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.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 71 | 74 | theorem rank_mul_le_left [StrongRankCondition R] (A : Matrix m n R) (B : Matrix n o R) :
(A * B).rank β€ A.rank := by |
rw [rank, rank, mulVecLin_mul]
exact Cardinal.toNat_le_toNat (LinearMap.rank_comp_le_left _ _) (rank_lt_aleph0 _ _)
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 77 | 81 | theorem rank_mul_le_right [StrongRankCondition R] (A : Matrix m n R) (B : Matrix n o R) :
(A * B).rank β€ B.rank := by |
rw [rank, rank, mulVecLin_mul]
exact finrank_le_finrank_of_rank_le_rank (LinearMap.lift_rank_comp_le_right _ _)
(rank_lt_aleph0 _ _)
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 89 | 93 | theorem rank_unit [StrongRankCondition R] [DecidableEq n] (A : (Matrix n n R)Λ£) :
(A : Matrix n n R).rank = Fintype.card n := by |
apply le_antisymm (rank_le_card_width (A : Matrix n n R)) _
have := rank_mul_le_left (A : Matrix n n R) (βAβ»ΒΉ : Matrix n n R)
rwa [β Units.val_mul, mul_inv_self, Units.val_one, rank_one] at this
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 96 | 99 | theorem rank_of_isUnit [StrongRankCondition R] [DecidableEq n] (A : Matrix n n R) (h : IsUnit A) :
A.rank = Fintype.card n := by |
obtain β¨A, rflβ© := h
exact rank_unit A
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 125 | 130 | theorem rank_submatrix_le [StrongRankCondition R] [Fintype m] (f : n β m) (e : n β m)
(A : Matrix m m R) : rank (A.submatrix f e) β€ rank A := by |
rw [rank, rank, mulVecLin_submatrix, LinearMap.range_comp, LinearMap.range_comp,
show LinearMap.funLeft R R e.symm = LinearEquiv.funCongrLeft R R e.symm from rfl,
LinearEquiv.range, Submodule.map_top]
exact Submodule.finrank_map_le _ _
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 133 | 136 | theorem rank_reindex [Fintype m] (eβ eβ : m β n) (A : Matrix m m R) :
rank (reindex eβ eβ A) = rank A := by |
rw [rank, rank, mulVecLin_reindex, LinearMap.range_comp, LinearMap.range_comp,
LinearEquiv.range, Submodule.map_top, LinearEquiv.finrank_map_eq]
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 140 | 142 | theorem rank_submatrix [Fintype m] (A : Matrix m m R) (eβ eβ : n β m) :
rank (A.submatrix eβ eβ) = rank A := by |
simpa only [reindex_apply] using rank_reindex eβ.symm eβ.symm A
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 217 | 220 | theorem ker_mulVecLin_conjTranspose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aα΄΄ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, conjTranspose_mul_self_mulVec_eq_zero]
| 1,847 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 255 | 264 | theorem ker_mulVecLin_transpose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aα΅ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, β mulVec_mulVec]
constructor
Β· intro h
replace h := congr_arg (dotProduct x) h
rwa [dotProduct_mulVec, dotProduct_zero, vecMul_transpose, dotProduct_self_eq_zero] at h
Β· intro h
rw [h, mulVec_zero]
| 1,847 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 60 | 62 | theorem rotation_trans (a b : circle) : (rotation a).trans (rotation b) = rotation (b * a) := by |
ext1
simp
| 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 65 | 71 | theorem rotation_ne_conjLIE (a : circle) : rotation a β conjLIE := by |
intro h
have h1 : rotation a 1 = conj 1 := LinearIsometryEquiv.congr_fun h 1
have hI : rotation a I = conj I := LinearIsometryEquiv.congr_fun h I
rw [rotation_apply, RingHom.map_one, mul_one] at h1
rw [rotation_apply, conj_I, β neg_one_mul, mul_left_inj' I_ne_zero, h1, eq_neg_self_iff] at hI
exact one_ne_z... | 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 90 | 93 | theorem LinearIsometry.re_apply_eq_re_of_add_conj_eq (f : β ββα΅’[β] β)
(hβ : β z, z + conj z = f z + conj (f z)) (z : β) : (f z).re = z.re := by |
simpa [ext_iff, add_re, add_im, conj_re, conj_im, β two_mul,
show (2 : β) β 0 by simp [two_ne_zero]] using (hβ z).symm
| 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 96 | 101 | theorem LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re {f : β ββα΅’[β] β}
(hβ : β z, (f z).re = z.re) (z : β) : (f z).im = z.im β¨ (f z).im = -z.im := by |
have hβ := f.norm_map z
simp only [Complex.abs_def, norm_eq_abs] at hβ
rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,
hβ, add_left_cancel_iff, mul_self_eq_mul_self_iff] at hβ
| 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 104 | 116 | theorem LinearIsometry.im_apply_eq_im {f : β ββα΅’[β] β} (h : f 1 = 1) (z : β) :
z + conj z = f z + conj (f z) := by |
have : βf z - 1β = βz - 1β := by rw [β f.norm_map (z - 1), f.map_sub, h]
apply_fun fun x => x ^ 2 at this
simp only [norm_eq_abs, β normSq_eq_abs] at this
rw [β ofReal_inj, β mul_conj, β mul_conj] at this
rw [RingHom.map_sub, RingHom.map_sub] at this
simp only [sub_mul, mul_sub, one_mul, mul_one] at this
... | 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 119 | 122 | theorem LinearIsometry.re_apply_eq_re {f : β ββα΅’[β] β} (h : f 1 = 1) (z : β) : (f z).re = z.re := by |
apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
intro z
apply LinearIsometry.im_apply_eq_im h
| 1,848 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 125 | 139 | theorem linear_isometry_complex_aux {f : β ββα΅’[β] β} (h : f 1 = 1) :
f = LinearIsometryEquiv.refl β β β¨ f = conjLIE := by |
have h0 : f I = I β¨ f I = -I := by
simp only [ext_iff, β and_or_left, neg_re, I_re, neg_im, neg_zero]
constructor
Β· rw [β I_re]
exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
Β· apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
intro z
rw [... | 1,848 |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 43 | 46 | theorem J_transpose : (J l R)α΅ = -J l R := by |
rw [J, fromBlocks_transpose, β neg_one_smul R (fromBlocks _ _ _ _ : Matrix (l β l) (l β l) R),
fromBlocks_smul, Matrix.transpose_zero, Matrix.transpose_one, transpose_neg]
simp [fromBlocks]
| 1,849 |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 52 | 55 | theorem J_squared : J l R * J l R = -1 := by |
rw [J, fromBlocks_multiply]
simp only [Matrix.zero_mul, Matrix.neg_mul, zero_add, neg_zero, Matrix.one_mul, add_zero]
rw [β neg_zero, β Matrix.fromBlocks_neg, β fromBlocks_one]
| 1,849 |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 59 | 62 | theorem J_inv : (J l R)β»ΒΉ = -J l R := by |
refine Matrix.inv_eq_right_inv ?_
rw [Matrix.mul_neg, J_squared]
exact neg_neg 1
| 1,849 |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 66 | 70 | theorem J_det_mul_J_det : det (J l R) * det (J l R) = 1 := by |
rw [β det_mul, J_squared, β one_smul R (-1 : Matrix _ _ R), smul_neg, β neg_smul, det_smul,
Fintype.card_sum, det_one, mul_one]
apply Even.neg_one_pow
exact even_add_self _
| 1,849 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Rin... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 114 | 132 | theorem exists_mulVec_eq_zero_iff_aux {K : Type*} [DecidableEq n] [Field K] {M : Matrix n n K} :
(β v β 0, M *α΅₯ v = 0) β M.det = 0 := by |
constructor
Β· rintro β¨v, hv, mul_eqβ©
contrapose! hv
exact eq_zero_of_mulVec_eq_zero hv mul_eq
Β· contrapose!
