Definition df-chsup 31286

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31385 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 31314. (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 30909	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30894	. . . . 5 class ℋ
4	3	cpw 4550	. . . 4 class 𝒫 ℋ
5	4	cpw 4550	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	6	cuni 4859	. . . . 5 class ∪ 𝑥
8		cort 30905	. . . . 5 class ⊥
9	7, 8	cfv 6481	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6481	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5172	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1541	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31309