image_addNat_Ioi — Mathlib

English

For all m ∈ ℕ and i ∈ Fin(n), the image of Ioi i under x ↦ x + m is Ioi(i + m).

Русский

Для всех m ∈ ℕ и i ∈ Fin(n) образ Ioi i при сдвиге на m равен Ioi(i + m).

LaTeX

$$$$ \\mathrm{Set.image}(\\\\mathrm{Fin.natAdd}(m))(\\\\mathrm{Set.Ioi} i) = \\\\mathrm{Set.Ioi}(\\\\mathrm{Fin.natAdd}(m)i). $$$$

Lean4

@[simp]
theorem image_addNat_Ioi (m) (i : Fin n) : (addNat · m) '' Ioi i = Ioi (i.addNat m) :=
  by
  rw [← preimage_addNat_Ioi_addNat, image_preimage_eq_of_subset]
  exact Ioi_subset_Ici_self.trans <| image_addNat_Ici m i ▸ image_subset_range _ _

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