Definition df-chj 31192

Description: Define Hilbert lattice join. See chjval 31234 for its value and chjcl 31239 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 31237. (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 30815	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30801	. . . 4 class ℋ
5	4	cpw 4604	. . 3 class 𝒫 ℋ
6	2	cv 1532	. . . . . 6 class 𝑥
7	3	cv 1532	. . . . . 6 class 𝑦
8	6, 7	cun 3942	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30812	. . . . 5 class ⊥
10	8, 9	cfv 6549	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6549	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7421	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1533	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31232