image_addNat_Icc — Mathlib

English

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

Русский

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

LaTeX

$$$$ \\mathrm{Set.image}(\\\\mathrm{Fin.natAdd}(m))(\\\\mathrm{Set.Icc}(i,j)) = \\\\mathrm{Set.Icc}(\\\\mathrm{Fin.natAdd}(m)i, \\\\mathrm{Fin.natAdd}(m)j). $$$$

Lean4

@[simp]
theorem image_addNat_Icc (m) (i j : Fin n) : (addNat · m) '' Icc i j = Icc (i.addNat m) (j.addNat m) :=
  by
  rw [← preimage_addNat_Icc_addNat, image_preimage_eq_of_subset]
  exact Icc_subset_Ici_self.trans <| image_addNat_Ici m i ▸ image_subset_range _ _

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