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 |
|---|---|---|---|---|---|---|
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
this : DecidableEq L := Classical.decEq L
⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = adjoin K {b | ∃... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n) | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
this : DecidableEq L := Classical.decEq L
ζ : L
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin K (rootS... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Field (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicField | instance : Field (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.528_0.xReI1DeVvechFQU | instance : Field (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Field (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance : Field (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.528_0.xReI1DeVvechFQU | instance : Field (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Algebra K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicField | instance algebra : Algebra K (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.532_0.xReI1DeVvechFQU | instance algebra : Algebra K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Algebra K (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.532_0.xReI1DeVvechFQU | instance algebra : Algebra K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Inhabited (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicField | instance : Inhabited (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.536_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Inhabited (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.536_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | letI := Classical.decEq (CyclotomicField n K) | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
⊢ IsCyclotomicExtens... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne' | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : Cyclo... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : Cyclo... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine ⟨?_, ?_⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, forall_eq] | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact ⟨ζ, hζ⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm] | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (al... | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ NoZeroSMulDivisors A (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' NoZeroSMulDivisors.of_algebraMap_injective _ | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Function.Injective ⇑(algebraMap A (CyclotomicField n K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)] | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Function.Injective ⇑(RingHom.comp (algebraMap K (CyclotomicField n K)) (algebraMap A K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _) | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ CommRing (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicRing | instance : CommRing (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.608_0.xReI1DeVvechFQU | instance : CommRing (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ CommRing ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.608_0.xReI1DeVvechFQU | instance : CommRing (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ IsDomain (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicRing | instance : IsDomain (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.612_0.xReI1DeVvechFQU | instance : IsDomain (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ IsDomain ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.612_0.xReI1DeVvechFQU | instance : IsDomain (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Inhabited (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | delta CyclotomicRing | instance : Inhabited (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.616_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Inhabited ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | infer_instance | instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.616_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
a : ℕ+
han : a ∈ {n}
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [mem_singleton_iff] at han | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
a : ℕ+
han : a = n
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | subst a | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := NeZero.of_noZeroSMulDivisors A K n | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n) | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨⟨μ, subset_adjoin _⟩, _⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : Cyclotomic... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : Cyclotomic... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : Cyclotomic... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rwa [← isRoot_cyclotomic_iff] at hμ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : Cyclotomic... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
⊢ x ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' subset_adjoin _ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
a : A
⊢ (algebraMap A ↥(adjoin A... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact Subalgebra.algebraMap_mem _ a | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_3
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y z : ↥(adjoin A {b | b ^ ↑n = 1... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact Subalgebra.add_mem _ hy hz | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_4
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y z : ↥(adjoin A {b | b ^ ↑n = 1... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact Subalgebra.mul_mem _ hy hz | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
... | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : C... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [isUnit_iff_ne_zero] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : C... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | apply map_ne_zero_of_mem_nonZeroDivisors | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case hg
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | apply adjoin_algebra_injective | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case h
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K)... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact hx | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
⊢ ∃ x_1,
x * (algebraMap... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K) | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rfl | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | have : IsLocalization (nonZeroDivisors A) K := inferInstance | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | replace := this.surj | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨⟨z, w⟩, hw⟩ := this k | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl) | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rintro y z ⟨a, ha⟩ ⟨b, hb⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [map_add, map_mul] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rintro y z ⟨a, ha⟩ ⟨b, hb⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
th... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [map_mul] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x y : CyclotomicRing n A K
h :
(algebraMap (Cyclot... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [adjoin_algebra_injective n A K h] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) ... | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
μ : CyclotomicField n K
h : IsPrimitiveRoot μ ↑n
⊢ CyclotomicRing n A K = ↥(adjoin A {μ}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h] | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.712_0.xReI1DeVvechFQU | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
μ : CyclotomicField n K
h : IsPrimitiveRoot μ ↑n
⊢ CyclotomicRing n A K = ↥(adjoin A {b | ∃ ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [CyclotomicRing] | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
... | Mathlib.NumberTheory.Cyclotomic.Basic.712_0.xReI1DeVvechFQU | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
⊢ IsCyclotomicExtension S K K | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩ | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne' | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
⊢ ∃ r, IsPrimitiveRo... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨r, _⟩ | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
... | Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
⊢ IsPrimitiveRoot r ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := h a ha | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
... | Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
this : NeZero ↑↑a
⊢ ... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rwa [coe_aeval_eq_eval, ← IsRoot.def, isRoot_cyclotomic_iff] at hr | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
... | Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ¬AffineIndependent 𝕜 f
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | rw [affineIndependent_iff] at h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ¬∀ (s : Finset ι) (w : ι → 𝕜), Finset.sum s w = 0 → ∑ e in s, w e • f e = 0 → ∀ e ∈ s, w e = 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | push_neg at h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ∃ s w, Finset.sum s w = 0 ∧ ∑ e in s, w e • f e = 0 ∧ ∃ e ∈ s, w e ≠ 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | let I : Finset ι := s.filter fun i ↦ 0 ≤ w i | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | let J : Finset ι := s.filter fun i ↦ w i < 0 | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | let p : E := centerMass I w f | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filte... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | simp only [mem_filter, h1', h2'.le, and_self, h2'] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | refine ⟨I, p, ?_, ?_⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi) | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | rw [← hp] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_) | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2.refine_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | linarith only [hI, hJI] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w... | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent se... | exact (mem_filter.mp hi').2.not_lt (mem_filter.mp hi).2 | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependen... | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ : ℝ
⊢ taylorWithin f (n + 1) s x₀ =
taylorWithin f n s x₀ +
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n +... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | dsimp only [taylorWithin] | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
| Mathlib.Analysis.Calculus.Taylor.75_0.INXnr4jrmq9RIjK | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ : ℝ
⊢ ∑ k in Finset.range (n + 1 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) =
∑ k in Finset.range ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [Finset.sum_range_succ] | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
| Mathlib.Analysis.Calculus.Taylor.75_0.INXnr4jrmq9RIjK | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ taylorWithinEval f (n + 1) s x₀ x =
taylorWithinEval f n s x₀ x + (((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval x) (taylorWithin f n s x₀) +
(PolynomialModule.eval (Polynomial.eval x (Polynomial.X - Polynomial.C x₀)))
((PolynomialModule.single ℝ (n + 1)) (taylorCo... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | congr | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, Polyno... | Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval (Polynomial.eval x (Polynomial.X - Polynomial.C x₀)))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀)) =
(((↑n + 1) * ↑n !)... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, Polyno... | Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (x - x₀) ^ (n + 1) • taylorCoeffWithin f (n + 1) s x₀ =
((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | dsimp only [taylorCoeffWithin] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, Polyno... | Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
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