Definition df-chj 31398

Description: Define Hilbert lattice join. See chjval 31440 for its value and chjcl 31445 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 31443. (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 31021	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31007	. . . 4 class ℋ
5	4	cpw 4556	. . 3 class 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	3	cv 1541	. . . . . 6 class 𝑦
8	6, 7	cun 3901	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31018	. . . . 5 class ⊥
10	8, 9	cfv 6500	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6500	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7370	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1542	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31438