Theorem ecovdi 8614

Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

Ref	Expression
ecovdi.1	⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ )
ecovdi.2	⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑀, 𝑁⟩] ∼ )
ecovdi.3	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
ecovdi.4	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝑊, 𝑋⟩] ∼ )
ecovdi.5	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑌, 𝑍⟩] ∼ )
ecovdi.6	⊢ (((𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) → ([⟨𝑊, 𝑋⟩] ∼ + [⟨𝑌, 𝑍⟩] ∼ ) = [⟨𝐾, 𝐿⟩] ∼ )
ecovdi.7	⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆))
ecovdi.8	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆))
ecovdi.9	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆))
ecovdi.10	⊢ 𝐻 = 𝐾
ecovdi.11	⊢ 𝐽 = 𝐿

Assertion

Ref	Expression
ecovdi	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑤,𝐶,𝑣,𝑢   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ∼ ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, · ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝐷,𝑤,𝑣,𝑢

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑀(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑁(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑊(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑋(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑌(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecovdi

Step	Hyp	Ref	Expression
1		ecovdi.1	. 2 ⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ )
2		oveq1 7282	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
3		oveq1 7282	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ))
4		oveq1 7282	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))
5	3, 4	oveq12d 7293	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
6	2, 5	eqeq12d 2754	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) ↔ (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))))
7		oveq1 7282	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
8	7	oveq2d 7291	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
9		oveq2 7283	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · 𝐵))
10	9	oveq1d 7290	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
11	8, 10	eqeq12d 2754	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) ↔ (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))))
12		oveq2 7283	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 7291	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + 𝐶)))
14		oveq2 7283	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · 𝐶))
15	14	oveq2d 7291	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
16	13, 15	eqeq12d 2754	. 2 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) ↔ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))))
17		ecovdi.10	. . . 4 ⊢ 𝐻 = 𝐾
18		ecovdi.11	. . . 4 ⊢ 𝐽 = 𝐿
19		opeq12 4806	. . . . 5 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
20	19	eceq1d 8537	. . . 4 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼ )
21	17, 18, 20	mp2an 689	. . 3 ⊢ [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼
22		ecovdi.2	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑀, 𝑁⟩] ∼ )
23	22	oveq2d 7291	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ))
24	23	adantl 482	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ))
25		ecovdi.7	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆))
26		ecovdi.3	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
27	25, 26	sylan2 593	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
28	24, 27	eqtrd 2778	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
29	28	3impb 1114	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
30		ecovdi.4	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝑊, 𝑋⟩] ∼ )
31		ecovdi.5	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑌, 𝑍⟩] ∼ )
32	30, 31	oveqan12d 7294	. . . . 5 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑊, 𝑋⟩] ∼ + [⟨𝑌, 𝑍⟩] ∼ ))
33		ecovdi.8	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆))
34		ecovdi.9	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆))
35		ecovdi.6	. . . . . 6 ⊢ (((𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆)) → ([⟨𝑊, 𝑋⟩] ∼ + [⟨𝑌, 𝑍⟩] ∼ ) = [⟨𝐾, 𝐿⟩] ∼ )
36	33, 34, 35	syl2an 596	. . . . 5 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑊, 𝑋⟩] ∼ + [⟨𝑌, 𝑍⟩] ∼ ) = [⟨𝐾, 𝐿⟩] ∼ )
37	32, 36	eqtrd 2778	. . . 4 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐾, 𝐿⟩] ∼ )
38	37	3impdi 1349	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐾, 𝐿⟩] ∼ )
39	21, 29, 38	3eqtr4a 2804	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )))
40	1, 6, 11, 16, 39	3ecoptocl 8598	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   × cxp 5587  (class class class)co 7275  [cec 8496   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-ov 7278  df-ec 8500  df-qs 8504

This theorem is referenced by:  distrsr  10847  axdistr  10914