Definition df-chj 27553

Description: Define Hilbert lattice join. See chjval 27595 for its value and chjcl 27600 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 27598. (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 27174	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chil 27160	. . . 4 class ℋ
5	4	cpw 4108	. . 3 class 𝒫 ℋ
6	2	cv 1474	. . . . . 6 class 𝑥
7	3	cv 1474	. . . . . 6 class 𝑦
8	6, 7	cun 3538	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 27171	. . . . 5 class ⊥
10	8, 9	cfv 5804	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 5804	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpt2 6551	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1475	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  27593