Definition df-chj 31290

Description: Define Hilbert lattice join. See chjval 31332 for its value and chjcl 31337 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 31335. (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 30913	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30899	. . . 4 class ℋ
5	4	cpw 4547	. . 3 class 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	3	cv 1540	. . . . . 6 class 𝑦
8	6, 7	cun 3895	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30910	. . . . 5 class ⊥
10	8, 9	cfv 6481	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6481	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7348	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1541	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31330