Definition df-chj 31334

Description: Define Hilbert lattice join. See chjval 31376 for its value and chjcl 31381 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 31379. (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 30957	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30943	. . . 4 class ℋ
5	4	cpw 4552	. . 3 class 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	3	cv 1540	. . . . . 6 class 𝑦
8	6, 7	cun 3897	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 30954	. . . . 5 class ⊥
10	8, 9	cfv 6490	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6490	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7358	. 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  31374