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.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 266 | 284 | theorem disjoint_lsingle_lsingle (s t : Set Ξ±) (hs : Disjoint s t) :
Disjoint (β¨ a β s, LinearMap.range (lsingle a : M ββ[R] Ξ± ββ M))
(β¨ a β t, LinearMap.range (lsingle a : M ββ[R] Ξ± ββ M)) := by |
-- Porting note: 2 placeholders are added to prevent timeout.
refine
(Disjoint.mono
(lsingle_range_le_ker_lapply s sαΆ ?_)
(lsingle_range_le_ker_lapply t tαΆ ?_))
?_
Β· apply disjoint_compl_right
Β· apply disjoint_compl_right
rw [disjoint_iff_inf_le]
refine le_trans (le_iInf fun i => ?_) ... | 1,319 |
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 287 | 289 | theorem span_single_image (s : Set M) (a : Ξ±) :
Submodule.span R (single a '' s) = (Submodule.span R s).map (lsingle a : M ββ[R] Ξ± ββ M) := by |
rw [β span_image]; rfl
| 1,319 |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 84 | 89 | theorem eq_top_of_unit_mem (x y : Ξ±) (hx : x β I) (h : y * x = 1) : I = β€ :=
eq_top_iff.2 fun z _ =>
calc
z = z * (y * x) := by | simp [h]
_ = z * y * x := Eq.symm <| mul_assoc z y x
_ β I := I.mul_mem_left _ hx
| 1,320 |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 106 | 110 | theorem unit_mul_mem_iff_mem {x y : Ξ±} (hy : IsUnit y) : y * x β I β x β I := by |
refine β¨fun h => ?_, fun h => I.mul_mem_left y hβ©
obtain β¨y', hy'β© := hy.exists_left_inv
have := I.mul_mem_left y' h
rwa [β mul_assoc, hy', one_mul] at this
| 1,320 |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 167 | 168 | theorem isCompactElement_top : CompleteLattice.IsCompactElement (β€ : Ideal Ξ±) := by |
simpa only [β span_singleton_one] using Submodule.singleton_span_isCompactElement 1
| 1,320 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 135 | 144 | theorem of_le (N : Submonoid R) (hβ : M β€ N) (hβ : β r β N, IsUnit (algebraMap R S r)) :
IsLocalization N S where
map_units' r := hβ r r.2
surj' s :=
have β¨β¨x, y, hyβ©, Hβ© := IsLocalization.surj M s
β¨β¨x, y, hβ hyβ©, Hβ©
exists_of_eq {x y} := by |
rw [IsLocalization.eq_iff_exists M]
rintro β¨c, hcβ©
exact β¨β¨c, hβ c.2β©, hcβ©
| 1,321 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 202 | 204 | theorem sec_spec' (z : S) :
algebraMap R S (IsLocalization.sec M z).1 = algebraMap R S (IsLocalization.sec M z).2 * z := by |
rw [mul_comm, sec_spec]
| 1,321 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 222 | 225 | theorem eq_zero_of_fst_eq_zero {z x} {y : M} (h : z * algebraMap R S y = algebraMap R S x)
(hx : x = 0) : z = 0 := by |
rw [hx, (algebraMap R S).map_zero] at h
exact (IsUnit.mul_left_eq_zero (IsLocalization.map_units S y)).1 h
| 1,321 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 230 | 237 | theorem map_eq_zero_iff (r : R) : algebraMap R S r = 0 β β m : M, βm * r = 0 := by |
constructor
Β· intro h
obtain β¨m, hmβ© := (IsLocalization.eq_iff_exists M S).mp ((algebraMap R S).map_zero.trans h.symm)
exact β¨m, by simpa using hm.symmβ©
Β· rintro β¨m, hmβ©
rw [β (IsLocalization.map_units S m).mul_right_inj, mul_zero, β RingHom.map_mul, hm,
RingHom.map_zero]
| 1,321 |
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Submonoid R) {S : Type*... | Mathlib/RingTheory/Localization/Away/Basic.lean | 58 | 61 | theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by |
convert IsLocalization.mk'_mul_mk'_eq_one (M := Submonoid.powers x) (S := S) _ 1
symm
apply IsLocalization.mk'_one
| 1,322 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 63 | 66 | theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a β’ b) := by |
rcases hb with β¨b', hbβ©
use a * b'
rw [β hb, (algebraMap R S).map_mul, Algebra.smul_def]
| 1,323 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 85 | 87 | theorem exists_integer_multiple (a : S) : β b : M, IsInteger R ((b : R) β’ a) := by |
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple'
| 1,323 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 91 | 103 | theorem exist_integer_multiples {ΞΉ : Type*} (s : Finset ΞΉ) (f : ΞΉ β S) :
β b : M, β i β s, IsLocalization.IsInteger R ((b : R) β’ f i) := by |
haveI := Classical.propDecidable
refine β¨β i β s, (sec M (f i)).2, fun i hi => β¨?_, ?_β©β©
Β· exact (β j β s.erase i, (sec M (f j)).2) * (sec M (f i)).1
rw [RingHom.map_mul, sec_spec', β mul_assoc, β (algebraMap R S).map_mul, β Algebra.smul_def]
