Metamath Proof Explorer

Theorem hst1a

Description: Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	hst1a	\|- ( S e. CHStates -> ( normh ` ( S ` ~H ) ) = 1 )

Proof

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	simp2bi	\|- ( S e. CHStates -> ( normh ` ( S ` ~H ) ) = 1 )

Step

Hyp

Ref

Expression

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 ) ) ) ) ) )

simp2bi

 |-  ( S e. CHStates -> ( normh ` ( S ` ~H ) ) = 1 )