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Theorem gaass 14751
Description: An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gaass.1  |-  X  =  ( Base `  G
)
gaass.2  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
gaass  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )
)  ->  ( ( A  .+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) )

Proof of Theorem gaass
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaass.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
2 gaass.2 . . . . . . . 8  |-  .+  =  ( +g  `  G )
3 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 14745 . . . . . . 7  |-  (  .(+)  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  .+  z
)  .(+)  x )  =  ( y  .(+)  ( z 
.(+)  x ) ) ) ) ) )
54simprbi 450 . . . . . 6  |-  (  .(+)  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  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x ) ) ) ) )
65simprd 449 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  ( ( ( 0g
`  G )  .(+)  x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y  .+  z
)  .(+)  x )  =  ( y  .(+)  ( z 
.(+)  x ) ) ) )
7 simpr 447 . . . . . 6  |-  ( ( ( ( 0g `  G )  .(+)  x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y  .+  z
)  .(+)  x )  =  ( y  .(+)  ( z 
.(+)  x ) ) )  ->  A. y  e.  X  A. z  e.  X  ( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
) )
87ralimi 2618 . . . . 5  |-  ( A. x  e.  Y  (
( ( 0g `  G )  .(+)  x )  =  x  /\  A. y  e.  X  A. z  e.  X  (
( y  .+  z
)  .(+)  x )  =  ( y  .(+)  ( z 
.(+)  x ) ) )  ->  A. x  e.  Y  A. y  e.  X  A. z  e.  X  ( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
) )
96, 8syl 15 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  A. y  e.  X  A. z  e.  X  ( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
) )
10 oveq2 5866 . . . . . 6  |-  ( x  =  C  ->  (
( y  .+  z
)  .(+)  x )  =  ( ( y  .+  z )  .(+)  C ) )
11 oveq2 5866 . . . . . . 7  |-  ( x  =  C  ->  (
z  .(+)  x )  =  ( z  .(+)  C ) )
1211oveq2d 5874 . . . . . 6  |-  ( x  =  C  ->  (
y  .(+)  ( z  .(+)  x ) )  =  ( y  .(+)  ( z  .(+)  C ) ) )
1310, 12eqeq12d 2297 . . . . 5  |-  ( x  =  C  ->  (
( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
)  <->  ( ( y 
.+  z )  .(+)  C )  =  ( y 
.(+)  ( z  .(+)  C ) ) ) )
14 oveq1 5865 . . . . . . 7  |-  ( y  =  A  ->  (
y  .+  z )  =  ( A  .+  z ) )
1514oveq1d 5873 . . . . . 6  |-  ( y  =  A  ->  (
( y  .+  z
)  .(+)  C )  =  ( ( A  .+  z )  .(+)  C ) )
16 oveq1 5865 . . . . . 6  |-  ( y  =  A  ->  (
y  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( z  .(+)  C ) ) )
1715, 16eqeq12d 2297 . . . . 5  |-  ( y  =  A  ->  (
( ( y  .+  z )  .(+)  C )  =  ( y  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  z )  .(+)  C )  =  ( A 
.(+)  ( z  .(+)  C ) ) ) )
18 oveq2 5866 . . . . . . 7  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
1918oveq1d 5873 . . . . . 6  |-  ( z  =  B  ->  (
( A  .+  z
)  .(+)  C )  =  ( ( A  .+  B )  .(+)  C ) )
20 oveq1 5865 . . . . . . 7  |-  ( z  =  B  ->  (
z  .(+)  C )  =  ( B  .(+)  C ) )
2120oveq2d 5874 . . . . . 6  |-  ( z  =  B  ->  ( A  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( B  .(+) 
C ) ) )
2219, 21eqeq12d 2297 . . . . 5  |-  ( z  =  B  ->  (
( ( A  .+  z )  .(+)  C )  =  ( A  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) ) )
2313, 17, 22rspc3v 2893 . . . 4  |-  ( ( C  e.  Y  /\  A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  Y  A. y  e.  X  A. z  e.  X  ( ( y 
.+  z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) )  ->  (
( A  .+  B
)  .(+)  C )  =  ( A  .(+)  ( B 
.(+)  C ) ) ) )
249, 23syl5 28 . . 3  |-  ( ( C  e.  Y  /\  A  e.  X  /\  B  e.  X )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
25243coml 1158 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
2625impcom 419 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )
)  ->  ( ( A  .+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362    GrpAct cga 14743
This theorem is referenced by:  gass  14755  gasubg  14756  galcan  14758  gacan  14759  gaorber  14762  gastacl  14763  gastacos  14764  galactghm  14783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ga 14744
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