Metamath Proof Explorer

Theorem crngring

Description: A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Assertion crngring 
|- ( R e. CRing -> R e. Ring )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
2 1 iscrng 
 |-  ( R e. CRing <-> ( R e. Ring /\ ( mulGrp ` R ) e. CMnd ) )
3 2 simplbi 
 |-  ( R e. CRing -> R e. Ring )