Icc_union_Ioo_eq_Ico — 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 Icc_union_Ioo_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ioo b c = Ico a c :=
  Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩)
    Ico_subset_Icc_union_Ioo

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