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Metamath Proof Explorer

Theorem nv0

Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007) (Revised by Mario Carneiro, 21-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses nv0.1 
|- X = ( BaseSet ` U )
nv0.4 
|- S = ( .sOLD ` U )
nv0.6 
|- Z = ( 0vec ` U )
Assertion nv0 
|- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = Z )

Proof

Step Hyp Ref Expression
1 nv0.1 
 |-  X = ( BaseSet ` U )
2 nv0.4 
 |-  S = ( .sOLD ` U )
3 nv0.6 
 |-  Z = ( 0vec ` U )
4 eqid 
 |-  ( 1st ` U ) = ( 1st ` U )
5 4 nvvc 
 |-  ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
6 eqid 
 |-  ( +v ` U ) = ( +v ` U )
7 6 vafval 
 |-  ( +v ` U ) = ( 1st ` ( 1st ` U ) )
8 2 smfval 
 |-  S = ( 2nd ` ( 1st ` U ) )
9 1 6 bafval 
 |-  X = ran ( +v ` U )
10 eqid 
 |-  ( GId ` ( +v ` U ) ) = ( GId ` ( +v ` U ) )
11 7 8 9 10 vc0 
 |-  ( ( ( 1st ` U ) e. CVecOLD /\ A e. X ) -> ( 0 S A ) = ( GId ` ( +v ` U ) ) )
12 5 11 sylan 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( GId ` ( +v ` U ) ) )
13 6 3 0vfval 
 |-  ( U e. NrmCVec -> Z = ( GId ` ( +v ` U ) ) )
14 13 adantr 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> Z = ( GId ` ( +v ` U ) ) )
15 12 14 eqtr4d 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = Z )