coe_int_isUnit_iff_isCoprime — Mathlib

English

If gcd(card α, n) = 1, then for any a ∈ α, (a^n)^{n⁻¹} = a.

Русский

Если gcd(|α|, n) = 1, то для любого a ∈ α выполнено (a^n)^{n⁻¹} = a.

LaTeX

$$$(a^n)^{n^{-1}} = a$$$

Lean4

theorem coe_int_isUnit_iff_isCoprime (n : ℤ) (m : ℕ) : IsUnit (n : ZMod m) ↔ IsCoprime (m : ℤ) n :=
  by
  rw [Int.isCoprime_iff_nat_coprime, Nat.coprime_comm, ← isUnit_iff_coprime, Associated.isUnit_iff]
  simpa only [eq_intCast, Int.cast_natCast] using (Int.associated_natAbs _).map (Int.castRingHom _)

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