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Theorem cmprtr2 25397
Description: Composite of two right translations. (cmprtr 25396 with a distinct variable condition relaxed.) (Contributed by FL, 1-Jan-2011.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
cmprtr2.1  |-  H  =  ( x  e.  X  |->  ( x G B ) )
Assertion
Ref Expression
cmprtr2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( x G ( B G A ) ) ) )
Distinct variable groups:    x, A    x, B    x, G    x, X
Allowed substitution hints:    F( x)    H( x)

Proof of Theorem cmprtr2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 trfun.2 . . . 4  |-  F  =  ( x  e.  X  |->  ( x G A ) )
2 oveq1 5865 . . . . 5  |-  ( x  =  y  ->  (
x G A )  =  ( y G A ) )
32cbvmptv 4111 . . . 4  |-  ( x  e.  X  |->  ( x G A ) )  =  ( y  e.  X  |->  ( y G A ) )
41, 3eqtri 2303 . . 3  |-  F  =  ( y  e.  X  |->  ( y G A ) )
5 trinv.1 . . 3  |-  X  =  ran  G
6 cmprtr2.1 . . . 4  |-  H  =  ( x  e.  X  |->  ( x G B ) )
7 oveq1 5865 . . . . 5  |-  ( x  =  y  ->  (
x G B )  =  ( y G B ) )
87cbvmptv 4111 . . . 4  |-  ( x  e.  X  |->  ( x G B ) )  =  ( y  e.  X  |->  ( y G B ) )
96, 8eqtri 2303 . . 3  |-  H  =  ( y  e.  X  |->  ( y G B ) )
104, 5, 9cmprtr 25396 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( y  e.  X  |->  ( y G ( B G A ) ) ) )
11 oveq1 5865 . . 3  |-  ( y  =  x  ->  (
y G ( B G A ) )  =  ( x G ( B G A ) ) )
1211cbvmptv 4111 . 2  |-  ( y  e.  X  |->  ( y G ( B G A ) ) )  =  ( x  e.  X  |->  ( x G ( B G A ) ) )
1310, 12syl6eq 2331 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( x G ( B G A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077   ran crn 4690    o. ccom 4693  (class class class)co 5858   GrpOpcgr 20853
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-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-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-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-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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