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 ∈ TopDRing → 𝑅 ∈ DivRing )

Proof

Step Hyp Ref Expression
1 eqid ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
2 eqid ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
3 1 2 istdrg ( 𝑅 ∈ TopDRing ↔ ( 𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) ∈ TopGrp ) )
4 3 simp2bi ( 𝑅 ∈ TopDRing → 𝑅 ∈ DivRing )