Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 50 | 56 | theorem Algebra.map_leftMulMatrix_localization {ΞΉ : Type*} [Fintype ΞΉ] [DecidableEq ΞΉ]
(b : Basis ΞΉ R S) (a : S) :
(algebraMap R Rβ).mapMatrix (leftMulMatrix b a) =
leftMulMatrix (b.localizationLocalization Rβ M Sβ) (algebraMap S Sβ a) := by |
ext i j
simp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, β map_mul,
Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_algebraMap]
| 2,048 |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 61 | 69 | theorem Algebra.norm_localization [Module.Free R S] [Module.Finite R S] (a : S) :
Algebra.norm Rβ (algebraMap S Sβ a) = algebraMap R Rβ (Algebra.norm R a) := by |
cases subsingleton_or_nontrivial R
Β· haveI : Subsingleton Rβ := Module.subsingleton R Rβ
simp [eq_iff_true_of_subsingleton]
let b := Module.Free.chooseBasis R S
letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [Algebra.norm_eq_matrix_det (b.localizationLocalization Rβ M Sβ),
Algebra.norm... | 2,048 |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 83 | 92 | theorem Algebra.trace_localization [Module.Free R S] [Module.Finite R S] (a : S) :
Algebra.trace Rβ Sβ (algebraMap S Sβ a) = algebraMap R Rβ (Algebra.trace R S a) := by |
cases subsingleton_or_nontrivial R
Β· haveI : Subsingleton Rβ := Module.subsingleton R Rβ
simp [eq_iff_true_of_subsingleton]
let b := Module.Free.chooseBasis R S
letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [Algebra.trace_eq_matrix_trace (b.localizationLocalization Rβ M Sβ),
Algebra.t... | 2,048 |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 101 | 109 | theorem Algebra.traceMatrix_localizationLocalization (b : Basis ΞΉ R S) :
Algebra.traceMatrix Rβ (b.localizationLocalization Rβ M Sβ) =
(algebraMap R Rβ).mapMatrix (Algebra.traceMatrix R b) := by |
have : Module.Finite R S := Module.Finite.of_basis b
have : Module.Free R S := Module.Free.of_basis b
ext i j : 2
simp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, traceMatrix_apply, traceForm_apply,
Basis.localizationLocalization_apply, β map_mul]
exact Algebra.trace_localization R M _
| 2,048 |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 115 | 119 | theorem Algebra.discr_localizationLocalization (b : Basis ΞΉ R S) :
Algebra.discr Rβ (b.localizationLocalization Rβ M Sβ) =
algebraMap R Rβ (Algebra.discr R b) := by |
rw [Algebra.discr_def, Algebra.discr_def, RingHom.map_det,
Algebra.traceMatrix_localizationLocalization]
| 2,048 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 65 | 69 | theorem norm_algebraMap [IsSeparable K L] (x : π K) :
norm K (algebraMap (π K) (π L) x) = x ^ finrank K L := by |
rw [RingOfIntegers.ext_iff, RingOfIntegers.coe_eq_algebraMap,
RingOfIntegers.algebraMap_norm_algebraMap, Algebra.norm_algebraMap,
RingOfIntegers.coe_eq_algebraMap, map_pow]
| 2,049 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 72 | 85 | theorem isUnit_norm_of_isGalois [IsGalois K L] {x : π L} : IsUnit (norm K x) β IsUnit x := by |
classical
refine β¨fun hx => ?_, IsUnit.map _β©
replace hx : IsUnit (algebraMap (π K) (π L) <| norm K x) := hx.map (algebraMap (π K) <| π L)
refine @isUnit_of_mul_isUnit_right (π L) _
β¨(univ \ {AlgEquiv.refl}).prod fun Ο : L ββ[K] L => Ο x,
prod_mem fun Ο _ => x.2.map (Ο : L β+* L).toIntAlgHomβ© _ ... | 2,049 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 90 | 99 | theorem dvd_norm [IsGalois K L] (x : π L) : x β£ algebraMap (π K) (π L) (norm K x) := by |
classical
have hint :
IsIntegral β€ (β Ο β univ.erase (AlgEquiv.refl : L ββ[K] L), Ο x) :=
IsIntegral.prod _ (fun Ο _ =>
((RingOfIntegers.isIntegral_coe x).map Ο))
refine β¨β¨_, hintβ©, ?_β©
ext
rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
simp [β Finset.mul_prod_erase _ _ (mem_univ Al... | 2,049 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 104 | 106 | theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
[IsScalarTower K F L] (x : π L) : norm K (norm F x) = norm K x := by |
rw [RingOfIntegers.ext_iff, coe_norm, coe_norm, coe_norm, Algebra.norm_norm]
| 2,049 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 111 | 126 | theorem isUnit_norm [CharZero K] {x : π F} : IsUnit (norm K x) β IsUnit x := by |
letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
let L := normalClosure K F (AlgebraicClosure F)
haveI : FiniteDimensional F L := FiniteDimensional.right K F L
haveI : IsAlgClosure K (AlgebraicClosure F) :=
IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F)
haveI : IsGalois F L := I... | 2,049 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 55 | 63 | theorem exists_mem_of_not_isUnit_aeval_prod [IsDomain R] {p : R[X]} {a : A}
(h : Β¬IsUnit (aeval a (Multiset.map (fun x : R => X - C x) p.roots).prod)) :
β k : R, k β Ο a β§ eval k p = 0 := by |
rw [β Multiset.prod_toList, AlgHom.map_list_prod] at h
replace h := mt List.prod_isUnit h
simp only [not_forall, exists_prop, aeval_C, Multiset.mem_toList, List.mem_map, aeval_X,
exists_exists_and_eq_and, Multiset.mem_map, AlgHom.map_sub] at h
rcases h with β¨r, r_mem, r_nuβ©
exact β¨r, by rwa [mem_iff, β I... | 2,050 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 81 | 91 | theorem subset_polynomial_aeval (a : A) (p : π[X]) : (eval Β· p) '' Ο a β Ο (aeval a p) := by |
rintro _ β¨k, hk, rflβ©
let q := C (eval k p) - p
have hroot : IsRoot q k := by simp only [q, eval_C, eval_sub, sub_self, IsRoot.def]
rw [β mul_div_eq_iff_isRoot, β neg_mul_neg, neg_sub] at hroot
have aeval_q_eq : ββ (eval k p) - aeval a p = aeval a q := by
simp only [q, aeval_C, AlgHom.map_sub, sub_left_i... | 2,050 |
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 114 | 125 | theorem finrank_endomorphism_eq_one {X : C} (isIso_iff_nonzero : β f : X βΆ X, IsIso f β f β 0)
[I : FiniteDimensional π (X βΆ X)] : finrank π (X βΆ X) = 1 := by |
have id_nonzero := (isIso_iff_nonzero (π X)).mp (by infer_instance)
refine finrank_eq_one (π X) id_nonzero ?_
intro f
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
have : FiniteDimensional π (End X) := I
obtain β¨c, nuβ© := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional π (End.of f)... | 2,051 |
