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Theorem xpccatid 14994
Description: The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpccat.t  |-  T  =  ( C  X.c  D )
xpccat.c  |-  ( ph  ->  C  e.  Cat )
xpccat.d  |-  ( ph  ->  D  e.  Cat )
xpccat.x  |-  X  =  ( Base `  C
)
xpccat.y  |-  Y  =  ( Base `  D
)
xpccat.i  |-  I  =  ( Id `  C
)
xpccat.j  |-  J  =  ( Id `  D
)
Assertion
Ref Expression
xpccatid  |-  ( ph  ->  ( T  e.  Cat  /\  ( Id `  T
)  =  ( x  e.  X ,  y  e.  Y  |->  <. (
I `  x ) ,  ( J `  y ) >. )
) )
Distinct variable groups:    x, y, I    x, J, y    x, C, y    ph, x, y   
x, X, y    x, D, y    x, Y, y
Allowed substitution hints:    T( x, y)

Proof of Theorem xpccatid
Dummy variables  f 
g  h  s  t  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccat.t . . . . 5  |-  T  =  ( C  X.c  D )
2 xpccat.x . . . . 5  |-  X  =  ( Base `  C
)
3 xpccat.y . . . . 5  |-  Y  =  ( Base `  D
)
41, 2, 3xpcbas 14984 . . . 4  |-  ( X  X.  Y )  =  ( Base `  T
)
54a1i 11 . . 3  |-  ( ph  ->  ( X  X.  Y
)  =  ( Base `  T ) )
6 eqidd 2442 . . 3  |-  ( ph  ->  ( Hom  `  T
)  =  ( Hom  `  T ) )
7 eqidd 2442 . . 3  |-  ( ph  ->  (comp `  T )  =  (comp `  T )
)
8 ovex 6115 . . . . 5  |-  ( C  X.c  D )  e.  _V
91, 8eqeltri 2511 . . . 4  |-  T  e. 
_V
109a1i 11 . . 3  |-  ( ph  ->  T  e.  _V )
11 biid 236 . . 3  |-  ( ( ( s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y
) )  /\  (
f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) )  <->  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )
12 eqid 2441 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
13 xpccat.i . . . . . 6  |-  I  =  ( Id `  C
)
14 xpccat.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
1514adantr 462 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  C  e.  Cat )
16 xp1st 6605 . . . . . . 7  |-  ( t  e.  ( X  X.  Y )  ->  ( 1st `  t )  e.  X )
1716adantl 463 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 1st `  t )  e.  X )
182, 12, 13, 15, 17catidcl 14616 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
I `  ( 1st `  t ) )  e.  ( ( 1st `  t
) ( Hom  `  C
) ( 1st `  t
) ) )
19 eqid 2441 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
20 xpccat.j . . . . . 6  |-  J  =  ( Id `  D
)
21 xpccat.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
2221adantr 462 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  D  e.  Cat )
23 xp2nd 6606 . . . . . . 7  |-  ( t  e.  ( X  X.  Y )  ->  ( 2nd `  t )  e.  Y )
2423adantl 463 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 2nd `  t )  e.  Y )
253, 19, 20, 22, 24catidcl 14616 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( J `  ( 2nd `  t ) )  e.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) )
26 opelxpi 4867 . . . . 5  |-  ( ( ( I `  ( 1st `  t ) )  e.  ( ( 1st `  t ) ( Hom  `  C ) ( 1st `  t ) )  /\  ( J `  ( 2nd `  t ) )  e.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) )  ->  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) ) >.  e.  (
( ( 1st `  t
) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) ) )
2718, 25, 26syl2anc 656 . . . 4  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>.  e.  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) ) )
28 eqid 2441 . . . . 5  |-  ( Hom  `  T )  =  ( Hom  `  T )
29 simpr 458 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  t  e.  ( X  X.  Y
) )
301, 4, 12, 19, 28, 29, 29xpchom 14986 . . . 4  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
t ( Hom  `  T
) t )  =  ( ( ( 1st `  t ) ( Hom  `  C ) ( 1st `  t ) )  X.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) ) )
3127, 30eleqtrrd 2518 . . 3  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>.  e.  ( t ( Hom  `  T )
t ) )
32 fvex 5698 . . . . . . . 8  |-  ( I `
 ( 1st `  t
) )  e.  _V
33 fvex 5698 . . . . . . . 8  |-  ( J `
 ( 2nd `  t
) )  e.  _V
3432, 33op1st 6584 . . . . . . 7  |-  ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )  =  ( I `  ( 1st `  t ) )
3534oveq1i 6100 . . . . . 6  |-  ( ( 1st `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  t
) ) ( 1st `  f ) )  =  ( ( I `  ( 1st `  t ) ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  t
