Definition df-chsup 30564

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 30663 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 30592. (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 30187	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30172	. . . . 5 class ℋ
4	3	cpw 4603	. . . 4 class 𝒫 ℋ
5	4	cpw 4603	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	6	cuni 4909	. . . . 5 class ∪ 𝑥
8		cort 30183	. . . . 5 class ⊥
9	7, 8	cfv 6544	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6544	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5232	. 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  30587