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.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 124 | 139 | theorem rank_submodule_le_one_iff (s : Submodule K V) [Module.Free K s] :
Module.rank K s ≤ 1 ↔ ∃ v₀ ∈ s, s ≤ K ∙ v₀ := by |
simp_rw [rank_le_one_iff, le_span_singleton_iff]
constructor
· rintro ⟨⟨v₀, hv₀⟩, h⟩
use v₀, hv₀
intro v hv
obtain ⟨r, hr⟩ := h ⟨v, hv⟩
use r
rwa [Subtype.ext_iff, coe_smul] at hr
· rintro ⟨v₀, hv₀, h⟩
use ⟨v₀, hv₀⟩
rintro ⟨v, hv⟩
obtain ⟨r, hr⟩ := h v hv
use r
rwa [Subt... | 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 262 | 274 | theorem eq_bot_of_rank_le_one (h : Module.rank F S ≤ 1) [Module.Free F S] : S = ⊥ := by |
nontriviality E
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := F) (M := S)
by_cases h1 : Module.rank F S = 1
· refine bot_unique fun x hx ↦ Algebra.mem_bot.2 ?_
rw [← b.mk_eq_rank'', eq_one_iff_unique, ← unique_iff_subsingleton_and_nonempty] at h1
obtain ⟨h1⟩ := h1
obtain ⟨y, hy⟩ := (bijective_alg... | 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 277 | 280 | theorem eq_bot_of_finrank_one (h : finrank F S = 1) [Module.Free F S] : S = ⊥ := by |
refine Subalgebra.eq_bot_of_rank_le_one ?_
rw [finrank, toNat_eq_one] at h
rw [h]
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 284 | 295 | theorem rank_eq_one_iff [Nontrivial E] [Module.Free F S] : Module.rank F S = 1 ↔ S = ⊥ := by |
refine ⟨fun h ↦ Subalgebra.eq_bot_of_rank_le_one h.le, ?_⟩
rintro rfl
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := F) (M := (⊥ : Subalgebra F E))
refine le_antisymm ?_ ?_
· have := lift_rank_range_le (Algebra.linearMap F E)
rwa [← one_eq_range, rank_self, lift_one, lift_le_one_iff] at this
· by_cont... | 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 299 | 301 | theorem finrank_eq_one_iff [Nontrivial E] [Module.Free F S] : finrank F S = 1 ↔ S = ⊥ := by |
rw [← Subalgebra.rank_eq_one_iff]
exact toNat_eq_iff one_ne_zero
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 304 | 308 | theorem bot_eq_top_iff_rank_eq_one [Nontrivial E] [Module.Free F E] :
(⊥ : Subalgebra F E) = ⊤ ↔ Module.rank F E = 1 := by |
haveI := Module.Free.of_equiv (Subalgebra.topEquiv (R := F) (A := E)).toLinearEquiv.symm
-- Porting note: removed `subalgebra_top_rank_eq_submodule_top_rank`
rw [← rank_top, Subalgebra.rank_eq_one_iff, eq_comm]
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 311 | 315 | theorem bot_eq_top_iff_finrank_eq_one [Nontrivial E] [Module.Free F E] :
(⊥ : Subalgebra F E) = ⊤ ↔ finrank F E = 1 := by |
haveI := Module.Free.of_equiv (Subalgebra.topEquiv (R := F) (A := E)).toLinearEquiv.symm
rw [← finrank_top, ← subalgebra_top_finrank_eq_submodule_top_finrank,
Subalgebra.finrank_eq_one_iff, eq_comm]
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 68 | 72 | theorem lift_rank_range_add_rank_ker (f : M →ₗ[R] M') :
lift.{u} (Module.rank R (LinearMap.range f)) + lift.{v} (Module.rank R (LinearMap.ker f)) =
lift.{v} (Module.rank R M) := by |
haveI := fun p : Submodule R M => Classical.decEq (M ⧸ p)
rw [← f.quotKerEquivRange.lift_rank_eq, ← lift_add, rank_quotient_add_rank]
| 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 75 | 78 | theorem rank_range_add_rank_ker (f : M →ₗ[R] M₁) :
Module.rank R (LinearMap.range f) + Module.rank R (LinearMap.ker f) = Module.rank R M := by |
haveI := fun p : Submodule R M => Classical.decEq (M ⧸ p)
rw [← f.quotKerEquivRange.rank_eq, rank_quotient_add_rank]
| 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 81 | 84 | theorem lift_rank_eq_of_surjective {f : M →ₗ[R] M'} (h : Surjective f) :
lift.{v} (Module.rank R M) =
lift.{u} (Module.rank R M') + lift.{v} (Module.rank R (LinearMap.ker f)) := by |
rw [← lift_rank_range_add_rank_ker f, ← rank_range_of_surjective f h]
| 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 86 | 88 | theorem rank_eq_of_surjective {f : M →ₗ[R] M₁} (h : Surjective f) :
Module.rank R M = Module.rank R M₁ + Module.rank R (LinearMap.ker f) := by |
rw [← rank_range_add_rank_ker f, ← rank_range_of_surjective f h]
| 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 91 | 109 | theorem exists_linearIndependent_of_lt_rank [StrongRankCondition R]
{s : Set M} (hs : LinearIndependent (ι := s) R Subtype.val) :
∃ t, s ⊆ t ∧ #t = Module.rank R M ∧ LinearIndependent (ι := t) R Subtype.val := by |
obtain ⟨t, ht, ht'⟩ := exists_set_linearIndependent R (M ⧸ Submodule.span R s)
choose sec hsec using Submodule.Quotient.mk_surjective (Submodule.span R s)
have hsec' : Submodule.Quotient.mk ∘ sec = id := funext hsec
have hst : Disjoint s (sec '' t) := by
rw [Set.disjoint_iff]
rintro _ ⟨hxs, ⟨x, hxt, rf... | 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 113 | 123 | theorem exists_linearIndependent_cons_of_lt_rank [StrongRankCondition R] {n : ℕ} {v : Fin n → M}
(hv : LinearIndependent R v) (h : n < Module.rank R M) :