intro h
have : Function.Injective (Matrix.toLin' M) := by
simpa only [β LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h
have :
M *
LinearMap.toMa... | 1,850 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Rin... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 135 | 167 | theorem exists_mulVec_eq_zero_iff' {A : Type*} (K : Type*) [DecidableEq n] [CommRing A]
[Nontrivial A] [Field K] [Algebra A K] [IsFractionRing A K] {M : Matrix n n A} :
(β v β 0, M *α΅₯ v = 0) β M.det = 0 := by |
have : (β v β 0, (algebraMap A K).mapMatrix M *α΅₯ v = 0) β _ :=
exists_mulVec_eq_zero_iff_aux
rw [β RingHom.map_det, IsFractionRing.to_map_eq_zero_iff] at this
refine Iff.trans ?_ this; constructor <;> rintro β¨v, hv, mul_eqβ©
Β· refine β¨fun i => algebraMap _ _ (v i), mt (fun h => funext fun i => ?_) hv, ?_β©
... | 1,850 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Rin... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 175 | 177 | theorem exists_vecMul_eq_zero_iff {A : Type*} [DecidableEq n] [CommRing A] [IsDomain A]
{M : Matrix n n A} : (β v β 0, v α΅₯* M = 0) β M.det = 0 := by |
simpa only [β M.det_transpose, β mulVec_transpose] using exists_mulVec_eq_zero_iff
| 1,850 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Rin... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 180 | 190 | theorem nondegenerate_iff_det_ne_zero {A : Type*} [DecidableEq n] [CommRing A] [IsDomain A]
{M : Matrix n n A} : Nondegenerate M β M.det β 0 := by |
rw [ne_eq, β exists_vecMul_eq_zero_iff]
push_neg
constructor
Β· intro hM v hv hMv
obtain β¨w, hwMvβ© := hM.exists_not_ortho_of_ne_zero hv
simp [dotProduct_mulVec, hMv, zero_dotProduct, ne_eq, not_true] at hwMv
Β· intro h v hv
refine not_imp_not.mp (h v) (funext fun i => ?_)
simpa only [dotProduct... | 1,850 |
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 66 | 74 | theorem Matrix.toLinearMapβ'Aux_stdBasis (f : Matrix n m R) (i : n) (j : m) :
f.toLinearMapβ'Aux Οβ Οβ (LinearMap.stdBasis Rβ (fun _ => Rβ) i 1)
(LinearMap.stdBasis Rβ (fun _ => Rβ) j 1) = f i j := by |
rw [Matrix.toLinearMapβ'Aux, mkβ'ββ_apply]
have : (β i', β j', (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j := by
simp_rw [mul_assoc, β Finset.mul_sum]
simp only [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true, mul_comm (f _ _)]
rw [β this]
exact Finset.sum_congr rfl f... | 1,851 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 98 | 100 | theorem polar_add (f g : M β R) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by |
simp only [polar, Pi.add_apply]
abel
| 1,852 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 103 | 104 | theorem polar_neg (f : M β R) (x y : M) : polar (-f) x y = -polar f x y := by |
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
| 1,852 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 107 | 108 | theorem polar_smul [Monoid S] [DistribMulAction S R] (f : M β R) (s : S) (x y : M) :
polar (s β’ f) x y = s β’ polar f x y := by | simp only [polar, Pi.smul_apply, smul_sub]
| 1,852 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 111 | 112 | theorem polar_comm (f : M β R) (x y : M) : polar f x y = polar f y x := by |
rw [polar, polar, add_comm, sub_sub, sub_sub, add_comm (f x) (f y)]
| 1,852 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 116 | 123 | theorem polar_add_left_iff {f : M β R} {x x' y : M} :
polar f (x + x') y = polar f x y + polar f x' y β
f (x + x' + y) + (f x + f x' + f y) = f (x + x') + f (x' + y) + f (y + x) := by |
simp only [β add_assoc]
simp only [polar, sub_eq_iff_eq_add, eq_sub_iff_add_eq, sub_add_eq_add_sub, add_sub]
simp only [add_right_comm _ (f y) _, add_right_comm _ (f x') (f x)]
rw [add_comm y x, add_right_comm _ _ (f (x + y)), add_comm _ (f (x + y)),
add_right_comm (f (x + y)), add_left_inj]
| 1,852 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 126 | 129 | theorem polar_comp {F : Type*} [CommRing S] [FunLike F R S] [AddMonoidHomClass F R S]
(f : M β R) (g : F) (x y : M) :
polar (g β f) x y = g (polar f x y) := by |
simp only [polar, Pi.smul_apply, Function.comp_apply, map_sub]
| 1,852 |
import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.Basic
universe uR uA uMβ uMβ
variable {R : Type uR} {A : Type uA} {Mβ : Type uMβ} {Mβ : Type uMβ}
open TensorProduct
open LinearMap (BilinForm)
namespace QuadraticForm
section CommRing
variable [CommRing R] [CommR... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean | 69 | 75 | theorem associated_tmul [Invertible (2 : A)] (Qβ : QuadraticForm A Mβ) (Qβ : QuadraticForm R Mβ) :
associated (R := A) (Qβ.tmul Qβ)
= (associated (R := A) Qβ).tmul (associated (R := R) Qβ) := by |
rw [QuadraticForm.tmul, tensorDistrib, BilinForm.tmul]
dsimp
have : Subsingleton (Invertible (2 : A)) := inferInstance
convert associated_left_inverse A ((associated_isSymm A Qβ).tmul (associated_isSymm R Qβ))
| 1,853 |
import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.Basic
universe uR uA uMβ uMβ
variable {R : Type uR} {A : Type uA} {Mβ : Type uMβ} {Mβ : Type uMβ}
open TensorProduct
open LinearMap (BilinForm)
namespace QuadraticForm
section CommRing
