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Theorem gaass 19328
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 . . . . . . 7 𝑋 = (Base‘𝐺)
2 gaass.2 . . . . . . 7 + = (+g𝐺)
3 eqid 2735 . . . . . . 7 (0g𝐺) = (0g𝐺)
41, 2, 3isga 19322 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 496 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
6 simpr 484 . . . . . 6 ((((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
76ralimi 3081 . . . . 5 (∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
85, 7simpl2im 503 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
9 oveq2 7439 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧) 𝑥) = ((𝑦 + 𝑧) 𝐶))
10 oveq2 7439 . . . . . . 7 (𝑥 = 𝐶 → (𝑧 𝑥) = (𝑧 𝐶))
1110oveq2d 7447 . . . . . 6 (𝑥 = 𝐶 → (𝑦 (𝑧 𝑥)) = (𝑦 (𝑧 𝐶)))
129, 11eqeq12d 2751 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) ↔ ((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶))))
13 oveq1 7438 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1413oveq1d 7446 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧) 𝐶) = ((𝐴 + 𝑧) 𝐶))
15 oveq1 7438 . . . . . 6 (𝑦 = 𝐴 → (𝑦 (𝑧 𝐶)) = (𝐴 (𝑧 𝐶)))
1614, 15eqeq12d 2751 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶)) ↔ ((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶))))
17 oveq2 7439 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
1817oveq1d 7446 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧) 𝐶) = ((𝐴 + 𝐵) 𝐶))
19 oveq1 7438 . . . . . . 7 (𝑧 = 𝐵 → (𝑧 𝐶) = (𝐵 𝐶))
2019oveq2d 7447 . . . . . 6 (𝑧 = 𝐵 → (𝐴 (𝑧 𝐶)) = (𝐴 (𝐵 𝐶)))
2118, 20eqeq12d 2751 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶)) ↔ ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2212, 16, 21rspc3v 3638 . . . 4 ((𝐶𝑌𝐴𝑋𝐵𝑋) → (∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
238, 22syl5 34 . . 3 ((𝐶𝑌𝐴𝑋𝐵𝑋) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
24233coml 1126 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑌) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2524impcom 407 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964   GrpAct cga 19320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ga 19321
This theorem is referenced by:  gass  19332  gasubg  19333  galcan  19335  gacan  19336  gaorber  19339  gastacl  19340  gastacos  19341  galactghm  19437
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