Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-!
# Sober spaces
A quasi... | Mathlib/Topology/Sober.lean | 148 | 150 | theorem genericPoint_spec [QuasiSober α] [IrreducibleSpace α] :
IsGenericPoint (genericPoint α) ⊤ := by |
simpa using (IrreducibleSpace.isIrreducible_univ α).genericPoint_spec
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 930 | 935 | theorem prod_filter_index [CommMonoid N] (g : α → M → N) :
(f.filter p).prod g = ∏ x ∈ (f.filter p).support, g x (f x) := by |
classical
refine Finset.prod_congr rfl fun x hx => ?_
rw [support_filter, Finset.mem_filter] at hx
rw [filter_apply_pos _ _ hx.2]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 93 | 99 | theorem dvd_map_of_isScalarTower' (R : Type*) {S : Type*} (K L : Type*) [CommRing R]
[CommRing S] [Field K] [CommRing L] [Algebra R S] [Algebra R K] [Algebra S L] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] (s : S) :
minpoly K (algebraMap S L s) ∣ map (algebraMap R K) (minpoly R ... |
apply minpoly.dvd K (algebraMap S L s)
rw [← map_aeval_eq_aeval_map, minpoly.aeval, map_zero]
rw [← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This module defin... | Mathlib/GroupTheory/GroupAction/Period.lean | 71 | 75 | theorem period_inv (g : G) (a : α) : period g⁻¹ a = period g a := by |
simp only [period_eq_minimalPeriod, Function.minimalPeriod_eq_minimalPeriod_iff,
isPeriodicPt_smul_iff]
intro n
rw [smul_eq_iff_eq_inv_smul, eq_comm, ← zpow_natCast, inv_zpow, inv_inv, zpow_natCast]
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 555 | 557 | theorem div_im (z w : K) : im (z / w) = im z * re w / normSq w - re z * im w / normSq w := by |
simp only [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, neg_mul, mul_neg, map_neg,
rclike_simps]
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 44 | 48 | theorem orbit.pairwiseDisjoint :
(Set.range fun x : X => orbit G x).PairwiseDisjoint id := by |
rintro s ⟨x, rfl⟩ t ⟨y, rfl⟩ h
contrapose! h
exact (orbit.eq_or_disjoint x y).resolve_right h
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 454 | 459 | theorem det_add_col_mul_row {A : Matrix m m α} (hA : IsUnit A.det) (u v : m → α) :
(A + col u * row v).det = A.det * (1 + row v * A⁻¹ * col u).det := by |
nth_rewrite 1 [← Matrix.mul_one A]
rwa [← Matrix.mul_nonsing_inv_cancel_left A (col u * row v),
← Matrix.mul_add, det_mul, ← Matrix.mul_assoc, det_one_add_mul_comm,
← Matrix.mul_assoc]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
/-!... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 177 | 181 | theorem isIdempotentComplete_iff_of_equivalence {D : Type*} [Category D] (ε : C ≌ D) :
IsIdempotentComplete C ↔ IsIdempotentComplete D := by |
constructor
· exact Equivalence.isIdempotentComplete ε
· exact Equivalence.isIdempotentComplete ε.symm
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 86 | 90 | theorem haveLebesgueDecomposition_spec (μ ν : Measure α) [h : HaveLebesgueDecomposition μ ν] :
Measurable (μ.rnDeriv ν) ∧
μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) := by |
rw [singularPart, rnDeriv, dif_pos h, dif_pos h]
exact Classical.choose_spec h.lebesgue_decomposition
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 394 | 396 | theorem isLeast_union_iff {a : α} {s t : Set α} :
IsLeast (s ∪ t) a ↔ IsLeast s a ∧ a ∈ lowerBounds t ∨ a ∈ lowerBounds s ∧ IsLeast t a := by |
simp [IsLeast, lowerBounds_union, or_and_right, and_comm (a := a ∈ t), and_assoc]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 468 | 482 | theorem lintegral_Lp_add_le_of_le_one {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hp0 : 0 ≤ p)
(hp1 : p ≤ 1) :
(∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
(2 : ℝ≥0∞) ^ (1 / p - 1) * ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) := by |
rcases eq_or_lt_of_le hp0 with (rfl | hp)
· simp only [Pi.add_apply, rpow_zero, lintegral_one, _root_.div_zero, zero_sub]
set_option tactic.skipAssignedInstances false in norm_num
rw [rpow_neg, rpow_one, ENNReal.inv_mul_cancel two_ne_zero two_ne_top]
calc
(∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤ ((∫⁻ a, ... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd... | Mathlib/Order/Heyting/Boundary.lean | 71 | 72 | theorem boundary_hnot_hnot (a : α) : ∂ (￢￢a) = ∂ (￢a) := by |
