Ico_union_Icc_eq_Icc — Mathlib

English

If a ≤ b and b < c, then Icc(a,b) ∪ Ioo(b,c) = Ioc(a,c).

Русский

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

LaTeX

$$$ a \le b \land b < c \implies Icc(a,b) \cup Ioo(b,c) = Ioc(a,c). $$$

Lean4

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

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