congr 2
refine _root_.trans ?_ (map_prod (Submonoid.subtype M) _... | 1,323 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 107 | 111 | theorem exist_integer_multiples_of_finite {ΞΉ : Type*} [Finite ΞΉ] (f : ΞΉ β S) :
β b : M, β i, IsLocalization.IsInteger R ((b : R) β’ f i) := by |
cases nonempty_fintype ΞΉ
obtain β¨b, hbβ© := exist_integer_multiples M Finset.univ f
exact β¨b, fun i => hb i (Finset.mem_univ _)β©
| 1,323 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 149 | 159 | theorem finsetIntegerMultiple_image [DecidableEq R] (s : Finset S) :
algebraMap R S '' finsetIntegerMultiple M s = commonDenomOfFinset M s β’ (s : Set S) := by |
delta finsetIntegerMultiple commonDenom
rw [Finset.coe_image]
ext
constructor
Β· rintro β¨_, β¨x, -, rflβ©, rflβ©
rw [map_integerMultiple]
exact Set.mem_image_of_mem _ x.prop
Β· rintro β¨x, hx, rflβ©
exact β¨_, β¨β¨x, hxβ©, s.mem_attach _, rflβ©, map_integerMultiple M s id _β©
| 1,323 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 37 | 47 | theorem exists_reduced_fraction (x : K) :
β (a : A) (b : nonZeroDivisors A), IsRelPrime a b β§ mk' K a b = x := by |
obtain β¨β¨b, b_nonzeroβ©, a, habβ© := exists_integer_multiple (nonZeroDivisors A) x
obtain β¨a', b', c', no_factor, rfl, rflβ© :=
UniqueFactorizationMonoid.exists_reduced_factors' a b
(mem_nonZeroDivisors_iff_ne_zero.mp b_nonzero)
obtain β¨_, b'_nonzeroβ© := mul_mem_nonZeroDivisors.mp b_nonzero
refine β¨a', ... | 1,324 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 70 | 72 | theorem mk'_num_den' (x : K) : algebraMap A K (num A x) / algebraMap A K (den A x) = x := by |
rw [β mk'_eq_div]
apply mk'_num_den
| 1,324 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 97 | 105 | theorem isInteger_of_isUnit_den {x : K} (h : IsUnit (den A x : A)) : IsInteger A x := by |
cases' h with d hd
have d_ne_zero : algebraMap A K (den A x) β 0 :=
IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2
use βdβ»ΒΉ * num A x
refine _root_.trans ?_ (mk'_num_den A x)
rw [map_mul, map_units_inv, hd]
apply mul_left_cancelβ d_ne_zero
rw [β mul_assoc, mul_inv_cancel d_ne_zero, ... | 1,324 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 53 | 61 | theorem mem_localizationLocalizationSubmodule {x : R} :
x β localizationLocalizationSubmodule M N β
β (y : N) (z : M), algebraMap R S x = y * algebraMap R S z := by |
rw [localizationLocalizationSubmodule, Submonoid.mem_comap, Submonoid.mem_sup]
constructor
Β· rintro β¨y, hy, _, β¨z, hz, rflβ©, eβ©
exact β¨β¨y, hyβ©, β¨z, hzβ©, e.symmβ©
Β· rintro β¨y, z, eβ©
exact β¨y, y.prop, _, β¨z, z.prop, rflβ©, e.symmβ©
| 1,325 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 66 | 70 | theorem localization_localization_map_units [IsLocalization N T]
(y : localizationLocalizationSubmodule M N) : IsUnit (algebraMap R T y) := by |
obtain β¨y', z, eqβ© := mem_localizationLocalizationSubmodule.mp y.prop
rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]
exact β¨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)β©
| 1,325 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 73 | 89 | theorem localization_localization_surj [IsLocalization N T] (x : T) :
β y : R Γ localizationLocalizationSubmodule M N,
x * algebraMap R T y.2 = algebraMap R T y.1 := by |
rcases IsLocalization.surj N x with β¨β¨y, sβ©, eqββ©
-- x = y / s
rcases IsLocalization.surj M y with β¨β¨z, tβ©, eqββ©
-- y = z / t
rcases IsLocalization.surj M (s : S) with β¨β¨z', t'β©, eqββ©
-- s = z' / t'
dsimp only at eqβ eqβ eqβ
refine β¨β¨z * t', z' * t, ?_β©, ?_β© -- x = y / s = (z * t') / (z' * t)
Β· rw [m... | 1,325 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 92 | 108 | theorem localization_localization_exists_of_eq [IsLocalization N T] (x y : R) :
algebraMap R T x = algebraMap R T y β
β c : localizationLocalizationSubmodule M N, βc * x = βc * y := by |
rw [IsScalarTower.algebraMap_apply R S T, IsScalarTower.algebraMap_apply R S T,
IsLocalization.eq_iff_exists N T]
rintro β¨z, eqββ©
rcases IsLocalization.surj M (z : S) with β¨β¨z', sβ©, eqββ©
dsimp only at eqβ
suffices (algebraMap R S) (x * z' : R) = (algebraMap R S) (y * z') by
obtain β¨c, eqβ : βc * (x *... | 1,325 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 125 | 133 | theorem localization_localization_isLocalization_of_has_all_units [IsLocalization N T]
(H : β x : S, IsUnit x β x β N) : IsLocalization (N.comap (algebraMap R S)) T := by |
