uniformContinuous_coeff — Mathlib

English

δ is independent of presentation, hence δ equals δ on any two presentations Q with P and P'.

Русский

δ независим от презентации; δ совпадает на любом двух представлениях Q с P и P'.

LaTeX

$$$δ Q P = δ Q P'$$$

Lean4

theorem uniformContinuous_coeff {uK : UniformSpace K} (d : ℤ) : UniformContinuous fun f : K⸨X⸩ ↦ f.coeff d :=
  by
  refine uniformContinuous_iff_eventually.mpr fun S hS ↦ eventually_iff_exists_mem.mpr ?_
  let γ : (ℤᵐ⁰)ˣ := Units.mk0 (exp (-(d + 1))) WithZero.coe_ne_zero
  use {P | Valued.v (P.snd - P.fst) < ↑γ}
  refine ⟨(Valued.hasBasis_uniformity K⸨X⸩ ℤᵐ⁰).mem_of_mem (by tauto), fun P hP ↦ ?_⟩
  rw [eq_coeff_of_valuation_sub_lt K (le_of_lt hP) (lt_add_one _)]
  exact mem_uniformity_of_eq hS rfl

Похожие утверждения