∃ (x : M), LinearIndependent R (Fin.cons x v) := by |
obtain ⟨t, h₁, h₂, h₃⟩ := exists_linearIndependent_of_lt_rank hv.to_subtype_range
have : range v ≠ t := by
refine fun e ↦ h.ne ?_
rw [← e, ← lift_injective.eq_iff, mk_range_eq_of_injective hv.injective] at h₂
simpa only [mk_fintype, Fintype.card_fin, lift_natCast, lift_id'] using h₂
obtain ⟨x, hx, hx... | 1,687 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 127 | 132 | theorem exists_linearIndependent_snoc_of_lt_rank [StrongRankCondition R] {n : ℕ} {v : Fin n → M}
(hv : LinearIndependent R v) (h : n < Module.rank R M) :
∃ (x : M), LinearIndependent R (Fin.snoc v x) := by |
simp only [Fin.snoc_eq_cons_rotate]
have ⟨x, hx⟩ := exists_linearIndependent_cons_of_lt_rank hv h
exact ⟨x, hx.comp _ (finRotate _).injective⟩
| 1,687 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 64 | 66 | theorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by |
dsimp only [IsOrtho]
rw [map_zero B, zero_apply]
| 1,688 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 73 | 74 | theorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by |
simp_rw [isOrtho_def, flip_apply]
| 1,688 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 91 | 98 | theorem isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :
B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by |
simp_rw [isOrthoᵢ_def]
constructor <;> intro h i j hij
· rw [flip_apply]
exact h j i (Ne.symm hij)
simp_rw [flip_apply] at h
exact h j i (Ne.symm hij)
| 1,688 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 246 | 252 | theorem isSymm_iff_eq_flip {B : LinearMap.BilinForm R M} : B.IsSymm ↔ B = B.flip := by |
constructor <;> intro h
· ext
rw [← h, flip_apply, RingHom.id_apply]
intro x y
conv_lhs => rw [h]
rfl
| 1,688 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 73 | 75 | theorem polar_eq_iInter {s : Set E} : B.polar s = ⋂ x ∈ s, { y : F | ‖B x y‖ ≤ 1 } := by |
ext
simp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq]
| 1,689 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 106 | 109 | theorem polar_zero : B.polar ({0} : Set E) = Set.univ := by |
refine Set.eq_univ_iff_forall.mpr fun y x hx => ?_
rw [Set.mem_singleton_iff.mp hx, map_zero, LinearMap.zero_apply, norm_zero]
exact zero_le_one
| 1,689 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 123 | 127 | theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
(B.polar s) := by |
rw [polar_eq_iInter]
refine isClosed_iInter fun x => isClosed_iInter fun _ => ?_
exact isClosed_le (WeakBilin.eval_continuous B.flip x).norm continuous_const
| 1,689 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.LinearAlgebra.Projection
#align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
set_option ... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 67 | 69 | theorem range_extend (hs : LinearIndependent K ((↑) : s → V)) :
range (Basis.extend hs) = hs.extend (subset_univ _) := by |
rw [coe_extend, Subtype.range_coe_subtype, setOf_mem_eq]
| 1,690 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.LinearAlgebra.Projection
#align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
set_option ... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 117 | 119 | theorem ofVectorSpace_apply_self (x : ofVectorSpaceIndex K V) : ofVectorSpace K V x = x := by |
unfold ofVectorSpace
exact Basis.mk_apply _ _ _
| 1,690 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.LinearAlgebra.Projection
#align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
set_option ... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 127 | 131 | theorem ofVectorSpaceIndex.linearIndependent :
LinearIndependent K ((↑) : ofVectorSpaceIndex K V → V) := by |
convert (ofVectorSpace K V).linearIndependent
ext x
rw [ofVectorSpace_apply_self]
| 1,690 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 67 | 68 | theorem of_subsingleton [Subsingleton G] : UniqueMul A B a0 b0 := by |
simp [UniqueMul, eq_iff_true_of_subsingleton]
| 1,691 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 71 | 75 | theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :
∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b := by |
rw [Finset.card_le_one_iff] at hA1 hB1
obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩
| 1,691 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 95 | 101 | theorem set_subsingleton (h : UniqueMul A B a0 b0) :
Set.Subsingleton { ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 } := by |
rintro ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0) ⟨x2, y2⟩
(hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0)
rcases h hx.1 hx.2.1 hx.2.2 with ⟨rfl, rfl⟩
rcases h hy.1 hy.2.1 hy.2.2 with ⟨rfl, rfl⟩
rfl
| 1,691 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 121 | 129 | theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
UniqueMul A B a0 b0 ↔ ((A ×ˢ B).filter (fun p ↦ p.1 * p.2 = a0 * b0)).card ≤ 1 := by |