convert localization_localization_isLocalization M N T using 1
dsimp [localizationLocalizationSubmodule]
congr
symm
rw [sup_eq_left]
rintro _ β¨x, hx, rflβ©
exact H _ (IsLocalization.map_units _ β¨x, hxβ©)
| 1,325 |
import Mathlib.RingTheory.Localization.LocalizationLocalization
#align_import ring_theory.localization.as_subring from "leanprover-community/mathlib"@"649ca66bf4d62796b5eefef966e622d91aa471f3"
namespace Localization
open nonZeroDivisors
variable {A : Type*} (K : Type*) [CommRing A] (S : Submonoid A) (hS : S β€ A... | Mathlib/RingTheory/Localization/AsSubring.lean | 31 | 32 | theorem map_isUnit_of_le (hS : S β€ Aβ°) (s : S) : IsUnit (algebraMap A K s) := by |
apply IsLocalization.map_units K (β¨s.1, hS s.2β© : Aβ°)
| 1,326 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 99 | 102 | theorem induction_on {Ξ² : LocalizedModule S M β Prop} (h : β (m : M) (s : S), Ξ² (mk m s)) :
β x : LocalizedModule S M, Ξ² x := by |
rintro β¨β¨m, sβ©β©
exact h m s
| 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 106 | 109 | theorem induction_onβ {Ξ² : LocalizedModule S M β LocalizedModule S M β Prop}
(h : β (m m' : M) (s s' : S), Ξ² (mk m s) (mk m' s')) : β x y, Ξ² x y := by |
rintro β¨β¨m, sβ©β© β¨β¨m', s'β©β©
exact h m m' s s'
| 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 120 | 121 | theorem liftOn_mk {Ξ± : Type*} {f : M Γ S β Ξ±} (wd : β (p p' : M Γ S), p β p' β f p = f p')
(m : M) (s : S) : liftOn (mk m s) f wd = f β¨m, sβ© := by | convert Quotient.liftOn_mk f wd β¨m, sβ©
| 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 132 | 135 | theorem liftOnβ_mk {Ξ± : Type*} (f : M Γ S β M Γ S β Ξ±)
(wd : β (p q p' q' : M Γ S), p β p' β q β q' β f p q = f p' q') (m m' : M)
(s s' : S) : liftOnβ (mk m s) (mk m' s') f wd = f β¨m, sβ© β¨m', s'β© := by |
convert Quotient.liftOnβ_mk f wd _ _
| 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 142 | 145 | theorem subsingleton (h : 0 β S) : Subsingleton (LocalizedModule S M) := by |
refine β¨fun a b β¦ ?_β©
induction a,b using LocalizedModule.induction_onβ
exact mk_eq.mpr β¨β¨0, hβ©, by simp only [Submonoid.mk_smul, zero_smul]β©
| 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 574 | 588 | theorem IsLocalizedModule.of_linearEquiv (e : M' ββ[R] M'') [hf : IsLocalizedModule S f] :
IsLocalizedModule S (e ββ f : M ββ[R] M'') where
map_units s := by |
rw [show algebraMap R (Module.End R M'') s = e ββ (algebraMap R (Module.End R M') s) ββ e.symm
by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp,
LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp]
exact (Module.End_isUnit_iff _).m... | 1,327 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 599 | 610 | theorem isLocalizedModule_iff_isLocalization {A Aβ} [CommSemiring A] [Algebra R A] [CommSemiring Aβ]
[Algebra A Aβ] [Algebra R Aβ] [IsScalarTower R A Aβ] :
IsLocalizedModule S (IsScalarTower.toAlgHom R A Aβ).toLinearMap β
IsLocalization (Algebra.algebraMapSubmonoid A S) Aβ := by |
rw [isLocalizedModule_iff, isLocalization_iff]
refine and_congr ?_ (and_congr (forall_congr' fun _ β¦ ?_) (forallβ_congr fun _ _ β¦ ?_))
Β· simp_rw [β (Algebra.lmul R Aβ).commutes, Algebra.lmul_isUnit_iff, Subtype.forall,
Algebra.algebraMapSubmonoid, β SetLike.mem_coe, Submonoid.coe_map,
Set.forall_mem_... | 1,327 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 52 | 52 | theorem powHalf_leftMoves (n) : (powHalf n).LeftMoves = PUnit := by | cases n <;> rfl
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 64 | 64 | theorem powHalf_moveLeft (n i) : (powHalf n).moveLeft i = 0 := by | cases n <;> cases i <;> rfl
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 85 | 86 | theorem birthday_half : birthday (powHalf 1) = 2 := by |
rw [birthday_def]; simp
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 90 | 95 | theorem numeric_powHalf (n) : (powHalf n).Numeric := by |
induction' n with n hn
Β· exact numeric_one
Β· constructor
Β· simpa using hn.moveLeft_lt default
Β· exact β¨fun _ => numeric_zero, fun _ => hnβ©
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 106 | 109 | theorem powHalf_le_one (n : β) : powHalf n β€ 1 := by |
induction' n with n hn
Β· exact le_rfl
Β· exact (powHalf_succ_le_powHalf n).trans hn
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 116 | 117 | theorem powHalf_pos (n : β) : 0 < powHalf n := by |
