Theorem gaass 18913

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 2738	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 18907	. . . . . 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 497	. . . . 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 485	. . . . . 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 3087	. . . . 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 504	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
9		oveq2 7275	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧) ⊕ 𝑥) = ((𝑦 + 𝑧) ⊕ 𝐶))
10		oveq2 7275	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ 𝐶))
11	10	oveq2d 7283	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦 ⊕ (𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)))
12	9, 11	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶))))
13		oveq1 7274	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
14	13	oveq1d 7282	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝑧) ⊕ 𝐶))
15		oveq1 7274	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)))
16	14, 15	eqeq12d 2754	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶))))
17		oveq2 7275	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
18	17	oveq1d 7282	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝐵) ⊕ 𝐶))
19		oveq1 7274	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ⊕ 𝐶) = (𝐵 ⊕ 𝐶))
20	19	oveq2d 7283	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
21	18, 20	eqeq12d 2754	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
22	12, 16, 21	rspc3v 3572	. . . 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 34	. . 3 ⊢ ((𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
24	23	3coml 1126	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
25	24	impcom 408	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3429   × cxp 5582  ⟶wf 6422  ‘cfv 6426  (class class class)co 7267  Basecbs 16922  +_gcplusg 16972  0_gc0g 17160  Grpcgrp 18587   GrpAct cga 18905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-map 8604  df-ga 18906

This theorem is referenced by:  gass  18917  gasubg  18918  galcan  18920  gacan  18921  gaorber  18924  gastacl  18925  gastacos  18926  galactghm  19022