Definition df-chj 29086

Description: Define Hilbert lattice join. See chjval 29128 for its value and chjcl 29133 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 29131. (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 28709	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 28695	. . . 4 class ℋ
5	4	cpw 4538	. . 3 class 𝒫 ℋ
6	2	cv 1532	. . . . . 6 class 𝑥
7	3	cv 1532	. . . . . 6 class 𝑦
8	6, 7	cun 3933	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 28706	. . . . 5 class ⊥
10	8, 9	cfv 6354	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6354	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7157	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1533	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  29126