rw [β lf_iff_lt numeric_zero (numeric_powHalf n), zero_lf_le]; simp
| 1,328 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 124 | 156 | theorem add_powHalf_succ_self_eq_powHalf (n) : powHalf (n + 1) + powHalf (n + 1) β powHalf n := by |
induction' n using Nat.strong_induction_on with n hn
constructor <;> rw [le_iff_forall_lf] <;> constructor
Β· rintro (β¨β¨β©β© | β¨β¨β©β©) <;> apply lf_of_lt
Β· calc
0 + powHalf n.succ β powHalf n.succ := zero_add_equiv _
_ < powHalf n := powHalf_succ_lt_powHalf n
Β· calc
powHalf n.succ + 0 ... | 1,328 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 126 | 128 | theorem linearIndependent_iff :
LinearIndependent R v β β l, Finsupp.total ΞΉ M R v l = 0 β l = 0 := by |
simp [LinearIndependent, LinearMap.ker_eq_bot']
| 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 131 | 151 | theorem linearIndependent_iff' :
LinearIndependent R v β
β s : Finset ΞΉ, β g : ΞΉ β R, β i β s, g i β’ v i = 0 β β i β s, g i = 0 :=
linearIndependent_iff.trans
β¨fun hf s g hg i his =>
have h :=
hf (β i β s, Finsupp.single i (g i)) <| by
simpa only [map_sum, Finsupp.total_single] u... |
{ rw [Finsupp.lapply_apply, Finsupp.single_eq_same] }
_ = β j β s, (Finsupp.lapply i : (ΞΉ ββ R) ββ[R] R) (Finsupp.single j (g j)) :=
Eq.symm <|
Finset.sum_eq_single i
(fun j _hjs hji => by rw [Finsupp.lapply_apply, Finsupp.single_eq_of_ne hji])
fun hn... | 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 154 | 164 | theorem linearIndependent_iff'' :
LinearIndependent R v β
β (s : Finset ΞΉ) (g : ΞΉ β R), (β i β s, g i = 0) β
β i β s, g i β’ v i = 0 β β i, g i = 0 := by |
classical
exact linearIndependent_iff'.trans
β¨fun H s g hg hv i => if his : i β s then H s g hv i his else hg i his, fun H s g hg i hi => by
convert
H s (fun j => if j β s then g j else 0) (fun j hj => if_neg hj)
(by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i
... | 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 167 | 171 | theorem not_linearIndependent_iff :
Β¬LinearIndependent R v β
β s : Finset ΞΉ, β g : ΞΉ β R, β i β s, g i β’ v i = 0 β§ β i β s, g i β 0 := by |
rw [linearIndependent_iff']
simp only [exists_prop, not_forall]
| 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 174 | 181 | theorem Fintype.linearIndependent_iff [Fintype ΞΉ] :
LinearIndependent R v β β g : ΞΉ β R, β i, g i β’ v i = 0 β β i, g i = 0 := by |
refine
β¨fun H g => by simpa using linearIndependent_iff'.1 H Finset.univ g, fun H =>
linearIndependent_iff''.2 fun s g hg hs i => H _ ?_ _β©
rw [β hs]
refine (Finset.sum_subset (Finset.subset_univ _) fun i _ hi => ?_).symm
rw [hg i hi, zero_smul]
| 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 186 | 189 | theorem Fintype.linearIndependent_iff' [Fintype ΞΉ] [DecidableEq ΞΉ] :
LinearIndependent R v β
LinearMap.ker (LinearMap.lsum R (fun _ β¦ R) β fun i β¦ LinearMap.id.smulRight (v i)) = β₯ := by |
simp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff]
| 1,329 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 192 | 194 | theorem Fintype.not_linearIndependent_iff [Fintype ΞΉ] :
Β¬LinearIndependent R v β β g : ΞΉ β R, β i, g i β’ v i = 0 β§ β i, g i β 0 := by |
simpa using not_iff_not.2 Fintype.linearIndependent_iff
| 1,329 |
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe w w' u u' v v'
variable {R : Type u} {R' : Type u'} {M Mβ : Type v} {M' : Type v'}
open Cardinal Submodule Function... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 79 | 84 | theorem cardinal_lift_le_rank {ΞΉ : Type w} {v : ΞΉ β M}
(hv : LinearIndependent R v) :
Cardinal.lift.{v} #ΞΉ β€ Cardinal.lift.{w} (Module.rank R M) := by |
rw [Module.rank]
refine le_trans ?_ (lift_le.mpr <| le_ciSup (bddAbove_range.{v, v} _) β¨_, hv.coe_rangeβ©)
exact lift_mk_le'.mpr β¨(Equiv.ofInjective _ hv.injective).toEmbeddingβ©
| 1,330 |
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe w w' u u' v v'
variable {R : Type u} {R' : Type u'} {M Mβ : Type v} {M' : Type v'}
open Cardinal Submodule Function... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 92 | 94 | theorem cardinal_le_rank {ΞΉ : Type v} {v : ΞΉ β M}
(hv : LinearIndependent R v) : #ΞΉ β€ Module.rank R M := by |
simpa using hv.cardinal_lift_le_rank
| 1,330 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 58 | 61 | theorem finrank_eq_of_rank_eq {n : β} (h : Module.rank R M = βn) : finrank R M = n := by |
