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Theorem xpcpropd 13984
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 2285 . . 3  |-  ( A  X.c  C )  =  ( A  X.c  C )
2 eqid 2285 . . 3  |-  ( Base `  A )  =  (
Base `  A )
3 eqid 2285 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2285 . . 3  |-  (  Hom  `  A )  =  (  Hom  `  A )
5 eqid 2285 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2285 . . 3  |-  (comp `  A )  =  (comp `  A )
7 eqid 2285 . . 3  |-  (comp `  C )  =  (comp `  C )
8 xpcpropd.a . . 3  |-  ( ph  ->  A  e.  V )
9 xpcpropd.c . . 3  |-  ( ph  ->  C  e.  V )
10 eqidd 2286 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  A
)  X.  ( Base `  C ) ) )
111, 2, 3xpcbas 13954 . . . . 5  |-  ( (
Base `  A )  X.  ( Base `  C
) )  =  (
Base `  ( A  X.c  C ) )
12 eqid 2285 . . . . 5  |-  (  Hom  `  ( A  X.c  C ) )  =  (  Hom  `  ( A  X.c  C ) )
131, 11, 4, 5, 12xpchomfval 13955 . . . 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 10 . . 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 2286 . . 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 13953 . 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 2285 . . 3  |-  ( B  X.c  D )  =  ( B  X.c  D )
18 eqid 2285 . . 3  |-  ( Base `  B )  =  (
Base `  B )
19 eqid 2285 . . 3  |-  ( Base `  D )  =  (
Base `  D )
20 eqid 2285 . . 3  |-  (  Hom  `  B )  =  (  Hom  `  B )
21 eqid 2285 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
22 eqid 2285 . . 3  |-  (comp `  B )  =  (comp `  B )
23 eqid 2285 . . 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 13601 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
28 xpcpropd.3 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
2928homfeqbas 13601 . . . 4  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3027, 29xpeq12d 4716 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  B
)  X.  ( Base `  D ) ) )
31263ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
32 xp1st 6151 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  u
)  e.  ( Base `  A ) )
33323ad2ant2 977 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  u
)  e.  ( Base `  A ) )
34 xp1st 6151 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  v
)  e.  ( Base `  A ) )
35343ad2ant3 978 . . . . . . 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 13602 . . . . . 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 976 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
38 xp2nd 6152 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  u
)  e.  ( Base `  C ) )
39383ad2ant2 977 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  u
)  e.  ( Base `  C ) )
40 xp2nd 6152 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  v
)  e.  ( Base `  C ) )
41403ad2ant3 978 . . . . . . 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 13602 . . . . . 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 4716 . . . . 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 5914 . . . 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 2329 . . 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 712 . . . . . . . . 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 712 . . . . . . . . 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 743 . . . . . . . . . . 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 6151 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5149, 50syl 15 . . . . . . . . . 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 6151 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
5351, 52syl 15 . . . . . . . . 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 6152 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5549, 54syl 15 . . . . . . . . . 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 6151 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
5755, 56syl 15 . . . . . . . . 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 735 . . . . . . . . . 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 6151 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  A ) )
6058, 59syl 15 . . . . . . . . 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 447 . . . . . . . . . . 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 6161 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6349, 62syl 15 . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . 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 5863 . . . . . . . . . . . . 13  |-  ( ( 1st `  x ) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( (  Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
6664, 65syl6eqr 2335 . . . . . . . . . . . 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 13956 . . . . . . . . . . . 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 2317 . . . . . . . . . . 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 2361 . . . . . . . . . 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 6151 . . . . . . . . . 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 15 . . . . . . . . 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 731 . . . . . . . . . . 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 13956 . . . . . . . . . . 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 2361 . . . . . . . . . 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 6151 . . . . . . . . . 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 15 . . . . . . . . 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 13613 . . . . . . . 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 712 . . . . . . . . 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 712 . . . . . . . . 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 6152 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
8251, 81syl 15 . . . . . . . . 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 6152 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
8455, 83syl 15 . . . . . . . . 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 6152 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
8658, 85syl 15 . . . . . . . . 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 6152 . . . . . . . . . 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 15 . . . . . . . . 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 6152 . . . . . . . . . 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 15 . . . . . . . . 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 13613 . . . . . . . 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 3806 . . . . . . 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 1146 . . . . . 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 5914 . . . . 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 1146 . . . 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 5914 . . 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 13953 . 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 2320 1  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   {ctp 3644   <.cop 3645    X. cxp 4689   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   1stc1st 6122   2ndc2nd 6123   ndxcnx 13147   Basecbs 13150    Hom chom 13221  compcco 13222    Homf chomf 13570  compfccomf 13571    X.c cxpc 13944
This theorem is referenced by:  curfpropd  14009  oppchofcl  14036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-hom 13234  df-cco 13235  df-homf 13574  df-comf 13575  df-xpc 13948
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