Definition df-chj 29093

Description: Define Hilbert lattice join. See chjval 29135 for its value and chjcl 29140 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 29138. (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 28716	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 28702	. . . 4 class ℋ
5	4	cpw 4497	. . 3 class 𝒫 ℋ
6	2	cv 1537	. . . . . 6 class 𝑥
7	3	cv 1537	. . . . . 6 class 𝑦
8	6, 7	cun 3879	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 28713	. . . . 5 class ⊥
10	8, 9	cfv 6324	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6324	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7137	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1538	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  29133