Definition df-chsup 31273

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31372 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 31301. (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 30896	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30881	. . . . 5 class ℋ
4	3	cpw 4553	. . . 4 class 𝒫 ℋ
5	4	cpw 4553	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1539	. . . . . 6 class 𝑥
7	6	cuni 4861	. . . . 5 class ∪ 𝑥
8		cort 30892	. . . . 5 class ⊥
9	7, 8	cfv 6486	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6486	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5176	. 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  31296