apply_fun toNat at h
rw [toNat_natCast] at h
exact mod_cast h
| 1,331 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 72 | 75 | theorem finrank_le_of_rank_le {n : β} (h : Module.rank R M β€ βn) : finrank R M β€ n := by |
rwa [β Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h
Β· exact h.trans_lt (nat_lt_aleph0 n)
Β· exact nat_lt_aleph0 n
| 1,331 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 78 | 81 | theorem finrank_lt_of_rank_lt {n : β} (h : Module.rank R M < βn) : finrank R M < n := by |
rwa [β Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_natCast] at h
Β· exact h.trans (nat_lt_aleph0 n)
Β· exact nat_lt_aleph0 n
| 1,331 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 84 | 89 | theorem lt_rank_of_lt_finrank {n : β} (h : n < finrank R M) : βn < Module.rank R M := by |
rwa [β Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_natCast]
Β· exact nat_lt_aleph0 n
Β· contrapose! h
rw [finrank, Cardinal.toNat_apply_of_aleph0_le h]
exact n.zero_le
| 1,331 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 92 | 93 | theorem one_lt_rank_of_one_lt_finrank (h : 1 < finrank R M) : 1 < Module.rank R M := by |
simpa using lt_rank_of_lt_finrank h
| 1,331 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 95 | 98 | theorem finrank_le_finrank_of_rank_le_rank
(h : lift.{w} (Module.rank R M) β€ Cardinal.lift.{v} (Module.rank R N))
(h' : Module.rank R N < β΅β) : finrank R M β€ finrank R N := by |
simpa only [toNat_lift] using toNat_le_toNat h (lift_lt_aleph0.mpr h')
| 1,331 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 61 | 63 | theorem of_subsingleton [Subsingleton M] (x y : M) : SameRay R x y := by |
rw [Subsingleton.elim x 0]
exact zero_left _
| 1,332 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 74 | 76 | theorem refl (x : M) : SameRay R x x := by |
nontriviality R
exact Or.inr (Or.inr <| β¨1, 1, zero_lt_one, zero_lt_one, rflβ©)
| 1,332 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 102 | 111 | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 β x = 0 β¨ z = 0) :
SameRay R x z := by |
rcases eq_or_ne x 0 with (rfl | hx); Β· exact zero_left z
rcases eq_or_ne z 0 with (rfl | hz); Β· exact zero_right x
rcases eq_or_ne y 0 with (rfl | hy);
Β· exact (hy rfl).elim (fun h => (hx h).elim) fun h => (hz h).elim
rcases hxy.exists_pos hx hy with β¨rβ, rβ, hrβ, hrβ, hββ©
rcases hyz.exists_pos hy hz with ... | 1,332 |
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 62 | 65 | theorem segment_eq_imageβ (x y : E) :
[x -[π] y] =
(fun p : π Γ π => p.1 β’ x + p.2 β’ y) '' { p | 0 β€ p.1 β§ 0 β€ p.2 β§ p.1 + p.2 = 1 } := by |
simp only [segment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
| 1,333 |
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 68 | 71 | theorem openSegment_eq_imageβ (x y : E) :
openSegment π x y =
(fun p : π Γ π => p.1 β’ x + p.2 β’ y) '' { p | 0 < p.1 β§ 0 < p.2 β§ p.1 + p.2 = 1 } := by |
simp only [openSegment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
| 1,333 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 75 | 80 | theorem starConvex_iff_segment_subset : StarConvex π x s β β β¦yβ¦, y β s β [x -[π] y] β s := by |
constructor
Β· rintro h y hy z β¨a, b, ha, hb, hab, rflβ©
exact h hy ha hb hab
Β· rintro h y hy a b ha hb hab
exact h hy β¨a, b, ha, hb, hab, rflβ©
| 1,334 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 93 | 99 | theorem starConvex_iff_pointwise_add_subset :
StarConvex π x s β β β¦a b : πβ¦, 0 β€ a β 0 β€ b β a + b = 1 β a β’ {x} + b β’ s β s := by |
refine
β¨?_, fun h y hy a b ha hb hab =>
h ha hb hab (add_mem_add (smul_mem_smul_set <| mem_singleton _) β¨_, hy, rflβ©)β©
rintro hA a b ha hb hab w β¨au, β¨u, rfl : u = x, rflβ©, bv, β¨v, hv, rflβ©, rflβ©
exact hA hv ha hb hab
| 1,334 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 121 | 125 | theorem StarConvex.union (hs : StarConvex π x s) (ht : StarConvex π x t) :
StarConvex π x (s βͺ t) := by |
rintro y (hy | hy) a b ha hb hab
Β· exact Or.inl (hs hy ha hb hab)
Β· exact Or.inr (ht hy ha hb hab)
| 1,334 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 128 | 133 | theorem starConvex_iUnion {ΞΉ : Sort*} {s : ΞΉ β Set E} (hs : β i, StarConvex π x (s i)) :
StarConvex π x (β i, s i) := by |
rintro y hy a b ha hb hab
rw [mem_iUnion] at hy β’
obtain β¨i, hyβ© := hy
