Metamath Proof Explorer


Theorem nvdir

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

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