# Metamath Proof Explorer

## Definition df-chsup

Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice CH , to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl . (Contributed by NM, 9-Dec-2003) (New usage is discouraged.)

Ref Expression
Assertion df-chsup ${⊢}{\bigvee }_{ℋ}=\left({x}\in 𝒫𝒫ℋ⟼\perp \left(\perp \left(\bigcup {x}\right)\right)\right)$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 chsup ${class}{\bigvee }_{ℋ}$
1 vx ${setvar}{x}$
2 chba ${class}ℋ$
3 2 cpw ${class}𝒫ℋ$
4 3 cpw ${class}𝒫𝒫ℋ$
5 cort ${class}\perp$
6 1 cv ${setvar}{x}$
7 6 cuni ${class}\bigcup {x}$
8 7 5 cfv ${class}\perp \left(\bigcup {x}\right)$
9 8 5 cfv ${class}\perp \left(\perp \left(\bigcup {x}\right)\right)$
10 1 4 9 cmpt ${class}\left({x}\in 𝒫𝒫ℋ⟼\perp \left(\perp \left(\bigcup {x}\right)\right)\right)$
11 0 10 wceq ${wff}{\bigvee }_{ℋ}=\left({x}\in 𝒫𝒫ℋ⟼\perp \left(\perp \left(\bigcup {x}\right)\right)\right)$