Definition df-chsup 27554

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 27653 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 27582. (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 27175	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chil 27160	. . . . 5 class ℋ
4	3	cpw 4108	. . . 4 class 𝒫 ℋ
5	4	cpw 4108	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1474	. . . . . 6 class 𝑥
7	6	cuni 4372	. . . . 5 class ∪ 𝑥
8		cort 27171	. . . . 5 class ⊥
9	7, 8	cfv 5804	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 5804	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 4643	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1475	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  27577