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Metamath Proof Explorer

Theorem nvdi

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 nvdi 
|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A 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 vcdi 
 |-  ( ( ( 1st ` U ) e. CVecOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A S C ) ) )
10 5 9 sylan 
 |-  ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B G C ) ) = ( ( A S B ) G ( A S C ) ) )