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Theorem xpcpropd 15801
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 2402 . . 3  |-  ( A  X.c  C )  =  ( A  X.c  C )
2 eqid 2402 . . 3  |-  ( Base `  A )  =  (
Base `  A )
3 eqid 2402 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2402 . . 3  |-  ( Hom  `  A )  =  ( Hom  `  A )
5 eqid 2402 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2402 . . 3  |-  (comp `  A )  =  (comp `  A )
7 eqid 2402 . . 3  |-  (comp `  C )  =  (comp `  C )
8 xpcpropd.a . . 3  |-  ( ph  ->  A  e.  V )
9 xpcpropd.c . . 3  |-  ( ph  ->  C  e.  V )
10 eqidd 2403 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  A
)  X.  ( Base `  C ) ) )
111, 2, 3xpcbas 15771 . . . . 5  |-  ( (
Base `  A )  X.  ( Base `  C
) )  =  (
Base `  ( A  X.c  C ) )
12 eqid 2402 . . . . 5  |-  ( Hom  `  ( A  X.c  C ) )  =  ( Hom  `  ( A  X.c  C ) )
131, 11, 4, 5, 12xpchomfval 15772 . . . 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 2403 . . 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 15770 . 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 2402 . . 3  |-  ( B  X.c  D )  =  ( B  X.c  D )
18 eqid 2402 . . 3  |-  ( Base `  B )  =  (
Base `  B )
19 eqid 2402 . . 3  |-  ( Base `  D )  =  (
Base `  D )
20 eqid 2402 . . 3  |-  ( Hom  `  B )  =  ( Hom  `  B )
21 eqid 2402 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
22 eqid 2402 . . 3  |-  (comp `  B )  =  (comp `  B )
23 eqid 2402 . . 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 15309 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
28 xpcpropd.3 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
2928homfeqbas 15309 . . . 4  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3027, 29xpeq12d 4848 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  B
)  X.  ( Base `  D ) ) )
31263ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( Hom f  `  A )  =  ( Hom f  `  B ) )
32 xp1st 6814 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  u
)  e.  ( Base `  A ) )
33323ad2ant2 1019 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  u
)  e.  ( Base `  A ) )
34 xp1st 6814 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  v
)  e.  ( Base `  A ) )
35343ad2ant3 1020 . . . . . . 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 15310 . . . . . 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 1018 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
38 xp2nd 6815 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  u
)  e.  ( Base `  C ) )
39383ad2ant2 1019 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  u
)  e.  ( Base `  C ) )
40 xp2nd 6815 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  v
)  e.  ( Base `  C ) )
41403ad2ant3 1020 . . . . . . 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 15310 . . . . . 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 4848 . . . . 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 6342 . . . 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 2455 . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 769 . . . . . . . . . . 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 6814 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5149, 50syl 17 . . . . . . . . . 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 6814 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
5351, 52syl 17 . . . . . . . . 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 6815 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5549, 54syl 17 . . . . . . . . . 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 6814 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
5755, 56syl 17 . . . . . . . . 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 761 . . . . . . . . . 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 6814 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  A ) )
6058, 59syl 17 . . . . . . . . 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 459 . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6349, 62syl 17 . . . . . . . . . . . . . 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 5853 . . . . . . . . . . . . 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 6281 . . . . . . . . . . . . 13  |-  ( ( 1st `  x ) ( Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( ( Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
6664, 65syl6eqr 2461 . . . . . . . . . . . 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 15773 . . . . . . . . . . . 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 2443 . . . . . . . . . . 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 2492 . . . . . . . . . 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 6814 . . . . . . . . . 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 17 . . . . . . . . 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 15773 . . . . . . . . . . 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 2492 . . . . . . . . . 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 6814 . . . . . . . . . 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 17 . . . . . . . . 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 15321 . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 6815 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
8251, 81syl 17 . . . . . . . . 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 6815 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
8455, 83syl 17 . . . . . . . . 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 6815 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
8658, 85syl 17 . . . . . . . . 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 6815 . . . . . . . . . 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 17 . . . . . . . . 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 6815 . . . . . . . . . 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 17 . . . . . . . . 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 15321 . . . . . . . 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 4167 . . . . . . 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 1192 . . . . . 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 6342 . . . . 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 1192 . . . 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 6342 . . 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 15770 . 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 2446 1  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {ctp 3976   <.cop 3978    X. cxp 4821   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783   ndxcnx 14838   Basecbs 14841   Hom chom 14920  compcco 14921   Hom f chomf 15280  compfccomf 15281    X.c cxpc 15761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-hom 14933  df-cco 14934  df-homf 15284  df-comf 15285  df-xpc 15765
This theorem is referenced by:  curfpropd  15826  oppchofcl  15853
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