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Theorem xpcval 14979
Description: Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
Hypotheses
Ref Expression
xpcval.t  |-  T  =  ( C  X.c  D )
xpcval.x  |-  X  =  ( Base `  C
)
xpcval.y  |-  Y  =  ( Base `  D
)
xpcval.h  |-  H  =  ( Hom  `  C
)
xpcval.j  |-  J  =  ( Hom  `  D
)
xpcval.o1  |-  .x.  =  (comp `  C )
xpcval.o2  |-  .xb  =  (comp `  D )
xpcval.c  |-  ( ph  ->  C  e.  V )
xpcval.d  |-  ( ph  ->  D  e.  W )
xpcval.b  |-  ( ph  ->  B  =  ( X  X.  Y ) )
xpcval.k  |-  ( ph  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
xpcval.o  |-  ( ph  ->  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
Assertion
Ref Expression
xpcval  |-  ( ph  ->  T  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  O >. } )
Distinct variable groups:    f, g, u, v, x, y, B    ph, f, g, u, v, x, y    C, f, g, u, v, x, y    D, f, g, u, v, x, y    f, K, g, x, y
Allowed substitution hints:    .xb ( x, y, v, u, f, g)    T( x, y, v, u, f, g)    .x. ( x, y, v, u, f, g)    H( x, y, v, u, f, g)    J( x, y, v, u, f, g)    K( v, u)    O( x, y, v, u, f, g)    V( x, y, v, u, f, g)    W( x, y, v, u, f, g)    X( x, y, v, u, f, g)    Y( x, y, v, u, f, g)

Proof of Theorem xpcval
Dummy variables  b  h  r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcval.t . 2  |-  T  =  ( C  X.c  D )
2 df-xpc 14974 . . . 4  |-  X.c  =  ( r  e.  _V , 
s  e.  _V  |->  [_ ( ( Base `  r
)  X.  ( Base `  s ) )  / 
b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
32a1i 11 . . 3  |-  ( ph  ->  X.c  =  ( r  e. 
_V ,  s  e. 
_V  |->  [_ ( ( Base `  r )  X.  ( Base `  s ) )  /  b ]_ [_ (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } ) )
4 fvex 5696 . . . . . 6  |-  ( Base `  r )  e.  _V
5 fvex 5696 . . . . . 6  |-  ( Base `  s )  e.  _V
64, 5xpex 6503 . . . . 5  |-  ( (
Base `  r )  X.  ( Base `  s
) )  e.  _V
76a1i 11 . . . 4  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( ( Base `  r
)  X.  ( Base `  s ) )  e. 
_V )
8 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
r  =  C )
98fveq2d 5690 . . . . . . 7  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( Base `  r )  =  ( Base `  C
) )
10 xpcval.x . . . . . . 7  |-  X  =  ( Base `  C
)
119, 10syl6eqr 2488 . . . . . 6  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( Base `  r )  =  X )
12 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
s  =  D )
1312fveq2d 5690 . . . . . . 7  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( Base `  s )  =  ( Base `  D
) )
14 xpcval.y . . . . . . 7  |-  Y  =  ( Base `  D
)
1513, 14syl6eqr 2488 . . . . . 6  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( Base `  s )  =  Y )
1611, 15xpeq12d 4860 . . . . 5  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( ( Base `  r
)  X.  ( Base `  s ) )  =  ( X  X.  Y
) )
17 xpcval.b . . . . . 6  |-  ( ph  ->  B  =  ( X  X.  Y ) )
1817adantr 465 . . . . 5  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  ->  B  =  ( X  X.  Y ) )
1916, 18eqtr4d 2473 . . . 4  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( ( Base `  r
)  X.  ( Base `  s ) )  =  B )
20 vex 2970 . . . . . . 7  |-  b  e. 
