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
|
|---|---|---|---|---|---|---|
AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_id
|
Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
|
@[simp]
lemma residueFieldMap_id (x : X) :
residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x)
|
X : LocallyRingedSpace
x : ↑X.toTopCat
⊢ residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x)
|
ext : 1
|
case hf
X : LocallyRingedSpace
x : ↑X.toTopCat
⊢ CommRingCat.Hom.hom (residueFieldMap (𝟙 X) x) = CommRingCat.Hom.hom (𝟙 (X.residueField x))
|
6d43b3b35eb00ee4
|
Multiset.map_univ_coe
|
Mathlib/Data/Multiset/Fintype.lean
|
theorem map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m
|
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
this : map Prod.fst (Finset.map m.coeEmbedding Finset.univ).val = m
⊢ map (fun x => x.fst) Finset.univ.val = m
|
simpa only [Finset.map_val, Multiset.coeEmbedding_apply, Multiset.map_map,
Function.comp_apply] using this
|
no goals
|
d3b22290796d9c51
|
FiniteField.roots_X_pow_card_sub_X
|
Mathlib/FieldTheory/Finite/Basic.lean
|
theorem roots_X_pow_card_sub_X : roots (X ^ q - X : K[X]) = Finset.univ.val
|
case h.e'_4
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
aux : X ^ q - X ≠ 0
this : (X ^ q - X).roots.toFinset = univ
⊢ derivative (X ^ q - X) = -1
|
rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
zero_mul, zero_sub]
|
no goals
|
f38f049b4c3caab9
|
HomologicalComplex.quasiIso_truncGEMap_iff
|
Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean
|
lemma quasiIso_truncGEMap_iff :
QuasiIso (truncGEMap φ e) ↔ ∀ (i : ι) (i' : ι') (_ : e.f i = i'), QuasiIsoAt φ i'
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{u_4, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
this : ∀ (i : ι) (i' : ι'), e.f i = i' → (QuasiIsoAt (truncGEMap φ e) i' ↔ QuasiIsoAt φ i')
h : ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ QuasiIsoAt (truncGEMap φ e) i'
|
rw [quasiIsoAt_iff_exactAt]
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{u_4, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
this : ∀ (i : ι) (i' : ι'), e.f i = i' → (QuasiIsoAt (truncGEMap φ e) i' ↔ QuasiIsoAt φ i')
h : ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ (L.truncGE e).ExactAt i'
case neg.hK
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{u_4, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
this : ∀ (i : ι) (i' : ι'), e.f i = i' → (QuasiIsoAt (truncGEMap φ e) i' ↔ QuasiIsoAt φ i')
h : ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ (K.truncGE e).ExactAt i'
|
80836fc97a82149a
|
IsClosed.exists_wbtw_isVisible
|
Mathlib/Analysis/Convex/Visible.lean
|
/-- If `s` is a closed set, then any point `x` sees some point of `s` in any direction where there
is something to see. -/
lemma IsClosed.exists_wbtw_isVisible (hs : IsClosed s) (hy : y ∈ s) (x : V) :
∃ z ∈ s, Wbtw ℝ x z y ∧ IsVisible ℝ s x z
|
V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed s
hy : y ∈ s
x : V
t : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s
ht₁ : 1 ∈ t
ht : BddBelow t
δ : ℝ := sInf t
hδ₁ : δ ≤ 1
hδ₀✝ : 0 ≤ δ
hδ : (lineMap x y) δ ∈ s
ε : ℝ
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
hε : (lineMap x ((lineMap x y) δ)) ε ∈ s
h : (lineMap x ((lineMap x y) δ)) ε ≠ (lineMap x y) δ
hδ₀ : δ = 0
⊢ False
|
simp [hδ₀] at h
|
no goals
|
b1e9329ed05e7997
|
CategoryTheory.OverPresheafAux.YonedaCollection.map₁_comp
|
Mathlib/CategoryTheory/Comma/Presheaf/Basic.lean
|
@[simp]
lemma map₁_comp {G H : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v} (η : F ⟶ G) (μ : G ⟶ H) :
YonedaCollection.map₁ (η ≫ μ) (X := X) =
YonedaCollection.map₁ μ (X := X) ∘ YonedaCollection.map₁ η (X := X)
|
case h.h'
C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G H : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
μ : G ⟶ H
x✝ : YonedaCollection F X
⊢ H.map (eqToHom ⋯) ((map₁ μ ∘ map₁ η) x✝).snd = (map₁ (η ≫ μ) x✝).snd
|
simp
|
no goals
|
576c452c575f7911
|
HasProd.hasProd_compl_iff
|
Mathlib/Topology/Algebra/InfiniteSum/Group.lean
|
theorem HasProd.hasProd_compl_iff {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
HasProd (f ∘ (↑) : ↑sᶜ → α) a₂ ↔ HasProd f (a₁ * a₂)
|
α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalGroup α
f : β → α
a₁ a₂ : α
s : Set β
hf : HasProd (s.mulIndicator f) a₁
h : HasProd f (a₁ * a₂)
⊢ HasProd (sᶜ.mulIndicator f) a₂
|
rw [Set.mulIndicator_compl]
|
α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalGroup α
f : β → α
a₁ a₂ : α
s : Set β
hf : HasProd (s.mulIndicator f) a₁
h : HasProd f (a₁ * a₂)
⊢ HasProd (f * (s.mulIndicator f)⁻¹) a₂
|
384979b83d123102
|
SimpleGraph.edgeDisjointTriangles_iff_mem_sym2_subsingleton
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
lemma edgeDisjointTriangles_iff_mem_sym2_subsingleton :
G.EdgeDisjointTriangles ↔
∀ ⦃e : Sym2 α⦄, ¬ e.IsDiag → {s ∈ G.cliqueSet 3 | e ∈ (s : Finset α).sym2}.Subsingleton
|
case mp
α : Type u_1
G : SimpleGraph α
this :
∀ (a b : α),
a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}
hG : G.EdgeDisjointTriangles
a b : α
hab : ¬a = b
⊢ {s | G.Adj (a, b).1 (a, b).2 ∧ ∃ c, G.Adj (a, b).1 c ∧ G.Adj (a, b).2 c ∧ s = {(a, b).1, (a, b).2, c}}.Subsingleton
|
rintro _ ⟨hab, c, hac, hbc, rfl⟩ _ ⟨-, d, had, hbd, rfl⟩
|
case mp.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
G : SimpleGraph α
this :
∀ (a b : α),
a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}
hG : G.EdgeDisjointTriangles
a b : α
hab✝ : ¬a = b
hab : G.Adj (a, b).1 (a, b).2
c : α
hac : G.Adj (a, b).1 c
hbc : G.Adj (a, b).2 c
d : α
had : G.Adj (a, b).1 d
hbd : G.Adj (a, b).2 d
⊢ {(a, b).1, (a, b).2, c} = {(a, b).1, (a, b).2, d}
|
dd5ab32164f7deff
|
orthogonalComplement_eq_orthogonalComplement
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem orthogonalComplement_eq_orthogonalComplement {L : Submodule 𝕜 E} [HasOrthogonalProjection K]
[HasOrthogonalProjection L] : Kᗮ = Lᗮ ↔ K = L :=
⟨fun h ↦ by simpa using congr(Submodule.orthogonal $(h)),
fun h ↦ congr(Submodule.orthogonal $(h))⟩
|
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K L : Submodule 𝕜 E
inst✝¹ : HasOrthogonalProjection K
inst✝ : HasOrthogonalProjection L
h : Kᗮ = Lᗮ
⊢ K = L
|
simpa using congr(Submodule.orthogonal $(h))
|
no goals
|
a29173c0d632433c
|
Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv
|
Mathlib/Analysis/Complex/Polynomial/Basic.lean
|
theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
(p.rootSet ℂ).toFinset.card =
(p.rootSet ℝ).toFinset.card +
(galActionHom p ℂ (restrict p ℂ
(AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card
|
case neg.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := ?neg.refine_1✝
⊢ (p.rootSet ℂ).toFinset.card =
(Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset).card +
(Finset.image (fun a => ↑a)
((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support).card
case neg.refine_1
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
⊢ Finset ℂ
|
on_goal 1 => let b : Finset ℂ := ?_
|
case neg.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := ?neg.refine_1✝
b : Finset ℂ := ?neg.refine_2.refine_1✝
⊢ (p.rootSet ℂ).toFinset.card =
(Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset).card +
(Finset.image (fun a => ↑a)
((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support).card
case neg.refine_2.refine_1
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := ?neg.refine_1✝
⊢ Finset ℂ
case neg.refine_1
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
⊢ Finset ℂ
|
8dbcfedb34311abf
|
Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem sup_ker_aeval_eq_ker_aeval_mul_of_coprime (f : M →ₗ[R] M) {p q : R[X]}
(hpq : IsCoprime p q) :
LinearMap.ker (aeval f p) ⊔ LinearMap.ker (aeval f q) = LinearMap.ker (aeval f (p * q))
|
case intro.intro
R : Type u
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
p q : R[X]
v : M
hv : v ∈ LinearMap.ker ((aeval f) (p * q))
p' q' : R[X]
hpq' : p' * p + q' * q = 1
h_eval₂_qpp' : ((aeval f) q * (aeval f) (p * p')) v = 0
h_eval₂_pqq' : ((aeval f) p * (aeval f) (q * q')) v = 0
