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.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section CommSemiring
variable [CommSemiring R] ... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 56 | 59 | theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : σ → A)
(p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false_iff]
| 1,295 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section CommSemiring
variable [CommSemiring R] ... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 62 | 65 | theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
(h : Function.Injective (algebraMap A B)) :
aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]
| 1,295 |
import Mathlib.LinearAlgebra.Span
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
universe uι u v
open Set
open Pointwise
namespace Submodule
variable {ι : Sort uι} {R M N P : Type*}
variabl... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 59 | 73 | theorem map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : Set M) (t : Set N) :
map₂ f (span R s) (span R t) = span R (Set.image2 (fun m n => f m n) s t) := by |
apply le_antisymm
· rw [map₂_le]
apply @span_induction' R M _ _ _ s
intro a ha
apply @span_induction' R N _ _ _ t
intro b hb
exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
all_goals intros; simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add,
Linear... | 1,296 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Opposites
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Submodule.Pointwise
import Mat... | Mathlib/Algebra/Algebra/Operations.lean | 88 | 90 | theorem le_one_toAddSubmonoid : 1 ≤ (1 : Submodule R A).toAddSubmonoid := by |
rintro x ⟨n, rfl⟩
exact ⟨n, map_natCast (algebraMap R A) n⟩
| 1,297 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a"
open DirectSum
variable {ι : Type*} {σ S R : Type*}
instance... | Mathlib/Algebra/DirectSum/Internal.lean | 56 | 59 | theorem SetLike.algebraMap_mem_graded [Zero ι] [CommSemiring S] [Semiring R] [Algebra S R]
(A : ι → Submodule S R) [SetLike.GradedOne A] (s : S) : algebraMap S R s ∈ A 0 := by |
rw [Algebra.algebraMap_eq_smul_one]
exact (A 0).smul_mem s <| SetLike.one_mem_graded _
| 1,298 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a"
open DirectSum
variable {ι : Type*} {σ S R : Type*}
instance... | Mathlib/Algebra/DirectSum/Internal.lean | 62 | 68 | theorem SetLike.natCast_mem_graded [Zero ι] [AddMonoidWithOne R] [SetLike σ R]
[AddSubmonoidClass σ R] (A : ι → σ) [SetLike.GradedOne A] (n : ℕ) : (n : R) ∈ A 0 := by |
induction' n with _ n_ih
· rw [Nat.cast_zero]
exact zero_mem (A 0)
· rw [Nat.cast_succ]
exact add_mem n_ih (SetLike.one_mem_graded _)
| 1,298 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a"
open DirectSum
variable {ι : Type*} {σ S R : Type*}
instance... | Mathlib/Algebra/DirectSum/Internal.lean | 74 | 80 | theorem SetLike.intCast_mem_graded [Zero ι] [AddGroupWithOne R] [SetLike σ R]
[AddSubgroupClass σ R] (A : ι → σ) [SetLike.GradedOne A] (z : ℤ) : (z : R) ∈ A 0 := by |
induction z
· rw [Int.ofNat_eq_coe, Int.cast_natCast]
exact SetLike.natCast_mem_graded _ _
· rw [Int.cast_negSucc]
exact neg_mem (SetLike.natCast_mem_graded _ _)
| 1,298 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 74 | 75 | theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := by |
simp_rw [annihilator, Module.mem_annihilator, Subtype.forall, Subtype.ext_iff]; rfl
| 1,299 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 82 | 96 | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by |
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
exact h ⟨x, hx⟩
· exact smul_zero _
· intro x y hx hy
rw [smul_add, hx, hy, zero_add]
· intro a x hx
... | 1,299 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 99 | 100 | theorem mem_annihilator_span_singleton (g : M) (r : R) :
r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by | simp [mem_annihilator_span]
| 1,299 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 426 | 426 | theorem one_eq_top : (1 : Ideal R) = ⊤ := by | erw [Submodule.one_eq_range, LinearMap.range_id]
| 1,299 |
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists Submodule.hasQuotient -- See `RingTheory.Ideal.QuotientOperations`
universe... | Mathlib/RingTheory/Ideal/Maps.lean | 90 | 95 | theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) :
I.map f ≤ I.comap g := by |
refine Ideal.span_le.2 ?_
rintro x ⟨x, hx, rfl⟩
