name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
CategoryTheory.eComp_eHomWhiskerLeft | Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean | /-- Whiskering commutes with the enriched composition. -/
@[reassoc]
lemma eComp_eHomWhiskerLeft (X Y : C) {Z Z' : C} (g : Z ⟶ Z') :
eComp V X Y Z ≫ eHomWhiskerLeft V X g =
_ ◁ eHomWhiskerLeft V Y g ≫ eComp V X Y Z' | V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X Y Z Z' : C
g : Z ⟶ Z'
⊢ (ρ_ (EnrichedCategory.Hom X Y ⊗ EnrichedCategory.Hom Y Z)).inv ≫
(EnrichedCategory.Hom X Y ⊗ EnrichedCategory.Hom Y Z) ◁ (eHomEquiv V) g ≫
... | simp [e_assoc'] | no goals | de11c142b5a620be |
IntermediateField.normalClosure_le_iff_of_normal | Mathlib/FieldTheory/Normal/Closure.lean | theorem normalClosure_le_iff_of_normal {K₁ K₂ : IntermediateField F L} [Normal F K₂] :
normalClosure F K₁ L ≤ K₂ ↔ K₁ ≤ K₂ | F : Type u_1
L : Type u_3
inst✝⁴ : Field F
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Normal F L
K₁ K₂ : IntermediateField F L
inst✝ : Normal F ↥K₂
⊢ normalClosure F (↥K₁) L ≤ K₂ ↔ K₁ ≤ K₂ | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩ | case refine_1
F : Type u_1
L : Type u_3
inst✝⁴ : Field F
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Normal F L
K₁ K₂ : IntermediateField F L
inst✝ : Normal F ↥K₂
h : normalClosure F (↥K₁) L ≤ K₂
⊢ K₁ ≤ K₂
case refine_2
F : Type u_1
L : Type u_3
inst✝⁴ : Field F
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Normal... | c18c94df9ca06e28 |
AlgebraicGeometry.IsLocalAtTarget.of_range_subset_iSup | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | lemma of_range_subset_iSup [P.RespectsRight @IsOpenImmersion] {ι : Type*} (U : ι → Y.Opens)
(H : Set.range f.base ⊆ (⨆ i, U i : Y.Opens)) (hf : ∀ i, P (f ∣_ U i)) : P f | case hf
P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
inst✝ : P.RespectsRight @IsOpenImmersion
ι : Type u_1
U : ι → Y.Opens
H : Set.range ⇑(ConcreteCategory.hom f.base) ⊆ ↑(⨆ i, U i)
hf : ∀ (i : ι), P (f ∣_ U i)
g : X ⟶ ↑(⨆ i, U i) := IsOpenImmersion.lift (⨆ i, U i).ι f ⋯
⊢ P (IsOpenImmersio... | rw [IsLocalAtTarget.iff_of_iSup_eq_top (P := P) (U := fun i : ι ↦ (⨆ i, U i).ι ⁻¹ᵁ U i)] | case hf
P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
inst✝ : P.RespectsRight @IsOpenImmersion
ι : Type u_1
U : ι → Y.Opens
H : Set.range ⇑(ConcreteCategory.hom f.base) ⊆ ↑(⨆ i, U i)
hf : ∀ (i : ι), P (f ∣_ U i)
g : X ⟶ ↑(⨆ i, U i) := IsOpenImmersion.lift (⨆ i, U i).ι f ⋯
⊢ ∀ (i : ι), P (IsO... | ee63d9b5b19cceb5 |
Polynomial.mkDerivation_one_eq_derivative | Mathlib/Algebra/Polynomial/Derivation.lean | lemma mkDerivation_one_eq_derivative (f : R[X]) : mkDerivation R (1 : R[X]) f = derivative f | R : Type u_1
inst✝ : CommSemiring R
f : R[X]
⊢ ((mkDerivation R) 1) f = derivative f | rw [mkDerivation_one_eq_derivative'] | R : Type u_1
inst✝ : CommSemiring R
f : R[X]
⊢ derivative' f = derivative f | 7860581f0f69ab53 |
Relation.comp_assoc | Mathlib/Logic/Relation.lean | theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r : α → β → Prop
p : β → γ → Prop
q : γ → δ → Prop
⊢ (r ∘r p) ∘r q = r ∘r p ∘r q | funext a d | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r : α → β → Prop
p : β → γ → Prop
q : γ → δ → Prop
a : α
d : δ
⊢ ((r ∘r p) ∘r q) a d = (r ∘r p ∘r q) a d | 1e716f779d42bb9c |
CategoryTheory.Localization.LeftBousfield.W_isoClosure | Mathlib/CategoryTheory/Localization/Bousfield.lean | lemma W_isoClosure : W P.isoClosure = W P | case h.mpr.intro.intro.intro.right.intro
C : Type u_1
inst✝ : Category.{u_3, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
hf : W P f
Z Z' : C
hZ' : P Z'
e : Z ≅ Z'
g : X ⟶ Z
a : Y ⟶ Z'
h : (fun g => f ≫ g) a = g ≫ e.hom
⊢ ∃ a, (fun g => f ≫ g) a = g | exact ⟨a ≫ e.inv, by simp only [reassoc_of% h, e.hom_inv_id, comp_id]⟩ | no goals | 905a2b1daf44ee1d |
MeasureTheory.addHaar_image_le_mul_of_det_lt | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
(hm : ENNReal.ofReal |A.det| < m) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
HC : IsCompact (⇑A '' closedBall 0 1)
⊢ T... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
HC : IsCompact (⇑A '' closedBall 0 1)
⊢ T... | 35c8118ffea35652 |
le_iff_exists_one_le_mul | Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean | @[to_additive] lemma le_iff_exists_one_le_mul [MulLeftMono α]
[MulLeftReflectLE α] : a ≤ b ↔ ∃ c, 1 ≤ c ∧ a * c = b :=
⟨exists_one_le_mul_of_le, by rintro ⟨c, hc, rfl⟩; exact le_mul_of_one_le_right' hc⟩
| α : Type u
inst✝⁴ : MulOneClass α
inst✝³ : Preorder α
inst✝² : ExistsMulOfLE α
a b : α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
⊢ (∃ c, 1 ≤ c ∧ a * c = b) → a ≤ b | rintro ⟨c, hc, rfl⟩ | case intro.intro
α : Type u
inst✝⁴ : MulOneClass α
inst✝³ : Preorder α
inst✝² : ExistsMulOfLE α
a : α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
c : α
hc : 1 ≤ c
⊢ a ≤ a * c | e9fc7aa0b63b7ad1 |
Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial | Mathlib/NumberTheory/RamificationInertia/Basic.lean | theorem FinrankQuotientMap.linearIndependent_of_nontrivial [IsDedekindDomain R]
(hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective f)
(f' : V'' →ₗ[R] V') {ι : Type*} {b : ι → V''} (hb' : LinearIndependent S (f' ∘ b)) :
LinearIndependent K (f ∘ b) | R : Type u
inst✝¹⁵ : CommRing R
S : Type v
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
K : Type u_1
inst✝¹² : Field K
inst✝¹¹ : Algebra R K
V : Type u_3
V' : Type u_4
V'' : Type u_5
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module R V
inst✝⁸ : Module K V
inst✝⁷ : IsScalarTower R K V
inst✝⁶ : AddCommGroup V'
inst✝⁵ : Module R V'... | refine Finset.sum_congr rfl ?_ | R : Type u
inst✝¹⁵ : CommRing R
S : Type v
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
K : Type u_1
inst✝¹² : Field K
inst✝¹¹ : Algebra R K
V : Type u_3
V' : Type u_4
V'' : Type u_5
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module R V
inst✝⁸ : Module K V
inst✝⁷ : IsScalarTower R K V
inst✝⁶ : AddCommGroup V'
inst✝⁵ : Module R V'... | 0f07a0e88e03e2b1 |
norm_cauchyPowerSeries_le | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | theorem norm_cauchyPowerSeries_le (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) :
‖cauchyPowerSeries f c R n‖ ≤
((2 * π)⁻¹ * ∫ θ : ℝ in (0)..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n :=
calc ‖cauchyPowerSeries f c R n‖
_ = (2 * π)⁻¹ * ‖∮ z in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ | case inl.zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
⊢ 0 ≤ ‖f c‖ | apply norm_nonneg | no goals | 1e003d11609d6cf7 |
WeierstrassCurve.Φ_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | lemma Φ_ne_zero [Nontrivial R] (n : ℤ) : W.Φ n ≠ 0 | case pos
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : Nontrivial R
n : ℤ
hn : n = 0
⊢ W.Φ n ≠ 0 | simpa only [hn, Φ_zero] using one_ne_zero | no goals | d331185d6009c3c3 |
mem_of_mem_permsOfList | Mathlib/Data/Fintype/Perm.lean | theorem mem_of_mem_permsOfList :
-- Porting note: was `∀ {x}` but need to capture the `x`
