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Theorem gaass 17730
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 2622 . . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
41, 2, 3isga 17724 . . . . . . 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 480 . . . . . 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 479 . . . . 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 477 . . . . . 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 2952 . . . . 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 17 . . . 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 6658 . . . . . 6 |- ( x = C -> ( ( y .+ z ) .(+) x ) = ( ( y .+ z ) .(+) C ) )
11 oveq2 6658 . . . . . . 7 |- ( x = C -> ( z .(+) x ) = ( z .(+) C ) )
1211oveq2d 6666 . . . . . 6 |- ( x = C -> ( y .(+) ( z .(+) x ) ) = ( y .(+) ( z .(+) C ) ) )
1310, 12eqeq12d 2637 . . . . 5 |- ( x = C -> ( ( ( y .+ z ) .(+) x ) = ( y .(+) ( z .(+) x ) ) <-> ( ( y .+ z ) .(+) C ) = ( y .(+) ( z .(+) C ) ) ) )
14 oveq1 6657 . . . . . . 7 |- ( y = A -> ( y .+ z ) = ( A .+ z ) )
1514oveq1d 6665 . . . . . 6 |- ( y = A -> ( ( y .+ z ) .(+) C ) = ( ( A .+ z ) .(+) C ) )
16 oveq1 6657 . . . . . 6 |- ( y = A -> ( y .(+) ( z .(+) C ) ) = ( A .(+) ( z .(+) C ) ) )
1715, 16eqeq12d 2637 . . . . 5 |- ( y = A -> ( ( ( y .+ z ) .(+) C ) = ( y .(+) ( z .(+) C ) ) <-> ( ( A .+ z ) .(+) C ) = ( A .(+) ( z .(+) C ) ) ) )
18 oveq2 6658 . . . . . . 7 |- ( z = B -> ( A .+ z ) = ( A .+ B ) )
1918oveq1d 6665 . . . . . 6 |- ( z = B -> ( ( A .+ z ) .(+) C ) = ( ( A .+ B ) .(+) C ) )
20 oveq1 6657 . . . . . . 7 |- ( z = B -> ( z .(+) C ) = ( B .(+) C ) )
2120oveq2d 6666 . . . . . 6 |- ( z = B -> ( A .(+) ( z .(+) C ) ) = ( A .(+) ( B .(+) C ) ) )
2219, 21eqeq12d 2637 . . . . 5 |- ( z = B -> ( ( ( A .+ z ) .(+) C ) = ( A .(+) ( z .(+) C ) ) <-> ( ( A .+ B ) .(+) C ) = ( A .(+) ( B .(+) C ) ) ) )
2313, 17, 22rspc3v 3325 . . . 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 34 . . 3 |- ( ( C e. Y /\ A e. X /\ B e. X ) -> ( .(+) e. ( G GrpAct Y ) -> ( ( A .+ B ) .(+) C ) = ( A .(+) ( B .(+) C ) ) ) )
25243coml 1272 . 2 |- ( ( A e. X /\ B e. X /\ C e. Y ) -> ( .(+) e. ( G GrpAct Y ) -> ( ( A .+ B ) .(+) C ) = ( A .(+) ( B .(+) C ) ) ) )
2625impcom 446 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   GrpAct cga 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ga 17723
This theorem is referenced by:  gass  17734  gasubg  17735  galcan  17737  gacan  17738  gaorber  17741  gastacl  17742  gastacos  17743  galactghm  17823
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