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
suppress_comp... | Mathlib/RepresentationTheory/FdRep.lean | 95 | 100 | theorem Iso.conj_Ο {V W : FdRep k G} (i : V β
W) (g : G) :
W.Ο g = (FdRep.isoToLinearEquiv i).conj (V.Ο g) := by |
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, β FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
| 2,052 |
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
suppress_comp... | Mathlib/RepresentationTheory/FdRep.lean | 113 | 114 | theorem forgetβ_Ο (V : FdRep k G) : ((forgetβ (FdRep k G) (Rep k G)).obj V).Ο = V.Ο := by |
ext g v; rfl
| 2,052 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 43 | 48 | theorem mul_average_left (g : G) : β(Finsupp.single g 1) * average k G = average k G := by |
simp only [mul_one, Finset.mul_sum, Algebra.mul_smul_comm, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G β MonoidAlgebra k G := fun x => Finsupp.single x 1
show β
(Fintype.card G : k) β’ β x : G, f (g * x) = β
(Fintype.card G : k) β’ β x : G, f x
rw [Function.B... | 2,053 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 54 | 59 | theorem mul_average_right (g : G) : average k G * β(Finsupp.single g 1) = average k G := by |
simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G β MonoidAlgebra k G := fun x => Finsupp.single x 1
show β
(Fintype.card G : k) β’ β x : G, f (x * g) = β
(Fintype.card G : k) β’ β x : G, f x
rw [Function.... | 2,053 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace Representation
namespace linHom... | Mathlib/RepresentationTheory/Invariants.lean | 133 | 139 | theorem mem_invariants_iff_comm {X Y : Rep k G} (f : X.V ββ[k] Y.V) (g : G) :
(linHom X.Ο Y.Ο) g f = f β f.comp (X.Ο g) = (Y.Ο g).comp f := by |
dsimp
erw [β ΟAut_apply_inv]
rw [β LinearMap.comp_assoc, β ModuleCat.comp_def, β ModuleCat.comp_def, Iso.inv_comp_eq,
ΟAut_apply_hom]
exact comm
| 2,053 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 100 | 103 | theorem dZero_ker_eq_invariants : LinearMap.ker (dZero A) = invariants A.Ο := by |
ext x
simp only [LinearMap.mem_ker, mem_invariants, β @sub_eq_zero _ _ _ x, Function.funext_iff]
rfl
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 401 | 403 | theorem map_one_of_isOneCocycle {f : G β A} (hf : IsOneCocycle f) :
f 1 = 0 := by |
simpa only [mul_one, one_smul, self_eq_add_right] using hf 1 1
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 405 | 407 | theorem map_one_fst_of_isTwoCocycle {f : G Γ G β A} (hf : IsTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by |
simpa only [one_smul, one_mul, mul_one, add_right_inj] using (hf 1 1 g).symm
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 409 | 411 | theorem map_one_snd_of_isTwoCocycle {f : G Γ G β A} (hf : IsTwoCocycle f) (g : G) :
f (g, 1) = g β’ f (1, 1) := by |
simpa only [mul_one, add_left_inj] using hf g 1 1
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 423 | 427 | theorem smul_map_inv_sub_map_inv_of_isTwoCocycle {f : G Γ G β A} (hf : IsTwoCocycle f) (g : G) :
g β’ f (gβ»ΒΉ, g) - f (g, gβ»ΒΉ) = f (1, 1) - f (g, 1) := by |
have := hf g gβ»ΒΉ g
simp only [mul_right_inv, mul_left_inv, map_one_fst_of_isTwoCocycle hf g] at this
exact sub_eq_sub_iff_add_eq_add.2 this.symm
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 524 | 526 | theorem map_one_of_isMulOneCocycle {f : G β M} (hf : IsMulOneCocycle f) :
f 1 = 1 := by |
simpa only [mul_one, one_smul, self_eq_mul_right] using hf 1 1
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 528 | 530 | theorem map_one_fst_of_isMulTwoCocycle {f : G Γ G β M} (hf : IsMulTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by |
simpa only [one_smul, one_mul, mul_one, mul_right_inj] using (hf 1 1 g).symm
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 532 | 534 | theorem map_one_snd_of_isMulTwoCocycle {f : G Γ G β M} (hf : IsMulTwoCocycle f) (g : G) :
f (g, 1) = g β’ f (1, 1) := by |
simpa only [mul_one, mul_left_inj] using hf g 1 1
| 2,054 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 546 | 551 | theorem smul_map_inv_div_map_inv_of_isMulTwoCocycle
{f : G Γ G β M} (hf : IsMulTwoCocycle f) (g : G) :
g β’ f (gβ»ΒΉ, g) / f (g, gβ»ΒΉ) = f (1, 1) / f (g, 1) := by |
have := hf g gβ»ΒΉ g
simp only [mul_right_inv, mul_left_inv, map_one_fst_of_isMulTwoCocycle hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm
| 2,054 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 54 | 55 | theorem char_mul_comm (V : FdRep k G) (g : G) (h : G) :
V.character (h * g) = V.character (g * h) := by | simp only [trace_mul_comm, character, map_mul]
| 2,055 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 59 | 60 | theorem char_one (V : FdRep k G) : V.character 1 = FiniteDimensional.finrank k V := by |
simp only [character, map_one, trace_one]
| 2,055 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 64 | 65 | theorem char_tensor (V W : FdRep k G) : (V β W).character = V.character * W.character := by |
ext g; convert trace_tensorProduct' (V.Ο g) (W.Ο g)
| 2,055 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 70 | 74 | theorem char_tensor' (V W : FdRep k G) :
character (Action.FunctorCategoryEquivalence.inverse.obj
(Action.FunctorCategoryEquivalence.functor.obj V β
Action.FunctorCategoryEquivalence.functor.obj W)) = V.character * W.character := by |
simp [β char_tensor]
| 2,055 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 77 | 78 | theorem char_iso {V W : FdRep k G} (i : V β
W) : V.character = W.character := by |
ext g; simp only [character, FdRep.Iso.conj_Ο i]; exact (trace_conj' (V.Ο g) _).symm
| 2,055 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 51 | 54 | theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
β c : K, f.HasEigenvalue c := by |
simp_rw [hasEigenvalue_iff_mem_spectrum]
exact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f
| 2,056 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 64 | 123 | theorem iSup_genEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
β¨ (ΞΌ : K) (k : β), f.genEigenspace ΞΌ k = β€ := by |
-- We prove the claim by strong induction on the dimension of the vector space.
