Definition df-chj 31381

Description: Define Hilbert lattice join. See chjval 31423 for its value and chjcl 31428 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 31426. (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 31004	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30990	. . . 4 class ℋ
5	4	cpw 4541	. . 3 class 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	3	cv 1541	. . . . . 6 class 𝑦
8	6, 7	cun 3887	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31001	. . . . 5 class ⊥
10	8, 9	cfv 6498	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6498	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7369	. 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  31421