Definition df-chsup 30316

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 30415 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 30344. (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 29939	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 29924	. . . . 5 class ℋ
4	3	cpw 4565	. . . 4 class 𝒫 ℋ
5	4	cpw 4565	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	6	cuni 4870	. . . . 5 class ∪ 𝑥
8		cort 29935	. . . . 5 class ⊥
9	7, 8	cfv 6501	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6501	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5193	. 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  30339