Metamath Proof Explorer

Theorem tdrgdrng

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

Ref Expression
Assertion tdrgdrng 
|- ( R e. TopDRing -> R e. DivRing )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
2 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
3 1 2 istdrg 
 |-  ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( ( mulGrp ` R ) |`s ( Unit ` R ) ) e. TopGrp ) )
4 3 simp2bi 
 |-  ( R e. TopDRing -> R e. DivRing )