induction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V
cases' n with n
-- If the vector space is 0-dimensional, the result is trivial.
Β· rw [β top_le_iff]
simp only [Submodule.finrank_eq_zero.1 ... | 2,056 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 132 | 192 | theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : β x β p, f x β p) :
p β β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ k = β¨ ΞΌ, β¨ k, p β f.genEigenspace ΞΌ k := by |
simp_rw [β (f.genEigenspace _).mono.directed_le.inf_iSup_eq]
refine le_antisymm (fun m hm β¦ ?_)
(le_inf_iff.mpr β¨iSup_le fun ΞΌ β¦ inf_le_left, iSup_mono fun ΞΌ β¦ inf_le_rightβ©)
classical
obtain β¨hmβ : m β p, hmβ : m β β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ kβ© := hm
obtain β¨m, hmβ, rflβ© := (mem_iSup_iff_exists_finsupp... | 2,056 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 194 | 197 | theorem eq_iSup_inf_genEigenspace [FiniteDimensional K V]
(h : β x β p, f x β p) (h' : β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ k = β€) :
p = β¨ ΞΌ, β¨ k, p β f.genEigenspace ΞΌ k := by |
rw [β inf_iSup_genEigenspace h, h', inf_top_eq]
| 2,056 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import linear_algebra.matrix.charpoly.eigs from "leanprover-community/mathlib"@"48dc6abe71248bd6f4bffc9703dc87bdd4e37d0b"
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {R : Type*} [Field R]
variable {A : Matrix... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 60 | 64 | theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.det = (Matrix.charpoly A).roots.prod := by |
rw [det_eq_sign_charpoly_coeff, β charpoly_natDegree_eq_dim A,
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic hAps, β mul_assoc,
β pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
| 2,057 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import linear_algebra.matrix.charpoly.eigs from "leanprover-community/mathlib"@"48dc6abe71248bd6f4bffc9703dc87bdd4e37d0b"
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {R : Type*} [Field R]
variable {A : Matrix... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 67 | 75 | theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.trace = (Matrix.charpoly A).roots.sum := by |
cases' isEmpty_or_nonempty n with h
Β· rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,
det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,
Multiset.empty_eq_zero, Multiset.sum_zero]
Β· rw [trace_eq_neg_charpoly_coeff, neg_eq_iff_eq_neg,
β Polynomial.sum_roots_eq... | 2,057 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 44 | 48 | theorem coeff_zero_mem_comap_of_root_mem_of_eval_mem {r : S} (hr : r β I) {p : R[X]}
(hp : p.evalβ f r β I) : p.coeff 0 β I.comap f := by |
rw [β p.divX_mul_X_add, evalβ_add, evalβ_C, evalβ_mul, evalβ_X] at hp
refine mem_comap.mpr ((I.add_mem_iff_right ?_).mp hp)
exact I.mul_mem_left _ hr
| 2,058 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 56 | 70 | theorem exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem {r : S}
(r_non_zero_divisor : β {x}, x * r = 0 β x = 0) (hr : r β I) {p : R[X]} :
p β 0 β p.evalβ f r = 0 β β i, p.coeff i β 0 β§ p.coeff i β I.comap f := by |
refine p.recOnHorner ?_ ?_ ?_
Β· intro h
contradiction
Β· intro p a coeff_eq_zero a_ne_zero _ _ hp
refine β¨0, ?_, coeff_zero_mem_comap_of_root_mem hr hpβ©
simp [coeff_eq_zero, a_ne_zero]
Β· intro p p_nonzero ih _ hp
rw [evalβ_mul, evalβ_X] at hp
obtain β¨i, hi, memβ© := ih p_nonzero (r_non_zero_d... | 2,058 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 | theorem injective_quotient_le_comap_map (P : Ideal R[X]) :
Function.Injective <|
Ideal.quotientMap
(Ideal.map (Polynomial.mapRingHom (Quotient.mk (P.comap (C : R β+* R[X])))) P)
(Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R β+* R[X]))))
le_comap_map := by |
refine quotientMap_injective' (le_of_eq ?_)
rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R β+* R[X]))))
(map_surjective (Ideal.Quotient.mk (P.comap (C : R β+* R[X]))) Ideal.Quotient.mk_surjective)]
refine le_antisymm (sup_le le_rfl ?_) (le_sup_of_le_left le_rfl)
refine fun p h... | 2,058 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 101 | 109 | theorem quotient_mk_maps_eq (P : Ideal R[X]) :
((Quotient.mk (map (mapRingHom (Quotient.mk (P.comap (C : R β+* R[X])))) P)).comp C).comp
(Quotient.mk (P.comap (C : R β+* R[X]))) =
(Ideal.quotientMap (map (mapRingHom (Quotient.mk (P.comap (C : R β+* R[X])))) P)
(mapRingHom (Quotient.mk (P.c... |
refine RingHom.ext fun x => ?_
repeat' rw [RingHom.coe_comp, Function.comp_apply]
rw [quotientMap_mk, coe_mapRingHom, map_C]
| 2,058 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 116 | 126 | theorem exists_nonzero_mem_of_ne_bot {P : Ideal R[X]} (Pb : P β β₯) (hP : β x : R, C x β P β x = 0) :
β p : R[X], p β P β§ Polynomial.map (Quotient.mk (P.comap (C : R β+* R[X]))) p β 0 := by |
