cv y) (NN)) (-> (br A (<) (cv y)) ph)))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) ph))) (expcnvlem1.1 A flclt ax-mp floor A ` nn0absclt ax-mp abs floor A ` ` nn0p1nnt ax-mp y cv nnret expcnvlem1.1 A flltp1t ax-mp expcnvlem1.1 A flclt ax-mp zre leabs expcnvlem1.1 A flclt ax-mp zre expcnvlem1.1 A flclt ax-mp floor A ` zcnt ax-mp abscl ax1re leadd1 mpbi expcnvlem1.1 expcnvlem1.1 A flclt ax-mp zre ax1re readdcl expcnvlem1.1 A flclt ax-mp floor A ` zcnt ax-mp abscl ax1re readdcl ltletr mp2an expcnvlem1.1 expcnvlem1.1 A flclt ax-mp floor A ` zcnt ax-mp abscl ax1re readdcl A abs floor A ` ` + 1 opr y cv ltletrt mp3an12 mpani syl ph imim1d r19.20i expcnvlem1.2 mprg x cv abs floor A ` ` + 1 opr <_ y cv breq1 ph imbi1d y NN biraldv NN rcla4ev mp2an))
+thm (expcnvlem2 () ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B))) (expcnvlem.3 (e. C (NN)))) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B)) (expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi 1 / B opr - 1 opr 1 / A opr - 1 opr C ltdivmult mpan2 mp3an expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl ltsubadd2 ax1re expcnvlem.2 pm3.26i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl expcnvlem.2 pm3.27i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i ltdiv23i 3bitr expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 0re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl expcnvlem.1 3simp1i expcnvlem.1 3simp2i recgt0i ltlei 1cn negid 1cn expcnvlem.1 3simp1i recn expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i reccl pncan3 3brtr4 ax1re renegcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl ax1re leadd2 mpbir 1 / A opr - 1 opr C bernneq mp3an 1cn expcnvlem.1 3simp1i recn expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i reccl pncan3 ^ C opreq1i expcnvlem.1 3simp1i recn expcnvlem.3 expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i A C recexpt mp3an eqtr breqtr ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl expcnvlem.1 3simp1i expcnvlem.3 A C reexpclt mp2an pm3.2i ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i expcnvlem.1 3simp1i expcnvlem.3 expcnvlem.1 3simp2i A C expgt0t mp3an pm3.2i 1 + 1 / A opr - 1 opr x. C opr opr A ^ C opr lerec2t mp2an mpbi expcnvlem.1 3simp1i expcnvlem.3 A C reexpclt mp2an ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl readdcl ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre remulcl lt01 expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.3 nnre expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi expcnvlem.3 nngt0 mulgt0i addgt0i gt0ne0i rereccl expcnvlem.2 pm3.26i lelttr mpan sylbi))
+thm (expcnvlem3 () ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B)))) (-> (e. C (NN)) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B))) (C C NN e. C 1 if 1 / B opr - 1 opr / 1 / A opr - 1 opr opr < breq2 C C NN e. C 1 if A ^ opreq2 < B breq1d imbi12d expcnvlem.1 expcnvlem.2 1nn C elimel expcnvlem2 dedth))
+thm (expcnvlem4 ((x y) (A x) (A y) (B x) (B y)) ((expcnvlem.1 (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1)))) (expcnvlem.2 (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B)))) (expcnvlem.2 pm3.26i expcnvlem.2 pm3.26i expcnvlem.2 pm3.27i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl ax1re resubcl expcnvlem.1 3simp3i expcnvlem.1 3simp1i expcnvlem.1 3simp2i A reclt1t mp2an mpbi ax1re expcnvlem.1 3simp1i expcnvlem.1 3simp1i expcnvlem.1 3simp2i gt0ne0i rereccl posdif mpbi gt0ne0i redivcl expcnvlem.1 expcnvlem.2 y cv expcnvlem3 x expcnvlem1))
+thm (expcnvlem5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))) (A A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if ^ y cv opreq1 < B breq1d x cv <_ y cv br imbi2d y NN biraldv x NN birexdv B B RR e. 0 < B br /\ B 1 if A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if ^ y cv opr < breq2 x cv <_ y cv br imbi2d y NN biraldv x NN birexdv A A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if RR eleq1 A A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if 0 < breq2 A A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if < 1 breq1 3anbi123d 1 / 2 opr A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if RR eleq1 1 / 2 opr A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if 0 < breq2 1 / 2 opr A RR e. 0 < A br A < 1 br /\/\ A 1 / 2 opr if < 1 breq1 3anbi123d 2re 2re 2pos gt0ne0i rereccl halfgt0 halflt1 3pm3.2i elimhyp B B RR e. 0 < B br /\ B 1 if RR eleq1 B B RR e. 0 < B br /\ B 1 if 0 < breq2 anbi12d 1 B RR e. 0 < B br /\ B 1 if RR eleq1 1 B RR e. 0 < B br /\ B 1 if 0 < breq2 anbi12d ax1re lt01 pm3.2i elimhyp x y expcnvlem4 dedth2h))
+thm (expcnvlem6 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<_) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))) (0re 0 A leloet mpan A B x y expcnvlem5 3exp1 1nn 0 < B br a1i 0 A = a1i y cv 0expt < B breq1d 0 A = adantl 0 A ^ y cv opreq1 < B breq1d y cv NN e. adantr bitr3d biimpd exp 1 <_ y cv br a1d com24 r19.21adv jcad x cv 1 <_ y cv breq1 A ^ y cv opr < B br imbi1d y NN biraldv NN rcla4ev syl6 B RR e. adantld A < 1 br a1d A RR e. a1i jaod sylbid 3imp1))
+thm (expcnvOLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((expcnv.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br F (~~>OLD) (0))) (abs A ` z cv w v expcnvlem6 A absclt abs A ` < 1 br adantr A absge0t abs A ` < 1 br adantr A CC e. abs A ` < 1 br pm3.27 3jca sylan x cv v cv A ^ opreq2 expcnv.1 A ^ v cv oprex fvopab4 - 0 opreq1d A CC e. adantl A v cv expclt A ^ v cv opr subid1t syl eqtrd abs fveq2d A v cv absexpt eqtrd < z cv breq1d w cv <_ v cv br imbi2d biraldva w NN birexdv abs A ` < 1 br z cv RR e. 0 < z cv br /\ ad2antll mpbird exp32 r19.21aiv A v cv expclt x cv v cv A ^ opreq2 expcnv.1 A ^ v cv oprex fvopab4 CC eleq1d A CC e. adantl mpbird exp r19.21aiv A ^ x cv oprex expcnv.1 fnopab2 jctil F NN CC v ffnfv sylibr 0cn F 0 z w v clim2OLD mpan2 syl abs A ` < 1 br adantr mpbird))
+thm (geoser ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z) (F w)) ((geoser.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (geoser.2 (e. A (CC)))) (-> (/\ (e. B (NN)) (=/= A (1))) (= (` (opr (+) (seq1) F) B) (opr (opr A (x.) (opr (1) (-) (opr A (^) B))) (/) (opr (1) (-) A)))) (1cn geoser.2 subcl 1 - A opr + seq1 F opr B ` divcan4t mp3an1 geoser.1 geoser.2 A x cv expclt mpan rgen B ser1cl2 geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi biimp syl2an geoser.2 mulid1 B NN e. a1i geoser.2 A B expclt mpan geoser.2 A A ^ B opr axmulcom mpan syl geoser.2 A B expp1t mpan eqtr4d - opreq12d geoser.2 A B expclt mpan geoser.2 1cn A 1 A ^ B opr subdit mp3an12 syl z cv 1 + seq1 F opr fveq2 x. 1 - A opr opreq1d z cv 1 + 1 opreq1 A ^ opreq2d A - opreq2d eqeq12d z cv w cv + seq1 F opr fveq2 x. 1 - A opr opreq1d z cv w cv + 1 opreq1 A ^ opreq2d A - opreq2d eqeq12d z cv w cv + 1 opr + seq1 F opr fveq2 x. 1 - A opr opreq1d z cv w cv + 1 opr + 1 opreq1 A ^ opreq2d A - opreq2d eqeq12d z cv B + seq1 F opr fveq2 x. 1 - A opr opreq1d z cv B + 1 opreq1 A ^ opreq2d A - opreq2d eqeq12d geoser.1 A ^ 1 oprex x cv 1 A ^ opreq2 ser112 geoser.2 A exp1t ax-mp eqtr x. 1 - A opr opreq1i geoser.2 1cn geoser.2 subdi geoser.2 mulid1 geoser.2 sqval df-2 A ^ opreq2i eqtr3 - opreq12i 3eqtr geoser.1 A ^ w cv + 1 opr oprex x cv w cv + 1 opr A ^ opreq2 ser1suc2 x. 1 - A opr opreq1d 1cn geoser.2 subcl + seq1 F opr w cv ` A ^ w cv + 1 opr opr 1 - A opr adddirt mp3an3 geoser.1 geoser.2 A x cv expclt mpan rgen w cv ser1cl2 w cv peano2nn geoser.2 A w cv + 1 opr expclt mpan syl sylanc w cv peano2nn geoser.2 A w cv + 1 opr expclt mpan 1cn geoser.2 A ^ w cv + 1 opr opr 1 A subdit mp3an23 3syl w cv peano2nn geoser.2 A w cv + 1 opr expclt mpan A ^ w cv + 1 opr opr ax1id 3syl w cv peano2nn geoser.2 A w cv + 1 opr expp1t mpan syl eqcomd - opreq12d eqtrd + seq1 F opr w cv ` x. 1 - A opr opr + opreq2d 3eqtrd + seq1 F opr w cv ` x. 1 - A opr opr A - A ^ w cv + 1 opr opr opr = adantr + seq1 F opr w cv ` x. 1 - A opr opr A - A ^ w cv + 1 opr opr opr + A ^ w cv + 1 opr opr - A ^ w cv + 1 opr + 1 opr opr opr opreq1 w cv NN e. adantl geoser.2 A A ^ w cv + 1 opr opr A ^ w cv + 1 opr + 1 opr opr npncant mp3an1 w cv peano2nn geoser.2 A w cv + 1 opr expclt mpan syl w cv peano2nn w cv + 1 opr peano2nn geoser.2 A w cv + 1 opr + 1 opr expclt mpan 3syl sylanc + seq1 F opr w cv ` x. 1 - A opr opr A - A ^ w cv + 1 opr opr opr = adantr 3eqtrd exp nnind 3eqtr4rd / 1 - A opr opreq1d A 1 =/= adantr eqtr3d))
+thm (geolimilemOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z) (G z)) ((geoser.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (geoser.2 (e. A (CC))) (geolimilem.3 (br (` (abs) A) (<) (1))) (geolimilem.4 (Fn G (NN))) (geolimilem.5 (-> (e. (cv z) (NN)) (= (` G (cv z)) (opr (-u (opr A (/) (opr (1) (-) A))) (x.) (` F (cv z))))))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A))) (geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl 0cn elisseti geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl negcl 0cn elisseti geoser.2 geolimilem.3 geoser.1 expcnvOLD mp2an geolimilem.4 geolimilem.5 climmulcOLD geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl negcl mul01 breqtr addex geoser.1 nnex x y A ^ x cv opr funopabex2 eqeltr seq1fn geoser.2 A z cv expclt mpan geoser.2 1cn A 1 A ^ z cv opr subdit mp3an12 syl geoser.2 mulid1 - A x. A ^ z cv opr opr opreq1i syl6eq / 1 - A opr opreq1d geoser.2 A z cv expclt mpan geoser.2 A A ^ z cv opr axmulcl mpan geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi A A x. A ^ z cv opr opr 1 - A opr divsubdirt mpan2 mp3an13 3syl geoser.2 A z cv expclt mpan geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi A A ^ z cv opr 1 - A opr div23t mpan2 mp3an13 syl A / 1 - A opr opr - opreq2d eqtrd geoser.2 A z cv expclt mpan geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl A / 1 - A opr opr A ^ z cv opr axmulcl mpan geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl A / 1 - A opr opr A / 1 - A opr opr x. A ^ z cv opr opr subnegt mpan 3syl 3eqtrd geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.1 geoser.2 z cv geoser mpan2 geolimilem.5 x cv z cv A ^ opreq2 geoser.1 A ^ z cv oprex fvopab4 A / 1 - A opr opr -u x. opreq2d geoser.2 A z cv expclt mpan geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl A / 1 - A opr opr A ^ z cv opr mulneg1t mpan syl 3eqtrd A / 1 - A opr opr + opreq2d 3eqtr4d climaddcxOLD geoser.2 1cn geoser.2 subcl geolimilem.3 geoser.2 abscl ax1re ltne ax-mp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq mto A 1 df-ne mpbir geoser.2 subid 1 - A opr eqeq1i geoser.2 1cn geoser.2 subcan2 0 1 - A opr eqcom 3bitr3 eqneqi mpbi divcl addid1 breqtr))
+thm (geolimiOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z)) ((geoser.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (geoser.2 (e. A (CC))) (geolimi.3 (br (` (abs) A) (<) (1)))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A))) (geoser.1 geoser.2 geolimi.3 A / 1 - A opr opr -u x. A ^ x cv opr oprex x y x cv NN e. y cv A / 1 - A opr opr -u x. A ^ x cv opr opr = /\ {<,>|} eqid fnopab2 x cv z cv A ^ opreq2 A / 1 - A opr opr -u x. opreq2d x y x cv NN e. y cv A / 1 - A opr opr -u x. A ^ x cv opr opr = /\ {<,>|} eqid A / 1 - A opr opr -u x. A ^ z cv opr oprex fvopab4 x cv z cv A ^ opreq2 geoser.1 A ^ z cv oprex fvopab4 A / 1 - A opr opr -u x. opreq2d eqtr4d geolimilemOLD))
+thm (geolimOLD ((x y) (A x) (A y)) ((geolim.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A)))) (A A CC e. abs A ` < 1 br /\ A 0 if ^ x cv opreq1 y cv eqeq2d x cv NN e. anbi2d x y biopabdv + seq1 opreq2d A A CC e. abs A ` < 1 br /\ A 0 if = id A A CC e. abs A ` < 1 br /\ A 0 if 1 - opreq2 / opreq12d ~~>OLD breq12d x y x cv NN e. y cv A CC e. abs A ` < 1 br /\ A 0 if ^ x cv opr = /\ {<,>|} eqid A A CC e. abs A ` < 1 br /\ A 0 if CC eleq1 A A CC e. abs A ` < 1 br /\ A 0 if abs fveq2 < 1 breq1d anbi12d 0 A CC e. abs A ` < 1 br /\ A 0 if CC eleq1 0 A CC e. abs A ` < 1 br /\ A 0 if abs fveq2 < 1 breq1d anbi12d 0cn abs0 lt01 eqbrtr pm3.2i elimhyp pm3.26i A A CC e. abs A ` < 1 br /\ A 0 if CC eleq1 A A CC e. abs A ` < 1 br /\ A 0 if abs fveq2 < 1 breq1d anbi12d 0 A CC e. abs A ` < 1 br /\ A 0 if CC eleq1 0 A CC e. abs A ` < 1 br /\ A 0 if abs fveq2 < 1 breq1d anbi12d 0cn abs0 lt01 eqbrtr pm3.2i elimhyp pm3.27i geolimiOLD dedth geolim.1 + seq1 opreq2i syl5eqbr))
+thm (geolimcilemOLD ((G z) (x y) (A x) (A y) (A z) (x z) (B x) (y z) (B y) (B z) (F z)) ((geolimci.1 (e. A (CC))) (geolimci.2 (e. B (CC))) (geolimci.3 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x)))))))) (geolimcilem.5 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))))) (-> (br (` (abs) A) (<) (1)) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))) (geolimci.1 geolimcilem.5 geolimOLD mpan geolimci.2 A / 1 - A opr oprex addex geolimci.3 nnex x y B x. A ^ x cv opr opr funopabex2 eqeltr seq1fn geolimci.2 geolimcilem.5 geolimci.1 A x cv expclt mpan fopab geolimci.3 nnex x y B x. A ^ x cv opr opr funopabex2 eqeltr x cv z cv A ^ opreq2 B x. opreq2d geolimci.3 B x. A ^ z cv opr oprex fvopab4 x cv z cv A ^ opreq2 geolimcilem.5 A ^ z cv oprex fvopab4 B x. opreq2d eqcomd eqtrd z cv ser1mulc climmulctOLD mpan syl))
+thm (geolimciOLD ((x y) (A x) (A y) (B x) (B y)) ((geolimci.1 (e. A (CC))) (geolimci.2 (e. B (CC))) (geolimci.3 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x))))))))) (-> (br (` (abs) A) (<) (1)) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))) (geolimci.1 geolimci.2 geolimci.3 x y x cv NN e. y cv A ^ x cv opr = /\ {<,>|} eqid geolimcilemOLD))
+thm (geolimcOLD ((x y) (A x) (A y) (B x) (B y)) ((geolimc.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x))))))))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))) (A A CC e. A 0 if abs fveq2 < 1 breq1d A A CC e. A 0 if ^ x cv opreq1 B x. opreq2d y cv eqeq2d x cv NN e. anbi2d x y biopabdv + seq1 opreq2d A A CC e. A 0 if = id A A CC e. A 0 if 1 - opreq2 / opreq12d B x. opreq2d ~~>OLD breq12d imbi12d B B CC e. B 0 if x. A CC e. A 0 if ^ x cv opr opreq1 y cv eqeq2d x cv NN e. anbi2d x y biopabdv + seq1 opreq2d B B CC e. B 0 if x. A CC e. A 0 if / 1 - A CC e. A 0 if opr opr opreq1 ~~>OLD breq12d abs A CC e. A 0 if ` < 1 br imbi2d 0cn A elimel 0cn B elimel x y x cv NN e. y cv B CC e. B 0 if x. A CC e. A 0 if ^ x cv opr opr = /\ {<,>|} eqid geolimciOLD dedth2h geolimc.1 + seq1 opreq2i ~~>OLD B x. A / 1 - A opr opr opr breq1i syl6ibr exp 3imp))
+thm (cvgratlem1 ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((cvgratlem1.1 (:--> F (NN) (RR))) (cvgratlem1.2 (e. A (RR))) (cvgratlem1.3 (e. B (NN))) (cvgratlem1.4 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))) (cvgratlem1.4 y cv 1 B + opreq2 F fveq2d y cv 1 A ^ opreq2 x. F B ` opreq1d < breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ imbi2d y cv z cv B + opreq2 F fveq2d y cv z cv A ^ opreq2 x. F B ` opreq1d < breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ imbi2d y cv z cv + 1 opr B + opreq2 F fveq2d y cv z cv + 1 opr A ^ opreq2 x. F B ` opreq1d < breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ imbi2d y cv C B + opreq2 F fveq2d y cv C A ^ opreq2 x. F B ` opreq1d < breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ imbi2d cvgratlem1.3 nnre leid cvgratlem1.3 x cv B B <_ breq2 x cv B + 1 opreq1 F fveq2d x cv B F fveq2 A x. opreq2d < breq12d imbi12d NN rcla4v ax-mp mpi cvgratlem1.2 recn A exp1t ax-mp x. F B ` opreq1i syl6breqr 0 < A br adantl F B + z cv opr + 1 opr ` A x. F B + z cv opr ` opr A x. A ^ z cv opr x. F B ` opr opr axlttrn cvgratlem1.3 B z cv nnaddclt mpan B + z cv opr peano2nn syl cvgratlem1.1 B + z cv opr + 1 opr ffvrni syl cvgratlem1.3 B z cv nnaddclt mpan cvgratlem1.1 B + z cv opr ffvrni syl cvgratlem1.2 A F B + z cv opr ` axmulrcl mpan syl cvgratlem1.2 A z cv reexpclt mpan cvgratlem1.3 cvgratlem1.1 B ffvrni ax-mp A ^ z cv opr F B ` axmulrcl mpan2 syl cvgratlem1.2 A A ^ z cv opr x. F B ` opr axmulrcl mpan syl syl3anc z cv nnnn0t cvgratlem1.3 nnre B z cv nn0addge1t mpan syl cvgratlem1.3 B z cv nnaddclt mpan x cv B + z cv opr B <_ breq2 x cv B + z cv opr + 1 opreq1 F fveq2d x cv B + z cv opr F fveq2 A x. opreq2d < breq12d imbi12d NN rcla4v syl mpid F B + z cv opr ` A ^ z cv opr x. F B ` opr A ltmul2t biimpd exp cvgratlem1.3 B z cv nnaddclt mpan cvgratlem1.1 B + z cv opr ffvrni syl cvgratlem1.2 A z cv reexpclt mpan cvgratlem1.3 cvgratlem1.1 B ffvrni ax-mp A ^ z cv opr F B ` axmulrcl mpan2 syl cvgratlem1.2 z cv NN e. a1i syl3anc imp3a syl2and exp4d com23 imp32 z cv nncnt cvgratlem1.3 nncn 1cn B z cv 1 axaddass mp3an13 syl F fveq2d cvgratlem1.2 recn A z cv expp1t mpan cvgratlem1.2 recn A z cv expclt mpan cvgratlem1.2 recn A ^ z cv opr A axmulcom mpan2 syl eqtrd x. F B ` opreq1d cvgratlem1.2 recn A z cv expclt mpan cvgratlem1.2 recn cvgratlem1.3 cvgratlem1.1 B ffvrni ax-mp recn A A ^ z cv opr F B ` axmulass mp3an13 syl eqtr2d < breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ adantr sylibd exp a2d nnind ax-mp cvgratlem1.3 cvgratlem1.4 B C nnaddclt mp2an cvgratlem1.1 B + C opr ffvrni ax-mp cvgratlem1.2 cvgratlem1.4 A C reexpclt mp2an cvgratlem1.3 cvgratlem1.1 B ffvrni ax-mp remulcl ltle syl))
+thm (cvgratlem2 ((A x) (B x) (F x)) ((cvgratlem1.1 (:--> F (NN) (RR))) (cvgratlem1.2 (e. A (RR))) (cvgratlem1.3 (e. B (NN))) (cvgratlem1.4 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (-> (br B (<) C) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem1.3 cvgratlem1.4 nnsub C - B opr C - B opr NN e. C - B opr 1 if B + opreq2 F fveq2d C - B opr C - B opr NN e. C - B opr 1 if A ^ opreq2 x. F B ` opreq1d <_ breq12d 0 < A br x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ imbi2d cvgratlem1.1 cvgratlem1.2 cvgratlem1.3 1nn C - B opr elimel x cvgratlem1 dedth cvgratlem1.3 nncn cvgratlem1.4 nncn pncan3 F fveq2i <_ A ^ C - B opr opr x. F B ` opr breq1i syl6ib sylbi impcom cvgratlem1.2 recn cvgratlem1.4 cvgratlem1.3 3pm3.2i A C B expsubt mpan cvgratlem1.2 gt0ne0 sylan x. F B ` opreq1d cvgratlem1.2 cvgratlem1.3 A B expgt0t mp3an12 cvgratlem1.2 cvgratlem1.3 A B reexpclt mp2an gt0ne0 cvgratlem1.3 cvgratlem1.1 B ffvrni ax-mp recn cvgratlem1.2 cvgratlem1.3 A B reexpclt mp2an recn cvgratlem1.2 recn cvgratlem1.4 A C expclt mp2an 3pm3.2i F B ` A ^ B opr A ^ C opr div13t mpan 3syl B < C br adantr eqtr4d x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. adantlr breqtrd exp))
+thm (cvgratlem3 ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((cvgratlem3.1 (:--> F (NN) (CC))) (cvgratlem3.2 (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y)))))))) (cvgratlem3.3 (e. A (RR))) (cvgratlem3.4 (e. B (NN))) (cvgratlem3.5 (e. C (NN)))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x)))))))) (-> (br B (<) C) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem3.2 cvgratlem3.1 y cv ffvrni F y cv ` absclt syl fopab cvgratlem3.3 cvgratlem3.4 cvgratlem3.5 x cvgratlem2 imp cvgratlem3.5 y cv C F fveq2 abs fveq2d cvgratlem3.2 abs F C ` fvex fvopab4 ax-mp cvgratlem3.4 y cv B F fveq2 abs fveq2d cvgratlem3.2 abs F B ` fvex fvopab4 ax-mp / A ^ B opr opreq1i x. A ^ C opr opreq1i 3brtr3g exp x cv peano2nn y cv x cv + 1 opr F fveq2 abs fveq2d cvgratlem3.2 abs F x cv + 1 opr ` fvex fvopab4 syl y cv x cv F fveq2 abs fveq2d cvgratlem3.2 abs F x cv ` fvex fvopab4 A x. opreq2d < breq12d B <_ x cv br imbi2d birala sylan2br))
+thm (cvgratlem1OLD ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((cvgratlem1OLD.1 (:--> F (NN) (RR)))) (-> (/\ (e. C (NN)) (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))) (y cv 1 B + opreq2 F fveq2d y cv 1 A ^ opreq2 x. F B ` opreq1d < breq12d A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ /\ imbi2d y cv z cv B + opreq2 F fveq2d y cv z cv A ^ opreq2 x. F B ` opreq1d < breq12d A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ /\ imbi2d y cv z cv + 1 opr B + opreq2 F fveq2d y cv z cv + 1 opr A ^ opreq2 x. F B ` opreq1d < breq12d A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ /\ imbi2d y cv C B + opreq2 F fveq2d y cv C A ^ opreq2 x. F B ` opreq1d < breq12d A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ /\ imbi2d B nnret B leidt syl x cv B B <_ breq2 x cv B + 1 opreq1 F fveq2d x cv B F fveq2 A x. opreq2d < breq12d imbi12d NN rcla4v mpid imp A RR e. 0 < A br /\ adantl A recnt A exp1t syl 0 < A br adantr x. F B ` opreq1d B NN e. x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. /\ adantr breqtrrd F B + z cv opr + 1 opr ` A x. F B + z cv opr ` opr A x. A ^ z cv opr x. F B ` opr opr axlttrn B z cv nnaddclt ancoms B + z cv opr peano2nn syl cvgratlem1OLD.1 B + z cv opr + 1 opr ffvrni syl A RR e. adantll A F B + z cv opr ` axmulrcl B z cv nnaddclt ancoms cvgratlem1OLD.1 B + z cv opr ffvrni syl sylan2 anassrs A A ^ z cv opr x. F B ` opr axmulrcl A RR e. z cv NN e. pm3.26 B NN e. adantr A ^ z cv opr F B ` axmulrcl A z cv reexpclt cvgratlem1OLD.1 B ffvrni syl2an sylanc syl3anc B z cv nn0addge1t B nnret z cv nnnn0t syl2an B z cv nnaddclt x cv B + z cv opr B <_ breq2 x cv B + z cv opr + 1 opreq1 F fveq2d x cv B + z cv opr F fveq2 A x. opreq2d < breq12d imbi12d NN rcla4v syl mpid ancoms A RR e. adantll F B + z cv opr ` A ^ z cv opr x. F B ` opr A ltmul2t biimpd exp B z cv nnaddclt ancoms A RR e. adantll cvgratlem1OLD.1 B + z cv opr ffvrni syl A ^ z cv opr F B ` axmulrcl A z cv reexpclt cvgratlem1OLD.1 B ffvrni syl2an A RR e. z cv NN e. pm3.26 B NN e. adantr syl3anc imp3a syl2and exp4d exp31 imp4a com12 com34 imp44 1cn B z cv 1 axaddass mp3an3 B nncnt z cv nncnt syl2an F fveq2d A CC e. adantll A z cv expp1t A ^ z cv opr A axmulcom A z cv expclt A CC e. z cv NN e. pm3.26 sylanc eqtrd B NN e. adantlr x. F B ` opreq1d A A ^ z cv opr F B ` axmulass 3expa A CC e. z cv NN e. pm3.26 A z cv expclt jca cvgratlem1OLD.1 B ffvrni recnd syl2an an1rs eqtr2d < breq12d A recnt sylanl1 ancoms 0 < A br adantrlr x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. adantrrr sylibd exp a2d nnind imp F B + C opr ` A ^ C opr x. F B ` opr ltlet B C nnaddclt ancoms cvgratlem1OLD.1 B + C opr ffvrni syl A RR e. adantrl A ^ C opr F B ` axmulrcl A C reexpclt ancoms cvgratlem1OLD.1 B ffvrni syl2an anasss sylanc 0 < A br adantrlr x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. adantrrr mpd))
+thm (cvgratlem2OLD ((A x) (B x) (F x)) ((cvgratlem1OLD.1 (:--> F (NN) (RR)))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (B C nnsubt cvgratlem1OLD.1 C - B opr A B x cvgratlem1OLD exp45 com3r imp4d C NN e. adantr B C pncan3t B nncnt C nncnt syl2an F fveq2d <_ A ^ C - B opr opr x. F B ` opr breq1d sylibd exp3a sylbid exp imp3a exp4a com12 com4l com12 imp42 A C B expsubt A recnt 0 < A br adantr B NN e. C NN e. B < C br /\ ad2antll C NN e. B < C br pm3.26 A RR e. 0 < A br /\ B NN e. /\ adantl A RR e. 0 < A br /\ B NN e. pm3.27 C NN e. B < C br /\ adantr 3jca A gt0ne0t B NN e. adantr C NN e. B < C br pm3.27 anim12i sylanc x. F B ` opreq1d F B ` A ^ B opr A ^ C opr div13t cvgratlem1OLD.1 B ffvrni recnd A RR e. 0 < A br /\ C NN e. ad2antlr A B reexpclt recnd 0 < A br adantlr C NN e. adantr A C reexpclt recnd 0 < A br adantlr B NN e. adantlr 3jca A ^ B opr gt0ne0t A B reexpclt 0 < A br adantlr A B expgt0t 3expa an1rs sylanc C NN e. adantr sylanc B < C br adantrr eqtr4d x NN B <_ x cv br F x cv + 1 opr ` < A x. F x cv ` opr br -> A.e. adantlrr breqtrd exp))
+thm (cvgratlem3OLD ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((cvgratlem3OLD.1 (:--> F (NN) (CC))) (cvgratlem3OLD.2 (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y))))))))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))) (cvgratlem3OLD.2 cvgratlem3OLD.1 y cv ffvrni F y cv ` absclt syl fopab A B x C cvgratlem2OLD imp y cv C F fveq2 abs fveq2d cvgratlem3OLD.2 abs F C ` fvex fvopab4 B < C br adantr A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br G x cv + 1 opr ` < A x. G x cv ` opr br -> A.e. /\ /\ adantl y cv B F fveq2 abs fveq2d cvgratlem3OLD.2 abs F B ` fvex fvopab4 / A ^ B opr opreq1d x. A ^ C opr opreq1d x NN B <_ x cv br G x cv + 1 opr ` < A x. G x cv ` opr br -> A.e. adantr A RR e. 0 < A br /\ adantl C NN e. B < C br /\ adantr 3brtr3d exp x cv peano2nn y cv x cv + 1 opr F fveq2 abs fveq2d cvgratlem3OLD.2 abs F x cv + 1 opr ` fvex fvopab4 syl y cv x cv F fveq2 abs fveq2d cvgratlem3OLD.2 abs F x cv ` fvex fvopab4 A x. opreq2d < breq12d B <_ x cv br imbi2d birala B NN e. anbi2i bicomi sylan2b))
+thm (cvgratlem4 ((A x) (B x) (F x)) ((cvgrat.1 (:--> F (NN) (CC)))) (-> (/\ (e. A (RR)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (br (0) (<) A)) (B nnret B leidt syl x cv B B <_ breq2 x cv B + 1 opreq1 F fveq2d abs fveq2d x cv B F fveq2 abs fveq2d A x. opreq2d < breq12d imbi12d NN rcla4v mpid A RR e. adantl B peano2nn cvgrat.1 B + 1 opr ffvrni F B + 1 opr ` absge0t 3syl A RR e. adantl 0re 0 abs F B + 1 opr ` ` A x. abs F B ` ` opr lelttrt mp3an1 B peano2nn cvgrat.1 B + 1 opr ffvrni F B + 1 opr ` absclt 3syl A RR e. adantl A abs F B ` ` axmulrcl cvgrat.1 B ffvrni F B ` absclt syl sylan2 sylanc mpand cvgrat.1 B ffvrni F B ` absge0t syl A RR e. adantl jctild A abs F B ` ` prodgt02t exp cvgrat.1 B ffvrni F B ` absclt syl sylan2 syld syld exp imp32))
+thm (cvgratlem5OLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (u x) (B x) (u y) (B y) (u v) (B v) (B u) (F x) (F y) (F v) (F u) (G y) (G v)) ((cvgrat.1 (:--> F (NN) (CC))) (cvgratlem5.2 (= G ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr A (^) (cv z)))))))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>OLD) (cv y)))) (A / 1 - A opr oprex B G v abs F B ` ` / A ^ B opr opr F y cvgcmp3cetOLD B NN e. x NN B <_ x cv br abs F x cv + 1 opr ` ` < A x. abs F x cv ` ` opr br -> A.e. pm3.26 A RR e. A < 1 br /\ adantl cvgrat.1 A B x cvgratlem4 A < 1 br adantlr 0re 0 A ltlet mpan imp A < 1 br adantlr A z cv reexpclt exp r19.21aiv cvgratlem5.2 RR fopab2 sylib 0 <_ A br adantr A v cv expge0t 3expa an1rs z cv v cv A ^ opreq2 cvgratlem5.2 A ^ v cv oprex fvopab4 A RR e. 0 <_ A br /\ adantl breqtrrd exp r19.21aiv jca A < 1 br adantlr cvgratlem5.2 geolimOLD A recnt A < 1 br 0 <_ A br ad2antll A absidt A < 1 br adantlr A RR e. A < 1 br pm3.27 0 <_ A br adantr eqbrtrd sylanc jca syldan B NN e. x NN B <_ x cv br abs F x cv + 1 opr ` ` < A x. abs F x cv ` ` opr br -> A.e. /\ adantr abs F B ` ` A ^ B opr redivclt cvgrat.1 B ffvrni F B ` absclt syl A RR e. 0 < A br 3ad2ant2 A B reexpclt 0 < A br 3adant3 A gt0ne0t B NN e. 3adant2 A B expne0t A recnt sylan 0 < A br 3adant3 mpbid syl3anc abs F B ` ` A ^ B opr divge0t cvgrat.1 B ffvrni F B ` absclt syl A RR e. 0 < A br 3ad2ant2 A B reexpclt 0 < A br 3adant3 jca cvgrat.1 B ffvrni F B ` absge0t syl A RR e. 0 < A br 3ad2ant2 A B expgt0t jca sylanc jca 3expa an1rs x NN B <_ x cv br abs F x cv + 1 opr ` ` < A x. abs F x cv ` ` opr br -> A.e. adantrr cvgrat.1 y u y cv NN e. u cv abs F y cv ` ` = /\ {<,>|} eqid A B x v cv cvgratlem3OLD imp z cv v cv A ^ opreq2 cvgratlem5.2 A ^ v cv oprex fvopab4 abs F B ` ` / A ^ B opr opr x. opreq2d A RR e. 0 < A br /\ B NN e. x NN B <_ x cv br abs F x cv + 1 opr ` ` < A x. abs F x cv ` ` opr br -> A.e. /\ /\ B < v cv br ad2antrl breqtrrd exp32 r19.21aiv cvgrat.1 jctil jca A < 1 br adantllr jca exp31 com23 imp mpd sylanc))
+thm (cvgratOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (F x) (F y)) ((cvgrat.1 (:--> F (NN) (CC)))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>OLD) (cv y)))) (cvgrat.1 z w z cv NN e. w cv A ^ z cv opr = /\ {<,>|} eqid B x y cvgratlem5OLD))
+thm (isumvalt ((f n) (n x) (n y) (F n) (f x) (f y) (F f) (x y) (F x) (F y) (N n) (N f) (N x) (N y)) () (-> (/\ (e. N A) (e. F B)) (= (opr N (sumoo) F) (U. ({|} x (br (opr (<,> N (+)) (seq) F) (~~>) (cv x)))))) (x CC abid2 axcnex eqeltr x visset x cv V N + <,> seq F opr climcl mpan x ss2abi ssexi uniex n cv N + opeq1 seq f cv opreq1d ~~> x cv breq1d x biabdv unieqd f cv F N + <,> seq opreq2 ~~> x cv breq1d x biabdv unieqd n f y x df-isum A B oprabval5))
+thm (isumclt ((x y) (F x) (F y) (N x) (N y)) ((isumclt.1 (e. F (V))) (isumclt.2 (e. N (ZZ)))) (-> (E. x (br (opr (<,> N (+)) (seq) F) (~~>) (cv x))) (e. (opr N (sumoo) F) (CC))) (x visset x cv V N + <,> seq F opr climcl mpan ancri x 19.22i x visset y visset N + <,> seq F opr climuni y cv CC e. adantrl x cv CC e. adantll x y gen2 jctir x CC N + <,> seq F opr ~~> x cv br df-reu x cv y cv CC eleq1 x cv y cv N + <,> seq F opr ~~> breq2 anbi12d eu4 bitr sylibr x CC N + <,> seq F opr ~~> x cv br reucl syl x visset x cv V N + <,> seq F opr climcl mpan pm4.71ri x biabi x CC N + <,> seq F opr ~~> x cv br df-rab eqtr4 unieqi CC eleq1i sylibr isumclt.2 isumclt.1 N ZZ F V x isumvalt mp2an CC eleq1i sylibr))
+thm (sumvaltOLD ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (e. F A) (= (` (sum1oo) F) (U. ({|} x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))))) (x CC abid2 axcnex eqeltr x visset + seq1 F opr climclOLD x ss2abi ssexi uniex y cv F + seq1 opreq2 ~~>OLD x cv breq1d x biabdv unieqd A V z fvopabg mpan2 y z x df-sumOLD F fveq1i syl5eq))
+thm (sumcltOLD ((x y) (F x) (F y)) () (-> (e. F A) (-> (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x))) (e. (` (sum1oo) F) (CC)))) (F A x sumvaltOLD CC eleq1d x visset + seq1 F opr climclOLD ancri x 19.22i x visset y visset + seq1 F opr climuniOLD y cv CC e. adantrl x cv CC e. adantll x y gen2 jctir x CC + seq1 F opr ~~>OLD x cv br df-reu x cv y cv CC eleq1 x cv y cv + seq1 F opr ~~>OLD breq2 anbi12d eu4 bitr sylibr x CC + seq1 F opr ~~>OLD x cv br reucl syl x visset + seq1 F opr climclOLD pm4.71ri x biabi x CC + seq1 F opr ~~>OLD x cv br df-rab eqtr4 unieqi CC eleq1i sylibr syl5bir))
+thm (sumval2tOLD ((F x) (A x)) () (-> (/\/\ (e. F B) (e. A C) (br (opr (+) (seq1) F) (~~>OLD) A)) (= (` (sum1oo) F) A)) (F B x sumvaltOLD x visset + seq1 F opr climclOLD pm4.71ri x biabi x CC + seq1 F opr ~~>OLD x cv br df-rab eqtr4 unieqi syl6eq A C e. + seq1 F opr ~~>OLD A br 3ad2ant1 A C e. + seq1 F opr ~~>OLD A br pm3.27 x cv A + seq1 F opr ~~>OLD breq2 CC reuuni2 A C + seq1 F opr climcltOLD A C + seq1 F opr climcltOLD exp impac x cv A + seq1 F opr ~~>OLD breq2 CC rcla4ev syl x + seq1 F opr climmoOLD x cv CC e. moani jctir x CC + seq1 F opr ~~>OLD x cv br reu5 sylibr sylanc mpbid F B e. 3adant1 eqtrd))
+thm (cvgsumtOLD ((x y) (F x) (F y) (A y)) () (-> (/\ (e. F A) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) (` (sum1oo) F))) (y visset F A y cv V sumval2tOLD mp3an2 + seq1 F opr ~~>OLD breq2d exp + seq1 F opr ~~>OLD y cv br + seq1 F opr ~~>OLD sum1oo F ` br ibibr sylibr y 19.23adv imp x cv y cv + seq1 F opr ~~>OLD breq2 cbvexv sylan2b))
+thm (geosum1 ((x y) (A x) (A y)) ((geosum.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (= (` (sum1oo) F) (opr A (/) (opr (1) (-) A)))) (geosum.1 geolimOLD geosum.1 nnex x y A ^ x cv opr funopabex2 eqeltr A / 1 - A opr oprex F V A / 1 - A opr opr V sumval2tOLD mp3an12 syl))
+thm (geosum0 ((x y) (A x) (A y)) ((geosum.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (= (opr (1) (+) (` (sum1oo) F)) (opr (1) (/) (opr (1) (-) A)))) (geosum.1 geosum1 1 + opreq2d A absclt ax1re abs A ` 1 ltnet mpan2 syl imp A 1 abs fveq2 0re ax1re lt01 ltlei ax1re absid ax-mp syl6eq nsyl A 1 df-ne sylibr 1cn A 1 A subcan2t mp3an2 anidms A subidt 1 - A opr eqeq1d 0 1 - A opr eqcom syl6bb bitr3d eqneqd abs A ` < 1 br adantr mpbid 1 - A opr dividt 1cn 1 A subclt mpan sylan + A / 1 - A opr opr opreq1d 1 - A opr A 1 - A opr divdirt 1cn 1 A subclt mpan A CC e. id 1cn 1 A subclt mpan 3jca sylan 1cn 1 A npcant mpan / 1 - A opr opreq1d 1 - A opr 0 =/= adantr eqtr3d eqtr3d syldan eqtrd))
+var (class U)
+thm (plffval ((f g) (f h) (f x) (f y) (F f) (g h) (g x) (g y) (F g) (h x) (h y) (F h) (x y) (F x) (F y) (G f) (G g) (G h) (G x) (G y)) ((plffval.1 (e. F (V))) (plffval.2 (e. G (V)))) (= (opr F (+f) G) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (` F (cv x)) (+) (` G (cv x))))))) (plffval.1 plffval.2 axcnex x y F x cv ` + G x cv ` opr funopabex2 f cv F x cv fveq1 + g cv x cv ` opreq1d y cv eqeq2d x cv CC e. anbi2d x y biopabdv g cv G x cv fveq1 F x cv ` + opreq2d y cv eqeq2d x cv CC e. anbi2d x y biopabdv f g h x y df-plf V V oprabval5 mp2an))
+thm (muffval ((f g) (f x) (f y) (f z) (C f) (g x) (g y) (g z) (C g) (x y) (x z) (C x) (y z) (C y) (C z) (F f) (F g) (F x) (F y) (F z)) ((muffval.1 (e. F (V)))) (-> (e. C D) (= (opr C (.f) F) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr C (x.) (` F (cv x)))))))) (muffval.1 axcnex x y C x. F x cv ` opr funopabex2 z cv C x. f cv x cv ` opreq1 y cv eqeq2d x cv CC e. anbi2d x y biopabdv f cv F x cv fveq1 C x. opreq2d y cv eqeq2d x cv CC e. anbi2d x y biopabdv z f g x y df-muf D V oprabval5 mpan2))
+thm (plfvalt ((x y) (A x) (A y) (F x) (F y) (G x) (G y)) ((plfvalt.1 (e. F (V))) (plfvalt.2 (e. G (V)))) (-> (e. A (CC)) (= (` (opr F (+f) G) A) (opr (` F A) (+) (` G A)))) (x cv A F fveq2 x cv A G fveq2 + opreq12d plfvalt.1 plfvalt.2 x y plffval F A ` + G A ` oprex fvopab4))
+thm (mufvalt ((x y) (A x) (A y) (C x) (C y) (F x) (F y)) ((mufvalt.1 (e. F (V)))) (-> (/\ (e. C D) (e. A (CC))) (= (` (opr C (.f) F) A) (opr C (x.) (` F A)))) (mufvalt.1 C D x y muffval A fveq1d x cv A F fveq2 C x. opreq2d x y x cv CC e. y cv C x. F x cv ` opr = /\ {<,>|} eqid C x. F A ` oprex fvopab4 sylan9eq))
+thm (efvalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (e^) A) (opr (1) (+) (` (sum1oo) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr A (^) (cv x)) (/) (` (!) (cv x)))))))))) (z cv A ^ x cv opreq1 / ! x cv ` opreq1d y cv eqeq2d x cv NN e. anbi2d x y biopabdv sum1oo fveq2d 1 + opreq2d z w x y df-ef 1 + sum1oo x y x cv NN e. y cv A ^ x cv opr / ! x cv ` opr = /\ {<,>|} ` oprex fvopab4))
+thm (eval ((x y)) () (= (e) (opr (1) (+) (` (sum1oo) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (df-e 1cn 1 x y efvalt ax-mp x cv 1expt / ! x cv ` opreq1d y cv eqeq2d pm5.32i x y biopabi sum1oo fveq2i 1 + opreq2i eqtr eqtr))
+thm (efcltlem1OLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F w)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (efcltlem1.2 (e. A (CC))) (efcltlem1.3 (=/= A (0)))) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x))) (efcltlem1.2 abscl efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z abs A ` 0 flget mp2an mpbi lt01 efcltlem1.2 abscl abs A ` flclt ax-mp zre ax1re addgegt0 mp2an gt0ne0i redivcl efcltlem1.2 abscl recn div1 efcltlem1.2 abscl abs A ` flleltt ax-mp pm3.27i eqbrtr efcltlem1.2 abscl ax1re efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre lt01 efcltlem1.2 absge0 efcltlem1.2 abscl 0z abs A ` 0 flget mp2an mpbi lt01 efcltlem1.2 abscl abs A ` flclt ax-mp zre ax1re addgegt0 mp2an ltdiv23i mpbi pm3.2i efcltlem1.2 abscl efcltlem1.2 absge0 abs A ` flge0nn0t mp2an floor abs A ` ` nn0p1nnt ax-mp efcltlem1.2 abscl efcltlem1.2 absge0 abs A ` flge0nn0t mp2an floor abs A ` ` nn0p1nnt ax-mp floor abs A ` ` + 1 opr w cv nnleltp1t mpan efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z abs A ` 0 flget mp2an mpbi lt01 efcltlem1.2 abscl abs A ` flclt ax-mp zre ax1re addgegt0 mp2an efcltlem1.2 abscl efcltlem1.3 efcltlem1.2 absgt0 mpbi pm3.2i floor abs A ` ` + 1 opr RR e. abs w cv + 1 opr ` RR e. abs A ` RR e. 0 < floor abs A ` ` + 1 opr br 0 < abs w cv + 1 opr ` br 0 < abs A ` br an6 floor abs A ` ` + 1 opr abs w cv + 1 opr ` abs A ` ltdiv2t sylbir mp3an3 mpanl12 w cv peano2nn w cv + 1 opr nncnt w cv + 1 opr absclt 3syl w cv peano2nn w cv + 1 opr nngt0t w cv + 1 opr absidt w cv + 1 opr nnret w cv + 1 opr nnnn0t w cv + 1 opr nn0ge0t syl sylanc breqtrrd syl sylanc w cv + 1 opr absidt w cv peano2nn w cv + 1 opr nnret syl w cv nnnn0t w cv peano2nn0 w cv + 1 opr nn0ge0t 3syl sylanc floor abs A ` ` + 1 opr < breq2d efcltlem1.2 abscl efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre efcltlem1.2 abscl abs A ` flclt ax-mp floor abs A ` ` peano2z ax-mp zre efcltlem1.2 absge0 efcltlem1.2 abscl 0z abs A ` 0 flget mp2an mpbi lt01 efcltlem1.2 abscl abs A ` flclt ax-mp zre ax1re addgegt0 mp2an gt0ne0i redivcl abs A ` / abs w cv + 1 opr ` opr abs A ` / floor abs A ` ` + 1 opr opr abs A ^ w cv opr / ! w cv ` opr ` ltmul1t mp3anl2 efcltlem1.2 abscl abs A ` abs w cv + 1 opr ` redivclt mp3an1 w cv peano2nn w cv + 1 opr nncnt w cv + 1 opr absclt 3syl abs w cv + 1 opr ` gt0ne0t w cv peano2nn w cv + 1 opr nncnt w cv + 1 opr absclt 3syl w cv peano2nn w cv + 1 opr nngt0t w cv + 1 opr absidt w cv + 1 opr nnret w cv + 1 opr nnnn0t w cv + 1 opr nn0ge0t syl sylanc breqtrrd syl sylanc sylanc A ^ w cv opr ! w cv ` divclt efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl syl3anc A ^ w cv opr / ! w cv ` opr absclt syl jca A ^ w cv opr / ! w cv ` opr absgt0t biimpa A ^ w cv opr ! w cv ` divclt efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl syl3anc A ^ w cv opr ! w cv ` divneq0bt biimpa efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl 3jca efcltlem1.3 efcltlem1.2 A w cv expne0t mpan mpbii sylanc sylanc sylanc w cv peano2nn y cv w cv + 1 opr A ^ opreq2 y cv w cv + 1 opr ! fveq2 / opreq12d efcltlem.1 A ^ w cv + 1 opr opr / ! w cv + 1 opr ` oprex fvopab4 syl efcltlem1.2 A w cv expp1t mpan w cv nnnn0t w cv facp1t syl / opreq12d A ^ w cv opr ! w cv ` A w cv + 1 opr divmuldivt efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl jca w cv peano2nn w cv + 1 opr nncnt syl efcltlem1.2 jctil jca w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl w cv peano2nn w cv + 1 opr nnne0t syl jca sylanc eqtr4d A ^ w cv opr / ! w cv ` opr A / w cv + 1 opr opr axmulcom A ^ w cv opr ! w cv ` divclt efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl syl3anc efcltlem1.2 A w cv + 1 opr divclt mp3an1 w cv peano2nn w cv + 1 opr nncnt syl w cv peano2nn w cv + 1 opr nnne0t syl sylanc sylanc 3eqtrd abs fveq2d A / w cv + 1 opr opr A ^ w cv opr / ! w cv ` opr absmult efcltlem1.2 A w cv + 1 opr divclt mp3an1 w cv peano2nn w cv + 1 opr nncnt syl w cv peano2nn w cv + 1 opr nnne0t syl sylanc A ^ w cv opr ! w cv ` divclt efcltlem1.2 A w cv expclt mpan w cv nnnn0t w cv facclt ! w cv ` nncnt 3syl w cv nnnn0t w cv facclt ! w cv ` nnne0t 3syl syl3anc sylanc efcltlem1.2 A w cv + 1 opr absdivt mp3an1 w cv peano2nn w cv + 1 opr nncnt syl w cv peano2nn w cv + 1 opr nnne0t syl sylanc x. abs A ^ w cv opr / ! w cv ` opr ` opreq1d 3eqtrd y cv w cv A ^ opreq2 y cv w cv ! fveq2 / opreq12d efcltlem.1 A ^ w cv opr / ! w cv ` oprex fvopab4 abs fveq2d abs A ` / floor abs A ` ` + 1 opr opr x. opreq2d < breq12d bitr4d 3bitr3d bitrd biimpd rgen pm3.2i efcltlem.1 A ^ y cv opr ! y cv ` divclt efcltlem1.2 A y cv expclt mpan y cv nnnn0t y cv facclt ! y cv ` nncnt 3syl y cv nnnn0t y cv facnzt syl syl3anc fopab abs A ` / floor abs A ` ` + 1 opr opr floor abs A ` ` + 1 opr w x cvgratOLD mp2an))
+thm (efcltlem2OLD ((y z) (A y) (A z)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))))) (-> (= A (0)) (br (opr (+) (seq1) F) (~~>OLD) (0))) (A 0 ^ y cv opreq1 / ! y cv ` opreq1d 0 eqeq1d y cv 0expt / ! y cv ` opreq1d y cv nnnn0t ! y cv ` div0t y cv facclt ! y cv ` nncnt syl y cv facnzt sylanc syl eqtrd syl5bir A ^ y cv opr / ! y cv ` opr 0 z cv eqeq2 syl6 pm5.32d y z biopabdv efcltlem.1 NN 0 y z fconstopab 3eqtr4g + seq1 opreq2d ser1clim0OLD syl6eqbr))
+thm (efcltlem3OLD ((x y) (x z) (A x) (y z) (A y) (A z) (F x)) ((efcltlem.1 (= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))) (A A CC e. A 0 =/= /\ A 1 if ^ y cv opreq1 / ! y cv ` opreq1d z cv eqeq2d y cv NN e. anbi2d y z biopabdv + seq1 opreq2d ~~>OLD x cv breq1d x biexdv y z y cv NN e. z cv A CC e. A 0 =/= /\ A 1 if ^ y cv opr / ! y cv ` opr = /\ {<,>|} eqid A A CC e. A 0 =/= /\ A 1 if CC eleq1 A A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1 A CC e. A 0 =/= /\ A 1 if CC eleq1 1 A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.26i A A CC e. A 0 =/= /\ A 1 if CC eleq1 A A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1 A CC e. A 0 =/= /\ A 1 if CC eleq1 1 A CC e. A 0 =/= /\ A 1 if 0 neeq1 anbi12d 1cn ax1ne0 pm3.2i elimhyp pm3.27i x efcltlem1OLD dedth efcltlem.1 + seq1 opreq2i ~~>OLD x cv breq1i x biex sylibr exp A 0 df-ne syl5ibr efcltlem.1 efcltlem2OLD 0cn elisseti x cv 0 + seq1 F opr ~~>OLD breq2 cla4ev syl pm2.61d2))
+thm (efclt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (CC)) (e. (` (e^) A) (CC))) (y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} eqid x efcltlem3OLD nnex y z A ^ y cv opr / ! y cv ` opr funopabex2 y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} V x sumcltOLD ax-mp 1cn 1 sum1oo y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} ` axaddcl mpan 3syl A y z efvalt CC eleq1d mpbird))
+thm (efcvgOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z)) ((efcvg.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr A (^) (cv x)) (/) (` (!) (cv x))))))))) (-> (e. A (CC)) (br (opr (+) (seq1) F) (~~>OLD) (opr (` (e^) A) (-) (1)))) (efcvg.1 z efcltlem3OLD A x y efvalt efcvg.1 sum1oo fveq2i 1 + opreq2i syl6eqr - 1 opreq1d efcvg.1 z efcltlem3OLD efcvg.1 nnex x y A ^ x cv opr / ! x cv ` opr funopabex2 eqeltr F V z sumcltOLD ax-mp 1cn 1 sum1oo F ` pncan2t mpan 3syl eqtr2d + seq1 F opr ~~>OLD breq2d efcvg.1 nnex x y A ^ x cv opr / ! x cv ` opr funopabex2 eqeltr F V z cvgsumtOLD mpan syl5bi mpd))
+thm (reefcl ((x y) (x z) (A x) (y z) (A y) (A z)) ((reefcl.1 (e. A (RR)))) (e. (` (e^) A) (RR)) (reefcl.1 recn y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} eqid x efcltlem3OLD ax-mp reefcl.1 recn A y z efvalt ax-mp + seq1 y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} opr ~~>OLD x cv br a1i nnex y z A ^ y cv opr / ! y cv ` opr funopabex2 x visset y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} V x cv V sumval2tOLD mp3an12 y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} eqid A ^ y cv opr ! y cv ` redivclt reefcl.1 A y cv reexpclt mpan y cv nnnn0t y cv facclt ! y cv ` nnret syl syl y cv nnnn0t y cv facnzt syl syl3anc fopab ser1re x visset + seq1 y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} opr climrecltOLD mpan eqeltrd ax1re 1 sum1oo y z y cv NN e. z cv A ^ y cv opr / ! y cv ` opr = /\ {<,>|} ` axaddrcl mpan syl eqeltrd x 19.23aiv ax-mp))
+thm (reefclt () () (-> (e. A (RR)) (e. (` (e^) A) (RR))) (A A RR e. A 0 if e^ fveq2 RR eleq1d 0re A elimel reefcl dedth))
+thm (erelem1 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (/\ (:--> F (NN) (RR)) (:--> G (NN) (RR))) (erelem1.1 2re 2re 2pos gt0ne0i rereccl 1 / 2 opr x cv reexpclt mpan 2re 2 1 / 2 opr ^ x cv opr axmulrcl mpan syl fopab erelem1.2 ! x cv ` rerecclt x cv nnnn0t x cv facclt syl ! x cv ` nnret syl x cv nnnn0t x cv facclt syl ! x cv ` nnne0t syl sylanc fopab pm3.2i))
+thm (erelem2OLD ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (opr (+) (seq1) F) (~~>OLD) (2)) (2cn 1cn elisseti x y x cv NN e. y cv 1 / 2 opr ^ x cv opr = /\ {<,>|} eqid 2cn 2re 2pos gt0ne0i reccl 0re 2re 2re 2pos gt0ne0i rereccl halfgt0 ltlei 2re 2re 2pos gt0ne0i rereccl absid ax-mp halflt1 eqbrtr geolimiOLD 2cn 2re 2pos gt0ne0i reccl 2times 2cn 2re 2pos gt0ne0i recid eqtr3 1cn 2cn 2re 2pos gt0ne0i reccl 2cn 2re 2pos gt0ne0i reccl subadd mpbir 1 / 2 opr / opreq2i 2cn 2re 2pos gt0ne0i reccl 1cn 2cn ax1ne0 2re 2pos gt0ne0i divneq0 divid eqtr breqtr addex erelem1.1 nnex x y 2 x. 1 / 2 opr ^ x cv opr opr funopabex2 eqeltr seq1fn 2cn x y x cv NN e. y cv 1 / 2 opr ^ x cv opr = /\ {<,>|} eqid 2cn 2re 2pos gt0ne0i reccl 1 / 2 opr x cv expclt mpan fopab erelem1.1 nnex x y 2 x. 1 / 2 opr ^ x cv opr opr funopabex2 eqeltr x cv z cv 1 / 2 opr ^ opreq2 2 x. opreq2d erelem1.1 2 x. 1 / 2 opr ^ z cv opr oprex fvopab4 x cv z cv 1 / 2 opr ^ opreq2 x y x cv NN e. y cv 1 / 2 opr ^ x cv opr = /\ {<,>|} eqid 1 / 2 opr ^ z cv oprex fvopab4 2 x. opreq2d eqtr4d z cv ser1mulc climmulcOLD 2cn mulid1 breqtr))
+thm (erelem3 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (-> (e. (cv z) (NN)) (/\ (br (0) (<_) (` G (cv z))) (br (` G (cv z)) (<_) (` F (cv z))))) (0re 0 1 / ! z cv ` opr ltlet mpan ! z cv ` rerecclt z cv nnnn0t z cv facclt syl ! z cv ` nnret syl z cv nnnn0t z cv facclt syl ! z cv ` nnne0t syl sylanc ! z cv ` recgt0t z cv nnnn0t z cv facclt syl ! z cv ` nnret syl z cv nnnn0t z cv facclt syl ! z cv ` nngt0t syl sylanc sylc x cv z cv ! fveq2 1 / opreq2d erelem1.2 1 / ! z cv ` oprex fvopab4 breqtrrd z cv faclbnd 2 ^ z cv opr / 2 opr ! z cv ` lerect 2re 2 z cv reexpclt mpan 2re 2re 2pos gt0ne0i 2 ^ z cv opr 2 redivclt mp3an23 syl z cv nnnn0t z cv facclt syl ! z cv ` nnret syl jca 2pos 2re 2 ^ z cv opr 2 divgt0t mpanl2 mpanr2 2re 2 z cv reexpclt mpan 2re 2pos 2 z cv expgt0t mp3an13 sylanc z cv nnnn0t z cv facclt syl ! z cv ` nngt0t syl jca sylanc mpbid 2cn 2 2 ^ z cv opr divrect mp3an1 2re 2 z cv reexpclt mpan recnd 2 ^ z cv opr gt0ne0t 2re 2 z cv reexpclt mpan 2re 2pos 2 z cv expgt0t mp3an13 sylanc sylanc 2re 2pos gt0ne0i 2cn 2 ^ z cv opr 2 recdivt mpanl2 mpanr2 2re 2 z cv reexpclt mpan recnd 2 ^ z cv opr gt0ne0t 2re 2 z cv reexpclt mpan 2re 2pos 2 z cv expgt0t mp3an13 sylanc sylanc 2cn 2re 2pos gt0ne0i 2 z cv recexpt mp3an13 2 x. opreq2d 3eqtr4d breqtrd x cv z cv ! fveq2 1 / opreq2d erelem1.2 1 / ! z cv ` oprex fvopab4 x cv z cv 1 / 2 opr ^ opreq2 2 x. opreq2d erelem1.1 2 x. 1 / 2 opr ^ z cv opr oprex fvopab4 3brtr4d jca))
+thm (erelem4OLD ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (2re elisseti erelem1.1 erelem1.2 erelem1 pm3.26i erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 erelem1.1 erelem1.2 erelem2OLD cvgcmpOLD))
+thm (erelem5 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (e. (sup (ran (opr (+) (seq1) G)) (RR) (<)) (RR)) (ltso + seq1 G opr ran supex erelem1.1 erelem1.2 erelem1 pm3.27i ser1re erelem1.1 erelem1.2 erelem4OLD climreclOLD))
+thm (erelem6 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (= (e) (opr (1) (+) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) (x y eval erelem1.2 sum1oo fveq2i erelem1.2 nnex x y 1 / ! x cv ` opr funopabex2 eqeltr erelem1.1 erelem1.2 erelem5 erelem1.1 erelem1.2 erelem4OLD G V + seq1 G opr ran RR < sup RR sumval2tOLD mp3an eqtr3 1 + opreq2i eqtr))
+thm (erelem7 ((x y)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (e. (e) (RR)) (erelem1.1 erelem1.2 erelem6 ax1re erelem1.1 erelem1.2 erelem5 readdcl eqeltr))
+thm (ege2le3lem1 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (br (` (opr (+) (seq1) G) (1)) (<_) (sup (ran (opr (+) (seq1) G)) (RR) (<))) (ltso + seq1 G opr ran supex erelem1.1 erelem1.2 erelem1 pm3.27i ser1re erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 pm3.26d z cv ser1mono erelem1.1 erelem1.2 erelem4OLD 1nn climubiOLD))
+thm (ege2le3lem2 ((x y) (x z) (y z) (F z) (G z)) ((erelem1.1 (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x)))))))) (erelem1.2 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))) (/\ (br (2) (<_) (e)) (br (e) (<_) (3))) (addex erelem1.2 nnex x y 1 / ! x cv ` opr funopabex2 eqeltr seq11 1nn x cv 1 ! fveq2 1 / opreq2d erelem1.2 1 / ! 1 ` oprex fvopab4 ax-mp fac1 1 / opreq2i 1cn div1 3eqtr eqtr2 erelem1.1 erelem1.2 ege2le3lem1 eqbrtr ax1re erelem1.1 erelem1.2 erelem5 ax1re leadd2 mpbi df-2 erelem1.1 erelem1.2 erelem6 3brtr4 2re elisseti erelem1.1 erelem1.2 erelem1 pm3.26i erelem1.1 erelem1.2 erelem1 pm3.27i erelem1.1 erelem1.2 z erelem3 erelem1.1 erelem1.2 erelem2OLD ltso + seq1 G opr ran supex erelem1.1 erelem1.2 erelem4OLD cvgcmpub erelem1.1 erelem1.2 erelem5 2re ax1re leadd2 mpbi erelem1.1 erelem1.2 erelem6 df-3 2cn 1cn addcom eqtr 3brtr4 pm3.2i))
+thm (ere ((x y)) () (e. (e) (RR)) (x y x cv NN e. y cv 2 x. 1 / 2 opr ^ x cv opr opr = /\ {<,>|} eqid x y x cv NN e. y cv 1 / ! x cv ` opr = /\ {<,>|} eqid erelem7))
+thm (ege2le3 ((x y)) () (/\ (br (2) (<_) (e)) (br (e) (<_) (3))) (x y x cv NN e. y cv 2 x. 1 / 2 opr ^ x cv opr opr = /\ {<,>|} eqid x y x cv NN e. y cv 1 / ! x cv ` opr = /\ {<,>|} eqid ege2le3lem2))
+thm (sinvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (sin) A) (opr (opr (` (e^) (opr (i) (x.) A)) (-) (` (e^) (opr (-u (i)) (x.) A))) (/) (opr (2) (x.) (i))))) (x cv A i x. opreq2 e^ fveq2d x cv A i -u x. opreq2 e^ fveq2d - opreq12d / 2 x. i opr opreq1d x y df-sin e^ i x. A opr ` - e^ i -u x. A opr ` opr / 2 x. i opr oprex fvopab4))
+thm (cosvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (cos) A) (opr (opr (` (e^) (opr (i) (x.) A)) (+) (` (e^) (opr (-u (i)) (x.) A))) (/) (2)))) (x cv A i x. opreq2 e^ fveq2d x cv A i -u x. opreq2 e^ fveq2d + opreq12d / 2 opreq1d x y df-cos e^ i x. A opr ` + e^ i -u x. A opr ` opr / 2 oprex fvopab4))
+thm (sinclt () () (-> (e. A (CC)) (e. (` (sin) A) (CC))) (A sinvalt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl jca e^ i x. A opr ` e^ i -u x. A opr ` subclt 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 e^ i x. A opr ` - e^ i -u x. A opr ` opr 2 x. i opr divclt mp3an23 3syl eqeltrd))
+thm (cosclt () () (-> (e. A (CC)) (e. (` (cos) A) (CC))) (A cosvalt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl jca e^ i x. A opr ` e^ i -u x. A opr ` axaddcl 2cn 2re 2pos gt0ne0i e^ i x. A opr ` + e^ i -u x. A opr ` opr 2 divclt mp3an23 3syl eqeltrd))
+thm (efivalt () () (-> (e. A (CC)) (= (` (e^) (opr (i) (x.) A)) (opr (` (cos) A) (+) (opr (i) (x.) (` (sin) A))))) (2cn 2re 2pos gt0ne0i e^ i x. A opr ` + e^ i -u x. A opr ` opr e^ i x. A opr ` - e^ i -u x. A opr ` opr 2 divdirt mpan2 mp3an3 e^ i x. A opr ` e^ i -u x. A opr ` axaddcl axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc e^ i x. A opr ` e^ i -u x. A opr ` subclt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc sylanc e^ i -u x. A opr ` e^ i x. A opr ` pncan3t axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl axicn i A axmulcl mpan i x. A opr efclt syl sylanc e^ i x. A opr ` + opreq2d e^ i x. A opr ` e^ i -u x. A opr ` e^ i x. A opr ` - e^ i -u x. A opr ` opr axaddass axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl e^ i x. A opr ` e^ i -u x. A opr ` subclt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc syl3anc axicn i A axmulcl mpan i x. A opr efclt syl e^ i x. A opr ` 2timest syl 3eqtr4d / 2 opreq1d axicn i A axmulcl mpan i x. A opr efclt syl 2cn 2re 2pos gt0ne0i 2 e^ i x. A opr ` divcan3t mp3an13 syl eqtr2d A cosvalt e^ i x. A opr ` e^ i -u x. A opr ` subclt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc axicn 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 i e^ i x. A opr ` - e^ i -u x. A opr ` opr 2 x. i opr div12t mpan2 mp3an13 syl A sinvalt i x. opreq2d e^ i x. A opr ` e^ i -u x. A opr ` subclt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc 2cn 2re 2pos gt0ne0i e^ i x. A opr ` - e^ i -u x. A opr ` opr 2 divrect mp3an23 syl axicn mulid2 / 2 x. i opr opreq1i axicn ine0 divid 1 / 2 opr x. opreq2i 1cn 2cn axicn axicn 2re 2pos gt0ne0i ine0 divmuldiv 2cn 2re 2pos gt0ne0i reccl mulid1 3eqtr3 eqtr3 e^ i x. A opr ` - e^ i -u x. A opr ` opr x. opreq2i syl6eqr 3eqtr4d + opreq12d 3eqtr4d))
+thm (ef0ALT ((x y)) () (= (` (e^) (0)) (1)) (0cn 0 x y efvalt ax-mp ! x cv ` div0t x cv nnnn0t x cv facclt ! x cv ` nncnt 3syl x cv nnnn0t x cv facclt ! x cv ` nnne0t 3syl sylanc x cv 0expt / ! x cv ` opreq1d x cv 0expt 3eqtr4d y cv eqeq2d pm5.32i x y biopabi sum1oo fveq2i 1 + opreq2i 0cn abs0 lt01 eqbrtr x y x cv NN e. y cv 0 ^ x cv opr = /\ {<,>|} eqid geosum0 mp2an 1cn subid1 1 / opreq2i 1cn div1 3eqtr 3eqtr))
+thm (sinnegt () () (-> (e. A (CC)) (= (` (sin) (-u A)) (-u (` (sin) A)))) (axicn i A mulneg12t mpan eqcomd e^ fveq2d axicn i A mul2negt mpan e^ fveq2d - opreq12d e^ i x. A opr ` e^ i -u x. A opr ` negdi3t axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc eqtr4d / 2 x. i opr opreq1d A negclt A -u sinvalt syl A sinvalt negeqd e^ i x. A opr ` e^ i -u x. A opr ` subclt axicn i A axmulcl mpan i x. A opr efclt syl axicn negcl i -u A axmulcl mpan i -u x. A opr efclt syl sylanc 2cn axicn mulcl 2cn axicn 2re 2pos gt0ne0i ine0 muln0 e^ i x. A opr ` - e^ i -u x. A opr ` opr 2 x. i opr divnegt mp3an23 syl eqtrd 3eqtr4d))
+thm (ruclem1 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br A (<) (opr (opr (opr (2) (x.) A) (+) B) (/) (3)))) (ruclem1.a ruclem1.b 2re ruclem1.a remulcl ltadd2 ruclem1.a recn mulid2 2 x. A opr + opreq2i df-3 x. A opreq1i 2cn 1cn ruclem1.a recn adddir eqtr2 eqtr3 < 2 x. A opr + B opr breq1i bitr 3re ruclem1.a remulcl 2re ruclem1.a remulcl ruclem1.b readdcl 3re 3pos ltdiv1i 3re recn ruclem1.a recn 3re 3pos gt0ne0i divcan3 < 2 x. A opr + B opr / 3 opr breq1i 3bitr))
+thm (ruclem2 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br (opr (opr (opr (2) (x.) A) (+) B) (/) (3)) (<) (opr (opr A (+) (opr (2) (x.) B)) (/) (3)))) (ruclem1.a ruclem1.b ruclem1.a ruclem1.b readdcl ltadd1 ruclem1.a recn 2times + B opreq1i ruclem1.a recn ruclem1.a recn ruclem1.b recn addass eqtr ruclem1.b recn 2times A + opreq2i ruclem1.a recn ruclem1.b recn ruclem1.b recn add12 eqtr < breq12i bitr4 2re ruclem1.a remulcl ruclem1.b readdcl ruclem1.a 2re ruclem1.b remulcl readdcl 3re 3pos ltdiv1i bitr))
+thm (ruclem3 () ((ruclem1.a (e. A (RR))) (ruclem1.b (e. B (RR)))) (<-> (br A (<) B) (br (opr (opr A (+) (opr (2) (x.) B)) (/) (3)) (<) B)) (ruclem1.a ruclem1.b 2re ruclem1.b remulcl ltadd1 ruclem1.b recn mulid2 + 2 x. B opr opreq1i df-3 2cn 1cn addcom eqtr x. B opreq1i 1cn 2cn ruclem1.b recn adddir eqtr2 eqtr3 A + 2 x. B opr opr < breq2i bitr ruclem1.a 2re ruclem1.b remulcl readdcl 3re ruclem1.b remulcl 3re 3pos ltdiv1i 3re recn ruclem1.b recn 3re 3pos gt0ne0i divcan3 A + 2 x. B opr opr / 3 opr < breq2i 3bitr))
+thm (ruclem4 () () (-> (/\ (= A C) (= B D)) (= (if (/\ (br (` (1st) A) (<) B) (br B (<) (` (2nd) A))) (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3)))) (if (/\ (br (` (1st) C) (<) D) (br D (<) (` (2nd) C))) (<,> (opr (opr (opr (2) (x.) D) (+) (` (2nd) C)) (/) (3)) (opr (opr D (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3)))))) (A C 1st fveq2 < B breq1d A C 2nd fveq2 B < breq2d anbi12d 2 x. B opr + 2nd A ` opr / 3 opr B + 2 x. 2nd A ` opr opr / 3 opr <,> 2 x. 1st A ` opr + 2nd A ` opr / 3 opr 1st A ` + 2 x. 2nd A ` opr opr / 3 opr <,> ifbid A C 2nd fveq2 2 x. B opr + opreq2d / 3 opreq1d A C 2nd fveq2 2 x. opreq2d B + opreq2d / 3 opreq1d jca 2 x. B opr + 2nd A ` opr / 3 opr 2 x. B opr + 2nd C ` opr / 3 opr B + 2 x. 2nd A ` opr opr / 3 opr B + 2 x. 2nd C ` opr opr / 3 opr opeq12 syl A C 1st fveq2 2 x. opreq2d A C 2nd fveq2 + opreq12d / 3 opreq1d A C 1st fveq2 A C 2nd fveq2 2 x. opreq2d + opreq12d / 3 opreq1d jca 2 x. 1st A ` opr + 2nd A ` opr / 3 opr 2 x. 1st C ` opr + 2nd C ` opr / 3 opr 1st A ` + 2 x. 2nd A ` opr opr / 3 opr 1st C ` + 2 x. 2nd C ` opr opr / 3 opr opeq12 syl jca 2 x. B opr + 2nd A ` opr / 3 opr B + 2 x. 2nd A ` opr opr / 3 opr <,> 2 x. B opr + 2nd C ` opr / 3 opr B + 2 x. 2nd C ` opr opr / 3 opr <,> 2 x. 1st A ` opr + 2nd A ` opr / 3 opr 1st A ` + 2 x. 2nd A ` opr opr / 3 opr <,> 2 x. 1st C ` opr + 2nd C ` opr / 3 opr 1st C ` + 2 x. 2nd C ` opr opr / 3 opr <,> 1st C ` < B br B < 2nd C ` br /\ ifeq12 syl eqtrd B D 1st C ` < breq2 B D < 2nd C ` breq1 anbi12d 2 x. B opr + 2nd C ` opr / 3 opr B + 2 x. 2nd C ` opr opr / 3 opr <,> 2 x. 1st C ` opr + 2nd C ` opr / 3 opr 1st C ` + 2 x. 2nd C ` opr opr / 3 opr <,> ifbid B D 2 x. opreq2 + 2nd C ` opreq1d / 3 opreq1d B D + 2 x. 2nd C ` opr opreq1 / 3 opreq1d jca 2 x. B opr + 2nd C ` opr / 3 opr 2 x. D opr + 2nd C ` opr / 3 opr B + 2 x. 2nd C ` opr opr / 3 opr D + 2 x. 2nd C ` opr opr / 3 opr opeq12 syl 1st C ` < D br D < 2nd C ` br /\ 2 x. 1st C ` opr + 2nd C ` opr / 3 opr 1st C ` + 2 x. 2nd C ` opr opr / 3 opr <,> ifeq1d eqtrd sylan9eq))
+thm (ruclem5 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (e. C (V)) (ruclem5.1 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> snex nnex ruclem5.0 NN V F RR fex mp2 F V NN 1 {} \ resexg ax-mp unex eqeltr))
+thm (ruclem6 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (= (|` C (\ (NN) ({} (1)))) (|` F (\ (NN) ({} (1))))) (ruclem5.1 C 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. NN 1 {} \ reseq1 ax-mp 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` NN 1 {} \ resundir 1 {} NN difdisj 1nn elisseti F 1 ` + 1 opr F 1 ` + 2 opr opex f1osn 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} 1 {} F 1 ` + 1 opr F 1 ` + 2 opr <,> {} f1ofn ax-mp 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} 1 {} NN 1 {} \ fnresdisj ax-mp mpbi F NN 1 {} \ residm uneq12i {/} F NN 1 {} \ |` uncom F NN 1 {} \ |` un0 3eqtr 3eqtr))
+thm (ruclem7 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))))) (= (` C (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) (1nn elisseti snid 1 1 {} C fvres ax-mp ruclem5.1 C 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. 1 {} reseq1 ax-mp 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` 1 {} resundir NN 1 {} \ 1 {} incom 1 {} NN difdisj eqtr NN 1 {} \ 1 {} F resdisj ax-mp 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} 1 {} |` uneq2i 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} 1 {} |` un0 eqtr 3eqtr 1 fveq1i 1nn elisseti snid 1 1 {} 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} fvres ax-mp 1nn elisseti F 1 ` + 1 opr F 1 ` + 2 opr opex fvsn 3eqtr eqtr3))
+thm (ruclem8 () ((ruclem5.0 (:--> F (NN) (RR))) (ruclem5.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem8.a (e. A (NN)))) (= (` C (opr A (+) (1))) (` F (opr A (+) (1)))) (ruclem8.a nngt0 0re ruclem8.a nnre ax1re ltadd1 1cn addid2 < A + 1 opr breq1i bitr mpbi ruclem8.a A peano2nn ax-mp A + 1 opr nngt1ne1t ax-mp A + 1 oprex 1 elsnc negbii bitr4 biimp ruclem8.a A peano2nn ax-mp jctil A + 1 opr NN 1 {} eldif sylibr A + 1 opr NN 1 {} \ C fvres A + 1 opr NN 1 {} \ F fvres ruclem5.0 ruclem5.1 ruclem6 A + 1 opr fveq1i syl5eq eqtr3d syl ax-mp))
+thm (ruclem9 ((x y) (x z) (y z) (A z)) ((ruclem9.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) A)))))) (e. D (V)) (reex reex xpex reex ruclem9.2 oprabex2))
+thm (ruclem10 () ((ruclem10.a (e. A (NN))) (ruclem.d (e. D (V))) (ruclem.c (e. C (V))) (ruclem10.3 (= G (o. (1st) (opr D (seq1) C))))) (= (` (1st) (` (opr D (seq1) C) A)) (` G A)) (ruclem10.3 A fveq1i fo1st 1st V V fof ax-mp 1st V V ffun ax-mp ruclem.d ruclem.c seq1fn pm3.2i ruclem10.a 1st D seq1 C opr NN A fvco2 mp2an eqtr2))
+thm (ruclem11 () ((ruclem11.a (e. A (NN))) (ruclem11.d (e. D (V))) (ruclem11.c (e. C (V))) (ruclem11.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` (2nd) (` (opr D (seq1) C) A)) (` H A)) (ruclem11.4 A fveq1i fo2nd 2nd V V fof ax-mp 2nd V V ffun ax-mp ruclem11.d ruclem11.c seq1fn pm3.2i ruclem11.a 2nd D seq1 C opr NN A fvco2 mp2an eqtr2))
+thm (ruclem12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) ((ruclem12.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))))) (= D ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (X. (RR) (RR))) (e. (cv v) (RR))) (= (cv u) (if (/\ (br (` (1st) (cv w)) (<) (cv v)) (br (cv v) (<) (` (2nd) (cv w)))) (<,> (opr (opr (opr (2) (x.) (cv v)) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (cv v) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv w))) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (` (1st) (cv w)) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3)))))))) (ruclem12.2 x cv w cv RR RR X. eleq1 y cv RR e. anbi1d y cv v cv RR eleq1 w cv RR RR X. e. anbi2d sylan9bb x cv w cv y cv v cv ruclem4 z cv eqeq2d anbi12d z cbvoprab12v z cv u cv 1st w cv ` < v cv br v cv < 2nd w cv ` br /\ 2 x. v cv opr + 2nd w cv ` opr / 3 opr v cv + 2 x. 2nd w cv ` opr opr / 3 opr <,> 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr <,> if eqeq1 w cv RR RR X. e. v cv RR e. /\ anbi2d w v cbvoprab3v 3eqtr))
+thm (ruclem13 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (D w) (D v) (D u)) ((ruclem13.0 (:--> F (NN) (RR))) (ruclem13.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem13.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))))) (:--> (opr D (seq1) C) (NN) (X. (RR) (RR))) (D seq1 C opr NN RR RR X. df-f ruclem13.2 ruclem9 ruclem13.0 ruclem13.1 ruclem5 seq1fn ruclem13.0 ruclem13.1 ruclem7 F 1 ` + 2 oprex F 1 ` + 1 opr RR RR opelxp ruclem13.0 1nn F NN RR 1 ffvrn mp2an ax1re readdcl ruclem13.0 1nn F NN RR 1 ffvrn mp2an 2re readdcl mpbir2an eqeltr ruclem13.0 NN 1 {} difss F NN RR NN 1 {} \ fssres mp2an ruclem13.0 ruclem13.1 ruclem6 C NN 1 {} \ |` F NN 1 {} \ |` NN 1 {} \ RR feq1 ax-mp mpbir D RR RR X. RR RR RR X. w v u foprval ruclem13.2 w v u ruclem12 2 x. v cv opr + 2nd w cv ` opr / 3 opr v cv + 2 x. 2nd w cv ` opr opr / 3 opr opex 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr opex 1st w cv ` < v cv br v cv < 2nd w cv ` br /\ ifex ruclem13.2 w v u ruclem12 V oprabval4g mp3an3 u cv eqeq2d pm5.32i w v u bioprabi eqtr4 2 x. v cv opr + 2nd w cv ` opr / 3 opr v cv + 2 x. 2nd w cv ` opr opr / 3 opr opex 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr opex 1st w cv ` < v cv br v cv < 2nd w cv ` br /\ ifex ruclem13.2 w v u ruclem12 V oprabval4g mp3an3 1st w cv ` < v cv br v cv < 2nd w cv ` br /\ 2 x. v cv opr + 2nd w cv ` opr / 3 opr v cv + 2 x. 2nd w cv ` opr opr / 3 opr <,> 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr <,> iftrue RR RR X. eleq1d 2 x. v cv opr + 2nd w cv ` opr / 3 opr RR v cv + 2 x. 2nd w cv ` opr opr / 3 opr RR opelxpi 2 x. v cv opr 2nd w cv ` axaddrcl 2re 2 v cv axmulrcl mpan sylan 3re 3re 3pos gt0ne0i 2 x. v cv opr + 2nd w cv ` opr 3 redivclt mp3an23 syl v cv 2 x. 2nd w cv ` opr axaddrcl 2re 2 2nd w cv ` axmulrcl mpan sylan2 3re 3re 3pos gt0ne0i v cv + 2 x. 2nd w cv ` opr opr 3 redivclt mp3an23 syl sylanc ancoms 1st w cv ` RR e. adantll syl5bir 1st w cv ` < v cv br v cv < 2nd w cv ` br /\ 2 x. v cv opr + 2nd w cv ` opr / 3 opr v cv + 2 x. 2nd w cv ` opr opr / 3 opr <,> 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr <,> iffalse RR RR X. eleq1d 2 x. 1st w cv ` opr + 2nd w cv ` opr / 3 opr RR 1st w cv ` + 2 x. 2nd w cv ` opr opr / 3 opr RR opelxpi 2 x. 1st w cv ` opr 2nd w cv ` axaddrcl 2re 2 1st w cv ` axmulrcl mpan sylan 3re 3re 3pos gt0ne0i 2 x. 1st w cv ` opr + 2nd w cv ` opr 3 redivclt mp3an23 syl 1st w cv ` 2 x. 2nd w cv ` opr axaddrcl 2re 2 2nd w cv ` axmulrcl mpan sylan2 3re 3re 3pos gt0ne0i 1st w cv ` + 2 x. 2nd w cv ` opr opr 3 redivclt mp3an23 syl sylanc syl5bir v cv RR e. adantrd pm2.61i w cv 1st w cv ` 2nd w cv ` <,> = adantll w cv RR RR elxp6 sylanb eqeltrd rgen2a mpbir2an ruclem13.2 ruclem9 ruclem13.0 ruclem13.1 ruclem5 RR RR X. RR seq1rn mp3an mpbir2an))
+thm (ruclem14 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` (opr D (seq1) C) (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))) (D seq1 C opr 1 ` eqid ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 seq11 ruclem.0 ruclem.1 ruclem7 eqtr eqtr))
+thm (ruclem15 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem15.a (e. A (NN)))) (= (` (opr D (seq1) C) (opr A (+) (1))) (if (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))))) (ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 A seq1p1 ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem13 ruclem15.a D seq1 C opr NN RR RR X. A ffvrn mp2an ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem.0 ruclem15.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an eqeltr 2 x. C A + 1 opr ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr C A + 1 opr ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr opex 2 x. 1st D seq1 C opr A ` ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr 1st D seq1 C opr A ` ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr opex 1st D seq1 C opr A ` ` < C A + 1 opr ` br C A + 1 opr ` < 2nd D seq1 C opr A ` ` br /\ ifex v cv eqid w cv D seq1 C opr A ` v cv v cv ruclem4 mpan2 D seq1 C opr A ` eqid D seq1 C opr A ` D seq1 C opr A ` v cv C A + 1 opr ` ruclem4 mpan ruclem.2 w v u ruclem12 oprabval2 mp2an ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 ruclem.0 ruclem.1 ruclem15.a ruclem8 < breq12i ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 < breq12i anbi12i 1st D seq1 C opr A ` ` < C A + 1 opr ` br C A + 1 opr ` < 2nd D seq1 C opr A ` ` br /\ G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ 2 x. C A + 1 opr ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr C A + 1 opr ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr <,> 2 x. 1st D seq1 C opr A ` ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr 1st D seq1 C opr A ` ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr <,> ifbi ax-mp ruclem.0 ruclem.1 ruclem15.a ruclem8 2 x. opreq2i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 + opreq12i / 3 opreq1i ruclem.0 ruclem.1 ruclem15.a ruclem8 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 2 x. opreq2i + opreq12i / 3 opreq1i 2 x. C A + 1 opr ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr C A + 1 opr ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr opeq12 mp2an ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 2 x. opreq2i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 + opreq12i / 3 opreq1i ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 ruclem15.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 2 x. opreq2i + opreq12i / 3 opreq1i 2 x. 1st D seq1 C opr A ` ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr 2 x. G A ` opr + H A ` opr / 3 opr 1st D seq1 C opr A ` ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr opeq12 mp2an 2 x. C A + 1 opr ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr C A + 1 opr ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr <,> 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr <,> 2 x. 1st D seq1 C opr A ` ` opr + 2nd D seq1 C opr A ` ` opr / 3 opr 1st D seq1 C opr A ` ` + 2 x. 2nd D seq1 C opr A ` ` opr opr / 3 opr <,> 2 x. G A ` opr + H A ` opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr <,> G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ ifeq12 mp2an eqtr 3eqtr))
+thm (ruclem16 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (= (` G (1)) (opr (` F (1)) (+) (1))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem14 1st fveq2i 1nn ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 F 1 ` + 1 oprex F 1 ` + 2 opr op1st 3eqtr3))
+thm (ruclem17 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (:--> G (NN) (RR)) (RR RR f1stres ruclem.0 ruclem.1 ruclem.2 ruclem13 1st RR RR X. |` RR RR X. RR D seq1 C opr NN fco mp2an ruclem.0 ruclem.1 ruclem.2 ruclem13 D seq1 C opr NN RR RR X. frn ax-mp D seq1 C opr RR RR X. 1st cores ax-mp ruclem.3 eqtr4 1st RR RR X. |` D seq1 C opr o. G NN RR feq1 ax-mp mpbi))
+thm (ruclem18 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)))) (G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr <,> 2 x. G A ` opr + H A ` opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr <,> iftrue ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq 1st fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 2 x. F A + 1 opr ` opr + H A ` opr / 3 oprex F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr op1st 3eqtr3g))
+thm (ruclem19 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` H (opr A (+) (1))) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3)))) (G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr <,> 2 x. G A ` opr + H A ` opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr <,> iftrue ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq 2nd fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 2 x. F A + 1 opr ` opr + H A ` opr / 3 oprex F A + 1 opr ` + 2 x. H A ` opr opr / 3 oprex op2nd 3eqtr3g))
+thm (ruclem20 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)))) (G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr <,> 2 x. G A ` opr + H A ` opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr <,> iffalse ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq 1st fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.3 ruclem10 2 x. G A ` opr + H A ` opr / 3 oprex G A ` + 2 x. H A ` opr opr / 3 opr op1st 3eqtr3g))
+thm (ruclem21 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` H (opr A (+) (1))) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)))) (G A ` < F A + 1 opr ` br F A + 1 opr ` < H A ` br /\ 2 x. F A + 1 opr ` opr + H A ` opr / 3 opr F A + 1 opr ` + 2 x. H A ` opr opr / 3 opr <,> 2 x. G A ` opr + H A ` opr / 3 opr G A ` + 2 x. H A ` opr opr / 3 opr <,> iffalse ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem15 syl5eq 2nd fveq2d ruclem18.a A peano2nn ax-mp ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 2 x. G A ` opr + H A ` opr / 3 oprex G A ` + 2 x. H A ` opr opr / 3 oprex op2nd 3eqtr3g))
+thm (ruclem22 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (e. (` G A) (RR)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 ruclem18.a G NN RR A ffvrn mp2an))
+thm (ruclem23 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (e. (` H A) (RR)) (ruclem18.a ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 ruclem.0 ruclem.1 ruclem.2 ruclem13 ruclem18.a D seq1 C opr NN RR RR X. A ffvrn mp2an D seq1 C opr A ` RR RR elxp6 pm3.27bd ax-mp pm3.27i eqeltrr))
+thm (ruclem24 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (-> (br (` G A) (<) (` H A)) (br (` G (opr A (+) (1))) (<) (` H (opr A (+) (1))))) (ruclem.0 ruclem18.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem2 biimp G A ` < F A + 1 opr ` br adantl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem18 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem19 < breq12d mpbird G A ` < H A ` br a1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem20 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem21 < breq12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem2 syl6rbbr biimpd pm2.61i))
+thm (ruclem25 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` G A) (<) (` H A)) (ruclem18.a w cv 1 G fveq2 w cv 1 H fveq2 < breq12d w cv v cv G fveq2 w cv v cv H fveq2 < breq12d w cv v cv + 1 opr G fveq2 w cv v cv + 1 opr H fveq2 < breq12d w cv A G fveq2 w cv A H fveq2 < breq12d ax1re ltplus1 df-2 breqtrr ax1re 2re ruclem.0 1nn F NN RR 1 ffvrn mp2an ltadd2 mpbi ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem16 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem14 2nd fveq2i 1nn ruclem.2 ruclem9 ruclem.0 ruclem.1 ruclem5 ruclem.4 ruclem11 F 1 ` + 1 oprex F 1 ` + 2 oprex op2nd 3eqtr3 3brtr4 v cv v cv NN e. v cv 1 if G fveq2 v cv v cv NN e. v cv 1 if H fveq2 < breq12d v cv v cv NN e. v cv 1 if + 1 opreq1 G fveq2d v cv v cv NN e. v cv 1 if + 1 opreq1 H fveq2d < breq12d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn v cv elimel ruclem24 dedth nnind ax-mp))
+thm (ruclem26 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` G A) (<) (` G (opr A (+) (1)))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem18.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an 2re ruclem.0 ruclem18.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an remulcl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 readdcl 3re 3re 3pos gt0ne0i redivcl lttr ruclem.0 ruclem18.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem1 sylan2b ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem18 breqtrrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem20 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem1 mpbi syl5breqr pm2.61i))
+thm (ruclem27 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem18.a (e. A (NN)))) (br (` H (opr A (+) (1))) (<) (` H A)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem19 ruclem.0 ruclem18.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem3 biimp G A ` < F A + 1 opr ` br adantl eqbrtrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem21 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem18.a ruclem23 ruclem3 mpbi syl6eqbr pm2.61i))
+thm (ruclem28 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN)))) (-. (/\ (br (` G (opr A (+) (1))) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H (opr A (+) (1)))))) (ruclem.0 ruclem28.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem23 ruclem1 biimp G A ` < F A + 1 opr ` br adantl ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem18 breqtrrd ruclem.0 ruclem28.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem22 ltnsym syl F A + 1 opr ` < H A + 1 opr ` br intnanrd ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem26 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem22 ruclem.0 ruclem28.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an lttr mpan ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem27 ruclem.0 ruclem28.a A peano2nn ax-mp F NN RR A + 1 opr ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem23 lttr mpan2 anim12i con3i pm2.61i))
+thm (ruclem29 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN)))) (-. (/\ (br (` G A) (<) (` F A)) (br (` F A) (<) (` H A)))) (ruclem28.a w cv 1 G fveq2 w cv 1 F fveq2 < breq12d w cv 1 F fveq2 w cv 1 H fveq2 < breq12d anbi12d negbid w cv v cv + 1 opr G fveq2 w cv v cv + 1 opr F fveq2 < breq12d w cv v cv + 1 opr F fveq2 w cv v cv + 1 opr H fveq2 < breq12d anbi12d negbid w cv A G fveq2 w cv A F fveq2 < breq12d w cv A F fveq2 w cv A H fveq2 < breq12d anbi12d negbid ruclem.0 1nn F NN RR 1 ffvrn mp2an ltplus1 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem16 breqtrr ruclem.0 1nn F NN RR 1 ffvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem22 ltnsym ax-mp F 1 ` < H 1 ` br intnanr v cv v cv NN e. v cv 1 if + 1 opreq1 G fveq2d v cv v cv NN e. v cv 1 if + 1 opreq1 F fveq2d < breq12d v cv v cv NN e. v cv 1 if + 1 opreq1 F fveq2d v cv v cv NN e. v cv 1 if + 1 opreq1 H fveq2d < breq12d anbi12d negbid ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn v cv elimel ruclem28 dedth nn1suc ax-mp))
+thm (ruclem30 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (-> (br (` G A) (<) (` G (opr A (+) B))) (br (` G A) (<) (` G (opr A (+) (opr B (+) (1)))))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem26 ruclem28.a nncn ruclem.b nncn 1cn addass G fveq2i breqtr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b B peano2nn ax-mp A B + 1 opr nnaddclt mp2an ruclem22 lttr mpan2))
+thm (ruclem31 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (-> (br (` H (opr A (+) B)) (<) (` H B)) (br (` H (opr (opr A (+) (1)) (+) B)) (<) (` H B))) (ruclem28.a nncn 1cn ruclem.b nncn add23 H fveq2i ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem27 eqbrtr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a A peano2nn ax-mp ruclem.b A + 1 opr B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem23 lttr mpan))
+thm (ruclem32 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v) (B w) (B v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem28.a (e. A (NN))) (ruclem.b (e. B (NN)))) (br (` G A) (<) (` H B)) (ruclem.b w cv 1 A + opreq2 G fveq2d G A ` < breq2d w cv v cv A + opreq2 G fveq2d G A ` < breq2d w cv v cv + 1 opr A + opreq2 G fveq2d G A ` < breq2d w cv B A + opreq2 G fveq2d G A ` < breq2d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem26 v cv v cv NN e. v cv 1 if A + opreq2 G fveq2d G A ` < breq2d v cv v cv NN e. v cv 1 if + 1 opreq1 A + opreq2d G fveq2d G A ` < breq2d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a 1nn v cv elimel ruclem30 dedth nnind ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem25 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 lttr mp2an ruclem28.a w cv 1 + B opreq1 H fveq2d < H B ` breq1d w cv v cv + B opreq1 H fveq2d < H B ` breq1d w cv v cv + 1 opr + B opreq1 H fveq2d < H B ` breq1d w cv A + B opreq1 H fveq2d < H B ` breq1d 1cn ruclem.b nncn addcom H fveq2i ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem27 eqbrtr v cv v cv NN e. v cv 1 if + B opreq1 H fveq2d < H B ` breq1d v cv v cv NN e. v cv 1 if + 1 opreq1 + B opreq1d H fveq2d < H B ` breq1d imbi12d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn v cv elimel ruclem.b ruclem31 dedth nnind ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem28.a ruclem.b A B nnaddclt mp2an ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.b ruclem23 lttr mp2an))
+thm (ruclem33 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C))))) (/\/\ (C_ (ran G) (RR)) (-. (= (ran G) ({/}))) (E.e. w (RR) (A.e. v (ran G) (br (cv v) (<_) (cv w))))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G NN RR frn ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G NN RR ffn ax-mp 1nn G NN 1 fnfvrn mp2an G 1 ` G ran n0i ax-mp ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G NN RR ffn ax-mp G NN v cv w fvelrn ax-mp G w cv ` v cv <_ H 1 ` breq1 G w cv ` H 1 ` ltlet w cv w cv NN e. w cv 1 if G fveq2 RR eleq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn w cv elimel ruclem22 dedth ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn ruclem23 jctir w cv w cv NN e. w cv 1 if G fveq2 < H 1 ` breq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn w cv elimel 1nn ruclem32 dedth sylc syl5bi com12 r19.23aiv sylbi rgen w cv H 1 ` v cv <_ breq2 v G ran biraldv RR rcla4ev mp2an 3pm3.2i))
+thm (ruclem34 ((S w) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<))))) (e. S (RR)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 suprcli ruclem.5 RR eleq1i mpbir))
+thm (ruclem35 ((S w) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<)))) (ruclem.a (e. A (NN)))) (/\ (br (` G A) (<) S) (br S (<) (` H A))) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem26 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G NN RR ffn ax-mp ruclem.a A peano2nn ax-mp G NN A + 1 opr fnfvrn mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 G A + 1 opr ` suprubi ax-mp ruclem.5 breqtrr ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem22 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ltletr mp2an ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem17 G NN RR ffn ax-mp G NN u cv w fvelrn ax-mp G w cv ` u cv H A + 1 opr ` < breq2 negbid G w cv ` H A + 1 opr ` ltnsymt w cv w cv NN e. w cv 1 if G fveq2 RR eleq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn w cv elimel ruclem22 dedth ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 jctir w cv w cv NN e. w cv 1 if G fveq2 < H A + 1 opr ` breq1d ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 1nn w cv elimel ruclem.a A peano2nn ax-mp ruclem32 dedth sylc syl5bi com12 r19.23aiv sylbi rgen ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 w v ruclem33 H A + 1 opr ` u suprnubi mp2an ruclem.5 H A + 1 opr ` < breq2i negbii mpbir ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 lenlt mpbir ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem27 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a A peano2nn ax-mp ruclem23 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem23 lelttr mp2an pm3.2i))
+thm (ruclem36 ((x y) (x z) (y z) (F z)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<)))) (ruclem.a (e. A (NN)))) (-. (= (` F A) S)) (ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.a ruclem29 ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem.a ruclem35 F A ` S G A ` < breq2 F A ` S < H A ` breq1 anbi12d mpbiri mto))
+thm (ruclem37 ((S w) (x y) (x z) (w x) (y z) (w y) (w z) (C w) (D w) (G w) (H w) (F z) (F w)) ((ruclem.0 (:--> F (NN) (RR))) (ruclem.1 (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (ruclem.2 (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (ruclem.3 (= G (o. (1st) (opr D (seq1) C)))) (ruclem.4 (= H (o. (2nd) (opr D (seq1) C)))) (ruclem.5 (= S (sup (ran G) (RR) (<))))) (-. (:-onto-> F (NN) (RR))) (w cv w cv NN e. w cv 1 if F fveq2 S eqeq1d negbid ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 1nn w cv elimel ruclem36 dedth nrex ruclem.0 F NN RR ffn ax-mp F NN S w fvelrn ax-mp mtbir ruclem.0 ruclem.1 ruclem.2 ruclem.3 ruclem.4 ruclem.5 ruclem34 F NN RR forn S eleq2d mpbiri mto))
+thm (ruclem38 ((F z) (x y) (x z) (y z)) ((ruclem38.0 (:--> F (NN) (RR)))) (-. (:-onto-> F (NN) (RR))) (ruclem38.0 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. eqid x y z x cv RR RR X. e. y cv RR e. /\ z cv 1st x cv ` < y cv br y cv < 2nd x cv ` br /\ 2 x. y cv opr + 2nd x cv ` opr / 3 opr y cv + 2 x. 2nd x cv ` opr opr / 3 opr <,> 2 x. 1st x cv ` opr + 2nd x cv ` opr / 3 opr 1st x cv ` + 2 x. 2nd x cv ` opr opr / 3 opr <,> if = /\ {<<,>,>|} eqid 1st x y z x cv RR RR X. e. y cv RR e. /\ z cv 1st x cv ` < y cv br y cv < 2nd x cv ` br /\ 2 x. y cv opr + 2nd x cv ` opr / 3 opr y cv + 2 x. 2nd x cv ` opr opr / 3 opr <,> 2 x. 1st x cv ` opr + 2nd x cv ` opr / 3 opr 1st x cv ` + 2 x. 2nd x cv ` opr opr / 3 opr <,> if = /\ {<<,>,>|} seq1 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. opr o. eqid 2nd x y z x cv RR RR X. e. y cv RR e. /\ z cv 1st x cv ` < y cv br y cv < 2nd x cv ` br /\ 2 x. y cv opr + 2nd x cv ` opr / 3 opr y cv + 2 x. 2nd x cv ` opr opr / 3 opr <,> 2 x. 1st x cv ` opr + 2nd x cv ` opr / 3 opr 1st x cv ` + 2 x. 2nd x cv ` opr opr / 3 opr <,> if = /\ {<<,>,>|} seq1 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. opr o. eqid 1st x y z x cv RR RR X. e. y cv RR e. /\ z cv 1st x cv ` < y cv br y cv < 2nd x cv ` br /\ 2 x. y cv opr + 2nd x cv ` opr / 3 opr y cv + 2 x. 2nd x cv ` opr opr / 3 opr <,> 2 x. 1st x cv ` opr + 2nd x cv ` opr / 3 opr 1st x cv ` + 2 x. 2nd x cv ` opr opr / 3 opr <,> if = /\ {<<,>,>|} seq1 1 F 1 ` + 1 opr F 1 ` + 2 opr <,> <,> {} F NN 1 {} \ |` u. opr o. ran RR < sup eqid ruclem37))
+thm (ruclem39 () () (-. (:-onto-> F (NN) (RR))) (F NN RR fof F F NN RR :--> F NN 0 {} X. if NN RR foeq1 negbid F F NN RR :--> F NN 0 {} X. if NN RR feq1 NN 0 {} X. F NN RR :--> F NN 0 {} X. if NN RR feq1 0re elisseti NN fconst 0re 0 RR snssi ax-mp NN 0 {} X. NN 0 {} RR fss mp2an elimhyp ruclem38 dedth syl F NN RR :-onto-> pm2.01 ax-mp))
+thm (ruc () () (br (NN) (~<) (RR)) (NN RR brsdom reex nnssre RR V NN ssdom2g mp2 f cv ruclem39 f cv NN RR df-f1o pm3.27bd mto f nex reex NN f bren mtbir mpbir2an))
+thm (nn0ennn ((x y)) () (br (NN0) (~~) (NN)) (nn0ex x cv nn0p1nnt y cv elnnnn0 pm3.27bd 1cn y cv 1 x cv subaddt mp3an2 x cv y cv - 1 opr eqcom y cv 1 + x cv opr eqcom 3bitr4g 1cn 1 x cv axaddcom mpan y cv eqeq2d y cv CC e. adantl bitrd y cv nncnt x cv nn0cnt syl2an ancoms en3))
+thm (nnenom ((x y) (x z) (y z)) () (br (NN) (~~) (om)) (nnex 0z x y y cv x cv + 1 opr = {<,>|} 0 rec om |` eqid z om2uzf1o z nn0z NN0 z ZZ 0 <_ z cv br {e.|} x y y cv x cv + 1 opr = {<,>|} 0 rec om |` om f1oeq3 ax-mp mpbir omex x y y cv x cv + 1 opr = {<,>|} 0 rec om |` NN0 f1oen ax-mp nn0ennn entr2))
+thm (xpnnen ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (br (X. (NN) (NN)) (~~) (NN)) (nnex nnex xpex x cv NN NN elxp5 x cv |^| |^| + x cv {} ran U. opr ^ 2 opr x cv {} ran U. nnaddclt x cv |^| |^| x cv {} ran U. nnaddclt 2nn x cv |^| |^| + x cv {} ran U. opr 2 nnexpclt mpan2 syl x cv |^| |^| NN e. x cv {} ran U. NN e. pm3.27 sylanc x cv x cv |^| |^| x cv {} ran U. <,> = adantl sylbi x cv z cv w cv <,> inteq inteqd z visset w cv op1stb syl6eq x cv z cv w cv <,> sneq rneqd unieqd z visset w visset op2nda syl6eq + opreq12d ^ 2 opreq1d x cv z cv w cv <,> sneq rneqd unieqd z visset w visset op2nda syl6eq + opreq12d y cv v cv u cv <,> inteq inteqd v visset u cv op1stb syl6eq y cv v cv u cv <,> sneq rneqd unieqd v visset u visset op2nda syl6eq + opreq12d ^ 2 opreq1d y cv v cv u cv <,> sneq rneqd unieqd v visset u visset op2nda syl6eq + opreq12d eqeqan12d z cv w cv v cv u cv nn0opth2t z cv nnnn0t w cv nnnn0t anim12i v cv nnnn0t u cv nnnn0t anim12i syl2an sylan9bbr x cv z cv w cv <,> y cv v cv u cv <,> eqeq12 z visset w visset u visset v cv opth syl6bb z cv NN e. w cv NN e. /\ v cv NN e. u cv NN e. /\ /\ adantl bitr4d exp43 com23 r19.23aivv r19.23advv imp x cv NN NN z w elxp2 y cv NN NN v u elxp2 syl2anb V dom2 ax-mp nnex nnex 1nn elisseti xpsnen ensymi nnex nnex xpex NN ssid 1nn 1 NN snssi ax-mp NN NN 1 {} NN ssxp mp2an NN NN X. V NN 1 {} X. ssdom2g mp2 NN NN 1 {} X. NN NN X. endomtr mp2an NN NN X. NN sbth mp2an))
+thm (xpomen () () (br (X. (om) (om)) (~~) (om)) (omex omex xpex nnenom nnenom nnex omex nnex omex xpen mp2an xpnnen entr3 nnenom entr))
+thm (znnenlem () () (-> (/\ (/\ (br (0) (<_) (cv x)) (-. (br (0) (<_) (cv y)))) (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ)))) (<-> (= (cv y) (cv x)) (= (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1))))) (y cv x cv = 2 x. x cv opr 2 -u x. y cv opr + 1 opr = pm5.21 0re y cv 0 ltnlet mpan2 x cv RR e. adantr 0re y cv 0 x cv ltletrt mp3an2 exp3a sylbird com23 imp3a y cv x cv ltnet syld ancoms x cv zret y cv zret syl2an impcom x cv y cv -u zneo y cv znegclt sylan2 y cv zcnt 2cn 2 y cv mulneg12t mpan syl x cv ZZ e. adantl + 1 opreq1d 2 x. x cv opr eqeq2d mtbird 0 <_ x cv br 0 <_ y cv br -. /\ adantl sylanc))
+thm (znnen ((x y)) () (br (ZZ) (~~) (NN)) (zex x cv elnn0z 2nn0 2 x cv nn0mulclt mpan sylbir 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iftrue NN0 eleq1d x cv ZZ e. adantl mpbird 2z 2 znegclt ax-mp 2 -u x cv zmulclt mpan 0 <_ x cv br -. adantr x cv zret x cv renegclt 2pos 2re 2 x cv -u axmulgt0 mpan mpani syl 0re x cv 0 ltnlet mpan2 x cv lt0neg1t bitr3d x cv recnt 2cn 2 x cv mulneg12t mpan syl 0 < breq2d 3imtr4d 2re renegcl 2 -u x cv axmulrcl mpan 0re 0 2 -u x. x cv opr ltlet mpan syl syld syl imp jca 2 -u x. x cv opr elnn0z biimpr 2 -u x. x cv opr peano2nn0 3syl 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iffalse NN0 eleq1d x cv ZZ e. adantl mpbird pm2.61an2 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iftrue 0 <_ y cv br 2 x. y cv opr 2 -u x. y cv opr + 1 opr iftrue eqeqan12d 2cn 2re 2pos gt0ne0i x cv y cv mulcant2 mp3an1 x cv zcnt y cv zcnt syl2an sylan9bb bicomd exp x y znnenlem x y equcom syl5bb 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iftrue 0 <_ y cv br 2 x. y cv opr 2 -u x. y cv opr + 1 opr iffalse eqeqan12d x cv ZZ e. y cv ZZ e. /\ adantr bitr4d exp y x znnenlem 0 <_ x cv br -. 0 <_ y cv br ancom x cv ZZ e. y cv ZZ e. ancom syl2anb 2 -u x. x cv opr + 1 opr 2 x. y cv opr eqcom syl6bbr 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iffalse 0 <_ y cv br 2 x. y cv opr 2 -u x. y cv opr + 1 opr iftrue eqeqan12d x cv ZZ e. y cv ZZ e. /\ adantr bitr4d exp 1cn 2 -u x. x cv opr 2 -u x. y cv opr 1 addcan2t mp3an3 2cn negcl 2 -u x cv axmulcl mpan 2cn negcl 2 -u y cv axmulcl mpan syl2an 2cn negcl 2cn 2re 2pos gt0ne0i negn0 x cv y cv mulcant2 mp3an1 bitr2d x cv zcnt y cv zcnt syl2an 0 <_ x cv br 2 x. x cv opr 2 -u x. x cv opr + 1 opr iffalse 0 <_ y cv br 2 x. y cv opr 2 -u x. y cv opr + 1 opr iffalse eqeqan12d bicomd sylan9bbr exp 4cases bicomd V dom2 ax-mp nn0ex nn0ssz NN0 V ZZ ssdomg mp2 ZZ NN0 sbth mp2an nn0ennn entr))
+thm (qnnen ((x y) (x z) (w x) (y z) (w y) (w z)) () (br (QQ) (~~) (NN)) (x y z x cv ZZ e. y cv NN e. /\ z cv x cv / y cv opr = /\ {<<,>,>|} ZZ NN X. QQ df-fo x cv / y cv oprex x y z x cv ZZ e. y cv NN e. /\ z cv x cv / y cv opr = /\ {<<,>,>|} eqid fnoprab2 x cv / y cv oprex x y z x cv ZZ e. y cv NN e. /\ z cv x cv / y cv opr = /\ {<<,>,>|} eqid w cv elrnoprab w cv x y elq bitr4 eqri mpbir2an zex nnex xpex x y z x cv ZZ e. y cv NN e. /\ z cv x cv / y cv opr = /\ {<<,>,>|} QQ fodom ax-mp znnen nnex enref zex nnex nnex nnex xpen mp2an xpnnen entr QQ ZZ NN X. NN domentr mp2an nnex nnssq NN V QQ ssdomg mp2 QQ NN sbth mp2an))
+thm (unbenlem ((m n) (m z) (m w) (m v) (m u) (m t) (A m) (n z) (n w) (n v) (n u) (n t) (A n) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t) (G m) (G n) (G v) (G u) (G t)) ((unbenlem.1 (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))))) (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (om))) (A G `' A " om entrt A V G `' A |` G `' A " f1oeng nnex A ssex 1z unbenlem.1 t om2uzf1o t nnz NN t ZZ 1 <_ t cv br {e.|} G om f1oeq3 ax-mp mpbir G om NN f1ocnv ax-mp G `' NN om f1of1 ax-mp G `' NN om A f1ores mpan sylc m NN n A m cv < n cv br E.e. A.e. adantr 1z unbenlem.1 v cv t om2uzuz t nnz syl6eleqr m cv G v cv ` < n cv breq1 n A birexdv NN rcla4v syl A NN C_ adantr u cv G `' A " G fvres n cv eqeq1d biimpa v cv om e. adantll 1z unbenlem.1 v cv u cv om2uzlt2 G `' A imassrn G dfdm4 1z unbenlem.1 t om2uzf1o t nnz NN t ZZ 1 <_ t cv br {e.|} G om f1oeq3 ax-mp mpbir G om NN f1of ax-mp G om NN fdm ax-mp eqtr3 sseqtr u cv sseli sylan2 G u cv ` n cv G v cv ` < breq2 sylan9bb syldan biimparc exp44 imp31 r19.22dva 1z unbenlem.1 t om2uzf1o t nnz NN t ZZ 1 <_ t cv br {e.|} G om f1oeq3 ax-mp mpbir G om NN f1ocnv ax-mp G `' NN om f1of1 ax-mp G `' NN om A f1ores mpan G `' A |` A G `' A " f1ocnv syl 1z unbenlem.1 t om2uzf1o t nnz NN t ZZ 1 <_ t cv br {e.|} G om f1oeq3 ax-mp mpbir G om NN f1ofun ax-mp G A funcnvres2 ax-mp G `' A |` `' G G `' A " |` G `' A " A f1oeq1 ax-mp sylib G G `' A " |` G `' A " A f1ofo G G `' A " |` G `' A " A forn syl n cv eleq2d G G `' A " |` G `' A " A f1ofn G G `' A " |` G `' A " n cv u fvelrn syl bitr3d syl biimpa syl5 exp4b com4l imp r19.23adv syld exp com3l imp r19.21aiv G `' A imassrn G dfdm4 1z unbenlem.1 t om2uzf1o t nnz NN t ZZ 1 <_ t cv br {e.|} G om f1oeq3 ax-mp mpbir G om NN f1of ax-mp G om NN fdm ax-mp eqtr3 sseqtr jctil G `' A " v u unbnnt syl sylanc))
+thm (unben ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (NN))) (x y y cv x cv + 1 opr = {<,>|} 1 rec om |` eqid A m n unbenlem omex nnenom ensymi A om NN entrt mpan2 syl))
+thm (infpnlem1 ((N j) (M j) (K j)) ((infpnlem.1 (= K (opr (` (!) N) (+) (1))))) (-> (/\ (e. N (NN)) (e. M (NN))) (-> (/\ (/\ (br (1) (<) M) (e. (opr K (/) M) (NN))) (A.e. j (NN) (-> (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) (br M (<_) (cv j))))) (/\ (br N (<) M) (A.e. j (NN) (-> (e. (opr M (/) (cv j)) (NN)) (\/ (= (cv j) (1)) (= (cv j) M))))))) (M N lenltt M nnret N nnret syl2an ancoms 1 < M br adantr N M facndivt K / M opr nnzt infpnlem.1 / M opreq1i ZZ eleq1i sylib nsyl N nnnn0t sylanl1 exp32 imp sylbird a3d exp imp3a j NN 1 < j cv br K / j cv opr NN e. /\ M <_ j cv br -> A.e. adantrd j cv M letri3t j cv nnret M nnret syl2an biimprd exp4b com3l imp32 N NN e. adantll 1 < j cv br K / j cv opr NN e. /\ imim2d com23 j cv M K nndivtrt exp 3com13 3expa N nnnn0t N facclt syl ! N ` peano2nn syl infpnlem.1 NN eleq1i sylibr K nncnt syl sylanl1 j cv <_ M br adantrl sylan2d exp4d com24 exp32 com24 imp31 com14 3imp com3l r19.20dva exp 1 < M br adantld imp3a M j primet N NN e. adantl sylibrd jcad))
+thm (infpnlem2 ((j k) (N j) (N k) (K j) (K k)) ((infpnlem.1 (= K (opr (` (!) N) (+) (1))))) (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j)))))))) (j cv K 1 < breq2 j cv K K / opreq2 NN eleq1d anbi12d NN rcla4ev N nnnn0t N facclt syl ! N ` peano2nn syl infpnlem.1 NN eleq1i sylibr N nnnn0t N facclt syl ! N ` nnge1t syl N nnnn0t N facclt syl 1nn jctil 1 ! N ` nnleltp1t syl mpbid infpnlem.1 syl6breqr 1nn N nnnn0t N facclt syl ! N ` peano2nn syl infpnlem.1 NN eleq1i sylibr K nncnt K nnne0t jca syl K dividt syl NN eleq1d mpbiri jca sylanc j cv k cv 1 < breq2 j cv k cv K / opreq2 NN eleq1d anbi12d nnwos syl infpnlem.1 j cv k infpnlem1 r19.22dva mpd))
+thm (infpn ((j k) (j n) (j m) (k n) (k m) (m n) (S j) (S k)) ((infpn.1 (= S ({e.|} n (NN) (/\ (br (1) (<) (cv n)) (A.e. m (NN) (-> (e. (opr (cv n) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv n)))))))))) (br S (~~) (NN)) (infpn.1 n NN 1 < n cv br m NN n cv / m cv opr NN e. m cv 1 = m cv n cv = \/ -> A.e. /\ ssrab eqsstr ! j cv ` + 1 opr eqid k m infpnlem2 j cv nnge1t k cv NN e. adantr ax1re 1 j cv k cv lelttrt mp3an1 j cv nnret k cv nnret syl2an mpand ancld m NN k cv / m cv opr NN e. m cv 1 = m cv k cv = \/ -> A.e. anim1d j cv < k cv br 1 < k cv br m NN k cv / m cv opr NN e. m cv 1 = m cv k cv = \/ -> A.e. anass syl6ib r19.22dva mpd infpn.1 k cv eleq2i n cv k cv 1 < breq2 n cv k cv / m cv opreq1 NN eleq1d n cv k cv m cv eqeq2 m cv 1 = orbi2d imbi12d m NN biraldv anbi12d NN elrab bitr j cv < k cv br anbi1i k cv NN e. 1 < k cv br m NN k cv / m cv opr NN e. m cv 1 = m cv k cv = \/ -> A.e. /\ j cv < k cv br anass 1 < k cv br m NN k cv / m cv opr NN e. m cv 1 = m cv k cv = \/ -> A.e. /\ j cv < k cv br ancom k cv NN e. anbi2i 3bitr birex2 sylibr rgen S j k unben mp2an))
+thm (resdomq () () (br (QQ) (~<) (RR)) (qnnen ruc QQ NN RR ensdomtr mp2an))
+thm (aleph1re () () (br (` (aleph) (1o)) (~<_) (RR)) (aleph0 omex nnenom ensymi eqbrtr ruc aleph {/} ` NN RR ensdomtr mp2an {/} RR alephnbtwn2 aleph {/} ` ~< RR br RR ~< aleph {/} suc ` br imnan mpbir ax-mp df-1o aleph fveq2i RR ~< breq2i mtbir aleph 1o fvex reex aleph 1o ` V RR V domtri mp2an mpbir))
+thm (infxpidmlem1 () ((infxpidmlem1.1 (e. A (V))) (infxpidmlem1.2 (e. B (V)))) (-> (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (-> (/\ (br (cv x) (~~) A) (br (cv x) (~~) B)) (br (cv x) (~~) (u. A B)))) (x cv A B u. sbth infxpidmlem1.1 A B ssun1 A V A B u. ssdomg mp2 x cv A A B u. endomtr mpan2 1o ~< x cv br x cv ~~ x cv x cv X. br /\ x cv ~~ B br ad2antrl A B u. A B X. x cv domentr A B unxpdom infxpidmlem1.1 A V 1o x cv sdomentr ax-mp infxpidmlem1.2 B V 1o x cv sdomentr ax-mp syl2an anandis x cv ~~ x cv x cv X. br adantlr x cv x cv x cv X. A B X. entrt infxpidmlem1.1 infxpidmlem1.2 xpex x cv ensym syl x visset infxpidmlem1.1 x visset infxpidmlem1.2 xpen sylan2 1o ~< x cv br adantll sylanc sylanc exp 1onn x visset 1o infsdomnn mpan2 sylan))
+thm (infxpidmlem2 ((f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem2.2 (e. B (V)))) (<-> (e. B H) (\/ (= B ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> B (X. (cv x) (cv x)) (cv x)))))) (infxpidmlem2.2 f cv B {/} eqeq1 f cv B t cv t cv X. t cv f1oeq1 om ~<_ t cv br t cv A C_ /\ anbi2d t biexdv orbi12d infxpidmlem.1 elab2 x cv t cv om ~<_ breq2 x cv t cv A sseq1 anbi12d x cv t cv x cv xpeq1 x cv t cv t cv xpeq2 eqtrd x cv x cv X. t cv t cv X. B x cv f1oeq2 syl x cv t cv B t cv t cv X. f1oeq3 bitrd anbi12d cbvexv B {/} = orbi2i bitr4))
+thm (infxpidmlem3 ((D x) (f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem2.2 (e. B (V))) (infxpidmlem3.3 (e. D (V)))) (-> (/\ (/\ (br (om) (~<_) D) (C_ D A)) (:-1-1-onto-> B (X. D D) D)) (e. B H)) (infxpidmlem3.3 x cv D om ~<_ breq2 x cv D A sseq1 anbi12d x cv D x cv xpeq1 x cv D D xpeq2 eqtrd x cv x cv X. D D X. B x cv f1oeq2 syl x cv D B D D X. f1oeq3 bitrd anbi12d cla4ev x om ~<_ x cv br x cv A C_ /\ B x cv x cv X. x cv :-1-1-onto-> /\ E. B {/} = olc infxpidmlem.1 infxpidmlem2.2 x infxpidmlem2 sylibr syl))
+thm (infxpidmlem4 ((f x) (g x) (t x) (A x) (f g) (f t) (A f) (g t) (A g) (A t) (H x) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))))) (-> (e. (cv g) H) (= (dom (cv g)) (X. (ran (cv g)) (ran (cv g))))) (infxpidmlem.1 g visset x infxpidmlem2 g cv {/} dmeq dm0 syl6eq g cv {/} rneq rn0 syl6eq g cv ran {/} g cv ran xpeq2 syl g cv ran xp0 syl6eq eqtr4d g cv x cv x cv X. x cv f1o2 g cv x cv x cv X. fndm g cv ran x cv g cv ran xpeq1 g cv ran x cv x cv xpeq2 eqtr2d sylan9eq g cv `' Fun 3adant2 sylbi om ~<_ x cv br x cv A C_ /\ adantl x 19.23aiv jaoi sylbi))
+thm (infxpidmlem5 ((y z) (f y) (g y) (t y) (A y) (f z) (g z) (t z) (A z) (f g) (f t) (A f) (g t) (A g) (A t) (C y) (C z) (C f) (C g) (C t) (H y) (H z) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))))) (-> (/\ (C_ C H) (e. (cv g) C)) (-> (/\ (e. (cv y) (ran (cv g))) (e. (cv z) (ran (cv g)))) (e. (<,> (cv y) (cv z)) (dom (U. C))))) (C H g cv ssel2 infxpidmlem.1 g infxpidmlem4 syl y cv z cv <,> eleq2d z visset y cv g cv ran g cv ran opelxp syl6bb g C g cv dom ssiun2 C g dmuni syl6ssr y cv z cv <,> sseld C H C_ adantl sylbird))
+thm (infxpidmlem6 ((f y) (g y) (t y) (A y) (f g) (f t) (A f) (g t) (A g) (A t) (B y) (B f) (B g) (B t) (C y) (C f) (C g) (C t) (H y) (H g)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C))))) (<-> (e. (cv y) B) (E.e. g C (e. (cv y) (ran (cv g))))) (infxpidmlem6.2 C g rnuni eqtr y cv eleq2i y cv g C g cv ran eliun bitr))
+thm (infxpidmlem7 ((x y) (x z) (w x) (f x) (g x) (h x) (v x) (t x) (A x) (y z) (w y) (f y) (g y) (h y) (v y) (t y) (A y) (w z) (f z) (g z) (h z) (v z) (t z) (A z) (f w) (g w) (h w) (v w) (t w) (A w) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C w) (C v) (C f) (C g) (C h) (C t) (H x) (H y) (H z) (H w) (H v) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C))))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (:-1-1-onto-> (U. C) (X. B B) B)) (g C g cv Fun g cv `' Fun /\ h C g cv h cv C_ h cv g cv C_ \/ A.e. r19.26 g C h fun11uni sylbir C H g cv ssel infxpidmlem.1 g visset x infxpidmlem2 {/} f10 g cv {/} {/} {/} f1eq1 mpbiri g cv {/} {/} df-f1 g cv {/} {/} ffun g cv `' Fun anim1i sylbi syl g cv x cv x cv X. x cv f1of1 g cv x cv x cv X. x cv df-f1 g cv x cv x cv X. x cv ffun g cv `' Fun anim1i sylbi syl om ~<_ x cv br x cv A C_ /\ adantl x 19.23aiv jaoi sylbi syl6 r19.21aiv sylan C H g cv ssel2 infxpidmlem.1 g infxpidmlem4 syl g C y cv g cv ran e. ra4e infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 sylibr exp ssrdv g cv ran B g cv ran B ssxp anidms syl C H C_ adantl eqsstrd y cv sseld exp r19.23adv C g dmuni y cv eleq2i y cv g C g cv dom eliun bitr syl5ib ssrdv g C h C g cv h cv C_ h cv g cv C_ \/ A.e. A.e. adantr B B relxp C H C_ g C h C g cv h cv C_ h cv g cv C_ \/ A.e. A.e. /\ a1i g cv w cv h cv sseq1 g cv w cv h cv sseq2 orbi12d h cv v cv w cv sseq2 h cv v cv w cv sseq1 orbi12d C C rcla42v w cv v cv rnss y cv sseld z cv v cv ran e. anim1d infxpidmlem.1 C v y z infxpidmlem5 syl9r w cv C e. adantrl v cv w cv rnss z cv sseld y cv w cv ran e. anim2d infxpidmlem.1 C w y z infxpidmlem5 syl9r v cv C e. adantrr jaod exp com3l syld com13 imp r19.23advv infxpidmlem.1 infxpidmlem6.2 y w infxpidmlem6 infxpidmlem.1 infxpidmlem6.2 z v infxpidmlem6 anbi12i z visset y cv B B opelxp w C v C y cv w cv ran e. z cv v cv ran e. reeanv 3bitr4 syl5ib relssdv eqssd jca C U. Fun C U. `' Fun C U. dom B B X. = an23 C U. B B X. df-fn C U. `' Fun anbi1i bitr4 sylib infxpidmlem6.2 eqcomi jctir C U. B B X. B f1o2 C U. B B X. Fn C U. `' Fun C U. ran B = df-3an bitr sylibr))
+thm (infxpidmlem8 ((x y) (f x) (g x) (h x) (t x) (A x) (f y) (g y) (h y) (t y) (A y) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B x) (B y) (B f) (B g) (B h) (B t) (C x) (C y) (C f) (C g) (C h) (C t) (H x) (H y) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem6.2 (= B (ran (U. C)))) (infxpidmlem8.3 (e. C (V)))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (e. (U. C) H)) (C H g cv ssel2 infxpidmlem.1 g visset x infxpidmlem2 biimp ord g cv x cv x cv X. x cv f1ofo g cv x cv x cv X. x cv forn syl eqcomd om ~<_ x cv br x cv A C_ /\ anim1i ancoms x 19.22i g visset g cv V rnexg ax-mp x cv g cv ran om ~<_ breq2 x cv g cv ran A sseq1 anbi12d ceqsexv sylib syl6 syl om g cv ran B domtr g C y cv g cv ran e. ra4e infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 sylibr exp ssrdv g visset g cv V rnexg ax-mp g cv ran V B ssdomg ax-mp syl sylan2 expcom C H C_ adantl g cv ran A C_ adantrd syld exp r19.23adv C uni0b C {/} {} g dfss3 g {/} elsn g C biral 3bitr negbii g C g cv {/} = rexnal bitr4 syl5ib C H g cv ssel2 infxpidmlem.1 g visset x infxpidmlem2 biimp ord g cv x cv x cv X. x cv f1ofo g cv x cv x cv X. x cv forn syl eqcomd om ~<_ x cv br x cv A C_ /\ anim1i ancoms x 19.22i g visset g cv V rnexg ax-mp x cv g cv ran om ~<_ breq2 x cv g cv ran A sseq1 anbi12d ceqsexv sylib syl6 syl om ~<_ g cv ran br g cv ran A C_ pm3.27 syl6 g cv {/} rneq rn0 syl6eq A 0ss g cv {/} = a1i eqsstrd pm2.61d2 y cv sseld exp r19.23adv infxpidmlem.1 infxpidmlem6.2 y g infxpidmlem6 syl5ib ssrdv jctird g C h C g cv h cv C_ h cv g cv C_ \/ A.e. A.e. adantr infxpidmlem.1 infxpidmlem6.2 g h infxpidmlem7 jctird infxpidmlem.1 infxpidmlem8.3 uniex infxpidmlem6.2 infxpidmlem8.3 uniex C U. V rnexg ax-mp eqeltr infxpidmlem3 syl6 C U. {/} = x om ~<_ x cv br x cv A C_ /\ C U. x cv x cv X. x cv :-1-1-onto-> /\ E. orc infxpidmlem.1 infxpidmlem8.3 uniex x infxpidmlem2 sylibr pm2.61d2))
+thm (infxpidmlem9 ((f z) (g z) (h z) (t z) (A z) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (H z) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (E.e. g H (A.e. h H (-. (C: (cv g) (cv h))))) (infxpidmlem.1 f f cv {/} = t om ~<_ t cv br t cv A C_ /\ f cv t cv t cv X. t cv :-1-1-onto-> /\ E. unab eqtr4 {/} f df-sn p0ex eqeltrr t A P~ om ~<_ t cv br f cv t cv t cv X. t cv :-1-1-onto-> /\ df-rex t visset A elpw om ~<_ t cv br f cv t cv t cv X. t cv :-1-1-onto-> /\ anbi1i t cv A C_ om ~<_ t cv br f cv t cv t cv X. t cv :-1-1-onto-> /\ ancom om ~<_ t cv br f cv t cv t cv X. t cv :-1-1-onto-> t cv A C_ an23 3bitr t biex bitr f biabi infxpidmlem.2 pwex t visset t visset xpex t visset t cv t cv X. V t cv V f mapex mp2an f cv t cv t cv X. t cv f1of om ~<_ t cv br adantl f ss2abi ssexi t abrexex2 eqeltrr unex eqeltr z g h zorn2 infxpidmlem.1 z cv U. ran eqid z visset g h infxpidmlem8 mpg))
+thm (infxpidmlem10 ((f y) (g y) (h y) (v y) (t y) (A y) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (H y) (H v) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (A.e. h H (-. (C: (cv g) (cv h)))) (-> (br (om) (~<_) A) (-. (= (cv g) ({/}))))) (g cv {/} v cv psseq1 v cv 0pss syl6bb bicon2d h cv v cv g cv psseq2 negbid H rcla4v impcom syl5bir com12 con3d exp r19.23adv infxpidmlem.1 v visset y visset infxpidmlem3 exp v cv y cv y cv X. y cv f1ofo v cv y cv y cv X. y cv forn syl eqcomd v cv {/} rneq rn0 syl6eq sylan9eq exp y visset infn0 nsyli com12 y cv A C_ adantr jcad v 19.22dv imp an1rs om y cv endom omex y visset omex y visset xpen anidms xpomen y visset y visset xpex y cv y cv X. V om om X. om enen1 mpan mpbii syl y visset y cv V om y cv y cv X. enen2 mpan mpbid y visset y cv y cv X. v bren sylib jca sylan y 19.23aiv infxpidmlem.2 om y domen v H v cv {/} = -. df-rex 3imtr4 syl5))
