Theorem gaass 19161

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 2733	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 19155	. . . . . 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 498	. . . . 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 486	. . . . . 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 3084	. . . . 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 505	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))
9		oveq2 7417	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧) ⊕ 𝑥) = ((𝑦 + 𝑧) ⊕ 𝐶))
10		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ 𝐶))
11	10	oveq2d 7425	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦 ⊕ (𝑧 ⊕ 𝑥)) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)))
12	9, 11	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)) ↔ ((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶))))
13		oveq1 7416	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
14	13	oveq1d 7424	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝑧) ⊕ 𝐶))
15		oveq1 7416	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)))
16	14, 15	eqeq12d 2749	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧) ⊕ 𝐶) = (𝑦 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶))))
17		oveq2 7417	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
18	17	oveq1d 7424	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧) ⊕ 𝐶) = ((𝐴 + 𝐵) ⊕ 𝐶))
19		oveq1 7416	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ⊕ 𝐶) = (𝐵 ⊕ 𝐶))
20	19	oveq2d 7425	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ⊕ (𝑧 ⊕ 𝐶)) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
21	18, 20	eqeq12d 2749	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧) ⊕ 𝐶) = (𝐴 ⊕ (𝑧 ⊕ 𝐶)) ↔ ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
22	12, 16, 21	rspc3v 3628	. . . 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 1128	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌) → ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶))))
25	24	impcom 409	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   × cxp 5675  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Grpcgrp 18819   GrpAct cga 19153

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-ga 19154

This theorem is referenced by:  gass  19165  gasubg  19166  galcan  19168  gacan  19169  gaorber  19172  gastacl  19173  gastacos  19174  galactghm  19272