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Theorem gagrpid 15063
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 2435 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2435 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
3 gagrpid.1 . . . . . 6  |-  .0.  =  ( 0g `  G )
41, 2, 3isga 15060 . . . . 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 451 . . . 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 450 . . 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 444 . . . 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 2773 . . 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 16 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  (  .0.  .(+)  x )  =  x )
10 oveq2 6081 . . . 4  |-  ( x  =  A  ->  (  .0.  .(+)  x )  =  (  .0.  .(+)  A ) )
11 id 20 . . . 4  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2449 . . 3  |-  ( x  =  A  ->  (
(  .0.  .(+)  x )  =  x  <->  (  .0.  .(+) 
A )  =  A ) )
1312rspccva 3043 . 2  |-  ( ( A. x  e.  Y  (  .0.  .(+)  x )  =  x  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )
149, 13sylan 458 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715   Grpcgrp 14677    GrpAct cga 15058
This theorem is referenced by:  gafo  15065  gass  15070  gasubg  15071  galcan  15073  gacan  15074  gaorber  15077  gastacl  15078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ga 15059
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