simp_rw [boundary, hnot_hnot_hnot, inf_comm]
|
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.D... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 109 | 111 | theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by |
rw [← SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 359 | 360 | theorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :
(p.append q).reverse = q.reverse.append p.reverse := by | simp [reverse]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.qu... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 161 | 182 | theorem QuasiCompact.affineProperty_isLocal : (QuasiCompact.affineProperty : _).IsLocal := by |
constructor
· apply AffineTargetMorphismProperty.respectsIso_mk <;> rintro X Y Z e _ _ H
exacts [@Homeomorph.compactSpace _ _ _ _ H (TopCat.homeoOfIso (asIso e.inv.1.base)), H]
· introv H
dsimp [affineProperty] at H ⊢
change CompactSpace ((Opens.map f.val.base).obj (Y.basicOpen r))
rw [Scheme.pre... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | 311 | 356 | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by |
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.left⟩,
← show _ = c.inr from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.right⟩]
dsimp [binaryCo... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 732 | 743 | theorem lift_type_lt {α : Type u} {β : Type v} {r s} [IsWellOrder α r] [IsWellOrder β s] :
lift.{max v w} (type r) < lift.{max u w} (type s) ↔ Nonempty (r ≺i s) := by |
haveI := @RelEmbedding.isWellOrder _ _ (@Equiv.ulift.{max v w} α ⁻¹'o r) r
(RelIso.preimage Equiv.ulift.{max v w} r) _
haveI := @RelEmbedding.isWellOrder _ _ (@Equiv.ulift.{max u w} β ⁻¹'o s) s
(RelIso.preimage Equiv.ulift.{max u w} s) _
exact ⟨fun ⟨f⟩ =>
⟨(f.equivLT (RelIso.preimage Equiv.ulift r).s... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 149 | 151 | theorem FiniteDimensional.finrank_prod [Module.Finite R M] [Module.Finite R M'] :
finrank R (M × M') = finrank R M + finrank R M' := by |
simp [finrank, rank_lt_aleph0 R M, rank_lt_aleph0 R M']
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.accumulate from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-!
# Accumulat... | Mathlib/Data/Set/Accumulate.lean | 31 | 32 | theorem mem_accumulate [LE α] {x : α} {z : β} : z ∈ Accumulate s x ↔ ∃ y ≤ x, z ∈ s y := by |
simp_rw [accumulate_def, mem_iUnion₂, exists_prop]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... | Mathlib/RingTheory/SimpleModule.lean | 435 | 440 | theorem second_iso {X Y : Submodule R M} (_ : X ⋖ X ⊔ Y) :
Iso (X,X ⊔ Y) (X ⊓ Y,Y) := by |
constructor
rw [sup_comm, inf_comm]
dsimp
exact (LinearMap.quotientInfEquivSupQuotient Y X).symm
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 479 | 489 | theorem Ideal.isIdempotentElem_iff_eq_bot_or_top_of_localRing {R} [CommRing R]
[IsNoetherianRing R] [LocalRing R] (I : Ideal R) :
IsIdempotentElem I ↔ I = ⊥ ∨ I = ⊤ := by |
constructor
· intro H
by_cases I = ⊤; · exact Or.inr ‹_›
refine Or.inl (eq_bot_iff.mpr ?_)
rw [← Ideal.iInf_pow_eq_bot_of_localRing I ‹_›]
apply le_iInf
rintro (_|n) <;> simp [H.pow_succ_eq]
· rintro (rfl | rfl) <;> simp [IsIdempotentElem]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 559 | 568 | theorem liftPropAt_iff_comp_inclusion (hG : LocalInvariantProp G G' P) {U V : Opens M} (hUV : U ≤ V)
(f : V → M') (x : U) :
LiftPropAt P f (Set.inclusion hUV x) ↔ LiftPropAt P (f ∘ Set.inclusion hUV : U → M') x := by |
simp only [LiftPropAt, liftPropWithinAt_iff']
congrm ?_ ∧ ?_
· simp_rw [continuousWithinAt_univ,
(TopologicalSpace.Opens.openEmbedding_of_le hUV).continuousAt_iff]
· apply hG.congr_iff
exact (TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq hUV).fun_comp
(chartAt H' (f (Set.inclusion ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,583 | 1,584 | theorem measurableSet_image (e : α ≃ᵐ β) {s : Set α} :
MeasurableSet (e '' s) ↔ MeasurableSet s := by | rw [image_eq_preimage, measurableSet_preimage]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 489 | 490 | theorem image_sub_const_Ioo : (fun x => x - a) '' Ioo b c = Ioo (b - a) (c - a) := by |
simp [sub_eq_neg_add]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 1,092 | 1,093 | theorem bex_eq_left {a : α} : (∃ (x : _) (_ : x = a), p x) ↔ p a := by |
