Theorem caovdirg 7629

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 3191	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
3		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	oveq1d 7425	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝑦)𝐺𝑧))
5		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	5	oveq1d 7425	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)))
7	4, 6	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧))))
8		oveq2 7418	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
9	8	oveq1d 7425	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝑧))
10		oveq1 7417	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
11	10	oveq2d 7426	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)))
12	9, 11	eqeq12d 2752	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧))))
13		oveq2 7418	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝐶))
14		oveq2 7418	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
15		oveq2 7418	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	14, 15	oveq12d 7428	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
17	13, 16	eqeq12d 2752	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
18	7, 12, 17	rspc3v 3622	. 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 506	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  (class class class)co 7410

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

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-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413

This theorem is referenced by:  caovdird  7630  srgdilem  20157  ringdilem  20214