Theorem hlmulid

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

	Ref	Expression
Hypotheses	hlmulf.1	|- X = ( BaseSet ` U )
	hlmulf.4	|- S = ( .sOLD ` U )
Assertion	hlmulid	|- ( ( U e. CHilOLD /\ A e. X ) -> ( 1 S A ) = A )

Proof

Step	Hyp	Ref	Expression
1		hlmulf.1	|- X = ( BaseSet ` U )
2		hlmulf.4	|- S = ( .sOLD ` U )
3		hlnv	|- ( U e. CHilOLD -> U e. NrmCVec )
4	1 2	nvsid	|- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
5	3 4	sylan	|- ( ( U e. CHilOLD /\ A e. X ) -> ( 1 S A ) = A )