exists_inf_eq_and_sup_eq — Mathlib

English

Given a ≤ b and complemented lattice, there exists b' with b ∧ b' = a and b ∨ b' = ⊤.

Русский

Пусть a ≤ b в композиции с допонерной решёткой; существует b' such that b ∧ b' = a и b ∨ b' = ⊤.

LaTeX

$$$\exists b'\; (b\wedge b')=a \land (b\vee b')=\top$$$

Lean4

theorem exists_inf_eq_and_sup_eq (hb : a ≤ b) (hc : b ≤ c) : ∃ b', b ⊓ b' = a ∧ b ⊔ b' = c :=
  by
  obtain ⟨d, hdisjoint, hcodisjoint⟩ := exists_isCompl b
  refine ⟨(d ⊔ a) ⊓ c, ?_, ?_⟩
  · simpa [← inf_assoc, ← inf_sup_assoc_of_le _ hb, hdisjoint.eq_bot] using hb.trans hc
  · simp [← sup_inf_assoc_of_le _ hc, ← sup_assoc, hcodisjoint.eq_top]

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