state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ deleteVerts G' (s ∩ G'.verts) = deleteVerts G' s | /-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s :=
by | Mathlib.Combinatorics.SimpleGraph.Subgraph.1294_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
x✝ : V
⊢ x✝ ∈ (deleteVerts G' (s ∩ G'.verts)).verts ↔ x✝ ∈ (deleteVerts G' s).verts | /-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s :=
by ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1294_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
x✝¹ x✝ : V
⊢ Adj (deleteVerts G' (s ∩ G'.verts)) x✝¹ x✝ ↔ Adj (deleteVerts G' s) x✝¹ x✝ | /-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s :=
by ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1294_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_set_right_eq : G'.deleteVerts (s ∩ G'.verts) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
R : Type u
inst✝ : CommSemiring R
x y z : R
H✝ : IsCoprime x y
a b : R
H : a * x + b * y = 1
⊢ b * y + a * x = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm, H] | @[symm]
theorem IsCoprime.symm (H : IsCoprime x y) : IsCoprime y x :=
let ⟨a, b, H⟩ := H
⟨b, a, by | Mathlib.RingTheory.Coprime.Basic.44_0.Ci6BN5Afffbdcdr | @[symm]
theorem IsCoprime.symm (H : IsCoprime x y) : IsCoprime y x | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
x✝ : IsCoprime x x
a b : R
h : a * x + b * x = 1
⊢ x * (a + b) = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [mul_comm, add_mul] | theorem isCoprime_self : IsCoprime x x ↔ IsUnit x :=
⟨fun ⟨a, b, h⟩ => isUnit_of_mul_eq_one x (a + b) <| by | Mathlib.RingTheory.Coprime.Basic.54_0.Ci6BN5Afffbdcdr | theorem isCoprime_self : IsCoprime x x ↔ IsUnit x | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsUnit x
b : R
hb : b * x = 1
⊢ b * x + 0 * x = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [zero_mul, add_zero] | theorem isCoprime_self : IsCoprime x x ↔ IsUnit x :=
⟨fun ⟨a, b, h⟩ => isUnit_of_mul_eq_one x (a + b) <| by rwa [mul_comm, add_mul], fun h =>
let ⟨b, hb⟩ := isUnit_iff_exists_inv'.1 h
⟨b, 0, by | Mathlib.RingTheory.Coprime.Basic.54_0.Ci6BN5Afffbdcdr | theorem isCoprime_self : IsCoprime x x ↔ IsUnit x | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
x✝ : IsCoprime 0 x
a b : R
H : a * 0 + b * x = 1
⊢ x * b = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [mul_zero, zero_add, mul_comm] at H | theorem isCoprime_zero_left : IsCoprime 0 x ↔ IsUnit x :=
⟨fun ⟨a, b, H⟩ => isUnit_of_mul_eq_one x b <| by | Mathlib.RingTheory.Coprime.Basic.60_0.Ci6BN5Afffbdcdr | theorem isCoprime_zero_left : IsCoprime 0 x ↔ IsUnit x | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsUnit x
b : R
hb : b * x = 1
⊢ 1 * 0 + b * x = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [one_mul, zero_add] | theorem isCoprime_zero_left : IsCoprime 0 x ↔ IsUnit x :=
⟨fun ⟨a, b, H⟩ => isUnit_of_mul_eq_one x b <| by rwa [mul_zero, zero_add, mul_comm] at H, fun H =>
let ⟨b, hb⟩ := isUnit_iff_exists_inv'.1 H
⟨1, b, by | Mathlib.RingTheory.Coprime.Basic.60_0.Ci6BN5Afffbdcdr | theorem isCoprime_zero_left : IsCoprime 0 x ↔ IsUnit x | Mathlib_RingTheory_Coprime_Basic |
R✝ : Type u
inst✝¹ : CommSemiring R✝
x y z : R✝
R : Type u_1
inst✝ : CommRing R
a b : ℤ
h : IsCoprime a b
⊢ IsCoprime ↑a ↑b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rcases h with ⟨u, v, H⟩ | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) := by
| Mathlib.RingTheory.Coprime.Basic.74_0.Ci6BN5Afffbdcdr | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) | Mathlib_RingTheory_Coprime_Basic |
case intro.intro
R✝ : Type u
inst✝¹ : CommSemiring R✝
x y z : R✝
R : Type u_1
inst✝ : CommRing R
a b u v : ℤ
H : u * a + v * b = 1
⊢ IsCoprime ↑a ↑b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | use u, v | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) := by
rcases h with ⟨u, v, H⟩
| Mathlib.RingTheory.Coprime.Basic.74_0.Ci6BN5Afffbdcdr | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) | Mathlib_RingTheory_Coprime_Basic |
case h
R✝ : Type u
inst✝¹ : CommSemiring R✝
x y z : R✝
R : Type u_1
inst✝ : CommRing R
a b u v : ℤ
H : u * a + v * b = 1
⊢ ↑u * ↑a + ↑v * ↑b = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw_mod_cast [H] | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) := by
rcases h with ⟨u, v, H⟩
use u, v
| Mathlib.RingTheory.Coprime.Basic.74_0.Ci6BN5Afffbdcdr | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) | Mathlib_RingTheory_Coprime_Basic |
case h
R✝ : Type u
inst✝¹ : CommSemiring R✝
x y z : R✝
R : Type u_1
inst✝ : CommRing R
a b u v : ℤ
H : u * a + v * b = 1
