Definition df-chj 31924

Description: Define Hilbert lattice join. See chjval 31966 for its value and chjcl 31971 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 31969. (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 31547	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31533	. . . 4 class ℋ
5	4	cpw 4559	. . 3 class 𝒫 ℋ
6	2	cv 1568	. . . . . 6 class 𝑥
7	3	cv 1568	. . . . . 6 class 𝑦
8	6, 7	cun 4151	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31544	. . . . 5 class ⊥
10	8, 9	cfv 6546	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6546	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7429	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1569	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is used by:  sshjval  31964