fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z := by
rw [mul_comm] at H
exact H.of_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_left_right | null |
IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y := by
rw [isCoprime_comm] at H ⊢
exact H.of_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_right_left | null |
IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z := by
rw [mul_comm] at H
exact H.of_mul_right_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_right_right | null |
IsCoprime.mul_left_iff : IsCoprime (x * y) z ↔ IsCoprime x z ∧ IsCoprime y z :=
⟨fun H => ⟨H.of_mul_left_left, H.of_mul_left_right⟩, fun ⟨H1, H2⟩ => H1.mul_left H2⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.mul_left_iff | null |
IsCoprime.mul_right_iff : IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z := by
rw [isCoprime_comm, IsCoprime.mul_left_iff, isCoprime_comm, @isCoprime_comm _ _ z] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.mul_right_iff | null |
IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z := by
obtain ⟨d, rfl⟩ := hdvd
exact IsCoprime.of_mul_left_left h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_isCoprime_of_dvd_left | null |
IsCoprime.of_isCoprime_of_dvd_right (h : IsCoprime z y) (hdvd : x ∣ y) : IsCoprime z x :=
(h.symm.of_isCoprime_of_dvd_left hdvd).symm
@[gcongr] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_isCoprime_of_dvd_right | null |
IsCoprime.mono (h₁ : x ∣ y) (h₂ : z ∣ w) (h : IsCoprime y w) : IsCoprime x z :=
h.of_isCoprime_of_dvd_left h₁ |>.of_isCoprime_of_dvd_right h₂ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.mono | null |
IsCoprime.isUnit_of_dvd (H : IsCoprime x y) (d : x ∣ y) : IsUnit x :=
let ⟨k, hk⟩ := d
isCoprime_self.1 <| IsCoprime.of_mul_right_left <| show IsCoprime x (x * k) from hk ▸ H | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.isUnit_of_dvd | null |
IsCoprime.isUnit_of_dvd' {a b x : R} (h : IsCoprime a b) (ha : x ∣ a) (hb : x ∣ b) :
IsUnit x :=
(h.of_isCoprime_of_dvd_left ha).isUnit_of_dvd hb | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.isUnit_of_dvd' | null |
IsCoprime.isRelPrime {a b : R} (h : IsCoprime a b) : IsRelPrime a b :=
fun _ ↦ h.isUnit_of_dvd' | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.isRelPrime | null |
IsCoprime.map (H : IsCoprime x y) {S : Type v} [CommSemiring S] (f : R →+* S) :
IsCoprime (f x) (f y) :=
let ⟨a, b, h⟩ := H
⟨f a, f b, by rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.map | null |
IsCoprime.of_add_mul_left_left (h : IsCoprime (x + y * z) y) : IsCoprime x y :=
let ⟨a, b, H⟩ := h
⟨a, a * z + b, by
simpa only [add_mul, mul_add, add_assoc, add_comm, add_left_comm, mul_assoc, mul_comm,
mul_left_comm] using H⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_add_mul_left_left | null |
IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y := by
rw [mul_comm] at h
exact h.of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_add_mul_right_left | null |
IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y := by
rw [isCoprime_comm] at h ⊢
exact h.of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_add_mul_left_right | null |
IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y := by
rw [mul_comm] at h
exact h.of_add_mul_left_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_add_mul_right_right | null |
IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y := by
rw [add_comm] at h
exact h.of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_add_left_left | null |
IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y := by
rw [add_comm] at h
exact h.of_add_mul_right_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_add_right_left | null |
IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y := by
rw [add_comm] at h