⊢ ↑1 = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact Int.cast_one | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) := by
rcases h with ⟨u, v, H⟩
use u, v
rw_mod_cast [H]
| Mathlib.RingTheory.Coprime.Basic.74_0.Ci6BN5Afffbdcdr | lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
IsCoprime (a : R) (b : R) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
p : Fin 2 → R
h : IsCoprime (p 0) (p 1)
⊢ p ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rintro rfl | /-- If a 2-vector `p` satisfies `IsCoprime (p 0) (p 1)`, then `p ≠ 0`. -/
theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 := by
| Mathlib.RingTheory.Coprime.Basic.81_0.Ci6BN5Afffbdcdr | /-- If a 2-vector `p` satisfies `IsCoprime (p 0) (p 1)`, then `p ≠ 0`. -/
theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
h : IsCoprime (OfNat.ofNat 0 0) (OfNat.ofNat 0 1)
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact not_isCoprime_zero_zero h | /-- If a 2-vector `p` satisfies `IsCoprime (p 0) (p 1)`, then `p ≠ 0`. -/
theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 := by
rintro rfl
| Mathlib.RingTheory.Coprime.Basic.81_0.Ci6BN5Afffbdcdr | /-- If a 2-vector `p` satisfies `IsCoprime (p 0) (p 1)`, then `p ≠ 0`. -/
theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
h : IsCoprime x y
⊢ x ≠ 0 ∨ y ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | apply not_or_of_imp | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 := by
| Mathlib.RingTheory.Coprime.Basic.87_0.Ci6BN5Afffbdcdr | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
case a
R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
h : IsCoprime x y
⊢ x = 0 → y ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rintro rfl rfl | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 := by
apply not_or_of_imp
| Mathlib.RingTheory.Coprime.Basic.87_0.Ci6BN5Afffbdcdr | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
case a
R : Type u
inst✝¹ : CommSemiring R
z : R
inst✝ : Nontrivial R
h : IsCoprime 0 0
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact not_isCoprime_zero_zero h | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 := by
apply not_or_of_imp
rintro rfl rfl
| Mathlib.RingTheory.Coprime.Basic.87_0.Ci6BN5Afffbdcdr | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
⊢ 1 * 1 + 0 * x = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [one_mul, zero_mul, add_zero] | theorem isCoprime_one_left : IsCoprime 1 x :=
⟨1, 0, by | Mathlib.RingTheory.Coprime.Basic.92_0.Ci6BN5Afffbdcdr | theorem isCoprime_one_left : IsCoprime 1 x | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
⊢ 0 * x + 1 * 1 = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [one_mul, zero_mul, zero_add] | theorem isCoprime_one_right : IsCoprime x 1 :=
⟨0, 1, by | Mathlib.RingTheory.Coprime.Basic.96_0.Ci6BN5Afffbdcdr | theorem isCoprime_one_right : IsCoprime x 1 | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x z
H2 : x ∣ y * z
⊢ x ∣ y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | let ⟨a, b, H⟩ := H1 | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by
| Mathlib.RingTheory.Coprime.Basic.100_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x z
H2 : x ∣ y * z
a b : R
H : a * x + b * z = 1
⊢ x ∣ y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm] | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by
let ⟨a, b, H⟩ := H1
| Mathlib.RingTheory.Coprime.Basic.100_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x z
H2 : x ∣ y * z
a b : R
H : a * x + b * z = 1
⊢ x ∣ y * a * x + b * (y * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact dvd_add (dvd_mul_left _ _) (H2.mul_left _) | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by
let ⟨a, b, H⟩ := H1
rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm]
| Mathlib.RingTheory.Coprime.Basic.100_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x y
H2 : x ∣ y * z
⊢ x ∣ z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | let ⟨a, b, H⟩ := H1 | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by
| Mathlib.RingTheory.Coprime.Basic.106_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x y
H2 : x ∣ y * z
a b : R
H : a * x + b * y = 1
⊢ x ∣ z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b] | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by
let ⟨a, b, H⟩ := H1
| Mathlib.RingTheory.Coprime.Basic.106_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x y
H2 : x ∣ y * z
a b : R
H : a * x + b * y = 1
⊢ x ∣ a * z * x + b * (y * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact dvd_add (dvd_mul_left _ _) (H2.mul_left _) | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by
let ⟨a, b, H⟩ := H1
rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b]
