Description: Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | hstcl | |- ( ( S e. CHStates /\ A e. CH ) -> ( S ` A ) e. ~H ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ishst | |- ( S e. CHStates <-> ( S : CH --> ~H /\ ( normh ` ( S ` ~H ) ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( ( ( S ` x ) .ih ( S ` y ) ) = 0 /\ ( S ` ( x vH y ) ) = ( ( S ` x ) +h ( S ` y ) ) ) ) ) ) |
|
| 2 | 1 | simp1bi | |- ( S e. CHStates -> S : CH --> ~H ) |
| 3 | 2 | ffvelrnda | |- ( ( S e. CHStates /\ A e. CH ) -> ( S ` A ) e. ~H ) |