Theorem ecovidi 6921

Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)

Hypotheses

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

Assertion

Ref	Expression
ecovidi	⊢ ((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 ecovidi

Step	Hyp	Ref	Expression
1		ecovidi.1	. 2 ⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ )
2		oveq1 6092	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
3		oveq1 6092	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ))
4		oveq1 6092	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ))
5	3, 4	oveq12d 6103	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) + ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
6	2, 5	eqeq12d 2253	. 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 6092	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
8	7	oveq2d 6101	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
9		oveq2 6093	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 · 𝐵))
10	9	oveq1d 6100	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ∼ ) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )))
11	8, 10	eqeq12d 2253	. 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 6093	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 6101	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 · (𝐵 + 𝐶)))
14		oveq2 6093	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 · 𝐶))
15	14	oveq2d 6101	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
16	13, 15	eqeq12d 2253	. 2 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ∼ )) ↔ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))))
17		ecovidi.10	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐻 = 𝐾)
18		ecovidi.11	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → 𝐽 = 𝐿)
19		opeq12 3906	. . . . 5 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
20	19	eceq1d 6843	. . . 4 ⊢ ((𝐻 = 𝐾 ∧ 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼ )
21	17, 18, 20	syl2anc 415	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → [⟨𝐻, 𝐽⟩] ∼ = [⟨𝐾, 𝐿⟩] ∼ )
22		ecovidi.2	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑀, 𝑁⟩] ∼ )
23	22	oveq2d 6101	. . . . . 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		ecovidi.7	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆))
26		ecovidi.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 2271	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
29	28	3impb 1230	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐻, 𝐽⟩] ∼ )
30		ecovidi.4	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝑊, 𝑋⟩] ∼ )
31		ecovidi.5	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ · [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑌, 𝑍⟩] ∼ )
32	30, 31	oveqan12d 6104	. . . . 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		ecovidi.8	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑊 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆))
34		ecovidi.9	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑌 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆))
35		ecovidi.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 2271	. . . 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 1334	. . 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	3eqtr4d 2281	. 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 6898	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3712   × cxp 4772  (class class class)co 6085  [cec 6805   / cqs 6806

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fv 5385  df-ov 6088  df-ec 6809  df-qs 6813

This theorem is used by:  distrnqg  7755  distrsrg  8127  axdistr  8242