image_natAdd_uIoo — 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_uIoo (m) (i j : Fin n) : natAdd m '' uIoo i j = uIoo (natAdd m i) (natAdd m j) := by
  simp [uIoo, ← (strictMono_natAdd m).monotone.map_max, ← (strictMono_natAdd m).monotone.map_min]

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