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.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 66 | 69 | theorem isUnit_of_isUnit_ι {m : M} (h : IsUnit (ι Q m)) : IsUnit (Q m) := by |
cases h.nonempty_invertible
letI := invertibleOfInvertibleι Q m
exact isUnit_of_invertible (Q m)
| 1,916 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
| Mathlib/RingTheory/Nakayama.lean | 52 | 61 | theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N : Submodule R M} (hN : N.FG)
(hIN : N ≤ I • N) (hIjac : I ≤ jacobson J) : N = J • N := by |
refine le_antisymm ?_ (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _)
intro n hn
cases' Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hN hIN with r hr
cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r (hIjac hr.1) with s hs
have : n = -(s * r - 1) • n := by
rw [neg_sub, s... | 1,917 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 109 | 111 | theorem eq_bot_of_le_smul_of_le_jacobson_bot (I : Ideal R) (N : Submodule R M) (hN : N.FG)
(hIN : N ≤ I • N) (hIjac : I ≤ jacobson ⊥) : N = ⊥ := by |
rw [eq_smul_of_le_smul_of_le_jacobson hN hIN hIjac, Submodule.bot_smul]
| 1,917 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 114 | 126 | theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
(hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N' ≤ N ⊔ I • N') : N ⊔ N' = N ⊔ J • N' := by |
have hNN' : N ⊔ N' = N ⊔ I • N' :=
le_antisymm (sup_le le_sup_left hNN)
(sup_le_sup_left (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _)
have h_comap := Submodule.comap_injective_of_surjective (LinearMap.range_eq_top.1 N.range_mkQ)
have : (I • N').map N.mkQ = N'.map N.mkQ := by
simpa onl... | 1,917 |
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tactic.RSuffices
#align_import ring_theory.artinian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
open Set Filter Po... | Mathlib/RingTheory/Artinian.lean | 175 | 195 | theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by |
refine by_contradiction fun hf => (RelEmbedding.wellFounded_iff_no_descending_seq.1
(wellFounded_submodule_lt (R := R) (M := M))).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
... | 1,918 |
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-community/mathlib"@"9822b65bfc4ac74537d77ae318d27df1df662471"
universe u v w w₁ w₂
section LieSubmodule
variable (R : Type u) (L : Type v) (M : Type ... | Mathlib/Algebra/Lie/Submodule.lean | 132 | 133 | theorem coe_toSubmodule_mk (p : Submodule R M) (h) :
(({ p with lie_mem := h } : LieSubmodule R L M) : Submodule R M) = p := by | cases p; rfl
| 1,919 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebra.lie.direct_sum from "leanprover-community/mathlib"@"c0cc689babd41c0e9d5f02429211ffbe2403472a"
universe u v w w₁
namespace DirectSum
open DF... | Mathlib/Algebra/Lie/DirectSum.lean | 130 | 136 | theorem lie_of_of_ne [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = 0 := by |
refine DFinsupp.ext fun k => ?_
rw [bracket_apply]
obtain rfl | hik := Decidable.eq_or_ne i k
· rw [of_eq_of_ne _ _ _ _ hij.symm, lie_zero, zero_apply]
· rw [of_eq_of_ne _ _ _ _ hik, zero_lie, zero_apply]
| 1,920 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebra.lie.direct_sum from "leanprover-community/mathlib"@"c0cc689babd41c0e9d5f02429211ffbe2403472a"
universe u v w w₁
namespace DirectSum
open DF... | Mathlib/Algebra/Lie/DirectSum.lean | 140 | 144 | theorem lie_of [DecidableEq ι] {i j : ι} (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = if hij : i = j then of L i ⁅x, hij.symm.recOn y⁆ else 0 := by |
obtain rfl | hij := Decidable.eq_or_ne i j
· simp only [lie_of_same L x y, dif_pos]
· simp only [lie_of_of_ne L hij x y, hij, dif_neg, dite_false]
| 1,920 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 62 | 81 | theorem lieIdeal_oper_eq_linear_span :
(↑⁅I, N⁆ : Submodule R M) =
Submodule.span R { m | ∃ (x : I) (n : N), ⁅(x : L), (n : M)⁆ = m } := by |
apply le_antisymm
· let s := { m : M | ∃ (x : ↥I) (n : ↥N), ⁅(x : L), (n : M)⁆ = m }
have aux : ∀ (y : L), ∀ m' ∈ Submodule.span R s, ⁅y, m'⁆ ∈ Submodule.span R s := by
intro y m' hm'
refine Submodule.span_induction (R := R) (M := M) (s := s)
(p := fun m' ↦ ⁅y, m'⁆ ∈ Submodule.span R s) hm'... | 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 84 | 93 | theorem lieIdeal_oper_eq_linear_span' :
(↑⁅I, N⁆ : Submodule R M) = Submodule.span R { m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m } := by |
rw [lieIdeal_oper_eq_linear_span]
congr
ext m
constructor
· rintro ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩
exact ⟨x, hx, n, hn, rfl⟩
