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Theorem xpcpropd 15324
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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
xpcpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
xpcpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  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 2460 . . 3  |-  ( A  X.c  C )  =  ( A  X.c  C )
2 eqid 2460 . . 3  |-  ( Base `  A )  =  (
Base `  A )
3 eqid 2460 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2460 . . 3  |-  ( Hom  `  A )  =  ( Hom  `  A )
5 eqid 2460 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2460 . . 3  |-  (comp `  A )  =  (comp `  A )
7 eqid 2460 . . 3  |-  (comp `  C )  =  (comp `  C )
8 xpcpropd.a . . 3  |-  ( ph  ->  A  e.  V )
9 xpcpropd.c . . 3  |-  ( ph  ->  C  e.  V )
10 eqidd 2461 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  A
)  X.  ( Base `  C ) ) )
111, 2, 3xpcbas 15294 . . . . 5  |-  ( (
Base `  A )  X.  ( Base `  C
) )  =  (
Base `  ( A  X.c  C ) )
12 eqid 2460 . . . . 5  |-  ( Hom  `  ( A  X.c  C ) )  =  ( Hom  `  ( A  X.c  C ) )
131, 11, 4, 5, 12xpchomfval 15295 . . . 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 2461 . . 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 15293 . 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 2460 . . 3  |-  ( B  X.c  D )  =  ( B  X.c  D )
18 eqid 2460 . . 3  |-  ( Base `  B )  =  (
Base `  B )
19 eqid 2460 . . 3  |-  ( Base `  D )  =  (
Base `  D )
20 eqid 2460 . . 3  |-  ( Hom  `  B )  =  ( Hom  `  B )
21 eqid 2460 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
22 eqid 2460 . . 3  |-  (comp `  B )  =  (comp `  B )
23 eqid 2460 . . 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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
2726homfeqbas 14941 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
28 xpcpropd.3 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
2928homfeqbas 14941 . . . 4  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3027, 29xpeq12d 5017 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  B
)  X.  ( Base `  D ) ) )
31263ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( Hom f  `  A )  =  ( Hom f  `  B ) )
32 xp1st 6804 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  u
)  e.  ( Base `  A ) )
33323ad2ant2 1013 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  u
)  e.  ( Base `  A ) )
34 xp1st 6804 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  v
)  e.  ( Base `  A ) )
35343ad2ant3 1014 . . . . . . 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 14942 . . . . . 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 1012 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
38 xp2nd 6805 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  u
)  e.  ( Base `  C ) )
39383ad2ant2 1013 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  u
)  e.  ( Base `  C ) )
40 xp2nd 6805 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  v
)  e.  ( Base `  C ) )
41403ad2ant3 1014 . . . . . . 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 14942 . . . . . 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 5017 . . . . 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 6336 . . . 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 2513 . . 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 731 . . . . . . . . 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 ) )  -> 
( Hom f  `  A )  =  ( Hom f  `  B ) )
47 xpcpropd.2 . . . . . . . . . 10  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
4847ad4antr 731 . . . . . . . . 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 766 . . . . . . . . . . 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 6804 . . . . . . . . . . 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 6804 . . . . . . . . . 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 6805 . . . . . . . . . . 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 6804 . . . . . . . . . 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 758 . . . . . . . . . 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 6804 . . . . . . . . . 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 461 . . . . . . . . . . 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 6811 . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . 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 6278 . . . . . . . . . . . . 13  |-  ( ( 1st `  x ) ( Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( ( Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
6664, 65syl6eqr 2519 . . . . . . . . . . . 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 15296 . . . . . . . . . . . 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 2501 . . . . . . . . . . 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 2550 . . . . . . . . . 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 6804 . . . . . . . . . 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 754 . . . . . . . . . . 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 15296 . . . . . . . . . . 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 2550 . . . . . . . . . 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 6804 . . . . . . . . . 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 14953 . . . . . . . 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 731 . . . . . . . . 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 ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
79 xpcpropd.4 . . . . . . . . . 10  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
8079ad4antr 731 . . . . . . . . 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 6805 . . . . . . . . . 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 6805 . . . . . . . . . 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 6805 . . . . . . . . . 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 6805 . . . . . . . . . 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 6805 . . . . . . . . . 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 14953 . . . . . . . 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 4214 . . . . . . 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 1186 . . . . . 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 6336 . . . . 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 1186 . . . 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 6336 . . 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 15293 . 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 2504 1  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {ctp 4024   <.cop 4026    X. cxp 4990   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   ndxcnx 14476   Basecbs 14479   Hom chom 14555  compcco 14556   Hom f chomf 14910  compfccomf 14911    X.c cxpc 15284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-hom 14568  df-cco 14569  df-homf 14914  df-comf 14915  df-xpc 15288
This theorem is referenced by:  curfpropd  15349  oppchofcl  15376
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