Definition df-chsup 31460

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31559 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 31488. (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 31083	. 2 class ∨ℋ
2		vx	. . 3 setvar 𝑥
3		chba 31068	. . . . 5 class ℋ
4	3	cpw 4554	. . . 4 class 𝒫 ℋ
5	4	cpw 4554	. . 3 class 𝒫 𝒫 ℋ
6	2	cv 1558	. . . . . 6 class 𝑥
7	6	cuni 4864	. . . . 5 class ∪ 𝑥
8		cort 31079	. . . . 5 class ⊥
9	7, 8	cfv 6517	. . . 4 class (⊥‘∪ 𝑥)
10	9, 8	cfv 6517	. . 3 class (⊥‘(⊥‘∪ 𝑥))
11	2, 5, 10	cmpt 5180	. 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
12	1, 11	wceq 1559	1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))

This definition is referenced by:  hsupval  31483