Theorem rngodir 37846

Description: Distributive law for the multiplication operation of a ring (right-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
rngodir	⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))

Proof of Theorem rngodir

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 37840	. . . 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		simp3 1138	. . . . 5 ⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
8	7	ralimi 3072	. . . 4 ⊢ (∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
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 ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
10		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
11	10	oveq1d 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝐴𝐺𝑦)𝐻𝑧))
12		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
13	12	oveq1d 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)))
14	11, 13	eqeq12d 2750	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)) ↔ ((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧))))
15		oveq2 7420	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
16	15	oveq1d 7427	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐺𝐵)𝐻𝑧))
17		oveq1 7419	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝑧) = (𝐵𝐻𝑧))
18	17	oveq2d 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)))
19	16, 18	eqeq12d 2750	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)) ↔ ((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧))))
20		oveq2 7420	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐺𝐵)𝐻𝐶))
21		oveq2 7420	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶))
22		oveq2 7420	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵𝐻𝑧) = (𝐵𝐻𝐶))
23	21, 22	oveq12d 7430	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))
24	20, 23	eqeq12d 2750	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)) ↔ ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶))))
25	14, 19, 24	rspc3v 3621	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶))))
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))) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶))))
27	6, 26	mpan9 506	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   × cxp 5663  ran crn 5666  ⟶wf 6536  ‘cfv 6540  (class class class)co 7412  1^st c1st 7993  2^nd c2nd 7994  AbelOpcablo 30490  RingOpscrngo 37835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7415  df-1st 7995  df-2nd 7996  df-rngo 37836

This theorem is referenced by:  rngo2  37848  rngolz  37863  rngonegmn1l  37882  rngosubdir  37887  prnc  38008