Theorem caovdird 6031

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

Hypotheses

Ref	Expression
caovdirg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
caovdird.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovdird.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovdird.4	⊢ (𝜑 → 𝐶 ∈ 𝐾)

Assertion

Ref	Expression
caovdird	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(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,𝐻,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdird

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (𝜑 → 𝜑)
2		caovdird.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovdird.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4		caovdird.4	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
5		caovdirg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
6	5	caovdirg 6030	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
7	1, 2, 3, 4, 6	syl13anc 1235	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  caovdilemd  6044  recexgt0sr  7735