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Theorem xpccatid 15583
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 15573 . . . 4  |-  ( X  X.  Y )  =  ( Base `  T
)
54a1i 11 . . 3  |-  ( ph  ->  ( X  X.  Y
)  =  ( Base `  T ) )
6 eqidd 2458 . . 3  |-  ( ph  ->  ( Hom  `  T
)  =  ( Hom  `  T ) )
7 eqidd 2458 . . 3  |-  ( ph  ->  (comp `  T )  =  (comp `  T )
)
8 ovex 6324 . . . . 5  |-  ( C  X.c  D )  e.  _V
91, 8eqeltri 2541 . . . 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 2457 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
13 xpccat.i . . . . . 6  |-  I  =  ( Id `  C
)
14 xpccat.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
1514adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  C  e.  Cat )
16 xp1st 6829 . . . . . . 7  |-  ( t  e.  ( X  X.  Y )  ->  ( 1st `  t )  e.  X )
1716adantl 466 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 1st `  t )  e.  X )
182, 12, 13, 15, 17catidcl 15098 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
I `  ( 1st `  t ) )  e.  ( ( 1st `  t
) ( Hom  `  C
) ( 1st `  t
) ) )
19 eqid 2457 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
20 xpccat.j . . . . . 6  |-  J  =  ( Id `  D
)
21 xpccat.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
2221adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  D  e.  Cat )
23 xp2nd 6830 . . . . . . 7  |-  ( t  e.  ( X  X.  Y )  ->  ( 2nd `  t )  e.  Y )
2423adantl 466 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 2nd `  t )  e.  Y )
253, 19, 20, 22, 24catidcl 15098 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( J `  ( 2nd `  t ) )  e.  ( ( 2nd `  t
) ( Hom  `  D
) ( 2nd `  t
) ) )
26 opelxpi 5040 . . . . 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 661 . . . 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 2457 . . . . 5  |-  ( Hom  `  T )  =  ( Hom  `  T )
29 simpr 461 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  t  e.  ( X  X.  Y
) )
301, 4, 12, 19, 28, 29, 29xpchom 15575 . . . 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 2548 . . 3  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>.  e.  ( t ( Hom  `  T )
t ) )
32 fvex 5882 . . . . . . . 8  |-  ( I `
 ( 1st `  t
) )  e.  _V
33 fvex 5882 . . . . . . . 8  |-  ( J `
 ( 2nd `  t
) )  e.  _V
3432, 33op1st 6807 . . . . . . 7  |-  ( 1st `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )  =  ( I `  ( 1st `  t ) )
3534oveq1i 6306 . . . . . 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 465 . . . . . . 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 1053 . . . . . . . 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 6829 . . . . . . . 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 2457 . . . . . . 7  |-  (comp `  C )  =  (comp `  C )
41 simpr1r 1054 . . . . . . . 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 1086 . . . . . . . . 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 15575 . . . . . . . . 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 2547 . . . . . . . 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 6829 . . . . . . . 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 15099 . . . . . 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 2510 . . . . 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 6808 . . . . . . 7  |-  ( 2nd `  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >. )  =  ( J `  ( 2nd `  t ) )
5150oveq1i 6306 . . . . . 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 465 . . . . . . 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 6830 . . . . . . . 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 2457 . . . . . . 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 6830 . . . . . . . 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 15099 . . . . . 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 2510 . . . . 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 4227 . . . 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 2457 . . . . 5  |-  (comp `  T )  =  (comp `  T )
6341, 31syldan 470 . . . . 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 15578 . . . 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 6836 . . . . 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 2508 . . 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 6307 . . . . . 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 1055 . . . . . . . 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 6829 . . . . . . . 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 1087 . . . . . . . . 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 15575 . . . . . . . . 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 2547 . . . . . . . 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 6829 . . . . . . . 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 15100 . . . . . 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 2510 . . . . 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 6307 . . . . . 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 6830 . . . . . . . 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 6830 . . . . . . . 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 15100 . . . . . 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 2510 . . . . 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 4227 . . . 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 15578 . . . 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 6836 . . . . 