Lemmas about units in ZMod.

def ZMod.unitsMap {n m : ℕ} (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ

unitsMap is a group homomorphism that maps units of ZMod m to units of ZMod n when n divides m.

Equations

• ZMod.unitsMap hm = Units.map ↑(ZMod.castHom hm (ZMod n))

theorem ZMod.unitsMap_def {n m : ℕ} (hm : n ∣ m) : unitsMap hm = Units.map ↑(castHom hm (ZMod n))

theorem ZMod.unitsMap_comp {n m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) : (unitsMap hm).comp (unitsMap hd) = unitsMap ⋯

@[simp]

theorem ZMod.unitsMap_self (n : ℕ) : unitsMap ⋯ = MonoidHom.id (ZMod n)ˣ

theorem ZMod.unitsMap_val {n m : ℕ} (h : n ∣ m) (a : (ZMod m)ˣ) : ↑((unitsMap h) a) = (↑a).cast

unitsMap_val shows that coercing from (ZMod m)ˣ to ZMod n gives the same result when going via (ZMod n)ˣ and ZMod m.

theorem ZMod.isUnit_cast_of_dvd {n m : ℕ} (hm : n ∣ m) (a : (ZMod m)ˣ) : IsUnit (↑a).cast

@[deprecated ZMod.isUnit_cast_of_dvd (since := "2024-12-16")]

theorem ZMod.IsUnit_cast_of_dvd {n m : ℕ} (hm : n ∣ m) (a : (ZMod m)ˣ) : IsUnit (↑a).cast

Alias of ZMod.isUnit_cast_of_dvd.

theorem ZMod.unitsMap_surjective {n m : ℕ} [hm : NeZero m] (h : n ∣ m) : Function.Surjective ⇑(unitsMap h)

theorem ZMod.not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) : ¬IsUnit ↑p

theorem ZMod.eq_unit_mul_divisor {N : ℕ} (a : ZMod N) : ∃ (d : ℕ), d ∣ N ∧ ∃ (u : ZMod N), IsUnit u ∧ a = u * ↑d

Any element of ZMod N has the form u * d where u is a unit and d is a divisor of N.

theorem ZMod.coe_int_mul_inv_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x ↑n) : ↑x * (↑x)⁻¹ = 1

theorem ZMod.coe_int_inv_mul_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x ↑n) : (↑x)⁻¹ * ↑x = 1

theorem ZMod.coe_int_mul_val_inv {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m ↑n) : ↑m * ↑(↑m)⁻¹.val = 1

theorem ZMod.coe_int_val_inv_mul {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m ↑n) : ↑(↑m)⁻¹.val * ↑m = 1

def ZMod.unitOfIsCoprime {m : ℕ} (n : ℤ) (h : IsCoprime n ↑m) : (ZMod m)ˣ

The unit of ZMod m associated with an integer prime to n.

Equations

• ZMod.unitOfIsCoprime n h = { val := ↑n, inv := (↑n)⁻¹, val_inv := ⋯, inv_val := ⋯ }

@[simp]

theorem ZMod.coe_unitOfIsCoprime {m : ℕ} (n : ℤ) (h : IsCoprime n ↑m) : ↑(unitOfIsCoprime n h) = ↑n

theorem ZMod.isUnit_inv {m : ℕ} {n : ℤ} (h : IsUnit ↑n) : IsUnit (↑n)⁻¹