∀ {l : List α} {f : Perm α}, f ∈ permsOfList l → (x : α ) → f x ≠ x → x ∈ l
| [], f, h, heq_iff_eq => by
have : f = 1 | α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
f : Equiv.Perm α
h✝ : f ∈ permsOfList (a :: l)
x : α
h : f ∈ flatMap (fun b => List.map (fun f => Equiv.swap a b * f) (permsOfList l)) l
hx : f x ≠ x
y : α
hy : y ∈ l
hy' : f ∈ List.map (fun f => Equiv.swap a y * f) (permsOfList l)
g : Equiv.Perm α
hg₁ : g ∈ permsOfLi... | rw [eq_inv_mul_iff_mul_eq.2 hg₂, mul_apply, swap_inv, swap_apply_def] | α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
f : Equiv.Perm α
h✝ : f ∈ permsOfList (a :: l)
x : α
h : f ∈ flatMap (fun b => List.map (fun f => Equiv.swap a b * f) (permsOfList l)) l
hx : f x ≠ x
y : α
hy : y ∈ l
hy' : f ∈ List.map (fun f => Equiv.swap a y * f) (permsOfList l)
g : Equiv.Perm α
hg₁ : g ∈ permsOfLi... | bda372e426bbfd7e |
Algebra.FormallyUnramified.isField_quotient_map_maximalIdeal | Mathlib/RingTheory/Unramified/LocalRing.lean | lemma FormallyUnramified.isField_quotient_map_maximalIdeal [FormallyUnramified R S] :
IsField (S ⧸ (maximalIdeal R).map (algebraMap R S)) | R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : FormallyUnramified R S
⊢ IsField (S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R)) | let mR := (maximalIdeal R).map (algebraMap R S) | R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : FormallyUnramified R S
mR : Ideal S := Ideal.map (algebraMap R S) (maximalIdeal R)
⊢ IsField (S ⧸ Ideal.map (algebr... | 347d7077afd8552b |
ZSpan.coe_floor_self | Mathlib/Algebra/Module/ZLattice/Basic.lean | theorem coe_floor_self (k : K) : (floor (Basis.singleton ι K) k : K) = ⌊k⌋ :=
Basis.ext_elem (Basis.singleton ι K) fun _ => by
rw [repr_floor_apply, Basis.singleton_repr, Basis.singleton_repr]
| ι : Type u_2
K : Type u_3
inst✝³ : NormedLinearOrderedField K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : Unique ι
k : K
x✝ : ι
⊢ ((Basis.singleton ι K).repr ↑(floor (Basis.singleton ι K) k)) x✝ = ((Basis.singleton ι K).repr ↑⌊k⌋) x✝ | rw [repr_floor_apply, Basis.singleton_repr, Basis.singleton_repr] | no goals | 5f4282b17751da8c |
Metric.glueDist_glued_points | Mathlib/Topology/MetricSpace/Gluing.lean | theorem glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :
glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε | X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
this : ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0
⊢ glueDist Φ Ψ ε (Sum.inl (Φ p)) (Sum.inr (Ψ p)) = ε | simp only [glueDist, this, zero_add] | no goals | d31a35a51f091ed3 |
MeasureTheory.countable_generateSetAlgebra | Mathlib/MeasureTheory/SetAlgebra.lean | theorem countable_generateSetAlgebra (h : 𝒜.Countable) :
(generateSetAlgebra 𝒜).Countable | α : Type u_1
𝒜 : Set (Set α)
h : 𝒜.Countable
ℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}
count_ℬ : ℬ.Countable
f : Set (Set (Set α)) → Set α := fun A => ⋃ a ∈ A, ⋂ t ∈ a, t
𝒞 : Set (Set (Set α)) := {a | a.Finite ∧ a ⊆ ℬ}
count_𝒞 : 𝒞.Countable
𝒟 : Set (Set (Set (Set α))) := {A | A.Finite ∧ A ⊆ 𝒞}
⊢ (generateS... | have count_𝒟 : 𝒟.Countable := countable_setOf_finite_subset (countable_coe_iff.1 count_𝒞) | α : Type u_1
𝒜 : Set (Set α)
h : 𝒜.Countable
ℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}
count_ℬ : ℬ.Countable
f : Set (Set (Set α)) → Set α := fun A => ⋃ a ∈ A, ⋂ t ∈ a, t
𝒞 : Set (Set (Set α)) := {a | a.Finite ∧ a ⊆ ℬ}
count_𝒞 : 𝒞.Countable
𝒟 : Set (Set (Set (Set α))) := {A | A.Finite ∧ A ⊆ 𝒞}
count_𝒟 : �... | 6d285f70f928df4c |
InfIrred.isPrimary | Mathlib/RingTheory/Lasker.lean | lemma _root_.InfIrred.isPrimary {I : Ideal R} (h : InfIrred I) : I.IsPrimary | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
h : InfIrred I
a b : R
hab : a * b ∈ I
f : ℕ → Ideal R := fun n => Submodule.colon I (span {b ^ n})
n m : ℕ
hnm : n ≤ m
⊢ Submodule.colon I (span {b ^ n}) ≤ Submodule.colon I (span {b ^ m}) | exact (Submodule.colon_mono le_rfl (Ideal.span_singleton_le_span_singleton.mpr
(pow_dvd_pow b hnm))) | no goals | 48120bc89c7755a4 |
AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | /-- A section over a compact open of a scheme is nilpotent if and only if its associated
basic open is empty. -/
lemma Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact {X : Scheme.{u}}
{U : X.Opens} (hU : IsCompact (U : Set X)) (f : Γ(X, U)) :
IsNilpotent f ↔ X.basicOpen f = ⊥ | X : Scheme
U : X.Opens
hU : IsCompact ↑U
f : ↑Γ(X, U)
hf : X.basicOpen f = ⊥
e : X.basicOpen f ≤ ⊥
this : Subsingleton ↑Γ(X, ⊥)
⊢ ((1 |_ ⊥) ⋯ |_ X.basicOpen f) e = 0 | rw [Subsingleton.eq_zero (1 |_ ⊥)] | X : Scheme
U : X.Opens
hU : IsCompact ↑U
f : ↑Γ(X, U)
hf : X.basicOpen f = ⊥
e : X.basicOpen f ≤ ⊥
this : Subsingleton ↑Γ(X, ⊥)
⊢ (0 |_ X.basicOpen f) e = 0 | e644cce7450265a8 |
DirectSum.coe_mul_apply | Mathlib/Algebra/DirectSum/Internal.lean | theorem coe_mul_apply [AddMonoid ι] [SetLike.GradedMonoid A]
[∀ (i : ι) (x : A i), Decidable (x ≠ 0)] (r r' : ⨁ i, A i) (n : ι) :
((r * r') n : R) =
∑ ij ∈ (r.support ×ˢ r'.support).filter (fun ij : ι × ι => ij.1 + ij.2 = n),
(r ij.1 * r' ij.2 : R) | ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁶ : DecidableEq ι
inst✝⁵ : Semiring R
inst✝⁴ : SetLike σ R
inst✝³ : AddSubmonoidClass σ R
A : ι → σ
inst✝² : AddMonoid ι
inst✝¹ : SetLike.GradedMonoid A
inst✝ : (i : ι) → (x : ↥(A i)) → Decidable (x ≠ 0)
r r' : ⨁ (i : ι), ↥(A i)
n : ι
⊢ ∑ i ∈ DFinsupp.support r ×ˢ DFinsupp.su... | simp_rw [coe_of_apply, apply_ite, ZeroMemClass.coe_zero, ← Finset.sum_filter, SetLike.coe_gMul] | no goals | 80e95d9bf28dfb76 |
CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality | Mathlib/CategoryTheory/Abelian/LeftDerived.lean | @[reassoc]
lemma ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality
{X Y : C} (f : X ⟶ Y) (P : ProjectiveResolution X) (Q : ProjectiveResolution Y)
(φ : P.complex ⟶ Q.complex) (comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f)
(F : C ⥤ D) [F.Additive] :
(P.isoLeftDerivedToHomotopyCategoryObj F)... | C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
... | erw [(F.mapHomotopyCategoryFactors (ComplexShape.down ℕ)).inv.naturality_assoc] | C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
... | c857b9d0776845d5 |
MeasureTheory.condExp_ae_eq_restrict_zero | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | theorem condExp_ae_eq_restrict_zero (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict s] 0) :
μ[f|m] =ᵐ[μ.restrict s] 0 | case pos
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᶠ[ae (μ.restrict s)] 0
hm : m ≤ m0
⊢ μ[f|m] =ᶠ[ae (μ.restrict s)] 0 | by_cases hμm : SigmaFinite (μ.trim hm) | case pos
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᶠ[ae (μ.restrict s)] 0
hm : m ≤ m0
hμm : SigmaFinite (μ.trim hm)
⊢ μ[f|m] =ᶠ[ae (μ.restrict s)] 0
case neg
α : Type u... | 1f7750ee72b97131 |
Set.Ico_mul_Ioc_subset' | Mathlib/Algebra/Order/Group/Pointwise/Interval.lean | theorem Ico_mul_Ioc_subset' (a b c d : α) : Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) | α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b c d : α
