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Theorem ltrcmp 25405
Description: Left translation expressed as a composite. (Contributed by FL, 3-Jul-2012.)
Hypotheses
Ref Expression
ltrdom.1  |-  F  =  ( x  e.  X  |->  ( A G x ) )
ltrdom.2  |-  X  =  ran  G
Assertion
Ref Expression
ltrcmp  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  F  =  ( G  o.  ( x  e.  X  |-> 
<. A ,  x >. ) ) )
Distinct variable groups:    x, A    x, G    x, X
Allowed substitution hint:    F( x)

Proof of Theorem ltrcmp
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 opelxpi 4721 . . . 4  |-  ( ( A  e.  X  /\  x  e.  X )  -> 
<. A ,  x >.  e.  ( X  X.  X
) )
21adantll 694 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  <. A ,  x >.  e.  ( X  X.  X ) )
3 eqidd 2284 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  |->  <. A ,  x >. )  =  ( x  e.  X  |->  <. A ,  x >. ) )
4 ltrdom.2 . . . . . . 7  |-  X  =  ran  G
54grpofo 20866 . . . . . 6  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
6 fof 5451 . . . . . 6  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
75, 6syl 15 . . . . 5  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
87adantr 451 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  G : ( X  X.  X ) --> X )
98feqmptd 5575 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  G  =  ( y  e.  ( X  X.  X
)  |->  ( G `  y ) ) )
10 fveq2 5525 . . . 4  |-  ( y  =  <. A ,  x >.  ->  ( G `  y )  =  ( G `  <. A ,  x >. ) )
11 df-ov 5861 . . . 4  |-  ( A G x )  =  ( G `  <. A ,  x >. )
1210, 11syl6eqr 2333 . . 3  |-  ( y  =  <. A ,  x >.  ->  ( G `  y )  =  ( A G x ) )
132, 3, 9, 12fmptco 5691 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( G  o.  ( x  e.  X  |->  <. A ,  x >. ) )  =  ( x  e.  X  |->  ( A G x ) ) )
14 ltrdom.1 . 2  |-  F  =  ( x  e.  X  |->  ( A G x ) )
1513, 14syl6reqr 2334 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  F  =  ( G  o.  ( x  e.  X  |-> 
<. A ,  x >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    e. cmpt 4077    X. cxp 4687   ran crn 4690    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (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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