Definition df-chj 28508

Description: Define Hilbert lattice join. See chjval 28550 for its value and chjcl 28555 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 28553. (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 28129	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chil 28115	. . . 4 class ℋ
5	4	cpw 4298	. . 3 class 𝒫 ℋ
6	2	cv 1630	. . . . . 6 class 𝑥
7	3	cv 1630	. . . . . 6 class 𝑦
8	6, 7	cun 3721	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 28126	. . . . 5 class ⊥
10	8, 9	cfv 6030	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6030	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpt2 6797	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1631	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  28548