Definition df-chj 31254

Description: Define Hilbert lattice join. See chjval 31296 for its value and chjcl 31301 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 31299. (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 30877	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30863	. . . 4 class ℋ
5	4	cpw 4551	. . 3 class 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	3	cv 1539	. . . . . 6 class 𝑦
8	6, 7	cun 3901	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30874	. . . . 5 class ⊥
10	8, 9	cfv 6482	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6482	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7351	. 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  31294