rw [SetLike.mem_coe, mem_comap, hf hx]
exact hx
| 1,300 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.RingTheory.Ideal.Maps
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R S : Type*} [CommSemiring R] [CommRing S] [Algebra R S]
... | Mathlib/Algebra/Algebra/Subalgebra/Operations.lean | 40 | 68 | theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
{ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' := by |
-- Porting note: needed to add this instance
let _i : Algebra { x // x ∈ S' } { x // x ∈ S' } := Algebra.id _
suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range by
obtain ⟨x, rfl⟩ := this
exact x.2
choose n hn using H
let s' : ι → S' := fun x => ⟨s x, hs x⟩
let l' : ι → S' := fun x => ⟨l... | 1,301 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 50 | 58 | theorem ideal_prod_eq (I : Ideal (R × S)) :
I = Ideal.prod (map (RingHom.fst R S) I : Ideal R) (map (RingHom.snd R S) I) := by |
apply Ideal.ext
rintro ⟨r, s⟩
rw [mem_prod, mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective,
mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
refine ⟨fun h => ⟨⟨_, ⟨h, rfl⟩⟩, ⟨_, ⟨h, rfl⟩⟩⟩, ?_⟩
rintro ⟨⟨⟨r, s'⟩, ⟨h₁, rfl⟩⟩, ⟨⟨r', s⟩, ⟨h₂, rfl⟩⟩⟩
simpa using I.add_mem (I... | 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 62 | 68 | theorem map_fst_prod (I : Ideal R) (J : Ideal S) : map (RingHom.fst R S) (prod I J) = I := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.1, fun h => ⟨⟨x, 0⟩, ⟨⟨h, Ideal.zero_mem _⟩, rfl⟩⟩⟩
| 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 72 | 78 | theorem map_snd_prod (I : Ideal R) (J : Ideal S) : map (RingHom.snd R S) (prod I J) = J := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩
| 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 82 | 85 | theorem map_prodComm_prod :
map ((RingEquiv.prodComm : R × S ≃+* S × R) : R × S →+* S × R) (prod I J) = prod J I := by |
refine Trans.trans (ideal_prod_eq _) ?_
simp [map_map]
| 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 103 | 105 | theorem prod.ext_iff {I I' : Ideal R} {J J' : Ideal S} :
prod I J = prod I' J' ↔ I = I' ∧ J = J' := by |
simp only [← idealProdEquiv_symm_apply, idealProdEquiv.symm.injective.eq_iff, Prod.mk.inj_iff]
| 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 108 | 118 | theorem isPrime_of_isPrime_prod_top {I : Ideal R} (h : (Ideal.prod I (⊤ : Ideal S)).IsPrime) :
I.IsPrime := by |
constructor
· contrapose! h
rw [h, prod_top_top, isPrime_iff]
simp [isPrime_iff, h]
· intro x y hxy
have : (⟨x, 1⟩ : R × S) * ⟨y, 1⟩ ∈ prod I ⊤ := by
rw [Prod.mk_mul_mk, mul_one, mem_prod]
exact ⟨hxy, trivial⟩
simpa using h.mem_or_mem this
| 1,302 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 121 | 126 | theorem isPrime_of_isPrime_prod_top' {I : Ideal S} (h : (Ideal.prod (⊤ : Ideal R) I).IsPrime) :
I.IsPrime := by |
apply isPrime_of_isPrime_prod_top (S := R)
rw [← map_prodComm_prod]
-- Note: couldn't synthesize the right instances without the `R` and `S` hints
exact map_isPrime_of_equiv (RingEquiv.prodComm (R := R) (S := S))
| 1,302 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 148 | 155 | theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.fst, Prod.ext rfl h.symm⟩
| 1,303 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 162 | 169 | theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.snd, Prod.ext h.symm rfl⟩
| 1,303 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 135 | 136 | theorem linearMap_vsub (f : P1 →ᵃ[k] P2) (p1 p2 : P1) : f.linear (p1 -ᵥ p2) = f p1 -ᵥ f p2 := by |
conv_rhs => rw [← vsub_vadd p1 p2, map_vadd, vadd_vsub]
| 1,304 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 162 | 169 | theorem ext_linear {f g : P1 →ᵃ[k] P2} (h₁ : f.linear = g.linear) {p : P1} (h₂ : f p = g p) :
f = g := by |
ext q
have hgl : g.linear (q -ᵥ p) = toFun g ((q -ᵥ p) +ᵥ q) -ᵥ toFun g q := by simp
have := f.map_vadd' q (q -ᵥ p)
rw [h₁, hgl, toFun_eq_coe, map_vadd, linearMap_vsub, h₂] at this
simp at this
exact this
| 1,304 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
#align_import linear_algebra.affine_space.affine_equiv from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
open Function Set
open Affine
-- Porting not... | Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | 80 | 86 | theorem toAffineMap_injective : Injective (toAffineMap : (P₁ ≃ᵃ[k] P₂) → P₁ →ᵃ[k] P₂) := by |
rintro ⟨e, el, h⟩ ⟨e', el', h'⟩ H
-- Porting note: added `AffineMap.mk.injEq`