+thm (infxpidmlem11 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (/\ (/\ (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x))) (/\ (br (cv x) (~~) (cv y)) (C_ (cv y) (\ A (cv x))))) (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y))) (E.e. h H (C: (cv g) (cv h)))) (h cv g cv u cv u. g cv psseq2 H rcla4ev g cv x cv x cv X. x cv u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv f1oun x cv y cv x cv x cv xpdisj2 x cv y cv x cv x cv xpdisj1 x cv y cv x cv y cv xpdisj1 jca x cv x cv X. y cv x cv X. y cv y cv X. undisj2 sylib jca x cv x cv X. x cv y cv X. y cv x cv X. y cv y cv X. u. undisj2 sylib ancri sylan2 x cv y cv x cv y cv xpun x cv x cv X. x cv y cv X. y cv x cv X. y cv y cv X. u. unass eqtr x cv y cv u. x cv y cv u. X. x cv x cv X. x cv y cv X. y cv x cv X. y cv y cv X. u. u. u. g cv u cv u. x cv y cv u. f1oeq2 ax-mp sylibr y cv A x cv \ x cv sslin x cv A difdisj syl6ss x cv y cv i^i ss0b sylib sylan2 om ~<_ x cv br x cv A C_ /\ adantll infxpidmlem.1 g visset u visset unex x visset y visset unex infxpidmlem3 x visset x cv y cv ssun1 x cv V x cv y cv u. ssdomg mp2 om x cv x cv y cv u. domtr mpan2 A x cv difss y cv A x cv \ A sstr mpan2 x cv A C_ anim2i x cv A y cv unss sylib anim12i anassrs sylan exp g cv x cv x cv X. x cv :-1-1-onto-> u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv :-1-1-onto-> /\ adantlr mpd x cv ~~ y cv br adantrl g cv u cv disjpss g cv ran x cv u cv ran y cv ineq12 {/} eqeq1d g cv x cv x cv X. x cv f1ofo g cv x cv x cv X. x cv forn syl u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv f1ofo u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv forn syl syl2an g cv x cv x cv X. x cv f1orel g cv u cv relin g cv u cv i^i relrn0 g cv u cv rnin g cv ran u cv ran i^i {/} g cv u cv i^i ran sseq2 mpbii g cv u cv i^i ran ss0 syl syl5bir 3syl u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv :-1-1-onto-> adantr sylbird imp om ~<_ x cv br x cv ~~ y cv br /\ adantr u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv f1orel u cv relrn0 syl u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv f1ofo u cv x cv y cv X. y cv x cv X. y cv y cv X. u. u. y cv forn syl {/} eqeq1d bitrd negbid y visset infn0 syl5bir om x cv y cv domentr syl5 imp g cv x cv x cv X. x cv :-1-1-onto-> adantll x cv y cv i^i {/} = adantlr sylanc exp43 y cv A x cv \ x cv sslin x cv A difdisj syl6ss x cv y cv i^i ss0b sylib syl5 com4t com23 x cv A C_ adantr imp43 sylanc exp42 com34 imp41))
+thm (infxpidmlem12 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((infxpidmlem.1 (= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (infxpidmlem.2 (e. A (V)))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)) (infxpidmlem.1 infxpidmlem.2 g h infxpidmlem9 infxpidmlem.1 infxpidmlem.2 h g infxpidmlem10 infxpidmlem.1 g visset x infxpidmlem2 g cv {/} = x om ~<_ x cv br x cv A C_ /\ g cv x cv x cv X. x cv :-1-1-onto-> /\ E. df-or bitr x cv y cv X. y cv x cv X. y cv y cv X. u. u. x cv y cv entrt x cv x cv x cv X. x cv y cv X. entrt x visset enref x visset x visset x visset y visset xpen mpan sylan2 om ~<_ x cv br adantll x cv x cv x cv X. y cv x cv X. entrt x visset enref x visset y visset x visset x visset xpen mpan2 sylan2 x cv x cv x cv X. y cv y cv X. entrt x visset y visset x visset y visset xpen anidms sylan2 jca om ~<_ x cv br adantll y visset x visset xpex y visset y visset xpex x infxpidmlem1 x cv ~~ y cv br adantr mpd x visset y visset xpex y visset x visset xpex y visset y visset xpex unex x infxpidmlem1 x cv ~~ y cv br adantr mp2and x visset y visset xpex y visset x visset xpex y visset y visset xpex unex unex x cv ensym syl om ~<_ x cv br x cv ~~ x cv x cv X. br /\ x cv ~~ y cv br pm3.27 sylanc x visset x cv x cv X. ensym sylanl2 y visset x cv y cv X. y cv x cv X. y cv y cv X. u. u. u bren sylib x visset x visset xpex g cv x cv f1oen om ~<_ x cv br anim2i x cv A C_ adantlr sylan y cv A x cv \ C_ adantrr infxpidmlem.1 infxpidmlem.2 x g y u h infxpidmlem11 exp u 19.23adv mpd exp y 19.23adv infxpidmlem.2 A V x cv difexg ax-mp x cv y domen syl5ib x visset infxpidmlem.2 A V x cv difexg ax-mp x cv V A x cv \ V domtri mp2an h H g cv h cv C: dfrex2 3imtr3g a3d A x cv sbth A x cv x cv X. x cv domentr x visset infxpidmlem.2 A V x cv difexg ax-mp unxpdom2 infxpidmlem.2 A x cv ssun2 A V x cv A u. ssdomg mp2 x cv A undif2 breqtrr A x cv A x cv \ u. x cv x cv X. domtr mpan syl 1onn x visset 1o infsdomnn mpan2 A x cv \ x cv sdomdom syl2an sylan x visset x cv V A ssdomg ax-mp syl2an A A X. x cv A entrt A A X. x cv x cv X. x cv entrt infxpidmlem.2 x visset infxpidmlem.2 x visset xpen anidms sylan x visset A ensym x cv x cv X. ~~ x cv br adantr sylanc expcom om ~<_ x cv br A x cv \ ~< x cv br /\ x cv A C_ ad2antlr mpd exp41 com24 imp31 x visset x visset xpex g cv x cv f1oen sylan2 syld x 19.23aiv g cv {/} = -. imim2i sylbi com13 syld com3r r19.23aiv ax-mp))
+thm (infxpidm ((f x) (A f) (A x)) ((infxpidm.1 (e. A (V)))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)) (f f cv {/} = x om ~<_ x cv br x cv A C_ /\ f cv x cv x cv X. x cv :-1-1-onto-> /\ E. \/ {|} eqid infxpidm.1 infxpidmlem12))
+thm (infunabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (u. A B) (~~) A)) (A +c B opr A A X. A domentr A +c B opr A 2o X. A A X. domtr infunabs.2 infunabs.1 infunabs.1 cdadom2 infunabs.1 xp2cda syl6breqr 2onn 2o omsdomnn ax-mp pm3.26i 2o om A domtr mpan 2ord elisseti infunabs.1 infunabs.1 xpdom2 syl syl2an infunabs.1 infxpidm B ~<_ A br adantl sylanc infunabs.1 infunabs.2 cdadom3 jctir A +c B opr A sbth syl ancoms infunabs.1 infunabs.2 uncdadom A B u. A +c B opr A domentr mpan syl infunabs.1 A B ssun1 A V A B u. ssdomg mp2 jctir A B u. A sbth syl))
+thm (infcdaabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (+c) B) (~~) A)) (A +c B opr A A X. A domentr A +c B opr A 2o X. A A X. domtr infunabs.2 infunabs.1 infunabs.1 cdadom2 infunabs.1 xp2cda syl6breqr 2onn 2o omsdomnn ax-mp pm3.26i 2o om A domtr mpan 2ord elisseti infunabs.1 infunabs.1 xpdom2 syl syl2an infunabs.1 infxpidm B ~<_ A br adantl sylanc infunabs.1 infunabs.2 cdadom3 jctir A +c B opr A sbth syl ancoms))
+thm (infcda () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (br (om) (~<_) A) (br (opr A (+c) B) (~~) (u. A B))) (infunabs.2 infunabs.1 B V A V entri3 mp2an A +c B opr A A B u. entrt infunabs.1 infunabs.2 infcdaabs infunabs.1 infunabs.2 infunabs infunabs.1 A B u. ensym syl sylanc exp B +c A opr B B A u. entrt infunabs.2 infunabs.1 infcdaabs infunabs.2 infunabs.1 infunabs infunabs.2 B A u. ensym syl sylanc infunabs.1 infunabs.2 cdacomen A +c B opr B +c A opr B A u. entrt mpan syl B A uncom syl6breq om A B domtr sylan exp31 pm2.43d jaod mpi))
+thm (infdif () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<) A)) (br (\ A B) (~~) A)) (A A B u. A B \ +c A B \ opr endomtr infunabs.1 infunabs.2 infunabs infunabs.1 A B u. ensym syl B A sdomdom sylan2 omex infunabs.2 om V B V entri2 mp2an infunabs.1 A B ssun1 A V A B u. ssdomg mp2 infunabs.1 infunabs.1 infunabs.2 unex A V A B u. V domtri mp2an mpbi A B u. B +c B opr A domsdomtr infunabs.1 A V B difexg ax-mp infunabs.2 infunabs.2 cdadom1 infunabs.1 A V B difexg ax-mp infunabs.2 uncdadom A B \ B u. A B \ +c B opr B +c B opr domtr mpan syl A B undif1 syl5eqbrr B +c B opr B A ensdomtr infunabs.2 B V domrefg ax-mp infunabs.2 infunabs.2 infcdaabs mpan2 sylan ancoms syl2an expcom mtoi exp om ~<_ A br adantl om ~<_ A br B ~< A br pm3.26 B ~< om br adantr A B +c B opr om domsdomtr A A B u. B +c B opr endomtr infunabs.1 infunabs.2 infunabs infunabs.1 A B u. ensym syl B A sdomdom sylan2 infunabs.1 A V B difexg ax-mp infunabs.2 infunabs.2 cdadom1 infunabs.1 A V B difexg ax-mp infunabs.2 uncdadom A B \ B u. A B \ +c B opr B +c B opr domtr mpan syl A B undif1 syl5eqbrr syl2an B B cdafi anidms syl2an an1rs exp con3d omex infunabs.1 om V A V domtri mp2an syl5ib mpd exp jaod mpi infunabs.1 A V B difexg ax-mp infunabs.2 A B \ V B V domtri mp2an bicon2i sylibr B A B \ sdomdom infunabs.2 infunabs.1 A V B difexg ax-mp infunabs.1 A V B difexg ax-mp cdadom2 A B undif1 infunabs.1 A V B difexg ax-mp infunabs.2 uncdadom eqbrtrr A B u. A B \ +c B opr A B \ +c A B \ opr domtr mpan syl 3syl sylanc om A A B \ +c A B \ opr domtr infunabs.1 A V B difexg ax-mp cdainf sylibr infunabs.1 A V B difexg ax-mp A B \ V domrefg ax-mp infunabs.1 A V B difexg ax-mp infunabs.1 A V B difexg ax-mp infcdaabs mpan2 syl A A B \ +c A B \ opr A B \ domentr exp om ~<_ A br adantl mpd syldan infunabs.1 A B difss A V A B \ ssdom2g mp2 jctil A B \ A sbth syl))
+thm (infxpabs () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (/\ (br (om) (~<_) A) (-. (= B ({/})))) (br B (~<_) A)) (br (X. A B) (~~) A)) (A B X. A sbth A B X. A A X. A domentr infunabs.2 infunabs.1 infunabs.1 xpdom2 infunabs.1 infxpidm syl2an ancoms infunabs.1 B xpdom3 syl2an an1rs))
+thm (infmap1 () ((infunabs.1 (e. A (V))) (infunabs.2 (e. B (V)))) (-> (/\ (/\ (br (2o) (~<_) A) (br (om) (~<_) B)) (br A (~<_) B)) (br (opr A (^m) B) (~~) (opr (2o) (^m) B))) (A ^m B opr 2o ^m B opr sbth A ^m B opr B ^m B opr 2o ^m B opr domtr infunabs.1 infunabs.2 infunabs.2 mapdom1 infunabs.2 infxpidm 2ord elisseti enref 2ord elisseti 2ord elisseti infunabs.2 infunabs.2 xpex infunabs.2 mapen mpan infunabs.2 canth2 B B P~ sdomdom ax-mp infunabs.2 pw2en B B P~ 2o ^m B opr domentr mp2an infunabs.2 2o ^m B oprex infunabs.2 mapdom1 ax-mp 2ord elisseti infunabs.2 infunabs.2 mapxpen B ^m B opr 2o ^m B opr ^m B opr 2o ^m B B X. opr domentr mp2an B ^m B opr 2o ^m B B X. opr 2o ^m B opr domentr mpan 3syl syl2an 2ord elisseti infunabs.1 infunabs.2 mapdom1 syl2an ancoms anassrs an1rs))
+thm (aleph1irr () () (br (` (aleph) (1o)) (~<_) (\ (RR) (QQ))) (aleph1re reex omex nnenom ensymi ruc om NN RR ensdomtr mp2an om RR sdomdom ax-mp resdomq reex qex infdif mp2an ensymi aleph 1o ` RR RR QQ \ domentr mp2an))
+thm (infmap2lem1 ((x z) (w x) (v x) (f x) (A x) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f) (B x) (B z) (B w) (B f) (B v) (R v) (R f)) ((infmap2lem.1 (e. A (V))) (infmap2lem.2 (e. B (V))) (infmap2lem.3 (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z))))))) (-> (/\ (C_ (cv f) R) (Fn (cv f) (dom R))) (-> (e. (cv v) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B)))) (/\ (C_ (cv v) A) (:-onto-> (` (cv f) (cv v)) B (cv v))))) (f cv R v cv f cv v cv ` <,> ssel infmap2lem.3 v cv f cv v cv ` <,> eleq2i v visset f cv v cv fvex z cv v cv A sseq1 z cv v cv ~~ B breq1 anbi12d z cv v cv w cv B foeq3 anbi12d w cv f cv v cv ` B v cv foeq1 v cv A C_ v cv ~~ B br /\ anbi2d opelopab bitr v cv A C_ v cv ~~ B br pm3.26 f cv v cv ` B v cv :-onto-> anim1i sylbi syl6 f cv R dom v cv fnopfv syl5 exp3a imp infmap2lem.3 dmeqi z w z cv A C_ z cv ~~ B br /\ w cv B z cv :-onto-> /\ dmopab z cv A C_ z cv ~~ B br w w cv B z cv :-onto-> E. anass w z cv A C_ z cv ~~ B br /\ w cv B z cv :-onto-> 19.42v infmap2lem.2 z cv ensym z visset B w bren w cv B z cv f1ofo w 19.22i sylbi syl pm4.71i z cv A C_ anbi2i 3bitr4 z biabi 3eqtr z cv x cv A sseq1 z cv x cv ~~ B breq1 anbi12d cbvabv eqtr v cv eleq2i syl5ibr))
+thm (infmap2lem2 ((x z) (w x) (v x) (u x) (f x) (A x) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B z) (B w) (B f) (B v) (B u) (R u) (R v) (R f)) ((infmap2lem.1 (e. A (V))) (infmap2lem.2 (e. B (V))) (infmap2lem.3 (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z))))))) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) (~<_) (opr A (^m) B)) (infmap2lem.3 A P~ A ^m B opr z w df-xp infmap2lem.1 pwex A ^m B oprex xpex eqeltrr z cv A C_ w cv B z cv :-onto-> pm3.26 z visset A elpw sylibr w cv B z cv fof w cv B z cv ffn syl z cv A C_ adantl w cv B z cv forn A sseq1d biimparc jca infmap2lem.1 infmap2lem.2 w cv elmap w cv B A df-f bitr sylibr jca z cv ~~ B br adantlr z w ssopab2i ssexi eqeltr R V f ac7g ax-mp A x df-pw infmap2lem.1 pwex eqeltrr x cv A C_ x cv ~~ B br pm3.26 x ss2abi ssexi infmap2lem.1 infmap2lem.2 infmap2lem.3 f v x infmap2lem1 f cv v cv ` B v cv A fss f cv v cv ` B v cv fof sylan ancoms infmap2lem.1 infmap2lem.2 f cv v cv ` elmap sylibr syl6 infmap2lem.1 infmap2lem.2 infmap2lem.3 f v x infmap2lem1 v cv A C_ f cv v cv ` B v cv :-onto-> pm3.27 syl6 infmap2lem.1 infmap2lem.2 infmap2lem.3 f u x infmap2lem1 u cv A C_ f cv u cv ` B u cv :-onto-> pm3.27 syl6 anim12d f cv v cv ` B v cv forn f cv u cv ` B u cv forn eqeqan12d f cv v cv ` f cv u cv ` rneq syl5bi v cv u cv f cv fveq2 f cv v cv ` B v cv :-onto-> f cv u cv ` B u cv :-onto-> /\ a1i impbid syl6 V dom2d mpi f 19.23aiv ax-mp))
+thm (infmap2 ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w)) ((infmap2.1 (e. A (V))) (infmap2.2 (e. B (V)))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (^m) B) (~~) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))))) (infmap2.1 infmap2.2 infxpabs infmap2.2 infmap2.1 xpcomen B A X. A B X. A entrt mpan syl infmap2.2 infmap2.1 xpex infmap2.1 infmap2.2 x ssenen A ^m B oprex x A ^m B opr abid2 infmap2.1 infmap2.2 x cv elmap x cv B A fssxp x cv B A ffun x visset fundmen x visset x cv dom ensym 3syl x cv B A fdm breqtrd jca sylbi x ss2abi eqsstr3 A ^m B opr V x x cv B A X. C_ x cv ~~ B br /\ {|} ssdomg mp2 A ^m B opr x x cv B A X. C_ x cv ~~ B br /\ {|} x x cv A C_ x cv ~~ B br /\ {|} domentr mpan 3syl infmap2.1 infmap2.2 z w z cv A C_ z cv ~~ B br /\ w cv B z cv :-onto-> /\ {<,>|} eqid x infmap2lem2 A ^m B opr x x cv A C_ x cv ~~ B br /\ {|} sbth mpan2 syl exp31 B {/} A ^m opreq2 infmap2.1 map0e syl6eq B {/} x cv ~~ breq2 x cv A C_ anbi2d x biabdv {/} x df-sn df1o2 x cv en0 x cv A C_ anbi2i A 0ss x cv {/} A sseq1 mpbiri pm4.71ri bitr4 x biabi 3eqtr4r syl6eq eqtr4d A ^m B oprex A ^m B opr V x x cv A C_ x cv ~~ B br /\ {|} eqeng ax-mp syl B ~<_ A br a1d pm2.61d2 imp))
+thm (alephadd () () (br (opr (` (aleph) A) (+c) (` (aleph) B)) (~~) (u. (` (aleph) A) (` (aleph) B))) (A aleph ndmfv B aleph ndmfv +c opreqan12d 0ex 0ex cdaval {/} {/} {} 1o {} xpundi {/} {} 1o {} u. xp0r eqtr3 eqtr syl6eq A aleph ndmfv B aleph dom e. -. adantr B aleph ndmfv A aleph dom e. -. adantl uneq12d {/} un0 syl6eq eqtr4d alephfnon aleph On fndm ax-mp A eleq2i negbii alephfnon aleph On fndm ax-mp B eleq2i negbii syl2anbr aleph A ` +c aleph B ` oprex aleph A ` +c aleph B ` opr V aleph A ` aleph B ` u. eqeng ax-mp syl exp A alephgeom aleph A fvex aleph A ` V om ssdom2g ax-mp aleph A fvex aleph B fvex infcda syl sylbi B alephgeom aleph B fvex aleph B ` V om ssdom2g ax-mp aleph B fvex aleph A fvex infcda aleph A fvex aleph B fvex cdacomen aleph A ` +c aleph B ` opr aleph B ` +c aleph A ` opr aleph B ` aleph A ` u. entrt mpan syl aleph B ` aleph A ` uncom syl6breq syl sylbi pm2.61ii))
+thm (alephexp1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (br (opr (` (aleph) A) (^m) (` (aleph) B)) (~~) (opr (2o) (^m) (` (aleph) B)))) (aleph A fvex aleph B fvex infmap1 A alephgeom aleph A fvex aleph A ` V om ssdom2g ax-mp sylbi 2onn 2o nnsdom ax-mp 2o om sdomdom ax-mp 2o om aleph A ` domtr mpan syl B alephgeom aleph B fvex aleph B ` V om ssdom2g ax-mp sylbi anim12i A B C_ adantr A B alephord3 aleph A fvex aleph A ` V aleph B ` ssdomg ax-mp syl6bi imp sylanc))
+thm (alephsuc3 ((A x)) () (-> (e. A (On)) (br (` (aleph) (suc A)) (~~) ({e.|} x (On) (br (cv x) (~~) (` (aleph) A))))) (A x alephsuc2 aleph A ` difeq1d A alephcard aleph A ` x cardval2 eqtr3 A On e. a1i x On x cv ~<_ aleph A ` br {e.|} difeq2d eqtrd x On x cv ~<_ aleph A ` br x cv ~< aleph A ` br difrab x cv aleph A ` bren2 x cv On e. a1i birabi eqtr4 syl6req aleph A suc fvex aleph A fvex infdif A sucelon A suc alephgeom bitr aleph A suc fvex aleph A suc ` V om ssdom2g ax-mp sylbi A alephordlem1 sylanc eqbrtrd aleph A suc fvex x On x cv ~~ aleph A ` br {e.|} ensym syl))
+thm (alephexp2 ((A x)) () (-> (e. A (On)) (br (opr (2o) (^m) (` (aleph) A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)))))) (A alephgeom aleph A fvex aleph A ` V om ssdom2g ax-mp sylbi aleph A fvex aleph A ` V domrefg ax-mp jctir aleph A fvex aleph A fvex x infmap2 syl A On e. A On e. pm3.2 pm2.43i A ssid jctir A A alephexp1 2o ^m aleph A ` oprex 2o ^m aleph A ` opr V aleph A ` ^m aleph A ` opr x x cv aleph A ` C_ x cv ~~ aleph A ` br /\ {|} enen1 mpan 3syl mpbid))
+thm (gch-kn ((A x)) () (-> (e. A (On)) (<-> (br (` (aleph) (suc A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A))))) (br (` (aleph) (suc A)) (~~) (opr (2o) (^m) (` (aleph) A))))) (A x alephexp2 aleph A ` x df-pw aleph A fvex pwex eqeltrr x cv aleph A ` C_ x cv ~~ aleph A ` br pm3.26 x ss2abi ssexi x x cv aleph A ` C_ x cv ~~ aleph A ` br /\ {|} V 2o ^m aleph A ` opr aleph A suc ` enen2 mpan syl bicomd))
+thm (hvmulcl () ((hvmulcl.1 (e. A (CC))) (hvmulcl.2 (e. B (H~)))) (e. (opr A (.s) B) (H~)) (hvmulcl.1 hvmulcl.2 A B ax-hvmulcl mp2an))
+thm (hvsubvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B)))) (A +v 1 -u .s B opr oprex x cv A +v 1 -u .s y cv opr opreq1 y cv B 1 -u .s opreq2 A +v opreq2d x y z df-hvsub oprabval2))
+thm (hvsubclt () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~))) (A B hvsubvalt A 1 -u .s B opr ax-hvaddcl 1cn negcl 1 -u B ax-hvmulcl mpan sylan2 eqeltrd))
+thm (hvaddcl () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (e. (opr A (+v) B) (H~)) (hvaddcl.1 hvaddcl.2 A B ax-hvaddcl mp2an))
+thm (hvcom () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (= (opr A (+v) B) (opr B (+v) A)) (hvaddcl.1 hvaddcl.2 A B ax-hvcom mp2an))
+thm (hvsubval () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B))) (hvaddcl.1 hvaddcl.2 A B hvsubvalt mp2an))
+thm (hvsubcl () ((hvaddcl.1 (e. A (H~))) (hvaddcl.2 (e. B (H~)))) (e. (opr A (-v) B) (H~)) (hvaddcl.1 hvaddcl.2 A B hvsubclt mp2an))
+thm (hvaddid2t () () (-> (e. A (H~)) (= (opr (0v) (+v) A) A)) (ax-hv0cl A 0v ax-hvcom mpan2 A ax-hvaddid eqtr3d))
+thm (hvmul0t () () (-> (e. A (CC)) (= (opr A (.s) (0v)) (0v))) (A mul01t .s 0v opreq1d ax-hv0cl 0v ax-hvmul0 ax-mp syl6eq 0cn ax-hv0cl A 0 0v ax-hvmulass mp3an23 eqtr3d ax-hv0cl 0v ax-hvmul0 ax-mp A .s opreq2i syl6req))
+thm (hvmul0ort () () (-> (/\ (e. A (CC)) (e. B (H~))) (<-> (= (opr A (.s) B) (0v)) (\/ (= A (0)) (= B (0v))))) (A .s B opr 0v 1 / A opr .s opreq2 A CC e. B H~ e. /\ A 0 =/= ad2antlr 1 / A opr A axmulcom A recclt A CC e. A 0 =/= pm3.26 sylanc A recidt eqtrd .s B opreq1d B H~ e. adantlr 1 / A opr A B ax-hvmulass A recclt B H~ e. adantlr A CC e. B H~ e. pm3.26 A 0 =/= adantr A CC e. B H~ e. pm3.27 A 0 =/= adantr syl3anc B ax-hvmulid A CC e. A 0 =/= ad2antlr 3eqtr3d A .s B opr 0v = adantlr A recclt 1 / A opr hvmul0t syl B H~ e. adantlr A .s B opr 0v = adantlr 3eqtr3d exp A 0 df-ne syl5ibr orrd exp A 0 .s B opreq1 0v eqeq1d B ax-hvmul0 syl5bir com12 A CC e. adantl B 0v A .s opreq2 0v eqeq1d A hvmul0t syl5bir com12 B H~ e. adantr jaod impbid))
+thm (hvsubidt () () (-> (e. A (H~)) (= (opr A (-v) A) (0v))) (A ax-hvmulid +v 1 -u .s A opr opreq1d 1cn 1cn negcl 1 1 -u A ax-hvdistr2 mp3an12 A A hvsubvalt anidms 3eqtr4rd 1cn negid .s A opreq1i syl6eq A ax-hvmul0 eqtrd))
+thm (hvnegidt () () (-> (e. A (H~)) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v))) (A A hvsubvalt anidms A hvsubidt eqtr3d))
+thm (hv2negt () () (-> (e. A (H~)) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A))) (ax-hv0cl 0v A hvsubvalt mpan 1cn negcl 1 -u A ax-hvmulcl mpan 1 -u .s A opr hvaddid2t syl eqtrd))
+thm (hvaddid2 () ((hvaddid2.1 (e. A (H~)))) (= (opr (0v) (+v) A) A) (hvaddid2.1 A hvaddid2t ax-mp))
+thm (hvnegid () ((hvaddid2.1 (e. A (H~)))) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v)) (hvaddid2.1 A hvnegidt ax-mp))
+thm (hv2neg () ((hvaddid2.1 (e. A (H~)))) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A)) (hvaddid2.1 A hv2negt ax-mp))
+thm (hvm1negt () () (-> (/\ (e. A (CC)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (.s) B)) (opr (-u A) (.s) B))) (1cn negcl 1 -u A B ax-hvmulass mp3an1 A mulm1t B H~ e. adantr .s B opreq1d eqtr3d))
+thm (hvadd23t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B))) (B C ax-hvcom A +v opreq2d A H~ e. 3adant1 A B C ax-hvass A C B ax-hvass 3com23 3eqtr4d))
+thm (hvadd12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C)))) (A B ax-hvcom +v C opreq1d C H~ e. 3adant3 A B C ax-hvass B A C ax-hvass 3com12 3eqtr3d))
+thm (hvadd4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D)))) (A B C hvadd23t +v D opreq1d 3expa D H~ e. adantrr A +v B opr C D ax-hvass 3expb A B ax-hvaddcl sylan A +v C opr B D ax-hvass 3expb A C ax-hvaddcl sylan an4s 3eqtr3d))
+thm (hvsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (-v) (opr C (+v) D)) (opr (opr A (-v) C) (+v) (opr B (-v) D)))) (1cn negcl 1 -u C D ax-hvdistr1 mp3an1 A H~ e. B H~ e. /\ adantl A +v B opr +v opreq2d 1cn negcl 1 -u C ax-hvmulcl mpan A H~ e. anim2i 1cn negcl 1 -u D ax-hvmulcl mpan B H~ e. anim2i anim12i an4s A 1 -u .s C opr B 1 -u .s D opr hvadd4t syl eqtr4d A +v B opr C +v D opr hvsubvalt A B ax-hvaddcl C D ax-hvaddcl syl2an A C hvsubvalt D H~ e. adantrr B H~ e. adantlr B D hvsubvalt C H~ e. adantrl A H~ e. adantll +v opreq12d 3eqtr4d))
+thm (hvaddsub12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (-v) C)) (opr B (+v) (opr A (-v) C)))) (A B 1 -u .s C opr hvadd12t 1cn negcl 1 -u C ax-hvmulcl mpan syl3an3 B C hvsubvalt A +v opreq2d A H~ e. 3adant1 A C hvsubvalt B +v opreq2d B H~ e. 3adant2 3eqtr4d))
+thm (hvsubcan1t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) B) A)) (A +v B opr B hvsubvalt A B ax-hvaddcl A H~ e. B H~ e. pm3.27 sylanc A B 1 -u .s B opr ax-hvass 3expb 1cn negcl 1 -u B ax-hvmulcl mpan ancli sylan2 eqtrd B hvnegidt A +v opreq2d A H~ e. adantl A ax-hvaddid B H~ e. adantr 3eqtrd))
+thm (hvpncan2t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) A) B)) (B A ax-hvcom -v A opreq1d eqcomd B A hvsubcan1t eqtrd ancoms))
+thm (hvaddsubasst () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (-v) C) (opr A (+v) (opr B (-v) C)))) (A B 1 -u .s C opr ax-hvass 1cn negcl 1 -u C ax-hvmulcl mpan syl3an3 A +v B opr C hvsubvalt A B ax-hvaddcl sylan 3impa B C hvsubvalt A H~ e. 3adant1 A +v opreq2d 3eqtr4d))
+thm (hvpncan3t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (+v) (opr B (-v) A)) B)) (A B A hvaddsubasst A H~ e. B H~ e. pm3.26 A H~ e. B H~ e. pm3.27 A H~ e. B H~ e. pm3.26 syl3anc A B hvpncan2t eqtr3d))
+thm (hvmulcomt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C)))) (A B axmulcom .s C opreq1d C H~ e. 3adant3 A B C ax-hvmulass B A C ax-hvmulass 3com12 3eqtr3d))
+thm (hvmulass () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr (opr A (x.) B) (.s) C) (opr A (.s) (opr B (.s) C))) (hvmulcom.1 hvmulcom.2 hvmulcom.3 A B C ax-hvmulass mp3an))
+thm (hvmulcom () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C))) (hvmulcom.1 hvmulcom.2 hvmulcom.3 A B C hvmulcomt mp3an))
+thm (hvmul2neg () ((hvmulcom.1 (e. A (CC))) (hvmulcom.2 (e. B (CC))) (hvmulcom.3 (e. C (H~)))) (= (opr (-u A) (.s) (opr (-u B) (.s) C)) (opr A (.s) (opr B (.s) C))) (hvmulcom.1 hvmulcom.2 mul2neg .s C opreq1i hvmulcom.1 negcl hvmulcom.2 negcl hvmulcom.3 hvmulass hvmulcom.1 hvmulcom.2 hvmulcom.3 hvmulass 3eqtr3))
+thm (hvsubdistr1t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C)))) (A B 1 -u .s C opr ax-hvdistr1 1cn negcl 1 -u C ax-hvmulcl mpan syl3an3 1cn negcl A 1 -u C hvmulcomt mp3an2 A .s B opr +v opreq2d B H~ e. 3adant2 eqtrd B C hvsubvalt A CC e. 3adant1 A .s opreq2d A .s B opr A .s C opr hvsubvalt A B ax-hvmulcl C H~ e. 3adant3 A C ax-hvmulcl B H~ e. 3adant2 sylanc 3eqtr4d))
+thm (hvsubdistr2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (-) B) (.s) C) (opr (opr A (.s) C) (-v) (opr B (.s) C)))) (A B -u C ax-hvdistr2 B negclt syl3an2 B mulm1t .s C opreq1d C H~ e. adantr 1cn negcl 1 -u B C ax-hvmulass mp3an1 eqtr3d A CC e. 3adant1 A .s C opr +v opreq2d eqtrd A B subnegt C H~ e. 3adant3 .s C opreq1d A .s C opr B .s C opr hvsubvalt A C ax-hvmulcl B CC e. 3adant2 B C ax-hvmulcl A CC e. 3adant1 sylanc 3eqtr4d))
+thm (hvdistr1 () ((hvdistr1.1 (e. A (CC))) (hvdistr1.2 (e. B (H~))) (hvdistr1.3 (e. C (H~)))) (= (opr A (.s) (opr B (+v) C)) (opr (opr A (.s) B) (+v) (opr A (.s) C))) (hvdistr1.1 hvdistr1.2 hvdistr1.3 A B C ax-hvdistr1 mp3an))
+thm (hvsubdistr1 () ((hvdistr1.1 (e. A (CC))) (hvdistr1.2 (e. B (H~))) (hvdistr1.3 (e. C (H~)))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C))) (hvdistr1.1 hvdistr1.2 hvdistr1.3 A B C hvsubdistr1t mp3an))
+thm (hvass () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (+v) B) (+v) C) (opr A (+v) (opr B (+v) C))) (hvass.1 hvass.2 hvass.3 A B C ax-hvass mp3an))
+thm (hvadd23 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B)) (hvass.1 hvass.2 hvass.3 A B C hvadd23t mp3an))
+thm (hvsubass () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (-v) B) (-v) C) (opr A (-v) (opr B (+v) C))) (hvass.1 hvass.2 hvsubcl hvass.3 hvsubval hvass.1 1cn negcl hvass.2 hvmulcl 1cn negcl hvass.3 hvmulcl hvass 1cn negcl hvass.2 hvass.3 hvdistr1 A +v opreq2i eqtr4 hvass.1 hvass.2 hvsubval +v 1 -u .s C opr opreq1i hvass.1 hvass.2 hvass.3 hvaddcl hvsubval 3eqtr4 eqtr))
+thm (hvsub23 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr (opr A (-v) B) (-v) C) (opr (opr A (-v) C) (-v) B)) (hvass.2 hvass.3 hvcom A -v opreq2i hvass.1 hvass.2 hvass.3 hvsubass hvass.1 hvass.3 hvass.2 hvsubass 3eqtr4))
+thm (hvadd12 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~)))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C))) (hvass.1 hvass.2 hvcom +v C opreq1i hvass.1 hvass.2 hvass.3 hvass hvass.2 hvass.1 hvass.3 hvass 3eqtr3))
+thm (hvadd4 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~))) (hvadd4.4 (e. D (H~)))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D))) (hvass.1 hvass.2 pm3.2i hvass.3 hvadd4.4 pm3.2i A B C D hvadd4t mp2an))
+thm (hvsubsub4 () ((hvass.1 (e. A (H~))) (hvass.2 (e. B (H~))) (hvass.3 (e. C (H~))) (hvadd4.4 (e. D (H~)))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D))) (hvass.1 1cn negcl hvass.2 hvmulcl 1cn negcl hvass.3 hvmulcl 1cn negcl 1cn negcl hvadd4.4 hvmulcl hvmulcl hvadd4 1cn negcl hvass.3 1cn negcl hvadd4.4 hvmulcl hvdistr1 A +v 1 -u .s B opr opr +v opreq2i 1cn negcl hvass.2 1cn negcl hvadd4.4 hvmulcl hvdistr1 A +v 1 -u .s C opr opr +v opreq2i 3eqtr4 hvass.1 1cn negcl hvass.2 hvmulcl hvaddcl hvass.3 1cn negcl hvadd4.4 hvmulcl hvaddcl hvsubval hvass.1 1cn negcl hvass.3 hvmulcl hvaddcl hvass.2 1cn negcl hvadd4.4 hvmulcl hvaddcl hvsubval 3eqtr4 hvass.1 hvass.2 hvsubval hvass.3 hvadd4.4 hvsubval -v opreq12i hvass.1 hvass.3 hvsubval hvass.2 hvadd4.4 hvsubval -v opreq12i 3eqtr4))
+thm (hvsubsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D)))) (A A H~ e. A 0v if -v B opreq1 -v C -v D opr opreq1d A A H~ e. A 0v if -v C opreq1 -v B -v D opr opreq1d eqeq12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 -v C -v D opr opreq1d B B H~ e. B 0v if -v D opreq1 A H~ e. A 0v if -v C opr -v opreq2d eqeq12d C C H~ e. C 0v if -v D opreq1 A H~ e. A 0v if -v B H~ e. B 0v if opr -v opreq2d C C H~ e. C 0v if A H~ e. A 0v if -v opreq2 -v B H~ e. B 0v if -v D opr opreq1d eqeq12d D D H~ e. D 0v if C H~ e. C 0v if -v opreq2 A H~ e. A 0v if -v B H~ e. B 0v if opr -v opreq2d D D H~ e. D 0v if B H~ e. B 0v if -v opreq2 A H~ e. A 0v if -v C H~ e. C 0v if opr -v opreq2d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel ax-hv0cl D elimel hvsubsub4 dedth4h))
+thm (hv2times () ((hv2times.1 (e. A (H~)))) (= (opr (2) (.s) A) (opr A (+v) A)) (1cn 1cn hv2times.1 1 1 A ax-hvdistr2 mp3an df-2 .s A opreq1i 1cn hv2times.1 hv2times.1 hvdistr1 3eqtr4 hv2times.1 hv2times.1 hvaddcl A +v A opr ax-hvmulid ax-mp eqtr))
+thm (hvnegdi () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A)) (hvnegdi.1 hvnegdi.2 hvsubval 1 -u .s opreq2i 1cn negcl hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvdistr1 1cn 1cn mul2neg 1cn mulid1 eqtr .s B opreq1i 1cn negcl 1cn negcl hvnegdi.2 hvmulass hvnegdi.2 B ax-hvmulid ax-mp 3eqtr3 +v 1 -u .s A opr opreq1i 1cn negcl hvnegdi.1 hvmulcl 1cn negcl 1cn negcl hvnegdi.2 hvmulcl hvmulcl hvcom hvnegdi.2 hvnegdi.1 hvsubval 3eqtr4 3eqtr))
+thm (hvsubeq0 () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (<-> (= (opr A (-v) B) (0v)) (= A B)) (hvnegdi.1 hvnegdi.2 hvsubval 0v eqeq1i A +v 1 -u .s B opr opr 0v +v B opreq1 sylbi hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvnegdi.2 hvadd23 hvnegdi.1 hvnegdi.2 1cn negcl hvnegdi.2 hvmulcl hvass hvnegdi.2 hvnegid A +v opreq2i hvnegdi.1 A ax-hvaddid ax-mp eqtr 3eqtr hvnegdi.2 hvaddid2 3eqtr3g A B -v B opreq1 hvnegdi.2 B hvsubidt ax-mp syl6eq impbi))
+thm (hvsubcan2 () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~)))) (= (opr (opr A (+v) B) (+v) (opr A (-v) B)) (opr (2) (.s) A)) (hvnegdi.1 hvnegdi.2 hvsubval A +v B opr +v opreq2i hvnegdi.1 hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvadd4 hvnegdi.1 hv2times eqcomi hvnegdi.2 hvnegid +v opreq12i eqtr 2cn hvnegdi.1 hvmulcl 2 .s A opr ax-hvaddid ax-mp 3eqtr))
+thm (hvaddcan () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~))) (hvaddcan.3 (e. C (H~)))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C)) (A +v B opr A +v C opr +v 1 -u .s A opr opreq1 hvnegdi.1 hvnegdi.2 1cn negcl hvnegdi.1 hvmulcl hvadd23 hvnegdi.1 hvnegid +v B opreq1i hvnegdi.2 hvaddid2 3eqtr hvnegdi.1 hvaddcan.3 1cn negcl hvnegdi.1 hvmulcl hvadd23 hvnegdi.1 hvnegid +v C opreq1i hvaddcan.3 hvaddid2 3eqtr 3eqtr3g B C A +v opreq2 impbi))
+thm (hvsubadd () ((hvnegdi.1 (e. A (H~))) (hvnegdi.2 (e. B (H~))) (hvaddcan.3 (e. C (H~)))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A)) (hvnegdi.1 hvnegdi.2 hvsubval C eqeq1i hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvadd12 hvnegdi.2 hvnegid A +v opreq2i hvnegdi.1 A ax-hvaddid ax-mp 3eqtr B +v C opr eqeq1i hvnegdi.2 hvnegdi.1 1cn negcl hvnegdi.2 hvmulcl hvaddcl hvaddcan.3 hvaddcan A B +v C opr eqcom 3bitr3 bitr))
+thm (hvnegdit () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A))) (A A H~ e. A 0v if -v B opreq1 1 -u .s opreq2d A A H~ e. A 0v if B -v opreq2 eqeq12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 1 -u .s opreq2d B B H~ e. B 0v if -v A H~ e. A 0v if opreq1 eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel hvnegdi dedth2h))
+thm (hvsubeq0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (-v) B) (0v)) (= A B))) (A A H~ e. A 0v if -v B opreq1 0v eqeq1d A A H~ e. A 0v if B eqeq1 bibi12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 0v eqeq1d B B H~ e. B 0v if A H~ e. A 0v if eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel hvsubeq0 dedth2h))
+thm (hvaddcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C))) (A A H~ e. A 0v if +v B opreq1 A A H~ e. A 0v if +v C opreq1 eqeq12d B C = bibi1d B B H~ e. B 0v if A H~ e. A 0v if +v opreq2 A H~ e. A 0v if +v C opr eqeq1d B B H~ e. B 0v if C eqeq1 bibi12d C C H~ e. C 0v if A H~ e. A 0v if +v opreq2 A H~ e. A 0v if +v B H~ e. B 0v if opr eqeq2d C C H~ e. C 0v if B H~ e. B 0v if eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel hvaddcan dedth3h))
+thm (hvaddcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) C) (opr B (+v) C)) (= A B))) (C A ax-hvcom B H~ e. 3adant3 C B ax-hvcom A H~ e. 3adant2 eqeq12d C A B hvaddcant bitr3d 3coml))
+thm (hvmulcant () () (-> (/\ (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (=/= A (0))) (<-> (= (opr A (.s) B) (opr A (.s) C)) (= B C))) (A 0 df-ne A 0 = B -v C opr 0v = biorf sylbi A CC e. B H~ e. C H~ e. /\/\ adantl A B -v C opr hvmul0ort B C hvsubclt sylan2 3impb A 0 =/= adantr bitr4d A B C hvsubdistr1t A 0 =/= adantr 0v eqeq1d A .s B opr A .s C opr hvsubeq0t A B ax-hvmulcl C H~ e. 3adant3 A C ax-hvmulcl B H~ e. 3adant2 sylanc A 0 =/= adantr 3bitrd B C hvsubeq0t A CC e. 3adant1 A 0 =/= adantr bitr3d))
+thm (hvmulcan2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (=/= C (0v))) (<-> (= (opr A (.s) C) (opr B (.s) C)) (= A B))) (C 0v df-ne C 0v = A - B opr 0 = biorf C 0v = A - B opr 0 = orcom syl6bb sylbi A CC e. B CC e. C H~ e. /\/\ adantl A - B opr C hvmul0ort A B subclt sylan 3impa C 0v =/= adantr bitr4d A B C hvsubdistr2t C 0v =/= adantr 0v eqeq1d bitr2d A .s C opr B .s C opr hvsubeq0t A C ax-hvmulcl B CC e. 3adant2 B C ax-hvmulcl A CC e. 3adant1 sylanc C 0v =/= adantr A B subeq0t C H~ e. 3adant3 C 0v =/= adantr 3bitr3d))
+thm (hvsubcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) (opr A (-v) C)) (= B C))) (A B hvsubvalt C H~ e. 3adant3 A C hvsubvalt B H~ e. 3adant2 eqeq12d A 1 -u .s B opr 1 -u .s C opr hvaddcant 1cn negcl 1 -u C ax-hvmulcl mpan syl3an3 1cn negcl 1 -u B ax-hvmulcl mpan syl3an2 1cn negcl 1cn ax1ne0 negn0 1 -u B C hvmulcant mpan2 mp3an1 A H~ e. 3adant1 3bitrd))
+thm (hvsubcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) C) (opr B (-v) C)) (= A B))) (1cn negcl 1cn ax1ne0 negn0 1 -u C -v A opr C -v B opr hvmulcant mpan2 mp3an1 C A hvsubclt B H~ e. 3adant3 C B hvsubclt A H~ e. 3adant2 sylanc C A hvnegdit B H~ e. 3adant3 C B hvnegdit A H~ e. 3adant2 eqeq12d C A B hvsubcant 3bitr3d 3coml))
+thm (hvsub0t () () (-> (e. A (H~)) (= (opr A (-v) (0v)) A)) (ax-hv0cl A 0v hvsubvalt mpan2 1cn negcl 1 -u hvmul0t ax-mp A +v opreq2i syl6eq A ax-hvaddid eqtrd))
+thm (hvsubaddt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A))) (A A H~ e. A 0v if -v B opreq1 C eqeq1d A A H~ e. A 0v if B +v C opr eqeq2 bibi12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 C eqeq1d B B H~ e. B 0v if +v C opreq1 A H~ e. A 0v if eqeq1d bibi12d C C H~ e. C 0v if A H~ e. A 0v if -v B H~ e. B 0v if opr eqeq2 C C H~ e. C 0v if B H~ e. B 0v if +v opreq2 A H~ e. A 0v if eqeq1d bibi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel hvsubadd dedth3h))
+thm (hvaddsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (<-> (= (opr A (+v) B) (opr C (+v) D)) (= (opr A (-v) C) (opr D (-v) B)))) (A +v B opr C +v D opr C +v B opr hvsubcan2t A B ax-hvaddcl C H~ e. D H~ e. /\ adantr C D ax-hvaddcl A H~ e. B H~ e. /\ adantl C B ax-hvaddcl ancoms D H~ e. adantrr A H~ e. adantll syl3anc A H~ e. B H~ e. pm3.27 C H~ e. anim2i ancoms A B C B hvsub4t syldan B hvsubidt A H~ e. C H~ e. ad2antlr A -v C opr +v opreq2d A C hvsubclt A -v C opr ax-hvaddid syl B H~ e. adantlr 3eqtrd D H~ e. adantrr C H~ e. D H~ e. pm3.26 B H~ e. anim1i C D C B hvsub4t syldan C hvsubidt D H~ e. B H~ e. ad2antll +v D -v B opr opreq1d D B hvsubclt D -v B opr hvaddid2t syl C H~ e. adantll 3eqtrd ancoms A H~ e. adantll eqeq12d bitr3d))
+thm (hicl () ((hicl.1 (e. A (H~))) (hicl.2 (e. B (H~)))) (e. (opr A (.i) B) (CC)) (hicl.1 hicl.2 A B ax-hicl mp2an))
+thm (axhis42 () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (br (0) (<) (opr A (.i) A))) (A ax-his4 A 0v df-ne sylan2br))
+thm (his5t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr B (.i) (opr A (.s) C)) (opr (` (*) A) (x.) (opr B (.i) C)))) (B A .s C opr ax-his1 A C ax-hvmulcl sylan2 3impb 3com12 A C B ax-his3 3com23 * fveq2d A C .i B opr cjmult C B ax-hicl sylan2 3impb 3com23 B C ax-his1 A CC e. 3adant1 * A ` x. opreq2d eqtr4d 3eqtrd))
+thm (his35 () ((his35.1 (e. A (CC))) (his35.2 (e. B (CC))) (his35.3 (e. C (H~))) (his35.4 (e. D (H~)))) (= (opr (opr A (.s) C) (.i) (opr B (.s) D)) (opr (opr A (x.) (` (*) B)) (x.) (opr C (.i) D))) (his35.2 his35.3 his35.4 B C D his5t mp3an A x. opreq2i his35.1 his35.3 his35.2 his35.4 hvmulcl A C B .s D opr ax-his3 mp3an his35.1 his35.2 cjcl his35.3 his35.4 hicl mulass 3eqtr4))
+thm (his7t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (.i) (opr B (+v) C)) (opr (opr A (.i) B) (+) (opr A (.i) C)))) (B C A ax-his2 * fveq2d B .i A opr C .i A opr cjaddt B A ax-hicl C A ax-hicl syl2an 3impdir eqtrd 3comr A B +v C opr ax-his1 B C ax-hvaddcl sylan2 3impb A B ax-his1 C H~ e. 3adant3 A C ax-his1 B H~ e. 3adant2 + opreq12d 3eqtr4d))
+thm (hiassdit () () (-> (/\ (/\ (e. A (CC)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr (opr A (.s) B) (+v) C) (.i) D) (opr (opr A (x.) (opr B (.i) D)) (+) (opr C (.i) D)))) (A .s B opr C D ax-his2 3expb A B ax-hvmulcl sylan A B D ax-his3 3expa C H~ e. adantrl + C .i D opr opreq1d eqtrd))
+thm (his2subt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (-v) B) (.i) C) (opr (opr A (.i) C) (-) (opr B (.i) C)))) (A 1 -u .s B opr C ax-his2 1cn negcl 1 -u B ax-hvmulcl mpan syl3an2 1cn negcl 1 -u B C ax-his3 mp3an1 B C ax-hicl B .i C opr mulm1t syl eqtrd A .i C opr + opreq2d A H~ e. 3adant1 eqtrd A B hvsubvalt .i C opreq1d C H~ e. 3adant3 A .i C opr B .i C opr subnegt A C ax-hicl B H~ e. 3adant2 B C ax-hicl A H~ e. 3adant1 sylanc 3eqtr4d))
+thm (hiret () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (e. (opr A (.i) B) (RR)) (= (opr A (.i) B) (opr B (.i) A)))) (A B ax-hicl A .i B opr cjrebt syl * A .i B opr ` A .i B opr eqcom syl6bb B A ax-his1 ancoms A .i B opr eqeq2d bitr4d))
+thm (hiidrclt () () (-> (e. A (H~)) (e. (opr A (.i) A) (RR))) (A .i A opr eqid A A hiret mpbiri anidms))
+thm (hi01t () () (-> (e. A (H~)) (= (opr (0v) (.i) A) (0))) (0cn ax-hv0cl 0 0v A ax-his3 mp3an12 ax-hv0cl 0v ax-hvmul0 ax-mp .i A opreq1i syl5eqr ax-hv0cl 0v A ax-hicl mpan 0v .i A opr mul02t syl eqtrd))
+thm (hi02t () () (-> (e. A (H~)) (= (opr A (.i) (0v)) (0))) (ax-hv0cl A 0v ax-his1 mpan2 A hi01t * fveq2d cj0 syl6eq eqtrd))
+thm (hiidge0t () () (-> (e. A (H~)) (br (0) (<_) (opr A (.i) A))) (A 0v = pm2.1 A axhis42 exp A 0v .i A opreq1 A hi01t sylan9eqr eqcomd exp orim12d mpi A hiidrclt 0re jctil 0 A .i A opr leloet syl mpbird))
+thm (his6t () () (-> (e. A (H~)) (<-> (= (opr A (.i) A) (0)) (= A (0v)))) (A axhis42 0 A .i A opr leltnetOLD A hiidrclt 0re jctil A hiidge0t sylc A 0v = -. adantr mpbid exp a3d A .i A opr 0 eqcom syl5ib A 0v .i A opreq1 0 eqeq1d A hi01t syl5bir com12 impbid))
+thm (his1 () ((his1.1 (e. A (H~))) (his1.2 (e. B (H~)))) (= (opr A (.i) B) (` (*) (opr B (.i) A))) (his1.1 his1.2 A B ax-his1 mp2an))
+thm (hial0 ((A x)) () (-> (e. A (H~)) (<-> (A.e. x (H~) (= (opr A (.i) (cv x)) (0))) (= A (0v)))) (x cv A A .i opreq2 0 eqeq1d H~ rcla4v A his6t sylibd A 0v .i x cv opreq1 x cv hi01t sylan9eq exp A H~ e. a1i r19.21adv impbid))
+thm (hisubcom () ((hisubcom.1 (e. A (H~))) (hisubcom.2 (e. B (H~))) (hisubcom.3 (e. C (H~))) (hisubcom.4 (e. D (H~)))) (= (opr (opr A (-v) B) (.i) (opr C (-v) D)) (opr (opr B (-v) A) (.i) (opr D (-v) C))) (hisubcom.2 hisubcom.1 hvnegdi hisubcom.4 hisubcom.3 hvnegdi .i opreq12i 1cn negcl 1cn negcl hisubcom.2 hisubcom.1 hvsubcl hisubcom.4 hisubcom.3 hvsubcl his35 ax1re renegcl 1 -u cjret ax-mp 1 -u x. opreq2i 1cn 1cn mul2neg 1cn mulid1 3eqtr x. B -v A opr .i D -v C opr opr opreq1i hisubcom.2 hisubcom.1 hvsubcl hisubcom.4 hisubcom.3 hvsubcl hicl mulid2 3eqtr eqtr3))
+thm (hi2eqt () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (.i) (opr A (-v) B)) (opr B (.i) (opr A (-v) B))) (= A B))) (A B A -v B opr his2subt A H~ e. B H~ e. pm3.26 A H~ e. B H~ e. pm3.27 A B hvsubclt syl3anc 0 eqeq1d A B hvsubclt A -v B opr his6t syl bitr3d A .i A -v B opr opr B .i A -v B opr opr subeq0t A B hvsubclt A A -v B opr ax-hicl syldan B A -v B opr ax-hicl A H~ e. B H~ e. pm3.27 A B hvsubclt sylanc sylanc A B hvsubeq0t 3bitr3d))
+thm (hial2eqt ((A x) (B x)) () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (A.e. x (H~) (= (opr A (.i) (cv x)) (opr B (.i) (cv x)))) (= A B))) (A B hvsubclt x cv A -v B opr A .i opreq2 x cv A -v B opr B .i opreq2 eqeq12d H~ rcla4v syl A B hi2eqt sylibd A B .i x cv opreq1 x cv H~ e. a1d r19.21aiv A H~ e. B H~ e. /\ a1i impbid))
+thm (orthcom () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (.i) B) (0)) (= (opr B (.i) A) (0)))) (B A ax-his1 ancoms 0 eqeq1d A .i B opr 0 * fveq2 cj0 syl6eq syl5bir A B ax-his1 0 eqeq1d B .i A opr 0 * fveq2 cj0 syl6eq syl5bir impbid))
+thm (normlem0 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~)))) (= (opr (opr F (-v) (opr S (.s) G)) (.i) (opr F (-v) (opr S (.s) G))) (opr (opr (opr F (.i) F) (+) (opr (-u (` (*) S)) (x.) (opr F (.i) G))) (+) (opr (opr (-u S) (x.) (opr G (.i) F)) (+) (opr (opr S (x.) (` (*) S)) (x.) (opr G (.i) G))))) (normlem1.2 normlem1.1 normlem1.3 hvmulcl hvsubval normlem1.1 mulm1 .s G opreq1i 1cn negcl normlem1.1 normlem1.3 hvmulass eqtr3 F +v opreq2i eqtr4 normlem1.2 normlem1.1 normlem1.3 hvmulcl hvsubval normlem1.1 mulm1 .s G opreq1i 1cn negcl normlem1.1 normlem1.3 hvmulass eqtr3 F +v opreq2i eqtr4 .i opreq12i normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl hvaddcl F S -u .s G opr F +v S -u .s G opr opr ax-his2 mp3an normlem1.2 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl F F S -u .s G opr his7t mp3an normlem1.1 negcl normlem1.2 normlem1.3 S -u F G his5t mp3an normlem1.1 cjneg x. F .i G opr opreq1i eqtr F .i F opr + opreq2i eqtr normlem1.1 negcl normlem1.3 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl hvaddcl S -u G F +v S -u .s G opr opr ax-his3 mp3an normlem1.3 normlem1.2 normlem1.1 negcl normlem1.3 hvmulcl G F S -u .s G opr his7t mp3an normlem1.1 negcl normlem1.3 normlem1.3 S -u G G his5t mp3an G .i F opr + opreq2i eqtr S -u x. opreq2i normlem1.1 negcl normlem1.3 normlem1.2 hicl normlem1.1 negcl cjcl normlem1.3 normlem1.3 hicl mulcl adddi normlem1.1 negcl normlem1.1 negcl cjcl normlem1.3 normlem1.3 hicl mulass normlem1.1 cjneg S -u x. opreq2i normlem1.1 normlem1.1 cjcl mul2neg eqtr x. G .i G opr opreq1i eqtr3 S -u x. G .i F opr opr + opreq2i eqtr 3eqtr + opreq12i 3eqtr))
+thm (normlem1 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem1.4 (e. R (RR))) (normlem1.5 (= (` (abs) S) (1)))) (= (opr (opr F (-v) (opr (opr S (x.) R) (.s) G)) (.i) (opr F (-v) (opr (opr S (x.) R) (.s) G))) (opr (opr (opr F (.i) F) (+) (opr (opr (` (*) S) (x.) (-u R)) (x.) (opr F (.i) G))) (+) (opr (opr (opr S (x.) (-u R)) (x.) (opr G (.i) F)) (+) (opr (opr R (^) (2)) (x.) (opr G (.i) G))))) (normlem1.1 normlem1.4 recn mulcl normlem1.2 normlem1.3 normlem0 normlem1.1 normlem1.4 recn cjmul normlem1.4 normlem1.4 recn cjreb mpbi * S ` x. opreq2i eqtr negeqi normlem1.1 cjcl normlem1.4 recn mulneg2 eqtr4 x. F .i G opr opreq1i F .i F opr + opreq2i normlem1.1 normlem1.4 recn mulneg2 eqcomi x. G .i F opr opreq1i normlem1.1 normlem1.4 recn cjmul S x. R opr x. opreq2i normlem1.1 normlem1.4 recn normlem1.1 cjcl normlem1.4 recn cjcl mul4 normlem1.5 ^ 2 opreq1i normlem1.1 absvalsq sq1 3eqtr3 normlem1.4 normlem1.4 recn cjreb mpbi R x. opreq2i x. opreq12i normlem1.4 recn normlem1.4 recn mulcl mulid2 eqtr 3eqtr normlem1.4 recn sqval eqtr4 x. G .i G opr opreq1i + opreq12i + opreq12i eqtr))
+thm (normlem2 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F))))))) (e. B (RR)) (normlem2.4 normlem1.1 cjcj eqcomi normlem1.3 normlem1.2 his1 x. opreq12i normlem1.1 cjcl normlem1.2 normlem1.3 hicl cjmul eqtr4 normlem1.2 normlem1.3 his1 * S ` x. opreq2i normlem1.1 normlem1.3 normlem1.2 hicl cjmul eqtr4 + opreq12i normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcom normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl cjadd 3eqtr4r normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl cjreb mpbir renegcl eqeltr))
+thm (normlem3 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem3.7 (e. R (RR)))) (= (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C) (opr (opr (opr F (.i) F) (+) (opr (opr (` (*) S) (x.) (-u R)) (x.) (opr F (.i) G))) (+) (opr (opr (opr S (x.) (-u R)) (x.) (opr G (.i) F)) (+) (opr (opr R (^) (2)) (x.) (opr G (.i) G))))) (normlem3.6 normlem3.5 normlem1.3 normlem1.3 hicl eqeltr normlem3.7 recn sqcl mulcl normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 recn normlem3.7 recn mulcl addcom normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl normlem3.7 recn mulneg1 normlem2.4 x. R opreq1i normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl normlem3.7 recn mulneg2 3eqtr4 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn negcl adddir normlem1.1 cjcl normlem1.2 normlem1.3 hicl normlem3.7 recn negcl mul23 normlem1.1 normlem1.3 normlem1.2 hicl normlem3.7 recn negcl mul23 + opreq12i 3eqtr normlem3.7 recn sqcl normlem3.5 normlem1.3 normlem1.3 hicl eqeltr mulcom normlem3.5 R ^ 2 opr x. opreq2i eqtr3 + opreq12i normlem1.1 cjcl normlem3.7 recn negcl mulcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem3.7 recn negcl mulcl normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn sqcl normlem1.3 normlem1.3 hicl mulcl addass 3eqtr + opreq12i normlem3.5 normlem1.3 normlem1.3 hicl eqeltr normlem3.7 recn sqcl mulcl normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 recn normlem3.7 recn mulcl addcl normlem3.6 normlem1.2 normlem1.2 hicl eqeltr addcom normlem1.2 normlem1.2 hicl normlem1.1 cjcl normlem3.7 recn negcl mulcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem3.7 recn negcl mulcl normlem1.3 normlem1.2 hicl mulcl normlem3.7 recn sqcl normlem1.3 normlem1.3 hicl mulcl addcl addass 3eqtr4))
+thm (normlem4 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem4.7 (e. R (RR))) (normlem4.8 (= (` (abs) S) (1)))) (= (opr (opr F (-v) (opr (opr S (x.) R) (.s) G)) (.i) (opr F (-v) (opr (opr S (x.) R) (.s) G))) (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C)) (normlem1.1 normlem1.2 normlem1.3 normlem4.7 normlem4.8 normlem1 normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 normlem4.7 normlem3 eqtr4))
+thm (normlem5 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem4.7 (e. R (RR))) (normlem4.8 (= (` (abs) S) (1)))) (br (0) (<_) (opr (opr (opr A (x.) (opr R (^) (2))) (+) (opr B (x.) R)) (+) C)) (normlem1.2 normlem1.1 normlem4.7 recn mulcl normlem1.3 hvmulcl hvsubcl F -v S x. R opr .s G opr opr hiidge0t ax-mp normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 normlem4.7 normlem4.8 normlem4 breqtr))
+thm (normlem6 ((A x) (B x) (C x)) ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem2.4 (= B (-u (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F)))))) (normlem3.5 (= A (opr G (.i) G))) (normlem3.6 (= C (opr F (.i) F))) (normlem6.7 (= (` (abs) S) (1)))) (br (` (abs) B) (<_) (opr (2) (x.) (opr (` (sqr) A) (x.) (` (sqr) C)))) (normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr x cv x cv RR e. x cv 0 if ^ 2 opreq1 A x. opreq2d x cv x cv RR e. x cv 0 if B x. opreq2 + opreq12d + C opreq1d 0 <_ breq2d normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem3.5 normlem3.6 0re x cv elimel normlem6.7 normlem5 dedth rgen discrlem ax-mp normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 sqrecl 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl 0re lesubadd2 mpbi 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl recn addid1 breqtr normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 sqege0 0re 4re 4pos ltlei normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr mulge0 mp2an 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl mulge0 mp2an normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 sqrecl 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl remulcl sqrle mp2an mpbi normlem1.1 normlem1.2 normlem1.3 normlem2.4 normlem2 absre 4re normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr remulcl 0re 4re 4pos ltlei normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr mulge0 mp2an sqrmuli sqr4 normlem3.5 normlem1.3 G hiidrclt ax-mp eqeltr normlem3.6 normlem1.2 F hiidrclt ax-mp eqeltr normlem1.3 G hiidge0t ax-mp normlem3.5 breqtrr normlem1.2 F hiidge0t ax-mp normlem3.6 breqtrr sqrmuli x. opreq12i eqtr2 3brtr4))
+thm (normlem7 () ((normlem1.1 (e. S (CC))) (normlem1.2 (e. F (H~))) (normlem1.3 (e. G (H~))) (normlem7.4 (= (` (abs) S) (1)))) (br (opr (opr (` (*) S) (x.) (opr F (.i) G)) (+) (opr S (x.) (opr G (.i) F))) (<_) (opr (2) (x.) (opr (` (sqr) (opr G (.i) G)) (x.) (` (sqr) (opr F (.i) F))))) (normlem1.1 normlem1.2 normlem1.3 * S ` x. F .i G opr opr + S x. G .i F opr opr opr -u eqid normlem2 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negreb mpbi leabs normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl absneg breqtrr normlem1.1 normlem1.2 normlem1.3 * S ` x. F .i G opr opr + S x. G .i F opr opr opr -u eqid G .i G opr eqid F .i F opr eqid normlem7.4 normlem6 normlem1.1 normlem1.2 normlem1.3 * S ` x. F .i G opr opr + S x. G .i F opr opr opr -u eqid normlem2 normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negreb mpbi normlem1.1 cjcl normlem1.2 normlem1.3 hicl mulcl normlem1.1 normlem1.3 normlem1.2 hicl mulcl addcl negcl abscl 2re normlem1.3 G hiidge0t ax-mp normlem1.3 G hiidrclt ax-mp sqrcl ax-mp normlem1.2 F hiidge0t ax-mp normlem1.2 F hiidrclt ax-mp sqrcl ax-mp remulcl remulcl letr mp2an))
+thm (normlem8 () ((normlem8.1 (e. A (H~))) (normlem8.2 (e. B (H~))) (normlem8.3 (e. C (H~))) (normlem8.4 (e. D (H~)))) (= (opr (opr A (+v) B) (.i) (opr C (+v) D)) (opr (opr (opr A (.i) C) (+) (opr B (.i) D)) (+) (opr (opr A (.i) D) (+) (opr B (.i) C)))) (normlem8.1 normlem8.3 normlem8.4 A C D his7t mp3an normlem8.2 normlem8.3 normlem8.4 B C D his7t mp3an + opreq12i normlem8.1 normlem8.2 normlem8.3 normlem8.4 hvaddcl A B C +v D opr ax-his2 mp3an normlem8.1 normlem8.3 hicl normlem8.2 normlem8.4 hicl normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl add42 3eqtr4))
+thm (normlem9 () ((normlem8.1 (e. A (H~))) (normlem8.2 (e. B (H~))) (normlem8.3 (e. C (H~))) (normlem8.4 (e. D (H~)))) (= (opr (opr A (-v) B) (.i) (opr C (-v) D)) (opr (opr (opr A (.i) C) (+) (opr B (.i) D)) (-) (opr (opr A (.i) D) (+) (opr B (.i) C)))) (normlem8.1 normlem8.2 hvsubval normlem8.3 normlem8.4 hvsubval .i opreq12i normlem8.1 1cn negcl normlem8.2 hvmulcl normlem8.3 1cn negcl normlem8.4 hvmulcl normlem8 1cn negcl normlem8.2 1cn negcl normlem8.4 hvmulcl 1 -u B 1 -u .s D opr ax-his3 mp3an 1cn negcl normlem8.2 normlem8.4 1 -u B D his5t mp3an 1 -u x. opreq2i ax1re renegcl 1 -u cjret ax-mp 1 -u x. opreq2i 1cn 1cn mul2neg 1cn mulid2 3eqtr x. B .i D opr opreq1i 1cn negcl 1cn negcl cjcl normlem8.2 normlem8.4 hicl mulass normlem8.2 normlem8.4 hicl mulid2 3eqtr3 3eqtr A .i C opr + opreq2i 1cn negcl normlem8.1 normlem8.4 1 -u A D his5t mp3an ax1re renegcl 1 -u cjret ax-mp x. A .i D opr opreq1i normlem8.1 normlem8.4 hicl mulm1 3eqtr 1cn negcl normlem8.2 normlem8.3 1 -u B C ax-his3 mp3an normlem8.2 normlem8.3 hicl mulm1 eqtr + opreq12i normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl negdi eqtr4 + opreq12i normlem8.1 normlem8.3 hicl normlem8.2 normlem8.4 hicl addcl normlem8.1 normlem8.4 hicl normlem8.2 normlem8.3 hicl addcl subneg eqtr4 3eqtr))
+thm (normlem7t () ((normlem7t.1 (e. A (H~))) (normlem7t.2 (e. B (H~)))) (-> (/\ (e. S (CC)) (= (` (abs) S) (1))) (br (opr (opr (` (*) S) (x.) (opr A (.i) B)) (+) (opr S (x.) (opr B (.i) A))) (<_) (opr (2) (x.) (opr (` (sqr) (opr B (.i) B)) (x.) (` (sqr) (opr A (.i) A)))))) (S S CC e. abs S ` 1 = /\ S 1 if * fveq2 x. A .i B opr opreq1d S S CC e. abs S ` 1 = /\ S 1 if x. B .i A opr opreq1 + opreq12d <_ 2 x. sqr B .i B opr ` x. sqr A .i A opr ` opr opr breq1d S S CC e. abs S ` 1 = /\ S 1 if CC eleq1 S S CC e. abs S ` 1 = /\ S 1 if abs fveq2 1 eqeq1d anbi12d 1 S CC e. abs S ` 1 = /\ S 1 if CC eleq1 1 S CC e. abs S ` 1 = /\ S 1 if abs fveq2 1 eqeq1d anbi12d 1cn 0re ax1re lt01 ltlei ax1re absid ax-mp pm3.2i elimhyp pm3.26i normlem7t.1 normlem7t.2 S S CC e. abs S ` 1 = /\ S 1 if CC eleq1 S S CC e. abs S ` 1 = /\ S 1 if abs fveq2 1 eqeq1d anbi12d 1 S CC e. abs S ` 1 = /\ S 1 if CC eleq1 1 S CC e. abs S ` 1 = /\ S 1 if abs fveq2 1 eqeq1d anbi12d 1cn 0re ax1re lt01 ltlei ax1re absid ax-mp pm3.2i elimhyp pm3.27i normlem7 dedth))
+thm (bcseq () ((normlem7t.1 (e. A (H~))) (normlem7t.2 (e. B (H~)))) (<-> (= (opr (opr A (.i) B) (x.) (opr B (.i) A)) (opr (opr A (.i) A) (x.) (opr B (.i) B))) (= (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B))) (A .i B opr x. B .i A opr opr A .i A opr x. B .i B opr opr x. B .i B opr opreq1 eqcomd normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 B .i B opr B .i B opr .s A opr A his5t mp3an normlem7t.2 B hiidrclt ax-mp B .i B opr cjret ax-mp normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 B .i B opr A A ax-his3 mp3an x. opreq12i normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 hicl mulcl mulcom normlem7t.1 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl mulcom x. B .i B opr opreq1i eqtr4 3eqtr normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.2 A .i B opr B .i B opr .s A opr B his5t mp3an normlem7t.2 normlem7t.1 his1 eqcomi normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 B .i B opr A B ax-his3 mp3an x. opreq12i normlem7t.2 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl mulcl mulcom normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulass normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulcl mulcom 3eqtr 3eqtr 3eqtr4g normlem7t.2 B hiidrclt ax-mp B .i B opr cjret ax-mp normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 A .i B opr B A ax-his3 mp3an x. opreq12i normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.1 B .i B opr A .i B opr .s B opr A his5t mp3an normlem7t.1 normlem7t.2 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 A .i B opr A .i B opr .s B opr B his5t mp3an normlem7t.2 normlem7t.1 his1 eqcomi normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 A .i B opr B B ax-his3 mp3an x. opreq12i normlem7t.2 normlem7t.1 hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl mulcl mulcom normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mul23 normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 hicl mulcl normlem7t.2 normlem7t.2 hicl mulcom 3eqtr 3eqtr 3eqtr4r A .i B opr x. B .i A opr opr A .i A opr x. B .i B opr opr = a1i + opreq12d - B .i B opr .s A opr .i A .i B opr .s B opr opr + A .i B opr .s B opr .i B .i B opr .s A opr opr opr opreq1d normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hicl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl hicl addcl subid syl6eq normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl normlem9 syl5eq normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hvsubcl B .i B opr .s A opr -v A .i B opr .s B opr opr his6t ax-mp sylib normlem7t.2 normlem7t.2 hicl normlem7t.1 hvmulcl normlem7t.1 normlem7t.2 hicl normlem7t.2 hvmulcl hvsubeq0 sylib B .i B opr .s A opr A .i B opr .s B opr .i A opreq1 normlem7t.1 normlem7t.1 hicl normlem7t.2 normlem7t.2 hicl mulcom normlem7t.2 normlem7t.2 hicl normlem7t.1 normlem7t.1 B .i B opr A A ax-his3 mp3an eqtr4 normlem7t.1 normlem7t.2 hicl normlem7t.2 normlem7t.1 A .i B opr B A ax-his3 mp3an eqcomi 3eqtr4g eqcomd impbi))
+thm (normlem9at () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (-v) B) (.i) (opr A (-v) B)) (opr (opr (opr A (.i) A) (+) (opr B (.i) B)) (-) (opr (opr A (.i) B) (+) (opr B (.i) A))))) (A A H~ e. A 0v if -v B opreq1 A A H~ e. A 0v if -v B opreq1 .i opreq12d A A H~ e. A 0v if = id A A H~ e. A 0v if = id .i opreq12d + B .i B opr opreq1d A A H~ e. A 0v if .i B opreq1 A A H~ e. A 0v if B .i opreq2 + opreq12d - opreq12d eqeq12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 .i opreq12d B B H~ e. B 0v if = id B B H~ e. B 0v if = id .i opreq12d A H~ e. A 0v if .i A H~ e. A 0v if opr + opreq2d B B H~ e. B 0v if A H~ e. A 0v if .i opreq2 B B H~ e. B 0v if .i A H~ e. A 0v if opreq1 + opreq12d - opreq12d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl A elimel ax-hv0cl B elimel normlem9 dedth2h))
+thm (normvalt ((x y) (A x) (A y)) () (-> (e. A (H~)) (= (` (norm) A) (` (sqr) (opr A (.i) A)))) (x cv A x cv A .i opreq12 anidms sqr fveq2d x y df-hnorm sqr A .i A opr fvex fvopab4))
+thm (normclt () () (-> (e. A (H~)) (e. (` (norm) A) (RR))) (A normvalt A .i A opr sqrclt A hiidrclt A hiidge0t sylanc eqeltrd))
+thm (normge0t () () (-> (e. A (H~)) (br (0) (<_) (` (norm) A))) (A .i A opr sqrge0t A hiidrclt A hiidge0t sylanc A normvalt breqtrrd))
+thm (normgt0t () () (-> (e. A (H~)) (<-> (-. (= A (0v))) (br (0) (<) (` (norm) A)))) (A .i A opr sqrgt0t A hiidrclt A 0v = -. adantr A axhis42 sylanc exp A 0v .i A opreq1 A hi01t sylan9eqr sqr fveq2d sqr0 syl6eq exp A .i A opr sqrclt A hiidrclt A hiidge0t sylanc 0re jctir sqr A .i A opr ` 0 lttri3t syl sqr A .i A opr ` < 0 br -. 0 < sqr A .i A opr ` br -. pm3.27 syl6bi syld con2d impbid A normvalt 0 < breq2d bitr4d))
+thm (norm0 () () (= (` (norm) (0v)) (0)) (ax-hv0cl 0v normvalt ax-mp ax-hv0cl 0v hi01t sqr fveq2d ax-mp sqr0 3eqtr))
+thm (norm-it () () (-> (e. A (H~)) (<-> (= (` (norm) A) (0)) (= A (0v)))) (A normgt0t 0 norm A ` leltnetOLD A normclt 0re jctil A normge0t sylc bitrd bicon4d 0 norm A ` eqcom syl6rbb))
+thm (normcl () ((normcl.1 (e. A (H~)))) (e. (` (norm) A) (RR)) (normcl.1 A normclt ax-mp))
+thm (normsq () ((normcl.1 (e. A (H~)))) (= (opr (` (norm) A) (^) (2)) (opr A (.i) A)) (normcl.1 A normvalt ax-mp ^ 2 opreq1i normcl.1 A hiidge0t ax-mp normcl.1 A hiidrclt ax-mp sqsqr ax-mp eqtr))
+thm (norm-i () ((normcl.1 (e. A (H~)))) (<-> (= (` (norm) A) (0)) (= A (0v))) (normcl.1 A norm-it ax-mp))
+thm (normsqt () () (-> (e. A (H~)) (= (opr (` (norm) A) (^) (2)) (opr A (.i) A))) (A A H~ e. A 0v if norm fveq2 ^ 2 opreq1d A A H~ e. A 0v if = id A A H~ e. A 0v if = id .i opreq12d eqeq12d ax-hv0cl A elimel normsq dedth))
+thm (normsub0 () ((normsub0.1 (e. A (H~))) (normsub0.2 (e. B (H~)))) (<-> (= (` (norm) (opr A (-v) B)) (0)) (= A B)) (normsub0.1 normsub0.2 hvsubcl norm-i normsub0.1 normsub0.2 hvsubeq0 bitr))
+thm (normsub0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (` (norm) (opr A (-v) B)) (0)) (= A B))) (A A H~ e. A 0v if -v B opreq1 norm fveq2d 0 eqeq1d A A H~ e. A 0v if B eqeq1 bibi12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 norm fveq2d 0 eqeq1d B B H~ e. B 0v if A H~ e. A 0v if eqeq2 bibi12d ax-hv0cl A elimel ax-hv0cl B elimel normsub0 dedth2h))
+thm (norm-ii () ((norm-ii.1 (e. A (H~))) (norm-ii.2 (e. B (H~)))) (br (` (norm) (opr A (+v) B)) (<_) (opr (` (norm) A) (+) (` (norm) B))) (ax1re 1cn cjreb mpbi x. B .i A opr opreq1i norm-ii.2 norm-ii.1 hicl mulid2 eqtr norm-ii.1 norm-ii.2 hicl mulid2 + opreq12i 1cn norm-ii.2 norm-ii.1 0re ax1re lt01 ltlei ax1re absid ax-mp normlem7 eqbrtrr ax1re 1cn cjreb mpbi x. B .i A opr opreq1i norm-ii.2 norm-ii.1 hicl mulid2 eqtr norm-ii.1 norm-ii.2 hicl mulid2 + opreq12i 1cn norm-ii.2 norm-ii.1 * 1 ` x. B .i A opr opr + 1 x. A .i B opr opr opr -u eqid normlem2 1cn cjcl norm-ii.2 norm-ii.1 hicl mulcl 1cn norm-ii.1 norm-ii.2 hicl mulcl addcl negreb mpbi eqeltrr 2re norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp remulcl remulcl norm-ii.1 A hiidrclt ax-mp norm-ii.2 B hiidrclt ax-mp readdcl leadd2 mpbi norm-ii.1 norm-ii.2 norm-ii.1 norm-ii.2 normlem8 norm-ii.1 norm-ii.2 hicl norm-ii.2 norm-ii.1 hicl addcom A .i A opr + B .i B opr opr + opreq2i eqtr norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn binom2 norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn sqcl 2cn norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp recn norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn mulcl mulcl norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp recn sqcl add23 norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqsqr ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqsqr ax-mp + opreq12i + 2 x. sqr A .i A opr ` x. sqr B .i B opr ` opr opr opreq1i 3eqtr 3brtr4 norm-ii.1 norm-ii.2 hvaddcl A +v B opr hiidge0t ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl sqege0 norm-ii.1 norm-ii.2 hvaddcl A +v B opr hiidrclt ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl sqrecl sqrle mp2an mpbi norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrge0 ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrge0 ax-mp norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp addge0 mp2an norm-ii.1 A hiidge0t ax-mp norm-ii.1 A hiidrclt ax-mp sqrcl ax-mp norm-ii.2 B hiidge0t ax-mp norm-ii.2 B hiidrclt ax-mp sqrcl ax-mp readdcl sqrsqe ax-mp breqtr norm-ii.1 norm-ii.2 hvaddcl A +v B opr normvalt ax-mp norm-ii.1 A normvalt ax-mp norm-ii.2 B normvalt ax-mp + opreq12i 3brtr4))
+thm (norm-iii () ((norm-iii.1 (e. A (CC))) (norm-iii.2 (e. B (H~)))) (= (` (norm) (opr A (.s) B)) (opr (` (abs) A) (x.) (` (norm) B))) (norm-iii.1 norm-iii.1 norm-iii.2 norm-iii.2 his35 sqr fveq2i norm-iii.1 cjmulrcl norm-iii.2 B hiidrclt ax-mp norm-iii.1 cjmulge0 norm-iii.2 B hiidge0t ax-mp sqrmuli eqtr norm-iii.1 norm-iii.2 hvmulcl A .s B opr normvalt ax-mp norm-iii.1 A absvalt ax-mp norm-iii.2 B normvalt ax-mp x. opreq12i 3eqtr4))
+thm (norm-iiit () () (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (norm) (opr A (.s) B)) (opr (` (abs) A) (x.) (` (norm) B)))) (A A CC e. A 0 if .s B opreq1 norm fveq2d A A CC e. A 0 if abs fveq2 x. norm B ` opreq1d eqeq12d B B H~ e. B 0v if A CC e. A 0 if .s opreq2 norm fveq2d B B H~ e. B 0v if norm fveq2 abs A CC e. A 0 if ` x. opreq2d eqeq12d 0cn A elimel ax-hv0cl B elimel norm-iii dedth2h))
+thm (normsub () ((normsub.1 (e. A (H~))) (normsub.2 (e. B (H~)))) (= (` (norm) (opr A (-v) B)) (` (norm) (opr B (-v) A))) (normsub.2 normsub.1 hvnegdi norm fveq2i 1cn negcl normsub.2 normsub.1 hvsubcl norm-iii eqtr3 1cn absneg 0re ax1re lt01 ltlei ax1re absid ax-mp eqtr x. norm B -v A opr ` opreq1i normsub.2 normsub.1 hvsubcl normcl recn mulid2 3eqtr))
+thm (normpyth () ((normsub.1 (e. A (H~))) (normsub.2 (e. B (H~)))) (-> (= (opr A (.i) B) (0)) (= (opr (` (norm) (opr A (+v) B)) (^) (2)) (opr (opr (` (norm) A) (^) (2)) (+) (opr (` (norm) B) (^) (2))))) (A .i B opr 0 = id normsub.1 normsub.2 A B orthcom mp2an biimp + opreq12d 0cn addid1 syl6eq A .i A opr + B .i B opr opr + opreq2d normsub.1 normsub.1 hicl normsub.2 normsub.2 hicl addcl addid1 syl6eq normsub.1 normsub.2 normsub.1 normsub.2 normlem8 syl5eq normsub.1 normsub.2 hvaddcl normsq normsub.1 normsq normsub.2 normsq + opreq12i 3eqtr4g))
+thm (normsubt () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (norm) (opr A (-v) B)) (` (norm) (opr B (-v) A)))) (A A H~ e. A 0v if -v B opreq1 norm fveq2d A A H~ e. A 0v if B -v opreq2 norm fveq2d eqeq12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 norm fveq2d B B H~ e. B 0v if -v A H~ e. A 0v if opreq1 norm fveq2d eqeq12d ax-hv0cl A elimel ax-hv0cl B elimel normsub dedth2h))
+thm (normnegt () () (-> (e. A (H~)) (= (` (norm) (opr (-u (1)) (.s) A)) (` (norm) A))) (ax-hv0cl 0v A normsubt mpan A hv2negt norm fveq2d A hvsub0t norm fveq2d 3eqtr3d))
+thm (normpytht () () (-> (/\ (e. A (H~)) (e. B (H~))) (-> (= (opr A (.i) B) (0)) (= (opr (` (norm) (opr A (+v) B)) (^) (2)) (opr (opr (` (norm) A) (^) (2)) (+) (opr (` (norm) B) (^) (2)))))) (A A H~ e. A 0v if .i B opreq1 0 eqeq1d A A H~ e. A 0v if +v B opreq1 norm fveq2d ^ 2 opreq1d A A H~ e. A 0v if norm fveq2 ^ 2 opreq1d + norm B ` ^ 2 opr opreq1d eqeq12d imbi12d B B H~ e. B 0v if A H~ e. A 0v if .i opreq2 0 eqeq1d B B H~ e. B 0v if A H~ e. A 0v if +v opreq2 norm fveq2d ^ 2 opreq1d B B H~ e. B 0v if norm fveq2 ^ 2 opreq1d norm A H~ e. A 0v if ` ^ 2 opr + opreq2d eqeq12d imbi12d ax-hv0cl A elimel ax-hv0cl B elimel normpyth dedth2h))
+thm (normpyct () () (-> (/\ (e. A (H~)) (e. B (H~))) (-> (= (opr A (.i) B) (0)) (br (` (norm) A) (<_) (` (norm) (opr A (+v) B))))) (A normclt norm A ` sqreclt syl recnd norm A ` ^ 2 opr ax0id syl B H~ e. adantr B normclt norm B ` sqege0t syl A H~ e. adantl 0re 0 norm B ` ^ 2 opr norm A ` ^ 2 opr leadd2t mp3an1 ancoms A normclt norm A ` sqreclt syl B normclt norm B ` sqreclt syl syl2an mpbid eqbrtrrd A .i B opr 0 = adantr A B normpytht imp breqtrrd exp norm A ` norm A +v B opr ` le2sqet A normclt B H~ e. adantr A B ax-hvaddcl A +v B opr normclt syl jca A normge0t B H~ e. adantr A B ax-hvaddcl A +v B opr normge0t syl jca sylc sylibrd))
+thm (norm3dif () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~)))) (br (` (norm) (opr A (-v) B)) (<_) (opr (` (norm) (opr A (-v) C)) (+) (` (norm) (opr C (-v) B)))) (norm3dif.1 1cn negcl norm3dif.3 hvmulcl norm3dif.3 1cn negcl norm3dif.2 hvmulcl hvaddcl hvass 1cn negcl norm3dif.3 hvmulcl norm3dif.3 1cn negcl norm3dif.2 hvmulcl hvass 1cn negcl norm3dif.3 hvmulcl norm3dif.3 hvcom norm3dif.3 norm3dif.3 hvsubval eqtr4 norm3dif.3 C hvsubidt ax-mp eqtr +v 1 -u .s B opr opreq1i ax-hv0cl 1cn negcl norm3dif.2 hvmulcl hvcom 1cn negcl norm3dif.2 hvmulcl 1 -u .s B opr ax-hvaddid ax-mp 3eqtr eqtr3 A +v opreq2i eqtr2 norm3dif.1 norm3dif.2 hvsubval norm3dif.1 norm3dif.3 hvsubval norm3dif.3 norm3dif.2 hvsubval +v opreq12i 3eqtr4 norm fveq2i norm3dif.1 norm3dif.3 hvsubcl norm3dif.3 norm3dif.2 hvsubcl norm-ii eqbrtr))
+thm (norm3adif () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~)))) (br (` (abs) (opr (` (norm) (opr A (-v) C)) (-) (` (norm) (opr B (-v) C)))) (<_) (` (norm) (opr A (-v) B))) (norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.2 norm3dif.3 hvsubcl normcl resubcl norm3dif.1 norm3dif.2 hvsubcl normcl absleOLD norm3dif.1 norm3dif.3 norm3dif.2 norm3dif norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.2 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.2 hvsubcl normcl lesubadd mpbir norm3dif.1 norm3dif.3 hvsubcl normcl recn norm3dif.2 norm3dif.3 hvsubcl normcl recn negdi3 norm3dif.2 norm3dif.3 norm3dif.1 norm3dif norm3dif.2 norm3dif.1 normsub + norm A -v C opr ` opreq1i breqtr norm3dif.2 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.1 norm3dif.2 hvsubcl normcl lesubadd mpbir eqbrtr mpbir2an))
+thm (norm3lem () ((norm3dif.1 (e. A (H~))) (norm3dif.2 (e. B (H~))) (norm3dif.3 (e. C (H~))) (norm3lem.4 (e. D (RR)))) (-> (/\ (br (` (norm) (opr A (-v) C)) (<) (opr D (/) (2))) (br (` (norm) (opr C (-v) B)) (<) (opr D (/) (2)))) (br (` (norm) (opr A (-v) B)) (<) D)) (norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.3 norm3dif.2 hvsubcl normcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl lt2add norm3dif.1 norm3dif.2 norm3dif.3 norm3dif norm3dif.1 norm3dif.2 hvsubcl normcl norm3dif.1 norm3dif.3 hvsubcl normcl norm3dif.3 norm3dif.2 hvsubcl normcl readdcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl norm3lem.4 2re 2re 2pos gt0ne0i redivcl readdcl lelttr mpan syl norm3lem.4 recn 2cn 2re 2pos gt0ne0i divcl 2times 2cn norm3lem.4 recn 2re 2pos gt0ne0i divcan2 eqtr3 syl6breq))
+thm (norm3lemt () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (RR)))) (-> (/\ (br (` (norm) (opr A (-v) C)) (<) (opr D (/) (2))) (br (` (norm) (opr C (-v) B)) (<) (opr D (/) (2)))) (br (` (norm) (opr A (-v) B)) (<) D))) (A A H~ e. A 0v if -v C opreq1 norm fveq2d < D / 2 opr breq1d norm C -v B opr ` < D / 2 opr br anbi1d A A H~ e. A 0v if -v B opreq1 norm fveq2d < D breq1d imbi12d B B H~ e. B 0v if C -v opreq2 norm fveq2d < D / 2 opr breq1d norm A H~ e. A 0v if -v C opr ` < D / 2 opr br anbi2d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 norm fveq2d < D breq1d imbi12d C C H~ e. C 0v if A H~ e. A 0v if -v opreq2 norm fveq2d < D / 2 opr breq1d C C H~ e. C 0v if -v B H~ e. B 0v if opreq1 norm fveq2d < D / 2 opr breq1d anbi12d norm A H~ e. A 0v if -v B H~ e. B 0v if opr ` < D br imbi1d D D RR e. D 2 if / 2 opreq1 norm A H~ e. A 0v if -v C H~ e. C 0v if opr ` < breq2d D D RR e. D 2 if / 2 opreq1 norm C H~ e. C 0v if -v B H~ e. B 0v if opr ` < breq2d anbi12d D D RR e. D 2 if norm A H~ e. A 0v if -v B H~ e. B 0v if opr ` < breq2 imbi12d ax-hv0cl A elimel ax-hv0cl B elimel ax-hv0cl C elimel 2re D elimel norm3lem dedth4h))
+thm (norm3adift () ((norm3adift.1 (e. C (H~)))) (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (abs) (opr (` (norm) (opr A (-v) C)) (-) (` (norm) (opr B (-v) C)))) (<_) (` (norm) (opr A (-v) B)))) (A A H~ e. A 0v if -v C opreq1 norm fveq2d - norm B -v C opr ` opreq1d abs fveq2d A A H~ e. A 0v if -v B opreq1 norm fveq2d <_ breq12d B B H~ e. B 0v if -v C opreq1 norm fveq2d norm A H~ e. A 0v if -v C opr ` - opreq2d abs fveq2d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 norm fveq2d <_ breq12d ax-hv0cl A elimel ax-hv0cl B elimel norm3adift.1 norm3adif dedth2h))
+thm (normpar () ((normpar.1 (e. A (H~))) (normpar.2 (e. B (H~)))) (= (opr (opr (` (norm) (opr A (-v) B)) (^) (2)) (+) (opr (` (norm) (opr A (+v) B)) (^) (2))) (opr (opr (2) (x.) (opr (` (norm) A) (^) (2))) (+) (opr (2) (x.) (opr (` (norm) B) (^) (2))))) (normpar.1 normpar.2 normpar.1 normpar.2 normlem9 normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl subneg eqtr normpar.1 normpar.2 normpar.1 normpar.2 normlem8 + opreq12i normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl negcl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl add42 normpar.1 normpar.2 hicl normpar.2 normpar.1 hicl addcl negid A .i A opr + B .i B opr opr + A .i A opr + B .i B opr opr opr + opreq2i normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl addcl addcl addid1 normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl normpar.1 normpar.1 hicl normpar.2 normpar.2 hicl add4 3eqtr 3eqtr normpar.1 normpar.2 hvsubcl normsq normpar.1 normpar.2 hvaddcl normsq + opreq12i normpar.1 normsq 2 x. opreq2i normpar.1 normpar.1 hicl 2times eqtr normpar.2 normsq 2 x. opreq2i normpar.2 normpar.2 hicl 2times eqtr + opreq12i 3eqtr4))
+thm (normpar2 () ((normpar2.1 (e. A (H~))) (normpar2.2 (e. B (H~))) (normpar2.3 (e. C (H~)))) (= (opr (` (norm) (opr A (-v) B)) (^) (2)) (opr (opr (opr (2) (x.) (opr (` (norm) (opr A (-v) C)) (^) (2))) (+) (opr (2) (x.) (opr (` (norm) (opr B (-v) C)) (^) (2)))) (-) (opr (` (norm) (opr (opr A (+v) B) (-v) (opr (2) (.s) C))) (^) (2)))) (4re recn normpar2.1 normpar2.3 hvsubcl normcl sqrecl recn mulcl 4re recn normpar2.2 normpar2.3 hvsubcl normcl sqrecl recn mulcl 2cn 2re 2pos gt0ne0i divdir 4re recn normpar2.1 normpar2.3 hvsubcl normcl sqrecl recn mulcl 4re recn normpar2.2 normpar2.3 hvsubcl normcl sqrecl recn mulcl addcom normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normpar2.1 normpar2.2 hvsubcl hvsubval normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubval +v 1 -u .s A -v B opr opr opreq1i normpar2.1 normpar2.2 hvcom normpar2.1 normpar2.2 hvnegdi +v opreq12i normpar2.2 normpar2.1 hvsubcan2 eqtr +v 1 -u .s 2 .s C opr opr opreq1i normpar2.1 normpar2.2 hvaddcl 1cn negcl 2cn normpar2.3 hvmulcl hvmulcl 1cn negcl normpar2.1 normpar2.2 hvsubcl hvmulcl hvadd23 2cn normpar2.2 hvmulcl 2cn normpar2.3 hvmulcl hvsubval 3eqtr4 3eqtr 2cn normpar2.2 normpar2.3 hvsubdistr1 eqtr4 norm fveq2i 2cn normpar2.2 normpar2.3 hvsubcl norm-iii 0re 2re 2pos ltlei 2re absid ax-mp x. norm B -v C opr ` opreq1i 3eqtr ^ 2 opreq1i 2cn normpar2.2 normpar2.3 hvsubcl normcl recn sqmul sq2 x. norm B -v C opr ` ^ 2 opr opreq1i 3eqtr normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubval +v A -v B opr opreq1i normpar2.1 normpar2.2 hvaddcl 1cn negcl 2cn normpar2.3 hvmulcl hvmulcl normpar2.1 normpar2.2 hvsubcl hvadd23 normpar2.1 normpar2.2 hvsubcan2 +v 1 -u .s 2 .s C opr opr opreq1i 2cn normpar2.1 hvmulcl 2cn normpar2.3 hvmulcl hvsubval eqtr4 3eqtr 2cn normpar2.1 normpar2.3 hvsubdistr1 eqtr4 norm fveq2i 2cn normpar2.1 normpar2.3 hvsubcl norm-iii 0re 2re 2pos ltlei 2re absid ax-mp x. norm A -v C opr ` opreq1i 3eqtr ^ 2 opreq1i 2cn normpar2.1 normpar2.3 hvsubcl normcl recn sqmul sq2 x. norm A -v C opr ` ^ 2 opr opreq1i 3eqtr + opreq12i eqtr4 normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normpar2.1 normpar2.2 hvsubcl normpar eqtr / 2 opreq1i 2cn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl sqrecl recn mulcl 2cn normpar2.1 normpar2.2 hvsubcl normcl sqrecl recn mulcl 2cn 2re 2pos gt0ne0i divdir 2cn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl sqrecl recn 2re 2pos gt0ne0i divcan3 2cn normpar2.1 normpar2.2 hvsubcl normcl sqrecl recn 2re 2pos gt0ne0i divcan3 + opreq12i 3eqtr 4re recn normpar2.1 normpar2.3 hvsubcl normcl sqrecl recn 2cn 2re 2pos gt0ne0i div23 4d2e2 x. norm A -v C opr ` ^ 2 opr opreq1i eqtr 4re recn normpar2.2 normpar2.3 hvsubcl normcl sqrecl recn 2cn 2re 2pos gt0ne0i div23 4d2e2 x. norm B -v C opr ` ^ 2 opr opreq1i eqtr + opreq12i 3eqtr3 2re normpar2.1 normpar2.3 hvsubcl normcl sqrecl remulcl 2re normpar2.2 normpar2.3 hvsubcl normcl sqrecl remulcl readdcl recn normpar2.1 normpar2.2 hvaddcl 2cn normpar2.3 hvmulcl hvsubcl normcl sqrecl recn normpar2.1 normpar2.2 hvsubcl normcl sqrecl recn subadd mpbir eqcomi))
+thm (polid2 () ((polid2.1 (e. A (H~))) (polid2.2 (e. B (H~))) (polid2.3 (e. C (H~))) (polid2.4 (e. D (H~)))) (= (opr A (.i) B) (opr (opr (opr (opr (opr A (+v) C) (.i) (opr D (+v) B)) (-) (opr (opr A (-v) C) (.i) (opr D (-v) B))) (+) (opr (i) (x.) (opr (opr (opr A (+v) (opr (i) (.s) C)) (.i) (opr D (+v) (opr (i) (.s) B))) (-) (opr (opr A (-v) (opr (i) (.s) C)) (.i) (opr D (-v) (opr (i) (.s) B)))))) (/) (4))) (4re recn polid2.1 polid2.2 hicl 4re 4pos gt0ne0i divcan3 2cn polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl subcl adddi 2t2e4 x. A .i B opr opreq1i 2cn 2cn polid2.1 polid2.2 hicl mulass eqtr3 polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl polid2.1 polid2.2 hicl A .i B opr C .i D opr A .i B opr ppncant mp3an polid2.1 polid2.2 hicl 2times eqtr4 2 x. opreq2i eqtr4 polid2.1 polid2.4 hicl polid2.3 polid2.2 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl A .i D opr + C .i B opr opr A .i B opr + C .i D opr opr A .i B opr + C .i D opr opr pnncant mp3an polid2.1 polid2.3 polid2.4 polid2.2 normlem8 polid2.1 polid2.3 polid2.4 polid2.2 normlem9 - opreq12i polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl addcl 2times 3eqtr4 polid2.1 axicn polid2.3 hvmulcl polid2.4 axicn polid2.2 hvmulcl normlem8 polid2.1 axicn polid2.3 hvmulcl polid2.4 axicn polid2.2 hvmulcl normlem9 - opreq12i polid2.1 polid2.4 hicl axicn polid2.3 hvmulcl axicn polid2.2 hvmulcl hicl addcl polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl A .i D opr + i .s C opr .i i .s B opr opr opr A .i i .s B opr opr + i .s C opr .i D opr opr A .i i .s B opr opr + i .s C opr .i D opr opr pnncant mp3an polid2.1 axicn polid2.2 hvmulcl hicl axicn polid2.3 hvmulcl polid2.4 hicl addcl 2times axicn polid2.1 polid2.2 i A B his5t mp3an cji x. A .i B opr opreq1i eqtr axicn polid2.3 polid2.4 i C D ax-his3 mp3an + opreq12i 2 x. opreq2i eqtr3 3eqtr i x. opreq2i 2cn axicn axicn negcl polid2.1 polid2.2 hicl mulcl axicn polid2.3 polid2.4 hicl mulcl addcl mul12 axicn axicn mulneg2 itimesi negeqi 1cn negneg 3eqtr x. A .i B opr opreq1i axicn axicn negcl polid2.1 polid2.2 hicl mulass polid2.1 polid2.2 hicl mulid2 3eqtr3 itimesi x. C .i D opr opreq1i axicn axicn polid2.3 polid2.4 hicl mulass polid2.3 polid2.4 hicl mulm1 3eqtr3 + opreq12i axicn axicn negcl polid2.1 polid2.2 hicl mulcl axicn polid2.3 polid2.4 hicl mulcl adddi polid2.1 polid2.2 hicl polid2.3 polid2.4 hicl subneg 3eqtr4 2 x. opreq2i eqtr3 eqtr + opreq12i 3eqtr4 / 4 opreq1i eqtr3))
+thm (polid () ((polid.1 (e. A (H~))) (polid.2 (e. B (H~)))) (= (opr A (.i) B) (opr (opr (opr (opr (` (norm) (opr A (+v) B)) (^) (2)) (-) (opr (` (norm) (opr A (-v) B)) (^) (2))) (+) (opr (i) (x.) (opr (opr (` (norm) (opr A (+v) (opr (i) (.s) B))) (^) (2)) (-) (opr (` (norm) (opr A (-v) (opr (i) (.s) B))) (^) (2))))) (/) (4))) (polid.1 polid.2 polid.2 polid.1 polid2 polid.1 polid.2 hvaddcl normsq polid.1 polid.2 hvsubcl normsq - opreq12i polid.1 axicn polid.2 hvmulcl hvaddcl normsq polid.1 axicn polid.2 hvmulcl hvsubcl normsq - opreq12i i x. opreq2i + opreq12i / 4 opreq1i eqtr4))
+thm (bcs () ((bcs.1 (e. A (H~))) (bcs.2 (e. B (H~)))) (br (` (abs) (opr A (.i) B)) (<_) (opr (` (norm) A) (x.) (` (norm) B))) (A .i B opr 0 abs fveq2 abs0 bcs.1 A normge0t ax-mp bcs.2 B normge0t ax-mp bcs.1 normcl bcs.2 normcl mulge0 mp2an eqbrtr syl6eqbr A .i B opr 0 df-ne bcs.1 bcs.2 hicl abslem2 bcs.2 bcs.1 his1 A .i B opr / abs A .i B opr ` opr x. opreq2i * A .i B opr / abs A .i B opr ` opr ` x. A .i B opr opr + opreq2i syl5req bcs.1 bcs.2 hicl abs00 eqneqi bcs.1 bcs.2 hicl bcs.1 bcs.2 hicl abscl recn divclz bcs.1 bcs.2 hicl bcs.1 bcs.2 hicl abscl recn divrecz abs fveq2d bcs.1 bcs.2 hicl abscl recn recclz bcs.1 bcs.2 hicl jctil A .i B opr 1 / abs A .i B opr ` opr absmult syl 1 / abs A .i B opr ` opr absidt bcs.1 bcs.2 hicl abscl rerecclz 0 1 / abs A .i B opr ` opr ltlet bcs.1 bcs.2 hicl abscl rerecclz 0re jctil bcs.1 bcs.2 hicl abs00 eqneqi bcs.1 bcs.2 hicl absgt0 bitr bcs.1 bcs.2 hicl abscl recgt0 sylbi sylc sylanc abs A .i B opr ` x. opreq2d eqtrd bcs.1 bcs.2 hicl abscl recn recidz 3eqtrd jca sylbir bcs.1 bcs.2 A .i B opr / abs A .i B opr ` opr normlem7t syl eqbrtrd sylbir bcs.1 normcl recn bcs.2 normcl recn mulcom bcs.2 B normvalt ax-mp bcs.1 A normvalt ax-mp x. opreq12i eqtr abs A .i B opr ` <_ breq2i 2pos bcs.1 bcs.2 hicl abscl bcs.2 B hiidge0t ax-mp bcs.2 B hiidrclt ax-mp sqrcl ax-mp bcs.1 A hiidge0t ax-mp bcs.1 A hiidrclt ax-mp sqrcl ax-mp remulcl 2re lemul2 ax-mp bitr sylibr pm2.61i))
+thm (hcau ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f)) () (<-> (e. F (Cauchy)) (/\ (:--> F (NN) (H~)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x)))))))))) (F Cauchy elisset F NN H~ ffn nnex NN V F fnex ax-mp syl x RR 0 < x cv br y NN z NN w NN y cv <_ z cv br y cv <_ w cv br /\ norm F z cv ` -v F w cv ` opr ` < x cv br -> A.e. A.e. E.e. -> A.e. adantr f cv F NN H~ feq1 f cv F z cv fveq1 f cv F w cv fveq1 -v opreq12d norm fveq2d < x cv breq1d y cv <_ z cv br y cv <_ w cv br /\ imbi2d w NN biraldv z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d f x y z w f cv df-hcau V elab2g pm5.21nii))
+thm (hcauseq ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (e. F (Cauchy)) (:--> F (NN) (H~))) (F x y z w hcau pm3.26bd))
+thm (hcaucvg ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (e. F (Cauchy)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x))))))))) (F x y z w hcau pm3.27bd))
+thm (seq1hcau ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) () (-> (:--> F (NN) (H~)) (<-> (e. F (Cauchy)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x)))))))))) (F x y z w hcau baib))
+thm (hlim ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (<-> (br F (~~>v) A) (/\ (/\ (:--> F (NN) (H~)) (e. A (H~))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x))))))))) (hlim.1 hlim.2 f cv F NN H~ feq1 w cv H~ e. anbi1d f cv F z cv fveq1 -v w cv opreq1d norm fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d w cv A H~ eleq1 F NN H~ :--> anbi2d w cv A F z cv ` -v opreq2 norm fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d f w x y z df-hlim brab))
+thm (hlimseq ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (:--> F (NN) (H~))) (hlim.1 hlim.2 x y z hlim pm3.26bd pm3.26d))
+thm (hlimvec ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (e. A (H~))) (hlim.1 hlim.2 x y z hlim pm3.26bd pm3.27d))
+thm (hlimconv ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((hlim.1 (e. F (V))) (hlim.2 (e. A (V)))) (-> (br F (~~>v) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x)))))))) (hlim.1 hlim.2 x y z hlim pm3.27bd))
+thm (hlim2 ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) () (-> (/\ (:--> F (NN) (H~)) (e. A (H~))) (<-> (br F (~~>v) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` F (cv z)) (-v) A)) (<) (cv x))))))))) (f cv F NN H~ feq1 w cv H~ e. anbi1d f cv F z cv fveq1 -v w cv opreq1d norm fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d w cv A H~ eleq1 F NN H~ :--> anbi2d w cv A F z cv ` -v opreq2 norm fveq2d < x cv breq1d y cv <_ z cv br imbi2d z NN biraldv y NN birexdv 0 < x cv br imbi2d x RR biraldv anbi12d f w x y z df-hlim V H~ brabg F NN H~ ffn nnex NN V F fnex ax-mp syl sylan F NN H~ :--> A H~ e. /\ x RR 0 < x cv br y NN z NN y cv <_ z cv br norm F z cv ` -v A opr ` < x cv br -> A.e. E.e. -> A.e. ibar bitr4d))
+thm (shex ((x y) (h x) (h y)) () (e. (SH) (V)) (h x y df-sh H~ h df-pw ax-hilex pwex eqeltrr h cv H~ C_ 0v h cv e. pm3.26 x h cv y h cv x cv +v y cv opr h cv e. A.e. A.e. x CC y h cv x cv .s y cv opr h cv e. A.e. A.e. /\ adantr h ss2abi ssexi eqeltr))
+thm (sh ((x y) (h x) (H x) (h y) (H y) (H h)) () (<-> (e. H (SH)) (/\ (/\ (C_ H (H~)) (e. (0v) H)) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H)))))) (H SH elisset ax-hilex H ssex 0v H e. x H y H x cv +v y cv opr H e. A.e. A.e. x CC y H x cv .s y cv opr H e. A.e. A.e. /\ ad2antll h cv H H~ sseq1 h cv H 0v eleq2 anbi12d h cv H x cv +v y cv opr eleq2 y raleqd x raleqd h cv H x cv .s y cv opr eleq2 y raleqd x CC biraldv anbi12d anbi12d h x y df-sh V elab2g pm5.21nii))
+thm (shss ((x y) (H x) (H y)) () (-> (e. H (SH)) (C_ H (H~))) (H x y sh pm3.26bd pm3.26d))
+thm (shelt () () (-> (/\ (e. H (SH)) (e. A H)) (e. A (H~))) (H shss A sseld imp))
+thm (shssi () ((shssi.1 (e. H (SH)))) (C_ H (H~)) (shssi.1 H shss ax-mp))
+thm (shel () ((shssi.1 (e. H (SH)))) (-> (e. A H) (e. A (H~))) (shssi.1 shssi A sseli))
+thm (sheli () ((shssi.1 (e. H (SH))) (sheli.1 (e. A H))) (e. A (H~)) (shssi.1 shssi sheli.1 sselii))
+thm (sh0 ((x y) (H x) (H y)) () (-> (e. H (SH)) (e. (0v) H)) (H x y sh pm3.26bd pm3.27d))
+thm (shaddclt ((x y) (H x) (H y) (A x) (A y) (B x) (B y)) () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B H)) (e. (opr A (+v) B) H))) (x cv A +v y cv opreq1 H eleq1d y cv B A +v opreq2 H eleq1d H H rcla42v H x y sh pm3.27bd pm3.26d syl5com))
+thm (shmulclt ((x y) (H x) (H y) (A x) (A y) (B x) (B y)) () (-> (e. H (SH)) (-> (/\ (e. A (CC)) (e. B H)) (e. (opr A (.s) B) H))) (x cv A .s y cv opreq1 H eleq1d y cv B A .s opreq2 H eleq1d CC H rcla42v H x y sh pm3.27bd pm3.27d syl5com))
+thm (shsubclt () () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B H)) (e. (opr A (-v) B) H))) (H shss A sseld H shss B sseld anim12d imp A B hvsubvalt syl 1cn negcl H 1 -u B shmulclt mpani A H e. anim2d H A 1 -u .s B opr shaddclt syld imp eqeltrd exp))
+thm (sh2 ((x y) (H x) (H y)) () (-> (C_ H (H~)) (<-> (e. H (SH)) (/\ (e. (0v) H) (/\ (A.e. x H (A.e. y H (e. (opr (cv x) (+v) (cv y)) H))) (A.e. x (CC) (A.e. y H (e. (opr (cv x) (.s) (cv y)) H))))))) (H H~ C_ 0v H e. x H y H x cv +v y cv opr H e. A.e. A.e. x CC y H x cv .s y cv opr H e. A.e. A.e. /\ anass baib H x y sh syl5bb))
+thm (closedsub ((f x) (h x) (H x) (f h) (H f) (H h)) () (<-> (e. H (CH)) (/\ (e. H (SH)) (A. f (A. x (-> (/\ (:--> (cv f) (NN) H) (br (cv f) (~~>v) (cv x))) (e. (cv x) H)))))) (H CH elisset H SH elisset f x f cv NN H :--> f cv ~~>v x cv br /\ x cv H e. -> A. A. adantr h cv H SH eleq1 h cv H f cv NN feq3 f cv ~~>v x cv br anbi1d h cv H x cv eleq2 imbi12d f x bi2aldv anbi12d h f x df-ch V elab2g pm5.21nii))
+thm (chsssh ((f x) (h x) (f h)) () (C_ (CH) (SH)) (h f x df-ch h SH f x f cv NN h cv :--> f cv ~~>v x cv br /\ x cv h cv e. -> A. A. ssab eqsstr))
+thm (chex () () (e. (CH) (V)) (shex chsssh ssexi))
+thm (chsh () () (-> (e. H (CH)) (e. H (SH))) (chsssh H sseli))
+thm (chshi () ((chshi.1 (e. H (CH)))) (e. H (SH)) (chshi.1 H chsh ax-mp))
+thm (ch0 () () (-> (e. H (CH)) (e. (0v) H)) (H chsh H sh0 syl))
+thm (chss () () (-> (e. H (CH)) (C_ H (H~))) (H chsh H shss syl))
+thm (chelt () () (-> (/\ (e. H (CH)) (e. A H)) (e. A (H~))) (H chss A sseld imp))
+thm (chssi () ((chssi.1 (e. H (CH)))) (C_ H (H~)) (chssi.1 chshi shssi))
+thm (chel () ((chssi.1 (e. H (CH)))) (-> (e. A H) (e. A (H~))) (chssi.1 chssi A sseli))
+thm (cheli () ((chssi.1 (e. H (CH))) (cheli.1 (e. A H))) (e. A (H~)) (chssi.1 chssi cheli.1 sselii))
+thm (chlim ((f x) (F x) (F f) (H x) (H f) (A x)) ((chlim.1 (e. A (V)))) (-> (e. H (CH)) (-> (/\ (:--> F (NN) H) (br F (~~>v) A)) (e. A H))) (H f x closedsub pm3.27bd F NN H ffn nnex NN V F fnex ax-mp syl F ~~>v A br adantr f cv F NN H feq1 f cv F ~~>v x cv breq1 anbi12d x cv H e. imbi1d x bialdv V cla4gv chlim.1 x cv A F ~~>v breq2 F NN H :--> anbi2d x cv A H eleq1 imbi12d cla4v syl6 syl pm2.43b syl))
+thm (hlim0 ((x y) (x z) (y z)) () (br (X. (NN) ({} (0v))) (~~>v) (0v)) (nnex 0v snex xpex ax-hv0cl elisseti x y z hlim ax-hv0cl elisseti NN fconst ax-hv0cl 0v H~ snssi ax-mp NN 0v {} X. NN 0v {} H~ fss mp2an ax-hv0cl pm3.2i ax-hv0cl elisseti z cv NN fvconst2 -v 0v opreq1d ax-hv0cl 0v hvsubidt ax-mp syl6eq norm fveq2d norm0 syl6eq < x cv breq1d biimprd 1 <_ z cv br a1d com3r r19.21aiv 1nn jctil y cv 1 <_ z cv breq1 norm NN 0v {} X. z cv ` -v 0v opr ` < x cv br imbi1d z NN biraldv NN rcla4ev syl x cv RR e. a1i rgen mpbir2an))
+thm (hlimcaui ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y) (F z) (F w) (F v)) ((hlimcau.1 (e. A (V))) (hlimcau.2 (e. F (V))) (hlimcaui.4 (br F (~~>v) A))) (e. F (Cauchy)) (F x y z w hcau hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp v cv x cv / 2 opr 0 < breq2 v cv x cv / 2 opr norm F z cv ` -v A opr ` < breq2 y cv <_ z cv br imbi2d z NN biraldv y NN birexdv imbi12d hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp hlimcaui.4 hlimcau.2 hlimcau.1 hlimvec ax-mp hlimcaui.4 F A v y z hlim2 mpbii mp2an vtoclri x cv x cv RR e. x cv 0 if / 2 opreq1 RR eleq1d 0re x cv elimel 2re 2re 2pos gt0ne0i redivcl dedth 0 < x cv br adantr x cv x cv RR e. x cv 0 if 0 < breq2 x cv x cv RR e. x cv 0 if / 2 opreq1 0 < breq2d imbi12d 0re x cv elimel 2re 2pos divgt0lem dedth imp sylc y cv <_ z cv br norm F z cv ` -v A opr ` < x cv / 2 opr br y cv <_ w cv br norm F w cv ` -v A opr ` < x cv / 2 opr br prth A F w cv ` normsubt < x cv / 2 opr breq1d norm F z cv ` -v A opr ` < x cv / 2 opr br anbi2d x cv RR e. z cv NN e. /\ adantl hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp z cv ffvrni F w cv ` H~ e. anim1i ancoms A H~ e. x cv RR e. /\ anim1i ancoms an4s ancoms F z cv ` F w cv ` A x cv norm3lemt syl sylbird hlimcaui.4 hlimcau.2 hlimcau.1 hlimseq ax-mp w cv ffvrni hlimcaui.4 hlimcau.2 hlimcau.1 hlimvec ax-mp jctil sylan2 syl9r r19.20dva r19.20dva z NN w y cv <_ z cv br norm F z cv ` -v A opr ` < x cv / 2 opr br -> y cv <_ w cv br norm F w cv ` -v A opr ` < x cv / 2 opr br -> raaan w cv z cv y cv <_ breq2 w cv z cv F fveq2 -v A opreq1d norm fveq2d < x cv / 2 opr breq1d imbi12d NN cbvralv z NN y cv <_ z cv br norm F z cv ` -v A opr ` < x cv / 2 opr br -> A.e. anbi2i z NN y cv <_ z cv br norm F z cv ` -v A opr ` < x cv / 2 opr br -> A.e. anidm 3bitr syl5ibr y NN r19.22sdv 0 < x cv br adantr mpd exp rgen mpbir2an))
+thm (hlimcau () ((hlimcau.1 (e. A (V))) (hlimcau.2 (e. F (V)))) (-> (br F (~~>v) A) (e. F (Cauchy))) (F F ~~>v A br F NN 0v {} X. if Cauchy eleq1 hlimcau.1 ax-hv0cl elisseti F ~~>v A br keepel hlimcau.2 nnex 0v snex xpex F ~~>v A br keepel F F ~~>v A br F NN 0v {} X. if ~~>v A breq1 A F ~~>v A br A 0v if F ~~>v A br F NN 0v {} X. if ~~>v breq2 NN 0v {} X. F ~~>v A br F NN 0v {} X. if ~~>v 0v breq1 0v F ~~>v A br A 0v if F ~~>v A br F NN 0v {} X. if ~~>v breq2 hlim0 elimhyp2v hlimcaui dedth))
+thm (hlimunii ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((hlimuni.1 (e. A (V))) (hlimuni.2 (e. B (V))) (hlimuni.3 (e. F (V))) (hlimunii.3 (/\ (br F (~~>v) A) (br F (~~>v) B)))) (= A B) (y cv w cv z nn2get rgen2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2pos divgt0lem hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2re 2pos gt0ne0i redivcl hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 x y z hlimconv ax-mp x cv norm A -v B opr ` / 2 opr 0 < breq2 x cv norm A -v B opr ` / 2 opr norm F z cv ` -v A opr ` < breq2 y cv <_ z cv br imbi2d z NN biraldv y NN birexdv imbi12d RR rcla4v mp2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl 2re 2re 2pos gt0ne0i redivcl hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 x w z hlimconv ax-mp x cv norm A -v B opr ` / 2 opr 0 < breq2 x cv norm A -v B opr ` / 2 opr norm F z cv ` -v B opr ` < breq2 w cv <_ z cv br imbi2d z NN biraldv w NN birexdv imbi12d RR rcla4v mp2 jca z NN y cv <_ z cv br norm F z cv ` -v A opr ` < norm A -v B opr ` / 2 opr br -> w cv <_ z cv br norm F z cv ` -v B opr ` < norm A -v B opr ` / 2 opr br -> r19.26 y cv <_ z cv br w cv <_ z cv br /\ norm F z cv ` -v A opr ` < norm A -v B opr ` / 2 opr br norm F z cv ` -v B opr ` < norm A -v B opr ` / 2 opr br /\ con3 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl ltnr hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimseq ax-mp z cv ffvrni hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp F z cv ` A normsubt mpan2 syl < norm A -v B opr ` / 2 opr breq1d norm F z cv ` -v B opr ` < norm A -v B opr ` / 2 opr br anbi1d hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimseq ax-mp z cv ffvrni hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl normcl hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp A B F z cv ` norm A -v B opr ` norm3lemt mpanl12 mpan2 syl sylbid mtoi syl5com y cv <_ z cv br norm F z cv ` -v A opr ` < norm A -v B opr ` / 2 opr br w cv <_ z cv br norm F z cv ` -v B opr ` < norm A -v B opr ` / 2 opr br prth syl5 r19.20i sylbir w NN r19.22si y NN r19.22si y NN w NN z NN y cv <_ z cv br norm F z cv ` -v A opr ` < norm A -v B opr ` / 2 opr br -> A.e. z NN w cv <_ z cv br norm F z cv ` -v B opr ` < norm A -v B opr ` / 2 opr br -> A.e. reeanv z NN y cv <_ z cv br w cv <_ z cv br /\ ralnex w NN birex w NN z NN y cv <_ z cv br w cv <_ z cv br /\ E.e. rexnal bitr y NN birex y NN w NN z NN y cv <_ z cv br w cv <_ z cv br /\ E.e. A.e. rexnal bitr 3imtr3 3syl mt2 hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubcl A -v B opr normgt0t ax-mp bicon1i mpbi hlimunii.3 pm3.26i hlimuni.3 hlimuni.1 hlimvec ax-mp hlimunii.3 pm3.27i hlimuni.3 hlimuni.2 hlimvec ax-mp hvsubeq0 mpbi))
+thm (hlimuni () ((hlimuni.1 (e. A (V))) (hlimuni.2 (e. B (V))) (hlimuni.3 (e. F (V)))) (-> (/\ (br F (~~>v) A) (br F (~~>v) B)) (= A B)) (A F ~~>v A br F ~~>v B br /\ A 0v if B eqeq1 B F ~~>v A br F ~~>v B br /\ B 0v if F ~~>v A br F ~~>v B br /\ A 0v if eqeq2 hlimuni.1 ax-hv0cl elisseti F ~~>v A br F ~~>v B br /\ keepel hlimuni.2 ax-hv0cl elisseti F ~~>v A br F ~~>v B br /\ keepel hlimuni.3 nnex 0v snex xpex F ~~>v A br F ~~>v B br /\ keepel A F ~~>v A br F ~~>v B br /\ A 0v if F ~~>v breq2 F ~~>v B br anbi1d B F ~~>v A br F ~~>v B br /\ B 0v if F ~~>v breq2 F ~~>v F ~~>v A br F ~~>v B br /\ A 0v if br anbi2d F F ~~>v A br F ~~>v B br /\ F NN 0v {} X. if ~~>v F ~~>v A br F ~~>v B br /\ A 0v if breq1 F F ~~>v A br F ~~>v B br /\ F NN 0v {} X. if ~~>v F ~~>v A br F ~~>v B br /\ B 0v if breq1 anbi12d 0v F ~~>v A br F ~~>v B br /\ A 0v if NN 0v {} X. ~~>v breq2 NN 0v {} X. ~~>v 0v br anbi1d 0v F ~~>v A br F ~~>v B br /\ B 0v if NN 0v {} X. ~~>v breq2 NN 0v {} X. ~~>v F ~~>v A br F ~~>v B br /\ A 0v if br anbi2d NN 0v {} X. F ~~>v A br F ~~>v B br /\ F NN 0v {} X. if ~~>v F ~~>v A br F ~~>v B br /\ A 0v if breq1 NN 0v {} X. F ~~>v A br F ~~>v B br /\ F NN 0v {} X. if ~~>v F ~~>v A br F ~~>v B br /\ B 0v if breq1 anbi12d hlim0 hlim0 pm3.2i elimhyp3v hlimunii dedth2v))
+thm (hlimreu ((x y) (F x) (F y) (H x) (H y)) ((hlimreu.1 (e. F (V)))) (<-> (E.e. x H (br F (~~>v) (cv x))) (E!e. x H (br F (~~>v) (cv x)))) (x visset y visset hlimreu.1 hlimuni x cv H e. y cv H e. /\ a1i rgen2 x H F ~~>v x cv br E.e. jctr x cv y cv F ~~>v breq2 H reu4 sylibr x H F ~~>v x cv br reurex impbi))
+thm (hlimeu ((x y) (F x) (F y)) ((hlimeu.1 (e. F (V)))) (<-> (E. x (br F (~~>v) (cv x))) (E! x (br F (~~>v) (cv x)))) (x visset y visset hlimeu.1 hlimuni x y gen2 x F ~~>v x cv br E. jctr x cv y cv F ~~>v breq2 eu4 sylibr x F ~~>v x cv br euex impbi))
+thm (chsscm ((f x) (h x) (f h) (C h)) ((cmh.1 (= C ({|} h (/\ (e. (cv h) (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))))))))) (C_ (CH) C) (f cv NN h cv :--> f cv ~~>v x cv br x cv h cv e. impexp f cv ~~>v x cv br x cv h cv e. ancr x cv H~ e. adantld f cv NN h cv :--> imim2i sylbi com12 x 19.20dv impcom x x cv H~ e. f cv ~~>v x cv br /\ x cv h cv e. f cv ~~>v x cv br /\ 19.22 syl x H~ f cv ~~>v x cv br df-rex x h cv f cv ~~>v x cv br df-rex 3imtr4g f cv x ax-hcompl syl5 exp com23 f 19.20i f Cauchy f cv NN h cv :--> x h cv f cv ~~>v x cv br E.e. -> df-ral sylibr h cv SH e. anim2i h cv f x closedsub cmh.1 eqabi 3imtr4 ssriv))
+thm (chcmh ((f x) (h x) (f h) (C h)) ((cmh.1 (= C ({|} h (/\ (e. (cv h) (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) (cv h)) (E.e. x (cv h) (br (cv f) (~~>v) (cv x)))))))))) (= (CH) C) (cmh.1 chsscm f Cauchy f cv NN h cv :--> x h cv f cv ~~>v x cv br E.e. -> df-ral f cv Cauchy e. x ax-17 f cv NN h cv :--> x ax-17 x h cv f cv ~~>v x cv br hbre1 hbim hbim x visset f visset hlimcau x h cv f cv ~~>v x cv br E.e. imim1i f cv ~~>v x cv br x 19.8a f visset x hlimeu sylib x f cv ~~>v x cv br x cv h cv e. eupick exp x h cv f cv ~~>v x cv br df-rex x x cv h cv e. f cv ~~>v x cv br exancom bitr syl5ib syl pm2.43a sylcom f cv NN h cv :--> imim2i f cv Cauchy e. f cv NN h cv :--> x h cv f cv ~~>v x cv br E.e. bi2.04 f cv NN h cv :--> f cv ~~>v x cv br x cv h cv e. impexp 3imtr4 19.21ai f 19.20i sylbi h cv SH e. anim2i cmh.1 eqabi h cv f x closedsub 3imtr4 ssriv eqssi))
+thm (ch2 ((f x) (h x) (H x) (f h) (H f) (H h) (g x) (f g) (g h)) () (<-> (e. H (CH)) (/\ (e. H (SH)) (A.e. f (Cauchy) (-> (:--> (cv f) (NN) H) (E.e. x H (br (cv f) (~~>v) (cv x))))))) (H CH elisset H SH elisset f Cauchy f cv NN H :--> x H f cv ~~>v x cv br E.e. -> A.e. adantr h cv H SH eleq1 h cv H f cv NN feq3 h cv H x f cv ~~>v x cv br rexeq imbi12d f Cauchy biraldv anbi12d h cv g cv SH eleq1 h cv g cv f cv NN feq3 h cv g cv x f cv ~~>v x cv br rexeq imbi12d f Cauchy biraldv anbi12d cbvabv chcmh V elab2g pm5.21nii))
+thm (chcompl ((f x) (H x) (H f) (F x) (F f)) () (-> (e. H (CH)) (-> (/\ (e. F (Cauchy)) (:--> F (NN) H)) (E.e. x H (br F (~~>v) (cv x))))) (H f x ch2 pm3.27bd f cv F NN H feq1 f cv F ~~>v x cv breq1 x H birexdv imbi12d Cauchy rcla4v com12 imp3a syl))
+thm (helch ((x y) (f x) (f y)) () (e. (H~) (CH)) (H~ f x closedsub H~ x y sh H~ ssid ax-hv0cl pm3.2i x cv y cv ax-hvaddcl rgen2 x cv y cv ax-hvmulcl rgen2a pm3.2i mpbir2an f visset x visset hlimvec f cv NN H~ :--> adantl f x gen2 mpbir2an))
+thm (helsh () () (e. (H~) (SH)) (helch chshi))
+thm (shsspwh () () (C_ (SH) (P~ (H~))) (SH pwuni helsh x cv shss rgen H~ SH x ssunieq mp2an H~ SH U. pweq ax-mp sseqtr4))
+thm (chsspwh () () (C_ (CH) (P~ (H~))) (chsssh shsspwh CH SH H~ P~ sstr mp2an))
+thm (hsn0elch ((x y) (f x) (f y)) () (e. ({} (0v)) (CH)) (0v {} f x closedsub 0v {} x y sh ax-hv0cl 0v H~ snssi ax-mp ax-hv0cl elisseti snid pm3.2i x cv 0v y cv 0v +v opreq12 ax-hv0cl hvaddid2 syl6eq x cv +v y cv oprex 0v elsnc sylibr x 0v elsn y 0v elsn syl2anb rgen2 y cv 0v x cv .s opreq2 x cv hvmul0t sylan9eqr x cv .s y cv oprex 0v elsnc sylibr y 0v elsn sylan2b rgen2a pm3.2i mpbir2an ax-hv0cl elisseti snid ax-hv0cl elisseti x visset f visset hlimuni 0v {} eleq1d ax-hv0cl elisseti f cv NN fconst2 hlim0 f cv NN 0v {} X. ~~>v 0v breq1 mpbiri sylbi sylan mpbii f x gen2 mpbir2an))
+thm (elch0 () () (<-> (e. A (0H)) (= A (0v))) (df-ch0 A eleq2i ax-hv0cl elisseti A elsnc2 bitr))
+thm (h0elch () () (e. (0H) (CH)) (df-ch0 hsn0elch eqeltr))
+thm (h0elsh () () (e. (0H) (SH)) (h0elch chshi))
+thm (ocvalt ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w)) () (-> (C_ H (H~)) (= (` (_|_) H) ({e.|} x (H~) (A.e. y H (= (opr (cv x) (.i) (cv y)) (0)))))) (ax-hilex H~ V H elpw2g ax-mp z cv H y x cv .i y cv opr 0 = raleq x H~ birabsdv z w x y df-oc z visset H~ elpw w cv x H~ y z cv x cv .i y cv opr 0 = A.e. {e.|} = anbi1i z w biopabi eqtr4 ax-hilex x y H x cv .i y cv opr 0 = A.e. rabex fvopab4 sylbir))
+thm (ocelt ((x y) (H x) (H y) (A x) (A y)) () (-> (C_ H (H~)) (<-> (e. A (` (_|_) H)) (/\ (e. A (H~)) (A.e. x H (= (opr A (.i) (cv x)) (0)))))) (H y x ocvalt A eleq2d y cv A .i x cv opreq1 0 eqeq1d x H biraldv H~ elrab syl6bb))
+thm (shocelt ((H x) (A x)) () (-> (e. H (SH)) (<-> (e. A (` (_|_) H)) (/\ (e. A (H~)) (A.e. x H (= (opr A (.i) (cv x)) (0)))))) (H shss H A x ocelt syl))
+thm (ocsh ((x y) (x z) (y z) (A x) (A y) (A z)) () (-> (C_ A (H~)) (e. (` (_|_) A) (SH))) (x H~ y A x cv .i y cv opr 0 = A.e. ssrab A x y ocvalt H~ sseq1d mpbiri A H~ y cv ssel y cv hi01t syl6 r19.21aiv ax-hv0cl jctil A 0v y ocelt mpbird jca x cv y cv z cv ax-his2 3expa x cv .i z cv opr 0 y cv .i z cv opr 0 + opreq12 0cn addid1 syl6eq sylan9eq exp ancoms A H~ z cv ssel2 sylan an1rs r19.20dva exp imdistand x cv y cv ax-hvaddcl z A x cv +v y cv opr .i z cv opr 0 = A.e. anim1i syl6 A x cv z ocelt A y cv z ocelt anbi12d x cv H~ e. z A x cv .i z cv opr 0 = A.e. y cv H~ e. z A y cv .i z cv opr 0 = A.e. an4 z A x cv .i z cv opr 0 = y cv .i z cv opr 0 = r19.26 x cv H~ e. y cv H~ e. /\ anbi2i bitr4 syl6bb A x cv +v y cv opr z ocelt 3imtr4d exp3a r19.21adv r19.21aiv y cv .i z cv opr 0 x cv x. opreq2 0 eqeq1d x cv mul01t syl5bir com12 z cv H~ e. y cv H~ e. ad2antrl x cv y cv z cv ax-his3 0 eqeq1d 3expa ancoms sylibrd A H~ z cv ssel2 sylan an1rs r19.20dva exp imdistand x cv y cv ax-hvmulcl z A x cv .s y cv opr .i z cv opr 0 = A.e. anim1i syl6 A y cv z ocelt x cv CC e. anbi2d x cv CC e. y cv H~ e. z A y cv .i z cv opr 0 = A.e. anass syl6bbr A x cv .s y cv opr z ocelt 3imtr4d exp3a r19.21adv r19.21aiv jca jca _|_ A ` x y sh sylibr))
+thm (shocsh () () (-> (e. A (SH)) (e. (` (_|_) A) (SH))) (A shss A ocsh syl))
+thm (ocss () () (-> (C_ A (H~)) (C_ (` (_|_) A) (H~))) (A ocsh _|_ A ` shss syl))
+thm (shocss () () (-> (e. A (SH)) (C_ (` (_|_) A) (H~))) (A shss A ocss syl))
+thm (occont ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A)))) (A B y cv ssel x cv .i y cv opr 0 = imim1d y 19.20dv y B x cv .i y cv opr 0 = df-ral y A x cv .i y cv opr 0 = df-ral 3imtr4g x cv H~ e. a1d r19.21aiv x H~ y B x cv .i y cv opr 0 = A.e. y A x cv .i y cv opr 0 = A.e. ss2rab sylibr A H~ C_ B H~ C_ /\ adantl B x y ocvalt A H~ C_ A B C_ ad2antlr A x y ocvalt B H~ C_ A B C_ ad2antll 3sstr4d exp))
+thm (occon2t () () (-> (/\ (C_ A (H~)) (C_ B (H~))) (-> (C_ A B) (C_ (` (_|_) (` (_|_) A)) (` (_|_) (` (_|_) B))))) (A B occont B ocss A ocss anim12i ancoms _|_ B ` _|_ A ` occont syl syld))
+thm (occon2 () ((occon2.1 (C_ A (H~))) (occon2.2 (C_ B (H~)))) (-> (C_ A B) (C_ (` (_|_) (` (_|_) A)) (` (_|_) (` (_|_) B)))) (occon2.1 occon2.2 A B occon2t mp2an))
+thm (oc0 () () (-> (e. H (SH)) (e. (0v) (` (_|_) H))) (H shocsh _|_ H ` sh0 syl))
+thm (ocorth ((H x) (A x) (B x)) () (-> (C_ H (H~)) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (opr A (.i) B) (0)))) (H B x ocelt B H~ e. x H B .i x cv opr 0 = A.e. pm3.27 syl6bi imp H H~ C_ A H e. /\ adantl x cv A B .i opreq2 0 eqeq1d H rcla4v H H~ C_ H H~ C_ B _|_ H ` e. /\ ad2antlr A B orthcom H H~ A ssel2 H ocss B sseld imp syl2an sylibrd mpd anandis exp))
+thm (shocorth () () (-> (e. H (SH)) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (opr A (.i) B) (0)))) (H shss H A B ocorth syl))
+thm (ococss ((x y) (A x) (A y)) () (-> (C_ A (H~)) (C_ A (` (_|_) (` (_|_) A)))) (A H~ y cv ssel A y cv x cv ocorth exp3a r19.21adv jcad A ocss _|_ A ` y cv x ocelt syl sylibrd ssrdv))
+thm (shococss () () (-> (e. A (SH)) (C_ A (` (_|_) (` (_|_) A)))) (A shss A ococss syl))
+thm (shorth () () (-> (e. H (SH)) (-> (C_ G (` (_|_) H)) (-> (/\ (e. A G) (e. B H)) (= (opr A (.i) B) (0))))) (H B A shocorth imp H shss B sseld H shocss A sseld anim12d imp B A orthcom syl mpbid G _|_ H ` A ssel B H e. anim1d imp A _|_ H ` e. B H e. ancom sylib sylan2 exp32))
+thm (ocin ((x y) (A x) (A y)) () (-> (e. A (SH)) (= (i^i A (` (_|_) A)) (0H))) (A x cv y shocelt y cv x cv x cv .i opreq2 0 eqeq1d A rcla4v com12 x cv his6t biimpd sylan9r syl6bi com23 imp3a x cv 0v A eleq1 x cv 0v _|_ A ` eleq1 anbi12d A sh0 A oc0 jca syl5bir com12 impbid x cv A _|_ A ` elin x cv elch0 3bitr4g eqrd))
+thm (chocval ((x y) (A x) (A y)) ((chocval.1 (e. A (CH)))) (= (` (_|_) A) ({e.|} x (H~) (A.e. y A (= (opr (cv x) (.i) (cv y)) (0))))) (chocval.1 chssi A x y ocvalt ax-mp))
+thm (chocuni () ((chocuni.1 (e. H (CH)))) (-> (/\ (/\ (e. A H) (e. B (` (_|_) H))) (/\ (e. C H) (e. D (` (_|_) H)))) (-> (/\ (= R (opr A (+v) B)) (= R (opr C (+v) D))) (/\ (= A C) (= B D)))) (A H~ e. B H~ e. pm3.26 D H~ e. anim1i A B A D hvsub4t syldan A hvsubidt B H~ e. D H~ e. ad2antll +v B -v D opr opreq1d B D hvsubclt B -v D opr hvaddid2t syl A H~ e. adantll 3eqtrd C H~ e. adantrl C D A D hvsub4t A H~ e. C H~ e. D H~ e. /\ pm3.27 C H~ e. D H~ e. pm3.27 A H~ e. anim2i sylanc D hvsubidt A H~ e. C H~ e. ad2antrr C -v A opr +v opreq2d C A hvsubclt C -v A opr ax-hvaddid syl ancoms D H~ e. adantrr 3eqtrd B H~ e. adantlr eqeq12d chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an chocuni.1 chshi H C A shsubclt ax-mp ancoms B -v D opr C -v A opr = a1d D _|_ H ` e. adantrr B _|_ H ` e. adantlr chocuni.1 chshi H shocsh ax-mp _|_ H ` B D shsubclt ax-mp B -v D opr C -v A opr _|_ H ` eleq1 biimpcd syl C H e. adantrl A H e. adantll jcad C A hvsubeq0t ancoms D H~ e. adantrr B H~ e. adantlr chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an C A eqcom syl6bb chocuni.1 chshi H ocin ax-mp C -v A opr eleq2i C -v A opr H _|_ H ` elin C -v A opr elch0 3bitr3 syl5bb sylibd chocuni.1 chshi H C A shsubclt ax-mp ancoms C -v A opr H B -v D opr eleq1a syl D _|_ H ` e. adantrr B _|_ H ` e. adantlr chocuni.1 chshi H shocsh ax-mp _|_ H ` B D shsubclt ax-mp B -v D opr C -v A opr = a1d C H e. adantrl A H e. adantll jcad B D hvsubeq0t C H~ e. adantrl A H~ e. adantll chocuni.1 A chel chocuni.1 chshi H shocsh ax-mp B shel anim12i chocuni.1 C chel chocuni.1 chshi H shocsh ax-mp D shel anim12i syl2an chocuni.1 chshi H ocin ax-mp B -v D opr eleq2i B -v D opr H _|_ H ` elin B -v D opr elch0 3bitr3 syl5bb sylibd jcad sylbid R A +v B opr C +v D opr eqtr2t -v A +v D opr opreq1d syl5))
+thm (occllem1 () ((occllem1.1 (e. A (H~))) (occllem1.2 (e. B (H~))) (occllem1.3 (e. S (H~)))) (br (` (abs) (opr (opr B (.i) S) (-) (opr A (.i) S))) (<_) (opr (` (norm) (opr B (-v) A)) (x.) (` (norm) S))) (occllem1.2 1cn negcl occllem1.1 hvmulcl occllem1.3 B 1 -u .s A opr S ax-his2 mp3an 1cn negcl occllem1.1 occllem1.3 1 -u A S ax-his3 mp3an B .i S opr + opreq2i eqtr2 occllem1.2 occllem1.3 hicl occllem1.1 occllem1.3 hicl subneg occllem1.1 occllem1.3 hicl mulm1 B .i S opr + opreq2i eqtr4 occllem1.2 occllem1.1 hvsubval .i S opreq1i 3eqtr4 abs fveq2i occllem1.2 occllem1.1 hvsubcl occllem1.3 bcs eqbrtr))
+thm (occllem2 () ((occllem2.1 (e. S (H~)))) (-> (/\ (e. A (H~)) (e. B (H~))) (br (` (abs) (opr (opr B (.i) S) (-) (opr A (.i) S))) (<_) (opr (` (norm) (opr B (-v) A)) (x.) (` (norm) S)))) (A A H~ e. A 0v if .i S opreq1 B .i S opr - opreq2d abs fveq2d A A H~ e. A 0v if B -v opreq2 norm fveq2d x. norm S ` opreq1d <_ breq12d B B H~ e. B 0v if .i S opreq1 - A H~ e. A 0v if .i S opr opreq1d abs fveq2d B B H~ e. B 0v if -v A H~ e. A 0v if opreq1 norm fveq2d x. norm S ` opreq1d <_ breq12d ax-hv0cl A elimel ax-hv0cl B elimel occllem2.1 occllem1 dedth2h))
+thm (occllem3 ((x y) (F x) (F y) (S x) (S y) (D x) (D y)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S))))))) (-> (e. D (NN)) (= (` G D) (opr (` F D) (.i) S))) (x cv D F fveq2 .i S opreq1d occllem3.1 F D ` .i S oprex fvopab4))
+thm (occllem4 ((x y) (F x) (F y) (S x) (S y)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S)))))) (occllem4.2 (e. S (H~)))) (-> (:--> F (NN) (H~)) (:--> G (NN) (CC))) (F NN H~ x cv ffvrn occllem4.2 jctir F x cv ` S ax-hicl syl exp r19.21aiv occllem3.1 CC fopab2 sylib))
