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Theorem evlfcllem 14011
Description: Lemma for evlfcl 14012. (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 2296 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2296 . . . 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 445 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
10 evlfcl.h . . . . 5  |-  ( ph  ->  ( H  e.  ( C  Func  D )  /\  Z  e.  ( Base `  C ) ) )
1110simpld 445 . . . 4  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
128simprd 449 . . . 4  |-  ( ph  ->  X  e.  ( Base `  C ) )
1310simprd 449 . . . 4  |-  ( ph  ->  Z  e.  ( Base `  C ) )
14 eqid 2296 . . . 4  |-  ( <. F ,  X >. ( 2nd `  E )
<. H ,  Z >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. H ,  Z >. )
15 evlfcl.q . . . . 5  |-  Q  =  ( C FuncCat  D )
16 eqid 2296 . . . . 5  |-  (comp `  Q )  =  (comp `  Q )
17 evlfcl.a . . . . . 6  |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X (  Hom  `  C
) Y ) ) )
1817simpld 445 . . . . 5  |-  ( ph  ->  A  e.  ( F N G ) )
19 evlfcl.b . . . . . 6  |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y (  Hom  `  C
) Z ) ) )
2019simpld 445 . . . . 5  |-  ( ph  ->  B  e.  ( G N H ) )
2115, 7, 16, 18, 20fuccocl 13854 . . . 4  |-  ( ph  ->  ( B ( <. F ,  G >. (comp `  Q ) H ) A )  e.  ( F N H ) )
22 eqid 2296 . . . . 5  |-  (comp `  C )  =  (comp `  C )
23 evlfcl.g . . . . . 6  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  Y  e.  ( Base `  C ) ) )
2423simprd 449 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
2517simprd 449 . . . . 5  |-  ( ph  ->  K  e.  ( X (  Hom  `  C
) Y ) )
2619simprd 449 . . . . 5  |-  ( ph  ->  L  e.  ( Y (  Hom  `  C
) Z ) )
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 13603 . . . 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 14009 . . 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 13853 . . . 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 5889 . . 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 13752 . . . . . . 7  |-  Rel  ( C  Func  D )
32 1st2ndbr 6185 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3331, 9, 32sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 13761 . . . . 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 5890 . . . 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 13841 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
377, 36, 4, 5, 6, 24, 13, 26nati 13845 . . . . . . . 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 5890 . . . . . . 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 2296 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
40 eqid 2296 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
414, 39, 33funcf1 13756 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
4241, 24ffvelrnd 5682 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Y )  e.  ( Base `  D
) )
4341, 13ffvelrnd 5682 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  Z )  e.  ( Base `  D
) )
4423simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
45 1st2ndbr 6185 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4631, 44, 45sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
474, 39, 46funcf1 13756 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
4847, 13ffvelrnd 5682 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( Base `  D
) )
494, 5, 40, 33, 24, 13funcf2 13758 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  F ) Z ) : ( Y (  Hom  `  C ) Z ) --> ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
5049, 26ffvelrnd 5682 . . . . . . . 8  |-  ( ph  ->  ( ( Y ( 2nd `  F ) Z ) `  L
)  e.  ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  F
) `  Z )
) )
517, 36, 4, 40, 13natcl 13843 . . . . . . . 8  |-  ( ph  ->  ( A `  Z
)  e.  ( ( ( 1st `  F
) `  Z )
(  Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
52 1st2ndbr 6185 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
5331, 11, 52sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
544, 39, 53funcf1 13756 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
5554, 13ffvelrnd 5682 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  H
) `  Z )  e.  ( Base `  D
) )
567, 20nat1st2nd 13841 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
577, 56, 4, 40, 13natcl 13843 . . . . . . . 8  |-  ( ph  ->  ( B `  Z
)  e.  ( ( ( 1st `  G
) `  Z )
(  Hom  `  D ) ( ( 1st `  H
) `  Z )
) )
