Definition df-chsup 30831

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 30930 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice C_ℋ, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 30859. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)

Assertion

Ref	Expression
df-chsup	⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

Detailed syntax breakdown of Definition df-chsup

Step	Hyp	Ref	Expression
1		chsup 30454	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30439	. . . . 5 class ℋ
4	3	cpw 4601	. . . 4 class 𝒫 ℋ
5	4	cpw 4601	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1538	. . . . . 6 class 𝑥
7	6	cuni 4907	. . . . 5 class ∪ 𝑥
8		cort 30450	. . . . 5 class ⊥
9	7, 8	cfv 6542	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6542	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5230	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1539	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  30854