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Theorem evlfcllem 15036
Description: Lemma for evlfcl 15037. (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 2443 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2443 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2443 . . . 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 459 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 evlfcl.h . . . . 5  |-  ( ph  ->  ( H  e.  ( C  Func  D )  /\  Z  e.  ( Base `  C ) ) )
1110simpld 459 . . . 4  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
128simprd 463 . . . 4  |-  ( ph  ->  X  e.  ( Base `  C ) )
1310simprd 463 . . . 4  |-  ( ph  ->  Z  e.  ( Base `  C ) )
14 eqid 2443 . . . 4  |-  ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. )
15 evlfcl.q . . . . 5  |-  Q  =  ( C FuncCat  D )
16 eqid 2443 . . . . 5  |-  (comp `  Q )  =  (comp `  Q )
17 evlfcl.a . . . . . 6  |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X ( Hom  `  C
) Y ) ) )
1817simpld 459 . . . . 5  |-  ( ph  ->  A  e.  ( F N G ) )
19 evlfcl.b . . . . . 6  |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y ( Hom  `  C
) Z ) ) )
2019simpld 459 . . . . 5  |-  ( ph  ->  B  e.  ( G N H ) )
2115, 7, 16, 18, 20fuccocl 14879 . . . 4  |-  ( ph  ->  ( B ( <. F ,  G >. (comp `  Q ) H ) A )  e.  ( F N H ) )
22 eqid 2443 . . . . 5  |-  (comp `  C )  =  (comp `  C )
23 evlfcl.g . . . . . 6  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  Y  e.  ( Base `  C ) ) )
2423simprd 463 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
2517simprd 463 . . . . 5  |-  ( ph  ->  K  e.  ( X ( Hom  `  C
) Y ) )
2619simprd 463 . . . . 5  |-  ( ph  ->  L  e.  ( Y ( Hom  `  C
) Z ) )
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 14628 . . . 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 15034 . . 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 14878 . . . 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 6111 . . 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 14777 . . . . . . 7  |-  Rel  ( C  Func  D )
32 1st2ndbr 6628 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3331, 9, 32sylancr 663 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 14786 . . . . 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 6112 . . . 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 14866 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
377, 36, 4, 5, 6, 24, 13, 26nati 14870 . . . . . . . 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 6112 . . . . . . 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 2443 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
40 eqid 2443 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
414, 39, 33funcf1 14781 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
4241, 24ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Y )  e.  ( Base `  D
) )
4341, 13ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Z )  e.  ( Base `  D
) )
4423simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
45 1st2ndbr 6628 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4631, 44, 45sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
474, 39, 46funcf1 14781 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
4847, 13ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( Base `  D
) )
494, 5, 40, 33, 24, 13funcf2 14783 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  F ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
5049, 26ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  F ) Z ) `  L
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
517, 36, 4, 40, 13natcl 14868 . . . . . . . 8  |-  ( ph  ->  ( A `  Z
)  e.  ( ( ( 1st `  F
) `  Z )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
52 1st2ndbr 6628 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
5331, 11, 52sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
544, 39, 53funcf1 14781 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
5554, 13ffvelrnd 5849 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  H
) `  Z )  e.  ( Base `  D
) )
567, 20nat1st2nd 14866 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
577, 56, 4, 40, 13natcl 14868 . . . . . . . 8  |-  ( ph  ->  ( B `  Z
)  e.  ( ( ( 1st `  G
) `  Z )
( Hom  `  D ) ( ( 1st `  H
) `  Z )
) )
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 14629 . . . . . . 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 5849 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( Base `  D
) )
607, 36, 4, 40, 24natcl 14868 . . . . . . . 8  |-  ( ph  ->  ( A `  Y
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Y )
) )
614, 5, 40, 46, 24, 13funcf2 14783 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  G ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  G
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6261, 26ffvelrnd 5849 . . . . . . . 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 14629 . . . . . . 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 2485 . . . . . 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 6111 . . . . 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 5849 . . . . . 6  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  D
) )
674, 5, 40, 33, 12, 24funcf2 14783 . . . . . . 7  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X ( Hom  `  C ) Y ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6867, 25ffvelrnd 5849 . . . . . 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 14628 . . . . . 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 14629 . . . . 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 14628 . . . . . 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 14629 . . . . 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 2483 . . . 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 2475 . . 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 2479 . 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 2443 . . . . 5  |-  ( Q  X.c  C )  =  ( Q  X.c  C )
7715fucbas 14875 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
7815, 7fuchom 14876 . . . . 5  |-  N  =  ( Hom  `  Q
)
79 eqid 2443 . . . . 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 15002 . . . 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 5700 . . 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 6099 . . 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 2493 . 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 6099 . . . . . 6  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
851, 2, 3, 4, 9, 12evlf1 15035 . . . . . 6  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
8684, 85syl5eqr 2489 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. F ,  X >. )  =  ( ( 1st `  F
) `  X )
)
87 df-ov 6099 . . . . . 6  |-  ( G ( 1st `  E
) Y )  =  ( ( 1st `  E
) `  <. G ,  Y >. )
881, 2, 3, 4, 44, 24evlf1 15035 . . . . . 6  |-  ( ph  ->  ( G ( 1st `  E ) Y )  =  ( ( 1st `  G ) `  Y
) )
8987, 88syl5eqr 2489 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. G ,  Y >. )  =  ( ( 1st `  G
) `  Y )
)
9086, 89opeq12d 4072 . . . 4  |-  ( ph  -> 
<. ( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >.  =  <. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. )
91 df-ov 6099 . . . . 5  |-  ( H ( 1st `  E
) Z )  =  ( ( 1st `  E
) `  <. H ,  Z >. )
921, 2, 3, 4, 11, 13evlf1 15035 . . . . 5  |-  ( ph  ->  ( H ( 1st `  E ) Z )  =  ( ( 1st `  H ) `  Z
) )
9391, 92syl5eqr 2489 . . . 4  |-  ( ph  ->  ( ( 1st `  E
) `  <. H ,  Z >. )  =  ( ( 1st `  H
) `  Z )
)
9490, 93oveq12d 6114 . . 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 6099 . . . 4  |-  ( B ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) L )  =  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )
96 eqid 2443 . . . . 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 15034 . . . 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 2489 . . 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 6099 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )
100 eqid 2443 . . . . 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 15034 . . . 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 2489 . . 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 6117 . 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 2485 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 369    = wceq 1369    e. wcel 1756   <.cop 3888   class class class wbr 4297   Rel wrel 4850   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Basecbs 14179   Hom chom 14254  compcco 14255   Catccat 14607    Func cfunc 14769   Nat cnat 14856   FuncCat cfuc 14857    X.c cxpc 14983   evalF cevlf 15024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-hom 14267  df-cco 14268  df-cat 14611  df-func 14773  df-nat 14858  df-fuc 14859  df-xpc 14987  df-evlf 15028
This theorem is referenced by:  evlfcl  15037
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