| Mathlib.RingTheory.Coprime.Basic.106_0.Ci6BN5Afffbdcdr | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x z
H2 : IsCoprime y z
a b : R
h1 : a * x + b * z = 1
c d : R
h2 : c * y + d * z = 1
⊢ a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z = (a * x + b * z) * (c * y + d * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | ring | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z :=
let ⟨a, b, h1⟩ := H1
let ⟨c, d, h2⟩ := H2
⟨a * c, a * x * d + b * c * y + b * d * z,
calc
a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z =
(a * x + b * z) * (c * y + d * z) :=
by | Mathlib.RingTheory.Coprime.Basic.112_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x z
H2 : IsCoprime y z
a b : R
h1 : a * x + b * z = 1
c d : R
h2 : c * y + d * z = 1
⊢ (a * x + b * z) * (c * y + d * z) = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [h1, h2, mul_one] | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z :=
let ⟨a, b, h1⟩ := H1
let ⟨c, d, h2⟩ := H2
⟨a * c, a * x * d + b * c * y + b * d * z,
calc
a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z =
(a * x + b * z) * (c * y + d * z) :=
by ri... | Mathlib.RingTheory.Coprime.Basic.112_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x y
H2 : IsCoprime x z
⊢ IsCoprime x (y * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm] at H1 H2 ⊢ | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) := by
| Mathlib.RingTheory.Coprime.Basic.124_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime y x
H2 : IsCoprime z x
⊢ IsCoprime (y * z) x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact H1.mul_left H2 | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) := by
rw [isCoprime_comm] at H1 H2 ⊢
| Mathlib.RingTheory.Coprime.Basic.124_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime x y
H1 : x ∣ z
H2 : y ∣ z
⊢ x * y ∣ z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain ⟨a, b, h⟩ := H | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
| Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [← mul_one z, ← h, mul_add] | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
| Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (a * x) + z * (b * y) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | apply dvd_add | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
| Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro.h₁
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (a * x) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm z, mul_assoc] | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· | Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro.h₁
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ a * (x * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact (mul_dvd_mul_left _ H2).mul_left _ | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· rw [mul_comm z, mul_assoc]
| Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro.h₂
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (b * y) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm b, ← mul_assoc] | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· rw [mul_comm z, mul_assoc]
exact (mul_dvd_mul_left _ H2).mul_left _
· | Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
case intro.intro.h₂
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * y * b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact (mul_dvd_mul_right H1 _).mul_right _ | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· rw [mul_comm z, mul_assoc]
exact (mul_dvd_mul_left _ H2).mul_left _
· rw [mul_comm b, ← mul_assoc]
| Mathlib.RingTheory.Coprime.Basic.129_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime (x * y) z
a b : R
h : a * (x * y) + b * z = 1
⊢ a * y * x + b * z = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [mul_right_comm, mul_assoc] | theorem IsCoprime.of_mul_left_left (H : IsCoprime (x * y) z) : IsCoprime x z :=
let ⟨a, b, h⟩ := H
⟨a * y, b, by | Mathlib.RingTheory.Coprime.Basic.139_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_left_left (H : IsCoprime (x * y) z) : IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime (x * y) z
⊢ IsCoprime y z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm] at H | theorem IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z := by
| Mathlib.RingTheory.Coprime.Basic.144_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime (y * x) z
⊢ IsCoprime y z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact H.of_mul_left_left | theorem IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z := by
rw [mul_comm] at H
