Theorem hlmulass

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

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

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	nvsass	|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A x. B ) S C ) = ( A S ( B S C ) ) )
5	3 4	sylan	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A x. B ) S C ) = ( A S ( B S C ) ) )