· rintro ⟨x, hx, n, hn, rfl⟩
exact ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩
| 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 96 | 100 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by |
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm
| 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 103 | 104 | theorem lie_coe_mem_lie (x : I) (m : N) : ⁅(x : L), (m : M)⁆ ∈ ⁅I, N⁆ := by |
rw [lieIdeal_oper_eq_span]; apply subset_lieSpan; use x, m
| 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 111 | 116 | theorem lie_comm : ⁅I, J⁆ = ⁅J, I⁆ := by |
suffices ∀ I J : LieIdeal R L, ⁅I, J⁆ ≤ ⁅J, I⁆ by exact le_antisymm (this I J) (this J I)
clear! I J; intro I J
rw [lieIdeal_oper_eq_span, lieSpan_le]; rintro x ⟨y, z, h⟩; rw [← h]
rw [← lie_skew, ← lie_neg, ← LieSubmodule.coe_neg]
apply lie_coe_mem_lie
| 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 119 | 121 | theorem lie_le_right : ⁅I, N⁆ ≤ N := by |
rw [lieIdeal_oper_eq_span, lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn]
exact N.lie_mem n.property
| 1,921 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 124 | 124 | theorem lie_le_left : ⁅I, J⁆ ≤ I := by | rw [lie_comm]; exact lie_le_right I J
| 1,921 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
| 1,922 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 136 | 141 | theorem ideal_oper_maxTrivSubmodule_eq_bot (I : LieIdeal R L) :
⁅I, maxTrivSubmodule R L M⁆ = ⊥ := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.lieIdeal_oper_eq_linear_span,
LieSubmodule.bot_coeSubmodule, Submodule.span_eq_bot]
rintro m ⟨⟨x, hx⟩, ⟨⟨m, hm⟩, rfl⟩⟩
exact hm x
| 1,922 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 312 | 315 | theorem LieSubmodule.trivial_lie_oper_zero [LieModule.IsTrivial L M] : ⁅I, N⁆ = ⊥ := by |
suffices ⁅I, N⁆ ≤ ⊥ from le_bot_iff.mp this
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
rintro m ⟨x, n, h⟩; rw [trivial_lie_zero] at h; simp [← h]
| 1,922 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 318 | 326 | theorem LieSubmodule.lie_abelian_iff_lie_self_eq_bot : IsLieAbelian I ↔ ⁅I, I⁆ = ⊥ := by |
simp only [_root_.eq_bot_iff, lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le,
LieSubmodule.bot_coe, Set.subset_singleton_iff, Set.mem_setOf_eq, exists_imp]
refine
⟨fun h z x y hz =>
hz.symm.trans
(((I : LieSubalgebra R L).coe_bracket x y).symm.trans
((coe_zero_iff_zero _ _).mpr (by ... | 1,922 |
import Mathlib.Algebra.Lie.Abelian
#align_import algebra.lie.tensor_product from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
universe u v w w₁ w₂ w₃
variable {R : Type u} [CommRing R]
open LieModule
namespace TensorProduct
open scoped TensorProduct
namespace... | Mathlib/Algebra/Lie/TensorProduct.lean | 115 | 122 | theorem coe_liftLie_eq_lift_coe (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) :
⇑(liftLie R L M N P f) = lift R L M N P f := by |
suffices (liftLie R L M N P f : M ⊗[R] N →ₗ[R] P) = lift R L M N P f by
rw [← this, LieModuleHom.coe_toLinearMap]
ext m n
simp only [liftLie, LinearEquiv.trans_apply, LieModuleEquiv.coe_to_linearEquiv,
coe_linearMap_maxTrivLinearMapEquivLieModuleHom, coe_maxTrivEquiv_apply,
coe_linearMap_maxTrivLinea... | 1,923 |
import Mathlib.Algebra.Lie.Abelian
#align_import algebra.lie.tensor_product from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
universe u v w w₁ w₂ w₃
variable {R : Type u} [CommRing R]
open LieModule
namespace TensorProduct
open scoped TensorProduct
namespace... | Mathlib/Algebra/Lie/TensorProduct.lean | 125 | 127 | theorem liftLie_apply (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (m : M) (n : N) :
liftLie R L M N P f (m ⊗ₜ n) = f m n := by |
simp only [coe_liftLie_eq_lift_coe, LieModuleHom.coe_toLinearMap, lift_apply]
| 1,923 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 82 | 85 | theorem derivedSeriesOfIdeal_add (k l : ℕ) : D (k + l) I = D k (D l I) := by |
induction' k with k ih
· rw [Nat.zero_add, derivedSeriesOfIdeal_zero]
· rw [Nat.succ_add k l, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ, ih]
| 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 89 | 97 | theorem derivedSeriesOfIdeal_le {I J : LieIdeal R L} {k l : ℕ} (h₁ : I ≤ J) (h₂ : l ≤ k) :
D k I ≤ D l J := by |
revert l; induction' k with k ih <;> intro l h₂
· rw [le_zero_iff] at h₂; rw [h₂, derivedSeriesOfIdeal_zero]; exact h₁
· have h : l = k.succ ∨ l ≤ k := by rwa [le_iff_eq_or_lt, Nat.lt_succ_iff] at h₂
cases' h with h h
· rw [h, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ]
