isCoatomic_dual_iff_isAtomic — Mathlib

English

For any α with an order bottom, IsCoatomic αᵒᵈ is equivalent to IsAtomic α.

Русский

Для любой частичной порядка с нулём, IsCoatomic αᵒᵈ эквивалентно IsAtomic α.

LaTeX

$$$ IsCoatomic (OrderDual\ α) \iff IsAtomic α $$$

Lean4

@[simp]
theorem isCoatomic_dual_iff_isAtomic [OrderBot α] : IsCoatomic αᵒᵈ ↔ IsAtomic α :=
  ⟨fun h => ⟨fun b => by apply h.eq_top_or_exists_le_coatom⟩, fun h => ⟨fun b => by apply h.eq_bot_or_exists_atom_le⟩⟩

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