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Theorem gaf 15072
Description: The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaf.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
gaf  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )

Proof of Theorem gaf
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaf.1 . . . 4  |-  X  =  ( Base `  G
)
2 eqid 2436 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2436 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 15068 . . 3  |-  (  .(+)  e.  ( G  GrpAct  Y )  <-> 
( ( G  e. 
Grp  /\  Y  e.  _V )  /\  (  .(+)  : ( X  X.  Y ) --> Y  /\  A. x  e.  Y  ( ( ( 0g `  G )  .(+)  x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y ( +g  `  G ) z ) 
.(+)  x )  =  ( y  .(+)  ( z  .(+)  x ) ) ) ) ) )
54simprbi 451 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  (  .(+)  : ( X  X.  Y ) --> Y  /\  A. x  e.  Y  ( (
( 0g `  G
)  .(+)  x )  =  x  /\  A. y  e.  X  A. z  e.  X  ( (
y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) ) ) )
65simpld 446 1  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685    GrpAct cga 15066
This theorem is referenced by:  gafo  15073  gass  15078  gasubg  15079  gacan  15082  gapm  15083  gastacos  15087  orbsta  15090  galactghm  15106  sylow2alem2  15252
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ga 15067
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