gauge_set_nonempty — Mathlib

English

The gauge is invariant under a sign flip when s is symmetric: gauge s x = gauge s (−x).

Русский

Гauge инвариантен относительно смены знака при симметричном s: gauge s x = gauge s (−x).

LaTeX

$$$$\\operatorname{gauge}(s, -x) = \\operatorname{gauge}(s, x).$$$$

Lean4

/-- If the given subset is `Absorbent` then the set we take an infimum over in `gauge` is nonempty,
which is useful for proving many properties about the gauge. -/
theorem gauge_set_nonempty (absorbs : Absorbent ℝ s) : {r : ℝ | 0 < r ∧ x ∈ r • s}.Nonempty :=
  let ⟨r, hr₁, hr₂⟩ := (absorbs x).exists_pos
  ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge rfl⟩

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