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Theorem xpcpropd 14307
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
xpcpropd.1  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
xpcpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
xpcpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
xpcpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
xpcpropd.a  |-  ( ph  ->  A  e.  V )
xpcpropd.b  |-  ( ph  ->  B  e.  V )
xpcpropd.c  |-  ( ph  ->  C  e.  V )
xpcpropd.d  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
xpcpropd  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )

Proof of Theorem xpcpropd
Dummy variables  f 
g  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( A  X.c  C )  =  ( A  X.c  C )
2 eqid 2438 . . 3  |-  ( Base `  A )  =  (
Base `  A )
3 eqid 2438 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2438 . . 3  |-  (  Hom  `  A )  =  (  Hom  `  A )
5 eqid 2438 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2438 . . 3  |-  (comp `  A )  =  (comp `  A )
7 eqid 2438 . . 3  |-  (comp `  C )  =  (comp `  C )
8 xpcpropd.a . . 3  |-  ( ph  ->  A  e.  V )
9 xpcpropd.c . . 3  |-  ( ph  ->  C  e.  V )
10 eqidd 2439 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  A
)  X.  ( Base `  C ) ) )
111, 2, 3xpcbas 14277 . . . . 5  |-  ( (
Base `  A )  X.  ( Base `  C
) )  =  (
Base `  ( A  X.c  C ) )
12 eqid 2438 . . . . 5  |-  (  Hom  `  ( A  X.c  C ) )  =  (  Hom  `  ( A  X.c  C ) )
131, 11, 4, 5, 12xpchomfval 14278 . . . 4  |-  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) ) ,  v  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) )
1413a1i 11 . . 3  |-  ( ph  ->  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) ) )
15 eqidd 2439 . . 3  |-  ( ph  ->  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 14276 . 2  |-  ( ph  ->  ( A  X.c  C )  =  { <. ( Base `  ndx ) ,  ( ( Base `  A
)  X.  ( Base `  C ) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  ( A  X.c  C ) ) >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
17 eqid 2438 . . 3  |-  ( B  X.c  D )  =  ( B  X.c  D )
18 eqid 2438 . . 3  |-  ( Base `  B )  =  (
Base `  B )
19 eqid 2438 . . 3  |-  ( Base `  D )  =  (
Base `  D )
20 eqid 2438 . . 3  |-  (  Hom  `  B )  =  (  Hom  `  B )
21 eqid 2438 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
22 eqid 2438 . . 3  |-  (comp `  B )  =  (comp `  B )
23 eqid 2438 . . 3  |-  (comp `  D )  =  (comp `  D )
24 xpcpropd.b . . 3  |-  ( ph  ->  B  e.  V )
25 xpcpropd.d . . 3  |-  ( ph  ->  D  e.  V )
26 xpcpropd.1 . . . . 5  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
2726homfeqbas 13924 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
28 xpcpropd.3 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
2928homfeqbas 13924 . . . 4  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3027, 29xpeq12d 4905 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  B
)  X.  ( Base `  D ) ) )
31263ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
32 xp1st 6378 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  u
)  e.  ( Base `  A ) )
33323ad2ant2 980 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  u
)  e.  ( Base `  A ) )
34 xp1st 6378 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  v
)  e.  ( Base `  A ) )
35343ad2ant3 981 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  v
)  e.  ( Base `  A ) )
362, 4, 20, 31, 33, 35homfeqval 13925 . . . . . 6  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  =  ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) ) )
37283ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
38 xp2nd 6379 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  u
)  e.  ( Base `  C ) )
39383ad2ant2 980 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  u
)  e.  ( Base `  C ) )
40 xp2nd 6379 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  v
)  e.  ( Base `  C ) )
41403ad2ant3 981 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  v
)  e.  ( Base `  C ) )
423, 5, 21, 37, 39, 41homfeqval 13925 . . . . . 6  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) )  =  ( ( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) )
4336, 42xpeq12d 4905 . . . . 5  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( ( 1st `  u ) (  Hom  `  A ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) )  =  ( ( ( 1st `  u ) (  Hom  `  B ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) )
4443mpt2eq3dva 6140 . . . 4  |-  ( ph  ->  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) ) )
4513, 44syl5eq 2482 . . 3  |-  ( ph  ->  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) ) )
4626ad4antr 714 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
47 xpcpropd.2 . . . . . . . . . 10  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
4847ad4antr 714 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(compf `  A )  =  (compf `  B ) )
49 simp-4r 745 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  x  e.  ( (
( Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) ) )
50 xp1st 6378 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5149, 50syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
52 xp1st 6378 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
5351, 52syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
54 xp2nd 6379 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5549, 54syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
56 xp1st 6378 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
