isOpen — Mathlib

English

The set of units in a normed ring with summable geometric series is open in the underlying ring.

Русский

Множество единиц в нормированном кольце с суммируемым геометрическим рядом открыто по совокупности hinter полем.

LaTeX

$$IsOpen( { x ∈ R | IsUnit x } )$$

Lean4

/-- The group of units of a normed ring with summable geometric series is an open subset
of the ring. -/
protected theorem isOpen : IsOpen {x : R | IsUnit x} :=
  by
  nontriviality R
  rw [Metric.isOpen_iff]
  rintro _ ⟨x, rfl⟩
  refine ⟨‖(↑x⁻¹ : R)‖⁻¹, _root_.inv_pos.mpr (Units.norm_pos x⁻¹), fun y hy ↦ ?_⟩
  rw [mem_ball_iff_norm] at hy
  exact (x.ofNearby y hy).isUnit

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