exact β¨i, hs i hy ha hb habβ©
| 1,334 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 136 | 139 | theorem starConvex_sUnion {S : Set (Set E)} (hS : β s β S, StarConvex π x s) :
StarConvex π x (ββ S) := by |
rw [sUnion_eq_iUnion]
exact starConvex_iUnion fun s => hS _ s.2
| 1,334 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import analysis.convex.basic from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
variable {π E F Ξ² : Type*}
open LinearMap Set
open scope... | Mathlib/Analysis/Convex/Basic.lean | 121 | 128 | theorem Directed.convex_iUnion {ΞΉ : Sort*} {s : ΞΉ β Set E} (hdir : Directed (Β· β Β·) s)
(hc : β β¦i : ΞΉβ¦, Convex π (s i)) : Convex π (β i, s i) := by |
rintro x hx y hy a b ha hb hab
rw [mem_iUnion] at hx hy β’
obtain β¨i, hxβ© := hx
obtain β¨j, hyβ© := hy
obtain β¨k, hik, hjkβ© := hdir i j
exact β¨k, hc (hik hx) (hjk hy) ha hb habβ©
| 1,335 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import analysis.convex.basic from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
variable {π E F Ξ² : Type*}
open LinearMap Set
open scope... | Mathlib/Analysis/Convex/Basic.lean | 131 | 134 | theorem DirectedOn.convex_sUnion {c : Set (Set E)} (hdir : DirectedOn (Β· β Β·) c)
(hc : β β¦A : Set Eβ¦, A β c β Convex π A) : Convex π (ββ c) := by |
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).convex_iUnion fun A => hc A.2
| 1,335 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.convex.body from "leanprover-community/mathlib"@"858a10cf68fd6c06872950fc58c4dcc68d465591"
open scoped Pointwise Topology NNReal
variable {V : Type*}
struc... | Mathlib/Analysis/Convex/Body.lean | 93 | 97 | theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : β x β K, - x β K) : 0 β K := by |
obtain β¨x, hxβ© := K.nonempty
rw [show 0 = (1/2 : β) β’ x + (1/2 : β) β’ (- x) by field_simp]
apply convex_iff_forall_pos.mp K.convex hx (h_symm x hx)
all_goals linarith
| 1,336 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 56 | 57 | theorem convexHull_eq_iInter : convexHull π s = β (t : Set E) (_ : s β t) (_ : Convex π t), t := by |
simp [convexHull, iInter_subtype, iInter_and]
| 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 62 | 63 | theorem mem_convexHull_iff : x β convexHull π s β β t, s β t β Convex π t β x β t := by |
simp_rw [convexHull_eq_iInter, mem_iInter]
| 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 94 | 100 | theorem convexHull_empty_iff : convexHull π s = β
β s = β
:= by |
constructor
Β· intro h
rw [β Set.subset_empty_iff, β h]
exact subset_convexHull π _
Β· rintro rfl
exact convexHull_empty
| 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 104 | 106 | theorem convexHull_nonempty_iff : (convexHull π s).Nonempty β s.Nonempty := by |
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
exact not_congr convexHull_empty_iff
| 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 127 | 131 | theorem convexHull_pair (x y : E) : convexHull π {x, y} = segment π x y := by |
refine (convexHull_min ?_ <| convex_segment _ _).antisymm
(segment_subset_convexHull (mem_insert _ _) <| subset_insert _ _ <| mem_singleton _)
rw [insert_subset_iff, singleton_subset_iff]
exact β¨left_mem_segment _ _ _, right_mem_segment _ _ _β©
| 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {π E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring π]
secti... | Mathlib/Analysis/Convex/Hull.lean | 144 | 158 | theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex π s) (x : E) :
Convex π (s \ {x}) β x β convexHull π (s \ {x}) := by |
constructor
Β· rintro hsx hx
rw [hsx.convexHull_eq] at hx
exact hx.2 (mem_singleton _)
rintro hx
suffices h : s \ {x} = convexHull π (s \ {x}) by
rw [h]
exact convex_convexHull π _
exact
Subset.antisymm (subset_convexHull π _) fun y hy =>
β¨convexHull_min diff_subset hs hy, by
... | 1,337 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Bornology.Absorbs
#align_import analysis.locally_convex.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set
open Pointwise Topology
... | Mathlib/Analysis/LocallyConvex/Basic.lean | 81 | 82 | theorem balanced_iff_closedBall_smul : Balanced π s β Metric.closedBall (0 : π) 1 β’ s β s := by |
simp [balanced_iff_smul_mem, smul_subset_iff]
| 1,338 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 81 | 82 | theorem mem_balancedCore_iff : x β balancedCore π s β β t, Balanced π t β§ t β s β§ x β t := by |
