Definition df-chj 31212

Description: Define Hilbert lattice join. See chjval 31254 for its value and chjcl 31259 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 31257. (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 30835	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30821	. . . 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 30832	. . . . 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  31252