image_affine_Ico — Mathlib

English

If a > 0, the affine map x ↦ a x + b sends Ioc(c, d) to Ioc(a c + b, a d + b).

Русский

Если a > 0, аффинное отображение x ↦ a x + b переводит Ioc(c, d) в Ioc(a c + b, a d + b).

LaTeX

$$$ (a \cdot \cdot + b)'' Ioc(c, d) = Ioc(a c + b, a d + b) $$$

Lean4

@[simp]
theorem image_affine_Ico (h : 0 < a) (b c d : K) : (a * · + b) '' Ico c d = Ico (a * c + b) (a * d + b) :=
  by
  suffices (· + b) '' ((a * ·) '' Ico c d) = Ico (a * c + b) (a * d + b) by rwa [Set.image_image] at this
  rw [image_mul_left_Ico h, image_add_const_Ico]

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