Definition df-chsup 31386

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31485 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 31414. (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 31009	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 30994	. . . . 5 class ℋ
4	3	cpw 4554	. . . 4 class 𝒫 ℋ
5	4	cpw 4554	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1540	. . . . . 6 class 𝑥
7	6	cuni 4863	. . . . 5 class ∪ 𝑥
8		cort 31005	. . . . 5 class ⊥
9	7, 8	cfv 6492	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6492	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5179	. 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  31409