variable [CommRing R] [CommR... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean | 77 | 82 | theorem polarBilin_tmul [Invertible (2 : A)] (Qβ : QuadraticForm A Mβ) (Qβ : QuadraticForm R Mβ) :
polarBilin (Qβ.tmul Qβ) = β
(2 : A) β’ (polarBilin Qβ).tmul (polarBilin Qβ) := by |
simp_rw [β two_nsmul_associated A, β two_nsmul_associated R, BilinForm.tmul, tmul_smul,
β smul_tmul', map_nsmul, associated_tmul]
rw [smul_comm (_ : A) (_ : β), β smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two,
one_smul]
| 1,853 |
import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.Basic
universe uR uA uMβ uMβ
variable {R : Type uR} {A : Type uA} {Mβ : Type uMβ} {Mβ : Type uMβ}
open TensorProduct
open LinearMap (BilinForm)
namespace QuadraticForm
section CommRing
variable [CommRing R] [CommR... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean | 95 | 99 | theorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R Mβ) :
associated (R := A) (Q.baseChange A) = (associated (R := R) Q).baseChange A := by |
dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]
rw [associated_tmul (QuadraticForm.sq (R := A)) Q, associated_sq]
exact rfl
| 1,853 |
import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.Basic
universe uR uA uMβ uMβ
variable {R : Type uR} {A : Type uA} {Mβ : Type uMβ} {Mβ : Type uMβ}
open TensorProduct
open LinearMap (BilinForm)
namespace QuadraticForm
section CommRing
variable [CommRing R] [CommR... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean | 101 | 105 | theorem polarBilin_baseChange [Invertible (2 : A)] (Q : QuadraticForm R Mβ) :
polarBilin (Q.baseChange A) = (polarBilin Q).baseChange A := by |
rw [QuadraticForm.baseChange, BilinForm.baseChange, polarBilin_tmul, BilinForm.tmul,
β LinearMap.map_smul, smul_tmul', β two_nsmul_associated R, coe_associatedHom, associated_sq,
smul_comm, β smul_assoc, two_smul, invOf_two_add_invOf_two, one_smul]
| 1,853 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uMβ uMβ uMβ uMβ
variable {R : Type uR} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 37 | 46 | theorem tmul_comp_tensorMap
{Qβ : QuadraticForm R Mβ} {Qβ : QuadraticForm R Mβ}
{Qβ : QuadraticForm R Mβ} {Qβ : QuadraticForm R Mβ}
(f : Qβ βqα΅’ Qβ) (g : Qβ βqα΅’ Qβ) :
(Qβ.tmul Qβ).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Qβ.tmul Qβ := by |
have hβ : Qβ = Qβ.comp f.toLinearMap := QuadraticForm.ext fun x => (f.map_app x).symm
have hβ : Qβ = Qβ.comp g.toLinearMap := QuadraticForm.ext fun x => (g.map_app x).symm
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext mβ mβ mβ' mβ'
simp [-associated_apply, hβ, hβ, associated_tmul]
| 1,854 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uMβ uMβ uMβ uMβ
variable {R : Type uR} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 79 | 85 | theorem tmul_comp_tensorComm (Qβ : QuadraticForm R Mβ) (Qβ : QuadraticForm R Mβ) :
(Qβ.tmul Qβ).comp (TensorProduct.comm R Mβ Mβ) = Qβ.tmul Qβ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext mβ mβ mβ' mβ'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
exact mul_comm _ _
| 1,854 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uMβ uMβ uMβ uMβ
variable {R : Type uR} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 114 | 121 | theorem tmul_comp_tensorAssoc
(Qβ : QuadraticForm R Mβ) (Qβ : QuadraticForm R Mβ) (Qβ : QuadraticForm R Mβ) :
(Qβ.tmul (Qβ.tmul Qβ)).comp (TensorProduct.assoc R Mβ Mβ Mβ) = (Qβ.tmul Qβ).tmul Qβ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext mβ mβ mβ' mβ' mβ'' mβ''
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
exact mul_assoc _ _ _
| 1,854 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uMβ uMβ uMβ uMβ
variable {R : Type uR} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 153 | 159 | theorem comp_tensorRId_eq (Qβ : QuadraticForm R Mβ) :
Qβ.comp (TensorProduct.rid R Mβ) = Qβ.tmul (sq (R := R)) := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext mβ mβ'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
simp [-associated_apply, one_mul]
| 1,854 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uMβ uMβ uMβ uMβ
variable {R : Type uR} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ} {Mβ : Type uMβ}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 186 | 192 | theorem comp_tensorLId_eq (Qβ : QuadraticForm R Mβ) :
Qβ.comp (TensorProduct.lid R Mβ) = (sq (R := R)).tmul Qβ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext mβ mβ'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
simp [-associated_apply, mul_one]
| 1,854 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import Mathlib.RingTheory.TensorProduct.Basic
variable {R A V : Type*}
variable [CommRing R] [CommRing A] [AddCommGroup V]
variable [Algebra R A] [Mod... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 104 | 113 | theorem toBaseChange_comp_involute (Q : QuadraticForm R V) :
(toBaseChange A Q).comp (involute : CliffordAlgebra (Q.baseChange A) ββ[A] _) =
(Algebra.TensorProduct.map (AlgHom.id _ _) involute).comp (toBaseChange A Q) := by |
ext v
show toBaseChange A Q (involute (ΞΉ (Q.baseChange A) (1 ββ[R] v)))
= (Algebra.TensorProduct.map (AlgHom.id _ _) involute :
A β[R] CliffordAlgebra Q ββ[A] _)
(toBaseChange A Q (ΞΉ (Q.baseChange A) (1 ββ[R] v)))
