Definition df-chsup 31641

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31740 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 31669. (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 31264	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31249	. . . . 5 class ℋ
4	3	cpw 4563	. . . 4 class 𝒫 ℋ
5	4	cpw 4563	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	6	cuni 4873	. . . . 5 class ∪ 𝑥
8		cort 31260	. . . . 5 class ⊥
9	7, 8	cfv 6538	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6538	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5193	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1570	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31664