⊢ ((aeval f) (q * q')) v + ((aeval f) (p * p')) v = v
|
rw [add_comm, mul_comm p p', mul_comm q q']
|
case intro.intro
R : Type u
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
p q : R[X]
v : M
hv : v ∈ LinearMap.ker ((aeval f) (p * q))
p' q' : R[X]
hpq' : p' * p + q' * q = 1
h_eval₂_qpp' : ((aeval f) q * (aeval f) (p * p')) v = 0
h_eval₂_pqq' : ((aeval f) p * (aeval f) (q * q')) v = 0
⊢ ((aeval f) (p' * p)) v + ((aeval f) (q' * q)) v = v
|
bb84be2e4bee1df0
|
Seminorm.continuous_iSup
|
Mathlib/Analysis/LocallyConvex/Barrelled.lean
|
theorem Seminorm.continuous_iSup
{ι : Sort*} {𝕜 E : Type*} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : ι → Seminorm 𝕜 E)
(hp : ∀ i, Continuous (p i)) (bdd : BddAbove (range p)) :
Continuous (⨆ i, p i)
|
ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : BarrelledSpace 𝕜 E
p : ι → Seminorm 𝕜 E
hp : ∀ (i : ι), Continuous ⇑(p i)
bdd : BddAbove (range p)
⊢ LowerSemicontinuous (⨆ i, ⇑(p i))
|
rw [Seminorm.bddAbove_range_iff] at bdd
|
ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : BarrelledSpace 𝕜 E
p : ι → Seminorm 𝕜 E
hp : ∀ (i : ι), Continuous ⇑(p i)
bdd : ∀ (x : E), BddAbove (range fun i => (p i) x)
⊢ LowerSemicontinuous (⨆ i, ⇑(p i))
|
5c60d0ed10836010
|
exists_seq_of_forall_finset_exists'
|
Mathlib/Data/Fintype/Basic.lean
|
theorem exists_seq_of_forall_finset_exists' {α : Type*} (P : α → Prop) (r : α → α → Prop)
[IsSymm α r] (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
∃ f : ℕ → α, (∀ n, P (f n)) ∧ Pairwise (r on f)
|
case intro.intro.inr.inr
α : Type u_4
P : α → Prop
r : α → α → Prop
inst✝ : IsSymm α r
h✝ : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
f : ℕ → α
hf : ∀ (n : ℕ), P (f n)
hf' : ∀ (m n : ℕ), m < n → r (f m) (f n)
m n : ℕ
hmn : m ≠ n
h : n < m
⊢ (r on f) m n
|
unfold Function.onFun
|
case intro.intro.inr.inr
α : Type u_4
P : α → Prop
r : α → α → Prop
inst✝ : IsSymm α r
h✝ : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
f : ℕ → α
hf : ∀ (n : ℕ), P (f n)
hf' : ∀ (m n : ℕ), m < n → r (f m) (f n)
m n : ℕ
hmn : m ≠ n
h : n < m
⊢ r (f m) (f n)
|
3491857fe7bb309d
|
AlgebraicGeometry.HasRingHomProperty.of_isLocalAtSource_of_isLocalAtTarget
|
Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
|
lemma of_isLocalAtSource_of_isLocalAtTarget [IsLocalAtTarget P] [IsLocalAtSource P] :
HasRingHomProperty P (fun f ↦ P (Spec.map (CommRingCat.ofHom f))) where
isLocal_ringHomProperty :=
isLocal_ringHomProperty_of_isLocalAtSource_of_isLocalAtTarget P
eq_affineLocally'
|
P : MorphismProperty Scheme
inst✝¹ : IsLocalAtTarget P
inst✝ : IsLocalAtSource P
Q : MorphismProperty Scheme := affineLocally fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
this : HasRingHomProperty Q fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
S : CommRingCat
X : Scheme
f : X ⟶ Spec S
hX : ∃ R, X = Spec R
⊢ P f ↔ Q f
|
obtain ⟨R, rfl⟩ := hX
|
case intro
P : MorphismProperty Scheme
inst✝¹ : IsLocalAtTarget P
inst✝ : IsLocalAtSource P
Q : MorphismProperty Scheme := affineLocally fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
this : HasRingHomProperty Q fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
S R : CommRingCat
f : Spec R ⟶ Spec S
⊢ P f ↔ Q f
|
b07ed824f3692719
|
Polynomial.coeff_mul_add_eq_of_natDegree_le
|
Mathlib/Algebra/Polynomial/Degree/Operations.lean
|
theorem coeff_mul_add_eq_of_natDegree_le {df dg : ℕ} {f g : R[X]}
(hdf : natDegree f ≤ df) (hdg : natDegree g ≤ dg) :
(f * g).coeff (df + dg) = f.coeff df * g.coeff dg
|
case h₀.mk.inr
R : Type u
inst✝ : Semiring R
df dg : ℕ
f g : R[X]
hdf : f.natDegree ≤ df
hdg : g.natDegree ≤ dg
df' dg' : ℕ
hmem : (df', dg') ∈ antidiagonal (df + dg)
hne : (df', dg') ≠ (df, dg)
hdf' : df' ≤ df
⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0
|
obtain h | hdg' := lt_or_le dg dg'
|
case h₀.mk.inr.inl
R : Type u
inst✝ : Semiring R
df dg : ℕ
f g : R[X]
hdf : f.natDegree ≤ df
hdg : g.natDegree ≤ dg
df' dg' : ℕ
hmem : (df', dg') ∈ antidiagonal (df + dg)
hne : (df', dg') ≠ (df, dg)
hdf' : df' ≤ df
h : dg < dg'
⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0
case h₀.mk.inr.inr
R : Type u
inst✝ : Semiring R
df dg : ℕ
f g : R[X]
hdf : f.natDegree ≤ df
hdg : g.natDegree ≤ dg
df' dg' : ℕ
hmem : (df', dg') ∈ antidiagonal (df + dg)
hne : (df', dg') ≠ (df, dg)
hdf' : df' ≤ df
hdg' : dg' ≤ dg
⊢ f.coeff (df', dg').1 * g.coeff (df', dg').2 = 0
|
cc639442faabee4c
|
MvPowerSeries.map.isLocalHom
|
Mathlib/RingTheory/MvPowerSeries/Inverse.lean
|
theorem map.isLocalHom : IsLocalHom (map σ f) :=
⟨by
rintro φ ⟨ψ, h⟩
replace h := congr_arg (constantCoeff σ S) h
rw [constantCoeff_map] at h
have : IsUnit (constantCoeff σ S ↑ψ) := isUnit_constantCoeff _ ψ.isUnit
rw [h] at this
rcases isUnit_of_map_unit f _ this with ⟨c, hc⟩
exact isUnit_of_mul_eq_one φ (invOfUnit φ c) (mul_invOfUnit φ c hc.symm)⟩
|
case intro
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : (constantCoeff σ S) ↑ψ = f ((constantCoeff σ R) φ)
⊢ IsUnit φ
|
have : IsUnit (constantCoeff σ S ↑ψ) := isUnit_constantCoeff _ ψ.isUnit
|
case intro
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : (constantCoeff σ S) ↑ψ = f ((constantCoeff σ R) φ)
this : IsUnit ((constantCoeff σ S) ↑ψ)
⊢ IsUnit φ
|
f085e8794e8de496
|
WithBot.add_lt_add_right
|
Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
|
protected lemma add_lt_add_right [LT α] [AddRightStrictMono α] (hz : z ≠ ⊥) :
x < y → x + z < y + z
|
α : Type u
inst✝² : Add α
x y z : WithBot α
inst✝¹ : LT α
inst✝ : AddRightStrictMono α
hz : z ≠ ⊥
⊢ x < y → x + z < y + z
|
lift z to α using hz
|
case intro
α : Type u
inst✝² : Add α
x y : WithBot α
inst✝¹ : LT α
inst✝ : AddRightStrictMono α
z : α
⊢ x < y → x + ↑z < y + ↑z
|
28ae5eb4d91ddc82
|
CategoryTheory.yonedaYonedaColimit_app_inv
|
Mathlib/CategoryTheory/Limits/Preserves/Yoneda.lean
|
theorem yonedaYonedaColimit_app_inv {X : C} : ((yonedaYonedaColimit F).app (op X)).inv =
(colimitObjIsoColimitCompEvaluation _ _).hom ≫
(colimit.post F (coyoneda.obj (op (yoneda.obj X))))
|
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : HasColimitsOfShape J (Type v₁)
inst✝ : HasColimitsOfShape J (Type (max u₁ v₁))
F : J ⥤ Cᵒᵖ ⥤ Type v₁
X : C
⊢ (colimitObjIsoColimitCompEvaluation (F ⋙ yoneda) (op (yoneda.obj X))).hom ≫
colimMap (whiskerLeft F (largeCurriedYonedaLemma.hom.app (op X))) ≫
(preservesColimitIso uliftFunctor.{u₁, v₁} (F.flip.obj (op X))).inv ≫
uliftFunctor.{u₁, v₁}.map (colimitObjIsoColimitCompEvaluation F (op X)).inv ≫
(yonedaOpCompYonedaObj (colimit F)).inv.app (op X) =
(colimitObjIsoColimitCompEvaluation (F ⋙ yoneda) (op (yoneda.obj X))).hom ≫
colimit.post F (coyoneda.obj (op (yoneda.obj X)))
|
simp only [Category.id_comp, Iso.cancel_iso_hom_left]
|
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : HasColimitsOfShape J (Type v₁)
inst✝ : HasColimitsOfShape J (Type (max u₁ v₁))
F : J ⥤ Cᵒᵖ ⥤ Type v₁
X : C
⊢ colimMap (whiskerLeft F (largeCurriedYonedaLemma.hom.app (op X))) ≫
(preservesColimitIso uliftFunctor.{u₁, v₁} (F.flip.obj (op X))).inv ≫
uliftFunctor.{u₁, v₁}.map (colimitObjIsoColimitCompEvaluation F (op X)).inv ≫
(yonedaOpCompYonedaObj (colimit F)).inv.app (op X) =
colimit.post F (coyoneda.obj (op (yoneda.obj X)))
|
874001b8ccf19d19
|
Bool.ofNat_le_ofNat
|
Mathlib/Data/Bool/Basic.lean
|
theorem ofNat_le_ofNat {n m : Nat} (h : n ≤ m) : ofNat n ≤ ofNat m
|
n m : ℕ
h : n ≤ m
⊢ (!decide (n = 0)) ≤ !decide (m = 0)
|
cases Nat.decEq n 0 with
| isTrue hn => rw [_root_.decide_eq_true hn]; exact Bool.false_le _
| isFalse hn =>
cases Nat.decEq m 0 with
| isFalse hm => rw [_root_.decide_eq_false hm]; exact Bool.le_true _
| isTrue hm => subst hm; have h := Nat.le_antisymm h (Nat.zero_le n); contradiction
|
no goals
|
a12bab163ba4b474
|
MvPolynomial.weightedHomogeneousComponent_eq_zero_of_not_mem
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
theorem weightedHomogeneousComponent_eq_zero_of_not_mem [DecidableEq M]
(φ : MvPolynomial σ R) (i : M) (hi : i ∉ Finset.image (weight w) φ.support) :
weightedHomogeneousComponent w i φ = 0
|
case h
