Theorem ecovdi 8846

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 7427	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
3		oveq1 7427	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ))
4		oveq1 7427	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))
5	3, 4	oveq12d 7438	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
6	2, 5	eqeq12d 2777	. 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 7427	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
8	7	oveq2d 7436	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
9		oveq2 7428	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · 𝐵))
10	9	oveq1d 7435	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
11	8, 10	eqeq12d 2777	. 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 7428	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 7436	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + 𝐶)))
14		oveq2 7428	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · 𝐶))
15	14	oveq2d 7436	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
16	13, 15	eqeq12d 2777	. 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 4835	. . . . 5 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
20	19	eceq1d 8758	. . . 4 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼ )
21	17, 18, 20	mp2an 705	. . 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 7436	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ))
24	23	adantl 487	. . . . 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 605	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
28	24, 27	eqtrd 2796	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
29	28	3impb 1132	. . 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 7439	. . . . 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 608	. . . . 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 2796	. . . 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 1369	. . 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 2822	. 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 8830	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   × cxp 5649  (class class class)co 7420  [cec 8715   / cqs 8716

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fv 6546  df-ov 7423  df-ec 8719  df-qs 8723

This theorem is used by:  distrsr  11176  axdistr  11243