simp_rw [balancedCore, mem_sUnion, mem_setOf_eq, and_assoc]
| 1,339 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 85 | 90 | theorem smul_balancedCore_subset (s : Set E) {a : π} (ha : βaβ β€ 1) :
a β’ balancedCore π s β balancedCore π s := by |
rintro x β¨y, hy, rflβ©
rw [mem_balancedCore_iff] at hy
rcases hy with β¨t, ht1, ht2, hyβ©
exact β¨t, β¨ht1, ht2β©, ht1 a ha (smul_mem_smul_set hy)β©
| 1,339 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 108 | 109 | theorem mem_balancedHull_iff : x β balancedHull π s β β r : π, βrβ β€ 1 β§ x β r β’ s := by |
simp [balancedHull]
| 1,339 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 114 | 118 | theorem Balanced.balancedHull_subset_of_subset (ht : Balanced π t) (h : s β t) :
balancedHull π s β t := by |
intros x hx
obtain β¨r, hr, y, hy, rflβ© := mem_balancedHull_iff.1 hx
exact ht.smul_mem hr (h hy)
| 1,339 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 97 | 103 | theorem IsExtreme.inter (hAB : IsExtreme π A B) (hAC : IsExtreme π A C) :
IsExtreme π A (B β© C) := by |
use Subset.trans inter_subset_left hAB.1
rintro xβ hxβA xβ hxβA x β¨hxB, hxCβ© hx
obtain β¨hxβB, hxβBβ© := hAB.2 hxβA hxβA hxB hx
obtain β¨hxβC, hxβCβ© := hAC.2 hxβA hxβA hxC hx
exact β¨β¨hxβB, hxβCβ©, hxβB, hxβCβ©
| 1,340 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 111 | 117 | theorem isExtreme_iInter {ΞΉ : Sort*} [Nonempty ΞΉ] {F : ΞΉ β Set E}
(hAF : β i : ΞΉ, IsExtreme π A (F i)) : IsExtreme π A (β i : ΞΉ, F i) := by |
obtain i := Classical.arbitrary ΞΉ
refine β¨iInter_subset_of_subset i (hAF i).1, fun xβ hxβA xβ hxβA x hxF hx β¦ ?_β©
simp_rw [mem_iInter] at hxF β’
have h := fun i β¦ (hAF i).2 hxβA hxβA (hxF i) hx
exact β¨fun i β¦ (h i).1, fun i β¦ (h i).2β©
| 1,340 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 120 | 123 | theorem isExtreme_biInter {F : Set (Set E)} (hF : F.Nonempty) (hA : β B β F, IsExtreme π A B) :
IsExtreme π A (β B β F, B) := by |
haveI := hF.to_subtype
simpa only [iInter_subtype] using isExtreme_iInter fun i : F β¦ hA _ i.2
| 1,340 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 126 | 127 | theorem isExtreme_sInter {F : Set (Set E)} (hF : F.Nonempty) (hAF : β B β F, IsExtreme π A B) :
IsExtreme π A (ββ F) := by | simpa [sInter_eq_biInter] using isExtreme_biInter hF hAF
| 1,340 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 36 | 37 | theorem mem_convexJoin : x β convexJoin π s t β β a β s, β b β t, x β segment π a b := by |
simp [convexJoin]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 57 | 57 | theorem convexJoin_empty_left (t : Set E) : convexJoin π β
t = β
:= by | simp [convexJoin]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 61 | 61 | theorem convexJoin_empty_right (s : Set E) : convexJoin π s β
= β
:= by | simp [convexJoin]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 65 | 66 | theorem convexJoin_singleton_left (t : Set E) (x : E) :
convexJoin π {x} t = β y β t, segment π x y := by | simp [convexJoin]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 70 | 71 | theorem convexJoin_singleton_right (s : Set E) (y : E) :
convexJoin π s {y} = β x β s, segment π x y := by | simp [convexJoin]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 75 | 75 | theorem convexJoin_singletons (x : E) : convexJoin π {x} {y} = segment π x y := by | simp
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 79 | 81 | theorem convexJoin_union_left (sβ sβ t : Set E) :
convexJoin π (sβ βͺ sβ) t = convexJoin π sβ t βͺ convexJoin π sβ t := by |
simp_rw [convexJoin, mem_union, iUnion_or, iUnion_union_distrib]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 85 | 87 | theorem convexJoin_union_right (s tβ tβ : Set E) :
convexJoin π s (tβ βͺ tβ) = convexJoin π s tβ βͺ convexJoin π s tβ := by |
simp_rw [convexJoin_comm s, convexJoin_union_left]
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 91 | 94 | theorem convexJoin_iUnion_left (s : ΞΉ β Set E) (t : Set E) :
convexJoin π (β i, s i) t = β i, convexJoin π (s i) t := by |
simp_rw [convexJoin, mem_iUnion, iUnion_exists]
exact iUnion_comm _
| 1,341 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ΞΉ : Sort*} {π E : Type*}
section OrderedSemiring
variable (π) [OrderedSemiring π] [AddCommMonoid E] [Module π E] {s t sβ sβ tβ tβ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 98 | 100 | theorem convexJoin_iUnion_right (s : Set E) (t : ΞΉ β Set E) :