exact LieSubmodule... | 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 116 | 124 | theorem derivedSeriesOfIdeal_add_le_add (J : LieIdeal R L) (k l : ℕ) :
D (k + l) (I + J) ≤ D k I + D l J := by |
let D₁ : LieIdeal R L →o LieIdeal R L :=
{ toFun := fun I => ⁅I, I⁆
monotone' := fun I J h => LieSubmodule.mono_lie I J I J h h }
have h₁ : ∀ I J : LieIdeal R L, D₁ (I ⊔ J) ≤ D₁ I ⊔ J := by
simp [D₁, LieSubmodule.lie_le_right, LieSubmodule.lie_le_left, le_sup_of_le_right]
rw [← D₁.iterate_sup_le_su... | 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 127 | 128 | theorem derivedSeries_of_bot_eq_bot (k : ℕ) : derivedSeriesOfIdeal R L k ⊥ = ⊥ := by |
rw [eq_bot_iff]; exact derivedSeriesOfIdeal_le_self ⊥ k
| 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 131 | 133 | theorem abelian_iff_derived_one_eq_bot : IsLieAbelian I ↔ derivedSeriesOfIdeal R L 1 I = ⊥ := by |
rw [derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero,
LieSubmodule.lie_abelian_iff_lie_self_eq_bot]
| 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 136 | 138 | theorem abelian_iff_derived_succ_eq_bot (I : LieIdeal R L) (k : ℕ) :
IsLieAbelian (derivedSeriesOfIdeal R L k I) ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ := by |
rw [add_comm, derivedSeriesOfIdeal_add I 1 k, abelian_iff_derived_one_eq_bot]
| 1,924 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 0 := by |
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| 1,925 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 49 | 50 | theorem lieCharacter_apply_lie' (χ : LieCharacter R L) (x y : L) : ⁅χ x, χ y⁆ = 0 := by |
rw [LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| 1,925 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 52 | 60 | theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 := by |
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, ←
LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
refine Submodule.span_induction h ?_ ?_ ?_ ?_
· rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lieCharacter_apply_lie
· exact χ.map_zero
· intro y z hy ... | 1,925 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 64 | 67 | theorem le_normalizer : N ≤ N.normalizer := by |
intro m hm
rw [mem_normalizer]
exact fun x => N.lie_mem hm
| 1,926 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 70 | 71 | theorem normalizer_inf : (N₁ ⊓ N₂).normalizer = N₁.normalizer ⊓ N₂.normalizer := by |
ext; simp [← forall_and]
| 1,926 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 75 | 78 | theorem monotone_normalizer : Monotone (normalizer : LieSubmodule R L M → LieSubmodule R L M) := by |
intro N₁ N₂ h m hm
rw [mem_normalizer] at hm ⊢
exact fun x => h (hm x)
| 1,926 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 82 | 83 | theorem comap_normalizer (f : M' →ₗ⁅R,L⁆ M) : N.normalizer.comap f = (N.comap f).normalizer := by |
ext; simp
| 1,926 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 86 | 87 | theorem top_lie_le_iff_le_normalizer (N' : LieSubmodule R L M) :
⁅(⊤ : LieIdeal R L), N⁆ ≤ N' ↔ N ≤ N'.normalizer := by | rw [lie_le_iff]; tauto
| 1,926 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 485 | 490 | theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k := by |
induction' k with k ih
· simpa
simp only [ucs_succ]
-- Porting note: `mono` makes no progress
apply monotone_normalizer ih
| 1,927 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 493 | 496 | theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ := by |
induction' k with k ih
· simp
· rwa [ucs_succ, ih]
| 1,927 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 504 | 508 | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by |
rw [← ucs_eq_self_of_normalizer_eq_self h k]
mono
simp
| 1,927 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 58 | 61 | theorem normalizer_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
H.toLieSubmodule.normalizer = H.toLieSubmodule := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, coe_normalizer_eq_normalizer,
IsCartanSubalgebra.self_normalizing, coe_toLieSubmodule]
| 1,928 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 65 | 69 | theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
H.toLieSubmodule.ucs k = H.toLieSubmodule := by |
induction' k with k ih
· simp
· simp [ih]
| 1,928 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 72 | 94 | theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by |
constructor
· intro h
have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
le_antisymm hk
(LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_sel... | 1,928 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 114 | 118 | theorem LieIdeal.normalizer_eq_top {R : Type u} {L : Type v} [CommRing R] [LieRing L]
[LieAlgebra R L] (I : LieIdeal R L) : (I : LieSubalgebra R L).normalizer = ⊤ := by |
ext x
simpa only [LieSubalgebra.mem_normalizer_iff, LieSubalgebra.mem_top, iff_true_iff] using
fun y hy => I.lie_mem hy
| 1,928 |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 61 | 69 | theorem lie_mem_weightSpace_of_mem_weightSpace {χ₁ χ₂ : H → R} {x : L} {m : M}
(hx : x ∈ rootSpace H χ₁) (hm : m ∈ weightSpace M χ₂) : ⁅x, m⁆ ∈ weightSpace M (χ₁ + χ₂) := by |
rw [weightSpace, LieSubmodule.mem_iInf]
intro y
replace hx : x ∈ weightSpaceOf L (χ₁ y) y := by
rw [rootSpace, weightSpace, LieSubmodule.mem_iInf] at hx; exact hx y
replace hm : m ∈ weightSpaceOf M (χ₂ y) y := by
rw [weightSpace, LieSubmodule.mem_iInf] at hm; exact hm y
exact lie_mem_maxGenEigenspace... | 1,929 |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 127 | 132 | theorem coe_rootSpaceWeightSpaceProduct_tmul (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃)
(x : rootSpace H χ₁) (m : weightSpace M χ₂) :
(rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ (x ⊗ₜ m) : M) = ⁅(x : L), (m : M)⁆ := by |
simp only [rootSpaceWeightSpaceProduct, rootSpaceWeightSpaceProductAux, coe_liftLie_eq_lift_coe,
AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, lift_apply, LinearMap.coe_mk, AddHom.coe_mk,
Submodule.coe_mk]
| 1,929 |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 135 | 141 | theorem mapsTo_toEnd_weightSpace_add_of_mem_rootSpace (α χ : H → R)
{x : L} (hx : x ∈ rootSpace H α) :
MapsTo (toEnd R L M x) (weightSpace M χ) (weightSpace M (α + χ)) := by |
intro m hm
let x' : rootSpace H α := ⟨x, hx⟩
let m' : weightSpace M χ := ⟨m, hm⟩
exact (rootSpaceWeightSpaceProduct R L H M α χ (α + χ) rfl (x' ⊗ₜ m')).property
| 1,929 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 82 | 86 | theorem exists_smul_add_of_span_sup_eq_top (y : L) : ∃ t : R, ∃ z ∈ I, y = t • x + z := by |
have hy : y ∈ (⊤ : Submodule R L) := Submodule.mem_top
simp only [← hxI, Submodule.mem_sup, Submodule.mem_span_singleton] at hy
obtain ⟨-, ⟨t, rfl⟩, z, hz, rfl⟩ := hy
exact ⟨t, z, hz, rfl⟩
| 1,930 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 89 | 102 | theorem lie_top_eq_of_span_sup_eq_top (N : LieSubmodule R L M) :
(↑⁅(⊤ : LieIdeal R L), N⁆ : Submodule R M) =
(N : Submodule R M).map (toEnd R L M x) ⊔ (↑⁅I, N⁆ : Submodule R M) := by |
simp only [lieIdeal_oper_eq_linear_span', Submodule.sup_span, mem_top, exists_prop,
true_and, Submodule.map_coe, toEnd_apply_apply]
refine le_antisymm (Submodule.span_le.mpr ?_) (Submodule.span_mono fun z hz => ?_)
· rintro z ⟨y, n, hn : n ∈ N, rfl⟩
obtain ⟨t, z, hz, rfl⟩ := exists_smul_add_of_span_sup_e... | 1,930 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 105 | 125 | theorem lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ}
(hxn : toEnd R L M x ^ n = 0) (hIM : lowerCentralSeries R L M i ≤ I.lcs M j) :
lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1) := by |
suffices
∀ l,
((⊤ : LieIdeal R L).lcs M (i + l) : Submodule R M) ≤
(I.lcs M j : Submodule R M).map (toEnd R L M x ^ l) ⊔
(I.lcs M (j + 1) : Submodule R M)
by simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using this n
intro l
induction' l with l ih
· simp on... | 1,930 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 128 | 140 | theorem isNilpotentOfIsNilpotentSpanSupEqTop (hnp : IsNilpotent <| toEnd R L M x)
(hIM : IsNilpotent R I M) : IsNilpotent R L M := by |
obtain ⟨n, hn⟩ := hnp
obtain ⟨k, hk⟩ := hIM
have hk' : I.lcs M k = ⊥ := by
simp only [← coe_toSubmodule_eq_iff, I.coe_lcs_eq, hk, bot_coeSubmodule]
suffices ∀ l, lowerCentralSeries R L M (l * n) ≤ I.lcs M l by
use k * n
simpa [hk'] using this k
intro l
induction' l with l ih
· simp
· exact ... | 1,930 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 165 | 170 | theorem LieAlgebra.isEngelian_of_subsingleton [Subsingleton L] : LieAlgebra.IsEngelian R L := by |
intro M _i1 _i2 _i3 _i4 _h
use 1
suffices (⊤ : LieIdeal R L) = ⊥ by simp [this]
haveI := (LieSubmodule.subsingleton_iff R L L).mpr inferInstance
apply Subsingleton.elim
| 1,930 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 173 | 183 | theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
(h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ := by |
intro M _i1 _i2 _i3 _i4 h'
letI : LieRingModule L M := LieRingModule.compLieHom M f
letI : LieModule R L M := compLieHom M f
have hnp : ∀ x, IsNilpotent (toEnd R L M x) := fun x => h' (f x)
