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Theorem cmprtr 25499
Description: Composite of two right translations. The terms  A and  B are constant. Don't use. See cmprtr2 25500. (Contributed by FL, 17-Oct-2010.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
cmprtr.1  |-  H  =  ( x  e.  X  |->  ( x G B ) )
Assertion
Ref Expression
cmprtr  |-  ( ( 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, F    x, G    x, X
Allowed substitution hint:    H( x)

Proof of Theorem cmprtr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 trinv.1 . . . . . . 7  |-  X  =  ran  G
21grpocl 20883 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  B  e.  X )  ->  (
x G B )  e.  X )
323expa 1151 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  B  e.  X
)  ->  ( x G B )  e.  X
)
43an32s 779 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  B  e.  X )  /\  x  e.  X
)  ->  ( x G B )  e.  X
)
543adantl2 1112 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  ( x G B )  e.  X
)
6 cmprtr.1 . . . 4  |-  H  =  ( x  e.  X  |->  ( x G B ) )
76a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  H  =  ( x  e.  X  |->  ( x G B ) ) )
8 trfun.2 . . . . 5  |-  F  =  ( x  e.  X  |->  ( x G A ) )
9 oveq1 5881 . . . . . 6  |-  ( x  =  y  ->  (
x G A )  =  ( y G A ) )
109cbvmptv 4127 . . . . 5  |-  ( x  e.  X  |->  ( x G A ) )  =  ( y  e.  X  |->  ( y G A ) )
118, 10eqtri 2316 . . . 4  |-  F  =  ( y  e.  X  |->  ( y G A ) )
1211a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( y  e.  X  |->  ( y G A ) ) )
13 oveq1 5881 . . 3  |-  ( y  =  ( x G B )  ->  (
y G A )  =  ( ( x G B ) G A ) )
145, 7, 12, 13fmptco 5707 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( x G B ) G A ) ) )
15 simpl1 958 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  G  e.  GrpOp
)
16 simpr 447 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
17 simpl3 960 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  B  e.  X )
18 simpl2 959 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
191grpoass 20886 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  B  e.  X  /\  A  e.  X )
)  ->  ( (
x G B ) G A )  =  ( x G ( B G A ) ) )
2015, 16, 17, 18, 19syl13anc 1184 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  ( (
x G B ) G A )  =  ( x G ( B G A ) ) )
2120mpteq2dva 4122 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
x  e.  X  |->  ( ( x G B ) G A ) )  =  ( x  e.  X  |->  ( x G ( B G A ) ) ) )
2214, 21eqtrd 2328 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    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   ran crn 4706    o. ccom 4709  (class class class)co 5874   GrpOpcgr 20869
This theorem is referenced by:  cmprtr2  25500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874
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