Theorem caovdirg 7359

Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)

Hypothesis

Ref	Expression
caovdirg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))

Assertion

Ref	Expression
caovdirg	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdirg

Step	Hyp	Ref	Expression
1		caovdirg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
2	1	ralrimivvva 3192	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
3		oveq1 7157	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	oveq1d 7165	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝑦)𝐺𝑧))
5		oveq1 7157	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	5	oveq1d 7165	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)))
7	4, 6	eqeq12d 2837	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧))))
8		oveq2 7158	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
9	8	oveq1d 7165	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝑧))
10		oveq1 7157	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
11	10	oveq2d 7166	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)))
12	9, 11	eqeq12d 2837	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧))))
13		oveq2 7158	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝐶))
14		oveq2 7158	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
15		oveq2 7158	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	14, 15	oveq12d 7168	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
17	13, 16	eqeq12d 2837	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
18	7, 12, 17	rspc3v 3635	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
19	2, 18	mpan9 509	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  (class class class)co 7150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153

This theorem is referenced by:  caovdird  7360  srgi  19255  ringi  19304