exact h.of_add_mul_left_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_add_left_right | null |
IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y := by
rw [add_comm] at h
exact h.of_add_mul_right_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.of_mul_add_right_right | null |
IsRelPrime.of_add_mul_left_left (h : IsRelPrime (x + y * z) y) : IsRelPrime x y :=
fun _ hx hy ↦ h (dvd_add hx <| dvd_mul_of_dvd_left hy z) hy | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_add_mul_left_left | null |
IsRelPrime.of_add_mul_right_left (h : IsRelPrime (x + z * y) y) : IsRelPrime x y :=
(mul_comm z y ▸ h).of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_add_mul_right_left | null |
IsRelPrime.of_add_mul_left_right (h : IsRelPrime x (y + x * z)) : IsRelPrime x y := by
rw [isRelPrime_comm] at h ⊢
exact h.of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_add_mul_left_right | null |
IsRelPrime.of_add_mul_right_right (h : IsRelPrime x (y + z * x)) : IsRelPrime x y :=
(mul_comm z x ▸ h).of_add_mul_left_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_add_mul_right_right | null |
IsRelPrime.of_mul_add_left_left (h : IsRelPrime (y * z + x) y) : IsRelPrime x y :=
(add_comm _ x ▸ h).of_add_mul_left_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_mul_add_left_left | null |
IsRelPrime.of_mul_add_right_left (h : IsRelPrime (z * y + x) y) : IsRelPrime x y :=
(add_comm _ x ▸ h).of_add_mul_right_left | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_mul_add_right_left | null |
IsRelPrime.of_mul_add_left_right (h : IsRelPrime x (x * z + y)) : IsRelPrime x y :=
(add_comm _ y ▸ h).of_add_mul_left_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_mul_add_left_right | null |
IsRelPrime.of_mul_add_right_right (h : IsRelPrime x (z * x + y)) : IsRelPrime x y :=
(add_comm _ y ▸ h).of_add_mul_right_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | IsRelPrime.of_mul_add_right_right | null |
isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z :=
⟨fun ⟨a, b, h⟩ => ⟨x • a, b, by rwa [smul_mul_assoc, ← mul_smul_comm]⟩, fun ⟨a, b, h⟩ =>
⟨x⁻¹ • a, b, by rwa [smul_mul_smul_comm, inv_mul_cancel, one_smul]⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_group_smul_left | null |
isCoprime_group_smul_right : IsCoprime y (x • z) ↔ IsCoprime y z :=
isCoprime_comm.trans <| (isCoprime_group_smul_left x z y).trans isCoprime_comm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_group_smul_right | null |
isCoprime_group_smul : IsCoprime (x • y) (x • z) ↔ IsCoprime y z :=
(isCoprime_group_smul_left x y (x • z)).trans (isCoprime_group_smul_right x y z) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_group_smul | null |
isCoprime_mul_unit_left_left (hu : IsUnit x) (y z : R) :
IsCoprime (x * y) z ↔ IsCoprime y z :=
let ⟨u, hu⟩ := hu
hu ▸ isCoprime_group_smul_left u y z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_left_left | null |
isCoprime_mul_unit_left_right (hu : IsUnit x) (y z : R) :
IsCoprime y (x * z) ↔ IsCoprime y z :=
let ⟨u, hu⟩ := hu
hu ▸ isCoprime_group_smul_right u y z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_left_right | null |
isCoprime_mul_unit_right_left (hu : IsUnit x) (y z : R) :
IsCoprime (y * x) z ↔ IsCoprime y z :=
mul_comm x y ▸ isCoprime_mul_unit_left_left hu y z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_right_left | null |
isCoprime_mul_unit_right_right (hu : IsUnit x) (y z : R) :
IsCoprime y (z * x) ↔ IsCoprime y z :=
mul_comm x z ▸ isCoprime_mul_unit_left_right hu y z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_right_right | null |
isCoprime_mul_units_left (hu : IsUnit u) (hv : IsUnit v) (y z : R) :
IsCoprime (u * y) (v * z) ↔ IsCoprime y z :=
Iff.trans
(isCoprime_mul_unit_left_left hu _ _)
(isCoprime_mul_unit_left_right hv _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_units_left | null |