rw [toBaseChange_ΞΉ, involute_ΞΉ, map_neg (toBaseChange A Q), toBaseChange_ΞΉ,
A... | 1,855 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import Mathlib.RingTheory.TensorProduct.Basic
variable {R A V : Type*}
variable [CommRing R] [CommRing A] [AddCommGroup V]
variable [Algebra R A] [Mod... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 124 | 137 | theorem toBaseChange_comp_reverseOp (Q : QuadraticForm R V) :
(toBaseChange A Q).op.comp reverseOp =
((Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)).toAlgHom.comp <|
(Algebra.TensorProduct.map
(AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))).comp
(toBaseChange A... |
ext v
show op (toBaseChange A Q (reverse (ΞΉ (Q.baseChange A) (1 ββ[R] v)))) =
Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)
(Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))
(toBaseChange A Q (ΞΉ (Q.baseChange A) (1 ββ[R] v))))
rw [toBaseChange_ΞΉ, re... | 1,855 |
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
#align_import l... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 88 | 93 | theorem toBilin'Aux_toMatrixAux [DecidableEq n] (Bβ : BilinForm Rβ (n β Rβ)) :
-- Porting note: had to hint the base ring even though it should be clear from context...
Matrix.toBilin'Aux (BilinForm.toMatrixAux (Rβ := Rβ)
(fun j => stdBasis Rβ (fun _ => Rβ) j 1) Bβ) = Bβ := by |
rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux,
toLinearMapβ'Aux_toMatrixβAux]
| 1,856 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
universe u v w wβ wβ
section Matrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/Matrix.lean | 69 | 72 | theorem Matrix.lieConj_apply (P A : Matrix n n R) (h : Invertible P) :
P.lieConj h A = P * A * Pβ»ΒΉ := by |
simp [LinearEquiv.conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
| 1,857 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
universe u v w wβ wβ
section Matrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/Matrix.lean | 76 | 79 | theorem Matrix.lieConj_symm_apply (P A : Matrix n n R) (h : Invertible P) :
(P.lieConj h).symm A = Pβ»ΒΉ * A * P := by |
simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
| 1,857 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 46 | 53 | theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}
(hf : f β B.skewAdjointSubmodule) (hg : g β B.skewAdjointSubmodule) :
β
f, gβ β B.skewAdjointSubmodule := by |
rw [mem_skewAdjointSubmodule] at *
have hfg : IsAdjointPair B B (f * g) (g * f) := by rw [β neg_mul_neg g f]; exact hf.mul hg
have hgf : IsAdjointPair B B (g * f) (f * g) := by rw [β neg_mul_neg f g]; exact hg.mul hf
change IsAdjointPair B B (f * g - g * f) (-(f * g - g * f)); rw [neg_sub]
exact hfg.sub hgf
| 1,858 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 77 | 80 | theorem skewAdjointLieSubalgebraEquiv_apply
(f : skewAdjointLieSubalgebra (B.complββ (Qβ := N) (Qβ' := N) βe βe)) :
β(skewAdjointLieSubalgebraEquiv B e f) = e.lieConj f := by |
simp [skewAdjointLieSubalgebraEquiv]
| 1,858 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 84 | 86 | theorem skewAdjointLieSubalgebraEquiv_symm_apply (f : skewAdjointLieSubalgebra B) :
β((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj f := by |
simp [skewAdjointLieSubalgebraEquiv]
| 1,858 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointMatrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 103 | 112 | theorem Matrix.isSkewAdjoint_bracket {A B : Matrix n n R} (hA : A β skewAdjointMatricesSubmodule J)
(hB : B β skewAdjointMatricesSubmodule J) : β
A, Bβ β skewAdjointMatricesSubmodule J := by |
simp only [mem_skewAdjointMatricesSubmodule] at *
change β
A, Bβα΅ * J = J * (-β
A, Bβ)
change Aα΅ * J = J * (-A) at hA
change Bα΅ * J = J * (-B) at hB
rw [Matrix.lie_transpose, LieRing.of_associative_ring_bracket,
LieRing.of_associative_ring_bracket, sub_mul, mul_assoc, mul_assoc, hA, hB, β mul_assoc,
β ... | 1,858 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointMatrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 142 | 145 | theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
(A : skewAdjointMatricesLieSubalgebra J) :
β(skewAdjointMatricesLieSubalgebraEquiv J P h A) = Pβ»ΒΉ * (A : Matrix n n R) * P := by |
simp [skewAdjointMatricesLieSubalgebraEquiv]
| 1,858 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w wβ
section SkewAdjointMatrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 170 | 176 | theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : RΛ£) (J A : Matrix n n R) :
A β skewAdjointMatricesLieSubalgebra (u β’ J) β A β skewAdjointMatricesLieSubalgebra J := by |
change A β skewAdjointMatricesSubmodule (u β’ J) β A β skewAdjointMatricesSubmodule J
simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
constructor <;> intro h
Β· simpa using congr_arg (fun B => uβ»ΒΉ β’ B) h
Β· simp [h]
| 1,858 |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 66 | 68 | theorem mem_unitaryGroup_iff : A β Matrix.unitaryGroup n Ξ± β A * star A = 1 := by |
refine β¨And.right, fun hA => β¨?_, hAβ©β©
simpa only [mul_eq_one_comm] using hA
| 1,859 |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 71 | 73 | theorem mem_unitaryGroup_iff' : A β Matrix.unitaryGroup n Ξ± β star A * A = 1 := by |
