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Theorem evlfcllem 15814
Description: Lemma for evlfcl 15815. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfcl.e  |-  E  =  ( C evalF  D )
evlfcl.q  |-  Q  =  ( C FuncCat  D )
evlfcl.c  |-  ( ph  ->  C  e.  Cat )
evlfcl.d  |-  ( ph  ->  D  e.  Cat )
evlfcl.n  |-  N  =  ( C Nat  D )
evlfcl.f  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  X  e.  ( Base `  C ) ) )
evlfcl.g  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  Y  e.  ( Base `  C ) ) )
evlfcl.h  |-  ( ph  ->  ( H  e.  ( C  Func  D )  /\  Z  e.  ( Base `  C ) ) )
evlfcl.a  |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X ( Hom  `  C
) Y ) ) )
evlfcl.b  |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y ( Hom  `  C
) Z ) ) )
Assertion
Ref Expression
evlfcllem  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) `  ( <. B ,  L >. (
<. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. )
<. A ,  K >. ) )  =  ( ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. ) ( <.
( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >. (comp `  D ) ( ( 1st `  E ) `
 <. H ,  Z >. ) ) ( (
<. F ,  X >. ( 2nd `  E )
<. G ,  Y >. ) `
 <. A ,  K >. ) ) )

Proof of Theorem evlfcllem
StepHypRef Expression
1 evlfcl.e . . . 4  |-  E  =  ( C evalF  D )
2 evlfcl.c . . . 4  |-  ( ph  ->  C  e.  Cat )
3 evlfcl.d . . . 4  |-  ( ph  ->  D  e.  Cat )
4 eqid 2402 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2402 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2402 . . . 4  |-  (comp `  D )  =  (comp `  D )
7 evlfcl.n . . . 4  |-  N  =  ( C Nat  D )
8 evlfcl.f . . . . 5  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  X  e.  ( Base `  C ) ) )
98simpld 457 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 evlfcl.h . . . . 5  |-  ( ph  ->  ( H  e.  ( C  Func  D )  /\  Z  e.  ( Base `  C ) ) )
1110simpld 457 . . . 4  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
128simprd 461 . . . 4  |-  ( ph  ->  X  e.  ( Base `  C ) )
1310simprd 461 . . . 4  |-  ( ph  ->  Z  e.  ( Base `  C ) )
14 eqid 2402 . . . 4  |-  ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. )
15 evlfcl.q . . . . 5  |-  Q  =  ( C FuncCat  D )
16 eqid 2402 . . . . 5  |-  (comp `  Q )  =  (comp `  Q )
17 evlfcl.a . . . . . 6  |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X ( Hom  `  C
) Y ) ) )
1817simpld 457 . . . . 5  |-  ( ph  ->  A  e.  ( F N G ) )
19 evlfcl.b . . . . . 6  |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y ( Hom  `  C
) Z ) ) )
2019simpld 457 . . . . 5  |-  ( ph  ->  B  e.  ( G N H ) )
2115, 7, 16, 18, 20fuccocl 15577 . . . 4  |-  ( ph  ->  ( B ( <. F ,  G >. (comp `  Q ) H ) A )  e.  ( F N H ) )
22 eqid 2402 . . . . 5  |-  (comp `  C )  =  (comp `  C )
23 evlfcl.g . . . . . 6  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  Y  e.  ( Base `  C ) ) )
2423simprd 461 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
2517simprd 461 . . . . 5  |-  ( ph  ->  K  e.  ( X ( Hom  `  C
) Y ) )
2619simprd 461 . . . . 5  |-  ( ph  ->  L  e.  ( Y ( Hom  `  C
) Z ) )
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 15299 . . . 4  |-  ( ph  ->  ( L ( <. X ,  Y >. (comp `  C ) Z ) K )  e.  ( X ( Hom  `  C
) Z ) )
281, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 21, 27evlf2val 15812 . . 3  |-  ( ph  ->  ( ( B (
<. F ,  G >. (comp `  Q ) H ) A ) ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. ) ( L ( <. X ,  Y >. (comp `  C ) Z ) K ) )  =  ( ( ( B ( <. F ,  G >. (comp `  Q ) H ) A ) `
 Z ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( X ( 2nd `  F
) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) ) ) )
2915, 7, 4, 6, 16, 18, 20, 13fuccoval 15576 . . . 4  |-  ( ph  ->  ( ( B (
<. F ,  G >. (comp `  Q ) H ) A ) `  Z
)  =  ( ( B `  Z ) ( <. ( ( 1st `  F ) `  Z
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( A `  Z ) ) )
3029oveq1d 6293 . . 3  |-  ( ph  ->  ( ( ( B ( <. F ,  G >. (comp `  Q ) H ) A ) `
 Z ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( X ( 2nd `  F
) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( X ( 2nd `  F
) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) ) ) )
31 relfunc 15475 . . . . . . 7  |-  Rel  ( C  Func  D )
32 1st2ndbr 6833 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3331, 9, 32sylancr 661 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 15484 . . . . 5  |-  ( ph  ->  ( ( X ( 2nd `  F ) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) )  =  ( ( ( Y ( 2nd `  F ) Z ) `  L
) ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  F ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
3534oveq2d 6294 . . . 4  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( X ( 2nd `  F
) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( ( Y ( 2nd `  F
) Z ) `  L ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  F ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
367, 18nat1st2nd 15564 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
377, 36, 4, 5, 6, 24, 13, 26nati 15568 . . . . . . . 8  |-  ( ph  ->  ( ( A `  Z ) ( <.