simp only [toAffineMap_mk, AffineMap.mk.injEq, Equiv.coe_inj,
LinearEquiv.toLinearMap_inj] at H
congr
exacts [H.1, H.2]
| 1,305 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 | theorem toAffineEquiv_injective : Injective (toAffineEquiv : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
rintro ⟨e, econt, einv_cont⟩ ⟨e', e'cont, e'inv_cont⟩ H
congr
| 1,306 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 84 | 87 | theorem coe_injective : Function.Injective ((↑) : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
intro e e' H
cases e
congr
| 1,306 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 61 | 64 | theorem AffineEquiv.pointReflection_midpoint_left (x y : P) :
pointReflection R (midpoint R x y) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 68 | 71 | theorem Equiv.pointReflection_midpoint_left (x y : P) :
(Equiv.pointReflection (midpoint R x y)) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 73 | 74 | theorem midpoint_comm (x y : P) : midpoint R x y = midpoint R y x := by |
rw [midpoint, ← lineMap_apply_one_sub, one_sub_invOf_two, midpoint]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 77 | 79 | theorem AffineEquiv.pointReflection_midpoint_right (x y : P) :
pointReflection R (midpoint R x y) y = x := by |
rw [midpoint_comm, AffineEquiv.pointReflection_midpoint_left]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 83 | 85 | theorem Equiv.pointReflection_midpoint_right (x y : P) :
(Equiv.pointReflection (midpoint R x y)) y = x := by |
rw [midpoint_comm, Equiv.pointReflection_midpoint_left]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 119 | 120 | theorem midpoint_vsub_right (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₂ = (⅟ 2 : R) • (p₁ -ᵥ p₂) := by |
rw [midpoint_comm, midpoint_vsub_left]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 129 | 130 | theorem right_vsub_midpoint (p₁ p₂ : P) : p₂ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₂ -ᵥ p₁) := by |
rw [midpoint_comm, left_vsub_midpoint]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 133 | 137 | theorem midpoint_vsub (p₁ p₂ p : P) :
midpoint R p₁ p₂ -ᵥ p = (⅟ 2 : R) • (p₁ -ᵥ p) + (⅟ 2 : R) • (p₂ -ᵥ p) := by |
rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg,
neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, ← vadd_vsub_assoc,
midpoint_comm, midpoint, lineMap_apply]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 140 | 143 | theorem vsub_midpoint (p₁ p₂ p : P) :
p -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p -ᵥ p₁) + (⅟ 2 : R) • (p -ᵥ p₂) := by |
rw [← neg_vsub_eq_vsub_rev, midpoint_vsub, neg_add, ← smul_neg, ← smul_neg, neg_vsub_eq_vsub_rev,
neg_vsub_eq_vsub_rev]
| 1,307 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 78 | 79 | theorem vectorSpan_empty : vectorSpan k (∅ : Set P) = (⊥ : Submodule k V) := by |
rw [vectorSpan_def, vsub_empty, Submodule.span_empty]
| 1,308 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 86 | 86 | theorem vectorSpan_singleton (p : P) : vectorSpan k ({p} : Set P) = ⊥ := by | simp [vectorSpan_def]
| 1,308 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 117 | 123 | theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty := by |
constructor
· contrapose
rw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty]
intro h
simp [h, spanPoints]
· exact fun h => h.mono (subset_spanPoints _ _)
| 1,308 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 128 | 132 | theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan {s : Set P} {p : P} {v : V}
(hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k s := by |
rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩
rw [hv2p, vadd_vadd]
exact ⟨p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl⟩
| 1,308 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 136 | 143 | theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints {s : Set P} {p1 p2 : P}
(hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s) : p1 -ᵥ p2 ∈ vectorSpan k s := by |
rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩
rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩
rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc]
have hv1v2 : v1 - v2 ∈ vectorSpan k s := (vectorSpan k s).sub_mem hv1 hv2
refine (vectorSpan k s).add_mem ?_ hv1v2
e... | 1,308 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.pointwise from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