refine β¨And.left, fun hA => β¨hA, ?_β©β©
rwa [mul_eq_one_comm] at hA
| 1,859 |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 76 | 80 | theorem det_of_mem_unitary {A : Matrix n n Ξ±} (hA : A β Matrix.unitaryGroup n Ξ±) :
A.det β unitary Ξ± := by |
constructor
Β· simpa [star, det_transpose] using congr_arg det hA.1
Β· simpa [star, det_transpose] using congr_arg det hA.2
| 1,859 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 55 | 57 | theorem charmatrix_apply_eq : charmatrix M i i = (X : R[X]) - C (M i i) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, map_apply,
diagonal_apply_eq]
| 1,860 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 62 | 64 | theorem charmatrix_apply_ne (h : i β j) : charmatrix M i j = -C (M i j) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, diagonal_apply_ne _ h,
map_apply, sub_eq_neg_self]
| 1,860 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 67 | 76 | theorem matPolyEquiv_charmatrix : matPolyEquiv (charmatrix M) = X - C M := by |
ext k i j
simp only [matPolyEquiv_coeff_apply, coeff_sub, Pi.sub_apply]
by_cases h : i = j
Β· subst h
rw [charmatrix_apply_eq, coeff_sub]
simp only [coeff_X, coeff_C]
split_ifs <;> simp
Β· rw [charmatrix_apply_ne _ _ _ h, coeff_X, coeff_neg, coeff_C, coeff_C]
split_ifs <;> simp [h]
| 1,860 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 79 | 83 | theorem charmatrix_reindex (e : n β m) :
charmatrix (reindex e e M) = reindex e e (charmatrix M) := by |
ext i j x
by_cases h : i = j
all_goals simp [h]
| 1,860 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 103 | 106 | theorem charpoly_reindex (e : n β m)
(M : Matrix n n R) : (reindex e e M).charpoly = M.charpoly := by |
unfold Matrix.charpoly
rw [charmatrix_reindex, Matrix.det_reindex_self]
| 1,860 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 134 | 154 | theorem aeval_self_charpoly (M : Matrix n n R) : aeval M M.charpoly = 0 := by |
-- We begin with the fact $Ο_M(t) I = adjugate (t I - M) * (t I - M)$,
-- as an identity in `Matrix n n R[X]`.
have h : M.charpoly β’ (1 : Matrix n n R[X]) = adjugate (charmatrix M) * charmatrix M :=
(adjugate_mul _).symm
-- Using the algebra isomorphism `Matrix n n R[X] ββ[R] Polynomial (Matrix n n R)`,
... | 1,860 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 87 | 91 | theorem quotient_map_C_eq_zero {I : Ideal R} :
β a β I, ((Quotient.mk (map (C : R β+* R[X]) I : Ideal R[X])).comp C) a = 0 := by |
intro a ha
rw [RingHom.comp_apply, Quotient.eq_zero_iff_mem]
exact mem_map_of_mem _ ha
| 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 94 | 107 | theorem evalβ_C_mk_eq_zero {I : Ideal R} :
β f β (map (C : R β+* R[X]) I : Ideal R[X]), evalβRingHom (C.comp (Quotient.mk I)) X f = 0 := by |
intro a ha
rw [β sum_monomial_eq a]
dsimp
rw [evalβ_sum]
refine Finset.sum_eq_zero fun n _ => ?_
dsimp
rw [evalβ_monomial (C.comp (Quotient.mk I)) X]
refine mul_eq_zero_of_left (Polynomial.ext fun m => ?_) (X ^ n)
erw [coeff_C]
by_cases h : m = 0
Β· simpa [h] using Quotient.eq_zero_iff_mem.2 ((mem... | 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 150 | 154 | theorem polynomialQuotientEquivQuotientPolynomial_symm_mk (I : Ideal R) (f : R[X]) :
I.polynomialQuotientEquivQuotientPolynomial.symm (Quotient.mk _ f) = f.map (Quotient.mk I) := by |
rw [polynomialQuotientEquivQuotientPolynomial, RingEquiv.symm_mk, RingEquiv.coe_mk,
Equiv.coe_fn_mk, Quotient.lift_mk, coe_evalβRingHom, evalβ_eq_eval_map, β Polynomial.map_map,
β evalβ_eq_eval_map, Polynomial.evalβ_C_X]
| 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 158 | 162 | theorem polynomialQuotientEquivQuotientPolynomial_map_mk (I : Ideal R) (f : R[X]) :
I.polynomialQuotientEquivQuotientPolynomial (f.map <| Quotient.mk I) =
Quotient.mk (map C I : Ideal R[X]) f := by |
apply (polynomialQuotientEquivQuotientPolynomial I).symm.injective
rw [RingEquiv.symm_apply_apply, polynomialQuotientEquivQuotientPolynomial_symm_mk]
| 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 175 | 194 | theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk I).comp C).range)
(hx : C x β I.map (Polynomial.mapRingHom ((Quotient.mk I).comp C).rangeRestrict)) : x = 0 := by |
let i := ((Quotient.mk I).comp C).rangeRestrict
have hi' : RingHom.ker (Polynomial.mapRingHom i) β€ I := by
refine fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => ?_
rw [mem_comap, β Quotient.eq_zero_iff_mem, β RingHom.comp_apply]
rw [RingHom.mem_ker, coe_mapRingHom] at hf
replace h... | 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 205 | 209 | theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i β I) :
(Ideal.Quotient.mk (Ideal.map (C : R β+* MvPolynomial Ο R) I :
Ideal (MvPolynomial Ο R))).comp C i = 0 := by |
simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]
exact Ideal.mem_map_of_mem _ hi
| 1,861 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 212 | 223 | theorem evalβ_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial Ο R}
(ha : a β (Ideal.map (C : R β+* MvPolynomial Ο R) I : Ideal (MvPolynomial Ο R))) :
evalβHom (C.comp (Ideal.Quotient.mk I)) X a = 0 := by |
rw [as_sum a]
rw [coe_evalβHom, evalβ_sum]
