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Theorem comfeqval 14975
Description: Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeqval.b  |-  B  =  ( Base `  C
)
comfeqval.h  |-  H  =  ( Hom  `  C
)
comfeqval.1  |-  .x.  =  (comp `  C )
comfeqval.2  |-  .xb  =  (comp `  D )
comfeqval.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
comfeqval.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
comfeqval.x  |-  ( ph  ->  X  e.  B )
comfeqval.y  |-  ( ph  ->  Y  e.  B )
comfeqval.z  |-  ( ph  ->  Z  e.  B )
comfeqval.f  |-  ( ph  ->  F  e.  ( X H Y ) )
comfeqval.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
comfeqval  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G (
<. X ,  Y >.  .xb 
Z ) F ) )

Proof of Theorem comfeqval
StepHypRef Expression
1 comfeqval.4 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
21oveqd 6294 . . 3  |-  ( ph  ->  ( <. X ,  Y >. (compf `  C ) Z )  =  ( <. X ,  Y >. (compf `  D ) Z ) )
32oveqd 6294 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >. (compf `  D ) Z ) F ) )
4 eqid 2441 . . 3  |-  (compf `  C
)  =  (compf `  C
)
5 comfeqval.b . . 3  |-  B  =  ( Base `  C
)
6 comfeqval.h . . 3  |-  H  =  ( Hom  `  C
)
7 comfeqval.1 . . 3  |-  .x.  =  (comp `  C )
8 comfeqval.x . . 3  |-  ( ph  ->  X  e.  B )
9 comfeqval.y . . 3  |-  ( ph  ->  Y  e.  B )
10 comfeqval.z . . 3  |-  ( ph  ->  Z  e.  B )
11 comfeqval.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
12 comfeqval.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
134, 5, 6, 7, 8, 9, 10, 11, 12comfval 14967 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
14 eqid 2441 . . 3  |-  (compf `  D
)  =  (compf `  D
)
15 eqid 2441 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2441 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
17 comfeqval.2 . . 3  |-  .xb  =  (comp `  D )
18 comfeqval.3 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1918homfeqbas 14963 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
205, 19syl5eq 2494 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
218, 20eleqtrd 2531 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
229, 20eleqtrd 2531 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
2310, 20eleqtrd 2531 . . 3  |-  ( ph  ->  Z  e.  ( Base `  D ) )
245, 6, 16, 18, 8, 9homfeqval 14964 . . . 4  |-  ( ph  ->  ( X H Y )  =  ( X ( Hom  `  D
) Y ) )
2511, 24eleqtrd 2531 . . 3  |-  ( ph  ->  F  e.  ( X ( Hom  `  D
) Y ) )
265, 6, 16, 18, 9, 10homfeqval 14964 . . . 4  |-  ( ph  ->  ( Y H Z )  =  ( Y ( Hom  `  D
) Z ) )
2712, 26eleqtrd 2531 . . 3  |-  ( ph  ->  G  e.  ( Y ( Hom  `  D
) Z ) )
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 14967 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  D ) Z ) F )  =  ( G ( <. X ,  Y >.  .xb  Z ) F ) )
293, 13, 283eqtr3d 2490 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G (
<. X ,  Y >.  .xb 
Z ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   <.cop 4016   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Hom chom 14580  compcco 14581   Hom f chomf 14935  compfccomf 14936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-homf 14939  df-comf 14940
This theorem is referenced by:  catpropd  14976  cidpropd  14977  oppccomfpropd  14994  monpropd  15004  funcpropd  15138  natpropd  15214  fucpropd  15215  xpcpropd  15346  hofpropd  15405
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