Metamath Proof Explorer


Theorem tdrgunit

Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses istrg.1 𝑀 = ( mulGrp ‘ 𝑅 )
istdrg.1 𝑈 = ( Unit ‘ 𝑅 )
Assertion tdrgunit ( 𝑅 ∈ TopDRing → ( 𝑀s 𝑈 ) ∈ TopGrp )

Proof

Step Hyp Ref Expression
1 istrg.1 𝑀 = ( mulGrp ‘ 𝑅 )
2 istdrg.1 𝑈 = ( Unit ‘ 𝑅 )
3 1 2 istdrg ( 𝑅 ∈ TopDRing ↔ ( 𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ( 𝑀s 𝑈 ) ∈ TopGrp ) )
4 3 simp3bi ( 𝑅 ∈ TopDRing → ( 𝑀s 𝑈 ) ∈ TopGrp )