this✝ : MulLeftMono α
this : MulRightMono α
⊢ Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) | rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩ | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b c d : α
this✝ : MulLeftMono α
this : MulRightMono α
y : α
hya : a ≤ y
hyb : y < b
z : α
hzc : c < z
hzd : z ≤ d
⊢ (fun x1 x2 => x1 * x2) y z ∈ Ioo (a * c) (b * d) | f4819a91e801aa24 |
CategoryTheory.CommSq.right_adjoint_hasLift_iff | Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | theorem right_adjoint_hasLift_iff : HasLift (sq.right_adjoint adj) ↔ HasLift sq | C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
G : C ⥤ D
F : D ⥤ C
A B : C
X Y : D
i : A ⟶ B
p : X ⟶ Y
u : G.obj A ⟶ X
v : G.obj B ⟶ Y
sq : CommSq u (G.map i) p v
adj : G ⊣ F
⊢ Nonempty ⋯.LiftStruct ↔ Nonempty sq.LiftStruct | exact Equiv.nonempty_congr (sq.rightAdjointLiftStructEquiv adj).symm | no goals | 81ecb5f9896b0d4a |
finrank_orthogonal_span_singleton | Mathlib/Analysis/InnerProductSpace/Projection.lean | 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 | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
_i : Fact (finrank 𝕜 E = n + 1)
v : E
hv : v ≠ 0
⊢ finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n | haveI : FiniteDimensional 𝕜 E := .of_fact_finrank_eq_succ n | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
_i : Fact (finrank 𝕜 E = n + 1)
v : E
hv : v ≠ 0
this : FiniteDimensional 𝕜 E
⊢ finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n | 1b86762dacd0365f |
Nat.binaryRec_eq | Mathlib/Data/Nat/BinaryRec.lean | theorem binaryRec_eq {z : motive 0} {f : ∀ b n, motive n → motive (bit b n)}
(b n) (h : f false 0 z = z ∨ (n = 0 → b = true)) :
binaryRec z f (bit b n) = f b n (binaryRec z f n) | motive : Nat → Sort u
z : motive 0
f : (b : Bool) → (n : Nat) → motive n → motive (bit b n)
b : Bool
n : Nat
h : f false 0 z = z ∨ (n = 0 → b = true)
h' : ¬bit b n = 0
⊢ ∀ (e : motive (bit ((bit b n).testBit 0) (bit b n >>> 1)) = motive (bit b n)),
e ▸ f ((bit b n).testBit 0) (bit b n >>> 1) (binaryRec z f (bit b n... | rw [testBit_bit_zero, bit_shiftRight_one] | motive : Nat → Sort u
z : motive 0
f : (b : Bool) → (n : Nat) → motive n → motive (bit b n)
b : Bool
n : Nat
h : f false 0 z = z ∨ (n = 0 → b = true)
h' : ¬bit b n = 0
⊢ ∀ (e : motive (bit b n) = motive (bit b n)), e ▸ f b n (binaryRec z f n) = f b n (binaryRec z f n) | 9ca2697a7905079f |
WeierstrassCurve.Jacobian.negY_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma negY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
W.negY P / P z ^ 3 = W.toAffine.negY (P x / P z ^ 2) (P y / P z ^ 3) | F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ W.negY P / P z ^ 3 = (toAffine W).negY (P x / P z ^ 2) (P y / P z ^ 3) | linear_combination (norm := (rw [negY, Affine.negY]; ring1))
-W.a₁ * P x / P z ^ 2 * div_self hPz - W.a₃ * div_self (pow_ne_zero 3 hPz) | no goals | 22a314b92fc5722d |
Lake.BuildKey.eq_of_quickCmp | Mathlib/.lake/packages/lean4/src/lean/lake/Lake/Build/Key.lean | theorem eq_of_quickCmp {k k' : BuildKey} :
quickCmp k k' = Ordering.eq → k = k' | case customTarget.moduleFacet
p t module✝ facet✝ : Name
⊢ (match customTarget p t with
| moduleFacet m f =>
match moduleFacet module✝ facet✝ with
| moduleFacet m' f' =>
match m.quickCmp m' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| x => Ordering.lt
... | all_goals (intro; contradiction) | no goals | 6e2940eeabc7d427 |
AlgebraicGeometry.Scheme.GlueData.ι_isoCarrier_inv | Mathlib/AlgebraicGeometry/Gluing.lean | theorem ι_isoCarrier_inv (i : D.J) :
(D_).ι i ≫ D.isoCarrier.inv = (D.ι i).base | D : GlueData
i : D.J
⊢ (D.toLocallyRingedSpaceGlueData.ι i ≫ D.isoLocallyRingedSpace.inv).base = (D.ι i).base | rw [D.ι_isoLocallyRingedSpace_inv i] | no goals | deea5fc46167b66c |
SetLike.natCast_mem_graded | Mathlib/Algebra/DirectSum/Internal.lean | 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 | ι : Type u_1
σ : Type u_2
R : Type u_4
inst✝⁴ : Zero ι
inst✝³ : AddMonoidWithOne R
inst✝² : SetLike σ R
inst✝¹ : AddSubmonoidClass σ R
A : ι → σ
inst✝ : GradedOne A
n : ℕ
⊢ ↑n ∈ A 0 | induction n with
| zero =>
rw [Nat.cast_zero]
exact zero_mem (A 0)
| succ _ n_ih =>
rw [Nat.cast_succ]
exact add_mem n_ih (SetLike.one_mem_graded _) | no goals | a490e68844ce179d |
Perfect.exists_nat_bool_injection | Mathlib/Topology/MetricSpace/Perfect.lean | theorem Perfect.exists_nat_bool_injection
(hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f | α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥... | intro l a | α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥... | ef9f9f33b845189a |
DFinsupp.lex_fibration | Mathlib/Data/DFinsupp/WellFounded.lean | theorem lex_fibration [∀ (i) (s : Set ι), Decidable (i ∈ s)] :
Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s)
fun x => piecewise x.2.1 x.2.2 x.1 | case neg
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
j : ι
h₁ : ¬(r j i ... | rfl | no goals | ce70f659976abe0e |
Set.Finite.exists_injOn_of_encard_le | Mathlib/Data/Set/Card.lean | theorem Finite.exists_injOn_of_encard_le [Nonempty β] {s : Set α} {t : Set β} (hs : s.Finite)
(hle : s.encard ≤ t.encard) : ∃ (f : α → β), s ⊆ f ⁻¹' t ∧ InjOn f s | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
s : Set α
t : Set β
hs : s.Finite
hle : s.encard ≤ t.encard
a : α
has : a ∈ s
b : β
hbt : b ∈ t
hle' : (s \ {a}).encard ≤ (t \ {b}).encard
f₀ : α → β
hinj : InjOn f₀ (s \ {a})
hf₀s : ∀ x ∈ s, ¬x = a → f₀ x ∈ t ∧ ¬f₀ x = b
x : α
hx : x ∈ s
h : ¬x = a
⊢ f₀ x ∈ t | exact (hf₀s x hx h).1 | no goals | 827f94f416b0baac |
CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight | Mathlib/CategoryTheory/Sites/CompatiblePlus.lean | theorem plusCompIso_whiskerRight {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
whiskerRight (J.plusMap η) F ≫ (J.plusCompIso F Q).hom =
(J.plusCompIso F P).hom ≫ J.plusMap (whiskerRight η F) | case w.h
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMultico... | dsimp [colimMap, IsColimit.map] | case w.h
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMultico... | 720f3f1cf176dc58 |
Polynomial.Monic.eq_X_pow_iff_natDegree_le_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | lemma eq_X_pow_iff_natDegree_le_natTrailingDegree (h₁ : p.Monic) :
p = X ^ p.natDegree ↔ p.natDegree ≤ p.natTrailingDegree | case refine_2.a
R : Type u
inst✝ : Semiring R
p : R[X]
h₁ : p.Monic
h : p.natDegree ≤ p.natTrailingDegree
n : ℕ
⊢ p.coeff n = (X ^ p.natDegree).coeff n | rw [coeff_X_pow] | case refine_2.a
R : Type u
inst✝ : Semiring R
p : R[X]
h₁ : p.Monic
h : p.natDegree ≤ p.natTrailingDegree
n : ℕ
⊢ p.coeff n = if n = p.natDegree then 1 else 0 | d8897cf4497f59d4 |
Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq | Mathlib/Topology/UniformSpace/Basic.lean | /-- Consider two functions `f` and `g` which coincide on a set `s` and are continuous there.
