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Theorem ltrcmp 25508
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 4737 . . . 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 2297 . . 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 20882 . . . . . 6  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
6 fof 5467 . . . . . 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 5591 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  G  =  ( y  e.  ( X  X.  X
)  |->  ( G `  y ) ) )
10 fveq2 5541 . . . 4  |-  ( y  =  <. A ,  x >.  ->  ( G `  y )  =  ( G `  <. A ,  x >. ) )
11 df-ov 5877 . . . 4  |-  ( A G x )  =  ( G `  <. A ,  x >. )
1210, 11syl6eqr 2346 . . 3  |-  ( y  =  <. A ,  x >.  ->  ( G `  y )  =  ( A G x ) )
132, 3, 9, 12fmptco 5707 . 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 2347 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 1632    e. wcel 1696   <.cop 3656    e. cmpt 4093    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869
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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