open Affine Pointwise
open Set
namespace AffineSubspace
variable {k : Type*} [Ring k]
variable {V P V₁ P₁ V₂ P₂ : Type*}
var... | Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 60 | 61 | theorem pointwise_vadd_bot (v : V) : v +ᵥ (⊥ : AffineSubspace k P) = ⊥ := by |
ext; simp [pointwise_vadd_eq_map, map_bot]
| 1,309 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.pointwise from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
open Affine Pointwise
open Set
namespace AffineSubspace
variable {k : Type*} [Ring k]
variable {V P V₁ P₁ V₂ P₂ : Type*}
var... | Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 64 | 67 | theorem pointwise_vadd_direction (v : V) (s : AffineSubspace k P) :
(v +ᵥ s).direction = s.direction := by |
rw [pointwise_vadd_eq_map, map_direction]
exact Submodule.map_id _
| 1,309 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.pointwise from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
open Affine Pointwise
open Set
namespace AffineSubspace
variable {k : Type*} [Ring k]
variable {V P V₁ P₁ V₂ P₂ : Type*}
var... | Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 74 | 79 | theorem map_pointwise_vadd (f : P₁ →ᵃ[k] P₂) (v : V₁) (s : AffineSubspace k P₁) :
(v +ᵥ s).map f = f.linear v +ᵥ s.map f := by |
erw [pointwise_vadd_eq_map, pointwise_vadd_eq_map, map_map, map_map]
congr 1
ext
exact f.map_vadd _ _
| 1,309 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 33 | 36 | theorem AffineSubspace.nonempty_map {E : AffineSubspace k P₁} [Ene : Nonempty E] {φ : P₁ →ᵃ[k] P₂} :
Nonempty (E.map φ) := by |
obtain ⟨x, hx⟩ := id Ene
exact ⟨⟨φ x, AffineSubspace.mem_map.mpr ⟨x, hx, rfl⟩⟩⟩
| 1,310 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 61 | 64 | theorem AffineMap.restrict.linear_aux {φ : P₁ →ᵃ[k] P₂} {E : AffineSubspace k P₁}
{F : AffineSubspace k P₂} (hEF : E.map φ ≤ F) : E.direction ≤ F.direction.comap φ.linear := by |
rw [← Submodule.map_le_iff_le_comap, ← AffineSubspace.map_direction]
exact AffineSubspace.direction_le hEF
| 1,310 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 73 | 78 | theorem AffineMap.restrict.injective {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Injective φ)
{E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty E] [Nonempty F]
(hEF : E.map φ ≤ F) : Function.Injective (AffineMap.restrict φ hEF) := by |
intro x y h
simp only [Subtype.ext_iff, Subtype.coe_mk, AffineMap.restrict.coe_apply] at h ⊢
exact hφ h
| 1,310 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 81 | 87 | theorem AffineMap.restrict.surjective (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁}
{F : AffineSubspace k P₂} [Nonempty E] [Nonempty F] (h : E.map φ = F) :
Function.Surjective (AffineMap.restrict φ (le_of_eq h)) := by |
rintro ⟨x, hx : x ∈ F⟩
rw [← h, AffineSubspace.mem_map] at hx
obtain ⟨y, hy, rfl⟩ := hx
exact ⟨⟨y, hy⟩, rfl⟩
| 1,310 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 104 | 105 | theorem dist_vadd_cancel_right (v₁ v₂ : V) (x : P) : dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂ := by |
rw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right]
| 1,311 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 114 | 116 | theorem dist_vadd_left (v : V) (x : P) : dist (v +ᵥ x) x = ‖v‖ := by |
-- porting note (#10745): was `simp [dist_eq_norm_vsub V _ x]`
rw [dist_eq_norm_vsub V _ x, vadd_vsub]
| 1,311 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 125 | 125 | theorem dist_vadd_right (v : V) (x : P) : dist x (v +ᵥ x) = ‖v‖ := by | rw [dist_comm, dist_vadd_left]
| 1,311 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 142 | 143 | theorem dist_vsub_cancel_left (x y z : P) : dist (x -ᵥ y) (x -ᵥ z) = dist y z := by |
rw [dist_eq_norm, vsub_sub_vsub_cancel_left, dist_comm, dist_eq_norm_vsub V]
| 1,311 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 36 | 41 | theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) := by |
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp [isClosed_singleton]
rw [← (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,
AffineSubspace.coe_direction_eq_vsub_set_right hx]
rfl
| 1,312 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 45 | 47 | theorem dist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
dist p₁ (homothety p₁ c p₂) = ‖c‖ * dist p₁ p₂ := by |
simp [homothety_def, norm_smul, ← dist_eq_norm_vsub, dist_comm]
