Definition df-chsup 31407

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31506 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 31435. (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 31030	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31015	. . . . 5 class ℋ
4	3	cpw 4536	. . . 4 class 𝒫 ℋ
5	4	cpw 4536	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1546	. . . . . 6 class 𝑥
7	6	cuni 4845	. . . . 5 class ∪ 𝑥
8		cort 31026	. . . . 5 class ⊥
9	7, 8	cfv 6492	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6492	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5160	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1547	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31430