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
w : σ → M
inst✝¹ : AddCommMonoid M
inst✝ : DecidableEq M
φ : MvPolynomial σ R
i : M
hi : ∀ (x : σ →₀ ℕ), ¬coeff x φ = 0 → ¬(weight w) x = i
⊢ ∀ d ∈ φ.support, (weight w) d ≠ i
|
exact fun m hm ↦ hi m (mem_support_iff.mp hm)
|
no goals
|
88ec2f0eca5bdfd4
|
Real.harm_mean_le_geom_mean_weighted
|
Mathlib/Analysis/MeanInequalities.lean
|
theorem harm_mean_le_geom_mean_weighted (w z : ι → ℝ) (hs : s.Nonempty) (hw : ∀ i ∈ s, 0 < w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) :
(∑ i ∈ s, w i / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ w i
|
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
⊢ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹ ≤ ∏ i ∈ s, z i ^ w i
|
rw [← inv_inv (∏ i ∈ s, z i ^ w i), inv_le_inv₀ p_pos p_pos₂, ← Finset.prod_inv_distrib]
|
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
⊢ ∏ x ∈ s, (z x ^ w x)⁻¹ ≤ ∏ i ∈ s, (z i)⁻¹ ^ w i
|
34ac124dd1bd80a1
|
IntermediateField.restrictScalars_injective
|
Mathlib/FieldTheory/IntermediateField/Basic.lean
|
theorem restrictScalars_injective :
Function.Injective (restrictScalars K : IntermediateField L' L → IntermediateField K L) :=
fun U V H => ext fun x => by rw [← mem_restrictScalars K, H, mem_restrictScalars]
|
K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field L'
inst✝³ : Algebra K L
inst✝² : Algebra K L'
inst✝¹ : Algebra L' L
inst✝ : IsScalarTower K L' L
U V : IntermediateField L' L
H : restrictScalars K U = restrictScalars K V
x : L
⊢ x ∈ U ↔ x ∈ V
|
rw [← mem_restrictScalars K, H, mem_restrictScalars]
|
no goals
|
3abce742204cc92a
|
frontier_closedBall'
|
Mathlib/Analysis/NormedSpace/Real.lean
|
theorem frontier_closedBall' (x : E) (r : ℝ) : frontier (closedBall x r) = sphere x r
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ frontier (closedBall x r) = sphere x r
|
rw [frontier, closure_closedBall, interior_closedBall' x r, closedBall_diff_ball]
|
no goals
|
a1ac7da3636e01d7
|
not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi
|
Mathlib/Analysis/Calculus/MeanValue.lean
|
theorem not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi (f : ℝ → ℝ) {a : ℝ}
(hf : Tendsto (deriv f) (𝓝[>] a) atTop) : ¬ DifferentiableWithinAt ℝ f (Ioi a) a
|
case h
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
⊢ False
|
have hdiff' : DifferentiableOn ℝ f (Ioc a b) := fun z hz => by
refine DifferentiableWithinAt.mono (t := Ioi a) ?_ Ioc_subset_Ioi_self
have : derivWithin f (Ioi a) z ≠ 0 := ne_of_gt <| by
simp_all only [mem_Ioo, and_imp, mem_Ioc, max_lt_iff]
exact differentiableWithinAt_of_derivWithin_ne_zero this
|
case h
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
⊢ False
|
d40ee29628b6175f
|
Nat.pair_unpair
|
Mathlib/Data/Nat/Pairing.lean
|
theorem pair_unpair (n : ℕ) : pair (unpair n).1 (unpair n).2 = n
|
n : ℕ
s : ℕ := n.sqrt
sm : s * s + (n - s * s) = n
⊢ pair
(if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt) else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).1
(if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt)
else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).2 =
n
|
split_ifs with h
|
case pos
n : ℕ
s : ℕ := n.sqrt
sm : s * s + (n - s * s) = n
h : n - n.sqrt * n.sqrt < n.sqrt
⊢ pair (n - n.sqrt * n.sqrt, n.sqrt).1 (n - n.sqrt * n.sqrt, n.sqrt).2 = n
case neg
n : ℕ
s : ℕ := n.sqrt
sm : s * s + (n - s * s) = n
h : ¬n - n.sqrt * n.sqrt < n.sqrt
⊢ pair (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt).1 (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt).2 = n
|
52459fe1e1a7ec3c
|
ordinaryHypergeometricSeries_apply_eq
|
Mathlib/Analysis/SpecialFunctions/OrdinaryHypergeometric.lean
|
theorem ordinaryHypergeometricSeries_apply_eq (x : 𝔸) (n : ℕ) :
(ordinaryHypergeometricSeries 𝔸 a b c n fun _ => x) =
((n !⁻¹ : 𝕂) * (ascPochhammer 𝕂 n).eval a * (ascPochhammer 𝕂 n).eval b *
((ascPochhammer 𝕂 n).eval c)⁻¹ ) • x ^ n
|
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
a b c : 𝕂
x : 𝔸
n : ℕ
⊢ ((ordinaryHypergeometricSeries 𝔸 a b c n) fun x_1 => x) =
((↑n !)⁻¹ * Polynomial.eval a (ascPochhammer 𝕂 n) * Polynomial.eval b (ascPochhammer 𝕂 n) *
(Polynomial.eval c (ascPochhammer 𝕂 n))⁻¹) •
x ^ n
|
rw [ordinaryHypergeometricSeries, ofScalars_apply_eq]
|
no goals
|
07c851f2e9c93820
|
HasFDerivAt.lim_real
|
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
theorem HasFDerivAt.lim_real (hf : HasFDerivAt f f' x) (v : E) :
Tendsto (fun c : ℝ => c • (f (x + c⁻¹ • v) - f x)) atTop (𝓝 (f' v))
|
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
hf : HasFDerivAt f f' x
v : E
⊢ ∀ (b : ℝ), ∃ i, ∀ (a : ℝ), i ≤ a → b ≤ ‖a‖
|
exact fun b => ⟨b, fun a ha => le_trans ha (le_abs_self _)⟩
|
no goals
|
dec1c6b98fdb342c
|
List.nodup_permutations
|
Mathlib/Data/List/Permutation.lean
|
theorem nodup_permutations (s : List α) (hs : Nodup s) : Nodup s.permutations
|
case cons.right.intro.mk.intro.mk.inr.inr
α : Type u_1
s : List α
x : α
l : List α
h : ∀ a' ∈ l, x ≠ a'
h' : Pairwise (fun x1 x2 => x1 ≠ x2) l
IH : l.permutations'.Nodup
as : List α
ha : as ~ l
bs : List α
hb : bs ~ l
H : as ≠ bs
a : List α
ha' : a ∈ permutations'Aux x as
hb' : a ∈ permutations'Aux x bs
n : ℕ
hn✝ : n < (permutations'Aux x as).length
hn' : insertIdx n x as = a
m : ℕ
hm✝ : m < (permutations'Aux x bs).length
hm' : insertIdx m x bs = a
hl : as.length = bs.length
hn : n ≤ as.length
hm : m ≤ bs.length
hx : (insertIdx n x as)[m] = x
hx' : (insertIdx m x bs)[n] = x
ht : m < n
⊢ x ∈ as
|
rw [← hx, getElem_insertIdx_of_lt ht]
|
case cons.right.intro.mk.intro.mk.inr.inr
α : Type u_1
s : List α
x : α
l : List α
h : ∀ a' ∈ l, x ≠ a'
h' : Pairwise (fun x1 x2 => x1 ≠ x2) l
IH : l.permutations'.Nodup
as : List α
ha : as ~ l
bs : List α
hb : bs ~ l
H : as ≠ bs
a : List α
ha' : a ∈ permutations'Aux x as
hb' : a ∈ permutations'Aux x bs
n : ℕ
hn✝ : n < (permutations'Aux x as).length
hn' : insertIdx n x as = a
m : ℕ
hm✝ : m < (permutations'Aux x bs).length
hm' : insertIdx m x bs = a
hl : as.length = bs.length
hn : n ≤ as.length
hm : m ≤ bs.length
hx : (insertIdx n x as)[m] = x
hx' : (insertIdx m x bs)[n] = x
ht : m < n
⊢ as[m] ∈ as
|
25ee05be5844251a
|
AlgebraicGeometry.Scheme.evaluation_ne_zero_iff_mem_basicOpen
|
Mathlib/AlgebraicGeometry/ResidueField.lean
|
lemma evaluation_ne_zero_iff_mem_basicOpen (x : X) (hx : x ∈ U) (f : Γ(X, U)) :
X.evaluation U x hx f ≠ 0 ↔ x ∈ X.basicOpen f
|
X : Scheme
U : X.Opens
x : ↑↑X.toPresheafedSpace
hx : x ∈ U
f : ↑Γ(X, U)
⊢ (ConcreteCategory.hom (X.evaluation U x hx)) f ≠ 0 ↔ x ∈ X.basicOpen f
|
simp
|
no goals
|
6c376f151db11a69
|
Rel.inter_dom_subset_preimage_image
|
Mathlib/Data/Rel.lean
|
theorem inter_dom_subset_preimage_image (s : Set α) : s ∩ r.dom ⊆ r.preimage (r.image s)
|
α : Type u_1
β : Type u_2
r : Rel α β
s : Set α
x : α
hx : x ∈ s ∧ x ∈ {x | ∃ y, r x y}
⊢ x ∈ r.preimage (r.image s)
|
rcases hx with ⟨hx, ⟨y, rxy⟩⟩
|
case intro.intro
α : Type u_1
β : Type u_2
r : Rel α β
s : Set α
x : α
hx : x ∈ s
y : β
rxy : r x y
⊢ x ∈ r.preimage (r.image s)
|
be39bf1537692879
|
CategoryTheory.GrothendieckTopology.Plus.inj_of_sep
|
Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
|
theorem inj_of_sep (P : Cᵒᵖ ⥤ D)
(hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
(X : C) : Function.Injective ((J.toPlus P).app (op X))
|
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ (I : S.Arrow), (ConcreteCategory.hom (P.map I.f.op)) x = (ConcreteCategory.hom (P.map I.f.op)) y) → x = y
X : C
x y : CD (P.obj (op X))
h : ∃ W h1 h2, (Meq.mk ⊤ x).refine h1 = (Meq.mk ⊤ y).refine h2
⊢ x = y
|
obtain ⟨W, h1, h2, hh⟩ := h
|
case intro.intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ (I : S.Arrow), (ConcreteCategory.hom (P.map I.f.op)) x = (ConcreteCategory.hom (P.map I.f.op)) y) → x = y
X : C
x y : CD (P.obj (op X))
W : J.Cover X
h1 h2 : W ⟶ ⊤
hh : (Meq.mk ⊤ x).refine h1 = (Meq.mk ⊤ y).refine h2
⊢ x = y
|
fc3a81a552d8f5a1
|
ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff
|
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
