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Theorem xpcval 16004
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 15999 . . . 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 5891 . . . . . 6  |-  ( Base `  r )  e.  _V
5 fvex 5891 . . . . . 6  |-  ( Base `  s )  e.  _V
64, 5xpex 6609 . . . . 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 762 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
r  =  C )
98fveq2d 5885 . . . . . . 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 764 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
s  =  D )
1312fveq2d 5885 . . . . . . 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 4879 . . . . 5  |-  ( (
ph  /\  ( r  =  C  /\  s  =  D ) )  -> 
( ( Base `  r
)  X.  ( Base `  s ) )  =  ( X  X.  Y
) )
17 xpcval.b . . . . . 6  |-  ( ph  ->  B  =  ( X  X.  Y ) )
1817adantr 466 . . . . 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 3090 . . . . . . 7  |-  b  e. 
_V
2120, 20mpt2ex 6884 . . . . . 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 462 . . . . . . 7  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  b  =  B )
24 simplrl 768 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  r  =  C )
2524fveq2d 5885 . . . . . . . . . 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 6322 . . . . . . . 8  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
( 1st `  u
) ( Hom  `  r
) ( 1st `  v
) )  =  ( ( 1st `  u
) H ( 1st `  v ) ) )
29 simplrr 769 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  s  =  D )
3029fveq2d 5885 . . . . . . . . . 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 6322 . . . . . . . 8  |-  ( ( ( ph  /\  (
r  =  C  /\  s  =  D )
)  /\  b  =  B )  ->  (
( 2nd `  u
) ( Hom  `  s
) ( 2nd `  v
) )  =  ( ( 2nd `  u
) J ( 2nd `  v ) ) )
3428, 33xpeq12d 4879 . . . . . . 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 6367 . . . . . 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 730 . . . . . 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 760 . . . . . . 7  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  b  =  B )
4039opeq2d 4197 . . . . . 6  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
41 simpr 462 . . . . . . 7  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  h  =  K )
4241opeq2d 4197 . . . . . 6  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  <. ( Hom  `  ndx ) ,  h >.  =  <. ( Hom  `  ndx ) ,  K >. )
4339, 39xpeq12d 4879 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
b  X.  b )  =  ( B  X.  B ) )
4441oveqd 6322 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
( 2nd `  x
) h y )  =  ( ( 2nd `  x ) K y ) )
4541fveq1d 5883 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  (
h `  x )  =  ( K `  x ) )
4624adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  r  =  C )
4746fveq2d 5885 . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . 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 6322 . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( r  =  C  /\  s  =  D ) )  /\  b  =  B )  /\  h  =  K )  ->  s  =  D )
5352fveq2d 5885 . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . 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 6322 . . . . . . . . . . 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 4198 . . . . . . . . . 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 6367 . . . . . . . . 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 6367 . . . . . . . 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 734 . . . . . . . 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 4197 . . . . . 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 4097 . . . . 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 3429 . . . 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 3429 . . 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 3096 . . . 4  |-  ( C  e.  V  ->  C  e.  _V )
7068, 69syl 17 . . 3  |-  ( ph  ->  C  e.  _V )
71 xpcval.d . . . 4  |-  ( ph  ->  D  e.  W )
72 elex 3096 . . . 4  |-  ( D  e.  W  ->  D  e.  _V )
7371, 72syl 17 . . 3  |-  ( ph  ->  D  e.  _V )
74 tpex 6604 . . . 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 6438 . 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 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   [_csb 3401   {ctp 4006   <.cop 4008    X. cxp 4852   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   ndxcnx 15072   Basecbs 15075   Hom chom 15154  compcco 15155    X.c cxpc 15995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-xpc 15999
This theorem is referenced by:  xpcbas  16005  xpchomfval  16006  xpccofval  16009  catcxpccl  16034  xpcpropd  16035
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