image_mul_const_uIcc — Mathlib

English

The preimage under inversion of the interval Ioo(a,0) is the interval Iio(a⁻¹) ∪ {0} on the right side, i.e., (Ioo(a,0))⁻¹ = Iio(a⁻¹).

Русский

Предобраз под инверсией от Ioo(a,0) равен Iio(a⁻¹).

LaTeX

$$$(\mathrm{Ioo}(a,0))^{-1} = \mathrm{Iio}(a^{-1})$$$

Lean4

@[simp]
theorem image_mul_const_uIcc (a b c : α) : (· * a) '' [[b, c]] = [[b * a, c * a]] :=
  if ha : a = 0 then by simp [ha]
  else
    calc
      (fun x => x * a) '' [[b, c]] = (· * a⁻¹) ⁻¹' [[b, c]] := (Units.mk0 a ha).mulRight.image_eq_preimage _
      _ = (fun x => x / a) ⁻¹' [[b, c]] := by simp only [div_eq_mul_inv]
      _ = [[b * a, c * a]] := preimage_div_const_uIcc ha _ _

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