Icc_union_Ioc_eq_Icc — Mathlib

English

If a ≤ b ≤ c, then Icc a b ∪ Ioc b c = Icc a c.

Русский

Если a ≤ b ≤ c, то Icc(a,b) ∪ Ioc(b,c) = Icc(a,c).

LaTeX

$$$(a \le b) \land (b \le c) \implies \mathrm{Icc}(a,b) \cup \mathrm{Ioc}(b,c) = \mathrm{Icc}(a,c)$$$

Lean4

@[simp]
theorem Icc_union_Ioc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Ioc b c = Icc a c :=
  Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩)
    Icc_subset_Icc_union_Ioc

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