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Metamath Proof Explorer

Theorem nvinv

Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of Kreyszig p. 51. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses nvinv.1 
|- X = ( BaseSet ` U )
nvinv.2 
|- G = ( +v ` U )
nvinv.4 
|- S = ( .sOLD ` U )
nvinv.5 
|- M = ( inv ` G )
Assertion nvinv 
|- ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S A ) = ( M ` A ) )

Proof

Step Hyp Ref Expression
1 nvinv.1 
 |-  X = ( BaseSet ` U )
2 nvinv.2 
 |-  G = ( +v ` U )
3 nvinv.4 
 |-  S = ( .sOLD ` U )
4 nvinv.5 
 |-  M = ( inv ` G )
5 eqid 
 |-  ( 1st ` U ) = ( 1st ` U )
6 5 nvvc 
 |-  ( U e. NrmCVec -> ( 1st ` U ) e. CVecOLD )
7 2 vafval 
 |-  G = ( 1st ` ( 1st ` U ) )
8 3 smfval 
 |-  S = ( 2nd ` ( 1st ` U ) )
9 1 2 bafval 
 |-  X = ran G
10 7 8 9 4 vcm 
 |-  ( ( ( 1st ` U ) e. CVecOLD /\ A e. X ) -> ( -u 1 S A ) = ( M ` A ) )
11 6 10 sylan 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S A ) = ( M ` A ) )