Theorem gaass 17553

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	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 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.

Step	Hyp	Ref	Expression
1		gaass.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
2		gaass.2	. . . . . . . 8 ⊢ + = (+g‘𝐺)
3		eqid 2610	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 17547	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simprbi 479	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
6	5	simprd 478	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
7		simpr 476	. . . . . 6 ⊢ ((((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
8	7	ralimi 2936	. . . . 5 ⊢ (∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
9	6, 8	syl 17	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
10		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧) ⊕ 𝑥) = ((𝑦 + 𝑧) ⊕ 𝐶))
11		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ 𝐶))
12	11	oveq2d 6565	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦 ⊕ (𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)))
13	10, 12	eqeq12d 2625	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶))))
14		oveq1 6556	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
15	14	oveq1d 6564	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝑧) ⊕ 𝐶))
16		oveq1 6556	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)))
17	15, 16	eqeq12d 2625	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶))))
18		oveq2 6557	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
19	18	oveq1d 6564	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝐵) ⊕ 𝐶))
20		oveq1 6556	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ⊕ 𝐶) = (𝐵 ⊕ 𝐶))
21	20	oveq2d 6565	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
22	19, 21	eqeq12d 2625	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
23	13, 17, 22	rspc3v 3296	. . . 4 ⊢ ((𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
24	9, 23	syl5 33	. . 3 ⊢ ((𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
25	24	3coml 1264	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
26	25	impcom 445	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245   GrpAct cga 17545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ga 17546

This theorem is referenced by:  gass  17557  gasubg  17558  galcan  17560  gacan  17561  gaorber  17564  gastacl  17565  gastacos  17566  galactghm  17646