simp only [exists_prop, exists_eq_left]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
import Mathlib.Order.Filter.CountableInter
#align_import topolog... | Mathlib/Topology/GDelta.lean | 231 | 239 | theorem IsGδ.setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (𝓤 Y)] (f : X → Y) :
IsGδ { x | ContinuousAt f x } := by |
obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toHasBasis, forall_prop_of_true,
setOf_forall, id]
refine .iInter fun k ↦ IsOpen.isGδ <| isOpen_iff_mem_nh... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,070 | 1,072 | theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by |
simp [ContinuousOn, continuousWithinAt_iff]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 486 | 489 | theorem nodup_aroots_iff_of_splits [Algebra F K] {f : F[X]} (hf : f ≠ 0)
(h : f.Splits (algebraMap F K)) : (f.aroots K).Nodup ↔ f.Separable := by |
rw [← (algebraMap F K).id_comp, ← splits_map_iff] at h
rw [nodup_roots_iff_of_splits (map_ne_zero hf) h, separable_map]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 675 | 715 | theorem eq_σ_comp_of_not_injective' {n : ℕ} {Δ' : SimplexCategory} (θ : mk (n + 1) ⟶ Δ')
(i : Fin (n + 1)) (hi : θ.toOrderHom (Fin.castSucc i) = θ.toOrderHom i.succ) :
∃ θ' : mk n ⟶ Δ', θ = σ i ≫ θ' := by |
use δ i.succ ≫ θ
ext1; ext1; ext1 x
simp only [len_mk, σ, mkHom, comp_toOrderHom, Hom.toOrderHom_mk, OrderHom.comp_coe,
OrderHom.coe_mk, Function.comp_apply]
by_cases h' : x ≤ Fin.castSucc i
· -- This was not needed before leanprover/lean4#2644
dsimp
rw [Fin.predAbove_of_le_castSucc i x h']
d... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 232 | 233 | theorem neg_mkRat (n d) : -mkRat n d = mkRat (-n) d := by |
if z : d = 0 then simp [z]; rfl else simp [← normalize_eq_mkRat z, neg_normalize]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,417 | 1,418 | theorem eraseMax.valid {t} (h : @Valid α _ t) : Valid (eraseMax t) := by |
rw [Valid.dual_iff, dual_eraseMax]; exact eraseMin.valid h.dual
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 390 | 419 | theorem liftR_map_last [lawful : LawfulMvFunctor F]
{α : TypeVec n} {ι ι'} (R : ι' → ι' → Prop)
(x : F (α ::: ι)) (f g : ι → ι') (hh : ∀ x : ι, R (f x) (g x)) :
LiftR' (RelLast' _ R) ((id ::: f) <$$> x) ((id ::: g) <$$> x) :=
let h : ι → { x : ι' × ι' // uncurry R x } := fun x => ⟨(f x, g x), hh x⟩
let ... |
dsimp [b]
apply eq_of_drop_last_eq
· dsimp
simp only [prod_map_id, dropFun_prod, dropFun_appendFun, dropFun_diag, TypeVec.id_comp,
dropFun_toSubtype]
erw [toSubtype_of_subtype_assoc, TypeVec.id_comp]
clear liftR_map_last q mvf lawful F x R f g hh h b c
ext (i x) : 2
in... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 235 | 237 | theorem map_filter {f : α ≃ β} {p : α → Prop} [DecidablePred p] :
(s.filter p).map f.toEmbedding = (s.map f.toEmbedding).filter (p ∘ f.symm) := by |
simp only [filter_map, Function.comp, Equiv.toEmbedding_apply, Equiv.symm_apply_apply]
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury G. Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Lemmas abou... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 308 | 309 | theorem min_le_of_mul_le_sq {a b c : M} (h : a * b ≤ c ^ 2) : min a b ≤ c := by |
simpa using min_le_max_of_mul_le_mul (h.trans_eq <| pow_two _)
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
/-!
# Exact sequences with free modules
This file proves resu... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 72 | 78 | theorem linearIndependent_shortExact {w : ι' → S.X₃} (hw : LinearIndependent R w) :
LinearIndependent R (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w)) := by |
apply linearIndependent_leftExact hS'.exact hv _ hS'.mono_f rfl
dsimp
convert hw
ext
apply Function.rightInverse_invFun ((epi_iff_surjective _).mp hS'.epi_g)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,607 | 1,612 | theorem isCompl_dualAnnihilator {W W' : Subspace K V₁} (h : IsCompl W W') :
IsCompl W.dualAnnihilator W'.dualAnnihilator := by |
rw [isCompl_iff, disjoint_iff, codisjoint_iff] at h ⊢
rw [← dualAnnihilator_inf_eq, ← dualAnnihilator_sup_eq, h.1, h.2, dualAnnihilator_top,
dualAnnihilator_bot]
exact ⟨rfl, rfl⟩
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
/-!
