Definition df-chj 31385

Description: Define Hilbert lattice join. See chjval 31427 for its value and chjcl 31432 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 31430. (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 31008	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30994	. . . 4 class ℋ
5	4	cpw 4554	. . 3 class 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	3	cv 1540	. . . . . 6 class 𝑦
8	6, 7	cun 3899	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31005	. . . . 5 class ⊥
10	8, 9	cfv 6492	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6492	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7360	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1541	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31425