theorem comp_right_hasFDerivWithinAt_iff {f : F → G} {s : Set F} {x : E} {f' : F →L[𝕜] G} :
HasFDerivWithinAt (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) (iso ⁻¹' s) x ↔
HasFDerivWithinAt f f' s (iso x)
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f'.comp ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
⊢ f = (f ∘ ⇑iso) ∘ ⇑iso.symm
|
rw [Function.comp_assoc, iso.self_comp_symm]
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f'.comp ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
⊢ f = f ∘ _root_.id
|
a55a7f049678bc87
|
Finset.map_subtype_embedding_Icc
|
Mathlib/Order/Interval/Finset/Defs.lean
|
theorem map_subtype_embedding_Icc (hp : ∀ ⦃a b x⦄, a ≤ x → x ≤ b → p a → p b → p x):
(Icc a b).map (Embedding.subtype p) = (Icc a b : Finset α)
|
α : Type u_1
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
⊢ map (Embedding.subtype p) (Finset.subtype p (Icc ↑a ↑b)) = Icc ↑a ↑b
|
refine Finset.subtype_map_of_mem fun x hx => ?_
|
α : Type u_1
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
x : α
hx : x ∈ Icc ↑a ↑b
⊢ p x
|
44baa1ad0b8aff61
|
Complex.continuousAt_cpow_zero_of_re_pos
|
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
theorem continuousAt_cpow_zero_of_re_pos {z : ℂ} (hz : 0 < z.re) :
ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) (0, z)
|
case refine_2.intro
z : ℂ
hz : 0 < z.re
hz₀ : z ≠ 0
C : ℝ
hC : |z.im| < C
⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (𝓝 (0, z)) fun x => rexp (-(x.1.arg * x.2.im))
|
refine ⟨Real.exp (π * C), eventually_map.2 ?_⟩
|
case refine_2.intro
z : ℂ
hz : 0 < z.re
hz₀ : z ≠ 0
C : ℝ
hC : |z.im| < C
⊢ ∀ᶠ (a : ℂ × ℂ) in 𝓝 (0, z), (fun x1 x2 => x1 ≤ x2) (rexp (-(a.1.arg * a.2.im))) (rexp (π * C))
|
b64130900d251c6e
|
DirichletCharacter.even_or_odd
|
Mathlib/NumberTheory/DirichletCharacter/Basic.lean
|
lemma even_or_odd [NoZeroDivisors S] : ψ.Even ∨ ψ.Odd
|
S : Type u_2
inst✝¹ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
inst✝ : NoZeroDivisors S
⊢ ψ (-1) ^ 2 = 1
|
rw [← map_pow _, neg_one_sq, map_one]
|
no goals
|
80d6617dc6d1d7a7
|
Algebra.Generators.map_toComp_ker
|
Mathlib/RingTheory/Generators.lean
|
lemma map_toComp_ker (Q : Generators S T) (P : Generators R S) :
P.ker.map (Q.toComp P).toAlgHom = RingHom.ker (Q.ofComp P).toAlgHom
|
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_2
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
this : DecidableEq (Q.vars →₀ ℕ) := Classical.decEq (Q.vars →₀ ℕ)
x : P.Ring
hx : (algebraMap P.Ring S) x = 0
⊢ (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R P.Ring Q.Ring
|
ext1
|
case hf
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_2
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
this : DecidableEq (Q.vars →₀ ℕ) := Classical.decEq (Q.vars →₀ ℕ)
x : P.Ring
hx : (algebraMap P.Ring S) x = 0
i✝ : P.vars
⊢ ((Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom) (X i✝) = (IsScalarTower.toAlgHom R P.Ring Q.Ring) (X i✝)
|
5cc741e4fe814640
|
Ordinal.blsub_nadd_of_mono
|
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
theorem blsub_nadd_of_mono {f : ∀ c < a ♯ b, Ordinal.{max u v}}
(hf : ∀ {i j} (hi hj), i ≤ j → f i hi ≤ f j hj) :
blsub.{u,v} _ f =
max (blsub.{u, v} a fun a' ha' => f (a' ♯ b) <| nadd_lt_nadd_right ha' b)
(blsub.{u, v} b fun b' hb' => f (a ♯ b') <| nadd_lt_nadd_left hb' a)
|
a b : Ordinal.{u}
f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v}
hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj
⊢ (a.blsub fun a' ha' => f (a' ♯ b) ⋯) ≤ (a ♯ b).blsub f
a b : Ordinal.{u}
f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v}
hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj
⊢ (b.blsub fun b' hb' => f (a ♯ b') ⋯) ≤ (a ♯ b).blsub f
|
all_goals
apply blsub_le_of_brange_subset.{u, u, v}
rintro c ⟨d, hd, rfl⟩
apply mem_brange_self
|
no goals
|
c7fd2bd8268739b9
|
Real.sInf_smul_of_nonpos
|
Mathlib/Data/Real/Pointwise.lean
|
theorem Real.sInf_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sInf (a • s) = a • sSup s
|
case inr.inl
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Module α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sInf 0 = 0
|
exact csInf_singleton 0
|
no goals
|
1b3d0160dc445b73
|
CategoryTheory.simple_of_cosimple
|
Mathlib/CategoryTheory/Simple.lean
|
theorem simple_of_cosimple (X : C) (h : ∀ {Z : C} (f : X ⟶ Z) [Epi f], IsIso f ↔ f ≠ 0) :
Simple X :=
⟨fun {Y} f I => by
classical
fconstructor
· intros
have hx := cokernel.π_of_epi f
by_contra h
subst h
exact (h _).mp (cokernel.π_of_zero _ _) hx
· intro hf
suffices Epi f by exact isIso_of_mono_of_epi _
apply Preadditive.epi_of_cokernel_zero
by_contra h'
exact cokernel_not_iso_of_nonzero hf ((h _).mpr h')⟩
|
case mpr.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
h : ∀ {Z : C} (f : X ⟶ Z) [inst : Epi f], IsIso f ↔ f ≠ 0
Y : C
f : Y ⟶ X
I : Mono f
hf : f ≠ 0
h' : ¬cokernel.π f = 0
⊢ False
|
exact cokernel_not_iso_of_nonzero hf ((h _).mpr h')
|
no goals
|
aea3a25865e3da1b
|
VitaliFamily.ae_eventually_measure_zero_of_singular
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0)
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ⟂ₘ μ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
⊢ μ s = 0
|
obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
|
case intro.intro.intro
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ μ s = 0
|
27b6e7c43d8d9d57
|
IsLocalization.nonZeroDivisors_le_comap
|
Mathlib/RingTheory/Localization/Defs.lean
|
theorem nonZeroDivisors_le_comap [IsLocalization M S] :
nonZeroDivisors R ≤ (nonZeroDivisors S).comap (algebraMap R S)
|
case intro.intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
a : R
ha : a ∈ nonZeroDivisors R
x : R
s c : ↥M
e : ↑c * x * a = 0
⊢ mk' S x s = 0
|
rw [mk'_eq_zero_iff]
|
case intro.intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
a : R
ha : a ∈ nonZeroDivisors R
x : R
s c : ↥M
e : ↑c * x * a = 0
⊢ ∃ m, ↑m * x = 0
|
4f7363ecf10ff4a6
|
ArithmeticFunction.cardFactors_eq_one_iff_prime
|
Mathlib/NumberTheory/ArithmeticFunction.lean
|
theorem cardFactors_eq_one_iff_prime {n : ℕ} : Ω n = 1 ↔ n.Prime
|
case succ
n : ℕ
h : Ω (n + 1) = 1
⊢ Nat.Prime (n + 1)
|
rcases List.length_eq_one.1 h with ⟨x, hx⟩
|
case succ.intro
n : ℕ
h : Ω (n + 1) = 1
x : ℕ
hx : (n + 1).primeFactorsList = [x]
⊢ Nat.Prime (n + 1)
|
f3f8948bdcda971a
|
symmDiff_eq_sup_sdiff_inf
|
Mathlib/Order/SymmDiff.lean
|
theorem symmDiff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b)
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a ∆ b = (a ⊔ b) \ (a ⊓ b)
|
simp [sup_sdiff, symmDiff]
|
no goals
|
b140439bd29480cc
|
AddCircle.isAddFundamentalDomain_of_ae_ball
|
Mathlib/MeasureTheory/Group/AddCircle.lean
|
theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
IsAddFundamentalDomain (AddSubgroup.zmultiples u) I
|
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ IsAddFundamentalDomain (↥G) I volume
|
refine IsAddFundamentalDomain.mk_of_measure_univ_le ?_ ?_ ?_ ?_
|
case refine_1
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ NullMeasurableSet I volume
case refine_2
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ ∀ (g : ↥G), g ≠ 0 → AEDisjoint volume (g +ᵥ I) I
case refine_3
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ ∀ (g : ↥G), QuasiMeasurePreserving (fun x => g +ᵥ x) volume volume
case refine_4
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ volume univ ≤ ∑' (g : ↥G), volume (g +ᵥ I)
|
3261dbc8487460b6
|
cfcₙ_apply_zero
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
|
@[simp] lemma cfcₙ_apply_zero {f : R → R} : cfcₙ f (0 : A) = 0
|
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
hf0 : f 0 = 0
⊢ cfcₙ f 0 = cfcₙ 0 0
|
apply cfcₙ_congr
|
case pos.hfg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
hf0 : f 0 = 0
⊢ Set.EqOn f 0 (σₙ R 0)
|
11ba940450e9b321
|
AlgebraicGeometry.IsLocalAtTarget.mk'
|
Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
|
/--
`P` is local at the target if
1. `P` respects isomorphisms.
2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`.
3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`.
