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Theorem comfeqval 14981
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 6312 . . 3  |-  ( ph  ->  ( <. X ,  Y >. (compf `  C ) Z )  =  ( <. X ,  Y >. (compf `  D ) Z ) )
32oveqd 6312 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >. (compf `  D ) Z ) F ) )
4 eqid 2467 . . 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 14973 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
14 eqid 2467 . . 3  |-  (compf `  D
)  =  (compf `  D
)
15 eqid 2467 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2467 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
17 comfeqval.2 . . 3  |-  .xb  =  (comp `  D )
18 comfeqval.3 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1918homfeqbas 14969 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
205, 19syl5eq 2520 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
218, 20eleqtrd 2557 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
229, 20eleqtrd 2557 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
2310, 20eleqtrd 2557 . . 3  |-  ( ph  ->  Z  e.  ( Base `  D ) )
245, 6, 16, 18, 8, 9homfeqval 14970 . . . 4  |-  ( ph  ->  ( X H Y )  =  ( X ( Hom  `  D
) Y ) )
2511, 24eleqtrd 2557 . . 3  |-  ( ph  ->  F  e.  ( X ( Hom  `  D
) Y ) )
265, 6, 16, 18, 9, 10homfeqval 14970 . . . 4  |-  ( ph  ->  ( Y H Z )  =  ( Y ( Hom  `  D
) Z ) )
2712, 26eleqtrd 2557 . . 3  |-  ( ph  ->  G  e.  ( Y ( Hom  `  D
) Z ) )
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 14973 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  D ) Z ) F )  =  ( G ( <. X ,  Y >.  .xb  Z ) F ) )
293, 13, 283eqtr3d 2516 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 1379    e. wcel 1767   <.cop 4039   ` cfv 5594  (class class class)co 6295   Basecbs 14507   Hom chom 14583  compcco 14584   Hom f chomf 14938  compfccomf 14939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-homf 14942  df-comf 14943
This theorem is referenced by:  catpropd  14982  cidpropd  14983  oppccomfpropd  15000  monpropd  15010  funcpropd  15144  natpropd  15220  fucpropd  15221  xpcpropd  15352  hofpropd  15411
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