| Mathlib.RingTheory.Coprime.Basic.144_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime x (y * z)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm] at H ⊢ | theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.149_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime (y * z) x
⊢ IsCoprime y x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact H.of_mul_left_left | theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y := by
rw [isCoprime_comm] at H ⊢
| Mathlib.RingTheory.Coprime.Basic.149_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime x (y * z)
⊢ IsCoprime x z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm] at H | theorem IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z := by
| Mathlib.RingTheory.Coprime.Basic.154_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
H : IsCoprime x (z * y)
⊢ IsCoprime x z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact H.of_mul_right_left | theorem IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z := by
rw [mul_comm] at H
| Mathlib.RingTheory.Coprime.Basic.154_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
⊢ IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm, IsCoprime.mul_left_iff, isCoprime_comm, @isCoprime_comm _ _ z] | theorem IsCoprime.mul_right_iff : IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z := by
| Mathlib.RingTheory.Coprime.Basic.163_0.Ci6BN5Afffbdcdr | theorem IsCoprime.mul_right_iff : IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime y z
hdvd : x ∣ y
⊢ IsCoprime x z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain ⟨d, rfl⟩ := hdvd | theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z := by
| Mathlib.RingTheory.Coprime.Basic.167_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
case intro
R : Type u
inst✝ : CommSemiring R
x z d : R
h : IsCoprime (x * d) z
⊢ IsCoprime x z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact IsCoprime.of_mul_left_left h | theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z := by
obtain ⟨d, rfl⟩ := hdvd
| Mathlib.RingTheory.Coprime.Basic.167_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝¹ : CommSemiring R
x y z : R
H : IsCoprime x y
S : Type v
inst✝ : CommSemiring S
f : R →+* S
a b : R
h : a * x + b * y = 1
⊢ f a * f x + f b * f y = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one] | theorem 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 | Mathlib.RingTheory.Coprime.Basic.186_0.Ci6BN5Afffbdcdr | theorem IsCoprime.map (H : IsCoprime x y) {S : Type v} [CommSemiring S] (f : R →+* S) :
IsCoprime (f x) (f y) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (x + y * z) y
a b : R
H : a * (x + y * z) + b * y = 1
⊢ a * x + (a * z + b) * y = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | simpa only [add_mul, mul_add, add_assoc, add_comm, add_left_comm, mul_assoc, mul_comm,
mul_left_comm] using H | theorem 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
| Mathlib.RingTheory.Coprime.Basic.192_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_left_left (h : IsCoprime (x + y * z) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (x + z * y) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm] at h | theorem IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.199_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (x + y * z) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_left_left | theorem IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y := by
rw [mul_comm] at h
| Mathlib.RingTheory.Coprime.Basic.199_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (y + x * z)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm] at h ⊢ | theorem IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.204_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (y + x * z) x
⊢ IsCoprime y x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_left_left | theorem IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y := by
rw [isCoprime_comm] at h ⊢
| Mathlib.RingTheory.Coprime.Basic.204_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (y + z * x)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm] at h | theorem IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.209_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (y + x * z)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_left_right | theorem IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y := by
rw [mul_comm] at h
| Mathlib.RingTheory.Coprime.Basic.209_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (y * z + x) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] at h | theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.214_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (x + y * z) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_left_left | theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y := by
