Theorem caovdir2d 7649

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 7648	. 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 7626	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆)
9	6, 8, 2	caovcomd 7629	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵)))
10	6, 3, 2	caovcomd 7629	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴))
11	6, 4, 2	caovcomd 7629	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
12	10, 11	oveq12d 7449	. 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
13	5, 9, 12	3eqtr4d 2785	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by: (None)