isCoprime_mul_units_right (hu : IsUnit u) (hv : IsUnit v) (y z : R) :
IsCoprime (y * u) (z * v) ↔ IsCoprime y z :=
Iff.trans
(isCoprime_mul_unit_right_left hu _ _)
(isCoprime_mul_unit_right_right hv _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_units_right | null |
isCoprime_mul_unit_left (hu : IsUnit x) (y z : R) :
IsCoprime (x * y) (x * z) ↔ IsCoprime y z :=
isCoprime_mul_units_left hu hu _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_left | null |
isCoprime_mul_unit_right (hu : IsUnit x) (y z : R) :
IsCoprime (y * x) (z * x) ↔ IsCoprime y z :=
isCoprime_mul_units_right hu hu _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | isCoprime_mul_unit_right | null |
add_mul_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + y * z) y :=
@of_add_mul_left_left R _ _ _ (-z) <| by simpa only [mul_neg, add_neg_cancel_right] using h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_left | null |
add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y := by
rw [mul_comm]
exact h.add_mul_left_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_left | null |
add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) := by
rw [isCoprime_comm]
exact h.symm.add_mul_left_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_right | null |
add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) := by
rw [isCoprime_comm]
exact h.symm.add_mul_right_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_right | null |
mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y := by
rw [add_comm]
exact h.add_mul_left_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_left | null |
mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y := by
rw [add_comm]
exact h.add_mul_right_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_left | null |
mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) := by
rw [add_comm]
exact h.add_mul_left_right z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_right | null |
mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) := by
rw [add_comm]
exact h.add_mul_right_right z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_right | null |
add_mul_left_left_iff {x y z : R} : IsCoprime (x + y * z) y ↔ IsCoprime x y :=
⟨of_add_mul_left_left, fun h => h.add_mul_left_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_left_iff | null |
add_mul_right_left_iff {x y z : R} : IsCoprime (x + z * y) y ↔ IsCoprime x y :=
⟨of_add_mul_right_left, fun h => h.add_mul_right_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_left_iff | null |
add_mul_left_right_iff {x y z : R} : IsCoprime x (y + x * z) ↔ IsCoprime x y :=
⟨of_add_mul_left_right, fun h => h.add_mul_left_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_right_iff | null |
add_mul_right_right_iff {x y z : R} : IsCoprime x (y + z * x) ↔ IsCoprime x y :=
⟨of_add_mul_right_right, fun h => h.add_mul_right_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_right_iff | null |
mul_add_left_left_iff {x y z : R} : IsCoprime (y * z + x) y ↔ IsCoprime x y :=
⟨of_mul_add_left_left, fun h => h.mul_add_left_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_left_iff | null |
mul_add_right_left_iff {x y z : R} : IsCoprime (z * y + x) y ↔ IsCoprime x y :=
⟨of_mul_add_right_left, fun h => h.mul_add_right_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_left_iff | null |
mul_add_left_right_iff {x y z : R} : IsCoprime x (x * z + y) ↔ IsCoprime x y :=
⟨of_mul_add_left_right, fun h => h.mul_add_left_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_right_iff | null |
mul_add_right_right_iff {x y z : R} : IsCoprime x (z * x + y) ↔ IsCoprime x y :=
⟨of_mul_add_right_right, fun h => h.mul_add_right_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_right_iff | null |
neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y := by
obtain ⟨a, b, h⟩ := h
use -a, b