# Independence with respect to a kernel and a measure
A family of sets of sets... | Mathlib/Probability/Independence/Kernel.lean | 410 | 446 | theorem IndepSets.indep {m1 m2 m : MeasurableSpace Ω} {κ : kernel α Ω} {μ : Measure α}
[IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
(hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
(hyp : IndepSets p1 p2 κ μ) :
Indep m1 m2 κ ... |
intros t1 t2 ht1 ht2
refine @induction_on_inter _ (fun t ↦ ∀ᵐ (a : α) ∂μ, κ a (t ∩ t2) = κ a t * κ a t2) _ m1 hpm1 hp1
?_ ?_ ?_ ?_ _ ht1
· simp only [Set.empty_inter, measure_empty, zero_mul, eq_self_iff_true,
Filter.eventually_true]
· intros t ht_mem_p1
have ht1 : MeasurableSet[m] t := by
... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 516 | 520 | theorem differentiableAt_apply (i : ι) (f : ∀ i, F' i) :
DifferentiableAt (𝕜:=𝕜) (fun f : ∀ i, F' i => f i) f := by |
have h := ((differentiableAt_pi (𝕜:=𝕜)
(Φ := fun (f : ∀ i, F' i) (i' : ι) => f i') (x:=f))).1
apply h; apply differentiableAt_id
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 476 | 477 | theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by |
rw [← inner_conj_symm, inner_smul_left, RingHom.map_mul, conj_conj, inner_conj_symm]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 72 | 72 | theorem edgeFinset_inj : G₁.edgeFinset = G₂.edgeFinset ↔ G₁ = G₂ := by | simp
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 205 | 207 | theorem of_mono_eq_zero {X Y : C} (f : X ⟶ Y) [Mono f] (h : f = 0) : IsZero X := by |
subst h
apply of_mono_zero X Y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,158 | 1,161 | theorem enum_zero_le' {o : Ordinal} (h0 : 0 < o) (a : o.out.α) :
@enum o.out.α (· < ·) _ 0 (by rwa [type_lt]) ≤ a := by |
rw [← not_lt]
apply enum_zero_le
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 352 | 356 | theorem mem_iSup_iff_exists_dfinsupp' (p : ι → Submodule R N) [∀ (i) (x : p i), Decidable (x ≠ 0)]
(x : N) : x ∈ iSup p ↔ ∃ f : Π₀ i, p i, (f.sum fun i xi => ↑xi) = x := by |
rw [mem_iSup_iff_exists_dfinsupp]
simp_rw [DFinsupp.lsum_apply_apply, DFinsupp.sumAddHom_apply,
LinearMap.toAddMonoidHom_coe, coeSubtype]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute... | Mathlib/Algebra/Field/Basic.lean | 29 | 29 | theorem add_div (a b c : α) : (a + b) / c = a / c + b / c := by | simp_rw [div_eq_mul_inv, add_mul]
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 259 | 259 | theorem re_ofReal_mul (r : ℝ) (z : ℂ) : (r * z).re = r * z.re := by | simp [ofReal']
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 900 | 900 | theorem dotProduct_neg : v ⬝ᵥ -w = -(v ⬝ᵥ w) := by | simp [dotProduct]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 324 | 342 | theorem mono_map_natDegree_eq {S F : Type*} [Semiring S]
[FunLike F R[X] S[X]] [AddMonoidHomClass F R[X] S[X]] {φ : F}
{p : R[X]} (k : ℕ) (fu : ℕ → ℕ) (fu0 : ∀ {n}, n ≤ k → fu n = 0)
(fc : ∀ {n m}, k ≤ n → n < m → fu n < fu m) (φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0)
(φ_mon_nat : ∀ n c, c ≠ 0 → (... |
refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)
p.natDegree (by simp [fu0]) ?_ ?_ _ rfl.le
· intro n r r0 _
rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]
· intro f g fg _ fk gk
rw [natDegree_add_eq_right_of_natDegree_lt fg, _root_.map_add]... |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Order.CompactlyGenerated.Basic
/-!
# Results about compactness properties for intervals in complete lattices
... | Mathlib/Order/CompactlyGenerated/Intervals.lean | 45 | 67 | theorem complementedLattice_of_complementedLattice_Iic
[IsModularLattice α] [IsCompactlyGenerated α]
{s : Set ι} {f : ι → α}
(h : ∀ i ∈ s, ComplementedLattice <| Iic (f i))
(h' : ⨆ i ∈ s, f i = ⊤) :
ComplementedLattice α := by |
apply complementedLattice_of_sSup_atoms_eq_top
have : ∀ i ∈ s, ∃ t : Set α, f i = sSup t ∧ ∀ a ∈ t, IsAtom a := fun i hi ↦ by
replace h := complementedLattice_iff_isAtomistic.mp (h i hi)
obtain ⟨u, hu, hu'⟩ := eq_sSup_atoms (⊤ : Iic (f i))
refine ⟨(↑) '' u, ?_, ?_⟩
· replace hu : f i = ↑(sSup u) :=... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 575 | 579 | theorem image_restrictPreimage :
t.restrictPreimage f '' (Subtype.val ⁻¹' s) = Subtype.val ⁻¹' (f '' s) := by |
delta Set.restrictPreimage
rw [← (Subtype.coe_injective).image_injective.eq_iff, ← image_comp, MapsTo.restrict_commutes,