) ) ( 1st `  f ) )
3614adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  C  e.  Cat )
37 simpr1l 1040 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
s  e.  ( X  X.  Y ) )
38 xp1st 6605 . . . . . . . 8  |-  ( s  e.  ( X  X.  Y )  ->  ( 1st `  s )  e.  X )
3937, 38syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  s
)  e.  X )
40 eqid 2441 . . . . . . 7  |-  (comp `  C )  =  (comp `  C )
41 simpr1r 1041 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
t  e.  ( X  X.  Y ) )
4241, 16syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  t
)  e.  X )
43 simpr31 1073 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
f  e.  ( s ( Hom  `  T
) t ) )
441, 4, 12, 19, 28, 37, 41xpchom 14986 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( s ( Hom  `  T ) t )  =  ( ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  t
) ) ) )
4543, 44eleqtrd 2517 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
f  e.  ( ( ( 1st `  s
) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  t
) ) ) )
46 xp1st 6605 . . . . . . . 8  |-  ( f  e.  ( ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  t
) ) )  -> 
( 1st `  f
)  e.  ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) ) )
4745, 46syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  f
)  e.  ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) ) )
482, 12, 13, 36, 39, 40, 42, 47catlid 14617 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( I `  ( 1st `  t ) ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  t
) ) ( 1st `  f ) )  =  ( 1st `  f
) )
4935, 48syl5eq 2485 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  t
) ) ( 1st `  f ) )  =  ( 1st `  f
) )
5032, 33op2nd 6585 . . . . . . 7  |-  ( 2nd `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )  =  ( J `  ( 2nd `  t ) )
5150oveq1i 6100 . . . . . 6  |-  ( ( 2nd `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) )  =  ( ( J `  ( 2nd `  t ) ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) )
5221adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  D  e.  Cat )
53 xp2nd 6606 . . . . . . . 8  |-  ( s  e.  ( X  X.  Y )  ->  ( 2nd `  s )  e.  Y )
5437, 53syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  s
)  e.  Y )
55 eqid 2441 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5641, 23syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  t
)  e.  Y )
57 xp2nd 6606 . . . . . . . 8  |-  ( f  e.  ( ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  t
) ) )  -> 
( 2nd `  f
)  e.  ( ( 2nd `  s ) ( Hom  `  D
) ( 2nd `  t
) ) )
5845, 57syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  f
)  e.  ( ( 2nd `  s ) ( Hom  `  D
) ( 2nd `  t
) ) )
593, 19, 20, 52, 54, 55, 56, 58catlid 14617 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( J `  ( 2nd `  t ) ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) )  =  ( 2nd `  f
) )
6051, 59syl5eq 2485 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) )  =  ( 2nd `  f
) )
6149, 60opeq12d 4064 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  t
) ) ( 1st `  f ) ) ,  ( ( 2nd `  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) ) >.  =  <. ( 1st `  f
) ,  ( 2nd `  f ) >. )
62 eqid 2441 . . . . 5  |-  (comp `  T )  =  (comp `  T )
6341, 31syldan 467 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) ) >.  e.  (
t ( Hom  `  T
) t ) )
641, 4, 28, 40, 55, 62, 37, 41, 41, 43, 63xpcco 14989 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. ( <. s ,  t >.
(comp `  T )
t ) f )  =  <. ( ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
( <. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  t ) ) ( 1st `  f ) ) ,  ( ( 2nd `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  t
) ) ( 2nd `  f ) ) >.
)
65 1st2nd2 6612 . . . . 5  |-  ( f  e.  ( ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  t
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  t
) ) )  -> 
f  =  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )
6645, 65syl 16 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
f  =  <. ( 1st `  f ) ,  ( 2nd `  f
) >. )
6761, 64, 663eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. ( <. s ,  t >.