have surj_id : Function.Surjective (LinearMap.id : M →ₗ[R] M) := Function.surjective_id
haveI : LieModule.IsNilpotent... | 1,930 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 63 | 65 | theorem map_toCotangent_ker : I.toCotangent.ker.map I.subtype = I ^ 2 := by |
rw [Ideal.toCotangent, Submodule.ker_mkQ, pow_two, Submodule.map_smul'' I ⊤ (Submodule.subtype I),
Algebra.id.smul_eq_mul, Submodule.map_subtype_top]
| 1,931 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 69 | 71 | theorem mem_toCotangent_ker {x : I} : x ∈ LinearMap.ker I.toCotangent ↔ (x : R) ∈ I ^ 2 := by |
rw [← I.map_toCotangent_ker]
simp
| 1,931 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 74 | 76 | theorem toCotangent_eq {x y : I} : I.toCotangent x = I.toCotangent y ↔ (x - y : R) ∈ I ^ 2 := by |
rw [← sub_eq_zero]
exact I.mem_toCotangent_ker
| 1,931 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 88 | 96 | theorem cotangent_subsingleton_iff : Subsingleton I.Cotangent ↔ IsIdempotentElem I := by |
constructor
· intro H
refine (pow_two I).symm.trans (le_antisymm (Ideal.pow_le_self two_ne_zero) ?_)
exact fun x hx => (I.toCotangent_eq_zero ⟨x, hx⟩).mp (Subsingleton.elim _ _)
· exact fun e =>
⟨fun x y =>
Quotient.inductionOn₂' x y fun x y =>
I.toCotangent_eq.mpr <| ((pow_two I)... | 1,931 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 122 | 128 | theorem cotangentIdeal_square (I : Ideal R) : I.cotangentIdeal ^ 2 = ⊥ := by |
rw [eq_bot_iff, pow_two I.cotangentIdeal, ← smul_eq_mul]
intro x hx
refine Submodule.smul_induction_on hx ?_ ?_
· rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩; apply (Submodule.Quotient.eq _).mpr _
rw [sub_zero, pow_two]; exact Ideal.mul_mem_mul hx hy
· intro x y hx hy; exact add_mem hx hy
| 1,931 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 132 | 136 | theorem to_quotient_square_range :
LinearMap.range I.cotangentToQuotientSquare = I.cotangentIdeal.restrictScalars R := by |
trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)
· rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
· rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]; ext; rfl
| 1,931 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 68 | 88 | theorem exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by |
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(... | 1,932 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 121 | 131 | theorem liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by |
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom... | 1,932 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 148 | 153 | theorem of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by |
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
| 1,932 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 188 | 196 | theorem comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by |
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars ... | 1,932 |
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [← forall_and, Function.Bijective]
| 1,933 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 62 | 63 | theorem KaehlerDifferential.one_smul_sub_smul_one_mem_ideal (a : S) :
(1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ KaehlerDifferential.ideal R S := by | simp [RingHom.mem_ker]
| 1,934 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 78 | 99 | theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) :
D.tensorProductTo (x * y) =
TensorProduct.lmul' (S := S) R x • D.tensorProductTo y +
TensorProduct.lmul' (S := S) R y • D.tensorProductTo x := by |
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x₁ x₂
refine TensorProduct.induction_on y ?_ ?_ ?_
· rw [mul_zero, map_... | 1,934 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 105 | 128 | theorem KaehlerDifferential.submodule_span_range_eq_ideal :
Submodule.span S (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
(KaehlerDifferential.ideal R S).restrictScalars S := by |
apply le_antisymm
· rw [Submodule.span_le]
rintro _ ⟨s, rfl⟩
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· rintro x (hx : _ = _)
have : x - TensorProduct.lmul' (S := S) R x ⊗ₜ[R] (1 : S) = x := by
rw [hx, TensorProduct.zero_tmul, sub_zero]
rw [← this]
clear this hx
... | 1,934 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 131 | 141 | theorem KaehlerDifferential.span_range_eq_ideal :
Ideal.span (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
KaehlerDifferential.ideal R S := by |
apply le_antisymm
· rw [Ideal.span_le]
rintro _ ⟨s, rfl⟩
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· change (KaehlerDifferential.ideal R S).restrictScalars S ≤ (Ideal.span _).restrictScalars S
