image_natAdd_Ioo — Mathlib

English

Let m ∈ ℕ and i, j ∈ Fin(n). The image of Ioo(i, j) under x ↦ x + m is Ioo(i + m, j + m).

Русский

Пусть m ∈ ℕ и i, j ∈ Fin(n). Образ Ioo(i, j) при сдвиге на m равен Ioo(i + m, j + m).

LaTeX

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

Lean4

@[simp]
theorem image_natAdd_Ioo (m) (i j : Fin n) : natAdd m '' Ioo i j = Ioo (natAdd m i) (natAdd m j) :=
  by
  rw [← preimage_natAdd_Ioo_natAdd, image_preimage_eq_of_subset]
  exact Ioo_subset_Ioi_self.trans <| image_natAdd_Ioi m i ▸ image_subset_range _ _

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