Theorem hlmul0

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

	Ref	Expression
Hypotheses	hlmul0.1	|- X = ( BaseSet ` U )
	hlmul0.4	|- S = ( .sOLD ` U )
	hlmul0.5	|- Z = ( 0vec ` U )
Assertion	hlmul0	|- ( ( U e. CHilOLD /\ A e. X ) -> ( 0 S A ) = Z )

Proof

Step	Hyp	Ref	Expression
1		hlmul0.1	|- X = ( BaseSet ` U )
2		hlmul0.4	|- S = ( .sOLD ` U )
3		hlmul0.5	|- Z = ( 0vec ` U )
4		hlnv	|- ( U e. CHilOLD -> U e. NrmCVec )
5	1 2 3	nv0	|- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = Z )
6	4 5	sylan	|- ( ( U e. CHilOLD /\ A e. X ) -> ( 0 S A ) = Z )