5755, 56syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
58 simpllr 737 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
y  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
59 xp1st 6378 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  A ) )
6058, 59syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  y
)  e.  ( Base `  A ) )
61 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )
62 1st2nd2 6388 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6349, 62syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6463fveq2d 5734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( (  Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
65 df-ov 6086 . . . . . . . . . . . . 13  |-  ( ( 1st `  x ) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( (  Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
6664, 65syl6eqr 2488 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( ( 1st `  x
) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) ) )
671, 11, 4, 5, 12, 51, 55xpchom 14279 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 1st `  x
) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
6866, 67eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
6961, 68eleqtrd 2514 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
70 xp1st 6378 . . . . . . . . . 10  |-  ( f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )  ->  ( 1st `  f )  e.  ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) ) )
7169, 70syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  f
)  e.  ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) ) )
72 simplr 733 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) )
731, 11, 4, 5, 12, 55, 58xpchom 14279 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y )  =  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A
) ( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) ) )
7472, 73eleqtrd 2514 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A
) ( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) ) )
75 xp1st 6378 . . . . . . . . . 10  |-  ( g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) ) )
7674, 75syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) ) )
772, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76comfeqval 13936 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) )  =  ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) )
7828ad4antr 714 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
79 xpcpropd.4 . . . . . . . . . 10  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
8079ad4antr 714 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(compf `  C )  =  (compf `  D ) )
81 xp2nd 6379 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
8251, 81syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
83 xp2nd 6379 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
8455, 83syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
85 xp2nd 6379 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
8658, 85syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  y
)  e.  ( Base `  C ) )
87 xp2nd 6379 . . . . . . . . . 10  |-  ( f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )  ->  ( 2nd `  f )  e.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C
) ( 2nd `  ( 2nd `  x ) ) ) )
8869, 87syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  f
)  e.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )
89 xp2nd 6379 . . . . . . . . . 10  |-  ( g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  ( 2nd `  x ) ) (  Hom  `  C )
( 2nd `  y
) ) )
9074, 89syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  ( 2nd `  x ) ) (  Hom  `  C )
( 2nd `  y
) ) )
913, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90comfeqval 13936 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) )  =  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) )
9277, 91opeq12d 3994 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
93923impa 1149 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( (
( Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
9493mpt2eq3dva 6140 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( ( ( Base `  A )  X.  ( Base `  C ) )  X.  ( ( Base `  A )  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  ->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
95943impa 1149 . . . 4  |-  ( (
ph  /\  x  e.  ( ( ( Base `  A )  X.  ( Base `  C ) )  X.  ( ( Base `  A )  X.  ( Base `  C ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  ->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
9695mpt2eq3dva 6140 . . 3  |-  ( ph  ->  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
9717, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96xpcval 14276 . 2  |-  ( ph  ->  ( B  X.c  D )  =  { <. ( Base `  ndx ) ,  ( ( Base `  A
)  X.  ( Base `  C ) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  ( A  X.c  C ) ) >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
9816, 97eqtr4d 2473 1  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {ctp 3818   <.cop 3819    X. cxp 4878   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   ndxcnx 13468   Basecbs 13471    Hom chom 13542  compcco 13543    Homf chomf 13893  compfccomf 13894    X.c cxpc 14267
This theorem is referenced by:  curfpropd  14332  oppchofcl  14359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-homf 13897  df-comf 13898  df-xpc 14271
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