Definition df-chj 31662

Description: Define Hilbert lattice join. See chjval 31704 for its value and chjcl 31709 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 31707. (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 31285	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31271	. . . 4 class ℋ
5	4	cpw 4562	. . 3 class 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	3	cv 1569	. . . . . 6 class 𝑦
8	6, 7	cun 3903	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31282	. . . . 5 class ⊥
10	8, 9	cfv 6536	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6536	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7412	. 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 referenced by:  sshjval  31702