(comp `  T )
t ) f )  =  f )
6834oveq2i 6101 . . . . . 6  |-  ( ( 1st `  g ) ( <. ( 1st `  t
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) )  =  ( ( 1st `  g
) ( <. ( 1st `  t ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( I `
 ( 1st `  t
) ) )
69 simpr2l 1042 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  u  e.  ( X  X.  Y ) )
70 xp1st 6605 . . . . . . . 8  |-  ( u  e.  ( X  X.  Y )  ->  ( 1st `  u )  e.  X )
7169, 70syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  u
)  e.  X )
72 simpr32 1074 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
g  e.  ( t ( Hom  `  T
) u ) )
731, 4, 12, 19, 28, 41, 69xpchom 14986 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( t ( Hom  `  T ) u )  =  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  u
) ) ) )
7472, 73eleqtrd 2517 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
g  e.  ( ( ( 1st `  t
) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  u
) ) ) )
75 xp1st 6605 . . . . . . . 8  |-  ( g  e.  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  u
) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) ) )
7674, 75syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) ) )
772, 12, 13, 36, 42, 40, 71, 76catrid 14618 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  g
) ( <. ( 1st `  t ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( I `
 ( 1st `  t
) ) )  =  ( 1st `  g
) )
7868, 77syl5eq 2485 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  g
) ( <. ( 1st `  t ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
)  =  ( 1st `  g ) )
7950oveq2i 6101 . . . . . 6  |-  ( ( 2nd `  g ) ( <. ( 2nd `  t
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) )  =  ( ( 2nd `  g
) ( <. ( 2nd `  t ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( J `
 ( 2nd `  t
) ) )
80 xp2nd 6606 . . . . . . . 8  |-  ( u  e.  ( X  X.  Y )  ->  ( 2nd `  u )  e.  Y )
8169, 80syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  u
)  e.  Y )
82 xp2nd 6606 . . . . . . . 8  |-  ( g  e.  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  u
) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  t ) ( Hom  `  D
) ( 2nd `  u
) ) )
8374, 82syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  t ) ( Hom  `  D
) ( 2nd `  u
) ) )
843, 19, 20, 52, 56, 55, 81, 83catrid 14618 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  g
) ( <. ( 2nd `  t ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( J `
 ( 2nd `  t
) ) )  =  ( 2nd `  g
) )
8579, 84syl5eq 2485 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  g
) ( <. ( 2nd `  t ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
)  =  ( 2nd `  g ) )
8678, 85opeq12d 4064 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  t ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
) ,  ( ( 2nd `  g ) ( <. ( 2nd `  t
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ) >.  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
871, 4, 28, 40, 55, 62, 41, 41, 69, 63, 72xpcco 14989 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( g ( <.
t ,  t >.
(comp `  T )
u ) <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. )  =  <. ( ( 1st `  g
) ( <. ( 1st `  t ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
) ,  ( ( 2nd `  g ) ( <. ( 2nd `  t
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ) >. )
88 1st2nd2 6612 . . . . 5  |-  ( g  e.  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  u
) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
8974, 88syl 16 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
9086, 87, 893eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( g ( <.
t ,  t >.
(comp `  T )
u ) <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. )  =  g
)
912, 12, 40, 36, 39, 42, 71, 47, 76catcocl 14619 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) )  e.  ( ( 1st `  s
) ( Hom  `  C
) ( 1st `  u
) ) )
923, 19, 55, 52, 54, 56, 81, 58, 83catcocl 14619 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) )  e.  ( ( 2nd `  s
) ( Hom  `  D
) ( 2nd `  u
) ) )
93 opelxpi 4867 . . . . 5  |-  ( ( ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) )  e.  ( ( 1st `  s
) ( Hom  `  C
) ( 1st `  u
) )  /\  (
( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) )  e.  ( ( 2nd `  s
) ( Hom  `  D
) ( 2nd `  u
) ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) ) >.  e.  ( ( ( 1st `  s ) ( Hom  `  C ) ( 1st `  u ) )  X.  ( ( 2nd `  s
) ( Hom  `  D
) ( 2nd `  u
) ) ) )
9491, 92, 93syl2anc 656 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) ) >.  e.  ( ( ( 1st `  s ) ( Hom  `  C ) ( 1st `  u ) )  X.  ( ( 2nd `  s
) ( Hom  `  D
) ( 2nd `  u
) ) ) )
951, 4, 28, 40, 55, 62, 37, 41, 69, 43, 72xpcco 14989 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( g ( <.
s ,  t >.
