Theorem gaass 17730

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 2622	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 17724	. . . . . . 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 480	. . . . . 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 479	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))
7		simpr 477	. . . . . 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 2952	. . . . 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 6658	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧) ⊕ 𝑥) = ((𝑦 + 𝑧) ⊕ 𝐶))
11		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ 𝐶))
12	11	oveq2d 6666	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦 ⊕ (𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)))
13	10, 12	eqeq12d 2637	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶))))
14		oveq1 6657	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
15	14	oveq1d 6665	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝑧) ⊕ 𝐶))
16		oveq1 6657	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)))
17	15, 16	eqeq12d 2637	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶))))
18		oveq2 6658	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
19	18	oveq1d 6665	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝐵) ⊕ 𝐶))
20		oveq1 6657	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ⊕ 𝐶) = (𝐵 ⊕ 𝐶))
21	20	oveq2d 6666	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
22	19, 21	eqeq12d 2637	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
23	13, 17, 22	rspc3v 3325	. . . 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 34	. . 3 ⊢ ((𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
25	24	3coml 1272	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
26	25	impcom 446	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422   GrpAct cga 17722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ga 17723

This theorem is referenced by:  gass  17734  gasubg  17735  galcan  17737  gacan  17738  gaorber  17741  gastacl  17742  gastacos  17743  galactghm  17823