Definition df-chj 31239

Description: Define Hilbert lattice join. See chjval 31281 for its value and chjcl 31286 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 31284. (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 30862	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30848	. . . 4 class ℋ
5	4	cpw 4563	. . 3 class 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	3	cv 1539	. . . . . 6 class 𝑦
8	6, 7	cun 3912	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30859	. . . . 5 class ⊥
10	8, 9	cfv 6511	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6511	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7389	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1540	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31279