| 1,312 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 57 | 58 | theorem dist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
dist (homothety p₁ c p₂) p₁ = ‖c‖ * dist p₁ p₂ := by | rw [dist_comm, dist_center_homothety]
| 1,312 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 68 | 72 | theorem dist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
dist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ := by |
rw [dist_comm p₁ p₂]
simp only [lineMap_apply, dist_eq_norm_vsub, vadd_vsub_vadd_cancel_right,
← sub_smul, norm_smul, vsub_eq_sub]
| 1,312 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 87 | 88 | theorem dist_lineMap_left (p₁ p₂ : P) (c : 𝕜) : dist (lineMap p₁ p₂ c) p₁ = ‖c‖ * dist p₁ p₂ := by |
simpa only [lineMap_apply_zero, dist_zero_right] using dist_lineMap_lineMap p₁ p₂ c 0
| 1,312 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 109 | 111 | theorem dist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
dist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ := by |
simpa only [lineMap_apply_one, dist_eq_norm'] using dist_lineMap_lineMap p₁ p₂ c 1
| 1,312 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 47 | 48 | theorem slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 := by |
rw [slope, sub_self, inv_zero, zero_smul]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 56 | 59 | theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := by |
rcases eq_or_ne a b with (rfl | hne)
· rw [sub_self, zero_smul, vsub_self]
· rw [slope, smul_inv_smul₀ (sub_ne_zero.2 hne.symm)]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 62 | 63 | theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := by |
rw [sub_smul_slope, vsub_vadd]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 67 | 69 | theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f := by |
ext a b
simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 73 | 75 | theorem slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b) :
slope (fun x => (x - a) • f x) a b = f b := by |
simp [slope, inv_smul_smul₀ (sub_ne_zero.2 h.symm)]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 78 | 79 | theorem eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b := by |
rw [← sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 92 | 93 | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by |
rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 102 | 116 | theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (f : k → PE) (a b c : k) :
((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c := by |
by_cases hab : a = b
· subst hab
rw [sub_self, zero_div, zero_smul, zero_add]
by_cases hac : a = c
· simp [hac]
· rw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul]
by_cases hbc : b = c;
· subst hbc
simp [sub_ne_zero.2 (Ne.symm hab)]
rw [add_comm]
simp_rw [slope, div_eq_inv_mul, mul... | 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 121 | 124 | theorem lineMap_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) :
lineMap (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c := by |
field_simp [sub_ne_zero.2 h.symm, ← sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c,
lineMap_apply_module]
| 1,313 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 129 | 135 | theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by |
obtain rfl | hab : a = b ∨ a ≠ b := Classical.em _; · simp
rw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b]
convert lineMap_slope_slope_sub_div_sub f b (lineMap a b r) a hab.symm using 2
rw [lineMap_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, ← sub_sub,
sub_sub_cancel]
| 1,313 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 52 | 54 | theorem lineMap_mono_left (ha : a ≤ a') (hr : r ≤ 1) : lineMap a b r ≤ lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_le_add_right (smul_le_smul_of_nonneg_left ha (sub_nonneg.2 hr)) _
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 57 | 59 | theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_lt_add_right (smul_lt_smul_of_pos_left ha (sub_pos.2 hr)) _
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 62 | 64 | theorem lineMap_mono_right (hb : b ≤ b') (hr : 0 ≤ r) : lineMap a b r ≤ lineMap a b' r := by |
simp only [lineMap_apply_module]
exact add_le_add_left (smul_le_smul_of_nonneg_left hb hr) _
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 67 | 69 | theorem lineMap_strict_mono_right (hb : b < b') (hr : 0 < r) : lineMap a b r < lineMap a b' r := by |