+thm (occllem5OLD ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (G z)) ((occllem3.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S))))))) (-> (A.e. z (NN) (= (opr (` F (cv z)) (.i) S) (0))) (br G (~~>OLD) (0))) (z cv x cv F fveq2 .i S opreq1d 0 eqeq1d NN rcla4v F x cv ` .i S opr 0 y cv eqeq2 syl6com pm5.32d x y biopabdv occllem3.1 NN 0 x y fconstopab 3eqtr4g clim0OLD syl6eqbr))
+thm (occllem6OLD ((w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (F z) (F w) (F v) (F u) (G w) (G u) (G v) (S x) (S y) (S z) (S w) (S v) (S u)) ((occllem6.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (` F (cv x)) (.i) S)))))) (occllem6.2 (e. A (H~))) (occllem6.3 (e. S (H~))) (occllem6.4 (e. F (V)))) (-> (-. (= S (0v))) (-> (br F (~~>v) A) (br G (~~>OLD) (opr A (.i) S)))) (occllem6.1 occllem6.3 occllem4 occllem6.2 occllem6.3 hicl jctir A H~ e. adantr S 0v = -. a1i z RR 0 < z cv br w NN v NN w cv <_ v cv br norm F v cv ` -v A opr ` < z cv br -> A.e. E.e. -> A.e. adantrd occllem6.3 normcl u cv norm S ` redivclt mp3an2 occllem6.3 normcl gt0ne0 sylan2 exp 0 < u cv br adantr occllem6.3 normcl u cv norm S ` divgt0t mpanl2 exp32 imp jcad com12 F NN H~ :--> adantr w NN v NN w cv <_ v cv br norm F v cv ` -v A opr ` < u cv / norm S ` opr br -> A.e. E.e. imim1d occllem6.3 normcl norm F v cv ` -v A opr ` norm S ` u cv ltmuldivt mp3anl2 F NN H~ v cv ffvrn occllem6.2 jctir F v cv ` A hvsubclt F v cv ` -v A opr normclt 3syl 0 < norm S ` br adantll u cv RR e. 0 < u cv br /\ adantlr u cv RR e. 0 < u cv br pm3.26 0 < norm S ` br F NN H~ :--> /\ v cv NN e. ad2antlr jca 0 < norm S ` br F NN H~ :--> pm3.26 u cv RR e. 0 < u cv br /\ v cv NN e. ad2antll sylanc F NN H~ v cv ffvrn occllem6.2 jctil occllem6.3 A F v cv ` occllem2 syl occllem6.1 v cv occllem3 - A .i S opr opreq1d abs fveq2d <_ norm F v cv ` -v A opr ` x. norm S ` opr breq1d F NN H~ :--> adantl mpbird 0 < norm S ` br adantll u cv RR e. adantr abs G v cv ` - A .i S opr opr ` norm F v cv ` -v A opr ` x. norm S ` opr u cv lelttrt 3expa G NN CC v cv ffvrn occllem6.1 occllem6.3 occllem4 sylan occllem6.2 occllem6.3 hicl jctir G v cv ` A .i S opr subclt G v cv ` - A .i S opr opr absclt 3syl F NN H~ v cv ffvrn occllem6.2 jctir F v cv ` A hvsubclt F v cv ` -v A opr normclt 3syl occllem6.3 normcl jctir norm F v cv ` -v A opr ` norm S ` axmulrcl syl jca 0 < norm S ` br adantll sylan mpand 0 < u cv br adantrr an1rs sylbird w cv <_ v cv br imim2d r19.20dva w NN r19.22sdv exp a2d syld exp4a occllem6.3 S normgt0t ax-mp sylanb A H~ e. adantrr z cv u cv / norm S ` opr 0 < breq2 z cv u cv / norm S ` opr norm F v cv ` -v A opr ` < breq2 w cv <_ v cv br imbi2d v NN biraldv w NN birexdv imbi12d RR rcla4v com12 imp3a syl5 r19.21adv exp imp3a jcad occllem6.4 occllem6.2 elisseti z w v hlim A .i S oprex G u w v climOLD 3imtr4g))
+thm (occllem7 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z)) ((occllem7.1 (e. A (H~))) (occllem7.2 (e. S (H~))) (occllem7.3 (e. F (V)))) (-> (/\ (br F (~~>v) A) (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0)))) (= (opr A (.i) S) (0))) (S 0v A .i opreq2 occllem7.1 A hi02t ax-mp syl6eq F ~~>v A br x NN F x cv ` .i S opr 0 = A.e. /\ a1d y z y cv NN e. z cv F y cv ` .i S opr = /\ {<,>|} eqid occllem7.1 occllem7.2 occllem7.3 occllem6OLD y z y cv NN e. z cv F y cv ` .i S opr = /\ {<,>|} eqid x occllem5OLD S 0v = -. a1i anim12d A .i S oprex 0re elisseti y z y cv NN e. z cv F y cv ` .i S opr = /\ {<,>|} climuniOLD syl6 pm2.61i))
+thm (occllem8 ((F x) (S x)) ((occllem8.1 (e. F (V)))) (-> (/\ (e. A (H~)) (e. S (H~))) (-> (/\ (br F (~~>v) A) (A.e. x (NN) (= (opr (` F (cv x)) (.i) S) (0)))) (= (opr A (.i) S) (0)))) (A A H~ e. A 0v if F ~~>v breq2 x NN F x cv ` .i S opr 0 = A.e. anbi1d A A H~ e. A 0v if .i S opreq1 0 eqeq1d imbi12d S S H~ e. S 0v if F x cv ` .i opreq2 0 eqeq1d x NN biraldv F ~~>v A H~ e. A 0v if br anbi2d S S H~ e. S 0v if A H~ e. A 0v if .i opreq2 0 eqeq1d imbi12d ax-hv0cl A elimel ax-hv0cl S elimel occllem8.1 x occllem7 dedth2h))
+thm (occl ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f)) ((occl.1 (C_ A (H~)))) (e. (` (_|_) A) (CH)) (_|_ A ` f x closedsub occl.1 A ocsh ax-mp f visset x visset hlimvec f cv NN _|_ A ` :--> adantl f visset x visset hlimvec f visset x cv y cv z occllem8 exp4b occl.1 y cv sseli syl5 com23 mpcom imp r19.20dva f cv NN _|_ A ` z cv ffvrn occl.1 A f cv z cv ` y ocelt ax-mp pm3.27bd syl exp r19.21aiv y A z NN f cv z cv ` .i y cv opr 0 = ralcom sylibr syl5 impcom jca occl.1 A x cv y ocelt ax-mp sylibr f x gen2 mpbir2an))
+thm (occlt () () (-> (C_ A (H~)) (e. (` (_|_) A) (CH))) (A A H~ C_ A H~ if _|_ fveq2 CH eleq1d A A H~ C_ A H~ if H~ sseq1 H~ A H~ C_ A H~ if H~ sseq1 H~ ssid elimhyp occl dedth))
+thm (shocclt () () (-> (e. A (SH)) (e. (` (_|_) A) (CH))) (A shss A occlt syl))
+thm (choclt () () (-> (e. A (CH)) (e. (` (_|_) A) (CH))) (A chsh A shocclt syl))
+thm (chocl () ((chocl.1 (e. A (CH)))) (e. (` (_|_) A) (CH)) (chocl.1 chssi occl))
+thm (projlem1 ((R z) (D z)) ((projlem1.1 (e. R (RR))) (projlem1.2 (e. D (RR)))) (-> (br (0) (<) D) (E.e. z (NN) (br (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z)) (<) (opr D (^) (2))))) (projlem1.2 gt0ne0 projlem1.2 sqegt0 projlem1.2 sqrecl gt0ne0 4re 2re projlem1.1 remulcl ax1re readdcl remulcl projlem1.2 sqrecl redivclz 4 x. 2 x. R opr + 1 opr opr / D ^ 2 opr opr z arch 3syl z cv nnret z cv z cv RR e. z cv 0 if 0 < breq2 0 < D ^ 2 opr br anbi1d z cv z cv RR e. z cv 0 if 4 x. 2 x. R opr + 1 opr opr / opreq2 < D ^ 2 opr breq1d z cv z cv RR e. z cv 0 if 4 x. 2 x. R opr + 1 opr opr / D ^ 2 opr opr < breq2 bibi12d imbi12d 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 0re z cv elimel projlem1.2 sqrecl ltdiv23 dedth z cv nngt0t sylani syl anabsi8 birexdva mpbird 3syl))
+thm (projlem2 ((R z) (D z)) ((projlem1.1 (e. R (RR))) (projlem1.2 (e. D (RR))) (projlem2.3 (br (0) (<_) R))) (-> (br (0) (<) D) (E.e. z (NN) (br (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) (cv z))) (<) D))) (projlem1.1 projlem1.2 z projlem1 sqr 4 x. 2 x. R opr + 1 opr opr / z cv opr ` D lt2sqet 4 x. 2 x. R opr + 1 opr opr / z cv opr sqrclt 4 x. 2 x. R opr + 1 opr opr z cv redivclt 3expa z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nnne0t sylanc 4 x. 2 x. R opr + 1 opr opr z cv divge0t z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc projlem1.2 jctir 0 < D br adantl 4 x. 2 x. R opr + 1 opr opr / z cv opr sqrge0t 4 x. 2 x. R opr + 1 opr opr z cv redivclt 3expa z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nnne0t sylanc 4 x. 2 x. R opr + 1 opr opr z cv divge0t z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc 0re projlem1.2 ltle anim12i ancoms sylc 4 x. 2 x. R opr + 1 opr opr / z cv opr sqsqrt 4 x. 2 x. R opr + 1 opr opr z cv redivclt 3expa z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nnne0t sylanc 4 x. 2 x. R opr + 1 opr opr z cv divge0t z cv nnret 4re 2re projlem1.1 remulcl ax1re readdcl remulcl jctil z cv nngt0t 0re 4re 2re projlem1.1 remulcl ax1re readdcl remulcl 4re 2re projlem1.1 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem2.3 2re projlem1.1 mulge0 mp2an lt01 2re projlem1.1 remulcl ax1re addgegt0 mp2an mulgt0i ltlei jctil sylanc sylanc < D ^ 2 opr breq1d 0 < D br adantl bitrd birexdva mpbird))
+thm (projlem3 () ((projlem3.1 (e. R (RR))) (projlem3.2 (e. D (NN))) (projlem3.3 (e. G (NN)))) (br (opr (opr (opr (2) (x.) (opr (opr R (+) (opr (1) (/) D)) (^) (2))) (+) (opr (2) (x.) (opr (opr R (+) (opr (1) (/) G)) (^) (2)))) (-) (opr (4) (x.) (opr R (^) (2)))) (<_) (opr (opr (opr (4) (x.) R) (+) (2)) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G)))) (2cn projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl readdcl sqrecl recn projlem3.1 projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl sqrecl recn adddi projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn binom2 projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addass eqtr projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn binom2 projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addass eqtr + opreq12i projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl projlem3.1 recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl add4 eqtr 2 x. opreq2i eqtr3 2cn 2cn projlem3.1 recn sqcl mulass 2t2e4 x. R ^ 2 opr opreq1i projlem3.1 recn sqcl 2times 2 x. opreq2i 3eqtr3 - opreq12i 2cn projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl addcl projlem3.1 recn sqcl projlem3.1 recn sqcl addcl subdi projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl projlem3.1 recn sqcl projlem3.1 recn sqcl addcl addsubass projlem3.1 recn sqcl projlem3.1 recn sqcl addcl 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl addcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl addcl addcl pncan3 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl recn sqcl 2cn projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl mulcl projlem3.3 nnre projlem3.3 nnne0 rereccl recn sqcl add4 projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn adddi 2 x. opreq2i 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl recn mulcl projlem3.1 recn projlem3.3 nnre projlem3.3 nnne0 rereccl recn mulcl adddi eqtr + 1 / D opr ^ 2 opr + 1 / G opr ^ 2 opr opr opreq1i eqtr4 3eqtr 2 x. opreq2i eqtr3 eqtr projlem3.2 nncn projlem3.2 nnne0 sqreci projlem3.2 nnlesq projlem3.2 nngt0 projlem3.2 nnne0 projlem3.2 nnre sqegt0 ax-mp projlem3.2 nnre projlem3.2 nnre sqrecl lerec mp2an mpbi eqbrtr projlem3.3 nncn projlem3.3 nnne0 sqreci projlem3.3 nnlesq projlem3.3 nngt0 projlem3.3 nnne0 projlem3.3 nnre sqegt0 ax-mp projlem3.3 nnre projlem3.3 nnre sqrecl lerec mp2an mpbi eqbrtr projlem3.2 nnre projlem3.2 nnne0 rereccl sqrecl projlem3.3 nnre projlem3.3 nnne0 rereccl sqrecl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl le2add mp2an projlem3.2 nnre projlem3.2 nnne0 rereccl sqrecl projlem3.3 nnre projlem3.3 nnne0 rereccl sqrecl readdcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl leadd2 mpbi 2pos 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl projlem3.2 nnre projlem3.2 nnne0 rereccl sqrecl projlem3.3 nnre projlem3.3 nnne0 rereccl sqrecl readdcl readdcl 2re projlem3.1 projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl remulcl remulcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl readdcl 2re lemul2 ax-mp mpbi 2cn 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulcl mulass 2t2e4 x. R x. 1 / D opr + 1 / G opr opr opr opreq1i 4re recn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulass eqtr4 eqtr3 + 2 x. 1 / D opr + 1 / G opr opr opr opreq1i 2cn 2cn projlem3.1 recn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn mulcl mulcl projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn adddi 4re projlem3.1 remulcl recn 2cn projlem3.2 nnre projlem3.2 nnne0 rereccl projlem3.3 nnre projlem3.3 nnne0 rereccl readdcl recn adddir 3eqtr4 breqtr eqbrtr))
+thm (projlem4 () ((projlem4.1 (e. R (RR))) (projlem4.2 (br (0) (<_) R)) (projlem4.3 (e. D (NN))) (projlem4.4 (e. G (NN))) (projlem4.5 (e. B (NN)))) (-> (/\ (br B (<_) D) (br B (<_) G)) (br (opr (opr (opr (4) (x.) R) (+) (2)) (x.) (opr (opr (1) (/) D) (+) (opr (1) (/) G))) (<_) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) B))) (projlem4.3 nnre projlem4.3 nnne0 rereccl projlem4.4 nnre projlem4.4 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl le2add projlem4.5 nngt0 projlem4.3 nngt0 projlem4.5 nnre projlem4.3 nnre lerec mp2an projlem4.5 nngt0 projlem4.4 nngt0 projlem4.5 nnre projlem4.4 nnre lerec mp2an syl2anb 2t2e4 x. 2 x. R opr + 1 opr opreq1i 2cn 2cn 2re projlem4.1 remulcl ax1re readdcl recn mul23 eqtr3 2cn 2re projlem4.1 remulcl recn 1cn adddi 2cn 2cn projlem4.1 recn mulass 2t2e4 x. R opreq1i eqtr3 2cn mulid1 + opreq12i eqtr2 2cn mulid1 x. opreq12i eqtr4 / B opreq1i 4re projlem4.1 remulcl 2re readdcl recn 2cn 1cn mulcl projlem4.5 nncn projlem4.5 nnne0 divass 2cn 1cn projlem4.5 nncn projlem4.5 nnne0 divass projlem4.5 nnre projlem4.5 nnne0 rereccl recn 2times eqtr 4 x. R opr + 2 opr x. opreq2i 3eqtr 4 x. R opr + 2 opr x. 1 / D opr + 1 / G opr opr opr <_ breq2i 0re 4re 4pos ltlei projlem4.2 4re projlem4.1 mulge0 mp2an 2pos 4re projlem4.1 remulcl 2re addgegt0 mp2an projlem4.3 nnre projlem4.3 nnne0 rereccl projlem4.4 nnre projlem4.4 nnne0 rereccl readdcl projlem4.5 nnre projlem4.5 nnne0 rereccl projlem4.5 nnre projlem4.5 nnne0 rereccl readdcl 4re projlem4.1 remulcl 2re readdcl lemul2 ax-mp bitr4 sylibr))
+thm (projlem5 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN))) (projlem5.10 (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D)))) (projlem5.11 (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))))) (br (opr (` (norm) (opr B (-v) C)) (^) (2)) (<) (opr (opr (opr (2) (x.) (opr (opr R (+) (opr (1) (/) D)) (^) (2))) (+) (opr (2) (x.) (opr (opr R (+) (opr (1) (/) G)) (^) (2)))) (-) (opr (4) (x.) (opr R (^) (2))))) (projlem5.2 projlem5.3 projlem5.1 normpar2 projlem5.10 projlem5.2 projlem5.1 hvsubcl B -v A opr normge0t ax-mp projlem5.5 0re projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.7 D nnrecgt0t ax-mp ltlei projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl addge0 mp2an projlem5.2 projlem5.1 hvsubcl normcl projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl lt2sqe mp2an mpbi 2pos projlem5.2 projlem5.1 hvsubcl normcl sqrecl projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl 2re ltmul2 ax-mp mpbi projlem5.11 projlem5.3 projlem5.1 hvsubcl C -v A opr normge0t ax-mp projlem5.5 0re projlem5.8 nnre projlem5.8 nnne0 rereccl projlem5.8 G nnrecgt0t ax-mp ltlei projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl addge0 mp2an projlem5.3 projlem5.1 hvsubcl normcl projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl lt2sqe mp2an mpbi 2pos projlem5.3 projlem5.1 hvsubcl normcl sqrecl projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl 2re ltmul2 ax-mp mpbi 2re projlem5.2 projlem5.1 hvsubcl normcl sqrecl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl sqrecl remulcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl lt2add mp2an 2re projlem5.2 projlem5.1 hvsubcl normcl sqrecl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl sqrecl remulcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl renegcl ltadd1 mpbi projlem5.6 4re projlem5.4 sqrecl remulcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl leneg mpbi projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl renegcl 4re projlem5.4 sqrecl remulcl renegcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl leadd2 mpbi 2re projlem5.2 projlem5.1 hvsubcl normcl sqrecl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl sqrecl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl renegcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl renegcl readdcl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl 4re projlem5.4 sqrecl remulcl renegcl readdcl ltletr mp2an 2re projlem5.2 projlem5.1 hvsubcl normcl sqrecl remulcl 2re projlem5.3 projlem5.1 hvsubcl normcl sqrecl remulcl readdcl recn projlem5.2 projlem5.3 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqrecl recn subneg 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl recn 4re projlem5.4 sqrecl remulcl recn subneg 3brtr4 eqbrtr))
+thm (projlem6 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN))) (projlem5.10 (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D)))) (projlem5.11 (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G))))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)))) (projlem5.4 projlem5.5 projlem5.7 projlem5.8 projlem5.9 projlem4 projlem5.1 projlem5.2 projlem5.3 projlem5.4 projlem5.5 projlem5.6 projlem5.7 projlem5.8 projlem5.7 projlem5.10 projlem5.11 projlem5 projlem5.4 projlem5.7 projlem5.8 projlem3 projlem5.2 projlem5.3 hvsubcl normcl sqrecl 2re projlem5.4 projlem5.7 nnre projlem5.7 nnne0 rereccl readdcl sqrecl remulcl 2re projlem5.4 projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl sqrecl remulcl readdcl 4re projlem5.4 sqrecl remulcl resubcl 4re projlem5.4 remulcl 2re readdcl projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl remulcl ltletr mp2an projlem5.2 projlem5.3 hvsubcl normcl sqrecl 4re projlem5.4 remulcl 2re readdcl projlem5.7 nnre projlem5.7 nnne0 rereccl projlem5.8 nnre projlem5.8 nnne0 rereccl readdcl remulcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl ltletr mpan syl 0re 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i ltlei 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl sqsqr ax-mp syl6breqr projlem5.2 projlem5.3 hvsubcl B -v C opr normge0t ax-mp 0re 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i ltlei 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl sqrge0 ax-mp projlem5.2 projlem5.3 hvsubcl normcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre projlem5.9 nnne0 redivcl 4re 2re projlem5.4 remulcl ax1re readdcl remulcl projlem5.9 nnre 4re 2re projlem5.4 remulcl ax1re readdcl 4pos 0re 2re 2pos ltlei projlem5.5 2re projlem5.4 mulge0 mp2an lt01 2re projlem5.4 remulcl ax1re addgegt0 mp2an mulgt0i projlem5.9 nngt0 divgt0i sqrlem24 lt2sqe mp2an sylibr))
+thm (projlem7 () ((projlem5.1 (e. A (H~))) (projlem5.2 (e. B (H~))) (projlem5.3 (e. C (H~))) (projlem5.4 (e. R (RR))) (projlem5.5 (br (0) (<_) R)) (projlem5.6 (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2)))) (projlem5.7 (e. D (NN))) (projlem5.8 (e. G (NN))) (projlem5.9 (e. N (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (B norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v C opreq1 norm fveq2d < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` breq1d N <_ D br N <_ G br /\ imbi2d C norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v opreq2 norm fveq2d < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` breq1d N <_ D br N <_ G br /\ imbi2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if 2 x. opreq2 + 1 opreq1d 4 x. opreq2d / N opreq1d sqr fveq2d norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if opr ` < breq2d N <_ D br N <_ G br /\ imbi2d projlem5.1 projlem5.2 projlem5.1 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ keepel projlem5.3 projlem5.1 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ keepel projlem5.4 0re norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ keepel R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if 0 <_ breq2 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if 0 <_ breq2 projlem5.5 0re leid keephyp B norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v C opreq1 -v 2 .s A opr opreq1d norm fveq2d ^ 2 opreq1d 4 x. R ^ 2 opr opr <_ breq2d C norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v opreq2 -v 2 .s A opr opreq1d norm fveq2d ^ 2 opreq1d 4 x. R ^ 2 opr opr <_ breq2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if ^ 2 opreq1 4 x. opreq2d <_ norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if opr -v 2 .s A opr opr ` ^ 2 opr breq1d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v A opreq1 -v 2 .s A opr opreq1d norm fveq2d ^ 2 opreq1d 4 x. 0 ^ 2 opr opr <_ breq2d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v opreq2 -v 2 .s A opr opreq1d norm fveq2d ^ 2 opreq1d 4 x. 0 ^ 2 opr opr <_ breq2d 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if ^ 2 opreq1 4 x. opreq2d <_ norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if +v norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if opr -v 2 .s A opr opr ` ^ 2 opr breq1d projlem5.6 0cn sqval 0cn mul01 eqtr 4 x. opreq2i 4re recn mul01 eqtr projlem5.1 projlem5.1 hvaddcl 2cn projlem5.1 hvmulcl hvsubcl normcl sqege0 eqbrtr keephyp3v projlem5.7 projlem5.8 projlem5.9 B norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opreq1 norm fveq2d < R + 1 / D opr opr breq1d norm C -v A opr ` < R + 1 / G opr opr br anbi1d C norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opreq1 norm fveq2d < R + 1 / G opr opr breq1d norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < R + 1 / D opr opr br anbi2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / D opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < breq2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / G opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opr ` < breq2d anbi12d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opreq1 norm fveq2d < 0 + 1 / D opr opr breq1d norm A -v A opr ` < 0 + 1 / G opr opr br anbi1d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opreq1 norm fveq2d < 0 + 1 / G opr opr breq1d norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < 0 + 1 / D opr opr br anbi2d 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / D opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < breq2d 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / G opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opr ` < breq2d anbi12d projlem5.7 nnre projlem5.7 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp norm fveq2i norm0 eqtr projlem5.7 nncn projlem5.7 nnne0 reccl addid2 3brtr4 projlem5.8 nnre projlem5.8 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp norm fveq2i norm0 eqtr projlem5.8 nncn projlem5.8 nnne0 reccl addid2 3brtr4 pm3.2i elimhyp3v pm3.26i B norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opreq1 norm fveq2d < R + 1 / D opr opr breq1d norm C -v A opr ` < R + 1 / G opr opr br anbi1d C norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opreq1 norm fveq2d < R + 1 / G opr opr breq1d norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < R + 1 / D opr opr br anbi2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / D opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < breq2d R norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / G opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opr ` < breq2d anbi12d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opreq1 norm fveq2d < 0 + 1 / D opr opr breq1d norm A -v A opr ` < 0 + 1 / G opr opr br anbi1d A norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opreq1 norm fveq2d < 0 + 1 / G opr opr breq1d norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < 0 + 1 / D opr opr br anbi2d 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / D opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ B A if -v A opr ` < breq2d 0 norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ R 0 if + 1 / G opr opreq1 norm norm B -v A opr ` < R + 1 / D opr opr br norm C -v A opr ` < R + 1 / G opr opr br /\ C A if -v A opr ` < breq2d anbi12d projlem5.7 nnre projlem5.7 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp norm fveq2i norm0 eqtr projlem5.7 nncn projlem5.7 nnne0 reccl addid2 3brtr4 projlem5.8 nnre projlem5.8 nngt0 recgt0i projlem5.1 A hvsubidt ax-mp norm fveq2i norm0 eqtr projlem5.8 nncn projlem5.8 nnne0 reccl addid2 3brtr4 pm3.2i elimhyp3v pm3.27i projlem6 dedth3v))
+thm (projlem8 ((u v) (u z) (u w) (A u) (v z) (v w) (A v) (w z) (A z) (A w) (H u) (H v) (H z) (H w) (S w) (S z)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. z (RR) (A.e. w S (br (cv w) (<_) (cv z))))) (projlem8.3 u RR v H u cv norm v cv -v A opr ` -u = E.e. ssrab eqsstr u cv norm 0v -v A opr ` -u norm v cv -v A opr ` -u eqeq1 v H birexdv RR elrab ax-hv0cl projlem8.1 hvsubcl normcl renegcl projlem8.2 H ch0 ax-mp norm 0v -v A opr ` -u eqid v cv 0v -v A opreq1 norm fveq2d negeqd norm 0v -v A opr ` -u eqeq2d H rcla4ev mp2an mpbir2an projlem8.3 eleqtrr norm 0v -v A opr ` -u S n0i ax-mp 0re projlem8.3 w cv eleq2i u cv w cv norm v cv -v A opr ` -u eqeq1 v H birexdv RR elrab bitr w cv norm v cv -v A opr ` -u <_ 0 breq1 projlem8.2 v cv chel projlem8.1 jctir v cv A hvsubclt v cv -v A opr normge0t v cv -v A opr normclt norm v cv -v A opr ` le0neg2t syl mpbid 3syl syl5bir com12 r19.23aiv w cv RR e. adantl sylbi rgen z cv 0 w cv <_ breq2 w S biraldv RR rcla4ev mp2an 3pm3.2i))
+thm (projlem9 ((u v) (u x) (u z) (u w) (A u) (v x) (v z) (v w) (A v) (x z) (w x) (A x) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (H w) (S w) (S z) (S x)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (e. (sup S (RR) (<)) (RR)) (ltso projlem8.1 projlem8.2 projlem8.3 z w projlem8 x sup3i supcli))
+thm (projlem10 ((u v) (u x) (A u) (v x) (A v) (A x) (H u) (H v) (H x) (S x) (B x)) ((projlem8.1 (e. A (H~))) (projlem8.2 (e. H (CH))) (projlem8.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))))) (-> (e. B H) (e. (-u (` (norm) (opr B (-v) A))) S)) (x cv B -v A opreq1 norm fveq2d negeqd S eleq1d projlem8.2 x cv chel projlem8.1 x cv A hvsubclt mpan2 syl x cv -v A opr normclt norm x cv -v A opr ` renegclt 3syl norm x cv -v A opr ` -u eqid v cv x cv -v A opreq1 norm fveq2d negeqd norm x cv -v A opr ` -u eqeq2d H rcla4ev mpan2 jca projlem8.3 norm x cv -v A opr ` -u eleq2i u cv norm x cv -v A opr ` -u norm v cv -v A opr ` -u eqeq1 v H birexdv RR elrab bitr sylibr vtoclga))
+thm (projlem11 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (e. R (RR)) (projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem9 renegcl eqeltr))
+thm (projlem12 ((u v) (u x) (u y) (A u) (v x) (v y) (A v) (x y) (A x) (A y) (H u) (H v) (H x) (H y) (S x) (S y) (B x)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (-> (e. B H) (br R (<_) (` (norm) (opr B (-v) A)))) (projlem11.1 projlem11.2 projlem11.3 B projlem10 projlem11.1 projlem11.2 projlem11.3 x y projlem8 norm B -v A opr ` -u suprubi syl projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn projlem11.1 projlem11.2 projlem11.3 projlem9 recn negcon2 mpbi syl6breq projlem11.2 B chel projlem11.1 B A hvsubclt mpan2 B -v A opr normclt 3syl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 jctil R norm B -v A opr ` lenegt syl mpbird))
+thm (projlem13 ((u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (H u) (H v) (H x) (H y) (H z) (S z) (S x) (S y) (R z)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (br (0) (<_) R) (projlem11.4 projlem11.1 projlem11.2 projlem11.3 projlem9 recn projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn negcon2 mpbir 0re renegcl projlem11.3 z cv eleq2i u cv z cv norm v cv -v A opr ` -u eqeq1 v H birexdv RR elrab bitr 0re renegcl z cv 0 -u lenltt mpan2 z cv norm v cv -v A opr ` -u <_ 0 -u breq1 projlem11.2 v cv chel projlem11.1 jctir v cv A hvsubclt v cv -v A opr normge0t 3syl projlem11.2 v cv chel projlem11.1 jctir v cv A hvsubclt v cv -v A opr normclt 3syl 0re jctil 0 norm v cv -v A opr ` lenegt syl mpbid syl5bir impcom syl5bi exp3a r19.23adv imp sylbi rgen ltso projlem11.1 projlem11.2 projlem11.3 x y projlem8 z sup3i 0 -u supnubi mp2an projlem11.1 projlem11.2 projlem11.3 projlem9 0re renegcl lenlt mpbir eqbrtrr 0re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 leneg mpbir))
+thm (projlem14 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem14.5 (e. C (NN))) (projlem14.6 (e. B H))) (br (opr R (-) (opr (1) (/) C)) (<) (` (norm) (opr B (-v) A))) (projlem14.5 C nnrecgt0t ax-mp projlem14.5 nnre projlem14.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltaddpos mpbi projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem14.5 nnre projlem14.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltsubadd mpbir projlem14.6 projlem11.1 projlem11.2 projlem11.3 projlem11.4 B projlem12 ax-mp projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem14.5 nnre projlem14.5 nnne0 rereccl resubcl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem11.2 projlem14.6 cheli projlem11.1 hvsubcl normcl ltletr mp2an))
+thm (projlem15 ((v w) (u w) (w z) (C w) (u v) (v z) (C v) (u z) (C u) (C z) (u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H y) (H z) (H w) (S w) (S z) (S x) (S y) (R z) (R w)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem15.5 (e. C (NN)))) (E.e. z H (br (` (norm) (opr (cv z) (-v) A)) (<) (opr R (+) (opr (1) (/) C)))) (projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl renegcl projlem11.4 projlem15.5 C nnrecgt0t ax-mp projlem15.5 nnre projlem15.5 nnne0 rereccl projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 ltaddpos mpbi eqbrtrr projlem11.1 projlem11.2 projlem11.3 projlem9 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl ltnegcon1 mpbi ltso projlem11.1 projlem11.2 projlem11.3 x y projlem8 w sup3i R + 1 / C opr opr -u suplubi mp2an w cv norm z cv -v A opr ` -u R + 1 / C opr opr -u < breq2 biimpd projlem11.2 z cv chel projlem11.1 jctir z cv A hvsubclt syl z cv -v A opr normclt projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem15.5 nnre projlem15.5 nnne0 rereccl readdcl norm z cv -v A opr ` R + 1 / C opr opr ltnegt mpan2 3syl biimprd syl9 com13 imp r19.22dva projlem11.3 w cv eleq2i u cv w cv norm z cv -v A opr ` -u eqeq1 z H birexdv v cv z cv -v A opreq1 norm fveq2d negeqd u cv eqeq2d H cbvrexv syl5bb RR elrab bitr pm3.27bd syl5com r19.23aiv ax-mp))
+thm (projlem16 ((u v) (v z) (C v) (u z) (C u) (C z) (u v) (u z) (A u) (v z) (A v) (A z) (H u) (H v) (H z) (S z) (R z)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem15.5 (e. C (NN)))) (E.e. z H (/\ (br (opr R (-) (opr (1) (/) C)) (<) (` (norm) (opr (cv z) (-v) A))) (br (` (norm) (opr (cv z) (-v) A)) (<) (opr R (+) (opr (1) (/) C))))) (projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem15.5 z projlem15 z cv z cv H e. z cv 0v if -v A opreq1 norm fveq2d R - 1 / C opr opr < breq2d projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem15.5 projlem11.2 H ch0 ax-mp z cv elimel projlem14 dedth norm z cv -v A opr ` < R + 1 / C opr opr br anim1i exp r19.22i ax-mp))
+thm (projlem17 ((u v) (u z) (u w) (f u) (A u) (v z) (v w) (f v) (A v) (w z) (f z) (A z) (f w) (A w) (A f) (H u) (H v) (H z) (H w) (H f) (S w) (S z) (R z) (R w) (R f)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<)))))) (E. f (/\ (:--> (cv f) (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` (cv f) (cv w)) (-v) A))) (br (` (norm) (opr (` (cv f) (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))) (nnex projlem11.2 elisseti z cv f cv w cv ` -v A opreq1 norm fveq2d R - 1 / w cv opr opr < breq2d z cv f cv w cv ` -v A opreq1 norm fveq2d < R + 1 / w cv opr opr breq1d anbi12d ac6 w cv w cv NN e. w cv 1 if 1 / opreq2 R - opreq2d < norm z cv -v A opr ` breq1d w cv w cv NN e. w cv 1 if 1 / opreq2 R + opreq2d norm z cv -v A opr ` < breq2d anbi12d z H birexdv projlem11.1 projlem11.2 projlem11.3 projlem11.4 1nn w cv elimel z projlem16 dedth mprg))
+thm (projlem18 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem18.5 (e. B H)) (projlem18.6 (e. C H))) (br (opr (4) (x.) (opr R (^) (2))) (<_) (opr (` (norm) (opr (opr B (+v) C) (-v) (opr (2) (.s) A))) (^) (2))) (2cn projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 recn sqmul sq2 x. R ^ 2 opr opreq1i eqtr2 2cn 2re 2pos gt0ne0i reccl projlem18.5 projlem18.6 projlem11.2 chshi H B C shaddclt ax-mp mp2an projlem11.2 chshi H 1 / 2 opr B +v C opr shmulclt ax-mp mp2an projlem11.1 projlem11.2 projlem11.3 projlem11.4 1 / 2 opr .s B +v C opr opr projlem12 ax-mp 2pos projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubcl normcl 2re lemul2 ax-mp mpbi 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubcl norm-iii 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulcl projlem11.1 hvsubdistr1 2cn 2re 2pos gt0ne0i recid .s B +v C opr opreq1i 2cn 2cn 2re 2pos gt0ne0i reccl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl hvmulass projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl B +v C opr ax-hvmulid ax-mp 3eqtr3 -v 2 .s A opr opreq1i eqtr norm fveq2i 0re 2re 2pos ltlei 2re absid ax-mp x. norm 1 / 2 opr .s B +v C opr opr -v A opr ` opreq1i 3eqtr3r breqtr 0re 2re 2pos ltlei projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem13 2re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 mulge0 mp2an projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl 2cn projlem11.1 hvmulcl hvsubcl B +v C opr -v 2 .s A opr opr normge0t ax-mp 2re projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 remulcl projlem11.2 projlem18.5 cheli projlem11.2 projlem18.6 cheli hvaddcl 2cn projlem11.1 hvmulcl hvsubcl normcl le2sqe mp2an mpbi eqbrtr))
+thm (projlem19 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem19.5 (e. B H)) (projlem19.6 (e. C H)) (projlem19.7 (e. D (NN))) (projlem19.8 (e. G (NN))) (projlem19.9 (e. N (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (projlem11.1 projlem11.2 projlem19.5 cheli projlem11.2 projlem19.6 cheli projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem13 projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem19.5 projlem19.6 projlem18 projlem19.7 projlem19.8 projlem19.9 projlem7))
+thm (projlem20 ((u v) (A u) (A v) (H u) (H v)) ((projlem11.1 (e. A (H~))) (projlem11.2 (e. H (CH))) (projlem11.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem11.4 (= R (-u (sup S (RR) (<))))) (projlem20.5 (e. N (NN)))) (-> (/\ (/\ (e. B H) (e. C H)) (/\ (e. D (NN)) (e. G (NN)))) (-> (/\ (br (` (norm) (opr B (-v) A)) (<) (opr R (+) (opr (1) (/) D))) (br (` (norm) (opr C (-v) A)) (<) (opr R (+) (opr (1) (/) G)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr B (-v) C)) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)))))) (B B H e. B 0v if -v A opreq1 norm fveq2d < R + 1 / D opr opr breq1d norm C -v A opr ` < R + 1 / G opr opr br anbi1d B B H e. B 0v if -v C opreq1 norm fveq2d < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` breq1d N <_ D br N <_ G br /\ imbi2d imbi12d C C H e. C 0v if -v A opreq1 norm fveq2d < R + 1 / G opr opr breq1d norm B H e. B 0v if -v A opr ` < R + 1 / D opr opr br anbi2d C C H e. C 0v if B H e. B 0v if -v opreq2 norm fveq2d < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` breq1d N <_ D br N <_ G br /\ imbi2d imbi12d D D NN e. D 1 if 1 / opreq2 R + opreq2d norm B H e. B 0v if -v A opr ` < breq2d norm C H e. C 0v if -v A opr ` < R + 1 / G opr opr br anbi1d D D NN e. D 1 if N <_ breq2 N <_ G br anbi1d norm B H e. B 0v if -v C H e. C 0v if opr ` < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` br imbi1d imbi12d G G NN e. G 1 if 1 / opreq2 R + opreq2d norm C H e. C 0v if -v A opr ` < breq2d norm B H e. B 0v if -v A opr ` < R + 1 / D NN e. D 1 if opr opr br anbi2d G G NN e. G 1 if N <_ breq2 N <_ D NN e. D 1 if br anbi2d norm B H e. B 0v if -v C H e. C 0v if opr ` < sqr 4 x. 2 x. R opr + 1 opr opr / N opr ` br imbi1d imbi12d projlem11.1 projlem11.2 projlem11.3 projlem11.4 projlem11.2 H ch0 ax-mp B elimel projlem11.2 H ch0 ax-mp C elimel 1nn D elimel 1nn G elimel projlem20.5 projlem19 dedth4h))
+thm (projlem21 ((A w) (D w) (F w) (R w)) ((projlem21.1 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (-> (e. D (NN)) (e. (` F D) H))) (F NN H D ffvrn projlem21.1 pm3.26bd sylan exp))
+thm (projlem22 ((A w) (D w) (F w) (R w)) ((projlem21.1 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (-> (e. D (NN)) (br (` (norm) (opr (` F D) (-v) A)) (<) (opr R (+) (opr (1) (/) D))))) (w cv D F fveq2 -v A opreq1d norm fveq2d w cv D 1 / opreq2 R + opreq2d < breq12d NN rcla4v projlem21.1 pm3.27bd R - 1 / w cv opr opr < norm F w cv ` -v A opr ` br norm F w cv ` -v A opr ` < R + 1 / w cv opr opr br pm3.27 w NN r19.20si syl syl5com))
+thm (projlem23 ((x y) (D x) (D y) (A x) (A y) (F x) (F y)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A)))))))) (-> (e. D (NN)) (= (` G D) (` (norm) (opr (` F D) (-v) A)))) (x cv D F fveq2 -v A opreq1d norm fveq2d projlem23.1 norm F D ` -v A opr fvex fvopab4))
+thm (projlem24 ((x y) (A x) (A y) (F x) (F y)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A))))))) (projlem24.2 (e. A (H~)))) (-> (:--> F (NN) (H~)) (:--> G (NN) (CC))) (F NN H~ x cv ffvrn projlem24.2 jctir F x cv ` A hvsubclt syl F x cv ` -v A opr normclt norm F x cv ` -v A opr ` recnt 3syl exp r19.21aiv projlem23.1 CC fopab2 sylib))
+thm (projlem25OLD ((x y) (u v) (u w) (u x) (u y) (A u) (v w) (v x) (v y) (A v) (w x) (w y) (A w) (A x) (A y) (u z) (F u) (v z) (F v) (w z) (F w) (x z) (F x) (y z) (F y) (F z) (G u) (G v) (G z) (G w)) ((projlem23.1 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A))))))) (projlem25.2 (e. A (H~))) (projlem25.3 (e. F (V)))) (-> (br F (~~>v) (cv z)) (br G (~~>OLD) (` (norm) (opr (cv z) (-v) A)))) (projlem25.3 z visset hlimseq projlem23.1 projlem25.2 projlem24 syl projlem25.3 z visset hlimvec projlem25.2 jctir z cv A hvsubclt z cv -v A opr normclt 3syl recnd jca projlem25.3 z visset w v u hlimconv abs norm F u cv ` -v A opr ` - norm z cv -v A opr ` opr ` norm F u cv ` -v z cv opr ` w cv lelttrt 3expa norm F u cv ` -v A opr ` norm z cv -v A opr ` resubclt F NN H~ u cv ffvrn projlem25.3 z visset hlimseq sylan projlem25.2 F u cv ` A hvsubclt mpan2 F u cv ` -v A opr normclt 3syl projlem25.3 z visset hlimvec u cv NN e. adantr projlem25.2 z cv A hvsubclt mpan2 z cv -v A opr normclt 3syl sylanc recnd norm F u cv ` -v A opr ` - norm z cv -v A opr ` opr absclt syl F u cv ` z cv hvsubclt F NN H~ u cv ffvrn projlem25.3 z visset hlimseq sylan projlem25.3 z visset hlimvec u cv NN e. adantr sylanc F u cv ` -v z cv opr normclt syl jca sylan an1rs projlem25.2 F u cv ` z cv norm3adift F NN H~ u cv ffvrn projlem25.3 z visset hlimseq sylan projlem25.3 z visset hlimvec u cv NN e. adantr sylanc w cv RR e. adantlr sylani anabsi5 projlem23.1 u cv projlem23 - norm z cv -v A opr ` opreq1d abs fveq2d < w cv breq1d F ~~>v z cv br w cv RR e. /\ norm F u cv ` -v z cv opr ` < w cv br ad2antlr mpbird exp v cv <_ u cv br imim2d r19.20dva v NN r19.22sdv 0 < w cv br imim2d r19.20dva mpd jca norm z cv -v A opr fvex G w v u climOLD sylibr))
+thm (projlem26OLD ((u w) (v w) (w x) (w y) (A w) (u v) (u x) (u y) (A u) (v x) (v y) (A v) (x y) (A x) (A y) (F w) (F x) (F y) (f w) (g w) (w z) (R w) (f g) (f z) (R f) (g z) (R g) (R z) (G f) (G g) (G z) (f x) (f ph) (g x) (g ph) (x z) (ph x) (ph z) (H u) (H v) (g y)) ((projlem26.1 (e. A (H~))) (projlem26.2 (e. H (CH))) (projlem26.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem26.4 (= R (-u (sup S (RR) (<))))) (projlem26.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem26.6 (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (norm) (opr (` F (cv x)) (-v) A)))))))) (-> ph (br G (~~>OLD) R)) (projlem26.5 pm3.26bd projlem26.2 chssi F NN H H~ fss mpan2 projlem26.6 projlem26.1 projlem24 3syl projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 recn jctir ax1re 1 z cv redivclt mp3an1 z cv RR e. 0 < z cv br pm3.26 z cv gt0ne0t sylanc 1 / z cv opr f arch syl ax1re 1 z cv f cv ltdiv23t mp3anl1 z cv RR e. 0 < z cv br pm3.26 f cv nnret anim12i z cv RR e. 0 < z cv br pm3.27 f cv nngt0t anim12i sylanc birexdva mpbid ph adantl f cv g cv lerect f cv nnret g cv nnret anim12i f cv nngt0t g cv nngt0t anim12i sylanc ancoms z cv RR e. 3adant3 1 / g cv opr 1 / f cv opr z cv lelttrt exp3a ax1re 1 f cv redivclt mp3an1 f cv nnret f cv nnne0t sylanc syl3an2 ax1re 1 g cv redivclt mp3an1 g cv nnret g cv nnne0t sylanc syl3an1 sylbid com23 imp3a 3exp com13 ph 0 < z cv br ad2antrl imp31 abs norm F g cv ` -v A opr ` - R opr ` 1 / g cv opr z cv axlttrn 3expa exp3a F NN H g cv ffvrn projlem26.5 pm3.26bd sylan projlem26.2 F g cv ` chel syl projlem26.1 jctir F g cv ` A hvsubclt F g cv ` -v A opr normclt 3syl projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 jctir norm F g cv ` -v A opr ` R resubclt syl recnd norm F g cv ` -v A opr ` - R opr absclt syl ax1re 1 g cv redivclt mp3an1 g cv nnret g cv nnne0t sylanc ph adantl jca z cv RR e. 0 < z cv br pm3.26 anim12i w cv g cv 1 / opreq2 R - opreq2d w cv g cv F fveq2 -v A opreq1d norm fveq2d < breq12d w cv g cv F fveq2 -v A opreq1d norm fveq2d w cv g cv 1 / opreq2 R + opreq2d < breq12d anbi12d NN rcla4v projlem26.5 pm3.27bd syl5com imp pm3.27d projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 norm F g cv ` -v A opr ` R 1 / g cv opr ltsubadd2t mp3an2 F NN H g cv ffvrn projlem26.5 pm3.26bd sylan projlem26.2 F g cv ` chel syl projlem26.1 jctir F g cv ` A hvsubclt F g cv ` -v A opr normclt 3syl ax1re 1 g cv redivclt mp3an1 g cv nnret g cv nnne0t sylanc ph adantl sylanc mpbird F NN H g cv ffvrn projlem26.5 pm3.26bd sylan projlem26.2 F g cv ` chel syl projlem26.1 jctir F g cv ` A hvsubclt F g cv ` -v A opr normclt 3syl recnd projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 recn jctir norm F g cv ` -v A opr ` R negdi3t syl w cv g cv 1 / opreq2 R - opreq2d w cv g cv F fveq2 -v A opreq1d norm fveq2d < breq12d w cv g cv F fveq2 -v A opreq1d norm fveq2d w cv g cv 1 / opreq2 R + opreq2d < breq12d anbi12d NN rcla4v projlem26.5 pm3.27bd syl5com imp pm3.26d projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 R norm F g cv ` -v A opr ` 1 / g cv opr ltsub23t mp3an1 F NN H g cv ffvrn projlem26.5 pm3.26bd sylan projlem26.2 F g cv ` chel syl projlem26.1 jctir F g cv ` A hvsubclt F g cv ` -v A opr normclt 3syl ax1re 1 g cv redivclt mp3an1 g cv nnret g cv nnne0t sylanc ph adantl sylanc mpbird eqbrtrd jca norm F g cv ` -v A opr ` - R opr 1 / g cv opr abslttOLD F NN H g cv ffvrn projlem26.5 pm3.26bd sylan projlem26.2 F g cv ` chel syl projlem26.1 jctir F g cv ` A hvsubclt F g cv ` -v A opr normclt 3syl projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 jctir norm F g cv ` -v A opr ` R resubclt syl ax1re 1 g cv redivclt mp3an1 g cv nnret g cv nnne0t sylanc ph adantl sylanc mpbird z cv RR e. 0 < z cv br /\ adantr sylc an1rs f cv NN e. adantlr syld projlem26.6 g cv projlem23 - R opreq1d abs fveq2d < z cv breq1d ph z cv RR e. 0 < z cv br /\ /\ f cv NN e. /\ adantl sylibrd exp4b com23 r19.21adv r19.22dva mpd exp32 r19.21aiv jca projlem26.1 projlem26.2 projlem26.3 projlem26.4 projlem11 elisseti G z f g climOLD sylibr))
+thm (projlem27 ((u v) (u w) (A u) (v w) (A v) (A w) (H u) (H v) (D w) (G w) (F w) (R w)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem27.6 (e. N (NN)))) (-> (/\ ph (/\ (e. D (NN)) (e. G (NN)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr (` F D) (-v) (` F G))) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N))))) (projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.6 F D ` F G ` D G projlem20 projlem27.5 D projlem21 projlem27.5 G projlem21 anim12d impac projlem27.5 D projlem22 projlem27.5 G projlem22 anim12d imp sylc))
+thm (projlem28 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (D x) (D y) (D z) (D w) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem28.6 (e. D (RR)))) (-> ph (-> (br (0) (<) D) (E.e. z (NN) (A.e. x (NN) (A.e. y (NN) (-> (/\ (br (cv z) (<_) (cv x)) (br (cv z) (<_) (cv y))) (br (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) D))))))) (z cv z cv NN e. z cv 1 if <_ x cv breq1 z cv z cv NN e. z cv 1 if <_ y cv breq1 anbi12d z cv z cv NN e. z cv 1 if 4 x. 2 x. R opr + 1 opr opr / opreq2 sqr fveq2d norm F x cv ` -v F y cv ` opr ` < breq2d imbi12d ph x cv NN e. y cv NN e. /\ /\ imbi2d projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 1nn z cv elimel x cv y cv projlem27 dedth exp3a com12 imp31 sqr 4 x. 2 x. R opr + 1 opr opr / z cv opr ` < D br adantlr projlem28.6 norm F x cv ` -v F y cv ` opr ` sqr 4 x. 2 x. R opr + 1 opr opr / z cv opr ` D axlttrn mp3an3 F x cv ` F y cv ` hvsubclt projlem27.5 x cv projlem21 imp projlem27.2 F x cv ` chel syl projlem27.5 y cv projlem21 imp projlem27.2 F y cv ` chel syl syl2an anandis F x cv ` -v F y cv ` opr normclt syl 4 x. 2 x. R opr + 1 opr opr / z cv opr sqrclt 4 x. 2 x. R opr + 1 opr opr z cv redivclt 3expa z cv nnret 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl remulcl jctil z cv nnne0t sylanc 4 x. 2 x. R opr + 1 opr opr z cv divge0t z cv nnret 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl remulcl jctil z cv nngt0t 0re 4re 4pos ltlei 0re 2re 2pos ltlei projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem13 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 mulge0 mp2an 0re ax1re lt01 ltlei 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re addge0 mp2an 4re 2re projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 remulcl ax1re readdcl mulge0 mp2an jctil sylanc sylanc syl2an exp4b com4r exp3a com4t com14 imp41 syld exp x y 19.21aivv x NN y NN z cv <_ x cv br z cv <_ y cv br /\ norm F x cv ` -v F y cv ` opr ` < D br -> r2al sylibr exp r19.22dva projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 projlem28.6 projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem13 z projlem2 syl5))
+thm (projlem29 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (g u) (H u) (g v) (H v) (g x) (g z) (H g) (H x) (H z) (g y) (g w) (F g) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (g ph) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (e. F (Cauchy))) (projlem27.5 pm3.26bd projlem27.2 chssi jctir F NN H H~ fss syl g cv g cv RR e. g cv 1 if 0 < breq2 g cv g cv RR e. g cv 1 if norm F x cv ` -v F y cv ` opr ` < breq2 z cv <_ x cv br z cv <_ y cv br /\ imbi2d y NN biraldv x NN biraldv z NN birexdv imbi12d ph imbi2d projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 ax1re g cv elimel z x y projlem28 dedth com12 r19.21aiv jca F g z x y hcau sylibr))
+thm (projlem30 ((u v) (u z) (u w) (A u) (v z) (v w) (A v) (w z) (A z) (A w) (H u) (H v) (H z) (F w) (F z) (R z) (R w) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))))) (-> ph (E.e. z H (br F (~~>v) (cv z)))) (projlem27.2 H F z chcompl ax-mp projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 projlem29 projlem27.5 pm3.26bd sylanc))
+thm (projlem31 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((projlem27.1 (e. A (H~))) (projlem27.2 (e. H (CH))) (projlem27.3 (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A)))))))) (projlem27.4 (= R (-u (sup S (RR) (<))))) (projlem27.5 (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (projlem31.6 (e. F (V)))) (-> ph (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))))) (projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 x projlem30 projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem11 elisseti norm x cv -v A opr fvex z y z cv NN e. y cv norm F z cv ` -v A opr ` = /\ {<,>|} climuniOLD projlem27.1 projlem27.2 projlem27.3 projlem27.4 projlem27.5 z y z cv NN e. y cv norm F z cv ` -v A opr ` = /\ {<,>|} eqid projlem26OLD z y z cv NN e. y cv norm F z cv ` -v A opr ` = /\ {<,>|} eqid projlem27.1 projlem31.6 x projlem25OLD syl2an <_ norm y cv -v A opr ` breq1d projlem27.1 projlem27.2 projlem27.3 projlem27.4 y cv projlem12 syl5bi r19.21aiv exp x H r19.22sdv mpd))
+thm (projlem ((x y) (f x) (u x) (v x) (w x) (H x) (f y) (u y) (v y) (w y) (H y) (f u) (f v) (f w) (H f) (u v) (u w) (H u) (v w) (H v) (H w) (A x) (A y) (A f) (A u) (A v) (A w)) ((projlem.1 (e. A (H~))) (projlem.2 (e. H (CH)))) (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A))))) (projlem.1 projlem.2 u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} eqid u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} RR < sup -u eqid f w projlem17 projlem.1 projlem.2 u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} eqid u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} RR < sup -u eqid f cv NN H :--> w NN u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} RR < sup -u - 1 / w cv opr opr < norm f cv w cv ` -v A opr ` br norm f cv w cv ` -v A opr ` < u RR v H u cv norm v cv -v A opr ` -u = E.e. {e.|} RR < sup -u + 1 / w cv opr opr br /\ A.e. /\ pm4.2 f visset x y projlem31 f 19.23aiv ax-mp))
+thm (pjthlem1 () ((pjthlem1.1 (e. A (H~))) (pjthlem1.2 (e. B (H~))) (pjthlem1.3 (e. D (H~))) (pjthlem1.4 (e. S (CC))) (pjthlem1.5 (= C (opr A (-v) B)))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)))) (pjthlem1.2 pjthlem1.1 normsub pjthlem1.5 norm fveq2i eqtr4 pjthlem1.2 pjthlem1.4 pjthlem1.3 hvmulcl hvaddcl pjthlem1.1 normsub pjthlem1.5 -v S .s D opr opreq1i pjthlem1.1 pjthlem1.2 pjthlem1.4 pjthlem1.3 hvmulcl hvsubass eqtr2 norm fveq2i eqtr <_ breq12i pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr C normge0t ax-mp pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr pjthlem1.4 pjthlem1.3 hvmulcl hvsubcl C -v S .s D opr opr normge0t ax-mp pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr normcl pjthlem1.5 pjthlem1.1 pjthlem1.2 hvsubcl eqeltr pjthlem1.4 pjthlem1.3 hvmulcl hvsubcl normcl le2sqe mp2an bitr))
+thm (pjthlem2 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D))))) (-> (-. (= D (0v))) (e. R (RR))) (pjthlem2.1 D axhis42 mpan pjthlem2.1 D hiidrclt ax-mp gt0ne0 pjthlem2.1 D hiidrclt ax-mp rerecclz 3syl pjthlem2.2 RR eleq1i sylibr))
+thm (pjthlem3 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D))))) (-> (-. (= D (0v))) (br (0) (<) R)) (pjthlem2.1 D axhis42 mpan pjthlem2.1 D hiidrclt ax-mp recgt0 syl pjthlem2.2 syl6breqr))
+thm (pjthlem4 () ((pjthlem2.1 (e. D (H~))) (pjthlem2.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem4.3 (e. C (H~))) (pjthlem4.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (e. S (CC))) (pjthlem2.1 pjthlem2.2 pjthlem2 recnd pjthlem4.3 pjthlem2.1 hicl jctir R C .i D opr axmulcl syl pjthlem4.4 CC eleq1i sylibr))
+thm (pjthlem5 () ((pjthlem5.1 (e. D (H~))) (pjthlem5.2 (e. C (H~))) (pjthlem5.3 (e. S (CC)))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (opr (` (norm) C) (^) (2)) (-) (` (*) (opr S (x.) (opr D (.i) C)))) (+) (opr (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (-) (opr S (x.) (opr D (.i) C))))) (pjthlem5.2 pjthlem5.3 pjthlem5.1 hvmulcl hvsubcl normsq pjthlem5.3 pjthlem5.2 pjthlem5.1 normlem0 pjthlem5.2 normcl sqrecl recn pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl cjcl subneg pjthlem5.2 normsq pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl cjmul pjthlem5.2 pjthlem5.1 his1 * S ` x. opreq2i eqtr4 negeqi pjthlem5.3 cjcl pjthlem5.2 pjthlem5.1 hicl mulneg1 eqtr4 + opreq12i eqtr2 pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl negcl pjthlem5.3 cjmulrcl recn pjthlem5.1 pjthlem5.1 hicl mulcl addcom pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulneg1 + S x. * S ` opr x. D .i D opr opr opreq1i pjthlem5.3 cjmulrcl recn pjthlem5.1 pjthlem5.1 hicl mulcl pjthlem5.3 pjthlem5.1 pjthlem5.2 hicl mulcl subneg 3eqtr4 + opreq12i 3eqtr))
+thm (pjthlem6 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr S (x.) (opr D (.i) C)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))))) (pjthlem6.1 pjthlem6.2 pjthlem2 recnd pjthlem6.3 pjthlem6.1 hicl pjthlem6.1 pjthlem6.3 hicl R C .i D opr D .i C opr axmulass mp3an23 syl pjthlem6.4 x. D .i C opr opreq1i pjthlem6.3 pjthlem6.1 hicl absvalsq pjthlem6.1 pjthlem6.3 his1 C .i D opr x. opreq2i eqtr4 R x. opreq2i 3eqtr4g))
+thm (pjthlem7 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))))) (pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.4 R RR e. a1i R recnt pjthlem6.3 pjthlem6.1 hicl R C .i D opr cjmult mpan2 syl R cjret x. * C .i D opr ` opreq1d eqtrd pjthlem6.4 * fveq2i syl5eq x. opreq12d R C .i D opr R * C .i D opr ` mul4t R recnt pjthlem6.3 pjthlem6.1 hicl jctir R recnt pjthlem6.3 pjthlem6.1 hicl cjcl jctir sylanc eqtrd x. D .i D opr opreq1d R R axmulcl R recnt R recnt sylanc pjthlem6.3 pjthlem6.1 hicl pjthlem6.3 pjthlem6.1 hicl cjcl mulcl pjthlem6.1 pjthlem6.1 hicl R x. R opr C .i D opr x. * C .i D opr ` opr D .i D opr mul23t mp3an23 syl eqtrd syl pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.1 pjthlem6.1 hicl R R D .i D opr mul23t mp3an3 R recnt R recnt sylanc syl pjthlem6.1 D axhis42 mpan pjthlem6.1 D hiidrclt ax-mp gt0ne0 syl pjthlem6.1 pjthlem6.1 hicl recclz pjthlem6.1 pjthlem6.1 hicl 1 / D .i D opr opr D .i D opr axmulcom mpan2 3syl pjthlem6.1 D axhis42 mpan pjthlem6.1 D hiidrclt ax-mp gt0ne0 pjthlem6.1 pjthlem6.1 hicl recidz 3syl eqtrd pjthlem6.2 x. D .i D opr opreq1i syl5eq x. R opreq1d pjthlem6.1 pjthlem6.2 pjthlem2 R recnt R mulid2t 3syl 3eqtrd x. C .i D opr x. * C .i D opr ` opr opreq1d eqtrd pjthlem6.3 pjthlem6.1 hicl absvalsq R x. opreq2i syl6eqr))
+thm (pjthlem8 () ((pjthlem6.1 (e. D (H~))) (pjthlem6.2 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem6.3 (e. C (H~))) (pjthlem6.4 (= S (opr R (x.) (opr C (.i) D))))) (-> (-. (= D (0v))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (` (norm) C) (^) (2)) (-) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2)))))) (pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem4 S S CC e. S 0 if .s D opreq1 C -v opreq2d norm fveq2d ^ 2 opreq1d S S CC e. S 0 if x. D .i C opr opreq1 * fveq2d norm C ` ^ 2 opr - opreq2d S S CC e. S 0 if = id S S CC e. S 0 if * fveq2 x. opreq12d x. D .i D opr opreq1d S S CC e. S 0 if x. D .i C opr opreq1 - opreq12d + opreq12d eqeq12d pjthlem6.1 pjthlem6.3 0cn S elimel pjthlem5 dedth syl pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem6 * fveq2d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl sqrecl R abs C .i D opr ` ^ 2 opr axmulrcl mpan2 R x. abs C .i D opr ` ^ 2 opr opr cjret 3syl eqtrd norm C ` ^ 2 opr - opreq2d pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem7 pjthlem6.1 pjthlem6.2 pjthlem6.3 pjthlem6.4 pjthlem6 - opreq12d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl sqrecl R abs C .i D opr ` ^ 2 opr axmulrcl mpan2 syl recnd R x. abs C .i D opr ` ^ 2 opr opr subidt syl eqtrd + opreq12d pjthlem6.1 pjthlem6.2 pjthlem2 pjthlem6.3 pjthlem6.1 hicl abscl sqrecl R abs C .i D opr ` ^ 2 opr axmulrcl mpan2 syl recnd pjthlem6.3 normcl sqrecl recn norm C ` ^ 2 opr R x. abs C .i D opr ` ^ 2 opr opr subclt mpan norm C ` ^ 2 opr - R x. abs C .i D opr ` ^ 2 opr opr opr ax0id 3syl 3eqtrd))
+thm (pjthlem9 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (-. (= D (0v))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2))))) (pjthlem9.3 pjthlem9.4 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.5 pjthlem4 S S CC e. S 0 if .s D opreq1 B +v opreq2d -v A opreq1d norm fveq2d norm B -v A opr ` <_ breq2d S S CC e. S 0 if .s D opreq1 C -v opreq2d norm fveq2d ^ 2 opreq1d norm C ` ^ 2 opr <_ breq2d bibi12d pjthlem9.1 pjthlem9.2 pjthlem9.3 0cn S elimel pjthlem9.6 pjthlem1 dedth syl))
+thm (pjthlem10 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) (0))) (R abs C .i D opr ` ^ 2 opr mulge0tOLD pjthlem9.3 pjthlem9.4 pjthlem2 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl sqrecl jctir 0 R ltlet pjthlem9.3 pjthlem9.4 pjthlem2 0re jctil pjthlem9.3 pjthlem9.4 pjthlem3 sylc pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl sqege0 jctir sylc norm B -v A opr ` <_ norm B +v S .s D opr opr -v A opr ` br adantr pjthlem9.1 pjthlem9.2 pjthlem9.3 pjthlem9.4 pjthlem9.5 pjthlem9.6 pjthlem9 biimpa pjthlem9.3 pjthlem9.4 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.5 pjthlem8 norm B -v A opr ` <_ norm B +v S .s D opr opr -v A opr ` br adantr breqtrd jca pjthlem9.3 pjthlem9.4 pjthlem2 pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl sqrecl jctir R abs C .i D opr ` ^ 2 opr axmulrcl syl pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr normcl sqrecl jctir R x. abs C .i D opr ` ^ 2 opr opr norm C ` ^ 2 opr lesub0t syl norm B -v A opr ` <_ norm B +v S .s D opr opr -v A opr ` br adantr mpbid))
+thm (pjthlem11 () ((pjthlem9.1 (e. A (H~))) (pjthlem9.2 (e. B (H~))) (pjthlem9.3 (e. D (H~))) (pjthlem9.4 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem9.5 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem9.6 (= C (opr A (-v) B)))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr C (.i) D) (0))) (pjthlem9.3 pjthlem9.4 pjthlem3 pjthlem9.3 pjthlem9.4 pjthlem2 0re jctir R 0 lttri2t syl 0 < R br R < 0 br olc syl5bir mpd norm B -v A opr ` <_ norm B +v S .s D opr opr -v A opr ` br adantr pjthlem9.1 pjthlem9.2 pjthlem9.3 pjthlem9.4 pjthlem9.5 pjthlem9.6 pjthlem10 pjthlem9.3 pjthlem9.4 pjthlem2 recnd pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl sqrecl recn R abs C .i D opr ` ^ 2 opr mul0ort mpan2 syl norm B -v A opr ` <_ norm B +v S .s D opr opr -v A opr ` br adantr mpbid ord mpd pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abscl recn sqe0 sylib pjthlem9.6 pjthlem9.1 pjthlem9.2 hvsubcl eqeltr pjthlem9.3 hicl abs00 sylib))
+thm (pjthlem12 ((A y) (B y) (D y) (S y) (H y)) ((pjthlem12.1 (e. A (H~))) (pjthlem12.2 (e. H (CH))) (pjthlem12.3 (e. B H)) (pjthlem12.4 (e. D H)) (pjthlem12.5 (= R (opr (1) (/) (opr D (.i) D)))) (pjthlem12.6 (= S (opr R (x.) (opr C (.i) D)))) (pjthlem12.7 (= C (opr A (-v) B)))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (-> (-. (= D (0v))) (= (opr C (.i) D) (0)))) (y cv B +v S .s D opr opr -v A opreq1 norm fveq2d norm B -v A opr ` <_ breq2d H rcla4v com12 pjthlem12.2 pjthlem12.4 cheli pjthlem12.5 pjthlem12.7 pjthlem12.1 pjthlem12.2 pjthlem12.3 cheli hvsubcl eqeltr pjthlem12.6 pjthlem4 pjthlem12.4 jctir pjthlem12.2 chshi H S D shmulclt ax-mp syl pjthlem12.3 jctil pjthlem12.2 chshi H B S .s D opr shaddclt ax-mp syl syl5 ancld pjthlem12.1 pjthlem12.2 pjthlem12.3 cheli pjthlem12.2 pjthlem12.4 cheli pjthlem12.5 pjthlem12.6 pjthlem12.7 pjthlem11 syl6))
+thm (pjthlem13 ((A y) (B y) (D y) (C y) (H y)) ((pjthlem13.1 (e. A (H~))) (pjthlem13.2 (e. H (CH))) (pjthlem13.3 (e. B H)) (pjthlem13.4 (e. D H)) (pjthlem13.5 (= C (opr A (-v) B)))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (= (opr C (.i) D) (0))) (pjthlem13.1 pjthlem13.2 pjthlem13.3 pjthlem13.4 1 / D .i D opr opr eqid 1 / D .i D opr opr x. C .i D opr opr eqid pjthlem13.5 y pjthlem12 D 0v C .i opreq2 pjthlem13.5 pjthlem13.1 pjthlem13.2 pjthlem13.3 cheli hvsubcl eqeltr ax-hv0cl his1 pjthlem13.5 pjthlem13.1 pjthlem13.2 pjthlem13.3 cheli hvsubcl eqeltr C hi01t ax-mp * fveq2i cj0 3eqtr syl6eq pm2.61d2))
+thm (pjthlem14 ((x z) (y z) (A z) (x y) (A x) (A y) (B z) (B x) (B y) (C z) (C x) (C y) (H z) (H x) (H y)) ((pjthlem14.1 (e. A (H~))) (pjthlem14.2 (e. H (CH))) (pjthlem14.3 (e. B H)) (pjthlem14.4 (= C (opr A (-v) B)))) (-> (A.e. z H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv z) (-v) A)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (x cv x cv H e. x cv 0v if C .i opreq2 0 eqeq1d z H norm B -v A opr ` <_ norm z cv -v A opr ` br A.e. imbi2d pjthlem14.1 pjthlem14.2 pjthlem14.3 pjthlem14.2 H ch0 ax-mp x cv elimel pjthlem14.4 z pjthlem13 dedth com12 r19.21aiv pjthlem14.2 chshi H C x shocelt ax-mp pjthlem14.4 pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli hvsubcl eqeltr mpbiran sylibr pjthlem14.3 jctil pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli 1cn negcl pjthlem14.2 pjthlem14.3 cheli hvmulcl hvadd12 pjthlem14.2 pjthlem14.3 cheli hvnegid A +v opreq2i pjthlem14.1 A ax-hvaddid ax-mp eqtr2 pjthlem14.1 pjthlem14.2 pjthlem14.3 cheli hvsubval B +v opreq2i 3eqtr4 pjthlem14.4 B +v opreq2i eqtr4 jctir x cv B +v y cv opreq1 A eqeq2d y cv C B +v opreq2 A eqeq2d H _|_ H ` rcla42ev syl))
+thm (pjth ((w z) (x z) (y z) (A z) (w x) (w y) (A w) (x y) (A x) (A y) (H z) (H w) (H x) (H y)) ((pjth.1 (e. A (H~))) (pjth.2 (e. H (CH)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) (pjth.1 pjth.2 z w projlem z cv z cv H e. z cv 0v if -v A opreq1 norm fveq2d <_ norm w cv -v A opr ` breq1d w H biraldv x H y _|_ H ` A x cv +v y cv opr = E.e. E.e. imbi1d pjth.1 pjth.2 pjth.2 H ch0 ax-mp z cv elimel A -v z cv H e. z cv 0v if opr eqid w x y pjthlem14 dedth r19.23aiv ax-mp))
+thm (pjtht ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (H H CH e. H H~ if _|_ fveq2 _|_ H ` _|_ H CH e. H H~ if ` y A x cv +v y cv opr = rexeq syl x rexeqd A A H~ e. A 0v if x cv +v y cv opr eqeq1 y _|_ H CH e. H H~ if ` birexdv x H CH e. H H~ if birexdv ax-hv0cl A elimel helch H elimel x y pjth dedth2h))
+thm (pjthu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H x) (H y) (H z) (H w)) ((pjthu.1 (e. H (CH)))) (-> (e. A (H~)) (E!e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (pjthu.1 H A x y pjtht mpan pjthu.1 x cv y cv z cv w cv A chocuni an4s x cv z cv = y cv w cv = pm3.26 syl6 exp imp3a A H~ e. adantl y w 19.23advv y cv w cv z cv +v opreq2 A eqeq2d _|_ H ` cbvrexv y _|_ H ` A x cv +v y cv opr = E.e. anbi2i y _|_ H ` w _|_ H ` A x cv +v y cv opr = A z cv +v w cv opr = reeanv y _|_ H ` w _|_ H ` A x cv +v y cv opr = A z cv +v w cv opr = /\ r2ex bitr3 bitr syl5ib exp32 r19.21adv r19.21aiv jca x cv z cv +v y cv opreq1 A eqeq2d y _|_ H ` birexdv H reu4 sylibr))
+thm (pjthu2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H x) (H y) (H z) (H w)) ((pjthu2.1 (e. H (CH)))) (-> (e. A (H~)) (E!e. y (` (_|_) H) (E.e. x H (= A (opr (cv x) (+v) (cv y)))))) (pjthu2.1 H A x y pjtht mpan y _|_ H ` x H A x cv +v y cv opr = rexcom sylibr pjthu2.1 x cv y cv z cv w cv A chocuni an4s x cv z cv = y cv w cv = pm3.27 syl6 expcom imp3a A H~ e. adantl x z 19.23advv x cv z cv +v w cv opreq1 A eqeq2d H cbvrexv x H A x cv +v y cv opr = E.e. anbi2i x H z H A x cv +v y cv opr = A z cv +v w cv opr = reeanv x H z H A x cv +v y cv opr = A z cv +v w cv opr = /\ r2ex bitr3 bitr syl5ib exp32 r19.21adv r19.21aiv jca y cv w cv x cv +v opreq2 A eqeq2d x H birexdv _|_ H ` reu4 sylibr))
+thm (pjthut ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E!e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (H H CH e. H H~ if x cv eleq2 H H CH e. H H~ if _|_ fveq2 _|_ H ` _|_ H CH e. H H~ if ` y A x cv +v y cv opr = rexeq syl anbi12d x bieudv x H y _|_ H ` A x cv +v y cv opr = E.e. df-reu x H CH e. H H~ if y _|_ H CH e. H H~ if ` A x cv +v y cv opr = E.e. df-reu 3bitr4g A H~ e. imbi2d helch H elimel A x y pjthu dedth imp))
+thm (pjmvalt ((x y) (x z) (w x) (f x) (h x) (H x) (y z) (w y) (f y) (h y) (H y) (w z) (f z) (h z) (H z) (f w) (h w) (H w) (f h) (H f) (H h)) () (-> (e. H (CH)) (= (` (Proj) H) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (U. ({e.|} z H (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (h cv H z cv eleq2 h cv H _|_ fveq2 _|_ h cv ` _|_ H ` w x cv z cv +v w cv opr = rexeq syl anbi12d z biabdv z h cv w _|_ h cv ` x cv z cv +v w cv opr = E.e. df-rab z H w _|_ H ` x cv z cv +v w cv opr = E.e. df-rab 3eqtr4g unieqd y cv eqeq2d x cv H~ e. anbi2d x y biopabdv h f x y z w df-pj ax-hilex x y z H w _|_ H ` x cv z cv +v w cv opr = E.e. {e.|} U. funopabex2 fvopab4))
+thm (pjvalt ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w) (A x) (A y) (A z) (A w)) () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (Proj) H) A) (U. ({e.|} x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))))) (H z w x y pjmvalt A fveq1d A H~ e. adantr z cv A x cv +v y cv opr eqeq1 y _|_ H ` birexdv x H birabsdv unieqd z w z cv H~ e. w cv x H y _|_ H ` z cv x cv +v y cv opr = E.e. {e.|} U. = /\ {<,>|} eqid V fvopab4g H CH x y _|_ H ` A x cv +v y cv opr = E.e. rabexg x H y _|_ H ` A x cv +v y cv opr = E.e. {e.|} V uniexg syl sylan2 ancoms eqtrd))
+thm (axpjclt ((x y) (H x) (H y) (A x) (A y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (Proj) H) A) H)) (H A x y pjvalt H A x y pjthut x H y _|_ H ` A x cv +v y cv opr = E.e. reucl syl eqeltrd))
+thm (pjhclt () () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (Proj) H) A) (H~))) (H chss A H~ e. adantr H A axpjclt sseldd))
+thm (omlsilem () ((omlsilem.1 (e. G (SH))) (omlsilem.2 (e. H (SH))) (omlsilem.3 (C_ G H)) (omlsilem.4 (= (i^i H (` (_|_) G)) (0H))) (omlsilem.5 (e. A H)) (omlsilem.6 (e. B G)) (omlsilem.7 (e. C (` (_|_) G)))) (-> (= A (opr B (+v) C)) (e. A G)) (omlsilem.6 omlsilem.2 omlsilem.5 sheli omlsilem.1 omlsilem.6 sheli omlsilem.1 G shocss ax-mp omlsilem.7 sselii hvsubadd B +v C opr A eqcom bitr omlsilem.5 omlsilem.3 omlsilem.6 sselii omlsilem.2 H A B shsubclt ax-mp mp2an A -v B opr C H eleq1 mpbii sylbir omlsilem.7 omlsilem.4 C eleq2i C H _|_ G ` elin C elch0 3bitr3 biimp mpan2 syl B +v opreq2d omlsilem.1 omlsilem.6 sheli B ax-hvaddid ax-mp syl6eq G eleq1d mpbiri A B +v C opr G eleq1 mpbird))
+thm (omlsi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((omlsi.1 (e. A (CH))) (omlsi.2 (e. B (SH))) (omlsi.3 (C_ A B)) (omlsi.4 (= (i^i B (` (_|_) A)) (0H)))) (= A B) (omlsi.3 omlsi.2 x cv shel omlsi.1 A x cv y z pjtht mpan syl x cv x cv B e. x cv 0v if y cv +v z cv opr eqeq1 x cv x cv B e. x cv 0v if A eleq1 imbi12d y cv y cv A e. y cv 0v if +v z cv opreq1 x cv B e. x cv 0v if eqeq2d x cv B e. x cv 0v if A e. imbi1d z cv z cv _|_ A ` e. z cv 0v if y cv A e. y cv 0v if +v opreq2 x cv B e. x cv 0v if eqeq2d x cv B e. x cv 0v if A e. imbi1d omlsi.1 chshi omlsi.2 omlsi.3 omlsi.4 omlsi.2 B sh0 ax-mp x cv elimel omlsi.1 A ch0 ax-mp y cv elimel omlsi.1 chshi A shocsh ax-mp _|_ A ` sh0 ax-mp z cv elimel omlsilem dedth3h 3exp imp r19.23adv exp r19.23adv mpd ssriv eqssi))
+thm (omls () ((omls.1 (e. A (CH))) (omls.2 (e. B (SH)))) (-> (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B)) (A A B C_ B _|_ A ` i^i 0H = /\ A 0H if B eqeq1 B A B C_ B _|_ A ` i^i 0H = /\ B 0H if A B C_ B _|_ A ` i^i 0H = /\ A 0H if eqeq2 omls.1 h0elch A B C_ B _|_ A ` i^i 0H = /\ keepel omls.2 h0elch chshi A B C_ B _|_ A ` i^i 0H = /\ keepel A A B C_ B _|_ A ` i^i 0H = /\ A 0H if B sseq1 A A B C_ B _|_ A ` i^i 0H = /\ A 0H if _|_ fveq2 B ineq2d 0H eqeq1d anbi12d B A B C_ B _|_ A ` i^i 0H = /\ B 0H if A B C_ B _|_ A ` i^i 0H = /\ A 0H if sseq2 B A B C_ B _|_ A ` i^i 0H = /\ B 0H if _|_ A B C_ B _|_ A ` i^i 0H = /\ A 0H if ` ineq1 0H eqeq1d anbi12d 0H A B C_ B _|_ A ` i^i 0H = /\ A 0H if 0H sseq1 0H A B C_ B _|_ A ` i^i 0H = /\ A 0H if _|_ fveq2 0H ineq2d 0H eqeq1d anbi12d 0H A B C_ B _|_ A ` i^i 0H = /\ B 0H if A B C_ B _|_ A ` i^i 0H = /\ A 0H if sseq2 0H A B C_ B _|_ A ` i^i 0H = /\ B 0H if _|_ A B C_ B _|_ A ` i^i 0H = /\ A 0H if ` ineq1 0H eqeq1d anbi12d 0H ssid h0elch chshi 0H ocin ax-mp pm3.2i elimhyp2v pm3.26i A A B C_ B _|_ A ` i^i 0H = /\ A 0H if B sseq1 A A B C_ B _|_ A ` i^i 0H = /\ A 0H if _|_ fveq2 B ineq2d 0H eqeq1d anbi12d B A B C_ B _|_ A ` i^i 0H = /\ B 0H if A B C_ B _|_ A ` i^i 0H = /\ A 0H if sseq2 B A B C_ B _|_ A ` i^i 0H = /\ B 0H if _|_ A B C_ B _|_ A ` i^i 0H = /\ A 0H if ` ineq1 0H eqeq1d anbi12d 0H A B C_ B _|_ A ` i^i 0H = /\ A 0H if 0H sseq1 0H A B C_ B _|_ A ` i^i 0H = /\ A 0H if _|_ fveq2 0H ineq2d 0H eqeq1d anbi12d 0H A B C_ B _|_ A ` i^i 0H = /\ B 0H if A B C_ B _|_ A ` i^i 0H = /\ A 0H if sseq2 0H A B C_ B _|_ A ` i^i 0H = /\ B 0H if _|_ A B C_ B _|_ A ` i^i 0H = /\ A 0H if ` ineq1 0H eqeq1d anbi12d 0H ssid h0elch chshi 0H ocin ax-mp pm3.2i elimhyp2v pm3.27i omlsi dedth2v))
+thm (ococ () ((ococ.1 (e. A (CH)))) (= (` (_|_) (` (_|_) A)) A) (ococ.1 ococ.1 chshi A shocsh ax-mp _|_ A ` shocsh ax-mp ococ.1 chshi A shococss ax-mp _|_ _|_ A ` ` _|_ A ` incom ococ.1 chshi A shocsh ax-mp _|_ A ` ocin ax-mp eqtr omlsi eqcomi))
+thm (ococt () () (-> (e. A (CH)) (= (` (_|_) (` (_|_) A)) A)) (A A CH e. A H~ if _|_ fveq2 _|_ fveq2d A A CH e. A H~ if = id eqeq12d helch A elimel ococ dedth))
+thm (dfch2 () () (= (CH) ({|} x (/\ (C_ (cv x) (H~)) (= (` (_|_) (` (_|_) (cv x))) (cv x))))) (x cv chss x cv ococt jca _|_ _|_ x cv ` ` x cv CH eleq1 x cv occlt _|_ x cv ` chss _|_ x cv ` occlt 3syl syl5bi impcom impbi biabri))
+thm (pjcl () ((pjcl.1 (e. H (CH)))) (-> (e. A (H~)) (e. (` (` (Proj) H) A) H)) (pjcl.1 H A axpjclt mpan))
+thm (pjhcl () ((pjcl.1 (e. H (CH)))) (-> (e. A (H~)) (e. (` (` (Proj) H) A) (H~))) (pjcl.1 H A pjhclt mpan))
+thm (pjcli () ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (e. (` (` (Proj) H) A) H) (pjcli.2 pjcli.1 A pjcl ax-mp))
+thm (pjhcli () ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (e. (` (` (Proj) H) A) (H~)) (pjcli.2 pjcli.1 A pjhcl ax-mp))
+thm (pjpj0 ((x y) (A x) (A y) (H x) (H y)) ((pjcli.1 (e. H (CH))) (pjcli.2 (e. A (H~)))) (= A (opr (` (` (Proj) H) A) (+v) (` (` (Proj) (` (_|_) H)) A))) (x H y _|_ H ` A x cv +v Proj _|_ H ` ` A ` opr = A Proj H ` A ` +v y cv opr = /\ r2ex x H y _|_ H ` A x cv +v Proj _|_ H ` ` A ` opr = A Proj H ` A ` +v y cv opr = reeanv bitr3 pjcli.1 chocl pjcli.2 _|_ H ` A y x pjvalt mp2an eqcomi pjcli.1 chocl pjcli.2 pjcli pjcli.2 pjcli.1 chocl A y x pjthu ax-mp y cv Proj _|_ H ` ` A ` +v x cv opreq1 A eqeq2d x _|_ _|_ H ` ` birexdv _|_ H ` reuuni2 mp2an mpbir pjcli.1 ococ _|_ _|_ H ` ` H x A x cv +v Proj _|_ H ` ` A ` opr = rexeq ax-mp pjcli.1 chocl chocl x cv chel pjcli.1 chocl pjcli.2 pjhcli x cv Proj _|_ H ` ` A ` ax-hvcom mpan2 A eqeq2d syl birexa bitr3 mpbir pjcli.1 pjcli.2 H A x y pjvalt mp2an eqcomi pjcli.1 pjcli.2 pjcli pjcli.2 pjcli.1 A x y pjthu ax-mp x cv Proj H ` A ` +v y cv opreq1 A eqeq2d y _|_ H ` birexdv H reuuni2 mp2an mpbir mpbir2an pjcli.1 x cv Proj _|_ H ` ` A ` Proj H ` A ` y cv A chocuni pjcli.1 chocl pjcli.2 pjcli x cv H e. jctr pjcli.1 pjcli.2 pjcli y cv _|_ H ` e. jctl syl2an imp pm3.26d x cv Proj H ` A ` +v Proj _|_ H ` ` A ` opreq1 A eqeq2d biimpcd x cv H e. y cv _|_ H ` e. /\ A Proj H ` A ` +v y cv opr = ad2antrl mpd x y 19.23aivv ax-mp))
+thm (axpjpjt () () (-> (/\ (e. H (CH)) (e. A (H~))) (= A (opr (` (` (Proj) H) A) (+v) (` (` (Proj) (` (_|_) H)) A)))) (H H CH e. H H~ if Proj fveq2 A fveq1d H H CH e. H H~ if _|_ fveq2 Proj fveq2d A fveq1d +v opreq12d A eqeq2d A A H~ e. A 0v if = id A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 A A H~ e. A 0v if Proj _|_ H CH e. H H~ if ` ` fveq2 +v opreq12d eqeq12d helch H elimel ax-hv0cl A elimel pjpj0 dedth2h))
+thm (pjpj () ((pjpj.1 (e. H (CH))) (pjpj.2 (e. A (H~)))) (= A (opr (` (` (Proj) H) A) (+v) (` (` (Proj) (` (_|_) H)) A))) (pjpj.1 pjpj.2 H A axpjpjt mp2an))
+thm (pjpjtht ((x y) (A x) (A y) (H x) (H y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y)))))) (x cv Proj H ` A ` +v y cv opreq1 A eqeq2d y cv Proj _|_ H ` ` A ` Proj H ` A ` +v opreq2 A eqeq2d H _|_ H ` rcla42ev H A axpjclt _|_ H ` A axpjclt H choclt sylan jca H A axpjpjt sylanc))
+thm (pjpjth ((x y) (A x) (A y) (H x) (H y)) ((pjpjth.1 (e. A (H~))) (pjpjth.2 (e. H (CH)))) (E.e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))) (pjpjth.2 pjpjth.1 H A x y pjpjtht mp2an))
+thm (pjopt () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (Proj) (` (_|_) H)) A) (opr A (-v) (` (` (Proj) H) A)))) (H A axpjpjt eqcomd A Proj H ` A ` Proj _|_ H ` ` A ` hvsubaddt H CH e. A H~ e. pm3.27 H A pjhclt _|_ H ` A pjhclt H choclt sylan syl3anc mpbird eqcomd))
+thm (pjpot () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (Proj) H) A) (opr A (-v) (` (` (Proj) (` (_|_) H)) A)))) (Proj _|_ H ` ` A ` Proj H ` A ` ax-hvcom _|_ H ` A pjhclt H choclt sylan H A pjhclt sylanc H A axpjpjt eqtr4d A Proj _|_ H ` ` A ` Proj H ` A ` hvsubaddt H CH e. A H~ e. pm3.27 _|_ H ` A pjhclt H choclt sylan H A pjhclt syl3anc mpbird eqcomd))
+thm (pjop () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (= (` (` (Proj) (` (_|_) H)) A) (opr A (-v) (` (` (Proj) H) A))) (pjop.1 pjop.2 H A pjopt mp2an))
+thm (pjpo () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (= (` (` (Proj) H) A) (opr A (-v) (` (` (Proj) (` (_|_) H)) A))) (pjop.1 pjop.2 H A pjpot mp2an))
+thm (pjoc1 () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (<-> (e. A H) (= (` (` (Proj) (` (_|_) H)) A) (0v))) (pjop.1 pjop.2 pjcli pjop.1 chshi H A Proj H ` A ` shsubclt ax-mp mpan2 pjop.1 pjop.2 pjop H eleq1i sylibr pjop.1 chocl pjop.2 pjcli jctir Proj _|_ H ` ` A ` H _|_ H ` elin sylibr pjop.1 chshi H ocin ax-mp Proj _|_ H ` ` A ` eleq2i sylib Proj _|_ H ` ` A ` elch0 sylib pjop.1 pjop.2 pjcli Proj _|_ H ` ` A ` 0v Proj H ` A ` +v opreq2 pjop.1 pjop.2 pjpj syl5eq pjop.1 pjop.2 pjhcli Proj H ` A ` ax-hvaddid ax-mp syl6eq H eleq1d mpbiri impbi))
+thm (pjch () ((pjop.1 (e. H (CH))) (pjop.2 (e. A (H~)))) (<-> (e. A H) (= (` (` (Proj) H) A) A)) (pjop.1 pjop.2 pjoc1 biimp Proj H ` A ` +v opreq2d pjop.1 pjop.2 pjpj syl5req pjop.1 pjop.2 pjhcli Proj H ` A ` ax-hvaddid ax-mp syl5eqr pjop.1 pjop.2 pjcli Proj H ` A ` A H eleq1 mpbii impbi))
+thm (pjoc1t () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (` (Proj) (` (_|_) H)) A) (0v)))) (H H CH e. H H~ if A eleq2 H H CH e. H H~ if _|_ fveq2 Proj fveq2d A fveq1d 0v eqeq1d bibi12d A A H~ e. A 0v if H CH e. H H~ if eleq1 A A H~ e. A 0v if Proj _|_ H CH e. H H~ if ` ` fveq2 0v eqeq1d bibi12d helch H elimel ax-hv0cl A elimel pjoc1 dedth2h))
+thm (pjoml ((A x) (B x)) ((pjoml.1 (e. A (CH))) (pjoml.2 (e. B (SH)))) (-> (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B)) (A B C_ B _|_ A ` i^i 0H = pm3.26 B _|_ A ` i^i 0H Proj _|_ A ` ` x cv ` eleq2 pjoml.2 B x cv Proj A ` x cv ` shsubclt ax-mp A B C_ x cv B e. pm3.27 A B Proj A ` x cv ` ssel2 pjoml.2 x cv shel pjoml.1 x cv pjcl syl sylan2 sylanc pjoml.2 x cv shel pjoml.1 A x cv pjopt mpan syl B eleq1d A B C_ adantl mpbird pjoml.2 x cv shel pjoml.1 chocl x cv pjcl syl A B C_ adantl jca Proj _|_ A ` ` x cv ` B _|_ A ` elin sylibr syl5bi exp3a com12 imp31 Proj _|_ A ` ` x cv ` elch0 sylib pjoml.2 x cv shel A B C_ B _|_ A ` i^i 0H = /\ adantl pjoml.1 A x cv pjoc1t mpan syl mpbird exp ssrdv eqssd))
+thm (pjococ () ((pjococ.1 (e. H (CH)))) (= (` (_|_) (` (_|_) H)) H) (pjococ.1 chshi H shococss ax-mp _|_ _|_ H ` ` _|_ H ` incom pjococ.1 chshi H shocsh ax-mp _|_ H ` ocin ax-mp eqtr pjococ.1 pjococ.1 chshi H shocsh ax-mp shssi occl chshi pjoml mp2an eqcomi))
+thm (pjoc2 () ((pjoc2.1 (e. H (CH))) (pjoc2.2 (e. A (H~)))) (<-> (e. A (` (_|_) H)) (= (` (` (Proj) H) A) (0v))) (pjoc2.1 chocl pjoc2.2 pjoc1 pjoc2.1 pjococ Proj fveq2i A fveq1i 0v eqeq1i bitr))
+thm (pjoc2t () ((pjoc2.1 (e. H (CH)))) (-> (e. A (H~)) (<-> (e. A (` (_|_) H)) (= (` (` (Proj) H) A) (0v)))) (A A H~ e. A 0v if _|_ H ` eleq1 A A H~ e. A 0v if Proj H ` fveq2 0v eqeq1d bibi12d pjoc2.1 ax-hv0cl A elimel pjoc2 dedth))
+thm (shsumvalt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (+H) B) ({e.|} x (H~) (E.e. y A (E.e. z B (= (cv x) (opr (cv y) (+v) (cv z)))))))) (ax-hilex x y A z B x cv y cv +v z cv opr = E.e. E.e. rabex w cv A y z v cv x cv y cv +v z cv opr = E.e. rexeq x H~ birabsdv v cv B z x cv y cv +v z cv opr = rexeq y A birexdv x H~ birabsdv w v u x y z df-shsum oprabval2))
+thm (shselt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (e. A (SH)) (e. B (SH))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y))))))) (A B z x y shsumvalt C eleq2d z cv C x cv +v y cv opr eqeq1 y B birexdv x A birexdv H~ elrab syl6bb A shss x cv sseld B shss y cv sseld im2anan9 x cv y cv ax-hvaddcl x cv +v y cv opr H~ C eleq1a syl syl6 r19.23advv pm4.71rd bitr4d))
+thm (shsel ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((shscl.1 (e. A (SH))) (shscl.2 (e. B (SH)))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (shscl.1 shscl.2 A B C x y shselt mp2an))
+thm (shscl ((x y) (x z) (w x) (f x) (g x) (v x) (u x) (A x) (y z) (w y) (f y) (g y) (v y) (u y) (A y) (w z) (f z) (g z) (v z) (u z) (A z) (f w) (g w) (v w) (u w) (A w) (f g) (f v) (f u) (A f) (g v) (g u) (A g) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B f) (B g) (B v) (B u)) ((shscl.1 (e. A (SH))) (shscl.2 (e. B (SH)))) (e. (opr A (+H) B) (SH)) (A +H B opr x y sh shscl.1 shscl.2 A B z x y shsumvalt mp2an z H~ x A y B z cv x cv +v y cv opr = E.e. E.e. ssrab eqsstr shscl.1 A sh0 ax-mp shscl.2 B sh0 ax-mp pm3.2i ax-hv0cl hvaddid2 eqcomi x cv 0v +v y cv opreq1 0v eqeq2d y cv 0v 0v +v opreq2 0v eqeq2d A B rcla42ev mp2an shscl.1 shscl.2 0v x y shsel mpbir pm3.2i f cv z cv +v v cv opr +v g cv opreq1 x cv +v y cv opr eqeq2d g cv w cv +v u cv opr z cv +v v cv opr +v opreq2 x cv +v y cv opr eqeq2d A B rcla42ev shscl.1 A z cv v cv shaddclt ax-mp shscl.2 B w cv u cv shaddclt ax-mp anim12i x cv z cv +v w cv opr = y cv v cv +v u cv opr = /\ adantrr x cv z cv +v w cv opr y cv v cv +v u cv opr +v opreq12 z cv v cv w cv u cv hvadd4t shscl.1 z cv shel shscl.1 v cv shel anim12i shscl.2 w cv shel shscl.2 u cv shel anim12i syl2an eqcomd sylan9eqr anasss sylanc w cv B e. x cv z cv +v w cv opr = u cv B e. y cv v cv +v u cv opr = an4 sylan2b an4s exp42 imp com3r exp32 imp r19.23aivv com3l r19.23aivv imp shscl.1 shscl.2 x cv z w shsel shscl.1 shscl.2 y cv v u shsel anbi12i shscl.1 shscl.2 x cv +v y cv opr f g shsel 3imtr4 rgen2 shscl.1 A x cv v cv shmulclt ax-mp shscl.2 B x cv u cv shmulclt ax-mp anim12i anandis y cv v cv +v u cv opr = adantlr y cv v cv +v u cv opr x cv .s opreq2 x cv v cv u cv ax-hvdistr1 3expb shscl.1 v cv shel shscl.2 u cv shel anim12i sylan2 sylan9eqr an1rs jca ancoms an4s an1rs f cv x cv .s v cv opr +v g cv opreq1 x cv .s y cv opr eqeq2d g cv x cv .s u cv opr x cv .s v cv opr +v opreq2 x cv .s y cv opr eqeq2d A B rcla42ev syl exp42 imp r19.23aivv impcom shscl.1 shscl.2 y cv v u shsel x cv CC e. anbi2i shscl.1 shscl.2 x cv .s y cv opr f g shsel 3imtr4 rgen2a pm3.2i mpbir2an))
+thm (shsclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (opr A (+H) B) (SH))) (A A SH e. A H~ if +H B opreq1 SH eleq1d B B SH e. B H~ if A SH e. A H~ if +H opreq2 SH eleq1d helsh A elimel helsh B elimel shscl dedth2h))
+thm (shscomt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (+H) B) (opr B (+H) A))) (A y cv shelt B z cv shelt anim12i an4s y cv z cv ax-hvcom syl x cv eqeq2d bi2rexdva y A z B x cv z cv +v y cv opr = rexcom syl6bb A B x cv y z shselt B A x cv z y shselt ancoms 3bitr4d eqrd))
+thm (shsvat ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (+v) D) (opr A (+H) B)))) (A B C +v D opr x y shselt C +v D opr eqid x cv C +v y cv opreq1 C +v D opr eqeq2d y cv D C +v opreq2 C +v D opr eqeq2d A B rcla42ev mpan2 syl5bir))
+thm (shsel1t () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (e. C A) (e. C (opr A (+H) B)))) (A C shelt C ax-hvaddid syl B SH e. adantlr B sh0 A SH e. adantl A B C 0v shsvat mpan2d imp eqeltrrd exp))
+thm (shsel2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (e. C B) (e. C (opr A (+H) B)))) (B A C shsel1t ancoms A B shscomt C eleq2d sylibrd))
+thm (shsvst () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (-v) D) (opr A (+H) B)))) (A B shsclt A +H B opr C D shsubclt syl A B C shsel1t A B D shsel2t syl2and))
+thm (shsub1t ((A x) (B x)) () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr A (+H) B))) (A B x cv shsel1t ssrdv))
+thm (shsub2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr B (+H) A))) (A B shsub1t A B shscomt sseqtrd))
+thm (choc0 ((x y)) () (= (` (_|_) (0H)) (H~)) (x cv H~ e. y 0H x cv .i y cv opr 0 = A.e. abai h0elch chshi 0H x cv y shocelt ax-mp x cv hi02t y 0H x cv .i y cv opr 0 = df-ral y cv elch0 x cv .i y cv opr 0 = imbi1i y bial ax-hv0cl elisseti y cv 0v x cv .i opreq2 0 eqeq1d ceqsalv 3bitr sylibr x cv H~ e. biantru 3bitr4 eqri))
+thm (choc1 ((x y)) () (= (` (_|_) (H~)) (0H)) (helch chshi H~ x cv y shocelt ax-mp pm3.27bd helch chshi H~ shocss ax-mp x cv sseli x cv y hial0 syl mpbid x cv elch0 sylibr ssriv h0elch chshi 0H shococss ax-mp choc0 _|_ fveq2i sseqtr eqssi))
+thm (chocnul ((x y) (x y)) () (= (` (_|_) ({/})) (H~)) (H~ 0ss {/} x cv y ocelt ax-mp {/} eqid {/} y x cv .i y cv opr 0 = rzal ax-mp mpbiran2 eqri))
+thm (shintcl ((x y) (x z) (A x) (y z) (A y) (A z)) ((shintcl.1 (/\ (C_ A (SH)) (-. (= A ({/})))))) (e. (|^| A) (SH)) (A |^| x y sh shintcl.1 pm3.27i A z n0 z cv A intss1 shintcl.1 pm3.26i z cv sseli z cv shss syl sstrd z 19.23aiv sylbi ax-mp ax-hv0cl elisseti A z elint2 shintcl.1 pm3.26i z cv sseli z cv sh0 syl mprgbir pm3.2i x cv A z cv elinti com12 y cv A z cv elinti com12 anim12d shintcl.1 pm3.26i z cv sseli z cv x cv y cv shaddclt syl syld com12 r19.21aiv x cv +v y cv oprex A z elint2 sylibr rgen2 y cv A z cv elinti com12 x cv CC e. anim2d shintcl.1 pm3.26i z cv sseli z cv x cv y cv shmulclt syl syld com12 r19.21aiv x cv .s y cv oprex A z elint2 sylibr rgen2a pm3.2i mpbir2an))
+thm (shintclt () () (-> (/\ (C_ A (SH)) (-. (= A ({/})))) (e. (|^| A) (SH))) (A A SH C_ A {/} = -. /\ A SH if inteq SH eleq1d A A SH C_ A {/} = -. /\ A SH if SH sseq1 A A SH C_ A {/} = -. /\ A SH if {/} eqeq1 negbid anbi12d SH A SH C_ A {/} = -. /\ A SH if SH sseq1 SH A SH C_ A {/} = -. /\ A SH if {/} eqeq1 negbid anbi12d SH ssid h0elsh 0H SH n0i ax-mp pm3.2i elimhyp shintcl dedth))
+thm (chintcl ((x y) (f x) (A x) (f y) (A y) (A f)) ((chintcl.1 (/\ (C_ A (CH)) (-. (= A ({/})))))) (e. (|^| A) (CH)) (A |^| f x closedsub chintcl.1 pm3.26i chsssh sstri chintcl.1 pm3.27i pm3.2i shintcl chintcl.1 pm3.26i y cv sseli x visset y cv f cv chlim syl exp3a com3r imp r19.20dva chintcl.1 pm3.27i f cv NN y fint x visset A y elint2 3imtr4g impcom f x gen2 mpbir2an))
+thm (chintclt () () (-> (/\ (C_ A (CH)) (-. (= A ({/})))) (e. (|^| A) (CH))) (A A CH C_ A {/} = -. /\ A CH if inteq CH eleq1d A A CH C_ A {/} = -. /\ A CH if CH sseq1 A A CH C_ A {/} = -. /\ A CH if {/} eqeq1 negbid anbi12d CH A CH C_ A {/} = -. /\ A CH if CH sseq1 CH A CH C_ A {/} = -. /\ A CH if {/} eqeq1 negbid anbi12d CH ssid h0elch 0H CH n0i ax-mp pm3.2i elimhyp chintcl dedth))
+thm (ococint ((A x)) () (-> (C_ A (H~)) (= (` (_|_) (` (_|_) A)) (|^| ({e.|} x (CH) (C_ A (cv x)))))) (_|_ x CH A x cv C_ {e.|} |^| ` _|_ A ` occont helch A H~ C_ jctl x cv H~ A sseq2 CH elrab sylibr H~ x CH A x cv C_ {e.|} intss1 x CH A x cv C_ {e.|} |^| ocss 3syl A ocss jca A x CH ssintub helch A H~ C_ jctl x cv H~ A sseq2 CH elrab sylibr H~ x CH A x cv C_ {e.|} intss1 syl A x CH A x cv C_ {e.|} |^| occont mpdan mpi sylc helch x cv H~ A sseq2 CH rcla4ev mpan x CH A x cv C_ rabn0 sylibr x CH A x cv C_ ssrab jctil x CH A x cv C_ {e.|} chintclt x CH A x cv C_ {e.|} |^| ococt 3syl sseqtrd A ocss _|_ A ` occlt syl A ococss jca x cv _|_ _|_ A ` ` A sseq2 CH elrab sylibr _|_ _|_ A ` ` x CH A x cv C_ {e.|} intss1 syl eqssd))
+thm (dfchsup2 ((x y) (x z) (y z)) () (= (\/H) ({<,>|} x y (/\ (C_ (cv x) (P~ (H~))) (= (cv y) (|^| ({e.|} z (CH) (C_ (U. (cv x)) (cv z)))))))) (x y df-chsup x cv H~ P~ uniss H~ unipw syl6ss x cv U. z ococint syl y cv eqeq2d pm5.32i x y biopabi eqtr))
+thm (spanvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (H~)) (= (` (span) A) (|^| ({e.|} x (SH) (C_ A (cv x)))))) (ax-hilex H~ V A elpw2g ax-mp ax-hilex H~ V A elpw2g ax-mp helsh x cv H~ A sseq2 SH rcla4ev mpan sylbi x SH A x cv C_ intexrab sylib y cv A x cv sseq1 x SH birabsdv inteqd y z x df-span y visset H~ elpw z cv x SH y cv x cv C_ {e.|} |^| = anbi1i y z biopabi eqtr4 V fvopab4g mpdan sylbir))
+thm (hsupval2t ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (P~ (H~))) (= (` (\/H) A) (|^| ({e.|} x (CH) (C_ (U. A) (cv x)))))) (z cv A unieq x cv sseq1d x CH birabsdv inteqd z y x dfchsup2 z visset H~ P~ elpw y cv x CH z cv U. x cv C_ {e.|} |^| = anbi1i z y biopabi eqtr4 V fvopab4g ax-hilex pwex H~ P~ V A elpw2g ax-mp biimpr A H~ P~ uniss H~ unipw syl6ss helch jctil x cv H~ A U. sseq2 CH elrab sylibr H~ x CH A U. x cv C_ {e.|} n0i syl x CH A U. x cv C_ {e.|} intex sylib sylanc))
+thm (hsupvalt ((A x)) () (-> (C_ A (P~ (H~))) (= (` (\/H) A) (` (_|_) (` (_|_) (U. A))))) (A x hsupval2t A H~ P~ uniss H~ unipw syl6ss A U. x ococint syl eqtr4d))
+thm (chsupval2t ((A x)) () (-> (C_ A (CH)) (= (` (\/H) A) (|^| ({e.|} x (CH) (C_ (U. A) (cv x)))))) (chsspwh A CH H~ P~ sstr2 mpi A x hsupval2t syl))
+thm (chsupvalt () () (-> (C_ A (CH)) (= (` (\/H) A) (` (_|_) (` (_|_) (U. A))))) (chsspwh A CH H~ P~ sstr2 mpi A hsupvalt syl))
+thm (spanclt ((A x)) () (-> (C_ A (H~)) (e. (` (span) A) (SH))) (A x spanvalt helsh x cv H~ A sseq2 SH rcla4ev mpan x SH A x cv C_ rabn0 sylibr x SH A x cv C_ ssrab jctil x SH A x cv C_ {e.|} shintclt syl eqeltrd))
+thm (elspanclt () () (-> (/\ (C_ A (H~)) (e. B (` (span) A))) (e. B (H~))) (span A ` B shelt A spanclt sylan))
+thm (shsupclt () () (-> (C_ A (P~ (H~))) (e. (` (span) (U. A)) (SH))) (A H~ P~ uniss H~ unipw syl6ss A U. spanclt syl))
+thm (hsupclt ((A x)) () (-> (C_ A (P~ (H~))) (e. (` (\/H) A) (CH))) (A x hsupval2t \/H A fvex A x hsupval2t V eleq1d mpbii x CH A U. x cv C_ {e.|} intex sylibr x CH A U. x cv C_ ssrab jctil x CH A U. x cv C_ {e.|} chintclt syl eqeltrd))
+thm (chsupclt () () (-> (C_ A (CH)) (e. (` (\/H) A) (CH))) (chsspwh A CH H~ P~ sstr2 mpi A hsupclt syl))
+thm (hsupss ((A x) (B x)) () (-> (/\ (C_ A (P~ (H~))) (C_ B (P~ (H~)))) (-> (C_ A B) (C_ (` (\/H) A) (` (\/H) B)))) (A x hsupval2t B H~ P~ C_ adantr B x hsupval2t A H~ P~ C_ adantl sseq12d A B uniss A U. B U. x cv sstr2 syl x cv CH e. a1d r19.21aiv x CH B U. x cv C_ A U. x cv C_ ss2rab sylibr x CH B U. x cv C_ {e.|} x CH A U. x cv C_ {e.|} intss syl syl5bir))
+thm (chsupss () () (-> (/\ (C_ A (CH)) (C_ B (CH))) (-> (C_ A B) (C_ (` (\/H) A) (` (\/H) B)))) (A B hsupss chsspwh A CH H~ P~ sstr2 mpi chsspwh B CH H~ P~ sstr2 mpi syl2an))
+thm (chsupid ((x y) (A x) (A y)) () (-> (e. A (CH)) (= (` (\/H) ({e.|} x (CH) (C_ (cv x) A))) A)) (A CH y intmin A CH x unisseq y cv sseq1d y CH birabsdv inteqd eqtr2d x CH x cv A C_ ssrab x CH x cv A C_ {e.|} y chsupval2t ax-mp syl5eq))
+thm (chsupsn ((A x)) () (-> (e. A (CH)) (= (` (\/H) ({} A)) A)) (A CH snssi A {} x chsupval2t syl A CH unisng A {} U. A eqimss syl ancli x cv A A {} U. sseq2 CH elrab sylibr A x CH A {} U. x cv C_ {e.|} intss1 syl A {} U. x CH ssintub A CH unisng x CH A {} U. x cv C_ {e.|} |^| sseq1d mpbii eqssd eqtrd))
+thm (hsupunss ((A x)) () (-> (C_ A (P~ (H~))) (C_ (U. A) (` (\/H) A))) (A U. x CH ssintub A x hsupval2t A U. sseq2d mpbiri))
+thm (spanss2 ((A x)) () (-> (C_ A (H~)) (C_ A (` (span) A))) (A x SH ssintub A x spanvalt A sseq2d mpbiri))
+thm (shsupunss () () (-> (C_ A (SH)) (C_ (U. A) (` (span) (U. A)))) (shsspwh A SH H~ P~ sstr mpan2 A H~ P~ uniss syl H~ unipw syl6ss A U. spanss2 syl))
+thm (chsupunss () () (-> (C_ A (CH)) (C_ (U. A) (` (\/H) A))) (chsspwh A CH H~ P~ sstr mpan2 A hsupunss syl))
+thm (spanid ((A x)) () (-> (e. A (SH)) (= (` (span) A) A)) (A shss A x spanvalt syl A SH x intmin eqtr4d))
+thm (spanss ((A x) (B x)) () (-> (/\ (C_ B (H~)) (C_ A B)) (C_ (` (span) A) (` (span) B))) (A B x cv sstr2 x cv SH e. a1d r19.21aiv x SH B x cv C_ A x cv C_ ss2rab sylibr x SH B x cv C_ {e.|} x SH A x cv C_ {e.|} intss syl B H~ C_ adantl A B H~ sstr ancoms A x spanvalt syl B x spanvalt A B C_ adantr 3sstr4d))
+thm (spanssoc () () (-> (C_ A (H~)) (C_ (` (span) A) (` (_|_) (` (_|_) A)))) (_|_ _|_ A ` ` A spanss A ocss _|_ A ` ocss syl A ococss sylanc A ocss _|_ A ` ocsh _|_ _|_ A ` ` spanid 3syl sseqtrd))
+thm (sshjvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) (_|_ _|_ A B u. ` fvex x cv A y cv uneq1 _|_ fveq2d _|_ fveq2d y cv B A uneq2 _|_ fveq2d _|_ fveq2d x y z df-chj x visset H~ elpw y visset H~ elpw anbi12i z cv _|_ _|_ x cv y cv u. ` ` = anbi1i x y z bioprabi eqtr4 oprabval2 ax-hilex H~ V A elpw2g ax-mp ax-hilex H~ V B elpw2g ax-mp syl2anbr))
+thm (shjvalt () () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) (A B sshjvalt A shss B shss syl2an))
+thm (chjvalt () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B))))) (A B shjvalt A chsh B chsh syl2an))
+thm (chjval () ((chjval.1 (e. A (CH))) (chjval.2 (e. B (CH)))) (= (opr A (vH) B) (` (_|_) (` (_|_) (u. A B)))) (chjval.1 chjval.2 A B chjvalt mp2an))
+thm (dfchj2 ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (vH) ({<<,>,>|} x y z (/\ (/\ (C_ (cv x) (H~)) (C_ (cv y) (H~))) (= (cv z) (|^| ({e.|} w (CH) (C_ (u. (cv x) (cv y)) (cv w)))))))) (x y z df-chj x cv H~ y cv unss x cv y cv u. w ococint sylbi z cv eqeq2d pm5.32i x y z bioprabi eqtr))
+thm (dfchj3 ((x y) (x z) (y z)) () (= (vH) ({<<,>,>|} x y z (/\ (/\ (C_ (cv x) (H~)) (C_ (cv y) (H~))) (= (cv z) (` (\/H) ({,} (cv x) (cv y))))))) (x y z df-chj x visset y visset H~ prsspw x cv y cv {,} hsupvalt sylbir x visset y visset unpr _|_ fveq2i _|_ fveq2i syl6req z cv eqeq2d pm5.32i x y z bioprabi eqtr))
+thm (sshjval3t ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (H~)) (C_ B (H~))) (= (opr A (vH) B) (` (\/H) ({,} A B)))) (\/H A B {,} fvex x cv A y cv preq1 \/H fveq2d y cv B A preq2 \/H fveq2d x y z dfchj3 x visset H~ elpw y visset H~ elpw anbi12i z cv \/H x cv y cv {,} ` = anbi1i x y z bioprabi eqtr4 oprabval2 ax-hilex H~ V A elpw2g ax-mp ax-hilex H~ V B elpw2g ax-mp syl2anbr))
+thm (sshjclt () () (-> (/\ (C_ A (H~)) (C_ B (H~))) (e. (opr A (vH) B) (CH))) (A B sshjvalt A H~ B unss A B u. ocss _|_ A B u. ` occlt syl sylbi eqeltrd))
+thm (shjclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (opr A (vH) B) (CH))) (A B sshjclt A shss B shss syl2an))
+thm (chjclt () () (-> (/\ (e. A (CH)) (e. B (CH))) (e. (opr A (vH) B) (CH))) (A B shjclt A chsh B chsh syl2an))
+thm (shjcomt () () (-> (/\ (e. A (SH)) (e. B (SH))) (= (opr A (vH) B) (opr B (vH) A))) (A B shjvalt B A shjvalt ancoms B A uncom _|_ fveq2i _|_ fveq2i syl6eq eqtr4d))
+thm (shincl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (i^i A B) (SH)) (shincl.1 elisseti shincl.2 elisseti intpr shincl.1 shincl.2 pm3.2i shincl.1 elisseti shincl.2 elisseti SH prss mpbi shincl.1 elisseti B prnz pm3.2i shintcl eqeltrr))
+thm (shscom () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (= (opr A (+H) B) (opr B (+H) A)) (shincl.1 shincl.2 A B shscomt mp2an))
+thm (shsva () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (+v) D) (opr A (+H) B))) (shincl.1 shincl.2 A B C D shsvat mp2an))
+thm (shsel1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (e. C A) (e. C (opr A (+H) B))) (shincl.1 shincl.2 A B C shsel1t mp2an))
+thm (shsel2 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (e. C B) (e. C (opr A (+H) B))) (shincl.1 shincl.2 A B C shsel2t mp2an))
+thm (shsvs () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (-> (/\ (e. C A) (e. D B)) (e. (opr C (-v) D) (opr A (+H) B))) (shincl.1 shincl.2 shscl A +H B opr C D shsubclt ax-mp shincl.1 shincl.2 C shsel1 shincl.1 shincl.2 D shsel2 syl2an))
+thm (shunss ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (u. A B) (opr A (+H) B)) (y cv x cv +v z cv opreq1 x cv eqeq2d z cv 0v x cv +v opreq2 x cv eqeq2d A B rcla42ev shincl.2 B sh0 ax-mp x cv A e. jctr shincl.1 x cv shel x cv ax-hvaddid syl eqcomd sylanc y cv 0v +v z cv opreq1 x cv eqeq2d z cv x cv 0v +v opreq2 x cv eqeq2d A B rcla42ev shincl.1 A sh0 ax-mp x cv B e. jctl shincl.2 x cv shel x cv hvaddid2t syl eqcomd sylanc jaoi x cv A B elun shincl.1 shincl.2 x cv y z shsel 3imtr4 ssriv))
+thm (shslej ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (opr A (+H) B) (opr A (vH) B)) (x cv y cv +v z cv opr _|_ _|_ A B u. ` ` eleq1 shincl.1 shssi shincl.2 shssi unssi A B u. ocss ax-mp _|_ A B u. ` ocsh ax-mp _|_ _|_ A B u. ` ` y cv z cv shaddclt ax-mp y cv A B elun1 shincl.1 shssi shincl.2 shssi unssi A B u. ococss ax-mp y cv sseli syl z cv B A elun2 shincl.1 shssi shincl.2 shssi unssi A B u. ococss ax-mp z cv sseli syl syl2an syl5bir com12 r19.23aivv shincl.1 shincl.2 x cv y z shsel shincl.1 shincl.2 A B shjvalt mp2an x cv eleq2i 3imtr4 ssriv))
+thm (shunssj () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ (u. A B) (opr A (vH) B)) (shincl.1 shincl.2 shunss shincl.1 shincl.2 shslej sstri))
+thm (shjcom () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (= (opr A (vH) B) (opr B (vH) A)) (shincl.1 shincl.2 A B shjcomt mp2an))
+thm (shsub1 ((A x) (B x)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr A (+H) B)) (shincl.1 shincl.2 x cv shsel1 ssriv))
+thm (shsub2 ((A x) (B x)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr B (+H) A)) (shincl.2 shincl.1 x cv shsel2 ssriv))
+thm (shub1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (C_ A (opr A (vH) B)) (shincl.1 shincl.2 shsub1 shincl.1 shincl.2 shslej sstri))
+thm (shjcl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (opr A (vH) B) (CH)) (shincl.1 shincl.2 A B shjclt mp2an))
+thm (shjshcl () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH)))) (e. (opr A (vH) B) (SH)) (shincl.1 shincl.2 shjcl chshi))
+thm (shlub () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shlub.1 (e. C (CH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (shincl.1 shssi shincl.2 shssi unssi shlub.1 chssi occon2 A C B unss shincl.1 shincl.2 A B shjvalt mp2an shlub.1 pjococ eqcomi sseq12i 3imtr4 shincl.1 shincl.2 shub1 A A vH B opr C sstr mpan B A ssun2 shincl.1 shincl.2 shunssj sstri B A vH B opr C sstr mpan jca impbi))
+thm (shless ((x y) (x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr A (+H) C) (opr B (+H) C))) (A B y cv ssel z C x cv y cv +v z cv opr = E.e. anim1d y 19.22dv y A z C x cv y cv +v z cv opr = E.e. df-rex y B z C x cv y cv +v z cv opr = E.e. df-rex 3imtr4g shincl.1 shless.1 x cv y z shsel shincl.2 shless.1 x cv y z shsel 3imtr4g ssrdv))
+thm (shlej1 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C))) (shincl.2 shless.1 shub1 A B B vH C opr sstr2 mpi shless.1 shincl.2 shub1 shless.1 shincl.2 shjcom sseqtr jctir shincl.1 shless.1 shincl.2 shless.1 shjcl shlub sylib))
+thm (shlej2 () ((shincl.1 (e. A (SH))) (shincl.2 (e. B (SH))) (shless.1 (e. C (SH)))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B))) (shincl.1 shincl.2 shless.1 shlej1 shless.1 shincl.1 shjcom shless.1 shincl.2 shjcom 3sstr4g))
+thm (shslejt () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ (opr A (+H) B) (opr A (vH) B))) (A A SH e. A H~ if +H B opreq1 A A SH e. A H~ if vH B opreq1 sseq12d B B SH e. B H~ if A SH e. A H~ if +H opreq2 B B SH e. B H~ if A SH e. A H~ if vH opreq2 sseq12d helsh A elimel helsh B elimel shslej dedth2h))
+thm (shinclt () () (-> (/\ (e. A (SH)) (e. B (SH))) (e. (i^i A B) (SH))) (A A SH e. A H~ if B ineq1 SH eleq1d B B SH e. B H~ if A SH e. A H~ if ineq2 SH eleq1d helsh A elimel helsh B elimel shincl dedth2h))
+thm (shub1t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr A (vH) B))) (A B shsub1t A B shslejt sstrd))
+thm (shub2t () () (-> (/\ (e. A (SH)) (e. B (SH))) (C_ A (opr B (vH) A))) (A B shub1t A B shjcomt sseqtrd))
+thm (shlubt () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (CH))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C))) (A A SH e. A H~ if C sseq1 B C C_ anbi1d A A SH e. A H~ if vH B opreq1 C sseq1d bibi12d B B SH e. B H~ if C sseq1 A SH e. A H~ if C C_ anbi2d B B SH e. B H~ if A SH e. A H~ if vH opreq2 C sseq1d bibi12d C C CH e. C H~ if A SH e. A H~ if sseq2 C C CH e. C H~ if B SH e. B H~ if sseq2 anbi12d C C CH e. C H~ if A SH e. A H~ if vH B SH e. B H~ if opr sseq2 bibi12d helsh A elimel helsh B elimel helch C elimel shlub dedth3h))
+thm (shlej1t () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (SH))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C)))) (A B B vH C opr sstr2 B C shub1t syl5com C B shub2t ancoms jctird A SH e. 3adant1 A C B vH C opr shlubt A SH e. B SH e. C SH e. 3simp1 A SH e. B SH e. C SH e. 3simp3 B C shjclt A SH e. 3adant1 syl3anc sylibd))
+thm (shlej2t () () (-> (/\/\ (e. A (SH)) (e. B (SH)) (e. C (SH))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B)))) (A B C shlej1t imp A C shjcomt B SH e. 3adant2 A B C_ adantr B C shjcomt A SH e. 3adant1 A B C_ adantr 3sstr3d exp))
+thm (shsidm ((x y) (x z) (A x) (y z) (A y) (A z)) ((shsidm.1 (e. A (SH)))) (= (opr A (+H) A) A) (shsidm.1 shsidm.1 x cv y z shsel x cv y cv +v z cv opr A eleq1 shsidm.1 A y cv z cv shaddclt ax-mp syl5bir com12 r19.23aivv sylbi ssriv shsidm.1 shsidm.1 shsub1 eqssi))
+thm (shslub () ((shslub.1 (e. A (SH))) (shslub.2 (e. B (SH))) (shslub.3 (e. C (SH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (+H) B) C)) (shslub.1 shslub.3 shslub.2 shless shslub.3 shslub.2 shscom syl6ss shslub.2 shslub.3 shslub.3 shless shslub.3 shsidm syl6ss sylan9ss shslub.1 shslub.2 shsub1 A A +H B opr C sstr mpan shslub.2 shslub.1 shsub1 shslub.2 shslub.1 shscom sseqtr B A +H B opr C sstr mpan jca impbi))
+thm (shlesb1 () ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (<-> (C_ A B) (= (opr A (+H) B) B)) (B ssid A B C_ biantrur shlesb1.2 shlesb1.1 shlesb1.2 shslub A +H B opr B eqss shlesb1.2 shlesb1.1 shsub1 shlesb1.1 shlesb1.2 shscom sseqtr4 mpbiran2 shlesb1.1 shlesb1.2 shscom B sseq1i bitr2 3bitr))
+thm (shsumval2 ((A x) (B x)) ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (= (opr A (+H) B) (|^| ({e.|} x (SH) (C_ (u. A B) (cv x))))) (A B ssun1 A B u. x SH ssintub sstri B A ssun2 A B u. x SH ssintub sstri pm3.2i shlesb1.1 shlesb1.2 x SH A B u. x cv C_ ssrab shlesb1.1 shlesb1.2 shscl shlesb1.1 shlesb1.2 shunss x cv A +H B opr A B u. sseq2 SH rcla4ev mp2an x SH A B u. x cv C_ rabn0 mpbir x SH A B u. x cv C_ {e.|} shintclt mp2an shslub mpbi x cv A +H B opr A B u. sseq2 SH elrab shlesb1.1 shlesb1.2 shscl shlesb1.1 shlesb1.2 shunss mpbir2an A +H B opr x SH A B u. x cv C_ {e.|} intss1 ax-mp eqssi))
+thm (shsumval3 ((A x) (B x)) ((shlesb1.1 (e. A (SH))) (shlesb1.2 (e. B (SH)))) (= (opr A (+H) B) (` (span) (u. A B))) (shlesb1.1 shlesb1.2 x shsumval2 shlesb1.1 shssi shlesb1.2 shssi unssi A B u. x spanvalt ax-mp eqtr4))
+thm (shmods ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((shmod.1 (e. A (SH))) (shmod.2 (e. B (SH))) (shmod.3 (e. C (SH)))) (-> (C_ A C) (C_ (i^i (opr A (+H) B) C) (opr A (+H) (i^i B C)))) (z cv x cv y cv hvsubaddt shmod.3 z cv shel shmod.1 x cv shel shmod.2 y cv shel syl3an x cv +v y cv opr z cv eqcom syl6bb 3expb shmod.1 shmod.3 shlesb1 biimp z cv -v x cv opr eleq2d shmod.3 shmod.1 z cv x cv shsvs shmod.3 shmod.1 shscom z cv -v x cv opr eleq2i sylib syl5bi z cv -v x cv opr y cv C eleq1 biimpd sylan9 y cv B e. anim2d y cv B C elin syl6ibr exp com13 ancoms anasss sylbird imp z cv x cv +v y cv opr A +H B C i^i opr eleq1 shmod.1 shmod.2 shmod.3 shincl x cv y cv shsva syl5bir exp3a com12 z cv C e. y cv B e. ad2antrl imp syld exp31 r19.23advv shmod.1 shmod.2 z cv x y shsel syl5ib com13 imp3a z cv A +H B opr C elin syl5ib ssrdv))
+thm (shmod () ((shmod.1 (e. A (SH))) (shmod.2 (e. B (SH))) (shmod.3 (e. C (SH)))) (-> (/\ (= (opr A (+H) B) (opr A (vH) B)) (C_ A C)) (C_ (i^i (opr A (vH) B) C) (opr A (vH) (i^i B C)))) (A +H B opr A vH B opr C ineq1 A +H B C i^i opr sseq1d shmod.1 shmod.2 shmod.3 shmods syl5bi imp shmod.1 shmod.2 shmod.3 shincl shslej A vH B opr C i^i A +H B C i^i opr A vH B C i^i opr sstr mpan2 syl))
+thm (sh0let () () (-> (e. A (SH)) (C_ (0H) A)) (A sh0 0v A snssi syl df-ch0 syl5ss))
+thm (ch0let () () (-> (e. A (CH)) (C_ (0H) A)) (A chsh A sh0let syl))
+thm (shle0t () () (-> (e. A (SH)) (<-> (C_ A (0H)) (= A (0H)))) (A sh0let A 0H C_ biantrud A 0H eqss syl6bbr))
+thm (chle0t () () (-> (e. A (CH)) (<-> (C_ A (0H)) (= A (0H)))) (A chsh A shle0t syl))
+thm (chnlen0 () () (-> (e. B (CH)) (-> (-. (C_ A B)) (-. (= A (0H))))) (A 0H B sseq1 B ch0let syl5bir com12 con3d))
+thm (ch0psst () () (-> (e. A (CH)) (<-> (C: (0H) A) (-. (= A (0H))))) (A chle0t negbid A ch0let A 0H C_ -. biantrurd bitr3d 0H A dfpss3 syl6rbbr))
+thm (orthin () () (-> (/\ (e. A (SH)) (e. B (SH))) (-> (C_ A (` (_|_) B)) (= (i^i A B) (0H)))) (B ocin A B i^i sseq2d A _|_ B ` B ssrin _|_ B ` B incom syl6ss syl5bi A SH e. adantl A B shinclt A B i^i sh0let syl jctird A B i^i 0H eqss syl6ibr))
+thm (shne0 ((A x)) ((sh0le.1 (e. A (SH)))) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v))))) (sh0le.1 A shle0t ax-mp negbii A 0H x nss bitr3 x A x cv 0H e. -. df-rex bitr4 x cv elch0 negbii x A birex bitr))
+thm (shs0 () ((sh0le.1 (e. A (SH)))) (= (opr A (+H) (0H)) A) (sh0le.1 h0elsh shsumval3 sh0le.1 A sh0let ax-mp 0H A ssequn2 mpbi span fveq2i sh0le.1 A spanid ax-mp 3eqtr))
+thm (shs00 () ((sh0le.1 (e. A (SH))) (shs00.2 (e. B (SH)))) (<-> (/\ (= A (0H)) (= B (0H))) (= (opr A (+H) B) (0H))) (A 0H B 0H +H opreq12 h0elsh shs0 syl6eq sh0le.1 shs00.2 shsub1 A +H B opr 0H A sseq2 mpbii sh0le.1 A shle0t ax-mp sylib shs00.2 sh0le.1 shsub2 A +H B opr 0H B sseq2 mpbii shs00.2 B shle0t ax-mp sylib jca impbi))
+thm (ch0le () ((ch0le.1 (e. A (CH)))) (C_ (0H) A) (ch0le.1 A ch0let ax-mp))
+thm (chle0 () ((ch0le.1 (e. A (CH)))) (<-> (C_ A (0H)) (= A (0H))) (ch0le.1 A chle0t ax-mp))
+thm (chne0 ((A x)) ((ch0le.1 (e. A (CH)))) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v))))) (ch0le.1 chshi x shne0))
+thm (chocin () ((ch0le.1 (e. A (CH)))) (= (i^i A (` (_|_) A)) (0H)) (ch0le.1 chshi A ocin ax-mp))
+thm (chj0 () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (0H)) A) (ch0le.1 h0elch chjval ch0le.1 ch0le 0H A ssequn2 mpbi _|_ fveq2i _|_ fveq2i ch0le.1 pjococ 3eqtr))
+thm (chm1 () ((ch0le.1 (e. A (CH)))) (= (i^i A (H~)) A) (A H~ inss1 A ssid ch0le.1 chssi ssini eqssi))
+thm (chjcl () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (e. (opr A (vH) B) (CH)) (ch0le.1 chshi chjcl.2 chshi shjcl))
+thm (chslej () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ (opr A (+H) B) (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shslej))
+thm (chsel ((x y) (B x) (B y) (C x) (C y) (x y) (A x) (A y)) ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (e. C (opr A (+H) B)) (E.e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (ch0le.1 chshi chjcl.2 chshi C x y shsel))
+thm (chincl () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (e. (i^i A B) (CH)) (ch0le.1 elisseti chjcl.2 elisseti intpr ch0le.1 chjcl.2 pm3.2i ch0le.1 elisseti chjcl.2 elisseti CH prss mpbi ch0le.1 elisseti B prnz pm3.2i chintcl eqeltrr))
+thm (chsscon3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A))) (ch0le.1 chssi chjcl.2 chssi A B occont mp2an chjcl.2 chssi occl chssi ch0le.1 chssi occl chssi _|_ B ` _|_ A ` occont mp2an ch0le.1 pjococ chjcl.2 pjococ 3sstr3g impbi))
+thm (chsscon1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ (` (_|_) A) B) (C_ (` (_|_) B) A)) (ch0le.1 chocl chjcl.2 chsscon3 ch0le.1 pjococ _|_ B ` sseq2i bitr))
+thm (chsscon2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A (` (_|_) B)) (C_ B (` (_|_) A))) (ch0le.1 chjcl.2 chocl chsscon3 chjcl.2 pjococ _|_ A ` sseq1i bitr))
+thm (chcon2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (= A (` (_|_) B)) (= B (` (_|_) A))) (ch0le.1 chjcl.2 chsscon2 chjcl.2 ch0le.1 chsscon1 anbi12i A _|_ B ` eqss B _|_ A ` eqss 3bitr4))
+thm (chcon3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (= A B) (= (` (_|_) B) (` (_|_) A))) (ch0le.1 chjcl.2 chsscon3 chjcl.2 ch0le.1 chsscon3 anbi12i A B eqss _|_ B ` _|_ A ` eqss 3bitr4))
+thm (chunssj () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ (u. A B) (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shunssj))
+thm (chjcom () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (opr A (vH) B) (opr B (vH) A)) (ch0le.1 chshi chjcl.2 chshi shjcom))
+thm (chub1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ A (opr A (vH) B)) (ch0le.1 chshi chjcl.2 chshi shub1))
+thm (chub2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (C_ A (opr B (vH) A)) (ch0le.1 chjcl.2 chub1 ch0le.1 chjcl.2 chjcom sseqtr))
+thm (chlub () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (ch0le.1 chshi chjcl.2 chshi chlub.1 shlub))
