not_isMin' — Mathlib

English

If a function is both monotone and antitone on a directed order, it must be constant.

Русский

Если функция монотонна и антимонотонна на направленном порядке, она константна.

LaTeX

$$$\forall \alpha,\beta [PartialOrder(\beta)], [Preorder(\alpha)], f: \alpha \to \beta, IsDirected(\alpha, \le) \Rightarrow (Monotone f \land Antitone f) \Rightarrow \forall a,b, f(a)=f(b).$$$

Lean4

protected theorem not_isMin' [IsDirected β (· ≥ ·)] {b : β} (hb : IsMax b) : ¬IsMin b := fun hb' ↦
  hb'.toDual.not_isMin hb.toDual

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