Definition df-chj 31513

Description: Define Hilbert lattice join. See chjval 31555 for its value and chjcl 31560 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 31558. (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 31136	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 31122	. . . 4 class ℋ
5	4	cpw 4555	. . 3 class 𝒫 ℋ
6	2	cv 1559	. . . . . 6 class 𝑥
7	3	cv 1559	. . . . . 6 class 𝑦
8	6, 7	cun 3902	. . . . 5 class (𝑥 ∪ 𝑦)
9		cort 31133	. . . . 5 class ⊥
10	8, 9	cfv 6521	. . . 4 class (⊥‘(𝑥 ∪ 𝑦))
11	10, 9	cfv 6521	. . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦)))
12	2, 3, 5, 5, 11	cmpo 7398	. 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
13	1, 12	wceq 1560	1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))

This definition is referenced by:  sshjval  31553