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Metamath Proof Explorer

Theorem nvsge0

Description: The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypotheses nvs.1 
|- X = ( BaseSet ` U )
nvs.4 
|- S = ( .sOLD ` U )
nvs.6 
|- N = ( normCV ` U )
Assertion nvsge0 
|- ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( N ` ( A S B ) ) = ( A x. ( N ` B ) ) )

Proof

Step Hyp Ref Expression
1 nvs.1 
 |-  X = ( BaseSet ` U )
2 nvs.4 
 |-  S = ( .sOLD ` U )
3 nvs.6 
 |-  N = ( normCV ` U )
4 recn 
 |-  ( A e. RR -> A e. CC )
5 4 adantr 
 |-  ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
6 1 2 3 nvs 
 |-  ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
7 5 6 syl3an2 
 |-  ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
8 absid 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
9 8 3ad2ant2 
 |-  ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( abs ` A ) = A )
10 9 oveq1d 
 |-  ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( ( abs ` A ) x. ( N ` B ) ) = ( A x. ( N ` B ) ) )
11 7 10 eqtrd 
 |-  ( ( U e. NrmCVec /\ ( A e. RR /\ 0 <_ A ) /\ B e. X ) -> ( N ` ( A S B ) ) = ( A x. ( N ` B ) ) )