monotone_of_continuous — Mathlib

English

If f is continuous and the domain is IsScott α D and the codomain β is Scott, then f is monotone.

Русский

Если f непрерывна и область имеет пространство IsScott α D, а кодом является Scott, то f монотонна.

LaTeX

$$$\\text{Continuous}(f) \\Rightarrow \\text{Monotone}(f)$$$

Lean4

theorem monotone_of_continuous [IsScott α D] (hf : Continuous f) : Monotone f := fun _ b hab ↦
  by
  by_contra h
  simpa only [mem_compl_iff, mem_preimage, mem_Iic, le_refl, not_true] using
    isUpperSet_of_isOpen (D := D) ((isOpen_compl_iff.2 isClosed_Iic).preimage hf) hab h

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