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