Definition df-chj 29087

Description: Define Hilbert lattice join. See chjval 29129 for its value and chjcl 29134 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 29132. (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 28710	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 28696	. . . 4 class ℋ
5	4	cpw 4539	. . 3 class 𝒫 ℋ
6	2	cv 1536	. . . . . 6 class 𝑥
7	3	cv 1536	. . . . . 6 class 𝑦
8	6, 7	cun 3934	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 28707	. . . . 5 class ⊥
10	8, 9	cfv 6355	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6355	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7158	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1537	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  29127