Then there is an open neighborhood of `s` on which `f` and `g` are uniformly close. -/
lemma exists_is_open_mem_uniformity_of_forall_mem_eq
[TopologicalSpace β] {r : Set (α × α)} {s : Set β}
{f g : β → α} (hf : ∀ x ∈ s,... | case intro.intro.intro
α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
r : Set (α × α)
s : Set β
f g : β → α
hf : ∀ x ∈ s, ContinuousAt f x
hg : ∀ x ∈ s, ContinuousAt g x
hfg : EqOn f g s
hr : r ∈ 𝓤 α
x : β
hx : x ∈ s
t : Set (α × α)
ht : t ∈ 𝓤 α
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
⊢ ∃ ... | have A : {z | (f x, f z) ∈ t} ∈ 𝓝 x := (hf x hx).preimage_mem_nhds (mem_nhds_left (f x) ht) | case intro.intro.intro
α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : TopologicalSpace β
r : Set (α × α)
s : Set β
f g : β → α
hf : ∀ x ∈ s, ContinuousAt f x
hg : ∀ x ∈ s, ContinuousAt g x
hfg : EqOn f g s
hr : r ∈ 𝓤 α
x : β
hx : x ∈ s
t : Set (α × α)
ht : t ∈ 𝓤 α
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
A : ... | 9202e3ace015df4e |
zero_le_two | Mathlib/Algebra/Order/Monoid/NatCast.lean | lemma zero_le_two [Preorder α] [ZeroLEOneClass α] [AddLeftMono α] :
(0 : α) ≤ 2 | α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : Preorder α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddLeftMono α
⊢ 0 ≤ 1 + 1 | exact add_nonneg zero_le_one zero_le_one | no goals | c96a41a8b678825b |
AnalyticAt.aeval_mvPolynomial | Mathlib/Analysis/Analytic/Polynomial.lean | theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (p : MvPolynomial σ A) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z | 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedCommRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
σ : Type u_5
f : E → σ → B
hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x => f x ... | simp_rw [map_mul, aeval_X] | 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedCommRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
σ : Type u_5
f : E → σ → B
hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x => f x ... | f79e0b8f632b4ffb |
LinearMap.IsSymmetric.add | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | theorem IsSymmetric.add {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
(T + S).IsSymmetric | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x y : E
⊢ inner ((T + S) x) y = inner x ((T + S) y) | rw [add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right, add_apply] | no goals | d0857d141203ead9 |
Orientation.oangle_sign_smul_add_right | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_sign_smul_add_right (x y : V) (r : ℝ) :
(o.oangle x (r • x + y)).sign = (o.oangle x y).sign | case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' => (x, r' • x + y... | have hx : (x, y) ∈ s := by
convert Set.mem_image_of_mem (fun r' : ℝ => (x, r' • x + y)) (Set.mem_univ 0)
simp | case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' => (x, r' • x + y... | 0c5bc13bb3049610 |
Set.prod_eq_prod_iff_of_nonempty | Mathlib/Data/Set/Prod.lean | theorem prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t).Nonempty) :
s ×ˢ t = s₁ ×ˢ t₁ ↔ s = s₁ ∧ t = t₁ | case mp
α : Type u_1
β : Type u_2
s s₁ : Set α
t t₁ : Set β
h : s.Nonempty ∧ t.Nonempty
heq : s ×ˢ t = s₁ ×ˢ t₁
h₁ : s₁.Nonempty ∧ t₁.Nonempty
⊢ s = s₁ ∧ t = t₁ | rw [← fst_image_prod s h.2, ← fst_image_prod s₁ h₁.2, heq, eq_self_iff_true, true_and, ←
snd_image_prod h.1 t, ← snd_image_prod h₁.1 t₁, heq] | no goals | 3e5bcc6932f4c938 |
MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub | Mathlib/MeasureTheory/Integral/Average.lean | theorem tendsto_integral_smul_of_tendsto_average_norm_sub
[CompleteSpace E]
{ι : Type*} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
(hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
(f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
(hg : Tendsto (fun i ↦ ∫ y, g i y ... | α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn ... | have : ∫ (y : α), g i y ∂μ = ∫ (y : α), 0 ∂μ := by congr; ext y; exact h'i y | α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn ... | 7e4f16c60f002777 |
exists_nhds_one_split4 | Mathlib/Topology/Algebra/Monoid.lean | theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | case intro.intro.intro.intro
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
V : Set M
V1 : V ∈ 𝓝 1
h' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W
⊢ ∃ V ∈ 𝓝 1, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * ... | use V, V1 | case right
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
V : Set M
V1 : V ∈ 𝓝 1
h' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W
⊢ ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | d4f4bd8c5fdcd322 |
Complex.re_balance | Mathlib/Data/Complex/BigOperators.lean | @[simp]
lemma re_balance [Fintype α] (f : α → ℂ) (a : α) : re (balance f a) = balance (re ∘ f) a | α : Type u_1
inst✝ : Fintype α
f : α → ℂ
a : α
⊢ (balance f a).re = balance (re ∘ f) a | simp [balance] | no goals | 570158a8c05c234f |
ApproximatesLinearOn.norm_fderiv_sub_le | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E →L[ℝ] E} {δ : ℝ≥0}
(hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : ∀ᵐ x ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ | case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ ... | apply ContinuousLinearMap.opNorm_le_bound _ δ.2 fun z => ?_ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, ... | 99ae4fc7a76b96c9 |
BoxIntegral.Prepartition.inf_splitMany | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | theorem inf_splitMany {I : Box ι} (π : Prepartition I) (s : Finset (ι × ℝ)) :
π ⊓ splitMany I s = π.biUnion fun J => splitMany J s | case insert
ι : Type u_1
I : Box ι
π : Prepartition I
p : ι × ℝ
s : Finset (ι × ℝ)
a✝ : p ∉ s
ihp : π ⊓ splitMany I s = π.biUnion fun J => splitMany J s
⊢ π ⊓ splitMany I (insert p s) = π.biUnion fun J => splitMany J (insert p s) | simp_rw [splitMany_insert, ← inf_assoc, ihp, inf_split, biUnion_assoc] | no goals | df100947daf62abf |
CategoryTheory.eqToHom_naturality | Mathlib/CategoryTheory/EqToHom.lean | theorem eqToHom_naturality {f g : β → C} (z : ∀ b, f b ⟶ g b) {j j' : β} (w : j = j') :
z j ≫ eqToHom (by simp [w]) = eqToHom (by simp [w]) ≫ z j' | C : Type u₁
inst✝ : Category.{v₁, u₁} C
β : Sort u_1
f g : β → C
z : (b : β) → f b ⟶ g b
j j' : β
w : j = j'
⊢ g j = g j' | simp [w] | no goals | 96faea952b54550a |
CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff' | Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.lean | lemma mem_distinguishedTriangles_iff' (T : Triangle Cᵒᵖ) :
T ∈ distinguishedTriangles C ↔
∃ (T' : Triangle C) (_ : T' ∈ distTriang C),
Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T')) | case mpr.intro.intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
T' : Triangle C
hT' : T' ∈ Pretriangulated.distinguishedTriangles
e : T ≅ (triangleOpEqu... | refine isomorphic_distinguished _ hT' _ ?_ | case mpr.intro.intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
T' : Triangle C
hT' : T' ∈ Pretriangulated.distinguishedTriangles
e : T ≅ (triangleOpEqu... | 0000737b9089f627 |
Polynomial.int_cyclotomic_rw | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | theorem int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
cyclotomic n ℤ = (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n h)).choose | n : ℕ
h : n ≠ 0
⊢ cyclotomic n ℤ = ⋯.choose | simp only [cyclotomic, h, dif_neg, not_false_iff] | n : ℕ
h : n ≠ 0
⊢ map (Int.castRingHom ℤ) ⋯.choose = ⋯.choose | 5c58602650d275b3 |
Nat.not_decide_mod_two_eq_one | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem not_decide_mod_two_eq_one (x : Nat)
: (!decide (x % 2 = 1)) = decide (x % 2 = 0) | case inr
x : Nat
p : x % 2 = 1
⊢ (!decide (x % 2 = 1)) = decide (x % 2 = 0) | simp [p] | no goals | a18ea6bc64c61fb4 |
groupCohomology.resolution.d_comp_ε | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | theorem d_comp_ε : (groupCohomology.resolution k G).d 1 0 ≫ ε k G = 0 | k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
x : ↑((resolution k G).X 1).V
⊢ (forget₂ToModuleCatHomotopyEquiv k G).hom.f 1 ≫
((ChainComplex.single₀ (ModuleCat k)).obj ((forget₂ (Rep k G) (ModuleCat k)).obj (Rep.trivial k G k))).d 1 0 =
0 | exact comp_zero | no goals | 187fb17c8f68cb1b |
List.nodup_finRange | Mathlib/Data/List/FinRange.lean | theorem nodup_finRange (n : ℕ) : (finRange n).Nodup | n : ℕ
⊢ (pmap Fin.mk (range n) ⋯).Nodup | exact (Pairwise.pmap (nodup_range n) _) fun _ _ _ _ => @Fin.ne_of_val_ne _ ⟨_, _⟩ ⟨_, _⟩ | no goals | 82aa54221459055c |
Stonean.epi_iff_surjective | Mathlib/Topology/Category/Stonean/Basic.lean | /--
A morphism in `Stonean` is an epi iff it is surjective.