-/
protected lemma mk' {P : MorphismProperty Scheme} [P.RespectsIso]
(restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U))
(of_sSup_eq_top :
∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ →
(∀ i, P (f ∣_ U i)) → P f) :
IsLocalAtTarget P
|
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top :
∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
⊢ IsLocalAtTarget P
|
refine ⟨inferInstance, fun {X Y} f 𝒰 ↦ ⟨?_, fun H ↦ of_sSup_eq_top f _ 𝒰.iSup_opensRange ?_⟩⟩
|
case refine_1
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top :
∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
⊢ P f → ∀ (i : 𝒰.1), P (Scheme.Cover.pullbackHom 𝒰 f i)
case refine_2
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top :
∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
H : ∀ (i : 𝒰.1), P (Scheme.Cover.pullbackHom 𝒰 f i)
⊢ ∀ (i : 𝒰.J), P (f ∣_ Scheme.Hom.opensRange (𝒰.map i))
|
30afa796e957c3d7
|
Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top
|
Mathlib/GroupTheory/Abelianization.lean
|
theorem Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top {N : Subgroup G} [N.Normal]
{H : Subgroup G} (hHN : N ⊔ H = ⊤) (hH : Subgroup.IsCommutative H) :
_root_.commutator G ≤ N
|
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : H.IsCommutative
⊢ _root_.commutator G ≤ N
|
rw [← quotient_commutative_iff_commutator_le]
|
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : H.IsCommutative
⊢ Std.Commutative fun x1 x2 => x1 * x2
|
4f1c7225fbfaf09a
|
Function.extend_smul
|
Mathlib/Algebra/Group/Action/Pi.lean
|
@[to_additive]
lemma extend_smul {M α β : Type*} [SMul M β] (r : M) (f : ι → α) (g : ι → β) (e : α → β) :
extend f (r • g) (r • e) = r • extend f g e
|
case h
ι : Type u_1
M : Type u_7
α : Type u_8
β : Type u_9
inst✝ : SMul M β
r : M
f : ι → α
g : ι → β
e : α → β
x : α
⊢ extend f (r • g) (r • e) x = (r • extend f g e) x
|
classical
simp only [extend_def, Pi.smul_apply]
split_ifs <;> rfl
|
no goals
|
6ccb80ef1a23d2f3
|
LinearMap.IsProj.eq_conj_prod_map'
|
Mathlib/LinearAlgebra/Projection.lean
|
theorem eq_conj_prod_map' {f : E →ₗ[R] E} (h : IsProj p f) :
f = (p.prodEquivOfIsCompl (ker f) h.isCompl).toLinearMap ∘ₗ
prodMap id 0 ∘ₗ (p.prodEquivOfIsCompl (ker f) h.isCompl).symm.toLinearMap
|
case hr.h
R : Type u_1
inst✝² : Ring R
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module R E
p : Submodule R E
f : E →ₗ[R] E
h : IsProj p f
x : ↥(ker f)
⊢ ((f ∘ₗ ↑(p.prodEquivOfIsCompl (ker f) ⋯)) ∘ₗ inr R ↥p ↥(ker f)) x =
((↑(p.prodEquivOfIsCompl (ker f) ⋯) ∘ₗ id.prodMap 0) ∘ₗ inr R ↥p ↥(ker f)) x
|
simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inr, coprod_apply, _root_.map_zero,
coe_subtype, zero_add, map_coe_ker, prodMap_apply, zero_apply, add_zero]
|
no goals
|
42e67a6dd540c5df
|
CategoryTheory.Functor.IsEventuallyConstantTo.coneπApp_eq
|
Mathlib/CategoryTheory/Limits/Constructions/EventuallyConstant.lean
|
lemma coneπApp_eq (j j' : J) (α : j' ⟶ i₀) (β : j' ⟶ j) :
h.coneπApp j = (h.isoMap α ⟨𝟙 _⟩).inv ≫ F.map β
|
case intro.intro.intro.intro
J : Type u_1
C : Type u_2
inst✝² : Category.{u_3, u_1} J
inst✝¹ : Category.{u_4, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝ : IsCofiltered J
j j' : J
α : j' ⟶ i₀
β : j' ⟶ j
s : J
γ : s ⟶ IsCofiltered.min i₀ j
δ : s ⟶ j'
h₁ : γ ≫ minToRight i₀ j = δ ≫ β
h₂ : γ ≫ minToLeft i₀ j = δ ≫ α
⊢ h.coneπApp j = (h.isoMap α ⋯).inv ≫ F.map β
|
dsimp [coneπApp]
|
case intro.intro.intro.intro
J : Type u_1
C : Type u_2
inst✝² : Category.{u_3, u_1} J
inst✝¹ : Category.{u_4, u_2} C
F : J ⥤ C
i₀ : J
h : F.IsEventuallyConstantTo i₀
inst✝ : IsCofiltered J
j j' : J
α : j' ⟶ i₀
β : j' ⟶ j
s : J
γ : s ⟶ IsCofiltered.min i₀ j
δ : s ⟶ j'
h₁ : γ ≫ minToRight i₀ j = δ ≫ β
h₂ : γ ≫ minToLeft i₀ j = δ ≫ α
⊢ (h.isoMap (minToLeft i₀ j) ⋯).inv ≫ F.map (minToRight i₀ j) = (h.isoMap α ⋯).inv ≫ F.map β
|
9049c30edd61763f
|
DirichletCharacter.unit_norm_eq_one
|
Mathlib/NumberTheory/DirichletCharacter/Bounds.lean
|
/-- The value at a unit of a Dirichlet character with target a normed field has norm `1`. -/
@[simp] lemma unit_norm_eq_one (a : (ZMod n)ˣ) : ‖χ a‖ = 1
|
F : Type u_1
inst✝ : NormedField F
n : ℕ
χ : DirichletCharacter F n
a : (ZMod n)ˣ
⊢ ‖χ ↑a‖ = 1
|
refine (pow_eq_one_iff_of_nonneg (norm_nonneg _) (Nat.card_pos (α := (ZMod n)ˣ)).ne').mp ?_
|
F : Type u_1
inst✝ : NormedField F
n : ℕ
χ : DirichletCharacter F n
a : (ZMod n)ˣ
⊢ ‖χ ↑a‖ ^ Nat.card (ZMod n)ˣ = 1
|
9709107840093d45
|
CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd
|
Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean
|
theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0
|
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst
|
simp
|
no goals
|
70e51031cc3995fa
|
NormedAddCommGroup.cauchy_series_of_le_geometric''
|
Mathlib/Analysis/SpecificLimits/Normed.lean
|
theorem NormedAddCommGroup.cauchy_series_of_le_geometric'' {C : ℝ} {u : ℕ → α} {N : ℕ} {r : ℝ}
(hr₀ : 0 < r) (hr₁ : r < 1) (h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n) :
CauchySeq fun n ↦ ∑ k ∈ range (n + 1), u k
|
α : Type u_1
inst✝ : SeminormedAddCommGroup α
C : ℝ
u : ℕ → α
N : ℕ
r : ℝ
hr₀ : 0 < r
hr₁ : r < 1
h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n
v : ℕ → α := fun n => if n < N then 0 else u n
hC : 0 ≤ C
this : ∀ n ≥ N, u n = v n
⊢ ℝ
|
exact C
|
no goals
|
eb1c22e866e16cfa
|
Ideal.finiteHeight_iff_lt
|
Mathlib/RingTheory/Ideal/Height.lean
|
lemma Ideal.finiteHeight_iff_lt {I : Ideal R} :
Ideal.FiniteHeight I ↔ I = ⊤ ∨ I.height < ⊤
|
case mpr.eq_top_or_height_ne_top.inr
R : Type u_1
inst✝ : CommRing R
I : Ideal R
h : I.height < ⊤
⊢ I = ⊤ ∨ I.height ≠ ⊤
|
exact Or.inr (ne_top_of_lt h)
|
no goals
|
0dedda41cd001cf1
|
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0
|
case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
fun n =>
MeasureTheory.SignedMeasure.findExistsOneDivLT s
(i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
hnn : ¬s ≤[A] 0
⊢ False
|
rw [restrict_le_restrict_iff _ _ A_meas] at hnn
|
case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
fun n =>
MeasureTheory.SignedMeasure.findExistsOneDivLT s
(i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
hnn : ¬∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ A → ↑s j ≤ ↑0 j
⊢ False
|
37ee21864bec50c9
|
MeasureTheory.Measure.lintegral_join
|
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
theorem lintegral_join {m : Measure (Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ x, f x ∂join m = ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m
|
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : α), f x ∂m.join = ∫⁻ (μ : Measure α), ∫⁻ (x : α), f x ∂μ ∂m
|
simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral,
join_apply (SimpleFunc.measurableSet_preimage _ _)]
|
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hf : Measurable f
⊢ ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * ∫⁻ (μ : Measure α), μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m =
∫⁻ (μ : Measure α), ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m
|
fa9f09aa020ca186
|
MeasurableSet.iUnion_of_monotone_of_frequently
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
theorem iUnion_of_monotone_of_frequently
{ι : Type*} [Preorder ι] [(atTop : Filter ι).IsCountablyGenerated] {s : ι → Set α}
(hsm : Monotone s) (hs : ∃ᶠ i in atTop, MeasurableSet (s i)) : MeasurableSet (⋃ i, s i)
|
α : Type u_1
inst✝² : MeasurableSpace α
ι : Type u_6
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hsm : Monotone s
hs : ∃ᶠ (i : ι) in atTop, MeasurableSet (s i)
⊢ MeasurableSet (⋃ i, s i)
|
rcases exists_seq_forall_of_frequently hs with ⟨x, hx, hxm⟩
|
case intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
ι : Type u_6
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hsm : Monotone s
hs : ∃ᶠ (i : ι) in atTop, MeasurableSet (s i)
x : ℕ → ι
hx : Tendsto x atTop atTop
hxm : ∀ (n : ℕ), MeasurableSet (s (x n))