(comp `  T )
u ) f )  =  <. ( ( 1st `  g ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) ) ,  ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) ) >. )
961, 4, 12, 19, 28, 37, 69xpchom 14986 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( s ( Hom  `  T ) u )  =  ( ( ( 1st `  s ) ( Hom  `  C
) ( 1st `  u
) )  X.  (
( 2nd `  s
) ( Hom  `  D
) ( 2nd `  u
) ) ) )
9794, 95, 963eltr4d 2522 . . 3  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( g ( <.
s ,  t >.
(comp `  T )
u ) f )  e.  ( s ( Hom  `  T )
u ) )
98 simpr2r 1043 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
v  e.  ( X  X.  Y ) )
99 xp1st 6605 . . . . . . . 8  |-  ( v  e.  ( X  X.  Y )  ->  ( 1st `  v )  e.  X )
10098, 99syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  v
)  e.  X )
101 simpr33 1075 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  h  e.  ( u
( Hom  `  T ) v ) )
1021, 4, 12, 19, 28, 69, 98xpchom 14986 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( u ( Hom  `  T ) v )  =  ( ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
103101, 102eleqtrd 2517 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  h  e.  ( (
( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
104 xp1st 6605 . . . . . . . 8  |-  ( h  e.  ( ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) )  -> 
( 1st `  h
)  e.  ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) ) )
105103, 104syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  h
)  e.  ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) ) )
1062, 12, 40, 36, 39, 42, 71, 47, 76, 100, 105catass 14620 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( ( 1st `  h ) ( <.
( 1st `  t
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  g ) ) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  f ) )  =  ( ( 1st `  h
) ( <. ( 1st `  s ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( ( 1st `  g ) ( <. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) ) ) )
1071, 4, 28, 40, 55, 62, 41, 69, 98, 72, 101xpcco 14989 . . . . . . . . 9  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( h ( <.
t ,  u >. (comp `  T ) v ) g )  =  <. ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) >.
)
108107fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) )  =  ( 1st `  <. (
( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) >.
) )
109 ovex 6115 . . . . . . . . 9  |-  ( ( 1st `  h ) ( <. ( 1st `  t
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  g ) )  e.  _V
110 ovex 6115 . . . . . . . . 9  |-  ( ( 2nd `  h ) ( <. ( 2nd `  t
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  g ) )  e.  _V
111109, 110op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  h ) ( <.
( 1st `  t
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  g ) ) ,  ( ( 2nd `  h ) ( <. ( 2nd `  t
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  g ) ) >. )  =  ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) )
112108, 111syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) )  =  ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) )
113112oveq1d 6105 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  f ) )  =  ( ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) (
<. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  f ) ) )
11495fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  (
g ( <. s ,  t >. (comp `  T ) u ) f ) )  =  ( 1st `  <. ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) ) >.
) )
115 ovex 6115 . . . . . . . . 9  |-  ( ( 1st `  g ) ( <. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) )  e.  _V
116 ovex 6115 . . . . . . . . 9  |-  ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) )  e.  _V
117115, 116op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  g ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) ) ,  ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) ) >. )  =  ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) )
118114, 117syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 1st `  (
g ( <. s ,  t >. (comp `  T ) u ) f ) )  =  ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) ) )
119118oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  h
) ( <. ( 1st `  s ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) )  =  ( ( 1st `  h
) ( <. ( 1st `  s ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( ( 1st `  g ) ( <. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) ) ) )
120106, 113, 1193eqtr4d 2483 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  f ) )  =  ( ( 1st `  h ) ( <. ( 1st `  s
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  ( g ( <. s ,  t
>. (comp `  T )
u ) f ) ) ) )
121 xp2nd 6606 . . . . . . . 8  |-  ( v  e.  ( X  X.  Y )  ->  ( 2nd `  v )  e.  Y )
12298, 121syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  v
)  e.  Y )
123 xp2nd 6606 . . . . . . . 8  |-  ( h  e.  ( ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) )  -> 
( 2nd `  h
)  e.  ( ( 2nd `  u ) ( Hom  `  D
) ( 2nd `  v
) ) )
124103, 123syl 16 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  h
)  e.  ( ( 2nd `  u ) ( Hom  `  D
) ( 2nd `  v
) ) )
1253, 19, 55, 52, 54, 56, 81, 58, 83, 122, 124catass 14620 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( ( 2nd `  h ) ( <.
