isNormal_enumOrd — Mathlib

English

If s is closed under nonempty suprema, then the enumerator function enumOrd s is normal.

Русский

Если s замкнуто по ненулевым верхин 성장, то перечисляющая функция enumOrd s является нормальной.

LaTeX

$$$$ \\text{If } H( t \\subseteq s , t.Nonempty , BddAbove t ) \\text{ implies } sSup(t) \\in s \\;\\text{ and } \\neg BddAbove(s) \\Rightarrow IsNormal(\\mathrm{enumOrd}(s)) $$$$

Lean4

/-- If `s` is closed under nonempty suprema, then its enumerator function is normal.
See also `enumOrd_isNormal_iff_isClosed`. -/
theorem isNormal_enumOrd (H : ∀ t ⊆ s, t.Nonempty → BddAbove t → sSup t ∈ s) (hs : ¬BddAbove s) :
    IsNormal (enumOrd s) :=
  by
  refine (isNormal_iff_strictMono_limit _).2 ⟨enumOrd_strictMono hs, fun o ho a ha ↦ ?_⟩
  trans ⨆ b : Iio o, enumOrd s b
  · refine enumOrd_le_of_forall_lt ?_ (fun b hb ↦ (enumOrd_strictMono hs (lt_succ b)).trans_le ?_)
    · have : Nonempty (Iio o) := ⟨0, ho.bot_lt⟩
      apply H _ _ (range_nonempty _) (bddAbove_of_small _)
      rintro _ ⟨c, rfl⟩
      exact enumOrd_mem hs c
    · exact Ordinal.le_iSup _ (⟨_, ho.succ_lt hb⟩ : Iio o)
  · exact Ordinal.iSup_le fun x ↦ ha _ x.2

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