rw [← KaehlerDifferential.submodule_span_range_eq_ideal, Ideal.span]
conv_rhs => rw ... | 1,934 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.discrete_valuation_ring.tfae from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variab... | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | 37 | 89 | theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
(h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
∃ n : ℕ, I = maximalIdeal R ^ n := by |
by_cases h : IsField R;
· exact ⟨0, by simp [letI := h.toField; (eq_bot_or_eq_top I).resolve_left hI]⟩
classical
obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
by_cases hI' : I = ⊤
· use 0; rw [pow_zero, hI', Ideal.one_eq_top]
have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
(SetLike.ext_iff.mp hx r).trans I... | 1,935 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.discrete_valuation_ring.tfae from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variab... | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | 92 | 150 | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [LocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal := by |
classical
by_cases ne_bot : maximalIdeal R = ⊥
· rw [ne_bot]; infer_instance
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximalIdeal R, a ≠ (0 : R) := by
by_contra! h'; apply ne_bot; rwa [eq_bot_iff]
have hle : Ideal.span {a} ≤ maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]
have : (Ideal.span {a}... | 1,935 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 40 | 80 | theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :
(adjoin R (s ∪ t)).toSubmodule.FG := by |
rcases fg_def.1 h1 with ⟨p, hp, hp'⟩
rcases fg_def.1 h2 with ⟨q, hq, hq'⟩
refine fg_def.2 ⟨p * q, hp.mul hq, le_antisymm ?_ ?_⟩
· rw [span_le, Set.mul_subset_iff]
intro x hx y hy
change x * y ∈ adjoin R (s ∪ t)
refine Subalgebra.mul_mem _ ?_ ?_
· have : x ∈ Subalgebra.toSubmodule (adjoin R s) :... | 1,936 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 129 | 137 | theorem FG.prod {S : Subalgebra R A} {T : Subalgebra R B} (hS : S.FG) (hT : T.FG) :
(S.prod T).FG := by |
obtain ⟨s, hs⟩ := fg_def.1 hS
obtain ⟨t, ht⟩ := fg_def.1 hT
rw [← hs.2, ← ht.2]
exact fg_def.2 ⟨LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1}),
Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))
(Set.Finite.image _ (Set.Finite.union ht.1 (Set.f... | 1,936 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 | theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S := by |
classical
obtain ⟨t, rfl⟩ := S.fg_of_noetherian
refine Finset.induction_on t ?_ ?_
· simpa using base
intro x t _ h
rw [Finset.coe_insert]
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
| 1,936 |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
namespace Algebra
| Mathlib/RingTheory/Adjoin/Tower.lean | 30 | 46 | theorem adjoin_restrictScalars (C D E : Type*) [CommSemiring C] [CommSemiring D] [CommSemiring E]
[Algebra C D] [Algebra C E] [Algebra D E] [IsScalarTower C D E] (S : Set E) :
(Algebra.adjoin D S).restrictScalars C =
(Algebra.adjoin ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) S).restrictScal... |
suffices
Set.range (algebraMap D E) =
Set.range (algebraMap ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) E) by
ext x
change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S)
rw [this]
ext x
constructor
· rintro ⟨y, hy⟩
exact ⟨⟨algebraMap D E y, ⟨y, ⟨Algebr... | 1,937 |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
namespace Algebra
theorem adjoin_restrictScalars (C D E : Typ... | Mathlib/RingTheory/Adjoin/Tower.lean | 49 | 58 | theorem adjoin_res_eq_adjoin_res (C D E F : Type*) [CommSemiring C] [CommSemiring D]
[CommSemiring E] [CommSemiring F] [Algebra C D] [Algebra C E] [Algebra C F] [Algebra D F]
[Algebra E F] [IsScalarTower C D F] [IsScalarTower C E F] {S : Set D} {T : Set E}
(hS : Algebra.adjoin C S = ⊤) (hT : Algebra.adjoin ... |
rw [adjoin_restrictScalars C E, adjoin_restrictScalars C D, ← hS, ← hT, ← Algebra.adjoin_image,
← Algebra.adjoin_image, ← AlgHom.coe_toRingHom, ← AlgHom.coe_toRingHom,
IsScalarTower.coe_toAlgHom, IsScalarTower.coe_toAlgHom, ← adjoin_union_eq_adjoin_adjoin, ←
adjoin_union_eq_adjoin_adjoin, Set.union_comm]... | 1,937 |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
section
open scoped Classical
theorem Algebra.fg_trans' {R S... | Mathlib/RingTheory/Adjoin/Tower.lean | 92 | 135 | theorem exists_subalgebra_of_fg (hAC : (⊤ : Subalgebra A C).FG) (hBC : (⊤ : Submodule B C).FG) :
