Definition df-chsup 31297

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31396 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 31325. (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 30920	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30905	. . . . 5 class ℋ
4	3	cpw 4580	. . . 4 class 𝒫 ℋ
5	4	cpw 4580	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	6	cuni 4888	. . . . 5 class ∪ 𝑥
8		cort 30916	. . . . 5 class ⊥
9	7, 8	cfv 6536	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6536	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5206	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1540	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31320