Definition df-chj 31276

Description: Define Hilbert lattice join. See chjval 31318 for its value and chjcl 31323 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 31321. (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 30899	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30885	. . . 4 class ℋ
5	4	cpw 4582	. . 3 class 𝒫 ℋ
6	2	cv 1538	. . . . . 6 class 𝑥
7	3	cv 1538	. . . . . 6 class 𝑦
8	6, 7	cun 3931	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30896	. . . . 5 class ⊥
10	8, 9	cfv 6542	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6542	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7416	. 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  31316