simp_rw [card_le_one_iff, mem_filter, mem_product]
refine ⟨fun h p1 p2 ⟨⟨ha1, hb1⟩, he1⟩ ⟨⟨ha2, hb2⟩, he2⟩ ↦ ?_, fun h a b ha hb he ↦ ?_⟩
· have h1 := h ha1 hb1 he1; have h2 := h ha2 hb2 he2
ext
· rw [h1.1, h2.1]
· rw [h1.2, h2.2]
· exact Prod.ext_iff.1 (@h (a, b) (a0, b0) ⟨⟨ha, hb⟩, he⟩ ⟨⟨ha0, hb0... | 1,691 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.Group.UniqueProds
#align_import algebra.monoid_algebra.no_zero_divisors from "leanprover-community/mathlib"@"3e067975886cf5801e597925328c335609511b1a"
open Finsupp
variable {R A : Type*} [Semiring R]
namespace MonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | 68 | 79 | theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 := by |
classical
simp_rw [mul_apply, sum, ← Finset.sum_product']
refine (Finset.sum_eq_single (a0, b0) ?_ ?_).trans (if_pos rfl) <;> simp_rw [Finset.mem_product]
· refine fun ab hab hne => if_neg (fun he => hne <| Prod.ext ?_ ?_)
exacts [(h hab.1 hab.2 he).1, (h hab.1 hab.2 he).2]
· refine fun hnmem => ite_eq_r... | 1,692 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
#align_import field_theory.finiteness from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
universe u v
open scoped Classical
open Cardinal
... | Mathlib/FieldTheory/Finiteness.lean | 32 | 43 | theorem iff_rank_lt_aleph0 : IsNoetherian K V ↔ Module.rank K V < ℵ₀ := by |
let b := Basis.ofVectorSpace K V
rw [← b.mk_eq_rank'', lt_aleph0_iff_set_finite]
constructor
· intro
exact (Basis.ofVectorSpaceIndex.linearIndependent K V).set_finite_of_isNoetherian
· intro hbfinite
refine
@isNoetherian_of_linearEquiv K (⊤ : Submodule K V) V _ _ _ _ _ (LinearEquiv.ofTop _ rfl)... | 1,693 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
#align_import field_theory.finiteness from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
universe u v
open scoped Classical
open Cardinal
... | Mathlib/FieldTheory/Finiteness.lean | 95 | 97 | theorem range_finsetBasis [IsNoetherian K V] :
Set.range (finsetBasis K V) = Basis.ofVectorSpaceIndex K V := by |
rw [finsetBasis, Basis.range_reindex, Basis.range_ofVectorSpace]
| 1,693 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
#align_import field_theory.finiteness from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
universe u v
open scoped Classical
open Cardinal
... | Mathlib/FieldTheory/Finiteness.lean | 103 | 112 | theorem iff_fg : IsNoetherian K V ↔ Module.Finite K V := by |
constructor
· intro h
exact
⟨⟨finsetBasisIndex K V, by
convert (finsetBasis K V).span_eq
simp⟩⟩
· rintro ⟨s, hs⟩
rw [IsNoetherian.iff_rank_lt_aleph0, ← rank_top, ← hs]
exact lt_of_le_of_lt (rank_span_le _) s.finite_toSet.lt_aleph0
| 1,693 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 59 | 63 | theorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :
Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V := by |
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq
| 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 81 | 108 | theorem rank_add_rank_split (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂)
(ce : V₁ →ₗ[K] V₃) (hde : ⊤ ≤ LinearMap.range db ⊔ LinearMap.range eb) (hgd : ker cd = ⊥)
(eq : db.comp cd = eb.comp ce) (eq₂ : ∀ d e, db d = eb e → ∃ c, cd c = d ∧ ce c = e) :
Module.rank K V + Module.rank K V₁ = Module.rank... |
have hf : Surjective (coprod db eb) := by rwa [← range_eq_top, range_coprod, eq_top_iff]
conv =>
rhs
rw [← rank_prod', rank_eq_of_surjective hf]
congr 1
apply LinearEquiv.rank_eq
let L : V₁ →ₗ[K] ker (coprod db eb) := by -- Porting note: this is needed to avoid a timeout
refine LinearMap.codRestr... | 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'l... |
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finr... | 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 171 | 193 | theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K := by |
constructor
· intro h
exact (finrank_span_eq_card h).symm
· intro hc
let f := Submodule.subtype (span K (Set.range b))
let b' : ι → span K (Set.range b) := fun i =>
⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩
have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h :... | 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 196 | 198 | theorem linearIndependent_iff_card_le_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι ≤ (Set.range b).finrank K := by |
rw [linearIndependent_iff_card_eq_finrank_span, (finrank_range_le_card _).le_iff_eq]
| 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 239 | 283 | theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) := by |
have aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K) := (rank_finsupp_self K ℕ).symm.trans_le
(Finsupp.lcoeFun.rank_le_of_injective <| by exact DFunLike.coe_injective)
refine max_le aleph0_le ?_
obtain card_K | card_K := le_or_lt #K ℵ₀
· exact card_K.trans aleph0_le
by_contra!