( 2nd `  t
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  g ) ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  f ) )  =  ( ( 2nd `  h
) ( <. ( 2nd `  s ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) ) ) )
126107fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) )  =  ( 2nd `  <. (
( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) >.
) )
127109, 110op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  h ) ( <.
( 1st `  t
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  g ) ) ,  ( ( 2nd `  h ) ( <. ( 2nd `  t
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  g ) ) >. )  =  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) )
128126, 127syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) )  =  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) )
129128oveq1d 6105 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) ) ( <.
( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  f ) )  =  ( ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) (
<. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  f ) ) )
13095fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  (
g ( <. s ,  t >. (comp `  T ) u ) f ) )  =  ( 2nd `  <. ( ( 1st `  g
) ( <. ( 1st `  s ) ,  ( 1st `  t
) >. (comp `  C
) ( 1st `  u
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) ) >.
) )
131115, 116op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  g ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) ) ,  ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) ) >. )  =  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) )
132130, 131syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( 2nd `  (
g ( <. s ,  t >. (comp `  T ) u ) f ) )  =  ( ( 2nd `  g
) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  u
) ) ( 2nd `  f ) ) )
133132oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  h
) ( <. ( 2nd `  s ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) )  =  ( ( 2nd `  h
) ( <. ( 2nd `  s ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) ) ) )
134125, 129, 1333eqtr4d 2483 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) ) ( <.
( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  f ) )  =  ( ( 2nd `  h ) ( <. ( 2nd `  s
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  ( g ( <. s ,  t
>. (comp `  T )
u ) f ) ) ) )
135120, 134opeq12d 4064 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( ( 1st `  (
h ( <. t ,  u >. (comp `  T
) v ) g ) ) ( <.
( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  f ) ) ,  ( ( 2nd `  ( h ( <. t ,  u >. (comp `  T )
v ) g ) ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  h ) ( <.
( 1st `  s
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  ( g ( <. s ,  t
>. (comp `  T )
u ) f ) ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  s ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) ) >. )
1362, 12, 40, 36, 42, 71, 100, 76, 105catcocl 14619 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) )  e.  ( ( 1st `  t
) ( Hom  `  C
) ( 1st `  v
) ) )
1373, 19, 55, 52, 56, 81, 122, 83, 124catcocl 14619 . . . . . . 7  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) )  e.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  v
) ) )
138 opelxpi 4867 . . . . . . 7  |-  ( ( ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) )  e.  ( ( 1st `  t
) ( Hom  `  C
) ( 1st `  v
) )  /\  (
( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) )  e.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  v
) ) )  ->  <. ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) >.  e.  ( ( ( 1st `  t ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
139136, 137, 138syl2anc 656 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  ->  <. ( ( 1st `  h
) ( <. ( 1st `  t ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  g ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  t ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  g ) ) >.  e.  ( ( ( 1st `  t ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
1401, 4, 12, 19, 28, 41, 98xpchom 14986 . . . . . 6  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( t ( Hom  `  T ) v )  =  ( ( ( 1st `  t ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  t
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
141139, 107, 1403eltr4d 2522 . . . . 5  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( h ( <.
t ,  u >. (comp `  T ) v ) g )  e.  ( t ( Hom  `  T
) v ) )
1421, 4, 28, 40, 55, 62, 37, 41, 98, 43, 141xpcco 14989 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( h (
<. t ,  u >. (comp `  T ) v ) g ) ( <.
s ,  t >.
