Definition df-chj 31969

Description: Define Hilbert lattice join. See chjval 32011 for its value and chjcl 32016 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 32014. (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 31592	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31578	. . . 4 class ℋ
5	4	cpw 4560	. . 3 class 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	3	cv 1569	. . . . . 6 class 𝑦
8	6, 7	cun 4152	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31589	. . . . 5 class ⊥
10	8, 9	cfv 6566	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6566	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7453	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1570	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is used by:  sshjval  32009