Definition df-chj 31913

Description: Define Hilbert lattice join. See chjval 31955 for its value and chjcl 31960 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 31958. (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 31536	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31522	. . . 4 class ℋ
5	4	cpw 4556	. . 3 class 𝒫 ℋ
6	2	cv 1568	. . . . . 6 class 𝑥
7	3	cv 1568	. . . . . 6 class 𝑦
8	6, 7	cun 4149	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31533	. . . . 5 class ⊥
10	8, 9	cfv 6538	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6538	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7422	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1569	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is used by:  sshjval  31953