Definition df-chj 30563

Description: Define Hilbert lattice join. See chjval 30605 for its value and chjcl 30610 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 30608. (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 30186	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30172	. . . 4 class ℋ
5	4	cpw 4603	. . 3 class 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	3	cv 1541	. . . . . 6 class 𝑦
8	6, 7	cun 3947	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30183	. . . . 5 class ⊥
10	8, 9	cfv 6544	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6544	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7411	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1542	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  30603