Definition df-chsup 29718

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 29817 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 29746. (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 29341	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 29326	. . . . 5 class ℋ
4	3	cpw 4539	. . . 4 class 𝒫 ℋ
5	4	cpw 4539	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1538	. . . . . 6 class 𝑥
7	6	cuni 4844	. . . . 5 class ∪ 𝑥
8		cort 29337	. . . . 5 class ⊥
9	7, 8	cfv 6458	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6458	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5164	. 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  29741