-/
lemma epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f | case intro.intro.intro
X Y : Stonean
f : X ⟶ Y
h✝ : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
... | apply_fun fun e => (e y).down at H | case intro.intro.intro
X Y : Stonean
f : X ⟶ Y
h✝ : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
... | 333cfe532dfd4152 |
Localization.mapPiEvalRingHom_bijective | Mathlib/RingTheory/Localization/Basic.lean | theorem mapPiEvalRingHom_bijective : Bijective (mapPiEvalRingHom S) | case refine_1.intro.intro.intro.intro
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
r₁ : (i : ι) → R i
s₁ : ↥T
r₂ : (i : ι) → R i
s₂ : ↥T
eq :
mk' (Localization S) ((Pi.evalRingHom R i) r₁) ⟨(Pi.eval... | rw [IsLocalization.eq] at eq ⊢ | case refine_1.intro.intro.intro.intro
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
r₁ : (i : ι) → R i
s₁ : ↥T
r₂ : (i : ι) → R i
s₂ : ↥T
eq :
∃ c,
↑c * (↑⟨(Pi.evalRingHom R i) ↑s₂, ⋯⟩ * (Pi.eval... | c584ed62bd4689bc |
AddCircle.coe_real_preimage_closedBall_inter_eq | Mathlib/Analysis/Normed/Group/AddCircle.lean | theorem coe_real_preimage_closedBall_inter_eq {x ε : ℝ} (s : Set ℝ)
(hs : s ⊆ closedBall x (|p| / 2)) :
(↑) ⁻¹' closedBall (x : AddCircle p) ε ∩ s = if ε < |p| / 2 then closedBall x ε ∩ s else s | case inr.inr
p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
⊢ ∀ (x_1 : ℝ), x_1 ∉ Icc (x + ↑z * p - ε) (x + ↑z * p + ε) ∩ s | rintro y ⟨⟨hy₁, hy₂⟩, hy₀⟩ | case inr.inr.intro.intro
p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
⊢ False | 1a0a569d82ddadab |
MeasureTheory.Measure.rnDeriv_smul_right' | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem rnDeriv_smul_right' (ν μ : Measure α) [SigmaFinite ν] [SigmaFinite μ]
{r : ℝ≥0} (hr : r ≠ 0) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ | α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.rnDeriv (r • μ) =ᶠ[ae (r • μ)] r⁻¹ • ν.rnDeriv μ | rw [← withDensity_eq_iff_of_sigmaFinite] | α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ (r • μ).withDensity (ν.rnDeriv (r • μ)) = (r • μ).withDensity (r⁻¹ • ν.rnDeriv μ)
case hf
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠... | c2338dd04dba20c8 |
MvPolynomial.rename_eval₂ | Mathlib/Algebra/MvPolynomial/Rename.lean | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | case h_X
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
k : σ → τ
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
(rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p) →
(rename k) (eval₂ C (g ∘ k) (p * X n)) = eval₂ C (⇑(rename k) ∘ g) ((re... | intros | case h_X
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
k : σ → τ
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
p✝ : MvPolynomial σ R
n✝ : σ
a✝ : (rename k) (eval₂ C (g ∘ k) p✝) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p✝)
⊢ (rename k) (eval₂ C (g ∘ k) (p✝ * X n✝)) = eval₂ C (⇑(rename k) ∘ g) ((rename k) ... | 39ce03b26de9f9fa |
IsLocalization.isMaximal_iff_isMaximal_disjoint | Mathlib/RingTheory/Jacobson/Ring.lean | theorem IsLocalization.isMaximal_iff_isMaximal_disjoint [H : IsJacobsonRing R] (J : Ideal S) :
J.IsMaximal ↔ (comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J | case mpr.refine_2.refine_1
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
J : Ideal S
h : (Ideal.comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J
I : Ideal S
hI : J < I
hI' : ↑(Ideal.comap (algebraMap R S) I) ⊆ ↑... | exact map_mono hI' | no goals | 8b3129c7e9d172bf |
ModularGroup.smul_eq_lcRow0_add | Mathlib/NumberTheory/Modular.lean | theorem smul_eq_lcRow0_add {p : Fin 2 → ℤ} (hp : IsCoprime (p 0) (p 1)) (hg : g 1 = p) :
↑(g • z) =
(lcRow0 p ↑(g : SL(2, ℝ)) : ℂ) / ((p 0 : ℂ) ^ 2 + (p 1 : ℂ) ^ 2) +
((p 1 : ℂ) * z - p 0) / (((p 0 : ℂ) ^ 2 + (p 1 : ℂ) ^ 2) * (p 0 * z + p 1)) | g : SL(2, ℤ)
z : ℍ
p : Fin 2 → ℤ
hp : IsCoprime (p 0) (p 1)
hg : ↑g 1 = p
nonZ1 : ↑(p 0) ^ 2 + ↑(p 1) ^ 2 ≠ 0
⊢ ↑(g • z) =
↑((lcRow0 p) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) / (↑(p 0) ^ 2 + ↑(p 1) ^ 2) +
(↑(p 1) * ↑z - ↑(p 0)) / ((↑(p 0) ^ 2 + ↑(p 1) ^ 2) * (↑(p 0) * ↑z + ↑(p 1))) | have : ((↑) : ℤ → ℝ) ∘ p ≠ 0 := fun h => hp.ne_zero (by ext i; simpa using congr_fun h i) | g : SL(2, ℤ)
z : ℍ
p : Fin 2 → ℤ
hp : IsCoprime (p 0) (p 1)
hg : ↑g 1 = p
nonZ1 : ↑(p 0) ^ 2 + ↑(p 1) ^ 2 ≠ 0
this : Int.cast ∘ p ≠ 0
⊢ ↑(g • z) =
↑((lcRow0 p) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) / (↑(p 0) ^ 2 + ↑(p 1) ^ 2) +
(↑(p 1) * ↑z - ↑(p 0)) / ((↑(p 0) ^ 2 + ↑(p 1) ^ 2) * (↑(p 0) * ↑z + ... | 98a60fcc566447f3 |
HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod {y : E}
(hf : HasFPowerSeriesWithinOnBall f p s x r) (hy : y ∈ EMetric.ball (0 : E) r)
(h'y : x + y ∈ insert x s) :
Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) | case h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f p s x r
hy : y... | simp only [norm_neg] | case h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f p s x r
hy : y... | 84f2046d5fc6e8a5 |
MeasureTheory.L1.edist_def | Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean | theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ (if 1 = ⊤ then eLpNormEssSup (↑↑f - ↑↑g) μ else ∫⁻ (a : α), ‖↑↑f a - ↑↑g a‖ₑ ∂μ) =
∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ | simp [edist_eq_enorm_sub] | no goals | fd4f8e4e94f651b9 |
RightDerivMeasurableAux.D_subset_differentiable_set | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem D_subset_differentiable_set {K : Set F} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } | F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
∀ (e p q : ℕ),
n e ≤ p →
n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2... | exact ⟨e, fun e' he' => M _ _ _ _ _ _ le_rfl hp le_rfl le_rfl he'⟩ | no goals | 807a865e58286923 |
CategoryTheory.IsKernelPair.mono_of_isIso_fst | Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | theorem mono_of_isIso_fst (h : IsKernelPair f a b) [IsIso a] : Mono f | case mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
R X Y : C
f : X ⟶ Y
a b : R ⟶ X
h : IsKernelPair f a b
inst✝ : IsIso a
l : X ⟶ (IsPullback.cone h).pt
h₁ : l ≫ (IsPullback.cone h).fst = 𝟙 X
h₂ : l ≫ (IsPullback.cone h).snd = 𝟙 X
⊢ Mono f | rw [IsPullback.cone_fst, ← IsIso.eq_comp_inv, Category.id_comp] at h₁ | case mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
R X Y : C
f : X ⟶ Y
a b : R ⟶ X
h : IsKernelPair f a b
inst✝ : IsIso a
l : X ⟶ (IsPullback.cone h).pt
h₁ : l = inv a
h₂ : l ≫ (IsPullback.cone h).snd = 𝟙 X
⊢ Mono f | 4902798598345cac |
mul_mul_mul_comm | Mathlib/Algebra/Group/Basic.lean | theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) | G : Type u_3
inst✝ : CommSemigroup G
a b c d : G
⊢ a * b * (c * d) = a * c * (b * d) | simp only [mul_left_comm, mul_assoc] | no goals | 356e02baff9749f3 |
LieModule.le_max_triv_iff_bracket_eq_bot | Mathlib/Algebra/Lie/Abelian.lean | theorem le_max_triv_iff_bracket_eq_bot {N : LieSubmodule R L M} :
N ≤ maxTrivSubmodule R L M ↔ ⁅(⊤ : LieIdeal R L), N⁆ = ⊥ | case refine_2
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
h : ⁅⊤, N⁆ = ⊥
m : M
hm : m ∈ N
⊢ m ∈ maxTrivSubmodule R L M | rw [mem_maxTrivSubmodule] | case refine_2
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
h : ⁅⊤, N⁆ = ⊥
m : M
hm : m ∈ N
⊢ ∀ (x : L), ⁅x, m⁆ = 0 | afdc682bd29e9457 |
IsPrimitiveRoot.minpoly_eq_pow | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) | n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
⊢ minpoly ℤ μ = minpoly ℤ (μ ^ p) | by_cases hn : n = 0 | case pos
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : n = 0
⊢ minpoly ℤ μ = minpoly ℤ (μ ^ p)
case neg
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : Cha... | 60585a9f1d93c365 |
EuclideanGeometry.eq_or_eq_reflection_of_dist_eq | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem eq_or_eq_reflection_of_dist_eq {n : ℕ} {s : Simplex ℝ P n} {p p₁ p₂ : P} {r : ℝ}
(hp₁ : p₁ ∈ affineSpan ℝ (insert p (Set.range s.points)))
(hp₂ : p₂ ∈ affineSpan ℝ (insert p (Set.range s.points))) (h₁ : ∀ i, dist (s.points i) p₁ = r)
(h₂ : ∀ i, dist (s.points i) p₂ = r) :
p₁ = p₂ ∨ p₁ = reflecti... | case intro.intro.intro.intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ =... | rw [h₁'] at hp₁ | case intro.intro.intro.intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ =... | 5c883a47f5f30942 |
AlgebraicGeometry.HasRingHomProperty.stalkMap | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | /-- Let `Q` be a property of ring maps that is stable under localization.
