Definition df-chsup 31398

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31497 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 31426. (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 31021	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31006	. . . . 5 class ℋ
4	3	cpw 4556	. . . 4 class 𝒫 ℋ
5	4	cpw 4556	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1541	. . . . . 6 class 𝑥
7	6	cuni 4865	. . . . 5 class ∪ 𝑥
8		cort 31017	. . . . 5 class ⊥
9	7, 8	cfv 6500	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6500	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5181	. 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  31421