toOrderIso — Mathlib

English

If e is an equivalence with monotone forward and monotone inverse maps, then e is an order isomorphism.

Русский

Если e — эквивалентность с монотононами вперед и назад, то e — порядковый изоморфизм.

LaTeX

$$$\text{toOrderIso}(e,h_1,h_2) : e \cong_o e$$$

Lean4

/-- If `e` is an equivalence with monotone forward and inverse maps, then `e` is an
order isomorphism. -/
def toOrderIso (e : α ≃ β) (h₁ : Monotone e) (h₂ : Monotone e.symm) : α ≃o β :=
  ⟨e, ⟨fun h => by simpa only [e.symm_apply_apply] using h₂ h, fun h => h₁ h⟩⟩

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