Then if the associated property of scheme morphisms holds for `f`, `Q` holds on all stalks. -/
lemma stalkMap
(hQ : ∀ {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (_ : Q f)
(J : Ideal S) (_ : J.IsPrime), Q (Localization.localR... | case inr.intro.intro.intro
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
hQ :
∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
Q f → ∀ (J : Ideal S) (x : J.IsPrime), Q (... | obtain ⟨V, hV, hx, e⟩ := Opens.isBasis_iff_nbhd.mp (isBasis_affine_open X)
(show x ∈ f ⁻¹ᵁ U from hfx) | case inr.intro.intro.intro.intro.intro.intro
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
hQ :
∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
Q f → ∀ (J : Ideal S) (x... | 0ddf35d0417ee27e |
CategoryTheory.Bicategory.adjointifyCounit_left_triangle | Mathlib/CategoryTheory/Bicategory/Adjunction/Basic.lean | theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :
leftZigzagIso η (adjointifyCounit η ε) = λ_ f ≪≫ (ρ_ f).symm | B : Type u
inst✝ : Bicategory B
a b : B
f : a ⟶ b
g : b ⟶ a
η : 𝟙 a ≅ f ≫ g
ε : g ≫ f ≅ 𝟙 b
⊢ 𝟙 (𝟙 a ≫ f) ⊗≫ f ◁ ε.inv ⊗≫ 𝟙 (f ≫ g) ▷ f ⊗≫ f ◁ ε.hom = 𝟙 (𝟙 a ≫ f) ⊗≫ f ◁ (ε.inv ≫ ε.hom) | bicategory | no goals | bfaa28356e9d9007 |
nsmul_one | Mathlib/Data/Nat/Cast/Defs.lean | @[simp] lemma nsmul_one {A} [AddMonoidWithOne A] : ∀ n : ℕ, n • (1 : A) = n
| 0 => by simp [zero_nsmul]
| n + 1 => by simp [succ_nsmul, nsmul_one n]
| A : Type u_2
inst✝ : AddMonoidWithOne A
⊢ 0 • 1 = ↑0 | simp [zero_nsmul] | no goals | 4588592ade33f2c9 |
MeasureTheory.tendsto_indicatorConstLp_set | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem tendsto_indicatorConstLp_set [hp₁ : Fact (1 ≤ p)] {β : Type*} {l : Filter β} {t : β → Set α}
{ht : ∀ b, MeasurableSet (t b)} {hμt : ∀ b, μ (t b) ≠ ∞} (hp : p ≠ ∞)
(h : Tendsto (fun b ↦ μ (t b ∆ s)) l (𝓝 0)) :
Tendsto (fun b ↦ indicatorConstLp p (ht b) (hμt b) c) l (𝓝 (indicatorConstLp p hs hμs c))... | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ⊤
c : E
hp₁ : Fact (1 ≤ p)
β : Type u_7
l : Filter β
t : β → Set α
ht : ∀ (b : β), MeasurableSet (t b)
hμt : ∀ (b : β), μ (t b) ≠ ⊤
hp : p ≠ ⊤
h : Tendsto (fun b => μ (t b ∆ s)... | rw [tendsto_iff_dist_tendsto_zero] | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ⊤
c : E
hp₁ : Fact (1 ≤ p)
β : Type u_7
l : Filter β
t : β → Set α
ht : ∀ (b : β), MeasurableSet (t b)
hμt : ∀ (b : β), μ (t b) ≠ ⊤
hp : p ≠ ⊤
h : Tendsto (fun b => μ (t b ∆ s)... | aecc359b1ac34770 |
Set.OrdConnected.image_hasDerivWithinAt | Mathlib/Analysis/Calculus/Darboux.lean | theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
(hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s) | case hs.intro.intro.intro.intro.intro.inl.intro.intro
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
hab : a ≤ b
this : Icc a b ⊆ s
c : ℝ
cmem : c ∈ Ioo a b
hc : f' c = m
⊢ m ∈ f' '' s | exact ⟨c, this <| Ioo_subset_Icc_self cmem, hc⟩ | no goals | 2ad4e8c6a69413a7 |
Vector.zipWith_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Zip.lean | theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : Vector α (n + m)} {l₂ : Vector β (n + m)} :
zipWith f l₁ l₂ = l₁' ++ l₂' ↔
∃ w x y z, l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zipWith f w y ∧ l₂' = zipWith f x z | case mk.mk.mk.mk.mpr.intro.mk.intro.mk.intro.mk.intro.mk.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
l₁' l₂' : Array γ
w : Array α
hw : w.size = l₁'.size
x : Array α
hx : x.size = l₂'.size
y : Array β
hy : y.size = l₁'.size
z : Array β
hz : z.size = l₂'.size
h₁ :
({ toArray := w, size_toArr... | exact ⟨w, x, y, z, by simpa [hw, hy] using ⟨w₁, w₂⟩⟩ | no goals | 5f549609bded35f4 |
List.length_of_sublistsLen | Mathlib/Data/List/Sublists.lean | theorem length_of_sublistsLen :
∀ {n} {l l' : List α}, l' ∈ sublistsLen n l → length l' = n
| 0, l, l', h => by simp_all
| n + 1, a :: l, l', h => by
rw [sublistsLen_succ_cons, mem_append, mem_map] at h
rcases h with (h | ⟨l', h, rfl⟩)
· exact length_of_sublistsLen h
· exact congr_arg (· + 1) (l... | α : Type u
n : ℕ
a : α
l l' : List α
h : l' ∈ sublistsLen (n + 1) l ∨ ∃ a_1 ∈ sublistsLen n l, a :: a_1 = l'
⊢ l'.length = n + 1 | rcases h with (h | ⟨l', h, rfl⟩) | case inl
α : Type u
n : ℕ
a : α
l l' : List α
h : l' ∈ sublistsLen (n + 1) l
⊢ l'.length = n + 1
case inr.intro.intro
α : Type u
n : ℕ
a : α
l l' : List α
h : l' ∈ sublistsLen n l
⊢ (a :: l').length = n + 1 | b61ba5e46726550c |
BoxIntegral.hasIntegralIndicatorConst | Mathlib/Analysis/BoxIntegral/Integrability.lean | theorem hasIntegralIndicatorConst (l : IntegrationParams) (hl : l.bRiemann = false)
{s : Set (ι → ℝ)} (hs : MeasurableSet s) (I : Box ι) (y : E) (μ : Measure (ι → ℝ))
[IsLocallyFiniteMeasure μ] :
HasIntegral.{u, v, v} I l (s.indicator fun _ => y) μ.toBoxAdditive.toSMul
((μ (s ∩ I)).toReal • y) | case intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : μ (s ∩ Box.Icc I) ≠ ⊤
B : μ (s ∩... | exact ⟨⟨r, hr₀⟩, hr⟩ | no goals | 14c4b865d3538e42 |
SimpleGraph.edgeDensity_add_edgeDensity_compl | Mathlib/Combinatorics/SimpleGraph/Density.lean | theorem edgeDensity_add_edgeDensity_compl (hs : s.Nonempty) (ht : t.Nonempty) (h : Disjoint s t) :
G.edgeDensity s t + Gᶜ.edgeDensity s t = 1 | α : Type u_4
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
s t : Finset α
inst✝ : DecidableEq α
hs : s.Nonempty
ht : t.Nonempty
h : Disjoint s t
⊢ G.edgeDensity s t + Gᶜ.edgeDensity s t = 1 | rw [edgeDensity_def, edgeDensity_def, div_add_div_same, div_eq_one_iff_eq] | α : Type u_4
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
s t : Finset α
inst✝ : DecidableEq α
hs : s.Nonempty
ht : t.Nonempty
h : Disjoint s t
⊢ ↑(#(G.interedges s t)) + ↑(#(Gᶜ.interedges s t)) = ↑(#s) * ↑(#t)
α : Type u_4
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
s t : Finset α
inst✝ : DecidableEq α
hs : s.None... | a46c2ef25e54d0ad |
gramSchmidt_mem_span | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | theorem gramSchmidt_mem_span (f : ι → E) :
∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
k : ι
hk : k ∈ Finset.Iio i
⊢ (inner (gramSchmidt 𝕜 f k) (f i) / ↑(‖gramSchmidt 𝕜 f k‖ ... | let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij | 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
k : ι
hk : k ∈ Finset.Iio i
hkj : k < j := LT.lt.trans_le (Finset.mem_Iio.mp hk) hij
