Theorem rngoass 34729

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 34722	. . . . 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 496	. . . 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 495	. . 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 1129	. . . . 5 ⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
8	7	ralimi 3126	. . . 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 3127	. . 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 7026	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
12	11	oveq1d 7034	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝑦)𝐻𝑧))
13		oveq1 7026	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝑦𝐻𝑧)))
14	12, 13	eqeq12d 2809	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧))))
15		oveq2 7027	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
16	15	oveq1d 7034	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝑧))
17		oveq1 7026	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝑧) = (𝐵𝐻𝑧))
18	17	oveq2d 7035	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝑧)))
19	16, 18	eqeq12d 2809	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧))))
20		oveq2 7027	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝐶))
21		oveq2 7027	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐻𝑧) = (𝐵𝐻𝐶))
22	21	oveq2d 7035	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝐶)))
23	20, 22	eqeq12d 2809	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))))
24	14, 19, 23	rspc3v 3573	. 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 507	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080  ∀wral 3104  ∃wrex 3105   × cxp 5444  ran crn 5447  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019  1^st c1st 7546  2^nd c2nd 7547  AbelOpcablo 28004  RingOpscrngo 34717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-ov 7022  df-1st 7548  df-2nd 7549  df-rngo 34718

This theorem is referenced by:  rngomndo  34758  rngoneglmul  34766  rngonegrmul  34767  zerdivemp1x  34770  isdrngo2  34781  crngm23  34825  crngm4  34826  prnc  34890