Theorem gaass 19541

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	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
2		gaass.2	. . . . . . 7 ⊢ + = (+g‘𝐺)
3		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 19535	. . . . . 6 ⊢ ( ⊕ ∈ (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 503	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
6		simpr 490	. . . . . 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)))
7	6	ralimi 3103	. . . . 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)))
8	5, 7	simpl2im 513	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
9		oveq2 7451	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧) ⊕ 𝑥) = ((𝑦 + 𝑧) ⊕ 𝐶))
10		oveq2 7451	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ 𝐶))
11	10	oveq2d 7459	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦 ⊕ (𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)))
12	9, 11	eqeq12d 2779	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶))))
13		oveq1 7450	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
14	13	oveq1d 7458	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝑧) ⊕ 𝐶))
15		oveq1 7450	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)))
16	14, 15	eqeq12d 2779	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶))))
17		oveq2 7451	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
18	17	oveq1d 7458	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝐵) ⊕ 𝐶))
19		oveq1 7450	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ⊕ 𝐶) = (𝐵 ⊕ 𝐶))
20	19	oveq2d 7459	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
21	18, 20	eqeq12d 2779	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
22	12, 16, 21	rspc3v 3595	. . . 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))))
23	8, 22	syl5 35	. . 3 ⊢ ((𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
24	23	3coml 1145	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
25	24	impcom 413	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   × cxp 5653  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  0_gc0g 17639  Grpcgrp 19173   GrpAct cga 19533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-map 8877  df-ga 19534

This theorem is used by:  gass  19545  gasubg  19546  galcan  19548  gacan  19549  gaorber  19552  gastacl  19553  gastacos  19554  galactghm  19648