rw [add_comm] at h
| Mathlib.RingTheory.Coprime.Basic.214_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (z * y + x) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] at h | theorem IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.219_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime (x + z * y) y
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_right_left | theorem IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y := by
rw [add_comm] at h
| Mathlib.RingTheory.Coprime.Basic.219_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (x * z + y)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] at h | theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.224_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (y + x * z)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_left_right | theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y := by
rw [add_comm] at h
| Mathlib.RingTheory.Coprime.Basic.224_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (z * x + y)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] at h | theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y := by
| Mathlib.RingTheory.Coprime.Basic.229_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime x (y + z * x)
⊢ IsCoprime x y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.of_add_mul_right_right | theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y := by
rw [add_comm] at h
| Mathlib.RingTheory.Coprime.Basic.229_0.Ci6BN5Afffbdcdr | theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Group G
inst✝² : MulAction G R
inst✝¹ : SMulCommClass G R R
inst✝ : IsScalarTower G R R
x : G
y z : R
x✝ : IsCoprime (x • y) z
a b : R
h : a * x • y + b * z = 1
⊢ x • a * y + b * z = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [smul_mul_assoc, ← mul_smul_comm] | theorem isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z :=
⟨fun ⟨a, b, h⟩ => ⟨x • a, b, by | Mathlib.RingTheory.Coprime.Basic.241_0.Ci6BN5Afffbdcdr | theorem isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Group G
inst✝² : MulAction G R
inst✝¹ : SMulCommClass G R R
inst✝ : IsScalarTower G R R
x : G
y z : R
x✝ : IsCoprime y z
a b : R
h : a * y + b * z = 1
⊢ x⁻¹ • a * x • y + b * z = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [smul_mul_smul, inv_mul_self, one_smul] | theorem 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 | Mathlib.RingTheory.Coprime.Basic.241_0.Ci6BN5Afffbdcdr | theorem isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (x + y * z + y * -z) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | simpa only [mul_neg, add_neg_cancel_right] using h | theorem 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 | Mathlib.RingTheory.Coprime.Basic.294_0.Ci6BN5Afffbdcdr | theorem add_mul_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + y * z) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (x + z * y) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [mul_comm] | theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y := by
| Mathlib.RingTheory.Coprime.Basic.298_0.Ci6BN5Afffbdcdr | theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (x + y * z) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.add_mul_left_left z | theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y := by
rw [mul_comm]
| Mathlib.RingTheory.Coprime.Basic.298_0.Ci6BN5Afffbdcdr | theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + x * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm] | theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) := by
| Mathlib.RingTheory.Coprime.Basic.303_0.Ci6BN5Afffbdcdr | theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (y + x * z) x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.symm.add_mul_left_left z | theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) := by
rw [isCoprime_comm]
| Mathlib.RingTheory.Coprime.Basic.303_0.Ci6BN5Afffbdcdr | theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + z * x) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [isCoprime_comm] | theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) := by
| Mathlib.RingTheory.Coprime.Basic.308_0.Ci6BN5Afffbdcdr | theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (y + z * x) x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.symm.add_mul_right_left z | theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) := by
rw [isCoprime_comm]
| Mathlib.RingTheory.Coprime.Basic.308_0.Ci6BN5Afffbdcdr | theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (y * z + x) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] | theorem mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y := by