simp only [lineMap_apply_module]
exact add_lt_add_left (smul_lt_smul_of_pos_left hb hr) _
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 77 | 80 | theorem lineMap_strict_mono_endpoints (ha : a < a') (hb : b < b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
lineMap a b r < lineMap a' b' r := by |
rcases h₀.eq_or_lt with (rfl | h₀); · simpa
exact (lineMap_mono_left ha.le h₁).trans_lt (lineMap_strict_mono_right hb h₀)
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 83 | 86 | theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' ↔ a < b := by |
simp only [lineMap_apply_module]
rw [← lt_sub_iff_add_lt, add_sub_assoc, ← sub_lt_iff_lt_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_lt_smul_iff_of_pos_left (sub_pos.2 h)]
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b := by |
simp only [lineMap_apply_module]
rw [← le_sub_iff_add_le, add_sub_assoc, ← sub_le_iff_le_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]
| 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 206 | 213 | theorem map_le_lineMap_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f a b := by |
rw [lineMap_apply, lineMap_apply, slope, slope, vsub_eq_sub, vsub_eq_sub, vsub_eq_sub,
vadd_eq_add, vadd_eq_add, smul_eq_mul, add_sub_cancel_right, smul_sub, smul_sub, smul_sub,
sub_le_iff_le_add, mul_inv_rev, mul_smul, mul_smul, ← smul_sub, ← smul_sub, ← smul_add,
smul_smul, ← mul_inv_rev, inv_smul_le_i... | 1,314 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 240 | 248 | theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b := by |
rw [← lineMap_apply_one_sub, ← lineMap_apply_one_sub _ _ r]
revert h; generalize 1 - r = r'; clear! r; intro h
simp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]
rw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub,
le_inv_smul_iff_of_pos h, smul_smul, mul_inv_can... | 1,314 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 53 | 57 | theorem to_affineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
f = g := by |
cases f
cases g
congr
| 1,315 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 108 | 111 | theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g := by |
ext a
exact ContinuousMap.congr_fun h a
| 1,315 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 | theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by |
ext
rfl
| 1,315 |
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 64 | 70 | theorem ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g := by |
rcases f with ⟨f_dom, f⟩
rcases g with ⟨g_dom, g⟩
obtain rfl : f_dom = g_dom := h
obtain rfl : f = g := LinearMap.ext fun x => h' rfl
rfl
| 1,316 |
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 151 | 157 | theorem mkSpanSingleton'_apply (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (c : R) (h) :
mkSpanSingleton' x y H ⟨c • x, h⟩ = c • y := by |
dsimp [mkSpanSingleton']
rw [← sub_eq_zero, ← sub_smul]
apply H
simp only [sub_smul, one_smul, sub_eq_zero]
apply Classical.choose_spec (mem_span_singleton.1 h)
| 1,316 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 64 | 112 | theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x := by |
obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom)
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x : E) - y ∈ s → f x ≤ c) ∧
∀ x : f.domain, (x : E) + y ∈ s → c ≤ f x := by
set Sp := f '' { x : f.domain | (x : E) + y ∈ s }
set Sn := f '' { x : f.do... | 1,317 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 115 | 139 | theorem exists_top (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x := by |
set S := { p : E →ₗ.[ℝ] ℝ | ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x }
have hSc : ∀ c, c ⊆ S → IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub := by
intro c hcs c_chain y hy
clear hp_nonneg hp_dense p
have cne : c.Nonempty := ⟨y, hy⟩
have hcd : DirectedOn (· ≤ ·) c := c_chain.directedOn
ref... | 1,317 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 77 | 85 | theorem IsClosable.leIsClosable {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f) :
g.IsClosable := by |
cases' hf with f' hf
have : g.graph.topologicalClosure ≤ f'.graph := by
rw [← hf]
exact Submodule.topologicalClosure_mono (le_graph_of_le hfg)
use g.graph.topologicalClosure.toLinearPMap
rw [Submodule.toLinearPMap_graph_eq]
exact fun _ hx hx' => f'.graph_fst_eq_zero_snd (this hx) hx'