+thm (chlubi () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C)) (ch0le.1 chjcl.2 chlub.1 chlub biimp))
+thm (chlej1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C))) (ch0le.1 chshi chjcl.2 chshi chlub.1 chshi shlej1))
+thm (chlej2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH)))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B))) (ch0le.1 chshi chjcl.2 chshi chlub.1 chshi shlej2))
+thm (chlej12 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chlub.1 (e. C (CH))) (chlej12.4 (e. D (CH)))) (-> (/\ (C_ A B) (C_ C D)) (C_ (opr A (vH) C) (opr B (vH) D))) (ch0le.1 chjcl.2 chlub.1 chlej1 chlub.1 chlej12.4 chjcl.2 chlej2 sylan9ss))
+thm (chlejb1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (C_ A B) (= (opr A (vH) B) B)) (B ssid ch0le.1 chjcl.2 chjcl.2 chlubi mpan2 chjcl.2 ch0le.1 chub2 jctir A vH B opr B eqss sylibr ch0le.1 chjcl.2 chub1 A A vH B opr B sstr2 A vH B opr B eqimss syl5 ax-mp impbi))
+thm (chdmm1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i A B)) (opr (` (_|_) A) (vH) (` (_|_) B))) (ch0le.1 chocl chjcl.2 chocl chub1 ch0le.1 ch0le.1 chocl chjcl.2 chocl chjcl chsscon1 mpbi chjcl.2 chocl ch0le.1 chocl chub2 chjcl.2 ch0le.1 chocl chjcl.2 chocl chjcl chsscon1 mpbi ssini ch0le.1 chocl chjcl.2 chocl chjcl ch0le.1 chjcl.2 chincl chsscon1 mpbi A B inss1 ch0le.1 chjcl.2 chincl ch0le.1 chsscon3 mpbi A B inss2 ch0le.1 chjcl.2 chincl chjcl.2 chsscon3 mpbi ch0le.1 chocl chjcl.2 chocl ch0le.1 chjcl.2 chincl chocl chlubi mp2an eqssi))
+thm (chdmm2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i (` (_|_) A) B)) (opr A (vH) (` (_|_) B))) (ch0le.1 chocl chjcl.2 chdmm1 ch0le.1 pjococ vH _|_ B ` opreq1i eqtr))
+thm (chdmm3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i A (` (_|_) B))) (opr (` (_|_) A) (vH) B)) (ch0le.1 chjcl.2 chocl chdmm1 chjcl.2 pjococ _|_ A ` vH opreq2i eqtr))
+thm (chdmm4 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (i^i (` (_|_) A) (` (_|_) B))) (opr A (vH) B)) (ch0le.1 chjcl.2 chocl chdmm2 chjcl.2 pjococ A vH opreq2i eqtr))
+thm (chdmj1 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr A (vH) B)) (i^i (` (_|_) A) (` (_|_) B))) (ch0le.1 chjcl.2 chdmm4 _|_ fveq2i ch0le.1 chocl chjcl.2 chocl chincl pjococ eqtr3))
+thm (chdmj2 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr (` (_|_) A) (vH) B)) (i^i A (` (_|_) B))) (ch0le.1 chocl chjcl.2 chdmj1 ch0le.1 pjococ _|_ B ` ineq1i eqtr))
+thm (chdmj3 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr A (vH) (` (_|_) B))) (i^i (` (_|_) A) B)) (ch0le.1 chjcl.2 chocl chdmj1 chjcl.2 pjococ _|_ A ` ineq2i eqtr))
+thm (chdmj4 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (= (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (i^i A B)) (ch0le.1 chjcl.2 chocl chdmj2 chjcl.2 pjococ A ineq2i eqtr))
+thm (chnle () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (-. (C_ B A)) (C: A (opr A (vH) B))) (ch0le.1 chjcl.2 chub1 A A vH B opr = -. biantrur chjcl.2 ch0le.1 chlejb1 B vH A opr A eqcom chjcl.2 ch0le.1 chjcom A eqeq2i 3bitr negbii A A vH B opr dfpss2 3bitr4))
+thm (chjass () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH))) (chjass.3 (e. C (CH)))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C))) (_|_ A ` _|_ B ` _|_ C ` inass ch0le.1 chjcl.2 chdmj1 _|_ C ` ineq1i chjcl.2 chjass.3 chdmj1 _|_ A ` ineq2i 3eqtr4 _|_ fveq2i ch0le.1 chjcl.2 chjcl chjass.3 chdmm4 ch0le.1 chjcl.2 chjass.3 chjcl chdmm4 3eqtr3))
+thm (chj00 () ((ch0le.1 (e. A (CH))) (chjcl.2 (e. B (CH)))) (<-> (/\ (= A (0H)) (= B (0H))) (= (opr A (vH) B) (0H))) (A 0H B 0H vH opreq12 h0elch chj0 syl6eq ch0le.1 chjcl.2 chub1 A vH B opr 0H A sseq2 mpbii ch0le.1 chle0 sylib chjcl.2 ch0le.1 chub2 A vH B opr 0H B sseq2 mpbii chjcl.2 chle0 sylib jca impbi))
+thm (chjo () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (` (_|_) A)) (H~)) (_|_ A ` A incom ch0le.1 chocin eqtr _|_ fveq2i ch0le.1 ch0le.1 chdmm2 choc0 3eqtr3))
+thm (chj1 () ((ch0le.1 (e. A (CH)))) (= (opr A (vH) (H~)) (H~)) (ch0le.1 helch chjcl chssi helch ch0le.1 chub2 eqssi))
+thm (chm0 () ((ch0le.1 (e. A (CH)))) (= (i^i A (0H)) (0H)) (A 0H inss2 ch0le.1 ch0le 0H ssid ssini eqssi))
+thm (shjshs () ((shjshs.1 (e. A (SH))) (shjshs.2 (e. B (SH)))) (= (opr A (vH) B) (` (_|_) (` (_|_) (opr A (+H) B)))) (shjshs.1 shjshs.2 A B shjvalt mp2an shjshs.1 shjshs.2 shunss shjshs.1 shssi shjshs.2 shssi unssi shjshs.1 shjshs.2 shscl shssi occon2 ax-mp eqsstr shjshs.1 shjshs.2 shslej shjshs.1 shjshs.2 shscl shssi shjshs.1 shjshs.2 shjcl chssi A +H B opr A vH B opr occont mp2an ax-mp shjshs.1 shjshs.2 shjcl shjshs.1 shjshs.2 shscl shssi occl chsscon1 mpbi eqssi))
+thm (shjshsel () ((shjshs.1 (e. A (SH))) (shjshs.2 (e. B (SH)))) (<-> (e. (opr A (+H) B) (CH)) (= (opr A (+H) B) (opr A (vH) B))) (A +H B opr ococt shjshs.1 shjshs.2 shjshs syl5req shjshs.1 shjshs.2 shjcl A +H B opr A vH B opr CH eleq1 mpbiri impbi))
+thm (chj0t () () (-> (e. A (CH)) (= (opr A (vH) (0H)) A)) (A A CH e. A 0H if vH 0H opreq1 A A CH e. A 0H if = id eqeq12d h0elch A elimel chj0 dedth))
+thm (chslejt () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ (opr A (+H) B) (opr A (vH) B))) (A B shslejt A chsh B chsh syl2an))
+thm (chinclt () () (-> (/\ (e. A (CH)) (e. B (CH))) (e. (i^i A B) (CH))) (A A CH e. A H~ if B ineq1 CH eleq1d B B CH e. B H~ if A CH e. A H~ if ineq2 CH eleq1d helch A elimel helch B elimel chincl dedth2h))
+thm (chsscon3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (C_ (` (_|_) B) (` (_|_) A)))) (A A CH e. A H~ if B sseq1 A A CH e. A H~ if _|_ fveq2 _|_ B ` sseq2d bibi12d B B CH e. B H~ if A CH e. A H~ if sseq2 B B CH e. B H~ if _|_ fveq2 _|_ A CH e. A H~ if ` sseq1d bibi12d helch A elimel helch B elimel chsscon3 dedth2h))
+thm (chsscon1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ (` (_|_) A) B) (C_ (` (_|_) B) A))) (_|_ A ` B chsscon3t A choclt sylan A ococt B CH e. adantr _|_ B ` sseq2d bitrd))
+thm (chsscon2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A (` (_|_) B)) (C_ B (` (_|_) A)))) (A _|_ B ` chsscon3t B choclt sylan2 B ococt A CH e. adantl _|_ A ` sseq1d bitrd))
+thm (chpsscon3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: A B) (C: (` (_|_) B) (` (_|_) A)))) (A B chsscon3t B A chsscon3t ancoms negbid anbi12d A B dfpss3 _|_ B ` _|_ A ` dfpss3 3bitr4g))
+thm (chpsscon1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: (` (_|_) A) B) (C: (` (_|_) B) A))) (_|_ A ` B chpsscon3t A choclt sylan A ococt B CH e. adantr _|_ B ` psseq2d bitrd))
+thm (chpsscon2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C: A (` (_|_) B)) (C: B (` (_|_) A)))) (A _|_ B ` chpsscon3t B choclt sylan2 B ococt A CH e. adantl _|_ A ` psseq1d bitrd))
+thm (chjcomt () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) B) (opr B (vH) A))) (A B shjcomt A chsh B chsh syl2an))
+thm (chub1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ A (opr A (vH) B))) (A B shub1t A chsh B chsh syl2an))
+thm (chub2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (C_ A (opr B (vH) A))) (A B chub1t A B chjcomt sseqtrd))
+thm (chlubt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (<-> (/\ (C_ A C) (C_ B C)) (C_ (opr A (vH) B) C))) (A B C shlubt B chsh syl3an2 A chsh syl3an1))
+thm (chlej1t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (C_ A B) (C_ (opr A (vH) C) (opr B (vH) C)))) (A B C shlej1t A chsh B chsh C chsh syl3an))
+thm (chlej2t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (C_ A B) (C_ (opr C (vH) A) (opr C (vH) B)))) (A B C shlej2t A chsh B chsh C chsh syl3an))
+thm (chlejb1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (= (opr A (vH) B) B))) (A A CH e. A 0H if B sseq1 A A CH e. A 0H if vH B opreq1 B eqeq1d bibi12d B B CH e. B 0H if A CH e. A 0H if sseq2 B B CH e. B 0H if A CH e. A 0H if vH opreq2 B B CH e. B 0H if = id eqeq12d bibi12d h0elch A elimel h0elch B elimel chlejb1 dedth2h))
+thm (chlejb2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (= (opr B (vH) A) B))) (A B chlejb1t A B chjcomt B eqeq1d bitrd))
+thm (chnlet () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (-. (C_ B A)) (C: A (opr A (vH) B)))) (A A CH e. A 0H if B sseq2 negbid A A CH e. A 0H if = id A A CH e. A 0H if vH B opreq1 psseq12d bibi12d B B CH e. B 0H if A CH e. A 0H if sseq1 negbid B B CH e. B 0H if A CH e. A 0H if vH opreq2 A CH e. A 0H if psseq2d bibi12d h0elch A elimel h0elch B elimel chnle dedth2h))
+thm (chabs1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (opr A (vH) (i^i A B)) A)) (A ssid A B inss1 pm3.2i A A B i^i A chlubt A CH e. B CH e. pm3.26 A B chinclt A CH e. B CH e. pm3.26 syl3anc mpbii A B chinclt A A B i^i chub1t syldan eqssd))
+thm (chabs2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (i^i A (opr A (vH) B)) A)) (A B chub1t A ssid jctil A A A vH B opr ssin sylib A A vH B opr inss1 jctil A A vH B opr i^i A eqss sylibr))
+thm (chabs1 () ((chabs.1 (e. A (CH))) (chabs.2 (e. B (CH)))) (= (opr A (vH) (i^i A B)) A) (chabs.1 chabs.2 A B chabs1t mp2an))
+thm (chabs2 () ((chabs.1 (e. A (CH))) (chabs.2 (e. B (CH)))) (= (i^i A (opr A (vH) B)) A) (chabs.1 chabs.2 A B chabs2t mp2an))
+thm (chjidmt () () (-> (e. A (CH)) (= (opr A (vH) A) A)) (A A chabs1t anidms A inidm A vH opreq2i syl5eqr))
+thm (chjidm () ((chjidm.1 (e. A (CH)))) (= (opr A (vH) A) A) (chjidm.1 A chjidmt ax-mp))
+thm (chdmm1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i A B)) (opr (` (_|_) A) (vH) (` (_|_) B)))) (A A CH e. A H~ if B ineq1 _|_ fveq2d A A CH e. A H~ if _|_ fveq2 vH _|_ B ` opreq1d eqeq12d B B CH e. B H~ if A CH e. A H~ if ineq2 _|_ fveq2d B B CH e. B H~ if _|_ fveq2 _|_ A CH e. A H~ if ` vH opreq2d eqeq12d helch A elimel helch B elimel chdmm1 dedth2h))
+thm (chdmm2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i (` (_|_) A) B)) (opr A (vH) (` (_|_) B)))) (_|_ A ` B chdmm1t A choclt sylan A ococt B CH e. adantr vH _|_ B ` opreq1d eqtrd))
+thm (chdmm3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i A (` (_|_) B))) (opr (` (_|_) A) (vH) B))) (A _|_ B ` chdmm1t B choclt sylan2 B ococt A CH e. adantl _|_ A ` vH opreq2d eqtrd))
+thm (chdmm4t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (i^i (` (_|_) A) (` (_|_) B))) (opr A (vH) B))) (A _|_ B ` chdmm2t B choclt sylan2 B ococt A CH e. adantl A vH opreq2d eqtrd))
+thm (chdmj1t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr A (vH) B)) (i^i (` (_|_) A) (` (_|_) B)))) (A B chdmm4t _|_ fveq2d _|_ A ` _|_ B ` chinclt A choclt B choclt syl2an _|_ A ` _|_ B ` i^i ococt syl eqtr3d))
+thm (chdmj2t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr (` (_|_) A) (vH) B)) (i^i A (` (_|_) B)))) (_|_ A ` B chdmj1t A choclt sylan A ococt _|_ B ` ineq1d B CH e. adantr eqtrd))
+thm (chdmj3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr A (vH) (` (_|_) B))) (i^i (` (_|_) A) B))) (A _|_ B ` chdmj1t B choclt sylan2 B ococt A CH e. adantl _|_ A ` ineq2d eqtrd))
+thm (chdmj4t () () (-> (/\ (e. A (CH)) (e. B (CH))) (= (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (i^i A B))) (A _|_ B ` chdmj2t B choclt sylan2 B ococt A CH e. adantl A ineq2d eqtrd))
+thm (chjasst () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C)))) (A A CH e. A H~ if vH B opreq1 vH C opreq1d A A CH e. A H~ if vH B vH C opr opreq1 eqeq12d B B CH e. B H~ if A CH e. A H~ if vH opreq2 vH C opreq1d B B CH e. B H~ if vH C opreq1 A CH e. A H~ if vH opreq2d eqeq12d C C CH e. C H~ if A CH e. A H~ if vH B CH e. B H~ if opr vH opreq2 C C CH e. C H~ if B CH e. B H~ if vH opreq2 A CH e. A H~ if vH opreq2d eqeq12d helch A elimel helch B elimel helch C elimel chjass dedth3h))
+thm (ledi () ((ledi.1 (e. A (CH))) (ledi.2 (e. B (CH))) (ledi.3 (e. C (CH)))) (C_ (opr (i^i A B) (vH) (i^i A C)) (i^i A (opr B (vH) C))) (A B inss1 A C inss1 ledi.1 ledi.2 chincl ledi.1 ledi.3 chincl ledi.1 chlubi mp2an A B inss2 A C inss2 ledi.1 ledi.2 chincl ledi.2 ledi.1 ledi.3 chincl ledi.3 chlej12 mp2an ssini))
+thm (spansn0 () () (= (` (span) ({} (0v))) (0H)) (df-ch0 span fveq2i h0elsh 0H spanid ax-mp eqtr3))
+thm (span0 () () (= (` (span) ({/})) (0H)) (h0elsh shssi 0H 0ss 0H {/} spanss mp2an h0elsh 0H spanid ax-mp sseqtr H~ 0ss {/} spanclt ax-mp span {/} ` sh0let ax-mp eqssi))
+thm (elspan ((A x) (B x)) ((elspan.1 (e. B (V)))) (-> (C_ A (H~)) (<-> (e. B (` (span) A)) (A.e. x (SH) (-> (C_ A (cv x)) (e. B (cv x)))))) (A x spanvalt B eleq2d elspan.1 x SH A x cv C_ elintrab syl6bb))
+thm (spanun ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((spanun.1 (C_ A (H~))) (spanun.2 (C_ B (H~)))) (= (` (span) (u. A B)) (opr (` (span) A) (+H) (` (span) B))) (spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shscl shssi spanun.1 A spanss2 ax-mp spanun.2 B spanss2 ax-mp A span A ` B span B ` unss12 mp2an spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shunss sstri span A ` +H span B ` opr A B u. spanss mp2an spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp shscl span A ` +H span B ` opr spanid ax-mp sseqtr spanun.1 A spanclt ax-mp spanun.2 B spanclt ax-mp x cv z w shsel z span A ` w span B ` x cv z cv +v w cv opr = r2ex bitr y SH A y cv C_ z cv y cv e. -> B y cv C_ w cv y cv e. -> /\ x cv z cv +v w cv opr = r19.27av spanun.1 z visset A y elspan ax-mp spanun.2 w visset B y elspan ax-mp anbi12i y SH A y cv C_ z cv y cv e. -> B y cv C_ w cv y cv e. -> r19.26 bitr4 sylanb A y cv C_ z cv y cv e. B y cv C_ w cv y cv e. prth A y cv B unss syl5ibr y cv z cv w cv shaddclt sylan9r x cv z cv +v w cv opr y cv eleq1 biimprd sylan9 exp42 imp4c r19.20i spanun.1 spanun.2 unssi x visset A B u. y elspan ax-mp sylibr syl z w 19.23aivv sylbi ssriv eqssi))
+thm (sshhococ () ((sshjococ.1 (C_ A (H~))) (sshjococ.2 (C_ B (H~)))) (= (opr A (vH) B) (opr (` (_|_) (` (_|_) A)) (vH) (` (_|_) (` (_|_) B)))) (sshjococ.1 A ococss ax-mp sshjococ.2 B ococss ax-mp A _|_ _|_ A ` ` B _|_ _|_ B ` ` unss12 mp2an sshjococ.1 sshjococ.2 unssi sshjococ.1 occl chocl chssi sshjococ.2 occl chocl chssi unssi occon2 ax-mp sshjococ.1 sshjococ.2 A B sshjvalt mp2an sshjococ.1 occl chocl sshjococ.2 occl chocl chjval 3sstr4 A B ssun1 sshjococ.1 sshjococ.2 unssi A B u. ococss ax-mp sstri sshjococ.1 sshjococ.2 A B sshjvalt mp2an sseqtr4 sshjococ.1 sshjococ.1 sshjococ.2 A B sshjclt mp2an chssi occon2 ax-mp B A ssun2 sshjococ.1 sshjococ.2 unssi A B u. ococss ax-mp sstri sshjococ.1 sshjococ.2 A B sshjvalt mp2an sseqtr4 sshjococ.2 sshjococ.1 sshjococ.2 A B sshjclt mp2an chssi occon2 ax-mp sshjococ.1 occl chocl sshjococ.2 occl chocl sshjococ.1 sshjococ.2 A B sshjclt mp2an chocl chocl chlubi mp2an sshjococ.1 sshjococ.2 A B sshjclt mp2an ococ sseqtr eqssi))
+thm (chne0t ((A x)) () (-> (e. A (CH)) (<-> (-. (= A (0H))) (E.e. x A (-. (= (cv x) (0v)))))) (A A CH e. A 0H if 0H eqeq1 negbid A A CH e. A 0H if x x cv 0v = -. rexeq bibi12d h0elch A elimel x chne0 dedth))
+thm (chsup0 () () (= (` (\/H) ({/})) (0H)) (0H {} 0ss CH 0ss h0elch 0H CH snssi ax-mp {/} 0H {} chsupss mp2an ax-mp h0elch 0H chsupsn ax-mp sseqtr CH 0ss {/} chsupclt ax-mp chle0 mpbi))
+thm (h1deot ((A x) (B x)) ((h1deot.1 (e. B (H~)))) (<-> (e. A (` (_|_) ({} B))) (/\ (e. A (H~)) (= (opr A (.i) B) (0)))) (h1deot.1 B H~ snssi ax-mp B {} A x ocelt ax-mp x B {} A .i x cv opr 0 = df-ral x B elsn A .i x cv opr 0 = imbi1i x bial h1deot.1 elisseti x cv B A .i opreq2 0 eqeq1d ceqsalv 3bitr A H~ e. anbi2i bitr))
+thm (h1det ((A x) (B x)) ((h1deot.1 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (/\ (e. A (H~)) (A.e. x (H~) (-> (= (opr B (.i) (cv x)) (0)) (= (opr A (.i) (cv x)) (0)))))) (h1deot.1 B H~ snssi ax-mp occl chssi _|_ B {} ` A x ocelt ax-mp h1deot.1 x cv h1deot h1deot.1 x cv B orthcom mpan2 pm5.32i bitr A .i x cv opr 0 = imbi1i x cv H~ e. B .i x cv opr 0 = A .i x cv opr 0 = impexp bitr x bial x _|_ B {} ` A .i x cv opr 0 = df-ral x H~ B .i x cv opr 0 = A .i x cv opr 0 = -> df-ral 3bitr4 A H~ e. anbi2i bitr))
+thm (h1did () () (-> (e. A (H~)) (e. A (` (_|_) (` (_|_) ({} A))))) (A H~ snssi A {} ococss syl A H~ _|_ _|_ A {} ` ` snssg mpbird))
+thm (h1dn0 () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (-. (= (` (_|_) (` (_|_) ({} A))) (0H)))) (_|_ _|_ A {} ` ` 0H A eleq2 A h1did syl5bi com12 A elch0 syl6ib con3d imp))
+thm (h1de2 ((A x) (B x)) ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (-> (e. A (` (_|_) (` (_|_) ({} B)))) (= (opr (opr B (.i) B) (.s) A) (opr (opr A (.i) B) (.s) B))) (h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.2 B .i B opr .s A opr A .i B opr .s B opr B his2subt mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.2 hicl mulcom h1de2.2 h1de2.2 hicl h1de2.1 h1de2.2 B .i B opr A B ax-his3 mp3an h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 A .i B opr B B ax-his3 mp3an 3eqtr4 h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.2 hicl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.2 hicl subeq0 mpbir eqtr h1de2.2 A x h1det h1de2.1 mpbiran h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl x cv B .i B opr .s A opr -v A .i B opr .s B opr opr B .i opreq2 0 eqeq1d x cv B .i B opr .s A opr -v A .i B opr .s B opr opr A .i opreq2 0 eqeq1d imbi12d H~ rcla4v ax-mp sylbi h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl h1de2.2 B .i B opr .s A opr -v A .i B opr .s B opr opr B orthcom mp2an h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl hvsubcl h1de2.1 B .i B opr .s A opr -v A .i B opr .s B opr opr A orthcom mp2an 3imtr4g mpi h1de2.2 h1de2.2 hicl h1de2.1 hvmulcl h1de2.1 h1de2.2 hicl h1de2.2 hvmulcl h1de2.1 B .i B opr .s A opr A .i B opr .s B opr A his2subt mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.1 B .i B opr A A ax-his3 mp3an h1de2.2 h1de2.2 hicl h1de2.1 h1de2.1 hicl mulcom eqtr h1de2.1 h1de2.2 hicl h1de2.2 h1de2.1 A .i B opr B A ax-his3 mp3an - opreq12i eqtr2 syl5eq h1de2.1 h1de2.1 hicl h1de2.2 h1de2.2 hicl mulcl h1de2.1 h1de2.2 hicl h1de2.2 h1de2.1 hicl mulcl subeq0 sylib eqcomd h1de2.1 h1de2.2 bcseq sylib))
+thm (h1de2b () ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (-> (-. (= B (0v))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (= A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B)))) (B .i B opr 0 df-ne h1de2.2 B his6t ax-mp negbii bitr h1de2.1 h1de2.2 h1de2 B .i B opr 0 =/= adantl 1 / B .i B opr opr .s opreq2d h1de2.2 h1de2.2 hicl recclz h1de2.2 h1de2.2 hicl h1de2.1 1 / B .i B opr opr B .i B opr A ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl 1cn divcan1z .s A opreq1d eqtr3d h1de2.1 A ax-hvmulid ax-mp syl6eq A _|_ _|_ B {} ` ` e. adantr h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl h1de2.2 1 / B .i B opr opr A .i B opr B ax-hvmulass mp3an23 syl h1de2.2 h1de2.2 hicl recclz h1de2.1 h1de2.2 hicl 1 / B .i B opr opr A .i B opr axmulcom mpan2 syl h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divrecz eqtr4d .s B opreq1d eqtr3d A _|_ _|_ B {} ` ` e. adantr 3eqtr3d exp A A .i B opr / B .i B opr opr .s B opr _|_ _|_ B {} ` ` eleq1 h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divclz h1de2.2 B h1did ax-mp h1de2.2 B H~ snssi ax-mp occl chocl chshi _|_ _|_ B {} ` ` A .i B opr / B .i B opr opr B shmulclt ax-mp mpan2 syl syl5bir com12 impbid sylbir))
+thm (h1de2ctlem ((A x) (B x)) ((h1de2.1 (e. A (H~))) (h1de2.2 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (E.e. x (CC) (= A (opr (cv x) (.s) B)))) (B 0v sneq _|_ fveq2d _|_ fveq2d A eleq2d h1de2.1 elisseti 0v elsnc hsn0elch ococ A eleq2i h1de2.2 B ax-hvmul0 ax-mp A eqeq2i 3bitr4r syl6rbbr 0cn x cv 0 .s B opreq1 A eqeq2d CC rcla4ev mpan syl6bir h1de2.1 h1de2.2 h1de2b x cv A .i B opr / B .i B opr opr .s B opreq1 A eqeq2d CC rcla4ev B .i B opr 0 df-ne h1de2.2 B his6t ax-mp negbii bitr h1de2.1 h1de2.2 hicl h1de2.2 h1de2.2 hicl divclz sylbir sylan exp sylbid pm2.61i A x cv .s B opr _|_ _|_ B {} ` ` eleq1 h1de2.2 B h1did ax-mp h1de2.2 B H~ snssi ax-mp occl chocl chshi _|_ _|_ B {} ` ` x cv B shmulclt ax-mp mpan2 syl5bir com12 r19.23aiv impbi))
+thm (h1de2ct ((A x) (B x)) ((h1de2ct.1 (e. B (H~)))) (<-> (e. A (` (_|_) (` (_|_) ({} B)))) (E.e. x (CC) (= A (opr (cv x) (.s) B)))) (h1de2ct.1 B H~ snssi ax-mp occl chocl A chel A x cv .s B opr H~ eleq1 h1de2ct.1 x cv B ax-hvmulcl mpan2 syl5bir com12 r19.23aiv A A H~ e. A 0v if _|_ _|_ B {} ` ` eleq1 A A H~ e. A 0v if x cv .s B opr eqeq1 x CC birexdv bibi12d ax-hv0cl A elimel h1de2ct.1 x h1de2ctlem dedth pm5.21nii))
+thm (spansn ((x y) (x z) (A x) (y z) (A y) (A z)) ((spansn.1 (e. A (H~)))) (= (` (span) ({} A)) (` (_|_) (` (_|_) ({} A)))) (spansn.1 A H~ snssi ax-mp A {} spanssoc ax-mp spansn.1 x cv z h1de2ct y cv z cv A shmulclt z cv .s A opr y cv x cv eleq1a com12 syl9r exp4a com3r r19.23aiv sylbi imp spansn.1 elisseti y cv snss syl5ibr exp r19.21aiv spansn.1 A H~ snssi ax-mp x visset A {} y elspan ax-mp sylibr ssriv eqssi))
+thm (elspansn ((A x) (B x)) ((spansn.1 (e. A (H~)))) (<-> (e. B (` (span) ({} A))) (E.e. x (CC) (= B (opr (cv x) (.s) A)))) (spansn.1 spansn B eleq2i spansn.1 B x h1de2ct bitr))
+thm (spansnt () () (-> (e. A (H~)) (= (` (span) ({} A)) (` (_|_) (` (_|_) ({} A))))) (A A H~ e. A 0v if sneq span fveq2d A A H~ e. A 0v if sneq _|_ fveq2d _|_ fveq2d eqeq12d ax-hv0cl A elimel spansn dedth))
+thm (spansncht () () (-> (e. A (H~)) (e. (` (span) ({} A)) (CH))) (A spansnt A H~ snssi A {} occlt _|_ A {} ` choclt 3syl eqeltrd))
+thm (spansnsht () () (-> (e. A (H~)) (e. (` (span) ({} A)) (SH))) (A spansncht span A {} ` chsh syl))
+thm (spansnch () ((spansnch.1 (e. A (H~)))) (e. (` (span) ({} A)) (CH)) (spansnch.1 A spansncht ax-mp))
+thm (spansnid () () (-> (e. A (H~)) (e. A (` (span) ({} A)))) (A h1did A spansnt eleqtrrd))
+thm (spansnmul () () (-> (/\ (e. A (H~)) (e. B (CC))) (e. (opr B (.s) A) (` (span) ({} A)))) (span A {} ` B A shmulclt imp an1s A H~ snssi A {} spanclt syl A spansnid jca sylan2 ancoms))
+thm (elspansnclt () () (-> (/\ (e. A (H~)) (e. B (` (span) ({} A)))) (e. B (H~))) (A {} B elspanclt A H~ snssi sylan))
+thm (elspansnt ((A x) (B x)) () (-> (e. A (H~)) (<-> (e. B (` (span) ({} A))) (E.e. x (CC) (= B (opr (cv x) (.s) A))))) (A A H~ e. A 0v if sneq span fveq2d B eleq2d A A H~ e. A 0v if x cv .s opreq2 B eqeq2d x CC birexdv bibi12d ax-hv0cl A elimel B x elspansn dedth))
+thm (elspansn2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (-. (= B (0v)))) (<-> (e. A (` (span) ({} B))) (= A (opr (opr (opr A (.i) B) (/) (opr B (.i) B)) (.s) B)))) (B spansnt A eleq2d A H~ e. B 0v = -. 3ad2ant2 A A H~ e. A 0v if _|_ _|_ B {} ` ` eleq1 A A H~ e. A 0v if = id A A H~ e. A 0v if .i B opreq1 / B .i B opr opreq1d .s B opreq1d eqeq12d bibi12d B 0v = -. imbi2d B B H~ e. B 0v if 0v eqeq1 negbid B B H~ e. B 0v if sneq _|_ fveq2d _|_ fveq2d A H~ e. A 0v if eleq2d B B H~ e. B 0v if A H~ e. A 0v if .i opreq2 B B H~ e. B 0v if .i B opreq1 B B H~ e. B 0v if B H~ e. B 0v if .i opreq2 eqtrd / opreq12d B B H~ e. B 0v if = id .s opreq12d A H~ e. A 0v if eqeq2d bibi12d imbi12d ax-hv0cl A elimel ax-hv0cl B elimel h1de2b dedth2h imp 3impa bitrd))
+thm (spansncol ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\/\ (e. A (H~)) (e. B (CC)) (=/= B (0))) (= (` (span) ({} (opr B (.s) A))) (` (span) ({} A)))) (y cv B A ax-hvmulass 3com13 3expa x cv eqeq2d biimprd y cv B axmulcl ancoms A H~ e. adantll jctild z cv y cv x. B opr .s A opreq1 x cv eqeq2d CC rcla4ev syl6 exp r19.23adv B 0 =/= 3adant3 z cv / B opr B A ax-hvmulass z cv B divclt 3expb A H~ e. adantlr B CC e. B 0 =/= pm3.26 z cv CC e. A H~ e. /\ adantl z cv CC e. A H~ e. pm3.27 B CC e. B 0 =/= /\ adantr syl3anc B z cv divcan1t 3exp com12 imp32 A H~ e. adantlr .s A opreq1d eqtr3d x cv eqeq2d biimprd z cv B divclt 3expb A H~ e. adantlr jctild y cv z cv / B opr .s B .s A opr opreq1 x cv eqeq2d CC rcla4ev syl6 exp43 com4l 3imp r19.23adv impbid B A ax-hvmulcl ancoms B .s A opr x cv y elspansnt syl B 0 =/= 3adant3 A x cv z elspansnt B CC e. B 0 =/= 3ad2ant1 3bitr4d eqrd))
+thm (spansneleqi () () (-> (e. A (H~)) (-> (= (` (span) ({} A)) (` (span) ({} B))) (e. A (` (span) ({} B))))) (span A {} ` span B {} ` A eleq2 A spansnid syl5bi com12))
+thm (spansneleq ((A x) (B x)) () (-> (/\ (e. B (H~)) (-. (= A (0v)))) (-> (e. A (` (span) ({} B))) (= (` (span) ({} A)) (` (span) ({} B))))) (B A x elspansnt A 0v = -. adantr A x cv .s B opr sneq span fveq2d B H~ e. A 0v = -. /\ x cv CC e. ad2antrr x cv 0 .s B opreq1 B ax-hvmul0 sylan9eqr exp A x cv .s B opr 0v eqeq1 biimprd sylan9 con3d exp com23 imp3a x cv 0 df-ne syl6ibr x cv CC e. adantr B x cv spansncol 3exp imp syld exp4b com23 imp43 eqtrd exp32 r19.23adv sylbid))
+thm (spansnsst ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (SH)) (e. B A)) (C_ (` (span) ({} B)) A)) (A H~ B ssel2 A shss sylan B x cv y elspansnt syl A y cv B shmulclt x cv y cv .s B opr A eleq1 biimprd syl9 com23 exp3a com23 imp r19.23adv sylbid ssrdv))
+thm (elspansn3t () () (-> (e. A (SH)) (-> (/\ (e. B A) (e. C (` (span) ({} B)))) (e. C A))) (A B spansnsst C sseld exp imp3a))
+thm (elspansn4t () () (-> (/\ (/\ (e. A (SH)) (e. B (H~))) (/\ (e. C (` (span) ({} B))) (-. (= C (0v))))) (<-> (e. B A) (e. C A))) (A B C elspansn3t exp3a com23 imp B H~ e. adantlr C 0v = -. adantrr A C B elspansn3t exp3a imp B spansnid C 0v = -. C span B {} ` e. ad2antll B C spansneleq imp eleqtrrd syl5 exp4b com24 exp4a com23 imp43 impbid))
+thm (elspansn5t () () (-> (e. A (SH)) (-> (/\ (/\ (e. B (H~)) (-. (e. B A))) (/\ (e. C (` (span) ({} B))) (e. C A))) (= C (0v)))) (A B C elspansn4t biimprd exp32 com34 imp3a exp C 0v = B A e. con1 syl8 com34 imp4c))
+thm (spansnss2t () () (-> (/\ (e. A (SH)) (e. B (H~))) (<-> (e. B A) (C_ (` (span) ({} B)) A))) (A B spansnsst exp B H~ e. adantr span B {} ` A B ssel B spansnid syl5com A SH e. adantl impbid))
+thm (spansnpj () ((spansnpj.1 (C_ A (H~))) (spansnpj.2 (e. B (H~)))) (C_ A (` (_|_) (` (span) ({} (` (` (Proj) (` (_|_) A)) B))))) (spansnpj.1 A ococss ax-mp spansnpj.1 occl chssi spansnpj.1 occl spansnpj.2 pjcli Proj _|_ A ` ` B ` _|_ A ` snssi ax-mp _|_ A ` Proj _|_ A ` ` B ` {} spanss mp2an spansnpj.1 occl chshi _|_ A ` spanid ax-mp sseqtr spansnpj.1 occl spansnpj.2 pjhcli spansnch spansnpj.1 occl chsscon3 mpbi sstri))
+thm (spanunsn ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((spanunsn.1 (e. A (CH))) (spanunsn.2 (e. B (H~)))) (= (` (span) (u. A ({} B))) (` (span) (u. A ({} (` (` (Proj) (` (_|_) A)) B))))) (spanunsn.1 chshi spanunsn.2 B H~ snssi ax-mp B {} spanclt ax-mp x cv y z shsel x cv y cv +v w cv .s B opr opr A +H span Proj _|_ A ` ` B ` {} ` opr eleq1 biimparc v cv y cv +v w cv .s Proj A ` B ` opr opr +v u cv opreq1 y cv +v w cv .s B opr opr eqeq2d u cv w cv .s Proj _|_ A ` ` B ` opr y cv +v w cv .s Proj A ` B ` opr opr +v opreq2 y cv +v w cv .s B opr opr eqeq2d A span Proj _|_ A ` ` B ` {} ` rcla42ev spanunsn.1 chshi A y cv w cv .s Proj A ` B ` opr shaddclt spanunsn.1 spanunsn.2 pjcli spanunsn.1 chshi A w cv Proj A ` B ` shmulclt ax-mp mpan2 sylan2i ax-mp spanunsn.1 chocl spanunsn.2 pjhcli Proj _|_ A ` ` B ` w cv spansnmul mpan y cv A e. adantl jca spanunsn.1 spanunsn.2 pjhcli spanunsn.1 chocl spanunsn.2 pjhcli w cv Proj A ` B ` Proj _|_ A ` ` B ` ax-hvdistr1 mp3an23 spanunsn.1 spanunsn.2 pjpj w cv .s opreq2i syl5eq y cv A e. adantl y cv +v opreq2d y cv w cv .s Proj A ` B ` opr w cv .s Proj _|_ A ` ` B ` opr ax-hvass 3expb spanunsn.1 y cv chel spanunsn.1 spanunsn.2 pjhcli w cv Proj A ` B ` ax-hvmulcl mpan2 spanunsn.1 chocl spanunsn.2 pjhcli w cv Proj _|_ A ` ` B ` ax-hvmulcl mpan2 jca syl2an eqtr4d sylanc spanunsn.1 chshi spanunsn.1 chocl spanunsn.2 pjhcli Proj _|_ A ` ` B ` H~ snssi ax-mp Proj _|_ A ` ` B ` {} spanclt ax-mp y cv +v w cv .s B opr opr v u shsel sylibr z cv w cv .s B opr y cv +v opreq2 x cv eqeq2d biimpa syl2an exp43 r19.23adv spanunsn.2 z cv w elspansn syl5ib r19.23adv r19.23aiv sylbi spanunsn.1 chshi spanunsn.1 chocl spanunsn.2 pjhcli Proj _|_ A ` ` B ` H~ snssi ax-mp Proj _|_ A ` ` B ` {} spanclt ax-mp x cv y z shsel x cv y cv +v w cv .s Proj _|_ A ` ` B ` opr opr A +H span B {} ` opr eleq1 biimparc v cv w cv -u .s Proj A ` B ` opr +v y cv opr +v u cv opreq1 y cv +v w cv .s Proj _|_ A ` ` B ` opr opr eqeq2d u cv w cv .s B opr w cv -u .s Proj A ` B ` opr +v y cv opr +v opreq2 y cv +v w cv .s Proj _|_ A ` ` B ` opr opr eqeq2d A span B {} ` rcla42ev spanunsn.1 chshi A w cv -u .s Proj A ` B ` opr y cv shaddclt w cv negclt spanunsn.1 spanunsn.2 pjcli spanunsn.1 chshi A w cv -u Proj A ` B ` shmulclt ax-mp mpan2 syl sylani ax-mp ancoms spanunsn.2 B w cv spansnmul mpan y cv A e. adantl jca spanunsn.1 spanunsn.2 pjhcli w cv Proj A ` B ` hvm1negt mpan2 w cv .s Proj A ` B ` opr +v opreq2d spanunsn.1 spanunsn.2 pjhcli w cv Proj A ` B ` ax-hvmulcl mpan2 w cv .s Proj A ` B ` opr hvnegidt syl w cv .s Proj A ` B ` opr w cv -u .s Proj A ` B ` opr ax-hvcom spanunsn.1 spanunsn.2 pjhcli w cv Proj A ` B ` ax-hvmulcl mpan2 w cv negclt spanunsn.1 spanunsn.2 pjhcli w cv -u Proj A ` B ` ax-hvmulcl mpan2 syl sylanc 3eqtr3d y cv A e. adantl +v y cv +v w cv .s Proj _|_ A ` ` B ` opr opr opreq1d y cv w cv .s Proj _|_ A ` ` B ` opr ax-hvaddcl spanunsn.1 y cv chel spanunsn.1 chocl spanunsn.2 pjhcli w cv Proj _|_ A ` ` B ` ax-hvmulcl mpan2 syl2an y cv +v w cv .s Proj _|_ A ` ` B ` opr opr hvaddid2t syl w cv -u .s Proj A ` B ` opr w cv .s Proj A ` B ` opr y cv w cv .s Proj _|_ A ` ` B ` opr hvadd4t w cv negclt spanunsn.1 spanunsn.2 pjhcli w cv -u Proj A ` B ` ax-hvmulcl mpan2 syl spanunsn.1 spanunsn.2 pjhcli w cv Proj A ` B ` ax-hvmulcl mpan2 jca y cv A e. adantl spanunsn.1 y cv chel spanunsn.1 chocl spanunsn.2 pjhcli w cv Proj _|_ A ` ` B ` ax-hvmulcl mpan2 anim12i sylanc 3eqtr3d spanunsn.1 spanunsn.2 pjhcli spanunsn.1 chocl spanunsn.2 pjhcli w cv Proj A ` B ` Proj _|_ A ` ` B ` ax-hvdistr1 mp3an23 spanunsn.1 spanunsn.2 pjpj w cv .s opreq2i syl5eq y cv A e. adantl w cv -u .s Proj A ` B ` opr +v y cv opr +v opreq2d eqtr4d sylanc spanunsn.1 chshi spanunsn.2 B H~ snssi ax-mp B {} spanclt ax-mp y cv +v w cv .s Proj _|_ A ` ` B ` opr opr v u shsel sylibr z cv w cv .s Proj _|_ A ` ` B ` opr y cv +v opreq2 x cv eqeq2d biimpa syl2an exp43 r19.23adv spanunsn.1 chocl spanunsn.2 pjhcli z cv w elspansn syl5ib r19.23adv r19.23aiv sylbi impbi eqri spanunsn.1 chssi spanunsn.2 B H~ snssi ax-mp spanun spanunsn.1 chshi A spanid ax-mp +H span B {} ` opreq1i eqtr spanunsn.1 chssi spanunsn.1 chocl spanunsn.2 pjhcli Proj _|_ A ` ` B ` H~ snssi ax-mp spanun spanunsn.1 chshi A spanid ax-mp +H span Proj _|_ A ` ` B ` {} ` opreq1i eqtr 3eqtr4))
+thm (h1datom ((x y) (A x) (A y) (B x) (B y)) ((h1datom.1 (e. A (CH))) (h1datom.2 (e. B (H~)))) (-> (C_ A (` (_|_) (` (_|_) ({} B)))) (\/ (= A (` (_|_) (` (_|_) ({} B)))) (= A (0H)))) (A _|_ _|_ B {} ` ` x cv ssel x cv y cv .s B opr 0v eqeq1 y cv 0 .s B opreq1 h1datom.2 B ax-hvmul0 ax-mp syl6eq syl5bir con3d y cv 0 df-ne syl6ibr y cv CC e. adantl y cv recclt h1datom.1 chshi A 1 / y cv opr x cv shmulclt ax-mp exp syl x cv y cv .s B opr = adantr x cv y cv .s B opr 1 / y cv opr .s opreq2 h1datom.2 1 / y cv opr y cv B ax-hvmulass mp3an3 y cv recclt y cv CC e. y cv 0 =/= pm3.26 sylanc y cv recclt y cv 1 / y cv opr axmulcom syldan y cv recidt eqtr3d .s B opreq1d eqtr3d h1datom.2 B ax-hvmulid ax-mp syl6eq sylan9eqr A eleq1d sylibd exp31 com23 imp syld com3r exp3a r19.23adv h1datom.2 x cv y h1de2ct syl5ib sylcom r19.23adv h1datom.1 x chne0 syl5ib B A snssi h1datom.2 B H~ snssi ax-mp h1datom.1 chssi occon2 syl h1datom.1 ococ syl6ss syl6 anc2li A _|_ _|_ B {} ` ` eqss syl6ibr con1d orrd))
+thm (h1datomt () () (-> (/\ (e. A (CH)) (e. B (H~))) (-> (C_ A (` (_|_) (` (_|_) ({} B)))) (\/ (= A (` (_|_) (` (_|_) ({} B)))) (= A (0H))))) (A A CH e. A 0H if _|_ _|_ B {} ` ` sseq1 A A CH e. A 0H if _|_ _|_ B {} ` ` eqeq1 A A CH e. A 0H if 0H eqeq1 orbi12d imbi12d B B H~ e. B 0v if sneq _|_ fveq2d _|_ fveq2d A CH e. A 0H if sseq2d B B H~ e. B 0v if sneq _|_ fveq2d _|_ fveq2d A CH e. A 0H if eqeq2d A CH e. A 0H if 0H = orbi1d imbi12d h0elch A elimel ax-hv0cl B elimel h1datom dedth2h))
+thm (hosmvalt ((x y) (f x) (g x) (h x) (S x) (f y) (g y) (h y) (S y) (f g) (f h) (S f) (g h) (S g) (S h) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+P) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (+v) (` T (cv x)))))))) (ax-hilex S x cv ` +v T x cv ` oprex x y x cv H~ e. y cv S x cv ` +v T x cv ` opr = /\ {<,>|} eqid fnopab2 H~ V x y x cv H~ e. y cv S x cv ` +v T x cv ` opr = /\ {<,>|} fnex mp2 f cv S x cv fveq1 +v g cv x cv ` opreq1d y cv eqeq2d x cv H~ e. anbi2d x y biopabdv g cv T x cv fveq1 S x cv ` +v opreq2d y cv eqeq2d x cv H~ e. anbi2d x y biopabdv f g h x y df-hosum f visset x cv f cv H~ H~ feq1 elab g visset x cv g cv H~ H~ feq1 elab anbi12i h cv x y x cv H~ e. y cv f cv x cv ` +v g cv x cv ` opr = /\ {<,>|} = anbi1i f g h bioprabi eqtr4 oprabval2 ax-hilex H~ V S H~ fex ax-mp x cv S H~ H~ feq1 elab3g ax-hilex H~ V T H~ fex ax-mp x cv T H~ H~ feq1 elab3g syl2anbr))
+thm (hodmvalt ((x y) (f x) (g x) (h x) (S x) (f y) (g y) (h y) (S y) (f g) (f h) (S f) (g h) (S g) (S h) (T x) (T y) (T f) (T g) (T h)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (-P) T) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` S (cv x)) (-v) (` T (cv x)))))))) (ax-hilex S x cv ` -v T x cv ` oprex x y x cv H~ e. y cv S x cv ` -v T x cv ` opr = /\ {<,>|} eqid fnopab2 H~ V x y x cv H~ e. y cv S x cv ` -v T x cv ` opr = /\ {<,>|} fnex mp2 f cv S x cv fveq1 -v g cv x cv ` opreq1d y cv eqeq2d x cv H~ e. anbi2d x y biopabdv g cv T x cv fveq1 S x cv ` -v opreq2d y cv eqeq2d x cv H~ e. anbi2d x y biopabdv f g h x y df-hodif f visset x cv f cv H~ H~ feq1 elab g visset x cv g cv H~ H~ feq1 elab anbi12i h cv x y x cv H~ e. y cv f cv x cv ` -v g cv x cv ` opr = /\ {<,>|} = anbi1i f g h bioprabi eqtr4 oprabval2 ax-hilex H~ V S H~ fex ax-mp x cv S H~ H~ feq1 elab3g ax-hilex H~ V T H~ fex ax-mp x cv T H~ H~ feq1 elab3g syl2anbr))
+thm (hosvalt ((x y) (S x) (S y) (T x) (T y) (A x) (A y)) () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (= (` (opr S (+P) T) A) (opr (` S A) (+v) (` T A)))) (S T x y hosmvalt A fveq1d x cv A S fveq2 x cv A T fveq2 +v opreq12d x y x cv H~ e. y cv S x cv ` +v T x cv ` opr = /\ {<,>|} eqid S A ` +v T A ` oprex fvopab4 sylan9eq))
+thm (hodvalt ((x y) (S x) (S y) (T x) (T y) (A x) (A y)) () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (= (` (opr S (-P) T) A) (opr (` S A) (-v) (` T A)))) (S T x y hodmvalt A fveq1d x cv A S fveq2 x cv A T fveq2 -v opreq12d x y x cv H~ e. y cv S x cv ` -v T x cv ` opr = /\ {<,>|} eqid S A ` -v T A ` oprex fvopab4 sylan9eq))
+thm (hosclt () () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (e. (` (opr S (+P) T) A) (H~))) (S T A hosvalt S H~ H~ A ffvrn T H~ H~ A ffvrn anim12i anandirs S A ` T A ` ax-hvaddcl syl eqeltrd))
+thm (hodclt () () (-> (/\ (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (e. A (H~))) (e. (` (opr S (-P) T) A) (H~))) (S T A hodvalt S H~ H~ A ffvrn T H~ H~ A ffvrn anim12i anandirs S A ` T A ` hvsubclt syl eqeltrd))
+thm (cmbrt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (C_H) B) (= A (opr (i^i A B) (vH) (i^i A (` (_|_) B)))))) (x cv A CH eleq1 y cv CH e. anbi1d x cv A = id x cv A y cv ineq1 x cv A _|_ y cv ` ineq1 vH opreq12d eqeq12d anbi12d y cv B CH eleq1 A CH e. anbi2d y cv B A ineq2 y cv B _|_ fveq2 A ineq2d vH opreq12d A eqeq2d anbi12d x y df-cm CH CH brabg A CH e. B CH e. /\ A A B i^i vH A _|_ B ` i^i opr = ibar bitr4d))
+thm (pjoml2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (= (opr A (vH) (i^i (` (_|_) A) B)) B)) (_|_ A ` B inss2 pjoml2.1 pjoml2.1 chocl pjoml2.2 chincl pjoml2.2 chlubi mpan2 pjoml2.1 pjoml2.1 chocl pjoml2.2 chincl chdmj1 B ineq2i B _|_ A ` _|_ _|_ A ` B i^i ` inass B _|_ A ` incom _|_ _|_ A ` B i^i ` ineq1i eqtr3 pjoml2.1 chocl pjoml2.2 chincl chshi _|_ A ` B i^i ocin ax-mp 3eqtr pjoml2.1 pjoml2.1 chocl pjoml2.2 chincl chjcl pjoml2.2 chshi pjoml mpan2 syl))
+thm (pjoml3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ B A) (= (i^i A (opr (` (_|_) A) (vH) B)) B)) (pjoml2.1 chocl pjoml2.2 chocl pjoml2 pjoml2.2 pjoml2.1 chsscon3 _|_ _|_ A ` vH _|_ _|_ A ` ` _|_ B ` i^i opr ` B eqcom pjoml2.1 pjoml2.1 chocl chocl pjoml2.2 chocl chincl chdmj2 pjoml2.1 chocl pjoml2.2 chdmm4 A ineq2i eqtr B eqeq1i pjoml2.2 pjoml2.1 chocl pjoml2.1 chocl chocl pjoml2.2 chocl chincl chjcl chcon2 3bitr3 3imtr4))
+thm (pjoml4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (= (opr A (vH) (i^i B (opr (` (_|_) A) (vH) (` (_|_) B)))) (opr A (vH) B)) (B _|_ A ` vH _|_ B ` opr inss1 pjoml2.2 pjoml2.1 chocl pjoml2.2 chocl chjcl chincl pjoml2.2 pjoml2.1 chlej2 ax-mp pjoml2.1 pjoml2.2 pjoml2.1 chocl pjoml2.2 chocl chjcl chincl chub1 pjoml2.1 pjoml2.2 chdmm1 B ineq1i _|_ A ` vH _|_ B ` opr B incom eqtr A B i^i vH opreq2i A B inss2 pjoml2.1 pjoml2.2 chincl pjoml2.2 pjoml2 ax-mp eqtr3 A B inss1 pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.2 pjoml2.1 chocl pjoml2.2 chocl chjcl chincl chlej1 ax-mp eqsstr3 pjoml2.1 pjoml2.2 pjoml2.1 pjoml2.2 pjoml2.1 chocl pjoml2.2 chocl chjcl chincl chjcl chlubi mp2an eqssi))
+thm (pjoml5 ((A x) (B x)) ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (E.e. x (CH) (/\ (C_ A (` (_|_) (cv x))) (= (opr A (vH) (cv x)) B)))) (pjoml2.1 pjoml2.2 pjoml2 pjoml2.1 pjoml2.2 chocl chub1 pjoml2.1 pjoml2.2 chdmm2 sseqtr4 jctil pjoml2.1 chocl pjoml2.2 chincl x cv _|_ A ` B i^i _|_ fveq2 A sseq2d x cv _|_ A ` B i^i A vH opreq2 B eqeq1d anbi12d CH rcla4ev mpan syl))
+thm (cmbr () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= A (opr (i^i A B) (vH) (i^i A (` (_|_) B))))) (pjoml2.1 pjoml2.2 A B cmbrt mp2an))
+thm (cmcmlem () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (br A (C_H) B) (br B (C_H) A)) (pjoml2.1 pjoml2.2 chdmj4 pjoml2.1 pjoml2.2 chdmj2 vH opreq12i A eqeq2i biimpr _|_ fveq2d pjoml2.1 chocl pjoml2.2 chocl chjcl pjoml2.1 chocl pjoml2.2 chjcl chdmj4 syl6req B ineq1d pjoml2.2 pjoml2.1 chocl chub2 B _|_ A ` vH B opr sseqin2 mpbi _|_ A ` vH _|_ B ` opr ineq2i _|_ A ` vH _|_ B ` opr _|_ A ` vH B opr B inass pjoml2.1 pjoml2.2 chdmm1 B ineq1i 3eqtr4r syl5eq A B i^i vH opreq2d A B inss2 pjoml2.1 pjoml2.2 chincl pjoml2.2 pjoml2 ax-mp A B incom _|_ A ` B incom vH opreq12i 3eqtr3g pjoml2.1 pjoml2.2 cmbr pjoml2.2 pjoml2.1 cmbr 3imtr4))
+thm (cmcm () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br B (C_H) A)) (pjoml2.1 pjoml2.2 cmcmlem pjoml2.2 pjoml2.1 cmcmlem impbi))
+thm (cmcm2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br A (C_H) (` (_|_) B))) (pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.2 chocl chincl chjcom pjoml2.2 pjococ A ineq2i A _|_ B ` i^i vH opreq2i eqtr4 A eqeq2i pjoml2.1 pjoml2.2 cmbr pjoml2.1 pjoml2.2 chocl cmbr 3bitr4))
+thm (cmcm3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br (` (_|_) A) (C_H) B)) (pjoml2.2 pjoml2.1 cmcm2 pjoml2.1 pjoml2.2 cmcm pjoml2.1 chocl pjoml2.2 cmcm 3bitr4))
+thm (cmcm4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (br (` (_|_) A) (C_H) (` (_|_) B))) (pjoml2.1 pjoml2.2 cmcm2 pjoml2.1 pjoml2.2 chocl cmcm3 bitr))
+thm (cmbr2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= A (i^i (opr A (vH) B) (opr A (vH) (` (_|_) B))))) (pjoml2.1 pjoml2.2 cmcm4 pjoml2.1 chocl pjoml2.2 chocl cmbr A A vH B opr A vH _|_ B ` opr i^i eqcom pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 chocl chjcl chincl pjoml2.1 chcon3 bitr pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 chocl chjcl chdmm1 pjoml2.1 pjoml2.2 chdmj1 pjoml2.1 pjoml2.2 chocl chdmj1 vH opreq12i eqtr _|_ A ` eqeq2i bitr2 3bitr))
+thm (cmcmi () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br B (C_H) A) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm mpbi))
+thm (cmcm2i () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br A (C_H) (` (_|_) B)) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm2 mpbi))
+thm (cmcm3i () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (cmcmi.1 (br A (C_H) B))) (br (` (_|_) A) (C_H) B) (cmcmi.1 pjoml2.1 pjoml2.2 cmcm3 mpbi))
+thm (cmbr3 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (= (i^i A (opr (` (_|_) A) (vH) B)) (i^i A B))) (pjoml2.1 pjoml2.2 cmcm pjoml2.2 pjoml2.1 cmbr2 bitr B B vH A opr B vH _|_ A ` opr i^i A ineq2 A B vH A opr B vH _|_ A ` opr inass pjoml2.2 pjoml2.1 chjcom A ineq2i pjoml2.1 pjoml2.2 chabs2 eqtr pjoml2.2 pjoml2.1 chocl chjcom ineq12i eqtr3 syl6req sylbi pjoml2.1 pjoml2.2 chdmm3 A ineq2i A _|_ A _|_ B ` i^i ` incom eqtr3 A B i^i eqeq1i _|_ A _|_ B ` i^i ` A i^i A B i^i vH A _|_ B ` i^i opreq1 sylbi A _|_ B ` inss1 pjoml2.1 pjoml2.2 chocl chincl pjoml2.1 pjoml2 ax-mp pjoml2.1 pjoml2.2 chocl chincl pjoml2.1 pjoml2.2 chocl chincl chocl pjoml2.1 chincl chjcom eqtr3 syl5eq pjoml2.1 pjoml2.2 cmbr sylibr impbi))
+thm (cmbr4 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (<-> (br A (C_H) B) (C_ (i^i A (opr (` (_|_) A) (vH) B)) B)) (pjoml2.1 pjoml2.2 cmbr3 A B inss2 A _|_ A ` vH B opr i^i A B i^i B sseq1 mpbiri A _|_ A ` vH B opr inss1 A _|_ A ` vH B opr i^i B C_ jctl A _|_ A ` vH B opr i^i A B ssin sylib pjoml2.2 pjoml2.1 chocl chub2 B _|_ A ` vH B opr A sslin ax-mp jctir A _|_ A ` vH B opr i^i A B i^i eqss sylibr impbi bitr))
+thm (lecm () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (-> (C_ A B) (br A (C_H) B)) (A B _|_ A ` vH B opr ssinss1 pjoml2.1 pjoml2.2 cmbr4 sylibr))
+thm (lecmi () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH))) (lecmi.1 (C_ A B))) (br A (C_H) B) (lecmi.1 pjoml2.1 pjoml2.2 lecm ax-mp))
+thm (cmj1 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br A (C_H) (opr A (vH) B)) (pjoml2.1 pjoml2.1 pjoml2.2 chjcl pjoml2.1 pjoml2.2 chub1 lecmi))
+thm (cmj2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br B (C_H) (opr A (vH) B)) (pjoml2.2 pjoml2.1 pjoml2.2 chjcl pjoml2.2 pjoml2.1 chub2 lecmi))
+thm (cmm1 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br A (C_H) (i^i A B)) (pjoml2.1 pjoml2.2 chincl pjoml2.1 pjoml2.1 pjoml2.2 chincl pjoml2.1 A B inss1 lecmi cmcmi))
+thm (cmm2 () ((pjoml2.1 (e. A (CH))) (pjoml2.2 (e. B (CH)))) (br B (C_H) (i^i A B)) (pjoml2.2 pjoml2.1 cmm1 B A incom breqtr))
+thm (cmid () ((cmid.1 (e. A (CH)))) (br A (C_H) A) (cmid.1 cmid.1 A ssid lecmi))
+thm (pjoml3t () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ B A) (= (i^i A (opr (` (_|_) A) (vH) B)) B))) (A A CH e. A H~ if B sseq2 A A CH e. A H~ if = id A A CH e. A H~ if _|_ fveq2 vH B opreq1d ineq12d B eqeq1d imbi12d B B CH e. B H~ if A CH e. A H~ if sseq1 B B CH e. B H~ if _|_ A CH e. A H~ if ` vH opreq2 A CH e. A H~ if ineq2d B B CH e. B H~ if = id eqeq12d imbi12d helch A elimel helch B elimel pjoml3 dedth2h))
+thm (fh1 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (i^i A (opr B (vH) C)) (opr (i^i A B) (vH) (i^i A C))) (fh1.1 fh1.2 chincl fh1.1 fh1.3 chincl chjcl fh1.1 fh1.2 fh1.3 chjcl chincl chshi fh1.1 fh1.2 fh1.3 ledi A B vH C opr incom fh1.1 fh1.2 chincl fh1.1 fh1.3 chincl chdmj1 fh1.1 fh1.2 chdmm1 fh1.1 fh1.3 chdmm1 ineq12i eqtr ineq12i B vH C opr A _|_ A ` vH _|_ B ` opr _|_ A ` vH _|_ C ` opr i^i inass fh1.1 fh1.2 fh1.4 cmcm2i fh1.1 fh1.2 chocl cmbr3 mpbi fh1.1 fh1.3 fh1.5 cmcm2i fh1.1 fh1.3 chocl cmbr3 mpbi ineq12i A _|_ A ` vH _|_ B ` opr _|_ A ` vH _|_ C ` opr inindi A _|_ B ` _|_ C ` inindi 3eqtr4 B vH C opr ineq2i B vH C opr A _|_ B ` _|_ C ` i^i in12 3eqtr fh1.2 fh1.3 chdmj1 B vH C opr ineq2i fh1.2 fh1.3 chjcl chocin eqtr3 A ineq2i fh1.1 chm0 eqtr 3eqtr omlsi eqcomi))
+thm (fh2 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (i^i B (opr A (vH) C)) (opr (i^i B A) (vH) (i^i B C))) (fh1.2 fh1.1 chincl fh1.2 fh1.3 chincl chjcl fh1.2 fh1.1 fh1.3 chjcl chincl chshi fh1.2 fh1.1 fh1.3 ledi fh1.2 fh1.1 chincl fh1.2 fh1.3 chincl chdmj1 fh1.2 fh1.1 chdmm1 _|_ B C i^i ` ineq1i eqtr B A vH C opr i^i ineq2i B A vH C opr _|_ B ` vH _|_ A ` opr _|_ B C i^i ` in4 fh1.2 fh1.1 fh1.1 fh1.2 fh1.4 cmcmi cmcm2i fh1.2 fh1.1 chocl cmbr3 mpbi B _|_ A ` incom eqtr A vH C opr _|_ B C i^i ` i^i ineq1i 3eqtr _|_ A ` B A vH C opr _|_ B C i^i ` in4 fh1.1 pjococ vH C opreq1i _|_ A ` ineq2i fh1.1 fh1.3 fh1.5 cmcm3i fh1.1 chocl fh1.3 cmbr3 mpbi eqtr3 B _|_ B C i^i ` i^i ineq1i 3eqtr _|_ A ` C B _|_ B C i^i ` i^i inass C B _|_ B C i^i ` in12 B C _|_ B C i^i ` inass eqtr4 fh1.2 fh1.3 chincl chocin eqtr _|_ A ` ineq2i fh1.1 chocl chm0 eqtr 3eqtr omlsi eqcomi))
+thm (fh3 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (opr A (vH) (i^i B C)) (i^i (opr A (vH) B) (opr A (vH) C))) (fh1.1 chocl fh1.2 chocl fh1.3 chocl fh1.1 chocl fh1.2 fh1.1 fh1.2 fh1.4 cmcm3i cmcm2i fh1.1 chocl fh1.3 fh1.1 fh1.3 fh1.5 cmcm3i cmcm2i fh1 fh1.2 fh1.3 chdmm1 _|_ A ` ineq2i fh1.1 fh1.2 chdmj1 fh1.1 fh1.3 chdmj1 vH opreq12i 3eqtr4r fh1.1 fh1.2 chjcl fh1.1 fh1.3 chjcl chdmm1 fh1.1 fh1.2 fh1.3 chincl chdmj1 3eqtr4 fh1.1 fh1.2 fh1.3 chincl chjcl fh1.1 fh1.2 chjcl fh1.1 fh1.3 chjcl chincl chcon3 mpbir))
+thm (fh4 () ((fh1.1 (e. A (CH))) (fh1.2 (e. B (CH))) (fh1.3 (e. C (CH))) (fh1.4 (br A (C_H) B)) (fh1.5 (br A (C_H) C))) (= (opr B (vH) (i^i A C)) (i^i (opr B (vH) A) (opr B (vH) C))) (fh1.1 chocl fh1.2 chocl fh1.3 chocl fh1.1 chocl fh1.2 fh1.1 fh1.2 fh1.4 cmcm3i cmcm2i fh1.1 chocl fh1.3 fh1.1 fh1.3 fh1.5 cmcm3i cmcm2i fh2 fh1.1 fh1.3 chdmm1 _|_ B ` ineq2i fh1.2 fh1.1 chdmj1 fh1.2 fh1.3 chdmj1 vH opreq12i 3eqtr4r fh1.2 fh1.1 chjcl fh1.2 fh1.3 chjcl chdmm1 fh1.2 fh1.1 fh1.3 chincl chdmj1 3eqtr4 fh1.2 fh1.1 fh1.3 chincl chjcl fh1.2 fh1.1 chjcl fh1.2 fh1.3 chjcl chincl chcon3 mpbir))
+thm (qlax1 () ((qlax1.1 (e. A (CH)))) (= A (` (_|_) (` (_|_) A))) (qlax1.1 ococ eqcomi))
+thm (qlax2 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) B) (opr B (vH) A)) (qlax.1 qlax.2 chjcom))
+thm (qlax3 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH))) (qlax.3 (e. C (CH)))) (= (opr (opr A (vH) B) (vH) C) (opr A (vH) (opr B (vH) C))) (qlax.1 qlax.2 qlax.3 chjass))
+thm (qlax4 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) (opr B (vH) (` (_|_) B))) (opr B (vH) (` (_|_) B))) (qlax.1 chj1 qlax.2 chjo A vH opreq2i qlax.2 chjo 3eqtr4))
+thm (qlax5 () ((qlax.1 (e. A (CH))) (qlax.2 (e. B (CH)))) (= (opr A (vH) (` (_|_) (opr (` (_|_) A) (vH) B))) A) (qlax.1 qlax.2 chdmj2 A vH opreq2i qlax.1 qlax.2 chocl chabs1 eqtr))
+thm (qlaxr1 () ((qlaxr1.1 (e. A (CH))) (qlaxr1.2 (e. B (CH))) (qlaxr1.3 (= A B))) (= B A) (qlaxr1.3 eqcomi))
+thm (qlaxr2 () ((qlaxr2.1 (e. A (CH))) (qlaxr2.2 (e. B (CH))) (qlaxr2.3 (e. C (CH))) (qlaxr2.4 (= A B)) (qlaxr2.5 (= B C))) (= A C) (qlaxr2.4 qlaxr2.5 eqtr))
+thm (qlaxr4 () ((qlaxr4.1 (e. A (CH))) (qlaxr4.2 (e. B (CH))) (qlaxr4.3 (= A B))) (= (` (_|_) A) (` (_|_) B)) (qlaxr4.3 _|_ fveq2i))
+thm (qlaxr5 () ((qlaxr5.1 (e. A (CH))) (qlaxr5.2 (e. B (CH))) (qlaxr5.3 (e. C (CH))) (qlaxr5.4 (= A B))) (= (opr A (vH) C) (opr B (vH) C)) (qlaxr5.4 vH C opreq1i))
+thm (qlaxr3 () ((qlaxr3.1 (e. A (CH))) (qlaxr3.2 (e. B (CH))) (qlaxr3.3 (e. C (CH))) (qlaxr3.4 (= (opr C (vH) (` (_|_) C)) (opr (` (_|_) (opr (` (_|_) A) (vH) (` (_|_) B))) (vH) (` (_|_) (opr A (vH) B)))))) (= A B) (qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl chshi qlaxr3.1 qlaxr3.2 chub1 A vH B opr _|_ A ` vH _|_ B ` opr incom qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 chocl qlaxr3.2 chocl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj1 cmcmi cmcm2i qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj2 cmcmi cmcm2i fh1 qlaxr3.3 chjo qlaxr3.4 eqtr3 choc0 qlaxr3.1 chocl qlaxr3.2 chocl chjcl qlaxr3.1 qlaxr3.2 chjcl chdmm1 3eqtr4 qlaxr3.1 chocl qlaxr3.2 chocl chjcl qlaxr3.1 qlaxr3.2 chjcl chincl h0elch chcon3 mpbir 3eqtr3 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 chocl chincl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 chocl chincl chj00 mpbir pm3.26i omlsi qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl chshi qlaxr3.2 qlaxr3.1 chub2 A vH B opr _|_ A ` vH _|_ B ` opr incom qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 chocl qlaxr3.2 chocl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.1 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj1 cmcmi cmcm2i qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 qlaxr3.2 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 qlaxr3.2 cmj2 cmcmi cmcm2i fh1 qlaxr3.3 chjo qlaxr3.4 eqtr3 choc0 qlaxr3.1 chocl qlaxr3.2 chocl chjcl qlaxr3.1 qlaxr3.2 chjcl chdmm1 3eqtr4 qlaxr3.1 chocl qlaxr3.2 chocl chjcl qlaxr3.1 qlaxr3.2 chjcl chincl h0elch chcon3 mpbir 3eqtr3 qlaxr3.1 qlaxr3.2 chjcl qlaxr3.1 chocl chincl qlaxr3.1 qlaxr3.2 chjcl qlaxr3.2 chocl chincl chj00 mpbir pm3.27i omlsi eqtr4))
+thm (osumlem1 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (/\ (/\ (/\ (e. C (H~)) (e. D (H~))) (e. R (H~))) (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv z) (H~))))) (osumlem1.4 R C +v D opr H~ eleq1 C D ax-hvaddcl syl5bir com12 imdistani osumlem1.1 C chel osumlem1.2 D chel anim12i sylan z cv x cv +v y cv opr H~ eleq1 x cv y cv ax-hvaddcl syl5bir com12 imdistani osumlem1.1 x cv chel osumlem1.1 chocl y cv chel anim12i sylan anim12i sylbi))
+thm (osumlem2 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (= (opr (opr (` (norm) (opr C (-v) (cv x))) (^) (2)) (+) (opr (` (norm) (opr D (-v) (cv y))) (^) (2))) (opr (` (norm) (opr R (-v) (cv z))) (^) (2)))) (osumlem1.4 C A e. D B e. /\ R C +v D opr = pm3.27 x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = pm3.27 -v opreqan12d osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 C D x cv y cv hvsub4t R H~ e. adantlr z cv H~ e. adantrr syl eqtrd norm fveq2d ^ 2 opreq1d C -v x cv opr D -v y cv opr normpytht osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 C x cv hvsubclt D y cv hvsubclt anim12i an4s R H~ e. adantlr z cv H~ e. adantrr syl osumlem1.1 chshi A C -v x cv opr D -v y cv opr shocorth ax-mp osumlem1.1 chshi A C x cv shsubclt ax-mp osumlem1.1 chocl chshi _|_ A ` D y cv shsubclt ax-mp osumlem1.3 D sseli sylan syl2an an4s R C +v D opr = adantlr z cv x cv +v y cv opr = adantrr sylc eqtr2d sylbi))
+thm (osumlem3 () ((osumlem1.1 (e. A (CH))) (osumlem1.2 (e. B (CH))) (osumlem1.3 (C_ B (` (_|_) A))) (osumlem1.4 (<-> ph (/\ (/\ (/\ (e. C A) (e. D B)) (= R (opr C (+v) D))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y)))))))) (-> ph (br (` (norm) (opr D (-v) (cv y))) (<_) (` (norm) (opr R (-v) (cv z))))) (osumlem1.4 osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 C x cv hvsubclt y cv H~ e. adantrr D H~ e. adantlr z cv H~ e. adantrr R H~ e. adantlr C -v x cv opr normclt syl norm C -v x cv opr ` sqege0t 3syl osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 0re C H~ e. D H~ e. /\ R H~ e. /\ x cv H~ e. y cv H~ e. /\ z cv H~ e. /\ /\ a1i C x cv hvsubclt y cv H~ e. adantrr D H~ e. adantlr z cv H~ e. adantrr R H~ e. adantlr C -v x cv opr normclt norm C -v x cv opr ` sqreclt 3syl D y cv hvsubclt x cv H~ e. adantrl C H~ e. adantll z cv H~ e. adantrr R H~ e. adantlr D -v y cv opr normclt norm D -v y cv opr ` sqreclt 3syl 3jca 0 norm C -v x cv opr ` ^ 2 opr norm D -v y cv opr ` ^ 2 opr leadd1t 3syl mpbid osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 D y cv hvsubclt x cv H~ e. adantrl C H~ e. adantll z cv H~ e. adantrr R H~ e. adantlr D -v y cv opr normclt norm D -v y cv opr ` sqreclt 3syl recnd norm D -v y cv opr ` ^ 2 opr addid2t 3syl osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem2 3brtr3d osumlem1.1 osumlem1.2 osumlem1.3 C A e. D B e. /\ R C +v D opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 D y cv hvsubclt x cv H~ e. adantrl C H~ e. adantll z cv H~ e. adantrr R H~ e. adantlr R z cv hvsubclt x cv H~ e. y cv H~ e. /\ adantrl C H~ e. D H~ e. /\ adantll jca norm D -v y cv opr ` norm R -v z cv opr ` le2sqet D -v y cv opr normclt R -v z cv opr normclt anim12i D -v y cv opr normge0t R -v z cv opr normge0t anim12i sylc 3syl mpbird sylbi))
+thm (osumlem4 ((u v) (u w) (A u) (v w) (A v) (A w) (B u) (B v) (B w) (F u) (F v) (F w) (G u) (G v) (G w) (H u) (H v) (H w) (u x) (v x) (w x) (u y) (v y) (w y) (u z) (v z) (w z)) ((osumlem4.1 (e. A (CH))) (osumlem4.2 (e. B (CH))) (osumlem4.3 (C_ B (` (_|_) A))) (osumlem4.4 (e. G (V))) (osumlem4.5 (e. H (V)))) (-> (/\ (/\ (/\ (:--> F (NN) A) (:--> G (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w)))))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (-> (br H (~~>v) (cv z)) (br G (~~>v) (cv y)))) (F NN A w cv ffvrn expcom G NN B w cv ffvrn expcom anim12d osumlem4.1 osumlem4.2 osumlem4.3 F w cv ` A e. G w cv ` B e. /\ H w cv ` F w cv ` +v G w cv ` opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem3 u cv RR e. adantr osumlem4.1 osumlem4.2 osumlem4.3 F w cv ` A e. G w cv ` B e. /\ H w cv ` F w cv ` +v G w cv ` opr = /\ x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ /\ pm4.2 osumlem1 norm G w cv ` -v y cv opr ` norm H w cv ` -v z cv opr ` u cv lelttrt 3exp G w cv ` y cv hvsubclt G w cv ` -v y cv opr normclt syl x cv H~ e. adantrl F w cv ` H~ e. adantll z cv H~ e. adantrr H w cv ` H~ e. adantlr H w cv ` z cv hvsubclt H w cv ` -v z cv opr normclt syl x cv H~ e. y cv H~ e. /\ adantrl F w cv ` H~ e. G w cv ` H~ e. /\ adantll sylc syl imp mpand exp41 com24 syl6 com4l imp41 norm H w cv ` -v z cv opr ` < u cv br norm G w cv ` -v y cv opr ` < u cv br v cv <_ w cv br imim2 syl6 r19.20dva exp31 com24 imp41 w NN v cv <_ w cv br norm H w cv ` -v z cv opr ` < u cv br -> v cv <_ w cv br norm G w cv ` -v y cv opr ` < u cv br -> r19.20 syl v NN r19.22sdv 0 < u cv br imim2d r19.20dva osumlem4.5 z visset u v w hlimconv syl5 osumlem4.2 chssi G NN B H~ fss mpan2 F NN A :--> w NN H w cv ` F w cv ` +v G w cv ` opr = A.e. ad2antlr osumlem4.1 chshi A shocss ax-mp y cv sseli x cv A e. z cv x cv +v y cv opr = ad2antlr anim12i jctild osumlem4.4 y visset u v w hlim syl6ibr))
+thm (osumlem5 ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B w) (B f) (B g) (H x) (H y) (H w) (H f) (H g)) ((osumlem5.1 (e. A (CH))) (osumlem5.2 (e. B (CH)))) (-> (:--> H (NN) (opr A (+H) B)) (E. f (E. g (/\ (/\ (:--> (cv f) (NN) A) (:--> (cv g) (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w))))))))) (H NN A +H B opr w cv ffvrn exp osumlem5.1 chshi osumlem5.2 chshi H w cv ` x y shsel syl6ib r19.21aiv nnex osumlem5.1 elisseti x cv f cv w cv ` +v y cv opreq1 H w cv ` eqeq2d y B birexdv ac6 nnex osumlem5.2 elisseti y cv g cv w cv ` f cv w cv ` +v opreq2 H w cv ` eqeq2d ac6 f cv NN A :--> anim2i f cv NN A :--> g cv NN B :--> w NN H w cv ` f cv w cv ` +v g cv w cv ` opr = A.e. anass g biex g f cv NN A :--> g cv NN B :--> w NN H w cv ` f cv w cv ` +v g cv w cv ` opr = A.e. /\ 19.42v bitr sylibr f 19.22i 3syl))
+thm (osumlem6 ((f w) (g w) (A w) (f g) (A f) (A g) (B w) (B f) (B g) (H w) (H f) (H g) (w x) (f x) (g x) (w y) (f y) (g y) (w z) (f z) (g z)) ((osumlem6.1 (e. A (CH))) (osumlem6.2 (e. B (CH))) (osumlem6.3 (C_ B (` (_|_) A))) (osumlem6.4 (e. H (V)))) (-> (/\ (/\ (:--> H (NN) (opr A (+H) B)) (br H (~~>v) (cv z))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (e. (cv y) B)) (osumlem6.1 osumlem6.2 H f g w osumlem5 osumlem6.2 y visset B g cv chlim ax-mp f cv NN A :--> g cv NN B :--> pm3.27 w NN H w cv ` f cv w cv ` +v g cv w cv ` opr = A.e. adantr x cv A e. y cv _|_ A ` e. /\ z cv x cv +v y cv opr = /\ H ~~>v z cv br ad2antll osumlem6.1 osumlem6.2 osumlem6.3 g visset osumlem6.4 f cv w x y z osumlem4 imp sylanc exp31 com23 f g 19.23aivv syl imp31))
+thm (osumlem7 ((x y) (x z) (h x) (A x) (y z) (h y) (A y) (h z) (A z) (A h) (B x) (B y) (B z) (B h)) ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH))) (osumlem7.3 (C_ B (` (_|_) A)))) (e. (opr A (+H) B) (CH)) (A +H B opr h z closedsub osumlem7.1 chshi osumlem7.2 chshi shscl h visset z visset hlimvec osumlem7.1 A z cv x y pjpjtht mpan syl h cv NN A +H B opr :--> adantl osumlem7.1 chshi osumlem7.2 chshi x cv y cv shsva x cv A e. y cv _|_ A ` e. pm3.26 h cv NN A +H B opr :--> h cv ~~>v z cv br /\ z cv x cv +v y cv opr = ad2antrl osumlem7.1 osumlem7.2 osumlem7.3 h visset z x y osumlem6 sylanc z cv x cv +v y cv opr A +H B opr eleq1 h cv NN A +H B opr :--> h cv ~~>v z cv br /\ x cv A e. y cv _|_ A ` e. /\ ad2antrr mpbird exp x y 19.23advv x A y _|_ A ` z cv x cv +v y cv opr = r2ex syl5ib mpd h z gen2 mpbir2an))
+thm (osumlem8 () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (-> (C_ B (` (_|_) A)) (e. (opr A (+H) B) (CH))) (A B _|_ A ` C_ A 0H if +H B opreq1 CH eleq1d B B _|_ A ` C_ B 0H if B _|_ A ` C_ A 0H if +H opreq2 CH eleq1d osumlem7.1 h0elch B _|_ A ` C_ keepel osumlem7.2 h0elch B _|_ A ` C_ keepel A B _|_ A ` C_ A 0H if _|_ fveq2 B sseq2d B B _|_ A ` C_ B 0H if _|_ B _|_ A ` C_ A 0H if ` sseq1 0H B _|_ A ` C_ A 0H if _|_ fveq2 0H sseq2d 0H B _|_ A ` C_ B 0H if _|_ B _|_ A ` C_ A 0H if ` sseq1 h0elch chssi choc0 sseqtr4 elimhyp2v osumlem7 dedth2v))
+thm (osum () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (-> (C_ A (` (_|_) B)) (= (opr A (+H) B) (opr A (vH) B))) (osumlem7.2 osumlem7.1 osumlem8 osumlem7.1 chshi osumlem7.2 chshi shscom CH eleq1i sylibr osumlem7.1 chshi osumlem7.2 chshi shunss osumlem7.1 chssi osumlem7.2 chssi unssi osumlem7.1 chshi osumlem7.2 chshi shscl shssi occon2 ax-mp osumlem7.1 osumlem7.2 chjval eqcomi A +H B opr CH e. a1i A +H B opr ococt sseq12d mpbii syl osumlem7.1 osumlem7.2 chslej jctil A +H B opr A vH B opr eqss sylibr))
+thm (osumcor () ((osumlem7.1 (e. A (CH))) (osumlem7.2 (e. B (CH)))) (= (opr (i^i A B) (+H) (i^i A (` (_|_) B))) (opr (i^i A B) (vH) (i^i A (` (_|_) B)))) (A B inss2 osumlem7.2 osumlem7.1 chocl chub2 sstri osumlem7.1 osumlem7.2 chdmm3 sseqtr4 osumlem7.1 osumlem7.2 chincl osumlem7.1 osumlem7.2 chocl chincl osum ax-mp))
+thm (osumcor2 () ((osumcor2.1 (e. A (CH))) (osumcor2.2 (e. B (CH)))) (-> (br A (C_H) B) (= (opr A (+H) B) (opr A (vH) B))) (osumcor2.1 osumcor2.2 cmcm2 osumcor2.1 osumcor2.2 chocl cmbr4 bitr osumcor2.1 osumcor2.1 chocl osumcor2.2 chocl chjcl chincl osumcor2.2 osum osumcor2.1 chocl osumcor2.2 chocl chjcom A ineq2i vH B opreq1i osumcor2.1 osumcor2.2 chocl osumcor2.1 chocl chjcl chincl osumcor2.2 chjcom eqtr osumcor2.2 osumcor2.1 pjoml4 osumcor2.2 osumcor2.1 chjcom 3eqtr A _|_ A ` vH _|_ B ` opr i^i +H B opr eqeq2i A _|_ A ` vH _|_ B ` opr inss1 osumcor2.1 osumcor2.1 chocl osumcor2.2 chocl chjcl chincl chshi osumcor2.1 chshi osumcor2.2 chshi shless ax-mp A _|_ A ` vH _|_ B ` opr i^i +H B opr A vH B opr A +H B opr sseq1 mpbii sylbi syl sylbi osumcor2.1 osumcor2.2 chslej jctil A +H B opr A vH B opr eqss sylibr))
+thm (spansnj () ((spansnj.1 (e. A (CH))) (spansnj.2 (e. B (H~)))) (= (opr A (+H) (` (span) ({} B))) (opr A (vH) (` (span) ({} B)))) (spansnj.1 chshi spansnj.2 spansnch chshi shjshs spansnj.1 chssi spansnj.1 chocl spansnj.2 pjhcli Proj _|_ A ` ` B ` H~ snssi ax-mp spanun spansnj.1 spansnj.2 spanunsn spansnj.1 chshi A spanid ax-mp +H span Proj _|_ A ` ` B ` {} ` opreq1i spansnj.1 chssi spansnj.2 spansnpj spansnj.1 spansnj.1 chocl spansnj.2 pjhcli spansnch osum ax-mp eqtr2 3eqtr4r spansnj.1 chssi spansnj.2 B H~ snssi ax-mp spanun eqtr spansnj.1 chshi A spanid ax-mp +H span B {} ` opreq1i eqtr2 spansnj.1 spansnj.1 chocl spansnj.2 pjhcli spansnch chjcl eqeltr ococ eqtr2))
+thm (spansnjt () () (-> (/\ (e. A (CH)) (e. B (H~))) (= (opr A (+H) (` (span) ({} B))) (opr A (vH) (` (span) ({} B))))) (A A CH e. A H~ if +H span B {} ` opreq1 A A CH e. A H~ if vH span B {} ` opreq1 eqeq12d B B H~ e. B 0v if sneq span fveq2d A CH e. A H~ if +H opreq2d B B H~ e. B 0v if sneq span fveq2d A CH e. A H~ if vH opreq2d eqeq12d helch A elimel ax-hv0cl B elimel spansnj dedth2h))
+thm (spansnsclt () () (-> (/\ (e. A (CH)) (e. B (H~))) (e. (opr A (+H) (` (span) ({} B))) (CH))) (A B spansnjt A span B {} ` chjclt B spansncht sylan2 eqeltrd))
+thm (spansncv ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((spansncv.1 (e. A (CH))) (spansncv.2 (e. B (CH))) (spansncv.3 (e. C (H~)))) (-> (/\ (C: A B) (C_ B (opr A (vH) (` (span) ({} C))))) (= B (opr A (vH) (` (span) ({} C))))) (A B C: B A vH span C {} ` opr C_ pm3.27 spansncv.1 spansncv.3 spansnch spansncv.2 chlubi A B pssss B A vH span C {} ` opr C_ adantr B A vH span C {} ` opr x cv ssel2 spansncv.1 spansncv.3 spansnj x cv eleq2i spansncv.1 spansncv.3 spansnch x cv y z chsel bitr3 y cv z cv hvpncan2t spansncv.1 y cv chel spansncv.3 spansnch z cv chel syl2an B eleq1d spansncv.2 chshi B y cv +v z cv opr y cv shsubclt ax-mp x cv y cv +v z cv opr B eleq1 biimpac A B y cv ssel2 A B pssss sylan syl2an exp43 com14 imp45 syl5bi imp anandis exp45 imp41 x cv A e. -. adantrr spansncv.3 C z cv spansneleq mpan imp B sseq1d spansncv.2 chshi B z cv spansnsst mpan syl5bi ancoms z cv 0v y cv +v opreq2 spansncv.1 y cv chel y cv ax-hvaddid syl sylan9eqr x cv eqeq2d y cv A x cv eleq1a z cv 0v = adantr sylbid exp com23 imp con3d imp sylan2 exp44 com12 imp41 A B C: x cv B e. /\ adantrl mpd exp43 r19.23aivv sylbi syl imp anandirs exp imp3a x 19.23adv A B x pssnel syl5 exp pm2.43d impcom sylanc eqssd))
+thm (spansncvt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (H~))) (-> (/\ (C: A B) (C_ B (opr A (vH) (` (span) ({} C))))) (= B (opr A (vH) (` (span) ({} C)))))) (A A CH e. A H~ if B psseq1 A A CH e. A H~ if vH span C {} ` opreq1 B sseq2d anbi12d A A CH e. A H~ if vH span C {} ` opreq1 B eqeq2d imbi12d B B CH e. B H~ if A CH e. A H~ if psseq2 B B CH e. B H~ if A CH e. A H~ if vH span C {} ` opr sseq1 anbi12d B B CH e. B H~ if A CH e. A H~ if vH span C {} ` opr eqeq1 imbi12d C C H~ e. C 0v if sneq span fveq2d A CH e. A H~ if vH opreq2d B CH e. B H~ if sseq2d A CH e. A H~ if B CH e. B H~ if C: anbi2d C C H~ e. C 0v if sneq span fveq2d A CH e. A H~ if vH opreq2d B CH e. B H~ if eqeq2d imbi12d helch A elimel helch B elimel ax-hv0cl C elimel spansncv dedth3h))
+thm (5oalem1 () ((5oalem1.1 (e. A (SH))) (5oalem1.2 (e. B (SH))) (5oalem1.3 (e. C (SH))) (5oalem1.4 (e. R (SH)))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R))) (e. (cv v) (opr B (+H) (i^i A (opr C (+H) R))))) (5oalem1.1 5oalem1.3 5oalem1.4 shscl shincl 5oalem1.2 x cv y cv shsva 5oalem1.1 5oalem1.3 5oalem1.4 shscl shincl 5oalem1.2 shscom x cv +v y cv opr eleq2i sylib x cv A e. y cv B e. pm3.26 v cv x cv +v y cv opr = z cv C e. x cv -v z cv opr R e. /\ ad2antll x cv z cv z cv hvaddsub12t 3expb anabsan2 z cv hvsubidt x cv +v opreq2d x cv ax-hvaddid sylan9eqr eqtr3d 5oalem1.1 x cv shel y cv B e. v cv x cv +v y cv opr = ad2antll 5oalem1.3 z cv shel x cv -v z cv opr R e. adantr syl2an 5oalem1.3 5oalem1.4 z cv x cv -v z cv opr shsva x cv A e. y cv B e. /\ v cv x cv +v y cv opr = /\ adantl eqeltrrd jca x cv A C +H R opr elin sylibr x cv A e. y cv B e. pm3.27 v cv x cv +v y cv opr = z cv C e. x cv -v z cv opr R e. /\ ad2antll sylanc v cv x cv +v y cv opr B +H A C +H R opr i^i opr eleq1 x cv A e. y cv B e. /\ z cv C e. x cv -v z cv opr R e. /\ ad2antlr mpbird))
+thm (5oalem2 () ((5oalem2.1 (e. A (SH))) (5oalem2.2 (e. B (SH))) (5oalem2.3 (e. C (SH))) (5oalem2.4 (e. D (SH)))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)))) (e. (opr (cv x) (-v) (cv z)) (i^i (opr A (+H) C) (opr B (+H) D)))) (5oalem2.1 5oalem2.3 x cv z cv shsvs x cv A e. y cv B e. pm3.26 z cv C e. w cv D e. pm3.26 syl2an x cv +v y cv opr z cv +v w cv opr = adantr 5oalem2.4 5oalem2.2 w cv y cv shsvs y cv B e. w cv D e. ancom 5oalem2.2 5oalem2.4 shscom w cv -v y cv opr eleq2i 3imtr4 x cv A e. y cv B e. pm3.27 z cv C e. w cv D e. pm3.27 syl2an x cv +v y cv opr z cv +v w cv opr = adantr x cv +v y cv opr z cv +v w cv opr -v z cv +v y cv opr opreq1 x cv H~ e. y cv H~ e. /\ z cv H~ e. w cv H~ e. /\ /\ adantl x cv H~ e. y cv H~ e. pm3.27 z cv H~ e. anim2i ancoms x cv y cv z cv y cv hvsub4t syldan y cv hvsubidt x cv -v z cv opr +v opreq2d x cv H~ e. z cv H~ e. ad2antlr x cv z cv hvsubclt x cv -v z cv opr ax-hvaddid syl y cv H~ e. adantlr 3eqtrd w cv H~ e. adantrr x cv +v y cv opr z cv +v w cv opr = adantr z cv w cv z cv y cv hvsub4t y cv H~ e. z cv H~ e. w cv H~ e. /\ pm3.27 z cv H~ e. w cv H~ e. pm3.26 y cv H~ e. anim1i ancoms sylanc z cv hvsubidt +v w cv -v y cv opr opreq1d y cv H~ e. w cv H~ e. ad2antrl w cv y cv hvsubclt w cv -v y cv opr hvaddid2t syl ancoms z cv H~ e. adantrl 3eqtrd x cv H~ e. adantll x cv +v y cv opr z cv +v w cv opr = adantr 3eqtr3d B +H D opr eleq1d 5oalem2.1 x cv shel 5oalem2.2 y cv shel anim12i 5oalem2.3 z cv shel 5oalem2.4 w cv shel anim12i anim12i sylan mpbird jca x cv -v z cv opr A +H C opr B +H D opr elin sylibr))
+thm (5oalem3 () ((5oalem3.1 (e. A (SH))) (5oalem3.2 (e. B (SH))) (5oalem3.3 (e. C (SH))) (5oalem3.4 (e. D (SH))) (5oalem3.5 (e. F (SH))) (5oalem3.6 (e. G (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))))) (5oalem3.1 5oalem3.2 5oalem3.5 5oalem3.6 x y f g 5oalem2 5oalem3.3 5oalem3.4 5oalem3.5 5oalem3.6 z w f g 5oalem2 anim12i an4s x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ f cv F e. g cv G e. /\ anandir sylanb 5oalem3.1 5oalem3.5 shscl 5oalem3.2 5oalem3.6 shscl shincl 5oalem3.3 5oalem3.5 shscl 5oalem3.4 5oalem3.6 shscl shincl x cv -v f cv opr z cv -v f cv opr shsvs syl x cv f cv z cv f cv hvsubsub4t anandirs f cv hvsubidt x cv -v z cv opr -v opreq2d x cv H~ e. z cv H~ e. /\ adantl x cv z cv hvsubclt x cv -v z cv opr hvsub0t syl f cv H~ e. adantr 3eqtrd 5oalem3.1 x cv shel y cv B e. adantr 5oalem3.3 z cv shel w cv D e. adantr anim12i 5oalem3.5 f cv shel g cv G e. adantr syl2an A +H F opr B +H G opr i^i +H C +H F opr D +H G opr i^i opr eleq1d x cv +v y cv opr f cv +v g cv opr = z cv +v w cv opr f cv +v g cv opr = /\ adantr mpbid))
+thm (5oalem4 () ((5oalem3.1 (e. A (SH))) (5oalem3.2 (e. B (SH))) (5oalem3.3 (e. C (SH))) (5oalem3.4 (e. D (SH))) (5oalem3.5 (e. F (SH))) (5oalem3.6 (e. G (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G)))))) (5oalem3.1 5oalem3.2 5oalem3.3 5oalem3.4 x y z w 5oalem2 x cv +v y cv opr f cv +v g cv opr z cv +v w cv opr eqtr3t sylan2 f cv F e. g cv G e. /\ adantlr 5oalem3.1 5oalem3.2 5oalem3.3 5oalem3.4 5oalem3.5 5oalem3.6 x y z w f g 5oalem3 jca x cv -v z cv opr A +H C opr B +H D opr i^i A +H F opr B +H G opr i^i +H C +H F opr D +H G opr i^i opr elin sylibr))
+thm (5oalem5 () ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)))) (/\ (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))) (5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.7 5oalem5.8 x y z w v u 5oalem4 f cv F e. g cv G e. /\ v cv R e. u cv S e. /\ pm3.27 x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ /\ anim2i x cv +v y cv opr v cv +v u cv opr = z cv +v w cv opr v cv +v u cv opr = /\ f cv +v g cv opr v cv +v u cv opr = pm3.26 syl2an x cv f cv z cv f cv hvsubsub4t anandirs f cv hvsubidt x cv -v z cv opr -v opreq2d x cv H~ e. z cv H~ e. /\ adantl x cv z cv hvsubclt x cv -v z cv opr hvsub0t syl f cv H~ e. adantr 3eqtrd 5oalem5.1 x cv shel y cv B e. adantr 5oalem5.3 z cv shel w cv D e. adantr anim12i 5oalem5.5 f cv shel g cv G e. adantr syl2an v cv R e. u cv S e. /\ adantrr x cv +v y cv opr v cv +v u cv opr = z cv +v w cv opr v cv +v u cv opr = /\ f cv +v g cv opr v cv +v u cv opr = /\ adantr f cv F e. g cv G e. /\ v cv R e. u cv S e. /\ pm3.26 x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ /\ anim2i x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ f cv F e. g cv G e. /\ anandir sylib f cv F e. g cv G e. /\ v cv R e. u cv S e. /\ pm3.27 x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ /\ adantl jca x cv +v y cv opr v cv +v u cv opr = z cv +v w cv opr v cv +v u cv opr = pm3.26 f cv +v g cv opr v cv +v u cv opr = anim1i x cv +v y cv opr v cv +v u cv opr = z cv +v w cv opr v cv +v u cv opr = pm3.27 f cv +v g cv opr v cv +v u cv opr = anim1i jca anim12i 5oalem5.1 5oalem5.2 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y f g v u 5oalem4 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 z w f g v u 5oalem4 anim12i an4s x cv A e. y cv B e. /\ f cv F e. g cv G e. /\ /\ z cv C e. w cv D e. /\ f cv F e. g cv G e. /\ /\ v cv R e. u cv S e. /\ anandir sylanb 5oalem5.1 5oalem5.5 shscl 5oalem5.2 5oalem5.6 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl 5oalem5.3 5oalem5.5 shscl 5oalem5.4 5oalem5.6 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl x cv -v f cv opr z cv -v f cv opr shsvs 3syl eqeltrrd jca x cv -v z cv opr A +H C opr B +H D opr i^i A +H R opr B +H S opr i^i +H C +H R opr D +H S opr i^i opr i^i A +H F opr B +H G opr i^i A +H R opr B +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i +H C +H F opr D +H G opr i^i C +H R opr D +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i opr elin sylibr))
+thm (5oalem6 () ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))))) (e. (cv h) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))))))) (h cv x cv +v y cv opr v cv +v u cv opr eqeq1 biimpcd h cv z cv +v w cv opr v cv +v u cv opr eqeq1 biimpcd anim12d h cv f cv +v g cv opr v cv +v u cv opr eqeq1 biimpcd anim12d exp3a com3l imp32 x cv A e. y cv B e. /\ z cv C e. w cv D e. /\ /\ f cv F e. g cv G e. /\ v cv R e. u cv S e. /\ /\ /\ anim2i an4s x cv A e. y cv B e. /\ h cv x cv +v y cv opr = z cv C e. w cv D e. /\ h cv z cv +v w cv opr = an4 f cv F e. g cv G e. /\ h cv f cv +v g cv opr = v cv R e. u cv S e. /\ h cv v cv +v u cv opr = an4 syl2anb 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y z w f g v u 5oalem5 syl 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.1 5oalem5.3 shscl 5oalem5.2 5oalem5.4 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl shscl shincl 5oalem5.1 5oalem5.5 shscl 5oalem5.2 5oalem5.6 shscl shincl 5oalem5.1 5oalem5.7 shscl 5oalem5.2 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl 5oalem5.3 5oalem5.5 shscl 5oalem5.4 5oalem5.6 shscl shincl 5oalem5.3 5oalem5.7 shscl 5oalem5.4 5oalem5.8 shscl shincl 5oalem5.5 5oalem5.7 shscl 5oalem5.6 5oalem5.8 shscl shincl shscl shincl shscl shincl x y h z 5oalem1 exp32 imp w cv D e. adantrr h cv z cv +v w cv opr = adantrr f cv F e. g cv G e. /\ h cv f cv +v g cv opr = /\ v cv R e. u cv S e. /\ h cv v cv +v u cv opr = /\ /\ adantr mpd))
+thm (5oalem7 ((A h) (A f) (A g) (A u) (A v) (A w) (A x) (A y) (A z) (f h) (g h) (h u) (h v) (h w) (h x) (h y) (h z) (f g) (f u) (f v) (f w) (f x) (f y) (f z) (g u) (g v) (g w) (g x) (g y) (g z) (u v) (u w) (u x) (u y) (u z) (v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (B h) (B f) (B g) (B u) (B v) (B w) (B x) (B y) (B z) (C h) (C f) (C g) (C u) (C v) (C w) (C x) (C y) (C z) (D h) (D f) (D g) (D u) (D v) (D w) (D x) (D y) (D z) (F h) (F f) (F g) (F u) (F v) (F w) (F x) (F y) (F z) (G h) (G f) (G g) (G u) (G v) (G w) (G x) (G y) (G z) (R h) (R f) (R g) (R u) (R v) (R w) (R x) (R y) (R z) (S h) (S f) (S g) (S u) (S v) (S w) (S x) (S y) (S z)) ((5oalem5.1 (e. A (SH))) (5oalem5.2 (e. B (SH))) (5oalem5.3 (e. C (SH))) (5oalem5.4 (e. D (SH))) (5oalem5.5 (e. F (SH))) (5oalem5.6 (e. G (SH))) (5oalem5.7 (e. R (SH))) (5oalem5.8 (e. S (SH)))) (C_ (i^i (i^i (opr A (+H) B) (opr C (+H) D)) (i^i (opr F (+H) G) (opr R (+H) S))) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))))) (x y f g z w x cv A e. y cv B e. /\ h cv x cv +v y cv opr = /\ z cv C e. w cv D e. /\ h cv z cv +v w cv opr = /\ /\ E. E. v u f cv F e. g cv G e. /\ h cv f cv +v g cv opr = /\ v cv R e. u cv S e. /\ h cv v cv +v u cv opr = /\ /\ E. E. ee4anv z w f g v u x cv A e. y cv B e. /\ h cv x cv +v y cv opr = /\ z cv C e. w cv D e. /\ h cv z cv +v w cv opr = /\ /\ f cv F e. g cv G e. /\ h cv f cv +v g cv opr = /\ v cv R e. u cv S e. /\ h cv v cv +v u cv opr = /\ /\ /\ E. E. exrot4 z w v u x cv A e. y cv B e. /\ h cv x cv +v y cv opr = /\ z cv C e. w cv D e. /\ h cv z cv +v w cv opr = /\ /\ f cv F e. g cv G e. /\ h cv f cv +v g cv opr = /\ v cv R e. u cv S e. /\ h cv v cv +v u cv opr = /\ /\ ee4anv f g bi2ex bitr x y bi2ex h cv A +H B opr C +H D opr i^i F +H G opr R +H S opr i^i elin 5oalem5.1 5oalem5.2 h cv x y shsel x A y B h cv x cv +v y cv opr = r2ex bitr 5oalem5.3 5oalem5.4 h cv z w shsel z C w D h cv z cv +v w cv opr = r2ex bitr anbi12i h cv A +H B opr C +H D opr elin x y z w x cv A e. y cv B e. /\ h cv x cv +v y cv opr = /\ z cv C e. w cv D e. /\ h cv z cv +v w cv opr = /\ ee4anv 3bitr4r 5oalem5.5 5oalem5.6 h cv f g shsel f F g G h cv f cv +v g cv opr = r2ex bitr 5oalem5.7 5oalem5.8 h cv v u shsel v R u S h cv v cv +v u cv opr = r2ex bitr anbi12i h cv F +H G opr R +H S opr elin f g v u f cv F e. g cv G e. /\ h cv f cv +v g cv opr = /\ v cv R e. u cv S e. /\ h cv v cv +v u cv opr = /\ ee4anv 3bitr4r anbi12i bitr4 3bitr4r 5oalem5.1 5oalem5.2 5oalem5.3 5oalem5.4 5oalem5.5 5oalem5.6 5oalem5.7 5oalem5.8 x y h z w f g v u 5oalem6 v u 19.23aivv f g 19.23aivv z w 19.23aivv x y 19.23aivv sylbi ssriv))
+thm (5oa () ((5oa.1 (e. A (CH))) (5oa.2 (e. B (CH))) (5oa.3 (e. C (CH))) (5oa.4 (e. D (CH))) (5oa.5 (e. F (CH))) (5oa.6 (e. G (CH))) (5oa.7 (e. R (CH))) (5oa.8 (e. S (CH))) (5oa.9 (C_ A (` (_|_) B))) (5oa.10 (C_ C (` (_|_) D))) (5oa.11 (C_ F (` (_|_) G))) (5oa.12 (C_ R (` (_|_) S)))) (C_ (i^i (i^i (opr A (vH) B) (opr C (vH) D)) (i^i (opr F (vH) G) (opr R (vH) S))) (opr B (vH) (i^i A (opr C (vH) (i^i (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))))))))) (5oa.9 5oa.1 5oa.2 osum ax-mp 5oa.10 5oa.3 5oa.4 osum ax-mp A +H B opr A vH B opr C +H D opr C vH D opr ineq12 mp2an 5oa.11 5oa.5 5oa.6 osum ax-mp 5oa.12 5oa.7 5oa.8 osum ax-mp F +H G opr F vH G opr R +H S opr R vH S opr ineq12 mp2an A +H B opr C +H D opr i^i A vH B opr C vH D opr i^i F +H G opr R +H S opr i^i F vH G opr R vH S opr i^i ineq12 mp2an eqcomi 5oa.1 chshi 5oa.2 chshi 5oa.3 chshi 5oa.4 chshi 5oa.5 chshi 5oa.6 chshi 5oa.7 chshi 5oa.8 chshi 5oalem7 eqsstr 5oa.2 chshi 5oa.1 chshi 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shscl shincl shslej 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shslej 5oa.1 5oa.3 chslej 5oa.2 5oa.4 chslej A +H C opr A vH C opr B +H D opr B vH D opr ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shslej 5oa.3 5oa.7 chslej 5oa.4 5oa.8 chslej C +H R opr C vH R opr D +H S opr D vH S opr ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.1 5oa.7 chslej 5oa.2 5oa.8 chslej A +H R opr A vH R opr B +H S opr B vH S opr ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri A +H C opr B +H D opr i^i A vH C opr B vH D opr i^i A +H R opr B +H S opr i^i +H C +H R opr D +H S opr i^i opr A vH R opr B vH S opr i^i vH C vH R opr D vH S opr i^i opr ss2in mp2an 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shslej 5oa.3 5oa.5 chslej 5oa.4 5oa.6 chslej C +H F opr C vH F opr D +H G opr D vH G opr ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shslej 5oa.5 5oa.7 chslej 5oa.6 5oa.8 chslej F +H R opr F vH R opr G +H S opr G vH S opr ss2in mp2an 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.3 5oa.7 chslej 5oa.4 5oa.8 chslej C +H R opr C vH R opr D +H S opr D vH S opr ss2in mp2an 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri C +H F opr D +H G opr i^i C vH F opr D vH G opr i^i C +H R opr D +H S opr i^i +H F +H R opr G +H S opr i^i opr C vH R opr D vH S opr i^i vH F vH R opr G vH S opr i^i opr ss2in mp2an 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shlej2 ax-mp 5oa.1 5oa.5 chslej 5oa.2 5oa.6 chslej A +H F opr A vH F opr B +H G opr B vH G opr ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shslej 5oa.5 5oa.7 chslej 5oa.6 5oa.8 chslej F +H R opr F vH R opr G +H S opr G vH S opr ss2in mp2an 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl shlej2 ax-mp 5oa.1 5oa.7 chslej 5oa.2 5oa.8 chslej A +H R opr A vH R opr B +H S opr B vH S opr ss2in mp2an 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shlej1 ax-mp sstri sstri A +H F opr B +H G opr i^i A vH F opr B vH G opr i^i A +H R opr B +H S opr i^i +H F +H R opr G +H S opr i^i opr A vH R opr B vH S opr i^i vH F vH R opr G vH S opr i^i opr ss2in mp2an 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shlej1 ax-mp sstri sstri A +H C opr B +H D opr i^i A +H R opr B +H S opr i^i +H C +H R opr D +H S opr i^i opr i^i A vH C opr B vH D opr i^i A vH R opr B vH S opr i^i vH C vH R opr D vH S opr i^i opr i^i A +H F opr B +H G opr i^i A +H R opr B +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i +H C +H F opr D +H G opr i^i C +H R opr D +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i opr A vH F opr B vH G opr i^i A vH R opr B vH S opr i^i vH F vH R opr G vH S opr i^i opr i^i vH C vH F opr D vH G opr i^i C vH R opr D vH S opr i^i vH F vH R opr G vH S opr i^i opr i^i opr ss2in mp2an 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl 5oa.1 5oa.3 chjcl 5oa.2 5oa.4 chjcl chincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shjcl chincl chshi 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shjshcl shincl 5oa.3 chshi shlej2 ax-mp sstri C +H A +H C opr B +H D opr i^i A +H R opr B +H S opr i^i +H C +H R opr D +H S opr i^i opr i^i A +H F opr B +H G opr i^i A +H R opr B +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i +H C +H F opr D +H G opr i^i C +H R opr D +H S opr i^i +H F +H R opr G +H S opr i^i opr i^i opr i^i opr C vH A vH C opr B vH D opr i^i A vH R opr B vH S opr i^i vH C vH R opr D vH S opr i^i opr i^i A vH F opr B vH G opr i^i A vH R opr B vH S opr i^i vH F vH R opr G vH S opr i^i opr i^i vH C vH F opr D vH G opr i^i C vH R opr D vH S opr i^i vH F vH R opr G vH S opr i^i opr i^i opr i^i opr A sslin ax-mp 5oa.1 chshi 5oa.3 chshi 5oa.1 chshi 5oa.3 chshi shscl 5oa.2 chshi 5oa.4 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.1 chshi 5oa.5 chshi shscl 5oa.2 chshi 5oa.6 chshi shscl shincl 5oa.1 chshi 5oa.7 chshi shscl 5oa.2 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl 5oa.3 chshi 5oa.5 chshi shscl 5oa.4 chshi 5oa.6 chshi shscl shincl 5oa.3 chshi 5oa.7 chshi shscl 5oa.4 chshi 5oa.8 chshi shscl shincl 5oa.5 chshi 5oa.7 chshi shscl 5oa.6 chshi 5oa.8 chshi shscl shincl shscl shincl shscl shincl shscl shincl 5oa.1 chshi 5oa.3 chshi 5oa.1 5oa.3 chjcl 5oa.2 5oa.4 chjcl chincl 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl shjcl chincl chshi 5oa.1 5oa.5 chjcl 5oa.2 5oa.6 chjcl chincl chshi 5oa.1 chshi 5oa.7 chshi shjshcl 5oa.2 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl 5oa.3 5oa.5 chjcl 5oa.4 5oa.6 chjcl chincl chshi 5oa.3 chshi 5oa.7 chshi shjshcl 5oa.4 chshi 5oa.8 chshi shjshcl shincl 5oa.5 chshi 5oa.7 chshi shjshcl 5oa.6 chshi 5oa.8 chshi shjshcl shincl shjshcl shincl shjshcl shincl shjshcl shincl 5oa.2 chshi shlej2 ax-mp sstri sstri))
+thm (3oalem1 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv v) (H~))) (/\ (e. (cv z) (H~)) (e. (cv w) (H~))))) (v cv x cv +v y cv opr H~ eleq1 x cv y cv ax-hvaddcl syl5bir com12 imdistani 3oalem1.1 x cv chel 3oalem1.3 y cv chel anim12i sylan 3oalem1.2 z cv chel 3oalem1.4 w cv chel anim12i v cv z cv +v w cv opr = adantr anim12i))
+thm (3oalem2 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (e. (cv v) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))))) (3oalem1.1 chshi 3oalem1.3 chshi 3oalem1.4 chshi 3oalem1.1 chshi 3oalem1.2 chshi shscl 3oalem1.3 chshi 3oalem1.4 chshi shscl shincl shscl shincl x cv y cv shsva x cv B e. y cv R e. pm3.26 v cv x cv +v y cv opr = z cv C e. w cv S e. /\ v cv z cv +v w cv opr = /\ ad2antll x cv B e. y cv R e. pm3.27 v cv x cv +v y cv opr = z cv C e. w cv S e. /\ v cv z cv +v w cv opr = /\ ad2antll 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 y cv w cv w cv hvaddsub12t 3expb anabsan2 w cv hvsubidt y cv +v opreq2d y cv ax-hvaddid sylan9eqr eqtr3d z cv H~ e. adantrl x cv H~ e. adantll v cv H~ e. adantlr syl 3oalem1.4 chshi 3oalem1.1 chshi 3oalem1.2 chshi shscl 3oalem1.3 chshi 3oalem1.4 chshi shscl shincl w cv y cv -v w cv opr shsva z cv C e. w cv S e. pm3.27 x cv B e. y cv R e. /\ v cv x cv +v y cv opr = /\ v cv z cv +v w cv opr = ad2antrl v cv x cv +v y cv opr z cv +v w cv opr eqtr2t -v x cv +v w cv opr opreq1d z cv C e. w cv S e. /\ adantrl x cv B e. y cv R e. /\ adantll 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 x cv H~ e. y cv H~ e. pm3.26 w cv H~ e. anim1i x cv y cv x cv w cv hvsub4t syldan x cv hvsubidt y cv H~ e. w cv H~ e. ad2antll +v y cv -v w cv opr opreq1d y cv w cv hvsubclt y cv -v w cv opr hvaddid2t syl x cv H~ e. adantll 3eqtrd z cv H~ e. adantrl v cv H~ e. adantlr syl 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem1 z cv w cv x cv w cv hvsub4t x cv H~ e. z cv H~ e. w cv H~ e. /\ pm3.27 z cv H~ e. w cv H~ e. pm3.27 x cv H~ e. anim2i sylanc w cv hvsubidt x cv H~ e. z cv H~ e. ad2antrr z cv -v x cv opr +v opreq2d z cv x cv hvsubclt z cv -v x cv opr ax-hvaddid syl ancoms w cv H~ e. adantrr 3eqtrd y cv H~ e. adantlr v cv H~ e. adantlr syl 3eqtr3d 3oalem1.2 chshi 3oalem1.1 chshi z cv x cv shsvs x cv B e. z cv C e. ancom 3oalem1.1 chshi 3oalem1.2 chshi shscom z cv -v x cv opr eleq2i 3imtr4 x cv B e. y cv R e. pm3.26 v cv x cv +v y cv opr = adantr z cv C e. w cv S e. pm3.26 v cv z cv +v w cv opr = adantr syl2an eqeltrd 3oalem1.3 chshi 3oalem1.4 chshi y cv w cv shsvs x cv B e. y cv R e. pm3.27 v cv x cv +v y cv opr = adantr z cv C e. w cv S e. pm3.27 v cv z cv +v w cv opr = adantr syl2an jca y cv -v w cv opr B +H C opr R +H S opr elin sylibr sylanc eqeltrrd jca y cv R S +H B +H C opr R +H S opr i^i opr elin sylibr sylanc v cv x cv +v y cv opr B +H R S +H B +H C opr R +H S opr i^i opr i^i opr eleq1 x cv B e. y cv R e. /\ z cv C e. w cv S e. /\ v cv z cv +v w cv opr = /\ ad2antlr mpbird))
+thm (3oalem3 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((3oalem1.1 (e. B (CH))) (3oalem1.2 (e. C (CH))) (3oalem1.3 (e. R (CH))) (3oalem1.4 (e. S (CH)))) (C_ (i^i (opr B (+H) R) (opr C (+H) S)) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S)))))) (3oalem1.1 3oalem1.3 v cv x y chsel x B y R v cv x cv +v y cv opr = r2ex bitr 3oalem1.2 3oalem1.4 v cv z w chsel z C w S v cv z cv +v w cv opr = r2ex bitr anbi12i v cv B +H R opr C +H S opr elin x y z w x cv B e. y cv R e. /\ v cv x cv +v y cv opr = /\ z cv C e. w cv S e. /\ v cv z cv +v w cv opr = /\ ee4anv 3bitr4 3oalem1.1 3oalem1.2 3oalem1.3 3oalem1.4 x y v z w 3oalem2 z w 19.23aivv x y 19.23aivv sylbi ssriv))
+thm (3oalem4 () ((3oalem4.3 (= R (i^i (` (_|_) B) (opr B (vH) A))))) (C_ R (` (_|_) B)) (3oalem4.3 _|_ B ` B vH A opr inss1 eqsstr))
+thm (3oalem5 () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (= (i^i (opr B (+H) R) (opr C (+H) S)) (i^i (opr B (vH) R) (opr C (vH) S))) (3oa.4 3oalem4 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.2 osum ax-mp 3oa.2 chshi 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi shscom 3oa.2 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chjcom 3eqtr4 3oa.5 3oalem4 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr 3oa.3 osum ax-mp 3oa.3 chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscom 3oa.3 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chjcom 3eqtr4 ineq12i))
+thm (3oalem6 () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (C_ (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))) (3oa.2 chshi 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shscl shincl shslej 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shslej 3oa.2 3oa.3 chslej B +H C opr B vH C opr R +H S opr ssrin ax-mp 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chslej R +H S opr R vH S opr B vH C opr sslin ax-mp sstri 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl 3oa.2 3oa.3 chjcl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chjcl chincl chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shlej2 ax-mp sstri S +H B +H C opr R +H S opr i^i opr S vH B vH C opr R vH S opr i^i opr R sslin ax-mp 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi 3oa.2 chshi 3oa.3 chshi shscl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr chshi 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chshi shscl shincl shscl shincl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr 3oa.2 3oa.3 chjcl 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr chjcl chincl chjcl chincl chshi 3oa.2 chshi shlej2 ax-mp sstri))
+thm (3oa () ((3oa.1 (e. A (CH))) (3oa.2 (e. B (CH))) (3oa.3 (e. C (CH))) (3oa.4 (= R (i^i (` (_|_) B) (opr B (vH) A)))) (3oa.5 (= S (i^i (` (_|_) C) (opr C (vH) A))))) (C_ (i^i (opr B (vH) R) (opr C (vH) S)) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))) (3oa.1 3oa.2 3oa.3 3oa.4 3oa.5 3oalem5 3oa.2 3oa.3 3oa.4 3oa.2 chocl 3oa.2 3oa.1 chjcl chincl eqeltr 3oa.5 3oa.3 chocl 3oa.3 3oa.1 chjcl chincl eqeltr 3oalem3 3oa.1 3oa.2 3oa.3 3oa.4 3oa.5 3oalem6 sstri eqsstr3))
+thm (pjorth () ((pjorth.1 (e. A (H~))) (pjorth.2 (e. B (H~)))) (-> (e. H (CH)) (= (opr (` (` (Proj) H) A) (.i) (` (` (Proj) (` (_|_) H)) B)) (0))) (H Proj H ` A ` Proj _|_ H ` ` B ` shocorth H chsh pjorth.1 H A axpjclt mpan2 H choclt pjorth.2 _|_ H ` B axpjclt mpan2 syl jca sylc))
+thm (pjch1t () () (-> (e. A (H~)) (= (` (` (Proj) (H~)) A) A)) (A A H~ e. A 0v if H~ eleq1 A A H~ e. A 0v if Proj H~ ` fveq2 A A H~ e. A 0v if = id eqeq12d bibi12d helch ax-hv0cl A elimel pjch dedth ibi))
+thm (pjot () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (Proj) (` (_|_) H)) A) (opr (` (` (Proj) (H~)) A) (-v) (` (` (Proj) H) A)))) (A pjch1t H CH e. adantl H A axpjpjt eqtr2d Proj H~ ` A ` Proj H ` A ` Proj _|_ H ` ` A ` hvsubaddt helch A pjcl H CH e. adantl H A pjhclt _|_ H ` A pjhclt H choclt sylan syl3anc mpbird eqcomd))
+thm (pjch0t () () (-> (e. A (H~)) (= (` (` (Proj) (0H)) A) (0v))) (helch H~ A pjot mpan choc1 Proj fveq2i A fveq1i syl5eqr helch A pjhcl Proj H~ ` A ` hvsubidt syl eqtrd))
+thm (pjidm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~)))) (= (` (` (Proj) H) (` (` (Proj) H) A)) (` (` (Proj) H) A)) (pjidm.1 pjidm.2 pjcli pjidm.1 pjidm.1 pjidm.2 pjhcli pjch mpbi))
+thm (pjadj () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (= (opr (` (` (Proj) H) A) (.i) B) (opr A (.i) (` (` (Proj) H) B))) (pjidm.1 pjadj.3 pjidm.2 H pjorth ax-mp * fveq2i cj0 eqtr pjidm.1 chocl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli his1 pjidm.1 pjidm.2 pjadj.3 H pjorth ax-mp 3eqtr4r Proj H ` A ` .i Proj H ` B ` opr + opreq2i pjidm.1 pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli pjidm.1 chocl pjadj.3 pjhcli Proj H ` A ` Proj H ` B ` Proj _|_ H ` ` B ` his7t mp3an pjidm.1 pjidm.2 pjhcli pjidm.1 chocl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli Proj H ` A ` Proj _|_ H ` ` A ` Proj H ` B ` ax-his2 mp3an 3eqtr4 pjidm.1 pjadj.3 pjpj Proj H ` A ` .i opreq2i pjidm.1 pjidm.2 pjpj .i Proj H ` B ` opreq1i 3eqtr4))
+thm (pjcomp () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (Proj) H) (opr A (+v) B)) A)) (pjidm.1 pjidm.2 pjadj.3 hvaddcl pjcli pjidm.1 chocl pjidm.2 pjadj.3 hvaddcl pjcli pjidm.1 pjidm.2 pjadj.3 hvaddcl pjpj A +v B opr eqid pjidm.1 Proj H ` A +v B opr ` Proj _|_ H ` ` A +v B opr ` A B A +v B opr chocuni mp2ani mpanl12 pm3.26d))
+thm (pjadd () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjadj.3 (e. B (H~)))) (= (` (` (Proj) H) (opr A (+v) B)) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) H) B))) (pjidm.1 pjidm.2 pjpj pjidm.1 pjadj.3 pjpj +v opreq12i pjidm.1 pjidm.2 pjhcli pjidm.1 chocl pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli pjidm.1 chocl pjadj.3 pjhcli hvadd4 eqtr Proj H ` fveq2i pjidm.1 pjidm.2 pjcli pjidm.1 pjadj.3 pjcli pjidm.1 chshi H Proj H ` A ` Proj H ` B ` shaddclt ax-mp mp2an pjidm.1 chocl pjidm.2 pjcli pjidm.1 chocl pjadj.3 pjcli pjidm.1 chocl chshi _|_ H ` Proj _|_ H ` ` A ` Proj _|_ H ` ` B ` shaddclt ax-mp mp2an pjidm.1 pjidm.1 pjidm.2 pjhcli pjidm.1 pjadj.3 pjhcli hvaddcl pjidm.1 chocl pjidm.2 pjhcli pjidm.1 chocl pjadj.3 pjhcli hvaddcl pjcomp mp2an eqtr))
+thm (pjinorm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~)))) (= (opr (` (` (Proj) H) A) (.i) A) (opr (` (norm) (` (` (Proj) H) A)) (^) (2))) (pjidm.1 pjidm.2 pjidm .i A opreq1i pjidm.1 pjidm.1 pjidm.2 pjhcli pjidm.2 pjadj eqtr3 pjidm.1 pjidm.2 pjhcli normsq eqtr4))
+thm (pjmul () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjmul.3 (e. C (CC)))) (= (` (` (Proj) H) (opr C (.s) A)) (opr C (.s) (` (` (Proj) H) A))) (pjidm.1 pjidm.2 pjpj C .s opreq2i pjmul.3 pjidm.1 pjidm.2 pjhcli pjidm.1 chocl pjidm.2 pjhcli hvdistr1 eqtr Proj H ` fveq2i pjmul.3 pjidm.1 pjidm.2 pjcli pjidm.1 chshi H C Proj H ` A ` shmulclt ax-mp mp2an pjmul.3 pjidm.1 chocl pjidm.2 pjcli pjidm.1 chocl chshi _|_ H ` C Proj _|_ H ` ` A ` shmulclt ax-mp mp2an pjidm.1 pjmul.3 pjidm.1 pjidm.2 pjhcli hvmulcl pjmul.3 pjidm.1 chocl pjidm.2 pjhcli hvmulcl pjcomp mp2an eqtr))
+thm (pjsub () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsub.3 (e. B (H~)))) (= (` (` (Proj) H) (opr A (-v) B)) (opr (` (` (Proj) H) A) (-v) (` (` (Proj) H) B))) (pjidm.1 pjidm.2 1cn negcl pjsub.3 hvmulcl pjadd pjidm.1 pjsub.3 1cn negcl pjmul Proj H ` A ` +v opreq2i eqtr pjidm.2 pjsub.3 hvsubval Proj H ` fveq2i pjidm.1 pjidm.2 pjhcli pjidm.1 pjsub.3 pjhcli hvsubval 3eqtr4))
+thm (pjsslem () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (= (opr (` (` (Proj) (` (_|_) H)) A) (-v) (` (` (Proj) (` (_|_) G)) A)) (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A))) (pjidm.1 pjidm.2 H A pjot mp2an pjsslem.1 pjidm.2 G A pjot mp2an -v opreq12i helch pjidm.2 pjcli pjidm.1 pjidm.2 pjhcli helch pjidm.2 pjcli pjsslem.1 pjidm.2 pjhcli hvsubsub4 helch pjidm.2 pjcli Proj H~ ` A ` hvsubidt ax-mp -v Proj H ` A ` -v Proj G ` A ` opr opreq1i 3eqtr pjidm.1 pjidm.2 pjhcli pjsslem.1 pjidm.2 pjhcli hvsubcl hv2neg pjidm.1 pjidm.2 pjhcli pjsslem.1 pjidm.2 pjhcli hvnegdi 3eqtr))
+thm (pjss2 () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H G) (= (` (` (Proj) H) (` (` (Proj) G) A)) (` (` (Proj) H) A))) (pjidm.1 pjsslem.1 chsscon3 pjsslem.1 chocl pjidm.2 pjcli _|_ G ` _|_ H ` Proj _|_ G ` ` A ` ssel mpi sylbi pjidm.1 chocl pjidm.2 pjcli jctil pjidm.1 chocl chshi _|_ H ` Proj _|_ H ` ` A ` Proj _|_ G ` ` A ` shsubclt ax-mp syl pjidm.1 pjidm.2 pjsslem.1 pjsslem _|_ H ` eleq1i pjidm.1 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl pjoc2 bitr pjidm.1 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjsub 0v eqeq1i pjidm.1 pjsslem.1 pjidm.2 pjhcli pjhcli pjidm.1 pjidm.1 pjidm.2 pjhcli pjhcli hvsubeq0 3bitr pjidm.1 pjidm.2 pjidm Proj H ` Proj G ` A ` ` eqeq2i bitr2 sylibr))
+thm (pjssm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H G) (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A))) (pjidm.1 pjidm.2 pjcli H G Proj H ` A ` ssel mpi pjsslem.1 pjidm.2 pjcli jctil pjsslem.1 chshi G Proj G ` A ` Proj H ` A ` shsubclt ax-mp syl pjidm.1 pjsslem.1 chsscon3 pjsslem.1 chocl pjidm.2 pjcli _|_ G ` _|_ H ` Proj _|_ G ` ` A ` ssel mpi pjidm.1 chocl pjidm.2 pjcli jctil pjidm.1 chocl chshi _|_ H ` Proj _|_ H ` ` A ` Proj _|_ G ` ` A ` shsubclt ax-mp syl sylbi pjidm.1 pjidm.2 pjsslem.1 pjsslem _|_ H ` eleq1i sylib jca Proj G ` A ` -v Proj H ` A ` opr G _|_ H ` elin pjsslem.1 pjidm.1 chocl chincl pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl pjch bitr3 sylib pjsslem.1 pjidm.1 chocl chincl pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjsub pjsslem.1 pjidm.1 chocl chincl pjsslem.1 pjidm.2 pjhcli pjhcli pjsslem.1 pjidm.1 chocl chincl pjidm.1 pjidm.2 pjhcli pjhcli hvsubval G _|_ H ` inss1 pjsslem.1 pjidm.1 chocl chincl pjidm.2 pjsslem.1 pjss2 ax-mp pjidm.1 chshi H shococss ax-mp G _|_ H ` inss2 pjsslem.1 pjidm.1 chocl chincl pjidm.1 chocl chsscon3 mpbi sstri pjidm.1 pjidm.2 pjcli sselii pjsslem.1 pjidm.1 chocl chincl pjidm.1 pjidm.2 pjhcli pjoc2 mpbi 1 -u .s opreq2i 1cn negcl 1 -u hvmul0t ax-mp eqtr +v opreq12i pjsslem.1 pjidm.1 chocl chincl pjidm.2 pjhcli Proj G _|_ H ` i^i ` A ` ax-hvaddid ax-mp eqtr 3eqtr syl5reqr))
+thm (pjssge0 () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A))) (Proj G ` A ` -v Proj H ` A ` opr Proj G _|_ H ` i^i ` A ` .i A opreq1 pjsslem.1 pjidm.1 chocl chincl pjidm.2 pjinorm syl6eq pjsslem.1 pjidm.1 chocl chincl pjidm.2 pjhcli normcl sqege0 syl5breqr))
+thm (pjdifnorm () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (<-> (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) (` (` (Proj) G) A)))) (pjsslem.1 pjidm.2 pjhcli normcl sqrecl pjidm.1 pjidm.2 pjhcli normcl sqrecl subge0 pjsslem.1 pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli pjidm.2 Proj G ` A ` Proj H ` A ` A his2subt mp3an pjsslem.1 pjidm.2 pjinorm pjidm.1 pjidm.2 pjinorm - opreq12i eqtr 0 <_ breq2i pjidm.1 pjidm.2 pjhcli Proj H ` A ` normge0t ax-mp pjsslem.1 pjidm.2 pjhcli Proj G ` A ` normge0t ax-mp pjidm.1 pjidm.2 pjhcli normcl pjsslem.1 pjidm.2 pjhcli normcl le2sqe mp2an 3bitr4))
+thm (pjcj () ((pjidm.1 (e. H (CH))) (pjidm.2 (e. A (H~))) (pjsslem.1 (e. G (CH)))) (-> (C_ H (` (_|_) G)) (= (` (` (Proj) (opr H (vH) G)) A) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A)))) (pjidm.1 pjidm.2 pjsslem.1 chocl pjssm A -v opreq2d pjidm.1 pjidm.2 pjhcli pjidm.2 pjsslem.1 chocl pjidm.2 pjhcli Proj H ` A ` A Proj _|_ G ` ` A ` hvaddsub12t mp3an pjsslem.1 pjidm.2 pjpo Proj H ` A ` +v opreq2i pjsslem.1 chocl pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvnegdi A +v opreq2i 3eqtr4 pjidm.2 pjsslem.1 chocl pjidm.2 pjhcli pjidm.1 pjidm.2 pjhcli hvsubcl hvsubval eqtr4 pjidm.1 pjsslem.1 chjcom pjsslem.1 pjidm.1 chdmm4 eqtr4 Proj fveq2i A fveq1i pjsslem.1 chocl pjidm.1 chocl chincl pjidm.2 pjop eqtr 3eqtr4g eqcomd))
+thm (pjadjt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (` (Proj) H) A) (.i) B) (opr A (.i) (` (` (Proj) H) B)))) (A A H~ e. A 0v if Proj H ` fveq2 .i B opreq1d A A H~ e. A 0v if .i Proj H ` B ` opreq1 eqeq12d B B H~ e. B 0v if Proj H ` A H~ e. A 0v if ` .i opreq2 B B H~ e. B 0v if Proj H ` fveq2 A H~ e. A 0v if .i opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjadj dedth2h))
+thm (pjaddt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (+v) B)) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) H) B)))) (A A H~ e. A 0v if +v B opreq1 Proj H ` fveq2d A A H~ e. A 0v if Proj H ` fveq2 +v Proj H ` B ` opreq1d eqeq12d B B H~ e. B 0v if A H~ e. A 0v if +v opreq2 Proj H ` fveq2d B B H~ e. B 0v if Proj H ` fveq2 Proj H ` A H~ e. A 0v if ` +v opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjadd dedth2h))
+thm (pjsubt () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (-v) B)) (opr (` (` (Proj) H) A) (-v) (` (` (Proj) H) B)))) (A A H~ e. A 0v if -v B opreq1 Proj H ` fveq2d A A H~ e. A 0v if Proj H ` fveq2 -v Proj H ` B ` opreq1d eqeq12d B B H~ e. B 0v if A H~ e. A 0v if -v opreq2 Proj H ` fveq2d B B H~ e. B 0v if Proj H ` fveq2 Proj H ` A H~ e. A 0v if ` -v opreq2d eqeq12d pjadjt.1 ax-hv0cl A elimel ax-hv0cl B elimel pjsub dedth2h))
+thm (pjmult () ((pjadjt.1 (e. H (CH)))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (` (Proj) H) (opr A (.s) B)) (opr A (.s) (` (` (Proj) H) B)))) (A A CC e. A 0 if .s B opreq1 Proj H ` fveq2d A A CC e. A 0 if .s Proj H ` B ` opreq1 eqeq12d B B H~ e. B 0v if A CC e. A 0 if .s opreq2 Proj H ` fveq2d B B H~ e. B 0v if Proj H ` fveq2 A CC e. A 0 if .s opreq2d eqeq12d pjadjt.1 ax-hv0cl B elimel 0cn A elimel pjmul dedth2h))
+thm (pjcjt2 () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (` (` (Proj) (opr H (vH) G)) A) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A))))) (H H CH e. H H~ if _|_ G ` sseq1 H H CH e. H H~ if vH G opreq1 Proj fveq2d A fveq1d H H CH e. H H~ if Proj fveq2 A fveq1d +v Proj G ` A ` opreq1d eqeq12d imbi12d G G CH e. G H~ if _|_ fveq2 H CH e. H H~ if sseq2d G G CH e. G H~ if H CH e. H H~ if vH opreq2 Proj fveq2d A fveq1d G G CH e. G H~ if Proj fveq2 A fveq1d Proj H CH e. H H~ if ` A ` +v opreq2d eqeq12d imbi12d A A H~ e. A 0v if Proj H CH e. H H~ if vH G CH e. G H~ if opr ` fveq2 A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 A A H~ e. A 0v if Proj G CH e. G H~ if ` fveq2 +v opreq12d eqeq12d H CH e. H H~ if _|_ G CH e. G H~ if ` C_ imbi2d helch H elimel ax-hv0cl A elimel helch G elimel pjcj dedth3h))
+thm (pj0 () ((pj0.1 (e. H (CH)))) (= (` (` (Proj) H) (0v)) (0v)) (pj0.1 chshi H oc0 ax-mp pj0.1 ax-hv0cl pjoc2 mpbi))
+thm (pjcht () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (` (Proj) H) A) A))) (H H CH e. H H~ if A eleq2 H H CH e. H H~ if Proj fveq2 A fveq1d A eqeq1d bibi12d A A H~ e. A 0v if H CH e. H H~ if eleq1 A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 A A H~ e. A 0v if = id eqeq12d bibi12d helch H elimel ax-hv0cl A elimel pjch dedth2h))
+thm (pjocin () ((pjocin.1 (e. G (CH))) (pjocin.2 (e. H (CH)))) (-> (e. A (` (_|_) (i^i G H))) (e. (` (` (Proj) G) A) (` (_|_) (i^i G H)))) (pjocin.1 pjocin.2 chincl chocl A chel pjocin.1 G A pjpot mpan syl pjocin.1 pjocin.2 chincl chocl A chel pjocin.1 chocl A pjcl G H inss1 pjocin.1 pjocin.2 chincl pjocin.1 chsscon3 mpbi Proj _|_ G ` ` A ` sseli 3syl pjocin.1 pjocin.2 chincl chocl chshi _|_ G H i^i ` A Proj _|_ G ` ` A ` shsubclt ax-mp mpdan eqeltrd))
+thm (pjin () ((pjocin.1 (e. G (CH))) (pjocin.2 (e. H (CH)))) (-> (e. A (i^i G H)) (e. (` (` (Proj) G) A) (i^i G H))) (pjocin.1 G A pjcht mpan biimpd G H inss1 A sseli pjocin.1 A chel syl G H inss1 A sseli sylc G H i^i eleq1d biimprd pm2.43i))
+thm (pjjs ((x y) (x z) (w x) (G x) (y z) (w y) (G y) (w z) (G z) (G w) (H x) (H y) (H z) (H w)) ((pjjs.1 (e. G (CH))) (pjjs.2 (e. H (SH)))) (-> (A.e. x (opr G (vH) H) (e. (` (` (Proj) (` (_|_) G)) (cv x)) H)) (= (opr G (vH) H) (opr G (+H) H))) (x cv w cv Proj _|_ G ` ` fveq2 H eleq1d G vH H opr rcla4v pjjs.1 w cv pjcl Proj _|_ G ` ` w cv ` H e. anim1i pjjs.1 G w cv axpjpjt mpan Proj _|_ G ` ` w cv ` H e. adantr jca pjjs.1 chshi pjjs.2 shjcl w cv chel sylan y cv Proj G ` w cv ` +v z cv opreq1 w cv eqeq2d z cv Proj _|_ G ` ` w cv ` Proj G ` w cv ` +v opreq2 w cv eqeq2d G H rcla42ev syl pjjs.1 chshi pjjs.2 w cv y z shsel sylibr exp syld com12 ssrdv pjjs.1 chshi pjjs.2 shslej jctir G vH H opr G +H H opr eqss sylibr))
+thm (pjfn ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w)) ((pjfn.1 (e. H (CH)))) (Fn (` (Proj) H) (H~)) (pjfn.1 elisseti z w _|_ H ` x cv z cv +v w cv opr = E.e. rabex uniex pjfn.1 H x y z w pjmvalt ax-mp fnopab2))
+thm (pjrn ((x y) (H x) (H y)) ((pjfn.1 (e. H (CH)))) (= (ran (` (Proj) H)) H) (x y x cv H~ e. y cv Proj H ` x cv ` = /\ rnopab pjfn.1 pjfn Proj H ` H~ x y fnopabfv mpbi rneqi y visset x cv y cv H~ eleq1 x cv y cv Proj H ` fveq2 y cv eqeq2d anbi12d cla4ev pjfn.1 y cv chel pjfn.1 y cv chel pjfn.1 H y cv pjcht mpan biimpd mpcom eqcomd sylanc y cv Proj H ` x cv ` H eleq1 pjfn.1 x cv pjcl syl5bir impcom x 19.23aiv impbi biabri 3eqtr4))
+thm (pjf () ((pjfn.1 (e. H (CH)))) (:--> (` (Proj) H) (H~) (H~)) (Proj H ` H~ H~ df-f pjfn.1 pjfn pjfn.1 pjrn pjfn.1 chssi eqsstr mpbir2an))
+thm (pjv () ((pjfn.1 (e. H (CH)))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (Proj) H) (opr A (+v) B)) A)) (pjfn.1 A B pjaddt pjfn.1 A chel pjfn.1 chocl B chel syl2an pjfn.1 A chel pjfn.1 H A pjcht mpan syl ibi pjfn.1 chocl B chel pjfn.1 B pjoc2t syl ibi +v opreqan12d pjfn.1 A chel A ax-hvaddid syl B _|_ H ` e. adantr 3eqtrd))
+thm (pjsumt () ((pjsumt.1 (e. G (CH))) (pjsumt.2 (e. H (CH)))) (-> (e. A (H~)) (-> (C_ G (` (_|_) H)) (= (` (` (Proj) (opr G (+H) H)) A) (opr (` (` (Proj) G) A) (+v) (` (` (Proj) H) A))))) (pjsumt.1 pjsumt.2 osum Proj fveq2d A fveq1d A H~ e. adantl pjsumt.1 pjsumt.2 G H A pjcjt2 mp3an12 imp eqtrd exp))
+thm (pj11 () ((pjsumt.1 (e. G (CH))) (pjsumt.2 (e. H (CH)))) (<-> (= (` (Proj) G) (` (Proj) H)) (= G H)) (Proj G ` Proj H ` rneq pjsumt.1 pjrn pjsumt.2 pjrn 3eqtr3g G H Proj fveq2 impbi))
+thm (pjoi0 () ((pjoi0.1 (e. G (CH))) (pjoi0.2 (e. H (CH))) (pjoi0.3 (e. A (H~)))) (-> (C_ G (` (_|_) H)) (= (opr (` (` (Proj) G) A) (.i) (` (` (Proj) H) A)) (0))) (pjoi0.1 pjrn pjoi0.2 pjrn _|_ fveq2i sseq12i pjoi0.1 pjfn pjoi0.3 Proj G ` H~ A fnfvrn mp2an pjoi0.2 pjfn pjoi0.3 Proj H ` H~ A fnfvrn mp2an pjoi0.2 pjrn pjoi0.2 eqeltr chshi Proj H ` ran Proj G ` ran Proj G ` A ` Proj H ` A ` shorth ax-mp mp2ani sylbir))
+thm (pjopyth () ((pjoi0.1 (e. G (CH))) (pjoi0.2 (e. H (CH))) (pjoi0.3 (e. A (H~)))) (-> (C_ G (` (_|_) H)) (= (opr (` (norm) (opr (` (` (Proj) G) A) (+v) (` (` (Proj) H) A))) (^) (2)) (opr (opr (` (norm) (` (` (Proj) G) A)) (^) (2)) (+) (opr (` (norm) (` (` (Proj) H) A)) (^) (2))))) (pjoi0.1 pjoi0.2 pjoi0.3 pjoi0 pjoi0.1 pjoi0.3 pjhcli pjoi0.2 pjoi0.3 pjhcli normpyth syl))
+thm (hoeq ((S x) (T x)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (<-> (A.e. x (H~) (= (` S (cv x)) (` T (cv x)))) (= S T)) (hoeq.1 S H~ H~ ffn ax-mp hoeq.2 T H~ H~ ffn ax-mp S H~ T H~ x eqfnfv mp2an H~ eqid mpbiran bicomi))
+thm (hoscl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (opr S (+P) T) A) (H~))) (hoeq.1 hoeq.2 S T A hosclt mpanl12))
+thm (hodcl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (opr S (-P) T) A) (H~))) (hoeq.1 hoeq.2 S T A hodclt mpanl12))
+thm (hoco () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (= (` (o. S T) A) (` S (` T A)))) (hoeq.1 S H~ H~ ffun ax-mp hoeq.2 T H~ H~ ffun ax-mp S T A fvco hoeq.2 T H~ H~ fdm ax-mp A eleq2i sylan2br mpanl12))
+thm (hococl () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (o. S T) A) (H~))) (hoeq.1 hoeq.2 A hoco hoeq.2 A ffvrni hoeq.1 T A ` ffvrni syl eqeltrd))
+thm (hocof () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (o. S T) (H~) (H~)) (hoeq.1 hoeq.2 S H~ H~ T H~ fco mp2an))
+thm (hocofn () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (Fn (o. S T) (H~)) (hoeq.1 hoeq.2 hocof S T o. H~ H~ ffn ax-mp))
+thm (hosf ((x y) (S x) (S y) (T x) (T y)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (opr S (+P) T) (H~) (H~)) (hoeq.1 hoeq.2 S T x y hosmvalt mp2an S x cv ` T x cv ` ax-hvaddcl hoeq.1 x cv ffvrni hoeq.2 x cv ffvrni sylanc fopab))
+thm (hodf ((x y) (S x) (S y) (T x) (T y)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (:--> (opr S (-P) T) (H~) (H~)) (hoeq.1 hoeq.2 S T x y hodmvalt mp2an S x cv ` T x cv ` hvsubclt hoeq.1 x cv ffvrni hoeq.2 x cv ffvrni sylanc fopab))
+thm (hosfn () ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (Fn (opr S (+P) T) (H~)) (hoeq.1 hoeq.2 hosf S +P T opr H~ H~ ffn ax-mp))
+thm (hoscom ((S x) (T x)) ((hoeq.1 (:--> S (H~) (H~))) (hoeq.2 (:--> T (H~) (H~)))) (= (opr S (+P) T) (opr T (+P) S)) (S x cv ` T x cv ` ax-hvcom hoeq.1 x cv ffvrni hoeq.2 x cv ffvrni sylanc hoeq.1 hoeq.2 S T x cv hosvalt mpanl12 hoeq.2 hoeq.1 T S x cv hosvalt mpanl12 3eqtr4d rgen hoeq.1 hoeq.2 hosf hoeq.2 hoeq.1 hosf x hoeq mpbi))