⊢ MeasurableSet (⋃ i, s i)
|
eaf9c9d5846defea
|
ProbabilityTheory.Kernel.densityProcess_fst_univ
|
Mathlib/Probability/Kernel/Disintegration/Density.lean
|
lemma densityProcess_fst_univ [IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) :
densityProcess κ (fst κ) n a x univ
= if fst κ a (countablePartitionSet n x) = 0 then 0 else 1
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
h : ¬(κ.fst a) (countablePartitionSet n x) = 0
this : countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ a) {p | p.1 ∈ countablePartitionSet n x}).toReal = 1
|
rw [this, ENNReal.div_self]
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
h : ¬(κ.fst a) (countablePartitionSet n x) = 0
this : countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}
⊢ ENNReal.toReal 1 = 1
case neg.h0
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
h : ¬(κ.fst a) (countablePartitionSet n x) = 0
this : countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}
⊢ (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ 0
case neg.hI
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
h : ¬(κ.fst a) (countablePartitionSet n x) = 0
this : countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}
⊢ (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ ⊤
|
148c86bd150ec8e7
|
Set.image2_swap
|
Mathlib/Data/Set/NAry.lean
|
theorem image2_swap (s : Set α) (t : Set β) : image2 f s t = image2 (fun a b => f b a) t s
|
case h
α : Type u_1
β : Type u_3
γ : Type u_5
f : α → β → γ
s : Set α
t : Set β
x✝ : γ
⊢ x✝ ∈ image2 f s t ↔ x✝ ∈ image2 (fun a b => f b a) t s
|
constructor <;> rintro ⟨a, ha, b, hb, rfl⟩ <;> exact ⟨b, hb, a, ha, rfl⟩
|
no goals
|
37c1bb72227ac3dc
|
lowerCentralSeries_antitone
|
Mathlib/GroupTheory/Nilpotent.lean
|
theorem lowerCentralSeries_antitone : Antitone (lowerCentralSeries G)
|
G : Type u_1
inst✝ : Group G
n : ℕ
x : G
hx : x ∈ closure {x | ∃ p ∈ lowerCentralSeries G n, ∃ q, p * q * p⁻¹ * q⁻¹ = x}
⊢ ∀ x ∈ {x | ∃ p ∈ lowerCentralSeries G n, ∃ q, p * q * p⁻¹ * q⁻¹ = x}, x ∈ lowerCentralSeries G n
|
rintro y ⟨z, hz, a, ha⟩
|
case intro.intro.intro
G : Type u_1
inst✝ : Group G
n : ℕ
x : G
hx : x ∈ closure {x | ∃ p ∈ lowerCentralSeries G n, ∃ q, p * q * p⁻¹ * q⁻¹ = x}
y z : G
hz : z ∈ lowerCentralSeries G n
a : G
ha : z * a * z⁻¹ * a⁻¹ = y
⊢ y ∈ lowerCentralSeries G n
|
db060d010fc2a818
|
Mathlib.Tactic.Monoidal.evalWhiskerRight_cons_whisker
|
Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.lean
|
theorem evalWhiskerRight_cons_whisker {f g h i j k : C}
{α : g ≅ f ⊗ h} {η : h ⟶ i} {ηs : f ⊗ i ⟶ j}
{η₁ : h ⊗ k ⟶ i ⊗ k} {η₂ : f ⊗ (h ⊗ k) ⟶ f ⊗ (i ⊗ k)} {ηs₁ : (f ⊗ i) ⊗ k ⟶ j ⊗ k}
{ηs₂ : f ⊗ (i ⊗ k) ⟶ j ⊗ k} {η₃ : f ⊗ (h ⊗ k) ⟶ j ⊗ k} {η₄ : (f ⊗ h) ⊗ k ⟶ j ⊗ k}
{η₅ : g ⊗ k ⟶ j ⊗ k}
(e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ k = η₁) (e_η₂ : f ◁ η₁ = η₂)
(e_ηs₁ : ηs ▷ k = ηs₁) (e_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
(e_η₃ : η₂ ≫ ηs₂ = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄)
(e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅) :
(α.hom ≫ (f ◁ η) ≫ ηs) ▷ k = η₅
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
f g h i j k : C
α : g ≅ f ⊗ h
η : h ⟶ i
ηs : f ⊗ i ⟶ j
η₁ : h ⊗ k ⟶ i ⊗ k
η₂ : f ⊗ h ⊗ k ⟶ f ⊗ i ⊗ k
ηs₁ : (f ⊗ i) ⊗ k ⟶ j ⊗ k
ηs₂ : f ⊗ i ⊗ k ⟶ j ⊗ k
η₃ : f ⊗ h ⊗ k ⟶ j ⊗ k
η₄ : (f ⊗ h) ⊗ k ⟶ j ⊗ k
η₅ : g ⊗ k ⟶ j ⊗ k
e_η₁ : ((Iso.refl h).hom ≫ η ≫ (Iso.refl i).hom) ▷ k = η₁
e_η₂ : f ◁ η₁ = η₂
e_ηs₁ : ηs ▷ k = ηs₁
e_ηs₂ : (α_ f i k).inv ≫ ηs₁ = ηs₂
e_η₃ : η₂ ≫ ηs₂ = η₃
e_η₄ : (α_ f h k).hom ≫ η₃ = η₄
e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅
⊢ (α.hom ≫ f ◁ η ≫ ηs) ▷ k = η₅
|
simp at e_η₁ e_η₅
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
f g h i j k : C
α : g ≅ f ⊗ h
η : h ⟶ i
ηs : f ⊗ i ⟶ j
η₁ : h ⊗ k ⟶ i ⊗ k
η₂ : f ⊗ h ⊗ k ⟶ f ⊗ i ⊗ k
ηs₁ : (f ⊗ i) ⊗ k ⟶ j ⊗ k
ηs₂ : f ⊗ i ⊗ k ⟶ j ⊗ k
η₃ : f ⊗ h ⊗ k ⟶ j ⊗ k
η₄ : (f ⊗ h) ⊗ k ⟶ j ⊗ k
η₅ : g ⊗ k ⟶ j ⊗ k
e_η₂ : f ◁ η₁ = η₂
e_ηs₁ : ηs ▷ k = ηs₁
e_ηs₂ : (α_ f i k).inv ≫ ηs₁ = ηs₂
e_η₃ : η₂ ≫ ηs₂ = η₃
e_η₄ : (α_ f h k).hom ≫ η₃ = η₄
e_η₅ : α.hom ▷ k ≫ η₄ = η₅
e_η₁ : η ▷ k = η₁
⊢ (α.hom ≫ f ◁ η ≫ ηs) ▷ k = η₅
|
07d7ee37bcc624a0
|
Subgroup.mem_sup
|
Mathlib/Algebra/Group/Subgroup/Lattice.lean
|
theorem mem_sup : x ∈ s ⊔ t ↔ ∃ y ∈ s, ∃ z ∈ t, y * z = x :=
⟨fun h => by
rw [sup_eq_closure] at h
refine Subgroup.closure_induction ?_ ?_ ?_ ?_ h
· rintro y (h | h)
· exact ⟨y, h, 1, t.one_mem, by simp⟩
· exact ⟨1, s.one_mem, y, h, by simp⟩
· exact ⟨1, s.one_mem, 1, ⟨t.one_mem, mul_one 1⟩⟩
· rintro _ _ _ _ ⟨y₁, hy₁, z₁, hz₁, rfl⟩ ⟨y₂, hy₂, z₂, hz₂, rfl⟩
exact ⟨_, mul_mem hy₁ hy₂, _, mul_mem hz₁ hz₂, by simp [mul_assoc, mul_left_comm]⟩
· rintro _ _ ⟨y, hy, z, hz, rfl⟩
exact ⟨_, inv_mem hy, _, inv_mem hz, mul_comm z y ▸ (mul_inv_rev z y).symm⟩, by
rintro ⟨y, hy, z, hz, rfl⟩; exact mul_mem_sup hy hz⟩
|
case refine_1.inr
C : Type u_2
inst✝ : CommGroup C
s t : Subgroup C
x : C
h✝ : x ∈ closure (↑s ∪ ↑t)
y : C
h : y ∈ ↑t
⊢ ∃ y_1 ∈ s, ∃ z ∈ t, y_1 * z = y
|
exact ⟨1, s.one_mem, y, h, by simp⟩
|
no goals
|
3e44d12937d16db9
|
Std.DHashMap.Internal.Raw₀.wfImp_insert
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean
|
theorem wfImp_insert [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α] {m : Raw₀ α β}
(h : Raw.WFImp m.1) {a : α} {b : β a} : Raw.WFImp (m.insert a b).1
|
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ Raw.WFImp (m.insert a b).val
|
rw [insert_eq_insertₘ]
|
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ Raw.WFImp (m.insertₘ a b).val
|
9846018d4fa634d0
|
mellin_cpow_smul
|
Mathlib/Analysis/MellinTransform.lean
|
theorem mellin_cpow_smul (f : ℝ → E) (s a : ℂ) :
mellin (fun t => (t : ℂ) ^ a • f t) s = mellin f (s + a)
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s a : ℂ
⊢ mellin (fun t => ↑t ^ a • f t) s = mellin f (s + a)
|
refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s a : ℂ
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • (fun t => ↑t ^ a • f t) t = ↑t ^ (s + a - 1) • f t
|
c3ddd64ee1f537d5
|
InnerProductSpace.Core.inner_self_ofReal_re
|
Mathlib/Analysis/InnerProductSpace/Defs.lean
|
theorem inner_self_ofReal_re (x : F) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫
|
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x : F
⊢ ↑(re ⟪x, x⟫_𝕜) = ⟪x, x⟫_𝕜
|
norm_num [ext_iff, inner_self_im]
|
no goals
|
0487fb94f5215f92
|
IsSemiprimaryRing.finite_of_isNoetherian_or_isArtinian
|
Mathlib/RingTheory/HopkinsLevitzki.lean
|
theorem finite_of_isNoetherian_or_isArtinian :
IsNoetherian R M ∨ IsArtinian R M → Module.Finite R₀ M
|
case refine_2
R₀ : Type u_1
R : Type u_2
M✝ : Type u
inst✝⁹ : Ring R₀
inst✝⁸ : Ring R
inst✝⁷ : Module R₀ R
inst✝⁶ : AddCommGroup M✝
inst✝⁵ : Module R₀ M✝
inst✝⁴ : Module R M✝
inst✝³ : IsScalarTower R₀ R M✝
inst✝² : IsSemiprimaryRing R
inst✝¹ : IsScalarTower R₀ R R
inst✝ : Module.Finite R₀ (R ⧸ Ring.jacobson R)
M : Type u
x✝³ : AddCommGroup M
x✝² : Module R₀ M
x✝¹ : Module R M
x✝ : IsScalarTower R₀ R M
hs :
(fun M [AddCommGroup M] [Module R₀ M] [Module R M] => IsNoetherian R M ∨ IsArtinian R M → Module.Finite R₀ M)
↥(Ring.jacobson R • ⊤)
hq :
(fun M [AddCommGroup M] [Module R₀ M] [Module R M] => IsNoetherian R M ∨ IsArtinian R M → Module.Finite R₀ M)
(M ⧸ Ring.jacobson R • ⊤)
h : IsNoetherian R M ∨ IsArtinian R M
N : Submodule R₀ M := Submodule.restrictScalars R₀ (Ring.jacobson R • ⊤)
this✝ : Module.Finite R₀ ↥N
this : Module.Finite R₀ (M ⧸ N)
⊢ Module.Finite R₀ M
|
exact .of_submodule_quotient N
|
no goals
|
19f8c33d4c3ff1de
|
FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
(hI : IsNoetherian R₁ I) :
IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K)
|
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
⊢ (↑J).FG
|
obtain ⟨s, hs⟩ := hI _ hJ
|
case neg.intro
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : R₁
I : FractionalIdeal R₁⁰ K
hI : ∀ J ≤ I, (↑J).FG
hx : ¬x = 0
h_gx : (algebraMap R₁ K) x ≠ 0
h_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≠ 0
