Metamath Proof Explorer

Theorem hstosum

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

		Ref	Expression
	Assertion	hstosum	\|- ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _\|_ ` B ) ) ) -> ( S ` ( A vH B ) ) = ( ( S ` A ) +h ( S ` B ) ) )

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

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 ) ) ) )

simprd

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