Definition df-chj 31289

Description: Define Hilbert lattice join. See chjval 31331 for its value and chjcl 31336 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 31334. (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 30912	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30898	. . . 4 class ℋ
5	4	cpw 4559	. . 3 class 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	3	cv 1539	. . . . . 6 class 𝑦
8	6, 7	cun 3909	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30909	. . . . 5 class ⊥
10	8, 9	cfv 6499	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6499	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7371	. 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  31329