Theorem caovdig 7637

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 3214	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
3		oveq1 7430	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝑦𝐹𝑧)))
4		oveq1 7430	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq1 7430	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	4, 5	oveq12d 7441	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)))
7	3, 6	eqeq12d 2782	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) ↔ (𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧))))
8		oveq1 7430	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
9	8	oveq2d 7439	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝑧)))
10		oveq2 7431	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
11	10	oveq1d 7438	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)))
12	9, 11	eqeq12d 2782	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧))))
13		oveq2 7431	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
14	13	oveq2d 7439	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝐶)))
15		oveq2 7431	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
16	15	oveq2d 7439	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
17	14, 16	eqeq12d 2782	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
18	7, 12, 17	rspc3v 3600	. 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 516	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  (class class class)co 7423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426

This theorem is used by:  caovdid  7638  caovdi  7642  srgdilem  20305  ringdilem  20362