image_comp, Subtype.image_preimage_coe, Subtype.image_preimage_coe, image_preimage_inter]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 692 | 693 | theorem lt_mul_of_lt_one_left (hb : b < 0) (h : a < 1) : b < a * b := by |
simpa only [one_mul] using mul_lt_mul_of_neg_right h hb
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 266 | 267 | theorem sqrt_lt' (hy : 0 < y) : √x < y ↔ x < y ^ 2 := by |
rw [← sqrt_lt_sqrt_iff_of_pos (pow_pos hy _), sqrt_sq hy.le]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 283 | 284 | theorem isCompact_preimage {s : Set Y} (h : X ≃ₜ Y) : IsCompact (h ⁻¹' s) ↔ IsCompact s := by |
rw [← image_symm]; exact h.symm.isCompact_image
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | Mathlib/RingTheory/Localization/Integral.lean | 129 | 141 | theorem integerNormalization_eq_zero_iff {p : K[X]} :
integerNormalization (nonZeroDivisors A) p = 0 ↔ p = 0 := by |
refine Polynomial.ext_iff.trans (Polynomial.ext_iff.trans ?_).symm
obtain ⟨⟨b, nonzero⟩, hb⟩ := integerNormalization_spec (nonZeroDivisors A) p
constructor <;> intro h i
· -- Porting note: avoided some defeq abuse
rw [coeff_zero, ← to_map_eq_zero_iff (K := K), hb i, h i, coeff_zero, smul_zero]
· have hi ... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.b... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 227 | 232 | theorem mul_basis_toMatrix [DecidableEq ι] [DecidableEq ι'] (b₁ : Basis ι R M) (b₂ : Basis ι' R M)
(b₃ : Basis κ R N) (A : Matrix κ ι R) :
A * b₁.toMatrix b₂ = LinearMap.toMatrix b₂ b₃ (toLin b₁ b₃ A) := by |
cases nonempty_fintype κ
have := linearMap_toMatrix_mul_basis_toMatrix b₂ b₁ b₃ (Matrix.toLin b₁ b₃ A)
rwa [LinearMap.toMatrix_toLin] at this
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,731 | 1,733 | theorem ULift.isClosed_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
IsClosed s ↔ IsClosed (ULift.up ⁻¹' s) := by |
rw [← isOpen_compl_iff, ← isOpen_compl_iff, isOpen_iff, preimage_compl]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "l... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 107 | 110 | theorem _root_.IsClosed.measure_eq_one_iff_eq_univ [OpensMeasurableSpace X] [IsProbabilityMeasure μ]
(hF : IsClosed F) :
μ F = 1 ↔ F = univ := by |
rw [← measure_univ (μ := μ), hF.measure_eq_univ_iff_eq]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 365 | 366 | theorem arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x := by |
rw [arccos, ← sin_pi_div_two_sub, arcsin_sin] <;> simp [sub_eq_add_neg] <;> linarith
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 279 | 280 | theorem orderOf_pow_dvd (n : ℕ) : orderOf (x ^ n) ∣ orderOf x := by |
rw [orderOf_dvd_iff_pow_eq_one, pow_right_comm, pow_orderOf_eq_one, one_pow]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 862 | 868 | theorem prod_sup_prod : prod p q₁ ⊔ prod p' q₁' = prod (p ⊔ p') (q₁ ⊔ q₁') := by |
refine le_antisymm
(sup_le (prod_mono le_sup_left le_sup_left) (prod_mono le_sup_right le_sup_right)) ?_
simp [SetLike.le_def]; intro xx yy hxx hyy
rcases mem_sup.1 hxx with ⟨x, hx, x', hx', rfl⟩
rcases mem_sup.1 hyy with ⟨y, hy, y', hy', rfl⟩
exact mem_sup.2 ⟨(x, y), ⟨hx, hy⟩, (x', y'), ⟨hx', hy'⟩, rfl⟩... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,185 | 1,188 | theorem le_ker_iff : I ≤ f.ker ↔ ∀ x, x ∈ I → f x = 0 := by |
constructor <;> intro h x hx
· specialize h hx; rw [mem_ker] at h; exact h
· rw [mem_ker]; apply h x hx
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 535 | 537 | theorem limit.map_pre' [HasLimitsOfShape K C] (F : J ⥤ C) {E₁ E₂ : K ⥤ J} (α : E₁ ⟶ E₂) :
limit.pre F E₂ = limit.pre F E₁ ≫ lim.map (whiskerRight α F) := by |
ext1; simp [← category.assoc]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 480 | 486 | theorem equiv_mul_left (h : H) (g : G) :
hHT.equiv (h * g) = (h * (hHT.equiv g).fst, (hHT.equiv g).snd) := by |
have : (hHT.equiv (h * g)).2 = (hHT.equiv g).2 := hHT.equiv_snd_eq_iff_rightCosetEquivalence.2 ?_
· ext
· rw [coe_mul, equiv_fst_eq_mul_inv, this, equiv_fst_eq_mul_inv, mul_assoc]
· rw [this]
· simp [RightCosetEquivalence, ← smul_smul]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 927 | 936 | theorem HasBasis.prod_same_index {p : ι → Prop} {sb : ι → Set β} (hla : la.HasBasis p sa)
(hlb : lb.HasBasis p sb) (h_dir : ∀ {i j}, p i → p j → ∃ k, p k ∧ sa k ⊆ sa i ∧ sb k ⊆ sb j) :
(la ×ˢ lb).HasBasis p fun i => sa i ×ˢ sb i := by |
simp only [hasBasis_iff, (hla.prod_pprod hlb).mem_iff]
refine fun t => ⟨?_, ?_⟩
· rintro ⟨⟨i, j⟩, ⟨hi, hj⟩, hsub : sa i ×ˢ sb j ⊆ t⟩
rcases h_dir hi hj with ⟨k, hk, ki, kj⟩