convexJoin π s (β i, t i) = β i, convexJoin π s (t i) := by |
simp_rw [convexJoin_comm s, convexJoin_iUnion_left]
| 1,341 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace β E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 32 | 35 | theorem norm_add (h : SameRay β x y) : βx + yβ = βxβ + βyβ := by |
rcases h.exists_eq_smul with β¨u, a, b, ha, hb, -, rfl, rflβ©
rw [β add_smul, norm_smul_of_nonneg (add_nonneg ha hb), norm_smul_of_nonneg ha,
norm_smul_of_nonneg hb, add_mul]
| 1,342 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace β E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 38 | 46 | theorem norm_sub (h : SameRay β x y) : βx - yβ = |βxβ - βyβ| := by |
rcases h.exists_eq_smul with β¨u, a, b, ha, hb, -, rfl, rflβ©
wlog hab : b β€ a generalizing a b with H
Β· rw [SameRay.sameRay_comm] at h
rw [norm_sub_rev, abs_sub_comm]
exact H b a hb ha h (le_of_not_le hab)
rw [β sub_nonneg] at hab
rw [β sub_smul, norm_smul_of_nonneg hab, norm_smul_of_nonneg ha, norm_s... | 1,342 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace β E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 49 | 52 | theorem norm_smul_eq (h : SameRay β x y) : βxβ β’ y = βyβ β’ x := by |
rcases h.exists_eq_smul with β¨u, a, b, ha, hb, -, rfl, rflβ©
simp only [norm_smul_of_nonneg, *, mul_smul]
rw [smul_comm, smul_comm b, smul_comm a b u]
| 1,342 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace β E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 59 | 65 | theorem norm_injOn_ray_left (hx : x β 0) : { y | SameRay β x y }.InjOn norm := by |
rintro y hy z hz h
rcases hy.exists_nonneg_left hx with β¨r, hr, rflβ©
rcases hz.exists_nonneg_left hx with β¨s, hs, rflβ©
rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr,
norm_of_nonneg hs] at h
rw [h]
| 1,342 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace β E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 68 | 69 | theorem norm_injOn_ray_right (hy : y β 0) : { x | SameRay β x y }.InjOn norm := by |
simpa only [SameRay.sameRay_comm] using norm_injOn_ray_left hy
| 1,342 |
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import algebra.module.projective from "leanprover-community/mathlib"@"405ea5cee7a7070ff8fb8dcb4cfb003532e34bce"
universe u v
open LinearMap ... | Mathlib/Algebra/Module/Projective.lean | 92 | 94 | theorem projective_def' :
Projective R P β β s : P ββ[R] P ββ R, Finsupp.total P P R id ββ s = .id := by |
simp_rw [projective_def, DFunLike.ext_iff, Function.LeftInverse, comp_apply, id_apply]
| 1,343 |
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import algebra.module.projective from "leanprover-community/mathlib"@"405ea5cee7a7070ff8fb8dcb4cfb003532e34bce"
universe u v
open LinearMap ... | Mathlib/Algebra/Module/Projective.lean | 98 | 116 | theorem projective_lifting_property [h : Projective R P] (f : M ββ[R] N) (g : P ββ[R] N)
(hf : Function.Surjective f) : β h : P ββ[R] M, f.comp h = g := by |
/-
Here's the first step of the proof.
Recall that `X ββ R` is Lean's way of talking about the free `R`-module
on a type `X`. The universal property `Finsupp.total` says that to a map
`X β N` from a type to an `R`-module, we get an associated R-module map
`(X ββ R) ββ N`. Apply this to a (noncomp... | 1,343 |
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import algebra.module.projective from "leanprover-community/mathlib"@"405ea5cee7a7070ff8fb8dcb4cfb003532e34bce"
universe u v
open LinearMap ... | Mathlib/Algebra/Module/Projective.lean | 156 | 163 | theorem Projective.of_basis {ΞΉ : Type*} (b : Basis ΞΉ R P) : Projective R P := by |
-- need P ββ (P ββ R) for definition of projective.
-- get it from `ΞΉ β (P ββ R)` coming from `b`.
use b.constr β fun i => Finsupp.single (b i) (1 : R)
intro m
simp only [b.constr_apply, mul_one, id, Finsupp.smul_single', Finsupp.total_single,
map_finsupp_sum]
exact b.total_repr m
| 1,343 |
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 64 | 69 | theorem Ideal.IsHomogeneous.mem_iff {I} (hI : Ideal.IsHomogeneous π I) {x} :
x β I β β i, (decompose π x i : A) β I := by |
classical
refine β¨fun hx i β¦ hI i hx, fun hx β¦ ?_β©
rw [β DirectSum.sum_support_decompose π x]
exact Ideal.sum_mem _ (fun i _ β¦ hx i)
| 1,344 |
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