sphere_zero_eq — Mathlib

English

The finrank (finite rank) of EuclideanSpace 𝕜 ι is equal to the cardinality of ι when ι is finite: finrank 𝕜 (EuclideanSpace 𝕜 ι) = card ι.

Русский

Порядок конечной размерности Евклидова пространства 𝕜-значимый, равен кардиналу множества ι: finrank 𝕜 (EuclideanSpace 𝕜 ι) = card ι.

LaTeX

$$$\operatorname{finrank}_{𝕜}(\mathrm{EuclideanSpace}\; 𝕜\; ι) = |ι|$$$

Lean4

theorem sphere_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
    Metric.sphere (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 = r ^ 2} :=
  by
  ext x
  have : (0 : ℝ) ≤ ∑ i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
  simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs, Real.sqrt_eq_iff_eq_sq this hr]

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