id_le_iterate_of_id_le — Mathlib

English

For a nonempty index set and a bounded above range, f preserves iSup, i.e., f (iSup g) = iSup (f ∘ g).

Русский

Для ненулевого множества индексов и ограниченного сверху диапазона f сохраняет iSup: f(iSup g) = iSup (f ∘ g).

LaTeX

$$$\\forall {ι} [Nonempty ι] {g : ι \\to α}, \\mathrm{IsNormal}(f) \\to \\mathrm{BddAbove}(\\mathrm{range}(g)) \\Rightarrow f(\\iSup_i g i) = \\iSup_i f(g i).$$$

Lean4

/-- If $x ≤ f x$ for all $x$ (we write this as `id ≤ f`), then the same is true for any iterate
`f^[n]` of `f`. -/
theorem id_le_iterate_of_id_le (h : id ≤ f) (n : ℕ) : id ≤ f^[n] := by
  simpa only [iterate_id] using monotone_id.iterate_le_of_le h n

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