Definition df-chsup 31382

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31481 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 31410. (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 31005	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30990	. . . . 5 class ℋ
4	3	cpw 4541	. . . 4 class 𝒫 ℋ
5	4	cpw 4541	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	6	cuni 4850	. . . . 5 class ∪ 𝑥
8		cort 31001	. . . . 5 class ⊥
9	7, 8	cfv 6498	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6498	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5166	. 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  31405