Theorem caovdir2d 7622

Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovdir2d.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
caovdir2d.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovdir2d.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovdir2d.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovdir2d.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
caovdir2d.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))

Assertion

Ref	Expression
caovdir2d	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(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,𝑆,𝑦,𝑧

Proof of Theorem caovdir2d

Step	Hyp	Ref	Expression
1		caovdir2d.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
2		caovdir2d.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
3		caovdir2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
4		caovdir2d.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
5	1, 2, 3, 4	caovdid 7621	. 2 ⊢ (𝜑 → (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
6		caovdir2d.com	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
7		caovdir2d.cl	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8	7, 3, 4	caovcld 7599	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆)
9	6, 8, 2	caovcomd 7602	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵)))
10	6, 3, 2	caovcomd 7602	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴))
11	6, 4, 2	caovcomd 7602	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
12	10, 11	oveq12d 7426	. 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
13	5, 9, 12	3eqtr4d 2782	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  (class class class)co 7408

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411

This theorem is referenced by: (None)