+thm (eigre () ((eigre.1 (e. A (H~))) (eigre.2 (e. B (CC)))) (-> (/\ (= (` T A) (opr B (.s) A)) (-. (= A (0v)))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR)))) (T A ` B .s A opr A .i opreq2 eigre.2 eigre.1 eigre.1 B A A his5t mp3an syl6eq T A ` B .s A opr .i A opreq1 eigre.2 eigre.1 eigre.1 B A A ax-his3 mp3an syl6eq eqeq12d eigre.1 A axhis42 mpan eigre.1 A hiidrclt ax-mp gt0ne0 eigre.2 cjcl eigre.2 eigre.1 eigre.1 hicl 3pm3.2i * B ` B A .i A opr mulcan2t mpan 3syl sylan9bb eigre.2 cjreb syl6bbr))
+thm (eigpos () ((eigpos.1 (e. A (H~))) (eigpos.2 (e. B (CC)))) (-> (/\ (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (/\ (= (` T A) (opr B (.s) A)) (-. (= A (0v))))) (/\ (e. B (RR)) (br (0) (<_) B))) (eigpos.1 eigpos.2 eigpos.1 hvmulcl A B .s A opr hiret mp2an T A ` B .s A opr = a1i T A ` B .s A opr A .i opreq2 RR eleq1d T A ` B .s A opr A .i opreq2 T A ` B .s A opr .i A opreq1 eqeq12d 3bitr4d A 0v = -. adantr eigpos.1 eigpos.2 T eigre bitrd biimpac 0 <_ A .i T A ` opr br adantlr eigpos.1 A hiidrclt ax-mp B A .i A opr prodge02t mpanl2 eigpos.1 eigpos.2 eigpos.1 hvmulcl A B .s A opr hiret mp2an T A ` B .s A opr = a1i T A ` B .s A opr A .i opreq2 RR eleq1d T A ` B .s A opr A .i opreq2 T A ` B .s A opr .i A opreq1 eqeq12d 3bitr4d A 0v = -. adantr eigpos.1 eigpos.2 T eigre bitrd biimpac 0 <_ A .i T A ` opr br adantlr eigpos.1 A axhis42 mpan A .i T A ` opr RR e. 0 <_ A .i T A ` opr br /\ T A ` B .s A opr = ad2antrr A .i T A ` opr RR e. 0 <_ A .i T A ` opr br pm3.27 T A ` B .s A opr = A 0v = -. /\ adantr T A ` B .s A opr A .i opreq2 A .i T A ` opr RR e. 0 <_ A .i T A ` opr br /\ A 0v = -. ad2antrl eigpos.1 eigpos.2 eigpos.1 hvmulcl A B .s A opr hiret mp2an T A ` B .s A opr = a1i T A ` B .s A opr A .i opreq2 RR eleq1d T A ` B .s A opr A .i opreq2 T A ` B .s A opr .i A opreq1 eqeq12d 3bitr4d A 0v = -. adantr eigpos.1 eigpos.2 T eigre bitrd biimpac 0 <_ A .i T A ` opr br adantlr eigpos.2 cjreb sylib x. A .i A opr opreq1d eigpos.2 eigpos.1 eigpos.1 B A A his5t mp3an syl5eq eqtrd breqtrd jca sylanc jca))
+thm (eigorth () ((eigorth.1 (e. A (H~))) (eigorth.2 (e. B (H~))) (eigorth.3 (e. C (CC))) (eigorth.4 (e. D (CC)))) (-> (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D)))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0)))) (T B ` D .s B opr A .i opreq2 eigorth.4 eigorth.1 eigorth.2 D A B his5t mp3an syl6eq T A ` C .s A opr .i B opreq1 eigorth.3 eigorth.1 eigorth.2 C A B ax-his3 mp3an syl6eq eqeqan12rd eigorth.4 cjcl eigorth.3 eigorth.1 eigorth.2 hicl * D ` C A .i B opr mulcan2t A .i B opr 0 df-ne sylan2br exp mp3an * D ` C eqcom syl6bb biimpcd con1d com12 A .i B opr 0 * D ` x. opreq2 A .i B opr 0 C x. opreq2 eigorth.3 mul01 eigorth.4 cjcl mul01 eqtr4 syl6eq eqtr4d C * D ` = -. a1i impbid sylan9bb))
+thm (hunid ((x y)) () (/\ (:-onto-> (|` (I) (H~)) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (|` (I) (H~)) (cv x)) (.i) (` (|` (I) (H~)) (cv y))) (opr (cv x) (.i) (cv y)))))) (H~ f1oi I H~ |` H~ H~ f1ofo ax-mp x cv H~ fvresi y cv H~ fvresi .i opreqan12d rgen2 pm3.2i))
+thm (hun ((x y) (x z) (w x) (T x) (y z) (w y) (T y) (w z) (T z) (T w) (A z) (A w) (B z) (B w)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) (` T B)) (opr A (.i) B))) (hun.1 pm3.27i x cv z cv T fveq2 .i T y cv ` opreq1d x cv z cv .i y cv opreq1 eqeq12d y cv w cv T fveq2 T z cv ` .i opreq2d y cv w cv z cv .i opreq2 eqeq12d H~ H~ cbvral2v mpbi z cv A T fveq2 .i T w cv ` opreq1d z cv A .i w cv opreq1 eqeq12d w cv B T fveq2 T A ` .i opreq2d w cv B A .i opreq2 eqeq12d H~ H~ rcla42v mpi))
+thm (hun1o ((x y) (x z) (w x) (T x) (y z) (w y) (T y) (w z) (T z) (T w)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) (:-1-1-onto-> T (H~) (H~)) (T H~ H~ df-f1o T H~ H~ z w f1fv hun.1 pm3.26i T H~ H~ fof ax-mp hun.1 z cv z cv hun anidms hun.1 w cv w cv hun anidms + opreqan12d hun.1 z cv w cv hun hun.1 w cv z cv hun ancoms + opreq12d - opreq12d T z cv ` T w cv ` normlem9at hun.1 pm3.26i T H~ H~ fof ax-mp z cv ffvrni hun.1 pm3.26i T H~ H~ fof ax-mp w cv ffvrni syl2an z cv w cv normlem9at 3eqtr4rd 0 eqeq1d T z cv ` T w cv ` hvsubclt T z cv ` -v T w cv ` opr his6t syl T z cv ` T w cv ` hvsubeq0t bitrd hun.1 pm3.26i T H~ H~ fof ax-mp z cv ffvrni hun.1 pm3.26i T H~ H~ fof ax-mp w cv ffvrni syl2an bitr2d z cv w cv hvsubclt z cv -v w cv opr his6t syl z cv w cv hvsubeq0t 3bitrd biimpd rgen2 mpbir2an hun.1 pm3.26i mpbir2an))
+thm (huncnv ((x y) (T x) (T y)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) (/\ (:-onto-> (`' T) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (`' T) (cv x)) (.i) (` (`' T) (cv y))) (opr (cv x) (.i) (cv y)))))) (hun.1 hun1o T H~ H~ f1ocnv ax-mp T `' H~ H~ f1ofo ax-mp hun.1 T `' x cv ` T `' y cv ` hun hun.1 hun1o T H~ H~ f1ocnv ax-mp T `' H~ H~ f1ofo ax-mp T `' H~ H~ fof ax-mp x cv ffvrni hun.1 hun1o T H~ H~ f1ocnv ax-mp T `' H~ H~ f1ofo ax-mp T `' H~ H~ fof ax-mp y cv ffvrni syl2an hun.1 hun1o T H~ H~ x cv f1ocnvfv2 mpan hun.1 hun1o T H~ H~ y cv f1ocnvfv2 mpan .i opreqan12d eqtr3d rgen2 pm3.2i))
+thm (hunicnv ((x y) (T x) (T y)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) B) (opr A (.i) (` (`' T) B)))) (hun.1 hun1o T H~ H~ B f1ocnvfv2 mpan A H~ e. adantl T A ` .i opreq2d hun.1 A T `' B ` hun hun.1 hun1o T H~ H~ f1ocnv ax-mp T `' H~ H~ f1of ax-mp B ffvrni sylan2 eqtr3d))
+thm (hunlin ((x y) (x z) (T x) (y z) (T y) (T z) (A z) (B z) (C z)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))) (hun.1 A .s B opr +v C opr z cv hunicnv A .s B opr C ax-hvaddcl A B ax-hvmulcl sylan sylan A B C T `' z cv ` hiassdit anassrs hun.1 hun1o T H~ H~ f1ocnv ax-mp T `' H~ H~ f1of ax-mp z cv ffvrni sylan2 A T B ` T C ` z cv hiassdit hun.1 pm3.26i T H~ H~ fof ax-mp C ffvrni sylanr1 hun.1 pm3.26i T H~ H~ fof ax-mp B ffvrni sylanl2 anassrs hun.1 B z cv hunicnv A x. opreq2d C H~ e. adantlr hun.1 C z cv hunicnv B H~ e. adantll + opreq12d A CC e. adantlll eqtr2d 3eqtrd exp r19.21aiv T A .s B opr +v C opr ` A .s T B ` opr +v T C ` opr z hial2eqt A .s B opr C ax-hvaddcl A B ax-hvmulcl sylan hun.1 pm3.26i T H~ H~ fof ax-mp A .s B opr +v C opr ffvrni syl A .s T B ` opr T C ` ax-hvaddcl A T B ` ax-hvmulcl hun.1 pm3.26i T H~ H~ fof ax-mp B ffvrni sylan2 hun.1 pm3.26i T H~ H~ fof ax-mp C ffvrni syl2an sylanc mpbid 3impa))
+thm (hunco ((x y) (S x) (S y) (x y) (T x) (T y)) ((hun.1 (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (hunco.2 (/\ (:-onto-> S (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` S (cv x)) (.i) (` S (cv y))) (opr (cv x) (.i) (cv y)))))))) (/\ (:-onto-> (o. S T) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (o. S T) (cv x)) (.i) (` (o. S T) (cv y))) (opr (cv x) (.i) (cv y)))))) (hunco.2 hun1o hun.1 hun1o S H~ H~ T H~ f1oco mp2an S T o. H~ H~ f1ofo ax-mp hunco.2 hun1o S H~ H~ f1ofun ax-mp hun.1 hun1o T H~ H~ f1of ax-mp S T H~ H~ x cv fvco3 mpanl12 hunco.2 hun1o S H~ H~ f1ofun ax-mp hun.1 hun1o T H~ H~ f1of ax-mp S T H~ H~ y cv fvco3 mpanl12 .i opreqan12d hunco.2 T x cv ` T y cv ` hun hun.1 hun1o T H~ H~ f1of ax-mp x cv ffvrni hun.1 hun1o T H~ H~ f1of ax-mp y cv ffvrni syl2an hun.1 x cv y cv hun 3eqtrd rgen2 pm3.2i))
+thm (hlin ((x y) (x z) (w x) (v x) (u x) (T x) (y z) (w y) (v y) (u y) (T y) (w z) (v z) (u z) (T z) (v w) (u w) (T w) (u v) (T v) (T u) (A w) (A v) (A u) (B v) (B u) (C u)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))) (hlin.1 pm3.27i x cv w cv .s y cv opreq1 +v z cv opreq1d T fveq2d x cv w cv .s T y cv ` opreq1 +v T z cv ` opreq1d eqeq12d y cv v cv w cv .s opreq2 +v z cv opreq1d T fveq2d y cv v cv T fveq2 w cv .s opreq2d +v T z cv ` opreq1d eqeq12d z cv u cv w cv .s v cv opr +v opreq2 T fveq2d z cv u cv T fveq2 w cv .s T v cv ` opr +v opreq2d eqeq12d CC H~ H~ cbvral3v mpbi w cv A .s v cv opreq1 +v u cv opreq1d T fveq2d w cv A .s T v cv ` opreq1 +v T u cv ` opreq1d eqeq12d v cv B A .s opreq2 +v u cv opreq1d T fveq2d v cv B T fveq2 A .s opreq2d +v T u cv ` opreq1d eqeq12d u cv C A .s B opr +v opreq2 T fveq2d u cv C T fveq2 A .s T B ` opr +v opreq2d eqeq12d CC H~ H~ rcla43v mpi))
+thm (hlin0 ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (= (` T (0v)) (0v)) (1cn ax-hv0cl ax-hv0cl hlin.1 1 0v 0v hlin mp3an 1cn ax-hv0cl hvmulcl 1 .s 0v opr ax-hvaddid ax-mp ax-hv0cl 0v ax-hvmulid ax-mp eqtr T fveq2i ax-hv0cl hlin.1 pm3.26i 0v ffvrni ax-mp T 0v ` ax-hvmulid ax-mp +v T 0v ` opreq1i 3eqtr3 -v T 0v ` opreq1i ax-hv0cl hlin.1 pm3.26i 0v ffvrni ax-mp T 0v ` hvsubidt ax-mp ax-hv0cl hlin.1 pm3.26i 0v ffvrni ax-mp ax-hv0cl hlin.1 pm3.26i 0v ffvrni ax-mp T 0v ` T 0v ` hvsubcan1t mp2an 3eqtr3r))
+thm (hlinadd ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (+v) B)) (opr (` T A) (+v) (` T B)))) (1cn hlin.1 1 A B hlin mp3an1 A ax-hvmulid +v B opreq1d T fveq2d B H~ e. adantr hlin.1 pm3.26i A ffvrni T A ` ax-hvmulid syl B H~ e. adantr +v T B ` opreq1d 3eqtr3d))
+thm (hlinmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (.s) (` T B)))) (ax-hv0cl hlin.1 A B 0v hlin mp3an3 A B ax-hvmulcl A .s B opr ax-hvaddid syl T fveq2d A T B ` ax-hvmulcl hlin.1 pm3.26i B ffvrni sylan2 A .s T B ` opr ax-hvaddid syl hlin.1 hlin0 A .s T B ` opr +v opreq2i syl5eq 3eqtr3d))
+thm (hlinaddmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (+v) (opr A (.s) C))) (opr (` T B) (+v) (opr A (.s) (` T C))))) (hlin.1 B A .s C opr hlinadd A C ax-hvmulcl sylan2 3impb 3com12 hlin.1 A C hlinmul B H~ e. 3adant2 T B ` +v opreq2d eqtrd))
+thm (hlinsub ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (-v) B)) (opr (` T A) (-v) (` T B)))) (1cn negcl hlin.1 1 -u A B hlinaddmul mp3an1 A B hvsubvalt T fveq2d T A ` T B ` hvsubvalt hlin.1 pm3.26i A ffvrni hlin.1 pm3.26i B ffvrni syl2an 3eqtr4d))
+thm (hlinsubmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (-v) (opr A (.s) C))) (opr (` T B) (-v) (opr A (.s) (` T C))))) (hlin.1 B A .s C opr hlinsub A C ax-hvmulcl sylan2 3impb 3com12 hlin.1 A C hlinmul T B ` -v opreq2d B H~ e. 3adant2 eqtrd))
+thm (hlinmulsub ((x y) (x z) (T x) (y z) (T y) (T z)) ((hlin.1 (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (-v) C)) (opr (opr A (.s) (` T B)) (-v) (` T C)))) (hlin.1 A .s B opr C hlinsub A B ax-hvmulcl sylan 3impa hlin.1 A B hlinmul C H~ e. 3adant3 -v T C ` opreq1d eqtrd))
+thm (hods ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (<-> (= (opr R (-P) S) T) (= (opr S (+P) T) R)) (R x cv ` S x cv ` T x cv ` hvsubaddt hods.1 x cv ffvrni hods.2 x cv ffvrni hods.3 x cv ffvrni syl3anc hods.1 hods.2 R S x cv hodvalt mpanl12 T x cv ` eqeq1d hods.2 hods.3 S T x cv hosvalt mpanl12 R x cv ` eqeq1d 3bitr4d birala hods.1 hods.2 hodf hods.3 x hoeq hods.2 hods.3 hosf hods.1 x hoeq 3bitr3))
+thm (hosass ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (opr (opr R (+P) S) (+P) T) (opr R (+P) (opr S (+P) T))) (R x cv ` S x cv ` T x cv ` ax-hvass hods.1 x cv ffvrni hods.2 x cv ffvrni hods.3 x cv ffvrni syl3anc hods.1 hods.2 hosf hods.3 R +P S opr T x cv hosvalt mpanl12 hods.1 hods.2 R S x cv hosvalt mpanl12 +v T x cv ` opreq1d eqtrd hods.1 hods.2 hods.3 hosf R S +P T opr x cv hosvalt mpanl12 hods.2 hods.3 S T x cv hosvalt mpanl12 R x cv ` +v opreq2d eqtrd 3eqtr4d rgen hods.1 hods.2 hosf hods.3 hosf hods.1 hods.2 hods.3 hosf hosf x hoeq mpbi))
+thm (hos12 () ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (opr R (+P) (opr S (+P) T)) (opr S (+P) (opr R (+P) T))) (hods.1 hods.2 hoscom +P T opreq1i hods.1 hods.2 hods.3 hosass hods.2 hods.1 hods.3 hosass 3eqtr3))
+thm (hosdir ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (o. (opr R (+P) S) T) (opr (o. R T) (+P) (o. S T))) (hods.3 x cv ffvrni hods.1 hods.2 R S T x cv ` hosvalt mpanl12 syl hods.1 hods.3 x cv hoco hods.2 hods.3 x cv hoco +v opreq12d eqtr4d hods.1 hods.2 hosf hods.3 x cv hoco hods.1 hods.3 hocof hods.2 hods.3 hocof R T o. S T o. x cv hosvalt mpanl12 3eqtr4d rgen hods.1 hods.2 hosf hods.3 hocof hods.1 hods.3 hocof hods.2 hods.3 hocof hosf x hoeq mpbi))
+thm (hoddir ((R x) (S x) (T x)) ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (= (o. (opr R (-P) S) T) (opr (o. R T) (-P) (o. S T))) (hods.3 x cv ffvrni hods.1 hods.2 R S T x cv ` hodvalt mpanl12 syl hods.1 hods.3 x cv hoco hods.2 hods.3 x cv hoco -v opreq12d eqtr4d hods.1 hods.2 hodf hods.3 x cv hoco hods.1 hods.3 hocof hods.2 hods.3 hocof R T o. S T o. x cv hodvalt mpanl12 3eqtr4d rgen hods.1 hods.2 hodf hods.3 hocof hods.1 hods.3 hocof hods.2 hods.3 hocof hodf x hoeq mpbi))
+thm (ho2co () ((hods.1 (:--> R (H~) (H~))) (hods.2 (:--> S (H~) (H~))) (hods.3 (:--> T (H~) (H~)))) (-> (e. A (H~)) (= (` (o. (o. R S) T) A) (` R (` S (` T A))))) (hods.1 hods.2 hocof hods.3 A hoco hods.3 A ffvrni hods.1 hods.2 T A ` hoco syl eqtrd))
+thm (ho0 () () (= (` (Proj) (0H)) (X. (H~) (0H))) (Proj 0H ` H~ 0v x fconstfv h0elch pjfn x cv pjch0t rgen mpbir2an ax-hv0cl elisseti Proj 0H ` H~ fconst2 mpbi df-ch0 0H 0v {} H~ xpeq2 ax-mp eqtr4))
+thm (ho1 () () (= (` (Proj) (H~)) (|` (I) (H~))) (helch pjfn H~ fnresi H~ eqid x cv pjch1t x cv H~ fvresi eqtr4d rgen pm3.2i Proj H~ ` H~ I H~ |` H~ x eqfnfv mpbiri mp2an))
+thm (hoid0 ((T x)) ((hoid0.1 (:--> T (H~) (H~)))) (= (opr T (+P) (` (Proj) (0H))) T) (hoid0.1 h0elch pjf T Proj 0H ` x cv hosvalt mpanl12 x cv pjch0t T x cv ` +v opreq2d hoid0.1 x cv ffvrni T x cv ` ax-hvaddid syl 3eqtrd rgen hoid0.1 h0elch pjf hosf hoid0.1 x hoeq mpbi))
+thm (hoid0r () ((hoid0.1 (:--> T (H~) (H~)))) (= (opr (` (Proj) (0H)) (+P) T) T) (h0elch pjf hoid0.1 hoscom hoid0.1 hoid0 eqtr))
+thm (hodid () ((hoid0.1 (:--> T (H~) (H~)))) (= (opr T (-P) T) (` (Proj) (0H))) (hoid0.1 hoid0 hoid0.1 hoid0.1 h0elch pjf hods mpbir))
+thm (hoid1 ((T x)) ((hoid0.1 (:--> T (H~) (H~)))) (= (o. T (` (Proj) (H~))) T) (hoid0.1 helch pjf x cv hoco x cv pjch1t T fveq2d eqtrd rgen hoid0.1 helch pjf hocof hoid0.1 x hoeq mpbi))
+thm (hoid1r ((T x)) ((hoid0.1 (:--> T (H~) (H~)))) (= (o. (` (Proj) (H~)) T) T) (helch pjf hoid0.1 x cv hoco hoid0.1 x cv ffvrni T x cv ` pjch1t syl eqtrd rgen helch pjf hoid0.1 hocof hoid0.1 x hoeq mpbi))
+thm (hodseq () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (+P) (opr T (-P) S)) T) (T -P S opr eqid hodseq.3 hodseq.2 hodseq.3 hodseq.2 hodf hods mpbi))
+thm (hods0 () ((hodseq.2 (:--> S (H~) (H~))) (hodseq.3 (:--> T (H~) (H~)))) (= (opr S (-P) T) (opr S (+P) (opr (` (Proj) (0H)) (-P) T))) (hodseq.3 hodseq.2 h0elch pjf hodseq.3 hodf hos12 hodseq.3 h0elch pjf hodseq S +P opreq2i hodseq.2 hoid0 3eqtr hodseq.2 hodseq.3 hodseq.2 h0elch pjf hodseq.3 hodf hosf hods mpbir))
+thm (hosdass () ((hosdass.1 (:--> R (H~) (H~))) (hosdass.2 (:--> S (H~) (H~))) (hosdass.3 (:--> T (H~) (H~)))) (= (opr (opr R (+P) S) (-P) T) (opr R (+P) (opr S (-P) T))) (hosdass.1 hosdass.2 h0elch pjf hosdass.3 hodf hosass hosdass.1 hosdass.2 hosf hosdass.3 hods0 hosdass.2 hosdass.3 hods0 R +P opreq2i 3eqtr4))
+thm (hosd () ((hosdass.1 (:--> R (H~) (H~))) (hosdass.2 (:--> S (H~) (H~))) (hosdass.3 (:--> T (H~) (H~)))) (= (opr (opr R (+P) S) (-P) T) (opr (opr R (-P) T) (+P) S)) (hosdass.1 hosdass.2 hoscom -P T opreq1i hosdass.2 hosdass.1 hosdass.3 hosdass hosdass.2 hosdass.1 hosdass.3 hodf hoscom 3eqtr))
+thm (hosd1 () ((hosd1.2 (:--> S (H~) (H~))) (hosd1.3 (:--> T (H~) (H~)))) (= (opr S (+P) T) (opr S (-P) (opr (` (Proj) (0H)) (-P) T))) (h0elch pjf hosd1.3 hodf hosd1.2 hosd1.3 hosf hoscom hosd1.2 hosd1.3 hosf h0elch pjf hosd1.3 hosdass hosd1.2 hosd1.3 hosf hoid0 -P T opreq1i hosd1.2 hosd1.3 hosd1.3 hosd hosd1.2 hosd1.3 hodf hosd1.3 hoscom hosd1.3 hosd1.2 hodseq eqtr 3eqtr eqtr3 eqtr hosd1.2 h0elch pjf hosd1.3 hodf hosd1.2 hosd1.3 hosf hods mpbir eqcomi))
+thm (hosd2 () ((hosd1.2 (:--> S (H~) (H~))) (hosd1.3 (:--> T (H~) (H~)))) (= (opr S (+P) T) (opr S (-P) (opr (opr T (-P) T) (-P) T))) (hosd1.2 hosd1.3 hosd1 hosd1.3 hodid -P T opreq1i S -P opreq2i eqtr4))
+thm (pjsdi ((H x) (S x) (T x)) ((pjsdi.1 (e. H (CH))) (pjsdi.2 (:--> S (H~) (H~))) (pjsdi.3 (:--> T (H~) (H~)))) (= (o. (` (Proj) H) (opr S (+P) T)) (opr (o. (` (Proj) H) S) (+P) (o. (` (Proj) H) T))) (pjsdi.1 S x cv ` T x cv ` pjaddt pjsdi.2 x cv ffvrni pjsdi.3 x cv ffvrni sylanc pjsdi.2 pjsdi.3 S T x cv hosvalt mpanl12 Proj H ` fveq2d pjsdi.1 pjf pjsdi.2 x cv hoco pjsdi.1 pjf pjsdi.3 x cv hoco +v opreq12d 3eqtr4d pjsdi.1 pjf pjsdi.2 pjsdi.3 hosf x cv hoco pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof Proj H ` S o. Proj H ` T o. x cv hosvalt mpanl12 3eqtr4d rgen pjsdi.1 pjf pjsdi.2 pjsdi.3 hosf hocof pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof hosf x hoeq mpbi))
+thm (pjddi ((H x) (S x) (T x)) ((pjsdi.1 (e. H (CH))) (pjsdi.2 (:--> S (H~) (H~))) (pjsdi.3 (:--> T (H~) (H~)))) (= (o. (` (Proj) H) (opr S (-P) T)) (opr (o. (` (Proj) H) S) (-P) (o. (` (Proj) H) T))) (pjsdi.1 S x cv ` T x cv ` pjsubt pjsdi.2 x cv ffvrni pjsdi.3 x cv ffvrni sylanc pjsdi.2 pjsdi.3 S T x cv hodvalt mpanl12 Proj H ` fveq2d pjsdi.1 pjf pjsdi.2 x cv hoco pjsdi.1 pjf pjsdi.3 x cv hoco -v opreq12d 3eqtr4d pjsdi.1 pjf pjsdi.2 pjsdi.3 hodf x cv hoco pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof Proj H ` S o. Proj H ` T o. x cv hodvalt mpanl12 3eqtr4d rgen pjsdi.1 pjf pjsdi.2 pjsdi.3 hodf hocof pjsdi.1 pjf pjsdi.2 hocof pjsdi.1 pjf pjsdi.3 hocof hodf x hoeq mpbi))
+thm (pjsdi2 () ((pjsdi2.1 (e. H (CH))) (pjsdi2.2 (:--> R (H~) (H~))) (pjsdi2.3 (:--> S (H~) (H~))) (pjsdi2.4 (:--> T (H~) (H~)))) (-> (= (o. R (opr S (+P) T)) (opr (o. R S) (+P) (o. R T))) (= (o. (o. (` (Proj) H) R) (opr S (+P) T)) (opr (o. (o. (` (Proj) H) R) S) (+P) (o. (o. (` (Proj) H) R) T)))) (R S +P T opr o. R S o. +P R T o. opr Proj H ` coeq2 pjsdi2.1 pjsdi2.2 pjsdi2.3 hocof pjsdi2.2 pjsdi2.4 hocof pjsdi syl6eq Proj H ` R S +P T opr coass Proj H ` R S coass Proj H ` R T coass +P opreq12i 3eqtr4g))
+thm (pjco () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (= (` (o. (` (Proj) G) (` (Proj) H)) A) (` (` (Proj) G) (` (` (Proj) H) A)))) (pjco.1 pjf pjco.2 pjf A hoco))
+thm (pjcocl () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (` (Proj) G) (` (Proj) H)) A) G)) (pjco.1 pjco.2 A pjco pjco.2 A pjhcl pjco.1 Proj H ` A ` pjcl syl eqeltrd))
+thm (pjcohcl () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (` (Proj) G) (` (Proj) H)) A) (H~))) (pjco.1 pjf pjco.2 pjf A hococl))
+thm (pjadjco () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (o. (` (Proj) G) (` (Proj) H)) A) (.i) B) (opr A (.i) (` (o. (` (Proj) H) (` (Proj) G)) B)))) (pjco.1 Proj H ` A ` B pjadjt pjco.2 A pjhcl sylan pjco.2 A Proj G ` B ` pjadjt pjco.1 B pjhcl sylan2 eqtrd pjco.1 pjco.2 A pjco .i B opreq1d B H~ e. adantr pjco.2 pjco.1 B pjco A .i opreq2d A H~ e. adantl 3eqtr4d))
+thm (pjcofn () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (Fn (o. (` (Proj) G) (` (Proj) H)) (H~)) (pjco.1 pjf pjco.2 pjf hocofn))
+thm (pjss1co ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (= (o. (` (Proj) H) (` (Proj) G)) (` (Proj) G))) (pjco.2 pjco.1 x cv pjco G H C_ adantl G H Proj G ` x cv ` ssel pjco.1 x cv pjcl syl5 pjco.1 x cv pjhcl pjco.2 H Proj G ` x cv ` pjcht mpan syl mpbidi imp eqtrd exp r19.21aiv pjco.2 pjf pjco.1 pjf hocof pjco.1 pjf x hoeq sylib Proj H ` Proj G ` o. Proj G ` rneq Proj H ` Proj G ` rnco Proj H ` Proj G ` o. ran Proj G ` ran Proj H ` ran sseq1 mpbii syl pjco.1 pjrn pjco.2 pjrn 3sstr3g impbi))
+thm (pjss2co ((x y) (H x) (H y) (G x) (G y)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) G))) (pjco.1 pjco.2 x cv pjco G H C_ adantl x cv x cv H~ e. x cv 0v if Proj H ` fveq2 Proj G ` fveq2d x cv x cv H~ e. x cv 0v if Proj G ` fveq2 eqeq12d G H C_ imbi2d pjco.1 ax-hv0cl x cv elimel pjco.2 pjss2 dedth impcom eqtrd exp r19.21aiv pjco.1 pjf pjco.2 pjf hocof pjco.1 pjf x hoeq sylib Proj G ` Proj H ` o. Proj G ` y cv fveq1 x cv .i opreq2d x cv H~ e. y cv H~ e. ad2antlr pjco.2 pjco.1 x cv y cv pjadjco Proj G ` Proj H ` o. Proj G ` = adantlr pjco.1 x cv y cv pjadjt Proj G ` Proj H ` o. Proj G ` = adantlr 3eqtr4d exp31 r19.21adv Proj H ` Proj G ` o. x cv ` Proj G ` x cv ` y hial2eqt pjco.2 pjco.1 x cv pjcohcl pjco.1 x cv pjhcl sylanc sylibd com12 r19.21aiv pjco.2 pjf pjco.1 pjf hocof pjco.1 pjf x hoeq sylib pjco.1 pjco.2 pjss1co sylibr impbi))
+thm (pjssmt () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (-> (C_ H G) (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)))) (A A H~ e. A 0v if Proj G ` fveq2 A A H~ e. A 0v if Proj H ` fveq2 -v opreq12d A A H~ e. A 0v if Proj G _|_ H ` i^i ` fveq2 eqeq12d H G C_ imbi2d pjco.2 ax-hv0cl A elimel pjco.1 pjssm dedth))
+thm (pjssge0t () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (-> (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)))) (A A H~ e. A 0v if Proj G ` fveq2 A A H~ e. A 0v if Proj H ` fveq2 -v opreq12d A A H~ e. A 0v if Proj G _|_ H ` i^i ` fveq2 eqeq12d A A H~ e. A 0v if Proj G ` fveq2 A A H~ e. A 0v if Proj H ` fveq2 -v opreq12d A A H~ e. A 0v if = id .i opreq12d 0 <_ breq2d imbi12d pjco.2 ax-hv0cl A elimel pjco.1 pjssge0 dedth))
+thm (pjdifnormt () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (e. A (H~)) (<-> (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) (` (` (Proj) G) A))))) (A A H~ e. A 0v if Proj G ` fveq2 A A H~ e. A 0v if Proj H ` fveq2 -v opreq12d A A H~ e. A 0v if = id .i opreq12d 0 <_ breq2d A A H~ e. A 0v if Proj H ` fveq2 norm fveq2d A A H~ e. A 0v if Proj G ` fveq2 norm fveq2d <_ breq12d bibi12d pjco.2 ax-hv0cl A elimel pjco.1 pjdifnorm dedth))
+thm (pjnormss ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (<-> (C_ G H) (A.e. x (H~) (br (` (norm) (` (` (Proj) G) (cv x))) (<_) (` (norm) (` (` (Proj) H) (cv x)))))) (pjco.2 pjco.1 x cv pjssmt pjco.2 pjco.1 x cv pjssge0t syld pjco.2 pjco.1 x cv pjdifnormt sylibd com12 r19.21aiv pjco.1 x cv pjhcl Proj G ` x cv ` normclt syl 0re jctir norm Proj G ` x cv ` ` 0 letri3t biimprd syl pjco.1 x cv pjhcl Proj G ` x cv ` normge0t syl sylan2i anabsi6 norm Proj H ` x cv ` ` 0 norm Proj G ` x cv ` ` <_ breq2 biimpac sylan2 exp32 imp pjco.2 x cv pjhcl Proj H ` x cv ` norm-it syl pjco.2 x cv pjoc2t bitr4d norm Proj G ` x cv ` ` <_ norm Proj H ` x cv ` ` br adantr pjco.1 x cv pjhcl Proj G ` x cv ` norm-it syl pjco.1 x cv pjoc2t bitr4d norm Proj G ` x cv ` ` <_ norm Proj H ` x cv ` ` br adantr 3imtr3d exp a2i pjco.2 chocl x cv chel syl5 pm2.43d x 19.20i x H~ norm Proj G ` x cv ` ` <_ norm Proj H ` x cv ` ` br df-ral _|_ H ` _|_ G ` x dfss2 3imtr4 pjco.1 pjco.2 chsscon3 sylibr impbi))
+thm (pjorthco ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) (0H)))) (pjco.1 pjco.2 chsscon2 H _|_ G ` Proj H ` x cv ` ssel sylbi pjco.2 x cv pjcl syl5com pjco.2 x cv pjhcl pjco.1 Proj H ` x cv ` pjoc2t syl sylibd impcom pjco.1 pjco.2 x cv pjco G _|_ H ` C_ adantl x cv pjch0t G _|_ H ` C_ adantl 3eqtr4d exp r19.21aiv pjco.1 pjf pjco.2 pjf hocof h0elch pjf x hoeq sylib))
+thm (pjscj ((H x) (G x)) ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (` (Proj) (opr G (vH) H)) (opr (` (Proj) G) (+P) (` (Proj) H)))) (pjco.1 pjco.2 G H x cv pjcjt2 mp3an12 impcom pjco.1 pjf pjco.2 pjf Proj G ` Proj H ` x cv hosvalt mpanl12 G _|_ H ` C_ adantl eqtr4d exp r19.21aiv pjco.1 pjco.2 chjcl pjf pjco.1 pjf pjco.2 pjf hosf x hoeq sylib))
+thm (pjssum () ((pjco.1 (e. G (CH))) (pjco.2 (e. H (CH)))) (-> (C_ G (` (_|_) H)) (= (` (Proj) (opr G (+H) H)) (opr (` (Proj) G) (+P) (` (Proj) H)))) (pjco.1 pjco.2 osum Proj fveq2d pjco.1 pjco.2 pjscj eqtrd))
+thm (pjidmco () ((pjidmco.1 (e. H (CH)))) (= (o. (` (Proj) H) (` (Proj) H)) (` (Proj) H)) (H ssid pjidmco.1 pjidmco.1 pjss2co mpbi))
+thm (pjocco () ((pjidmco.1 (e. H (CH)))) (= (o. (` (Proj) H) (` (Proj) (` (_|_) H))) (` (Proj) (0H))) (pjidmco.1 chssi H ococss ax-mp pjidmco.1 pjidmco.1 chocl pjorthco ax-mp))
+thm (pjtot ((H x)) ((pjidmco.1 (e. H (CH)))) (= (opr (` (Proj) H) (+P) (` (Proj) (` (_|_) H))) (` (Proj) (H~))) (pjidmco.1 H x cv axpjpjt mpan x cv pjch1t pjidmco.1 pjf pjidmco.1 chocl pjf Proj H ` Proj _|_ H ` ` x cv hosvalt mpanl12 3eqtr4rd rgen pjidmco.1 pjf pjidmco.1 chocl pjf hosf helch pjf x hoeq mpbi))
+thm (pjoc () ((pjidmco.1 (e. H (CH)))) (= (opr (` (Proj) (H~)) (-P) (` (Proj) H)) (` (Proj) (` (_|_) H))) (pjidmco.1 pjtot helch pjf pjidmco.1 pjf pjidmco.1 chocl pjf hods mpbir))
+thm (pjin1 () ((pjin1.1 (e. G (CH))) (pjin1.2 (e. H (CH)))) (= (` (Proj) (i^i G H)) (o. (` (Proj) G) (` (Proj) (i^i G H)))) (G H inss1 pjin1.1 pjin1.2 chincl pjin1.1 pjss1co mpbi eqcomi))
+thm (pjin2 () ((pjin1.1 (e. G (CH))) (pjin1.2 (e. H (CH)))) (<-> (/\ (= (` (Proj) G) (o. (` (Proj) G) (` (Proj) H))) (= (` (Proj) H) (o. (` (Proj) H) (` (Proj) G)))) (= (` (Proj) G) (` (Proj) H))) (G H eqss pjin1.1 pjin1.2 pjss2co Proj G ` Proj H ` o. Proj G ` eqcom bitr pjin1.2 pjin1.1 pjss2co Proj H ` Proj G ` o. Proj H ` eqcom bitr anbi12i bitr2 G H Proj fveq2 sylbi Proj G ` Proj H ` Proj G ` coeq2 pjin1.1 pjidmco syl5eqr Proj G ` Proj H ` Proj H ` coeq2 pjin1.2 pjidmco syl6req jca impbi))
+thm (pjin3 () ((pjin3.1 (e. F (CH))) (pjin3.2 (e. G (CH))) (pjin3.3 (e. H (CH)))) (<-> (/\ (= (` (Proj) F) (o. (` (Proj) F) (` (Proj) G))) (= (` (Proj) F) (o. (` (Proj) F) (` (Proj) H)))) (= (` (Proj) F) (o. (` (Proj) F) (` (Proj) (i^i G H))))) (F G H ssin pjin3.1 pjin3.2 pjss2co Proj F ` Proj G ` o. Proj F ` eqcom bitr pjin3.1 pjin3.3 pjss2co Proj F ` Proj H ` o. Proj F ` eqcom bitr anbi12i pjin3.1 pjin3.2 pjin3.3 chincl pjss2co Proj F ` Proj G H i^i ` o. Proj F ` eqcom bitr 3bitr3))
+thm (pjclem1 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (br G (C_H) H) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) (i^i G H)))) (pjclem1.1 pjclem1.2 cmbr G G H i^i vH G _|_ H ` i^i opr Proj fveq2 sylbi G H inss2 pjclem1.2 pjclem1.1 chocl chub2 pjclem1.1 pjclem1.2 chdmm3 sseqtr4 sstri pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 chocl chincl pjscj ax-mp Proj G ` eqeq2i Proj G ` Proj G H i^i ` +P Proj G _|_ H ` i^i ` opr Proj H ` coeq2 pjclem1.2 pjclem1.1 pjclem1.2 chincl pjf pjclem1.1 pjclem1.2 chocl chincl pjf pjsdi G H inss2 pjclem1.1 pjclem1.2 chincl pjclem1.2 pjss1co mpbi pjclem1.2 pjclem1.1 chocl chub2 pjclem1.1 pjclem1.2 chdmm3 sseqtr4 pjclem1.2 pjclem1.1 pjclem1.2 chocl chincl pjorthco ax-mp +P opreq12i pjclem1.1 pjclem1.2 chincl pjf hoid0 3eqtr Proj H ` Proj G ` o. eqeq2i Proj H ` Proj G ` o. Proj G H i^i ` Proj G ` coeq2 G H inss1 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjss1co mpbi syl6eq sylbi syl sylbi syl pjclem1.1 pjclem1.2 cmcm3 pjclem1.1 chocl pjclem1.2 cmbr bitr _|_ G ` _|_ G ` H i^i vH _|_ G ` _|_ H ` i^i opr Proj fveq2 _|_ G ` H inss2 pjclem1.2 pjclem1.1 chub2 pjclem1.1 pjclem1.2 chdmm4 sseqtr4 sstri pjclem1.1 chocl pjclem1.2 chincl pjclem1.1 chocl pjclem1.2 chocl chincl pjscj ax-mp Proj _|_ G ` ` eqeq2i Proj _|_ G ` ` Proj _|_ G ` H i^i ` +P Proj _|_ G ` _|_ H ` i^i ` opr Proj H ` coeq2 pjclem1.2 pjclem1.1 chocl pjclem1.2 chincl pjf pjclem1.1 chocl pjclem1.2 chocl chincl pjf pjsdi _|_ G ` H inss2 pjclem1.1 chocl pjclem1.2 chincl pjclem1.2 pjss1co mpbi pjclem1.2 pjclem1.1 chub2 pjclem1.1 pjclem1.2 chdmm4 sseqtr4 pjclem1.2 pjclem1.1 chocl pjclem1.2 chocl chincl pjorthco ax-mp +P opreq12i pjclem1.1 chocl pjclem1.2 chincl pjf hoid0 3eqtr Proj H ` Proj _|_ G ` ` o. eqeq2i Proj H ` Proj _|_ G ` ` o. Proj _|_ G ` H i^i ` Proj G ` coeq2 pjclem1.1 pjclem1.2 chocl chub1 pjclem1.1 pjclem1.2 chdmm2 sseqtr4 pjclem1.1 pjclem1.1 chocl pjclem1.2 chincl pjorthco ax-mp syl6eq sylbi syl sylbi syl sylbi +P opreq12d pjclem1.1 pjtot Proj H ` coeq2i pjclem1.2 pjclem1.1 pjf pjclem1.1 chocl pjf pjsdi pjclem1.2 pjf hoid1 3eqtr3r Proj G ` coeq2i pjclem1.1 pjclem1.2 pjf pjclem1.1 pjf hocof pjclem1.2 pjf pjclem1.1 chocl pjf hocof pjsdi eqtr2 pjclem1.1 pjclem1.2 chincl pjf hoid0 3eqtr3g))
+thm (pjclem2 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (br G (C_H) H) (= (o. (` (Proj) G) (` (Proj) H)) (o. (` (Proj) H) (` (Proj) G)))) (pjclem1.1 pjclem1.2 pjclem1 pjclem1.1 pjclem1.2 cmcm pjclem1.2 pjclem1.1 pjclem1 sylbi G H incom Proj fveq2i syl6reqr eqtrd))
+thm (pjclem3 () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (= (o. (` (Proj) G) (` (Proj) H)) (o. (` (Proj) H) (` (Proj) G))) (= (o. (` (Proj) G) (` (Proj) (` (_|_) H))) (o. (` (Proj) (` (_|_) H)) (` (Proj) G)))) (Proj G ` Proj H ` o. Proj H ` Proj G ` o. Proj H~ ` Proj G ` o. -P opreq2 pjclem1.1 pjf hoid1 pjclem1.1 pjf hoid1r eqtr4 -P Proj G ` Proj H ` o. opreq1i syl5eq pjclem1.1 helch pjf pjclem1.2 pjf pjddi helch pjf pjclem1.2 pjf pjclem1.1 pjf hoddir 3eqtr4g pjclem1.2 pjoc Proj G ` coeq2i pjclem1.2 pjoc Proj G ` coeq1i 3eqtr3g))
+thm (pjclem4a () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (e. A (i^i G H)) (= (` (o. (` (Proj) G) (` (Proj) H)) A) A)) (A G H elin pjclem1.2 A chel A G e. adantl pjclem1.1 pjclem1.2 A pjco syl pjclem1.2 A chel pjclem1.2 H A pjcht mpan syl ibi Proj H ` A ` A G eleq1 pjclem1.1 Proj H ` A ` chel pjclem1.1 G Proj H ` A ` pjcht mpan syl ibi syl6bir Proj H ` A ` A Proj G ` Proj H ` A ` ` eqeq2 sylibd syl impcom eqtrd sylbi))
+thm (pjclem4 ((x y) (G x) (G y) (H x) (H y)) ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (-> (= (o. (` (Proj) G) (` (Proj) H)) (o. (` (Proj) H) (` (Proj) G))) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) (i^i G H)))) (pjclem1.1 pjclem1.2 chincl Proj G ` Proj H ` o. x cv ` x cv -v Proj G ` Proj H ` o. x cv ` opr pjv pjclem1.1 pjclem1.2 x cv pjcocl Proj G ` Proj H ` o. Proj H ` Proj G ` o. = adantl Proj G ` Proj H ` o. Proj H ` Proj G ` o. x cv fveq1 H eleq1d pjclem1.2 pjclem1.1 x cv pjcocl syl5bir imp jca Proj G ` Proj H ` o. x cv ` G H elin sylibr pjclem1.1 pjclem1.2 x cv pjcohcl x cv Proj G ` Proj H ` o. x cv ` hvsubclt mpdan Proj G ` Proj H ` o. Proj H ` Proj G ` o. = adantl x cv H~ e. y cv G H i^i e. pm3.26 pjclem1.1 pjclem1.2 x cv pjcohcl y cv G H i^i e. adantr pjclem1.1 pjclem1.2 chincl y cv chel x cv H~ e. adantl 3jca Proj G ` Proj H ` o. Proj H ` Proj G ` o. = adantl x cv Proj G ` Proj H ` o. x cv ` y cv his2subt syl Proj G ` Proj H ` o. Proj H ` Proj G ` o. x cv fveq1 .i y cv opreq1d pjclem1.2 pjclem1.1 x cv y cv pjadjco pjclem1.1 pjclem1.2 chincl y cv chel sylan2 pjclem1.1 pjclem1.2 y cv pjclem4a x cv .i opreq2d x cv H~ e. adantl eqtrd sylan9eq - Proj G ` Proj H ` o. x cv ` .i y cv opr opreq1d pjclem1.1 pjclem1.2 x cv pjcohcl pjclem1.1 pjclem1.2 chincl y cv chel anim12i Proj G ` Proj H ` o. Proj H ` Proj G ` o. = adantl Proj G ` Proj H ` o. x cv ` y cv ax-hicl Proj G ` Proj H ` o. x cv ` .i y cv opr subidt 3syl eqtr3d eqtrd exp32 imp r19.21aiv jca pjclem1.1 pjclem1.2 chincl chshi G H i^i x cv -v Proj G ` Proj H ` o. x cv ` opr y shocelt ax-mp sylibr sylanc Proj G ` Proj H ` o. x cv ` x cv Proj G ` Proj H ` o. x cv ` hvaddsub12t pjclem1.1 pjclem1.2 x cv pjcohcl x cv H~ e. id pjclem1.1 pjclem1.2 x cv pjcohcl syl3anc pjclem1.1 pjclem1.2 x cv pjcohcl Proj G ` Proj H ` o. x cv ` hvsubidt syl x cv +v opreq2d x cv ax-hvaddid 3eqtrd Proj G H i^i ` fveq2d Proj G ` Proj H ` o. Proj H ` Proj G ` o. = adantl eqtr3d exp r19.21aiv pjclem1.1 pjf pjclem1.2 pjf hocof pjclem1.1 pjclem1.2 chincl pjf x hoeq sylib))
+thm (pjc () ((pjclem1.1 (e. G (CH))) (pjclem1.2 (e. H (CH)))) (<-> (br G (C_H) H) (= (o. (` (Proj) G) (` (Proj) H)) (o. (` (Proj) H) (` (Proj) G)))) (pjclem1.1 pjclem1.2 pjclem2 pjclem1.1 pjclem1.2 pjclem4 pjclem1.1 pjclem1.2 pjclem3 pjclem1.1 pjclem1.2 chocl pjclem4 syl +P opreq12d pjclem1.2 pjtot Proj G ` coeq2i pjclem1.1 pjclem1.2 pjf pjclem1.2 chocl pjf pjsdi pjclem1.1 pjf hoid1 3eqtr3r G H inss2 pjclem1.2 pjclem1.1 chocl chub2 sstri pjclem1.1 pjclem1.2 chdmm3 sseqtr4 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 chocl chincl pjscj ax-mp 3eqtr4g pjclem1.1 pjclem1.1 pjclem1.2 chincl pjclem1.1 pjclem1.2 chocl chincl chjcl pj11 sylib pjclem1.1 pjclem1.2 cmbr sylibr impbi))
+thm (pjcohocl () ((pjcohocl.1 (e. H (CH))) (pjcohocl.2 (:--> T (H~) (H~)))) (-> (e. A (H~)) (e. (` (o. (` (Proj) H) T) A) H)) (pjcohocl.1 pjf pjcohocl.2 A hoco pjcohocl.2 A ffvrni pjcohocl.1 T A ` pjcl syl eqeltrd))
+thm (pjadj2co () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) A) (.i) B) (opr A (.i) (` (o. (o. (` (Proj) H) (` (Proj) G)) (` (Proj) F)) B)))) (pjadj2co.1 pjadj2co.2 Proj H ` A ` B pjadjco pjadj2co.3 A pjhcl sylan pjadj2co.3 A Proj G ` Proj F ` o. B ` pjadjt pjadj2co.2 pjadj2co.1 B pjcohcl sylan2 eqtrd pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf A hoco .i B opreq1d B H~ e. adantr pjadj2co.3 pjf pjadj2co.2 pjf pjadj2co.1 pjf hocof B hoco Proj H ` Proj G ` Proj F ` coass B fveq1i syl5eq A .i opreq2d A H~ e. adantl 3eqtr4d))
+thm (pj2cocl () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (e. A (H~)) (e. (` (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) A) F)) (pjadj2co.1 pjf pjadj2co.2 pjf pjadj2co.3 pjf A ho2co pjadj2co.3 A pjhcl pjadj2co.2 Proj H ` A ` pjhcl pjadj2co.1 Proj G ` Proj H ` A ` ` pjcl 3syl eqeltrd))
+thm (pj3lem1 () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (e. A (i^i (i^i F G) H)) (= (` (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) A) A)) (A F G H i^i elin pjadj2co.1 A chel A G H i^i e. adantr pjadj2co.1 pjf pjadj2co.2 pjf pjadj2co.3 pjf hocof A hoco syl pjadj2co.2 pjadj2co.3 A pjclem4a Proj G ` Proj H ` o. A ` A F eleq1 pjadj2co.1 Proj G ` Proj H ` o. A ` chel pjadj2co.1 F Proj G ` Proj H ` o. A ` pjcht mpan syl ibi syl6bir Proj G ` Proj H ` o. A ` A Proj F ` Proj G ` Proj H ` o. A ` ` eqeq2 sylibd syl impcom eqtrd sylbi F G H inass A eleq2i Proj F ` Proj G ` Proj H ` coass A fveq1i A eqeq1i 3imtr4))
+thm (pj3s ((x y) (F x) (F y) (G x) (G y) (H x) (H y)) ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) H) (` (Proj) G)) (` (Proj) F))) (C_ (ran (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H))) G)) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (` (Proj) (i^i (i^i F G) H)))) (pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl Proj F ` Proj G ` o. Proj H ` o. x cv ` x cv -v Proj F ` Proj G ` o. Proj H ` o. x cv ` opr pjv pjadj2co.1 pjadj2co.2 pjadj2co.3 x cv pj2cocl Proj F ` Proj G ` o. Proj H ` o. ran G C_ adantl Proj F ` Proj G ` o. Proj H ` o. ran G Proj F ` Proj G ` o. Proj H ` o. x cv ` ssel pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocofn Proj F ` Proj G ` o. Proj H ` o. H~ x cv fnfvrn mpan syl5 imp jca Proj F ` Proj G ` o. Proj H ` o. x cv ` F G elin sylibr Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = adantll Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. x cv fveq1 H eleq1d pjadj2co.3 pjadj2co.2 pjadj2co.1 x cv pj2cocl syl5bir imp Proj F ` Proj G ` o. Proj H ` o. ran G C_ adantlr jca Proj F ` Proj G ` o. Proj H ` o. x cv ` F G i^i H elin sylibr pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl x cv Proj F ` Proj G ` o. Proj H ` o. x cv ` hvsubclt mpdan Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. ran G C_ /\ adantl x cv H~ e. y cv F G i^i H i^i e. pm3.26 pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl y cv F G i^i H i^i e. adantr pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl y cv chel x cv H~ e. adantl 3jca Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. ran G C_ /\ adantl x cv Proj F ` Proj G ` o. Proj H ` o. x cv ` y cv his2subt syl Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. x cv fveq1 Proj F ` Proj G ` o. Proj H ` o. ran G C_ adantr .i y cv opreq1d pjadj2co.3 pjadj2co.2 pjadj2co.1 x cv y cv pjadj2co pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl y cv chel sylan2 pjadj2co.1 pjadj2co.2 pjadj2co.3 y cv pj3lem1 x cv .i opreq2d x cv H~ e. adantl eqtrd sylan9eq - Proj F ` Proj G ` o. Proj H ` o. x cv ` .i y cv opr opreq1d pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl y cv chel anim12i Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. ran G C_ /\ adantl Proj F ` Proj G ` o. Proj H ` o. x cv ` y cv ax-hicl Proj F ` Proj G ` o. Proj H ` o. x cv ` .i y cv opr subidt 3syl eqtr3d eqtrd exp32 imp r19.21aiv jca pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl chshi F G i^i H i^i x cv -v Proj F ` Proj G ` o. Proj H ` o. x cv ` opr y shocelt ax-mp sylibr sylanc Proj F ` Proj G ` o. Proj H ` o. x cv ` x cv Proj F ` Proj G ` o. Proj H ` o. x cv ` hvaddsub12t pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl x cv H~ e. id pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl syl3anc pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl Proj F ` Proj G ` o. Proj H ` o. x cv ` hvsubidt syl x cv +v opreq2d x cv ax-hvaddid 3eqtrd Proj F G i^i H i^i ` fveq2d Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. ran G C_ /\ adantl eqtr3d exp r19.21aiv pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocof pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl pjf x hoeq sylib))
+thm (pj3 () ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) H) (` (Proj) G)) (` (Proj) F))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) G) (` (Proj) F)) (` (Proj) H)))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (` (Proj) (i^i (i^i F G) H)))) (pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3s Proj G ` Proj F ` Proj H ` o. rnco pjadj2co.2 pjrn sseqtr Proj G ` Proj F ` Proj H ` coass Proj F ` Proj G ` o. Proj H ` o. Proj G ` Proj F ` o. Proj H ` o. Proj G ` Proj F ` Proj H ` o. o. eqeq1 mpbiri rneqd G sseq1d mpbiri sylan2))
+thm (pj3cor1 ((x y) (F x) (F y) (G x) (G y) (H x) (H y)) ((pjadj2co.1 (e. F (CH))) (pjadj2co.2 (e. G (CH))) (pjadj2co.3 (e. H (CH)))) (-> (/\ (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) H) (` (Proj) G)) (` (Proj) F))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) G) (` (Proj) F)) (` (Proj) H)))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) H) (` (Proj) F)) (` (Proj) G)))) (Proj F ` Proj G ` o. Proj H ` o. Proj G ` Proj F ` o. Proj H ` o. y cv fveq1 x cv .i opreq2d Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = adantl x cv H~ e. y cv H~ e. ad2antlr pjadj2co.1 pjadj2co.2 chincl pjadj2co.3 chincl x cv y cv pjadjt Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. Proj G ` Proj F ` o. Proj H ` o. = /\ adantlr pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3 x cv fveq1d .i y cv opreq1d x cv H~ e. y cv H~ e. ad2antlr pjadj2co.1 pjadj2co.2 pjadj2co.3 pj3 y cv fveq1d x cv .i opreq2d x cv H~ e. y cv H~ e. ad2antlr 3eqtr4d pjadj2co.3 pjadj2co.1 pjadj2co.2 x cv y cv pjadj2co Proj F ` Proj G ` o. Proj H ` o. Proj H ` Proj G ` o. Proj F ` o. = Proj F ` Proj G ` o. Proj H ` o. Proj G ` Proj F ` o. Proj H ` o. = /\ adantlr 3eqtr4d exp31 r19.21adv Proj F ` Proj G ` o. Proj H ` o. x cv ` Proj H ` Proj F ` o. Proj G ` o. x cv ` y hial2eqt pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf x cv hococl pjadj2co.3 pjf pjadj2co.1 pjf hocof pjadj2co.2 pjf x cv hococl sylanc sylibd com12 r19.21aiv pjadj2co.1 pjf pjadj2co.2 pjf hocof pjadj2co.3 pjf hocof pjadj2co.3 pjf pjadj2co.1 pjf hocof pjadj2co.2 pjf hocof x hoeq sylib))
+thm (pjopytht () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (opr (` (norm) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A))) (^) (2)) (opr (opr (` (norm) (` (` (Proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (Proj) G) A)) (^) (2)))))) (H H CH e. H H~ if _|_ G ` sseq1 H H CH e. H H~ if Proj fveq2 A fveq1d +v Proj G ` A ` opreq1d norm fveq2d ^ 2 opreq1d H H CH e. H H~ if Proj fveq2 A fveq1d norm fveq2d ^ 2 opreq1d + norm Proj G ` A ` ` ^ 2 opr opreq1d eqeq12d imbi12d G G CH e. G H~ if _|_ fveq2 H CH e. H H~ if sseq2d G G CH e. G H~ if Proj fveq2 A fveq1d Proj H CH e. H H~ if ` A ` +v opreq2d norm fveq2d ^ 2 opreq1d G G CH e. G H~ if Proj fveq2 A fveq1d norm fveq2d ^ 2 opreq1d norm Proj H CH e. H H~ if ` A ` ` ^ 2 opr + opreq2d eqeq12d imbi12d A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 A A H~ e. A 0v if Proj G CH e. G H~ if ` fveq2 +v opreq12d norm fveq2d ^ 2 opreq1d A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 norm fveq2d ^ 2 opreq1d A A H~ e. A 0v if Proj G CH e. G H~ if ` fveq2 norm fveq2d ^ 2 opreq1d + opreq12d eqeq12d H CH e. H H~ if _|_ G CH e. G H~ if ` C_ imbi2d helch H elimel helch G elimel ax-hv0cl A elimel pjopyth dedth3h))
+thm (pjnorm () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) A)) (pjnorm.1 pjnorm.2 pjhcli pjnorm.1 chocl pjnorm.2 pjhcli pm3.2i pjnorm.1 pjnorm.2 pjnorm.2 H pjorth ax-mp Proj H ` A ` Proj _|_ H ` ` A ` normpyct mp2 pjnorm.1 pjnorm.2 pjpj norm fveq2i breqtrr))
+thm (pjpyth () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (= (opr (` (norm) A) (^) (2)) (opr (opr (` (norm) (` (` (Proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (Proj) (` (_|_) H)) A)) (^) (2)))) (pjnorm.1 pjnorm.2 pjpj norm fveq2i ^ 2 opreq1i pjnorm.1 chshi H shococss ax-mp pjnorm.1 pjnorm.1 chocl pjnorm.2 pjopyth ax-mp eqtr))
+thm (pjnel () ((pjnorm.1 (e. H (CH))) (pjnorm.2 (e. A (H~)))) (<-> (-. (e. A H)) (br (` (norm) (` (` (Proj) H) A)) (<) (` (norm) A))) (pjnorm.1 pjnorm.2 pjnorm norm Proj H ` A ` ` norm A ` = -. biantrur pjnorm.1 pjnorm.2 pjoc1 pjnorm.1 pjnorm.2 pjhcli normcl sqrecl recn 0cn sqcl pjnorm.1 chocl pjnorm.2 pjhcli normcl sqrecl recn addcan 0 eqid 0cn sqe0 mpbir norm Proj H ` A ` ` ^ 2 opr + opreq2i pjnorm.1 pjnorm.2 pjhcli normcl sqrecl recn addid1 eqtr2 pjnorm.1 pjnorm.2 pjpyth eqeq12i pjnorm.1 chocl pjnorm.2 pjhcli norm-i norm Proj _|_ H ` ` A ` ` 0 eqcom bitr3 0re leid pjnorm.1 chocl pjnorm.2 pjhcli Proj _|_ H ` ` A ` normge0t ax-mp 0re pjnorm.1 chocl pjnorm.2 pjhcli normcl sqe11 mp2an bitr4 3bitr4r pjnorm.1 pjnorm.2 pjhcli Proj H ` A ` normge0t ax-mp pjnorm.2 A normge0t ax-mp pjnorm.1 pjnorm.2 pjhcli normcl pjnorm.2 normcl sqe11 mp2an 3bitr negbii pjnorm.1 pjnorm.2 pjhcli normcl pjnorm.2 normcl ltlen 3bitr4))
+thm (pjnormt () () (-> (/\ (e. H (CH)) (e. A (H~))) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) A))) (H H CH e. H H~ if Proj fveq2 A fveq1d norm fveq2d <_ norm A ` breq1d A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 norm fveq2d A A H~ e. A 0v if norm fveq2 <_ breq12d helch H elimel ax-hv0cl A elimel pjnorm dedth2h))
+thm (pjnelt () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (-. (e. A H)) (br (` (norm) (` (` (Proj) H) A)) (<) (` (norm) A)))) (H H CH e. H H~ if A eleq2 negbid H H CH e. H H~ if Proj fveq2 A fveq1d norm fveq2d < norm A ` breq1d bibi12d A A H~ e. A 0v if H CH e. H H~ if eleq1 negbid A A H~ e. A 0v if Proj H CH e. H H~ if ` fveq2 norm fveq2d A A H~ e. A 0v if norm fveq2 < breq12d bibi12d helch H elimel ax-hv0cl A elimel pjnel dedth2h))
+thm (pjelt () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (norm) (` (` (Proj) H) A)) (` (norm) A)))) (H A pjnelt bicon1d H A pjnormt norm Proj H ` A ` ` < norm A ` br -. biantrurd bitr3d norm Proj H ` A ` ` norm A ` eqleltt H A pjhclt Proj H ` A ` normclt syl A normclt H CH e. adantl sylanc bitr4d))
+thm (pjs14 () ((pjs14.1 (e. G (CH))) (pjs14.2 (e. H (CH)))) (-> (e. A (H~)) (br (` (norm) (` (o. (` (Proj) H) (` (Proj) G)) A)) (<_) (` (norm) (` (` (Proj) G) A)))) (pjs14.2 pjs14.1 A pjco norm fveq2d pjs14.1 A pjhcl pjs14.2 H Proj G ` A ` pjnormt mpan syl eqbrtrd))
+thm (stelt ((x y) (f x) (f y) (S x) (S y) (S f)) () (<-> (e. S (States)) (/\ (/\ (:--> S (CH) (RR)) (A.e. x (CH) (/\ (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1))))) (/\ (= (` S (H~)) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+) (` S (cv y)))))))))) (S States elisset chex CH V S RR fex ax-mp x CH 0 <_ S x cv ` br S x cv ` <_ 1 br /\ A.e. S H~ ` 1 = x CH y CH x cv _|_ y cv ` C_ S x cv vH y cv opr ` S x cv ` + S y cv ` opr = -> A.e. A.e. /\ ad2antll f cv S CH RR feq1 f cv S x cv fveq1 0 <_ breq2d f cv S x cv fveq1 <_ 1 breq1d anbi12d x CH biraldv anbi12d f cv S H~ fveq1 1 eqeq1d f cv S x cv vH y cv opr fveq1 f cv S x cv fveq1 f cv S y cv fveq1 + opreq12d eqeq12d x cv _|_ y cv ` C_ imbi2d y CH biraldv x CH biraldv anbi12d anbi12d f x y df-st V elab2g pm5.21nii))
+thm (stclt ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (e. (` S A) (RR)))) (S CH RR A ffvrn S x y stelt pm3.26bd pm3.26d sylan exp))
+thm (stge0t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (0) (<_) (` S A)))) (x cv A S fveq2 0 <_ breq2d CH rcla4v S x y stelt pm3.26bd pm3.27d 0 <_ S x cv ` br S x cv ` <_ 1 br pm3.26 x CH r19.20si syl syl5com))
+thm (stle1t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (` S A) (<_) (1)))) (x cv A S fveq2 <_ 1 breq1d CH rcla4v S x y stelt pm3.26bd pm3.27d 0 <_ S x cv ` br S x cv ` <_ 1 br pm3.27 x CH r19.20si syl syl5com))
+thm (sthil ((x y) (S x) (S y)) () (-> (e. S (States)) (= (` S (H~)) (1))) (S x y stelt pm3.27bd pm3.26d))
+thm (stjt ((x y) (A x) (A y) (S x) (S y) (B y)) () (-> (e. S (States)) (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (C_ A (` (_|_) B))) (= (` S (opr A (vH) B)) (opr (` S A) (+) (` S B))))) (x cv A _|_ y cv ` sseq1 x cv A vH y cv opreq1 S fveq2d x cv A S fveq2 + S y cv ` opreq1d eqeq12d imbi12d y cv B _|_ fveq2 A sseq2d y cv B A vH opreq2 S fveq2d y cv B S fveq2 S A ` + opreq2d eqeq12d imbi12d CH CH rcla42v S x y stelt pm3.27bd pm3.27d syl5com imp3a))
+thm (sto1 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (= (opr (` S A) (+) (` S (` (_|_) A))) (1))) (sto1.1 sto1.1 chocl pm3.2i sto1.1 chshi A shococss ax-mp S A _|_ A ` stjt mp2ani S sthil sto1.1 chjo S fveq2i syl5eq eqtr3d))
+thm (sto2 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (= (` S (` (_|_) A)) (opr (1) (-) (` S A)))) (sto1.1 S sto1 1cn 1 S A ` S _|_ A ` ` subaddt mp3an1 sto1.1 S A stclt mpi recnd sto1.1 chocl S _|_ A ` stclt mpi recnd sylanc mpbird eqcomd))
+thm (stge1 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (<-> (br (1) (<_) (` S A)) (= (` S A) (1)))) (sto1.1 S A stle1t mpi 1 <_ S A ` br anim1i exp sto1.1 S A stclt mpi ax1re jctir S A ` 1 letri3t syl sylibrd ax1re leid S A ` 1 1 <_ breq2 mpbiri S States e. a1i impbid))
+thm (stle0 () ((sto1.1 (e. A (CH)))) (-> (e. S (States)) (<-> (br (` S A) (<_) (0)) (= (` S A) (0)))) (sto1.1 S A stge0t mpi S A ` <_ 0 br anim2i expcom sto1.1 S A stclt mpi 0re jctir S A ` 0 letri3t syl sylibrd 0re leid S A ` 0 <_ 0 breq1 mpbiri S States e. a1i impbid))
+thm (stle () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (C_ A B) (br (` S A) (<_) (` S B)))) (S A _|_ B ` stjt stle.2 chshi B shococss ax-mp A B _|_ _|_ B ` ` sstr2 mpi stle.1 stle.2 chocl pm3.2i jctil syl5 imp stle.1 stle.2 chocl chjcl S A vH _|_ B ` opr stle1t mpi stle.2 S sto1 breqtrrd A B C_ adantr eqbrtrrd stle.1 S A stclt mpi stle.2 S B stclt mpi stle.2 chocl S _|_ B ` stclt mpi 3jca A B C_ adantr S A ` S B ` S _|_ B ` ` leadd1t syl mpbird exp))
+thm (stles () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (C_ A B) (-> (= (` S A) (1)) (= (` S B) (1))))) (stle.2 S B stle1t mpi A B C_ S A ` 1 = /\ adantr stle.1 stle.2 S stle imp S A ` 1 = adantrr S A ` 1 <_ S B ` breq1 S States e. A B C_ ad2antrr mpbid jca stle.2 S B stclt mpi ax1re jctir A B C_ S A ` 1 = /\ adantr S B ` 1 letri3t syl mpbird exp32))
+thm (stji1 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (= (` S (opr (` (_|_) A) (vH) (i^i A B))) (opr (` S (` (_|_) A)) (+) (` S (i^i A B))))) (stle.1 chocl stle.1 stle.2 chincl pm3.2i A B inss1 stle.1 stle.2 chincl stle.1 chsscon3 mpbi S _|_ A ` A B i^i stjt mp2ani))
+thm (stm1 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (` S A) (1)))) (S A B i^i ` 1 <_ S A ` breq1 A B inss1 stle.1 stle.2 chincl stle.1 S stle mpi syl5bi com12 stle.1 S stge1 sylibd))
+thm (stm1r () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (` S B) (1)))) (stle.2 stle.1 S stm1 A B incom S fveq2i 1 eqeq1i syl5ib))
+thm (stm1add () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (` S (i^i A B)) (1)) (= (opr (` S A) (+) (` S B)) (2)))) (stle.1 stle.2 S stm1 stle.1 stle.2 S stm1r jcad S A ` 1 S B ` 1 + opreq12 df-2 syl6eqr syl6))
+thm (stadd () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH)))) (-> (e. S (States)) (-> (= (opr (` S A) (+) (` S B)) (2)) (= (` S A) (1)))) (S A ` S B ` axaddrcl stle.1 S A stclt mpi stle.2 S B stclt mpi sylanc 2re jctir S A ` + S B ` opr 2 ltnet syl con2d stle.2 S B stle1t mpi S B ` 1 S A ` leadd2t stle.2 S B stclt mpi ax1re S States e. a1i stle.1 S A stclt mpi syl3anc mpbid S A ` < 1 br adantr S A ` 1 1 ltadd1t biimpd stle.1 S A stclt mpi ax1re S States e. a1i ax1re S States e. a1i syl3anc imp S A ` + S B ` opr S A ` + 1 opr 1 + 1 opr lelttrt S A ` S B ` axaddrcl stle.1 S A stclt mpi stle.2 S B stclt mpi sylanc stle.1 S A stclt mpi ax1re jctir S A ` 1 axaddrcl syl ax1re ax1re readdcl S States e. a1i syl3anc S A ` < 1 br adantr mp2and df-2 syl6breqr exp con3d stle.1 S A stle1t mpi stle.1 S A stclt mpi ax1re jctir S A ` 1 leloet syl mpbid ord syld syld))
+thm (stm1add3 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH))) (stm1add3.3 (e. C (CH)))) (-> (e. S (States)) (-> (= (` S (i^i (i^i A B) C)) (1)) (= (opr (opr (` S A) (+) (` S B)) (+) (` S C)) (3)))) (stle.1 stle.2 chincl stm1add3.3 S stm1 stle.1 stle.2 S stm1add syld stle.1 stle.2 chincl stm1add3.3 S stm1r jcad S A ` + S B ` opr 2 S C ` 1 + opreq12 df-3 syl6eqr syl6))
+thm (stadd3 () ((stle.1 (e. A (CH))) (stle.2 (e. B (CH))) (stm1add3.3 (e. C (CH)))) (-> (e. S (States)) (-> (= (opr (opr (` S A) (+) (` S B)) (+) (` S C)) (3)) (= (` S A) (1)))) (S A ` S B ` S C ` axaddass stle.1 S A stclt mpi recnd stle.2 S B stclt mpi recnd stm1add3.3 S C stclt mpi recnd syl3anc 3 eqeq1d S A ` S B ` + S C ` opr axaddrcl stle.1 S A stclt mpi S B ` S C ` axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc sylanc 3re jctir S A ` + S B ` + S C ` opr opr 3 ltnet syl con2d S B ` S C ` axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc stle.2 S B stclt mpi ax1re jctir S B ` 1 axaddrcl syl ax1re ax1re readdcl S States e. a1i stm1add3.3 S C stle1t mpi S C ` 1 S B ` leadd2t stm1add3.3 S C stclt mpi ax1re S States e. a1i stle.2 S B stclt mpi syl3anc mpbid stle.2 S B stle1t mpi S B ` 1 1 leadd1t stle.2 S B stclt mpi ax1re S States e. a1i ax1re S States e. a1i syl3anc mpbid letrd S B ` + S C ` opr 1 + 1 opr S A ` leadd2t S B ` S C ` axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc ax1re ax1re readdcl S States e. a1i stle.1 S A stclt mpi syl3anc mpbid S A ` < 1 br adantr S A ` 1 1 + 1 opr ltadd1t biimpd stle.1 S A stclt mpi ax1re S States e. a1i ax1re ax1re readdcl S States e. a1i syl3anc imp S A ` + S B ` + S C ` opr opr S A ` + 1 + 1 opr opr 1 + 1 + 1 opr opr lelttrt S A ` S B ` + S C ` opr axaddrcl stle.1 S A stclt mpi S B ` S C ` axaddrcl stle.2 S B stclt mpi stm1add3.3 S C stclt mpi sylanc sylanc stle.1 S A stclt mpi ax1re ax1re readdcl jctir S A ` 1 + 1 opr axaddrcl syl ax1re ax1re ax1re readdcl readdcl S States e. a1i syl3anc S A ` < 1 br adantr mp2and df-3 df-2 + 1 opreq1i 1cn 1cn 1cn addass eqtr eqtr2 syl6breq exp con3d stle.1 S A stle1t mpi stle.1 S A stclt mpi ax1re jctir S A ` 1 leloet syl mpbid ord syld syld sylbid))
+thm (st0 () () (-> (e. S (States)) (= (` S (0H)) (0))) (helch S sto2 S sthil 1 - opreq2d eqtrd choc1 S fveq2i 1cn subid 3eqtr3g))
+thm (strlem1 ((u x) (A u) (A x) (B u) (B x)) ((strlem1.1 (e. A (CH))) (strlem1.2 (e. B (CH)))) (-> (-. (C_ A B)) (E.e. u (\ A B) (= (` (norm) (cv u)) (1)))) (A B ssdif0 negbii A B \ x n0 bitr u cv 1 / norm x cv ` opr .s x cv opr norm fveq2 1 eqeq1d A B \ rcla4ev norm x cv ` rerecclt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc recnd strlem1.1 chshi A 1 / norm x cv ` opr x cv shmulclt ax-mp exp syl x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl recnd strlem1.2 chshi B norm x cv ` 1 / norm x cv ` opr .s x cv opr shmulclt ax-mp exp syl norm x cv ` recidt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl recnd strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc .s x cv opreq1d norm x cv ` 1 / norm x cv ` opr x cv ax-hvmulass x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl recnd norm x cv ` rerecclt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc recnd x cv A B eldifi strlem1.1 x cv chel syl syl3anc x cv A B eldifi strlem1.1 x cv chel x cv ax-hvmulid 3syl 3eqtr3d B eleq1d sylibd con3d anim12d x cv A B eldif 1 / norm x cv ` opr .s x cv opr A B eldif 3imtr4g pm2.43i 1 / norm x cv ` opr x cv norm-iiit norm x cv ` rerecclt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc recnd x cv A B eldifi strlem1.1 x cv chel syl sylanc 1 / norm x cv ` opr absidt norm x cv ` rerecclt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc 1 norm x cv ` divge0t x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl ax1re jctil strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv normgt0t 3syl mpbid 0re ax1re lt01 ltlei jctil sylanc sylanc x. norm x cv ` opreq1d 1 / norm x cv ` opr norm x cv ` axmulcom norm x cv ` rerecclt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc recnd x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl recnd sylanc norm x cv ` recidt x cv A B eldifi strlem1.1 x cv chel x cv normclt 3syl recnd strlem1.2 B ch0 ax-mp 0v A B eldifn mt2 x cv 0v A B \ eleq1 mtbiri con2i x cv A B eldifi strlem1.1 x cv chel x cv norm-it 3syl mtbird norm x cv ` 0 df-ne sylibr sylanc eqtrd 3eqtrd sylanc x 19.23aiv sylbi))
+thm (strlem2 ((x y) (C x) (C y) (u x) (u y)) ((strlem2.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2)))))))) (-> (e. C (CH)) (= (` S C) (opr (` (norm) (` (` (Proj) C) (cv u))) (^) (2)))) (x cv C Proj fveq2 u cv fveq1d norm fveq2d ^ 2 opreq1d strlem2.1 norm Proj C ` u cv ` ` ^ 2 oprex fvopab4))
+thm (strlem3a ((x z) (w x) (u x) (w z) (u z) (u w) (S z) (S w) (x y) (u y) (y z) (w y)) ((strlem3a.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2)))))))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (e. S (States))) (x cv u cv pjhclt norm u cv ` 1 = adantrr Proj x cv ` u cv ` normclt norm Proj x cv ` u cv ` ` sqreclt 3syl expcom r19.21aiv strlem3a.1 RR fopab2 sylib z cv u cv pjhclt norm u cv ` 1 = adantrr Proj z cv ` u cv ` normclt norm Proj z cv ` u cv ` ` sqege0t 3syl strlem3a.1 z cv strlem2 u cv H~ e. norm u cv ` 1 = /\ adantr breqtrrd strlem3a.1 z cv strlem2 u cv H~ e. norm u cv ` 1 = /\ adantr z cv u cv pjnormt norm u cv ` 1 = adantrr u cv H~ e. norm u cv ` 1 = pm3.27 z cv CH e. adantl breqtrd norm Proj z cv ` u cv ` ` 1 le2sqet z cv u cv pjhclt norm u cv ` 1 = adantrr Proj z cv ` u cv ` normclt syl ax1re jctir z cv u cv pjhclt norm u cv ` 1 = adantrr Proj z cv ` u cv ` normge0t syl 0re ax1re lt01 ltlei jctir sylc mpbid sq1 syl6breq eqbrtrd jca expcom r19.21aiv jca u cv pjch1t norm fveq2d ^ 2 opreq1d norm u cv ` 1 ^ 2 opreq1 sq1 syl6eq sylan9eq helch strlem3a.1 H~ strlem2 ax-mp syl5eq z cv w cv u cv pjcjt2 imp norm fveq2d ^ 2 opreq1d z cv w cv u cv pjopytht imp eqtrd z cv w cv chjclt u cv H~ e. 3adant3 z cv _|_ w cv ` C_ adantr strlem3a.1 z cv vH w cv opr strlem2 syl z cv CH e. w cv CH e. u cv H~ e. 3simpa z cv _|_ w cv ` C_ adantr strlem3a.1 z cv strlem2 strlem3a.1 w cv strlem2 + opreqan12d syl 3eqtr4d 3exp1 com3r norm u cv ` 1 = adantr r19.21adv r19.21aiv jca jca S z w stelt sylibr))
+thm (strlem3 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (e. S (States))) (strlem3.2 strlem3.1 strlem3a u cv A B eldifi strlem3.3 u cv chel syl sylan sylbi))
+thm (strlem4 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (= (` S A) (1))) (strlem3.2 norm u cv ` 1 norm Proj A ` u cv ` ` eqeq2 strlem3.3 u cv chel strlem3.3 A u cv pjcht mpan syl ibi norm fveq2d syl5bi impcom u cv A B eldifi sylan sylbi ^ 2 opreq1d sq1 syl6eq strlem3.3 strlem3.1 A strlem2 ax-mp syl5eq))
+thm (strlem5 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (br (` S B) (<) (1))) (strlem3.2 norm u cv ` 1 norm Proj B ` u cv ` ` < breq2 u cv A B eldif strlem3.4 B u cv pjnelt mpan biimpa strlem3.3 u cv chel sylan sylbi syl5bi impcom u cv A B eldifi strlem3.3 u cv chel norm Proj B ` u cv ` ` 1 lt2sqet strlem3.4 u cv pjhcl Proj B ` u cv ` normclt syl ax1re jctir strlem3.4 u cv pjhcl Proj B ` u cv ` normge0t syl 0re ax1re lt01 ltlei jctir sylc 3syl norm u cv ` 1 = adantr mpbid strlem3.4 strlem3.1 B strlem2 ax-mp syl5eqbr sq1 syl6breq sylbi))
+thm (strlem6 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((strlem3.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (strlem3.2 (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1))))) (strlem3.3 (e. A (CH))) (strlem3.4 (e. B (CH)))) (-> ph (-. (-> (= (` S A) (1)) (= (` S B) (1))))) (strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem4 strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem5 strlem3.1 strlem3.2 strlem3.3 strlem3.4 strlem3 strlem3.4 S B stclt mpi ax1re S B ` 1 ltnet mpan2 3syl mpd jca S A ` 1 = S B ` 1 = annim sylib))
+thm (str ((x y) (u x) (f x) (A x) (u y) (f y) (A y) (f u) (A u) (A f) (B x) (B y) (B u) (B f)) ((str.1 (e. A (CH))) (str.2 (e. B (CH)))) (-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B)) (f States f cv A ` 1 = f cv B ` 1 = -> dfral2 str.1 str.2 u strlem1 f cv x y x cv CH e. y cv norm Proj x cv ` u cv ` ` ^ 2 opr = /\ {<,>|} A fveq1 1 eqeq1d f cv x y x cv CH e. y cv norm Proj x cv ` u cv ` ` ^ 2 opr = /\ {<,>|} B fveq1 1 eqeq1d imbi12d negbid States rcla4ev x y x cv CH e. y cv norm Proj x cv ` u cv ` ` ^ 2 opr = /\ {<,>|} eqid u cv A B \ e. norm u cv ` 1 = /\ pm4.2 str.1 str.2 strlem3 x y x cv CH e. y cv norm Proj x cv ` u cv ` ` ^ 2 opr = /\ {<,>|} eqid u cv A B \ e. norm u cv ` 1 = /\ pm4.2 str.1 str.2 strlem6 sylanc exp r19.23aiv syl con1i sylbi))
+thm (strb ((A f) (B f)) ((str.1 (e. A (CH))) (str.2 (e. B (CH)))) (<-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B)) (str.1 str.2 f str str.1 str.2 f cv stles com12 r19.21aiv impbi))
+thm (large ((A f)) ((large.1 (e. A (CH)))) (<-> (-. (= A (0H))) (E.e. f (States) (= (` (cv f) A) (1)))) (f States f cv A ` 1 = ralnex ax1ne0 1 0 df-ne mpbi f cv st0 1 eqeq1d 0 1 eqcom syl6bb mtbiri f cv 0H ` 1 = f cv A ` 1 = mtt syl birala large.1 h0elch f strb large.1 chle0 3bitr bitr3 bicon1i))
+thm (jplem1 () ((jplem1.1 (e. A (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (opr (` (norm) (` (` (Proj) A) (cv u))) (^) (2)) (1)))) (jplem1.1 A u cv pjelt mpan norm u cv ` 1 norm Proj A ` u cv ` ` eqeq2 sylan9bb norm Proj A ` u cv ` ` 1 sqe11t jplem1.1 u cv pjhcl Proj A ` u cv ` normclt syl ax1re jctir jplem1.1 u cv pjhcl Proj A ` u cv ` normge0t syl 0re ax1re lt01 ltlei jctir sylc sq1 norm Proj A ` u cv ` ` ^ 2 opr eqeq2i syl5bbr norm u cv ` 1 = adantr bitr4d))
+thm (jplem2 ((u x) (x y) (u y) (A x) (A y)) ((jp.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (jp.2 (e. A (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (` S A) (1)))) (jp.2 u jplem1 jp.2 jp.1 A strlem2 ax-mp 1 eqeq1i syl6bbr))
+thm (jp ((u x) (x y) (u y) (A x) (A y) (B x) (B y)) ((jp.1 (= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (Proj) (cv x)) (cv u))) (^) (2))))))) (jp.2 (e. A (CH))) (jp.3 (e. B (CH)))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (/\ (= (` S A) (1)) (= (` S B) (1))) (= (` S (i^i A B)) (1)))) (jp.1 jp.2 jplem2 jp.1 jp.3 jplem2 anbi12d u cv A B elin syl5bb jp.1 jp.2 jp.3 chincl jplem2 bitr3d))
+thm (golem1 () ((golem1.1 (e. A (CH))) (golem1.2 (e. B (CH))) (golem1.3 (e. C (CH))) (golem1.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (golem1.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (golem1.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (golem1.7 (= D (opr (` (_|_) B) (vH) (i^i B A)))) (golem1.8 (= R (opr (` (_|_) C) (vH) (i^i C B)))) (golem1.9 (= S (opr (` (_|_) A) (vH) (i^i A C))))) (-> (e. (cv f) (States)) (= (opr (opr (` (cv f) F) (+) (` (cv f) G)) (+) (` (cv f) H)) (opr (opr (` (cv f) D) (+) (` (cv f) R)) (+) (` (cv f) S)))) (f cv _|_ A ` ` f cv _|_ B ` ` f cv _|_ C ` ` axaddass golem1.1 chocl f cv _|_ A ` stclt mpi recnd golem1.2 chocl f cv _|_ B ` stclt mpi recnd golem1.3 chocl f cv _|_ C ` stclt mpi recnd syl3anc f cv _|_ A ` ` f cv _|_ B ` ` + f cv _|_ C ` ` opr axaddcom golem1.1 chocl f cv _|_ A ` stclt mpi recnd f cv _|_ B ` ` f cv _|_ C ` ` axaddcl golem1.2 chocl f cv _|_ B ` stclt mpi recnd golem1.3 chocl f cv _|_ C ` stclt mpi recnd sylanc sylanc eqtrd + f cv A B i^i ` + f cv B C i^i ` opr + f cv C A i^i ` opr opreq1d f cv _|_ A ` ` + f cv _|_ B ` ` opr f cv A B i^i ` + f cv B C i^i ` opr f cv _|_ C ` ` f cv C A i^i ` add4t f cv _|_ A ` ` f cv _|_ B ` ` axaddcl golem1.1 chocl f cv _|_ A ` stclt mpi recnd golem1.2 chocl f cv _|_ B ` stclt mpi recnd sylanc f cv A B i^i ` f cv B C i^i ` axaddcl golem1.1 golem1.2 chincl f cv A B i^i stclt mpi recnd golem1.2 golem1.3 chincl f cv B C i^i stclt mpi recnd sylanc jca golem1.3 chocl f cv _|_ C ` stclt mpi recnd golem1.3 golem1.1 chincl f cv C A i^i stclt mpi recnd jca sylanc f cv _|_ B ` ` + f cv _|_ C ` ` opr f cv A B i^i ` + f cv B C i^i ` opr f cv _|_ A ` ` f cv C A i^i ` add4t f cv _|_ B ` ` f cv _|_ C ` ` axaddcl golem1.2 chocl f cv _|_ B ` stclt mpi recnd golem1.3 chocl f cv _|_ C ` stclt mpi recnd sylanc f cv A B i^i ` f cv B C i^i ` axaddcl golem1.1 golem1.2 chincl f cv A B i^i stclt mpi recnd golem1.2 golem1.3 chincl f cv B C i^i stclt mpi recnd sylanc jca golem1.1 chocl f cv _|_ A ` stclt mpi recnd golem1.3 golem1.1 chincl f cv C A i^i stclt mpi recnd jca sylanc 3eqtr4d f cv _|_ A ` ` f cv A B i^i ` f cv _|_ B ` ` f cv B C i^i ` add4t golem1.1 chocl f cv _|_ A ` stclt mpi recnd golem1.1 golem1.2 chincl f cv A B i^i stclt mpi recnd jca golem1.2 chocl f cv _|_ B ` stclt mpi recnd golem1.2 golem1.3 chincl f cv B C i^i stclt mpi recnd jca sylanc + f cv _|_ C ` ` + f cv C A i^i ` opr opreq1d f cv _|_ B ` ` f cv A B i^i ` f cv _|_ C ` ` f cv B C i^i ` add4t golem1.2 chocl f cv _|_ B ` stclt mpi recnd golem1.1 golem1.2 chincl f cv A B i^i stclt mpi recnd jca golem1.3 chocl f cv _|_ C ` stclt mpi recnd golem1.2 golem1.3 chincl f cv B C i^i stclt mpi recnd jca sylanc + f cv _|_ A ` ` + f cv C A i^i ` opr opreq1d 3eqtr4d golem1.1 golem1.2 f cv stji1 golem1.2 golem1.3 f cv stji1 + opreq12d golem1.3 golem1.1 f cv stji1 + opreq12d golem1.2 golem1.1 f cv stji1 B A incom f cv fveq2i f cv _|_ B ` ` + opreq2i syl6eq golem1.3 golem1.2 f cv stji1 C B incom f cv fveq2i f cv _|_ C ` ` + opreq2i syl6eq + opreq12d golem1.1 golem1.3 f cv stji1 A C incom f cv fveq2i f cv _|_ A ` ` + opreq2i syl6eq + opreq12d 3eqtr4d golem1.4 f cv fveq2i golem1.5 f cv fveq2i + opreq12i golem1.6 f cv fveq2i + opreq12i golem1.7 f cv fveq2i golem1.8 f cv fveq2i + opreq12i golem1.9 f cv fveq2i + opreq12i 3eqtr4g))
+thm (golem2 () ((golem1.1 (e. A (CH))) (golem1.2 (e. B (CH))) (golem1.3 (e. C (CH))) (golem1.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (golem1.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (golem1.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (golem1.7 (= D (opr (` (_|_) B) (vH) (i^i B A)))) (golem1.8 (= R (opr (` (_|_) C) (vH) (i^i C B)))) (golem1.9 (= S (opr (` (_|_) A) (vH) (i^i A C))))) (-> (e. (cv f) (States)) (-> (= (` (cv f) (i^i (i^i F G) H)) (1)) (= (` (cv f) D) (1)))) (golem1.4 golem1.1 chocl golem1.1 golem1.2 chincl chjcl eqeltr golem1.5 golem1.2 chocl golem1.2 golem1.3 chincl chjcl eqeltr golem1.6 golem1.3 chocl golem1.3 golem1.1 chincl chjcl eqeltr f cv stm1add3 golem1.1 golem1.2 golem1.3 golem1.4 golem1.5 golem1.6 golem1.7 golem1.8 golem1.9 f golem1 3 eqeq1d sylibd golem1.7 golem1.2 chocl golem1.2 golem1.1 chincl chjcl eqeltr golem1.8 golem1.3 chocl golem1.3 golem1.2 chincl chjcl eqeltr golem1.9 golem1.1 chocl golem1.1 golem1.3 chincl chjcl eqeltr f cv stadd3 syld))
+thm (goeq ((F f) (G f) (H f) (D f)) ((goeq.1 (e. A (CH))) (goeq.2 (e. B (CH))) (goeq.3 (e. C (CH))) (goeq.4 (= F (opr (` (_|_) A) (vH) (i^i A B)))) (goeq.5 (= G (opr (` (_|_) B) (vH) (i^i B C)))) (goeq.6 (= H (opr (` (_|_) C) (vH) (i^i C A)))) (goeq.7 (= D (opr (` (_|_) B) (vH) (i^i B A))))) (C_ (i^i (i^i F G) H) D) (goeq.4 goeq.1 chocl goeq.1 goeq.2 chincl chjcl eqeltr goeq.5 goeq.2 chocl goeq.2 goeq.3 chincl chjcl eqeltr chincl goeq.6 goeq.3 chocl goeq.3 goeq.1 chincl chjcl eqeltr chincl goeq.7 goeq.2 chocl goeq.2 goeq.1 chincl chjcl eqeltr f str goeq.1 goeq.2 goeq.3 goeq.4 goeq.5 goeq.6 goeq.7 _|_ C ` vH C B i^i opr eqid _|_ A ` vH A C i^i opr eqid f golem2 mprg))
+thm (stcltr1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltr1.3 (e. B (CH)))) (-> ph (-> (-> (= (` S A) (1)) (= (` S B) (1))) (C_ A B))) (stcltr1.1 stcltr1.2 stcltr1.3 x cv A S fveq2 1 eqeq1d S y cv ` 1 = imbi1d x cv A y cv sseq1 imbi12d y cv B S fveq2 1 eqeq1d S A ` 1 = imbi2d y cv B A sseq2 imbi12d CH CH rcla42v mp2an S States e. adantl sylbi))
+thm (stcltr2 ((x y) (A x) (A y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH)))) (-> ph (-> (= (` S A) (1)) (= A (H~)))) (stcltr1.1 helch stcltr1.2 stcltr1 S A ` 1 = S H~ ` 1 = ax-1 syl5 A H~ eqss stcltr1.2 chssi mpbiran syl6ibr))
+thm (stcltrlem1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltrlem1.3 (e. B (CH)))) (-> ph (-> (= (` S B) (1)) (= (` S (opr (` (_|_) A) (vH) (i^i A B))) (1)))) (stcltr1.1 pm3.26bd stcltr1.2 stcltrlem1.3 S stji1 syl S B ` 1 = adantr stcltr1.1 stcltrlem1.3 stcltr2 imp B H~ A ineq2 stcltr1.2 chm1 syl6eq syl S fveq2d S _|_ A ` ` + opreq2d stcltr1.1 pm3.26bd S B ` 1 = adantr S _|_ A ` ` S A ` axaddcom stcltr1.2 chocl S _|_ A ` stclt mpi recnd stcltr1.2 S A stclt mpi recnd sylanc stcltr1.2 S sto1 eqtrd syl 3eqtrd exp))
+thm (stcltrlem2 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((stcltr1.1 (<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y)))))))) (stcltr1.2 (e. A (CH))) (stcltrlem1.3 (e. B (CH)))) (-> ph (C_ B (opr (` (_|_) A) (vH) (i^i A B)))) (stcltr1.1 stcltr1.2 stcltrlem1.3 stcltrlem1 stcltr1.1 stcltrlem1.3 stcltr1.2 chocl stcltr1.2 stcltrlem1.3 chincl chjcl stcltr1 mpd))
+thm (stcltrth ((x y) (s x) (A x) (s y) (A y) (A s) (B x) (B y) (B s)) ((stcltrth.1 (e. A (CH))) (stcltrth.2 (e. B (CH))) (stcltrth.3 (E.e. s (States) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` (cv s) (cv x)) (1)) (= (` (cv s) (cv y)) (1))) (C_ (cv x) (cv y)))))))) (C_ B (opr (` (_|_) A) (vH) (i^i A B))) (stcltrth.3 s cv States e. x CH y CH s cv x cv ` 1 = s cv y cv ` 1 = -> x cv y cv C_ -> A.e. A.e. /\ pm4.2 stcltrth.1 stcltrth.2 stcltrlem2 exp r19.23aiv ax-mp))
+thm (cvbrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (<o) B) (/\ (C: A B) (-. (E.e. x (CH) (/\ (C: A (cv x)) (C: (cv x) B))))))) (y cv A CH eleq1 z cv CH e. anbi1d y cv A z cv psseq1 y cv A x cv psseq1 x cv z cv C: anbi1d x CH birexdv negbid anbi12d anbi12d z cv B CH eleq1 A CH e. anbi2d z cv B A psseq2 z cv B x cv psseq2 A x cv C: anbi2d x CH birexdv negbid anbi12d anbi12d y z x df-cv CH CH brabg A CH e. B CH e. /\ A B C: x CH A x cv C: x cv B C: /\ E.e. -. /\ ibar bitr4d))
+thm (cvbr2t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (<o) B) (/\ (C: A B) (A.e. x (CH) (-> (/\ (C: A (cv x)) (C_ (cv x) B)) (= (cv x) B)))))) (A B x cvbrt A x cv C: x cv B C_ /\ x cv B = iman A x cv C: x cv B C_ x cv B = -. anass x cv B dfpss2 A x cv C: anbi2i bitr4 negbii bitr x CH biral x CH A x cv C: x cv B C: /\ ralnex bitr A B C: anbi2i syl6bbr))
+thm (cvcon3t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (<o) B) (br (` (_|_) B) (<o) (` (_|_) A)))) (A B chpsscon3t A x cv chpsscon3t B CH e. adantlr x cv B chpsscon3t ancoms A CH e. adantll anbi12d y cv _|_ x cv ` _|_ B ` psseq2 y cv _|_ x cv ` _|_ A ` psseq1 anbi12d CH rcla4ev x cv choclt sylan exp ancomsd A CH e. B CH e. /\ adantl sylbid exp r19.23adv B y cv chpsscon1t A CH e. adantll y cv A chpsscon2t ancoms B CH e. adantlr anbi12d x cv _|_ y cv ` A psseq2 x cv _|_ y cv ` B psseq1 anbi12d CH rcla4ev y cv choclt sylan exp ancomsd A CH e. B CH e. /\ adantl sylbid exp r19.23adv impbid negbid anbi12d A B x cvbrt _|_ B ` _|_ A ` y cvbrt B choclt A choclt syl2an ancoms 3bitr4d))
+thm (cvpsst ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br A (<o) B) (C: A B))) (A B x cvbrt A B C: x CH A x cv C: x cv B C: /\ E.e. -. pm3.26 syl6bi))
+thm (cvnbtwnt ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (<o) B) (-. (/\ (C: A C) (C: C B))))) (A B x cvbrt x cv C A psseq2 x cv C B psseq1 anbi12d CH rcla4ev exp con3d com12 A B C: adantl syl6bi com23 imp 3impa))
+thm (cvnbtwn2t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (<o) B) (-> (/\ (C: A C) (C_ C B)) (= C B)))) (A B C cvnbtwnt A C C: C B C_ /\ C B = iman A C C: C B C_ C B = -. anass C B dfpss2 A C C: anbi2i bitr4 negbii bitr2 syl6ib))
+thm (cvnbtwn3t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (<o) B) (-> (/\ (C_ A C) (C: C B)) (= C A)))) (A B C cvnbtwnt A C C_ C B C: /\ A C = iman C A eqcom A C C_ C B C: /\ imbi2i A C dfpss2 C B C: anbi1i A C C_ A C = -. C B C: an23 bitr negbii 3bitr4r syl6ib))
+thm (cvnbtwn4t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (<o) B) (-> (/\ (C_ A C) (C_ C B)) (\/ (= C A) (= C B))))) (A B C cvnbtwnt A C C_ C B C_ /\ C A = C B = \/ iman A C C_ C B C_ A C = -. C B = -. an4 C A = C B = ioran C A eqcom negbii C B = -. anbi1i bitr A C C_ C B C_ /\ anbi2i A C dfpss2 C B dfpss2 anbi12i 3bitr4 negbii bitr2 syl6ib))
+thm (cvnsymt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br A (<o) B) (-. (br B (<o) A)))) (A B cvpsst B A cvpsst ancoms B A pssn2lp B A C: A B C: imnan mpbir syl6 con2d syld))
+thm (cvnreft () () (-> (e. A (CH)) (-. (br A (<o) A))) (A A cvnsymt anidms pm2.01d))
+thm (cvntrt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (/\ (br A (<o) B) (br B (<o) C)) (-. (br A (<o) C)))) (A B cvpsst C CH e. 3adant3 B C cvpsst A CH e. 3adant1 anim12d A C B cvnbtwnt 3com23 con2d syld))
+thm (spansncv2t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (H~))) (-> (-. (C_ (` (span) ({} B)) A)) (br A (<o) (opr A (vH) (` (span) ({} B)))))) (A x cv B spansncvt 3exp com23 imp r19.21aiv A A vH span B {} ` opr C: anim2i expcom A span B {} ` chnlet B spansncht sylan2 A span B {} ` chjclt B spansncht sylan2 A A vH span B {} ` opr x cvbr2t syldan 3imtr4d))
+thm (mdbr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))))) (y cv A CH eleq1 z cv CH e. anbi1d y cv A x cv vH opreq2 z cv ineq1d y cv A z cv ineq1 x cv vH opreq2d eqeq12d x cv z cv C_ imbi2d x CH biraldv anbi12d z cv B CH eleq1 A CH e. anbi2d z cv B x cv sseq2 z cv B x cv vH A opr ineq2 z cv B A ineq2 x cv vH opreq2d eqeq12d imbi12d x CH biraldv anbi12d y z x df-md CH CH brabg A CH e. B CH e. /\ x CH x cv B C_ x cv vH A opr B i^i x cv vH A B i^i opr = -> A.e. ibar bitr4d))
+thm (mdi ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (MH) B) (-> (C_ C B) (= (i^i (opr C (vH) A) B) (opr C (vH) (i^i A B)))))) (A B x mdbr biimpd x cv C B sseq1 x cv C vH A opreq1 B ineq1d x cv C vH A B i^i opreq1 eqeq12d imbi12d CH rcla4v sylan9 3impa))
+thm (mdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))))) (A B x mdbr x cv B C_ x cv x cv vH A opr C_ iba x cv x cv vH A opr B ssin syl6bb x cv A chub1t ancoms syl5bi com12 A x cv chub2t A x cv vH A opr B ssrin syl jctird B CH e. adantlr x cv A B i^i x cv vH A opr B i^i chlubt A CH e. B CH e. /\ x cv CH e. pm3.27 A B chinclt x cv CH e. adantr x cv vH A opr B chinclt x cv A chjclt ancoms sylan an1rs syl3anc sylibd x cv vH A B i^i opr x cv vH A opr B i^i C_ x cv vH A opr B i^i x cv vH A B i^i opr C_ iba x cv vH A opr B i^i x cv vH A B i^i opr eqss syl6rbbr syl6 pm5.74d biraldva bitrd))
+thm (mdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (= (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B)))))) (A B y mdbr x cv B chinclt x cv B inss2 y cv x cv B i^i B sseq1 y cv x cv B i^i vH A opreq1 B ineq1d y cv x cv B i^i vH A B i^i opreq1 eqeq12d imbi12d CH rcla4v mpii syl exp com3l r19.21adv x cv B dfss biimp vH A opreq1d B ineq1d x cv B dfss biimp vH A B i^i opreq1d eqeq12d biimprcd x CH r19.20si x cv y cv B sseq1 x cv y cv vH A opreq1 B ineq1d x cv y cv vH A B i^i opreq1 eqeq12d imbi12d CH cbvralv sylib B CH e. a1i impbid A CH e. adantl bitrd))
+thm (mdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (C_ (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B)))))) (A B y mdbr2 x cv B chinclt x cv B inss2 y cv x cv B i^i B sseq1 y cv x cv B i^i vH A opreq1 B ineq1d y cv x cv B i^i vH A B i^i opreq1 sseq12d imbi12d CH rcla4v mpii syl exp com3l r19.21adv x cv B dfss biimp vH A opreq1d B ineq1d x cv B dfss biimp vH A B i^i opreq1d sseq12d biimprcd x CH r19.20si x cv y cv B sseq1 x cv y cv vH A opreq1 B ineq1d x cv y cv vH A B i^i opreq1 sseq12d imbi12d CH cbvralv sylib B CH e. a1i impbid A CH e. adantl bitrd))
+thm (dmdbr ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B))))))) (_|_ A ` _|_ B ` y mdbr A choclt B choclt syl2an x cv choclt _|_ x cv ` _|_ B ` C_ _|_ x cv ` vH _|_ A ` opr _|_ B ` i^i _|_ x cv ` vH _|_ A ` _|_ B ` i^i opr = -> imim1i com12 A CH e. B CH e. /\ adantl B x cv chsscon3t biimpd A CH e. adantll _|_ x cv ` vH _|_ A ` opr B chdmm3t _|_ x cv ` _|_ A ` chjclt x cv choclt A choclt syl2an sylan x cv A chdmj4t B CH e. adantr vH B opreq1d eqtrd anasss x cv _|_ A ` _|_ B ` i^i chdmj2t _|_ A ` _|_ B ` chinclt A choclt B choclt syl2an sylan2 A B chdmm4t x cv CH e. adantl x cv ineq2d eqtrd eqeq12d ancoms _|_ x cv ` vH _|_ A ` opr _|_ B ` i^i _|_ x cv ` vH _|_ A ` _|_ B ` i^i opr _|_ fveq2 syl5bi imim12d syld exp com23 y cv _|_ x cv ` _|_ B ` sseq1 y cv _|_ x cv ` vH _|_ A ` opreq1 _|_ B ` ineq1d y cv _|_ x cv ` vH _|_ A ` _|_ B ` i^i opreq1 eqeq12d imbi12d CH rcla4v com12 syl5 r19.21adv y cv choclt B _|_ y cv ` C_ _|_ y cv ` A i^i vH B opr _|_ y cv ` A vH B opr i^i = -> imim1i com12 A CH e. B CH e. /\ adantl B y cv chsscon2t biimprd A CH e. adantll _|_ y cv ` A i^i B chdmj1t _|_ y cv ` A chinclt y cv choclt sylan sylan y cv A chdmm2t B CH e. adantr _|_ B ` ineq1d eqtrd anasss y cv A vH B opr chdmm2t A B chjclt sylan2 A B chdmj1t y cv CH e. adantl y cv vH opreq2d eqtrd eqeq12d ancoms _|_ y cv ` A i^i vH B opr _|_ y cv ` A vH B opr i^i _|_ fveq2 syl5bi imim12d syld exp com23 x cv _|_ y cv ` B sseq2 x cv _|_ y cv ` A ineq1 vH B opreq1d x cv _|_ y cv ` A vH B opr ineq1 eqeq12d imbi12d CH rcla4v com12 syl5 r19.21adv impbid bitrd))
+thm (dmdi ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (= (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B)))))) (A B x dmdbr biimpd x cv C B sseq2 x cv C A ineq1 vH B opreq1d x cv C A vH B opr ineq1 eqeq12d imbi12d CH rcla4v sylan9 3impa))
+thm (dmdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (C_ (i^i (cv x) (opr A (vH) B)) (opr (i^i (cv x) A) (vH) B)))))) (A B x dmdbr x cv A inss1 x cv A i^i B x cv chlubt biimpd mpani x cv A chinclt ancoms B CH e. adantlr A CH e. B CH e. pm3.27 x cv CH e. adantr A CH e. B CH e. /\ x cv CH e. pm3.27 syl3anc x cv A inss2 x cv A i^i A B chlej1t mpi x cv A chinclt ancoms B CH e. adantlr A CH e. B CH e. pm3.26 x cv CH e. adantr A CH e. B CH e. pm3.27 x cv CH e. adantr syl3anc jctird x cv A i^i vH B opr x cv A vH B opr ssin syl6ib x cv A i^i vH B opr x cv A vH B opr i^i eqss baib syl6 pm5.74d biraldva bitrd))
+thm (dmdi2 () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (C_ (i^i C (opr A (vH) B)) (opr (i^i C A) (vH) B))))) (A B C dmdi C A i^i vH B opr C A vH B opr i^i eqimss2 syl8))
+thm (dmdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (= (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (i^i (opr (cv x) (vH) B) (opr A (vH) B)))))) (A B y dmdbr B x cv chub2t ancoms x cv B chjclt y cv x cv vH B opr B sseq2 y cv x cv vH B opr A ineq1 vH B opreq1d y cv x cv vH B opr A vH B opr ineq1 eqeq12d imbi12d CH rcla4v syl mpid exp com3l r19.21adv B x cv chlejb2t biimpa A ineq1d vH B opreq1d B x cv chlejb2t biimpa A vH B opr ineq1d eqeq12d biimpd exp com23 r19.20dva x cv y cv B sseq2 x cv y cv A ineq1 vH B opreq1d x cv y cv A vH B opr ineq1 eqeq12d imbi12d CH cbvralv syl6ib impbid A CH e. adantl bitrd))
+thm (dmdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))) (A B y dmdbr2 B x cv chub2t ancoms x cv B chjclt y cv x cv vH B opr B sseq2 y cv x cv vH B opr A vH B opr ineq1 y cv x cv vH B opr A ineq1 vH B opreq1d sseq12d imbi12d CH rcla4v syl mpid exp com3l r19.21adv B x cv chlejb2t biimpa A vH B opr ineq1d B x cv chlejb2t biimpa A ineq1d vH B opreq1d sseq12d biimpd exp com23 r19.20dva x cv y cv B sseq2 x cv y cv A vH B opr ineq1 x cv y cv A ineq1 vH B opreq1d sseq12d imbi12d CH cbvralv syl6ib impbid A CH e. adantl bitrd))
+thm (dmdi4 ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (C_ (i^i (opr C (vH) B) (opr A (vH) B)) (opr (i^i (opr C (vH) B) A) (vH) B)))) (A B x dmdbr4 biimpd x cv C vH B opreq1 A vH B opr ineq1d x cv C vH B opreq1 A ineq1d vH B opreq1d sseq12d CH rcla4v sylan9 3impa))
+thm (dmdbr5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ (cv x) (opr A (vH) B)) (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))))) (A B x dmdbr4 x cv x cv vH B opr A vH B opr ssin x cv x cv vH B opr A vH B opr i^i x cv vH B opr A i^i vH B opr sstr2 sylbi x cv B chub1t ancoms sylan exp com23 r19.20dva A CH e. adantl sylbid y cv B chjclt ancoms A CH e. adantll A B chjclt y cv CH e. adantr jca y cv vH B opr A vH B opr chinclt y cv vH B opr A vH B opr inss2 y cv vH B opr A vH B opr i^i CH e. y cv vH B opr A vH B opr i^i A vH B opr C_ y cv vH B opr A vH B opr i^i y cv vH B opr A vH B opr i^i vH B opr A i^i vH B opr C_ -> pm2.27 mpii 3syl B y cv chub2t A CH e. adantll B A chub2t ancoms y cv CH e. adantr jca B y cv vH B opr A vH B opr ssin sylib B y cv vH B opr A vH B opr i^i chlejb2t A CH e. B CH e. pm3.27 y cv CH e. adantr y cv vH B opr A vH B opr chinclt y cv B chjclt ancoms A CH e. adantll A B chjclt y cv CH e. adantr sylanc sylanc mpbid A ineq1d A B chabs2t A vH B opr A incom syl5eq y cv vH B opr ineq2d y cv vH B opr A vH B opr A inass syl5eq y cv CH e. adantr eqtrd vH B opreq1d y cv vH B opr A vH B opr i^i sseq2d sylibd exp com23 x cv y cv vH B opr A vH B opr i^i A vH B opr sseq1 x cv y cv vH B opr A vH B opr i^i = id x cv y cv vH B opr A vH B opr i^i vH B opreq1 A ineq1d vH B opreq1d sseq12d imbi12d CH rcla4v com12 syl5 r19.21adv A B y dmdbr4 sylibrd impbid))
+thm (mddmd ((x y) (A x) (A y)) () (-> (e. A (CH)) (<-> (A.e. x (CH) (br A (MH) (cv x))) (A.e. x (CH) (br (` (_|_) A) (MH) (` (_|_) (cv x)))))) (A y cv x mdbr A x cv chjcomt y cv ineq1d A vH x cv opr y cv incom syl5reqr y cv CH e. adantlr A y cv i^i x cv chjcomt A y cv chinclt sylan A y cv incom vH x cv opreq1i syl5reqr eqeq12d y cv A vH x cv opr i^i y cv A i^i vH x cv opr eqcom syl6bb x cv y cv C_ imbi2d biraldva bitrd biraldva x cv y cv A MH breq2 CH cbvralv syl5bb y CH x CH x cv y cv C_ y cv A i^i vH x cv opr y cv A vH x cv opr i^i = -> ralcom syl6bb A x cv y dmdbr biraldva bitr4d))
+thm (mdsublat2 ((A x) (B x) (C x) (D x)) () (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. C (CH)) (e. D (CH)))) (-> (/\ (/\ (/\ (C_ C A) (C_ D B)) (= (i^i A B) (0H))) (br A (MH) B)) (br C (MH) D))) (x cv D B sstr2 com12 C A C_ A B i^i 0H = ad2antlr A CH e. B CH e. /\ C CH e. D CH e. /\ /\ x cv CH e. ad2antlr x cv vH C opr D i^i x cv vH A opr B i^i x cv vH A B i^i opr sstr2 x cv vH C opr D i^i x cv vH A B i^i opr x cv vH C D i^i opr sstr2 syl6 com23 C A x cv chlej2t 3expa x cv vH C opr x cv vH A opr D B ss2in exp syl6 exp ancoms D CH e. adantrr B CH e. adantlr imp43 A B i^i 0H = adantrr A B i^i 0H x cv vH opreq2 x cv chj0t sylan9eqr C CH e. D CH e. /\ adantl x cv C D i^i chub1t ancoms C D chinclt sylan A B i^i 0H = adantrr eqsstrd A CH e. B CH e. /\ adantll anassrs C A C_ D B C_ /\ adantrl sylc an1rs imim12d r19.20dva A B x mdbr2 C CH e. D CH e. /\ C A C_ D B C_ /\ A B i^i 0H = /\ ad2antll C D x mdbr2 A CH e. B CH e. /\ C A C_ D B C_ /\ A B i^i 0H = /\ ad2antlr 3imtr4d exp imp3a))
+thm (ssmd1 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br A (MH) B))) (A B x mdbr2 x cv vH A opr B inss1 A B C_ a1i A B dfss biimp x cv vH opreq2d sseqtrd x cv B C_ a1d x cv CH e. a1d r19.21aiv syl5bir))
+thm (ssmd2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br B (MH) A))) (x cv vH B opr A inss2 A CH e. x cv CH e. /\ a1i A x cv chub2t sstrd A B C_ adantrl A B C_ x cv CH e. pm3.26 A B sseqin2 sylib A CH e. adantl x cv vH opreq2d sseqtr4d x cv A C_ a1d exp32 r19.21adv B CH e. adantr B A x mdbr2 ancoms bicomd sylibd))
+thm (ssdmd1 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) A) (MH) (` (_|_) B)))) (A B chsscon3t _|_ B ` _|_ A ` ssmd2 B choclt A choclt syl2an ancoms sylbid))
+thm (ssdmd2 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) B) (MH) (` (_|_) A)))) (A B chsscon3t _|_ B ` _|_ A ` ssmd1 B choclt A choclt syl2an ancoms sylbid))
+thm (mdsublat ((x y) (A x) (A y) (B x) (B y)) ((mdsublat.1 (e. A (CH))) (mdsublat.2 (e. B (CH)))) (-> (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))) (br A (MH) B)) (A B inss2 mdsublat.1 mdsublat.2 chincl mdsublat.2 y cv A B i^i B chlubt mp3an23 biimpd mpan2i mdsublat.2 mdsublat.1 chub2 y cv vH A B i^i opr B A vH B opr sstr mpan2 syl6 mdsublat.1 mdsublat.2 chincl A B i^i y cv chub2t mpan jctild mdsublat.1 mdsublat.2 chincl y cv A B i^i chjclt mpan2 jctild A B inss2 mdsublat.1 mdsublat.2 chincl mdsublat.2 y cv A B i^i B chlubt mp3an23 biimpd mpan2i jcad mdsublat.1 mdsublat.2 chincl mdsublat.1 y cv A B i^i A chjasst mp3an23 mdsublat.1 mdsublat.2 chincl mdsublat.1 chjcom mdsublat.1 mdsublat.2 chabs1 eqtr y cv vH opreq2i syl6eq B ineq1d mdsublat.1 mdsublat.2 chincl mdsublat.1 mdsublat.2 chincl y cv A B i^i A B i^i chjasst mp3an23 mdsublat.1 mdsublat.2 chincl chjidm y cv vH opreq2i syl6eq eqeq12d biimpd imim12d y cv vH A B i^i opr CH e. A B i^i y cv vH A B i^i opr C_ y cv vH A B i^i opr A vH B opr C_ /\ /\ y cv vH A B i^i opr B C_ y cv vH A B i^i opr vH A opr B i^i y cv vH A B i^i opr vH A B i^i opr = impexp y cv vH A B i^i opr CH e. A B i^i y cv vH A B i^i opr C_ y cv vH A B i^i opr A vH B opr C_ /\ y cv vH A B i^i opr B C_ y cv vH A B i^i opr vH A opr B i^i y cv vH A B i^i opr vH A B i^i opr = -> impexp bitr2 syl5ib x cv y cv vH A B i^i opr A B i^i sseq2 x cv y cv vH A B i^i opr A vH B opr sseq1 anbi12d x cv y cv vH A B i^i opr B sseq1 x cv y cv vH A B i^i opr vH A opreq1 B ineq1d x cv y cv vH A B i^i opr vH A B i^i opreq1 eqeq12d imbi12d imbi12d CH rcla4v com12 syl5com r19.21aiv mdsublat.1 mdsublat.2 A B y mdbr mp2an sylibr))
+thm (cvmd ((A x) (B x)) ((mdsublat.1 (e. A (CH))) (mdsublat.2 (e. B (CH)))) (-> (br (i^i A B) (<o) B) (br A (MH) B)) (mdsublat.1 mdsublat.2 chincl mdsublat.2 A B i^i B x cv cvnbtwn4t mp3an12 impcom mdsublat.1 mdsublat.2 chincl mdsublat.1 chjcom mdsublat.1 mdsublat.2 chabs1 eqtr B ineq1i mdsublat.1 mdsublat.2 chincl chjidm eqtr4 x cv A B i^i vH A opreq1 B ineq1d x cv A B i^i vH A B i^i opreq1 eqeq12d mpbiri B vH A opr B incom mdsublat.2 mdsublat.1 chabs2 mdsublat.2 mdsublat.1 chabs1 B A incom B vH opreq2i eqtr3 3eqtr x cv B vH A opreq1 B ineq1d x cv B vH A B i^i opreq1 eqeq12d mpbiri jaoi syl6 A B i^i x cv C_ x cv A vH B opr C_ x cv B C_ anass mdsublat.2 mdsublat.1 chub2 x cv B A vH B opr sstr mpan2 pm4.71ri A B i^i x cv C_ anbi2i bitr4 syl5ib exp4b r19.21aiv mdsublat.1 mdsublat.2 x mdsublat syl))
+thm (mdexch ((A x) (B x) (C x)) ((mdexch.1 (e. A (CH))) (mdexch.2 (e. B (CH))) (mdexch.3 (e. C (CH)))) (-> (/\/\ (br A (MH) B) (br C (MH) (opr A (vH) B)) (C_ (i^i C (opr A (vH) B)) A)) (/\ (br (opr C (vH) A) (MH) B) (= (i^i (opr C (vH) A) B) (i^i A B)))) (mdexch.3 mdexch.1 C A x cv chjasst mp3an12 mdexch.3 mdexch.1 chjcl x cv C vH A opr chjcomt mpan2 mdexch.1 A x cv chjclt mpan mdexch.3 A vH x cv opr C chjcomt mpan2 syl 3eqtr4d B ineq1d A vH x cv opr vH C opr A vH B opr B inass A vH B opr B incom mdexch.1 mdexch.2 chjcom B ineq2i mdexch.2 mdexch.1 chabs2 3eqtr A vH x cv opr vH C opr ineq2i eqtr syl6eqr x cv B C_ C MH A vH B opr br C A vH B opr i^i A C_ /\ ad2antll mdexch.2 mdexch.1 x cv B A chlej2t mp3an23 mdexch.1 A x cv chjclt mpan mdexch.3 mdexch.1 mdexch.2 chjcl C A vH B opr A vH x cv opr mdi mp3an12 syl com23 syld imp31 C A vH B opr i^i A C_ adantrr mdexch.1 A x cv chjclt mpan mdexch.3 mdexch.1 mdexch.2 chjcl chincl mdexch.1 C A vH B opr i^i A A vH x cv opr chlej2t mp3an12 syl imp mdexch.1 A x cv chjclt mpan mdexch.1 A vH x cv opr A chjcomt mpan2 syl mdexch.1 mdexch.1 A A x cv chjasst mp3an12 mdexch.1 chjidm vH x cv opreq1i syl5reqr mdexch.1 A x cv chjcomt mpan 3eqtrd C A vH B opr i^i A C_ adantr sseqtrd x cv B C_ adantlr C MH A vH B opr br adantrl eqsstrd A vH x cv opr vH C opr A vH B opr i^i x cv vH A opr B ssrin syl eqsstrd A MH B br adantrl mdexch.1 mdexch.2 A B x cv mdi mp3an12 com23 imp31 mdexch.1 mdexch.3 chub2 A C vH A opr B ssrin ax-mp mdexch.1 mdexch.2 chincl mdexch.3 mdexch.1 chjcl mdexch.2 chincl A B i^i C vH A opr B i^i x cv chlej2t mp3an12 mpi x cv B C_ A MH B br ad2antll eqsstrd C MH A vH B opr br C A vH B opr i^i A C_ /\ adantrr sstrd exp31 com3r 3impb r19.21aiv mdexch.3 mdexch.1 chjcl mdexch.2 C vH A opr B x mdbr2 mp2an sylibr mdexch.1 mdexch.2 chub1 mdexch.3 mdexch.1 mdexch.2 chjcl mdexch.1 C A vH B opr A mdi mp3an mpi mdexch.3 mdexch.1 mdexch.2 chjcl chincl mdexch.1 chlejb1 biimp mdexch.1 mdexch.3 mdexch.1 mdexch.2 chjcl chincl chjcom syl5eq sylan9eq B ineq1d A vH C opr A vH B opr B inass syl5eqr mdexch.3 mdexch.1 chjcom mdexch.1 mdexch.2 chjcom B ineq2i B A vH B opr incom mdexch.2 mdexch.1 chabs2 3eqtr3r ineq12i syl5eq A MH B br 3adant1 jca))
+thm (cvmdt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br (i^i A B) (<o) B) (br A (MH) B))) (A A CH e. A H~ if B ineq1 <o B breq1d A A CH e. A H~ if MH B breq1 imbi12d B B CH e. B H~ if A CH e. A H~ if ineq2 B B CH e. B H~ if = id <o breq12d B B CH e. B H~ if A CH e. A H~ if MH breq2 imbi12d helch A elimel helch B elimel cvmd dedth2h))
+thm (cvdmdt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br B (<o) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B)))) (B A vH B opr cvcon3t A CH e. B CH e. pm3.27 A B chjclt sylanc A B chdmj1t <o _|_ B ` breq1d bitrd _|_ A ` _|_ B ` cvmdt A choclt B choclt syl2an sylbid))
+thm (elat ((A x)) () (<-> (e. A (Atoms)) (/\ (e. A (CH)) (br (0H) (<o) A))) (x df-at A eleq2i x cv A 0H <o breq2 CH elrab bitr))
+thm (elat2 ((A x)) () (<-> (e. A (Atoms)) (/\ (e. A (CH)) (/\ (-. (= A (0H))) (A.e. x (CH) (-> (C_ (cv x) A) (\/ (= (cv x) A) (= (cv x) (0H)))))))) (A elat h0elch 0H A x cvbr2t mpan A ch0psst x cv ch0psst x cv A = imbi1d x cv A C_ imbi2d 0H x cv C: x cv A C_ x cv A = impexp 0H x cv C: x cv A C_ x cv A = bi2.04 bitr x cv A = x cv 0H = orcom x cv 0H = x cv A = df-or bitr x cv A C_ imbi2i 3bitr4g birala A CH e. a1i anbi12d bitr2d pm5.32i bitr4))
+thm (elatcv0 () () (-> (e. A (CH)) (<-> (e. A (Atoms)) (br (0H) (<o) A))) (A elat baib))
+thm (atcv0 () () (-> (e. A (Atoms)) (br (0H) (<o) A)) (A elat pm3.27bd))
+thm (atssch () () (C_ (Atoms) (CH)) (x df-at x CH 0H <o x cv br ssrab eqsstr))
+thm (atelch () () (-> (e. A (Atoms)) (e. A (CH))) (atssch A sseli))
+thm (atn0 ((A x)) () (-> (e. A (Atoms)) (-. (= A (0H)))) (A x elat2 A 0H = -. x CH x cv A C_ x cv A = x cv 0H = \/ -> A.e. pm3.26 A CH e. adantl sylbi))
+thm (atss ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (Atoms))) (-> (C_ A B) (\/ (= A B) (= A (0H))))) (x cv A B sseq1 x cv A B eqeq1 x cv A 0H eqeq1 orbi12d imbi12d CH rcla4v B 0H = -. adantld B CH e. adantld imp B x elat2 sylan2b))
+thm (atsseq () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (C_ A B) (= A B))) (A atn0 B Atoms e. A B C_ ad2antll A B atss A atelch sylan imp ord mt3d exp A B eqimss A Atoms e. B Atoms e. /\ a1i impbid))
+thm (atcveq0 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (br A (<o) B) (= A (0H)))) (A B cvpsst B atelch sylan2 A ch0let B Atoms e. adantr jctild B atcv0 A CH e. adantr h0elch 0H B A cvnbtwn3t mp3an1 B atelch sylan mpd ancoms syld A 0H <o B breq1 B atcv0 syl5bir com12 A CH e. adantl impbid))
+thm (h1dat ((A x)) () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (e. (` (_|_) (` (_|_) ({} A))) (Atoms))) (A H~ snssi A {} occlt _|_ A {} ` choclt 3syl A 0v = -. adantr A h1dn0 x cv A h1datomt expcom r19.21aiv A 0v = -. adantr jca jca _|_ _|_ A {} ` ` x elat2 sylibr))
+thm (spansnat () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (e. (` (span) ({} A)) (Atoms))) (A spansnt A 0v = -. adantr A h1dat eqeltrd))
+thm (sh1dle () () (-> (/\ (e. A (SH)) (e. B A)) (C_ (` (_|_) (` (_|_) ({} B))) A)) (A B shelt B spansnt syl A B spansnsst eqsstr3d))
+thm (ch1dle () () (-> (/\ (e. A (CH)) (e. B A)) (C_ (` (_|_) (` (_|_) ({} B))) A)) (A B sh1dle A chsh sylan))
+thm (atom1d ((x y) (A x) (A y)) () (<-> (e. A (Atoms)) (E.e. x (H~) (/\ (-. (= (cv x) (0v))) (= A (` (span) ({} (cv x))))))) (A y elat2 A x chne0t A CH e. x ax-17 y CH y cv A C_ y cv A = y cv 0H = \/ -> A.e. x ax-17 x H~ x cv 0v = -. A _|_ _|_ x cv {} ` ` = /\ hbre1 hbim x H~ x cv 0v = -. A _|_ _|_ x cv {} ` ` = /\ ra4e A x cv chelt x cv 0v = -. adantrr y CH y cv A C_ y cv A = y cv 0H = \/ -> A.e. adantrr x cv A e. x cv 0v = -. pm3.27 A CH e. y CH y cv A C_ y cv A = y cv 0H = \/ -> A.e. ad2antrl x cv h1dn0 A x cv chelt sylan anasss y CH y cv A C_ y cv A = y cv 0H = \/ -> A.e. adantrr A x cv ch1dle A x cv chelt x cv H~ snssi x cv {} occlt syl _|_ x cv {} ` choclt 3syl y cv _|_ _|_ x cv {} ` ` A sseq1 y cv _|_ _|_ x cv {} ` ` A eqeq1 y cv _|_ _|_ x cv {} ` ` 0H eqeq1 orbi12d imbi12d CH rcla4v syl mpid exp imp32 x cv 0v = -. adantrlr ord mt3d eqcomd jca sylanc exp44 r19.23ad sylbid imp32 sylbi A _|_ _|_ x cv {} ` ` Atoms eleq1 x cv h1dat syl5bir exp3a com3l imp3a r19.23aiv impbi x cv spansnt A eqeq2d x cv 0v = -. anbi2d birexa bicomi bitr))
+thm (chcv1t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (-. (C_ B A)) (br A (<o) (opr A (vH) B)))) (B span x cv {} ` A sseq1 negbid B span x cv {} ` A vH opreq2 A <o breq2d imbi12d A x cv spansncv2t syl5bir com12 x cv 0v = -. adantld exp r19.23adv imp B x atom1d sylan2b A B chjclt A A vH B opr cvpsst syldan A B chnlet sylibrd B atelch sylan2 impbid))
+thm (chcv2t () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (C: A (opr A (vH) B)) (br A (<o) (opr A (vH) B)))) (A B chnlet B atelch sylan2 A B chcv1t bitr3d))
+thm (chjatom ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (Atoms))) (= (opr A (+H) B) (opr A (vH) B))) (B span x cv {} ` A +H opreq2 B span x cv {} ` A vH opreq2 eqeq12d A x cv spansnjt syl5bir exp3a x cv 0v = -. adantl com3l r19.23adv B x atom1d syl5ib imp))
+thm (shatomic ((x y) (A x) (A y)) ((shatomic.1 (e. A (SH)))) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A))) (shatomic.1 y shne0 x cv _|_ _|_ y cv {} ` ` A sseq1 Atoms rcla4ev y cv h1dat shatomic.1 y cv shel sylan shatomic.1 A y cv sh1dle mpan y cv 0v = -. adantr sylanc exp r19.23aiv sylbi))
+thm (hatomic ((A x)) ((hatomic.1 (e. A (CH)))) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A))) (hatomic.1 chshi x shatomic))
+thm (hatomict ((A x)) () (-> (e. A (CH)) (-> (-. (= A (0H))) (E.e. x (Atoms) (C_ (cv x) A)))) (A A CH e. A 0H if 0H eqeq1 negbid A A CH e. A 0H if x cv sseq2 x Atoms birexdv imbi12d h0elch A elimel x hatomic dedth))
+thm (shatomistic ((x y) (A x) (A y)) ((shatomistic.1 (e. A (SH)))) (= A (` (span) (U. ({e.|} x (Atoms) (C_ (cv x) A))))) (shatomistic.1 y cv shel y cv spansnsht span y cv {} ` spanid 3syl y cv 0v = -. adantr y cv spansnat shatomistic.1 y cv shel sylan shatomistic.1 A y cv spansnsst mpan y cv 0v = -. adantr jca x cv span y cv {} ` A sseq1 Atoms elrab sylibr span y cv {} ` x Atoms x cv A C_ {e.|} elssuni atssch chsssh sstri Atoms SH x x cv A C_ rabss ax-mp x Atoms x cv A C_ {e.|} x SH x cv A C_ {e.|} uniss ax-mp shatomistic.1 A SH x unisseq ax-mp shatomistic.1 shssi eqsstr3 sstri x Atoms x cv A C_ {e.|} U. span y cv {} ` spanss mpan 3syl eqsstr3d shatomistic.1 y cv shel y cv spansnid syl y cv 0v = -. adantr sseldd exp atssch chsssh sstri Atoms SH x x cv A C_ rabss ax-mp x Atoms x cv A C_ {e.|} x SH x cv A C_ {e.|} uniss ax-mp shatomistic.1 A SH x unisseq ax-mp shatomistic.1 shssi eqsstr3 sstri x Atoms x cv A C_ {e.|} U. spanclt ax-mp span x Atoms x cv A C_ {e.|} U. ` sh0 ax-mp y cv 0v span x Atoms x cv A C_ {e.|} U. ` eleq1 mpbiri pm2.61d2 ssriv shatomistic.1 A SH x unisseq ax-mp shatomistic.1 shssi eqsstr3 atssch chsssh sstri Atoms SH x x cv A C_ rabss ax-mp x Atoms x cv A C_ {e.|} x SH x cv A C_ {e.|} uniss ax-mp x SH x cv A C_ {e.|} U. x Atoms x cv A C_ {e.|} U. spanss mp2an shatomistic.1 A SH x unisseq ax-mp span fveq2i shatomistic.1 A spanid ax-mp eqtr3 sseqtr eqssi))
+thm (hatomistic ((x y) (A x) (A y)) ((hatomistic.1 (e. A (CH)))) (= A (` (\/H) ({e.|} x (Atoms) (C_ (cv x) A)))) (x Atoms x cv A C_ ssrab atssch sstri x Atoms x cv A C_ {e.|} chsupclt ax-mp hatomistic.1 chshi y cv atelch y cv A C_ anim1i x cv y cv A sseq1 Atoms elrab x cv y cv A sseq1 CH elrab 3imtr4 ssriv x Atoms x cv A C_ ssrab atssch sstri x CH x cv A C_ ssrab x Atoms x cv A C_ {e.|} x CH x cv A C_ {e.|} chsupss mp2an ax-mp hatomistic.1 A x chsupid ax-mp sseqtr x cv y cv A sseq1 Atoms elrab y cv x Atoms x cv A C_ {e.|} elssuni sylbir x Atoms x cv A C_ ssrab atssch sstri x Atoms x cv A C_ {e.|} chsupunss ax-mp y cv x Atoms x cv A C_ {e.|} U. \/H x Atoms x cv A C_ {e.|} ` sstr2 mpi syl exp y cv atn0 y cv \/H x Atoms x cv A C_ {e.|} ` C_ adantr y cv atelch y cv chle0t syl y cv \/H x Atoms x cv A C_ {e.|} ` _|_ \/H x Atoms x cv A C_ {e.|} ` ` ssin x Atoms x cv A C_ ssrab atssch sstri x Atoms x cv A C_ {e.|} chsupclt ax-mp chocin y cv sseq2i bitr2 syl5bbr biimpa exp32 imp mtod exp syld y cv A C_ y cv _|_ \/H x Atoms x cv A C_ {e.|} ` ` C_ imnan sylib y cv A _|_ \/H x Atoms x cv A C_ {e.|} ` ` ssin negbii sylib nrex hatomistic.1 x Atoms x cv A C_ ssrab atssch sstri x Atoms x cv A C_ {e.|} chsupclt ax-mp chocl chincl y hatomic mt3 omlsi eqcomi))
+thm (chpssat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (C: A B) (E.e. x (Atoms) (/\ (C_ (cv x) B) (-. (C_ (cv x) A))))) (A B dfpss3 pm3.27bd x cv B C_ x cv A C_ iman x Atoms biral x Atoms x cv B C_ x cv A C_ ss2rab x Atoms x cv B C_ ssrab atssch sstri x Atoms x cv A C_ ssrab atssch sstri x Atoms x cv B C_ {e.|} x Atoms x cv A C_ {e.|} chsupss mp2an chpssat.2 x hatomistic chpssat.1 x hatomistic 3sstr4g sylbir sylbir con3i x Atoms x cv B C_ x cv A C_ -. /\ dfrex2 sylibr syl))
+thm (chrelat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (C: A B) (E.e. x (Atoms) (/\ (C: A (opr A (vH) (cv x))) (C_ (opr A (vH) (cv x)) B)))) (chpssat.1 chpssat.2 x chpssat chpssat.1 A x cv chnlet mpan A B C_ adantl A B C_ x cv B C_ ibar chpssat.1 chpssat.2 A x cv B chlubt mp3an13 sylan9bb anbi12d A B pssss x cv atelch syl2an x cv B C_ x cv A C_ -. ancom syl5bb birexdva mpbid))
+thm (chrelat2 ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B))))) (A B nssinpss chpssat.1 chpssat.2 chincl chpssat.1 x chrelat x cv atelch chpssat.1 chpssat.2 chincl chpssat.1 A B i^i x cv A chlubt mp3an13 A B i^i A C_ x cv A C_ pm3.27 syl6bir A B i^i A B i^i vH x cv opr C: adantld chpssat.1 chpssat.2 chincl A B i^i x cv chnlet mpan x cv A B ssin negbii syl5bb chpssat.1 chpssat.2 chincl chpssat.1 A B i^i x cv A chlubt mp3an13 anbi12d x cv B C_ x cv A C_ pm3.21 x cv A C_ x cv B C_ /\ -. x cv A C_ ianor x cv A C_ x cv B C_ /\ pm4.13 x cv A C_ -. orbi1i x cv A C_ x cv B C_ /\ x cv A C_ -. orcom x cv A C_ x cv A C_ x cv B C_ /\ imor bitr4 bitr3 bitr2 sylib con2i A B i^i A C_ adantrl syl6bir jcad syl r19.22i syl sylbi x cv A B sstr2 com12 x cv Atoms e. a1d r19.21aiv x cv A C_ x cv B C_ iman x Atoms biral x Atoms x cv A C_ x cv B C_ -. /\ ralnex bitr sylib con2i impbi))
+thm (cvat ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (br A (<o) B) (E.e. x (Atoms) (= (opr A (vH) (cv x)) B))) (chpssat.1 chpssat.2 A B cvpsst mp2an chpssat.1 chpssat.2 x chrelat syl x cv atelch chpssat.1 A x cv chjclt mpan syl chpssat.1 chpssat.2 A B A vH x cv opr cvnbtwn2t mp3an12 syl com12 r19.22dv mpd))
+thm (cvbr3 ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (<-> (br A (<o) B) (/\ (C: A B) (E.e. x (Atoms) (= (opr A (vH) (cv x)) B)))) (chpssat.1 chpssat.2 A B cvpsst mp2an chpssat.1 chpssat.2 x cvat jca chpssat.1 A x cv chcv2t mpan A vH x cv opr B = adantr A vH x cv opr B A psseq2 x cv Atoms e. adantl A vH x cv opr B A <o breq2 x cv Atoms e. adantl 3bitr3d biimpd exp com3r r19.23adv imp impbi))
+thm (cvexchlem ((A x) (B x)) ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B))) (chpssat.1 chpssat.2 chincl chpssat.2 A B i^i B cvpsst mp2an chpssat.1 chpssat.2 chincl chpssat.2 x chpssat syl chpssat.1 A x cv chcv1t mpan biimpa x cv A B ssin x cv A C_ x cv B C_ ancom bitr3 baibr negbid biimpar sylan2 A B i^i <o B br adantrr chpssat.1 chpssat.1 chpssat.2 chincl A A B i^i x cv chjasst mp3an12 chpssat.1 chpssat.2 chabs1 vH x cv opreq1i syl5reqr x cv B C_ x cv A B i^i C_ -. /\ A B i^i <o B br /\ adantr chpssat.1 chpssat.2 chincl A B i^i x cv chnlet mpan chpssat.1 chpssat.2 chincl chpssat.2 A B i^i x cv B chlubt mp3an13 A B inss2 x cv B C_ biantrur syl5bb anbi12d x cv B C_ x cv A B i^i C_ -. ancom syl5bb chpssat.1 chpssat.2 chincl A B i^i x cv chjclt mpan chpssat.1 chpssat.2 chincl chpssat.2 A B i^i B A B i^i vH x cv opr cvnbtwn2t mp3an12 syl com23 sylbid imp32 A vH opreq2d eqtr3d x cv atelch sylan breqtrd exp32 r19.23aiv mpcom))
+thm (cvexch () ((chpssat.1 (e. A (CH))) (chpssat.2 (e. B (CH)))) (<-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B))) (chpssat.1 chpssat.2 cvexchlem chpssat.2 chocl chpssat.1 chocl cvexchlem chpssat.1 chpssat.2 chdmj1 _|_ A ` _|_ B ` incom eqtr <o _|_ A ` breq1i chpssat.1 chpssat.2 chdmm1 chpssat.1 chocl chpssat.2 chocl chjcom eqtr _|_ B ` <o breq2i 3imtr4 chpssat.1 chpssat.1 chpssat.2 chjcl A A vH B opr cvcon3t mp2an chpssat.1 chpssat.2 chincl chpssat.2 A B i^i B cvcon3t mp2an 3imtr4 impbi))
+thm (chrelat2t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B)))))) (A A CH e. A H~ if B sseq1 negbid A A CH e. A H~ if x cv sseq2 x cv B C_ -. anbi1d x Atoms birexdv bibi12d B B CH e. B H~ if A CH e. A H~ if sseq2 negbid B B CH e. B H~ if x cv sseq2 negbid x cv A CH e. A H~ if C_ anbi2d x Atoms birexdv bibi12d helch A elimel helch B elimel x chrelat2 dedth2h))
+thm (chrelat3t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B))))) (A B x chrelat2t x Atoms x cv A C_ x cv B C_ -. /\ dfrex2 syl6bb bicon4d x cv A C_ x cv B C_ iman x Atoms biral syl6bbr))
+thm (chrelat3 ((A x) (B x)) ((chrelat3.1 (e. A (CH))) (chrelat3.2 (e. B (CH)))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B)))) (chrelat3.1 chrelat3.2 A B x chrelat3t mp2an))
+thm (chrelat4 ((A x) (B x)) ((chrelat3.1 (e. A (CH))) (chrelat3.2 (e. B (CH)))) (<-> (= A B) (A.e. x (Atoms) (<-> (C_ (cv x) A) (C_ (cv x) B)))) (chrelat3.1 chrelat3.2 x chrelat3 chrelat3.2 chrelat3.1 x chrelat3 anbi12i A B eqss x cv A C_ x cv B C_ bi x Atoms biral x Atoms x cv A C_ x cv B C_ -> x cv B C_ x cv A C_ -> r19.26 bitr 3bitr4))
+thm (cvexcht () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B)))) (A A CH e. A H~ if B ineq1 <o B breq1d A A CH e. A H~ if = id A A CH e. A H~ if vH B opreq1 <o breq12d bibi12d B B CH e. B H~ if A CH e. A H~ if ineq2 B B CH e. B H~ if = id <o breq12d B B CH e. B H~ if A CH e. A H~ if vH opreq2 A CH e. A H~ if <o breq2d bibi12d helch A elimel helch B elimel cvexch dedth2h))
+thm (cvp () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (= (i^i A B) (0H)) (br A (<o) (opr A (vH) B)))) (A B i^i B atcveq0 A B chinclt B atelch sylan2 A CH e. B Atoms e. pm3.27 sylanc A B cvexcht B atelch sylan2 bitr3d))
+thm (atnssm0 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (-. (C_ B A)) (= (i^i A B) (0H)))) (A B chcv1t A B cvp bicomd bitrd))
+thm (atnem0 () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (-. (= A B)) (= (i^i A B) (0H)))) (B A atsseq B A eqcom syl6bb ancoms negbid A B atnssm0 A atelch sylan bitr3d))
+thm (atcv0eq () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (br (0H) (<o) (opr A (vH) B)) (= A B))) (A B atnem0 A B cvp A atelch sylan A atcv0 B Atoms e. adantr A <o A vH B opr br biantrurd 3bitrd A B chjclt h0elch 0H A A vH B opr cvntrt mp3an1 syldan A atelch B atelch syl2an sylbid a3d A B vH B opreq1 B atelch B chjidmt syl sylan9eq eqcomd Atoms eleq1d exp ibd A vH B opr atcv0 syl6com A Atoms e. adantl impbid))
+thm (atcv1 () () (-> (/\ (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (br A (<o) (opr B (vH) C))) (<-> (= A (0H)) (= B C))) (A 0H <o B vH C opr breq1 B C atcv0eq sylan9bbr biimpd exp com23 A CH e. 3adant1 imp A B vH C opr atcveq0 B C vH C opreq1 C atelch C chjidmt syl sylan9eq eqcomd Atoms eleq1d exp ibd impcom sylan2 biimpd exp32 com34 imp B Atoms e. 3adant2 imp impbid))
+thm (atexch () () (-> (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (C_ B (opr A (vH) C)) (= (i^i A B) (0H))) (C_ C (opr A (vH) B)))) (C A chub2t ancoms C atelch sylan2 B Atoms e. 3adant2 B A vH C opr C_ A B i^i 0H = /\ adantr A B cvp A B chjclt B atelch sylan2 A A vH B opr cvpsst syldan sylbid C Atoms e. 3adant3 B A vH C opr C_ adantld A C chub1t B CH e. 3adant2 B A vH C opr C_ a1d ancrd A B A vH C opr chlubt A CH e. B CH e. C CH e. 3simp1 A CH e. B CH e. C CH e. 3simp2 A C chjclt B CH e. 3adant2 syl3anc sylibd A CH e. id B atelch C atelch syl3an A B i^i 0H = adantrd jcad imp A B cvp A B chjclt B atelch sylan2 A A vH B opr cvpsst syldan sylbid C Atoms e. 3adant3 A C chub1t B CH e. 3adant2 B A vH C opr C_ a1d ancrd A B A vH C opr chlubt A CH e. B CH e. C CH e. 3simp1 A CH e. B CH e. C CH e. 3simp2 A C chjclt B CH e. 3adant2 syl3anc sylibd A CH e. id B atelch C atelch syl3an anim12d ancomsd A A vH B opr A vH C opr psssstr syl6 A C chcv2t B Atoms e. 3adant2 sylibd A CH e. B CH e. C CH e. 3simp1 A C chjclt B CH e. 3adant2 A B chjclt C CH e. 3adant3 3jca A CH e. id B atelch C atelch syl3an A A vH C opr A vH B opr cvnbtwn2t syl syld imp mpd sseqtr4d exp))
+thm (atoml ((A x) (B x)) ((atoml.1 (e. A (CH)))) (-> (e. B (Atoms)) (e. (i^i (opr A (vH) B) (` (_|_) A)) (u. (Atoms) ({} (0H))))) (B atelch atoml.1 A B chjclt mpan syl atoml.1 chocl A vH B opr _|_ A ` chinclt mpan2 A vH B opr _|_ A ` i^i x hatomict 3syl imp atoml.1 chocl _|_ A ` x cv pjoml3t mpan imp atoml.1 pjococ vH x cv opreq1i _|_ A ` ineq1i _|_ _|_ A ` ` vH x cv opr _|_ A ` incom eqtr3 syl5eq x cv atelch A vH B opr _|_ A ` inss2 x cv A vH B opr _|_ A ` i^i _|_ A ` sstr mpan2 syl2an B Atoms e. adantll A vH B opr _|_ A ` i^i 0H = -. adantrr atoml.1 A B chub1t mpan x cv CH e. adantr atoml.1 A x cv A vH B opr chlubt mp3an1 atoml.1 A B chjclt mpan sylan2 biimpd ancoms mpand B atelch x cv atelch syl2an imp A vH B opr _|_ A ` inss1 x cv A vH B opr _|_ A ` i^i A vH B opr sstr mpan2 sylan2 A vH B opr _|_ A ` i^i 0H = -. adantrr atoml.1 A B A vH x cv opr chlubt mp3an1 biimpd exp3a com23 imp B atelch x cv atelch atoml.1 A x cv chjclt mpan syl anim12i x cv A vH B opr _|_ A ` i^i C_ A vH B opr _|_ A ` i^i 0H = -. /\ adantr atoml.1 A x cv B atexch mp3an1 B Atoms e. x cv Atoms e. pm3.22 x cv A vH B opr _|_ A ` i^i C_ A vH B opr _|_ A ` i^i 0H = -. /\ adantr A vH B opr _|_ A ` inss1 x cv A vH B opr _|_ A ` i^i A vH B opr sstr mpan2 x cv Atoms e. adantl x cv chsh atoml.1 chshi x cv A orthin mpan2 syl imp A x cv incom syl5eq x cv atelch A vH B opr _|_ A ` inss2 x cv A vH B opr _|_ A ` i^i _|_ A ` sstr mpan2 syl2an jca B Atoms e. adantll A vH B opr _|_ A ` i^i 0H = -. adantrr sylc jca x cv atelch atoml.1 A x cv chub1t mpan syl B Atoms e. x cv A vH B opr _|_ A ` i^i C_ A vH B opr _|_ A ` i^i 0H = -. /\ ad2antlr sylc eqssd _|_ A ` ineq1d eqtr3d Atoms eleq1d exp43 com24 imp31 ibd exp com23 r19.23adv mpd exp con1d orrd A vH B opr _|_ A ` i^i Atoms 0H {} elun _|_ A fvex A vH B opr inex2 0H elsnc A vH B opr _|_ A ` i^i Atoms e. orbi2i bitr sylibr))
