Definition df-chj 31342

Description: Define Hilbert lattice join. See chjval 31384 for its value and chjcl 31389 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 31387. (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 30965	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30951	. . . 4 class ℋ
5	4	cpw 4622	. . 3 class 𝒫 ℋ
6	2	cv 1536	. . . . . 6 class 𝑥
7	3	cv 1536	. . . . . 6 class 𝑦
8	6, 7	cun 3974	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30962	. . . . 5 class ⊥
10	8, 9	cfv 6573	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6573	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7450	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1537	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31382