refine Finset.sum_eq_zero fun n _ => ?_
simp only [evalβ_monomial, Function.comp_apply, RingHom.coe_comp]
refine mul_eq_zero_of_left ?_ _
suffices coeff n a β I by
rw [β @Ideal.mk_ker R _ I, RingHom.mem_ker] at this
simp only [this, C_0]
exact mem_map_C_... | 1,861 |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 125 | 129 | theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 β jacobson I) :
β s, s * r - 1 β I := by |
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
| 1,862 |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 134 | 143 | theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I β β M : Set (Ideal R), (β J β M, IsMaximal J β¨ J = β€) β§ I = sInf M := by |
use fun hI => β¨{ J : Ideal R | I β€ J β§ J.IsMaximal }, β¨fun _ hJ => Or.inl hJ.right, hI.symmβ©β©
rintro β¨M, hM, hInfβ©
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
Β· exact (mem_sInf.1 hx) β¨le_sInf_iff.1 (le_of_eq hInf) I hI, is_maxβ©
Β· exac... | 1,862 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Tactic.TFAE
#align_import ring_theory.ideal.local_ring from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
un... | Mathlib/RingTheory/Ideal/LocalRing.lean | 93 | 96 | theorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a β¨ IsUnit b := by |
rcases h with β¨u, huβ©
rw [β Units.inv_mul_eq_one, mul_add] at hu
apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right
| 1,863 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Tactic.TFAE
#align_import ring_theory.ideal.local_ring from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
un... | Mathlib/RingTheory/Ideal/LocalRing.lean | 136 | 138 | theorem le_maximalIdeal {J : Ideal R} (hJ : J β β€) : J β€ maximalIdeal R := by |
rcases Ideal.exists_le_maximal J hJ with β¨M, hM1, hM2β©
rwa [β eq_maximalIdeal hM1]
| 1,863 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Module.Simple
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.RingTheory.Ideal.LocalRing
#align_import topology.algebra.module.finite_dimension from "leanprove... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 77 | 127 | theorem unique_topology_of_t2 {t : TopologicalSpace π} (hβ : @TopologicalAddGroup π t _)
(hβ : @ContinuousSMul π π _ hnorm.toUniformSpace.toTopologicalSpace t) (hβ : @T2Space π t) :
t = hnorm.toUniformSpace.toTopologicalSpace := by |
-- Let `π£β` denote the topology on `π` induced by the norm, and `π£` be any T2 vector
-- topology on `π`. To show that `π£β = π£`, it suffices to show that they have the same
-- neighborhoods of 0.
refine TopologicalAddGroup.ext hβ inferInstance (le_antisymm ?_ ?_)
Β· -- To show `π£ β€ π£β`, we have to show... | 1,864 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Module.Simple
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.RingTheory.Ideal.LocalRing
#align_import topology.algebra.module.finite_dimension from "leanprove... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 132 | 173 | theorem LinearMap.continuous_of_isClosed_ker (l : E ββ[π] π)
(hl : IsClosed (LinearMap.ker l : Set E)) :
Continuous l := by |
-- `l` is either constant or surjective. If it is constant, the result is trivial.
by_cases H : finrank π (LinearMap.range l) = 0
Β· rw [Submodule.finrank_eq_zero, LinearMap.range_eq_bot] at H
rw [H]
exact continuous_zero
Β· -- In the case where `l` is surjective, we factor it as `Ο : (E β§Έ l.ker) ββ[π]... | 1,864 |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover-community/mathlib"@"3397560e65278e5f31acefcdea63138bd53d1cd4"
variable {π E F G : Type*} [NontriviallyNormedField π] [NormedAddCommGroup E] [NormedS... | Mathlib/Analysis/NormedSpace/Complemented.lean | 39 | 43 | theorem ker_closedComplemented_of_finiteDimensional_range (f : E βL[π] F)
[FiniteDimensional π (range f)] : (ker f).ClosedComplemented := by |
set f' : E βL[π] range f := f.codRestrict _ (LinearMap.mem_range_self (f : E ββ[π] F))
rcases f'.exists_right_inverse_of_surjective (f : E ββ[π] F).range_rangeRestrict with β¨g, hgβ©
simpa only [f', ker_codRestrict] using f'.closedComplemented_ker_of_rightInverse g (ext_iff.1 hg)
| 1,865 |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover-community/mathlib"@"3397560e65278e5f31acefcdea63138bd53d1cd4"
variable {π E F G : Type*} [NontriviallyNormedField π] [NormedAddCommGroup E] [NormedS... | Mathlib/Analysis/NormedSpace/Complemented.lean | 139 | 143 | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace π]
[FiniteDimensional π (E β§Έ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented := by |
obtain β¨q, hqβ© : β q, IsCompl p q := p.exists_isCompl
haveI : FiniteDimensional π q := (p.quotientEquivOfIsCompl q hq).finiteDimensional
exact .of_isCompl_isClosed hq hp q.closed_of_finiteDimensional
| 1,865 |
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.NormedSpace.Complemented
#align_import analysis.calculus.implicit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/Analysis/Calculus/Implicit.lean | 201 | 214 | theorem implicitFunction_hasStrictFDerivAt (g'inv : G βL[π] E)
(hg'inv : Ο.rightDeriv.comp g'inv = ContinuousLinearMap.id π G)
(hg'invf : Ο.leftDeriv.comp g'inv = 0) :
HasStrictFDerivAt (Ο.implicitFunction (Ο.leftFun Ο.pt)) g'inv (Ο.rightFun Ο.pt) := by |
