This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nvmval2

Description: Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013) (New usage is discouraged.)

Ref Expression
Hypotheses nvmval.1 
|- X = ( BaseSet ` U )
nvmval.2 
|- G = ( +v ` U )
nvmval.4 
|- S = ( .sOLD ` U )
nvmval.3 
|- M = ( -v ` U )
Assertion nvmval2 
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( ( -u 1 S B ) G A ) )

Proof

Step Hyp Ref Expression
1 nvmval.1 
 |-  X = ( BaseSet ` U )
2 nvmval.2 
 |-  G = ( +v ` U )
3 nvmval.4 
 |-  S = ( .sOLD ` U )
4 nvmval.3 
 |-  M = ( -v ` U )
5 1 2 3 4 nvmval 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A G ( -u 1 S B ) ) )
6 neg1cn 
 |-  -u 1 e. CC
7 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ -u 1 e. CC /\ B e. X ) -> ( -u 1 S B ) e. X )
8 6 7 mp3an2 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) e. X )
9 8 3adant2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) e. X )
10 1 2 nvcom 
 |-  ( ( U e. NrmCVec /\ A e. X /\ ( -u 1 S B ) e. X ) -> ( A G ( -u 1 S B ) ) = ( ( -u 1 S B ) G A ) )
11 9 10 syld3an3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G ( -u 1 S B ) ) = ( ( -u 1 S B ) G A ) )
12 5 11 eqtrd 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( ( -u 1 S B ) G A ) )