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.e... | 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 288 | 300 | theorem rank_fun_infinite {ι : Type v} [hι : Infinite ι] : Module.rank K (ι → K) = #(ι → K) := by |
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ι → K)
obtain ⟨e⟩ := lift_mk_le'.mp ((aleph0_le_mk_iff.mpr hι).trans_eq (lift_uzero #ι).symm)
have := LinearMap.lift_rank_le_of_injective _ <|
LinearMap.funLeft_injective_of_surjective K K _ (invFun_surjective e.injective)
rw [lift_umax.{u,v}, li... | 1,694 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 304 | 311 | theorem rank_dual_eq_card_dual_of_aleph0_le_rank' {V : Type*} [AddCommGroup V] [Module K V]
(h : ℵ₀ ≤ Module.rank K V) : Module.rank Kᵐᵒᵖ (V →ₗ[K] K) = #(V →ₗ[K] K) := by |
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := K) (M := V)
rw [← b.mk_eq_rank'', aleph0_le_mk_iff] at h
have e := (b.constr Kᵐᵒᵖ (M' := K)).symm.trans
(LinearEquiv.piCongrRight fun _ ↦ MulOpposite.opLinearEquiv Kᵐᵒᵖ)
rw [e.rank_eq, e.toEquiv.cardinal_eq]
apply rank_fun_infinite
| 1,694 |
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import linear_algebra.finite_dimensional from "leanprover-community/mathlib"@"e95e4f92c8f8da3c7f693c3ec948bcf9b6683f51"
universe u v v' w
open Cardina... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 123 | 126 | theorem finite_of_finite [Finite K] [FiniteDimensional K V] : Finite V := by |
cases nonempty_fintype K
haveI := fintypeOfFintype K V
infer_instance
| 1,695 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 108 | 116 | theorem mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ a : K, a • w = v := by |
rw [mk_eq_mk_iff K v w hv hw]
constructor
· rintro ⟨a, ha⟩
exact ⟨a, ha⟩
· rintro ⟨a, ha⟩
refine ⟨Units.mk0 a fun c => hv.symm ?_, ha⟩
rwa [c, zero_smul] at ha
| 1,696 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 137 | 139 | theorem submodule_eq (v : ℙ K V) : v.submodule = K ∙ v.rep := by |
conv_lhs => rw [← v.mk_rep]
rfl
| 1,696 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 142 | 144 | theorem finrank_submodule (v : ℙ K V) : finrank K v.submodule = 1 := by |
rw [submodule_eq]
exact finrank_span_singleton v.rep_nonzero
| 1,696 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (K V : Type*) [Field K] [AddCommGroup V] [Module K V]
namespace Projectivization
open scoped LinearAlgebra.Projectivi... | Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 155 | 158 | theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ := by |
rw [eq_top_iff, SetLike.le_def]
intro x _hx
exact subset_span _ (Set.mem_univ x)
| 1,697 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 48 | 58 | theorem independent_iff : Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f) := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh⟩
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert hh.units_smul a
ext i
exact (ha i).symm
· convert Independent.mk _ _ h
· simp only [mk_rep, Function.comp_apply]
· intro i
apply rep_nonzero
| 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 63 | 72 | theorem independent_iff_completeLattice_independent :
Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨f, hf, hi⟩
simp only [submodule_mk]
exact (CompleteLattice.independent_iff_linearIndependent_of_ne_zero (R := K) hf).mpr hi
· rw [independent_iff]
refine h.linearIndependent (Projectivization.submodule ∘ f) (fun i => ?_) fun i => ?_
· simpa only [Function.comp_a... | 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 84 | 94 | theorem dependent_iff : Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f) := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh1⟩
contrapose! hh1
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert hh1.units_smul a⁻¹
ext i
simp only [← ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]
· convert Dependent.mk _ _ h
· simp on... | 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 98 | 99 | theorem dependent_iff_not_independent : Dependent f ↔ ¬Independent f := by |
rw [dependent_iff, independent_iff]
| 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 103 | 104 | theorem independent_iff_not_dependent : Independent f ↔ ¬Dependent f := by |
rw [dependent_iff_not_independent, Classical.not_not]
| 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 109 | 114 | theorem dependent_pair_iff_eq (u v : ℙ K V) : Dependent ![u, v] ↔ u = v := by |
rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2,
Function.comp_apply, Matrix.cons_val_one, Matrix.head_cons, Ne]
