Theorem ecovdi 6648

Description: Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6649 will be more helpful. (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 5884	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
3		oveq1 5884	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ))
4		oveq1 5884	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))
5	3, 4	oveq12d 5895	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
6	2, 5	eqeq12d 2192	. 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 5884	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
8	7	oveq2d 5893	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
9		oveq2 5885	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · 𝐵))
10	9	oveq1d 5892	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
11	8, 10	eqeq12d 2192	. 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 5885	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 5893	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + 𝐶)))
14		oveq2 5885	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · 𝐶))
15	14	oveq2d 5893	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
16	13, 15	eqeq12d 2192	. 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 3782	. . . . 5 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
20	19	eceq1d 6573	. . . 4 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼ )
21	17, 18, 20	mp2an 426	. . 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 5893	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ))
24	23	adantl 277	. . . . 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 286	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑀, 𝑁⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
28	24, 27	eqtrd 2210	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
29	28	3impb 1199	. . 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 5896	. . . . 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 289	. . . . 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 2210	. . . 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 1293	. . 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 2236	. 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 6626	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3597   × cxp 4626  (class class class)co 5877  [cec 6535   / cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fv 5226  df-ov 5880  df-ec 6539  df-qs 6543

This theorem is referenced by: (None)