Theorem rngodi 37928

Description: Distributive law for the multiplication operation of a ring (left-distributivity). (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
rngodi	⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))

Proof of Theorem rngodi

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

Step	Hyp	Ref	Expression
1		ringi.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3		ringi.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	rngoi 37923	. . . 4 ⊢ (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 495	. . 3 ⊢ (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 494	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
7		simp2 1137	. . . . 5 ⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
8	7	ralimi 3073	. . . 4 ⊢ (∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
9	8	2ralimi 3110	. . 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 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
10		oveq1 7412	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝑦𝐺𝑧)))
11		oveq1 7412	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
12		oveq1 7412	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
13	11, 12	oveq12d 7423	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)))
14	10, 13	eqeq12d 2751	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ↔ (𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧))))
15		oveq1 7412	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
16	15	oveq2d 7421	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝑧)))
17		oveq2 7413	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
18	17	oveq1d 7420	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)))
19	16, 18	eqeq12d 2751	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧))))
20		oveq2 7413	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
21	20	oveq2d 7421	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝐶)))
22		oveq2 7413	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶))
23	22	oveq2d 7421	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
24	21, 23	eqeq12d 2751	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
25	14, 19, 24	rspc3v 3617	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
26	9, 25	syl5 34	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
27	6, 26	mpan9 506	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   × cxp 5652  ran crn 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  AbelOpcablo 30525  RingOpscrngo 37918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-1st 7988  df-2nd 7989  df-rngo 37919

This theorem is referenced by:  rngorz  37947  rngonegmn1r  37966  rngosubdi  37969