normalize_smul — Mathlib

English

For any real r, normalization scales by the sign of r: normalize(r x) = sign(r) · normalize(x).

Русский

Для любого вещественного r нормирование масштабирует на знак r: normalize(r x) = sign(r) · normalize(x).

LaTeX

$$$\\operatorname{normalize}(r x) = \\operatorname{sign}(r)\\,\\operatorname{normalize}(x)$$$

Lean4

theorem normalize_smul (r : ℝ) (x : V) : normalize (r • x) = (SignType.sign r : ℝ) • normalize x :=
  by
  rcases lt_trichotomy 0 r with (h_pos | rfl | h_neg)
  · simp [normalize_smul_of_pos, h_pos]
  · simp
  · simp [normalize_smul_of_neg, h_neg]

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