have := Ο.hasStrictFDerivAt.to_localInverse
simp only [prodFun] at this
convert this.comp (Ο.rightFun Ο.pt) ((hasStrictFDerivAt_const _ _).prod (hasStrictFDerivAt_id _))
-- Porting note: added parentheses to help `simp`
simp only [ContinuousLinearMap.ext_iff, (ContinuousLinearMap.comp_apply)] at hg'inv hg'in... | 1,866 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.FiniteDimension
variable {π E A B : Type*} [NontriviallyNormedField π] [NormedAddCommGroup E] [NormedSpace π E]
[CommSemiring A] {z : E} {... | Mathlib/Analysis/Analytic/Polynomial.lean | 26 | 32 | theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt π f z) (p : A[X]) :
AnalyticAt π (fun x β¦ aeval (f x) p) z := by |
refine p.induction_on (fun k β¦ ?_) (fun p q hp hq β¦ ?_) fun p i hp β¦ ?_
Β· simp_rw [aeval_C]; apply analyticAt_const
Β· simp_rw [aeval_add]; exact hp.add hq
Β· convert hp.mul hf
simp_rw [pow_succ, aeval_mul, β mul_assoc, aeval_X]
| 1,867 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.FiniteDimension
variable {π E A B : Type*} [NontriviallyNormedField π] [NormedAddCommGroup E] [NormedSpace π E]
[CommSemiring A] {z : E} {... | Mathlib/Analysis/Analytic/Polynomial.lean | 47 | 52 | theorem AnalyticAt.aeval_mvPolynomial (hf : β i, AnalyticAt π (f Β· i) z) (p : MvPolynomial Ο A) :
AnalyticAt π (fun x β¦ aeval (f x) p) z := by |
apply p.induction_on (fun k β¦ ?_) (fun p q hp hq β¦ ?_) fun p i hp β¦ ?_ -- `refine` doesn't work
Β· simp_rw [aeval_C]; apply analyticAt_const
Β· simp_rw [map_add]; exact hp.add hq
Β· simp_rw [map_mul, aeval_X]; exact hp.mul (hf i)
| 1,867 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 163 | 176 | theorem ContinuousLinearMap.continuous_det : Continuous fun f : E βL[π] E => f.det := by |
change Continuous fun f : E βL[π] E => LinearMap.det (f : E ββ[π] E)
-- Porting note: this could be easier with `det_cases`
by_cases h : β s : Finset E, Nonempty (Basis (β₯s) π E)
Β· rcases h with β¨s, β¨bβ©β©
haveI : FiniteDimensional π E := FiniteDimensional.of_fintype_basis b
simp_rw [LinearMap.det_eq... | 1,868 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 189 | 192 | theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace β E']
[FiniteDimensional β E'] : 0 < lipschitzExtensionConstant E' := by |
rw [lipschitzExtensionConstant]
exact zero_lt_one.trans_le (le_max_right _ _)
| 1,868 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 198 | 220 | theorem LipschitzOnWith.extend_finite_dimension {Ξ± : Type*} [PseudoMetricSpace Ξ±] {E' : Type*}
[NormedAddCommGroup E'] [NormedSpace β E'] [FiniteDimensional β E'] {s : Set Ξ±} {f : Ξ± β E'}
{K : ββ₯0} (hf : LipschitzOnWith K f s) :
β g : Ξ± β E', LipschitzWith (lipschitzExtensionConstant E' * K) g β§ EqOn f g s ... |
/- This result is already known for spaces `ΞΉ β β`. We use a continuous linear equiv between
`E'` and such a space to transfer the result to `E'`. -/
let ΞΉ : Type _ := Basis.ofVectorSpaceIndex β E'
let A := (Basis.ofVectorSpace β E').equivFun.toContinuousLinearEquiv
have LA : LipschitzWith βA.toContinuousL... | 1,868 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 223 | 229 | theorem LinearMap.exists_antilipschitzWith [FiniteDimensional π E] (f : E ββ[π] F)
(hf : LinearMap.ker f = β₯) : β K > 0, AntilipschitzWith K f := by |
cases subsingleton_or_nontrivial E
Β· exact β¨1, zero_lt_one, AntilipschitzWith.of_subsingletonβ©
Β· rw [LinearMap.ker_eq_bot] at hf
let e : E βL[π] LinearMap.range f := (LinearEquiv.ofInjective f hf).toContinuousLinearEquiv
exact β¨_, e.nnnorm_symm_pos, e.antilipschitzβ©
| 1,868 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 235 | 241 | theorem LinearMap.injective_iff_antilipschitz [FiniteDimensional π E] (f : E ββ[π] F) :
Injective f β β K > 0, AntilipschitzWith K f := by |
constructor
Β· rw [β LinearMap.ker_eq_bot]
exact f.exists_antilipschitzWith
Β· rintro β¨K, -, Hβ©
exact H.injective
| 1,868 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 246 | 255 | theorem ContinuousLinearMap.isOpen_injective [FiniteDimensional π E] :
IsOpen { L : E βL[π] F | Injective L } := by |
rw [isOpen_iff_eventually]
rintro Οβ hΟβ
rcases Οβ.injective_iff_antilipschitz.mp hΟβ with β¨K, K_pos, Hβ©
have : βαΆ Ο in π Οβ, βΟ - Οβββ < Kβ»ΒΉ := eventually_nnnorm_sub_lt _ <| inv_pos_of_pos K_pos
filter_upwards [this] with Ο hΟ
apply Ο.injective_iff_antilipschitz.mpr
exact β¨(Kβ»ΒΉ - βΟ - Οβββ)β»ΒΉ, inv_pos_... | 1,868 |
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
import Mathlib.Analysis.NormedSpace.FiniteDimension
open Set
open scoped NNReal
namespace ApproximatesLinearOn
| Mathlib/Analysis/Calculus/InverseFunctionTheorem/FiniteDimensional.lean | 27 | 47 | theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSpace β E]
{F : Type*} [NormedAddCommGroup F] [NormedSpace β F] [FiniteDimensional β F] {s : Set E}
{f : E β F} {f' : E βL[β] F} {c : ββ₯0} (hf : ApproximatesLinearOn f (f' : E βL[β] F) s c)
(hc : Subsingleton E β¨ lipschitzExtensio... |
-- the difference `f - f'` is Lipschitz on `s`. It can be extended to a Lipschitz function `u`
-- on the whole space, with a slightly worse Lipschitz constant. Then `f' + u` will be the
-- desired homeomorphism.
