Metamath Proof Explorer

Theorem ringgrp

Description: A ring is a group. (Contributed by NM, 15-Sep-2011)

Ref Expression
Assertion ringgrp 
|- ( R e. Ring -> R e. Grp )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` R ) = ( Base ` R )
2 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
3 eqid 
 |-  ( +g ` R ) = ( +g ` R )
4 eqid 
 |-  ( .r ` R ) = ( .r ` R )
5 1 2 3 4 isring 
 |-  ( R e. Ring <-> ( R e. Grp /\ ( mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
6 5 simp1bi 
 |-  ( R e. Ring -> R e. Grp )