exact ⟨k, hk, (Set.prod_mono ki kj).trans hsub⟩
· rintro ⟨i, hi, h⟩
exact ⟨⟨i, i⟩, ⟨hi, hi⟩, h⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 520 | 522 | theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) :
arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by |
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,341 | 1,343 | theorem map_sub [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i) :
f (update m i (x - y)) = f (update m i x) - f (update m i y) := by |
rw [sub_eq_add_neg, sub_eq_add_neg, MultilinearMap.map_add, map_neg]
|
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_s... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 142 | 152 | theorem gramSchmidt_mem_span (f : ι → E) :
∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) := by |
intro j i hij
rw [gramSchmidt_def 𝕜 f i]
simp_rw [orthogonalProjection_singleton]
refine Submodule.sub_mem _ (subset_span (mem_image_of_mem _ hij))
(Submodule.sum_mem _ fun k hk => ?_)
let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij
exact smul_mem _ _
(span_mono (image_subset f <| Iic_subset... |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 220 | 220 | theorem sqrt_zero : √0 = 0 := by | simp [Real.sqrt]
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from... | Mathlib/Data/Finsupp/BigOperators.lean | 114 | 128 | theorem Finset.support_sum_eq [AddCommMonoid M] (s : Finset (ι →₀ M))
(hs : (s : Set (ι →₀ M)).PairwiseDisjoint Finsupp.support) :
(s.sum id).support = Finset.sup s Finsupp.support := by |
classical
suffices s.1.Pairwise (_root_.Disjoint on Finsupp.support) by
convert Multiset.support_sum_eq s.1 this
exact (Finset.sum_val _).symm
obtain ⟨l, hl, hn⟩ : ∃ l : List (ι →₀ M), l.toFinset = s ∧ l.Nodup := by
refine ⟨s.toList, ?_, Finset.nodup_toList _⟩
simp
subst hl
rwa [List.toFinset... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | Mathlib/Probability/Independence/Conditional.lean | 656 | 676 | theorem iCondIndepFun_iff_condexp_inter_preimage_eq_mul {β : ι → Type*}
(m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) (hf : ∀ i, Measurable (f i)) :
iCondIndepFun m' hm' m f μ ↔
∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
(μ⟦⋂ i ∈ S, f i ⁻¹' ... |
rw [iCondIndepFun_iff]
swap
· exact hf
refine ⟨fun h s sets h_sets ↦ ?_, fun h s sets h_sets ↦ ?_⟩
· refine h s (g := fun i ↦ f i ⁻¹' (sets i)) (fun i hi ↦ ?_)
exact ⟨sets i, h_sets i hi, rfl⟩
· classical
let g := fun i ↦ if hi : i ∈ s then (h_sets i hi).choose else Set.univ
specialize h s (set... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 892 | 896 | theorem Orthonormal.inner_finsupp_eq_zero {v : ι → E} (hv : Orthonormal 𝕜 v) {s : Set ι} {i : ι}
(hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ Finsupp.supported 𝕜 𝕜 s) :
⟪Finsupp.total ι E 𝕜 v l, v i⟫ = 0 := by |
rw [Finsupp.mem_supported'] at hl
simp only [hv.inner_left_finsupp, hl i hi, map_zero]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 256 | 275 | theorem sup_le_of_le_directed {α : Type*} [SemilatticeSup α] [OrderBot α] (s : Set α)
(hs : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s) (t : Finset α) :
(∀ x ∈ t, ∃ y ∈ s, x ≤ y) → ∃ x ∈ s, t.sup id ≤ x := by |
classical
induction' t using Finset.induction_on with a r _ ih h
· simpa only [forall_prop_of_true, and_true_iff, forall_prop_of_false, bot_le, not_false_iff,
sup_empty, forall_true_iff, not_mem_empty]
· intro h
have incs : (r : Set α) ⊆ ↑(insert a r) := by
rw [Finset.coe_subset]
... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.finite... | Mathlib/RingTheory/FinitePresentation.lean | 452 | 455 | theorem of_surjective (f : A →+* B) (hf : Surjective f) (hker : f.ker.FG) :
f.FinitePresentation := by |
rw [← f.comp_id]
exact (id A).comp_surjective hf hker
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Convex.Gauge
#align_import analysis.loca... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 52 | 60 | theorem nhds_basis_abs_convex :
(𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ 𝓝 (0 : E) ∧ Balanced 𝕜 s ∧ Convex ℝ s) id := by |
refine
(LocallyConvexSpace.convex_basis_zero ℝ E).to_hasBasis (fun s hs => ?_) fun s hs =>
⟨s, ⟨hs.1, hs.2.2⟩, rfl.subset⟩
refine ⟨convexHull ℝ (balancedCore 𝕜 s), ?_, convexHull_min (balancedCore_subset s) hs.2⟩
refine ⟨Filter.mem_of_superset (balancedCore_mem_nhds_zero hs.1) (subset_convexHull ℝ _),... |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 430 | 432 | theorem lcm_one_right (i : ℤ) : lcm i 1 = natAbs i := by |
