units_inv_smul — Mathlib

English

The scaling by the inverse of a unit equals scaling by the unit itself on rays: u^{-1} • v = u • v.

Русский

Умножение на обратную единицу эквивалентно умножению на саму единицу на лучах: u^{-1} • v = u • v.

LaTeX

$$$u^{-1}\cdot v = u\cdot v$ for unit $u$.$$

Lean4

/-- Scaling by an inverse unit is the same as scaling by itself. -/
@[simp]
theorem units_inv_smul (u : Rˣ) (v : Module.Ray R M) : u⁻¹ • v = u • v :=
  have := mul_self_pos.2 u.ne_zero
  calc
    u⁻¹ • v = (u * u) • u⁻¹ • v := Eq.symm <| (u⁻¹ • v).units_smul_of_pos _ (by exact this)
    _ = u • v := by rw [mul_smul, smul_inv_smul]

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