( ( 1st `  F
) `  Y ) ,  ( ( 1st `  F ) `  Z
) >. (comp `  D
) ( ( 1st `  G ) `  Z
) ) ( ( Y ( 2nd `  F
) Z ) `  L ) )  =  ( ( ( Y ( 2nd `  G
) Z ) `  L ) ( <.
( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  G ) `  Z
) ) ( A `
 Y ) ) )
3837oveq2d 6294 . . . . . . 7  |-  ( ph  ->  ( ( B `  Z ) ( <.
( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Z ) ( <. ( ( 1st `  F ) `  Y
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  G
) `  Z )
) ( ( Y ( 2nd `  F
) Z ) `  L ) ) )  =  ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( ( Y ( 2nd `  G ) Z ) `
 L ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  G ) `  Z
) ) ( A `
 Y ) ) ) )
39 eqid 2402 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
40 eqid 2402 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
414, 39, 33funcf1 15479 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
4241, 24ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Y )  e.  ( Base `  D
) )
4341, 13ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Z )  e.  ( Base `  D
) )
4423simpld 457 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
45 1st2ndbr 6833 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4631, 44, 45sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
474, 39, 46funcf1 15479 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
4847, 13ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( Base `  D
) )
494, 5, 40, 33, 24, 13funcf2 15481 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  F ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
5049, 26ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  F ) Z ) `  L
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
517, 36, 4, 40, 13natcl 15566 . . . . . . . 8  |-  ( ph  ->  ( A `  Z
)  e.  ( ( ( 1st `  F
) `  Z )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
52 1st2ndbr 6833 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
5331, 11, 52sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
544, 39, 53funcf1 15479 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
5554, 13ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  H
) `  Z )  e.  ( Base `  D
) )
567, 20nat1st2nd 15564 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
577, 56, 4, 40, 13natcl 15566 . . . . . . . 8  |-  ( ph  ->  ( B `  Z
)  e.  ( ( ( 1st `  G
) `  Z )
( Hom  `  D ) ( ( 1st `  H
) `  Z )
) )
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 15300 . . . . . . 7  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  Y
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( Y ( 2nd `  F
) Z ) `  L ) )  =  ( ( B `  Z ) ( <.
( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Z ) ( <. ( ( 1st `  F ) `  Y
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  G
) `  Z )
) ( ( Y ( 2nd `  F
) Z ) `  L ) ) ) )
5947, 24ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( Base `  D
) )
607, 36, 4, 40, 24natcl 15566 . . . . . . . 8  |-  ( ph  ->  ( A `  Y
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Y )
) )
614, 5, 40, 46, 24, 13funcf2 15481 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  G ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  G
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6261, 26ffvelrnd 6010 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  G ) Z ) `  L
)  e.  ( ( ( 1st `  G
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6339, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57catass 15300 . . . . . . 7  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Y ) )  =  ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( ( Y ( 2nd `  G ) Z ) `
 L ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  G ) `  Z
) ) ( A `
 Y ) ) ) )
6438, 58, 633eqtr4d 2453 . . . . . 6  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  Y
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( Y ( 2nd `  F
) Z ) `  L ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Y ) ) )
6564oveq1d 6293 . . . . 5  |-  ( ph  ->  ( ( ( ( B `  Z ) ( <. ( ( 1st `  F ) `  Z
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( A `  Z ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  F ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  F
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) )  =  ( ( ( ( B `  Z ) ( <. ( ( 1st `  G ) `  Y
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Y ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
6641, 12ffvelrnd 6010 . . . . . 6  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  D
) )
674, 5, 40, 33, 12, 24funcf2 15481 . . . . . . 7  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X ( Hom  `  C ) Y ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6867, 25ffvelrnd 6010 . . . . . 6  |-  ( ph  ->  ( ( X ( 2nd `  F ) Y ) `  K
)  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6939, 40, 6, 3, 43, 48, 55, 51, 57catcocl 15299 . . . . . 6  |-  ( ph  ->  ( ( B `  Z ) ( <.
( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) )  e.  ( ( ( 1st `  F ) `
 Z ) ( Hom  `  D )
( ( 1st `  H
) `  Z )
) )
7039, 40, 6, 3, 66, 42, 43, 68, 50, 55, 69catass 15300 . . . . 5  |-  ( ph  ->  ( ( ( ( B `  Z ) ( <. ( ( 1st `  F ) `  Z
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( A `  Z ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  F ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  F
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( ( Y ( 2nd `  F
) Z ) `  L ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  F ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
7139, 40, 6, 3, 59, 48, 55, 62, 57catcocl 15299 . . . . . 6  |-  ( ph  ->  ( ( B `  Z ) ( <.
( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) )  e.  ( ( ( 1st `  G ) `  Y
) ( Hom  `  D
) ( ( 1st `  H ) `  Z
) ) )
7239, 40, 6, 3, 66, 42, 59, 68, 60, 55, 71catass 15300 . . . . 5  |-  ( ph  ->  ( ( ( ( B `  Z ) ( <. ( ( 1st `  G ) `  Y
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  Y ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Y ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( X ( 2nd `  F
) Y ) `  K ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
7365, 70, 723eqtr3d 2451 . . . 4  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( ( Y ( 2nd `  F
) Z ) `  L ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  F ) `  Z
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) )  =  ( ( ( B `  Z ) ( <. ( ( 1st `  G ) `  Y
) ,  ( ( 1st `  G ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
7435, 73eqtrd 2443 . . 3  |-  ( ph  ->  ( ( ( B `
 Z ) (
<. ( ( 1st `  F
) `  Z ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( A `
 Z ) ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Z ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) ( ( X ( 2nd `  F
) Z ) `  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
7528, 30, 743eqtrd 2447 . 2  |-  ( ph  ->  ( ( B (
<. F ,  G >. (comp `  Q ) H ) A ) ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. ) ( L ( <. X ,  Y >. (comp `  C ) Z ) K ) )  =  ( ( ( B `
 Z ) (
<. ( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
76 eqid 2402 . . . . 5  |-  ( Q  X.c  C )  =  ( Q  X.c  C )
7715fucbas 15573 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
7815, 7fuchom 15574 . . . . 5  |-  N  =  ( Hom  `  Q
)
79 eqid 2402 . . . . 5  |-  (comp `  ( Q  X.c  C )
)  =  (comp `  ( Q  X.c  C )
)
8076, 77, 4, 78, 5, 9, 12, 44, 24, 16, 22, 79, 11, 13, 18, 25, 20, 26xpcco2 15780 . . . 4  |-  ( ph  ->  ( <. B ,  L >. ( <. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. )
<. A ,  K >. )  =  <. ( B (
<. F ,  G >. (comp `  Q ) H ) A ) ,  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) >. )
8180fveq2d 5853 . . 3  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) `  ( <. B ,  L >. (
<. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. )
<. A ,  K >. ) )  =  ( (
<. F ,  X >. ( 2nd `  E )
<. H ,  Z >. ) `
 <. ( B (
<. F ,  G >. (comp `  Q ) H ) A ) ,  ( L ( <. X ,  Y >. (comp `  C
) Z ) K ) >. ) )
82 df-ov 6281 . . 3  |-  ( ( B ( <. F ,  G >. (comp `  Q
) H ) A ) ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) ( L (
<. X ,  Y >. (comp `  C ) Z ) K ) )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) `  <. ( B ( <. F ,  G >. (comp `  Q
) H ) A ) ,  ( L ( <. X ,  Y >. (comp `  C ) Z ) K )
>. )
8381, 82syl6eqr 2461 . 2  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) `  ( <. B ,  L >. (
<. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. )
<. A ,  K >. ) )  =  ( ( B ( <. F ,  G >. (comp `  Q
) H ) A ) ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) ( L (
<. X ,  Y >. (comp `  C ) Z ) K ) ) )
84 df-ov 6281 . . . . . 6  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
851, 2, 3, 4, 9, 12evlf1 15813 . . . . . 6  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
8684, 85syl5eqr 2457 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. F ,  X >. )  =  ( ( 1st `  F
) `  X )
)
87 df-ov 6281 . . . . . 6  |-  ( G ( 1st `  E