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 2508 . . 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 15101 . . . . 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 15101 . . . . 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 5040 . . . . 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 661 . . . 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 15578 . . . 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 15575 . . . 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 2560 . . 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 1056 . . . . . . . 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 6829 . . . . . . . 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 1088 . . . . . . . . 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 15575 . . . . . . . . 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 2547 . . . . . . . 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 6829 . . . . . . . 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 15102 . . . . . 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 15578 . . . . . . . . 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 5876 . . . . . . . 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 6324 . . . . . . . . 9  |-  ( ( 1st `  h ) ( <. ( 1st `  t
) ,  ( 1st `  u ) >. (comp `  C ) ( 1st `  v ) ) ( 1st `  g ) )  e.  _V
110 ovex 6324 . . . . . . . . 9  |-  ( ( 2nd `  h ) ( <. ( 2nd `  t
) ,  ( 2nd `  u ) >. (comp `  D ) ( 2nd `  v ) ) ( 2nd `  g ) )  e.  _V
111109, 110op1st 6807 . . . . . . . 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 2514 . . . . . . 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 6311 . . . . . 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 5876 . . . . . . . 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 6324 . . . . . . . . 9  |-  ( ( 1st `  g ) ( <. ( 1st `  s
) ,  ( 1st `  t ) >. (comp `  C ) ( 1st `  u ) ) ( 1st `  f ) )  e.  _V
116 ovex 6324 . . . . . . . . 9  |-  ( ( 2nd `  g ) ( <. ( 2nd `  s
) ,  ( 2nd `  t ) >. (comp `  D ) ( 2nd `  u ) ) ( 2nd `  f ) )  e.  _V
117115, 116op1st 6807 . . . . . . . 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 2514 . . . . . . 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 6312 . . . . . 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 2508 . . . . 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 6830 . . . . . . . 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 6830 . . . . . . . 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 15102 . . . . . 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 5876 . . . . . . . 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 6808 . . . . . . . 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 2514 . . . . . . 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 6311 . . . . . 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 5876 . . . . . . . 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 6808 . . . . . . . 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 2514 . . . . . . 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 6312 . . . . . 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 2508 . . . . 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 4227 . . . 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 15101 . . . . . . 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 15101 . . . . . . 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 5040 . . . . . . 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 661 . . . . . 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 15575 . . . . . 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 2560 . . . . 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 15578 . . . 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 15578 . . . 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 2508 . . 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 15097 . 2  |-  ( ph  ->  ( T  e.  Cat  /\  ( Id `  T
)  =  ( t  e.  ( X  X.  Y )  |->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. ) ) )
146 vex 3112 . . . . . . . 8  |-  x  e. 
_V
147 vex 3112 . . . . . . . 8  |-  y  e. 
_V
148146, 147op1std 6809 . . . . . . 7  |-  ( t  =  <. x ,  y
>.  ->  ( 1st `  t
)  =  x )
149148fveq2d 5876 . . . . . 6  |-  ( t  =  <. x ,  y
>.  ->  ( I `  ( 1st `  t ) )  =  ( I `
 x ) )
150146, 147op2ndd 6810 . . . . . . 7  |-  ( t  =  <. x ,  y
>.  ->  ( 2nd `  t
)  =  y )
151150fveq2d 5876 . . . . . 6  |-  ( t  =  <. x ,  y
>.  ->  ( J `  ( 2nd `  t ) )  =  ( J `
 y ) )
152149, 151opeq12d 4227 . . . . 5  |-  ( t  =  <. x ,  y
>.  ->  <. ( I `  ( 1st `  t ) ) ,  ( J `
 ( 2nd `  t
) ) >.  =  <. ( I `  x ) ,  ( J `  y ) >. )
153152mpt2mpt 6393 . . . 4  |-  ( t  e.  ( X  X.  Y )  |->  <. (
I `  ( 1st `  t ) ) ,  ( J `  ( 2nd `  t ) )
>. )  =  (
x  e.  X , 
y  e.  Y  |->  <.
( I `  x
) ,  ( J `
 y ) >.
)
154153eqeq2i 2475 . . 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 694 . 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    |-> cmpt 4515    X. cxp 5006   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   Basecbs 14643   Hom chom 14722  compcco 14723   Catccat 15080   Idccid 15081    X.c cxpc 15563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-hom 14735  df-cco 14736  df-cat 15084  df-cid 15085  df-xpc 15567
This theorem is referenced by:  xpcid  15584  xpccat  15585
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