Definition df-chj 31735

Description: Define Hilbert lattice join. See chjval 31777 for its value and chjcl 31782 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 31780. (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 31358	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31344	. . . 4 class ℋ
5	4	cpw 4564	. . 3 class 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	3	cv 1569	. . . . . 6 class 𝑦
8	6, 7	cun 3904	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31355	. . . . 5 class ⊥
10	8, 9	cfv 6540	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6540	. . 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 1570	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is used by:  sshjval  31775