Theorem caovdirg 7644

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 3201	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
3		oveq1 7433	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	oveq1d 7441	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝑦)𝐺𝑧))
5		oveq1 7433	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	5	oveq1d 7441	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)))
7	4, 6	eqeq12d 2744	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧))))
8		oveq2 7434	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
9	8	oveq1d 7441	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝑧))
10		oveq1 7433	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
11	10	oveq2d 7442	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)))
12	9, 11	eqeq12d 2744	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧))))
13		oveq2 7434	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝐶))
14		oveq2 7434	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
15		oveq2 7434	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
16	14, 15	oveq12d 7444	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
17	13, 16	eqeq12d 2744	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
18	7, 12, 17	rspc3v 3627	. 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 505	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  (class class class)co 7426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429

This theorem is referenced by:  caovdird  7645  srgdilem  20139  ringdilem  20196