obtain β¨u, hu, ufβ© :
β u : E β F, LipschitzWith (lipschitzExtensionConstant F * c) u β§ EqOn (f ... | 1,869 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {π : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 35 | 43 | theorem contDiffOn_clm_apply {n : ββ} {f : E β F βL[π] G} {s : Set E} [FiniteDimensional π F] :
ContDiffOn π n f s β β y, ContDiffOn π n (fun x => f x y) s := by |
refine β¨fun h y => h.clm_apply contDiffOn_const, fun h => ?_β©
let d := finrank π F
have hd : d = finrank π (Fin d β π) := (finrank_fin_fun π).symm
let eβ := ContinuousLinearEquiv.ofFinrankEq hd
let eβ := (eβ.arrowCongr (1 : G βL[π] G)).trans (ContinuousLinearEquiv.piRing (Fin d))
rw [β id_comp f, β eβ... | 1,870 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {π : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 46 | 48 | theorem contDiff_clm_apply_iff {n : ββ} {f : E β F βL[π] G} [FiniteDimensional π F] :
ContDiff π n f β β y, ContDiff π n fun x => f x y := by |
simp_rw [β contDiffOn_univ, contDiffOn_clm_apply]
| 1,870 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {π : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 60 | 62 | theorem contDiff_succ_iff_fderiv_apply [FiniteDimensional π E] {n : β} {f : E β F} :
ContDiff π (n + 1 : β) f β Differentiable π f β§ β y, ContDiff π n fun x => fderiv π f x y := by |
rw [contDiff_succ_iff_fderiv, contDiff_clm_apply_iff]
| 1,870 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {π : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 71 | 75 | theorem contDiffOn_succ_iff_fderiv_apply [FiniteDimensional π E] {n : β} {f : E β F} {s : Set E}
(hs : UniqueDiffOn π s) :
ContDiffOn π (n + 1 : β) f s β
DifferentiableOn π f s β§ β y, ContDiffOn π n (fun x => fderivWithin π f s x y) s := by |
rw [contDiffOn_succ_iff_fderivWithin hs, contDiffOn_clm_apply]
| 1,870 |
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 49 | 62 | theorem isConformalMap_complex_linear {map : β βL[β] E} (nonzero : map β 0) :
IsConformalMap (map.restrictScalars β) := by |
have minorβ : βmap 1β β 0 := by simpa only [ext_ring_iff, Ne, norm_eq_zero] using nonzero
refine β¨βmap 1β, minorβ, β¨βmap 1ββ»ΒΉ β’ ((map : β ββ[β] E) : β ββ[β] E), ?_β©, ?_β©
Β· intro x
simp only [LinearMap.smul_apply]
have : x = x β’ (1 : β) := by rw [smul_eq_mul, mul_one]
nth_rw 1 [this]
rw [LinearMap... | 1,871 |
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 78 | 91 | theorem IsConformalMap.is_complex_or_conj_linear (h : IsConformalMap g) :
(β map : β βL[β] β, map.restrictScalars β = g) β¨
β map : β βL[β] β, map.restrictScalars β = g βL βconjCLE := by |
rcases h with β¨c, -, li, rflβ©
obtain β¨li, rflβ© : β li' : β ββα΅’[β] β, li'.toLinearIsometry = li :=
β¨li.toLinearIsometryEquiv rfl, by ext1; rflβ©
rcases linear_isometry_complex li with β¨a, rfl | rflβ©
-- let rot := c β’ (a : β) β’ ContinuousLinearMap.id β β,
Β· refine Or.inl β¨c β’ (a : β) β’ ContinuousLinearMap.i... | 1,871 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 49 | 62 | theorem HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) :
HasStrictDerivAt (fun x : β => (e x).re) e'.re z := by |
have A : HasStrictFDerivAt ((β) : β β β) ofRealCLM z := ofRealCLM.hasStrictFDerivAt
have B :
HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : β βL[β] β).restrictScalars β)
(ofRealCLM z) :=
h.hasStrictFDerivAt.restrictScalars β
have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM... | 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 68 | 81 | theorem HasDerivAt.real_of_complex (h : HasDerivAt e e' z) :
HasDerivAt (fun x : β => (e x).re) e'.re z := by |
have A : HasFDerivAt ((β) : β β β) ofRealCLM z := ofRealCLM.hasFDerivAt
have B :
HasFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : β βL[β] β).restrictScalars β)
(ofRealCLM z) :=
h.hasFDerivAt.restrictScalars β
have C : HasFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasFDerivAt
-- Porting note... | 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 84 | 89 | theorem ContDiffAt.real_of_complex {n : ββ} (h : ContDiffAt β n e z) :
ContDiffAt β n (fun x : β => (e x).re) z := by |
have A : ContDiffAt β n ((β) : β β β) z := ofRealCLM.contDiff.contDiffAt
have B : ContDiffAt β n e z := h.restrict_scalars β
have C : ContDiffAt β n re (e z) := reCLM.contDiff.contDiffAt
exact C.comp z (B.comp z A)
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 99 | 103 | theorem HasStrictDerivAt.complexToReal_fderiv' {f : β β E} {x : β} {f' : E}
(h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (reCLM.smulRight f' + I β’ imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasStrictFDerivAt.restrictScalars β
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 106 | 108 | theorem HasDerivAt.complexToReal_fderiv' {f : β β E} {x : β} {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (reCLM.smulRight f' + I β’ imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars β
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 111 | 115 | theorem HasDerivWithinAt.complexToReal_fderiv' {f : β β E} {s : Set β} {x : β} {f' : E}
(h : HasDerivWithinAt f f' s x) :
HasFDerivWithinAt f (reCLM.smulRight f' + I β’ imCLM.smulRight f') s x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasFDerivWithinAt.restrictScalars β
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 118 | 120 | theorem HasStrictDerivAt.complexToReal_fderiv {f : β β β} {f' x : β} (h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (f' β’ (1 : β βL[β] β)) x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasStrictFDerivAt.restrictScalars β
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 123 | 125 | theorem HasDerivAt.complexToReal_fderiv {f : β β β} {f' x : β} (h : HasDerivAt f f' x) :
HasFDerivAt f (f' β’ (1 : β βL[β] β)) x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivAt.restrictScalars β
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 128 | 130 | theorem HasDerivWithinAt.complexToReal_fderiv {f : β β β} {s : Set β} {f' x : β}
(h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' β’ (1 : β βL[β] β)) s x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivWithinAt.restrictScalars β
| 1,872 |
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