) Y )  =  ( ( 1st `  E
) `  <. G ,  Y >. )
881, 2, 3, 4, 44, 24evlf1 15813 . . . . . 6  |-  ( ph  ->  ( G ( 1st `  E ) Y )  =  ( ( 1st `  G ) `  Y
) )
8987, 88syl5eqr 2457 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. G ,  Y >. )  =  ( ( 1st `  G
) `  Y )
)
9086, 89opeq12d 4167 . . . 4  |-  ( ph  -> 
<. ( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >.  =  <. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. )
91 df-ov 6281 . . . . 5  |-  ( H ( 1st `  E
) Z )  =  ( ( 1st `  E
) `  <. H ,  Z >. )
921, 2, 3, 4, 11, 13evlf1 15813 . . . . 5  |-  ( ph  ->  ( H ( 1st `  E ) Z )  =  ( ( 1st `  H ) `  Z
) )
9391, 92syl5eqr 2457 . . . 4  |-  ( ph  ->  ( ( 1st `  E
) `  <. H ,  Z >. )  =  ( ( 1st `  H
) `  Z )
)
9490, 93oveq12d 6296 . . 3  |-  ( ph  ->  ( <. ( ( 1st `  E ) `  <. F ,  X >. ) ,  ( ( 1st `  E ) `  <. G ,  Y >. ) >. (comp `  D )
( ( 1st `  E
) `  <. H ,  Z >. ) )  =  ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  G ) `
 Y ) >.
(comp `  D )
( ( 1st `  H
) `  Z )
) )
95 df-ov 6281 . . . 4  |-  ( B ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) L )  =  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )
96 eqid 2402 . . . . 5  |-  ( <. G ,  Y >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. )
971, 2, 3, 4, 5, 6, 7, 44, 11, 24, 13, 96, 20, 26evlf2val 15812 . . . 4  |-  ( ph  ->  ( B ( <. G ,  Y >. ( 2nd `  E )
<. H ,  Z >. ) L )  =  ( ( B `  Z
) ( <. (
( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) )
9895, 97syl5eqr 2457 . . 3  |-  ( ph  ->  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )  =  ( ( B `  Z
) ( <. (
( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) )
99 df-ov 6281 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )
100 eqid 2402 . . . . 5  |-  ( <. F ,  X >. ( 2nd `  E )
<. G ,  Y >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
1011, 2, 3, 4, 5, 6, 7, 9, 44, 12, 24, 100, 18, 25evlf2val 15812 . . . 4  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  Y >. ) K )  =  ( ( A `  Y
) ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  G ) `  Y
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
10299, 101syl5eqr 2457 . . 3  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )  =  ( ( A `  Y
) ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >. (comp `  D
) ( ( 1st `  G ) `  Y
) ) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
10394, 98, 102oveq123d 6299 . 2  |-  ( ph  ->  ( ( ( <. G ,  Y >. ( 2nd `  E )
<. H ,  Z >. ) `
 <. B ,  L >. ) ( <. (
( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >. (comp `  D ) ( ( 1st `  E ) `
 <. H ,  Z >. ) ) ( (
<. F ,  X >. ( 2nd `  E )
<. G ,  Y >. ) `
 <. A ,  K >. ) )  =  ( ( ( B `  Z ) ( <.
( ( 1st `  G
) `  Y ) ,  ( ( 1st `  G ) `  Z
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( Y ( 2nd `  G
) Z ) `  L ) ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. (comp `  D
) ( ( 1st `  H ) `  Z
) ) ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.
(comp `  D )
( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) ) )
10475, 83, 1033eqtr4d 2453 1  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. ) `  ( <. B ,  L >. (
<. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. )
<. A ,  K >. ) )  =  ( ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. ) ( <.
( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >. (comp `  D ) ( ( 1st `  E ) `
 <. H ,  Z >. ) ) ( (
<. F ,  X >. ( 2nd `  E )
<. G ,  Y >. ) `
 <. A ,  K >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3978   class class class wbr 4395   Rel wrel 4828   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   Hom chom 14920  compcco 14921   Catccat 15278    Func cfunc 15467   Nat cnat 15554   FuncCat cfuc 15555    X.c cxpc 15761   evalF cevlf 15802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-hom 14933  df-cco 14934  df-cat 15282  df-func 15471  df-nat 15556  df-fuc 15557  df-xpc 15765  df-evlf 15806
This theorem is referenced by:  evlfcl  15815
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