Definition df-chj 31603

Description: Define Hilbert lattice join. See chjval 31645 for its value and chjcl 31650 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 31648. (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 31226	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31212	. . . 4 class ℋ
5	4	cpw 4567	. . 3 class 𝒫 ℋ
6	2	cv 1566	. . . . . 6 class 𝑥
7	3	cv 1566	. . . . . 6 class 𝑦
8	6, 7	cun 3911	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31223	. . . . 5 class ⊥
10	8, 9	cfv 6537	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6537	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7413	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1567	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31643