Theorem rngoass 38410

Description: Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringi.1	⊢ 𝐺 = (1st ‘𝑅)
ringi.2	⊢ 𝐻 = (2nd ‘𝑅)
ringi.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngoass	⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Proof of Theorem rngoass

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringi.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngoi 38403	. . . . 5 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
5	4	simprd 499	. . . 4 ⊢ (𝑅 ∈ RingOps → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
6	5	simpld 498	. . 3 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
7		simp1 1150	. . . . 5 ⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
8	7	ralimi 3100	. . . 4 ⊢ (∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧 ∈ 𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
9	8	2ralimi 3133	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
10	6, 9	syl 17	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
11		oveq1 7403	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
12	11	oveq1d 7411	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝑦)𝐻𝑧))
13		oveq1 7403	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝑦𝐻𝑧)))
14	12, 13	eqeq12d 2779	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧))))
15		oveq2 7404	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
16	15	oveq1d 7411	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝑧))
17		oveq1 7403	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝑧) = (𝐵𝐻𝑧))
18	17	oveq2d 7412	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝑧)))
19	16, 18	eqeq12d 2779	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧))))
20		oveq2 7404	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝐶))
21		oveq2 7404	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐻𝑧) = (𝐵𝐻𝐶))
22	21	oveq2d 7412	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝐶)))
23	20, 22	eqeq12d 2779	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))))
24	14, 19, 23	rspc3v 3598	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))))
25	10, 24	mpan9 514	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   × cxp 5646  ran crn 5649  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969  AbelOpcablo 30754  RingOpscrngo 38398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-1st 7970  df-2nd 7971  df-rngo 38399

This theorem is referenced by:  rngomndo  38439  rngoneglmul  38447  rngonegrmul  38448  zerdivemp1x  38451  isdrngo2  38462  crngm23  38506  crngm4  38507  prnc  38571