+thm (atcvatlem ((A x) (B x) (C x)) ((atoml.1 (e. A (CH)))) (-> (/\ (/\ (e. B (Atoms)) (e. C (Atoms))) (/\ (-. (= A (0H))) (C: A (opr B (vH) C)))) (-> (-. (C_ B A)) (e. A (Atoms)))) (x cv B atnem0 x cv B A sseq1 biimpcd con3d imp A B vH C opr C: adantrl syl5bi x cv B cvp x cv B chjcomt B atelch sylan2 x cv <o breq2d bitrd x cv atelch sylan sylibd ancoms C Atoms e. adantlr imp B x cv chub1t B atelch x cv atelch syl2an C Atoms e. 3adant3 x cv A C_ B A C_ -. /\ A B vH C opr C: /\ adantr x cv A B vH C opr sstr A B vH C opr pssss sylan2 B A C_ -. adantlr B Atoms e. x cv Atoms e. C Atoms e. /\/\ adantl x cv B atnem0 x cv B A sseq1 biimpcd con3d imp syl5bi ancoms C Atoms e. 3adant3 imp B x cv incom syl5eq A B vH C opr C: adantrr B x cv C atexch B atelch syl3an1 x cv A C_ B A C_ -. /\ A B vH C opr C: /\ adantr mp2and jca B CH e. x cv CH e. C CH e. 3simp1 B CH e. x cv CH e. C CH e. 3simp3 B x cv chjclt C CH e. 3adant3 3jca B atelch x cv atelch C atelch syl3an B C B vH x cv opr chlubt syl x cv A C_ B A C_ -. /\ A B vH C opr C: /\ adantr mpbid B C chub1t x cv CH e. 3adant2 x cv A B vH C opr sstr A B vH C opr pssss sylan2 B A C_ -. adantlr anim12i B x cv B vH C opr chlubt B CH e. x cv CH e. C CH e. 3simp1 B CH e. x cv CH e. C CH e. 3simp2 B C chjclt x cv CH e. 3adant2 syl3anc x cv A C_ B A C_ -. /\ A B vH C opr C: /\ adantr mpbid B atelch x cv atelch C atelch 3anim123i sylan eqssd anassrs A psseq2d exp ibd exp32 3expa an1rs com34 imp45 B CH e. x cv CH e. pm3.27 B x cv chjclt jca B atelch x cv atelch syl2an atoml.1 x cv B vH x cv opr A cvnbtwn3t mp3an3 exp4a com23 imp4a syl C Atoms e. adantlr imp A B vH C opr C: B A C_ -. /\ adantrr mp2and Atoms eleq1d biimprcd exp4c pm2.43b imp exp4d exp r19.23adv atoml.1 x hatomic syl5 imp32))
+thm (atcvat () ((atoml.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C: A (opr B (vH) C))) (e. A (Atoms)))) (atoml.1 B C A chlubt 3comr B vH C opr A ssnpss syl6bi con2d B A C_ C A C_ ianor syl6ib mp3an1 B atelch C atelch syl2an imp A 0H = -. adantrl atoml.1 B C atcvatlem C B chjcomt C atelch B atelch syl2an A psseq2d A 0H = -. anbi2d atoml.1 C B atcvatlem exp sylbird ancoms imp jaod mpd exp))
+thm (atcvat2 () ((atoml.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A (<o) (opr B (vH) C))) (e. A (Atoms)))) (atoml.1 A B C atcv1 mp3anl1 negbid B C chjclt B atelch C atelch syl2an atoml.1 A B vH C opr cvpsst mpan syl atoml.1 B C atcvat exp3a com23 syld imp sylbird exp com23 imp3a))
+thm (atcvat2t () () (-> (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A (<o) (opr B (vH) C))) (e. A (Atoms)))) (A A CH e. A 0H if <o B vH C opr breq1 B C = -. anbi2d A A CH e. A 0H if Atoms eleq1 imbi12d B Atoms e. C Atoms e. /\ imbi2d h0elch A elimel B C atcvat2 dedth imp 3impb))
+thm (atcvat3 () ((atcvat3.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) (e. (i^i A (opr B (vH) C)) (Atoms)))) (atcvat3.1 A C chcv1t mpan biimpa B Atoms e. adantll B A vH C opr C_ adantrr B C chjcomt A vH opreq2d atcvat3.1 A C B chjasst mp3an1 ancoms eqtr4d B A vH C opr C_ adantr B A vH C opr A vH C opr chlej2t B CH e. C CH e. pm3.26 atcvat3.1 A C chjclt mpan B CH e. adantl atcvat3.1 A C chjclt mpan B CH e. adantl syl3anc imp eqsstrd atcvat3.1 A C chjclt mpan A vH C opr chjidmt syl B CH e. B A vH C opr C_ ad2antlr sseqtrd atcvat3.1 C B vH C opr A chlej2t mp3an3 B CH e. C CH e. pm3.27 B C chjclt jca C B chub2t ancoms sylc B A vH C opr C_ adantr eqssd B atelch C atelch anim12i sylan A <o breq2d C A C_ -. adantrl mpbird exp B C chjclt atcvat3.1 jctil B atelch C atelch syl2an A B vH C opr cvexcht syl sylibrd B C = -. adantr A B vH C opr i^i B C atcvat2t B C chjclt atcvat3.1 jctil A B vH C opr chinclt syl B atelch C atelch syl2an B Atoms e. C Atoms e. pm3.26 B Atoms e. C Atoms e. pm3.27 syl3anc exp3a imp syld exp4b imp4c))
+thm (atcvat4 ((A x) (B x) (C x)) ((atcvat3.1 (e. A (CH)))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C_ B (opr A (vH) C))) (E.e. x (Atoms) (/\ (C_ (cv x) A) (C_ B (opr C (vH) (cv x))))))) (B C C vH x cv opr sseq1 C x cv chub1t C atelch x cv atelch syl2an syl5bir exp3a impcom x cv A C_ anim2d exp3a com23 r19.22dv atcvat3.1 x hatomic syl5 exp B A vH C opr C_ a1i com4l imp4a B Atoms e. adantl atcvat3.1 C A chlejb2t mpan2 biimpa B sseq2d biimpa anasss exp B CH e. adantl B C chub2t jctird B atelch C atelch syl2an B Atoms e. C Atoms e. pm3.26 jctild imp anassrs x cv B A sseq1 x cv B C vH opreq2 B sseq2d anbi12d Atoms rcla4ev syl A 0H = -. adantrl exp31 atcvat3.1 B C atcvat3 C A B vH C opr i^i B atexch C atelch B Atoms e. B C = -. C A C_ -. /\ B A vH C opr C_ /\ ad2antlr atcvat3.1 B C atcvat3 imp B Atoms e. C Atoms e. pm3.26 B C = -. C A C_ -. /\ B A vH C opr C_ /\ adantr 3jca A B vH C opr inss2 B Atoms e. C Atoms e. /\ a1i B C chjcomt B atelch C atelch syl2an sseqtrd B C = -. C A C_ -. /\ B A vH C opr C_ /\ adantr atcvat3.1 A C atnssm0 mpan B Atoms e. adantl C A B vH C opr i^i chinclt B CH e. C CH e. pm3.27 B C chjclt atcvat3.1 A B vH C opr chinclt mpan syl sylanc B atelch C atelch syl2an C A B vH C opr i^i i^i chle0t syl A B vH C opr inss1 A B vH C opr i^i A C sslin ax-mp C A incom sseqtr A C i^i 0H C A B vH C opr i^i i^i sseq2 mpbii syl5bi sylbid imp B C = -. adantrl B A vH C opr C_ adantrr jca sylc A B vH C opr inss1 jctil exp jcad x cv A B vH C opr i^i A sseq1 x cv A B vH C opr i^i C vH opreq2 B sseq2d anbi12d Atoms rcla4ev syl6 exp3a B C = C A C_ ioran syl5ib A 0H = -. B A vH C opr C_ pm3.27 syl7 ecase3d))
+thm (atdmd () () (-> (/\ (e. A (Atoms)) (e. B (CH))) (br (` (_|_) A) (MH) (` (_|_) B))) (B A cvp B A chjcomt A atelch sylan2 B <o breq2d bitrd ancoms A B cvdmdt A atelch sylan sylbid B A chcv1t B A cvp bitr4d ancoms bicon1d A B ssdmd1 A atelch sylan sylbid pm2.61d))
+thm (atmd ((A x) (B x)) () (-> (/\ (e. A (Atoms)) (e. B (CH))) (br A (MH) B)) (A x cv atdmd exp r19.21aiv A atelch A x mddmd syl mpbird B CH e. adantr x cv B A MH breq2 CH rcla4v A Atoms e. adantl mpd))
+thm (atmd2 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (br A (MH) B)) (A B cvp A B cvexcht A B cvmdt sylbird B atelch sylan2 sylbid A B chcv1t A B cvp bitr4d bicon1d B A ssmd2 ancoms B atelch sylan2 sylbid pm2.61d))
+var (set p)
+thm (mdsymlem1 ((A p) (B p)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (e. (cv p) (CH)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (C_ (cv p) A)) (mdsymlem1.1 p cv A chub2t mpan2 mdsymlem1.3 syl6ssr mdsymlem1.1 mdsymlem1.2 chjcom p cv sseq2i biimp anim12i p cv C B vH A opr ssin sylib _|_ B ` MH _|_ A ` br adantrl B C i^i A C_ adantlr mdsymlem1.1 A p cv chub1t mpan mdsymlem1.3 syl6ssr _|_ B ` MH _|_ A ` br adantr mdsymlem1.3 p cv CH e. a1i mdsymlem1.1 A p cv chjclt mpan eqeltrd mdsymlem1.2 mdsymlem1.1 B A C dmdi mp3an12 syl imp mpd B C i^i A C_ adantlr mdsymlem1.3 p cv CH e. a1i mdsymlem1.1 A p cv chjclt mpan eqeltrd mdsymlem1.2 B C chinclt mpan mdsymlem1.1 B C i^i A chlejb1t mpan2 3syl biimpa C B incom vH A opreq1i syl5eq _|_ B ` MH _|_ A ` br adantr eqtr3d p cv A vH B opr C_ adantrr sseqtrd))
+thm (mdsymlem2 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (e. (cv p) (Atoms)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (-> (-. (= B (0H))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (q cv p cv vH r cv opreq1 p cv sseq2d q cv p cv A sseq1 r cv B C_ anbi1d anbi12d Atoms rcla4ev p cv Atoms e. B C i^i A C_ pm3.26 _|_ B ` MH _|_ A ` br p cv A vH B opr C_ /\ r cv Atoms e. r cv B C_ /\ ad2antll p cv r cv chub1t p cv atelch r cv atelch syl2an B C i^i A C_ adantlr _|_ B ` MH _|_ A ` br p cv A vH B opr C_ /\ adantlr mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 mdsymlem1 p cv atelch B C i^i A C_ anim1i sylan r cv Atoms e. adantr jca r cv B C_ anim1i p cv p cv vH r cv opr C_ p cv A C_ r cv B C_ anass sylib anasss sylanc exp32 r19.22dv mdsymlem1.2 r hatomic syl5))
+thm (mdsymlem3 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (-. (C_ (i^i B C) A))) (C_ (cv p) (opr A (vH) B))) (-. (= A (0H)))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))) (p cv atelch mdsymlem1.3 p cv CH e. a1i mdsymlem1.1 A p cv chjclt mpan eqeltrd mdsymlem1.2 B C chinclt mpan syl mdsymlem1.1 B C i^i A r chrelat2t mpan2 3syl biimpa p cv A vH B opr C_ A 0H = -. ad2antll mdsymlem1.1 r cv p cv q atcvat4 exp4b com34 com23 imp4b r cv B C ssin mdsymlem1.3 r cv sseq2i biimp r cv B C_ adantl sylbir sylan2 r cv A C_ -. adantrr com12 B C i^i A C_ -. adantlr p cv A vH B opr C_ adantlr imp q cv r cv atnem0 ancoms q cv r cv A sseq1 biimpcd con3d imp r cv B C i^i C_ adantrl syl5bi p cv Atoms e. adantll r cv p cv vH q cv opr C_ adantr p cv q cv chjcomt p cv atelch q cv atelch syl2an r cv Atoms e. adantlr r cv sseq2d q cv r cv p cv atexch q cv atelch syl3an1 3com13 3expa exp3a sylbid imp syld exp3a exp31 com24 imp3a com24 imp4b anasss q cv A C_ r cv p cv vH q cv opr C_ pm3.26 q cv Atoms e. adantl p cv Atoms e. r cv Atoms e. r cv B C i^i C_ r cv A C_ -. /\ /\ /\ a1i r cv B C ssin r cv B C_ r cv C C_ pm3.26 sylbir r cv Atoms e. r cv A C_ -. ad2antrl p cv Atoms e. adantl jctird jcad exp3a B C i^i A C_ -. adantlr p cv A vH B opr C_ adantlr A 0H = -. adantlr r19.22dv mpd exp32 r19.22dv mpd))
+thm (mdsymlem4 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (e. (cv p) (Atoms)) (-> (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (/\ (/\ (-. (= A (0H))) (-. (= B (0H)))) (C_ (cv p) (opr A (vH) B)))) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 r q mdsymlem2 exp31 com4t exp com23 A 0H = -. a1d imp44 com3l mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 r q mdsymlem3 anasss exp31 com3r ancoms B 0H = -. adantlr _|_ B ` MH _|_ A ` br adantl com3l pm2.61d r Atoms q Atoms p cv q cv vH r cv opr C_ q cv A C_ r cv B C_ /\ /\ rexcom syl6ib))
+thm (mdsymlem5 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) (-> (-. (= (cv q) (cv p))) (-> (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))) (-> (/\ (/\ (e. (cv c) (CH)) (C_ A (cv c))) (e. (cv p) (Atoms))) (-> (C_ (cv p) (cv c)) (C_ (cv p) (opr (i^i (cv c) B) (vH) A))))))) (q cv p cv atnem0 p cv q cv vH r cv opr C_ anbi2d r cv Atoms e. 3adant3 q cv p cv r cv atexch q cv atelch syl3an1 sylbid exp3a 3com23 3expa c cv CH e. adantrl q cv A C_ r cv B C_ /\ adantrd imp32 A c cv C_ adantrl p cv c cv C_ adantrr q cv A c cv vH A opr sstr mdsymlem1.1 A c cv chub2t mpan sylan2 p cv c cv c cv vH A opr sstr mdsymlem1.1 c cv A chub1t mpan2 sylan2 anim12i anandirs ancoms q cv CH e. p cv CH e. /\ adantll q cv p cv c cv vH A opr chlubt mdsymlem1.1 c cv A chjclt mpan2 syl3an3 3expa q cv A C_ p cv c cv C_ /\ adantr mpbid A c cv C_ adantrl mdsymlem1.1 A c cv chlejb2t mpan biimpa q cv CH e. p cv CH e. /\ adantll q cv A C_ p cv c cv C_ /\ adantrr sseqtrd exp45 anasss q cv atelch p cv atelch c cv CH e. anim1i ancoms syl2an r cv Atoms e. adantlr q cv A C_ r cv B C_ pm3.26 p cv q cv vH r cv opr C_ q cv p cv = -. ad2antlr syl7 imp44 sstrd q cv A C_ r cv B C_ pm3.27 p cv q cv vH r cv opr C_ q cv p cv = -. ad2antlr A c cv C_ p cv c cv C_ ad2antlr q cv Atoms e. r cv Atoms e. /\ c cv CH e. p cv Atoms e. /\ /\ adantl jca r cv c cv B ssin sylib r cv c cv B i^i q cv chlej1t mdsymlem1.2 c cv B chinclt mpan2 syl3an2 3comr 3expa q cv A C_ adantr r cv vH q cv opr c cv B i^i vH q cv opr c cv B i^i vH A opr sstr2 mdsymlem1.1 q cv A c cv B i^i chlej2t mp3an2 mdsymlem1.2 c cv B chinclt mpan2 sylan2 r cv CH e. adantlr imp syl5com q cv r cv chjcomt c cv CH e. q cv A C_ ad2antll c cv B i^i vH A opr sseq1d sylibrd syld p cv q cv vH r cv opr C_ adantrl p cv q cv vH r cv opr c cv B i^i vH A opr sstr2 q cv CH e. r cv CH e. /\ c cv CH e. /\ q cv A C_ ad2antrl syld exp32 q cv atelch r cv atelch anim12i sylan p cv Atoms e. adantrr imp31 r cv B C_ adantrr anasss q cv p cv = -. adantrr A c cv C_ adantrl p cv c cv C_ adantrr mpd exp32 exp4d exp32 com34 imp4c com24))
+thm (mdsymlem6 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (br (` (_|_) B) (MH) (` (_|_) A))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r c mdsymlem5 q cv p cv A sseq1 p cv A c cv B i^i vH A opr sstr2 mdsymlem1.2 c cv B chinclt mpan2 mdsymlem1.1 A c cv B i^i chub2t mpan syl syl5 syl6bi imp3a p cv c cv C_ a1i com13 A c cv C_ adantrr p cv Atoms e. adantrr r cv B C_ adantlr p cv q cv vH r cv opr C_ adantll com12 exp3a pm2.61d2 r19.23aivv com12 p cv A vH B opr C_ imim2d com34 imp4b mdsymlem1.1 mdsymlem1.2 chjcom p cv sseq2i p cv c cv C_ anbi2i p cv c cv B vH A opr ssin bitr syl5ibr exp r19.20dva c cv B vH A opr i^i c cv B i^i vH A opr p chrelat3t mdsymlem1.2 mdsymlem1.1 chjcl c cv B vH A opr chinclt mpan2 mdsymlem1.2 c cv B chinclt mpan2 mdsymlem1.1 c cv B i^i A chjclt mpan2 syl sylanc A c cv C_ adantr sylibrd exp com3r r19.21aiv mdsymlem1.2 mdsymlem1.1 B A c dmdbr2 mp2an sylibr))
+thm (mdsymlem7 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))))) (mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem4 exp4d com13 r19.21adv mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem6 A 0H = -. B 0H = -. /\ a1i impbid))
+thm (mdsymlem8 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((mdsymlem1.1 (e. A (CH))) (mdsymlem1.2 (e. B (CH))) (mdsymlem1.3 (= C (opr A (vH) (cv p))))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (br (` (_|_) A) (MH) (` (_|_) B)))) (mdsymlem1.1 mdsymlem1.2 chjcom p cv sseq2i q cv r cv chjcomt q cv atelch r cv atelch syl2an p cv sseq2d q cv A C_ r cv B C_ ancom q cv Atoms e. r cv Atoms e. /\ a1i anbi12d bi2rexa q Atoms r Atoms p cv r cv vH q cv opr C_ r cv B C_ q cv A C_ /\ /\ rexcom bitr imbi12i p Atoms biral A 0H = -. B 0H = -. /\ a1i mdsymlem1.1 mdsymlem1.2 mdsymlem1.3 q r mdsymlem7 mdsymlem1.2 mdsymlem1.1 B vH p cv opr eqid r q mdsymlem7 ancoms 3bitr4d))
+thm (mdsym ((A x) (B x)) ((mdsym.1 (e. A (CH))) (mdsym.2 (e. B (CH)))) (<-> (br A (MH) B) (br B (MH) A)) (mdsym.2 chocl mdsym.1 chocl _|_ B ` vH x cv opr eqid mdsymlem8 mdsym.1 pjococ mdsym.2 pjococ MH breq12i mdsym.2 pjococ mdsym.1 pjococ MH breq12i 3bitr3g _|_ B ` 0H = _|_ A ` 0H = oran mdsym.1 chssi _|_ B ` 0H _|_ fveq2 mdsym.2 pjococ choc0 3eqtr3g A sseq2d mpbiri mdsym.1 mdsym.2 A B ssmd1 mp2an mdsym.1 mdsym.2 A B ssmd2 mp2an jca A MH B br B MH A br pm5.1 3syl mdsym.2 chssi _|_ A ` 0H _|_ fveq2 mdsym.1 pjococ choc0 3eqtr3g B sseq2d mpbiri mdsym.2 mdsym.1 B A ssmd2 mp2an mdsym.2 mdsym.1 B A ssmd1 mp2an jca A MH B br B MH A br pm5.1 3syl jaoi sylbir pm2.61i))
+thm (sumdmdi ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))) (A +H B opr A vH B opr x cv ineq2 x cv CH e. B x cv C_ /\ adantr x cv chsh x cv y cv w cv shsubclt exp3a syl B x cv w cv ssel2 syl7 exp4a com23 imp41 z cv A e. adantlr y cv z cv +v w cv opr = adantr y cv w cv z cv hvsubaddt w cv z cv ax-hvcom y cv eqeq1d z cv +v w cv opr y cv eqcom syl6bb y cv H~ e. 3adant1 bitrd 3com23 x cv y cv chelt B x cv C_ adantlr sumdmdi.1 z cv chel sumdmdi.2 w cv chel syl3an 3expa y cv -v w cv opr z cv x cv eleq1 syl6bir imp mpbid x cv CH e. B x cv C_ /\ y cv x cv e. /\ z cv A e. /\ w cv B e. /\ y cv z cv +v w cv opr = pm3.27 jca exp31 r19.22dv w B z cv x cv e. y cv z cv +v w cv opr = r19.42v syl6ib r19.22dva z cv x cv A elin z cv x cv e. z cv A e. ancom bitr w B y cv z cv +v w cv opr = E.e. anbi1i z cv A e. z cv x cv e. w B y cv z cv +v w cv opr = E.e. anass bitr birex2 syl6ibr x cv chsh sumdmdi.1 chshi x cv A shinclt mpan2 syl B x cv C_ y cv x cv e. ad2antll sumdmdi.2 chshi x cv A i^i B y cv z w shselt mpan2 syl sylibrd sumdmdi.1 sumdmdi.2 y cv z w chsel syl5ib exp imp3a y cv x cv A +H B opr elin syl5ib ssrdv A +H B opr A vH B opr = adantl eqsstr3d sumdmdi.1 x cv A chinclt mpan2 sumdmdi.2 x cv A i^i B chslejt mpan2 syl A +H B opr A vH B opr = B x cv C_ ad2antrl sstrd exp32 r19.21aiv sumdmdi.1 sumdmdi.2 A B x dmdbr2 mp2an sylibr))
+thm (cmmd () ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (br A (C_H) B) (br A (MH) B)) (sumdmdi.1 sumdmdi.2 cmcm4 sumdmdi.1 chocl sumdmdi.2 chocl osumcor2 sylbi sumdmdi.1 chocl sumdmdi.2 chocl sumdmdi syl sumdmdi.1 ococ sumdmdi.2 ococ 3brtr3g))
+thm (sumdmdlem ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C y) (C z) (C w)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (/\ (e. C (H~)) (-. (e. C (opr A (+H) B)))) (= (i^i (opr B (+H) (` (span) ({} C))) A) (i^i B A))) (C spansnsht sumdmdi.2 chshi B span C {} ` y cv z w shselt mpan syl y cv z cv w cv hvsubaddt z cv +v w cv opr y cv eqcom syl6bb sumdmdi.1 y cv chel sumdmdi.2 z cv chel C w cv elspansnclt syl3an 3expa y cv -v z cv opr w cv A +H B opr eleq1 sumdmdi.1 chshi sumdmdi.2 chshi y cv z cv shsvs syl5bi com12 C H~ e. w cv span C {} ` e. /\ adantr sylbird exp32 com4r imp31 C A +H B opr e. -. adantrr sumdmdi.1 chshi sumdmdi.2 chshi shscl A +H B opr C w cv elspansn5t ax-mp exp32 y cv z cv +v w cv opr = y cv A e. z cv B e. /\ /\ adantl mpdd w cv 0v z cv +v opreq2 z cv ax-hvaddid sylan9eqr sumdmdi.2 z cv chel sylan y cv eqeq2d y cv A e. adantll biimpac y cv B A elin biimpr ancoms y cv z cv B eleq1 biimparc sylan2 anassrs exp y cv z cv +v w cv opr = w cv 0v = ad2antrl mpd exp32 imp w cv span C {} ` e. a1d C H~ e. C A +H B opr e. -. /\ adantr mpdd exp com23 exp32 com4l imp4c exp4a com23 com4l exp3a r19.23advv sylbid com23 imp4b y cv B +H span C {} ` opr A elin syl5ib ssrdv C spansnsht sumdmdi.2 chshi B span C {} ` shsub1t mpan B B +H span C {} ` opr A ssrin 3syl C A +H B opr e. -. adantr eqssd))
+thm (sumdmdlem2 ((x y) (A x) (A y) (B x) (B y)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (-> (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) (= (opr A (+H) B) (opr A (vH) B))) (y cv spansnsht sumdmdi.2 chshi span y cv {} ` B shsub2t mpan2 syl y cv spansnid sseldd x Atoms x cv vH B opr A vH B opr i^i x cv vH B opr A i^i vH B opr C_ A.e. y cv A +H B opr e. -. ad2antrl y cv spansnat x cv span y cv {} ` vH B opreq1 A vH B opr ineq1d x cv span y cv {} ` vH B opreq1 A ineq1d vH B opreq1d sseq12d Atoms rcla4v syl sumdmdi.2 B y cv spansnjt B span y cv {} ` chjcomt y cv spansncht sylan2 eqtrd mpan A vH B opr ineq1d sumdmdi.2 B y cv spansnjt B span y cv {} ` chjcomt y cv spansncht sylan2 eqtrd mpan A ineq1d vH B opreq1d sseq12d y cv 0v = -. adantr sylibrd com12 exp3a imp B ssid y cv 0v sneq span fveq2d spansn0 syl6eq B +H opreq2d sumdmdi.2 chshi shs0 syl6eq A vH B opr ineq1d B A vH B opr inss1 B ssid sumdmdi.2 sumdmdi.1 chub2 ssini eqssi syl6eq y cv 0v sneq span fveq2d spansn0 syl6eq B +H opreq2d sumdmdi.2 chshi shs0 syl6eq A ineq1d vH B opreq1d sumdmdi.2 sumdmdi.1 chincl sumdmdi.2 chjcom sumdmdi.2 sumdmdi.1 chabs1 eqtr syl6eq sseq12d mpbiri pm2.61d2 y cv A +H B opr e. -. adantrr sumdmdi.2 chshi sumdmdi.1 chshi shsub2 sumdmdi.1 sumdmdi.2 y cv sumdmdlem vH B opreq1d sumdmdi.2 sumdmdi.1 chincl sumdmdi.2 chjcom sumdmdi.2 sumdmdi.1 chabs1 eqtr syl6eq A +H B opr sseq1d mpbiri x Atoms x cv vH B opr A vH B opr i^i x cv vH B opr A i^i vH B opr C_ A.e. adantl sstrd y cv sseld y cv B +H span y cv {} ` opr A vH B opr elin syl5ibr mpand exp32 com34 y cv A +H B opr e. pm2.18 syl8 sumdmdi.1 sumdmdi.2 chjcl y cv chel syl5 pm2.43d ssrdv sumdmdi.1 sumdmdi.2 chslej jctil A +H B opr A vH B opr eqss sylibr))
+thm (sumdmd ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))) (sumdmdi.1 sumdmdi.2 sumdmdi sumdmdi.1 sumdmdi.2 A B x dmdbr4 mp2an x cv atelch x cv vH B opr A vH B opr i^i x cv vH B opr A i^i vH B opr C_ imim1i r19.20i2 sylbi sumdmdi.1 sumdmdi.2 x sumdmdlem2 syl impbi))
+thm (dmdbr4at ((A x) (B x)) ((sumdmdi.1 (e. A (CH))) (sumdmdi.2 (e. B (CH)))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))) (sumdmdi.1 sumdmdi.2 A B x dmdbr4 mp2an x cv atelch x cv vH B opr A vH B opr i^i x cv vH B opr A i^i vH B opr C_ imim1i r19.20i2 sylbi sumdmdi.1 sumdmdi.2 x sumdmdlem2 sumdmdi.1 sumdmdi.2 sumdmd sylib impbi))
+thm (cdjreu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((cdjreu.1 (e. A (SH))) (cdjreu.2 (e. B (SH)))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (E!e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))) (cdjreu.1 cdjreu.2 C x y shsel biimp x cv y cv z cv w cv hvaddsub4t cdjreu.1 x cv shel cdjreu.2 y cv shel anim12i cdjreu.1 z cv shel cdjreu.2 w cv shel anim12i syl2an an4s A B i^i 0H = adantll x cv -v z cv opr w cv -v y cv opr B eleq1 cdjreu.2 B w cv y cv shsubclt ax-mp ancoms syl5bir com12 x cv A e. z cv A e. /\ adantl cdjreu.1 A x cv z cv shsubclt ax-mp y cv B e. w cv B e. /\ adantr jctild A B i^i 0H = adantll A B i^i 0H x cv -v z cv opr eleq2 x cv -v z cv opr A B elin syl5bbr x cv A e. z cv A e. /\ y cv B e. w cv B e. /\ ad2antll sylibd x cv z cv hvsubeq0t x cv -v z cv opr elch0 syl5bb cdjreu.1 x cv shel cdjreu.1 z cv shel syl2an A B i^i 0H = y cv B e. w cv B e. /\ ad2antlr sylibd sylbid C x cv +v y cv opr z cv +v w cv opr eqtr2t syl5 exp r19.23advv y B w B C x cv +v y cv opr = C z cv +v w cv opr = reeanv syl5ibr exp r19.21aivv anim12i x cv z cv +v y cv opreq1 C eqeq2d y B birexdv y cv w cv z cv +v opreq2 C eqeq2d B cbvrexv syl6bb A reu4 sylibr))
addfile ./zfc/set_mm.ghi
hunk ./zfc/set_mm.ghi 1
+# Converted from set.mm - Version of 22-Oct-2005
+
+# Original by Norm Megill
+
+# This file (specifically, the version of this file with the above date)
+# is placed in the public domain per the Creative Commons Public Domain
+# Dedication.  http://creativecommons.org/licenses/publicdomain/
+
+param (SET_MM_AX zfc/set_mm_ax () "")
+var ( wff ph )
+var ( wff ps )
+var ( wff ch )
+var ( wff th )
+var ( wff ta )
+stmt (a1i () (ph) (-> ps ph))
+stmt (a2i () ((-> ph (-> ps ch))) (-> (-> ph ps) (-> ph ch)))
+stmt (syl () ((-> ph ps) (-> ps ch)) (-> ph ch))
+stmt (com12 () ((-> ph (-> ps ch))) (-> ps (-> ph ch)))
+stmt (a1d () ((-> ph ps)) (-> ph (-> ch ps)))
+stmt (a2d () ((-> ph (-> ps (-> ch th)))) (-> ph (-> (-> ps ch) (-> ps th))))
+stmt (imim2 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps))))
+stmt (imim1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch))))
+stmt (imim2i () ((-> ph ps)) (-> (-> ch ph) (-> ch ps)))
+stmt (imim1i () ((-> ph ps)) (-> (-> ps ch) (-> ph ch)))
+stmt (imim12i () ((-> ph ps) (-> ch th)) (-> (-> ps ch) (-> ph th)))
+stmt (3syl () ((-> ph ps) (-> ps ch) (-> ch th)) (-> ph th))
+stmt (syl5 () ((-> ph (-> ps ch)) (-> th ps)) (-> ph (-> th ch)))
+stmt (syl6 () ((-> ph (-> ps ch)) (-> ch th)) (-> ph (-> ps th)))
+stmt (syl7 () ((-> ph (-> ps (-> ch th))) (-> ta ch)) (-> ph (-> ps (-> ta th))))
+stmt (syl8 () ((-> ph (-> ps (-> ch th))) (-> th ta)) (-> ph (-> ps (-> ch ta))))
+stmt (imim2d () ((-> ph (-> ps ch))) (-> ph (-> (-> th ps) (-> th ch))))
+stmt (syld () ((-> ph (-> ps ch)) (-> ph (-> ch th))) (-> ph (-> ps th)))
+stmt (imim1d () ((-> ph (-> ps ch))) (-> ph (-> (-> ch th) (-> ps th))))
+stmt (imim12d () ((-> ph (-> ps ch)) (-> ph (-> th ta))) (-> ph (-> (-> ch th) (-> ps ta))))
+stmt (pm2.04 () () (-> (-> ph (-> ps ch)) (-> ps (-> ph ch))))
+stmt (pm2.83 () () (-> (-> ph (-> ps ch)) (-> (-> ph (-> ch th)) (-> ph (-> ps th)))))
+stmt (com23 () ((-> ph (-> ps (-> ch th)))) (-> ph (-> ch (-> ps th))))
+stmt (com13 () ((-> ph (-> ps (-> ch th)))) (-> ch (-> ps (-> ph th))))
+stmt (com3l () ((-> ph (-> ps (-> ch th)))) (-> ps (-> ch (-> ph th))))
+stmt (com3r () ((-> ph (-> ps (-> ch th)))) (-> ch (-> ph (-> ps th))))
+stmt (com34 () ((-> ph (-> ps (-> ch (-> th ta))))) (-> ph (-> ps (-> th (-> ch ta)))))
+stmt (com24 () ((-> ph (-> ps (-> ch (-> th ta))))) (-> ph (-> th (-> ch (-> ps ta)))))
+stmt (com14 () ((-> ph (-> ps (-> ch (-> th ta))))) (-> th (-> ps (-> ch (-> ph ta)))))
+stmt (com4l () ((-> ph (-> ps (-> ch (-> th ta))))) (-> ps (-> ch (-> th (-> ph ta)))))
+stmt (com4t () ((-> ph (-> ps (-> ch (-> th ta))))) (-> ch (-> th (-> ph (-> ps ta)))))
+stmt (com4r () ((-> ph (-> ps (-> ch (-> th ta))))) (-> th (-> ph (-> ps (-> ch ta)))))
+stmt (a1dd () ((-> ph (-> ps ch))) (-> ph (-> ps (-> th ch))))
+stmt (mp2 () (ph ps (-> ph (-> ps ch))) ch)
+stmt (mpi () (ps (-> ph (-> ps ch))) (-> ph ch))
+stmt (mpii () (ch (-> ph (-> ps (-> ch th)))) (-> ph (-> ps th)))
+stmt (mpd () ((-> ph ps) (-> ph (-> ps ch))) (-> ph ch))
+stmt (mpdd () ((-> ph (-> ps ch)) (-> ph (-> ps (-> ch th)))) (-> ph (-> ps th)))
+stmt (mpid () ((-> ph ch) (-> ph (-> ps (-> ch th)))) (-> ph (-> ps th)))
+stmt (mpdi () ((-> ps ch) (-> ph (-> ps (-> ch th)))) (-> ph (-> ps th)))
+stmt (mpcom () ((-> ps ph) (-> ph (-> ps ch))) (-> ps ch))
+stmt (syldd () ((-> ph (-> ps (-> ch th))) (-> ph (-> ps (-> th ta)))) (-> ph (-> ps (-> ch ta))))
+stmt (sylcom () ((-> ph (-> ps ch)) (-> ps (-> ch th))) (-> ph (-> ps th)))
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+stmt (sbi2 () () (-> (-> ([/] (cv y) x ph) ([/] (cv y) x ps)) ([/] (cv y) x (-> ph ps))))
+stmt (sbim () () (<-> ([/] (cv y) x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (sbor () () (<-> ([/] (cv y) x (\/ ph ps)) (\/ ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (sb19.21 () ((-> ph (A. x ph))) (<-> ([/] (cv y) x (-> ph ps)) (-> ph ([/] (cv y) x ps))))
+stmt (sban () () (<-> ([/] (cv y) x (/\ ph ps)) (/\ ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (sbbi () () (<-> ([/] (cv y) x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (sblbis () ((<-> ([/] (cv y) x ph) ps)) (<-> ([/] (cv y) x (<-> ch ph)) (<-> ([/] (cv y) x ch) ps)))
+stmt (sbrbis () ((<-> ([/] (cv y) x ph) ps)) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ([/] (cv y) x ch))))
+stmt (sbrbif () ((-> ch (A. x ch)) (<-> ([/] (cv y) x ph) ps)) (<-> ([/] (cv y) x (<-> ph ch)) (<-> ps ch)))
+stmt (sbea4 () () (-> ([/] (cv y) x ph) (E. x ph)))
+stmt (sbia4 () () (-> (A. x (-> ph ps)) (-> ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (sbba4 () () (-> (A. x (<-> ph ps)) (<-> ([/] (cv y) x ph) ([/] (cv y) x ps))))
+stmt (bisbd () ((-> ph (A. x ph)) (-> ph (<-> ps ch))) (-> ph (<-> ([/] (cv y) x ps) ([/] (cv y) x ch))))
+stmt (sbequ5 () () (<-> ([/] (cv w) z (A. x (= (cv x) (cv y)))) (A. x (= (cv x) (cv y)))))
+stmt (sbequ6 () () (<-> ([/] (cv w) z (-. (A. x (= (cv x) (cv y))))) (-. (A. x (= (cv x) (cv y))))))
+stmt (sbt () (ph) ([/] (cv y) x ph))
+stmt (equsb1 () () ([/] (cv y) x (= (cv x) (cv y))))
+stmt (equsb2 () () ([/] (cv y) x (= (cv y) (cv x))))
+stmt (sbequ8 () () (<-> ([/] (cv y) x ph) ([/] (cv y) x (-> (= (cv x) (cv y)) ph))))
+stmt (sbied () ((-> ph (A. x ph)) (-> ph (-> ch (A. x ch))) (-> ph (-> (= (cv x) (cv y)) (<-> ps ch)))) (-> ph (<-> ([/] (cv y) x ps) ch)))
+stmt (sbie () ((-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (<-> ([/] (cv y) x ph) ps))
+stmt (hbsb4 () ((-> ph (A. z ph))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph)))))
+stmt (hbsb4t () () (-> (A. x (A. z (-> ph (A. z ph)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> ([/] (cv y) x ph) (A. z ([/] (cv y) x ph))))))
+stmt (dvelimf2 () ((-> ph (A. x ph)) (-> ps (A. z ps)) (-> (= (cv z) (cv y)) (<-> ph ps))) (-> (-. (A. x (= (cv x) (cv y)))) (-> ps (A. x ps))))
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+stmt (sbco () () (<-> ([/] (cv y) x ([/] (cv x) y ph)) ([/] (cv y) x ph)))
+stmt (sbid2 () ((-> ph (A. x ph))) (<-> ([/] (cv y) x ([/] (cv x) y ph)) ph))
+stmt (sbidm () () (<-> ([/] (cv y) x ([/] (cv y) x ph)) ([/] (cv y) x ph)))
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+stmt (sbco3 () () (<-> ([/] (cv z) y ([/] (cv y) x ph)) ([/] (cv z) x ([/] (cv x) y ph))))
+stmt (sbcom () () (<-> ([/] (cv y) z ([/] (cv y) x ph)) ([/] (cv y) x ([/] (cv y) z ph))))
+stmt (sb5rf () ((-> ph (A. y ph))) (<-> ph (E. y (/\ (= (cv y) (cv x)) ([/] (cv y) x ph)))))
+stmt (sb6rf () ((-> ph (A. y ph))) (<-> ph (A. y (-> (= (cv y) (cv x)) ([/] (cv y) x ph)))))
+stmt (sb8 () ((-> ph (A. y ph))) (<-> (A. x ph) (A. y ([/] (cv y) x ph))))
+stmt (sb8e () ((-> ph (A. y ph))) (<-> (E. x ph) (E. y ([/] (cv y) x ph))))
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+stmt (equid2 ((x y)) () (= (cv x) (cv x)))
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+stmt (a4b1 ((ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (-> (A. x ph) ps))
+stmt (a4w ((ph x)) ((-> (= (cv x) (cv y)) (-> ph ps))) (-> ph (E. x ps)))
+stmt (a4w1 ((ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps)) ps) (E. x ph))
+stmt (equvin ((x z) (y z)) () (<-> (= (cv x) (cv y)) (E. z (/\ (= (cv x) (cv z)) (= (cv z) (cv y))))))
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+stmt (a16gb ((x y)) () (-> (A. x (= (cv x) (cv y))) (<-> ph (A. z ph))))
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+stmt (bi2exdv ((ph x) (ph y)) ((-> ph (<-> ps ch))) (-> ph (<-> (E. x (E. y ps)) (E. x (E. y ch)))))
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+stmt (19.9rv ((ph x)) () (<-> ph (E. x ph)))
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+stmt (19.21adv ((ph x) (ps x)) ((-> ph (-> ps ch))) (-> ph (-> ps (A. x ch))))
+stmt (19.20dv ((ph x)) ((-> ph (-> ps ch))) (-> ph (-> (A. x ps) (A. x ch))))
+stmt (19.22dv ((ph x)) ((-> ph (-> ps ch))) (-> ph (-> (E. x ps) (E. x ch))))
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+stmt (19.22dvv ((ph x) (ph y)) ((-> ph (-> ps ch))) (-> ph (-> (E. x (E. y ps)) (E. x (E. y ch)))))
+stmt (19.23v ((ps x)) () (<-> (A. x (-> ph ps)) (-> (E. x ph) ps)))
+stmt (19.23vv ((ps x) (ps y)) () (<-> (A. x (A. y (-> ph ps))) (-> (E. x (E. y ph)) ps)))
+stmt (19.23aiv ((ps x)) ((-> ph ps)) (-> (E. x ph) ps))
+stmt (19.23aivv ((ps x) (ps y)) ((-> ph ps)) (-> (E. x (E. y ph)) ps))
+stmt (19.23adv ((ch x) (ph x)) ((-> ph (-> ps ch))) (-> ph (-> (E. x ps) ch)))
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+stmt (19.28v ((ph x)) () (<-> (A. x (/\ ph ps)) (/\ ph (A. x ps))))
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+stmt (19.42v ((ph x)) () (<-> (E. x (/\ ph ps)) (/\ ph (E. x ps))))
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+stmt (19.42vv ((ph x) (ph y)) () (<-> (E. x (E. y (/\ ph ps))) (/\ ph (E. x (E. y ps)))))
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+var ( set f )
+var ( set g )
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+stmt (bimod () ((-> ph (A. x ph)) (-> ph (<-> ps ch))) (-> ph (<-> (E* x ps) (E* x ch))))
+stmt (bimo () ((<-> ps ch)) (<-> (E* x ps) (E* x ch)))
+stmt (hbmo1 () () (-> (E* x ph) (A. x (E* x ph))))
+stmt (hbmo () ((-> ph (A. x ph))) (-> (E* y ph) (A. x (E* y ph))))
+stmt (cbvmo ((x y)) ((-> ph (A. y ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (E* x ph) (E* y ps)))
+stmt (eu5 ((x y) (ph y)) () (<-> (E! x ph) (/\ (E. x ph) (E* x ph))))
+stmt (eu4 ((x y) (ph y) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (E! x ph) (/\ (E. x ph) (A. x (A. y (-> (/\ ph ps) (= (cv x) (cv y))))))))
+stmt (eumo () () (-> (E! x ph) (E* x ph)))
+stmt (eumoi () ((E! x ph)) (E* x ph))
+stmt (exmoeu () () (<-> (E. x ph) (-> (E* x ph) (E! x ph))))
+stmt (exmoeu2 () () (-> (E. x ph) (<-> (E* x ph) (E! x ph))))
+stmt (moabs () () (<-> (E* x ph) (-> (E. x ph) (E* x ph))))
+stmt (exmo () () (\/ (E. x ph) (E* x ph)))
+stmt (immo ((x y) (ph y) (ps y)) () (-> (A. x (-> ph ps)) (-> (E* x ps) (E* x ph))))
+stmt (moimv ((x y) (ph x) (ph y) (ps y)) () (-> (E* x (-> ph ps)) (-> ph (E* x ps))))
+stmt (euimmo () () (-> (A. x (-> ph ps)) (-> (E! x ps) (E* x ph))))
+stmt (euim () () (-> (/\ (E. x ph) (A. x (-> ph ps))) (-> (E! x ps) (E! x ph))))
+stmt (moan () () (-> (E* x ph) (E* x (/\ ps ph))))
+stmt (moani () ((E* x ph)) (E* x (/\ ps ph)))
+stmt (moor () () (-> (E* x (\/ ph ps)) (E* x ph)))
+stmt (mooran1 () () (-> (\/ (E* x ph) (E* x ps)) (E* x (/\ ph ps))))
+stmt (mooran2 () () (-> (E* x (\/ ph ps)) (/\ (E* x ph) (E* x ps))))
+stmt (moanim ((x y) (ph y) (ps y)) ((-> ph (A. x ph))) (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))))
+stmt (euan () ((-> ph (A. x ph))) (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))))
+stmt (moanimv ((ph x)) () (<-> (E* x (/\ ph ps)) (-> ph (E* x ps))))
+stmt (euanv ((ph x)) () (<-> (E! x (/\ ph ps)) (/\ ph (E! x ps))))
+stmt (mopick ((x y) (ph y) (ps y)) () (-> (/\ (E* x ph) (E. x (/\ ph ps))) (-> ph ps)))
+stmt (eupick () () (-> (/\ (E! x ph) (E. x (/\ ph ps))) (-> ph ps)))
+stmt (eupickb () () (-> (/\/\ (E! x ph) (E! x ps) (E. x (/\ ph ps))) (<-> ph ps)))
+stmt (mopick2 () () (-> (/\/\ (E* x ph) (E. x (/\ ph ps)) (E. x (/\ ph ch))) (E. x (/\/\ ph ps ch))))
+stmt (moexex () ((-> ph (A. y ph))) (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))))
+stmt (moexexv ((ph y)) () (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps)))))
+stmt (2moex () () (-> (E* x (E. y ph)) (A. y (E* x ph))))
+stmt (2euex () () (-> (E! x (E. y ph)) (E. y (E! x ph))))
+stmt (2eumo () () (-> (E! x (E* y ph)) (E* x (E! y ph))))
+stmt (2eu2ex () () (-> (E! x (E! y ph)) (E. x (E. y ph))))
+stmt (2moswap () () (-> (A. x (E* y ph)) (-> (E* x (E. y ph)) (E* y (E. x ph)))))
+stmt (2euswap () () (-> (A. x (E* y ph)) (-> (E! x (E. y ph)) (E! y (E. x ph)))))
+stmt (2exeu () () (-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y ph))))
+stmt (2mo ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ([/] (cv z) x ([/] (cv w) y ph))) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))))
+stmt (2mos ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (x z) (w x) (y z) (w y)) ((-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps))) (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ps) (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))))
+stmt (2eu1 () () (-> (A. x (E* y ph)) (<-> (E! x (E! y ph)) (/\ (E! x (E. y ph)) (E! y (E. x ph))))))
+stmt (2eu2 () () (-> (E! y (E. x ph)) (<-> (E! x (E! y ph)) (E! x (E. y ph)))))
+stmt (2eu3 () () (-> (A. x (A. y (\/ (E* x ph) (E* y ph)))) (<-> (/\ (E! x (E! y ph)) (E! y (E! x ph))) (/\ (E! x (E. y ph)) (E! y (E. x ph))))))
+stmt (2eu4 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))))
+stmt (2eu5 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E! y ph)) (A. x (E* y ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))))
+stmt (2eu6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E. z (E. w (A. x (A. y (<-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))))
+stmt (2eu7 () () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y (/\ (E. x ph) (E. y ph))))))
+stmt (2eu8 () () (<-> (E! x (E! y (/\ (E. x ph) (E. y ph)))) (E! x (E! y (/\ (E! x ph) (E. y ph))))))
+stmt (exists1 ((x y)) () (<-> (E! x (= (cv x) (cv x))) (A. x (= (cv x) (cv y)))))
+stmt (exists2 ((x y)) () (-> (/\ (E. x ph) (E. x (-. ph))) (-. (E! x (= (cv x) (cv x))))))
+var ( set t )
+stmt (axun ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (A. y (-> (E. x (/\ (e. (cv y) (cv x)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))))
+stmt (axpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (A. y (-> (A. x (-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) (e. (cv y) (cv x))))))
+stmt (axreg ((x y) (x z) (y z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))))
+stmt (axinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))
+stmt (axac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. x (A. y (A. z (-> (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))))
+stmt (axext ((x y) (x z) (y z)) () (E. z (-> (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y))) (= (cv x) (cv y)))))
+stmt (zfext2 ((x z) (w z) (w x) (y z) (w y)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))))
+stmt (bm1.1 ((x y) (x z) (y z) (ph z)) ((-> ph (A. x ph))) (-> (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (E! x (A. y (<-> (e. (cv y) (cv x)) ph)))))
+var ( class A )
+var ( class B )
+var ( class C )
+var ( class D )
+var ( class R )
+var ( class S )
+var ( class T )
+stmt (abid () () (<-> (e. (cv x) ({|} x ph)) ph))
+stmt (hbab1 ((x y)) () (-> (e. (cv y) ({|} x ph)) (A. x (e. (cv y) ({|} x ph)))))
+stmt (hbab ((x z)) ((-> ph (A. x ph))) (-> (e. (cv z) ({|} y ph)) (A. x (e. (cv z) ({|} y ph)))))
+stmt (dfcleq ((A x) (B x) (x y) (x z) (y z)) () (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))))
+stmt (eqrd ((A x) (B x) (ph x)) ((-> ph (<-> (e. (cv x) A) (e. (cv x) B)))) (-> ph (= A B)))
+stmt (eqri ((A x) (B x)) ((<-> (e. (cv x) A) (e. (cv x) B))) (= A B))
+stmt (eqid ((A x)) () (= A A))
+stmt (eqcom ((A x) (B x)) () (<-> (= A B) (= B A)))
+stmt (eqcoms () ((-> (= A B) ph)) (-> (= B A) ph))
+stmt (eqcomi () ((= A B)) (= B A))
+stmt (eqcomd () ((-> ph (= A B))) (-> ph (= B A)))
+stmt (eqeq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (= A C) (= B C))))
+stmt (eqeq1i () ((= A B)) (<-> (= A C) (= B C)))
+stmt (eqeq1d () ((-> ph (= A B))) (-> ph (<-> (= A C) (= B C))))
+stmt (eqeq2 () () (-> (= A B) (<-> (= C A) (= C B))))
+stmt (eqeq2i () ((= A B)) (<-> (= C A) (= C B)))
+stmt (eqeq2d () ((-> ph (= A B))) (-> ph (<-> (= C A) (= C B))))
+stmt (eqeq12 () () (-> (/\ (= A B) (= C D)) (<-> (= A C) (= B D))))
+stmt (eqeq12i () ((= A B) (= C D)) (<-> (= A C) (= B D)))
+stmt (eqeq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (<-> (= A C) (= B D))))
+stmt (eqeqan12d () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ph ps) (<-> (= A C) (= B D))))
+stmt (eqeqan12rd () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ps ph) (<-> (= A C) (= B D))))
+stmt (eqtrt () () (-> (/\ (= A B) (= B C)) (= A C)))
+stmt (eqtr2t () () (-> (/\ (= A B) (= A C)) (= B C)))
+stmt (eqtr3t () () (-> (/\ (= A C) (= B C)) (= A B)))
+stmt (eqtr () ((= A B) (= B C)) (= A C))
+stmt (eqtr2 () ((= A B) (= B C)) (= C A))
+stmt (eqtr3 () ((= A B) (= A C)) (= B C))
+stmt (eqtr4 () ((= A B) (= C B)) (= A C))
+stmt (3eqtr () ((= A B) (= B C) (= C D)) (= A D))
+stmt (3eqtr3 () ((= A B) (= A C) (= B D)) (= C D))
+stmt (3eqtr3r () ((= A B) (= A C) (= B D)) (= D C))
+stmt (3eqtr4 () ((= A B) (= C A) (= D B)) (= C D))
+stmt (3eqtr4r () ((= A B) (= C A) (= D B)) (= D C))
+stmt (eqtrd () ((-> ph (= A B)) (-> ph (= B C))) (-> ph (= A C)))
+stmt (eqtr2d () ((-> ph (= A B)) (-> ph (= B C))) (-> ph (= C A)))
+stmt (eqtr3d () ((-> ph (= A B)) (-> ph (= A C))) (-> ph (= B C)))
+stmt (eqtr4d () ((-> ph (= A B)) (-> ph (= C B))) (-> ph (= A C)))
+stmt (3eqtrd () ((-> ph (= A B)) (-> ph (= B C)) (-> ph (= C D))) (-> ph (= A D)))
+stmt (3eqtr3d () ((-> ph (= A B)) (-> ph (= A C)) (-> ph (= B D))) (-> ph (= C D)))
+stmt (3eqtr4d () ((-> ph (= A B)) (-> ph (= C A)) (-> ph (= D B))) (-> ph (= C D)))
+stmt (3eqtr4rd () ((-> ph (= A B)) (-> ph (= C A)) (-> ph (= D B))) (-> ph (= D C)))
+stmt (syl5eq () ((-> ph (= A B)) (= C A)) (-> ph (= C B)))
+stmt (syl5req () ((-> ph (= A B)) (= C A)) (-> ph (= B C)))
+stmt (syl5eqr () ((-> ph (= A B)) (= A C)) (-> ph (= C B)))
+stmt (syl5reqr () ((-> ph (= A B)) (= A C)) (-> ph (= B C)))
+stmt (syl6eq () ((-> ph (= A B)) (= B C)) (-> ph (= A C)))
+stmt (syl6req () ((-> ph (= A B)) (= B C)) (-> ph (= C A)))
+stmt (syl6eqr () ((-> ph (= A B)) (= C B)) (-> ph (= A C)))
+stmt (syl6reqr () ((-> ph (= A B)) (= C B)) (-> ph (= C A)))
+stmt (sylan9eq () ((-> ph (= A B)) (-> ps (= B C))) (-> (/\ ph ps) (= A C)))
+stmt (sylan9eqr () ((-> ph (= A B)) (-> ps (= B C))) (-> (/\ ps ph) (= A C)))
+stmt (3eqtr3g () ((-> ph (= A B)) (= A C) (= B D)) (-> ph (= C D)))
+stmt (3eqtr4g () ((-> ph (= A B)) (= C A) (= D B)) (-> ph (= C D)))
+var ( class F )
+var ( class G )
+stmt (eq2tr () ((-> (= A C) (= D F)) (-> (= B D) (= C G))) (<-> (/\ (= A C) (= B F)) (/\ (= B D) (= A G))))
+stmt (eleq1 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. A C) (e. B C))))
+stmt (eleq2 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. C A) (e. C B))))
+stmt (eleq12 () () (-> (/\ (= A B) (= C D)) (<-> (e. A C) (e. B D))))
+stmt (eleq1i () ((= A B)) (<-> (e. A C) (e. B C)))
+stmt (eleq2i () ((= A B)) (<-> (e. C A) (e. C B)))
+stmt (eleq12i () ((= A B) (= C D)) (<-> (e. A C) (e. B D)))
+stmt (eleq1d () ((-> ph (= A B))) (-> ph (<-> (e. A C) (e. B C))))
+stmt (eleq2d () ((-> ph (= A B))) (-> ph (<-> (e. C A) (e. C B))))
+stmt (eleq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (<-> (e. A C) (e. B D))))
+stmt (eleq1a () () (-> (e. A B) (-> (= C A) (e. C B))))
+stmt (eqeltr () ((= A B) (e. B C)) (e. A C))
+stmt (eqeltrr () ((= A B) (e. A C)) (e. B C))
+stmt (eleqtr () ((e. A B) (= B C)) (e. A C))
+stmt (eleqtrr () ((e. A B) (= C B)) (e. A C))
+stmt (eqeltrd () ((-> ph (= A B)) (-> ph (e. B C))) (-> ph (e. A C)))
+stmt (eqeltrrd () ((-> ph (= A B)) (-> ph (e. A C))) (-> ph (e. B C)))
+stmt (eleqtrd () ((-> ph (e. A B)) (-> ph (= B C))) (-> ph (e. A C)))
+stmt (eleqtrrd () ((-> ph (e. A B)) (-> ph (= C B))) (-> ph (e. A C)))
+stmt (syl6eleqr () ((-> ph (e. A B)) (= C B)) (-> ph (e. A C)))
+stmt (cleqf ((A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))))
+stmt (nelneq () () (-> (/\ (e. A C) (-. (e. B C))) (-. (= A B))))
+stmt (nelneq2 () () (-> (/\ (e. A B) (-. (e. A C))) (-. (= B C))))
+stmt (hblem ((A y) (x y) (x z) (y z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A)))) (-> (e. (cv z) A) (A. x (e. (cv z) A))))
+stmt (hbeq ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (-> (= A B) (A. x (= A B))))
+stmt (hbel ((A y) (w y) (A w) (B z) (w z) (B w) (x y) (w x) (x z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B)))) (-> (e. A B) (A. x (e. A B))))
+stmt (hbeleq ((x y) (x z) (y z) (A y) (A z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A)))) (-> (= (cv y) A) (A. x (= (cv y) A))))
+stmt (eqab ((A x) (x y) (A y) (ph y)) () (<-> (= A ({|} x ph)) (A. x (<-> (e. (cv x) A) ph))))
+stmt (eqabr ((A x)) () (<-> (= ({|} x ph) A) (A. x (<-> ph (e. (cv x) A)))))
+stmt (eqabi () ((= A ({|} x ph))) (<-> (e. (cv x) A) ph))
+stmt (eqabri () ((= ({|} x ph) A)) (<-> ph (e. (cv x) A)))
+stmt (eqabd () ((-> ph (= A ({|} x ps)))) (-> ph (<-> (e. (cv x) A) ps)))
+stmt (eq2ab ((ph y) (ps y) (x y)) () (<-> (= ({|} x ph) ({|} x ps)) (A. x (<-> ph ps))))
+stmt (biabri ((A x)) ((<-> (e. (cv x) A) ph)) (= A ({|} x ph)))
+stmt (biabi () ((<-> ph ps)) (= ({|} x ph) ({|} x ps)))
+stmt (biabd () ((-> ph (A. x ph)) (-> ph (<-> ps ch))) (-> ph (= ({|} x ps) ({|} x ch))))
+stmt (biabdv ((ph x)) ((-> ph (<-> ps ch))) (-> ph (= ({|} x ps) ({|} x ch))))
+stmt (biabrdv ((A x) (ph x)) ((-> ph (<-> (e. (cv x) A) ps))) (-> ph (= A ({|} x ps))))
+stmt (biabldv ((A x) (ph x)) ((-> ph (<-> ps (e. (cv x) A)))) (-> ph (= ({|} x ps) A)))
+stmt (abid2 ((A x)) () (= ({|} x (e. (cv x) A)) A))
+stmt (clelab ((A x) (x y) (A y) (ph y)) () (<-> (e. A ({|} x ph)) (E. x (/\ (= (cv x) A) ph))))
+stmt (sbab ((A z) (x z) (y z)) () (-> (= (cv x) (cv y)) (= A ({|} z ([/] (cv y) x (e. (cv z) A))))))
+stmt (sbabel ((v w) (A w) (A v) (w x) (x z) (v x) (v z) (y z) (v y) (ph v)) ((-> (e. (cv w) A) (A. x (e. (cv w) A)))) (<-> ([/] (cv y) x (e. ({|} z ph) A)) (e. ({|} z ([/] (cv y) x ph)) A)))
+stmt (neeq1 () () (-> (= A B) (<-> (=/= A C) (=/= B C))))
+stmt (neeq2 () () (-> (= A B) (<-> (=/= C A) (=/= C B))))
+stmt (neeq1i () ((= A B)) (<-> (=/= A C) (=/= B C)))
+stmt (neeq2i () ((= A B)) (<-> (=/= C A) (=/= C B)))
+stmt (neeq1d () ((-> ph (= A B))) (-> ph (<-> (=/= A C) (=/= B C))))
+stmt (neeq2d () ((-> ph (= A B))) (-> ph (<-> (=/= C A) (=/= C B))))
+stmt (eqneqi () ((<-> (= A B) (= C D))) (<-> (=/= A B) (=/= C D)))
+stmt (eqneqd () ((-> ph (<-> (= A B) (= C D)))) (-> ph (<-> (=/= A B) (=/= C D))))
+stmt (necom () () (<-> (=/= A B) (=/= B A)))
+stmt (neleq1 () () (-> (= A B) (<-> (e/ A C) (e/ B C))))
+stmt (neleq2 () () (-> (= A B) (<-> (e/ C A) (e/ C B))))
+stmt (ralnex () () (<-> (A.e. x A (-. ph)) (-. (E.e. x A ph))))
+stmt (rexnal () () (<-> (E.e. x A (-. ph)) (-. (A.e. x A ph))))
+stmt (dfral2 () () (<-> (A.e. x A ph) (-. (E.e. x A (-. ph)))))
+stmt (dfrex2 () () (<-> (E.e. x A ph) (-. (A.e. x A (-. ph)))))
+stmt (biralda () ((-> ph (A. x ph)) (-> (/\ ph (e. (cv x) A)) (<-> ps ch))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))))
+stmt (birexda () ((-> ph (A. x ph)) (-> (/\ ph (e. (cv x) A)) (<-> ps ch))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))))
+stmt (biraldva ((ph x)) ((-> (/\ ph (e. (cv x) A)) (<-> ps ch))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))))
+stmt (birexdva ((ph x)) ((-> (/\ ph (e. (cv x) A)) (<-> ps ch))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))))
+stmt (birald () ((-> ph (A. x ph)) (-> ph (<-> ps ch))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))))
+stmt (birexd () ((-> ph (A. x ph)) (-> ph (<-> ps ch))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))))
+stmt (biraldv ((ph x)) ((-> ph (<-> ps ch))) (-> ph (<-> (A.e. x A ps) (A.e. x A ch))))
+stmt (birexdv ((ph x)) ((-> ph (<-> ps ch))) (-> ph (<-> (E.e. x A ps) (E.e. x A ch))))
+stmt (biraldv2 ((ph x)) ((-> ph (<-> (-> (e. (cv x) A) ps) (-> (e. (cv x) B) ch)))) (-> ph (<-> (A.e. x A ps) (A.e. x B ch))))
+stmt (birexdv2 ((ph x)) ((-> ph (<-> (/\ (e. (cv x) A) ps) (/\ (e. (cv x) B) ch)))) (-> ph (<-> (E.e. x A ps) (E.e. x B ch))))
+stmt (biral () ((<-> ph ps)) (<-> (A.e. x A ph) (A.e. x A ps)))
+stmt (birex () ((<-> ph ps)) (<-> (E.e. x A ph) (E.e. x A ps)))
+stmt (bi2ral () ((<-> ph ps)) (<-> (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps))))
+stmt (bi2rex () ((<-> ph ps)) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))))
+stmt (biral2 () ((<-> (-> (e. (cv x) A) ph) (-> (e. (cv x) B) ps))) (<-> (A.e. x A ph) (A.e. x B ps)))
+stmt (birex2 () ((<-> (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) ps))) (<-> (E.e. x A ph) (E.e. x B ps)))
+stmt (birala () ((-> (e. (cv x) A) (<-> ph ps))) (<-> (A.e. x A ph) (A.e. x A ps)))
+stmt (birexa () ((-> (e. (cv x) A) (<-> ph ps))) (<-> (E.e. x A ph) (E.e. x A ps)))
+stmt (bi2rexa ((x y) (A y)) ((-> (/\ (e. (cv x) A) (e. (cv y) B)) (<-> ph ps))) (<-> (E.e. x A (E.e. y B ph)) (E.e. x A (E.e. y B ps))))
+stmt (r2al ((x y) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A. x (A. y (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)))))
+stmt (bi2ralda ((x y) (A y)) ((-> ph (A. x ph)) (-> ph (A. y ph)) (-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))))
+stmt (bi2raldva ((x y) (ph x) (ph y) (A y)) ((-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch))) (-> ph (<-> (A.e. x A (A.e. y B ps)) (A.e. x A (A.e. y B ch)))))
+stmt (bi2rexdva ((x y) (ph x) (ph y) (A y)) ((-> (/\ ph (/\ (e. (cv x) A) (e. (cv y) B))) (<-> ps ch))) (-> ph (<-> (E.e. x A (E.e. y B ps)) (E.e. x A (E.e. y B ch)))))
+stmt (ralinexa () () (<-> (A.e. x A (-> ph (-. ps))) (-. (E.e. x A (/\ ph ps)))))
+stmt (rexanali () () (<-> (E.e. x A (/\ ph (-. ps))) (-. (A.e. x A (-> ph ps)))))
+stmt (risset ((A x) (B x)) () (<-> (e. A B) (E.e. x B (= (cv x) A))))
+stmt (hbral ((x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> ph (A. x ph))) (-> (A.e. y A ph) (A. x (A.e. y A ph))))
+stmt (hbra1 () () (-> (A.e. x A ph) (A. x (A.e. x A ph))))
+stmt (hbrex ((x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> ph (A. x ph))) (-> (E.e. y A ph) (A. x (E.e. y A ph))))
+stmt (hbre1 () () (-> (E.e. x A ph) (A. x (E.e. x A ph))))
+stmt (r3al ((x y) (x z) (y z) (A y) (A z) (B z)) () (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A. x (A. y (A. z (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph))))))
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+stmt (rexex () () (-> (E.e. x A ph) (E. x ph)))
+stmt (ra4 () () (-> (A.e. x A ph) (-> (e. (cv x) A) ph)))
+stmt (ra4e () () (-> (/\ (e. (cv x) A) ph) (E.e. x A ph)))
+stmt (ra42 () () (-> (A.e. x A (A.e. y B ph)) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)))
+stmt (rspec () ((A.e. x A ph)) (-> (e. (cv x) A) ph))
+stmt (rgen () ((-> (e. (cv x) A) ph)) (A.e. x A ph))
+stmt (rgen2 ((y z) (A y) (A z) (x z)) ((-> (/\ (e. (cv x) A) (e. (cv y) A)) ph)) (A.e. x A (A.e. y A ph)))
+stmt (mprg () ((-> (A.e. x A ph) ps) (-> (e. (cv x) A) ph)) ps)
+stmt (mprgbir () ((<-> ph (A.e. x A ps)) (-> (e. (cv x) A) ps)) ph)
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+stmt (r19.20i () ((-> (e. (cv x) A) (-> ph ps))) (-> (A.e. x A ph) (A.e. x A ps)))
+stmt (r19.20si () ((-> ph ps)) (-> (A.e. x A ph) (A.e. x A ps)))
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+stmt (r19.20dva ((ph x)) ((-> (/\ ph (e. (cv x) A)) (-> ps ch))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))))
+stmt (r19.20sdv ((ph x)) ((-> ph (-> ps ch))) (-> ph (-> (A.e. x A ps) (A.e. x A ch))))
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+stmt (r19.21ai () ((-> ph (A. x ph)) (-> ph (-> (e. (cv x) A) ps))) (-> ph (A.e. x A ps)))
+stmt (r19.21aiv ((ph x)) ((-> ph (-> (e. (cv x) A) ps))) (-> ph (A.e. x A ps)))
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+stmt (rgen2a ((x y) (A y)) ((-> (/\ (e. (cv x) A) (e. (cv y) B)) ph)) (A.e. x A (A.e. y B ph)))
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+stmt (r19.21bi () ((-> ph (A.e. x A ps))) (-> (/\ ph (e. (cv x) A)) ps))
+stmt (rspec2 () ((A.e. x A (A.e. y B ph))) (-> (/\ (e. (cv x) A) (e. (cv y) B)) ph))
+stmt (rspec3 () ((A.e. x A (A.e. y B (A.e. z C ph)))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. (cv z) C)) ph))
+stmt (r19.21be () ((-> ph (A.e. x A ps))) (A.e. x A (-> ph ps)))
+stmt (nrex () ((-> (e. (cv x) A) (-. ps))) (-. (E.e. x A ps)))
+stmt (nrexdv ((ph x)) ((-> (/\ ph (e. (cv x) A)) (-. ps))) (-> ph (-. (E.e. x A ps))))
+stmt (r19.22 () () (-> (A.e. x A (-> ph ps)) (-> (E.e. x A ph) (E.e. x A ps))))
+stmt (r19.22i () ((-> (e. (cv x) A) (-> ph ps))) (-> (E.e. x A ph) (E.e. x A ps)))
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+stmt (r19.22si () ((-> ph ps)) (-> (E.e. x A ph) (E.e. x A ps)))
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+stmt (r19.22dva ((ph x)) ((-> (/\ ph (e. (cv x) A)) (-> ps ch))) (-> ph (-> (E.e. x A ps) (E.e. x A ch))))
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+stmt (r19.26 () () (<-> (A.e. x A (/\ ph ps)) (/\ (A.e. x A ph) (A.e. x A ps))))
+stmt (r19.26-2 () () (<-> (A.e. x A (A.e. y B (/\ ph ps))) (/\ (A.e. x A (A.e. y B ph)) (A.e. x A (A.e. y B ps)))))
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+stmt (r19.15 () () (-> (A.e. x A (<-> ph ps)) (<-> (A.e. x A ph) (A.e. x A ps))))
+stmt (r19.27av ((ps x)) () (-> (/\ (A.e. x A ph) ps) (A.e. x A (/\ ph ps))))
+stmt (r19.28av ((ph x)) () (-> (/\ ph (A.e. x A ps)) (A.e. x A (/\ ph ps))))
+stmt (r19.29 () () (-> (/\ (A.e. x A ph) (E.e. x A ps)) (E.e. x A (/\ ph ps))))
+stmt (r19.29r () () (-> (/\ (E.e. x A ph) (A.e. x A ps)) (E.e. x A (/\ ph ps))))
+stmt (r19.32v ((ph x)) () (<-> (A.e. x A (\/ ph ps)) (\/ ph (A.e. x A ps))))
+stmt (r19.35 () () (<-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) (E.e. x A ps))))
+stmt (r19.36av ((ps x)) () (-> (E.e. x A (-> ph ps)) (-> (A.e. x A ph) ps)))
+stmt (r19.37av ((ph x)) () (-> (E.e. x A (-> ph ps)) (-> ph (E.e. x A ps))))
+stmt (r19.40 () () (-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) (E.e. x A ps))))
+stmt (r19.41v ((ps x)) () (<-> (E.e. x A (/\ ph ps)) (/\ (E.e. x A ph) ps)))
+stmt (r19.42v ((ph x)) () (<-> (E.e. x A (/\ ph ps)) (/\ ph (E.e. x A ps))))
+stmt (r19.43 () () (<-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) (E.e. x A ps))))
+stmt (r19.44av ((ps x)) () (-> (E.e. x A (\/ ph ps)) (\/ (E.e. x A ph) ps)))
+stmt (r19.45av ((ph x)) () (-> (E.e. x A (\/ ph ps)) (\/ ph (E.e. x A ps))))
+stmt (hbreu1 () () (-> (E!e. x A ph) (A. x (E!e. x A ph))))
+stmt (rabid () () (<-> (e. (cv x) ({e.|} x A ph)) (/\ (e. (cv x) A) ph)))
+stmt (rabid2 ((A x)) () (<-> (= A ({e.|} x A ph)) (A.e. x A ph)))
+stmt (rabswap () () (= ({e.|} x A (e. (cv x) B)) ({e.|} x B (e. (cv x) A))))
+stmt (hbrab1 ((x y)) () (-> (e. (cv y) ({e.|} x A ph)) (A. x (e. (cv y) ({e.|} x A ph)))))
+stmt (hbrab ((x y) (x z) (y z) (A z)) ((-> ph (A. x ph)) (-> (e. (cv z) A) (A. x (e. (cv z) A)))) (-> (e. (cv z) ({e.|} y A ph)) (A. x (e. (cv z) ({e.|} y A ph)))))
+stmt (ralcom ((x y) (B x) (A y)) () (<-> (A.e. x A (A.e. y B ph)) (A.e. y B (A.e. x A ph))))
+stmt (rexcom ((x y) (B x) (A y)) () (<-> (E.e. x A (E.e. y B ph)) (E.e. y B (E.e. x A ph))))
+stmt (ralcom2 ((y z) (A y) (A z) (x z) (A x)) () (-> (A.e. x A (A.e. y A ph)) (A.e. y A (A.e. x A ph))))
+stmt (ralcom3 () () (<-> (A.e. x A (-> (e. (cv x) B) ph)) (A.e. x B (-> (e. (cv x) A) ph))))
+stmt (reeanv ((ph y) (ps x) (x y) (A y) (B x)) () (<-> (E.e. x A (E.e. y B (/\ ph ps))) (/\ (E.e. x A ph) (E.e. y B ps))))
+stmt (bireudva ((ph x)) ((-> (/\ ph (e. (cv x) A)) (<-> ps ch))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))))
+stmt (bireudv ((ph x)) ((-> ph (<-> ps ch))) (-> ph (<-> (E!e. x A ps) (E!e. x A ch))))
+stmt (bireua () ((-> (e. (cv x) A) (<-> ph ps))) (<-> (E!e. x A ph) (E!e. x A ps)))
+stmt (bireu () ((<-> ph ps)) (<-> (E!e. x A ph) (E!e. x A ps)))
+stmt (raleqf ((A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))))
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+stmt (reueqf ((A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))))
+stmt (raleq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ph))))
+stmt (rexeq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ph))))
+stmt (reueq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ph))))
+stmt (raleqd ((A x) (B x)) ((-> (= A B) (<-> ph ps))) (-> (= A B) (<-> (A.e. x A ph) (A.e. x B ps))))
+stmt (rexeqd ((A x) (B x)) ((-> (= A B) (<-> ph ps))) (-> (= A B) (<-> (E.e. x A ph) (E.e. x B ps))))
+stmt (reueqd ((A x) (B x)) ((-> (= A B) (<-> ph ps))) (-> (= A B) (<-> (E!e. x A ph) (E!e. x B ps))))
+stmt (cbvralf ((A z) (x y) (x z) (y z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) A) (A. y (e. (cv z) A))) (-> ph (A. y ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (A.e. x A ph) (A.e. y A ps)))
+stmt (cbvral ((x y) (x z) (A x) (y z) (A y) (A z)) ((-> ph (A. y ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (A.e. x A ph) (A.e. y A ps)))
+stmt (cbvrex ((x y) (A x) (A y)) ((-> ph (A. y ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (E.e. x A ph) (E.e. y A ps)))
+stmt (cbvralv ((ph y) (ps x) (x y) (A x) (A y)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (A.e. x A ph) (A.e. y A ps)))
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+stmt (cbvral3v ((ph w) (ps z) (v x) (ch x) (ch v) (u y) (th y) (th u) (w x) (A x) (A w) (x y) (B x) (w y) (v y) (B y) (v w) (B w) (B v) (x z) (u x) (C x) (y z) (C y) (w z) (v z) (u z) (C z) (u w) (C w) (u v) (C v) (C u)) ((-> (= (cv x) (cv w)) (<-> ph ch)) (-> (= (cv y) (cv v)) (<-> ch th)) (-> (= (cv z) (cv u)) (<-> th ps))) (<-> (A.e. x A (A.e. y B (A.e. z C ph))) (A.e. w A (A.e. v B (A.e. u C ps)))))
+stmt (reurex () () (-> (E!e. x A ph) (E.e. x A ph)))
+stmt (reu2 ((x y) (A x) (A y) (ph y)) () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ([/] (cv y) x ph)) (= (cv x) (cv y))))))))
+stmt (reu5 () () (<-> (E!e. x A ph) (/\ (E.e. x A ph) (E* x (/\ (e. (cv x) A) ph)))))
+stmt (reu4 ((x y) (A x) (A y) (ph y) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (<-> (E!e. x A ph) (/\ (E.e. x A ph) (A.e. x A (A.e. y A (-> (/\ ph ps) (= (cv x) (cv y))))))))
+stmt (2reuswap ((x y) (A x) (A y)) () (-> (A.e. x A (E* y (/\ (e. (cv y) A) ph))) (-> (E!e. x A (E.e. y A ph)) (E!e. y A (E.e. x A ph)))))
+stmt (birabi () ((-> (e. (cv x) A) (<-> ps ch))) (= ({e.|} x A ps) ({e.|} x A ch)))
+stmt (birabdv ((ph x)) ((-> ph (-> (e. (cv x) A) (<-> ps ch)))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))))
+stmt (birabsdv ((ph x)) ((-> ph (<-> ps ch))) (-> ph (= ({e.|} x A ps) ({e.|} x A ch))))
+stmt (rabeqf ((A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))))
+stmt (rabeq ((x y) (A x) (A y) (B x) (B y)) () (-> (= A B) (= ({e.|} x A ph) ({e.|} x B ph))))
+stmt (eqrabi () ((= A ({e.|} x B ph))) (<-> (e. (cv x) A) (/\ (e. (cv x) B) ph)))
+stmt (visset () () (e. (cv x) (V)))
+stmt (isset ((A x)) () (<-> (e. A (V)) (E. x (= (cv x) A))))
+stmt (isseti ((A x)) ((e. A (V))) (E. x (= (cv x) A)))
+stmt (issetri ((A x)) ((E. x (= (cv x) A))) (e. A (V)))
+stmt (elisset ((A x) (B x)) () (-> (e. A B) (e. A (V))))
+stmt (elisseti () ((e. A B)) (e. A (V)))
+stmt (elex ((A x)) () (-> (e. A B) (E. x (= (cv x) A))))
+stmt (ralv () () (<-> (A.e. x (V) ph) (A. x ph)))
+stmt (rexv () () (<-> (E.e. x (V) ph) (E. x ph)))
+stmt (rabab () () (= ({e.|} x (V) ph) ({|} x ph)))
+stmt (ralcom4 ((x y) (A y)) () (<-> (A.e. x A (A. y ph)) (A. y (A.e. x A ph))))
+stmt (rexcom4 ((x y) (A y)) () (<-> (E.e. x A (E. y ph)) (E. y (E.e. x A ph))))
+stmt (ceqsalg ((A x)) ((-> ps (A. x ps)) (-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (<-> (A. x (-> (= (cv x) A) ph)) ps)))
+stmt (ceqsal ((A x)) ((-> ps (A. x ps)) (e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> (A. x (-> (= (cv x) A) ph)) ps))
+stmt (ceqsalv ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> (A. x (-> (= (cv x) A) ph)) ps))
+stmt (gencl ((ps x)) ((<-> th (E. x (/\ ch (= A B)))) (-> (= A B) (<-> ph ps)) (-> ch ph)) (-> th ps))
+stmt (2gencl ((x y) (R x) (ps x) (C y) (S y) (ch y)) ((<-> (e. C S) (E. x (/\ (e. (cv x) R) (= A C)))) (<-> (e. D S) (E. y (/\ (e. (cv y) R) (= B D)))) (-> (= A C) (<-> ph ps)) (-> (= B D) (<-> ps ch)) (-> (/\ (e. (cv x) R) (e. (cv y) R)) ph)) (-> (/\ (e. C S) (e. D S)) ch))
+stmt (3gencl ((x y) (x z) (y z) (D y) (D z) (F z) (R x) (R y) (S y) (S z) (ps x) (ch y) (th z)) ((<-> (e. D S) (E. x (/\ (e. (cv x) R) (= A D)))) (<-> (e. F S) (E. y (/\ (e. (cv y) R) (= B F)))) (<-> (e. G S) (E. z (/\ (e. (cv z) R) (= C G)))) (-> (= A D) (<-> ph ps)) (-> (= B F) (<-> ps ch)) (-> (= C G) (<-> ch th)) (-> (/\/\ (e. (cv x) R) (e. (cv y) R) (e. (cv z) R)) ph)) (-> (/\/\ (e. D S) (e. F S) (e. G S)) th))
+stmt (cgsex2g ((x y) (ps x) (ps y) (A x) (A y) (B x) (B y)) ((-> (/\ (= (cv x) A) (= (cv y) B)) ch) (-> ch (<-> ph ps))) (-> (/\ (e. A C) (e. B D)) (<-> (E. x (E. y (/\ ch ph))) ps)))
+stmt (cgsex4g ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (ps x) (ps y) (ps z) (ps w)) ((-> (/\ (/\ (= (cv x) A) (= (cv y) B)) (/\ (= (cv z) C) (= (cv w) D))) ch) (-> ch (<-> ph ps))) (-> (/\ (/\ (e. A R) (e. B S)) (/\ (e. C R) (e. D S))) (<-> (E. x (E. y (E. z (E. w (/\ ch ph))))) ps)))
+stmt (ceqsex ((A x)) ((-> ps (A. x ps)) (e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> (E. x (/\ (= (cv x) A) ph)) ps))
+stmt (ceqsexv ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> (E. x (/\ (= (cv x) A) ph)) ps))
+stmt (gencbvex ((ps x) (ph y) (th x) (ch y) (A y)) ((e. A (V)) (-> (= A (cv y)) (<-> ph ps)) (-> (= A (cv y)) (<-> ch th)) (<-> th (E. x (/\ ch (= A (cv y)))))) (<-> (E. x (/\ ch ph)) (E. y (/\ th ps))))
+stmt (gencbval ((ps x) (ph y) (th x) (ch y) (A y)) ((e. A (V)) (-> (= A (cv y)) (<-> ph ps)) (-> (= A (cv y)) (<-> ch th)) (<-> th (E. x (/\ ch (= A (cv y)))))) (<-> (A. x (-> ch ph)) (A. y (-> th ps))))
+stmt (vtoclf ((A x)) ((-> ps (A. x ps)) (e. A (V)) (-> (= (cv x) A) (<-> ph ps)) ph) ps)
+stmt (vtocl ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps)) ph) ps)
+stmt (vtocl2 ((x y) (A x) (A y) (B x) (B y) (ps x) (ps y)) ((e. A (V)) (e. B (V)) (-> (/\ (= (cv x) A) (= (cv y) B)) (<-> ph ps)) ph) ps)
+stmt (vtocl3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (ps x) (ps y) (ps z)) ((e. A (V)) (e. B (V)) (e. C (V)) (-> (/\/\ (= (cv x) A) (= (cv y) B) (= (cv z) C)) (<-> ph ps)) ph) ps)
+stmt (vtoclb ((A x) (ch x) (th x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ch)) (-> (= (cv x) A) (<-> ps th)) (<-> ph ps)) (<-> ch th))
+stmt (vtoclgf ((A y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> ps (A. x ps)) (-> (= (cv x) A) (<-> ph ps)) ph) (-> (e. A B) ps))
+stmt (vtoclg ((x y) (A x) (A y) (ps x)) ((-> (= (cv x) A) (<-> ph ps)) ph) (-> (e. A B) ps))
+stmt (vtoclbg ((A x) (ch x) (th x)) ((-> (= (cv x) A) (<-> ph ch)) (-> (= (cv x) A) (<-> ps th)) (<-> ph ps)) (-> (e. A B) (<-> ch th)))
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+stmt (vtocl2g ((A x) (A y) (B y) (ps x) (ch y)) ((-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) ph) (-> (/\ (e. A C) (e. B D)) ch))
+stmt (vtoclga ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps)) (-> (e. (cv x) B) ph)) (-> (e. A B) ps))
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+stmt (vtocl3ga ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z) (C z) (D x) (D y) (D z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch y) (th z)) ((-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\/\ (e. (cv x) D) (e. (cv y) R) (e. (cv z) S)) ph)) (-> (/\/\ (e. A D) (e. B R) (e. C S)) th))
+stmt (vtocleg ((A x) (ph x)) ((-> (= (cv x) A) ph)) (-> (e. A B) ph))
+stmt (vtocle ((A x) (ph x)) ((e. A (V)) (-> (= (cv x) A) ph)) ph)
+stmt (vtoclef ((A x)) ((-> ph (A. x ph)) (e. A (V)) (-> (= (cv x) A) ph)) ph)
+stmt (vtoclri ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps)) (A.e. x B ph)) (-> (e. A B) ps))
+stmt (cla4gf ((x y) (A y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> ps (A. x ps)) (-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (-> (A. x ph) ps)))
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+stmt (cla4egv ((ps x) (x y) (A x) (A y)) ((-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (-> ps (E. x ph))))
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+stmt (cla4v ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (-> (A. x ph) ps))
+stmt (cla4ev ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (-> ps (E. x ph)))
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+stmt (rcla4v ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (-> (A.e. x B ph) ps)))
+stmt (rcla4va ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (/\ (e. A B) (A.e. x B ph)) ps))
+stmt (rcla4ev ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (/\ (e. A B) ps) (E.e. x B ph)))
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+stmt (alexeq ((A x) (x y) (A y) (ph y)) ((e. A (V))) (<-> (A. x (-> (= (cv x) A) ph)) (E. x (/\ (= (cv x) A) ph))))
+stmt (ceqex ((A x) (x y) (A y) (ph y)) () (-> (= (cv x) A) (<-> ph (E. x (/\ (= (cv x) A) ph)))))
+stmt (ceqsexg ((x y) (A x) (A y)) ((-> ps (A. x ps)) (-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)))
+stmt (ceqsexgv ((A x) (ps x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (<-> (E. x (/\ (= (cv x) A) ph)) ps)))
+stmt (ceqsrexv ((A x) (B x) (ps x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (<-> (E.e. x B (/\ (= (cv x) A) ph)) ps)))
+stmt (clel2 ((A x) (B x)) ((e. A (V))) (<-> (e. A B) (A. x (-> (= (cv x) A) (e. (cv x) B)))))
+stmt (clel3g ((A x) (B x)) () (-> (e. B C) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x)))))))
+stmt (clel3 ((A x) (B x)) ((e. B (V))) (<-> (e. A B) (E. x (/\ (= (cv x) B) (e. A (cv x))))))
+stmt (clel4 ((A x) (B x)) ((e. B (V))) (<-> (e. A B) (A. x (-> (= (cv x) B) (e. A (cv x))))))
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+stmt (elab ((ps x) (A x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> (e. A ({|} x ph)) ps))
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+stmt (elabg ((ps x) (x y) (A x) (A y)) ((-> (= (cv x) A) (<-> ph ps))) (-> (e. A B) (<-> (e. A ({|} x ph)) ps)))
+stmt (elab2g ((ps x) (A x)) ((-> (= (cv x) A) (<-> ph ps)) (= B ({|} x ph))) (-> (e. A C) (<-> (e. A B) ps)))
+stmt (elab2 ((ps x) (A x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps)) (= B ({|} x ph))) (<-> (e. A B) ps))
+stmt (elab3g ((ps x) (A x)) ((-> ps (e. A (V))) (-> (= (cv x) A) (<-> ph ps))) (<-> (e. A ({|} x ph)) ps))
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+stmt (cbvab ((x y) (x z) (y z) (ph z) (ps z)) ((-> ph (A. y ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv y)) (<-> ph ps))) (= ({|} x ph) ({|} y ps)))
+stmt (cbvabv ((x y) (ph y) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (= ({|} x ph) ({|} y ps)))
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+stmt (eueq ((x y) (A x) (A y)) () (<-> (e. A (V)) (E! x (= (cv x) A))))
+stmt (eueq1 ((A x)) ((e. A (V))) (E! x (= (cv x) A)))
+stmt (eueq2 ((ph x) (A x) (B x)) ((e. A (V)) (e. B (V))) (E! x (\/ (/\ ph (= (cv x) A)) (/\ (-. ph) (= (cv x) B)))))
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+stmt (moeq ((A x)) () (E* x (= (cv x) A)))
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+stmt (mosub ((x y) (A x) (A y)) ((E* x ph)) (E* x (E. y (/\ (= (cv y) A) ph))))
+stmt (mo2icl ((x y) (A x) (A y) (ph y)) () (-> (A. x (-> ph (= (cv x) A))) (E* x ph)))
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+stmt (ru ((x y)) () (e/ ({|} x (e/ (cv x) (cv x))) (V)))
+stmt (vsbcint ((ps x) (ps y) (x y) (A x)) ((-> (= (cv x) A) (<-> ph ps))) (-> (= (cv y) A) (<-> ([/] (cv y) x ph) ps)))
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+stmt (dfsbcq () () (-> (= A B) (<-> ([/] A x ph) ([/] B x ph))))
+stmt (sbceq1 () () (-> (= (cv x) A) (<-> ph ([/] A x ph))))
+stmt (a4sbc ((ph y) (A y) (x y)) () (-> (e. A B) (-> (A. x ph) ([/] A x ph))))
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+stmt (hbsbcv ((A x) (x y) (A y)) ((e. A (V))) (-> ([/] A x ph) (A. x ([/] A x ph))))
+stmt (sbcco ((x z) (A z) (y z) (ph y) (ph z)) () (-> (e. A B) (<-> ([/] A y ([/] (cv y) x ph)) ([/] A x ph))))
+stmt (sbcco2 ((x y) (ph y) (A y)) ((-> (= (cv x) (cv y)) (= A B))) (<-> ([/] (cv x) y ([/] B x ph)) ([/] A x ph)))
+stmt (sbc5g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph)))))
+stmt (sbc6g ((x y) (A x) (A y)) () (-> (e. A B) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))))
+stmt (sbc5 ((A x)) ((e. A (V))) (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))))
+stmt (sbc6 ((A x)) ((e. A (V))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph))))
+stmt (sbc2or ((A x)) () (\/ (<-> ([/] A x ph) (E. x (/\ (= (cv x) A) ph))) (<-> ([/] A x ph) (A. x (-> (= (cv x) A) ph)))))
+stmt (sbcie ((A x) (ps x)) ((e. A (V)) (-> (= (cv x) A) (<-> ph ps))) (<-> ([/] A x ph) ps))
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+stmt (elabs2 ((x y)) () (<-> (e. A ({|} x ph)) (/\ (e. A (V)) ([/] A x ph))))
+stmt (elabs () ((e. A (V))) (<-> (e. A ({|} x ph)) ([/] A x ph)))
+stmt (sbcn ((x y) (A y) (ph y)) () (-> (e. A B) (<-> ([/] A x (-. ph)) (-. ([/] A x ph)))))
+stmt (sbcim ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ([/] A x ph) ([/] A x ps)))))
+stmt (sbcan ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (/\ ph ps)) (/\ ([/] A x ph) ([/] A x ps)))))
+stmt (sbcor ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (\/ ph ps)) (\/ ([/] A x ph) ([/] A x ps)))))
+stmt (sbcbi ((x y) (A y) (ph y) (ps y)) () (-> (e. A B) (<-> ([/] A x (<-> ph ps)) (<-> ([/] A x ph) ([/] A x ps)))))
+stmt (sbcal ((x z) (A x) (A z) (x y) (y z) (ph z)) () (-> (e. A B) (<-> ([/] A y (A. x ph)) (A. x ([/] A y ph)))))
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+stmt (sbcel1 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. A C) (<-> ([/] A x (e. (cv x) B)) (e. A B))))
+stmt (sbcel2 ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B C) (<-> ([/] B x (e. A (cv x))) (e. A B))))
+stmt (bisbcdv ((ph x)) ((-> ph (<-> ps ch))) (-> (/\ (e. A B) ph) (<-> ([/] A x ps) ([/] A x ch))))
+stmt (sbcgf ((A y) (ph y) (x y)) ((-> ph (A. x ph))) (-> (e. A B) (<-> ([/] A x ph) ph)))
+stmt (sbc19.21g () ((-> ph (A. x ph))) (-> (e. A B) (<-> ([/] A x (-> ph ps)) (-> ph ([/] A x ps)))))
+stmt (ra4sbcf ((A y) (x y) (B x) (B y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A)))) (-> (e. A B) (-> (A.e. x B ph) ([/] A x ph))))
+stmt (ra5 () ((-> ph (A. x ph))) (-> (A.e. x A (-> ph ps)) (-> ph (A.e. x A ps))))
+stmt (axrep ((ph w) (x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (-> (E. y (A. z (-> ph (= (cv z) (cv y))))) (A. z (<-> (e. (cv z) (cv x)) (E. x (/\ (e. (cv x) (cv y)) (A. y ph))))))))
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+stmt (zfrep2 ((x y) (x z) (w x) (y z) (w y) (w z)) ((-> ph (A. z ph))) (-> (A. x (E. z (A. y (-> ph (= (cv y) (cv z)))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))))
+stmt (zfrep3 ((x y) (x z) (w x) (y z) (w y) (w z)) ((-> ph (A. z ph))) (-> (A. x (-> (e. (cv x) (cv w)) (E. z (A. y (-> ph (= (cv y) (cv z))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) (cv w)) ph)))))))
+stmt (zfrepclf ((y z) (A y) (v y) (A z) (v z) (A v) (ph z) (ph v) (A w) (v w) (x y) (x z) (v x) (w x)) ((-> (e. (cv w) A) (A. x (e. (cv w) A))) (e. A (V)) (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))))
+stmt (zfrep3cl ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) ((e. A (V)) (-> (e. (cv x) A) (E. z (A. y (-> ph (= (cv y) (cv z))))))) (E. z (A. y (<-> (e. (cv y) (cv z)) (E. x (/\ (e. (cv x) A) ph))))))
+stmt (zfrep4 ((ph y) (ph z) (y z) (ps z) (x y) (x z)) ((e. ({|} x ph) (V)) (-> ph (E. z (A. y (-> ps (= (cv y) (cv z))))))) (e. ({|} y (E. x (/\ ph ps))) (V)))
+stmt (zfaus ((x y) (A x) (x z) (A y) (y z) (A z) (ph y) (ph z)) ((e. A (V))) (E. y (A. x (<-> (e. (cv x) (cv y)) (/\ (e. (cv x) A) ph)))))
+stmt (bm1.3ii ((ph x) (x z) (ph z) (x y) (y z)) ((E. x (A. y (-> ph (e. (cv y) (cv x)))))) (E. x (A. y (<-> (e. (cv y) (cv x)) ph))))
+stmt (nalset ((x y) (x z) (y z)) () (-. (E. x (A. y (e. (cv y) (cv x))))))
+stmt (nvelv ((x y)) () (-. (e. (V) (V))))
+stmt (vnex () () (-. (E. x (= (cv x) (V)))))
+stmt (dfin5 ((A x) (B x)) () (= (i^i A B) ({e.|} x A (e. (cv x) B))))
+stmt (dfss () () (<-> (C_ A B) (= A (i^i A B))))
+stmt (dfdif2 ((A x) (B x)) () (= (\ A B) ({e.|} x A (-. (e. (cv x) B)))))
+stmt (eldif ((A x) (B x) (C x)) () (<-> (e. A (\ B C)) (/\ (e. A B) (-. (e. A C)))))
+stmt (dfss2 ((A x) (B x)) () (<-> (C_ A B) (A. x (-> (e. (cv x) A) (e. (cv x) B)))))
+stmt (dfss3 ((A x) (B x)) () (<-> (C_ A B) (A.e. x A (e. (cv x) B))))
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+stmt (sseli () ((C_ A B)) (-> (e. C A) (e. C B)))
+stmt (sselii () ((C_ A B) (e. C A)) (e. C B))
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+stmt (sseldd () ((-> ph (C_ A B)) (-> ph (e. C A))) (-> ph (e. C B)))
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+stmt (eqssi () ((C_ A B) (C_ B A)) (= A B))
+stmt (eqssd () ((-> ph (C_ A B)) (-> ph (C_ B A))) (-> ph (= A B)))
+stmt (ssid () () (C_ A A))
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+stmt (sseqtr () ((C_ A B) (= B C)) (C_ A C))
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+stmt (eqsstrd () ((-> ph (= A B)) (-> ph (C_ B C))) (-> ph (C_ A C)))
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+stmt (3sstr4 () ((C_ A B) (= C A) (= D B)) (C_ C D))
+stmt (3sstr3g () ((-> ph (C_ A B)) (= A C) (= B D)) (-> ph (C_ C D)))
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+stmt (3sstr4d () ((-> ph (C_ A B)) (-> ph (= C A)) (-> ph (= D B))) (-> ph (C_ C D)))
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+stmt (syl6ssr () ((-> ph (C_ A B)) (= C B)) (-> ph (C_ A C)))
+stmt (eqimss () () (-> (= A B) (C_ A B)))
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+stmt (ss2abi () ((-> ph ps)) (C_ ({|} x ph) ({|} x ps)))
+stmt (ss2rab () () (<-> (C_ ({e.|} x A ph) ({e.|} x A ps)) (A.e. x A (-> ph ps))))
+stmt (ss2rabi () ((-> (e. (cv x) A) (-> ph ps))) (C_ ({e.|} x A ph) ({e.|} x A ps)))
+stmt (rabss ((A x) (B x)) () (-> (C_ A B) (C_ ({e.|} x A ph) ({e.|} x B ph))))
+stmt (ssab ((A x)) () (C_ ({|} x (/\ (e. (cv x) A) ph)) A))
+stmt (ssrab ((A x)) () (C_ ({e.|} x A ph) A))
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+stmt (psseq1 () () (-> (= A B) (<-> (C: A C) (C: B C))))
+stmt (psseq2 () () (-> (= A B) (<-> (C: C A) (C: C B))))
+stmt (psseq1i () ((= A B)) (<-> (C: A C) (C: B C)))
+stmt (psseq2i () ((= A B)) (<-> (C: C A) (C: C B)))
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+stmt (psseq1d () ((-> ph (= A B))) (-> ph (<-> (C: A C) (C: B C))))
+stmt (psseq2d () ((-> ph (= A B))) (-> ph (<-> (C: C A) (C: C B))))
+stmt (psseq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (<-> (C: A C) (C: B D))))
+stmt (pssss () () (-> (C: A B) (C_ A B)))
+stmt (pssssd () ((-> ph (C: A B))) (-> ph (C_ A B)))
+stmt (sspss () () (<-> (C_ A B) (\/ (C: A B) (= A B))))
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+stmt (ssnpss () () (-> (C_ A B) (-. (C: B A))))
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+stmt (difeq1i () ((= A B)) (= (\ A C) (\ B C)))
+stmt (difeq2i () ((= A B)) (= (\ C A) (\ C B)))
+stmt (difeq12i () ((= A B) (= C D)) (= (\ A C) (\ B D)))
+stmt (difeq1d () ((-> ph (= A B))) (-> ph (= (\ A C) (\ B C))))
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+stmt (eldifi () () (-> (e. A (\ B C)) (e. A B)))
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+stmt (elndif () () (-> (e. A B) (-. (e. A (\ C B)))))
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+stmt (difdif ((A x) (B x)) () (= (\ A (\ B A)) A))
+stmt (difss ((A x) (B x)) () (C_ (\ A B) A))
+stmt (ddif ((A x)) () (= (\ (V) (\ (V) A)) A))
+stmt (ssconb ((A x) (B x) (C x)) () (-> (/\ (C_ A C) (C_ B C)) (<-> (C_ A (\ C B)) (C_ B (\ C A)))))
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+stmt (unidm ((A x)) () (= (u. A A) A))
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+stmt (uneq2 () () (-> (= A B) (= (u. C A) (u. C B))))
+stmt (uneq1i () ((= A B)) (= (u. A C) (u. B C)))
+stmt (uneq2i () ((= A B)) (= (u. C A) (u. C B)))
+stmt (uneq12i () ((= A B) (= C D)) (= (u. A C) (u. B D)))
+stmt (uneq1d () ((-> ph (= A B))) (-> ph (= (u. A C) (u. B C))))
+stmt (uneq2d () ((-> ph (= A B))) (-> ph (= (u. C A) (u. C B))))
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+stmt (uneq12 () () (-> (/\ (= A B) (= C D)) (= (u. A C) (u. B D))))
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+stmt (unass ((A x) (B x) (C x)) () (= (u. (u. A B) C) (u. A (u. B C))))
+stmt (un12 () () (= (u. A (u. B C)) (u. B (u. A C))))
+stmt (un23 () () (= (u. (u. A B) C) (u. (u. A C) B)))
+stmt (un4 () () (= (u. (u. A B) (u. C D)) (u. (u. A C) (u. B D))))
+stmt (unundi () () (= (u. A (u. B C)) (u. (u. A B) (u. A C))))
+stmt (unundir () () (= (u. (u. A B) C) (u. (u. A C) (u. B C))))
+stmt (ssun1 ((A x) (B x)) () (C_ A (u. A B)))
+stmt (ssun2 () () (C_ A (u. B A)))
+stmt (ssun3 () () (-> (C_ A B) (C_ A (u. B C))))
+stmt (ssun4 () () (-> (C_ A B) (C_ A (u. C B))))
+stmt (elun1 () () (-> (e. A B) (e. A (u. B C))))
+stmt (elun2 () () (-> (e. A B) (e. A (u. C B))))
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+stmt (unss2 () () (-> (C_ A B) (C_ (u. C A) (u. C B))))
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+stmt (unssi () ((C_ A C) (C_ B C)) (C_ (u. A B) C))
+stmt (ssun () () (-> (\/ (C_ A B) (C_ A C)) (C_ A (u. B C))))
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+stmt (ineq2 () () (-> (= A B) (= (i^i C A) (i^i C B))))
+stmt (ineq12 () () (-> (/\ (= A B) (= C D)) (= (i^i A C) (i^i B D))))
+stmt (ineq1i () ((= A B)) (= (i^i A C) (i^i B C)))
+stmt (ineq2i () ((= A B)) (= (i^i C A) (i^i C B)))
+stmt (ineq12i () ((= A B) (= C D)) (= (i^i A C) (i^i B D)))
+stmt (ineq1d () ((-> ph (= A B))) (-> ph (= (i^i A C) (i^i B C))))
+stmt (ineq2d () ((-> ph (= A B))) (-> ph (= (i^i C A) (i^i C B))))
+stmt (ineq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (= (i^i A C) (i^i B D))))
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+stmt (birabrdv ((ph x) (A x) (B x)) ((-> ph (-> (e. (cv x) B) (<-> (e. (cv x) A) ch)))) (-> ph (= (i^i B A) ({e.|} x B ch))))
+stmt (inidm ((A x)) () (= (i^i A A) A))
+stmt (inass ((A x) (B x) (C x)) () (= (i^i (i^i A B) C) (i^i A (i^i B C))))
+stmt (in12 () () (= (i^i A (i^i B C)) (i^i B (i^i A C))))
+stmt (in23 () () (= (i^i (i^i A B) C) (i^i (i^i A C) B)))
+stmt (in4 () () (= (i^i (i^i A B) (i^i C D)) (i^i (i^i A C) (i^i B D))))
+stmt (inindi () () (= (i^i A (i^i B C)) (i^i (i^i A B) (i^i A C))))
+stmt (inindir () () (= (i^i (i^i A B) C) (i^i (i^i A C) (i^i B C))))
+stmt (sseqin2 () () (<-> (C_ A B) (= (i^i B A) A)))
+stmt (inss1 ((A x) (B x)) () (C_ (i^i A B) A))
+stmt (inss2 () () (C_ (i^i A B) B))
+stmt (ssin () () (<-> (/\ (C_ A B) (C_ A C)) (C_ A (i^i B C))))
+stmt (ssini () ((C_ A B) (C_ A C)) (C_ A (i^i B C)))
+stmt (ssrin ((A x) (B x) (C x)) () (-> (C_ A B) (C_ (i^i A C) (i^i B C))))
+stmt (sslin () () (-> (C_ A B) (C_ (i^i C A) (i^i C B))))
+stmt (ss2in () () (-> (/\ (C_ A B) (C_ C D)) (C_ (i^i A C) (i^i B D))))
+stmt (ssinss1 () () (-> (C_ A C) (C_ (i^i A B) C)))
+stmt (nssinpss () () (<-> (-. (C_ A B)) (C: (i^i A B) A)))
+stmt (nsspssun () () (<-> (-. (C_ A B)) (C: B (u. A B))))
+stmt (dfss4 ((A x) (B x)) () (<-> (C_ A B) (= (\ B (\ B A)) A)))
+stmt (dfun2 ((A x) (B x)) () (= (u. A B) (\ (V) (\ (\ (V) A) B))))
+stmt (dfin2 ((A x) (B x)) () (= (i^i A B) (\ A (\ (V) B))))
+stmt (difin ((A x) (B x)) () (= (\ A (i^i A B)) (\ A B)))
+stmt (dfun3 () () (= (u. A B) (\ (V) (i^i (\ (V) A) (\ (V) B)))))
+stmt (dfin3 () () (= (i^i A B) (\ (V) (u. (\ (V) A) (\ (V) B)))))
+stmt (dfin4 () () (= (i^i A B) (\ A (\ A B))))
+stmt (invdif () () (= (i^i A (\ (V) B)) (\ A B)))
+stmt (indif () () (= (i^i A (\ A B)) (\ A B)))
+stmt (indi ((A x) (B x) (C x)) () (= (i^i A (u. B C)) (u. (i^i A B) (i^i A C))))
+stmt (undi ((A x) (B x) (C x)) () (= (u. A (i^i B C)) (i^i (u. A B) (u. A C))))
+stmt (indir () () (= (i^i (u. A B) C) (u. (i^i A C) (i^i B C))))
+stmt (undir () () (= (u. (i^i A B) C) (i^i (u. A C) (u. B C))))
+stmt (unineq ((A x) (B x) (C x)) () (<-> (/\ (= (u. A C) (u. B C)) (= (i^i A C) (i^i B C))) (= A B)))
+stmt (difundi () () (= (\ A (u. B C)) (i^i (\ A B) (\ A C))))
+stmt (difundir () () (= (\ (u. A B) C) (u. (\ A C) (\ B C))))
+stmt (difindi () () (= (\ A (i^i B C)) (u. (\ A B) (\ A C))))
+stmt (difindir () () (= (\ (i^i A B) C) (i^i (\ A C) (\ B C))))
+stmt (undm () () (= (\ (V) (u. A B)) (i^i (\ (V) A) (\ (V) B))))
+stmt (indm () () (= (\ (V) (i^i A B)) (u. (\ (V) A) (\ (V) B))))
+stmt (difun1 () () (= (\ A (u. B C)) (\ (\ A B) C)))
+stmt (dif23 () () (= (\ (\ A B) C) (\ (\ A C) B)))
+stmt (symdif1 () () (= (u. (\ A B) (\ B A)) (\ (u. A B) (i^i A B))))
+stmt (symdif2 ((A x) (B x)) () (= (u. (\ A B) (\ B A)) ({|} x (-. (<-> (e. (cv x) A) (e. (cv x) B))))))
+stmt (unab ((x y) (ph y) (ps y)) () (= (u. ({|} x ph) ({|} x ps)) ({|} x (\/ ph ps))))
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+stmt (uniiun ((x y) (A x) (A y)) () (= (U. A) (U_ x A (cv x))))
+stmt (intiin ((x y) (A x) (A y)) () (= (|^| A) (|^|_ x A (cv x))))
+stmt (iunid ((x y) (A x) (A y)) () (= (U_ x A ({} (cv x))) A))
+stmt (iun0 ((x y) (A y)) () (= (U_ x A ({/})) ({/})))
+stmt (0iun ((x y) (A y)) () (= (U_ x ({/}) A) ({/})))
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+stmt (iunun ((x y) (A y) (B y) (C y)) () (= (U_ x A (u. B C)) (u. (U_ x A B) (U_ x A C))))
+stmt (iunxun ((x y) (A y) (B y) (C y)) () (= (U_ x (u. A B) C) (u. (U_ x A C) (U_ x B C))))
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+stmt (iinpw ((x y) (A x) (A y)) () (= (P~ (|^| A)) (|^|_ x A (P~ (cv x)))))
+stmt (iunpwss ((x y) (A x) (A y)) () (C_ (U_ x A (P~ (cv x))) (P~ (U. A))))
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+stmt (dftr3 ((x y) (A x) (A y)) () (<-> (Tr A) (A.e. x A (C_ (cv x) A))))
+stmt (dftr4 ((A x)) () (<-> (Tr A) (C_ A (P~ A))))
+stmt (treq () () (-> (= A B) (<-> (Tr A) (Tr B))))
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+stmt (trel3 () () (-> (Tr A) (-> (/\/\ (e. B C) (e. C D) (e. D A)) (e. B A))))
+stmt (trss ((A x) (B x)) () (-> (Tr A) (-> (e. B A) (C_ B A))))
+stmt (trin ((A x) (B x)) () (-> (/\ (Tr A) (Tr B)) (Tr (i^i A B))))
+stmt (tr0 () () (Tr ({/})))
+stmt (trv () () (Tr (V)))
+stmt (breq () () (-> (= R S) (<-> (br A R B) (br A S B))))
+stmt (breq1 () () (-> (= A B) (<-> (br A R C) (br B R C))))
+stmt (breq2 () () (-> (= A B) (<-> (br C R A) (br C R B))))
+stmt (breq12 () () (-> (/\ (= A B) (= C D)) (<-> (br A R C) (br B R D))))
+stmt (breqi () ((= R S)) (<-> (br A R B) (br A S B)))
+stmt (breq1i () ((= A B)) (<-> (br A R C) (br B R C)))
+stmt (breq2i () ((= A B)) (<-> (br C R A) (br C R B)))
+stmt (breq12i () ((= A B) (= C D)) (<-> (br A R C) (br B R D)))
+stmt (breq1d () ((-> ph (= A B))) (-> ph (<-> (br A R C) (br B R C))))
+stmt (breq2d () ((-> ph (= A B))) (-> ph (<-> (br C R A) (br C R B))))
+stmt (breq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (<-> (br A R C) (br B R D))))
+stmt (breqan12d () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ph ps) (<-> (br A R C) (br B R D))))
+stmt (breqan12rd () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ps ph) (<-> (br A R C) (br B R D))))
+stmt (eqbrtr () ((= A B) (br B R C)) (br A R C))
+stmt (eqbrtrd () ((-> ph (= A B)) (-> ph (br B R C))) (-> ph (br A R C)))
+stmt (eqbrtrr () ((= A B) (br A R C)) (br B R C))
+stmt (eqbrtrrd () ((-> ph (= A B)) (-> ph (br A R C))) (-> ph (br B R C)))
+stmt (breqtr () ((br A R B) (= B C)) (br A R C))
+stmt (breqtrd () ((-> ph (br A R B)) (-> ph (= B C))) (-> ph (br A R C)))
+stmt (breqtrr () ((br A R B) (= C B)) (br A R C))
+stmt (breqtrrd () ((-> ph (br A R B)) (-> ph (= C B))) (-> ph (br A R C)))
+stmt (3brtr3 () ((br A R B) (= A C) (= B D)) (br C R D))
+stmt (3brtr4 () ((br A R B) (= C A) (= D B)) (br C R D))
+stmt (3brtr3d () ((-> ph (br A R B)) (-> ph (= A C)) (-> ph (= B D))) (-> ph (br C R D)))
+stmt (3brtr4d () ((-> ph (br A R B)) (-> ph (= C A)) (-> ph (= D B))) (-> ph (br C R D)))
+stmt (3brtr3g () ((-> ph (br A R B)) (= A C) (= B D)) (-> ph (br C R D)))
+stmt (3brtr4g () ((-> ph (br A R B)) (= C A) (= D B)) (-> ph (br C R D)))
+stmt (syl5eqbr () ((-> ph (br A R B)) (= C A)) (-> ph (br C R B)))
+stmt (syl5eqbrr () ((-> ph (br A R B)) (= A C)) (-> ph (br C R B)))
+stmt (syl5breq () ((-> ph (= A B)) (br C R A)) (-> ph (br C R B)))
+stmt (syl5breqr () ((-> ph (= B A)) (br C R A)) (-> ph (br C R B)))
+stmt (syl6eqbr () ((-> ph (= A B)) (br B R C)) (-> ph (br A R C)))
+stmt (syl6breq () ((-> ph (br A R B)) (= B C)) (-> ph (br A R C)))
+stmt (syl6breqr () ((-> ph (br A R B)) (= C B)) (-> ph (br A R C)))
+stmt (ssbrd () ((-> ph (C_ A B))) (-> ph (-> (br C A D) (br C B D))))
+stmt (hbbr ((A y) (B y) (R y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) R) (A. x (e. (cv y) R))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (br A R B) (A. x (br A R B))))
+stmt (brab1 ((x z) (y z) (R z)) () (<-> (br (cv x) R (cv y)) (e. (cv x) ({|} z (br (cv z) R (cv y))))))
+stmt (brprc () () (-> (-. (e. B (V))) (<-> (br A R B) (br A R A))))
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+stmt (opabss ((x z) (R x) (R z) (y z) (R y)) () (C_ ({<,>|} x y (br (cv x) R (cv y))) R))
+stmt (biopabd ((x z) (y z) (ph z) (ps z) (ch z)) ((-> ph (A. x ph)) (-> ph (A. y ph)) (-> ph (<-> ps ch))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))))
+stmt (biopabdv ((x y) (ph x) (ph y)) ((-> ph (<-> ps ch))) (-> ph (= ({<,>|} x y ps) ({<,>|} x y ch))))
+stmt (biopabi ((x y) (x z) (y z) (ph z) (ps z)) ((<-> ph ps)) (= ({<,>|} x y ph) ({<,>|} x y ps)))
+stmt (cbvopab ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph v) (ps v)) ((-> ph (A. z ph)) (-> ph (A. w ph)) (-> ps (A. x ps)) (-> ps (A. y ps)) (-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} z w ps)))
+stmt (cbvopabv ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps y)) ((-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} z w ps)))
+stmt (cbvopab1 ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((-> ph (A. z ph)) (-> ps (A. x ps)) (-> (= (cv x) (cv z)) (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} z y ps)))
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+stmt (cbvopab1v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps x) (ps w)) ((-> (= (cv x) (cv z)) (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} z y ps)))
+stmt (cbvopab2v ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w) (ps y) (ps w)) ((-> (= (cv y) (cv z)) (<-> ph ps))) (= ({<,>|} x y ph) ({<,>|} x z ps)))
+stmt (elopab ((x y) (x z) (A x) (y z) (A y) (A z) (ph z)) () (<-> (e. A ({<,>|} x y ph)) (E. x (E. y (/\ (= A (<,> (cv x) (cv y))) ph)))))
+stmt (hbopab ((x y) (x z) (w x) (y z) (w y) (w z)) ((-> ph (A. z ph))) (-> (e. (cv w) ({<,>|} x y ph)) (A. z (e. (cv w) ({<,>|} x y ph)))))
+stmt (hbopab1 ((x y) (w x) (w y) (x z) (ph w)) () (-> (e. (cv z) ({<,>|} x y ph)) (A. x (e. (cv z) ({<,>|} x y ph)))))
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+stmt (brabg ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (= R ({<,>|} x y ph))) (-> (/\ (e. A C) (e. B D)) (<-> (br A R B) ch)))
+stmt (opelopab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((e. A (V)) (e. B (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch))) (<-> (e. (<,> A B) ({<,>|} x y ph)) ch))
+stmt (brab ((x y) (A x) (A y) (B x) (B y) (ch x) (ch y)) ((e. A (V)) (e. B (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (= R ({<,>|} x y ph))) (<-> (br A R B) ch))
+stmt (ssopab2 ((ph z) (ps z) (x y) (x z) (y z)) () (<-> (C_ ({<,>|} x y ph) ({<,>|} x y ps)) (A. x (A. y (-> ph ps)))))
+stmt (ssopab2i ((x y)) ((-> ph ps)) (C_ ({<,>|} x y ph) ({<,>|} x y ps)))
+stmt (unopab ((x z) (y z) (ph z) (ps z)) () (= (u. ({<,>|} x y ph) ({<,>|} x y ps)) ({<,>|} x y (\/ ph ps))))
+stmt (epelc ((x y) (A x) (A y) (B x) (B y)) ((e. A (V)) (e. B (V))) (<-> (br A (E) B) (e. A B)))
+stmt (epel () () (<-> (br (cv x) (E) (cv y)) (e. (cv x) (cv y))))
+stmt (ideqg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (<-> (br A (I) B) (= A B))))
+stmt (ideq () ((e. A (V)) (e. B (V))) (<-> (br A (I) B) (= A B)))
+stmt (poss ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (C_ A B) (-> (Po R B) (Po R A))))
+stmt (poeq1 ((x y) (x z) (R x) (y z) (R y) (R z) (S x) (S y) (S z) (A x) (A y) (A z)) () (-> (= R S) (<-> (Po R A) (Po S A))))
+stmt (poeq2 () () (-> (= A B) (<-> (Po R A) (Po R B))))
+stmt (pocl ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) () (-> (Po R A) (-> (/\/\ (e. B A) (e. C A) (e. D A)) (/\ (-. (br B R B)) (-> (/\ (br B R C) (br C R D)) (br B R D))))))
+stmt (poirr () () (-> (/\ (Po R A) (e. B A)) (-. (br B R B))))
+stmt (potr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))))
+stmt (po2nr () () (-> (/\ (Po R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))))
+stmt (po3nr () () (-> (/\ (Po R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))))
+stmt (po0 ((x y) (x z) (R x) (y z) (R y) (R z)) () (Po R ({/})))
+stmt (sopo ((x y) (R x) (R y) (A x) (A y)) () (-> (Or R A) (Po R A)))
+stmt (soss ((x y) (R x) (R y) (A x) (A y) (B x) (B y)) () (-> (C_ A B) (-> (Or R B) (Or R A))))
+stmt (soeq1 ((x y) (R x) (R y) (S x) (S y) (A x) (A y)) () (-> (= R S) (<-> (Or R A) (Or S A))))
+stmt (soeq2 () () (-> (= A B) (<-> (Or R A) (Or R B))))
+stmt (sonr () () (-> (/\ (Or R A) (e. B A)) (-. (br B R B))))
+stmt (sotr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-> (/\ (br B R C) (br C R D)) (br B R D))))
+stmt (solin ((x y) (A x) (A y) (B x) (B y) (C x) (C y) (R x) (R y)) () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (\/\/ (br B R C) (= B C) (br C R B))))
+stmt (so2nr () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (-. (/\ (br B R C) (br C R B)))))
+stmt (so3nr () () (-> (/\ (Or R A) (/\/\ (e. B A) (e. C A) (e. D A))) (-. (/\/\ (br B R C) (br C R D) (br D R B)))))
+stmt (sotric () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (br B R C) (-. (\/ (= B C) (br C R B))))))
+stmt (sotrieq () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (-. (\/ (br B R C) (br C R B))))))
+stmt (sotrieq2 () () (-> (/\ (Or R A) (/\ (e. B A) (e. C A))) (<-> (= B C) (/\ (-. (br B R C)) (-. (br C R B))))))
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+stmt (so ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) ((-> (/\/\ (e. (cv x) A) (e. (cv y) A) (e. (cv z) A)) (/\ (<-> (br (cv x) R (cv y)) (-. (\/ (= (cv x) (cv y)) (br (cv y) R (cv x))))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))))) (Or R A))
+stmt (so0 ((x y) (R x) (R y)) () (Or R ({/})))
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+stmt (supex ((x y) (x z) (A x) (y z) (A y) (A z) (R x) (R y) (R z) (B x) (B y) (B z)) ((Or R A)) (e. (sup B A R) (V)))
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+stmt (rdgfnon ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (Fn (rec F A) (On)))
+stmt (rdgval ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (-> (e. (cv g) (On)) (= (` (rec F A) (cv g)) (` ({<,>|} w z (\/\/ (/\ (= (cv w) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv w) ({/})) (Lim (dom (cv w))))) (= (cv z) (` F (` (cv w) (U. (dom (cv w))))))) (/\ (Lim (dom (cv w))) (= (cv z) (U. (ran (cv w))))))) (|` (rec F A) (cv g))))))
+stmt (rdg0 ((g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. A (V))) (= (` (rec F A) ({/})) A))
+stmt (rdgsuc ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. B (On))) (= (` (rec F A) (suc B)) (` F (` (rec F A) B))))
+stmt (rdglim ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. B (On))) (-> (Lim B) (= (` (rec F A) B) (U. (" (rec F A) B)))))
+stmt (rdg0t ((A x) (F x)) () (-> (e. A C) (= (` (rec F A) ({/})) A)))
+stmt (rdgsuct () () (-> (e. B (On)) (= (` (rec F A) (suc B)) (` F (` (rec F A) B)))))
+stmt (rdgsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B))) (-> (e. (cv z) D) (A. x (e. (cv z) D))) (= F (rec ({<,>|} x y (= (cv y) C)) A)) (-> (= (cv x) (` F B)) (= C D))) (-> (/\ (e. B (On)) (e. D R)) (= (` F (suc B)) D)))
+stmt (rdgsucopabn ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B))) (-> (e. (cv z) D) (A. x (e. (cv z) D))) (= F (rec ({<,>|} x y (= (cv y) C)) A)) (-> (= (cv x) (` F B)) (= C D))) (-> (-. (e. D (V))) (= (` F (suc B)) ({/}))))
+stmt (rdglimt () () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. (" (rec F A) B)))))
+stmt (rdglim2 ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U. ({|} y (E.e. x B (= (cv y) (` (rec F A) (cv x)))))))))
+stmt (rdglim2a ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) () (-> (/\ (e. B C) (Lim B)) (= (` (rec F A) B) (U_ x B (` (rec F A) (cv x))))))
+stmt (frfnom () () (Fn (|` (rec F A) (om)) (om)))
+stmt (fr0t () () (-> (e. A B) (= (` (|` (rec F A) (om)) ({/})) A)))
+stmt (frsuct () () (-> (e. B (om)) (= (` (|` (rec F A) (om)) (suc B)) (` F (` (|` (rec F A) (om)) B)))))
+stmt (frsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B))) (-> (e. (cv z) D) (A. x (e. (cv z) D))) (= F (|` (rec ({<,>|} x y (= (cv y) C)) A) (om))) (-> (= (cv x) (` F B)) (= C D))) (-> (/\ (e. B (om)) (e. D R)) (= (` F (suc B)) D)))
+stmt (tz7.48lem ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w)) ((Fn F (On))) (-> (/\ (C_ A (On)) (A.e. x A (A.e. y (cv x) (-. (= (` F (cv x)) (` F (cv y))))))) (Fun (`' (|` F A)))))
+stmt (tz7.48-1 ((x y) (F x) (F y) (A x) (A y)) ((Fn F (On))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (C_ (ran F) A)))
+stmt (tz7.48-2 ((x y) (F x) (F y) (A x) (A y)) ((Fn F (On))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (Fun (`' F))))
+stmt (tz7.48-3 ((F x) (A x)) ((Fn F (On))) (-> (A.e. x (On) (e. (` F (cv x)) (\ A (" F (cv x))))) (-. (e. A (V)))))
+stmt (tz7.49 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((Fn F (On)) (e. A (V))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (/\/\ (A.e. y (cv x) (-. (= (\ A (" F (cv y))) ({/})))) (= (" F (cv x)) A) (Fun (`' (|` F (cv x))))))))
+stmt (tz7.49c ((x y) (F x) (F y) (A x) (A y)) ((Fn F (On)) (e. A (V))) (-> (A.e. x (On) (-> (-. (= (\ A (" F (cv x))) ({/}))) (e. (` F (cv x)) (\ A (" F (cv x)))))) (E.e. x (On) (:-1-1-onto-> (|` F (cv x)) (cv x) A))))
+stmt (abianfplem ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((e. A (V)) (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x)))) (-> (e. (cv v) (On)) (-> (= (` F (cv x)) (cv x)) (= (` G (cv v)) (cv x)))))
+stmt (abianfp ((x y) (v x) (A x) (v y) (A y) (A v) (x z) (w x) (u x) (F x) (y z) (w y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (G y) (G v) (G u)) ((e. A (V)) (= G (rec ({<,>|} z w (= (cv w) (` F (cv z)))) (cv x)))) (<-> (E.e. x A (= (` F (cv x)) (cv x))) (E.e. x A (A.e. v (On) (/\ (e. (` G (cv v)) A) (-> (-. (= (` F (` G (cv v))) (` G (cv v)))) (A.e. u (cv v) (-. (= (` G (cv v)) (` G (cv u)))))))))))
+stmt (opreq () () (-> (= F G) (= (opr A F B) (opr A G B))))
+stmt (opreq1 () () (-> (= A B) (= (opr A F C) (opr B F C))))
+stmt (opreq2 () () (-> (= A B) (= (opr C F A) (opr C F B))))
+stmt (opreq12 () () (-> (/\ (= A B) (= C D)) (= (opr A F C) (opr B F D))))
+stmt (opreq1i () ((= A B)) (= (opr A F C) (opr B F C)))
+stmt (opreq2i () ((= A B)) (= (opr C F A) (opr C F B)))
+stmt (opreq12i () ((= A B) (= C D)) (= (opr A F C) (opr B F D)))
+stmt (opreq1d () ((-> ph (= A B))) (-> ph (= (opr A F C) (opr B F C))))
+stmt (opreq2d () ((-> ph (= A B))) (-> ph (= (opr C F A) (opr C F B))))
+stmt (opreq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (= (opr A F C) (opr B F D))))
+stmt (opreqan12d () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ph ps) (= (opr A F C) (opr B F D))))
+stmt (opreqan12rd () ((-> ph (= A B)) (-> ps (= C D))) (-> (/\ ps ph) (= (opr A F C) (opr B F D))))
+stmt (hbopr ((F y) (A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) F) (A. x (e. (cv y) F))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (e. (cv y) (opr A F B)) (A. x (e. (cv y) (opr A F B)))))
+stmt (oprex () () (e. (opr A F B) (V)))
+stmt (oprprc1 () ((Rel (dom F))) (-> (-. (e. A (V))) (= (opr A F B) ({/}))))
+stmt (oprprc2 () () (-> (-. (e. B (V))) (= (opr A F B) (opr A F A))))
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+stmt (reloprab ((x z) (w x) (w z) (y z) (w y) (ph w)) () (Rel ({<<,>,>|} x y z ph)))
+stmt (hboprab1 ((x y) (x z) (v x) (y z) (v y) (v z) (w x) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. x (e. (cv w) ({<<,>,>|} x y z ph)))))
+stmt (hboprab2 ((x y) (x z) (v x) (y z) (v y) (v z) (w y) (ph v)) () (-> (e. (cv w) ({<<,>,>|} x y z ph)) (A. y (e. (cv w) ({<<,>,>|} x y z ph)))))
+stmt (bioprabd ((x z) (w x) (w z) (y z) (w y) (ph w) (ps w) (ch w)) ((-> ph (A. x ph)) (-> ph (A. y ph)) (-> ph (A. z ph)) (-> ph (<-> ps ch))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))))
+stmt (bioprabdv ((x z) (ph x) (ph z) (y z) (ph y)) ((-> ph (<-> ps ch))) (-> ph (= ({<<,>,>|} x y z ps) ({<<,>,>|} x y z ch))))
+stmt (bioprabi ((x y) (x z) (w x) (y z) (w y) (w z) (ph w) (ps w)) ((<-> ph ps)) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)))
+stmt (cbvoprab12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u) (ps u)) ((-> ph (A. w ph)) (-> ph (A. v ph)) (-> ps (A. x ps)) (-> ps (A. y ps)) (-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)))
+stmt (cbvoprab12v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps x) (ps y)) ((-> (/\ (= (cv x) (cv w)) (= (cv y) (cv v))) (<-> ph ps))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} w v z ps)))
+stmt (cbvoprab3v ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (ph w) (ph v) (ps z) (ps v)) ((-> (= (cv z) (cv w)) (<-> ph ps))) (= ({<<,>,>|} x y z ph) ({<<,>,>|} x y w ps)))
+stmt (dmoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (dom ({<<,>,>|} x y z ph)) ({<,>|} x y (E. z ph))))
+stmt (dmoprabss ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (C_ (dom ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph))) (X. A B)))
+stmt (rnoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) () (= (ran ({<<,>,>|} x y z ph)) ({|} z (E. x (E. y ph)))))
+stmt (reldmoprab ((x y) (x z) (y z)) () (Rel (dom ({<<,>,>|} x y z ph))))
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+stmt (ssoprab2i ((ph w) (ps w) (x y) (x z) (w x) (y z) (w y) (w z)) ((-> ph ps)) (C_ ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps)))
+stmt (funoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) ((E* z ph)) (Fun ({<<,>,>|} x y z ph)))
+stmt (fnoprab ((x y) (x z) (y z) (ph z)) ((-> ph (E! z ps))) (Fn ({<<,>,>|} x y z (/\ ph ps)) ({<,>|} x y ph)))
+stmt (fnoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (Fn F (X. A B)))
+stmt (dmoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (= (dom F) (X. A B)))
+stmt (ffnoprval ((x y) (w x) (A x) (w y) (A y) (A w) (B x) (B y) (B w) (C x) (C y) (C w) (F x) (F y) (F w)) () (<-> (:--> F (X. A B) C) (/\ (Fn F (X. A B)) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))))
+stmt (eqfnoprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z) (D z) (F x) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (Fn F (X. A B)) (Fn G (X. C D))) (<-> (= F G) (/\ (= (X. A B) (X. C D)) (A.e. x A (A.e. y B (= (opr (cv x) F (cv y)) (opr (cv x) G (cv y)))))))))
+stmt (fnoprval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) () (<-> (Fn F (X. A B)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y))))))))
+stmt (foprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (F x) (F y) (F z)) () (<-> (:--> F (X. A B) C) (/\ (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))))) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C))))))
+stmt (oprabex ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V)) (-> (/\ (e. (cv x) A) (e. (cv y) B)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (e. F (V)))
+stmt (oprabex2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. A (V)) (e. B (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (e. F (V)))
+stmt (oprabex3 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z)) ((e. H (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R)))))))))) (e. F (V)))
+stmt (oprabval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((e. C (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E! z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\ (e. A R) (e. B S)) (<-> (= (opr A F B) C) th)))
+stmt (oprabvalig ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (D x) (D y) (D z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\/\ (e. A R) (e. B S) (e. C D)) (-> th (= (opr A F B) C))))
+stmt (oprabvali ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((e. C (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\ (e. A R) (e. B S)) (-> th (= (opr A F B) C))))
+stmt (oprabval2g ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (H x) (H y) (H z) (G x) (R z) (S x) (S y) (S z)) ((-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S)))
+stmt (oprabval2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (R z) (S x) (S y) (S z)) ((e. S (V)) (-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)))
+stmt (oprabval5 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (G x) (R z) (S x) (S y) (S z)) ((e. S (V)) (-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (= (cv z) R)))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S)))
+stmt (oprabval3 ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z) (S x) (S y) (S z) (S w) (S v) (S u) (S f)) ((e. S (V)) (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= R S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R)))))))))) (-> (/\ (/\ (e. A H) (e. B H)) (/\ (e. C H) (e. D H))) (= (opr (<,> A B) F (<,> C D)) S)))
+stmt (oprabval4g ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C z) (C w) (C v) (C u) (C f) (D w) (D v)) ((= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. C D)) (= (opr (cv x) F (cv y)) C)))
+stmt (elrnoprab ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C z) (D x) (D y) (D w) (D v) (D u) (F w) (F v) (F u)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (<-> (e. D (ran F)) (E.e. x A (E.e. y B (= D C)))))
+stmt (oprvalex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (e. ({|} z (E.e. x A (E.e. y B (= (cv z) (opr (cv x) F (cv y)))))) (V)))
+stmt (oprssdm ((x y) (S x) (S y) (F x) (F y)) ((-. (e. ({/}) S)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (C_ (X. S S) (dom F)))
+stmt (ndmoprg () ((= (dom F) (X. S S))) (-> (/\ (e. B C) (-. (/\ (e. A S) (e. B S)))) (= (opr A F B) ({/}))))
+stmt (ndmoprcl ((A x) (B x) (F x) (S x)) ((= (dom F) (X. S S)) (-> (/\ (e. A S) (e. (cv x) S)) (e. (opr A F (cv x)) S)) (e. ({/}) S)) (e. (opr A F B) S))
+stmt (ndmopr () ((e. B (V)) (= (dom F) (X. S S))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) ({/}))))
+stmt (ndmoprrcl () ((e. B (V)) (= (dom F) (X. S S)) (-. (e. ({/}) S))) (-> (e. (opr A F B) S) (/\ (e. A S) (e. B S))))
+stmt (ndmoprcom () ((e. B (V)) (= (dom F) (X. S S)) (e. A (V))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) (opr B F A))))
+stmt (ndmoprass () ((e. B (V)) (= (dom F) (X. S S)) (e. C (V)) (-. (e. ({/}) S))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr (opr A F B) F C) (opr A F (opr B F C)))))
+stmt (ndmoprdistr () ((e. B (V)) (= (dom F) (X. S S)) (e. C (V)) (-. (e. ({/}) S)) (= (dom G) (X. S S))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))))
+stmt (ndmord () ((e. B (V)) (= (dom F) (X. S S)) (e. A (V)) (C_ R (X. S S)) (-. (e. ({/}) S)) (-> (/\/\ (e. A S) (e. B S) (e. C S)) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))
+stmt (ndmordi () ((e. A (V)) (= (dom F) (X. S S)) (C_ R (X. S S)) (-. (e. ({/}) S)) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) (-> (br (opr C F A) R (opr C F B)) (br A R B)))
+stmt (caoprcl ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y)) ((-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (-> (/\ (e. A S) (e. B S)) (e. (opr A F B) S)))
+stmt (caoprcom ((x y) (F x) (F y) (A x) (A y) (B x) (B y)) ((e. A (V)) (e. B (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (= (opr A F B) (opr B F A)))
+stmt (caoprass ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr A F (opr B F C))))
+stmt (caoprcan ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. C (V)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (-> (/\ (e. A S) (e. B S)) (-> (= (opr A F B) (opr A F C)) (= B C))))
