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Theorem cmperltr 25409
Description: A right and left translation expressed as a composite. Note that  x and  y can't be the same. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cmperltr.1  |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )
cmperltr.2  |-  E  =  ( y  e.  X  |->  ( A G y ) )
cmperltr.3  |-  H  =  ( x  e.  X  |->  ( x G B ) )
cmperltr.4  |-  X  =  ran  G
Assertion
Ref Expression
cmperltr  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( E  o.  H ) )
Distinct variable groups:    x, A, y    x, B, y    x, E    x, G, y    x, X, y
Allowed substitution hints:    E( y)    F( x, y)    H( x, y)

Proof of Theorem cmperltr
StepHypRef Expression
1 cmperltr.1 . . 3  |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )
2 cmperltr.4 . . . . . . . 8  |-  X  =  ran  G
32grpoass 20870 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  x  e.  X  /\  B  e.  X )
)  ->  ( ( A G x ) G B )  =  ( A G ( x G B ) ) )
433exp2 1169 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( x  e.  X  ->  ( B  e.  X  ->  ( ( A G x ) G B )  =  ( A G ( x G B ) ) ) ) ) )
54com34 77 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  (
x  e.  X  -> 
( ( A G x ) G B )  =  ( A G ( x G B ) ) ) ) ) )
653imp1 1164 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  ( ( A G x ) G B )  =  ( A G ( x G B ) ) )
76mpteq2dva 4106 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
x  e.  X  |->  ( ( A G x ) G B ) )  =  ( x  e.  X  |->  ( A G ( x G B ) ) ) )
81, 7syl5eq 2327 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( x  e.  X  |->  ( A G ( x G B ) ) ) )
92grpocl 20867 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  B  e.  X )  ->  (
x G B )  e.  X )
1093expa 1151 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  B  e.  X
)  ->  ( x G B )  e.  X
)
1110an32s 779 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  B  e.  X )  /\  x  e.  X
)  ->  ( x G B )  e.  X
)
12113adantl2 1112 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  ( x G B )  e.  X
)
13 cmperltr.3 . . . 4  |-  H  =  ( x  e.  X  |->  ( x G B ) )
1413a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  H  =  ( x  e.  X  |->  ( x G B ) ) )
15 cmperltr.2 . . . 4  |-  E  =  ( y  e.  X  |->  ( A G y ) )
1615a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  E  =  ( y  e.  X  |->  ( A G y ) ) )
17 oveq2 5866 . . 3  |-  ( y  =  ( x G B )  ->  ( A G y )  =  ( A G ( x G B ) ) )
1812, 14, 16, 17fmptco 5691 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( E  o.  H )  =  ( x  e.  X  |->  ( A G ( x G B ) ) ) )
198, 18eqtr4d 2318 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( E  o.  H ) )
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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