rw [Int.lcm]
apply Nat.lcm_one_right
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"0... | Mathlib/Algebra/Squarefree/Basic.lean | 147 | 152 | theorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :
Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) := by |
refine ⟨fun h p hp hp' ↦ hp.not_unit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩
have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)
obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀
exact h p irr ((mul_dvd_mul dvd dvd).trans hd)
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 225 | 226 | theorem dist_le_opNorm (x y : E) : dist (f x) (f y) ≤ ‖f‖ * dist x y := by |
simp_rw [dist_eq_norm, ← map_sub, f.le_opNorm]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
#align_import algebra.direct_sum.ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Algebra/DirectSum/Ring.lean | 327 | 331 | theorem mul_eq_sum_support_ghas_mul [∀ (i : ι) (x : A i), Decidable (x ≠ 0)] (a a' : ⨁ i, A i) :
a * a' =
∑ ij ∈ DFinsupp.support a ×ˢ DFinsupp.support a',
DirectSum.of _ _ (GradedMonoid.GMul.mul (a ij.fst) (a' ij.snd)) := by |
simp only [mul_eq_dfinsupp_sum, DFinsupp.sum, Finset.sum_product]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 193 | 195 | theorem expSeries_apply_eq_div (x : 𝔸) (n : ℕ) : (expSeries 𝕂 𝔸 n fun _ => x) = x ^ n / n ! := by |
rw [div_eq_mul_inv, ← (Nat.cast_commute n ! (x ^ n)).inv_left₀.eq, ← smul_eq_mul,
expSeries_apply_eq, inv_natCast_smul_eq 𝕂 𝔸]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.N... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 288 | 288 | theorem div_eq_zero_iff : a / b = 0 ↔ a = 0 ∨ b = 0 := by | simp [div_eq_mul_inv]
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Adam Topaz
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fi... | Mathlib/CategoryTheory/FintypeCat.lean | 211 | 213 | theorem incl_mk_nat_card (n : ℕ) : Fintype.card (incl.obj (mk n)) = n := by |
convert Finset.card_fin n
apply Fintype.ofEquiv_card
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87... | Mathlib/Algebra/Group/Indicator.lean | 295 | 299 | theorem mulIndicator_const_preimage_eq_union (U : Set α) (s : Set M) (a : M) [Decidable (a ∈ s)]
[Decidable ((1 : M) ∈ s)] : (U.mulIndicator fun _ => a) ⁻¹' s =
(if a ∈ s then U else ∅) ∪ if (1 : M) ∈ s then Uᶜ else ∅ := by |
rw [mulIndicator_preimage, preimage_one, preimage_const]
split_ifs <;> simp [← compl_eq_univ_diff]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 157 | 159 | theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by |
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 220 | 222 | theorem leadingCoeff_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.toPoly.leadingCoeff = P.c := by |
rw [of_b_eq_zero ha hb, leadingCoeff_linear hc]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 838 | 840 | theorem toNNReal_pow {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : (x ^ n).toNNReal = x.toNNReal ^ n := by |
rw [← coe_inj, NNReal.coe_pow, Real.coe_toNNReal _ (pow_nonneg hx _),
Real.coe_toNNReal x hx]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
... | Mathlib/RingTheory/Prime.lean | 65 | 67 | theorem Prime.neg {p : α} (hp : Prime p) : Prime (-p) := by |
obtain ⟨h1, h2, h3⟩ := hp
exact ⟨neg_ne_zero.mpr h1, by rwa [IsUnit.neg_iff], by simpa [neg_dvd] using h3⟩
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-commu... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 60 | 65 | theorem PiToModule.fromEnd_apply_single_one [DecidableEq ι] (f : Module.End R M) (i : ι) :
PiToModule.fromEnd R b f (Pi.single i 1) = f (b i) := by |
rw [PiToModule.fromEnd_apply]
congr
convert Fintype.total_apply_single (S := R) R b i (1 : R)
rw [one_smul]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 965 | 973 | theorem MapsTo.restrict_surjective_iff (h : MapsTo f s t) :
Surjective (MapsTo.restrict _ _ _ h) ↔ SurjOn f s t := by |
refine ⟨fun h' b hb ↦ ?_, fun h' ⟨b, hb⟩ ↦ ?_⟩
· obtain ⟨⟨a, ha⟩, ha'⟩ := h' ⟨b, hb⟩
replace ha' : f a = b := by simpa [Subtype.ext_iff] using ha'
rw [← ha']
exact mem_image_of_mem f ha
· obtain ⟨a, ha, rfl⟩ := h' hb
exact ⟨⟨a, ha⟩, rfl⟩
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.modeq from "leanprover-communit... | Mathlib/Order/Filter/ModEq.lean | 29 | 30 | theorem frequently_mod_eq {d n : ℕ} (h : d < n) : ∃ᶠ m in atTop, m % n = d := by |
simpa only [Nat.ModEq, mod_eq_of_lt h] using frequently_modEq h.ne_bot d