obtain β¨m, hmβ© := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)
refine β¨m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp ?_)β©
refine
(injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk
(P.comap (C : R β+* R[X]))))).mp
?_ _ pp0
refine map_injective _ ... | 2,058 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 139 | 149 | theorem comap_eq_of_scalar_tower_quotient [Algebra R S] [Algebra (R β§Έ p) (S β§Έ P)]
[IsScalarTower R (R β§Έ p) (S β§Έ P)] (h : Function.Injective (algebraMap (R β§Έ p) (S β§Έ P))) :
comap (algebraMap R S) P = p := by |
ext x
rw [mem_comap, β Quotient.eq_zero_iff_mem, β Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,
IsScalarTower.algebraMap_apply R (R β§Έ p) (S β§Έ P), Quotient.algebraMap_eq]
constructor
Β· intro hx
exact (injective_iff_map_eq_zero (algebraMap (R β§Έ p) (S β§Έ P))).mp h _ hx
Β· intro hx
rw [hx, RingHom... | 2,058 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 116 | 119 | theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal β€ := by |
cases x
rfl
| 2,059 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 123 | 126 | theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod β€ x.asIdeal := by |
cases x
rfl
| 2,059 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 147 | 149 | theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s := by |
ext x
exact (Submodule.gi R R).gc s x.asIdeal
| 2,059 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 164 | 169 | theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
(vanishingIdeal t : Set R) = { f : R | β x : PrimeSpectrum R, x β t β f β x.asIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, Submodule.mem_iInf]
apply forall_congr'; intro x
rw [Submodule.mem_iInf]
| 2,059 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 172 | 174 | theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
f β vanishingIdeal t β β x : PrimeSpectrum R, x β t β f β x.asIdeal := by |
rw [β SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
| 2,059 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 178 | 179 | theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by | simp [vanishingIdeal]
| 2,059 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Algebra.Ring.Subring.Basic
#align_import algebraic_geometry.struct... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 108 | 118 | theorem IsFraction.eq_mk' {U : Opens (PrimeSpectrum.Top R)} {f : β x : U, Localizations R x}
(hf : IsFraction f) :
β r s : R,
β x : U,
β hs : s β x.1.asIdeal,
f x =
IsLocalization.mk' (Localization.AtPrime _) r
(β¨s, hsβ© : (x : PrimeSpectrum.Top R).asIdeal.primeC... |
rcases hf with β¨r, s, hβ©
refine β¨r, s, fun x => β¨(h x).1, (IsLocalization.mk'_eq_iff_eq_mul.mpr ?_).symmβ©β©
exact (h x).2.symm
| 2,060 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Topology.NoetherianSpace
#align_import algebraic_geometry.prime_spectrum.noetherian from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
namespace PrimeSpectrum
open Submodule
variable (R : Type u) [CommR... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean | 27 | 54 | theorem exists_primeSpectrum_prod_le (I : Ideal R) :
β Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) β€ I := by |
-- Porting note: Need to specify `P` explicitly
refine IsNoetherian.induction
(P := fun I => β Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) β€ I)
(fun (M : Ideal R) hgt => ?_) I
by_cases h_prM : M.IsPrime
Β· use {β¨M, h_prMβ©}
rw [Multiset.map_singleton, Multiset.prod_singleton]
by_c... | 2,061 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Topology.NoetherianSpace
#align_import algebraic_geometry.prime_spectrum.noetherian from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
namespace PrimeSpectrum
open Submodule
variable (R : Type u) [CommR... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean | 60 | 97 | theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : Β¬IsField A) {I : Ideal A}
(h_nzI : I β β₯) :
β Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) β€ I β§ Multiset.prod (Z.map asIdeal) β β₯ := by |
revert h_nzI
-- Porting note: Need to specify `P` explicitly
refine IsNoetherian.induction (P := fun I => I β β₯ β β Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) β€ I β§ Multiset.prod (Z.map asIdeal) β β₯)
(fun (M : Ideal A) hgt => ?_) I
intro h_nzM
have hA_nont : Nontrivial A := IsDom... | 2,061 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 38 | 40 | theorem isOpen_imageOfDf : IsOpen (imageOfDf f) := by |
rw [imageOfDf, setOf_exists fun i (x : PrimeSpectrum R) => coeff f i β x.asIdeal]
exact isOpen_iUnion fun i => isOpen_basicOpen
| 2,062 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 54 | 66 | theorem imageOfDf_eq_comap_C_compl_zeroLocus :
imageOfDf f = PrimeSpectrum.comap (C : R β+* R[X]) '' (zeroLocus {f})αΆ := by |
ext x
refine β¨fun hx => β¨β¨map C x.asIdeal, isPrime_map_C_of_isPrime x.IsPrimeβ©, β¨?_, ?_β©β©, ?_β©
