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Metamath Proof Explorer

Theorem matplusg

Description: The matrix ring has the same addition as its underlying group. (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 matplusg 
|- ( ( N e. Fin /\ R e. V ) -> ( +g ` G ) = ( +g ` A ) )

Proof

Step Hyp Ref Expression
1 matbas.a 
 |-  A = ( N Mat R )
2 matbas.g 
 |-  G = ( R freeLMod ( N X. N ) )
3 plusgid 
 |-  +g = Slot ( +g ` ndx )
4 plusgndxnmulrndx 
 |-  ( +g ` ndx ) =/= ( .r ` ndx )
5 3 4 setsnid 
 |-  ( +g ` G ) = ( +g ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
6 eqid 
 |-  ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
7 1 2 6 matval 
 |-  ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
8 7 fveq2d 
 |-  ( ( N e. Fin /\ R e. V ) -> ( +g ` A ) = ( +g ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) ) )
9 5 8 eqtr4id 
 |-  ( ( N e. Fin /\ R e. V ) -> ( +g ` G ) = ( +g ` A ) )