| Mathlib.RingTheory.Coprime.Basic.313_0.Ci6BN5Afffbdcdr | theorem mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (x + y * z) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.add_mul_left_left z | theorem mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y := by
rw [add_comm]
| Mathlib.RingTheory.Coprime.Basic.313_0.Ci6BN5Afffbdcdr | theorem mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (z * y + x) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] | theorem mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y := by
| Mathlib.RingTheory.Coprime.Basic.318_0.Ci6BN5Afffbdcdr | theorem mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (x + z * y) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.add_mul_right_left z | theorem mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y := by
rw [add_comm]
| Mathlib.RingTheory.Coprime.Basic.318_0.Ci6BN5Afffbdcdr | theorem mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (x * z + y) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] | theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) := by
| Mathlib.RingTheory.Coprime.Basic.323_0.Ci6BN5Afffbdcdr | theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + x * z) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.add_mul_left_right z | theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) := by
rw [add_comm]
| Mathlib.RingTheory.Coprime.Basic.323_0.Ci6BN5Afffbdcdr | theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (z * x + y) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [add_comm] | theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) := by
| Mathlib.RingTheory.Coprime.Basic.328_0.Ci6BN5Afffbdcdr | theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + z * x) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact h.add_mul_right_right z | theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) := by
rw [add_comm]
| Mathlib.RingTheory.Coprime.Basic.328_0.Ci6BN5Afffbdcdr | theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) | Mathlib_RingTheory_Coprime_Basic |
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
⊢ IsCoprime (-x) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain ⟨a, b, h⟩ := h | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y := by
| Mathlib.RingTheory.Coprime.Basic.365_0.Ci6BN5Afffbdcdr | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y | Mathlib_RingTheory_Coprime_Basic |
case intro.intro
R : Type u
inst✝ : CommRing R
x y a b : R
h : a * x + b * y = 1
⊢ IsCoprime (-x) y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | use -a, b | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y := by
obtain ⟨a, b, h⟩ := h
| Mathlib.RingTheory.Coprime.Basic.365_0.Ci6BN5Afffbdcdr | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y | Mathlib_RingTheory_Coprime_Basic |
case h
R : Type u
inst✝ : CommRing R
x y a b : R
h : a * x + b * y = 1
⊢ -a * -x + b * y = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rwa [neg_mul_neg] | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y := by
obtain ⟨a, b, h⟩ := h
use -a, b
| Mathlib.RingTheory.Coprime.Basic.365_0.Ci6BN5Afffbdcdr | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
⊢ a ^ 2 + b ^ 2 ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | intro h' | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
| Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by rw [pow_two]; exact mul_self_nonneg _)).mp h' | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
| Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
⊢ 0 ≤ a ^ 2 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [pow_two] | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
⊢ 0 ≤ a * a | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact mul_self_nonneg _ | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
⊢ 0 ≤ b ^ 2 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | rw [pow_two] | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
⊢ 0 ≤ b * b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact mul_self_nonneg _ | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by rw [pow_two]; | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
case intro
R : Type u_1
inst✝ : LinearOrderedCommRing R
a b : R
h : IsCoprime a b
h' : a ^ 2 + b ^ 2 = 0
ha : a ^ 2 = 0
hb : b ^ 2 = 0
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain rfl := pow_eq_zero ha | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by rw [pow_two]; exact mul_s... | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
case intro
R : Type u_1
