complementedLattice_Icc — Mathlib

English

The open-interval diamond isomorphism is established by a pair of monotone maps between Ioo intervals in a modular lattice.

Русский

Установлено diamond-изоморфизм между открытыми интервалами через пары монотонных отображений.

LaTeX

$$$\text{InfIooOrderIsoIooSup}(a,b): Ioo(a\wedge b,a) \simeq_o Ioo(b,a\vee b).$$$

Lean4

instance complementedLattice_Icc [Fact (a ≤ b)] : ComplementedLattice (Set.Icc a b) where
  exists_isCompl := fun ⟨x, ha, hb⟩ => by
    simp_rw [Set.Icc.isCompl_iff]
    obtain ⟨y, rfl, rfl⟩ := exists_inf_eq_and_sup_eq ha hb
    exact ⟨⟨y, inf_le_right, le_sup_right⟩, rfl, rfl⟩

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