padicValuation_le_one — Mathlib

English

For any prime p and any integer x, v_p^{ℤ}(x) ≤ 1.

Русский

Для любого p и целого x, v_p^{ℤ}(x) ≤ 1.

LaTeX

$$v_p^{ℤ}(x) \\le 1$$

Lean4

theorem padicValuation_le_one (p : ℕ) [Fact p.Prime] (x : ℤ) : Int.padicValuation p x ≤ 1 :=
  by
  simp only [← Rat.padicValuation_cast, Rat.padicValuation, Valuation.coe_mk, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
    Rat.intCast_eq_zero, padicValRat.of_int]
  split_ifs
  · simp
  · rw [← le_log_iff_exp_le] <;> simp_all

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