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Metamath Proof Explorer

Theorem resv0g

Description: 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018)

Ref Expression
Hypotheses resvbas.1 
|- H = ( G |`v A )
resv0g.2 
|- .0. = ( 0g ` G )
Assertion resv0g 
|- ( A e. V -> .0. = ( 0g ` H ) )

Proof

Step Hyp Ref Expression
1 resvbas.1 
 |-  H = ( G |`v A )
2 resv0g.2 
 |-  .0. = ( 0g ` G )
3 eqidd 
 |-  ( A e. V -> ( Base ` G ) = ( Base ` G ) )
4 eqid 
 |-  ( Base ` G ) = ( Base ` G )
5 1 4 resvbas 
 |-  ( A e. V -> ( Base ` G ) = ( Base ` H ) )
6 eqid 
 |-  ( +g ` G ) = ( +g ` G )
7 1 6 resvplusg 
 |-  ( A e. V -> ( +g ` G ) = ( +g ` H ) )
8 7 oveqdr 
 |-  ( ( A e. V /\ ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
9 3 5 8 grpidpropd 
 |-  ( A e. V -> ( 0g ` G ) = ( 0g ` H ) )
10 2 9 eqtrid 
 |-  ( A e. V -> .0. = ( 0g ` H ) )