Metamath Proof Explorer


Theorem nvmval

Description: Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007) (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 nvmval
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A G ( -u 1 S B ) ) )

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 2 nvgrp
 |-  ( U e. NrmCVec -> G e. GrpOp )
6 1 2 bafval
 |-  X = ran G
7 eqid
 |-  ( inv ` G ) = ( inv ` G )
8 eqid
 |-  ( /g ` G ) = ( /g ` G )
9 6 7 8 grpodivval
 |-  ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A ( /g ` G ) B ) = ( A G ( ( inv ` G ) ` B ) ) )
10 5 9 syl3an1
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A ( /g ` G ) B ) = ( A G ( ( inv ` G ) ` B ) ) )
11 1 2 4 8 nvm
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A ( /g ` G ) B ) )
12 1 2 3 7 nvinv
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) = ( ( inv ` G ) ` B ) )
13 12 3adant2
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) = ( ( inv ` G ) ` B ) )
14 13 oveq2d
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G ( -u 1 S B ) ) = ( A G ( ( inv ` G ) ` B ) ) )
15 10 11 14 3eqtr4d
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A G ( -u 1 S B ) ) )