+stmt (caoprord ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B)))))
+stmt (caoprord2 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (-> (e. C S) (<-> (br A R B) (br (opr A F C) R (opr B F C)))))
+stmt (caoprord3 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (e. D (V))) (-> (/\ (/\ (e. B S) (e. C S)) (= (opr A F B) (opr C F D))) (<-> (br A R C) (br D R B))))
+stmt (caoprdistr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C))))
+stmt (caopr32 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr (opr A F C) F B)))
+stmt (caopr12 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr A F (opr B F C)) (opr B F (opr A F C))))
+stmt (caopr31 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr (opr C F B) F A)))
+stmt (caopr13 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr A F (opr B F C)) (opr C F (opr B F A))))
+stmt (caopr4 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr B F D))))
+stmt (caopr411 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr C F B) F (opr A F D))))
+stmt (caopr42 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr D F B))))
+stmt (caoprdistrr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (= (opr (opr A F B) G C) (opr (opr A G C) F (opr B G C))))
+stmt (caoprdilem ((x y) (x z) (H x) (y z) (H y) (H z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))) (e. D (V)) (e. H (V)) (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z))))) (= (opr (opr (opr A G C) F (opr B G D)) G H) (opr (opr A G (opr C G H)) F (opr B G (opr D G H)))))
+stmt (caoprlem2 ((x y) (x z) (H x) (y z) (H y) (H z) (R x) (R y) (R z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))) (e. D (V)) (e. H (V)) (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z)))) (e. R (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr (opr (opr A G C) F (opr B G D)) G H) F (opr (opr (opr A G D) F (opr B G C)) G R)) (opr (opr A G (opr (opr C G H) F (opr D G R))) F (opr B G (opr (opr C G R) F (opr D G H))))))
+stmt (caoprmo ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (S w) (S v) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B S) (= (dom F) (X. S S)) (-. (e. ({/}) S)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (e. (cv x) S) (= (opr (cv x) F B) (cv x)))) (E* w (= (opr A F (cv w)) B)))
+stmt (1stval ((x y) (A x) (A y)) () (= (` (1st) A) (U. (dom ({} A)))))
+stmt (2ndval ((x y) (A x) (A y)) () (= (` (2nd) A) (U. (ran ({} A)))))
+stmt (op1st () ((e. A (V))) (= (` (1st) (<,> A B)) A))
+stmt (op2nd () ((e. A (V)) (e. B (V))) (= (` (2nd) (<,> A B)) B))
+stmt (op1stg ((A x) (B x)) () (-> (e. A C) (= (` (1st) (<,> A B)) A)))
+stmt (op2ndg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (= (` (2nd) (<,> A B)) B)))
+stmt (elxp6 () () (<-> (e. A (X. B C)) (/\ (= A (<,> (` (1st) A) (` (2nd) A))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C)))))
+stmt (fo1st ((x y)) () (:-onto-> (1st) (V) (V)))
+stmt (fo2nd ((x y)) () (:-onto-> (2nd) (V) (V)))
+stmt (f1stres ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (:--> (|` (1st) (X. A B)) (X. A B) A))
+stmt (1st2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv x))) A) (` (1st) A)))
+stmt (df1st2 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (u x) (u y) (u z) (u w) (u v)) () (= ({<<,>,>|} x y z (= (cv z) (cv x))) (|` (1st) (X. (V) (V)))))
+stmt (1ord () () (e. (1o) (On)))
+stmt (2ord () () (e. (2o) (On)))
+stmt (df1o2 () () (= (1o) ({} ({/}))))
+stmt (df2o2 () () (= (2o) ({,} ({/}) ({} ({/})))))
+stmt (0ne1oOLD () () (-. (= ({/}) (1o))))
+stmt (1ne0 () () (-. (= (1o) ({/}))))
+stmt (ordgt0ge1 () () (-> (Ord A) (<-> (e. ({/}) A) (C_ (1o) A))))
+stmt (ordge1n0 () () (-> (Ord A) (<-> (C_ (1o) A) (-. (= A ({/}))))))
+stmt (el1o () () (<-> (e. A (1o)) (= A ({/}))))
+stmt (0lt1o () () (e. ({/}) (1o)))
+stmt (fnoa ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (+o) (X. (On) (On))))
+stmt (fnom ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Fn (.o) (X. (On) (On))))
+stmt (oav ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A z) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (` (rec ({<,>|} x y (= (cv y) (suc (cv x)))) A) B))))
+stmt (omv ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (.o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+o) A))) ({/})) B))))
+stmt (oe0lem () ((-> (/\ ph (= A ({/}))) ps) (-> (/\ (/\ (e. A (On)) ph) (e. ({/}) A)) ps)) (-> (/\ (e. A (On)) ph) ps))
+stmt (oev ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B z) (B w) (B v)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (^o) B) (if (= A ({/})) (\ (1o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B)))))
+stmt (oevn0 ((x y) (A x) (A y)) () (-> (/\ (/\ (e. A (On)) (e. B (On))) (e. ({/}) A)) (= (opr A (^o) B) (` (rec ({<,>|} x y (= (cv y) (opr (cv x) (.o) A))) (1o)) B))))
+stmt (oa0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (+o) ({/})) A)))
+stmt (om0 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (opr A (.o) ({/})) ({/}))))
+stmt (om0x ((x y) (A x) (A y)) () (= (opr A (.o) ({/})) ({/})))
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+stmt (oe0m0 () () (= (opr ({/}) (^o) ({/})) (1o)))
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+stmt (ecopqsi () ((e. R (V)) (= S (/. (X. A A) R))) (-> (/\ (e. B A) (e. C A)) (e. ([] (<,> B C) R) S)))
+stmt (qsex ((x y) (A x) (A y) (R x) (R y)) ((e. A (V))) (e. (/. A R) (V)))
+stmt (snec ((x y) (A x) (A y) (R x) (R y)) ((e. A (V))) (= ({} ([] A R)) (/. ({} A) R)))
+stmt (ecqs () ((e. A (V)) (e. R (V))) (= ([] A R) (U. (/. ({} A) R))))
+stmt (0nelqs ((R x) (A x)) ((Er R) (= (dom R) A)) (-. (e. ({/}) (/. A R))))
+stmt (ecelqsdm () ((e. B (V)) (Er R) (= (dom R) A)) (-> (e. ([] B R) (/. A R)) (e. B A)))
+stmt (ecid ((A y)) ((e. A (V))) (= ([] A (`' (E))) A))
+stmt (qsid ((x y) (A x) (A y)) ((e. A (V))) (= (/. A (`' (E))) A))
+stmt (ectocl ((A x) (B x) (R x) (ps x)) ((= S (/. B R)) (-> (= ([] (cv x) R) A) (<-> ph ps)) (-> (e. (cv x) B) ph)) (-> (e. A S) ps))
+stmt (ecoptocl ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (ps x) (ps y) (ps z)) ((= S (/. (X. B C) R)) (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps)) (-> (/\ (e. (cv x) B) (e. (cv y) C)) ph)) (-> (e. A S) ps))
+stmt (2ecoptocl ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w) (S z) (S w) (R x) (R y) (R z) (R w) (ps x) (ps y) (ch z) (ch w)) ((= S (/. (X. C D) R)) (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps)) (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch)) (-> (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (/\ (e. (cv z) C) (e. (cv w) D))) ph)) (-> (/\ (e. A S) (e. B S)) ch))
+stmt (3ecoptocl ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B z) (B w) (B v) (B u) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S z) (S w) (S v) (S u) (R x) (R y) (R z) (R w) (R v) (R u) (ps x) (ps y) (ch z) (ch w) (th v) (th u)) ((= S (/. (X. D D) R)) (-> (= ([] (<,> (cv x) (cv y)) R) A) (<-> ph ps)) (-> (= ([] (<,> (cv z) (cv w)) R) B) (<-> ps ch)) (-> (= ([] (<,> (cv v) (cv u)) R) C) (<-> ch th)) (-> (/\/\ (/\ (e. (cv x) D) (e. (cv y) D)) (/\ (e. (cv z) D) (e. (cv w) D)) (/\ (e. (cv v) D) (e. (cv u) D))) ph)) (-> (/\/\ (e. A S) (e. B S) (e. C S)) th))
+stmt (brecop ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (S x) (S y) (S z) (S w) (S v) (S u) (H x) (H y) (G z) (G w) (G v) (G u) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u)) ((e. S (V)) (Er S) (= (dom S) (X. G G)) (= H (/. (X. G G) S)) (= R ({<,>|} x y (/\ (/\ (e. (cv x) H) (e. (cv y) H)) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) ([] (<,> (cv z) (cv w)) S)) (= (cv y) ([] (<,> (cv v) (cv u)) S))) ph)))))))) (-> (/\ (/\ (/\ (e. (cv z) G) (e. (cv w) G)) (/\ (e. A G) (e. B G))) (/\ (/\ (e. (cv v) G) (e. (cv u) G)) (/\ (e. C G) (e. D G)))) (-> (/\ (= ([] (<,> (cv z) (cv w)) S) ([] (<,> A B) S)) (= ([] (<,> (cv v) (cv u)) S) ([] (<,> C D) S))) (<-> ph ps)))) (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) ps)))
+var ( class Q )
+stmt (brecop2 () ((e. S (V)) (e. B (V)) (e. C (V)) (e. D (V)) (Er S) (= (dom S) (X. G G)) (= H (/. (X. G G) S)) (C_ R (X. H H)) (C_ Q (X. G G)) (-. (e. ({/}) G)) (= (dom F) (X. G G)) (-> (/\ (/\ (e. A G) (e. B G)) (/\ (e. C G) (e. D G))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C))))) (<-> (br ([] (<,> A B) S) R ([] (<,> C D) S)) (br (opr A F D) Q (opr B F C))))
+var ( set s )
+var ( set r )
+stmt (ecopopreq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (D x) (D y) (D z) (D w) (D v) (D u) (x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (S x) (S y) (S z) (S w) (S v) (S u)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v)))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C)))))
+stmt (ecopoprdm ((f g) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (F y) (w z) (v z) (u z) (f z) (g z) (F z) (v w) (u w) (f w) (g w) (F w) (u v) (f v) (g v) (F v) (f u) (g u) (F u) (F f) (F g) (R f) (R g) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (= (dom R) (X. S S)))
+stmt (ecopoprsym ((f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B f) (B g) (B h) (B t) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (F w) (u v) (f v) (g v) (h v) (t v) (F v) (f u) (g u) (h u) (t u) (F u) (F f) (F g) (F h) (F t) (R f) (R g) (R h) (R t) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (e. B (V))) (-> (br A R B) (br B R A)))
+stmt (ecopoprtrn ((f g) (f h) (f t) (f s) (f r) (A f) (g h) (g t) (g s) (g r) (A g) (h t) (h s) (h r) (A h) (s t) (r t) (A t) (r s) (A s) (A r) (B f) (B g) (B h) (B t) (B s) (B r) (C f) (C g) (C h) (C t) (C s) (C r) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (F w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (F v) (f u) (g u) (h u) (t u) (s u) (r u) (F u) (F f) (F g) (F h) (F t) (F s) (F r) (R f) (R g) (R h) (R t) (R s) (R r) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t) (S s) (S r)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))) (e. B (V)) (e. C (V))) (-> (/\ (br A R B) (br B R C)) (br A R C)))
+stmt (ecopoprer ((f g) (f h) (g h) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (F z) (v w) (u w) (f w) (g w) (h w) (F w) (u v) (f v) (g v) (h v) (F v) (f u) (g u) (h u) (F u) (F f) (F g) (F h) (R f) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (Er R))
+stmt (eceqopreq ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((e. B (V)) (e. C (V)) (e. D (V)) (Er R) (= (dom R) (X. S S)) (= (dom F) (X. S S)) (-. (e. ({/}) S)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C))))) (-> (/\ (e. A S) (e. C S)) (<-> (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C)))))
+stmt (th3qlem1 ((x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (A x) (A y) (A z) (A w) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((Er R) (= (dom R) S) (-> (/\ (/\ (e. (cv y) S) (e. (cv w) S)) (/\ (e. (cv z) S) (e. (cv v) S))) (-> (/\ (br (cv y) R (cv w)) (br (cv z) R (cv v))) (br (opr (cv y) F (cv z)) R (opr (cv w) F (cv v)))))) (-> (/\ (e. A (/. S R)) (e. B (/. S R))) (E* x (E. y (E. z (/\ (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R))))))))
+stmt (th3qlem2 ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (-> (/\ (e. A (/. (X. S S) R)) (e. B (/. (X. S S) R))) (E* z (E. w (E. v (E. u (E. t (/\ (/\ (= A ([] (<,> (cv w) (cv v)) R)) (= B ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))))
+stmt (th3qcor ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))) (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))))) (Fun G))
+stmt (th3q ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))) (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) G ([] (<,> C D) R)) ([] (opr (<,> A B) F (<,> C D)) R))))
+var ( set a )
+var ( set b )
+var ( set c )
+var ( set d )
+var ( class L )
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+var ( class W )
+var ( class X )
+var ( class Y )
+var ( class Z )
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+stmt (unir1 () () (= (U. (" (R1) (On))) (V)))
+stmt (rankval ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x))))))))
+stmt (rankvalg ((x y) (A x) (A y)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))))))
+stmt (rankval2 ((A x)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (C_ A (` (R1) (cv x))))))))
+stmt (rankon ((A x)) () (e. (` (rank) A) (On)))
+stmt (rankid ((x y) (A x) (A y)) ((e. A (V))) (e. A (` (R1) (suc (` (rank) A)))))
+stmt (rankr1lem () ((e. A (V))) (-> (e. B (On)) (-> (-. (e. A (` (R1) B))) (C_ B (` (rank) A)))))
+stmt (rankr1 ((x y) (A x) (A y) (B x) (B y)) ((e. A (V))) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B))))))
+stmt (rankr1g ((A x) (B x)) () (-> (e. A C) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B)))))))
+stmt (ssrankr1 () ((e. A (V))) (-> (e. B (On)) (<-> (C_ B (` (rank) A)) (-. (e. A (` (R1) B))))))
+stmt (rankr1a () ((e. A (V))) (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (` (rank) A) B))))
+stmt (r1val2 ((A x)) () (-> (e. A (On)) (= (` (R1) A) ({|} x (e. (` (rank) (cv x)) A)))))
+stmt (r1val3 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ ({|} y (e. (` (rank) (cv y)) (cv x))))))))
+stmt (rankel () ((e. B (V))) (-> (e. A B) (e. (` (rank) A) (` (rank) B))))
+stmt (rankval3 ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) A) (|^| ({e.|} x (On) (A.e. y A (e. (` (rank) (cv y)) (cv x)))))))
+stmt (bndrank ((x y) (A x) (A y)) () (-> (E.e. x (On) (A.e. y A (C_ (` (rank) (cv y)) (cv x)))) (e. A (V))))
+stmt (unbndrank ((x y) (A x) (A y)) () (-> (-. (e. A (V))) (A.e. x (On) (E.e. y A (e. (cv x) (` (rank) (cv y)))))))
+stmt (rankpw () ((e. A (V))) (= (` (rank) (P~ A)) (suc (` (rank) A))))
+stmt (r1pw ((A x) (B x)) () (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) (suc B))))))
+stmt (r1pwcl ((A x) (A y) (x y) (B x) (B y)) () (-> (Lim B) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) B)))))
+stmt (rankss () ((e. B (V))) (-> (C_ A B) (C_ (` (rank) A) (` (rank) B))))
+stmt (ranksn ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) ({} A)) (suc (` (rank) A))))
+stmt (rankuni ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (` (rank) (U. A)) (U_ x A (` (rank) (cv x)))))
+stmt (rankuniss ((A x)) ((e. A (V))) (C_ (` (rank) (U. A)) (` (rank) A)))
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+stmt (rankpr () ((e. A (V)) (e. B (V))) (= (` (rank) ({,} A B)) (suc (u. (` (rank) A) (` (rank) B)))))
+stmt (r1rankid () () (-> (e. A B) (C_ A (` (R1) (` (rank) A)))))
+stmt (rankonid ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (On)) (= (` (rank) A) A)))
+stmt (rankr1id ((x y) (A x) (A y)) () (<-> (e. A (On)) (= (` (rank) (` (R1) A)) A)))
+stmt (ranklon ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) A) (U_ x A (suc (` (rank) (cv x))))))
+stmt (scottex ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) (V)))
+stmt (scott0 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (= A ({/})) (= ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) ({/}))))
+stmt (scottexs ((x y) (x z) (y z) (ph y) (ph z)) () (e. ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)))
+stmt (scott0s ((x y) (x z) (y z) (ph y) (ph z)) () (<-> (E. x ph) (-. (= ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) ({/})))))
+stmt (cplem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w) (D w)) ((= C ({e.|} y B (A.e. z B (C_ (` (rank) (cv y)) (` (rank) (cv z)))))) (= D (U_ x A C))) (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B D) ({/}))))))
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+stmt (cp ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (E. w (A.e. x (cv z) (-> (E. y ph) (E.e. y (cv w) ph)))))
+stmt (bnd ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (-> (A.e. x (cv z) (E. y ph)) (E. w (A.e. x (cv z) (E.e. y (cv w) ph)))))
+stmt (bnd2 ((ph z) (ph w) (ph v) (w z) (v z) (v w) (x z) (w x) (v x) (A x) (A z) (A w) (A v) (x y) (B x) (y z) (w y) (v y) (B y) (B z) (B w) (B v)) ((e. A (V))) (-> (A.e. x A (E.e. y B ph)) (E. z (/\ (C_ (cv z) B) (A.e. x A (E.e. y (cv z) ph))))))
+stmt (kardex ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (e. ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V)))
+stmt (karden ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C z) (D z)) ((e. A (V)) (e. B (V)) (= C ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) (= D ({|} x (/\ (br (cv x) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (<-> (= C D) (br A (~~) B)))
+stmt (aceq1 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (y z) (w y) (v y) (u y) (t y) (h y) (w z) (v z) (u z) (t z) (h z) (v w) (u w) (t w) (h w) (u v) (t v) (h v) (t u) (h u) (h t)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x)))))))))))
+stmt (aceq0 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v)))))))))))
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+stmt (aceq3lem ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (f x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (f y) (w z) (v z) (u z) (t z) (h z) (g z) (f z) (v w) (u w) (t w) (h w) (g w) (f w) (u v) (t v) (h v) (g v) (f v) (t u) (h u) (g u) (f u) (h t) (g t) (f t) (g h) (f h) (f g) (F t) (F h) (F g)) ((= F ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u))))))))) (-> (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))) (E. f (/\ (C_ (cv f) (cv y)) (Fn (cv f) (dom (cv y)))))))
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+stmt (aceq4 ((f x) (f z) (f y) (f w) (f v) (x z) (x y) (w x) (v x) (y z) (w z) (v z) (w y) (v y) (v w)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (/\ (Fn (cv f) (cv x)) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))))))
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+stmt (aceq5lem3 ((u w) (t w) (h w) (t u) (h u) (h t) (A w)) ((= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (<-> (e. (X. ({} (cv w)) (cv w)) A) (/\ (-. (= (cv w) ({/}))) (e. (cv w) (cv h)))))
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+stmt (kmlem5 () () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) ({/}))))
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+stmt (kmlem13 ((x y) (x z) (v x) (y z) (v y) (v z)) () (<-> (-> (-. (E.e. z (cv x) (A.e. v (cv z) ph))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (\/ (E.e. z (cv x) (A.e. v (cv z) ph)) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))
+stmt (kmlem14 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (A. z (E. v (A. u (/\ (e. (cv y) (cv x)) ph)))))))
+stmt (kmlem15 ((x y) (x z) (v x) (u x) (y z) (v y) (u y) (v z) (u z) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (/\ (-. (e. (cv y) (cv x))) ch) (A. z (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps))))))
+stmt (kmlem16 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (\/ (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (/\ (-. (e. (cv y) (cv x))) ch))) (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) ph) (/\ (-. (e. (cv y) (cv x))) ps))))))))
+stmt (aceqkm ((x y) (x z) (w x) (v x) (u x) (f x) (t x) (h x) (y z) (w y) (v y) (u y) (f y) (t y) (h y) (w z) (v z) (u z) (f z) (t z) (h z) (v w) (u w) (f w) (t w) (h w) (u v) (f v) (t v) (h v) (f u) (t u) (h u) (f t) (f h) (h t)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))))))))))
+stmt (ackm ((x y) (x z) (v x) (u x) (f x) (y z) (v y) (u y) (f y) (v z) (u z) (f z) (u v) (f v) (f u)) () (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))))))))))
+stmt (numthlem ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (B x) (B y) (B f) (F x) (F y) (F f) (G x) (G y) (G f)) ((e. A (V)) (= B ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. B)) (= G ({<,>|} f y (= (cv y) (` (cv g) (\ A (ran (cv f)))))))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))))
+stmt (numth ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h)) ((e. A (V))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A))))
+stmt (numth2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (E.e. x (On) (br (cv x) (~~) A)))
+stmt (numthcor ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (br A (~<) (cv x)))))
+stmt (weth ((x y) (x z) (w x) (f x) (A x) (y z) (w y) (f y) (A y) (w z) (f z) (A z) (f w) (A w) (A f)) ((e. A (V))) (E. x (We (cv x) A)))
+stmt (zornlem1 ((w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D u) (D v) (D f) (D t) (R x) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (e. (` F (cv x)) D)))
+stmt (zornlem2 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (-> (e. (cv y) (cv x)) (br (` F (cv y)) R (` F (cv x))))))
+stmt (zornlem3 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (Po R A) (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/})))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y)))))))
+stmt (zornlem4 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (Po R A) (We (cv w) A)) (E.e. x (On) (= D ({/})))))
+stmt (zornlem5 ((u x) (v x) (f x) (t x) (s x) (H x) (u v) (f u) (t u) (s u) (H u) (f v) (t v) (s v) (H v) (f t) (f s) (H f) (s t) (H t) (H s) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (s x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (s y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (s w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h s) (A h) (t z) (f t) (g t) (t u) (t v) (s t) (A t) (f z) (g z) (u z) (v z) (s z) (A z) (f g) (f u) (f v) (f s) (A f) (g u) (g v) (g s) (A g) (u v) (s u) (A u) (s v) (A v) (A s) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F s) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (C_ (" F (cv x)) A)))
+stmt (zornlem6 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (Po R A) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (Or R (" F (cv x))))))
+stmt (zornlem7 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (/\ (Po R A) (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a)))))))) (E.e. a A (A.e. b A (-. (br (cv a) R (cv b)))))))
+stmt (zorn2lem ((x y) (x z) (y z) (w x) (w y)) () (<-> (br (cv z) ({<,>|} x y (C: (cv x) (cv y))) (cv w)) (C: (cv z) (cv w))))
+var ( set q )
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+stmt (zorn2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A x) (A y) (A z) (A w) (A v) (A u)) ((e. A (V))) (-> (A. z (-> (/\ (C_ (cv z) A) (A.e. x (cv z) (A.e. y (cv z) (\/ (C_ (cv x) (cv y)) (C_ (cv y) (cv x)))))) (e. (U. (cv z)) A))) (E.e. x A (A.e. y A (-. (C: (cv x) (cv y)))))))
+stmt (fodom ((F f) (A f) (B f)) ((e. A (V))) (-> (:-onto-> F A B) (br B (~<_) A)))
+stmt (fodomg ((A x) (B x) (F x)) () (-> (e. A C) (-> (:-onto-> F A B) (br B (~<_) A))))
+stmt (fodomb ((f g) (f z) (A f) (g z) (A g) (A z) (B f) (B g) (B z)) ((e. A (V)) (e. B (V))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A))))
+stmt (imadomg () () (-> (e. A B) (-> (Fun F) (br (" F A) (~<_) A))))
+stmt (fnrndomg () () (-> (e. A B) (-> (Fn F A) (br (ran F) (~<_) A))))
+stmt (htalem ((x y) (A x) (A y) (R x) (R y)) ((e. A (V)) (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x)))))))) (-> (/\ (We R A) (-. (= A ({/})))) (e. B A)))
+stmt (hta ((v z) (A z) (A v) (x y) (R x) (x z) (v x) (R y) (y z) (v y) (R z) (R v) (ph y) (ph z) (ph v)) ((= A ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x)))))))) (-> (We R A) (-> ph ([/] B x ph))))
+stmt (oncardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A))))))
+stmt (oncardon ((A x)) () (-> (e. A (On)) (e. (` (card) A) (On))))
+stmt (oncardid ((A x)) () (-> (e. A (On)) (br (` (card) A) (~~) A)))
+stmt (cardonle ((A x)) () (-> (e. A (On)) (C_ (` (card) A) A)))
+stmt (card0 () () (= (` (card) ({/})) ({/})))
+stmt (cardnn () () (-> (e. A (om)) (= (` (card) A) A)))
+stmt (cardom () () (= (` (card) (om)) (om)))
+stmt (cardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A)))))
+stmt (cardon ((A x)) () (e. (` (card) A) (On)))
+stmt (cardid ((A x)) () (br (` (card) A) (~~) A))
+stmt (oncard ((x y) (A x) (A y)) () (<-> (E. x (= A (` (card) (cv x)))) (= A (` (card) A))))
+stmt (cardne ((A x) (B x)) () (-> (e. A (` (card) B)) (-. (br A (~~) B))))
+stmt (carden () () (-> (/\ (e. A C) (e. B D)) (<-> (= (` (card) A) (` (card) B)) (br A (~~) B))))
+stmt (cardeq0 () () (-> (e. A B) (<-> (= (` (card) A) ({/})) (= A ({/})))))
+stmt (cardsn () () (-> (e. A B) (= (` (card) ({} A)) (1o))))
+stmt (carddomi () () (-> (e. A C) (-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))))
+stmt (carddom () () (-> (/\ (e. A C) (e. B D)) (<-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B))))
+stmt (cardsdom () () (-> (/\ (e. A C) (e. B D)) (<-> (e. (` (card) A) (` (card) B)) (br A (~<) B))))
+stmt (domtri () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (~<_) B) (-. (br B (~<) A)))))
+stmt (entri () () (-> (/\ (e. A C) (e. B D)) (\/\/ (br A (~<) B) (br A (~~) B) (br B (~<) A))))
+stmt (entri2 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<) A))))
+stmt (entri3 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<_) A))))
+stmt (sucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B C)) (<-> (br A (~<) B) (br (suc A) (~<_) B))))
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+stmt (unxpdom ((x y) (A x) (A y) (B y)) () (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B))))
+stmt (unxpdom2 () ((e. A (V)) (e. B (V))) (-> (/\ (br (1o) (~<) A) (br B (~<_) A)) (br (u. A B) (~<_) (X. A A))))
+stmt (sucxpdom ((A x)) () (-> (br (1o) (~<) A) (br (suc A) (~<_) (X. A A))))
+stmt (sdomel () () (-> (/\ (e. A (On)) (e. B (On))) (-> (br A (~<) B) (e. A B))))
+stmt (sdomsdomcard () () (<-> (br A (~<) B) (br A (~<) (` (card) B))))
+stmt (cardidm () () (= (` (card) (` (card) A)) (` (card) A)))
+stmt (canth3 () () (-> (e. A B) (e. (` (card) A) (` (card) (P~ A)))))
+stmt (cardlim ((A x)) () (<-> (C_ (om) (` (card) A)) (Lim (` (card) A))))
+stmt (cardsdomel () () (-> (e. A (On)) (<-> (br A (~<) B) (e. A (` (card) B)))))
+stmt (iscard ((A x)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x A (br (cv x) (~<) A)))))
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+stmt (cardval2 ((A x)) () (= (` (card) A) ({e.|} x (On) (br (cv x) (~<) A))))
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+stmt (carduni ((x y) (A x) (A y) (B y)) () (-> (e. A B) (-> (A.e. x A (= (` (card) (cv x)) (cv x))) (= (` (card) (U. A)) (U. A)))))
+stmt (cardiun ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (-> (A.e. x A (= (` (card) B) B)) (= (` (card) (U_ x A B)) (U_ x A B)))))
+stmt (cardmin ((x y) (A x) (A y) (B y)) () (-> (e. A B) (= (` (card) (|^| ({e.|} x (On) (br A (~<) (cv x))))) (|^| ({e.|} x (On) (br A (~<) (cv x)))))))
+stmt (cardprc ((x y)) () (-. (e. ({|} x (= (` (card) (cv x)) (cv x))) (V))))
+stmt (alephfnon ((x y) (x z) (y z)) () (Fn (aleph) (On)))
+stmt (aleph0 ((x y) (x z) (y z)) () (= (` (aleph) ({/})) (om)))
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+stmt (alephon ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. (` (aleph) A) (On)))
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+stmt (alephcard ((x y) (A x) (A y)) () (= (` (card) (` (aleph) A)) (` (aleph) A)))
+stmt (alephnbtwn ((A x) (B x)) () (-> (= (` (card) B) B) (-. (/\ (e. (` (aleph) A) B) (e. B (` (aleph) (suc A)))))))
+stmt (alephnbtwn2 () () (-. (/\ (br (` (aleph) A) (~<) B) (br B (~<) (` (aleph) (suc A))))))
+stmt (alephsucpw () () (br (` (aleph) (suc A)) (~<_) (P~ (` (aleph) A))))
+stmt (aleph1 () () (br (` (aleph) (1o)) (~<_) (opr (2o) (^m) (` (aleph) ({/})))))
+stmt (alephordlem1 ((x y) (A x) (A y)) () (-> (e. A (On)) (br (` (aleph) A) (~<) (` (aleph) (suc A)))))
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+stmt (alephord () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B)))))
+stmt (alephord2 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))))
+stmt (alephord2i () () (-> (e. B (On)) (-> (e. A B) (e. (` (aleph) A) (` (aleph) B)))))
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+stmt (alephsucdom () () (-> (e. B (On)) (<-> (br A (~<_) (` (aleph) B)) (br A (~<) (` (aleph) (suc B))))))
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+stmt (alephgeom () () (<-> (e. A (On)) (C_ (om) (` (aleph) A))))
+stmt (alephislim () () (<-> (e. A (On)) (Lim (` (aleph) A))))
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+stmt (leaddnnegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) A) (br B (<_) (opr B (+) A)))))
+stmt (subge02t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<_) B) (br (opr A (-) B) (<_) A))))
+stmt (ltsubpost () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br (opr B (-) A) (<) B))))
+stmt (posdift () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A)))))
+stmt (nnegdift () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (br (0) (<_) (opr B (-) A)))))
+stmt (ltnegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (-u B) (<) (-u A)))))
+stmt (ltnegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<) B) (br (-u B) (<) A))))
+stmt (lenegt () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) B) (br (-u B) (<_) (-u A)))))
+stmt (lenegcon1t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A))))
+stmt (lenegcon2t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (br A (<_) (-u B)) (br B (<_) (-u A)))))
+stmt (lesub1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (<-> (br A (<_) B) (br (opr A (-) C) (<_) (opr B (-) C)))))
+stmt (ltaddpos () ((e. A (RR)) (e. B (RR))) (<-> (br (0) (<) A) (br B (<) (opr B (+) A))))
+stmt (posdif () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (0) (<) (opr B (-) A))))
+stmt (ltnegcon1 () ((e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<) B) (br (-u B) (<) A)))
+stmt (lenegcon1 () ((e. A (RR)) (e. B (RR))) (<-> (br (-u A) (<_) B) (br (-u B) (<_) A)))
+stmt (lt0neg1t () () (-> (e. A (RR)) (<-> (br A (<) (0)) (br (0) (<) (-u A)))))
+stmt (lt0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (-u A) (<) (0)))))
+stmt (le0neg1t () () (-> (e. A (RR)) (<-> (br A (<_) (0)) (br (0) (<_) (-u A)))))
+stmt (le0neg2t () () (-> (e. A (RR)) (<-> (br (0) (<_) A) (br (-u A) (<_) (0)))))
+stmt (lesub0t () () (-> (/\ (e. A (RR)) (e. B (RR))) (<-> (/\ (br (0) (<_) A) (br B (<_) (opr B (-) A))) (= A (0)))))
+stmt (mulge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (br (0) (<_) (opr A (x.) B))))
+stmt (mulge0tOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (br (0) (<_) (opr A (x.) B)))))
+stmt (ltsqt () () (-> (e. A (RR)) (-> (-. (= A (0))) (br (0) (<) (opr A (x.) A)))))
+stmt (lt01 () () (br (0) (<) (1)))
+stmt (eqneg () ((e. A (CC))) (<-> (= A (-u A)) (= A (0))))
+stmt (eqnegt () () (-> (e. A (CC)) (<-> (= A (-u A)) (= A (0)))))
+stmt (negeq0t () () (-> (e. A (CC)) (<-> (= A (0)) (= (-u A) (0)))))
+stmt (negne0 () ((e. A (CC))) (<-> (=/= A (0)) (=/= (-u A) (0))))
+stmt (negn0 () ((e. A (CC)) (=/= A (0))) (=/= (-u A) (0)))
+stmt (elimgt0 () () (br (0) (<) (if (br (0) (<) A) A (1))))
+stmt (elimge0 () () (br (0) (<_) (if (br (0) (<_) A) A (0))))
+stmt (ltplus1t () () (-> (e. A (RR)) (br A (<) (opr A (+) (1)))))
+stmt (ltplus1 () ((e. A (RR))) (br A (<) (opr A (+) (1))))
+stmt (recgt0i () ((e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr (1) (/) A)))
+stmt (recgt0 () ((e. A (RR))) (-> (br (0) (<) A) (br (0) (<) (opr (1) (/) A))))
+stmt (leplus1t () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (br A (<_) (opr B (+) (1)))))
+stmt (plus1let () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (opr A (+) (1)) (<_) B)) (br A (<_) B)))
+stmt (prodgt0lem () ((e. A (RR)) (e. B (RR)) (br (0) (<) A)) (-> (br (0) (<) (opr A (x.) B)) (br (0) (<) B)))
+stmt (prodgt0 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B))) (br (0) (<) B)))
+stmt (prodge0 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B))) (br (0) (<_) B)))
+stmt (divgt0lem () ((e. A (RR)) (e. B (RR)) (br (0) (<) B)) (-> (br (0) (<) A) (br (0) (<) (opr A (/) B))))
+stmt (divgt0 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (/) B))))
+stmt (divgt0i () ((e. A (RR)) (e. B (RR)) (br (0) (<) A) (br (0) (<) B)) (br (0) (<) (opr A (/) B)))
+stmt (divge0 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<) B)) (br (0) (<_) (opr A (/) B))))
+stmt (ltmul1i () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C))))
+stmt (ltmul1 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))))
+stmt (ltdiv1i () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C))))
+stmt (ltdiv1 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))))
+stmt (ltmuldiv () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br (opr A (x.) C) (<) B) (br A (<) (opr B (/) C)))))
+stmt (prodgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) (opr A (x.) B)))) (br (0) (<) B)))
+stmt (prodgt02tOLD () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) (opr B (x.) A)))) (br (0) (<) B)))
+stmt (prodgt02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) B) (br (0) (<) (opr A (x.) B)))) (br (0) (<) A)))
+stmt (prodge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) B)))
+stmt (prodge02t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) B) (br (0) (<_) (opr A (x.) B)))) (br (0) (<_) A)))
+stmt (ltmul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (x.) C) (<) (opr B (x.) C)))))
+stmt (ltmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))))
+stmt (lemul1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))))
+stmt (lemul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))))
+stmt (ltmul2 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br A (<) B) (br (opr C (x.) A) (<) (opr C (x.) B)))))
+stmt (lemul1 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr A (x.) C) (<_) (opr B (x.) C)))))
+stmt (lemul2 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (br (0) (<) C) (<-> (br A (<_) B) (br (opr C (x.) A) (<_) (opr C (x.) B)))))
+stmt (lemul1it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr A (x.) C) (<_) (opr B (x.) C))))
+stmt (lemul2it () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<_) C) (br A (<_) B))) (br (opr C (x.) A) (<_) (opr C (x.) B))))
+stmt (mulgt1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (1) (<) A) (br (1) (<) B))) (br (1) (<) (opr A (x.) B))))
+stmt (recgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr (1) (/) A))))
+stmt (divgt0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (br (0) (<) (opr A (/) B))))
+stmt (divge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<) B))) (br (0) (<_) (opr A (/) B))))
+stmt (ltdiv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<) B) (br (opr A (/) C) (<) (opr B (/) C)))))
+stmt (lediv1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) C)) (<-> (br A (<_) B) (br (opr A (/) C) (<_) (opr B (/) C)))))
+stmt (gt0divt () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br (0) (<) B)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) B)))))
+stmt (ltmuldivt () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (x.) B) (<) C) (br A (<) (opr C (/) B)))))
+stmt (ltmuldiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) A)) (<-> (br (opr A (x.) B) (<) C) (br B (<) (opr C (/) A)))))
+stmt (ltdivmult () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr B (x.) C)))))
+stmt (ltdivmul2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (br (0) (<) B)) (<-> (br (opr A (/) B) (<) C) (br A (<) (opr C (x.) B)))))
+stmt (ltreci () ((e. A (RR)) (e. B (RR)) (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A))))
+stmt (ltrec () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))))
+stmt (lerec () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<) A) (br (0) (<) B)) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))))
+stmt (lt2sq () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))))
+stmt (le2sq () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B)))))
+stmt (sq11 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (x.) A) (opr B (x.) B)) (= A B))))
+stmt (ltrect () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<) B) (br (opr (1) (/) B) (<) (opr (1) (/) A)))))
+stmt (lerect () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) B) (br (opr (1) (/) B) (<_) (opr (1) (/) A)))))
+stmt (lt2sqt () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (<-> (br A (<) B) (br (opr A (x.) A) (<) (opr B (x.) B)))))
+stmt (ltdiv2t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\/\ (br (0) (<) A) (br (0) (<) B) (br (0) (<) C))) (<-> (br A (<) B) (br (opr C (/) B) (<) (opr C (/) A)))))
+stmt (ltrec1t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br (opr (1) (/) A) (<) B) (br (opr (1) (/) B) (<) A))))
+stmt (lerec2t () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<) A) (br (0) (<) B))) (<-> (br A (<_) (opr (1) (/) B)) (br B (<_) (opr (1) (/) A)))))
+stmt (ltdiv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))))
+stmt (lediv23t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. C (RR))) (/\ (br (0) (<) B) (br (0) (<) C))) (<-> (br (opr A (/) B) (<_) C) (br (opr A (/) C) (<_) B))))
+stmt (ltdiv23 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\ (br (0) (<) B) (br (0) (<) C)) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B))))
+stmt (ltdiv23i () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br (0) (<) C)) (<-> (br (opr A (/) B) (<) C) (br (opr A (/) C) (<) B)))
+stmt (reclt1t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (<-> (br A (<) (1)) (br (1) (<) (opr (1) (/) A)))))
+stmt (recgt1t () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (/\ (br (0) (<) (opr (1) (/) A)) (br (opr (1) (/) A) (<) (1)))))
+stmt (le2sqt () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B))))))
+stmt (halfpos () ((e. A (RR))) (<-> (br (0) (<) A) (br (opr A (/) (opr (1) (+) (1))) (<) A)))
+stmt (ledivp1 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<_) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<_) B)))
+stmt (ltdivp1 () ((e. A (RR)) (e. B (RR)) (e. C (RR))) (-> (/\/\ (br (0) (<_) A) (br (0) (<_) C) (br A (<) (opr B (/) (opr C (+) (1))))) (br (opr A (x.) C) (<) B)))
+stmt (posex ((A x) (B x)) ((e. A (RR)) (e. B (RR)) (br (0) (<) A) (br (0) (<) B)) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (/\ (br (cv x) (<) A) (br (cv x) (<) B)))))
+stmt (max1 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br A (<_) (if (br A (<_) B) B A))))
+stmt (max1ALT () () (-> (e. A (RR)) (br A (<_) (if (br A (<_) B) B A))))
+stmt (max2 () () (-> (/\ (e. A (RR)) (e. B (RR))) (br B (<_) (if (br A (<_) B) B A))))
+stmt (peano5nn ((x y) (A x) (A y)) ((e. A (V))) (-> (/\ (e. (1) A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ (NN) A)))
+stmt (nnssre () () (C_ (NN) (RR)))
+stmt (nnsscn () () (C_ (NN) (CC)))
+stmt (nnret () () (-> (e. A (NN)) (e. A (RR))))
+stmt (nncnt () () (-> (e. A (NN)) (e. A (CC))))
+stmt (nnre () ((e. A (NN))) (e. A (RR)))
+stmt (nncn () ((e. A (NN))) (e. A (CC)))
+stmt (nnex () () (e. (NN) (V)))
+stmt (1nn ((x y)) () (e. (1) (NN)))
+stmt (peano2nn ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (NN)) (e. (opr A (+) (1)) (NN))))
+stmt (dfnn2 ((x y) (x z) (y z)) () (= (NN) (|^| ({|} x (/\/\ (C_ (cv x) (RR)) (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))))
+stmt (nnind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((-> (= (cv x) (1)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (e. (cv y) (NN)) (-> ch th))) (-> (e. A (NN)) ta))
+stmt (nn1suc ((x y) (x z) (y z) (A x) (A y) (A z) (ps x) (ch x) (th x) (th z) (ph y) (ph z)) ((-> (= (cv x) (1)) (<-> ph ps)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph ch)) (-> (= (cv x) A) (<-> ph th)) ps (-> (e. (cv y) (NN)) ch)) (-> (e. A (NN)) th))
+stmt (nnaddclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (+) B) (NN))))
+stmt (nnmulclt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (x.) B) (NN))))
+stmt (nn2get ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (E.e. x (NN) (/\ (br A (<_) (cv x)) (br B (<_) (cv x))))))
+stmt (nnge1t ((x y) (A x) (A y)) () (-> (e. A (NN)) (br (1) (<_) A)))
+stmt (nngt1ne1t () () (-> (e. A (NN)) (<-> (br (1) (<) A) (-. (= A (1))))))
+stmt (nnle1eq1t () () (-> (e. A (NN)) (<-> (br A (<_) (1)) (= A (1)))))
+stmt (nngt0t () () (-> (e. A (NN)) (br (0) (<) A)))
+stmt (lt1nnn () () (-> (/\ (e. A (RR)) (br A (<) (1))) (-. (e. A (NN)))))
+stmt (0nnn () () (-. (e. (0) (NN))))
+stmt (nnne0t () () (-> (e. A (NN)) (=/= A (0))))
+stmt (nngt0 () ((e. A (NN))) (br (0) (<) A))
+stmt (nnne0 () ((e. A (NN))) (=/= A (0)))
+stmt (nnrecgt0t () () (-> (e. A (NN)) (br (0) (<) (opr (1) (/) A))))
+stmt (nnleltp1t ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<_) B) (br A (<) (opr B (+) (1))))))
+stmt (nnltp1let () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (br (opr A (+) (1)) (<_) B))))
+stmt (nnsub ((x y) (A x) (A y) (B x) (B y)) ((e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (e. (opr B (-) A) (NN))))
+stmt (nnsubt () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (br A (<) B) (e. (opr B (-) A) (NN)))))
+stmt (nnaddm1clt () () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr (opr A (+) B) (-) (1)) (NN))))
+stmt (nndivt ((A x) (B x)) () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (E.e. x (NN) (= (opr A (x.) (cv x)) B)) (e. (opr B (/) A) (NN)))))
+stmt (nndivtrt () () (-> (/\ (/\/\ (e. A (NN)) (e. B (NN)) (e. C (CC))) (/\ (e. (opr B (/) A) (NN)) (e. (opr C (/) B) (NN)))) (e. (opr C (/) A) (NN))))
+stmt (2re () () (e. (2) (RR)))
+stmt (2cn () () (e. (2) (CC)))
+stmt (3re () () (e. (3) (RR)))
+stmt (4re () () (e. (4) (RR)))
+stmt (5re () () (e. (5) (RR)))
+stmt (6re () () (e. (6) (RR)))
+stmt (7re () () (e. (7) (RR)))
+stmt (8re () () (e. (8) (RR)))
+stmt (9re () () (e. (9) (RR)))
+stmt (2pos () () (br (0) (<) (2)))
+stmt (3pos () () (br (0) (<) (3)))
+stmt (4pos () () (br (0) (<) (4)))
+stmt (5pos () () (br (0) (<) (5)))
+stmt (6pos () () (br (0) (<) (6)))
+stmt (7pos () () (br (0) (<) (7)))
+stmt (8pos () () (br (0) (<) (8)))
+stmt (9pos () () (br (0) (<) (9)))
+stmt (2nn () () (e. (2) (NN)))
+stmt (2p2e4 () () (= (opr (2) (+) (2)) (4)))
+stmt (2times () ((e. A (CC))) (= (opr (2) (x.) A) (opr A (+) A)))
+stmt (2timest () () (-> (e. A (CC)) (= (opr (2) (x.) A) (opr A (+) A))))
+stmt (times2 () ((e. A (CC))) (= (opr A (x.) (2)) (opr A (+) A)))
+stmt (3p2e5 () () (= (opr (3) (+) (2)) (5)))
+stmt (3p3e6 () () (= (opr (3) (+) (3)) (6)))
+stmt (4p2e6 () () (= (opr (4) (+) (2)) (6)))
+stmt (4p3e7 () () (= (opr (4) (+) (3)) (7)))
+stmt (4p4e8 () () (= (opr (4) (+) (4)) (8)))
+stmt (5p2e7 () () (= (opr (5) (+) (2)) (7)))
+stmt (5p3e8 () () (= (opr (5) (+) (3)) (8)))
+stmt (5p4e9 () () (= (opr (5) (+) (4)) (9)))
+stmt (6p2e8 () () (= (opr (6) (+) (2)) (8)))
+stmt (6p3e9 () () (= (opr (6) (+) (3)) (9)))
+stmt (7p2e9 () () (= (opr (7) (+) (2)) (9)))
+stmt (2t2e4 () () (= (opr (2) (x.) (2)) (4)))
+stmt (3t2e6 () () (= (opr (3) (x.) (2)) (6)))
+stmt (3t3e9 () () (= (opr (3) (x.) (3)) (9)))
+stmt (4t2e8 () () (= (opr (4) (x.) (2)) (8)))
+stmt (4d2e2 () () (= (opr (4) (/) (2)) (2)))
+stmt (1lt2 () () (br (1) (<) (2)))
+stmt (halfgt0 () () (br (0) (<) (opr (1) (/) (2))))
+stmt (halflt1 () () (br (opr (1) (/) (2)) (<) (1)))
+stmt (halfpost () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (opr A (/) (2)) (<) A))))
+stmt (halfpos2t () () (-> (e. A (RR)) (<-> (br (0) (<) A) (br (0) (<) (opr A (/) (2))))))
+stmt (2halvest () () (-> (e. A (CC)) (= (opr (opr A (/) (2)) (+) (opr A (/) (2))) A)))
+stmt (nominpos ((x y)) () (-. (E.e. x (RR) (/\ (br (0) (<) (cv x)) (-. (E.e. y (RR) (/\ (br (0) (<) (cv y)) (br (cv y) (<) (cv x)))))))))
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+stmt (lbcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) S)))
+stmt (lble ((w x) (w y) (S w) (x y) (S x) (S y) (A w) (A y)) () (-> (/\/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y)))) (e. A S)) (br (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))) (<_) A)))
+stmt (lbinfm ((w x) (w y) (w z) (v w) (S w) (x y) (x z) (v x) (S x) (y z) (v y) (S y) (v z) (S z) (S v)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (= (sup S (RR) (`' (<))) (U. ({e.|} x S (A.e. y S (br (cv x) (<_) (cv y))))))))
+stmt (lbinfmcl ((x y) (S x) (S y)) () (-> (/\ (C_ S (RR)) (E.e. x S (A.e. y S (br (cv x) (<_) (cv y))))) (e. (sup S (RR) (`' (<))) S)))
+stmt (sup2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (=/= A ({/})) (E.e. x (RR) (A.e. y A (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))))
+stmt (sup3 ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z)))))))))
+stmt (infm3lem ((x y)) () (-> (e. (cv x) (RR)) (E.e. y (RR) (= (cv x) (-u (cv y))))))
+stmt (infm3 ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y)))))))))
+stmt (suprcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. (sup A (RR) (<)) (RR))))
+stmt (suprub ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (e. B A)) (br B (<_) (sup A (RR) (<)))))
+stmt (suprlub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (br B (<) (sup A (RR) (<))))) (E.e. z A (br B (<) (cv z)))))
+stmt (suprnub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z)))))) (-. (br B (<) (sup A (RR) (<))))))
+stmt (suprleub ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B)))) (br (sup A (RR) (<)) (<_) B)))
+stmt (sup3i ((x y) (x z) (A x) (y z) (A y) (A z)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (E.e. x (RR) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))))
+stmt (suprcli ((x y) (x z) (A x) (y z) (A y) (A z)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (e. (sup A (RR) (<)) (RR)))
+stmt (suprubi ((x y) (A x) (A y)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (-> (e. B A) (br B (<_) (sup A (RR) (<)))))
+stmt (suprlubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (-> (/\ (e. B (RR)) (br B (<) (sup A (RR) (<)))) (E.e. z A (br B (<) (cv z)))))
+stmt (suprnubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (-> (/\ (e. B (RR)) (A.e. z A (-. (br B (<) (cv z))))) (-. (br B (<) (sup A (RR) (<))))))
+stmt (suprleubi ((B z) (x y) (x z) (A x) (y z) (A y) (A z)) ((/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x)))))) (-> (/\ (e. B (RR)) (A.e. z A (br (cv z) (<_) B))) (br (sup A (RR) (<)) (<_) B)))
+stmt (reuunineg ((x y) (x z) (y z) (ph y) (ph z) (ps x) (ps z)) ((-> (= (cv x) (-u (cv y))) (<-> ph ps))) (-> (E!e. x (RR) ph) (= (U. ({e.|} x (RR) ph)) (-u (U. ({e.|} y (RR) ps))))))
+stmt (infmsup ((x y) (x z) (w x) (v x) (u x) (t x) (A x) (y z) (w y) (v y) (u y) (t y) (A y) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (= (sup A (RR) (`' (<))) (-u (sup ({e.|} z (RR) (e. (-u (cv z)) A)) (RR) (<))))))
+stmt (infmrcl ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (e. (sup A (RR) (`' (<))) (RR))))
+stmt (nnunb ((x y) (x z) (y z)) () (-. (E.e. x (RR) (A.e. y (NN) (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x)))))))
+stmt (arch ((x y) (A x) (A y)) () (-> (e. A (RR)) (E.e. x (NN) (br A (<) (cv x)))))
+stmt (nnreclt ((A x)) () (-> (/\ (e. A (RR)) (br (0) (<) A)) (E.e. x (NN) (br (opr (1) (/) (cv x)) (<) A))))
+stmt (avril1 ((F x)) () (-. (/\ (br A (P~ (RR)) (` (i) (1))) (br F ({/}) (opr (0) (x.) (1))))))
+stmt (elnn0 () () (<-> (e. A (NN0)) (\/ (e. A (NN)) (= A (0)))))
+stmt (nnssnn0 () () (C_ (NN) (NN0)))
+stmt (nn0ssre () () (C_ (NN0) (RR)))
+stmt (nn0sscn () () (C_ (NN0) (CC)))
+stmt (nn0ex () () (e. (NN0) (V)))
+stmt (nnnn0t () () (-> (e. A (NN)) (e. A (NN0))))
+stmt (nnnn0 () ((e. N (NN))) (e. N (NN0)))
+stmt (nn0ret () () (-> (e. A (NN0)) (e. A (RR))))
+stmt (nn0cnt () () (-> (e. A (NN0)) (e. A (CC))))
+stmt (nn0re () ((e. A (NN0))) (e. A (RR)))
+stmt (nn0cn () ((e. A (NN0))) (e. A (CC)))
+stmt (0nn0 () () (e. (0) (NN0)))
+stmt (1nn0 () () (e. (1) (NN0)))
+stmt (2nn0 () () (e. (2) (NN0)))
+stmt (lt0nnn0 () () (-> (/\ (e. A (RR)) (br A (<) (0))) (-. (e. A (NN0)))))
+stmt (nn0ge0t () () (-> (e. N (NN0)) (br (0) (<_) N)))
+stmt (nn0ge0 () ((e. N (NN0))) (br (0) (<_) N))
+stmt (nn0addclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0))))
+stmt (nn0addcl () ((e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0)))
+stmt (nn0mulcl () ((e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0)))
+stmt (nn0mulclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0))))
+stmt (peano2nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN0))))
+stmt (nnnn0addclt () () (-> (/\ (e. M (NN)) (e. N (NN0))) (e. (opr M (+) N) (NN))))
+stmt (nn0p1nnt () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN))))
+stmt (nn0ltp1let () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))))
+stmt (nn0leltp1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))))
+stmt (nn0ltlem1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))))
+stmt (nn0addge1t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N))))
+stmt (nn0addge2t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A))))
+stmt (nn0addge1 () ((e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N)))
+stmt (nn0addge2 () ((e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A)))
+stmt (nn0le2x () ((e. N (NN0))) (br N (<_) (opr (2) (x.) N)))
+stmt (nn0lele2x () ((e. M (NN0)) (e. N (NN0))) (-> (br N (<_) M) (br N (<_) (opr (2) (x.) M))))
+stmt (elz ((N x)) () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/\/ (= N (0)) (e. N (NN)) (e. (-u N) (NN))))))
+stmt (nnnegz () () (-> (e. N (NN)) (e. (-u N) (ZZ))))
+stmt (zret () () (-> (e. N (ZZ)) (e. N (RR))))
+stmt (zcnt () () (-> (e. N (ZZ)) (e. N (CC))))
+stmt (zre () ((e. A (ZZ))) (e. A (RR)))
+stmt (zssre () () (C_ (ZZ) (RR)))
+stmt (zsscn () () (C_ (ZZ) (CC)))
+stmt (zex () () (e. (ZZ) (V)))
+stmt (elnnz () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (0) (<) N))))
+stmt (0z () () (e. (0) (ZZ)))
+stmt (elnn0z () () (<-> (e. N (NN0)) (/\ (e. N (ZZ)) (br (0) (<_) N))))
+stmt (elznn0nn () () (<-> (e. N (ZZ)) (\/ (e. N (NN0)) (/\ (e. N (RR)) (e. (-u N) (NN))))))
+stmt (elznn0 () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN0)) (e. (-u N) (NN0))))))
+stmt (elznn () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN)) (e. (-u N) (NN0))))))
+stmt (nnssz () () (C_ (NN) (ZZ)))
+stmt (nn0ssz () () (C_ (NN0) (ZZ)))
+stmt (nnzt () () (-> (e. N (NN)) (e. N (ZZ))))
+stmt (nn0zt () () (-> (e. N (NN0)) (e. N (ZZ))))
+stmt (elnnz1 () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (1) (<_) N))))
+stmt (znnnlt1t () () (-> (e. N (ZZ)) (<-> (-. (e. N (NN))) (br N (<) (1)))))
+stmt (nnz () () (= (NN) ({e.|} x (ZZ) (br (1) (<_) (cv x)))))
+stmt (nn0z () () (= (NN0) ({e.|} x (ZZ) (br (0) (<_) (cv x)))))
+stmt (1z () () (e. (1) (ZZ)))
+stmt (2z () () (e. (2) (ZZ)))
+stmt (nn0subt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))))
+stmt (nn0sub2t () () (-> (/\/\ (e. M (NN0)) (e. N (NN0)) (br M (<_) N)) (e. (opr N (-) M) (NN0))))
+stmt (znegclt () () (-> (e. N (ZZ)) (e. (-u N) (ZZ))))
+stmt (nn0negz () () (-> (e. N (NN0)) (e. (-u N) (ZZ))))
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+stmt (peano2z () () (-> (e. N (ZZ)) (e. (opr N (+) (1)) (ZZ))))
+stmt (zsubclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (-) N) (ZZ))))
+stmt (peano2zr () () (-> (e. N (ZZ)) (e. (opr N (-) (1)) (ZZ))))
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+stmt (elnn0nn () () (<-> (e. N (NN0)) (/\ (e. N (CC)) (e. (opr N (+) (1)) (NN)))))
+stmt (elnnnn0 () () (<-> (e. N (NN)) (/\ (e. N (CC)) (e. (opr N (-) (1)) (NN0)))))
+stmt (elnnnn0b () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (0) (<) N))))
+stmt (znnsubt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (e. (opr N (-) M) (NN)))))
+stmt (znn0subt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0)))))
+stmt (znn0sub2t () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. (opr N (-) M) (NN0))))
+stmt (zmulclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (x.) N) (ZZ))))
+stmt (zltp1let () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N))))
+stmt (zleltp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1))))))
+stmt (zlem1ltt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))))
+stmt (zltlem1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))))
+stmt (nn0lem1ltt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))))
+stmt (nnlem1ltt () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N))))
+stmt (nnltlem1t () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1))))))
+stmt (z2get ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (E.e. k (ZZ) (/\ (br M (<_) (cv k)) (br N (<_) (cv k))))))
+stmt (recnzt () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (-. (e. (opr (1) (/) A) (ZZ)))))
+stmt (btwnnzt () () (-> (/\/\ (e. A (ZZ)) (br A (<) B) (br B (<) (opr A (+) (1)))) (-. (e. B (ZZ)))))
+stmt (gtndivt () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br B (<) A)) (-. (e. (opr B (/) A) (ZZ)))))
+stmt (halfnz () () (-. (e. (opr (1) (/) (2)) (ZZ))))
+stmt (primet ((A x)) () (-> (e. A (NN)) (<-> (A.e. x (NN) (-> (e. (opr A (/) (cv x)) (NN)) (\/ (= (cv x) (1)) (= (cv x) A)))) (A.e. x (NN) (-> (/\/\ (br (1) (<) (cv x)) (br (cv x) (<_) A) (e. (opr A (/) (cv x)) (NN))) (= (cv x) A))))))
+stmt (sqznn () () (-> (/\ (e. A (ZZ)) (=/= A (0))) (e. (opr A (x.) A) (NN))))
+stmt (zneo () () (-> (/\ (e. A (ZZ)) (e. B (ZZ))) (-. (= (opr (2) (x.) A) (opr (opr (2) (x.) B) (+) (1))))))
+stmt (peano2uz2 ((A x) (B x)) () (-> (/\ (e. A (ZZ)) (e. B ({e.|} x (ZZ) (br A (<_) (cv x))))) (e. (opr B (+) (1)) ({e.|} x (ZZ) (br A (<_) (cv x))))))
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+stmt (peano5uz ((x y) (A x) (A y) (k x) (N x) (k y) (N y) (N k)) ((e. A (V)) (e. N (ZZ))) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A)))
+stmt (peano5uzt ((A x) (k x) (N x) (N k)) ((e. A (V))) (-> (e. N (ZZ)) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A))))
+stmt (uzind ((j w) (N j) (N w) (j ps) (ch j) (j th) (j ta) (k w) (k ph) (ph w) (j k) (M j) (M k) (M w)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<_) (cv k))) (-> ch th))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) ta))
+stmt (uzind2 ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((-> (= (cv j) (opr M (+) (1))) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<) (cv k))) (-> ch th))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) ta))
+stmt (uzind3 ((j k) (N j) (N k) (j ps) (ch j) (j th) (j ta) (m ph) (j m) (M j) (k m) (M m) (M k)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv m)) (<-> ph ch)) (-> (= (cv j) (opr (cv m) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\ (e. M (ZZ)) (e. (cv m) ({e.|} k (ZZ) (br M (<_) (cv k))))) (-> ch th))) (-> (/\ (e. M (ZZ)) (e. N ({e.|} k (ZZ) (br M (<_) (cv k))))) ta))
+stmt (uzindOLD ((x z) (w x) (A x) (w z) (A z) (A w) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (w y) (ph y) (ph w) (x y) (B x) (y z) (B y) (B z) (B w)) ((-> (= (cv x) B) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (/\ (/\ (e. (cv y) (ZZ)) (e. B (ZZ))) (br B (<_) (cv y))) (-> ch th))) (-> (/\ (/\ (e. A (ZZ)) (e. B (ZZ))) (br B (<_) A)) ta))
+stmt (uzind3OLD ((x z) (A x) (A z) (ps x) (ch x) (th x) (ta x) (ph y) (x y) (B x) (y z) (B y) (B z)) ((-> (= (cv x) B) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (/\ (e. B (ZZ)) (e. (cv y) ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> ch th))) (-> (/\ (e. B (ZZ)) (e. A ({e.|} z (ZZ) (br B (<_) (cv z))))) ta))
+stmt (uzind3iOLD ((k n) (N n) (N k) (n ps) (ch n) (n th) (n ta) (j ph) (j n) (M n) (j k) (M j) (M k)) ((e. M (ZZ)) (-> (= (cv n) M) (<-> ph ps)) (-> (= (cv n) (cv j)) (<-> ph ch)) (-> (= (cv n) (opr (cv j) (+) (1))) (<-> ph th)) (-> (= (cv n) N) (<-> ph ta)) ps (-> (e. (cv j) ({e.|} k (ZZ) (br M (<_) (cv k)))) (-> ch th))) (-> (e. N ({e.|} k (ZZ) (br M (<_) (cv k)))) ta))
+stmt (uzwoOLD ((x y) (w x) (v x) (u x) (A x) (w y) (v y) (u y) (A y) (v w) (u w) (A w) (u v) (A v) (A u) (y z) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (uzwo2OLD ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (nnwo ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (nnwoOLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (nnwof ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (A z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (nnwos ((x y) (x z) (y z) (ph y) (ph z) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (-> (E.e. x (NN) ph) (E.e. x (NN) (/\ ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y))))))))
+stmt (indstr ((x y) (ph y) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps)) (-> (e. (cv x) (NN)) (-> (A.e. y (NN) (-> (br (cv y) (<) (cv x)) ps)) ph))) (-> (e. (cv x) (NN)) ph))
+stmt (nn0ind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((-> (= (cv x) (0)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (e. (cv y) (NN0)) (-> ch th))) (-> (e. A (NN0)) ta))
+stmt (nn0indALT ((x y) (x z) (y z) (A x) (A z) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (ph y) (ph z)) ((-> (= (cv x) (0)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (e. (cv y) (NN0)) (-> ch th))) (-> (e. A (NN0)) ta))
+stmt (btwnz ((x z) (A x) (A z) (A y)) () (-> (e. A (RR)) (/\ (E.e. x (ZZ) (br (cv x) (<) A)) (E.e. y (ZZ) (br A (<) (cv y))))))
+stmt (uzwo3lem1 ((v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((= R ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> (e. B (RR)) (E!e. x R (A.e. y R (br (cv x) (<_) (cv y))))))
+stmt (uzwo3lem2 ((x y) (x z) (w x) (v x) (t x) (y z) (w y) (v y) (t y) (w z) (v z) (t z) (v w) (t w) (t v) (A x) (A y) (B t) (S y) (S t) (R w) (v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((= R ({e.|} z (ZZ) (br B (<_) (cv z)))) (= S (U. ({e.|} w R (A.e. v R (br (cv w) (<_) (cv v))))))) (-> (/\ (e. B (RR)) (/\ (C_ A R) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (uzwo3 ((x y) (u x) (t x) (A x) (u y) (t y) (A y) (t u) (A u) (A t) (w z) (v z) (u z) (t z) (B z) (v w) (u w) (t w) (B w) (u v) (t v) (B v) (B u) (B t)) () (-> (/\ (e. B (RR)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y))))))
+stmt (zmin ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y))))))))
+stmt (zmax ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))))
+stmt (zbtwnre ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (br (cv x) (<) (opr A (+) (1)))))))
+stmt (rebtwnz ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))))
+stmt (flvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1)))))))))
+stmt (flclt ((A z)) () (-> (e. A (RR)) (e. (` (floor) A) (ZZ))))
+stmt (flleltt ((x y) (A x) (A y)) () (-> (e. A (RR)) (/\ (br (` (floor) A) (<_) A) (br A (<) (opr (` (floor) A) (+) (1))))))
+stmt (fllet () () (-> (e. A (RR)) (br (` (floor) A) (<_) A)))
+stmt (flltp1t () () (-> (e. A (RR)) (br A (<) (opr (` (floor) A) (+) (1)))))
+stmt (flget () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br B (<_) A) (br B (<_) (` (floor) A)))))
+stmt (flltt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br A (<) B) (br (` (floor) A) (<) B))))
+stmt (flidt () () (-> (e. A (ZZ)) (= (` (floor) A) A)))
+stmt (flval2t ((x y) (A x) (A y)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x))))))))))
+stmt (flval3t ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (= (` (floor) A) (sup ({e.|} x (ZZ) (br (cv x) (<_) A)) (RR) (<)))))
+stmt (flbit ((A x) (B x)) () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (= (` (floor) A) B) (/\ (br B (<_) A) (br A (<) (opr B (+) (1)))))))
+stmt (flge0nn0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (floor) A) (NN0))))
+stmt (flge1nnt () () (-> (/\ (e. A (RR)) (br (1) (<_) A)) (e. (` (floor) A) (NN))))
+stmt (fladdzt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (= (` (floor) (opr A (+) B)) (opr (` (floor) A) (+) B))))
+stmt (btwnzge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (ZZ))) (/\ (br B (<_) A) (br A (<) (opr B (+) (1))))) (<-> (br (0) (<_) A) (br (0) (<_) B))))
+stmt (elq ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (QQ)) (E.e. x (ZZ) (E.e. y (NN) (= A (opr (cv x) (/) (cv y)))))))
+stmt (znq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (/) B) (QQ))))
+stmt (qret ((x y) (A x) (A y)) () (-> (e. A (QQ)) (e. A (RR))))
+stmt (zqt ((x y) (A x) (A y)) () (-> (e. A (ZZ)) (e. A (QQ))))
+stmt (zssq () () (C_ (ZZ) (QQ)))
+stmt (nn0ssq () () (C_ (NN0) (QQ)))
+stmt (nnssq () () (C_ (NN) (QQ)))
+stmt (qssre () () (C_ (QQ) (RR)))
+stmt (qsscn () () (C_ (QQ) (CC)))
+stmt (nnqt () () (-> (e. A (NN)) (e. A (QQ))))
+stmt (qcnt () () (-> (e. A (QQ)) (e. A (CC))))
+stmt (qex () () (e. (QQ) (V)))
+stmt (qaddclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (+) B) (QQ))))
+stmt (qnegclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (QQ)) (e. (-u A) (QQ))))
+stmt (qmulclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (x.) B) (QQ))))
+stmt (qsubclt () () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (-) B) (QQ))))
+stmt (qrecclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (QQ)) (=/= A (0))) (e. (opr (1) (/) A) (QQ))))
+stmt (qdivclt () () (-> (/\ (/\ (e. A (QQ)) (e. B (QQ))) (=/= B (0))) (e. (opr A (/) B) (QQ))))
+stmt (qrevaddclt () () (-> (e. B (QQ)) (<-> (/\ (e. A (CC)) (e. (opr A (+) B) (QQ))) (e. A (QQ)))))
+stmt (nnrecqt () () (-> (e. A (NN)) (e. (opr (1) (/) A) (QQ))))
+stmt (qbtwnre ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (br A (<) B)) (E.e. x (QQ) (/\ (br A (<) (cv x)) (br (cv x) (<) B)))))
+stmt (monoord ((y z) (A y) (A z) (B y) (x y) (x z) (F x) (F y) (F z)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1)))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (` F B))))
+stmt (om2uz0 ((x y) (C x) (C y)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (= (` G ({/})) C))
+stmt (om2uzsuc ((x y) (w x) (v x) (w y) (v y) (v w) (G w) (G v) (A w) (A v) (C x) (C y) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A (om)) (= (` G (suc A)) (opr (` G A) (+) (1)))))
+stmt (om2uzuz ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A (om)) (e. (` G A) ({e.|} z (ZZ) (br C (<_) (cv z))))))
+stmt (om2uzlt ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (B z) (B w) (B v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (/\ (e. A (om)) (e. B (om))) (-> (e. A B) (br (` G A) (<) (` G B)))))
+stmt (om2uzlt2 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (br (` G A) (<) (` G B)))))
+stmt (om2uzran ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G z) (G w) (G v) (G u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (= (ran G) ({e.|} z (ZZ) (br C (<_) (cv z)))))
+stmt (om2uzf1o ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (:-1-1-onto-> G (om) ({e.|} z (ZZ) (br C (<_) (cv z)))))
+stmt (uzrdgval ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) B) (` (rec F A) (` (`' G) B)))))
+stmt (uzrdgini ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) C) A)))
+stmt (uzrdgsuc ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) (opr B (+) (1))) (` F (` (o. (rec F A) (`' G)) B)))))
+stmt (uzrdginip1 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) (opr C (+) (1))) (` F A))))
+stmt (uzrdgfnuz ((x y) (x z) (y z) (G z) (A z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (Fn (o. (rec F A) (`' G)) ({e.|} z (ZZ) (br C (<_) (cv z)))))
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+stmt (seq1lem2 () () (-> (e. A (NN)) (e. (opr A (+) (1)) (\ (NN) ({} (1))))))
+stmt (seq1rval ((A z) (S z) (S w) (w z) (F z) (F w)) ((= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))) (e. A (V))) (= (` H A) (<,> (opr (` (1st) A) (+) (1)) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1)))))))
+stmt (seq1val ((x y) (f x) (g x) (h x) (G x) (f y) (g y) (h y) (G y) (f g) (f h) (G f) (g h) (G g) (G h) (H x) (H y) (H f) (H g) (H h) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (f z) (g z) (h z) (F z) (f w) (g w) (h w) (F w) (F f) (F g) (F h) (S x) (S y) (S z) (S w) (S f) (S g) (S h)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (= (opr S (seq1) F) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x))))))))
+stmt (seq1fnlem ((x y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (Fn (opr S (seq1) F) (NN)))
+stmt (seq1val2 ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) A) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A)))))
+stmt (seq11lem ((w z) (F z) (F w) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (= (` (opr S (seq1) F) (1)) (` F (1))))
+stmt (seq1suclem ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))))
+stmt (seq11 ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (= (` (opr S (seq1) F) (1)) (` F (1))))
+stmt (seq1p1 ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1)))))))
+stmt (seq1m1 () ((e. S (V)) (e. F (V))) (-> (/\ (e. N (NN)) (br (1) (<) N)) (= (` (opr S (seq1) F) N) (opr (` (opr S (seq1) F) (opr N (-) (1))) S (` F N)))))
+stmt (seq1fn ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (Fn (opr S (seq1) F) (NN)))
+stmt (seq1rn ((x y) (x z) (w x) (v x) (S x) (y z) (w y) (v y) (S y) (w z) (v z) (S z) (v w) (S w) (S v) (F x) (F y) (F z) (F w) (F v) (C x) (C y) (C z) (C w) (C v) (D x) (D y) (D z) (D w) (D v)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) (C_ (ran (opr S (seq1) F)) C)))
+stmt (seq1rn2 () ((e. S (V)) (e. F (V))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (C_ (ran (opr S (seq1) F)) C)))
+stmt (seq1cl () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. A (NN)) (:--> F (NN) C) (:--> S (X. C C) C)) (e. (` (opr S (seq1) F) A) C)))
+stmt (seq1res ((x y) (x z) (S x) (y z) (S y) (S z) (F x) (F y) (F z)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) (|` F (NN))) (opr S (seq1) F)))
+stmt (ser1ft ((F f)) () (-> (:--> F (NN) (CC)) (:--> (opr (+) (seq1) F) (NN) (CC))))
+stmt (ser1f () ((:--> F (NN) (CC))) (:--> (opr (+) (seq1) F) (NN) (CC)))
+stmt (ser1cl () ((:--> F (NN) (CC))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (CC))))
+stmt (ser1recl ((A x) (x y) (F x) (F y)) ((:--> F (NN) (RR))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (RR))))
+stmt (ser1re ((F x)) ((:--> F (NN) (RR))) (:--> (opr (+) (seq1) F) (NN) (RR)))
+stmt (ser1cl2 ((x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (A.e. x (NN) (e. A (CC)))) (-> (e. B (NN)) (e. (` (opr (+) (seq1) F) B) (CC))))
+stmt (ser1f2 ((x y) (x z) (y z) (A y) (F z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (A.e. x (NN) (e. A (CC)))) (:--> (opr (+) (seq1) F) (NN) (CC)))
+stmt (ser112 ((B y) (x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (e. B (V)) (-> (= (cv x) (1)) (= A B))) (= (` (opr (+) (seq1) F) (1)) B))
+stmt (ser1suc2 ((B y) (x y) (C x) (C y) (x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (e. C (V)) (-> (= (cv x) (opr B (+) (1))) (= A C))) (-> (e. B (NN)) (= (` (opr (+) (seq1) F) (opr B (+) (1))) (opr (` (opr (+) (seq1) F) B) (+) C))))
+stmt (ser1mono ((A y) (x y) (F x) (F y)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) F) A) (<_) (` (opr (+) (seq1) F) (opr A (+) (1))))))
+stmt (ser1mulc ((x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z) (A x) (A y) (A z) (B y) (B z)) ((e. A (CC)) (:--> F (NN) (CC)) (e. G (V)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x)))))) (-> (e. B (NN)) (= (` (opr (+) (seq1) G) B) (opr A (x.) (` (opr (+) (seq1) F) B)))))
+stmt (ser1add2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (e. H (V)) (-> (/\/\ (e. (cv k) (NN)) (e. N (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))))
+stmt (ser1add ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (e. H (V)) (-> (/\ (e. (cv k) (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))))
+stmt (ser1addOLD ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N m)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (e. H (V)) (-> (e. (cv k) (NN)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N)))))
+stmt (ser1dir ((F k) (G k) (H k) (R k) (S k) (T k)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (:--> H (NN) (CC)) (e. R (V)) (Fn S (NN)) (Fn T (NN)) (-> (e. (cv k) (NN)) (= (` R (cv k)) (opr (opr (` F (cv k)) (+) (` G (cv k))) (x.) (` H (cv k))))) (-> (e. (cv k) (NN)) (= (` S (cv k)) (opr (` F (cv k)) (x.) (` H (cv k))))) (-> (e. (cv k) (NN)) (= (` T (cv k)) (opr (` G (cv k)) (x.) (` H (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) R) N) (opr (` (opr (+) (seq1) S) N) (+) (` (opr (+) (seq1) T) N)))))
+stmt (shftfval ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (F x) (F y) (F w) (F f) (F g) (B x) (B y)) ((e. F (V))) (-> (e. A B) (= (opr F (shift) A) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A))))))))
+stmt (shftfn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((e. F (V))) (-> (e. A B) (Fn (opr F (shift) A) (CC))))
+stmt (shftres () ((e. F (V))) (-> (/\ (e. A C) (C_ B (CC))) (Fn (|` (opr F (shift) A) B) B)))
+stmt (shftresvalt () ((e. F (V))) (-> (e. B C) (= (` (|` (opr F (shift) A) C) B) (` (opr F (shift) A) B))))
+stmt (shftvalt ((x y) (A x) (A y) (F x) (F y) (B x) (B y) (C x) (C y)) ((e. F (V))) (-> (/\ (e. A C) (e. B (CC))) (= (` (opr F (shift) A) B) (` F (opr B (-) A)))))
+stmt (shftval2t () ((e. F (V))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (` (opr F (shift) (opr A (-) B)) (opr A (+) C)) (` F (opr B (+) C)))))
+stmt (shftval3t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (opr A (-) B)) A) (` F B))))
+stmt (shftval4t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (-u A)) B) (` F (opr A (+) B)))))
+stmt (shftval5t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) A) (opr B (+) A)) (` F B))))
+stmt (shftf ((A x) (F x) (B x) (C x)) ((e. F (V))) (-> (/\/\ (e. A D) (C_ B (CC)) (A.e. x B (e. (` F (opr (cv x) (-) A)) C))) (:--> (|` (opr F (shift) A) B) B C)))
+stmt (2shft ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr F (shift) A) (shift) B) (opr F (shift) (opr A (+) B)))))
+stmt (shftcan2t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) (-u A)) (shift) A) B) (` F B))))
+stmt (shftcan1t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) A) (shift) (-u A)) B) (` F B))))
+stmt (shftidt () ((e. F (V))) (-> (e. A (CC)) (= (` (opr F (shift) (0)) A) (` F A))))
+stmt (seq1shftid ((j k) (j n) (S j) (k n) (S k) (S n) (F j) (F k) (F n)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) (opr F (shift) (0))) (opr S (seq1) F)))
+stmt (uzvalt ((j k) (j y) (N j) (k y) (N k) (N y)) () (-> (e. N (ZZ)) (= (` (ZZ>) N) ({e.|} k (ZZ) (br N (<_) (cv k))))))
+stmt (eluz1t ((N k) (M k)) () (-> (e. M (ZZ)) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N)))))
+stmt (eluz2t ((j k) (j y) (N j) (k y) (N k) (N y) (M k)) () (<-> (e. N (` (ZZ>) M)) (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N))))
+stmt (eluz1 () ((e. M (ZZ))) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N))))
+stmt (eluzelz () () (-> (e. N (` (ZZ>) M)) (e. N (ZZ))))
+stmt (eluzel2 () () (-> (e. N (` (ZZ>) M)) (e. M (ZZ))))
+stmt (eluzle () () (-> (e. N (` (ZZ>) M)) (br M (<_) N)))
+stmt (eluzt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. N (` (ZZ>) M)) (br M (<_) N))))
+stmt (uzidt () () (-> (e. M (ZZ)) (e. M (` (ZZ>) M))))
+stmt (uztrn () () (-> (/\ (e. M (` (ZZ>) K)) (e. K (` (ZZ>) N))) (e. M (` (ZZ>) N))))
+stmt (uzssz ((j k) (j y) (M j) (k y) (M k) (M y)) () (C_ (` (ZZ>) M) (ZZ)))
+stmt (uzss ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (C_ (` (ZZ>) N) (` (ZZ>) M))))
+stmt (uzssOLD ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (C_ (` (ZZ>) N) (` (ZZ>) M))))
+stmt (eluzp1m1t () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. (opr N (-) (1)) (` (ZZ>) M))))
+stmt (eluzp1lt () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (br M (<) N)))
+stmt (eluzp1p1t () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) (opr M (+) (1))))))
+stmt (nnuz () () (= (NN) (` (ZZ>) (1))))
+stmt (nn0uz () () (= (NN0) (` (ZZ>) (0))))
+stmt (raluz ((M x)) () (-> (e. M (ZZ)) (<-> (A.e. x (` (ZZ>) M) ph) (A.e. x (ZZ) (-> (br M (<_) (cv x)) ph)))))
+stmt (raluz2 ((M x)) () (<-> (A.e. x (` (ZZ>) M) ph) (-> (e. M (ZZ)) (A.e. x (ZZ) (-> (br M (<_) (cv x)) ph)))))
+stmt (rexuz ((M x)) () (-> (e. M (ZZ)) (<-> (E.e. x (` (ZZ>) M) ph) (E.e. x (ZZ) (/\ (br M (<_) (cv x)) ph)))))
+stmt (rexuz2 ((M x)) () (<-> (E.e. x (` (ZZ>) M) ph) (/\ (e. M (ZZ)) (E.e. x (ZZ) (/\ (br M (<_) (cv x)) ph)))))
+stmt (peano2uz () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) M))))
+stmt (uzind4 ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (e. (cv k) (` (ZZ>) M)) (-> ch th))) (-> (e. N (` (ZZ>) M)) ta))
+stmt (uzind4i ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((e. M (ZZ)) (-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) ps (-> (e. (cv k) (` (ZZ>) M)) (-> ch th))) (-> (e. N (` (ZZ>) M)) ta))
+stmt (uzwo ((h t) (h n) (h m) (M h) (n t) (m t) (M t) (m n) (M n) (M m) (h j) (h k) (S h) (j t) (j k) (j m) (j n) (S j) (k t) (S t) (k m) (k n) (S k) (S m) (S n)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E.e. j S (A.e. k S (br (cv j) (<_) (cv k))))))
+stmt (uzinfm ((j k) (M j) (M k)) ((e. M (ZZ))) (= (sup (` (ZZ>) M) (RR) (`' (<))) M))
+stmt (nninfm () () (= (sup (NN) (RR) (`' (<))) (1)))
+stmt (nn0infm () () (= (sup (NN0) (RR) (`' (<))) (0)))
+stmt (seq0fval ((f g) (f h) (S f) (g h) (S g) (S h) (F f) (F g) (F h)) ((e. S (V)) (e. F (V))) (= (opr S (seq0) F) (|` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0))))
+stmt (seq0valt () ((e. S (V)) (e. F (V))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) N) (` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) N))))
+stmt (seqzfval ((g h) (g k) (g x) (S g) (h k) (h x) (S h) (k x) (S k) (S x) (F g) (F h) (F k) (F x) (A k) (M g) (M h) (M k) (M x)) ((e. S (V)) (e. F (V))) (-> (e. M A) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) ({e.|} k (ZZ) (br M (<_) (cv k)))))))
+stmt (seqzfval2 ((S k) (F k) (M k)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (` (ZZ>) M)))))
+stmt (seqzfn () ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (Fn (opr (<,> M S) (seq) F) (` (ZZ>) M))))
+stmt (seqzvalt ((S k) (F k) (A k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M A) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) N))))
+stmt (seq1seqz ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) F) (opr (<,> (1) S) (seq) F)))
+stmt (seq0seqz ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq0) F) (opr (<,> (0) S) (seq) F)))
+stmt (seq1seq02t () ((e. S (V)) (e. F (V))) (-> (e. N (NN)) (= (` (opr S (seq1) (opr F (shift) (1))) N) (` (opr (opr S (seq0) F) (shift) (1)) N))))
+stmt (seq1seq0t () ((e. S (V)) (e. F (V))) (-> (e. N (NN)) (= (` (opr S (seq1) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) N))))
+stmt (seq1seq0 ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) F) (|` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) (NN))))
+stmt (seq0fn () ((e. S (V)) (e. F (V))) (Fn (opr S (seq0) F) (NN0)))
+stmt (seqz1 ((S k) (F k) (M k)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (` (opr (<,> M S) (seq) F) M) (` F M))))
+stmt (seqzp1 ((S k) (F k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))))
+stmt (seqzp1OLD ((S k) (F k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1)))))))
+stmt (seqzm1 () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (` (opr (<,> M S) (seq) F) N) (opr (` (opr (<,> M S) (seq) F) (opr N (-) (1))) S (` F N)))))
+stmt (seq00 () ((e. S (V)) (e. F (V))) (= (` (opr S (seq0) F) (0)) (` F (0))))
+stmt (seq0p1 () ((e. S (V)) (e. F (V))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) (opr N (+) (1))) (opr (` (opr S (seq0) F) N) S (` F (opr N (+) (1)))))))
+stmt (seq01 () ((e. S (V)) (e. F (V))) (= (` (opr S (seq0) F) (1)) (opr (` F (0)) S (` F (1)))))
+stmt (seqzval2t () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u M))) (shift) M) N))))
+stmt (seqzrn ((j k) (j n) (j v) (j w) (B j) (k n) (k v) (k w) (B k) (n v) (n w) (B n) (v w) (B v) (B w) (C j) (C k) (C n) (C v) (C w) (F j) (F k) (F n) (F v) (F w) (M j) (M k) (M n) (M v) (M w) (S j) (S k) (S n) (S v) (S w)) ((e. S (V)) (e. F (V))) (-> (/\ (/\ (e. M (ZZ)) (e. (` F M) C)) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) (C_ (ran (opr (<,> M S) (seq) F)) C)))
+stmt (seqzrn2 () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)))
+stmt (seqzcl () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. N (` (ZZ>) M)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C)))
+stmt (seqzrnlemOLD () ((e. S (V)) (e. F (V)) (e. M (ZZ)) (= D (` (ZZ>) M))) (-> (e. N D) (e. (opr N (+) (1)) (\ D ({} M)))))
+stmt (seqzrnOLD ((j k) (j n) (j v) (j w) (B j) (k n) (k v) (k w) (B k) (n v) (n w) (B n) (v w) (B v) (B w) (C j) (C k) (C n) (C v) (C w) (D j) (D k) (D n) (F j) (F k) (F n) (F v) (F w) (M j) (M k) (M n) (M v) (M w) (S j) (S k) (S n) (S v) (S w)) ((e. S (V)) (e. F (V)) (e. M (ZZ)) (= D (` (ZZ>) M))) (-> (/\/\ (e. (` F M) C) (:--> (|` F (\ D ({} M))) (\ D ({} M)) B) (:--> S (X. C B) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)))
+stmt (seqzrn2OLD () ((e. S (V)) (e. F (V)) (e. M (ZZ)) (= D (` (ZZ>) M))) (-> (/\ (:--> F D C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C)))
+stmt (seqzclOLD () ((e. S (V)) (e. F (V)) (e. M (ZZ)) (= D (` (ZZ>) M))) (-> (/\/\ (e. N D) (:--> F D C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C)))
+stmt (seqzres ((j k) (j n) (k n) (D j) (D k) (D n) (F j) (F k) (F n) (M j) (M k) (M n) (S j) (S k) (S n)) ((e. S (V)) (e. F (V)) (e. M (ZZ)) (= D (` (ZZ>) M))) (= (opr (<,> M S) (seq) (|` F D)) (opr (<,> M S) (seq) F)))
+stmt (serzcl1 () ((:--> F (` (ZZ>) M) (CC))) (-> (e. N (` (ZZ>) M)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))))
+stmt (serzcl () ((e. M (ZZ)) (:--> F (` (ZZ>) M) (CC))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC))))
+stmt (serzf ((M k) (F k)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (CC))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (CC)))
+stmt (serzrecl ((j k) (F j) (F k) (M j) (M k) (N j)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (RR))))
+stmt (serzref ((F k) (M k)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (RR)))
+stmt (dfseq0 ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (seq0) ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g))))))
+stmt (serzmulc ((j k) (j m) (C j) (k m) (C k) (C m) (F j) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((e. M (ZZ)) (e. C (CC)) (:--> F (` (ZZ>) M) (CC)) (e. G (V)) (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (= (` G (cv k)) (opr C (x.) (` F (cv k)))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M (+)) (seq) G) N) (opr C (x.) (` (opr (<,> M (+)) (seq) F) N)))))
+stmt (ser0mulc ((j k) (F j) (F k) (G j) (G k) (C j) (C k)) ((e. C (CC)) (:--> F (NN0) (CC)) (e. G (V)) (-> (e. (cv k) (NN0)) (= (` G (cv k)) (opr C (x.) (` F (cv k)))))) (-> (e. N (NN0)) (= (` (opr (+) (seq0) G) N) (opr C (x.) (` (opr (+) (seq0) F) N)))))
+stmt (expnn0valt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CC)) (e. B (NN0))) (= (opr A (^) B) (if (= B (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} A))) B)))))
+stmt (expnn00t () () (-> (e. A (CC)) (= (opr A (^) (0)) (1))))
+stmt (expvalt () () (-> (/\ (e. A (CC)) (e. B (NN))) (= (opr A (^) B) (` (opr (x.) (seq1) (X. (NN) ({} A))) B))))
+stmt (expp1t () () (-> (/\ (e. A (CC)) (e. B (NN))) (= (opr A (^) (opr B (+) (1))) (opr (opr A (^) B) (x.) A))))
+stmt (exp1t () () (-> (e. A (CC)) (= (opr A (^) (1)) A)))
+stmt (1expt ((x y) (A x) (A y)) () (-> (e. A (NN)) (= (opr (1) (^) A) (1))))
+stmt (expcllem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (F x) (F y) (F z) (F w)) ((C_ F (CC)) (-> (/\ (e. (cv x) F) (e. (cv y) F)) (e. (opr (cv x) (x.) (cv y)) F))) (-> (/\ (e. A F) (e. B (NN))) (e. (opr A (^) B) F)))
+stmt (nnexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (NN)) (e. B (NN))) (e. (opr A (^) B) (NN))))
+stmt (nn0expclt ((x y) (A x) (A y)) () (-> (/\ (e. A (NN0)) (e. B (NN))) (e. (opr A (^) B) (NN0))))
+stmt (zexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (^) B) (ZZ))))
+stmt (qexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (QQ)) (e. B (NN))) (e. (opr A (^) B) (QQ))))
+stmt (reexpclt ((x y) (A x) (A y)) () (-> (/\ (e. A (RR)) (e. B (NN))) (e. (opr A (^) B) (RR))))
+stmt (expclt ((x y) (A x) (A y)) () (-> (/\ (e. A (CC)) (e. B (NN))) (e. (opr A (^) B) (CC))))
+stmt (expnn0p1t () () (-> (/\ (e. A (CC)) (e. B (NN0))) (= (opr A (^) (opr B (+) (1))) (opr (opr A (^) B) (x.) A))))
+stmt (expnn0clt () () (-> (/\ (e. A (CC)) (e. B (NN0))) (e. (opr A (^) B) (CC))))
+stmt (expeq0t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (NN))) (<-> (= (opr A (^) B) (0)) (= A (0)))))
+stmt (expne0t () () (-> (/\ (e. A (CC)) (e. B (NN))) (<-> (=/= A (0)) (=/= (opr A (^) B) (0)))))
+stmt (expgt0t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (0) (<) A)) (br (0) (<) (opr A (^) B))))
+stmt (0expt () () (-> (e. A (NN)) (= (opr (0) (^) A) (0))))
+stmt (expge0t () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (0) (<_) A)) (br (0) (<_) (opr A (^) B))))
+stmt (expgt1t ((x y) (A x) (A y) (B x)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (1) (<) A)) (br (1) (<) (opr A (^) B))))
+stmt (expge1t () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (1) (<_) A)) (br (1) (<_) (opr A (^) B))))
+stmt (mulexpt ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (NN))) (= (opr (opr A (x.) B) (^) C) (opr (opr A (^) C) (x.) (opr B (^) C)))))
+stmt (recexpt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) B) (opr (1) (/) (opr A (^) B)))))
+stmt (expaddt ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (= (opr A (^) (opr B (+) C)) (opr (opr A (^) B) (x.) (opr A (^) C)))))
+stmt (expmult ((x y) (A x) (A y) (B x) (B y) (C x)) () (-> (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (= (opr A (^) (opr B (x.) C)) (opr (opr A (^) B) (^) C))))
+stmt (expsubt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (NN)) (e. C (NN))) (/\ (=/= A (0)) (br C (<) B))) (= (opr A (^) (opr B (-) C)) (opr (opr A (^) B) (/) (opr A (^) C)))))
+stmt (divexpt () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (NN))) (=/= B (0))) (= (opr (opr A (/) B) (^) C) (opr (opr A (^) C) (/) (opr B (^) C)))))
+stmt (expordit () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (/\ (br (1) (<) A) (br B (<) C))) (br (opr A (^) B) (<) (opr A (^) C))))
+stmt (expcant () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (br (1) (<) A)) (<-> (= (opr A (^) B) (opr A (^) C)) (= B C))))
+stmt (expordt () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN)) (e. N (NN))) (br (1) (<) A)) (<-> (br M (<) N) (br (opr A (^) M) (<) (opr A (^) N)))))
+stmt (explt1t () () (-> (/\ (/\/\ (e. A (RR)) (e. B (NN)) (e. C (NN))) (/\ (br (0) (<) A) (br A (<) (1)))) (<-> (br B (<) C) (br (opr A (^) C) (<) (opr A (^) B)))))
+stmt (sqvalt () () (-> (e. A (CC)) (= (opr A (^) (2)) (opr A (x.) A))))
+stmt (sqclt () () (-> (e. A (CC)) (e. (opr A (^) (2)) (CC))))
+stmt (sqval () ((e. A (CC))) (= (opr A (^) (2)) (opr A (x.) A)))
+stmt (sqcl () ((e. A (CC))) (e. (opr A (^) (2)) (CC)))
+stmt (sqe0 () ((e. A (CC))) (<-> (= (opr A (^) (2)) (0)) (= A (0))))
+stmt (sqmul () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (x.) B) (^) (2)) (opr (opr A (^) (2)) (x.) (opr B (^) (2)))))
+stmt (sqdiv () ((e. A (CC)) (e. B (CC)) (=/= B (0))) (= (opr (opr A (/) B) (^) (2)) (opr (opr A (^) (2)) (/) (opr B (^) (2)))))
+stmt (sqreci () ((e. A (CC)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) (2)) (opr (1) (/) (opr A (^) (2)))))
+stmt (sqreclt () () (-> (e. A (RR)) (e. (opr A (^) (2)) (RR))))
+stmt (sqrecl () ((e. A (RR))) (e. (opr A (^) (2)) (RR)))
+stmt (lt2sqe () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2))))))
+stmt (le2sqe () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2))))))
+stmt (sqe11 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B))))
+stmt (sqegt0 () ((e. A (RR))) (-> (=/= A (0)) (br (0) (<) (opr A (^) (2)))))
+stmt (sqege0 () ((e. A (RR))) (br (0) (<_) (opr A (^) (2))))
+stmt (sqe11t () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B)))))
+stmt (lt2sqet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2)))))))
+stmt (le2sqet () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2)))))))
+stmt (sqege0t () () (-> (e. A (RR)) (br (0) (<_) (opr A (^) (2)))))
+stmt (sumsqne0 () ((e. A (RR)) (e. B (RR))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr (opr A (^) (2)) (+) (opr B (^) (2))) (0))))
+stmt (sq1 () () (= (opr (1) (^) (2)) (1)))
+stmt (sq2 () () (= (opr (2) (^) (2)) (4)))
+stmt (cu2 () () (= (opr (2) (^) (3)) (8)))
+stmt (binom2 () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (^) (2)) (opr (opr (opr A (^) (2)) (+) (opr (2) (x.) (opr A (x.) B))) (+) (opr B (^) (2)))))
+stmt (binom2a () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (x.) (opr A (-) B)) (opr (opr A (^) (2)) (-) (opr B (^) (2)))))
+stmt (bernneq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) B)) (<_) (opr (opr (1) (+) A) (^) B))))
+stmt (discrlem1 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (-u (opr B (/) (opr (2) (x.) A)))) (br (0) (<) A)) (<-> (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))))
+stmt (discrlem2 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (-u (opr B (/) (opr (2) (x.) A)))) (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)))) (-> (br (0) (<) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))))
+stmt (discrlem3 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (opr (opr C (+) (1)) (/) (-u B))) (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)))) (-> (= (0) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))))
+stmt (discrlem ((A x) (B x) (C x)) ((e. A (RR)) (e. B (RR)) (e. C (RR)) (A.e. x (RR) (br (0) (<_) (opr (opr (opr A (x.) (opr (cv x) (^) (2))) (+) (opr B (x.) (cv x))) (+) C)))) (-> (br (0) (<_) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0))))
+stmt (nnsqcl () ((e. A (NN))) (e. (opr A (^) (2)) (NN)))
+stmt (nnlesq () ((e. A (NN))) (br A (<_) (opr A (^) (2))))
+stmt (nneo ((x y) (A x) (A y)) ((e. A (NN))) (<-> (e. (opr A (/) (2)) (NN)) (-. (e. (opr (opr A (+) (1)) (/) (2)) (NN)))))
+stmt (nnesq () ((e. A (NN))) (<-> (e. (opr A (/) (2)) (NN)) (e. (opr (opr A (^) (2)) (/) (2)) (NN))))
+stmt (nn0le2sqet () ((e. A (NN0)) (e. B (NN0))) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B))))
+stmt (nn0opthlem1 () ((e. A (NN0)) (e. C (NN0))) (<-> (br A (<) C) (br (opr (opr A (x.) A) (+) (opr (2) (x.) A)) (<) (opr C (x.) C))))
+stmt (nn0opthlem2 () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (-> (/\ (br B (<_) A) (br D (<_) C)) (-> (br A (<) C) (-. (= (opr (opr A (x.) A) (+) B) (opr (opr C (x.) C) (+) D))))))
+stmt (nn0opth () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (<-> (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) (/\ (= A C) (= B D))))
+stmt (nn0opth2 () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D))))
+stmt (nn0opth2t () () (-> (/\ (/\ (e. A (NN0)) (e. B (NN0))) (/\ (e. C (NN0)) (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D)))))
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+stmt (sqr0 () () (= (` (sqr) (0)) (0)))
+stmt (sqrlem1 () ((e. A (RR)) (br (0) (<) A)) (br A (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A))))
+stmt (sqrlem2 () ((e. A (RR)) (br (0) (<) A)) (br (opr (opr A (/) (opr (1) (+) A)) (x.) (opr A (/) (opr (1) (+) A))) (<) A))
+stmt (sqrlem3 () ((e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr A (/) (opr (1) (+) A))))
+stmt (sqrlem4 ((A x) (S x) (D x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (<-> (e. D S) (/\ (e. D (RR)) (/\ (br (0) (<_) D) (br (opr D (x.) D) (<_) A)))))
+stmt (sqrlem5 ((A x) (S x) (D x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (/\ (e. D (RR)) (/\ (br (0) (<) D) (br (opr D (x.) D) (<) A))) (e. D S)))
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+stmt (eluzfzt () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))) (e. K (opr M (...) N))))
+stmt (elfzel2 () ((e. N (V))) (-> (e. K (opr M (...) N)) (e. N (ZZ))))
+stmt (elfzel1 () () (-> (e. K (opr M (...) N)) (e. M (ZZ))))
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+stmt (eluzfz1t () () (-> (e. N (` (ZZ>) M)) (e. M (opr M (...) N))))
+stmt (eluzfz2t () () (-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))))
+stmt (eluzfz2tOLD () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. N (opr M (...) N))))
+stmt (eluzfz2b () () (<-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N))))
+stmt (elfz3t () () (-> (e. N (ZZ)) (e. N (opr N (...) N))))
+stmt (elfz1eqt () () (-> (e. K (opr N (...) N)) (= K N)))
+stmt (elfzle3 () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) N))) (br M (<_) N)))
+stmt (elfznn0t () () (-> (e. K (opr (0) (...) N)) (e. K (NN0))))
+stmt (fzaddelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (+) K) (opr (opr M (+) K) (...) (opr N (+) K))))))
+stmt (fzsubelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (-) K) (opr (opr M (-) K) (...) (opr N (-) K))))))
+stmt (fzss1t ((M k) (N k) (K k)) () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (C_ (opr K (...) N) (opr M (...) N))))
+stmt (fzss1tOLD ((M k) (N k) (K k)) () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (br M (<_) K))) (C_ (opr K (...) N) (opr M (...) N))))
+stmt (fzss2tOLD ((M k) (N k) (K k)) () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. K (ZZ)) (br K (<_) N))) (C_ (opr M (...) K) (opr M (...) N))))
+stmt (fzssp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr M (...) N) (opr M (...) (opr N (+) (1))))))
+stmt (fzp1sst () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr (opr M (+) (1)) (...) N) (opr M (...) N))))
+stmt (elfzp1 () () (-> (e. N (` (ZZ>) M)) (<-> (e. K (opr M (...) (opr N (+) (1)))) (\/ (e. K (opr M (...) N)) (= K (opr N (+) (1)))))))
+stmt (fznn0t () () (-> (e. N (NN0)) (<-> (e. K (opr (0) (...) N)) (/\ (e. K (NN0)) (br K (<_) N)))))
+stmt (fsequb ((j k) (j m) (j n) (j x) (j y) (F j) (k m) (k n) (k x) (k y) (F k) (m n) (m x) (m y) (F m) (n x) (n y) (F n) (x y) (F x) (F y) (M j) (M k) (M m) (M n) (M x) (M y) (N j) (N k) (N n) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (E.e. x (RR) (A.e. k (opr M (...) N) (br (` F (cv k)) (<) (cv x))))))
+stmt (fsumvalt ((f x) (x z) (M x) (f z) (M f) (M z) (N x) (N f) (N z) (F x) (F f) (F z)) () (-> (/\/\ (e. F (V)) (e. M A) (e. N B)) (= (opr (<,> M N) (sum) F) (if (br M (<_) N) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)) (0)))))
+stmt (fsumalt () () (= (opr (<,> M N) (sum) F) (` (sum) (<,> (<,> M N) F))))
+stmt (fsumval2t () ((e. F (V))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M N) (sum) F) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)))))
+stmt (fsumval2tOLD () ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M N) (sum) F) (` (opr (+) (seq0) (opr F (shift) (-u M))) (opr N (-) M)))))
+stmt (fsumval3t () ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br N (<) M)) (= (opr (<,> M N) (sum) F) (0))))
+stmt (fsum1 () ((e. F (V))) (-> (e. M (ZZ)) (= (opr (<,> M M) (sum) F) (` F M))))
+stmt (fsump1 () ((e. F (V))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M (opr N (+) (1))) (sum) F) (opr (opr (<,> M N) (sum) F) (+) (` F (opr N (+) (1)))))))
+stmt (fsump1OLD () ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M (opr N (+) (1))) (sum) F) (opr (opr (<,> M N) (sum) F) (+) (` F (opr N (+) (1)))))))
+stmt (fsumserz ((F j) (F k) (j k) (M j) (M k) (N j)) ((e. F (V))) (-> (e. N (` (ZZ>) M)) (= (opr (<,> M N) (sum) F) (` (opr (<,> M (+)) (seq) F) N))))
+stmt (fsumserzOLD ((F j) (F k) (j k) (M j) (M k) (N j)) ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (opr (<,> M N) (sum) F) (` (opr (<,> M (+)) (seq) F) N))))
+stmt (fsumser0 () ((e. F (V))) (-> (e. N (NN0)) (= (opr (<,> (0) N) (sum) F) (` (opr (+) (seq0) F) N))))
+stmt (fsumm1 () ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (opr (<,> M N) (sum) F) (opr (opr (<,> M (opr N (-) (1))) (sum) F) (+) (` F N)))))
+stmt (fsumclt ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((e. F (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (opr (<,> M N) (sum) F) (CC))))
+stmt (fsumcltOLD ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((e. F (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (opr (<,> M N) (sum) F) (CC))))
+stmt (fsumreclt ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m)) ((e. F (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (e. (opr (<,> M N) (sum) F) (RR))))
+stmt (fsumnn0cl ((F k) (N k)) ((e. F (V)) (-> (/\/\ (e. (cv k) (NN0)) (e. N (NN0)) (br (cv k) (<_) N)) (e. (` F (cv k)) (CC)))) (-> (e. N (NN0)) (e. (opr (<,> (0) N) (sum) F) (CC))))
+stmt (fsum1p ((j k) (j m) (F j) (k m) (F k) (F m) (M j) (M k) (M m) (N j) (N k)) ((e. F (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (opr (<,> M N) (sum) F) (opr (` F M) (+) (opr (<,> (opr M (+) (1)) N) (sum) F)))))
+stmt (fsumsplit ((j k) (j m) (F j) (k m) (F k) (F m) (M j) (M k) (M m) (K j) (N j) (N k) (N m)) ((e. F (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1))))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (opr (<,> M N) (sum) F) (opr (opr (<,> M K) (sum) F) (+) (opr (<,> (opr K (+) (1)) N) (sum) F)))))
+stmt (fsumadd ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (H j) (H k) (H m)) ((e. F (V)) (e. G (V)) (e. H (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (= (opr (<,> M N) (sum) H) (opr (opr (<,> M N) (sum) F) (+) (opr (<,> M N) (sum) G)))))
+stmt (fsumeq ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((e. F (V)) (e. G (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (cv k))))) (= (opr (<,> M N) (sum) G) (opr (<,> M N) (sum) F))))
+stmt (fsummulc ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (A j) (A k) (A m)) ((e. F (V)) (e. G (V)) (e. A (CC))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (x.) (` F (cv k))))))) (= (opr (<,> M N) (sum) G) (opr A (x.) (opr (<,> M N) (sum) F)))))
+stmt (fsumshft1 ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((e. F (V)) (e. G (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (opr (cv k) (+) (1)))))) (= (opr (<,> M N) (sum) G) (opr (<,> (opr M (+) (1)) (opr N (+) (1))) (sum) F))))
+stmt (fsumshftm1 ((j k) (M j) (M k) (N j) (N k) (F j) (F k) (G j) (G k)) ((e. F (V)) (e. G (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (= (` G (cv k)) (` F (opr (cv k) (-) (1)))))) (= (opr (<,> M N) (sum) G) (opr (<,> (opr M (-) (1)) (opr N (-) (1))) (sum) F))))
+stmt (fsumabs ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m)) ((e. F (V)) (e. G (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (` (abs) (` F (cv k))))))) (br (` (abs) (opr (<,> M N) (sum) F)) (<_) (opr (<,> M N) (sum) G))))
+stmt (fsumub ((j k) (j m) (M j) (k m) (M k) (M m) (N j) (N k) (F j) (F k) (F m) (G j) (G k) (G m) (A j) (A k) (A m)) ((e. F (V)) (e. G (V)) (e. A (RR))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (RR)) (br (` F (cv k)) (<_) A)))) (br (opr (<,> M N) (sum) F) (<_) (opr (opr (opr N (-) M) (+) (1)) (x.) A))))
+stmt (serz0 ((j k) (M j) (M k) (N j)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M (+)) (seq) (X. (` (ZZ>) M) ({} (0)))) N) (0))))
+stmt (serzcmp ((j k) (j m) (F j) (k m) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR)) (:--> G (` (ZZ>) M) (RR)) (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (br (` G (cv k)) (<_) (` F (cv k))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (br (` (opr (<,> M (+)) (seq) G) N) (<_) (` (opr (<,> M (+)) (seq) F) N))))
+stmt (serzcmp0 ((F k) (M k)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR)) (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (br (0) (<_) (` F (cv k))))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (br (0) (<_) (` (opr (<,> M (+)) (seq) F) N))))
+stmt (ser0split ((F k) (M k) (N k)) ((e. F (V))) (-> (/\ (/\/\ (e. M (NN0)) (e. N (NN0)) (br (0) (<) M)) (A.e. k (opr (0) (...) (opr M (+) N)) (e. (` F (cv k)) (CC)))) (= (` (opr (opr (+) (seq0) F) (shift) (-u M)) N) (opr (` (opr (+) (seq0) F) (opr M (-) (1))) (+) (` (opr (+) (seq0) (opr F (shift) (-u M))) N)))))
+stmt (serzsplit ((F k) (M k) (N k)) ((e. F (V))) (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1))))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` (opr (<,> M (+)) (seq) F) K) (+) (` (opr (<,> (opr K (+) (1)) (+)) (seq) F) N)))))
+stmt (climrelOLD ((x y) (x z) (f x) (w x) (y z) (f y) (w y) (f z) (w z) (f w)) () (Rel (~~>OLD)))
+stmt (climrel ((j k) (j x) (j y) (f j) (k x) (k y) (f k) (x y) (f x) (f y)) () (Rel (~~>)))
+stmt (clim ((j k) (j x) (j y) (f j) (F j) (k x) (k y) (f k) (F k) (x y) (f x) (F x) (f y) (F y) (F f) (A j) (A k) (A x) (A y) (A f)) () (-> (/\ (e. F C) (e. A D)) (<-> (br F (~~>) A) (/\ (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))))
+stmt (climuz ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) () (-> (/\ (e. F C) (e. A D)) (<-> (br F (~~>) A) (/\ (e. A (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))))
+stmt (climcl ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) () (-> (/\ (e. A C) (br F (~~>) A)) (e. A (CC))))
+stmt (climcvgc1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x) (B j) (B k) (B x)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))
+stmt (climcvg1 ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))
+stmt (climcvg1z ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))
+stmt (clim1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((e. F (V))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))))
+stmt (clim1z ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((e. F (V))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))
+stmt (clim1a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x)) ((e. F (V))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))
+stmt (clim1az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x)) ((e. F (V))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (` (ZZ>) (cv m)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))
+stmt (climcvg2 ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) ((e. M (ZZ))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))
+stmt (climcvg2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))
+stmt (climcvgc2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))
+stmt (clim2az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (` (ZZ>) (cv m)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))
+stmt (clim3a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (N j) (N k) (N m) (N n) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))
+stmt (clim3 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))
+stmt (climcvg4 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) ((e. M (ZZ))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))
+stmt (climcvgc4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))
+stmt (climOLD ((x y) (x z) (w x) (f x) (F x) (y z) (w y) (f y) (F y) (w z) (f z) (F z) (f w) (F w) (F f) (A x) (A y) (A z) (A w) (A f)) ((e. A (V))) (<-> (br F (~~>OLD) A) (/\ (/\ (:--> F (NN) (CC)) (e. A (CC))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x))))))))))
+stmt (climseqOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((e. A (V))) (-> (br F (~~>OLD) A) (:--> F (NN) (CC))))
+stmt (climclOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((e. A (V))) (-> (br F (~~>OLD) A) (e. A (CC))))
+stmt (climcvgOLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((e. A (V))) (-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x)))))))))
+stmt (climcvg2OLD ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z)) ((e. A (V))) (-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x)))))))))
+stmt (climfvclOLD () ((e. A (V))) (-> (/\ (br F (~~>OLD) A) (e. B (NN))) (e. (` F B) (CC))))
+stmt (climcltOLD ((F x) (A x)) () (-> (/\ (e. A B) (br F (~~>OLD) A)) (e. A (CC))))
+stmt (clim2OLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w)) () (-> (/\ (:--> F (NN) (CC)) (e. A (CC))) (<-> (br F (~~>OLD) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) A)) (<) (cv x))))))))))
+stmt (climconst ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (N j) (N k) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (= (` F (cv k)) A)))) (br F (~~>) A)))
+stmt (climconst2 ((A k)) () (-> (e. A (CC)) (br (X. (ZZ) ({} A)) (~~>) A)))
+stmt (climconstOLD ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (CC))) (br (X. (NN) ({} A)) (~~>OLD) A))
+stmt (clim0 () () (br (X. (ZZ) ({} (0))) (~~>) (0)))
+stmt (clim0OLD () () (br (X. (NN) ({} (0))) (~~>OLD) (0)))
+stmt (climunii ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (F j) (F k) (F m)) ((e. A (V)) (e. B (V)) (/\ (br F (~~>) A) (br F (~~>) B))) (= A B))
+stmt (climuni () ((e. A (V)) (e. B (V))) (-> (/\ (br F (~~>) A) (br F (~~>) B)) (= A B)))
+stmt (climuniiOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((e. A (V)) (e. B (V)) (/\ (br F (~~>OLD) A) (br F (~~>OLD) B))) (= A B))
+stmt (climuniOLD () ((e. A (V)) (e. B (V))) (-> (/\ (br F (~~>OLD) A) (br F (~~>OLD) B)) (= A B)))
+stmt (climmoOLD ((x y) (F x) (F y)) () (E* x (br F (~~>OLD) (cv x))))
+stmt (climshiftOLD ((y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (w x) (v x) (u x) (F x) (F y) (F w) (F v) (F u) (x z) (G x) (G y) (G z) (G w) (G v) (B x) (B y) (B z) (B w) (B v) (B u)) ((e. A (V))) (-> (/\/\ (:--> G (NN) (CC)) (e. B (NN0)) (A.e. x (NN) (-> (br B (<) (cv x)) (= (` G (cv x)) (` F (opr (cv x) (-) B)))))) (-> (br F (~~>OLD) A) (br G (~~>OLD) A))))
+stmt (climreOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (G x) (G y) (G z) (G w) (A y) (A z) (A w)) ((e. A (V)) (br F (~~>OLD) A) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x)))))) (br G (~~>OLD) (` (Re) A)))
+stmt (climrecl ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((e. A (V)) (e. M (ZZ)) (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (e. (` F (cv k)) (RR))) (br F (~~>) A)) (e. A (RR)))
+stmt (climreclOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F x) (F y) (F z)) ((e. A (V)) (:--> F (NN) (RR)) (br F (~~>OLD) A)) (e. A (RR)))
+stmt (climrecltOLD () ((e. A (V))) (-> (/\ (:--> F (NN) (RR)) (br F (~~>OLD) A)) (e. A (RR))))