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 13604 . . . . . . 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 5682 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( Base `  D
) )
607, 36, 4, 40, 24natcl 13843 . . . . . . . 8  |-  ( ph  ->  ( A `  Y
)  e.  ( ( ( 1st `  F
) `  Y )
(  Hom  `  D ) ( ( 1st `  G
) `  Y )
) )
614, 5, 40, 46, 24, 13funcf2 13758 . . . . . . . . 9  |-  ( ph  ->  ( Y ( 2nd `  G ) Z ) : ( Y (  Hom  `  C ) Z ) --> ( ( ( 1st `  G
) `  Y )
(  Hom  `  D ) ( ( 1st `  G
) `  Z )
) )
6261, 26ffvelrnd 5682 . . . . . . . 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 13604 . . . . . . 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 2338 . . . . . 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 5889 . . . . 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 5682 . . . . . 6  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  D
) )
674, 5, 40, 33, 12, 24funcf2 13758 . . . . . . 7  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X (  Hom  `  C ) Y ) --> ( ( ( 1st `  F
) `  X )
(  Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
6867, 25ffvelrnd 5682 . . . . . 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 13603 . . . . . 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 13604 . . . . 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 13603 . . . . . 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 13604 . . . . 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 2336 . . . 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 2328 . . 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 2332 . 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 2296 . . . . 5  |-  ( Q  X.c  C )  =  ( Q  X.c  C )
7715fucbas 13850 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
7815, 7fuchom 13851 . . . . 5  |-  N  =  (  Hom  `  Q
)
79 eqid 2296 . . . . 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 13977 . . . 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 5545 . . 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 5877 . . 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 2346 . 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 5877 . . . . . 6  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
851, 2, 3, 4, 9, 12evlf1 14010 . . . . . 6  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
8684, 85syl5eqr 2342 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. F ,  X >. )  =  ( ( 1st `  F
) `  X )
)
87 df-ov 5877 . . . . . 6  |-  ( G ( 1st `  E
) Y )  =  ( ( 1st `  E
) `  <. G ,  Y >. )
881, 2, 3, 4, 44, 24evlf1 14010 . . . . . 6  |-  ( ph  ->  ( G ( 1st `  E ) Y )  =  ( ( 1st `  G ) `  Y
) )
8987, 88syl5eqr 2342 . . . . 5  |-  ( ph  ->  ( ( 1st `  E
) `  <. G ,  Y >. )  =  ( ( 1st `  G
) `  Y )
)
9086, 89opeq12d 3820 . . . 4  |-  ( ph  -> 
<. ( ( 1st `  E
) `  <. F ,  X >. ) ,  ( ( 1st `  E
) `  <. G ,  Y >. ) >.  =  <. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  G ) `  Y
) >. )
91 df-ov 5877 . . . . 5  |-  ( H ( 1st `  E
) Z )  =  ( ( 1st `  E
) `  <. H ,  Z >. )
921, 2, 3, 4, 11, 13evlf1 14010 . . . . 5  |-  ( ph  ->  ( H ( 1st `  E ) Z )  =  ( ( 1st `  H ) `  Z
) )
9391, 92syl5eqr 2342 . . . 4  |-  ( ph  ->  ( ( 1st `  E
) `  <. H ,  Z >. )  =  ( ( 1st `  H
) `  Z )
)
9490, 93oveq12d 5892 . . 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 5877 . . . 4  |-  ( B ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) L )  =  ( ( <. G ,  Y >. ( 2nd `  E
) <. H ,  Z >. ) `  <. B ,  L >. )
96 eqid 2296 . . . . 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 14009 . . . 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 2342 . . 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 5877 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. ) `  <. A ,  K >. )
100 eqid 2296 . . . . 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 14009 . . . 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 2342 . . 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 5895 . 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 2338 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 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Func cfunc 13744   Nat cnat 13831   FuncCat cfuc 13832    X.c cxpc 13958   evalF cevlf 13999
This theorem is referenced by:  evlfcl  14012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-func 13748  df-nat 13833  df-fuc 13834  df-xpc 13962  df-evlf 14003
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