Theorem rngodir 35343

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 35337	. . . 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 499	. . 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 498	. 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 1135	. . . . 5 ⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
8	7	ralimi 3128	. . . 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 3129	. . 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 7142	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
11	10	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝐴𝐺𝑦)𝐻𝑧))
12		oveq1 7142	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
13	12	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)))
14	11, 13	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)) ↔ ((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧))))
15		oveq2 7143	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
16	15	oveq1d 7150	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐺𝐵)𝐻𝑧))
17		oveq1 7142	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝑧) = (𝐵𝐻𝑧))
18	17	oveq2d 7151	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)))
19	16, 18	eqeq12d 2814	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴𝐺𝑦)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝑦𝐻𝑧)) ↔ ((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧))))
20		oveq2 7143	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐺𝐵)𝐻𝐶))
21		oveq2 7143	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶))
22		oveq2 7143	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵𝐻𝑧) = (𝐵𝐻𝐶))
23	21, 22	oveq12d 7153	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))
24	20, 23	eqeq12d 2814	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴𝐺𝐵)𝐻𝑧) = ((𝐴𝐻𝑧)𝐺(𝐵𝐻𝑧)) ↔ ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶))))
25	14, 19, 24	rspc3v 3584	. . 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 510	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   × cxp 5517  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670  AbelOpcablo 28327  RingOpscrngo 35332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-1st 7671  df-2nd 7672  df-rngo 35333

This theorem is referenced by:  rngo2  35345  rngolz  35360  rngonegmn1l  35379  rngosubdir  35384  prnc  35505