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Theorem gagrpid 14797
Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gagrpid.1  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
gagrpid  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )

Proof of Theorem gagrpid
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2316 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
3 gagrpid.1 . . . . . 6  |-  .0.  =  ( 0g `  G )
41, 2, 3isga 14794 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  <-> 
( ( G  e. 
Grp  /\  Y  e.  _V )  /\  (  .(+)  : ( ( Base `  G )  X.  Y
) --> Y  /\  A. x  e.  Y  (
(  .0.  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) ) ) ) )
54simprbi 450 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  (  .(+)  : ( ( Base `  G
)  X.  Y ) --> Y  /\  A. x  e.  Y  ( (  .0.  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) ) ) )
65simprd 449 . . 3  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  ( (  .0.  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) ) )
7 simpl 443 . . . 4  |-  ( ( (  .0.  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) )  -> 
(  .0.  .(+)  x )  =  x )
87ralimi 2652 . . 3  |-  ( A. x  e.  Y  (
(  .0.  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) )  ->  A. x  e.  Y  (  .0.  .(+)  x )  =  x )
96, 8syl 15 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  (  .0.  .(+)  x )  =  x )
10 oveq2 5908 . . . 4  |-  ( x  =  A  ->  (  .0.  .(+)  x )  =  (  .0.  .(+)  A ) )
11 id 19 . . . 4  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2330 . . 3  |-  ( x  =  A  ->  (
(  .0.  .(+)  x )  =  x  <->  (  .0.  .(+) 
A )  =  A ) )
1312rspccva 2917 . 2  |-  ( ( A. x  e.  Y  (  .0.  .(+)  x )  =  x  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )
149, 13sylan 457 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822    X. cxp 4724   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   0gc0g 13449   Grpcgrp 14411    GrpAct cga 14792
This theorem is referenced by:  gafo  14799  gass  14804  gasubg  14805  galcan  14807  gacan  14808  gaorber  14811  gastacl  14812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-ga 14793
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