Definition df-chsup 31700

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31799 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 31728. (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 31323	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31308	. . . . 5 class ℋ
4	3	cpw 4567	. . . 4 class 𝒫 ℋ
5	4	cpw 4567	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1569	. . . . . 6 class 𝑥
7	6	cuni 4877	. . . . 5 class ∪ 𝑥
8		cort 31319	. . . . 5 class ⊥
9	7, 8	cfv 6543	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6543	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5197	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1570	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is used by:  hsupval  31723