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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 → M ↾ 𝑠 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 ∧ M ↾ 𝑠 U ∈ TopGrp
4 3 simp3bi ⊢ R ∈ TopDRing → M ↾ 𝑠 U ∈ TopGrp