rwa [neg_mul_neg] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_left | null |
neg_left_iff (x y : R) : IsCoprime (-x) y ↔ IsCoprime x y :=
⟨fun h => neg_neg x ▸ h.neg_left, neg_left⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_left_iff | null |
neg_right {x y : R} (h : IsCoprime x y) : IsCoprime x (-y) :=
h.symm.neg_left.symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_right | null |
neg_right_iff (x y : R) : IsCoprime x (-y) ↔ IsCoprime x y :=
⟨fun h => neg_neg y ▸ h.neg_right, neg_right⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_right_iff | null |
neg_neg {x y : R} (h : IsCoprime x y) : IsCoprime (-x) (-y) :=
h.neg_left.neg_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_neg | null |
neg_neg_iff (x y : R) : IsCoprime (-x) (-y) ↔ IsCoprime x y :=
(neg_left_iff _ _).trans (neg_right_iff _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_neg_iff | null |
abs_left_iff (x y : R) : IsCoprime |x| y ↔ IsCoprime x y := by
cases le_or_gt 0 x with
| inl h => rw [abs_of_nonneg h]
| inr h => rw [abs_of_neg h, IsCoprime.neg_left_iff] | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_left_iff | null |
abs_left {x y : R} (h : IsCoprime x y) : IsCoprime |x| y := abs_left_iff _ _ |>.2 h | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_left | null |
abs_right_iff (x y : R) : IsCoprime x |y| ↔ IsCoprime x y := by
rw [isCoprime_comm, IsCoprime.abs_left_iff, isCoprime_comm] | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_right_iff | null |
abs_right {x y : R} (h : IsCoprime x y) : IsCoprime x |y| := abs_right_iff _ _ |>.2 h | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_right | null |
abs_abs_iff (x y : R) : IsCoprime |x| |y| ↔ IsCoprime x y :=
(abs_left_iff _ _).trans (abs_right_iff _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_abs_iff | null |
abs_abs {x y : R} (h : IsCoprime x y) : IsCoprime |x| |y| := h.abs_left.abs_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | abs_abs | null |
sq_add_sq_ne_zero {R : Type*} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R]
{a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff_of_nonneg (sq_nonneg _) (sq_nonneg _)).mp h'
obtain rfl := pow_eq_zero ha
obtain rfl := pow_eq_zero hb
exact not_isCoprime_zero_zero h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | sq_add_sq_ne_zero | null |
@[simp]
Nat.isCoprime_iff {m n : ℕ} : IsCoprime m n ↔ m = 1 ∨ n = 1 := by
refine ⟨fun ⟨a, b, H⟩ => ?_, fun h => ?_⟩
· simp_rw [Nat.add_eq_one_iff, mul_eq_one, mul_eq_zero] at H
exact H.symm.imp (·.1.2) (·.2.2)
· obtain rfl | rfl := h
· exact isCoprime_one_left
· exact isCoprime_one_right | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | Nat.isCoprime_iff | `IsCoprime` is not a useful definition for `Nat`; consider using `Nat.Coprime` instead. |
PNat.isCoprime_iff {m n : ℕ+} : IsCoprime (m : ℕ) n ↔ m = 1 ∨ n = 1 := by simp | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | PNat.isCoprime_iff | `IsCoprime` is not a useful definition for `PNat`; consider using `Nat.Coprime` instead. |
@[simp]
Semifield.isCoprime_iff {R : Type*} [Semifield R] {m n : R} :
IsCoprime m n ↔ m ≠ 0 ∨ n ≠ 0 := by
obtain rfl | hn := eq_or_ne n 0
· simp [isCoprime_zero_right]
suffices IsCoprime m n by simpa [hn]
refine ⟨0, n⁻¹, ?_⟩
simp [inv_mul_cancel₀ hn] | lemma | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | Semifield.isCoprime_iff | `IsCoprime` is not a useful definition if an inverse is available. |
add_mul_left_left (h : IsRelPrime x y) (z : R) : IsRelPrime (x + y * z) y :=
@of_add_mul_left_left R _ _ _ (-z) <| by simpa only [mul_neg, add_neg_cancel_right] using h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_left | null |
add_mul_right_left (h : IsRelPrime x y) (z : R) : IsRelPrime (x + z * y) y :=
mul_comm z y ▸ h.add_mul_left_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_left | null |
