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Theorem comfeqval 14646
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 6107 . . 3  |-  ( ph  ->  ( <. X ,  Y >. (compf `  C ) Z )  =  ( <. X ,  Y >. (compf `  D ) Z ) )
32oveqd 6107 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >. (compf `  D ) Z ) F ) )
4 eqid 2442 . . 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 14638 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
14 eqid 2442 . . 3  |-  (compf `  D
)  =  (compf `  D
)
15 eqid 2442 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2442 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
17 comfeqval.2 . . 3  |-  .xb  =  (comp `  D )
18 comfeqval.3 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1918homfeqbas 14634 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
205, 19syl5eq 2486 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
218, 20eleqtrd 2518 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
229, 20eleqtrd 2518 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
2310, 20eleqtrd 2518 . . 3  |-  ( ph  ->  Z  e.  ( Base `  D ) )
245, 6, 16, 18, 8, 9homfeqval 14635 . . . 4  |-  ( ph  ->  ( X H Y )  =  ( X ( Hom  `  D
) Y ) )
2511, 24eleqtrd 2518 . . 3  |-  ( ph  ->  F  e.  ( X ( Hom  `  D
) Y ) )
265, 6, 16, 18, 9, 10homfeqval 14635 . . . 4  |-  ( ph  ->  ( Y H Z )  =  ( Y ( Hom  `  D
) Z ) )
2712, 26eleqtrd 2518 . . 3  |-  ( ph  ->  G  e.  ( Y ( Hom  `  D
) Z ) )
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 14638 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  D ) Z ) F )  =  ( G ( <. X ,  Y >.  .xb  Z ) F ) )
293, 13, 283eqtr3d 2482 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 1369    e. wcel 1756   <.cop 3882   ` cfv 5417  (class class class)co 6090   Basecbs 14173   Hom chom 14248  compcco 14249   Hom f chomf 14603  compfccomf 14604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-homf 14607  df-comf 14608
This theorem is referenced by:  catpropd  14647  cidpropd  14648  oppccomfpropd  14665  monpropd  14675  funcpropd  14809  natpropd  14885  fucpropd  14886  xpcpropd  15017  hofpropd  15076
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