Definition df-chj 31296

Description: Define Hilbert lattice join. See chjval 31338 for its value and chjcl 31343 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 31341. (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 30919	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30905	. . . 4 class ℋ
5	4	cpw 4580	. . 3 class 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	3	cv 1539	. . . . . 6 class 𝑦
8	6, 7	cun 3929	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30916	. . . . 5 class ⊥
10	8, 9	cfv 6536	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6536	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7412	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1540	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31336