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Metamath Proof Explorer

Theorem matinvg

Description: The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015)

Ref Expression
Hypotheses matbas.a 
|- A = ( N Mat R )
matbas.g 
|- G = ( R freeLMod ( N X. N ) )
Assertion matinvg 
|- ( ( N e. Fin /\ R e. V ) -> ( invg ` G ) = ( invg ` A ) )

Proof

Step Hyp Ref Expression
1 matbas.a 
 |-  A = ( N Mat R )
2 matbas.g 
 |-  G = ( R freeLMod ( N X. N ) )
3 eqidd 
 |-  ( ( N e. Fin /\ R e. V ) -> ( Base ` G ) = ( Base ` G ) )
4 1 2 matbas 
 |-  ( ( N e. Fin /\ R e. V ) -> ( Base ` G ) = ( Base ` A ) )
5 1 2 matplusg 
 |-  ( ( N e. Fin /\ R e. V ) -> ( +g ` G ) = ( +g ` A ) )
6 5 oveqdr 
 |-  ( ( ( N e. Fin /\ R e. V ) /\ ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` A ) y ) )
7 3 4 6 grpinvpropd 
 |-  ( ( N e. Fin /\ R e. V ) -> ( invg ` G ) = ( invg ` A ) )