Β· rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]
cases' hx with i hi
exact fun a => hi (mem_map_C_iff.mp a i)
Β· ext x
refine β¨fun h => ?_, fun h => subset_span (mem_image_of_mem C.1 h)β©
rw [β... | 2,062 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 74 | 79 | theorem isOpenMap_comap_C : IsOpenMap (PrimeSpectrum.comap (C : R β+* R[X])) := by |
rintro U β¨s, zβ©
rw [β compl_compl U, β z, β iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,
image_iUnion]
simp_rw [β imageOfDf_eq_comap_C_compl_zeroLocus]
exact isOpen_iUnion fun f => isOpen_imageOfDf
| 2,062 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spectrum.maximal from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
noncomputable section
open scoped Classical
universe u v
variable (R : Typ... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 65 | 69 | theorem toPrimeSpectrum_range :
Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } := by |
simp only [isClosed_singleton_iff_isMaximal]
ext β¨x, _β©
exact β¨fun β¨y, hyβ© => hy βΈ y.IsMaximal, fun hx => β¨β¨x, hxβ©, rflβ©β©
| 2,063 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spectrum.maximal from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
noncomputable section
open scoped Classical
universe u v
variable (R : Typ... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 92 | 117 | theorem iInf_localization_eq_bot : (β¨
v : MaximalSpectrum R,
Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = β₯ := by |
ext x
rw [Algebra.mem_bot, Algebra.mem_iInf]
constructor
Β· contrapose
intro hrange hlocal
let denom : Ideal R := (Submodule.span R {1} : Submodule R K).colon (Submodule.span R {x})
have hdenom : (1 : R) β denom := by
intro hdenom
rcases Submodule.mem_span_singleton.mp
(Submodule... | 2,063 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 76 | 78 | theorem coe_inv_of_nonzero {J : FractionalIdeal Rββ° K} (h : J β 0) :
(βJβ»ΒΉ : Submodule Rβ K) = IsLocalization.coeSubmodule K β€ / (J : Submodule Rβ K) := by |
simp_rw [inv_nonzero _ h, coe_one, coe_mk, IsLocalization.coeSubmodule_top]
| 2,064 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 87 | 92 | theorem inv_anti_mono (hI : I β 0) (hJ : J β 0) (hIJ : I β€ J) : Jβ»ΒΉ β€ Iβ»ΒΉ := by |
-- Porting note: in Lean3, introducing `x` would just give `x β Jβ»ΒΉ β x β Iβ»ΒΉ`, but
-- in Lean4, it goes all the way down to the subtypes
intro x
simp only [val_eq_coe, mem_coe, mem_inv_iff hJ, mem_inv_iff hI]
exact fun h y hy => h y (hIJ hy)
| 2,064 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 108 | 122 | theorem right_inverse_eq (I J : FractionalIdeal Rββ° K) (h : I * J = 1) : J = Iβ»ΒΉ := by |
have hI : I β 0 := ne_zero_of_mul_eq_one I J h
suffices h' : I * (1 / I) = 1 from
congr_arg Units.inv <| @Units.ext _ _ (Units.mkOfMulEqOne _ _ h) (Units.mkOfMulEqOne _ _ h') rfl
apply le_antisymm
Β· apply mul_le.mpr _
intro x hx y hy
rw [mul_comm]
exact (mem_div_iff_of_nonzero hI).mp hy x hx
... | 2,064 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 136 | 137 | theorem map_inv (I : FractionalIdeal Rββ° K) (h : K ββ[Rβ] K') :
Iβ»ΒΉ.map (h : K ββ[Rβ] K') = (I.map h)β»ΒΉ := by | rw [inv_eq, map_div, map_one, inv_eq]
| 2,064 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 148 | 150 | theorem spanSingleton_div_spanSingleton (x y : K) :
spanSingleton Rββ° x / spanSingleton Rββ° y = spanSingleton Rββ° (x / y) := by |
rw [div_spanSingleton, mul_comm, spanSingleton_mul_spanSingleton, div_eq_mul_inv]
| 2,064 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 153 | 155 | theorem spanSingleton_div_self {x : K} (hx : x β 0) :
spanSingleton Rββ° x / spanSingleton Rββ° x = 1 := by |
rw [spanSingleton_div_spanSingleton, div_self hx, spanSingleton_one]
| 2,064 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 38 | 74 | theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
(hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x β P) (hxP2 : x β P ^ 2)
(hxQ : β Q : Ideal R, IsPrime Q β Q β P β x β Q) : P = Ideal.span {x} := by |
letI := Classical.decEq (Ideal R)
have hx0 : x β 0 := by
rintro rfl
exact hxP2 (zero_mem _)
by_cases hP0 : P = β₯
Β· subst hP0
-- Porting note: was `simpa using hxP2` but that hypothesis didn't even seem relevant in Lean 3
rwa [eq_comm, span_singleton_eq_bot, β mem_bot]
have hspan0 : span ({x} ... | 2,065 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 78 | 102 | theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {R A : Type*}
[CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A]
(I : (FractionalIdeal S A)Λ£) {v : A} (hv : v β (βIβ»ΒΉ : FractionalIdeal S A))
(h : Submodule.comap (Algebra.linearMap R A) ((I : Submodul... |
have hinv := I.mul_inv
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})
have hJ : IsLocalization.coeSubmodule A J = βI * Submodule.span R {v} := by
-- Porting note: had to insert `val_eq_coe` into this rewrite.
