Definition df-chsup 31293

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31392 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 31321. (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 30916	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30901	. . . . 5 class ℋ
4	3	cpw 4549	. . . 4 class 𝒫 ℋ
5	4	cpw 4549	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	6	cuni 4858	. . . . 5 class ∪ 𝑥
8		cort 30912	. . . . 5 class ⊥
9	7, 8	cfv 6486	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6486	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5174	. 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  31316