eq_iff_atom_le_iff — Mathlib

English

In the presence of OrderBot, IsCoatomistic(OrderDual α) iff IsAtomistic α.

Русский

При наличии OrderBot: IsCoatomistic(OrderDual α) ⇔ IsAtomistic α.

LaTeX

$$$[OrderBot \\alpha] \\Rightarrow (\\mathrm{IsCoatomistic}(\\alpha^\\mathrm{op}) \\iff \\mathrm{IsAtomistic}(\\alpha))$$$

Lean4

theorem eq_iff_atom_le_iff {α} [BooleanAlgebra α] [IsAtomic α] {x y : α} : x = y ↔ ∀ a, IsAtom a → (a ≤ x ↔ a ≤ y) :=
  by
  refine ⟨fun h => h ▸ by simp, fun h => ?_⟩
  exact
    le_antisymm (le_iff_atom_le_imp.2 fun a ha hx => (h a ha).1 hx) (le_iff_atom_le_imp.2 fun a ha hy => (h a ha).2 hy)

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