add_mul_left_right (h : IsRelPrime x y) (z : R) : IsRelPrime x (y + x * z) :=
(h.symm.add_mul_left_left z).symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_right | null |
add_mul_right_right (h : IsRelPrime x y) (z : R) : IsRelPrime x (y + z * x) :=
(h.symm.add_mul_right_left z).symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_right | null |
mul_add_left_left (h : IsRelPrime x y) (z : R) : IsRelPrime (y * z + x) y :=
add_comm x _ ▸ h.add_mul_left_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_left | null |
mul_add_right_left (h : IsRelPrime x y) (z : R) : IsRelPrime (z * y + x) y :=
add_comm x _ ▸ h.add_mul_right_left z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_left | null |
mul_add_left_right (h : IsRelPrime x y) (z : R) : IsRelPrime x (x * z + y) :=
add_comm y _ ▸ h.add_mul_left_right z | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_right | null |
mul_add_right_right (h : IsRelPrime x y) (z : R) : IsRelPrime x (z * x + y) :=
add_comm y _ ▸ h.add_mul_right_right z
variable {z} | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_right | null |
add_mul_left_left_iff : IsRelPrime (x + y * z) y ↔ IsRelPrime x y :=
⟨of_add_mul_left_left, fun h ↦ h.add_mul_left_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_left_iff | null |
add_mul_right_left_iff : IsRelPrime (x + z * y) y ↔ IsRelPrime x y :=
⟨of_add_mul_right_left, fun h ↦ h.add_mul_right_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_left_iff | null |
add_mul_left_right_iff : IsRelPrime x (y + x * z) ↔ IsRelPrime x y :=
⟨of_add_mul_left_right, fun h ↦ h.add_mul_left_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_left_right_iff | null |
add_mul_right_right_iff : IsRelPrime x (y + z * x) ↔ IsRelPrime x y :=
⟨of_add_mul_right_right, fun h ↦ h.add_mul_right_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | add_mul_right_right_iff | null |
mul_add_left_left_iff {x y z : R} : IsRelPrime (y * z + x) y ↔ IsRelPrime x y :=
⟨of_mul_add_left_left, fun h ↦ h.mul_add_left_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_left_iff | null |
mul_add_right_left_iff {x y z : R} : IsRelPrime (z * y + x) y ↔ IsRelPrime x y :=
⟨of_mul_add_right_left, fun h ↦ h.mul_add_right_left z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_left_iff | null |
mul_add_left_right_iff {x y z : R} : IsRelPrime x (x * z + y) ↔ IsRelPrime x y :=
⟨of_mul_add_left_right, fun h ↦ h.mul_add_left_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_left_right_iff | null |
mul_add_right_right_iff {x y z : R} : IsRelPrime x (z * x + y) ↔ IsRelPrime x y :=
⟨of_mul_add_right_right, fun h ↦ h.mul_add_right_right z⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | mul_add_right_right_iff | null |
neg_left (h : IsRelPrime x y) : IsRelPrime (-x) y := fun _ ↦ (h <| dvd_neg.mp ·) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_left | null |
neg_right (h : IsRelPrime x y) : IsRelPrime x (-y) := h.symm.neg_left.symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_right | null |
protected neg_neg (h : IsRelPrime x y) : IsRelPrime (-x) (-y) := h.neg_left.neg_right | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_neg | null |
neg_left_iff (x y : R) : IsRelPrime (-x) y ↔ IsRelPrime x y :=
⟨fun h ↦ neg_neg x ▸ h.neg_left, neg_left⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_left_iff | null |
neg_right_iff (x y : R) : IsRelPrime x (-y) ↔ IsRelPrime x y :=
⟨fun h ↦ neg_neg y ▸ h.neg_right, neg_right⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_right_iff | null |
neg_neg_iff (x y : R) : IsRelPrime (-x) (-y) ↔ IsRelPrime x y :=
(neg_left_iff _ _).trans (neg_right_iff _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Action.Units",
"Mathlib.Algebra.Group.Nat.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Logic.Basic",
"Mathlib.Tactic.Ring"