_V
2120, 20mpt2ex 6645 . . . . . 6  |-  ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  e.  _V
2221a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  e.  _V )
23 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  b  =  B )
24 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  r  =  C )
2524fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  ( Hom  `  r )  =  ( Hom  `  C
) )
26 xpcval.h . . . . . . . . . 10  |-  H  =  ( Hom  `  C
)
2725, 26syl6eqr 2488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  ( Hom  `  r )  =  H )
2827oveqd 6103 . . . . . . . 8  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  =  ( ( 1st `  u
) H ( 1st `  v ) ) )
29 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  s  =  D )
3029fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  ( Hom  `  s )  =  ( Hom  `  D
) )
31 xpcval.j . . . . . . . . . 10  |-  J  =  ( Hom  `  D
)
3230, 31syl6eqr 2488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  ( Hom  `  s )  =  J )
3332oveqd 6103 . . . . . . . 8  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) )  =  ( ( 2nd `  u
) J ( 2nd `  v ) ) )
3428, 33xpeq12d 4860 . . . . . . 7  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) )  =  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
3523, 23, 34mpt2eq123dv 6143 . . . . . 6  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
36 xpcval.k . . . . . . 7  |-  ( ph  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
3736ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
3835, 37eqtr4d 2473 . . . . 5  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  =  K )
39 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  b  =  B )
4039opeq2d 4061 . . . . . 6  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
41 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  h  =  K )
4241opeq2d 4061 . . . . . 6  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. ( Hom  `  ndx ) ,  h >.  =  <. ( Hom  `  ndx ) ,  K >. )
4339, 39xpeq12d 4860 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
b  X.  b )  =  ( B  X.  B ) )
4441oveqd 6103 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
( 2nd `  x
) h y )  =  ( ( 2nd `  x ) K y ) )
4541fveq1d 5688 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
h `  x )  =  ( K `  x ) )
4624adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  r  =  C )
4746fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (comp `  r )  =  (comp `  C ) )
48 xpcval.o1 . . . . . . . . . . . . . 14  |-  .x.  =  (comp `  C )
4947, 48syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (comp `  r )  =  .x.  )
5049oveqd 6103 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) )  =  (
<. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) )
5150oveqd 6103 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) )  =  ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) )
5229adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  s  =  D )
5352fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (comp `  s )  =  (comp `  D ) )
54 xpcval.o2 . . . . . . . . . . . . . 14  |-  .xb  =  (comp `  D )
5553, 54syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (comp `  s )  =  .xb  )
5655oveqd 6103 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  s )
( 2nd `  y
) )  =  (
<. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>.  .xb  ( 2nd `  y
) ) )
5756oveqd 6103 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) )  =  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) )
5851, 57opeq12d 4062 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
5944, 45, 58mpt2eq123dv 6143 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
6043, 39, 59mpt2eq123dv 6143 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
x  e.  ( b  X.  b ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
61 xpcval.o . . . . . . . . 9  |-  ( ph  ->  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
6261ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  O  =  ( x  e.  ( B  X.  B
) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
6360, 62eqtr4d 2473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
x  e.  ( b  X.  b ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  O )
6463opeq2d 4061 . . . . . 6  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. (comp ` 
ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >.  =  <. (comp `  ndx ) ,  O >. )
6540, 42, 64tpeq123d 3964 . . . . 5  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  { <. (
Base `  ndx ) ,  b >. ,  <. ( Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  K >. , 
<. (comp `  ndx ) ,  O >. } )
6622, 38, 65csbied2 3310 . . . 4  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  [_ (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  K >. , 
<. (comp `  ndx ) ,  O >. } )
677, 19, 66csbied2 3310 . . 3  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  ->  [_ ( ( Base `  r
)  X.  ( Base `  s ) )  / 
b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  K >. , 
<. (comp `  ndx ) ,  O >. } )
68 xpcval.c . . . 4  |-  ( ph  ->  C  e.  V )
69 elex 2976 . . . 4  |-  ( C  e.  V  ->  C  e.  _V )
7068, 69syl 16 . . 3  |-  ( ph  ->  C  e.  _V )
71 xpcval.d . . . 4  |-  ( ph  ->  D  e.  W )
72 elex 2976 . . . 4  |-  ( D  e.  W  ->  D  e.  _V )
7371, 72syl 16 . . 3  |-  ( ph  ->  D  e.  _V )
74 tpex 6374 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  O >. }  e.  _V
7574a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  K >. , 
<. (comp `  ndx ) ,  O >. }  e.  _V )
763, 67, 70, 73, 75ovmpt2d 6213 . 2  |-  ( ph  ->  ( C  X.c  D )  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  O >. } )
771, 76syl5eq 2482 1  |-  ( ph  ->  T  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  O >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   [_csb 3283   {ctp 3876   <.cop 3878    X. cxp 4833   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   ndxcnx 14163   Basecbs 14166   Hom chom 14241  compcco 14242    X.c cxpc 14970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-xpc 14974
This theorem is referenced by:  xpcbas  14980  xpchomfval  14981  xpccofval  14984  catcxpccl  15009  xpcpropd  15010
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