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Theorem evlfcllem 14318
Description: Lemma for evlfcl 14319. (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 2436 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2436 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2436 . . . 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 446 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 evlfcl.h . . . . 5  |-  ( ph  ->  ( H  e.  ( C  Func  D )  /\  Z  e.  ( Base `  C ) ) )
1110simpld 446 . . . 4  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
128simprd 450 . . . 4  |-  ( ph  ->  X  e.  ( Base `  C ) )
1310simprd 450 . . . 4  |-  ( ph  ->  Z  e.  ( Base `  C ) )
14 eqid 2436 . . . 4  |-  ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. )
15 evlfcl.q . . . . 5  |-  Q  =  ( C FuncCat  D )
16 eqid 2436 . . . . 5  |-  (comp `  Q )  =  (comp `  Q )
17 evlfcl.a . . . . . 6  |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X (  Hom  `  C
) Y ) ) )
1817simpld 446 . . . . 5  |-  ( ph  ->  A  e.  ( F N G ) )
19 evlfcl.b . . . . . 6  |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y (  Hom  `  C
) Z ) ) )
2019simpld 446 . . . . 5  |-  ( ph  ->  B  e.  ( G N H ) )
2115, 7, 16, 18, 20fuccocl 14161 . . . 4  |-  ( ph  ->  ( B ( <. F ,  G >. (comp `  Q ) H ) A )  e.  ( F N H ) )
22 eqid 2436 . . . . 5  |-  (comp `  C )  =  (comp `  C )
23 evlfcl.g . . . . . 6  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  Y  e.  ( Base `  C ) ) )
2423simprd 450 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
2517simprd 450 . . . . 5  |-  ( ph  ->  K  e.  ( X (  Hom  `  C
) Y ) )
2619simprd 450 . . . . 5  |-  ( ph  ->  L  e.  ( Y (  Hom  `  C
) Z ) )
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 13910 . . . 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 14316 . . 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 14160 . . . 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 6096 . . 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 14059 . . . . . . 7  |-  Rel  ( C  Func  D )
32 1st2ndbr 6396 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3331, 9, 32sylancr 645 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 14068 . . . . 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 6097 . . . 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 14148 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
377, 36, 4, 5, 6, 24, 13, 26nati 14152 . . . . . . . 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 6097 . . . . . . 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 2436 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
40 eqid 2436 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
414, 39, 33funcf1 14063 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
4241, 24ffvelrnd 5871 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Y )  e.  ( Base `  D
) )
4341, 13ffvelrnd 5871 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Z )  e.  ( Base `  D
) )
4423simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
45 1st2ndbr 6396 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4631, 44, 45sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
474, 39, 46funcf1 14063 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
4847, 13ffvelrnd 5871 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( Base `  D
) )
494, 5, 40, 33, 24, 13funcf2 14065 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  F ) Z ) : ( Y (  Hom  `  C ) Z ) --> ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
5049, 26ffvelrnd 5871 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  F ) Z ) `  L
)  e.  ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
517, 36, 4, 40, 13natcl 14150 . . . . . . . 8  |-  ( ph  ->  ( A `  Z
)  e.  ( ( ( 1st `  F
) `  Z )
(  Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
52 1st2ndbr 6396 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
5331, 11, 52sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
544, 39, 53funcf1 14063 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
5554, 13ffvelrnd 5871 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  H
) `  Z )  e.  ( Base `  D
) )
567, 20nat1st2nd 14148 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
577, 56, 4, 40, 13natcl 14150 . . . . . . . 8  |-  ( ph  ->  ( B `  Z
)  e.  ( ( ( 1st `  G
) `  Z )
(  Hom  `  D ) ( ( 1st `  H
) `  Z )
) )
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 13911 . . . . . . 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 5871 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( Base `  D
) )
607, 36, 4, 40, 24natcl 14150 . . . . . . . 8  |-  ( ph  ->  ( A `  Y
)  e.  ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  G
) `  Y )
) )
614, 5, 40, 46, 24, 13funcf2 14065 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  G ) Z ) : ( Y (  Hom  `  C ) Z ) --> ( ( ( 1st `  G
) `  Y )
(  Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6261, 26ffvelrnd 5871 . . . . . . . 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 13911 . . . . . . 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 2478 . . . . . 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 6096 . . . . 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 5871 . . . . . 6  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  D
) )
674, 5, 40, 33, 12, 24funcf2 14065 . . . . . . 7  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X (  Hom  `  C ) Y ) --> ( ( ( 1st `  F
) `  X )
(  Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6867, 25ffvelrnd 5871 . . . . . 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 13910 . . . . . 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 13911 . . . . 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 13910 . . . . . 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 13911 . . . . 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 2476 . . . 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 2468 . . 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 2472 . 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 2436 . . . . 5  |-  ( Q  X.c  C )  =  ( Q  X.c  C )
7715fucbas 14157 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
7815, 7fuchom 14158 . . . . 5  |-  N  =  (  Hom  `  Q
)
79 eqid 2436 . . . . 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 14284 . . . 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 5732 . . 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 6084 . . 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 2486 . 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 6084 . . . . . 6  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
851, 2, 3, 4, 9, 12evlf1 14317 . . . . . 6  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
8684, 85syl5eqr 2482 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. F ,  X >. )  =  ( ( 1st `  F
) `  X )
)
87 df-ov 6084 . . . . . 6  |-  ( G ( 1st `  E
) Y )  =  ( ( 1st `  E
) `  <. G ,  Y >. )
881, 2, 3, 4, 44, 24evlf1 14317 . . . . . 6  |-  ( ph  ->  ( G ( 1st `  E ) Y )  =  ( ( 1st `  G ) `  Y
) )
8987, 88syl5eqr 2482 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. G ,  Y >. )  =  ( ( 1st `  G
) `  Y )
)
9086, 89opeq12d 3992 . . . 4  |-  ( ph  -> 
<. ( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >.  =  <. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. )
91 df-ov 6084 . . . . 5  |-  ( H ( 1st `  E
) Z )  =  ( ( 1st `  E
) `  <. H ,  Z >. )
921, 2, 3, 4, 11, 13evlf1 14317 . . . . 5  |-  ( ph  ->  ( H ( 1st `  E ) Z )  =  ( ( 1st `  H ) `  Z
) )
9391, 92syl5eqr 2482 . . . 4  |-  ( ph  ->  ( ( 1st `  E
) `  <. H ,  Z >. )  =  ( ( 1st `  H
) `  Z )
)
9490, 93oveq12d 6099 . . 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 6084 . . . 4  |-  ( B ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) L )  =  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )
96 eqid 2436 . . . . 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 14316 . . . 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 2482 . . 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 6084 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )
100 eqid 2436 . . . . 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 14316 . . . 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 2482 . . 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 6102 . 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 2478 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889    Func cfunc 14051   Nat cnat 14138   FuncCat cfuc 14139    X.c cxpc 14265   evalF cevlf 14306
This theorem is referenced by:  evlfcl  14319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-hom 13553  df-cco 13554  df-cat 13893  df-func 14055  df-nat 14140  df-fuc 14141  df-xpc 14269  df-evlf 14310
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