Definition df-chsup 28725

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 28824 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 28753. (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 28346	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 28331	. . . . 5 class ℋ
4	3	cpw 4378	. . . 4 class 𝒫 ℋ
5	4	cpw 4378	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1657	. . . . . 6 class 𝑥
7	6	cuni 4658	. . . . 5 class ∪ 𝑥
8		cort 28342	. . . . 5 class ⊥
9	7, 8	cfv 6123	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6123	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 4952	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1658	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  28748