Metamath Proof Explorer

Theorem domnnzr

Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015)

Ref Expression
Assertion domnnzr 
|- ( R e. Domn -> R e. NzRing )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` R ) = ( Base ` R )
2 eqid 
 |-  ( .r ` R ) = ( .r ` R )
3 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
4 1 2 3 isdomn 
 |-  ( R e. Domn <-> ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) ) )
5 4 simplbi 
 |-  ( R e. Domn -> R e. NzRing )