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 89 | 92 | theorem IsClosable.existsUnique {f : E →ₗ.[R] F} (hf : f.IsClosable) :
∃! f' : E →ₗ.[R] F, f.graph.topologicalClosure = f'.graph := by |
refine exists_unique_of_exists_of_unique hf fun _ _ hy₁ hy₂ => eq_of_eq_graph ?_
rw [← hy₁, ← hy₂]
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 103 | 104 | theorem closure_def {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose := by |
simp [closure, hf]
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 107 | 107 | theorem closure_def' {f : E →ₗ.[R] F} (hf : ¬f.IsClosable) : f.closure = f := by | simp [closure, hf]
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 112 | 115 | theorem IsClosable.graph_closure_eq_closure_graph {f : E →ₗ.[R] F} (hf : f.IsClosable) :
f.graph.topologicalClosure = f.closure.graph := by |
rw [closure_def hf]
exact hf.choose_spec
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 119 | 124 | theorem le_closure (f : E →ₗ.[R] F) : f ≤ f.closure := by |
by_cases hf : f.IsClosable
· refine le_of_le_graph ?_
rw [← hf.graph_closure_eq_closure_graph]
exact (graph f).le_topologicalClosure
rw [closure_def' hf]
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 127 | 132 | theorem IsClosable.closure_mono {f g : E →ₗ.[R] F} (hg : g.IsClosable) (h : f ≤ g) :
f.closure ≤ g.closure := by |
refine le_of_le_graph ?_
rw [← (hg.leIsClosable h).graph_closure_eq_closure_graph]
rw [← hg.graph_closure_eq_closure_graph]
exact Submodule.topologicalClosure_mono (le_graph_of_le h)
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 136 | 138 | theorem IsClosable.closure_isClosed {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosed := by |
rw [IsClosed, ← hf.graph_closure_eq_closure_graph]
exact f.graph.isClosed_topologicalClosure
| 1,318 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 169 | 179 | theorem closureHasCore (f : E →ₗ.[R] F) : f.closure.HasCore f.domain := by |
refine ⟨f.le_closure.1, ?_⟩
congr
ext x y hxy
· simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]
intro hx
exact f.le_closure.1 hx
let z : f.closure.domain := ⟨y.1, f.le_closure.1 y.2⟩
have hyz : (y : E) = z := by simp
rw [f.le_closure.2 hyz]
exact domRestrict_apply (hxy.t... | 1,318 |
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 | 63 | 69 | theorem sum_smul_index_linearMap' [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂]
[Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂} :
((c • v).sum fun a => h a) = c • v.sum fun a => h a := by |
rw [Finsupp.sum_smul_index', Finsupp.smul_sum]
· simp only [map_smul]
· intro i
exact (h i).map_zero
| 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 | 133 | 136 | theorem LinearEquiv.finsuppUnique_symm_apply [Unique α] (m : M) :
(LinearEquiv.finsuppUnique R M α).symm m = Finsupp.single default m := by |
ext; simp [LinearEquiv.finsuppUnique, Equiv.funUnique, single, Pi.single,
equivFunOnFinite, Function.update]
| 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 | 234 | 234 | theorem lapply_comp_lsingle_same (a : α) : lapply a ∘ₗ lsingle a = (.id : M →ₗ[R] M) := by | ext; simp
| 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 | 237 | 238 | theorem lapply_comp_lsingle_of_ne (a a' : α) (h : a ≠ a') :
lapply a ∘ₗ lsingle a' = (0 : M →ₗ[R] M) := by | ext; simp [h.symm]
| 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 | 245 | 252 | theorem lsingle_range_le_ker_lapply (s t : Set α) (h : Disjoint s t) :
⨆ a ∈ s, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) ≤
⨅ a ∈ t, ker (lapply a : (α →₀ M) →ₗ[R] M) := by |
refine iSup_le fun a₁ => iSup_le fun h₁ => range_le_iff_comap.2 ?_
simp only [(ker_comp _ _).symm, eq_top_iff, SetLike.le_def, mem_ker, comap_iInf, mem_iInf]
intro b _ a₂ h₂
have : a₁ ≠ a₂ := fun eq => h.le_bot ⟨h₁, eq.symm ▸ h₂⟩
exact single_eq_of_ne this
| 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 | 255 | 257 | theorem iInf_ker_lapply_le_bot : ⨅ a, ker (lapply a : (α →₀ M) →ₗ[R] M) ≤ ⊥ := by |
simp only [SetLike.le_def, mem_iInf, mem_ker, mem_bot, lapply_apply]
exact fun a h => Finsupp.ext h
| 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 | 260 | 263 | theorem iSup_lsingle_range : ⨆ a, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) = ⊤ := by |
refine eq_top_iff.2 <| SetLike.le_def.2 fun f _ => ?_
rw [← sum_single f]
exact sum_mem fun a _ => Submodule.mem_iSup_of_mem a ⟨_, rfl⟩
| 1,319 |
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