supIrred_Iic — Mathlib

English

The SupIrred of Iic a is definable and computable via Iic a.

Русский

SupIrred для Iic a определяется через Iic a.

LaTeX

$$SupIrred(LowerSet.Iic a)$$

Lean4

@[simp]
theorem supIrred_Iic (a : α) : SupIrred (Iic a) :=
  by
  refine ⟨fun h ↦ Iic_ne_bot h.eq_bot, fun s t hst ↦ ?_⟩
  have := mem_Iic_iff.2 (le_refl a)
  rw [← hst] at this
  exact
    this.imp (fun ha ↦ (le_sup_left.trans_eq hst).antisymm <| Iic_le.2 ha) fun ha ↦
      (le_sup_right.trans_eq hst).antisymm <| Iic_le.2 ha

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