⊢ (i... | 2db3f8db7f4c82a7 |
SimpleGraph.isClique_iff_induce_eq | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem isClique_iff_induce_eq : G.IsClique s ↔ G.induce s = ⊤ | α : Type u_1
G : SimpleGraph α
s : Set α
⊢ G.IsClique s ↔ induce s G = ⊤ | rw [isClique_iff] | α : Type u_1
G : SimpleGraph α
s : Set α
⊢ s.Pairwise G.Adj ↔ induce s G = ⊤ | 6f8b149e1d0a0705 |
inner_eq_norm_mul_iff | Mathlib/Analysis/InnerProductSpace/Basic.lean | theorem inner_eq_norm_mul_iff {x y : E} :
⟪x, y⟫ = (‖x‖ : 𝕜) * ‖y‖ ↔ (‖y‖ : 𝕜) • x = (‖x‖ : 𝕜) • y | case inr.ha
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
⊢ ↑‖x‖ ≠ 0 | rwa [Ne, ofReal_eq_zero, norm_eq_zero] | no goals | 7e2c5381c23d5d8c |
WittVector.teichmuller_mul_aux₁ | Mathlib/RingTheory/WittVector/Teichmuller.lean | theorem teichmuller_mul_aux₁ {R : Type*} (x y : MvPolynomial R ℚ) :
teichmullerFun p (x * y) = teichmullerFun p x * teichmullerFun p y | case a
p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_3
x y : MvPolynomial R ℚ
⊢ ghostMap (teichmullerFun p (x * y)) = ghostMap (teichmullerFun p x * teichmullerFun p y) | rw [RingHom.map_mul] | case a
p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_3
x y : MvPolynomial R ℚ
⊢ ghostMap (teichmullerFun p (x * y)) = ghostMap (teichmullerFun p x) * ghostMap (teichmullerFun p y) | c8848d9ac0120b29 |
Computation.Results.len_unique | Mathlib/Data/Seq/Computation.lean | theorem Results.len_unique {s : Computation α} {a b m n} (h1 : Results s a m) (h2 : Results s b n) :
m = n | α : Type u
s : Computation α
a b : α
m n : ℕ
h1 : s.Results a m
h2 : s.Results b n
⊢ m = n | haveI := h1.terminates | α : Type u
s : Computation α
a b : α
m n : ℕ
h1 : s.Results a m
h2 : s.Results b n
this : s.Terminates
⊢ m = n | 840300a45bb1690f |
MonoidHom.transfer_center_eq_pow | Mathlib/GroupTheory/Transfer.lean | theorem transfer_center_eq_pow [FiniteIndex (center G)] (g : G) :
transfer (MonoidHom.id (center G)) g = ⟨g ^ (center G).index, (center G).pow_index_mem g⟩ :=
transfer_eq_pow (id (center G)) g fun k _ hk => by rw [← mul_right_inj, ← hk.comm,
mul_inv_cancel_right]
| G : Type u_1
inst✝¹ : Group G
inst✝ : (center G).FiniteIndex
g : G
k : ℕ
x✝ : G
hk : x✝⁻¹ * g ^ k * x✝ ∈ center G
⊢ x✝⁻¹ * g ^ k * x✝ = g ^ k | rw [← mul_right_inj, ← hk.comm,
mul_inv_cancel_right] | no goals | 622773158d764980 |
Filter.HasBasis.liminf_eq_ciSup_ciInf | Mathlib/Order/LiminfLimsup.lean | theorem HasBasis.liminf_eq_ciSup_ciInf {v : Filter ι}
{p : ι' → Prop} {s : ι' → Set ι} [Countable (Subtype p)] [Nonempty (Subtype p)]
(hv : v.HasBasis p s) {f : ι → α} (hs : ∀ (j : Subtype p), (s j).Nonempty)
(H : ∃ (j : Subtype p), BddBelow (range (fun (i : s j) ↦ f i))) :
liminf f v = ⨆ (j : Subtype p... | case neg
α : Type u_1
ι : Type u_4
ι' : Type u_5
inst✝² : ConditionallyCompleteLinearOrder α
v : Filter ι
p : ι' → Prop
s : ι' → Set ι
inst✝¹ : Countable (Subtype p)
inst✝ : Nonempty (Subtype p)
hv : v.HasBasis p s
f : ι → α
hs : ∀ (j : Subtype p), (s ↑j).Nonempty
j0 : Subtype p
hj0 : BddBelow (range fun i => f ↑i)
m :... | have Z : ∃ n, (exists_surjective_nat (Subtype p)).choose n ∈ m ∨ ∀ j, j ∉ m := by
rcases (exists_surjective_nat (Subtype p)).choose_spec j0 with ⟨n, rfl⟩
exact ⟨n, Or.inl hj0⟩ | case neg
α : Type u_1
ι : Type u_4
ι' : Type u_5
inst✝² : ConditionallyCompleteLinearOrder α
v : Filter ι
p : ι' → Prop
s : ι' → Set ι
inst✝¹ : Countable (Subtype p)
inst✝ : Nonempty (Subtype p)
hv : v.HasBasis p s
f : ι → α
hs : ∀ (j : Subtype p), (s ↑j).Nonempty
j0 : Subtype p
hj0 : BddBelow (range fun i => f ↑i)
m :... | abe131dbac054323 |
PythagoreanTriple.classified | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem classified : h.IsClassified | case neg
x y z : ℤ
h : PythagoreanTriple x y z
h0 : ¬x.gcd y = 0
⊢ ⋯.IsPrimitiveClassified | apply h.normalize.isPrimitiveClassified_of_coprime | case neg
x y z : ℤ
h : PythagoreanTriple x y z
h0 : ¬x.gcd y = 0
⊢ (x / ↑(x.gcd y)).gcd (y / ↑(x.gcd y)) = 1 | b7ab9a71544358c0 |
Filter.isTopologicalBasis_Iic_principal | Mathlib/Topology/Filter.lean | theorem isTopologicalBasis_Iic_principal :
IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) :=
{ exists_subset_inter | case intro.intro
α : Type u_2
s t : Set α
l : Filter α
hl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t
⊢ ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t | exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩ | no goals | 87676d3a7fd616f3 |
Ideal.IsHomogeneous.radical_eq | Mathlib/RingTheory/GradedAlgebra/Radical.lean | theorem Ideal.IsHomogeneous.radical_eq {I : Ideal A} (hI : I.IsHomogeneous 𝒜) :
I.radical = InfSet.sInf { J | Ideal.IsHomogeneous 𝒜 J ∧ I ≤ J ∧ J.IsPrime } | case a
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
⊢ ∀ x ∈ {J | I ≤ J ∧ J.IsPrime}, ∃ y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ J.IsPrime}, y ≤ ... | rintro J ⟨HJ₁, HJ₂⟩ | case a.intro
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
J : Ideal A
HJ₁ : I ≤ J
HJ₂ : J.IsPrime
⊢ ∃ y ∈ {J | IsHomogeneous 𝒜 J ∧ I ≤ J ∧ J... | 40171ff4505ac007 |
ProbabilityTheory.hasFiniteIntegral_compProd_iff | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | theorem hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) | α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝² : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
f : β × γ → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂η ... | exact h1f.enorm.lintegral_kernel_prod_right'' | no goals | 09df794e8ec71c81 |
LinearMap.split_surjective_of_localization_maximal | Mathlib/RingTheory/LocalProperties/Projective.lean | theorem LinearMap.split_surjective_of_localization_maximal
(f : M →ₗ[R] N) [Module.FinitePresentation R N]
(H : ∀ (I : Ideal R) (_ : I.IsMaximal),
∃ (g : _ →ₗ[Localization.AtPrime I] _),
(LocalizedModule.map I.primeCompl f).comp g = LinearMap.id) :
∃ (g : N →ₗ[R] M), f.comp g = LinearMap.id | case h.e.h.a.mpr
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f✝ : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f✝ ∘ₗ g = id
I : Ideal R
... | rintro ⟨g, rfl⟩ | case h.e.h.a.mpr.intro
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = id
I : Idea... | d80210efccbecc76 |
PerfectPairing.IsPerfectCompl.left_top_iff | Mathlib/LinearAlgebra/PerfectPairing/Basic.lean | @[simp]
lemma left_top_iff :
