Metamath Proof Explorer

Theorem trgtgp

Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Assertion trgtgp 
|- ( R e. TopRing -> R e. TopGrp )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
2 1 istrg 
 |-  ( R e. TopRing <-> ( R e. TopGrp /\ R e. Ring /\ ( mulGrp ` R ) e. TopMnd ) )
3 2 simp1bi 
 |-  ( R e. TopRing -> R e. TopGrp )