Theorem hldir

Description: Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hldi.1	|- X = ( BaseSet ` U )
	hldi.2	|- G = ( +v ` U )
	hldi.4	|- S = ( .sOLD ` U )
Assertion	hldir	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A + B ) S C ) = ( ( A S C ) G ( B S C ) ) )

Proof

Step	Hyp	Ref	Expression
1		hldi.1	|- X = ( BaseSet ` U )
2		hldi.2	|- G = ( +v ` U )
3		hldi.4	|- S = ( .sOLD ` U )
4		hlnv	|- ( U e. CHilOLD -> U e. NrmCVec )
5	1 2 3	nvdir	|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A + B ) S C ) = ( ( A S C ) G ( B S C ) ) )
6	4 5	sylan	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A + B ) S C ) = ( ( A S C ) G ( B S C ) ) )