-- Arguably this is because `Subtype.ext_iff` is ... | 2,065 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 109 | 168 | theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
[Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S β€ Rβ°)
(hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
Submodule.IsPrincipal (I : Submodule R A) := by |
have hinv' := hinv
rw [Subtype.ext_iff, val_eq_coe, coe_mul] at hinv
let s := hf.toFinset
haveI := Classical.decEq (Ideal R)
have coprime : β M β s, β M' β s.erase M, M β M' = β€ := by
simp_rw [Finset.mem_erase, hf.mem_toFinset]
rintro M hM M' β¨hne, hM'β©
exact Ideal.IsMaximal.coprime_of_ne hM hM' ... | 2,065 |
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.Data.Int.Associated
import Mathlib.LinearAlgebra.FreeModule.Determinant
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathli... | Mathlib/RingTheory/Ideal/Norm.lean | 70 | 74 | theorem cardQuot_apply (S : Submodule R M) [h : Fintype (M β§Έ S)] :
cardQuot S = Fintype.card (M β§Έ S) := by |
-- Porting note: original proof was AddSubgroup.index_eq_card _
suffices Fintype (M β§Έ S.toAddSubgroup) by convert AddSubgroup.index_eq_card S.toAddSubgroup
convert h
| 2,066 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
| Mathlib/RingTheory/FractionalIdeal/Norm.lean | 36 | 51 | theorem absNorm_div_norm_eq_absNorm_div_norm {I : FractionalIdeal Rβ° K} (a : Rβ°) (Iβ : Ideal R)
(h : a β’ (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) Iβ) :
(Ideal.absNorm I.num : β) / |Algebra.norm β€ (I.den:R)| =
(Ideal.absNorm Iβ : β) / |Algebra.norm β€ (a:R)| := by |
rw [div_eq_div_iff]
Β· replace h := congr_arg (I.den β’ Β·) h
have h' := congr_arg (a β’ Β·) (den_mul_self_eq_num I)
dsimp only at h h'
rw [smul_comm] at h
rw [h, Submonoid.smul_def, Submonoid.smul_def, β Submodule.ideal_span_singleton_smul,
β Submodule.ideal_span_singleton_smul, β Submodule.map_s... | 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 78 | 82 | theorem absNorm_eq' {I : FractionalIdeal Rβ° K} (a : Rβ°) (Iβ : Ideal R)
(h : a β’ (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) Iβ) :
absNorm I = (Ideal.absNorm Iβ : β) / |Algebra.norm β€ (a:R)| := by |
rw [absNorm, β absNorm_div_norm_eq_absNorm_div_norm a Iβ h, MonoidWithZeroHom.coe_mk,
ZeroHom.coe_mk]
| 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 84 | 84 | theorem absNorm_nonneg (I : FractionalIdeal Rβ° K) : 0 β€ absNorm I := by | dsimp [absNorm]; positivity
| 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 88 | 88 | theorem absNorm_one : absNorm (1 : FractionalIdeal Rβ° K) = 1 := by | convert absNorm.map_one'
| 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 90 | 95 | theorem absNorm_eq_zero_iff [NoZeroDivisors K] {I : FractionalIdeal Rβ° K} :
absNorm I = 0 β I = 0 := by |
refine β¨fun h β¦ zero_of_num_eq_bot zero_not_mem_nonZeroDivisors ?_, fun h β¦ h βΈ absNorm_botβ©
rw [absNorm_eq, div_eq_zero_iff] at h
refine Ideal.absNorm_eq_zero_iff.mp <| Nat.cast_eq_zero.mp <| h.resolve_right ?_
simpa [Algebra.norm_eq_zero_iff] using nonZeroDivisors.coe_ne_zero _
| 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 97 | 100 | theorem coeIdeal_absNorm (Iβ : Ideal R) :
absNorm (Iβ : FractionalIdeal Rβ° K) = Ideal.absNorm Iβ := by |
rw [absNorm_eq' 1 Iβ (by rw [one_smul]; rfl), OneMemClass.coe_one, _root_.map_one, abs_one,
Int.cast_one, _root_.div_one]
| 2,067 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free β€ R] [Module.Finite β€ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 106 | 128 | theorem abs_det_basis_change [NoZeroDivisors K] {ΞΉ : Type*} [Fintype ΞΉ]
[DecidableEq ΞΉ] (b : Basis ΞΉ β€ R) (I : FractionalIdeal Rβ° K) (bI : Basis ΞΉ β€ I) :
|(b.localizationLocalization β β€β° K).det ((β) β bI)| = absNorm I := by |
have := IsFractionRing.nontrivial R K
let bβ : Basis ΞΉ β K := b.localizationLocalization β β€β° K
let bI.num : Basis ΞΉ β€ I.num := bI.map
((equivNum (nonZeroDivisors.coe_ne_zero _)).restrictScalars β€)
rw [absNorm_eq, β Ideal.natAbs_det_basis_change b I.num bI.num, Int.cast_natAbs, Int.cast_abs,
Int.cast... | 2,067 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
variable (K : Type*) [Field K] [NumberField K]
namespace NumberField
open scoped nonZeroDivisors
section Basis
open Module
-- This is necessary to avoid several time... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 87 | 90 | theorem mem_span_basisOfFractionalIdeal {I : (FractionalIdeal (π K)β° K)Λ£} {x : K} :
x β Submodule.span β€ (Set.range (basisOfFractionalIdeal K I)) β x β (I : Set K) := by |
rw [basisOfFractionalIdeal, (fractionalIdealBasis K I.1).ofIsLocalizedModule_span β β€β° _]
simp
| 2,068 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
variable (K : Type*) [Field K] [NumberField K]
namespace NumberField
open scoped nonZeroDivisors
section Basis
open Module
-- This is necessary to avoid several time... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 93 | 96 | theorem fractionalIdeal_rank (I : (FractionalIdeal (π K)β° K)Λ£) :
finrank β€ I = finrank β€ (π K) := by |
rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
finrank_eq_card_basis (basisOfFractionalIdeal K I)]
| 2,068 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 97 | 99 | theorem int_valuation_ne_zero (x : R) (hx : x β 0) : v.intValuationDef x β 0 := by |
rw [intValuationDef, if_neg hx]
exact WithZero.coe_ne_zero
| 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 108 | 110 | theorem int_valuation_zero_le (x : nonZeroDivisors R) : 0 < v.intValuationDef x := by |
rw [v.intValuationDef_if_neg (nonZeroDivisors.coe_ne_zero x)]
exact WithZero.zero_lt_coe _
| 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 114 | 120 | theorem int_valuation_le_one (x : R) : v.intValuationDef x β€ 1 := by |
rw [intValuationDef]
by_cases hx : x = 0
Β· rw [if_pos hx]; exact WithZero.zero_le 1
Β· rw [if_neg hx, β WithZero.coe_one, β ofAdd_zero, WithZero.coe_le_coe, ofAdd_le,
Right.neg_nonpos_iff]
exact Int.natCast_nonneg _
| 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 124 | 134 | theorem int_valuation_lt_one_iff_dvd (r : R) :