+stmt (climge0 ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((e. A (V)) (e. M (ZZ)) (-> (/\ (e. (cv k) (ZZ)) (br M (<_) (cv k))) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k))))) (br F (~~>) A)) (br (0) (<_) A))
+stmt (climaddOLD ((x y) (v x) (f x) (g x) (t x) (u x) (F x) (v y) (f y) (g y) (t y) (u y) (F y) (f v) (g v) (t v) (u v) (F v) (f g) (f t) (f u) (F f) (g t) (g u) (F g) (t u) (F t) (F u) (G x) (G y) (G v) (G u) (G g) (G f) (G t) (h x) (A x) (h y) (A y) (h v) (A v) (h t) (A t) (f h) (g h) (h u) (A h) (A f) (A g) (A u) (B y) (B v) (B t) (B h) (B u) (B f) (B g) (H x) (H y) (H v) (H t) (H h) (H f) (H u)) ((e. A (V)) (e. B (V)) (br F (~~>OLD) A) (br G (~~>OLD) B) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr (` F (cv x)) (+) (` G (cv x)))))) (br H (~~>OLD) (opr A (+) B)))
+stmt (climadd ((f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (F k) (F t) (F u) (G f) (G g) (G k) (G t) (G u) (H f) (h k) (H h) (k v) (H k) (H t) (H u) (H v) (N f) (N g) (N h) (N k) (N t) (N u) (N v)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (CC)) (e. B (CC)) (br F (~~>) A) (br G (~~>) B) (e. N (ZZ)) (-> (/\ (e. (cv k) (ZZ)) (br N (<_) (cv k))) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))) (br H (~~>) (opr A (+) B)))
+stmt (climaddc ((C k) (F k) (G k) (N k)) ((e. F (V)) (e. G (V)) (e. A (CC)) (e. C (CC)) (br F (~~>) A) (e. N (ZZ)) (-> (/\ (e. (cv k) (ZZ)) (br N (<_) (cv k))) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C))))) (br G (~~>) (opr A (+) C)))
+stmt (climaddtOLD ((F x) (G x) (A x) (B x) (H x)) ((e. A (V)) (e. B (V)) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr (` F (cv x)) (+) (` G (cv x)))))) (-> (/\ (br F (~~>OLD) A) (br G (~~>OLD) B)) (br H (~~>OLD) (opr A (+) B))))
+stmt (climaddcOLD ((A k) (F k) (G k)) ((e. A (CC)) (e. B (V)) (br F (~~>OLD) B) (Fn G (NN)) (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (+) (` F (cv k)))))) (br G (~~>OLD) (opr A (+) B)))
+stmt (climaddcxOLD ((j k) (j m) (j x) (A j) (k m) (k x) (A k) (m x) (A m) (A x) (B j) (B m) (B x) (F j) (F k) (F m) (F x) (G j) (G k) (G m) (G x)) ((e. A (CC)) (e. B (V)) (br F (~~>OLD) B) (Fn G (NN)) (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (+) (` F (cv k)))))) (br G (~~>OLD) (opr A (+) B)))
+stmt (climmulcOLD ((j k) (j m) (j y) (j z) (A j) (k m) (k y) (k z) (A k) (m y) (m z) (A m) (y z) (A y) (A z) (B j) (B m) (B y) (B z) (F j) (F k) (F m) (F y) (G j) (G k) (G m) (G z)) ((e. A (CC)) (e. B (V)) (br F (~~>OLD) B) (Fn G (NN)) (-> (e. (cv k) (NN)) (= (` G (cv k)) (opr A (x.) (` F (cv k)))))) (br G (~~>OLD) (opr A (x.) B)))
+stmt (climmulctxOLD ((F x) (G x) (A x) (B x)) ((e. A (CC)) (e. B (V)) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x)))))) (-> (br F (~~>OLD) B) (br G (~~>OLD) (opr A (x.) B))))
+stmt (climmulctOLD ((F x) (G x) (A x) (B x)) ((e. B (V)) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (opr A (x.) (` F (cv x)))))) (-> (/\ (e. A (CC)) (br F (~~>OLD) B)) (br G (~~>OLD) (opr A (x.) B))))
+stmt (clim2serz ((j k) (F j) (F k) (M j) (M k) (N k)) ((e. F (V)) (e. A (V)) (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N) (br (opr (<,> M (+)) (seq) F) (~~>) A) (:--> F (` (ZZ>) M) (CC))) (br (opr (<,> (opr N (+) (1)) (+)) (seq) F) (~~>) (opr A (-) (` (opr (<,> M (+)) (seq) F) N))))
+stmt (climcjOLD ((j x) (k x) (n x) (A x) (j k) (j n) (A j) (k n) (A k) (A n) (F x) (F j) (F k) (F n) (G x) (G j) (G k) (G n)) ((e. A (V)) (br F (~~>OLD) A) (Fn G (NN)) (-> (e. (cv n) (NN)) (= (` G (cv n)) (` (*) (` F (cv n)))))) (br G (~~>OLD) (` (*) A)))
+stmt (climubiOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A y) (A z) (A w) (B y) (B z) (B w)) ((e. A (V)) (:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1))))) (br F (~~>OLD) A) (e. B (NN))) (br (` F B) (<_) A))
+stmt (climubOLD ((F x)) ((e. A (V)) (:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1))))) (br F (~~>OLD) A)) (-> (e. B (NN)) (br (` F B) (<_) A)))
+stmt (climsupOLD ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1))))) (E.e. y (RR) (A.e. x (NN) (br (` F (cv x)) (<_) (cv y))))) (br F (~~>OLD) (sup (ran F) (RR) (<))))
+stmt (climcau ((x y) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (F z) (v w) (F w) (F v) (A y) (A z) (A w) (A v)) ((e. A (V)) (br F (~~>OLD) A)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) (<) (cv x))))))))
+stmt (caucvglem1 ((v y) (u y) (A y) (u v) (A v) (A u) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S z) (S w)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (<-> (e. A S) (/\ (e. A (RR)) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br A (<) (` F (cv y)))))))))
+stmt (caucvglem2 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. x (RR) (A.e. f S (br (cv f) (<_) (cv x))))))
+stmt (caucvglem3 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (e. (sup S (RR) (<)) (RR)))
+stmt (caucvglem4 ((v y) (u y) (t y) (A y) (u v) (t v) (A v) (t u) (A u) (A t) (x y) (x z) (w x) (u x) (v x) (t x) (f x) (F x) (y z) (w y) (u y) (v y) (t y) (f y) (F y) (w z) (u z) (v z) (t z) (f z) (F z) (u w) (v w) (t w) (f w) (F w) (u v) (t u) (f u) (F u) (t v) (f v) (F v) (f t) (F t) (F f) (S x) (S z) (S w) (S t) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (e. A (RR)) (-> (br A (<) (sup S (RR) (<))) (e. A S))))
+stmt (caucvglem5 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (-u (opr (cv x) (/) (2))) (<_) (opr (sup S (RR) (<)) (-) (` F (cv w)))))))
+stmt (caucvglem6 ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S x) (S z) (S w)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (opr (sup S (RR) (<)) (-) (` F (cv w))) (<_) (opr (cv x) (/) (2))))))
+stmt (caucvgOLD ((x y) (x z) (w x) (u x) (v x) (t x) (F x) (y z) (w y) (u y) (v y) (t y) (F y) (w z) (u z) (v z) (t z) (F z) (u w) (v w) (t w) (F w) (u v) (t u) (F u) (t v) (F v) (F t) (S x) (S z) (S w) (S t)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (br F (~~>OLD) (sup S (RR) (<))))
+stmt (caucvg2OLD ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (E.e. x (RR) (br F (~~>OLD) (cv x))))
+stmt (caucvg3lemOLD ((u x) (v x) (F x) (u v) (F u) (F v) (x y) (x z) (w x) (G x) (y z) (w y) (u y) (v y) (G y) (w z) (u z) (v z) (G z) (u w) (v w) (G w) (G u) (G v) (t x) (H x) (t y) (H y) (t z) (H z) (t w) (H w) (H t) (R x) (t u) (R u) (t v) (R v) (R t)) ((:--> F (NN) (CC)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x))))) (Fn R (NN)) (-> (e. (cv x) (NN)) (= (` R (cv x)) (opr (i) (x.) (` H (cv x)))))) (E.e. x (CC) (br F (~~>OLD) (cv x))))
+stmt (caucvg3OLD ((x y) (x z) (w x) (u x) (v x) (t x) (h x) (F x) (y z) (w y) (u y) (v y) (t y) (h y) (F y) (w z) (u z) (v z) (t z) (h z) (F z) (u w) (v w) (t w) (h w) (F w) (u v) (t u) (h u) (F u) (t v) (h v) (F v) (h t) (F t) (F h)) ((:--> F (NN) (CC)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z)))))))) (E.e. x (CC) (br F (~~>OLD) (cv x))))
+stmt (caucvg3aOLD ((v x) (u x) (F x) (u v) (F v) (F u) (x y) (x z) (w x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u) (D x) (D u) (D v) (R x) (R z) (R w) (S z) (S w)) ((:--> F (NN) (CC)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))) (= R ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` G (cv y)))))))) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` H (cv y)))))))) (Fn D (NN)) (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` H (cv x)))))) (br F (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))))
+stmt (ser1f0 ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A k) (A m) (A n)) ((:--> F (NN) (CC)) (e. A (CC)) (br (opr (+) (seq1) F) (~~>) A)) (br F (~~>) (0)))
+stmt (ser1const ((j k) (A j) (A k) (N j)) ((e. A (CC))) (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} A))) N) (opr N (x.) A))))
+stmt (ser10 () () (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} (0)))) N) (0))))
+stmt (ser1clim0OLD ((x y) (x z) (y z)) () (br (opr (+) (seq1) (X. (NN) ({} (0)))) (~~>OLD) (0)))
+stmt (ser1trunclem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (F x) (F y) (F z) (C y)) ((Fn F (NN)) (-> (e. (cv x) (NN)) (= (` F (cv x)) (if (br (cv x) (<_) A) B (0))))) (-> (/\/\ (e. A (NN)) (e. B (CC)) (e. C (NN))) (= (` (opr (+) (seq1) F) C) (if (br C (<_) A) (opr C (x.) B) (opr A (x.) B)))))
+stmt (ser1truncOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) ((Fn F (NN)) (-> (e. (cv x) (NN)) (= (` F (cv x)) (if (br (cv x) (<_) A) B (0))))) (-> (/\ (e. A (NN)) (e. B (CC))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (x.) B))))
+stmt (ser1trunc2OLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (if (br (cv x) (<_) A) B (0))))))) (-> (/\ (e. A (NN)) (e. B (CC))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (x.) B))))
+stmt (ser1cmp ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (` G (cv x)) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) G) A) (<_) (` (opr (+) (seq1) F) A))))
+stmt (ser1cmp0 ((F x)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (0) (<_) (` (opr (+) (seq1) F) A))))
+stmt (ser1cmp2lem ((A z) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F z) (G x) (G z) (S z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (-> (/\ (e. A (NN)) (e. B (NN))) (-> (br A (<_) B) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A))))))
+stmt (ser1cmp2 ((A z) (x y) (w x) (x z) (B x) (w y) (y z) (B y) (w z) (B w) (B z) (F x) (F w) (F z) (G x) (G w) (G z) (S w) (S z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A)))))
+stmt (cvgcmp2lemOLD ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (S x) (S z) (S w) (H x) (H z)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) A) (e. B (NN)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr S (+) (` (opr (+) (seq1) F) (cv x)))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmp2OLD ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) A) (e. B (NN)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmp2clemOLD ((A x) (B x) (F x) (G x) (S x) (C x)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) A) (e. B (NN)) (e. C (RR)) (br (0) (<_) C) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x))))) (Fn S (NN)) (-> (e. (cv x) (NN)) (= (` S (cv x)) (opr C (x.) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmp2cOLD ((x z) (A x) (A z) (w x) (B x) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (C x) (C z) (C w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) A) (e. B (NN)) (e. C (RR)) (br (0) (<_) C) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmpOLD ((A x) (F x) (G x)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmpALTOLD ((x y) (A x) (A y) (F x) (G x) (G y)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (cvgcmpub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (G x) (G y) (G z) (G w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A) (e. B (V)) (br (opr (+) (seq1) G) (~~>OLD) B)) (br B (<_) A))
+stmt (cvgcmpubexOLD ((x y) (A x) (A y) (F x) (G x) (G y)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)) (E.e. y (RR) (/\ (br (cv y) (<_) A) (br (opr (+) (seq1) G) (~~>OLD) (cv y)))))
+stmt (cvgcmp3OLD ((A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((:--> F (NN) (RR)) (:--> T (NN) (CC)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (e. A (V)) (br (opr (+) (seq1) F) (~~>OLD) A) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x))))) (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v)))))))) (Fn C (NN)) (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x))))) (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v)))))))) (Fn D (NN)) (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x))))) (e. B (NN)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (` F (cv x))))) (br (opr (+) (seq1) T) (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))))
+stmt (cvgcmp3cOLD ((Q x) (A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((:--> F (NN) (RR)) (:--> T (NN) (CC)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (e. A (V)) (br (opr (+) (seq1) F) (~~>OLD) A) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x))))) (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v)))))))) (Fn C (NN)) (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x))))) (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v)))))))) (Fn D (NN)) (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x))))) (e. B (NN)) (e. Q (RR)) (br (0) (<_) Q) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (opr Q (x.) (` F (cv x)))))) (br (opr (+) (seq1) T) (~~>OLD) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<))))))
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+stmt (cvgcmp3cetlem1OLD ((A z) (x z) (B x) (B z) (F x) (F z) (x y) (G x) (y z) (G y) (G z) (C x) (C z) (ph y) (ph z)) ((e. A (V)) (e. B (NN)) (:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) A) (<-> ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (-> ph (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))))
+stmt (cvgcmp3cetlem2OLD ((x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (G x) (G y) (G z) (C x) (C y) (C z) (ph y) (ph z)) ((e. A (V)) (e. B (NN)) (<-> ph (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)))) (-> (/\ ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))))
+stmt (cvgcmp3cetOLD ((A y) (x y) (B x) (B y) (F x) (F y) (G x) (G y) (C x) (C y)) ((e. A (V))) (-> (/\ (e. B (NN)) (/\ (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>OLD) A)) (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (E. y (br (opr (+) (seq1) G) (~~>OLD) (cv y)))))
+stmt (expcnvlem1 ((ph x) (x y) (A x) (A y)) ((e. A (RR)) (-> (e. (cv y) (NN)) (-> (br A (<) (cv y)) ph))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) ph))))
+stmt (expcnvlem2 () ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B)) (e. C (NN))) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B)))
+stmt (expcnvlem3 () ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (-> (e. C (NN)) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B))))
+stmt (expcnvlem4 ((x y) (A x) (A y) (B x) (B y)) ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B)))))
+stmt (expcnvlem5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))))
+stmt (expcnvlem6 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<_) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))))
+stmt (expcnvOLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br F (~~>OLD) (0))))
+stmt (geoser ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z) (F w)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))) (e. A (CC))) (-> (/\ (e. B (NN)) (=/= A (1))) (= (` (opr (+) (seq1) F) B) (opr (opr A (x.) (opr (1) (-) (opr A (^) B))) (/) (opr (1) (-) A)))))
+stmt (geolimilemOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))) (e. A (CC)) (br (` (abs) A) (<) (1)) (Fn G (NN)) (-> (e. (cv z) (NN)) (= (` G (cv z)) (opr (-u (opr A (/) (opr (1) (-) A))) (x.) (` F (cv z)))))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A))))
+stmt (geolimiOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x)))))) (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A))))
+stmt (geolimOLD ((x y) (A x) (A y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>OLD) (opr A (/) (opr (1) (-) A)))))
+stmt (geolimcilemOLD ((G z) (x y) (A x) (A y) (A z) (x z) (B x) (y z) (B y) (B z) (F z)) ((e. A (CC)) (e. B (CC)) (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (-> (br (` (abs) A) (<) (1)) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))))
+stmt (geolimciOLD ((x y) (A x) (A y) (B x) (B y)) ((e. A (CC)) (e. B (CC)) (= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x)))))))) (-> (br (` (abs) A) (<) (1)) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))))
+stmt (geolimcOLD ((x y) (A x) (A y) (B x) (B y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr B (x.) (opr A (^) (cv x)))))))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>OLD) (opr B (x.) (opr A (/) (opr (1) (-) A))))))
+stmt (cvgratlem1 ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((:--> F (NN) (RR)) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))))
+stmt (cvgratlem2 ((A x) (B x) (F x)) ((:--> F (NN) (RR)) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (-> (br B (<) C) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))))
+stmt (cvgratlem3 ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((:--> F (NN) (CC)) (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y))))))) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x)))))))) (-> (br B (<) C) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))))
+stmt (cvgratlem1OLD ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((:--> F (NN) (RR))) (-> (/\ (e. C (NN)) (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B)))))
+stmt (cvgratlem2OLD ((A x) (B x) (F x)) ((:--> F (NN) (RR))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C))))))
+stmt (cvgratlem3OLD ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((:--> F (NN) (CC)) (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y)))))))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C))))))
+stmt (cvgratlem4 ((A x) (B x) (F x)) ((:--> F (NN) (CC))) (-> (/\ (e. A (RR)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (br (0) (<) A)))
+stmt (cvgratlem5OLD ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (u x) (B x) (u y) (B y) (u v) (B v) (B u) (F x) (F y) (F v) (F u) (G y) (G v)) ((:--> F (NN) (CC)) (= G ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr A (^) (cv z))))))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>OLD) (cv y)))))
+stmt (cvgratOLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (F x) (F y)) ((:--> F (NN) (CC))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>OLD) (cv y)))))
+stmt (isumvalt ((f n) (n x) (n y) (F n) (f x) (f y) (F f) (x y) (F x) (F y) (N n) (N f) (N x) (N y)) () (-> (/\ (e. N A) (e. F B)) (= (opr N (sumoo) F) (U. ({|} x (br (opr (<,> N (+)) (seq) F) (~~>) (cv x)))))))
+stmt (isumclt ((x y) (F x) (F y) (N x) (N y)) ((e. F (V)) (e. N (ZZ))) (-> (E. x (br (opr (<,> N (+)) (seq) F) (~~>) (cv x))) (e. (opr N (sumoo) F) (CC))))
+stmt (sumvaltOLD ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (e. F A) (= (` (sum1oo) F) (U. ({|} x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))))))
+stmt (sumcltOLD ((x y) (F x) (F y)) () (-> (e. F A) (-> (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x))) (e. (` (sum1oo) F) (CC)))))
+stmt (sumval2tOLD ((F x) (A x)) () (-> (/\/\ (e. F B) (e. A C) (br (opr (+) (seq1) F) (~~>OLD) A)) (= (` (sum1oo) F) A)))
+stmt (cvgsumtOLD ((x y) (F x) (F y) (A y)) () (-> (/\ (e. F A) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))) (br (opr (+) (seq1) F) (~~>OLD) (` (sum1oo) F))))
+stmt (geosum1 ((x y) (A x) (A y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (= (` (sum1oo) F) (opr A (/) (opr (1) (-) A)))))
+stmt (geosum0 ((x y) (A x) (A y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr A (^) (cv x))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (= (opr (1) (+) (` (sum1oo) F)) (opr (1) (/) (opr (1) (-) A)))))
+var ( class U )
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+stmt (muffval ((f g) (f x) (f y) (f z) (C f) (g x) (g y) (g z) (C g) (x y) (x z) (C x) (y z) (C y) (C z) (F f) (F g) (F x) (F y) (F z)) ((e. F (V))) (-> (e. C D) (= (opr C (.f) F) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr C (x.) (` F (cv x)))))))))
+stmt (plfvalt ((x y) (A x) (A y) (F x) (F y) (G x) (G y)) ((e. F (V)) (e. G (V))) (-> (e. A (CC)) (= (` (opr F (+f) G) A) (opr (` F A) (+) (` G A)))))
+stmt (mufvalt ((x y) (A x) (A y) (C x) (C y) (F x) (F y)) ((e. F (V))) (-> (/\ (e. C D) (e. A (CC))) (= (` (opr C (.f) F) A) (opr C (x.) (` F A)))))
+stmt (efvalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (e^) A) (opr (1) (+) (` (sum1oo) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr A (^) (cv x)) (/) (` (!) (cv x)))))))))))
+stmt (eval ((x y)) () (= (e) (opr (1) (+) (` (sum1oo) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x))))))))))
+stmt (efcltlem1OLD ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F w)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))) (e. A (CC)) (=/= A (0))) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x))))
+stmt (efcltlem2OLD ((y z) (A y) (A z)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (-> (= A (0)) (br (opr (+) (seq1) F) (~~>OLD) (0))))
+stmt (efcltlem3OLD ((x y) (x z) (A x) (y z) (A y) (A z) (F x)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq1) F) (~~>OLD) (cv x)))))
+stmt (efclt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (CC)) (e. (` (e^) A) (CC))))
+stmt (efcvgOLD ((x y) (x z) (A x) (y z) (A y) (A z) (F z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (opr A (^) (cv x)) (/) (` (!) (cv x)))))))) (-> (e. A (CC)) (br (opr (+) (seq1) F) (~~>OLD) (opr (` (e^) A) (-) (1)))))
+stmt (reefcl ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (RR))) (e. (` (e^) A) (RR)))
+stmt (reefclt () () (-> (e. A (RR)) (e. (` (e^) A) (RR))))
+stmt (erelem1 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (/\ (:--> F (NN) (RR)) (:--> G (NN) (RR))))
+stmt (erelem2OLD ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (opr (+) (seq1) F) (~~>OLD) (2)))
+stmt (erelem3 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (-> (e. (cv z) (NN)) (/\ (br (0) (<_) (` G (cv z))) (br (` G (cv z)) (<_) (` F (cv z))))))
+stmt (erelem4OLD ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (opr (+) (seq1) G) (~~>OLD) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (erelem5 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (e. (sup (ran (opr (+) (seq1) G)) (RR) (<)) (RR)))
+stmt (erelem6 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (= (e) (opr (1) (+) (sup (ran (opr (+) (seq1) G)) (RR) (<)))))
+stmt (erelem7 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (e. (e) (RR)))
+stmt (ege2le3lem1 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (` (opr (+) (seq1) G) (1)) (<_) (sup (ran (opr (+) (seq1) G)) (RR) (<))))
+stmt (ege2le3lem2 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (/\ (br (2) (<_) (e)) (br (e) (<_) (3))))
+stmt (ere ((x y)) () (e. (e) (RR)))
+stmt (ege2le3 ((x y)) () (/\ (br (2) (<_) (e)) (br (e) (<_) (3))))
+stmt (sinvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (sin) A) (opr (opr (` (e^) (opr (i) (x.) A)) (-) (` (e^) (opr (-u (i)) (x.) A))) (/) (opr (2) (x.) (i))))))
+stmt (cosvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (cos) A) (opr (opr (` (e^) (opr (i) (x.) A)) (+) (` (e^) (opr (-u (i)) (x.) A))) (/) (2)))))
+stmt (sinclt () () (-> (e. A (CC)) (e. (` (sin) A) (CC))))
+stmt (cosclt () () (-> (e. A (CC)) (e. (` (cos) A) (CC))))
+stmt (efivalt () () (-> (e. A (CC)) (= (` (e^) (opr (i) (x.) A)) (opr (` (cos) A) (+) (opr (i) (x.) (` (sin) A))))))
+stmt (ef0ALT ((x y)) () (= (` (e^) (0)) (1)))
+stmt (sinnegt () () (-> (e. A (CC)) (= (` (sin) (-u A)) (-u (` (sin) A)))))
+stmt (ruclem1 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br A (<) (opr (opr (opr (2) (x.) A) (+) B) (/) (3)))))
+stmt (ruclem2 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (opr (opr (opr (2) (x.) A) (+) B) (/) (3)) (<) (opr (opr A (+) (opr (2) (x.) B)) (/) (3)))))
+stmt (ruclem3 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (opr (opr A (+) (opr (2) (x.) B)) (/) (3)) (<) B)))
+stmt (ruclem4 () () (-> (/\ (= A C) (= B D)) (= (if (/\ (br (` (1st) A) (<) B) (br B (<) (` (2nd) A))) (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3)))) (if (/\ (br (` (1st) C) (<) D) (br D (<) (` (2nd) C))) (<,> (opr (opr (opr (2) (x.) D) (+) (` (2nd) C)) (/) (3)) (opr (opr D (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3)))))))
+stmt (ruclem5 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (e. C (V)))
+stmt (ruclem6 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (= (|` C (\ (NN) ({} (1)))) (|` F (\ (NN) ({} (1))))))
+stmt (ruclem7 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (= (` C (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))))
+stmt (ruclem8 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (e. A (NN))) (= (` C (opr A (+) (1))) (` F (opr A (+) (1)))))
+stmt (ruclem9 ((x y) (x z) (y z) (A z)) ((= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) A))))) (e. D (V)))
+stmt (ruclem10 () ((e. A (NN)) (e. D (V)) (e. C (V)) (= G (o. (1st) (opr D (seq1) C)))) (= (` (1st) (` (opr D (seq1) C) A)) (` G A)))
+stmt (ruclem11 () ((e. A (NN)) (e. D (V)) (e. C (V)) (= H (o. (2nd) (opr D (seq1) C)))) (= (` (2nd) (` (opr D (seq1) C) A)) (` H A)))
+stmt (ruclem12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) ((= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (= D ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (X. (RR) (RR))) (e. (cv v) (RR))) (= (cv u) (if (/\ (br (` (1st) (cv w)) (<) (cv v)) (br (cv v) (<) (` (2nd) (cv w)))) (<,> (opr (opr (opr (2) (x.) (cv v)) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (cv v) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv w))) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (` (1st) (cv w)) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3)))))))))
+stmt (ruclem13 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (D w) (D v) (D u)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (:--> (opr D (seq1) C) (NN) (X. (RR) (RR))))
+stmt (ruclem14 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (= (` (opr D (seq1) C) (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2)))))
+stmt (ruclem15 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (= (` (opr D (seq1) C) (opr A (+) (1))) (if (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))))))
+stmt (ruclem16 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (= (` G (1)) (opr (` F (1)) (+) (1))))
+stmt (ruclem17 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (:--> G (NN) (RR)))
+stmt (ruclem18 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)))))
+stmt (ruclem19 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` H (opr A (+) (1))) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3)))))
+stmt (ruclem20 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)))))
+stmt (ruclem21 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` H (opr A (+) (1))) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)))))
+stmt (ruclem22 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (e. (` G A) (RR)))
+stmt (ruclem23 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (e. (` H A) (RR)))
+stmt (ruclem24 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (br (` G A) (<) (` H A)) (br (` G (opr A (+) (1))) (<) (` H (opr A (+) (1))))))
+stmt (ruclem25 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` G A) (<) (` H A)))
+stmt (ruclem26 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` G A) (<) (` G (opr A (+) (1)))))
+stmt (ruclem27 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` H (opr A (+) (1))) (<) (` H A)))
+stmt (ruclem28 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-. (/\ (br (` G (opr A (+) (1))) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H (opr A (+) (1)))))))
+stmt (ruclem29 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-. (/\ (br (` G A) (<) (` F A)) (br (` F A) (<) (` H A)))))
+stmt (ruclem30 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (-> (br (` G A) (<) (` G (opr A (+) B))) (br (` G A) (<) (` G (opr A (+) (opr B (+) (1)))))))
+stmt (ruclem31 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (-> (br (` H (opr A (+) B)) (<) (` H B)) (br (` H (opr (opr A (+) (1)) (+) B)) (<) (` H B))))
+stmt (ruclem32 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v) (B w) (B v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (br (` G A) (<) (` H B)))
+stmt (ruclem33 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (/\/\ (C_ (ran G) (RR)) (-. (= (ran G) ({/}))) (E.e. w (RR) (A.e. v (ran G) (br (cv v) (<_) (cv w))))))
+stmt (ruclem34 ((S w) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<)))) (e. S (RR)))
+stmt (ruclem35 ((S w) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<))) (e. A (NN))) (/\ (br (` G A) (<) S) (br S (<) (` H A))))
+stmt (ruclem36 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<))) (e. A (NN))) (-. (= (` F A) S)))
+stmt (ruclem37 ((S w) (x y) (x z) (w x) (y z) (w y) (w z) (C w) (D w) (G w) (H w) (F z) (F w)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<)))) (-. (:-onto-> F (NN) (RR))))
+stmt (ruclem38 ((F z) (x y) (x z) (y z)) ((:--> F (NN) (RR))) (-. (:-onto-> F (NN) (RR))))
+stmt (ruclem39 () () (-. (:-onto-> F (NN) (RR))))
+stmt (ruc () () (br (NN) (~<) (RR)))
+stmt (nn0ennn ((x y)) () (br (NN0) (~~) (NN)))
+stmt (nnenom ((x y) (x z) (y z)) () (br (NN) (~~) (om)))
+stmt (xpnnen ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (br (X. (NN) (NN)) (~~) (NN)))
+stmt (xpomen () () (br (X. (om) (om)) (~~) (om)))
+stmt (znnenlem () () (-> (/\ (/\ (br (0) (<_) (cv x)) (-. (br (0) (<_) (cv y)))) (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ)))) (<-> (= (cv y) (cv x)) (= (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1))))))
+stmt (znnen ((x y)) () (br (ZZ) (~~) (NN)))
+stmt (qnnen ((x y) (x z) (w x) (y z) (w y) (w z)) () (br (QQ) (~~) (NN)))
+stmt (unbenlem ((m n) (m z) (m w) (m v) (m u) (m t) (A m) (n z) (n w) (n v) (n u) (n t) (A n) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t) (G m) (G n) (G v) (G u) (G t)) ((= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (om))))
+stmt (unben ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (NN))))
+stmt (infpnlem1 ((N j) (M j) (K j)) ((= K (opr (` (!) N) (+) (1)))) (-> (/\ (e. N (NN)) (e. M (NN))) (-> (/\ (/\ (br (1) (<) M) (e. (opr K (/) M) (NN))) (A.e. j (NN) (-> (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) (br M (<_) (cv j))))) (/\ (br N (<) M) (A.e. j (NN) (-> (e. (opr M (/) (cv j)) (NN)) (\/ (= (cv j) (1)) (= (cv j) M))))))))
+stmt (infpnlem2 ((j k) (N j) (N k) (K j) (K k)) ((= K (opr (` (!) N) (+) (1)))) (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j)))))))))
+stmt (infpn ((j k) (j n) (j m) (k n) (k m) (m n) (S j) (S k)) ((= S ({e.|} n (NN) (/\ (br (1) (<) (cv n)) (A.e. m (NN) (-> (e. (opr (cv n) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv n))))))))) (br S (~~) (NN)))
+stmt (resdomq () () (br (QQ) (~<) (RR)))
+stmt (aleph1re () () (br (` (aleph) (1o)) (~<_) (RR)))
+stmt (infxpidmlem1 () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (-> (/\ (br (cv x) (~~) A) (br (cv x) (~~) B)) (br (cv x) (~~) (u. A B)))))
+stmt (infxpidmlem2 ((f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. B (V))) (<-> (e. B H) (\/ (= B ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> B (X. (cv x) (cv x)) (cv x)))))))
+stmt (infxpidmlem3 ((D x) (f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. B (V)) (e. D (V))) (-> (/\ (/\ (br (om) (~<_) D) (C_ D A)) (:-1-1-onto-> B (X. D D) D)) (e. B H)))
+stmt (infxpidmlem4 ((f x) (g x) (t x) (A x) (f g) (f t) (A f) (g t) (A g) (A t) (H x) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (-> (e. (cv g) H) (= (dom (cv g)) (X. (ran (cv g)) (ran (cv g))))))
+stmt (infxpidmlem5 ((y z) (f y) (g y) (t y) (A y) (f z) (g z) (t z) (A z) (f g) (f t) (A f) (g t) (A g) (A t) (C y) (C z) (C f) (C g) (C t) (H y) (H z) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (-> (/\ (C_ C H) (e. (cv g) C)) (-> (/\ (e. (cv y) (ran (cv g))) (e. (cv z) (ran (cv g)))) (e. (<,> (cv y) (cv z)) (dom (U. C))))))
+stmt (infxpidmlem6 ((f y) (g y) (t y) (A y) (f g) (f t) (A f) (g t) (A g) (A t) (B y) (B f) (B g) (B t) (C y) (C f) (C g) (C t) (H y) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C)))) (<-> (e. (cv y) B) (E.e. g C (e. (cv y) (ran (cv g))))))
+stmt (infxpidmlem7 ((x y) (x z) (w x) (f x) (g x) (h x) (v x) (t x) (A x) (y z) (w y) (f y) (g y) (h y) (v y) (t y) (A y) (w z) (f z) (g z) (h z) (v z) (t z) (A z) (f w) (g w) (h w) (v w) (t w) (A w) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C w) (C v) (C f) (C g) (C h) (C t) (H x) (H y) (H z) (H w) (H v) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C)))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (:-1-1-onto-> (U. C) (X. B B) B)))
+stmt (infxpidmlem8 ((x y) (f x) (g x) (h x) (t x) (A x) (f y) (g y) (h y) (t y) (A y) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B x) (B y) (B f) (B g) (B h) (B t) (C x) (C y) (C f) (C g) (C h) (C t) (H x) (H y) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C))) (e. C (V))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (e. (U. C) H)))
+stmt (infxpidmlem9 ((f z) (g z) (h z) (t z) (A z) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (H z) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (E.e. g H (A.e. h H (-. (C: (cv g) (cv h))))))
+stmt (infxpidmlem10 ((f y) (g y) (h y) (v y) (t y) (A y) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (H y) (H v) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (A.e. h H (-. (C: (cv g) (cv h)))) (-> (br (om) (~<_) A) (-. (= (cv g) ({/}))))))
+stmt (infxpidmlem11 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (/\ (/\ (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x))) (/\ (br (cv x) (~~) (cv y)) (C_ (cv y) (\ A (cv x))))) (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y))) (E.e. h H (C: (cv g) (cv h)))))
+stmt (infxpidmlem12 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)))
+stmt (infxpidm ((f x) (A f) (A x)) ((e. A (V))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A)))
+stmt (infunabs () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (u. A B) (~~) A)))
+stmt (infcdaabs () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (+c) B) (~~) A)))
+stmt (infcda () ((e. A (V)) (e. B (V))) (-> (br (om) (~<_) A) (br (opr A (+c) B) (~~) (u. A B))))
+stmt (infdif () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<) A)) (br (\ A B) (~~) A)))
+stmt (infxpabs () ((e. A (V)) (e. B (V))) (-> (/\ (/\ (br (om) (~<_) A) (-. (= B ({/})))) (br B (~<_) A)) (br (X. A B) (~~) A)))
+stmt (infmap1 () ((e. A (V)) (e. B (V))) (-> (/\ (/\ (br (2o) (~<_) A) (br (om) (~<_) B)) (br A (~<_) B)) (br (opr A (^m) B) (~~) (opr (2o) (^m) B))))
+stmt (aleph1irr () () (br (` (aleph) (1o)) (~<_) (\ (RR) (QQ))))
+stmt (infmap2lem1 ((x z) (w x) (v x) (f x) (A x) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f) (B x) (B z) (B w) (B f) (B v) (R v) (R f)) ((e. A (V)) (e. B (V)) (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)))))) (-> (/\ (C_ (cv f) R) (Fn (cv f) (dom R))) (-> (e. (cv v) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B)))) (/\ (C_ (cv v) A) (:-onto-> (` (cv f) (cv v)) B (cv v))))))
+stmt (infmap2lem2 ((x z) (w x) (v x) (u x) (f x) (A x) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B z) (B w) (B f) (B v) (B u) (R u) (R v) (R f)) ((e. A (V)) (e. B (V)) (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)))))) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) (~<_) (opr A (^m) B)))
+stmt (infmap2 ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w)) ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (^m) B) (~~) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))))))
+stmt (alephadd () () (br (opr (` (aleph) A) (+c) (` (aleph) B)) (~~) (u. (` (aleph) A) (` (aleph) B))))
+stmt (alephexp1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (br (opr (` (aleph) A) (^m) (` (aleph) B)) (~~) (opr (2o) (^m) (` (aleph) B)))))
+stmt (alephsuc3 ((A x)) () (-> (e. A (On)) (br (` (aleph) (suc A)) (~~) ({e.|} x (On) (br (cv x) (~~) (` (aleph) A))))))
+stmt (alephexp2 ((A x)) () (-> (e. A (On)) (br (opr (2o) (^m) (` (aleph) A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A)))))))
+stmt (gch-kn ((A x)) () (-> (e. A (On)) (<-> (br (` (aleph) (suc A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A))))) (br (` (aleph) (suc A)) (~~) (opr (2o) (^m) (` (aleph) A))))))
+stmt (hvmulcl () ((e. A (CC)) (e. B (H~))) (e. (opr A (.s) B) (H~)))
+stmt (hvsubvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B)))))
+stmt (hvsubclt () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~))))
+stmt (hvaddcl () ((e. A (H~)) (e. B (H~))) (e. (opr A (+v) B) (H~)))
+stmt (hvcom () ((e. A (H~)) (e. B (H~))) (= (opr A (+v) B) (opr B (+v) A)))
+stmt (hvsubval () ((e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B))))
+stmt (hvsubcl () ((e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~)))
+stmt (hvaddid2t () () (-> (e. A (H~)) (= (opr (0v) (+v) A) A)))
+stmt (hvmul0t () () (-> (e. A (CC)) (= (opr A (.s) (0v)) (0v))))
+stmt (hvmul0ort () () (-> (/\ (e. A (CC)) (e. B (H~))) (<-> (= (opr A (.s) B) (0v)) (\/ (= A (0)) (= B (0v))))))
+stmt (hvsubidt () () (-> (e. A (H~)) (= (opr A (-v) A) (0v))))
+stmt (hvnegidt () () (-> (e. A (H~)) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v))))
+stmt (hv2negt () () (-> (e. A (H~)) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A))))
+stmt (hvaddid2 () ((e. A (H~))) (= (opr (0v) (+v) A) A))
+stmt (hvnegid () ((e. A (H~))) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v)))
+stmt (hv2neg () ((e. A (H~))) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A)))
+stmt (hvm1negt () () (-> (/\ (e. A (CC)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (.s) B)) (opr (-u A) (.s) B))))
+stmt (hvadd23t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B))))
+stmt (hvadd12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C)))))
+stmt (hvadd4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D)))))
+stmt (hvsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (+v) B) (-v) (opr C (+v) D)) (opr (opr A (-v) C) (+v) (opr B (-v) D)))))
+stmt (hvaddsub12t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (-v) C)) (opr B (+v) (opr A (-v) C)))))
+stmt (hvsubcan1t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) B) A)))
+stmt (hvpncan2t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (-v) A) B)))
+stmt (hvaddsubasst () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (-v) C) (opr A (+v) (opr B (-v) C)))))
+stmt (hvpncan3t () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (+v) (opr B (-v) A)) B)))
+stmt (hvmulcomt () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C)))))
+stmt (hvmulass () ((e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (x.) B) (.s) C) (opr A (.s) (opr B (.s) C))))
+stmt (hvmulcom () ((e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr A (.s) (opr B (.s) C)) (opr B (.s) (opr A (.s) C))))
+stmt (hvmul2neg () ((e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (-u A) (.s) (opr (-u B) (.s) C)) (opr A (.s) (opr B (.s) C))))
+stmt (hvsubdistr1t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C)))))
+stmt (hvsubdistr2t () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (-) B) (.s) C) (opr (opr A (.s) C) (-v) (opr B (.s) C)))))
+stmt (hvdistr1 () ((e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (+v) C)) (opr (opr A (.s) B) (+v) (opr A (.s) C))))
+stmt (hvsubdistr1 () ((e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (-v) C)) (opr (opr A (.s) B) (-v) (opr A (.s) C))))
+stmt (hvass () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr A (+v) (opr B (+v) C))))
+stmt (hvadd23 () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr (opr A (+v) C) (+v) B)))
+stmt (hvsubass () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (-v) B) (-v) C) (opr A (-v) (opr B (+v) C))))
+stmt (hvsub23 () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (-v) B) (-v) C) (opr (opr A (-v) C) (-v) B)))
+stmt (hvadd12 () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (+v) (opr B (+v) C)) (opr B (+v) (opr A (+v) C))))
+stmt (hvadd4 () ((e. A (H~)) (e. B (H~)) (e. C (H~)) (e. D (H~))) (= (opr (opr A (+v) B) (+v) (opr C (+v) D)) (opr (opr A (+v) C) (+v) (opr B (+v) D))))
+stmt (hvsubsub4 () ((e. A (H~)) (e. B (H~)) (e. C (H~)) (e. D (H~))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D))))
+stmt (hvsubsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr A (-v) B) (-v) (opr C (-v) D)) (opr (opr A (-v) C) (-v) (opr B (-v) D)))))
+stmt (hv2times () ((e. A (H~))) (= (opr (2) (.s) A) (opr A (+v) A)))
+stmt (hvnegdi () ((e. A (H~)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A)))
+stmt (hvsubeq0 () ((e. A (H~)) (e. B (H~))) (<-> (= (opr A (-v) B) (0v)) (= A B)))
+stmt (hvsubcan2 () ((e. A (H~)) (e. B (H~))) (= (opr (opr A (+v) B) (+v) (opr A (-v) B)) (opr (2) (.s) A)))
+stmt (hvaddcan () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C)))
+stmt (hvsubadd () ((e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A)))
+stmt (hvnegdit () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (-u (1)) (.s) (opr A (-v) B)) (opr B (-v) A))))
+stmt (hvsubeq0t () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (= (opr A (-v) B) (0v)) (= A B))))
+stmt (hvaddcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) B) (opr A (+v) C)) (= B C))))
+stmt (hvaddcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (+v) C) (opr B (+v) C)) (= A B))))
+stmt (hvmulcant () () (-> (/\ (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (=/= A (0))) (<-> (= (opr A (.s) B) (opr A (.s) C)) (= B C))))
+stmt (hvmulcan2t () () (-> (/\ (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (=/= C (0v))) (<-> (= (opr A (.s) C) (opr B (.s) C)) (= A B))))
+stmt (hvsubcant () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) (opr A (-v) C)) (= B C))))
+stmt (hvsubcan2t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) C) (opr B (-v) C)) (= A B))))
+stmt (hvsub0t () () (-> (e. A (H~)) (= (opr A (-v) (0v)) A)))
+stmt (hvsubaddt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (<-> (= (opr A (-v) B) C) (= (opr B (+v) C) A))))
+stmt (hvaddsub4t () () (-> (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (<-> (= (opr A (+v) B) (opr C (+v) D)) (= (opr A (-v) C) (opr D (-v) B)))))
+stmt (hicl () ((e. A (H~)) (e. B (H~))) (e. (opr A (.i) B) (CC)))
+stmt (axhis42 () () (-> (/\ (e. A (H~)) (-. (= A (0v)))) (br (0) (<) (opr A (.i) A))))
+stmt (his5t () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr B (.i) (opr A (.s) C)) (opr (` (*) A) (x.) (opr B (.i) C)))))
+stmt (his35 () ((e. A (CC)) (e. B (CC)) (e. C (H~)) (e. D (H~))) (= (opr (opr A (.s) C) (.i) (opr B (.s) D)) (opr (opr A (x.) (` (*) B)) (x.) (opr C (.i) D))))
+stmt (his7t () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr A (.i) (opr B (+v) C)) (opr (opr A (.i) B) (+) (opr A (.i) C)))))
+stmt (hiassdit () () (-> (/\ (/\ (e. A (CC)) (e. B (H~))) (/\ (e. C (H~)) (e. D (H~)))) (= (opr (opr (opr A (.s) B) (+v) C) (.i) D) (opr (opr A (x.) (opr B (.i) D)) (+) (opr C (.i) D)))))
+stmt (his2subt () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (-v) B) (.i) C) (opr (opr A (.i) C) (-) (opr B (.i) C)))))
+stmt (hiret () () (-> (/\ (e. A (H~)) (e. B (H~))) (<-> (e. (opr A (.i) B) (RR)) (= (opr A (.i) B) (opr B (.i) A)))))
+stmt (hiidrclt () () (-> (e. A (H~)) (e. (opr A (.i) A) (RR))))
+stmt (hi01t () () (-> (e. A (H~)) (= (opr (0v) (.i) A) (0))))
+stmt (hi02t () () (-> (e. A (H~)) (= (opr A (.i) (0v)) (0))))
+stmt (hiidge0t () () (-> (e. A (H~)) (br (0) (<_) (opr A (.i) A))))
+stmt (his6t () () (-> (e. A (H~)) (<-> (= (opr A (.i) A) (0)) (= A (0v)))))
+stmt (his1 () ((e. A (H~)) (e. B (H~))) (= (opr A (.i) B) (` (*) (opr B (.i) A))))
+stmt (hial0 ((A x)) () (-> (e. A (H~)) (<-> (A.e. x (H~) (= (opr A (.i) (cv x)) (0))) (= A (0v)))))
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+stmt (5oalem3 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))))))
+stmt (5oalem4 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G)))))))
+stmt (5oalem5 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)))) (/\ (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))))
+stmt (5oalem6 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))))) (e. (cv h) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))))))))
+stmt (5oalem7 ((A h) (A f) (A g) (A u) (A v) (A w) (A x) (A y) (A z) (f h) (g h) (h u) (h v) (h w) (h x) (h y) (h z) (f g) (f u) (f v) (f w) (f x) (f y) (f z) (g u) (g v) (g w) (g x) (g y) (g z) (u v) (u w) (u x) (u y) (u z) (v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (B h) (B f) (B g) (B u) (B v) (B w) (B x) (B y) (B z) (C h) (C f) (C g) (C u) (C v) (C w) (C x) (C y) (C z) (D h) (D f) (D g) (D u) (D v) (D w) (D x) (D y) (D z) (F h) (F f) (F g) (F u) (F v) (F w) (F x) (F y) (F z) (G h) (G f) (G g) (G u) (G v) (G w) (G x) (G y) (G z) (R h) (R f) (R g) (R u) (R v) (R w) (R x) (R y) (R z) (S h) (S f) (S g) (S u) (S v) (S w) (S x) (S y) (S z)) ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (C_ (i^i (i^i (opr A (+H) B) (opr C (+H) D)) (i^i (opr F (+H) G) (opr R (+H) S))) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))))))
+stmt (5oa () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (e. D (CH)) (e. F (CH)) (e. G (CH)) (e. R (CH)) (e. S (CH)) (C_ A (` (_|_) B)) (C_ C (` (_|_) D)) (C_ F (` (_|_) G)) (C_ R (` (_|_) S))) (C_ (i^i (i^i (opr A (vH) B) (opr C (vH) D)) (i^i (opr F (vH) G) (opr R (vH) S))) (opr B (vH) (i^i A (opr C (vH) (i^i (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))))))))))
+stmt (3oalem1 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((e. B (CH)) (e. C (CH)) (e. R (CH)) (e. S (CH))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (e. (cv v) (H~))) (/\ (e. (cv z) (H~)) (e. (cv w) (H~))))))
+stmt (3oalem2 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((e. B (CH)) (e. C (CH)) (e. R (CH)) (e. S (CH))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) S)) (= (cv v) (opr (cv z) (+v) (cv w))))) (e. (cv v) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))))))
+stmt (3oalem3 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((e. B (CH)) (e. C (CH)) (e. R (CH)) (e. S (CH))) (C_ (i^i (opr B (+H) R) (opr C (+H) S)) (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S)))))))
+stmt (3oalem4 () ((= R (i^i (` (_|_) B) (opr B (vH) A)))) (C_ R (` (_|_) B)))
+stmt (3oalem5 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= R (i^i (` (_|_) B) (opr B (vH) A))) (= S (i^i (` (_|_) C) (opr C (vH) A)))) (= (i^i (opr B (+H) R) (opr C (+H) S)) (i^i (opr B (vH) R) (opr C (vH) S))))
+stmt (3oalem6 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= R (i^i (` (_|_) B) (opr B (vH) A))) (= S (i^i (` (_|_) C) (opr C (vH) A)))) (C_ (opr B (+H) (i^i R (opr S (+H) (i^i (opr B (+H) C) (opr R (+H) S))))) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))))
+stmt (3oa () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= R (i^i (` (_|_) B) (opr B (vH) A))) (= S (i^i (` (_|_) C) (opr C (vH) A)))) (C_ (i^i (opr B (vH) R) (opr C (vH) S)) (opr B (vH) (i^i R (opr S (vH) (i^i (opr B (vH) C) (opr R (vH) S)))))))
+stmt (pjorth () ((e. A (H~)) (e. B (H~))) (-> (e. H (CH)) (= (opr (` (` (Proj) H) A) (.i) (` (` (Proj) (` (_|_) H)) B)) (0))))
+stmt (pjch1t () () (-> (e. A (H~)) (= (` (` (Proj) (H~)) A) A)))
+stmt (pjot () () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (Proj) (` (_|_) H)) A) (opr (` (` (Proj) (H~)) A) (-v) (` (` (Proj) H) A)))))
+stmt (pjch0t () () (-> (e. A (H~)) (= (` (` (Proj) (0H)) A) (0v))))
+stmt (pjidm () ((e. H (CH)) (e. A (H~))) (= (` (` (Proj) H) (` (` (Proj) H) A)) (` (` (Proj) H) A)))
+stmt (pjadj () ((e. H (CH)) (e. A (H~)) (e. B (H~))) (= (opr (` (` (Proj) H) A) (.i) B) (opr A (.i) (` (` (Proj) H) B))))
+stmt (pjcomp () ((e. H (CH)) (e. A (H~)) (e. B (H~))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (Proj) H) (opr A (+v) B)) A)))
+stmt (pjadd () ((e. H (CH)) (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (+v) B)) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) H) B))))
+stmt (pjinorm () ((e. H (CH)) (e. A (H~))) (= (opr (` (` (Proj) H) A) (.i) A) (opr (` (norm) (` (` (Proj) H) A)) (^) (2))))
+stmt (pjmul () ((e. H (CH)) (e. A (H~)) (e. C (CC))) (= (` (` (Proj) H) (opr C (.s) A)) (opr C (.s) (` (` (Proj) H) A))))
+stmt (pjsub () ((e. H (CH)) (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (-v) B)) (opr (` (` (Proj) H) A) (-v) (` (` (Proj) H) B))))
+stmt (pjsslem () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (= (opr (` (` (Proj) (` (_|_) H)) A) (-v) (` (` (Proj) (` (_|_) G)) A)) (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A))))
+stmt (pjss2 () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (-> (C_ H G) (= (` (` (Proj) H) (` (` (Proj) G) A)) (` (` (Proj) H) A))))
+stmt (pjssm () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (-> (C_ H G) (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A))))
+stmt (pjssge0 () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (-> (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A))))
+stmt (pjdifnorm () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (<-> (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) (` (` (Proj) G) A)))))
+stmt (pjcj () ((e. H (CH)) (e. A (H~)) (e. G (CH))) (-> (C_ H (` (_|_) G)) (= (` (` (Proj) (opr H (vH) G)) A) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A)))))
+stmt (pjadjt () ((e. H (CH))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (` (Proj) H) A) (.i) B) (opr A (.i) (` (` (Proj) H) B)))))
+stmt (pjaddt () ((e. H (CH))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (+v) B)) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) H) B)))))
+stmt (pjsubt () ((e. H (CH))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` (` (Proj) H) (opr A (-v) B)) (opr (` (` (Proj) H) A) (-v) (` (` (Proj) H) B)))))
+stmt (pjmult () ((e. H (CH))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` (` (Proj) H) (opr A (.s) B)) (opr A (.s) (` (` (Proj) H) B)))))
+stmt (pjcjt2 () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (` (` (Proj) (opr H (vH) G)) A) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A))))))
+stmt (pj0 () ((e. H (CH))) (= (` (` (Proj) H) (0v)) (0v)))
+stmt (pjcht () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (` (Proj) H) A) A))))
+stmt (pjocin () ((e. G (CH)) (e. H (CH))) (-> (e. A (` (_|_) (i^i G H))) (e. (` (` (Proj) G) A) (` (_|_) (i^i G H)))))
+stmt (pjin () ((e. G (CH)) (e. H (CH))) (-> (e. A (i^i G H)) (e. (` (` (Proj) G) A) (i^i G H))))
+stmt (pjjs ((x y) (x z) (w x) (G x) (y z) (w y) (G y) (w z) (G z) (G w) (H x) (H y) (H z) (H w)) ((e. G (CH)) (e. H (SH))) (-> (A.e. x (opr G (vH) H) (e. (` (` (Proj) (` (_|_) G)) (cv x)) H)) (= (opr G (vH) H) (opr G (+H) H))))
+stmt (pjfn ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w)) ((e. H (CH))) (Fn (` (Proj) H) (H~)))
+stmt (pjrn ((x y) (H x) (H y)) ((e. H (CH))) (= (ran (` (Proj) H)) H))
+stmt (pjf () ((e. H (CH))) (:--> (` (Proj) H) (H~) (H~)))
+stmt (pjv () ((e. H (CH))) (-> (/\ (e. A H) (e. B (` (_|_) H))) (= (` (` (Proj) H) (opr A (+v) B)) A)))
+stmt (pjsumt () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (-> (C_ G (` (_|_) H)) (= (` (` (Proj) (opr G (+H) H)) A) (opr (` (` (Proj) G) A) (+v) (` (` (Proj) H) A))))))
+stmt (pj11 () ((e. G (CH)) (e. H (CH))) (<-> (= (` (Proj) G) (` (Proj) H)) (= G H)))
+stmt (pjoi0 () ((e. G (CH)) (e. H (CH)) (e. A (H~))) (-> (C_ G (` (_|_) H)) (= (opr (` (` (Proj) G) A) (.i) (` (` (Proj) H) A)) (0))))
+stmt (pjopyth () ((e. G (CH)) (e. H (CH)) (e. A (H~))) (-> (C_ G (` (_|_) H)) (= (opr (` (norm) (opr (` (` (Proj) G) A) (+v) (` (` (Proj) H) A))) (^) (2)) (opr (opr (` (norm) (` (` (Proj) G) A)) (^) (2)) (+) (opr (` (norm) (` (` (Proj) H) A)) (^) (2))))))
+stmt (hoeq ((S x) (T x)) ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (= (` S (cv x)) (` T (cv x)))) (= S T)))
+stmt (hoscl () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (opr S (+P) T) A) (H~))))
+stmt (hodcl () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (opr S (-P) T) A) (H~))))
+stmt (hoco () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (= (` (o. S T) A) (` S (` T A)))))
+stmt (hococl () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (o. S T) A) (H~))))
+stmt (hocof () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (o. S T) (H~) (H~)))
+stmt (hocofn () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (Fn (o. S T) (H~)))
+stmt (hosf ((x y) (S x) (S y) (T x) (T y)) ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (opr S (+P) T) (H~) (H~)))
+stmt (hodf ((x y) (S x) (S y) (T x) (T y)) ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (opr S (-P) T) (H~) (H~)))
+stmt (hosfn () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (Fn (opr S (+P) T) (H~)))
+stmt (hoscom ((S x) (T x)) ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+P) T) (opr T (+P) S)))
+stmt (eigre () ((e. A (H~)) (e. B (CC))) (-> (/\ (= (` T A) (opr B (.s) A)) (-. (= A (0v)))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR)))))
+stmt (eigpos () ((e. A (H~)) (e. B (CC))) (-> (/\ (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (/\ (= (` T A) (opr B (.s) A)) (-. (= A (0v))))) (/\ (e. B (RR)) (br (0) (<_) B))))
+stmt (eigorth () ((e. A (H~)) (e. B (H~)) (e. C (CC)) (e. D (CC))) (-> (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D)))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0)))))
+stmt (hunid ((x y)) () (/\ (:-onto-> (|` (I) (H~)) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (|` (I) (H~)) (cv x)) (.i) (` (|` (I) (H~)) (cv y))) (opr (cv x) (.i) (cv y)))))))
+stmt (hun ((x y) (x z) (w x) (T x) (y z) (w y) (T y) (w z) (T z) (T w) (A z) (A w) (B z) (B w)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) (` T B)) (opr A (.i) B))))
+stmt (hun1o ((x y) (x z) (w x) (T x) (y z) (w y) (T y) (w z) (T z) (T w)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (:-1-1-onto-> T (H~) (H~)))
+stmt (huncnv ((x y) (T x) (T y)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (/\ (:-onto-> (`' T) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (`' T) (cv x)) (.i) (` (`' T) (cv y))) (opr (cv x) (.i) (cv y)))))))
+stmt (hunicnv ((x y) (T x) (T y)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` T A) (.i) B) (opr A (.i) (` (`' T) B)))))
+stmt (hunlin ((x y) (x z) (T x) (y z) (T y) (T z) (A z) (B z) (C z)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))))
+stmt (hunco ((x y) (S x) (S y) (x y) (T x) (T y)) ((/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))) (/\ (:-onto-> S (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` S (cv x)) (.i) (` S (cv y))) (opr (cv x) (.i) (cv y))))))) (/\ (:-onto-> (o. S T) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` (o. S T) (cv x)) (.i) (` (o. S T) (cv y))) (opr (cv x) (.i) (cv y)))))))
+stmt (hlin ((x y) (x z) (w x) (v x) (u x) (T x) (y z) (w y) (v y) (u y) (T y) (w z) (v z) (u z) (T z) (v w) (u w) (T w) (u v) (T v) (T u) (A w) (A v) (A u) (B v) (B u) (C u)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C)))))
+stmt (hlin0 ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (= (` T (0v)) (0v)))
+stmt (hlinadd ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (+v) B)) (opr (` T A) (+v) (` T B)))))
+stmt (hlinmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\ (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (.s) (` T B)))))
+stmt (hlinaddmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (+v) (opr A (.s) C))) (opr (` T B) (+v) (opr A (.s) (` T C))))))
+stmt (hlinsub ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (` T (opr A (-v) B)) (opr (` T A) (-v) (` T B)))))
+stmt (hlinsubmul ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr B (-v) (opr A (.s) C))) (opr (` T B) (-v) (opr A (.s) (` T C))))))
+stmt (hlinmulsub ((x y) (x z) (T x) (y z) (T y) (T z)) ((/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z))))))))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (-v) C)) (opr (opr A (.s) (` T B)) (-v) (` T C)))))
+stmt (hods ((R x) (S x) (T x)) ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (= (opr R (-P) S) T) (= (opr S (+P) T) R)))
+stmt (hosass ((R x) (S x) (T x)) ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (+P) S) (+P) T) (opr R (+P) (opr S (+P) T))))
+stmt (hos12 () ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr R (+P) (opr S (+P) T)) (opr S (+P) (opr R (+P) T))))
+stmt (hosdir ((R x) (S x) (T x)) ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. (opr R (+P) S) T) (opr (o. R T) (+P) (o. S T))))
+stmt (hoddir ((R x) (S x) (T x)) ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. (opr R (-P) S) T) (opr (o. R T) (-P) (o. S T))))
+stmt (ho2co () ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (= (` (o. (o. R S) T) A) (` R (` S (` T A))))))
+stmt (ho0 () () (= (` (Proj) (0H)) (X. (H~) (0H))))
+stmt (ho1 () () (= (` (Proj) (H~)) (|` (I) (H~))))
+stmt (hoid0 ((T x)) ((:--> T (H~) (H~))) (= (opr T (+P) (` (Proj) (0H))) T))
+stmt (hoid0r () ((:--> T (H~) (H~))) (= (opr (` (Proj) (0H)) (+P) T) T))
+stmt (hodid () ((:--> T (H~) (H~))) (= (opr T (-P) T) (` (Proj) (0H))))
+stmt (hoid1 ((T x)) ((:--> T (H~) (H~))) (= (o. T (` (Proj) (H~))) T))
+stmt (hoid1r ((T x)) ((:--> T (H~) (H~))) (= (o. (` (Proj) (H~)) T) T))
+stmt (hodseq () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+P) (opr T (-P) S)) T))
+stmt (hods0 () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (-P) T) (opr S (+P) (opr (` (Proj) (0H)) (-P) T))))
+stmt (hosdass () ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (+P) S) (-P) T) (opr R (+P) (opr S (-P) T))))
+stmt (hosd () ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (+P) S) (-P) T) (opr (opr R (-P) T) (+P) S)))
+stmt (hosd1 () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+P) T) (opr S (-P) (opr (` (Proj) (0H)) (-P) T))))
+stmt (hosd2 () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr S (+P) T) (opr S (-P) (opr (opr T (-P) T) (-P) T))))
+stmt (pjsdi ((H x) (S x) (T x)) ((e. H (CH)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. (` (Proj) H) (opr S (+P) T)) (opr (o. (` (Proj) H) S) (+P) (o. (` (Proj) H) T))))
+stmt (pjddi ((H x) (S x) (T x)) ((e. H (CH)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (o. (` (Proj) H) (opr S (-P) T)) (opr (o. (` (Proj) H) S) (-P) (o. (` (Proj) H) T))))
+stmt (pjsdi2 () ((e. H (CH)) (:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (= (o. R (opr S (+P) T)) (opr (o. R S) (+P) (o. R T))) (= (o. (o. (` (Proj) H) R) (opr S (+P) T)) (opr (o. (o. (` (Proj) H) R) S) (+P) (o. (o. (` (Proj) H) R) T)))))
+stmt (pjco () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (= (` (o. (` (Proj) G) (` (Proj) H)) A) (` (` (Proj) G) (` (` (Proj) H) A)))))
+stmt (pjcocl () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (e. (` (o. (` (Proj) G) (` (Proj) H)) A) G)))
+stmt (pjcohcl () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (e. (` (o. (` (Proj) G) (` (Proj) H)) A) (H~))))
+stmt (pjadjco () ((e. G (CH)) (e. H (CH))) (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr (` (o. (` (Proj) G) (` (Proj) H)) A) (.i) B) (opr A (.i) (` (o. (` (Proj) H) (` (Proj) G)) B)))))
+stmt (pjcofn () ((e. G (CH)) (e. H (CH))) (Fn (o. (` (Proj) G) (` (Proj) H)) (H~)))
+stmt (pjss1co ((H x) (G x)) ((e. G (CH)) (e. H (CH))) (<-> (C_ G H) (= (o. (` (Proj) H) (` (Proj) G)) (` (Proj) G))))
+stmt (pjss2co ((x y) (H x) (H y) (G x) (G y)) ((e. G (CH)) (e. H (CH))) (<-> (C_ G H) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) G))))
+stmt (pjssmt () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (-> (C_ H G) (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)))))
+stmt (pjssge0t () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (-> (= (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (` (` (Proj) (i^i G (` (_|_) H))) A)) (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)))))
+stmt (pjdifnormt () ((e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (<-> (br (0) (<_) (opr (opr (` (` (Proj) G) A) (-v) (` (` (Proj) H) A)) (.i) A)) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) (` (` (Proj) G) A))))))
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+stmt (pjorthco ((H x) (G x)) ((e. G (CH)) (e. H (CH))) (-> (C_ G (` (_|_) H)) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) (0H)))))
+stmt (pjscj ((H x) (G x)) ((e. G (CH)) (e. H (CH))) (-> (C_ G (` (_|_) H)) (= (` (Proj) (opr G (vH) H)) (opr (` (Proj) G) (+P) (` (Proj) H)))))
+stmt (pjssum () ((e. G (CH)) (e. H (CH))) (-> (C_ G (` (_|_) H)) (= (` (Proj) (opr G (+H) H)) (opr (` (Proj) G) (+P) (` (Proj) H)))))
+stmt (pjidmco () ((e. H (CH))) (= (o. (` (Proj) H) (` (Proj) H)) (` (Proj) H)))
+stmt (pjocco () ((e. H (CH))) (= (o. (` (Proj) H) (` (Proj) (` (_|_) H))) (` (Proj) (0H))))
+stmt (pjtot ((H x)) ((e. H (CH))) (= (opr (` (Proj) H) (+P) (` (Proj) (` (_|_) H))) (` (Proj) (H~))))
+stmt (pjoc () ((e. H (CH))) (= (opr (` (Proj) (H~)) (-P) (` (Proj) H)) (` (Proj) (` (_|_) H))))
+stmt (pjin1 () ((e. G (CH)) (e. H (CH))) (= (` (Proj) (i^i G H)) (o. (` (Proj) G) (` (Proj) (i^i G H)))))
+stmt (pjin2 () ((e. G (CH)) (e. H (CH))) (<-> (/\ (= (` (Proj) G) (o. (` (Proj) G) (` (Proj) H))) (= (` (Proj) H) (o. (` (Proj) H) (` (Proj) G)))) (= (` (Proj) G) (` (Proj) H))))
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+stmt (pjclem1 () ((e. G (CH)) (e. H (CH))) (-> (br G (C_H) H) (= (o. (` (Proj) G) (` (Proj) H)) (` (Proj) (i^i G H)))))
+stmt (pjclem2 () ((e. G (CH)) (e. H (CH))) (-> (br G (C_H) H) (= (o. (` (Proj) G) (` (Proj) H)) (o. (` (Proj) H) (` (Proj) G)))))
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+stmt (pjclem4a () ((e. G (CH)) (e. H (CH))) (-> (e. A (i^i G H)) (= (` (o. (` (Proj) G) (` (Proj) H)) A) A)))
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+stmt (pjcohocl () ((e. H (CH)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (o. (` (Proj) H) T) A) H)))
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+stmt (pj2cocl () ((e. F (CH)) (e. G (CH)) (e. H (CH))) (-> (e. A (H~)) (e. (` (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) A) F)))
+stmt (pj3lem1 () ((e. F (CH)) (e. G (CH)) (e. H (CH))) (-> (e. A (i^i (i^i F G) H)) (= (` (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) A) A)))
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+stmt (pj3 () ((e. F (CH)) (e. G (CH)) (e. H (CH))) (-> (/\ (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) H) (` (Proj) G)) (` (Proj) F))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (o. (o. (` (Proj) G) (` (Proj) F)) (` (Proj) H)))) (= (o. (o. (` (Proj) F) (` (Proj) G)) (` (Proj) H)) (` (Proj) (i^i (i^i F G) H)))))
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+stmt (pjopytht () () (-> (/\/\ (e. H (CH)) (e. G (CH)) (e. A (H~))) (-> (C_ H (` (_|_) G)) (= (opr (` (norm) (opr (` (` (Proj) H) A) (+v) (` (` (Proj) G) A))) (^) (2)) (opr (opr (` (norm) (` (` (Proj) H) A)) (^) (2)) (+) (opr (` (norm) (` (` (Proj) G) A)) (^) (2)))))))
+stmt (pjnorm () ((e. H (CH)) (e. A (H~))) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) A)))
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+stmt (pjnel () ((e. H (CH)) (e. A (H~))) (<-> (-. (e. A H)) (br (` (norm) (` (` (Proj) H) A)) (<) (` (norm) A))))
+stmt (pjnormt () () (-> (/\ (e. H (CH)) (e. A (H~))) (br (` (norm) (` (` (Proj) H) A)) (<_) (` (norm) A))))
+stmt (pjnelt () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (-. (e. A H)) (br (` (norm) (` (` (Proj) H) A)) (<) (` (norm) A)))))
+stmt (pjelt () () (-> (/\ (e. H (CH)) (e. A (H~))) (<-> (e. A H) (= (` (norm) (` (` (Proj) H) A)) (` (norm) A)))))
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+stmt (stelt ((x y) (f x) (f y) (S x) (S y) (S f)) () (<-> (e. S (States)) (/\ (/\ (:--> S (CH) (RR)) (A.e. x (CH) (/\ (br (0) (<_) (` S (cv x))) (br (` S (cv x)) (<_) (1))))) (/\ (= (` S (H~)) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (= (` S (opr (cv x) (vH) (cv y))) (opr (` S (cv x)) (+) (` S (cv y)))))))))))
+stmt (stclt ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (e. (` S A) (RR)))))
+stmt (stge0t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (0) (<_) (` S A)))))
+stmt (stle1t ((x y) (A x) (A y) (S x) (S y)) () (-> (e. S (States)) (-> (e. A (CH)) (br (` S A) (<_) (1)))))
+stmt (sthil ((x y) (S x) (S y)) () (-> (e. S (States)) (= (` S (H~)) (1))))
+stmt (stjt ((x y) (A x) (A y) (S x) (S y) (B y)) () (-> (e. S (States)) (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (C_ A (` (_|_) B))) (= (` S (opr A (vH) B)) (opr (` S A) (+) (` S B))))))
+stmt (sto1 () ((e. A (CH))) (-> (e. S (States)) (= (opr (` S A) (+) (` S (` (_|_) A))) (1))))
+stmt (sto2 () ((e. A (CH))) (-> (e. S (States)) (= (` S (` (_|_) A)) (opr (1) (-) (` S A)))))
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+stmt (chrelat ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (-> (C: A B) (E.e. x (Atoms) (/\ (C: A (opr A (vH) (cv x))) (C_ (opr A (vH) (cv x)) B)))))
+stmt (chrelat2 ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B))))))
+stmt (cvat ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (-> (br A (<o) B) (E.e. x (Atoms) (= (opr A (vH) (cv x)) B))))
+stmt (cvbr3 ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br A (<o) B) (/\ (C: A B) (E.e. x (Atoms) (= (opr A (vH) (cv x)) B)))))
+stmt (cvexchlem ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B))))
+stmt (cvexch () ((e. A (CH)) (e. B (CH))) (<-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B))))
+stmt (chrelat2t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (-. (C_ A B)) (E.e. x (Atoms) (/\ (C_ (cv x) A) (-. (C_ (cv x) B)))))))
+stmt (chrelat3t ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B))))))
+stmt (chrelat3 ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (C_ A B) (A.e. x (Atoms) (-> (C_ (cv x) A) (C_ (cv x) B)))))
+stmt (chrelat4 ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (= A B) (A.e. x (Atoms) (<-> (C_ (cv x) A) (C_ (cv x) B)))))
+stmt (cvexcht () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (i^i A B) (<o) B) (br A (<o) (opr A (vH) B)))))
+stmt (cvp () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (= (i^i A B) (0H)) (br A (<o) (opr A (vH) B)))))
+stmt (atnssm0 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (<-> (-. (C_ B A)) (= (i^i A B) (0H)))))
+stmt (atnem0 () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (-. (= A B)) (= (i^i A B) (0H)))))
+stmt (atcv0eq () () (-> (/\ (e. A (Atoms)) (e. B (Atoms))) (<-> (br (0H) (<o) (opr A (vH) B)) (= A B))))
+stmt (atcv1 () () (-> (/\ (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (br A (<o) (opr B (vH) C))) (<-> (= A (0H)) (= B C))))
+stmt (atexch () () (-> (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (C_ B (opr A (vH) C)) (= (i^i A B) (0H))) (C_ C (opr A (vH) B)))))
+stmt (atoml ((A x) (B x)) ((e. A (CH))) (-> (e. B (Atoms)) (e. (i^i (opr A (vH) B) (` (_|_) A)) (u. (Atoms) ({} (0H))))))
+stmt (atcvatlem ((A x) (B x) (C x)) ((e. A (CH))) (-> (/\ (/\ (e. B (Atoms)) (e. C (Atoms))) (/\ (-. (= A (0H))) (C: A (opr B (vH) C)))) (-> (-. (C_ B A)) (e. A (Atoms)))))
+stmt (atcvat () ((e. A (CH))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C: A (opr B (vH) C))) (e. A (Atoms)))))
+stmt (atcvat2 () ((e. A (CH))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A (<o) (opr B (vH) C))) (e. A (Atoms)))))
+stmt (atcvat2t () () (-> (/\/\ (e. A (CH)) (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= B C)) (br A (<o) (opr B (vH) C))) (e. A (Atoms)))))
+stmt (atcvat3 () ((e. A (CH))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (/\ (-. (= B C)) (-. (C_ C A))) (C_ B (opr A (vH) C))) (e. (i^i A (opr B (vH) C)) (Atoms)))))
+stmt (atcvat4 ((A x) (B x) (C x)) ((e. A (CH))) (-> (/\ (e. B (Atoms)) (e. C (Atoms))) (-> (/\ (-. (= A (0H))) (C_ B (opr A (vH) C))) (E.e. x (Atoms) (/\ (C_ (cv x) A) (C_ B (opr C (vH) (cv x))))))))
+stmt (atdmd () () (-> (/\ (e. A (Atoms)) (e. B (CH))) (br (` (_|_) A) (MH) (` (_|_) B))))
+stmt (atmd ((A x) (B x)) () (-> (/\ (e. A (Atoms)) (e. B (CH))) (br A (MH) B)))
+stmt (atmd2 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (br A (MH) B)))
+var ( set p )
+stmt (mdsymlem1 ((A p) (B p)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (/\ (e. (cv p) (CH)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (C_ (cv p) A)))
+stmt (mdsymlem2 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (/\ (e. (cv p) (Atoms)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (-> (-. (= B (0H))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))))
+stmt (mdsymlem3 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (-. (C_ (i^i B C) A))) (C_ (cv p) (opr A (vH) B))) (-. (= A (0H)))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))))
+stmt (mdsymlem4 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (e. (cv p) (Atoms)) (-> (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (/\ (/\ (-. (= A (0H))) (-. (= B (0H)))) (C_ (cv p) (opr A (vH) B)))) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))))
+stmt (mdsymlem5 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) (-> (-. (= (cv q) (cv p))) (-> (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))) (-> (/\ (/\ (e. (cv c) (CH)) (C_ A (cv c))) (e. (cv p) (Atoms))) (-> (C_ (cv p) (cv c)) (C_ (cv p) (opr (i^i (cv c) B) (vH) A))))))))
+stmt (mdsymlem6 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (br (` (_|_) B) (MH) (` (_|_) A))))
+stmt (mdsymlem7 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))))))
+stmt (mdsymlem8 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (br (` (_|_) A) (MH) (` (_|_) B)))))
+stmt (mdsym ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br B (MH) A)))
+stmt (sumdmdi ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (CH)) (e. B (CH))) (-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))))
+stmt (cmmd () ((e. A (CH)) (e. B (CH))) (-> (br A (C_H) B) (br A (MH) B)))
+stmt (sumdmdlem ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C y) (C z) (C w)) ((e. A (CH)) (e. B (CH))) (-> (/\ (e. C (H~)) (-. (e. C (opr A (+H) B)))) (= (i^i (opr B (+H) (` (span) ({} C))) A) (i^i B A))))
+stmt (sumdmdlem2 ((x y) (A x) (A y) (B x) (B y)) ((e. A (CH)) (e. B (CH))) (-> (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) (= (opr A (+H) B) (opr A (vH) B))))
+stmt (sumdmd ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B))))
+stmt (dmdbr4at ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))))
+stmt (cdjreu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (SH)) (e. B (SH))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (E!e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y)))))))
addfile ./zfc/set_mm_ax.ghi
hunk ./zfc/set_mm_ax.ghi 1
+# Converted from set.mm - Version of 22-Oct-2005
+
+# Original by Norm Megill
+
+# This file (specifically, the version of this file with the above date)
+# is placed in the public domain per the Creative Commons Public Domain
+# Dedication.  http://creativecommons.org/licenses/publicdomain/
+
+kind (wff)
+var (wff ph)
+var (wff ps)
+var (wff ch)
+var (wff th)
+var (wff ta)
+term (wff (-. wff))
+term (wff (-> wff wff))
+stmt (ax-1 () () (-> ph (-> ps ph)))
+stmt (ax-2 () () (-> (-> ph (-> ps ch)) (-> (-> ph ps) (-> ph ch))))
+stmt (ax-3 () () (-> (-> (-. ph) (-. ps)) (-> ps ph)))
+stmt (ax-mp () (ph (-> ph ps)) ps)
+term (wff (<-> wff wff))
+stmt (df-bi () () (-. (-> (-> (<-> ph ps) (-. (-> (-> ph ps) (-. (-> ps ph))))) (-. (-> (-. (-> (-> ph ps) (-. (-> ps ph)))) (<-> ph ps))))))
+term (wff (\/ wff wff))
+term (wff (/\ wff wff))
+stmt (df-or () () (<-> (\/ ph ps) (-> (-. ph) ps)))
+stmt (df-an () () (<-> (/\ ph ps) (-. (-> ph (-. ps)))))
+var (wff et)
+term (wff (\/\/ wff wff wff))
+term (wff (/\/\ wff wff wff))
+stmt (df-3or () () (<-> (\/\/ ph ps ch) (\/ (\/ ph ps) ch)))
+stmt (df-3an () () (<-> (/\/\ ph ps ch) (/\ (/\ ph ps) ch)))
+var (wff ze)
+kind (set)
+var (set x)
+var (set y)
+var (set z)
+var (set w)
+var (set v)
+var (set u)
+term (wff (A. set wff))
+stmt (ax-4 () () (-> (A. x ph) ph))
+stmt (ax-5 () () (-> (A. x (-> (A. x ph) ps)) (-> (A. x ph) (A. x ps))))
+stmt (ax-6 () () (-> (-. (A. x (-. (A. x ph)))) ph))
+stmt (ax-7 () () (-> (A. x (A. y ph)) (A. y (A. x ph))))
+stmt (ax-gen () (ph) (A. x ph))
+term (wff (E. set wff))
+stmt (df-ex () () (<-> (E. x ph) (-. (A. x (-. ph)))))
+kind (class)
+term (class (cv set))
+term (wff (= class class))
+stmt (ax-8 () () (-> (= (cv x) (cv y)) (-> (= (cv x) (cv z)) (= (cv y) (cv z)))))
+stmt (ax-9 () () (-. (A. x (-. (= (cv x) (cv y))))))
+stmt (ax-10 () () (-> (A. x (= (cv x) (cv y))) (-> (A. x ph) (A. y ph))))
+stmt (ax-11 () () (-> (-. (A. x (= (cv x) (cv y)))) (-> (= (cv x) (cv y)) (-> ph (A. x (-> (= (cv x) (cv y)) ph))))))
+stmt (ax-12 () () (-> (-. (A. z (= (cv z) (cv x)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> (= (cv x) (cv y)) (A. z (= (cv x) (cv y)))))))
+term (wff (e. class class))
+stmt (ax-13 () () (-> (= (cv x) (cv y)) (-> (e. (cv x) (cv z)) (e. (cv y) (cv z)))))
+stmt (ax-14 () () (-> (= (cv x) (cv y)) (-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))))
+stmt (ax-15 () () (-> (-. (A. z (= (cv z) (cv x)))) (-> (-. (A. z (= (cv z) (cv y)))) (-> (e. (cv x) (cv y)) (A. z (e. (cv x) (cv y)))))))
+term (wff ([/] class set wff))
+stmt (df-sb () () (<-> ([/] (cv y) x ph) (/\ (-> (= (cv x) (cv y)) ph) (E. x (/\ (= (cv x) (cv y)) ph)))))
+stmt (ax-16 ((x y)) () (-> (A. x (= (cv x) (cv y))) (-> ph (A. x ph))))
+stmt (ax-17 ((ph x)) () (-> ph (A. x ph)))
+var (set f)
+var (set g)
+term (wff (E! set wff))
+term (wff (E* set wff))
+stmt (df-eu ((x y) (ph y)) () (<-> (E! x ph) (E. y (A. x (<-> ph (= (cv x) (cv y)))))))
+stmt (df-mo () () (<-> (E* x ph) (-> (E. x ph) (E! x ph))))
+var (set t)
+stmt (ax-ext ((x y) (x z) (y z)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y))))
+stmt (ax-rep ((x y) (x z) (w x) (y z) (w y) (w z)) () (-> (A. w (E. y (A. z (-> (A. y ph) (= (cv z) (cv y)))))) (E. y (A. z (<-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv w) (cv x)) (A. y ph))))))))
+stmt (ax-un ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))))
+stmt (ax-pow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv x)))) (e. (cv z) (cv y))))))
+stmt (ax-reg ((x y) (x z) (y z)) () (-> (E. y (e. (cv y) (cv x))) (E. y (/\ (e. (cv y) (cv x)) (A. z (-> (e. (cv z) (cv y)) (-. (e. (cv z) (cv x)))))))))
+stmt (ax-inf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv y)))))))))
+stmt (ax-ac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v))))))))))
+term (class ({|} set wff))
+var (class A)
+var (class B)
+var (class C)
+var (class D)
+var (class R)
+var (class S)
+var (class T)
+stmt (df-clab () () (<-> (e. (cv x) ({|} y ph)) ([/] (cv x) y ph)))
+stmt (df-cleq ((A x) (B x) (x y) (x z) (y z)) ((-> (A. x (<-> (e. (cv x) (cv y)) (e. (cv x) (cv z)))) (= (cv y) (cv z)))) (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B)))))
+stmt (df-clel ((A x) (B x)) () (<-> (e. A B) (E. x (/\ (= (cv x) A) (e. (cv x) B)))))
+var (class F)
+var (class G)
+term (wff (=/= class class))
+term (wff (e/ class class))
+stmt (df-ne () () (<-> (=/= A B) (-. (= A B))))
+stmt (df-nel () () (<-> (e/ A B) (-. (e. A B))))
+term (wff (A.e. set class wff))
+term (wff (E.e. set class wff))
+term (wff (E!e. set class wff))
+term (class ({e.|} set class wff))
+stmt (df-ral () () (<-> (A.e. x A ph) (A. x (-> (e. (cv x) A) ph))))
+stmt (df-rex () () (<-> (E.e. x A ph) (E. x (/\ (e. (cv x) A) ph))))
+stmt (df-reu () () (<-> (E!e. x A ph) (E! x (/\ (e. (cv x) A) ph))))
+stmt (df-rab () () (= ({e.|} x A ph) ({|} x (/\ (e. (cv x) A) ph))))
+term (class (V))
+stmt (df-v () () (= (V) ({|} x (= (cv x) (cv x)))))
+stmt (df-sbc () () (<-> ([/] A x ph) (e. A ({|} x ph))))
+term (class (\ class class))
+term (class (u. class class))
+term (class (i^i class class))
+term (wff (C_ class class))
+term (wff (C: class class))
+stmt (df-dif ((A x) (B x)) () (= (\ A B) ({|} x (/\ (e. (cv x) A) (-. (e. (cv x) B))))))
+stmt (df-un ((A x) (B x)) () (= (u. A B) ({|} x (\/ (e. (cv x) A) (e. (cv x) B)))))
+stmt (df-in ((A x) (B x)) () (= (i^i A B) ({|} x (/\ (e. (cv x) A) (e. (cv x) B)))))
+stmt (df-ss () () (<-> (C_ A B) (= (i^i A B) A)))
+stmt (df-pss () () (<-> (C: A B) (/\ (C_ A B) (=/= A B))))
+term (class ({/}))
+stmt (df-nul () () (= ({/}) (\ (V) (V))))
+term (class (if wff class class))
+stmt (df-if ((ph x) (A x) (B x)) () (= (if ph A B) ({|} x (\/ (/\ (e. (cv x) A) ph) (/\ (e. (cv x) B) (-. ph))))))
+var (wff si)
+var (wff rh)
+term (class (P~ class))
+stmt (df-pw ((A x)) () (= (P~ A) ({|} x (C_ (cv x) A))))
+term (class ({} class))
+term (class ({,} class class))
+term (class (<,> class class))
+stmt (df-sn ((A x)) () (= ({} A) ({|} x (= (cv x) A))))
+stmt (df-pr () () (= ({,} A B) (u. ({} A) ({} B))))
+term (class ({,,} class class class))
+stmt (df-tp () () (= ({,,} A B C) (u. ({,} A B) ({} C))))
+stmt (df-op () () (= (<,> A B) ({,} ({} A) ({,} A B))))
+term (class (U. class))
+stmt (df-uni ((x y) (A x) (A y)) () (= (U. A) ({|} x (E. y (/\ (e. (cv x) (cv y)) (e. (cv y) A))))))
+term (class (|^| class))
+stmt (df-int ((x y) (A x) (A y)) () (= (|^| A) ({|} x (A. y (-> (e. (cv y) A) (e. (cv x) (cv y)))))))
+term (class (U_ set class class))
+term (class (|^|_ set class class))
+stmt (df-iun ((x y) (A y) (B y)) () (= (U_ x A B) ({|} y (E.e. x A (e. (cv y) B)))))
+stmt (df-iin ((x y) (A y) (B y)) () (= (|^|_ x A B) ({|} y (A.e. x A (e. (cv y) B)))))
+term (wff (Tr class))
+stmt (df-tr () () (<-> (Tr A) (C_ (U. A) A)))
+term (wff (br class class class))
+term (class ({<,>|} set set wff))
+term (class (E))
+term (class (I))
+term (wff (Po class class))
+term (wff (Or class class))
+term (class (sup class class class))
+term (wff (Fr class class))
+term (wff (We class class))
+stmt (df-br () () (<-> (br A R B) (e. (<,> A B) R)))
+stmt (df-opab ((x z) (y z) (ph z)) () (= ({<,>|} x y ph) ({|} z (E. x (E. y (/\ (= (cv z) (<,> (cv x) (cv y))) ph))))))
+stmt (df-eprel ((x y)) () (= (E) ({<,>|} x y (e. (cv x) (cv y)))))
+stmt (df-id ((x y)) () (= (I) ({<,>|} x y (= (cv x) (cv y)))))
+stmt (df-po ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) () (<-> (Po R A) (A.e. x A (A.e. y A (A.e. z A (/\ (-. (br (cv x) R (cv x))) (-> (/\ (br (cv x) R (cv y)) (br (cv y) R (cv z))) (br (cv x) R (cv z)))))))))
+stmt (df-so ((x y) (R x) (R y) (A x) (A y)) () (<-> (Or R A) (/\ (Po R A) (A.e. x A (A.e. y A (\/\/ (br (cv x) R (cv y)) (= (cv x) (cv y)) (br (cv y) R (cv x))))))))
+stmt (df-sup ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z) (B x) (B y) (B z)) () (= (sup B A R) (U. ({e.|} x A (/\ (A.e. y B (-. (br (cv x) R (cv y)))) (A.e. y A (-> (br (cv y) R (cv x)) (E.e. z B (br (cv y) R (cv z))))))))))
+stmt (df-fr ((x y) (x z) (R x) (y z) (R y) (R z) (A x) (A y) (A z)) () (<-> (Fr R A) (A. x (-> (/\ (C_ (cv x) A) (-. (= (cv x) ({/})))) (E.e. y (cv x) (A.e. z (cv x) (-. (br (cv z) R (cv y)))))))))
+stmt (df-we () () (<-> (We R A) (/\ (Fr R A) (Or R A))))
+term (wff (Ord class))
+term (class (On))
+term (wff (Lim class))
+term (class (suc class))
+stmt (df-ord () () (<-> (Ord A) (/\ (Tr A) (We (E) A))))
+stmt (df-on () () (= (On) ({|} x (Ord (cv x)))))
+stmt (df-lim () () (<-> (Lim A) (/\/\ (Ord A) (-. (= A ({/}))) (= A (U. A)))))
+stmt (df-suc () () (= (suc A) (u. A ({} A))))
+term (class (om))
+stmt (df-om ((x y)) () (= (om) ({|} x (/\ (Ord (cv x)) (A. y (-> (Lim (cv y)) (e. (cv x) (cv y))))))))
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+var (set j)
+var (set k)
+var (set m)
+var (set n)
+var (class H)
+var (class J)
+var (class K)
+var (class M)
+var (class N)
+term (class (X. class class))
+term (class (`' class))
+term (class (dom class))
+term (class (ran class))
+term (class (|` class class))
+term (class (" class class))
+term (class (o. class class))
+term (wff (Rel class))
+term (wff (Fun class))
+term (wff (Fn class class))
+term (wff (:--> class class class))
+term (wff (:-1-1-> class class class))
+term (wff (:-onto-> class class class))
+term (wff (:-1-1-onto-> class class class))
+term (class (` class class))
+term (wff (Isom class class class class class))
+stmt (df-xp ((x y) (A x) (A y) (B x) (B y)) () (= (X. A B) ({<,>|} x y (/\ (e. (cv x) A) (e. (cv y) B)))))
+stmt (df-rel () () (<-> (Rel A) (C_ A (X. (V) (V)))))
+stmt (df-cnv ((x y) (A x) (A y)) () (= (`' A) ({<,>|} x y (br (cv y) A (cv x)))))
+stmt (df-co ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (= (o. A B) ({<,>|} x y (E. z (/\ (br (cv x) B (cv z)) (br (cv z) A (cv y)))))))
+stmt (df-dm ((x y) (A x) (A y)) () (= (dom A) ({|} x (E. y (br (cv x) A (cv y))))))
+stmt (df-rn () () (= (ran A) (dom (`' A))))
+stmt (df-res () () (= (|` A B) (i^i A (X. B (V)))))
+stmt (df-ima () () (= (" A B) (ran (|` A B))))
+stmt (df-fun () () (<-> (Fun A) (/\ (Rel A) (C_ (o. A (`' A)) (I)))))
+stmt (df-fn () () (<-> (Fn A B) (/\ (Fun A) (= (dom A) B))))
+stmt (df-f () () (<-> (:--> F A B) (/\ (Fn F A) (C_ (ran F) B))))
+stmt (df-f1 () () (<-> (:-1-1-> F A B) (/\ (:--> F A B) (Fun (`' F)))))
+stmt (df-fo () () (<-> (:-onto-> F A B) (/\ (Fn F A) (= (ran F) B))))
+stmt (df-f1o () () (<-> (:-1-1-onto-> F A B) (/\ (:-1-1-> F A B) (:-onto-> F A B))))
+stmt (df-fv ((A x) (F x)) () (= (` F A) (U. ({|} x (= (" F ({} A)) ({} (cv x)))))))
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+term (class (rec class class))
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+term (class (opr class class class))
+term (class ({<<,>,>|} set set set wff))
+stmt (df-opr () () (= (opr A F B) (` F (<,> A B))))
+stmt (df-oprab ((w x) (w y) (w z) (ph w)) () (= ({<<,>,>|} x y z ph) ({|} w (E. x (E. y (E. z (/\ (= (cv w) (<,> (<,> (cv x) (cv y)) (cv z))) ph)))))))
+term (class (1st))
+term (class (2nd))
+stmt (df-1st ((x y)) () (= (1st) ({<,>|} x y (= (cv y) (U. (dom ({} (cv x))))))))
+stmt (df-2nd ((x y)) () (= (2nd) ({<,>|} x y (= (cv y) (U. (ran ({} (cv x))))))))
+term (class (1o))
+term (class (2o))
+term (class (+o))
+term (class (.o))
+term (class (^o))
+stmt (df-1o () () (= (1o) (suc ({/}))))
+stmt (df-2o () () (= (2o) (suc (1o))))
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+stmt (df-omul ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (= (.o) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (On)) (e. (cv y) (On))) (= (cv z) (` (rec ({<,>|} w v (= (cv v) (opr (cv w) (+o) (cv x)))) ({/})) (cv y)))))))
+stmt (df-oexp ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (= (^o) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (On)) (e. (cv y) (On))) (= (cv z) (if (= (cv x) ({/})) (\ (1o) (cv y)) (` (rec ({<,>|} w v (= (cv v) (opr (cv w) (.o) (cv x)))) (1o)) (cv y))))))))
+term (wff (Er class))
+term (class ([] class class))
+term (class (/. class class))
+stmt (df-er () () (<-> (Er R) (C_ (u. (`' R) (o. R R)) R)))
+stmt (df-ec () () (= ([] A R) (" R ({} A))))
+stmt (df-qs ((x y) (A x) (A y) (R x) (R y)) () (= (/. A R) ({|} y (E.e. x A (= (cv y) ([] (cv x) R))))))
+var (class Q)
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+var (set b)
+var (set c)
+var (set d)
+var (class L)
+var (class W)
+var (class X)
+var (class Y)
+var (class Z)
+term (class (^m))
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+term (class (~~))
+term (class (~<_))
+term (class (~<))
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+stmt (df-dom ((x y) (f x) (f y)) () (= (~<_) ({<,>|} x y (E. f (:-1-1-> (cv f) (cv x) (cv y))))))
+stmt (df-sdom () () (= (~<) (\ (~<_) (~~))))
+term (class (R1))
+term (class (rank))
+stmt (df-r1 ((x y)) () (= (R1) (rec ({<,>|} x y (= (cv y) (P~ (cv x)))) ({/}))))
+stmt (df-rank ((x y) (x z) (y z)) () (= (rank) ({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (e. (cv x) (` (R1) (suc (cv z))))))))))
+var (set q)
+term (class (card))
+term (class (aleph))
+term (class (cf))
+stmt (df-card ((x y) (x z) (y z)) () (= (card) ({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (br (cv z) (~~) (cv x))))))))
+stmt (df-aleph ((x y) (x z) (y z)) () (= (aleph) (rec ({<,>|} x y (= (cv y) (|^| ({e.|} z (On) (br (cv x) (~<) (cv z)))))) (om))))
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+term (class (+c))
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+term (class (N.))
+term (class (+N))
+term (class (.N))
+term (class (<N))
+term (class (+pQ))
+term (class (.pQ))
+term (class (~Q))
+term (class (Q.))
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+term (class (.Q))
+term (class (*Q))
+term (class (<Q))
+term (class (P.))
+term (class (1P))
+term (class (+P.))
+term (class (.P))
+term (class (<P))
+term (class (+pR))
+term (class (.pR))
+term (class (~R))
+term (class (R.))
+term (class (0R))
+term (class (1R))
+term (class (-1R))
+term (class (+R))
+term (class (.R))
+term (class (<R))
+stmt (df-ni () () (= (N.) (\ (om) ({} ({/})))))
+stmt (df-pli () () (= (+N) (|` (+o) (X. (N.) (N.)))))
+stmt (df-mi () () (= (.N) (|` (.o) (X. (N.) (N.)))))
+stmt (df-lti () () (= (<N) (i^i (E) (X. (N.) (N.)))))
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+stmt (df-enq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (~Q) ({<,>|} x y (/\ (/\ (e. (cv x) (X. (N.) (N.))) (e. (cv y) (X. (N.) (N.)))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) (.N) (cv u)) (opr (cv w) (.N) (cv v))))))))))))
+stmt (df-nq () () (= (Q.) (/. (X. (N.) (N.)) (~Q))))
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+stmt (df-rq ((x y)) () (= (*Q) ({<,>|} x y (/\ (e. (cv x) (Q.)) (= (opr (cv x) (.Q) (cv y)) (1Q))))))
+stmt (df-ltq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (<Q) ({<,>|} x y (/\ (/\ (e. (cv x) (Q.)) (e. (cv y) (Q.))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) ([] (<,> (cv z) (cv w)) (~Q))) (= (cv y) ([] (<,> (cv v) (cv u)) (~Q)))) (br (opr (cv z) (.N) (cv u)) (<N) (opr (cv w) (.N) (cv v))))))))))))
+stmt (df-1q () () (= (1Q) ([] (<,> (1o) (1o)) (~Q))))
+stmt (df-np ((x y) (x z) (y z)) () (= (P.) ({|} x (/\ (/\ (C: ({/}) (cv x)) (C: (cv x) (Q.))) (A.e. y (cv x) (/\ (A. z (-> (br (cv z) (<Q) (cv y)) (e. (cv z) (cv x)))) (E.e. z (cv x) (br (cv y) (<Q) (cv z)))))))))
+stmt (df-1p () () (= (1P) ({|} x (br (cv x) (<Q) (1Q)))))
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+stmt (df-mp ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (.P) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (P.)) (e. (cv y) (P.))) (= (cv z) ({|} w (E.e. v (cv x) (E.e. u (cv y) (= (cv w) (opr (cv v) (.Q) (cv u)))))))))))
+stmt (df-ltp ((x y)) () (= (<P) ({<,>|} x y (/\ (/\ (e. (cv x) (P.)) (e. (cv y) (P.))) (C: (cv x) (cv y))))))
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+stmt (df-0r () () (= (0R) ([] (<,> (1P) (1P)) (~R))))
+stmt (df-1r () () (= (1R) ([] (<,> (opr (1P) (+P.) (1P)) (1P)) (~R))))
+stmt (df-m1r () () (= (-1R) ([] (<,> (1P) (opr (1P) (+P.) (1P))) (~R))))
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+term (class (+))
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+stmt (df-1 () () (= (1) (<,> (1R) (0R))))
+stmt (df-i () () (= (i) (<,> (0R) (1R))))
+stmt (df-r () () (= (RR) (X. (R.) ({} (0R)))))
+stmt (df-plus ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (+) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (cv w) (+R) (cv u)) (opr (cv v) (+R) (cv f)))))))))))))
+stmt (df-mul ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (x.) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) (<,> (opr (opr (cv w) (.R) (cv u)) (+R) (opr (-1R) (.R) (opr (cv v) (.R) (cv f)))) (opr (opr (cv v) (.R) (cv u)) (+R) (opr (cv w) (.R) (cv f))))))))))))))
+stmt (df-lt ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (<RR) ({<,>|} x y (/\ (/\ (e. (cv x) (RR)) (e. (cv y) (RR))) (E. z (E. w (/\ (/\ (= (cv x) (<,> (cv z) (0R))) (= (cv y) (<,> (cv w) (0R)))) (br (cv z) (<R) (cv w)))))))))
+term (class (-))
+term (class (-u class))
+term (class (/))
+term (class (<_))
+term (class (NN))
+term (class (NN0))
+term (class (ZZ))
+term (class (QQ))
+stmt (df-sub ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (-) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (= (cv z) (U. ({e.|} w (CC) (= (opr (cv y) (+) (cv w)) (cv x)))))))))
+stmt (df-neg () () (= (-u A) (opr (0) (-) A)))
+stmt (df-div ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (/) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (\ (CC) ({} (0))))) (= (cv z) (U. ({e.|} w (CC) (= (opr (cv y) (x.) (cv w)) (cv x)))))))))
+term (class (+oo))
+term (class (-oo))
+term (class (RR*))
+term (class (<))
+stmt (df-pnf () () (= (+oo) (CC)))
+stmt (df-mnf () () (= (-oo) ({} (CC))))
+stmt (df-xr () () (= (RR*) (u. (RR) ({,} (+oo) (-oo)))))
+stmt (df-ltxr () () (= (<) (u. (u. (i^i (<RR) (X. (RR) (RR))) ({} (<,> (-oo) (+oo)))) (u. (X. (RR) ({} (+oo))) (X. ({} (-oo)) (RR))))))
+stmt (df-le () () (= (<_) (\ (X. (RR*) (RR*)) (`' (<)))))
+stmt (df-n ((x y)) () (= (NN) (|^| ({|} x (/\ (e. (1) (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x))))))))
+term (class (2))
+term (class (3))
+term (class (4))
+term (class (5))
+term (class (6))
+term (class (7))
+term (class (8))
+term (class (9))
+stmt (df-2 () () (= (2) (opr (1) (+) (1))))
+stmt (df-3 () () (= (3) (opr (2) (+) (1))))
+stmt (df-4 () () (= (4) (opr (3) (+) (1))))
+stmt (df-5 () () (= (5) (opr (4) (+) (1))))
+stmt (df-6 () () (= (6) (opr (5) (+) (1))))
+stmt (df-7 () () (= (7) (opr (6) (+) (1))))
+stmt (df-8 () () (= (8) (opr (7) (+) (1))))
+stmt (df-9 () () (= (9) (opr (8) (+) (1))))
+stmt (df-n0 () () (= (NN0) (u. (NN) ({} (0)))))
+stmt (df-z () () (= (ZZ) ({e.|} n (RR) (\/\/ (= (cv n) (0)) (e. (cv n) (NN)) (e. (-u (cv n)) (NN))))))
+term (class (floor))
+stmt (df-fl ((x y) (x z) (y z)) () (= (floor) ({<,>|} x y (/\ (e. (cv x) (RR)) (= (cv y) (U. ({e.|} z (ZZ) (/\ (br (cv z) (<_) (cv x)) (br (cv x) (<) (opr (cv z) (+) (1)))))))))))
+stmt (df-q ((m n) (m x) (n x)) () (= (QQ) ({|} x (E.e. m (ZZ) (E.e. n (NN) (= (cv x) (opr (cv m) (/) (cv n))))))))
+term (class (seq1))
+stmt (df-seq1 ((x y) (x z) (w x) (f x) (g x) (h x) (y z) (w y) (f y) (g y) (h y) (w z) (f z) (g z) (h z) (f w) (g w) (h w) (f g) (f h) (g h)) () (= (seq1) ({<<,>,>|} f g h (= (cv h) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (2nd) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) (cv f) (` (cv g) (opr (` (1st) (cv z)) (+) (1))))))) (<,> (1) (` (cv g) (1)))) (`' (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (cv x))))))))))
+term (class (shift))
+stmt (df-shft ((x y) (x z) (f x) (g x) (y z) (f y) (g y) (f z) (g z) (f g)) () (= (shift) ({<<,>,>|} f x g (= (cv g) ({<,>|} y z (/\ (e. (cv y) (CC)) (= (cv z) (` (cv f) (opr (cv y) (-) (cv x))))))))))
+term (class (ZZ>))
+stmt (df-uz ((j k) (j y) (k y)) () (= (ZZ>) ({<,>|} j y (/\ (e. (cv j) (ZZ)) (= (cv y) ({e.|} k (ZZ) (br (cv j) (<_) (cv k))))))))
+term (class (limsup))
+stmt (df-limsup ((k x) (k y) (k z) (x y) (x z) (y z)) () (= (limsup) ({<,>|} x y (= (cv y) (sup ({|} z (E.e. k (ZZ) (/\ (C_ (ran (|` (cv x) (` (ZZ>) (cv k)))) (RR*)) (= (cv z) (sup (ran (|` (cv x) (` (ZZ>) (cv k)))) (RR*) (<)))))) (RR*) (`' (<)))))))
+term (class (seq))
+term (class (seq0))
+stmt (df-seqz ((g h) (g k) (g x) (h k) (h x) (k x)) () (= (seq) ({<<,>,>|} x g h (= (cv h) (|` (opr (opr (` (2nd) (cv x)) (seq1) (opr (cv g) (shift) (opr (1) (-) (` (1st) (cv x))))) (shift) (opr (` (1st) (cv x)) (-) (1))) ({e.|} k (ZZ) (br (` (1st) (cv x)) (<_) (cv k))))))))
+stmt (df-seq0 ((f g) (f h) (g h)) () (= (seq0) ({<<,>,>|} f g h (= (cv h) (|` (opr (opr (cv f) (seq1) (opr (cv g) (shift) (1))) (shift) (-u (1))) (NN0))))))
+term (class (^))
+stmt (df-exp ((x y) (x z) (y z)) () (= (^) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (NN0))) (= (cv z) (if (= (cv y) (0)) (1) (` (opr (x.) (seq1) (X. (NN) ({} (cv x)))) (cv y))))))))
+term (class (sqr))
+stmt (df-sqr ((x y) (x z) (y z)) () (= (sqr) ({<,>|} x y (/\ (/\ (e. (cv x) (RR)) (br (0) (<_) (cv x))) (= (cv y) (sup ({e.|} z (RR) (/\ (br (0) (<_) (cv z)) (br (opr (cv z) (x.) (cv z)) (<_) (cv x)))) (RR) (<)))))))
+term (class (Re))
+term (class (Im))
+term (class (*))
+term (class (abs))
+stmt (df-re ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (Re) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (U. ({e.|} z (RR) (E.e. w (RR) (= (cv x) (opr (cv z) (+) (opr (cv w) (x.) (i))))))))))))
+stmt (df-im ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (Im) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (U. ({e.|} w (RR) (E.e. z (RR) (= (cv x) (opr (cv z) (+) (opr (cv w) (x.) (i))))))))))))
+stmt (df-cj ((x y)) () (= (*) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (` (Re) (cv x)) (-) (opr (` (Im) (cv x)) (x.) (i))))))))
+stmt (df-abs ((x y)) () (= (abs) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` (sqr) (opr (cv x) (x.) (` (*) (cv x)))))))))
+term (class (!))
+stmt (df-fac () () (= (!) (u. (opr (x.) (seq1) (|` (I) (NN))) ({} (<,> (0) (1))))))
+term (class (C.))
+stmt (df-bc ((k n) (m n) (k m)) () (= (C.) ({<<,>,>|} n k m (/\ (/\ (e. (cv n) (NN0)) (e. (cv k) (ZZ))) (= (cv m) (if (/\ (br (0) (<_) (cv k)) (br (cv k) (<_) (cv n))) (opr (` (!) (cv n)) (/) (opr (` (!) (opr (cv n) (-) (cv k))) (x.) (` (!) (cv k)))) (0)))))))
+term (class (...))
+stmt (df-fz ((m n) (k m) (m z) (k n) (n z) (k z)) () (= (...) ({<<,>,>|} m n z (/\ (/\ (e. (cv m) (ZZ)) (e. (cv n) (ZZ))) (= (cv z) ({e.|} k (ZZ) (/\ (br (cv m) (<_) (cv k)) (br (cv k) (<_) (cv n)))))))))
+term (class (sum))
+stmt (df-fsum ((f x) (x z) (f z)) () (= (sum) ({<<,>,>|} x f z (= (cv z) (if (br (` (1st) (cv x)) (<_) (` (2nd) (cv x))) (` (opr (+) (seq0) (opr (cv f) (shift) (-u (` (1st) (cv x))))) (opr (` (2nd) (cv x)) (-) (` (1st) (cv x)))) (0))))))
+term (class (~~>))
+term (class (~~>OLD))
+stmt (df-climOLD ((x y) (x z) (f x) (w x) (y z) (f y) (w y) (f z) (w z) (f w)) () (= (~~>OLD) ({<,>|} f w (/\ (/\ (:--> (cv f) (NN) (CC)) (e. (cv w) (CC))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (abs) (opr (` (cv f) (cv z)) (-) (cv w))) (<) (cv x)))))))))))
+stmt (df-clim ((j k) (j x) (j y) (f j) (k x) (k y) (f k) (x y) (f x) (f y)) () (= (~~>) ({<,>|} f y (/\ (e. (cv y) (CC)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) (CC)) (br (` (abs) (opr (` (cv f) (cv k)) (-) (cv y))) (<) (cv x))))))))))))
+term (class (sum1oo))
+term (class (sumoo))
+stmt (df-sumOLD ((x y) (x z) (y z)) () (= (sum1oo) ({<,>|} x y (= (cv y) (U. ({|} z (br (opr (+) (seq1) (cv x)) (~~>OLD) (cv z))))))))
+stmt (df-isum ((f n) (n y) (n z) (f y) (f z) (y z)) () (= (sumoo) ({<<,>,>|} n f y (= (cv y) (U. ({|} z (br (opr (<,> (cv n) (+)) (seq) (cv f)) (~~>) (cv z))))))))
+var (class U)
+term (class (+f))
+term (class (.f))
+stmt (df-plf ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (+f) ({<<,>,>|} f g h (= (cv h) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (` (cv f) (cv x)) (+) (` (cv g) (cv x))))))))))
+stmt (df-muf ((f g) (f x) (f y) (f z) (g x) (g y) (g z) (x y) (x z) (y z)) () (= (.f) ({<<,>,>|} z f g (= (cv g) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (cv z) (x.) (` (cv f) (cv x))))))))))
+term (class (e^))
+term (class (e))
+term (class (sin))
+term (class (cos))
+term (class (pi))
+stmt (df-ef ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (e^) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (1) (+) (` (sum1oo) ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr (opr (cv x) (^) (cv z)) (/) (` (!) (cv z)))))))))))))
+stmt (df-e () () (= (e) (` (e^) (1))))
+stmt (df-sin ((x y)) () (= (sin) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (opr (` (e^) (opr (i) (x.) (cv x))) (-) (` (e^) (opr (-u (i)) (x.) (cv x)))) (/) (opr (2) (x.) (i))))))))
+stmt (df-cos ((x y)) () (= (cos) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (opr (opr (` (e^) (opr (i) (x.) (cv x))) (+) (` (e^) (opr (-u (i)) (x.) (cv x)))) (/) (2)))))))
+stmt (df-pi () () (= (pi) (sup ({e.|} x (RR) (/\ (br (0) (<) (cv x)) (= (` (sin) (cv x)) (0)))) (RR) (`' (<)))))
+term (class (H~))
+term (class (+v))
+term (class (.s))
+term (class (0v))
+term (class (-v))
+term (class (.i))
+term (class (norm))
+term (class (Cauchy))
+term (class (~~>v))
+term (class (SH))
+term (class (CH))
+term (class (_|_))
+term (class (+H))
+term (class (span))
+term (class (vH))
+term (class (\/H))
+term (class (0H))
+term (class (C_H))
+term (class (Proj))
+term (class (+P))
+term (class (-P))
+term (class (States))
+term (class (Atoms))
+term (class (<o))
+term (class (MH))
+stmt (ax-hilex () () (e. (H~) (V)))
+stmt (ax-hvaddcl () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (+v) B) (H~))))
+stmt (ax-hvcom () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (+v) B) (opr B (+v) A))))
+stmt (ax-hvass () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (+v) C) (opr A (+v) (opr B (+v) C)))))
+stmt (ax-hv0cl () () (e. (0v) (H~)))
+stmt (ax-hvaddid () () (-> (e. A (H~)) (= (opr A (+v) (0v)) A)))
+stmt (ax-hvmulcl () () (-> (/\ (e. A (CC)) (e. B (H~))) (e. (opr A (.s) B) (H~))))
+stmt (ax-hvmulid () () (-> (e. A (H~)) (= (opr (1) (.s) A) A)))
+stmt (ax-hvmulass () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (x.) B) (.s) C) (opr A (.s) (opr B (.s) C)))))
+stmt (ax-hvdistr1 () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr A (.s) (opr B (+v) C)) (opr (opr A (.s) B) (+v) (opr A (.s) C)))))
+stmt (ax-hvdistr2 () () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (H~))) (= (opr (opr A (+) B) (.s) C) (opr (opr A (.s) C) (+v) (opr B (.s) C)))))
+stmt (ax-hvmul0 () () (-> (e. A (H~)) (= (opr (0) (.s) A) (0v))))
+stmt (df-hvsub ((x y) (x z) (y z)) () (= (-v) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (H~)) (e. (cv y) (H~))) (= (cv z) (opr (cv x) (+v) (opr (-u (1)) (.s) (cv y))))))))
+stmt (ax-hicl () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (.i) B) (CC))))
+stmt (ax-his1 () () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (.i) B) (` (*) (opr B (.i) A)))))
+stmt (ax-his2 () () (-> (/\/\ (e. A (H~)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (+v) B) (.i) C) (opr (opr A (.i) C) (+) (opr B (.i) C)))))
+stmt (ax-his3 () () (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (opr (opr A (.s) B) (.i) C) (opr A (x.) (opr B (.i) C)))))
+stmt (ax-his4 () () (-> (/\ (e. A (H~)) (=/= A (0v))) (br (0) (<) (opr A (.i) A))))
+stmt (df-hnorm ((x y)) () (= (norm) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (` (sqr) (opr (cv x) (.i) (cv x))))))))
+stmt (df-hcau ((x y) (x z) (w x) (f x) (y z) (w y) (f y) (w z) (f z) (f w)) () (= (Cauchy) ({|} f (/\ (:--> (cv f) (NN) (H~)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (A.e. w (NN) (-> (/\ (br (cv y) (<_) (cv z)) (br (cv y) (<_) (cv w))) (br (` (norm) (opr (` F (cv z)) (-v) (` F (cv w)))) (<) (cv x))))))))))))
+stmt (df-hlim ((x y) (x z) (f x) (y z) (f y) (f z)) () (= (~~>v) ({<,>|} f w (/\ (/\ (:--> (cv f) (NN) (H~)) (e. (cv w) (H~))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<_) (cv z)) (br (` (norm) (opr (` (cv f) (cv z)) (-v) (cv w))) (<) (cv x)))))))))))
+stmt (ax-hcompl ((F x)) () (-> (e. F (Cauchy)) (E.e. x (H~) (br F (~~>v) (cv x)))))
+stmt (df-sh ((x y) (h x) (h y)) () (= (SH) ({|} h (/\ (/\ (C_ (cv h) (H~)) (e. (0v) (cv h))) (/\ (A.e. x (cv h) (A.e. y (cv h) (e. (opr (cv x) (+v) (cv y)) (cv h)))) (A.e. x (CC) (A.e. y (cv h) (e. (opr (cv x) (.s) (cv y)) (cv h)))))))))
+stmt (df-ch ((f x) (h x) (f h)) () (= (CH) ({|} h (/\ (e. (cv h) (SH)) (A. f (A. x (-> (/\ (:--> (cv f) (NN) (cv h)) (br (cv f) (~~>v) (cv x))) (e. (cv x) (cv h)))))))))
+stmt (df-oc ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (_|_) ({<,>|} x y (/\ (C_ (cv x) (H~)) (= (cv y) ({e.|} z (H~) (A.e. w (cv x) (= (opr (cv z) (.i) (cv w)) (0)))))))))
+stmt (df-ch0 () () (= (0H) ({} (0v))))
+stmt (df-pj ((f h) (h x) (h y) (h z) (h w) (f x) (f y) (f z) (f w) (x y) (x z) (w x) (y z) (w y) (w z)) () (= (Proj) ({<,>|} h f (/\ (e. (cv h) (CH)) (= (cv f) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (U. ({e.|} z (cv h) (E.e. w (` (_|_) (cv h)) (= (cv x) (opr (cv z) (+v) (cv w))))))))))))))
+stmt (df-shsum ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (+H) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (SH)) (e. (cv y) (SH))) (= (cv z) ({e.|} w (H~) (E.e. v (cv x) (E.e. u (cv y) (= (cv w) (opr (cv v) (+v) (cv u)))))))))))
+stmt (df-span ((x y) (x z) (y z)) () (= (span) ({<,>|} x y (/\ (C_ (cv x) (H~)) (= (cv y) (|^| ({e.|} z (SH) (C_ (cv x) (cv z)))))))))
+stmt (df-chj ((x y) (x z) (y z)) () (= (vH) ({<<,>,>|} x y z (/\ (/\ (C_ (cv x) (H~)) (C_ (cv y) (H~))) (= (cv z) (` (_|_) (` (_|_) (u. (cv x) (cv y)))))))))
+stmt (df-chsup ((x y)) () (= (\/H) ({<,>|} x y (/\ (C_ (cv x) (P~ (H~))) (= (cv y) (` (_|_) (` (_|_) (U. (cv x)))))))))
+stmt (df-hosum ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (+P) ({<<,>,>|} f g h (/\ (/\ (:--> (cv f) (H~) (H~)) (:--> (cv g) (H~) (H~))) (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` (cv f) (cv x)) (+v) (` (cv g) (cv x)))))))))))
+stmt (df-hodif ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (-P) ({<<,>,>|} f g h (/\ (/\ (:--> (cv f) (H~) (H~)) (:--> (cv g) (H~) (H~))) (= (cv h) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (opr (` (cv f) (cv x)) (-v) (` (cv g) (cv x)))))))))))
+stmt (df-cm ((x y)) () (= (C_H) ({<,>|} x y (/\ (/\ (e. (cv x) (CH)) (e. (cv y) (CH))) (= (cv x) (opr (i^i (cv x) (cv y)) (vH) (i^i (cv x) (` (_|_) (cv y)))))))))
+stmt (df-st ((f x) (f y) (x y)) () (= (States) ({|} f (/\ (/\ (:--> (cv f) (CH) (RR)) (A.e. x (CH) (/\ (br (0) (<_) (` (cv f) (cv x))) (br (` (cv f) (cv x)) (<_) (1))))) (/\ (= (` (cv f) (H~)) (1)) (A.e. x (CH) (A.e. y (CH) (-> (C_ (cv x) (` (_|_) (cv y))) (= (` (cv f) (opr (cv x) (vH) (cv y))) (opr (` (cv f) (cv x)) (+) (` (cv f) (cv y))))))))))))
+stmt (df-cv ((x y) (x z) (y z)) () (= (<o) ({<,>|} x y (/\ (/\ (e. (cv x) (CH)) (e. (cv y) (CH))) (/\ (C: (cv x) (cv y)) (-. (E.e. z (CH) (/\ (C: (cv x) (cv z)) (C: (cv z) (cv y))))))))))
+stmt (df-md ((x y) (x z) (y z)) () (= (MH) ({<,>|} x y (/\ (/\ (e. (cv x) (CH)) (e. (cv y) (CH))) (A.e. z (CH) (-> (C_ (cv z) (cv y)) (= (i^i (opr (cv z) (vH) (cv x)) (cv y)) (opr (cv z) (vH) (i^i (cv x) (cv y))))))))))
+stmt (df-at () () (= (Atoms) ({e.|} x (CH) (br (0H) (<o) (cv x)))))
+var (set p)
+# 1466775 Metamath steps converted to 435691 Ghilbert steps
}
