isOpen_closedBall — Mathlib

English

For r ≠ 0, the closed ball { x : v(x) ≤ r } is open as well as closed.

Русский

При r ≠ 0 замкованный шар { x : v(x) ≤ r } является открытым и замкнутым.

LaTeX

$$IsOpen( {x ∈ R | v(x) ≤ r} )$$

Lean4

theorem isOpen_closedBall {r : ValueGroupWithZero R} (hr : r ≠ 0) : IsOpen {x | v x ≤ r} :=
  by
  rw [isOpen_iff_mem_nhds]
  intro x hx
  rw [mem_nhds_iff']
  simp only [setOf_subset_setOf]
  exact ⟨Units.mk0 _ hr, fun y hy => (sub_add_cancel y x).symm ▸ le_trans ((v).map_add _ _) (max_le (le_of_lt hy) hx)⟩

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