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Theorem comfeqval 13861
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 6037 . . 3  |-  ( ph  ->  ( <. X ,  Y >. (compf `  C ) Z )  =  ( <. X ,  Y >. (compf `  D ) Z ) )
32oveqd 6037 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >. (compf `  D ) Z ) F ) )
4 eqid 2387 . . 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 13853 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  C ) Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
14 eqid 2387 . . 3  |-  (compf `  D
)  =  (compf `  D
)
15 eqid 2387 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2387 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
17 comfeqval.2 . . 3  |-  .xb  =  (comp `  D )
18 comfeqval.3 . . . . . 6  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1918homfeqbas 13849 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
205, 19syl5eq 2431 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
218, 20eleqtrd 2463 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
229, 20eleqtrd 2463 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
2310, 20eleqtrd 2463 . . 3  |-  ( ph  ->  Z  e.  ( Base `  D ) )
245, 6, 16, 18, 8, 9homfeqval 13850 . . . 4  |-  ( ph  ->  ( X H Y )  =  ( X (  Hom  `  D
) Y ) )
2511, 24eleqtrd 2463 . . 3  |-  ( ph  ->  F  e.  ( X (  Hom  `  D
) Y ) )
265, 6, 16, 18, 9, 10homfeqval 13850 . . . 4  |-  ( ph  ->  ( Y H Z )  =  ( Y (  Hom  `  D
) Z ) )
2712, 26eleqtrd 2463 . . 3  |-  ( ph  ->  G  e.  ( Y (  Hom  `  D
) Z ) )
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 13853 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >. (compf `  D ) Z ) F )  =  ( G ( <. X ,  Y >.  .xb  Z ) F ) )
293, 13, 283eqtr3d 2427 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 1649    e. wcel 1717   <.cop 3760   ` cfv 5394  (class class class)co 6020   Basecbs 13396    Hom chom 13467  compcco 13468    Homf chomf 13818  compfccomf 13819
This theorem is referenced by:  catpropd  13862  cidpropd  13863  oppccomfpropd  13880  monpropd  13890  funcpropd  14024  natpropd  14100  fucpropd  14101  xpcpropd  14232  hofpropd  14291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-homf 13822  df-comf 13823
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