J : FractionalIdeal R₁⁰ K
hJ : J * spanSingleton R₁⁰ ((algebraMap R₁ K) x) ≤ I
s : Finset K
hs : span R₁ ↑s = ↑(J * spanSingleton R₁⁰ ((algebraMap R₁ K) x))
⊢ (↑J).FG
|
4cee4bbb43b9cbc0
|
Rel.abs_edgeDensity_sub_edgeDensity_le_two_mul_sub_sq
|
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
theorem abs_edgeDensity_sub_edgeDensity_le_two_mul_sub_sq (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁)
(hδ₀ : 0 ≤ δ) (hδ₁ : δ < 1) (hs₂ : (1 - δ) * #s₁ ≤ #s₂)
(ht₂ : (1 - δ) * #t₁ ≤ #t₂) :
|(edgeDensity r s₂ t₂ : 𝕜) - edgeDensity r s₁ t₁| ≤ 2 * δ - δ ^ 2
|
case inr.inr
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝¹ : LinearOrderedField 𝕜
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
δ : 𝕜
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
hs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs₂' : s₂.Nonempty
ht₂' : t₂.Nonempty
⊢ |↑(edgeDensity r s₂ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2
|
have hr : 2 * δ - δ ^ 2 = 1 - (1 - δ) * (1 - δ) := by ring
|
case inr.inr
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝¹ : LinearOrderedField 𝕜
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
δ : 𝕜
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
hs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs₂' : s₂.Nonempty
ht₂' : t₂.Nonempty
hr : 2 * δ - δ ^ 2 = 1 - (1 - δ) * (1 - δ)
⊢ |↑(edgeDensity r s₂ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2
|
80e245cc464aeb05
|
MvPolynomial.sum_weightedHomogeneousComponent
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
theorem sum_weightedHomogeneousComponent :
(finsum fun m => weightedHomogeneousComponent w m φ) = φ
|
case a.h₀
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
d : σ →₀ ℕ
m : M
a✝ : m ∈ ⋯.toFinset
hm' : m ≠ (weight w) d
⊢ (if (weight w) d = m then coeff d φ else 0) = 0
|
rw [if_neg hm'.symm]
|
no goals
|
109812e04e99df7b
|
IsDedekindDomain.HeightOneSpectrum.intValuation.map_add_le_max'
|
Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
|
theorem intValuation.map_add_le_max' (x y : R) :
v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors ⊓
(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors
⊢ nmin ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors
|
exact min_le_left _ _
|
no goals
|
f44842bea4e820a0
|
NonUnitalAlgebra.adjoin_induction₂
|
Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
|
theorem adjoin_induction₂ {s : Set A} {p : ∀ x y, x ∈ adjoin R s → y ∈ adjoin R s → Prop}
(mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_adjoin R hx) (subset_adjoin R hy))
(zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
(add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
(add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
(mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
(mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz))
(smul_left : ∀ r x y hx hy, p x y hx hy → p (r • x) y (SMulMemClass.smul_mem r hx) hy)
(smul_right : ∀ r x y hx hy, p x y hx hy → p x (r • y) hx (SMulMemClass.smul_mem r hy))
{x y : A} (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) :
p x y hx hy
|
case mem.add
R : Type u
A : Type v
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
s : Set A
p : (x y : A) → x ∈ adjoin R s → y ∈ adjoin R s → Prop
mem_mem : ∀ (x y : A) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : A) (hx : x ∈ adjoin R s), p 0 x ⋯ hx
zero_right : ∀ (x : A) (hx : x ∈ adjoin R s), p x 0 hx ⋯
add_left :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
smul_left : ∀ (r : R) (x y : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s), p x y hx hy → p (r • x) y ⋯ hy
smul_right : ∀ (r : R) (x y : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s), p x y hx hy → p x (r • y) hx ⋯
x y z : A
hz : z ∈ s
x✝ y✝ : A
hx✝ : x✝ ∈ adjoin R s
hy✝ : y✝ ∈ adjoin R s
h₁ : p x✝ z hx✝ ⋯
h₂ : p y✝ z hy✝ ⋯
⊢ p (x✝ + y✝) z ⋯ ⋯
|
exact add_left _ _ _ _ _ _ h₁ h₂
|
no goals
|
65484542bb46a13c
|
HomologicalComplex.isIso_pOpcycles
|
Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
|
lemma isIso_pOpcycles : IsIso (K.pOpcycles j)
|
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
hi : c.prev j = i
h : K.d i j = 0
inst✝ : K.HasHomology j
⊢ IsIso (K.pOpcycles j)
|
obtain rfl := hi
|
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
j : ι
inst✝ : K.HasHomology j
h : K.d (c.prev j) j = 0
⊢ IsIso (K.pOpcycles j)
|
4467dd650f7a1532
|
CategoryTheory.IsVanKampenColimit.of_iso
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
theorem IsVanKampenColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (H : IsVanKampenColimit c)
(e : c ≅ c') : IsVanKampenColimit c'
|
case h
J : Type v'
inst✝¹ : Category.{u', v'} J
C : Type u
inst✝ : Category.{v, u} C
F : J ⥤ C
c c' : Cocone F
H : IsVanKampenColimit c
e : c ≅ c'
F' : J ⥤ C
c'' : Cocone F'
α : F' ⟶ F
f : c''.pt ⟶ c'.pt
h : α ≫ c'.ι = c''.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
this : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι
j : J
⊢ IsPullback (c''.ι.app j) (α.app j) (f ≫ e.inv.hom) (c.ι.app j) ↔ IsPullback (c''.ι.app j) (α.app j) f (c'.ι.app j)
|
conv_lhs => rw [← Category.comp_id (α.app j)]
|
case h
J : Type v'
inst✝¹ : Category.{u', v'} J
C : Type u
inst✝ : Category.{v, u} C
F : J ⥤ C
c c' : Cocone F
H : IsVanKampenColimit c
e : c ≅ c'
F' : J ⥤ C
c'' : Cocone F'
α : F' ⟶ F
f : c''.pt ⟶ c'.pt
h : α ≫ c'.ι = c''.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
this : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι
j : J
⊢ IsPullback (c''.ι.app j) (α.app j ≫ 𝟙 (F.obj j)) (f ≫ e.inv.hom) (c.ι.app j) ↔
IsPullback (c''.ι.app j) (α.app j) f (c'.ι.app j)
|
256639f190b732be
|
Option.lawfulBEq_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean
|
theorem lawfulBEq_iff : LawfulBEq (Option α) ↔ LawfulBEq α
|
case mpr.eq_of_beq
α : Type u_1
inst✝ : BEq α
h : LawfulBEq α
⊢ ∀ {a b : Option α}, (a == b) = true → a = b
|
intro a b h
|
case mpr.eq_of_beq
α : Type u_1
inst✝ : BEq α
h✝ : LawfulBEq α
a b : Option α
h : (a == b) = true
⊢ a = b
|
84ae32359a69ec10
|
AlgebraicGeometry.morphismRestrict_app
|
Mathlib/AlgebraicGeometry/Restrict.lean
|
theorem morphismRestrict_app {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V : U.toScheme.Opens) :
(f ∣_ U).app V = f.app (U.ι ''ᵁ V) ≫
X.presheaf.map (eqToHom (image_morphismRestrict_preimage f U V)).op
|
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
V : (↑U).Opens
this :
Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =
(Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫
X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op
e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V
e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V
⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op
|
simp only [Opens.toScheme_presheaf_obj, Hom.app_eq_appLE, eqToHom_op, Hom.appLE_map]
|
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
V : (↑U).Opens
this :
Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =
(Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫
X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op
e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V
e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V
⊢ Hom.appLE (f ∣_ U) V ((f ∣_ U) ⁻¹ᵁ V) ⋯ = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯
|
9d85b88464a65d5b
|
Affine.Triangle.mem_circumsphere_of_two_zsmul_oangle_eq
|
Mathlib/Geometry/Euclidean/Angle/Sphere.lean
|
theorem mem_circumsphere_of_two_zsmul_oangle_eq {t : Triangle ℝ P} {p : P} {i₁ i₂ i₃ : Fin 3}
(h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
(h : (2 : ℤ) • ∡ (t.points i₁) p (t.points i₃) =
(2 : ℤ) • ∡ (t.points i₁) (t.points i₂) (t.points i₃)) : p ∈ t.circumsphere
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) = 2 • ∡ (t.points i₁) (t.points i₂) (t.points i₃)
t'p : Fin 3 → P := Function.update t.points i₂ p
h₁ : t'p i₁ = t.points i₁
h₂ : t'p i₂ = p
