Definition df-chj 31399

Description: Define Hilbert lattice join. See chjval 31441 for its value and chjcl 31446 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to C_ℋ; see sshjcl 31444. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
df-chj	⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-chj

Step	Hyp	Ref	Expression
1		chj 31022	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31008	. . . 4 class ℋ
5	4	cpw 4542	. . 3 class 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	3	cv 1541	. . . . . 6 class 𝑦
8	6, 7	cun 3888	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31019	. . . . 5 class ⊥
10	8, 9	cfv 6493	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6493	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7363	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1542	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31439