exists_one_lt — Mathlib

English

If v1 is equivalent to v2, then v2 is equivalent to v1.

Русский

Если v1 эквивалентна v2, то и v2 эквивалентна v1.

LaTeX

$$$ (v_1.IsEquiv v_2) \\Rightarrow (v_2.IsEquiv v_1) $$$

Lean4

theorem exists_one_lt {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation K Γ₀} [hv : v.IsNontrivial] :
    ∃ x : K, v x ≠ 0 ∧ 1 < v x := by
  obtain ⟨x, h0, h1⟩ := hv.exists_lt_one
  use x⁻¹
  simp [one_lt_inv₀ (zero_lt_iff.mpr h0), h0, h1]

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