Metamath Proof Explorer

Theorem hstorth

Description: Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	hstorth	\|- ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _\|_ ` B ) ) ) -> ( ( S ` A ) .ih ( S ` B ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		hstel2	\|- ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _\|_ ` B ) ) ) -> ( ( ( S ` A ) .ih ( S ` B ) ) = 0 /\ ( S ` ( A vH B ) ) = ( ( S ` A ) +h ( S ` B ) ) ) )
2	1	simpld	\|- ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _\|_ ` B ) ) ) -> ( ( S ` A ) .ih ( S ` B ) ) = 0 )

Step

Hyp

Ref

Expression

hstel2

 |-  ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _|_ ` B ) ) ) -> ( ( ( S ` A ) .ih ( S ` B ) ) = 0 /\ ( S ` ( A vH B ) ) = ( ( S ` A ) +h ( S ` B ) ) ) )

simpld

 |-  ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _|_ ` B ) ) ) -> ( ( S ` A ) .ih ( S ` B ) ) = 0 )