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Math... | Mathlib/Combinatorics/Quiver/Covering.lean | 153 | 163 | theorem Prefunctor.symmetrifyStar (u : U) :
φ.symmetrify.star u =
(Quiver.symmetrifyStar _).symm ∘ Sum.map (φ.star u) (φ.costar u) ∘
Quiver.symmetrifyStar u := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [Equiv.eq_symm_comp]
ext ⟨v, f | g⟩ <;>
-- porting note (#10745): was `simp [Quiver.symmetrifyStar]`
simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>
erw [Equiv.sigmaSumDistrib_apply, Equiv.sigmaSumDistrib_apply] <;>... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 320 | 322 | theorem nthRoots_zero_right {R} [CommRing R] [IsDomain R] (n : ℕ) :
nthRoots n (0 : R) = Multiset.replicate n 0 := by |
rw [nthRoots, C.map_zero, sub_zero, roots_pow, roots_X, Multiset.nsmul_singleton]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 323 | 324 | theorem map_subtype_range_inclusion {p p' : Submodule R M} (h : p ≤ p') :
map p'.subtype (range <| inclusion h) = p := by | simp [range_inclusion, map_comap_eq, h]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,155 | 1,159 | theorem finrank_orthogonal_span_singleton {n : ℕ} [_i : Fact (finrank 𝕜 E = n + 1)] {v : E}
(hv : v ≠ 0) : finrank 𝕜 (𝕜 ∙ v)ᗮ = n := by |
haveI : FiniteDimensional 𝕜 E := .of_fact_finrank_eq_succ n
exact Submodule.finrank_add_finrank_orthogonal' <| by
simp [finrank_span_singleton hv, _i.elim, add_comm]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 215 | 223 | theorem dvd_content_iff_C_dvd {p : R[X]} {r : R} : r ∣ p.content ↔ C r ∣ p := by |
rw [C_dvd_iff_dvd_coeff]
constructor
· intro h i
apply h.trans (content_dvd_coeff _)
· intro h
rw [content, Finset.dvd_gcd_iff]
intro i _
apply h i
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,680 | 1,681 | theorem flip_bijective_iff₁ [FiniteDimensional K V₁] : Bijective B.flip ↔ Bijective B := by |
simp_rw [Bijective, flip_injective_iff₁, flip_surjective_iff₁, and_comm]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 385 | 394 | theorem weightedHomogeneousComponent_eq_zero [SemilatticeSup M] [OrderBot M]
(h : weightedTotalDegree w φ < n) : weightedHomogeneousComponent w n φ = 0 := by |
classical
rw [weightedHomogeneousComponent_apply, sum_eq_zero]
intro d hd
rw [Finset.mem_filter] at hd
exfalso
apply lt_irrefl n
nth_rw 1 [← hd.2]
exact lt_of_le_of_lt (le_weightedTotalDegree w hd.1) h
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Abelian.Subobject
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib... | Mathlib/CategoryTheory/Abelian/Generator.lean | 35 | 52 | theorem has_injective_coseparator [HasLimits C] [EnoughInjectives C] (G : C) (hG : IsSeparator G) :
∃ G : C, Injective G ∧ IsCoseparator G := by |
haveI : WellPowered C := wellPowered_of_isDetector G hG.isDetector
haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small _ _
let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)
refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩
refin... |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 518 | 520 | theorem blimsup_eq_limsup_subtype {f : Filter β} {u : β → α} {p : β → Prop} :
blimsup u f p = limsup (u ∘ ((↑) : { x | p x } → β)) (comap (↑) f) := by |
rw [blimsup_eq_limsup, limsup, limsup, ← map_map, map_comap_setCoe_val]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 80 | 83 | theorem Ioc_mul_Ico_subset' (a b c d : α) : Ioc a b * Ico c d ⊆ Ioo (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_lt_mul_of_lt_of_le hya hzc, mul_lt_mul_of_le_of_lt hyb hzd⟩
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import pr... | Mathlib/Probability/Martingale/Convergence.lean | 201 | 209 | theorem Submartingale.exists_ae_trim_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ.trim (sSup_le fun m ⟨n, hn⟩ => hn ▸ ℱ.le _ : ⨆ n, ℱ n ≤ m0),
∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by |
letI := (⨆ n, ℱ n)
rw [ae_iff, trim_measurableSet_eq]
· exact hf.exists_ae_tendsto_of_bdd hbdd
· exact MeasurableSet.compl <| measurableSet_exists_tendsto
fun n => (hf.stronglyMeasurable n).measurable.mono (le_sSup ⟨n, rfl⟩) le_rfl
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 46 | 50 | theorem inversion_mem_perpBisector_inversion_iff' (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c ∧ x ≠ c := by |
rcases eq_or_ne x c with rfl | hx
· simp [*]
· simp [inversion_mem_perpBisector_inversion_iff hR hx hy, hx]
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.