simp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply,
← mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, Classical.... | 1,698 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 119 | 120 | theorem independent_pair_iff_neq (u v : ℙ K V) : Independent ![u, v] ↔ u ≠ v := by |
rw [independent_iff_not_dependent, dependent_pair_iff_eq u v]
| 1,698 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 69 | 69 | theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by | simp [eigenspace]
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 98 | 101 | theorem hasEigenvalue_of_hasEigenvector {f : End R M} {μ : R} {x : M} (h : HasEigenvector f μ x) :
HasEigenvalue f μ := by |
rw [HasEigenvalue, Submodule.ne_bot_iff]
use x; exact h
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 104 | 105 | theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x := by |
rw [eigenspace, LinearMap.mem_ker, LinearMap.sub_apply, algebraMap_end_apply, sub_eq_zero]
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 113 | 115 | theorem HasEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasEigenvector μ v) (n : ℕ) :
(f ^ n) v = μ ^ n • v := by |
induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 138 | 144 | theorem HasEigenvalue.mem_spectrum {f : End R M} {μ : R} (hμ : HasEigenvalue f μ) :
μ ∈ spectrum R f := by |
refine spectrum.mem_iff.mpr fun h_unit => ?_
set f' := LinearMap.GeneralLinearGroup.toLinearEquiv h_unit.unit
rcases hμ.exists_hasEigenvector with ⟨v, hv⟩
refine hv.2 ((LinearMap.ker_eq_bot'.mp f'.ker) v (?_ : μ • v - f v = 0))
rw [hv.apply_eq_smul, sub_self]
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 147 | 149 | theorem hasEigenvalue_iff_mem_spectrum [FiniteDimensional K V] {f : End K V} {μ : K} :
f.HasEigenvalue μ ↔ μ ∈ spectrum K f := by |
rw [spectrum.mem_iff, IsUnit.sub_iff, LinearMap.isUnit_iff_ker_eq_bot, HasEigenvalue, eigenspace]
| 1,699 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 154 | 163 | theorem eigenspace_div (f : End K V) (a b : K) (hb : b ≠ 0) :
eigenspace f (a / b) = LinearMap.ker (b • f - algebraMap K (End K V) a) :=
calc
eigenspace f (a / b) = eigenspace f (b⁻¹ * a) := by | rw [div_eq_mul_inv, mul_comm]
_ = LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id) := by rw [eigenspace]; rfl
_ = LinearMap.ker (f - b⁻¹ • a • LinearMap.id) := by rw [smul_smul]
_ = LinearMap.ker (f - b⁻¹ • algebraMap K (End K V) a) := rfl
_ = LinearMap.ker (b • (f - b⁻¹ • algebraMap K (End K V) a)) := by
... | 1,699 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 86 | 88 | theorem isSimpleModule_iff_isAtom : IsSimpleModule R m ↔ IsAtom m := by |
rw [← Set.isSimpleOrder_Iic_iff_isAtom]
exact m.mapIic.isSimpleOrder_iff
| 1,700 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 91 | 94 | theorem isSimpleModule_iff_isCoatom : IsSimpleModule R (M ⧸ m) ↔ IsCoatom m := by |
rw [← Set.isSimpleOrder_Ici_iff_isCoatom]
apply OrderIso.isSimpleOrder_iff
exact Submodule.comapMkQRelIso m
| 1,700 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 97 | 101 | theorem covBy_iff_quot_is_simple {A B : Submodule R M} (hAB : A ≤ B) :
A ⋖ B ↔ IsSimpleModule R (B ⧸ Submodule.comap B.subtype A) := by |
set f : Submodule R B ≃o Set.Iic B := B.mapIic with hf
rw [covBy_iff_coatom_Iic hAB, isSimpleModule_iff_isCoatom, ← OrderIso.isCoatom_iff f, hf]
simp [-OrderIso.isCoatom_iff, Submodule.map_comap_subtype, inf_eq_right.2 hAB]
| 1,700 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 125 | 127 | theorem toSpanSingleton_surjective {m : M} (hm : m ≠ 0) :
Function.Surjective (toSpanSingleton R M m) := by |
rw [← range_eq_top, ← span_singleton_eq_range, span_singleton_eq_top R hm]
| 1,700 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 129 | 132 | theorem ker_toSpanSingleton_isMaximal {m : M} (hm : m ≠ 0) :
Ideal.IsMaximal (ker (toSpanSingleton R M m)) := by |
rw [Ideal.isMaximal_def, ← isSimpleModule_iff_isCoatom]
exact congr (quotKerEquivOfSurjective _ <| toSpanSingleton_surjective R hm)
| 1,700 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 81 | 83 | theorem conjugate_i (g : G) (v : V) : (conjugate π g : W → V) (i v) = v := by |
rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, mul_left_inv,
← one_def, one_smul]
| 1,701 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 125 | 127 | theorem equivariantProjection_apply (v : W) :
π.equivariantProjection G v = ⅟(Fintype.card G : k) • ∑ g : G, π.conjugate g v := by |
simp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply]
