Definition df-chsup 31572

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31671 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 31600. (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 31195	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31180	. . . . 5 class ℋ
4	3	cpw 4558	. . . 4 class 𝒫 ℋ
5	4	cpw 4558	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1562	. . . . . 6 class 𝑥
7	6	cuni 4868	. . . . 5 class ∪ 𝑥
8		cort 31191	. . . . 5 class ⊥
9	7, 8	cfv 6525	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6525	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5186	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1563	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31595