Theorem caovdir2d 6018

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 6017	. 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 5995	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆)
9	6, 8, 2	caovcomd 5998	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵)))
10	6, 3, 2	caovcomd 5998	. . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴))
11	6, 4, 2	caovcomd 5998	. . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
12	10, 11	oveq12d 5860	. 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
13	5, 9, 12	3eqtr4d 2208	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  addcmpblnq  7308  ltanqg  7341  addcmpblnq0  7384  mulasssrg  7699  mulgt0sr  7719  mulextsr1lem  7721