∃ B₀ : Subalgebra A B, B₀.FG ∧ (⊤ : Submodule B₀ C).FG := by |
cases' hAC with x hx
cases' hBC with y hy
have := hy
simp_rw [eq_top_iff', mem_span_finset] at this
choose f hf using this
let s : Finset B := Finset.image₂ f (x ∪ y * y) y
have hxy :
∀ xi ∈ x, xi ∈ span (Algebra.adjoin A (↑s : Set B)) (↑(insert 1 y : Finset C) : Set C) :=
fun xi hxi =>
hf xi... | 1,937 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 48 | 49 | theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e := by |
simp [expand, eval₂]
| 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 65 | 66 | theorem expand_monomial (r : R) : expand R p (monomial q r) = monomial (q * p) r := by |
simp_rw [← smul_X_eq_monomial, AlgHom.map_smul, AlgHom.map_pow, expand_X, mul_comm, pow_mul]
| 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 80 | 80 | theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by | simp [expand]
| 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 95 | 97 | theorem derivative_expand (f : R[X]) : Polynomial.derivative (expand R p f) =
expand R p (Polynomial.derivative f) * (p * (X ^ (p - 1) : R[X])) := by |
rw [coe_expand, derivative_eval₂_C, derivative_pow, C_eq_natCast, derivative_X, mul_one]
| 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 100 | 117 | theorem coeff_expand {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff n = if p ∣ n then f.coeff (n / p) else 0 := by |
simp only [expand_eq_sum]
simp_rw [coeff_sum, ← pow_mul, C_mul_X_pow_eq_monomial, coeff_monomial, sum]
split_ifs with h
· rw [Finset.sum_eq_single (n / p), Nat.mul_div_cancel' h, if_pos rfl]
· intro b _ hb2
rw [if_neg]
intro hb3
apply hb2
rw [← hb3, Nat.mul_div_cancel_left b hp]
... | 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 121 | 123 | theorem coeff_expand_mul {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (n * p) = f.coeff n := by |
rw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]
| 1,938 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 127 | 128 | theorem coeff_expand_mul' {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (p * n) = f.coeff n := by | rw [mul_comm, coeff_expand_mul hp]
| 1,938 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 49 | 51 | theorem charmatrix_apply_natDegree [Nontrivial R] (i j : n) :
(charmatrix M i j).natDegree = ite (i = j) 1 0 := by |
by_cases h : i = j <;> simp [h, ← degree_eq_iff_natDegree_eq_of_pos (Nat.succ_pos 0)]
| 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 54 | 56 | theorem charmatrix_apply_natDegree_le (i j : n) :
(charmatrix M i j).natDegree ≤ ite (i = j) 1 0 := by |
split_ifs with h <;> simp [h, natDegree_X_le]
| 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 61 | 78 | theorem charpoly_sub_diagonal_degree_lt :
(M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1) := by |
rw [charpoly, det_apply', ← insert_erase (mem_univ (Equiv.refl n)),
sum_insert (not_mem_erase (Equiv.refl n) univ), add_comm]
simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl]
rw [← mem_degreeLT]
apply Submodule.sum_mem... | 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 81 | 86 | theorem charpoly_coeff_eq_prod_coeff_of_le {k : ℕ} (h : Fintype.card n - 1 ≤ k) :
M.charpoly.coeff k = (∏ i : n, (X - C (M i i))).coeff k := by |
apply eq_of_sub_eq_zero; rw [← coeff_sub]
apply Polynomial.coeff_eq_zero_of_degree_lt
apply lt_of_lt_of_le (charpoly_sub_diagonal_degree_lt M) ?_
rw [Nat.cast_le]; apply h
| 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 89 | 93 | theorem det_of_card_zero (h : Fintype.card n = 0) (M : Matrix n n R) : M.det = 1 := by |
rw [Fintype.card_eq_zero_iff] at h
suffices M = 1 by simp [this]
ext i
exact h.elim i
| 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 96 | 119 | theorem charpoly_degree_eq_dim [Nontrivial R] (M : Matrix n n R) :
M.charpoly.degree = Fintype.card n := by |
by_cases h : Fintype.card n = 0
· rw [h]
unfold charpoly
rw [det_of_card_zero]
· simp
· assumption
rw [← sub_add_cancel M.charpoly (∏ i : n, (X - C (M i i)))]
-- Porting note: added `↑` in front of `Fintype.card n`
have h1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n) := by
rw [... | 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 127 | 145 | theorem charpoly_monic (M : Matrix n n R) : M.charpoly.Monic := by |
nontriviality R -- Porting note: was simply `nontriviality`
by_cases h : Fintype.card n = 0
· rw [charpoly, det_of_card_zero h]
apply monic_one
have mon : (∏ i : n, (X - C (M i i))).Monic := by
apply monic_prod_of_monic univ fun i : n => X - C (M i i)
simp [monic_X_sub_C]