(comp `  T )
v ) f )  =  <. ( ( 1st `  ( h ( <.
t ,  u >. (comp `  T ) v ) g ) ) (
<. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  f ) ) ,  ( ( 2nd `  ( h ( <. t ,  u >. (comp `  T )
v ) g ) ) ( <. ( 2nd `  s ) ,  ( 2nd `  t
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  f ) ) >.
)
1431, 4, 28, 40, 55, 62, 37, 69, 98, 97, 101xpcco 14989 . . . 4  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( h ( <.
s ,  u >. (comp `  T ) v ) ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) )  =  <. (
( 1st `  h
) ( <. ( 1st `  s ) ,  ( 1st `  u
) >. (comp `  C
) ( 1st `  v
) ) ( 1st `  ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) ) ,  ( ( 2nd `  h
) ( <. ( 2nd `  s ) ,  ( 2nd `  u
) >. (comp `  D
) ( 2nd `  v
) ) ( 2nd `  ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) ) >. )
144135, 142, 1433eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( (
s  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y ) )  /\  ( u  e.  ( X  X.  Y )  /\  v  e.  ( X  X.  Y ) )  /\  ( f  e.  ( s ( Hom  `  T
) t )  /\  g  e.  ( t
( Hom  `  T ) u )  /\  h  e.  ( u ( Hom  `  T ) v ) ) ) )  -> 
( ( h (
<. t ,  u >. (comp `  T ) v ) g ) ( <.
s ,  t >.
(comp `  T )
v ) f )  =  ( h (
<. s ,  u >. (comp `  T ) v ) ( g ( <.
s ,  t >.
(comp `  T )
u ) f ) ) )
1455, 6, 7, 10, 11, 31, 67, 90, 97, 144iscatd2 14615 . 2  |-  ( ph  ->  ( T  e.  Cat  /\  ( Id `  T
)  =  ( t  e.  ( X  X.  Y )  |->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ) )
146 vex 2973 . . . . . . . 8  |-  x  e. 
_V
147 vex 2973 . . . . . . . 8  |-  y  e. 
_V
148146, 147op1std 6586 . . . . . . 7  |-  ( t  =  <. x ,  y
>.  ->  ( 1st `  t
)  =  x )
149148fveq2d 5692 . . . . . 6  |-  ( t  =  <. x ,  y
>.  ->  ( I `  ( 1st `  t ) )  =  ( I `
 x ) )
150146, 147op2ndd 6587 . . . . . . 7  |-  ( t  =  <. x ,  y
>.  ->  ( 2nd `  t
)  =  y )
151150fveq2d 5692 . . . . . 6  |-  ( t  =  <. x ,  y
>.  ->  ( J `  ( 2nd `  t ) )  =  ( J `
 y ) )
152149, 151opeq12d 4064 . . . . 5  |-  ( t  =  <. x ,  y
>.  ->  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >.  =  <. ( I `  x ) ,  ( J `  y ) >. )
153152mpt2mpt 6181 . . . 4  |-  ( t  e.  ( X  X.  Y )  |->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. )  =  (
x  e.  X , 
y  e.  Y  |->  <.
( I `  x
) ,  ( J `
 y ) >.
)
154153eqeq2i 2451 . . 3  |-  ( ( Id `  T )  =  ( t  e.  ( X  X.  Y
)  |->  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )  <->  ( Id `  T )  =  ( x  e.  X ,  y  e.  Y  |->  <. ( I `  x ) ,  ( J `  y )
>. ) )
155154anbi2i 689 . 2  |-  ( ( T  e.  Cat  /\  ( Id `  T )  =  ( t  e.  ( X  X.  Y
)  |->  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )
)  <->  ( T  e. 
Cat  /\  ( Id `  T )  =  ( x  e.  X , 
y  e.  Y  |->  <.
( I `  x
) ,  ( J `
 y ) >.
) ) )
156145, 155sylib 196 1  |-  ( ph  ->  ( T  e.  Cat  /\  ( Id `  T
)  =  ( x  e.  X ,  y  e.  Y  |->  <. (
I `  x ) ,  ( J `  y ) >. )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880    e. cmpt 4347    X. cxp 4834   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599    X.c cxpc 14974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-xpc 14978
This theorem is referenced by:  xpcid  14995  xpccat  14996
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