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Theorem gaass 15105
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 2442 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 15099 . . . . . . 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 452 . . . . . 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 451 . . . . 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 449 . . . . . 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 2787 . . . . 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 6118 . . . . . 6  |-  ( x  =  C  ->  (
( y  .+  z
)  .(+)  x )  =  ( ( y  .+  z )  .(+)  C ) )
11 oveq2 6118 . . . . . . 7  |-  ( x  =  C  ->  (
z  .(+)  x )  =  ( z  .(+)  C ) )
1211oveq2d 6126 . . . . . 6  |-  ( x  =  C  ->  (
y  .(+)  ( z  .(+)  x ) )  =  ( y  .(+)  ( z  .(+)  C ) ) )
1310, 12eqeq12d 2456 . . . . 5  |-  ( x  =  C  ->  (
( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
)  <->  ( ( y 
.+  z )  .(+)  C )  =  ( y 
.(+)  ( z  .(+)  C ) ) ) )
14 oveq1 6117 . . . . . . 7  |-  ( y  =  A  ->  (
y  .+  z )  =  ( A  .+  z ) )
1514oveq1d 6125 . . . . . 6  |-  ( y  =  A  ->  (
( y  .+  z
)  .(+)  C )  =  ( ( A  .+  z )  .(+)  C ) )
16 oveq1 6117 . . . . . 6  |-  ( y  =  A  ->  (
y  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( z  .(+)  C ) ) )
1715, 16eqeq12d 2456 . . . . 5  |-  ( y  =  A  ->  (
( ( y  .+  z )  .(+)  C )  =  ( y  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  z )  .(+)  C )  =  ( A 
.(+)  ( z  .(+)  C ) ) ) )
18 oveq2 6118 . . . . . . 7  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
1918oveq1d 6125 . . . . . 6  |-  ( z  =  B  ->  (
( A  .+  z
)  .(+)  C )  =  ( ( A  .+  B )  .(+)  C ) )
20 oveq1 6117 . . . . . . 7  |-  ( z  =  B  ->  (
z  .(+)  C )  =  ( B  .(+)  C ) )
2120oveq2d 6126 . . . . . 6  |-  ( z  =  B  ->  ( A  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( B  .(+) 
C ) ) )
2219, 21eqeq12d 2456 . . . . 5  |-  ( z  =  B  ->  (
( ( A  .+  z )  .(+)  C )  =  ( A  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) ) )
2313, 17, 22rspc3v 3067 . . . 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 31 . . 3  |-  ( ( C  e.  Y  /\  A  e.  X  /\  B  e.  X )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
25243coml 1161 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
2625impcom 421 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   _Vcvv 2962    X. cxp 4905   -->wf 5479   ` cfv 5483  (class class class)co 6110   Basecbs 13500   +g cplusg 13560   0gc0g 13754   Grpcgrp 14716    GrpAct cga 15097
This theorem is referenced by:  gass  15109  gasubg  15110  galcan  15112  gacan  15113  gaorber  15116  gastacl  15117  gastacos  15118  galactghm  15137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-ga 15098
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