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Theorem gaass 17651
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 + = (+g𝐺)
Assertion
Ref Expression
gaass (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((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 + = (+g𝐺)
3 eqid 2621 . . . . . . . 8 (0g𝐺) = (0g𝐺)
41, 2, 3isga 17645 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 480 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
65simprd 479 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
7 simpr 477 . . . . . 6 ((((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
87ralimi 2947 . . . . 5 (∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
96, 8syl 17 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
10 oveq2 6612 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧) 𝑥) = ((𝑦 + 𝑧) 𝐶))
11 oveq2 6612 . . . . . . 7 (𝑥 = 𝐶 → (𝑧 𝑥) = (𝑧 𝐶))
1211oveq2d 6620 . . . . . 6 (𝑥 = 𝐶 → (𝑦 (𝑧 𝑥)) = (𝑦 (𝑧 𝐶)))
1310, 12eqeq12d 2636 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) ↔ ((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶))))
14 oveq1 6611 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1514oveq1d 6619 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧) 𝐶) = ((𝐴 + 𝑧) 𝐶))
16 oveq1 6611 . . . . . 6 (𝑦 = 𝐴 → (𝑦 (𝑧 𝐶)) = (𝐴 (𝑧 𝐶)))
1715, 16eqeq12d 2636 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶)) ↔ ((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶))))
18 oveq2 6612 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
1918oveq1d 6619 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧) 𝐶) = ((𝐴 + 𝐵) 𝐶))
20 oveq1 6611 . . . . . . 7 (𝑧 = 𝐵 → (𝑧 𝐶) = (𝐵 𝐶))
2120oveq2d 6620 . . . . . 6 (𝑧 = 𝐵 → (𝐴 (𝑧 𝐶)) = (𝐴 (𝐵 𝐶)))
2219, 21eqeq12d 2636 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶)) ↔ ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2313, 17, 22rspc3v 3309 . . . 4 ((𝐶𝑌𝐴𝑋𝐵𝑋) → (∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
249, 23syl5 34 . . 3 ((𝐶𝑌𝐴𝑋𝐵𝑋) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
25243coml 1269 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑌) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2625impcom 446 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343   GrpAct cga 17643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-ga 17644
This theorem is referenced by:  gass  17655  gasubg  17656  galcan  17658  gacan  17659  gaorber  17662  gastacl  17663  gastacos  17664  galactghm  17744
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