| 1,701 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 129 | 133 | theorem equivariantProjection_condition (v : V) : (π.equivariantProjection G) (i v) = v := by |
rw [equivariantProjection_apply]
simp only [conjugate_i π i h]
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_smul_cast k, smul_smul,
Invertible.invOf_mul_self, one_smul]
| 1,701 |
import Mathlib.RingTheory.SimpleModule
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.simple from "leanprover-community/mathlib"@"f430769b562e0cedef59ee1ed968d67e0e0c86ba"
universe u v w
variable {R : Type u} {M : Type v} {N : Type w} [Ring R] [TopologicalSpace R] [Topological... | Mathlib/Topology/Algebra/Module/Simple.lean | 28 | 34 | theorem LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M) := by |
rcases l.surjective_or_eq_zero with (hl | rfl)
· exact l.ker.isClosed_or_dense_of_isCoatom (LinearMap.isCoatom_ker_of_surjective hl)
· rw [LinearMap.ker_zero]
left
exact isClosed_univ
| 1,702 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 46 | 47 | theorem rank_zero [Nontrivial K] : rank (0 : V →ₗ[K] V') = 0 := by |
rw [rank, LinearMap.range_zero, rank_bot]
| 1,703 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 52 | 55 | theorem rank_comp_le_left (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') : rank (f.comp g) ≤ rank f := by |
refine rank_le_of_submodule _ _ ?_
rw [LinearMap.range_comp]
exact LinearMap.map_le_range
| 1,703 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 58 | 60 | theorem lift_rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') :
Cardinal.lift.{v'} (rank (f.comp g)) ≤ Cardinal.lift.{v''} (rank g) := by |
rw [rank, rank, LinearMap.range_comp]; exact lift_rank_map_le _ _
| 1,703 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 72 | 73 | theorem rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'₁) : rank (f.comp g) ≤ rank g := by |
simpa only [Cardinal.lift_id] using lift_rank_comp_le_right g f
| 1,703 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 79 | 81 | theorem rank_comp_le (g : V →ₗ[K] V') (f : V' →ₗ[K] V'₁) :
rank (f.comp g) ≤ min (rank f) (rank g) := by |
simpa only [Cardinal.lift_id] using lift_rank_comp_le g f
| 1,703 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 53 | 56 | theorem FiniteDimensional.rank_linearMap :
Module.rank S (M →ₗ[R] N) = lift.{w} (Module.rank R M) * lift.{v} (Module.rank S N) := by |
rw [(linearMapEquivFun R S M N).rank_eq, rank_fun_eq_lift_mul,
← finrank_eq_card_chooseBasisIndex, ← finrank_eq_rank R, lift_natCast]
| 1,704 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 59 | 61 | theorem FiniteDimensional.finrank_linearMap :
finrank S (M →ₗ[R] N) = finrank R M * finrank S N := by |
simp_rw [finrank, rank_linearMap, toNat_mul, toNat_lift]
| 1,704 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 66 | 68 | theorem FiniteDimensional.rank_linearMap_self :
Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) := by |
rw [rank_linearMap, rank_self, lift_one, mul_one]
| 1,704 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 70 | 71 | theorem FiniteDimensional.finrank_linearMap_self : finrank S (M →ₗ[R] S) = finrank R M := by |
rw [finrank_linearMap, finrank_self, mul_one]
| 1,704 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 113 | 119 | theorem Matrix.rank_vecMulVec {K m n : Type u} [CommRing K] [Fintype n]
[DecidableEq n] (w : m → K) (v : n → K) : (Matrix.vecMulVec w v).toLin'.rank ≤ 1 := by |
nontriviality K
rw [Matrix.vecMulVec_eq, Matrix.toLin'_mul]
refine le_trans (LinearMap.rank_comp_le_left _ _) ?_
refine (LinearMap.rank_le_domain _).trans_eq ?_
rw [rank_fun', Fintype.card_unit, Nat.cast_one]
| 1,704 |
import Mathlib.Algebra.Module.Submodule.Localization
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.OreLocalization.OreSet
open Cardinal nonZeroDivisors
section CommRing
universe u u' v v'
variable {R : Type u} (S : Type u') {M : T... | Mathlib/LinearAlgebra/Dimension/Localization.lean | 85 | 93 | theorem exists_set_linearIndependent_of_isDomain [IsDomain R] :
∃ s : Set M, #s = Module.rank R M ∧ LinearIndependent (ι := s) R Subtype.val := by |
obtain ⟨w, hw⟩ :=
IsLocalizedModule.linearIndependent_lift R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl
(Module.Free.chooseBasis (FractionRing R) (LocalizedModule R⁰ M)).linearIndependent
refine ⟨Set.range w, ?_, (linearIndependent_subtype_range hw.injective).mpr hw⟩
apply Cardinal.lift_injective.{max ... | 1,705 |
import Mathlib.Algebra.Module.Submodule.Localization
