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Theorem gaass 16149
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 2467 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 16143 . . . . . . 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 464 . . . . . 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 463 . . . . 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 461 . . . . . 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 2857 . . . . 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 6293 . . . . . 6  |-  ( x  =  C  ->  (
( y  .+  z
)  .(+)  x )  =  ( ( y  .+  z )  .(+)  C ) )
11 oveq2 6293 . . . . . . 7  |-  ( x  =  C  ->  (
z  .(+)  x )  =  ( z  .(+)  C ) )
1211oveq2d 6301 . . . . . 6  |-  ( x  =  C  ->  (
y  .(+)  ( z  .(+)  x ) )  =  ( y  .(+)  ( z  .(+)  C ) ) )
1310, 12eqeq12d 2489 . . . . 5  |-  ( x  =  C  ->  (
( ( y  .+  z )  .(+)  x )  =  ( y  .(+)  ( z  .(+)  x )
)  <->  ( ( y 
.+  z )  .(+)  C )  =  ( y 
.(+)  ( z  .(+)  C ) ) ) )
14 oveq1 6292 . . . . . . 7  |-  ( y  =  A  ->  (
y  .+  z )  =  ( A  .+  z ) )
1514oveq1d 6300 . . . . . 6  |-  ( y  =  A  ->  (
( y  .+  z
)  .(+)  C )  =  ( ( A  .+  z )  .(+)  C ) )
16 oveq1 6292 . . . . . 6  |-  ( y  =  A  ->  (
y  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( z  .(+)  C ) ) )
1715, 16eqeq12d 2489 . . . . 5  |-  ( y  =  A  ->  (
( ( y  .+  z )  .(+)  C )  =  ( y  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  z )  .(+)  C )  =  ( A 
.(+)  ( z  .(+)  C ) ) ) )
18 oveq2 6293 . . . . . . 7  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
1918oveq1d 6300 . . . . . 6  |-  ( z  =  B  ->  (
( A  .+  z
)  .(+)  C )  =  ( ( A  .+  B )  .(+)  C ) )
20 oveq1 6292 . . . . . . 7  |-  ( z  =  B  ->  (
z  .(+)  C )  =  ( B  .(+)  C ) )
2120oveq2d 6301 . . . . . 6  |-  ( z  =  B  ->  ( A  .(+)  ( z  .(+)  C ) )  =  ( A  .(+)  ( B  .(+) 
C ) ) )
2219, 21eqeq12d 2489 . . . . 5  |-  ( z  =  B  ->  (
( ( A  .+  z )  .(+)  C )  =  ( A  .(+)  ( z  .(+)  C )
)  <->  ( ( A 
.+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) ) )
2313, 17, 22rspc3v 3226 . . . 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 32 . . 3  |-  ( ( C  e.  Y  /\  A  e.  X  /\  B  e.  X )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
25243coml 1203 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )  ->  (  .(+)  e.  ( G  GrpAct  Y )  -> 
( ( A  .+  B )  .(+)  C )  =  ( A  .(+)  ( B  .(+)  C )
) ) )
2625impcom 430 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  Y )
)  ->  ( ( A  .+  B )  .(+)  C )  =  ( A 
.(+)  ( B  .(+)  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   0gc0g 14698   Grpcgrp 15730    GrpAct cga 16141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-ga 16142
This theorem is referenced by:  gass  16153  gasubg  16154  galcan  16156  gacan  16157  gaorber  16160  gastacl  16161  gastacos  16162  galactghm  16242
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