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Theorem comfeqval 13611
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  ->  (  Homf 
`  C )  =  (  Homf 
`  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 5875 . . 3  |-  ( ph  ->  ( <. X ,  Y >. (compf `  C ) Z )  =  ( <. X ,  Y >. (compf `  D ) Z ) )
32oveqd 5875 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >. (compf `  D ) Z ) F ) )
4 eqid 2283 . . 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 13603 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
14 eqid 2283 . . 3  |-  (compf `  D
)  =  (compf `  D
)
15 eqid 2283 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2283 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
17 comfeqval.2 . . 3  |-  .xb  =  (comp `  D )
18 comfeqval.3 . . . . . 6  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1918homfeqbas 13599 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
205, 19syl5eq 2327 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
218, 20eleqtrd 2359 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
229, 20eleqtrd 2359 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
2310, 20eleqtrd 2359 . . 3  |-  ( ph  ->  Z  e.  ( Base `  D ) )
245, 6, 16, 18, 8, 9homfeqval 13600 . . . 4  |-  ( ph  ->  ( X H Y )  =  ( X (  Hom  `  D
) Y ) )
2511, 24eleqtrd 2359 . . 3  |-  ( ph  ->  F  e.  ( X (  Hom  `  D
) Y ) )
265, 6, 16, 18, 9, 10homfeqval 13600 . . . 4  |-  ( ph  ->  ( Y H Z )  =  ( Y (  Hom  `  D
) Z ) )
2712, 26eleqtrd 2359 . . 3  |-  ( ph  ->  G  e.  ( Y (  Hom  `  D
) Z ) )
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 13603 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  D ) Z ) F )  =  ( G ( <. X ,  Y >.  .xb  Z ) F ) )
293, 13, 283eqtr3d 2323 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G (
<. X ,  Y >.  .xb 
Z ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220    Homf chomf 13568  compfccomf 13569
This theorem is referenced by:  catpropd  13612  cidpropd  13613  oppccomfpropd  13630  monpropd  13640  funcpropd  13774  natpropd  13850  fucpropd  13851  xpcpropd  13982  hofpropd  14041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homf 13572  df-comf 13573
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