diam_smul₀ — Mathlib

English

The smul ball lemma expresses that scaling commutes with the ball up to the appropriate radius: c·ball x r = ball (c·x) (|c|·r).

Русский

Лемма масштабирования для шара: c·ball(x,r) = ball(c·x, |c|·r).

LaTeX

$$$c \cdot \mathrm{ball}(x,r) = \mathrm{ball}(c\cdot x, \lVert c\rVert \cdot r).$$$

Lean4

theorem diam_smul₀ (c : 𝕜) (x : Set E) : diam (c • x) = ‖c‖ * diam x := by
  simp_rw [diam, ediam_smul₀, ENNReal.toReal_smul, NNReal.smul_def, coe_nnnorm, smul_eq_mul]

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