nnreal_trichotomy — Mathlib

English

For every real r, either r = 0 or there exists x ≥ 0 with x > 0 and r = x or r = -x.

Русский

Для каждого r ∈ ℝ либо r = 0, либо существует x ≥ 0, x > 0 и r = x или r = -x.

LaTeX

$$$ r = 0 \lor \exists x \in \mathbb{R}_{\ge 0},\; 0 < x \land (r = x \lor r = -x). $$$

Lean4

/-- Every real number is either zero, positive or negative, phrased using `ℝ≥0`. -/
theorem nnreal_trichotomy (r : ℝ) : r = 0 ∨ ∃ x : ℝ≥0, 0 < x ∧ (r = x ∨ r = -x) :=
  by
  obtain ⟨x, hx⟩ := nnreal_dichotomy r
  rw [or_iff_not_imp_left]
  aesop (add simp pos_iff_ne_zero)

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