Definition df-chj 31912

Description: Define Hilbert lattice join. See chjval 31954 for its value and chjcl 31959 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 31957. (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 31535	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31521	. . . 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 31532	. . . . 5 class ⊥
10	8, 9	cfv 6538	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6538	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7422	. 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  31952