Theorem caovdig 6228

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 2625	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
3		oveq1 6056	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝑦𝐹𝑧)))
4		oveq1 6056	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq1 6056	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
6	4, 5	oveq12d 6067	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)))
7	3, 6	eqeq12d 2247	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) ↔ (𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧))))
8		oveq1 6056	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
9	8	oveq2d 6065	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝑧)))
10		oveq2 6057	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
11	10	oveq1d 6064	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)))
12	9, 11	eqeq12d 2247	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧))))
13		oveq2 6057	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
14	13	oveq2d 6065	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝐶)))
15		oveq2 6057	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
16	15	oveq2d 6065	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
17	14, 16	eqeq12d 2247	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
18	7, 12, 17	rspc3v 2936	. 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 281	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  (class class class)co 6049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by:  caovdid  6229  caovdi  6233  srgdilem  14105  ringdilem  14148