Theorem nvsass

Description: Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvscl.1	|- X = ( BaseSet ` U )
	nvscl.4	|- S = ( .sOLD ` U )
Assertion	nvsass	|- ( ( U e. NrmCVec /\ ( 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		nvscl.1	|- X = ( BaseSet ` U )
2		nvscl.4	|- S = ( .sOLD ` U )
3		eqid	|- ( 1st ` U ) = ( 1st ` U )
4	3	nvvc	|- ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
5		eqid	|- ( +v ` U ) = ( +v ` U )
6	5	vafval	|- ( +v ` U ) = ( 1st ` ( 1st ` U ) )
7	2	smfval	|- S = ( 2nd ` ( 1st ` U ) )
8	1 5	bafval	|- X = ran ( +v ` U )
9	6 7 8	vcass	|- ( ( ( 1st ` U ) e. CVecOLD /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A x. B ) S C ) = ( A S ( B S C ) ) )
10	4 9	sylan	|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. CC /\ C e. X ) ) -> ( ( A x. B ) S C ) = ( A S ( B S C ) ) )