padicValNat_eq_zero_of_mem_Ioo — Mathlib

English

If p and q are primes and p ≠ q, then padicValNat p (q^n) = 0 for every n.

Русский

Если p и q — простые, и p ≠ q, то v_p(q^n) = 0 для любого n.

LaTeX

$$$\\forall p,q,n \\in \\mathbb{N},\\ p,q\\text{ primes},\\ p \\neq q \\Rightarrow \\operatorname{padicValNat}(p, q^n)=0$$$

Lean4

/-- The `p`-adic valuation of `m` equals zero if it is between `p * k` and `p * (k + 1)` for
some `k`. -/
theorem padicValNat_eq_zero_of_mem_Ioo {m k : ℕ} (hm : m ∈ Set.Ioo (p * k) (p * (k + 1))) : padicValNat p m = 0 :=
  padicValNat.eq_zero_of_not_dvd <| not_dvd_of_lt_of_lt_mul_succ hm.1 hm.2

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