Definition df-chj 31792

Description: Define Hilbert lattice join. See chjval 31834 for its value and chjcl 31839 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 31837. (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 31415	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31401	. . . 4 class ℋ
5	4	cpw 4557	. . 3 class 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	3	cv 1569	. . . . . 6 class 𝑦
8	6, 7	cun 3897	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31412	. . . . 5 class ⊥
10	8, 9	cfv 6533	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6533	. . 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 1570	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is used by:  sshjval  31832