Definition df-chsup 29652

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 29751 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 29680. (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 29275	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 29260	. . . . 5 class ℋ
4	3	cpw 4538	. . . 4 class 𝒫 ℋ
5	4	cpw 4538	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	6	cuni 4844	. . . . 5 class ∪ 𝑥
8		cort 29271	. . . . 5 class ⊥
9	7, 8	cfv 6430	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6430	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5161	. 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  29675