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.OreLocalization.OreSet
open Cardinal nonZeroDivisors
section CommRing
universe u u' v v'
variable {R : Type u} (S : Type u') {M : T... | Mathlib/LinearAlgebra/Dimension/Localization.lean | 96 | 102 | theorem rank_quotient_add_rank_of_isDomain [IsDomain R] (M' : Submodule R M) :
Module.rank R (M ⧸ M') + Module.rank R M' = Module.rank R M := by |
apply lift_injective.{max u v}
rw [lift_add, ← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (M'.toLocalized R⁰) le_rfl,
← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl,
← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (M'.toLocalizedQuotient R⁰) le_r... | 1,705 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 72 | 81 | theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔
∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 := by |
constructor
· exact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _)
· intro h s w hw hs i hi
rw [Finset.weightedVSub_indicator_subset _ _ (Finset.subset_univ s)] at hs
rw [← Finset.sum_indicator_subset _ (Finset.subset_univ s)] at hw
replace h := h ((↑s : Set ι).indicator w) hw hs i
... | 1,706 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 86 | 134 | theorem affineIndependent_iff_linearIndependent_vsub (p : ι → P) (i1 : ι) :
AffineIndependent k p ↔ LinearIndependent k fun i : { x // x ≠ i1 } => (p i -ᵥ p i1 : V) := by |
classical
constructor
· intro h
rw [linearIndependent_iff']
intro s g hg i hi
set f : ι → k := fun x => if hx : x = i1 then -∑ y ∈ s, g y else g ⟨x, hx⟩ with hfdef
let s2 : Finset ι := insert i1 (s.map (Embedding.subtype _))
have hfg : ∀ x : { x // x ≠ i1 }, g x = f x := by
... | 1,706 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 139 | 158 | theorem affineIndependent_set_iff_linearIndependent_vsub {s : Set P} {p₁ : P} (hp₁ : p₁ ∈ s) :
AffineIndependent k (fun p => p : s → P) ↔
LinearIndependent k (fun v => v : (fun p => (p -ᵥ p₁ : V)) '' (s \ {p₁}) → V) := by |
rw [affineIndependent_iff_linearIndependent_vsub k (fun p => p : s → P) ⟨p₁, hp₁⟩]
constructor
· intro h
have hv : ∀ v : (fun p => (p -ᵥ p₁ : V)) '' (s \ {p₁}), (v : V) +ᵥ p₁ ∈ s \ {p₁} := fun v =>
(vsub_left_injective p₁).mem_set_image.1 ((vadd_vsub (v : V) p₁).symm ▸ v.property)
let f : (fun p : ... | 1,706 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem mem_space_iff : x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ convexHull 𝕜 (s : Set E) := by |
simp [space]
| 1,707 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 91 | 93 | theorem convexHull_subset_space (hs : s ∈ K.faces) : convexHull 𝕜 ↑s ⊆ K.space := by |
convert subset_biUnion_of_mem hs
rfl
| 1,707 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 110 | 119 | theorem disjoint_or_exists_inter_eq_convexHull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
Disjoint (convexHull 𝕜 (s : Set E)) (convexHull 𝕜 ↑t) ∨
∃ u ∈ K.faces, convexHull 𝕜 (s : Set E) ∩ convexHull 𝕜 ↑t = convexHull 𝕜 ↑u := by |
classical
by_contra! h
refine h.2 (s ∩ t) (K.down_closed hs inter_subset_left fun hst => h.1 <|
disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans ?_) ?_
· rw [← coe_inter, hst, coe_empty, convexHull_empty]
rfl
· rw [coe_inter, convexHull_inter_convexHull hs ht]
| 1,707 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 158 | 162 | theorem vertices_eq : K.vertices = ⋃ k ∈ K.faces, (k : Set E) := by |
ext x
refine ⟨fun h => mem_biUnion h <| mem_coe.2 <| mem_singleton_self x, fun h => ?_⟩
obtain ⟨s, hs, hx⟩ := mem_iUnion₂.1 h
exact K.down_closed hs (Finset.singleton_subset_iff.2 <| mem_coe.1 hx) (singleton_ne_empty _)
| 1,707 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis
import Mathlib.RingTheory.Ideal.Basic
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace Ideal
variable {ι R S : Type*} [CommSemiring R] [CommRing S] [IsDomai... | Mathlib/RingTheory/Ideal/Basis.lean | 35 | 39 | theorem basisSpanSingleton_apply (b : Basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basisSpanSingleton b hx i : S) = x * b i := by |
simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,
Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,
LinearEquiv.restrictScalars_apply, Algebra.coe_lmul_eq_mul, LinearMap.mul_apply']
| 1,708 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Basis
#align_import ring_theory.algebra_tower from "leanprover-community/mathlib"@"94825b2b0b982306be14d891c4f063a1eca4f370"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type ... | Mathlib/RingTheory/AlgebraTower.lean | 108 | 121 | theorem linearIndependent_smul {ι : Type v₁} {b : ι → S} {ι' : Type w₁} {c : ι' → A}
