Definition df-chj 28741

Description: Define Hilbert lattice join. See chjval 28783 for its value and chjcl 28788 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 28786. (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 28362	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 28348	. . . 4 class ℋ
5	4	cpw 4378	. . 3 class 𝒫 ℋ
6	2	cv 1600	. . . . . 6 class 𝑥
7	3	cv 1600	. . . . . 6 class 𝑦
8	6, 7	cun 3789	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 28359	. . . . 5 class ⊥
10	8, 9	cfv 6135	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6135	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpt2 6924	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1601	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  28781