] | Mathlib/RingTheory/Coprime/Basic.lean | neg_neg_iff | null |
iSup_iInf_eq_top_iff_pairwise {t : Finset ι} (h : t.Nonempty) (I : ι → Ideal R) :
(⨆ i ∈ t, ⨅ (j) (_ : j ∈ t) (_ : j ≠ i), I j) = ⊤ ↔
(t : Set ι).Pairwise fun i j => I i ⊔ I j = ⊤ := by
haveI : DecidableEq ι := Classical.decEq ι
rw [eq_top_iff_one, Submodule.mem_iSup_finset_iff_exists_sum]
refine h.cons_induction ?_ ?_ <;> clear t h
· simp only [Finset.sum_singleton, Finset.coe_singleton, Set.pairwise_singleton, iff_true]
refine fun a => ⟨fun i => if h : i = a then ⟨1, ?_⟩ else 0, ?_⟩
· simp [h]
· simp only [dif_pos, Submodule.coe_mk]
intro a t hat h ih
rw [Finset.coe_cons,
Set.pairwise_insert_of_symmetric fun i j (h : I i ⊔ I j = ⊤) ↦ (sup_comm _ _).trans h]
constructor
· rintro ⟨μ, hμ⟩
rw [Finset.sum_cons] at hμ
refine ⟨ih.mp ⟨Pi.single h.choose ⟨μ a, ?a1⟩ + fun i => ⟨μ i, ?a2⟩, ?a3⟩, fun b hb ab => ?a4⟩
case a1 =>
have := Submodule.coe_mem (μ a)
rw [mem_iInf] at this ⊢
intro i
specialize this i
rw [mem_iInf, mem_iInf] at this ⊢
intro hi _
apply this (Finset.subset_cons _ hi)
rintro rfl
exact hat hi
case a2 =>
have := Submodule.coe_mem (μ i)
simp only [mem_iInf] at this ⊢
intro j hj ij
exact this _ (Finset.subset_cons _ hj) ij
case a3 =>
rw [← @if_pos _ _ h.choose_spec R (μ a) 0, ← Finset.sum_pi_single', ← Finset.sum_add_distrib]
at hμ
convert hμ
rename_i i _
rw [Pi.add_apply, Submodule.coe_add, Submodule.coe_mk]
by_cases hi : i = h.choose
· rw [hi, Pi.single_eq_same, Pi.single_eq_same, Submodule.coe_mk]
· rw [Pi.single_eq_of_ne hi, Pi.single_eq_of_ne hi, Submodule.coe_zero]
case a4 =>
rw [eq_top_iff_one, Submodule.mem_sup]
rw [add_comm] at hμ
refine ⟨_, ?_, _, ?_, hμ⟩
· refine sum_mem _ fun x hx => ?_
have := Submodule.coe_mem (μ x)
simp only [mem_iInf] at this
apply this _ (Finset.mem_cons_self _ _)
rintro rfl
exact hat hx
... | theorem | RingTheory | [
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.RingTheory.Ideal.BigOperators",
"Mathlib.RingTheory.Ideal.Operations"
] | Mathlib/RingTheory/Coprime/Ideal.lean | iSup_iInf_eq_top_iff_pairwise | A finite family of ideals is pairwise coprime (that is, any two of them generate the whole ring)
iff when taking all the possible intersections of all but one of these ideals, the resulting family
of ideals still generate the whole ring.
For example with three ideals : `I ⊔ J = I ⊔ K = J ⊔ K = ⊤ ↔ (I ⊓ J) ⊔ (I ⊓ K) ⊔ (J ⊓ K) = ⊤`.
When ideals are all of the form `I i = R ∙ s i`, this is equivalent to the
`exists_sum_eq_one_iff_pairwise_coprime` lemma. |
Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
refine Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, ?_⟩)
rwa [mul_comm m, mul_comm n, eq_comm]
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Int.isCoprime_iff_gcd_eq_one | null |
@[simp, norm_cast]
Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
alias ⟨IsCoprime.natCoprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
@[deprecated (since := "2025-08-25")]
alias IsCoprime.nat_coprime := IsCoprime.natCoprime | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Nat.isCoprime_iff_coprime | null |
Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) :=
mod_cast h.isCoprime.intCast | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Nat.Coprime.cast | null |
ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | ne_zero_or_ne_zero_of_nat_coprime | null |
IsCoprime.prod_left (h : ∀ i ∈ t, IsCoprime (s i) x) : IsCoprime (∏ i ∈ t, s i) x := by
induction t using Finset.cons_induction with
| empty => apply isCoprime_one_left
| cons b t hbt ih =>
rw [Finset.prod_cons]
rw [Finset.forall_mem_cons] at h
exact h.1.mul_left (ih h.2) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.prod_left | null |
IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.prod_right | null |
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