rw [← sub_add_cancel (∏ ... | 1,939 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 260 | 276 | theorem matPolyEquiv_eq_X_pow_sub_C {K : Type*} (k : ℕ) [Field K] (M : Matrix n n K) :
matPolyEquiv ((expand K k : K[X] →+* K[X]).mapMatrix (charmatrix (M ^ k))) =
X ^ k - C (M ^ k) := by |
-- Porting note: `i` and `j` are used later on, but were not mentioned in mathlib3
ext m i j
rw [coeff_sub, coeff_C, matPolyEquiv_coeff_apply, RingHom.mapMatrix_apply, Matrix.map_apply,
AlgHom.coe_toRingHom, DMatrix.sub_apply, coeff_X_pow]
by_cases hij : i = j
· rw [hij, charmatrix_apply_eq, AlgHom.map_s... | 1,939 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 55 | 69 | theorem traceAux_eq : traceAux R b = traceAux R c :=
LinearMap.ext fun f =>
calc
Matrix.trace (LinearMap.toMatrix b b f) =
Matrix.trace (LinearMap.toMatrix b b ((LinearMap.id.comp f).comp LinearMap.id)) := by |
rw [LinearMap.id_comp, LinearMap.comp_id]
_ = Matrix.trace (LinearMap.toMatrix c b LinearMap.id * LinearMap.toMatrix c c f *
LinearMap.toMatrix b c LinearMap.id) := by
rw [LinearMap.toMatrix_comp _ c, LinearMap.toMatrix_comp _ c]
_ = Matrix.trace (LinearMap.toMatrix c c f * Linear... | 1,940 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 84 | 89 | theorem trace_eq_matrix_trace_of_finset {s : Finset M} (b : Basis s R M) (f : M →ₗ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by |
have : ∃ s : Finset M, Nonempty (Basis s R M) := ⟨s, ⟨b⟩⟩
rw [trace, dif_pos this, ← traceAux_def]
congr 1
apply traceAux_eq
| 1,940 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 92 | 95 | theorem trace_eq_matrix_trace (f : M →ₗ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by |
rw [trace_eq_matrix_trace_of_finset R b.reindexFinsetRange, ← traceAux_def, ← traceAux_def,
traceAux_eq R b b.reindexFinsetRange]
| 1,940 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 116 | 119 | theorem trace_conj (g : M →ₗ[R] M) (f : (M →ₗ[R] M)ˣ) :
trace R M (↑f * g * ↑f⁻¹) = trace R M g := by |
rw [trace_mul_comm]
simp
| 1,940 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 138 | 150 | theorem trace_eq_contract_of_basis [Finite ι] (b : Basis ι R M) :
LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M := by |
classical
cases nonempty_fintype ι
apply Basis.ext (Basis.tensorProduct (Basis.dualBasis b) b)
rintro ⟨i, j⟩
simp only [Function.comp_apply, Basis.tensorProduct_apply, Basis.coe_dualBasis, coe_comp]
rw [trace_eq_matrix_trace R b, toMatrix_dualTensorHom]
by_cases hij : i = j
· rw [hij]
... | 1,940 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 155 | 157 | theorem trace_eq_contract_of_basis' [Fintype ι] [DecidableEq ι] (b : Basis ι R M) :
LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquivOfBasis b).symm.toLinearMap := by |
simp [LinearEquiv.eq_comp_toLinearMap_symm, trace_eq_contract_of_basis b]
| 1,940 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 43 | 46 | theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by |
rw [PiToModule.fromMatrix_apply, Fintype.total_apply, Matrix.mulVec_single]
simp_rw [mul_one]
| 1,941 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 60 | 65 | theorem PiToModule.fromEnd_apply_single_one [DecidableEq ι] (f : Module.End R M) (i : ι) :
PiToModule.fromEnd R b f (Pi.single i 1) = f (b i) := by |
rw [PiToModule.fromEnd_apply]
congr
convert Fintype.total_apply_single (S := R) R b i (1 : R)
rw [one_smul]
| 1,941 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 68 | 75 | theorem PiToModule.fromEnd_injective (hb : Submodule.span R (Set.range b) = ⊤) :
Function.Injective (PiToModule.fromEnd R b) := by |
intro x y e
ext m
obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.total R R b) := by
rw [(Fintype.range_total R b).trans hb]
exact Submodule.mem_top
exact (LinearMap.congr_fun e m : _)
| 1,941 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 100 | 111 | theorem Matrix.represents_iff' {A : Matrix ι ι R} {f : Module.End R M} :
A.Represents b f ↔ ∀ j, ∑ i : ι, A i j • b i = f (b j) := by |
constructor
· intro h i
have := LinearMap.congr_fun h (Pi.single i 1)
rwa [PiToModule.fromEnd_apply_single_one, PiToModule.fromMatrix_apply_single_one] at this
· intro h
-- Porting note: was `ext`
refine LinearMap.pi_ext' (fun i => LinearMap.ext_ring ?_)
simp_rw [LinearMap.comp_apply, LinearM... | 1,941 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 114 | 121 | theorem Matrix.Represents.mul {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A * A').Represents b (f * f') := by |
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl
| 1,941 |
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