Definition df-md 30642

Description: Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 30656 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
df-md	⊢ 𝑀ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-md

Step	Hyp	Ref	Expression
1		cmd 29328	. 2 class 𝑀ℋ
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1538	. . . . . 6 class 𝑥
4		cch 29291	. . . . . 6 class Cℋ
5	3, 4	wcel 2106	. . . . 5 wff 𝑥 ∈ Cℋ
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1538	. . . . . 6 class 𝑦
8	7, 4	wcel 2106	. . . . 5 wff 𝑦 ∈ Cℋ
9	5, 8	wa 396	. . . 4 wff (𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )
10		vz	. . . . . . . 8 setvar 𝑧
11	10	cv 1538	. . . . . . 7 class 𝑧
12	11, 7	wss 3887	. . . . . 6 wff 𝑧 ⊆ 𝑦
13		chj 29295	. . . . . . . . 9 class ∨ℋ
14	11, 3, 13	co 7275	. . . . . . . 8 class (𝑧 ∨ℋ 𝑥)
15	14, 7	cin 3886	. . . . . . 7 class ((𝑧 ∨ℋ 𝑥) ∩ 𝑦)
16	3, 7	cin 3886	. . . . . . . 8 class (𝑥 ∩ 𝑦)
17	11, 16, 13	co 7275	. . . . . . 7 class (𝑧 ∨ℋ (𝑥 ∩ 𝑦))
18	15, 17	wceq 1539	. . . . . 6 wff ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))
19	12, 18	wi 4	. . . . 5 wff (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦)))
20	19, 10, 4	wral 3064	. . . 4 wff ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦)))
21	9, 20	wa 396	. . 3 wff ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))
22	21, 2, 6	copab 5136	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}
23	1, 22	wceq 1539	1 wff 𝑀ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}

This definition is referenced by:  mdbr  30656