v.intValuationDef r < 1 β v.asIdeal β£ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
Β· simp [hr]
Β· rw [β WithZero.coe_one, β ofAdd_zero, WithZero.coe_lt_coe, ofAdd_lt, neg_lt_zero, β
Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
have h : (Ideal.span {r} : Ideal R) β 0 := by
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
exact h... | 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 139 | 147 | theorem int_valuation_le_pow_iff_dvd (r : R) (n : β) :
v.intValuationDef r β€ Multiplicative.ofAdd (-(n : β€)) β v.asIdeal ^ n β£ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
Β· simp_rw [hr, Ideal.dvd_span_singleton, zero_le', Submodule.zero_mem]
Β· rw [WithZero.coe_le_coe, ofAdd_le, neg_le_neg_iff, Int.ofNat_le, Ideal.dvd_span_singleton, β
Associates.le_singleton_iff,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr)
... | 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 156 | 160 | theorem IntValuation.map_one' : v.intValuationDef 1 = 1 := by |
rw [v.intValuationDef_if_neg (zero_ne_one.symm : (1 : R) β 0), Ideal.span_singleton_one, β
Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
Associates.count_zero (by apply v.associates_irreducible), Int.ofNat_zero, neg_zero, ofAdd_zero,
WithZero.coe_one]
| 2,069 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 164 | 175 | theorem IntValuation.map_mul' (x y : R) :
v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y := by |
simp only [intValuationDef]
by_cases hx : x = 0
Β· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
Β· by_cases hy : y = 0
Β· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
Β· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), β WithZero.coe_mul, WithZero.coe_inj, β
ofAdd_add, β Ideal.span_singl... | 2,069 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 93 | 102 | theorem valuationOfNeZeroToFun_eq (x : KΛ£) :
(v.valuationOfNeZeroToFun x : β€ββ) = v.valuation (x : K) := by |
rw [show v.valuation (x : K) = _ * _ by rfl]
rw [Units.val_inv_eq_inv_val]
change _ = ite _ _ _ * (ite _ _ _)β»ΒΉ
simp_rw [IsLocalization.toLocalizationMap_sec, SubmonoidClass.coe_subtype,
if_neg <| IsLocalization.sec_fst_ne_zero le_rfl x.ne_zero,
if_neg (nonZeroDivisors.coe_ne_zero _),
valuationOfNe... | 2,070 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 120 | 131 | theorem valuation_of_unit_eq (x : RΛ£) :
v.valuationOfNeZero (Units.map (algebraMap R K : R β* K) x) = 1 := by |
rw [β WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]
constructor
Β· exact v.valuation_le_one x
Β· cases' x with x _ hx _
change Β¬v.valuation (algebraMap R K x) < 1
apply_fun v.intValuation at hx
rw [map_one, map_mul] at hx
rw [not_lt, β hx, β mul_one <| v.valuation _, va... | 2,070 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 150 | 155 | theorem valuation_of_unit_mod_eq (n : β) (x : RΛ£) :
v.valuationOfNeZeroMod n (Units.map (algebraMap R K : R β* K) x : K/n) = 1 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [valuationOfNeZeroMod, MonoidHom.comp_apply, β QuotientGroup.coe_mk',
QuotientGroup.map_mk' (G := KΛ£) (N := MonoidHom.range (powMonoidHom n)),
valuation_of_unit_eq, QuotientGroup.mk_one, map_one]
| 2,070 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 37 | 59 | theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal {I : Ideal R} (hI : I β β₯)
(hM : Module.IsTorsionBySet R M I) :
DirectSum.IsInternal fun p : (factors I).toFinset =>
torsionBySet R M (p ^ (factors I).count βp : Ideal R) := by |
let P := factors I
have prime_of_mem := fun p (hp : p β P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
Β· convert hM
rw [β Finset.inf_eq_iInf, IsDedekindDomain.inf_prime_pow_eq_prod, β Finset.prod_multiset_count,
β assoc... | 2,071 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 65 | 72 | theorem isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (β€ : Submodule R M).annihilator).toFinset =>
torsionBySet R M (p ^ (factors (β€ : Submodule R M).annihilator).count βp : Ideal R) := by |
have hM' := Module.isTorsionBySet_annihilator_top R M
have hI := Submodule.annihilator_top_inter_nonZeroDivisors hM
refine isInternal_prime_power_torsion_of_is_torsion_by_ideal ?_ hM'
rw [β Set.nonempty_iff_ne_empty] at hI; rw [Submodule.ne_bot_iff]
obtain β¨x, H, hxβ© := hI; exact β¨x, H, nonZeroDivisors.ne_ze... | 2,071 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 75 | 84 | theorem Submodule.isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (β€ : Submodule R M).annihilator).toFinset =>
torsionBy R M
(IsPrincipal.generator (p : Ideal R) ^
(factors (β€ : Submodule R M).annihilator).coun... |
convert isInternal_prime_power_torsion hM
ext p : 1
rw [β torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, β Ideal.span_singleton_pow,
Ideal.span_singleton_generator]
| 2,072 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 89 | 98 | theorem Submodule.exists_isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
β (ΞΉ : Type u) (_ : Fintype ΞΉ) (_ : DecidableEq ΞΉ) (p : ΞΉ β R) (_ : β i, Irreducible <| p i)
(e : ΞΉ β β), DirectSum.IsInternal fun i => torsionBy R M <| p i ^ e i := by |
refine β¨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hMβ©
Β· exact Finset.fintypeCoeSort _
Β· rintro β¨p, hpβ©
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime_of_prime hP
exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
| 2,072 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 110 | 121 | theorem _root_.Ideal.torsionOf_eq_span_pow_pOrder (x : M) :
torsionOf R M x = span {p ^ pOrder hM x} := by |
dsimp only [pOrder]
rw [β (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, β
Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
have prop :
(fun n : β => p ^ n β’ x = 0) = fun n : β =>
(Associates.mk <| generator <| torsionOf R M x) β£ Associates.mk p ^ n := by
... | 2,072 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 124 | 148 | theorem p_pow_smul_lift {x y : M} {k : β} (hM' : Module.IsTorsionBy R M (p ^ pOrder hM y))
(h : p ^ k β’ x β R β y) : β a : R, p ^ k β’ x = p ^ k β’ a β’ y := by |
-- Porting note: needed to make `smul_smul` work below.
