Definition df-chj 29672

Description: Define Hilbert lattice join. See chjval 29714 for its value and chjcl 29719 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 29717. (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 29295	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 29281	. . . 4 class ℋ
5	4	cpw 4533	. . 3 class 𝒫 ℋ
6	2	cv 1538	. . . . . 6 class 𝑥
7	3	cv 1538	. . . . . 6 class 𝑦
8	6, 7	cun 3885	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 29292	. . . . 5 class ⊥
10	8, 9	cfv 6433	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6433	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7277	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1539	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  29712