inst✝ : LinearOrderedCommRing R
b : R
hb : b ^ 2 = 0
h : IsCoprime 0 b
h' : 0 ^ 2 + b ^ 2 = 0
ha : 0 ^ 2 = 0
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | obtain rfl := pow_eq_zero hb | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by rw [pow_two]; exact mul_s... | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
case intro
R : Type u_1
inst✝ : LinearOrderedCommRing R
ha hb : 0 ^ 2 = 0
h : IsCoprime 0 0
h' : 0 ^ 2 + 0 ^ 2 = 0
⊢ False | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupPower.Ring
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.Gro... | exact not_isCoprime_zero_zero h | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 := by
intro h'
obtain ⟨ha, hb⟩ := (add_eq_zero_iff'
--Porting TODO: replace with sq_nonneg when that file is ported
(by rw [pow_two]; exact mul_self_nonneg _)
(by rw [pow_two]; exact mul_s... | Mathlib.RingTheory.Coprime.Basic.393_0.Ci6BN5Afffbdcdr | theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | Mathlib_RingTheory_Coprime_Basic |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g✝ : β → γ
f g : α → β
s : Set β
h : ∀ (x : α), f x = g x
⊢ f ⁻¹' s = g ⁻¹' s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | congr with x | theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
| Mathlib.Data.Set.Image.68_0.IJFiTzmYGOCpPSd | theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g✝ : β → γ
f g : α → β
s : Set β
h : ∀ (x : α), f x = g x
x : α
⊢ x ∈ f ⁻¹' s ↔ x ∈ g ⁻¹' s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [h] | theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
| Mathlib.Data.Set.Image.68_0.IJFiTzmYGOCpPSd | theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : α → β
g : β → γ
b : β
s : Set β
inst✝ : Decidable (b ∈ s)
⊢ (fun x => b) ⁻¹' s = if b ∈ s then univ else ∅ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | split_ifs with hb | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
| Mathlib.Data.Set.Image.147_0.IJFiTzmYGOCpPSd | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ | Mathlib_Data_Set_Image |
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : α → β
g : β → γ
b : β
s : Set β
inst✝ : Decidable (b ∈ s)
hb : b ∈ s
⊢ (fun x => b) ⁻¹' s = univ
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : α → β
g : β → γ
b : β
s : Set β
inst✝ : Decidable (b ∈ s)
hb : b ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb] | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
| Mathlib.Data.Set.Image.147_0.IJFiTzmYGOCpPSd | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g : β → γ
inst✝ : Nonempty β
f : α → β
hf : ∀ (b : β), f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ
⊢ ∃ b, f = const α b | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf' | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
| Mathlib.Data.Set.Image.153_0.IJFiTzmYGOCpPSd | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b | Mathlib_Data_Set_Image |
case inl.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g : β → γ
inst✝ : Nonempty β
f : α → β
hf : ∀ (b : β), f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ
b : β
hb : f ⁻¹' {b} = univ
⊢ ∃ b, f = const α b | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩ | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
... | Mathlib.Data.Set.Image.153_0.IJFiTzmYGOCpPSd | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b | Mathlib_Data_Set_Image |
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g : β → γ
inst✝ : Nonempty β
f : α → β
hf : ∀ (b : β), f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ
hf' : ¬∃ b, f ⁻¹' {b} = univ
⊢ ∃ b, f = const α b | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
... | Mathlib.Data.Set.Image.153_0.IJFiTzmYGOCpPSd | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b | Mathlib_Data_Set_Image |
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g : β → γ
inst✝ : Nonempty β
f : α → β
hf : ∀ (b : β), f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ
hf' : ¬∃ b, f ⁻¹' {b} = univ
this : ∀ (x : α) (b : β), f x ≠ b
⊢ ∃ b, f = const α b | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩ | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
... | Mathlib.Data.Set.Image.153_0.IJFiTzmYGOCpPSd | /-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
g : β → γ
f : α → α
n : ℕ
⊢ preimage f^[n] = (preimage f)^[n] | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | induction' n with n ih | theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
| Mathlib.Data.Set.Image.171_0.IJFiTzmYGOCpPSd | theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] | Mathlib_Data_Set_Image |
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