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Theorem gaass 15033
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 2408 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 15027 . . . . . . 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 451 . . . . . 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 450 . . . . 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 448 . . . . . 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 2745 . . . . 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 16 . . . 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 6052 . . . . . 6  |-  ( x  =  C  ->  (
( y  .+  z
)  .(+)  x )  =  ( ( y  .+  z )  .(+)  C ) )
11 oveq2 6052 . . . . . . 7  |-  ( x  =  C  ->  (
z  .(+)  x )  =  ( z  .(+)  C ) )
1211oveq2d 6060 . . . . . 6  |-  ( x  =  C  ->  (
y  .(+)  ( z  .(+)  x ) )  =  ( y  .(+)  ( z  .(+)  C ) ) )
1310, 12eqeq12d 2422 . . . . 5  |-  ( x  =  C  ->  (
( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
)  <->  ( ( y 
.+  z )  .(+)  C )  =  ( y 
.(+)  ( z  .(+)  C ) ) ) )
14 oveq1 6051 . . . . . . 7  |-  ( y  =  A  ->  (
y  .+  z )  =  ( A  .+  z ) )
1514oveq1d 6059 . . . . . 6  |-  ( y  =  A  ->  (
( y  .+  z
)  .(+)  C )  =  ( ( A  .+  z )  .(+)  C ) )
16 oveq1 6051 . . . . . 6  |-  ( y  =  A  ->  (
y  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( z  .(+)  C ) ) )
1715, 16eqeq12d 2422 . . . . 5  |-  ( y  =  A  ->  (
( ( y  .+  z )  .(+)  C )  =  ( y  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  z )  .(+)  C )  =  ( A 
.(+)  ( z  .(+)  C ) ) ) )
18 oveq2 6052 . . . . . . 7  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
1918oveq1d 6059 . . . . . 6  |-  ( z  =  B  ->  (
( A  .+  z
)  .(+)  C )  =  ( ( A  .+  B )  .(+)  C ) )
20 oveq1 6051 . . . . . . 7  |-  ( z  =  B  ->  (
z  .(+)  C )  =  ( B  .(+)  C ) )
2120oveq2d 6060 . . . . . 6  |-  ( z  =  B  ->  ( A  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( B  .(+) 
C ) ) )
2219, 21eqeq12d 2422 . . . . 5  |-  ( z  =  B  ->  (
( ( A  .+  z )  .(+)  C )  =  ( A  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) ) )
2313, 17, 22rspc3v 3025 . . . 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 30 . . 3  |-  ( ( C  e.  Y  /\  A  e.  X  /\  B  e.  X )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
25243coml 1160 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
2625impcom 420 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920    X. cxp 4839   -->wf 5413   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488   0gc0g 13682   Grpcgrp 14644    GrpAct cga 15025
This theorem is referenced by:  gass  15037  gasubg  15038  galcan  15040  gacan  15041  gaorber  15044  gastacl  15045  gastacos  15046  galactghm  15065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-ga 15026
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