letI : MulAction R M := MulActionWithZero.toMulAction
by_cases hk : k β€ pOrder hM y
Β· let f :=
((R β p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans
(quotTorsionOfEquivSpanSingleton R M y)
Β· have : f.symm β¨p ^ k β’ x, hβ© β
... | 2,072 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 153 | 165 | theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
{k : β} (f : (R β§Έ R β p ^ k) ββ[R] M β§Έ R β z) :
β x : M, p ^ k β’ x = 0 β§ Submodule.Quotient.mk (p := span R {z}) x = f 1 := by |
have f1 := mk_surjective (R β z) (f 1)
have : p ^ k β’ f1.choose β R β z := by
rw [β Quotient.mk_eq_zero, mk_smul, f1.choose_spec, β f.map_smul]
convert f.map_zero; change _ β’ Submodule.Quotient.mk _ = _
rw [β mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
exact Submodule.mem_span_si... | 2,072 |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ΞΉ : Type} [DecidableEq ΞΉ] (p : ΞΉ β β) (n : ΞΉ β β) :
(β¨ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 91 | 100 | theorem finite_of_fg_torsion [AddCommGroup M] [Module β€ M] [Module.Finite β€ M]
(hM : Module.IsTorsion β€ M) : _root_.Finite M := by |
rcases Module.equiv_directSum_of_isTorsion hM with β¨ΞΉ, _, p, h, e, β¨lβ©β©
haveI : β i : ΞΉ, NeZero (p i ^ e i).natAbs := fun i =>
β¨Int.natAbs_ne_zero.mpr <| pow_ne_zero (e i) (h i).ne_zeroβ©
haveI : β i : ΞΉ, _root_.Finite <| β€ β§Έ Submodule.span β€ {p i ^ e i} := fun i =>
Finite.of_equiv _ (p i ^ e i).quotientS... | 2,073 |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ΞΉ : Type} [DecidableEq ΞΉ] (p : ΞΉ β β) (n : ΞΉ β β) :
(β¨ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 114 | 126 | theorem equiv_free_prod_directSum_zmod [hG : AddGroup.FG G] :
β (n : β) (ΞΉ : Type) (_ : Fintype ΞΉ) (p : ΞΉ β β) (_ : β i, Nat.Prime <| p i) (e : ΞΉ β β),
Nonempty <| G β+ (Fin n ββ β€) Γ β¨ i : ΞΉ, ZMod (p i ^ e i) := by |
obtain β¨n, ΞΉ, fΞΉ, p, hp, e, β¨fβ©β© :=
@Module.equiv_free_prod_directSum _ _ _ _ _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG)
refine β¨n, ΞΉ, fΞΉ, fun i => (p i).natAbs, fun i => ?_, e, β¨?_β©β©
Β· rw [β Int.prime_iff_natAbs_prime, β irreducible_iff_prime]; exact hp i
exact
f.toAddEquiv.trans
((AddEquiv.re... | 2,073 |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ΞΉ : Type} [DecidableEq ΞΉ] (p : ΞΉ β β) (n : ΞΉ β β) :
(β¨ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 131 | 143 | theorem equiv_directSum_zmod_of_finite [Finite G] :
β (ΞΉ : Type) (_ : Fintype ΞΉ) (p : ΞΉ β β) (_ : β i, Nat.Prime <| p i) (e : ΞΉ β β),
Nonempty <| G β+ β¨ i : ΞΉ, ZMod (p i ^ e i) := by |
cases nonempty_fintype G
obtain β¨n, ΞΉ, fΞΉ, p, hp, e, β¨fβ©β© := equiv_free_prod_directSum_zmod G
cases' n with n
Β· have : Unique (Fin Nat.zero ββ β€) :=
{ uniq := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]; trivial }
exact β¨ΞΉ, fΞΉ, p, hp, e, β¨f.trans AddEquiv.uniqueProdβ©β©
Β· haveI := @Fintyp... | 2,073 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 61 | 63 | theorem coe_toPrincipalIdeal (x : KΛ£) :
(toPrincipalIdeal R K x : FractionalIdeal Rβ° K) = spanSingleton _ (x : K) := by |
simp only [toPrincipalIdeal]; rfl
| 2,074 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 67 | 69 | theorem toPrincipalIdeal_eq_iff {I : (FractionalIdeal Rβ° K)Λ£} {x : KΛ£} :
toPrincipalIdeal R K x = I β spanSingleton Rβ° (x : K) = I := by |
simp only [toPrincipalIdeal]; exact Units.ext_iff
| 2,074 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 72 | 79 | theorem mem_principal_ideals_iff {I : (FractionalIdeal Rβ° K)Λ£} :
I β (toPrincipalIdeal R K).range β β x : K, spanSingleton Rβ° x = I := by |
simp only [MonoidHom.mem_range, toPrincipalIdeal_eq_iff]
constructor <;> rintro β¨x, hxβ©
Β· exact β¨x, hxβ©
Β· refine β¨Units.mk0 x ?_, hxβ©
rintro rfl
simp [I.ne_zero.symm] at hx
| 2,074 |
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