(hb : LinearIndependent R b) (hc : LinearIndependent S c) :
LinearIndependent R fun p : ι × ι' => b p.1 • c p.2 := by |
rw [linearIndependent_iff'] at hb hc; rw [linearIndependent_iff'']; rintro s g hg hsg ⟨i, k⟩
by_cases hik : (i, k) ∈ s
· have h1 : ∑ i ∈ s.image Prod.fst ×ˢ s.image Prod.snd, g i • b i.1 • c i.2 = 0 := by
rw [← hsg]
exact
(Finset.sum_subset Finset.subset_product fun p _ hp =>
show... | 1,709 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 91 | 99 | theorem Matrix.vecMul_stdBasis (M : Matrix m n R) (i j) :
(LinearMap.stdBasis R (fun _ ↦ R) i 1 ᵥ* M) j = M i j := by |
have : (∑ i', (if i = i' then 1 else 0) * M i' j) = M i j := by
simp_rw [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true]
simp only [vecMul, dotProduct]
convert this
split_ifs with h <;> simp only [stdBasis_apply]
· rw [h, Function.update_same]
· rw [Function.update_noteq (Ne.symm h), Pi.zero_ap... | 1,710 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 102 | 110 | theorem range_vecMulLinear (M : Matrix m n R) :
LinearMap.range M.vecMulLinear = span R (range M) := by |
letI := Classical.decEq m
simp_rw [range_eq_map, ← iSup_range_stdBasis, Submodule.map_iSup, range_eq_map, ←
Ideal.span_singleton_one, Ideal.span, Submodule.map_span, image_image, image_singleton,
Matrix.vecMulLinear_apply, iSup_span, range_eq_iUnion, iUnion_singleton_eq_range,
LinearMap.stdBasis, coe_s... | 1,710 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 112 | 123 | theorem Matrix.vecMul_injective_iff {R : Type*} [CommRing R] {M : Matrix m n R} :
Function.Injective M.vecMul ↔ LinearIndependent R (fun i ↦ M i) := by |
rw [← coe_vecMulLinear]
simp only [← LinearMap.ker_eq_bot, Fintype.linearIndependent_iff, Submodule.eq_bot_iff,
LinearMap.mem_ker, vecMulLinear_apply]
refine ⟨fun h c h0 ↦ congr_fun <| h c ?_, fun h c h0 ↦ funext <| h c ?_⟩
· rw [← h0]
ext i
simp [vecMul, dotProduct]
· rw [← h0]
ext j
sim... | 1,710 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 173 | 176 | theorem Matrix.toLinearMapRight'_one :
Matrix.toLinearMapRight' (1 : Matrix m m R) = LinearMap.id := by |
ext
simp [LinearMap.one_apply, stdBasis_apply]
| 1,710 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 25 | 29 | theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
[FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
(RingHom.ker f).IsRadical ↔ IsReduced S := by |
simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker]
rfl
| 1,711 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 32 | 35 | theorem isRadical_iff_span_singleton [CommSemiring R] :
IsRadical y ↔ (Ideal.span ({y} : Set R)).IsRadical := by |
simp_rw [IsRadical, ← Ideal.mem_span_singleton]
exact forall_swap.trans (forall_congr' fun r => exists_imp.symm)
| 1,711 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 123 | 126 | theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by |
obtain ⟨k, hk⟩ := hnp
use k
simp [← p.mapQ_pow, hk]
| 1,711 |
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 81 | 91 | theorem isNoetherian_submodule {N : Submodule R M} :
IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by |
refine ⟨fun ⟨hn⟩ => fun s hs =>
have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs
Submodule.map_comap_eq_self this ▸ (hn _).map _,
fun h => ⟨fun s => ?_⟩⟩
have f := (Submodule.equivMapOfInjective N.subtype Subtype.val_injective s).symm
have h₁ := h (s.map N.subtype) (Submodule.map_sub... | 1,712 |
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 136 | 139 | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by |
constructor <;> intro h
· exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl)
· exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl).symm
| 1,712 |
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 59 | 65 | theorem IsAssociatedPrime.map_of_injective (h : IsAssociatedPrime I M) (hf : Function.Injective f) :
IsAssociatedPrime I M' := by |
obtain ⟨x, rfl⟩ := h.2
refine ⟨h.1, ⟨f x, ?_⟩⟩
ext r
rw [Submodule.mem_annihilator_span_singleton, Submodule.mem_annihilator_span_singleton, ←
map_smul, ← f.map_zero, hf.eq_iff]
| 1,713 |
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 74 | 78 | theorem not_isAssociatedPrime_of_subsingleton [Subsingleton M] : ¬IsAssociatedPrime I M := by |
rintro ⟨hI, x, hx⟩
apply hI.ne_top
rwa [Subsingleton.elim x 0, Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot]
at hx
| 1,713 |
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