one_of_mem_closedBall — Mathlib

English

Let b be an affine basis; coord_i is smooth for each i.

Русский

Пусть b — аффинная база; coord_i гладкая для каждого i.

LaTeX

$$$ \operatorname{ContDiff}_{\mathbb{k}}(\top)(b.coord_i). $$$

Lean4

theorem one_of_mem_closedBall (hx : x ∈ closedBall c f.rIn) : f x = 1 :=
  by
  apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn
  simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, ← div_eq_inv_mul,
    div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx

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