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Theorem evlfcllem 15344
Description: Lemma for evlfcl 15345. (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 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2467 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2467 . . . 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 2467 . . . 4  |-  ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. )
15 evlfcl.q . . . . 5  |-  Q  =  ( C FuncCat  D )
16 eqid 2467 . . . . 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 15187 . . . 4  |-  ( ph  ->  ( B ( <. F ,  G >. (comp `  Q ) H ) A )  e.  ( F N H ) )
22 eqid 2467 . . . . 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 14936 . . . 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 15342 . . 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 15186 . . . 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 6297 . . 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 15085 . . . . . . 7  |-  Rel  ( C  Func  D )
32 1st2ndbr 6830 . . . . . . 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 15094 . . . . 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 6298 . . . 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 15174 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
377, 36, 4, 5, 6, 24, 13, 26nati 15178 . . . . . . . 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 6298 . . . . . . 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 2467 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
40 eqid 2467 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
414, 39, 33funcf1 15089 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
4241, 24ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Y )  e.  ( Base `  D
) )
4341, 13ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Z )  e.  ( Base `  D
) )
4423simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
45 1st2ndbr 6830 . . . . . . . . . . 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 15089 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
4847, 13ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( Base `  D
) )
494, 5, 40, 33, 24, 13funcf2 15091 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  F ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
5049, 26ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  F ) Z ) `  L
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
517, 36, 4, 40, 13natcl 15176 . . . . . . . 8  |-  ( ph  ->  ( A `  Z
)  e.  ( ( ( 1st `  F
) `  Z )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
52 1st2ndbr 6830 . . . . . . . . . . 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 15089 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
5554, 13ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  H
) `  Z )  e.  ( Base `  D
) )
567, 20nat1st2nd 15174 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
577, 56, 4, 40, 13natcl 15176 . . . . . . . 8  |-  ( ph  ->  ( B `  Z
)  e.  ( ( ( 1st `  G
) `  Z )
( Hom  `  D ) ( ( 1st `  H
) `  Z )
) )
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 14937 . . . . . . 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 6020 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( Base `  D
) )
607, 36, 4, 40, 24natcl 15176 . . . . . . . 8  |-  ( ph  ->  ( A `  Y
)  e.  ( ( ( 1st `  F
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Y )
) )
614, 5, 40, 46, 24, 13funcf2 15091 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  G ) Z ) : ( Y ( Hom  `  C ) Z ) --> ( ( ( 1st `  G
) `  Y )
( Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6261, 26ffvelrnd 6020 . . . . . . . 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 14937 . . . . . . 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 2518 . . . . . 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 6297 . . . . 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 6020 . . . . . 6  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  D
) )
674, 5, 40, 33, 12, 24funcf2 15091 . . . . . . 7  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X ( Hom  `  C ) Y ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6867, 25ffvelrnd 6020 . . . . . 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 14936 . . . . . 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 14937 . . . . 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 14936 . . . . . 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 14937 . . . . 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 2516 . . . 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 2508 . . 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 2512 . 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 2467 . . . . 5  |-  ( Q  X.c  C )  =  ( Q  X.c  C )
7715fucbas 15183 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
7815, 7fuchom 15184 . . . . 5  |-  N  =  ( Hom  `  Q
)
79 eqid 2467 . . . . 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 15310 . . . 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 5868 . . 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 6285 . . 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 2526 . 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 6285 . . . . . 6  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
851, 2, 3, 4, 9, 12evlf1 15343 . . . . . 6  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
8684, 85syl5eqr 2522 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. F ,  X >. )  =  ( ( 1st `  F
) `  X )
)
87 df-ov 6285 . . . . . 6  |-  ( G ( 1st `  E
) Y )  =  ( ( 1st `  E
) `  <. G ,  Y >. )
881, 2, 3, 4, 44, 24evlf1 15343 . . . . . 6  |-  ( ph  ->  ( G ( 1st `  E ) Y )  =  ( ( 1st `  G ) `  Y
) )
8987, 88syl5eqr 2522 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. G ,  Y >. )  =  ( ( 1st `  G
) `  Y )
)
9086, 89opeq12d 4221 . . . 4  |-  ( ph  -> 
<. ( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >.  =  <. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. )
91 df-ov 6285 . . . . 5  |-  ( H ( 1st `  E
) Z )  =  ( ( 1st `  E
) `  <. H ,  Z >. )
921, 2, 3, 4, 11, 13evlf1 15343 . . . . 5  |-  ( ph  ->  ( H ( 1st `  E ) Z )  =  ( ( 1st `  H ) `  Z
) )
9391, 92syl5eqr 2522 . . . 4  |-  ( ph  ->  ( ( 1st `  E
) `  <. H ,  Z >. )  =  ( ( 1st `  H
) `  Z )
)
9490, 93oveq12d 6300 . . 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 6285 . . . 4  |-  ( B ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) L )  =  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )
96 eqid 2467 . . . . 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 15342 . . . 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 2522 . . 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 6285 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )
100 eqid 2467 . . . . 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 15342 . . . 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 2522 . . 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 6303 . 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 2518 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 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915    Func cfunc 15077   Nat cnat 15164   FuncCat cfuc 15165    X.c cxpc 15291   evalF cevlf 15332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-hom 14575  df-cco 14576  df-cat 14919  df-func 15081  df-nat 15166  df-fuc 15167  df-xpc 15295  df-evlf 15336
This theorem is referenced by:  evlfcl  15345
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