Definition df-chsup 31397

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31496 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 31425. (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 31020	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31005	. . . . 5 class ℋ
4	3	cpw 4542	. . . 4 class 𝒫 ℋ
5	4	cpw 4542	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	6	cuni 4851	. . . . 5 class ∪ 𝑥
8		cort 31016	. . . . 5 class ⊥
9	7, 8	cfv 6492	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6492	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5167	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1542	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31420