Metamath Proof Explorer


Theorem mat0

Description: The matrix ring has the same zero 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 mat0
|- ( ( N e. Fin /\ R e. V ) -> ( 0g ` G ) = ( 0g ` 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 grpidpropd
 |-  ( ( N e. Fin /\ R e. V ) -> ( 0g ` G ) = ( 0g ` A ) )