Theorem caovdig 7342

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
caovdig.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))

Assertion

Ref	Expression
caovdig	⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

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 caovdig

Step	Hyp	Ref	Expression
1		caovdig.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
2	1	ralrimivvva 3157	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
3		oveq1 7142	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝑦𝐹𝑧)))
4		oveq1 7142	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq1 7142	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	4, 5	oveq12d 7153	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)))
7	3, 6	eqeq12d 2814	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) ↔ (𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧))))
8		oveq1 7142	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
9	8	oveq2d 7151	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝑧)))
10		oveq2 7143	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
11	10	oveq1d 7150	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)))
12	9, 11	eqeq12d 2814	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧))))
13		oveq2 7143	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
14	13	oveq2d 7151	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝐶)))
15		oveq2 7143	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
16	15	oveq2d 7151	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
17	14, 16	eqeq12d 2814	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
18	7, 12, 17	rspc3v 3584	. 2 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
19	2, 18	mpan9 510	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  (class class class)co 7135

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-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138

This theorem is referenced by:  caovdid  7343  caovdi  7347  srgi  19254  ringi  19306