Definition df-chj 31338

Description: Define Hilbert lattice join. See chjval 31380 for its value and chjcl 31385 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 31383. (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 30961	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30947	. . . 4 class ℋ
5	4	cpw 4604	. . 3 class 𝒫 ℋ
6	2	cv 1535	. . . . . 6 class 𝑥
7	3	cv 1535	. . . . . 6 class 𝑦
8	6, 7	cun 3960	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30958	. . . . 5 class ⊥
10	8, 9	cfv 6562	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6562	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7432	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1536	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31378