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Theorem grplactf1o 14893
Description: The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
grplact.1  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
grplact.2  |-  X  =  ( Base `  G
)
grplact.3  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
grplactf1o  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
) : X -1-1-onto-> X )
Distinct variable groups:    g, a, A    G, a, g    .+ , a,
g    X, a, g
Allowed substitution hints:    F( g, a)

Proof of Theorem grplactf1o
StepHypRef Expression
1 grplact.1 . . 3  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
2 grplact.2 . . 3  |-  X  =  ( Base `  G
)
3 grplact.3 . . 3  |-  .+  =  ( +g  `  G )
4 eqid 2438 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
51, 2, 3, 4grplactcnv 14892 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( ( inv g `  G
) `  A )
) ) )
65simpld 447 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
) : X -1-1-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4269   `'ccnv 4880   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534   Grpcgrp 14690   inv gcminusg 14691
This theorem is referenced by:  eqgen  14998  dchrsum2  21057  sumdchr2  21059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818
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