p.IsPerfectCompl ⊤ V ↔ V = ⊤ | case refine_1
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : PerfectPairing R M N
V : Submodule R N
h : p.IsPerfectCompl ⊤ V
⊢ V = ⊤ | exact eq_top_of_isCompl_bot <| by simpa using h.isCompl_right | no goals | 318990f5a095ed69 |
AlgebraicGeometry.isIntegral_of_irreducibleSpace_of_isReduced | Mathlib/AlgebraicGeometry/Properties.lean | theorem isIntegral_of_irreducibleSpace_of_isReduced [IsReduced X] [H : IrreducibleSpace X] :
IsIntegral X | X : Scheme
inst✝ : IsReduced X
H : IrreducibleSpace ↑↑X.toPresheafedSpace
⊢ IsIntegral X | constructor | case nonempty
X : Scheme
inst✝ : IsReduced X
H : IrreducibleSpace ↑↑X.toPresheafedSpace
⊢ autoParam (Nonempty ↑↑X.toPresheafedSpace) _auto✝
case component_integral
X : Scheme
inst✝ : IsReduced X
H : IrreducibleSpace ↑↑X.toPresheafedSpace
⊢ autoParam (∀ (U : X.Opens) [inst : Nonempty ↑↑(↑U).toPresheafedSpace], IsDomain... | 6a2b6ae754d9eaf7 |
CategoryTheory.Subobject.factorThru_eq_zero | Mathlib/CategoryTheory/Subobject/FactorThru.lean | theorem factorThru_eq_zero [HasZeroMorphisms C] {X Y : C} {P : Subobject Y} {f : X ⟶ Y}
{h : Factors P f} : P.factorThru f h = 0 ↔ f = 0 | case mpr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasZeroMorphisms C
X Y : C
P : Subobject Y
h : P.Factors 0
⊢ P.factorThru 0 h = 0 | ext | case mpr.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasZeroMorphisms C
X Y : C
P : Subobject Y
h : P.Factors 0
⊢ P.factorThru 0 h ≫ P.arrow = 0 ≫ P.arrow | 985c057fb37bcf56 |
EuclideanGeometry.eq_or_eq_reflection_of_dist_eq | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem eq_or_eq_reflection_of_dist_eq {n : ℕ} {s : Simplex ℝ P n} {p p₁ p₂ : P} {r : ℝ}
(hp₁ : p₁ ∈ affineSpan ℝ (insert p (Set.range s.points)))
(hp₂ : p₂ ∈ affineSpan ℝ (insert p (Set.range s.points))) (h₁ : ∀ i, dist (s.points i) p₁ = r)
(h₂ : ∀ i, dist (s.points i) p₂ = r) :
p₁ = p₂ ∨ p₁ = reflecti... | case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P :... | rw [Simplex.orthogonalProjectionSpan] at hp | case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
p p₁ p₂ : P
r : ℝ
h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r
h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r
span_s : AffineSubspace ℝ P :... | 5c883a47f5f30942 |
Rat.AbsoluteValue.one_lt_of_not_bounded | Mathlib/NumberTheory/Ostrowski.lean | /-- If `f n > 1` for some `n` then `f n > 1` for all `n ≥ 2` -/
lemma one_lt_of_not_bounded (notbdd : ¬ ∀ n : ℕ, f n ≤ 1) {n₀ : ℕ} (hn₀ : 1 < n₀) : 1 < f n₀ | case inl
f : AbsoluteValue ℚ ℝ
n₀ : ℕ
hn₀ : 1 < n₀
h : f ↑n₀ ≤ 1
h_ineq1 : ∀ {m : ℕ}, 1 ≤ m → f ↑m ≤ ↑n₀ * (logb ↑n₀ ↑m + 1)
⊢ f ↑0 ≤ 1 | simp | no goals | 2244dd101a1df19f |
padicValNat.padicValNat_eq_maxPowDiv | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem padicValNat_eq_maxPowDiv : @padicValNat = @maxPowDiv | case h.h
p n : ℕ
⊢ padicValNat p n = p.maxPowDiv n | by_cases h : 1 < p ∧ 0 < n | case pos
p n : ℕ
h : 1 < p ∧ 0 < n
⊢ padicValNat p n = p.maxPowDiv n
case neg
p n : ℕ
h : ¬(1 < p ∧ 0 < n)
⊢ padicValNat p n = p.maxPowDiv n | 0b878b2767fb869e |
Module.End.genEigenspace_inf_le_add | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma genEigenspace_inf_le_add
(f₁ f₂ : End R M) (μ₁ μ₂ : R) (k₁ k₂ : ℕ∞) (h : Commute f₁ f₂) :
(f₁.genEigenspace μ₁ k₁) ⊓ (f₂.genEigenspace μ₂ k₂) ≤
(f₁ + f₂).genEigenspace (μ₁ + μ₂) (k₁ + k₂) | R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f₁ f₂ : End R M
μ₁ μ₂ : R
k₁ k₂ : ℕ∞
h : Commute f₁ f₂
m : M
l₁ : ℕ
hlk₁ : ↑l₁ ≤ k₁
hl₁ : ((f₁ - μ₁ • 1) ^ l₁) m = 0
l₂ : ℕ
hlk₂ : ↑l₂ ≤ k₂
hl₂ : ((f₂ - μ₂ • 1) ^ l₂) m = 0
⊢ f₁ + f₂ - (μ₁ • 1 + μ₂ • 1) = f₁ - μ₁ • 1 + (f₂ - μ₂ • 1) | exact add_sub_add_comm f₁ f₂ (μ₁ • 1) (μ₂ • 1) | no goals | 4aaf1ee24d6a3721 |
norm_jacobiTheta₂_term_fderiv_ge | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | lemma norm_jacobiTheta₂_term_fderiv_ge (n : ℤ) (z τ : ℂ) :
π * |n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖ | n : ℤ
z τ : ℂ
this : ‖(jacobiTheta₂_term_fderiv n z τ) (0, 1)‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖
⊢ π * ↑|n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖ | refine le_trans ?_ this | n : ℤ
z τ : ℂ
this : ‖(jacobiTheta₂_term_fderiv n z τ) (0, 1)‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖
⊢ π * ↑|n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ ‖(jacobiTheta₂_term_fderiv n z τ) (0, 1)‖ | 956d060d78984c99 |
IsPrimitiveRoot.minpoly_eq_pow | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) | case neg.intro.inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducibl... | replace hunit := degree_eq_zero_of_isUnit hunit | case neg.intro.inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducibl... | 60585a9f1d93c365 |
Set.sdiff_singleton_wcovBy | Mathlib/Order/Cover.lean | @[simp] lemma sdiff_singleton_wcovBy (s : Set α) (a : α) : s \ {a} ⩿ s | case neg
α : Type u_1
s : Set α
a : α
ha : a ∉ s
⊢ s \ {a} ⩿ s | simp [ha] | no goals | 655eed34228b7954 |
Left.sign_neg | Mathlib/Data/Sign.lean | theorem Left.sign_neg [AddLeftStrictMono α] (a : α) : sign (-a) = -sign a | case pos
α : Type u_1
inst✝³ : AddGroup α
inst✝² : Preorder α
inst✝¹ : DecidableRel fun x1 x2 => x1 < x2
inst✝ : AddLeftStrictMono α
a : α
h : ¬a < 0
h✝ : 0 < a
⊢ -1 = -1 | simp | no goals | c7aaf8f9187179cd |
Subadditive.eventually_div_lt_of_div_lt | Mathlib/Analysis/Subadditive.lean | theorem eventually_div_lt_of_div_lt {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / n < L) :
∀ᶠ p in atTop, u p / p < L | u : ℕ → ℝ
h : Subadditive u
L : ℝ
n : ℕ
hn : n ≠ 0
hL : u n / ↑n < L
⊢ ∀ᶠ (p : ℕ) in atTop, u p / ↑p < L | refine .atTop_of_arithmetic hn fun r _ => ?_ | u : ℕ → ℝ
h : Subadditive u
L : ℝ
n : ℕ
hn : n ≠ 0
hL : u n / ↑n < L
r : ℕ
x✝ : r < n
⊢ ∀ᶠ (a : ℕ) in atTop, u (n * a + r) / ↑(n * a + r) < L | 9f51e9f68e34b75e |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
X.presheaf.map i ≫ H.invApp _ (unop V) =
invApp f (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) | C : Type u
inst✝ : Category.{v, u} C
X Y : PresheafedSpace C
f : X ⟶ Y
H : IsOpenImmersion f
U V : (Opens ↑↑X)ᵒᵖ
i : U ⟶ V
⊢ X.presheaf.map (i ≫ eqToHom ⋯) =
X.presheaf.map (eqToHom ⋯ ≫ ((Opens.map f.base).map ((opensFunctor f).map i.unop)).op) | congr 1 | no goals | d59e5d3543c98b92 |
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