h₃ : t'p i₃ = t.points i₃
ha : AffineIndependent ℝ t'p
t' : Triangle ℝ P := { points := t'p, independent := ha }
⊢ p ∈ Simplex.circumsphere t
|
have h₁' : t'.points i₁ = t.points i₁ := h₁
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
p : P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h : 2 • ∡ (t.points i₁) p (t.points i₃) = 2 • ∡ (t.points i₁) (t.points i₂) (t.points i₃)
t'p : Fin 3 → P := Function.update t.points i₂ p
h₁ : t'p i₁ = t.points i₁
h₂ : t'p i₂ = p
h₃ : t'p i₃ = t.points i₃
ha : AffineIndependent ℝ t'p
t' : Triangle ℝ P := { points := t'p, independent := ha }
h₁' : t'.points i₁ = t.points i₁
⊢ p ∈ Simplex.circumsphere t
|
c3015bbdfd99892f
|
QuasispectrumRestricts.algebraMap_image
|
Mathlib/Algebra/Algebra/Quasispectrum.lean
|
theorem algebraMap_image (h : QuasispectrumRestricts a f) :
algebraMap R S '' quasispectrum R a = quasispectrum S a
|
case refine_1
R : Type u_3
S : Type u_4
A : Type u_5
inst✝⁸ : Semifield R
inst✝⁷ : Field S
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Algebra R S
a : A
f : S → R
inst✝² : IsScalarTower S A A
inst✝¹ : SMulCommClass S A A
inst✝ : IsScalarTower R S A
h : QuasispectrumRestricts a f
⊢ ⇑(algebraMap R S) '' quasispectrum R a ⊆ quasispectrum S a
|
simpa only [quasispectrum.preimage_algebraMap] using
(quasispectrum S a).image_preimage_subset (algebraMap R S)
|
no goals
|
fbd832ca15e540c9
|
CategoryTheory.Limits.isFiltered_costructuredArrow_yoneda_of_preservesFiniteLimits
|
Mathlib/CategoryTheory/Limits/Preserves/Presheaf.lean
|
theorem isFiltered_costructuredArrow_yoneda_of_preservesFiniteLimits
[PreservesFiniteLimits A] : IsFiltered (CostructuredArrow yoneda A)
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteColimits C
A : Cᵒᵖ ⥤ Type v
inst✝ : PreservesFiniteLimits A
⊢ IsFiltered (CostructuredArrow yoneda A)
|
suffices IsCofiltered A.Elements from
IsFiltered.of_equivalence (CategoryOfElements.costructuredArrowYonedaEquivalence _)
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteColimits C
A : Cᵒᵖ ⥤ Type v
inst✝ : PreservesFiniteLimits A
⊢ IsCofiltered A.Elements
|
6a5d20ed7fadbfba
|
List.sublist_eq_map_getElem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Pairwise.lean
|
theorem sublist_eq_map_getElem {l l' : List α} (h : l' <+ l) : ∃ is : List (Fin l.length),
l' = is.map (l[·]) ∧ is.Pairwise (· < ·)
|
case slnil
α : Type u_1
l l' : List α
⊢ ∃ is, [] = map (fun x => [][x]) is ∧ Pairwise (fun x1 x2 => x1 < x2) is
|
exact ⟨[], by simp⟩
|
no goals
|
1127e51537a4a512
|
ContinuousLinearMap.opNorm_le_of_shell'
|
Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean
|
theorem opNorm_le_of_shell' {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜}
(hc : ‖c‖ < 1) (hf : ∀ x, ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C
|
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
ε C : ℝ
ε_pos : 0 < ε
hC : 0 ≤ C
c : 𝕜
hc : 1 < ‖c⁻¹‖
hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
h0 : ¬c = 0
⊢ ∀ (x : E), ε / ‖c⁻¹‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
|
rwa [norm_inv, div_eq_mul_inv, inv_inv]
|
no goals
|
6fc0364de4dbd55b
|
AlgebraicIndependent.adjoin_of_disjoint
|
Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean
|
theorem adjoin_of_disjoint {s t : Set ι} (h : Disjoint s t) :
AlgebraicIndependent (adjoin R (x '' s)) fun i : t ↦ x i
|
ι : Type u_1
R : Type u_2
A : Type u_3
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
s t : Set ι
h : Disjoint s t
e : MvPolynomial (↑t ⊕ ↑s) R ≃ₐ[R] MvPolynomial ↑t ↥(adjoin R (range (x ∘ Subtype.val))) :=
(sumAlgEquiv R ↑t ↑s).trans (mapAlgEquiv ↑t ⋯.aevalEquiv)
this :
(AlgHom.restrictScalars R (aeval fun i => x ↑i)).comp ↑e = (aeval x).comp (rename (Sum.elim Subtype.val Subtype.val))
x✝ :
∀ {R : Type u_2} {S₁ S₂ : Type u_3} {σ : Type u_1} [inst : CommSemiring S₂] [inst_1 : SMul R S₁]
[inst_2 : SMulZeroClass R S₂] [inst_3 : SMulZeroClass S₁ S₂] [inst_4 : IsScalarTower R S₁ S₂],
IsScalarTower R S₁ (MvPolynomial σ S₂)
⊢ Injective ⇑((AlgHom.restrictScalars R (aeval fun i => x ↑i)).comp ↑e)
|
rw [this, AlgHom.coe_comp]
|
ι : Type u_1
R : Type u_2
A : Type u_3
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
s t : Set ι
h : Disjoint s t
e : MvPolynomial (↑t ⊕ ↑s) R ≃ₐ[R] MvPolynomial ↑t ↥(adjoin R (range (x ∘ Subtype.val))) :=
(sumAlgEquiv R ↑t ↑s).trans (mapAlgEquiv ↑t ⋯.aevalEquiv)
this :
(AlgHom.restrictScalars R (aeval fun i => x ↑i)).comp ↑e = (aeval x).comp (rename (Sum.elim Subtype.val Subtype.val))
x✝ :
∀ {R : Type u_2} {S₁ S₂ : Type u_3} {σ : Type u_1} [inst : CommSemiring S₂] [inst_1 : SMul R S₁]
[inst_2 : SMulZeroClass R S₂] [inst_3 : SMulZeroClass S₁ S₂] [inst_4 : IsScalarTower R S₁ S₂],
IsScalarTower R S₁ (MvPolynomial σ S₂)
⊢ Injective (⇑(aeval x) ∘ ⇑(rename (Sum.elim Subtype.val Subtype.val)))
|
5c3dc9406b742efb
|
DirichletCharacter.LSeriesSummable_iff
|
Mathlib/NumberTheory/LSeries/Dirichlet.lean
|
/-- The L-series of a Dirichlet character mod `N > 0` converges absolutely at `s` if and only if
`re s > 1`. -/
lemma LSeriesSummable_iff {N : ℕ} (hN : N ≠ 0) (χ : DirichletCharacter ℂ N) {s : ℂ} :
LSeriesSummable ↗χ s ↔ 1 < s.re
|
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
s : ℂ
H : LSeriesSummable (fun n => χ ↑n) s
⊢ 1 < s.re
|
by_contra! h
|
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
s : ℂ
H : LSeriesSummable (fun n => χ ↑n) s
h : s.re ≤ 1
⊢ False
|
def54cbe0d797bc5
|
Polynomial.count_roots_le_one
|
Mathlib/FieldTheory/Separable.lean
|
theorem count_roots_le_one [DecidableEq R] {p : R[X]} (hsep : Separable p) (x : R) :
p.roots.count x ≤ 1
|
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hsep : p.Separable
x : R
⊢ rootMultiplicity x p ≤ 1
|
exact rootMultiplicity_le_one_of_separable hsep x
|
no goals
|
aa12e9ef6c5df2f0
|
FiniteMultiplicity.lt_multiplicity_of_lt_emultiplicity
|
Mathlib/RingTheory/Multiplicity.lean
|
theorem FiniteMultiplicity.lt_multiplicity_of_lt_emultiplicity (hfin : FiniteMultiplicity a b)
{n : ℕ} (h : n < emultiplicity a b) : n < multiplicity a b
|
α : Type u_1
inst✝ : Monoid α
a b : α
hfin : FiniteMultiplicity a b
n : ℕ
h : ↑n < emultiplicity a b
⊢ n < multiplicity a b
|
rw [emultiplicity_eq_multiplicity hfin] at h
|
α : Type u_1
inst✝ : Monoid α
a b : α
hfin : FiniteMultiplicity a b
n : ℕ
h : ↑n < ↑(multiplicity a b)
⊢ n < multiplicity a b
|
b7570eeb016c79e0
|
List.sublist_replicate_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem sublist_replicate_iff : l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = replicate n a
|
case cons.mp
α✝ : Type u_1
a b : α✝
l : List α✝
ih : ∀ {m : Nat}, l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = replicate n a
m : Nat
w : b :: l <+ replicate m a
⊢ ∃ n, n ≤ m ∧ b :: l = replicate n a
|
cases m with
| zero => simp at w
| succ m =>
simp [replicate_succ] at w
cases w with
| cons _ w =>
obtain ⟨n, le, rfl⟩ := ih.1 (sublist_of_cons_sublist w)
obtain rfl := (mem_replicate.1 (mem_of_cons_sublist w)).2
exact ⟨n+1, Nat.add_le_add_right le 1, rfl⟩
| cons₂ _ w =>
obtain ⟨n, le, rfl⟩ := ih.1 w
refine ⟨n+1, Nat.add_le_add_right le 1, by simp [replicate_succ]⟩
|
no goals
|
b36e6729c3d1c9f6
|
lipschitzWith_iff_dist_le_mul
|
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{f : α → β} : LipschitzWith K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y
|
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
f : α → β
⊢ LipschitzWith K f ↔ ∀ (x y : α), dist (f x) (f y) ≤ ↑K * dist x y
|
simp only [LipschitzWith, edist_nndist, dist_nndist]
|
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
f : α → β
⊢ (∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)) ↔ ∀ (x y : α), ↑(nndist (f x) (f y)) ≤ ↑K * ↑(nndist x y)
|
b97f0204d09f7d2f
|
Int.image_Ico_emod
|
Mathlib/Data/Int/Interval.lean
|
theorem image_Ico_emod (n a : ℤ) (h : 0 ≤ a) : (Ico n (n + a)).image (· % a) = Ico 0 a
|
case inr.h.mp.intro.intro
n a : ℤ
h : 0 ≤ a
ha : 0 < a
i : ℤ
left✝ : n ≤ i ∧ i < n + a
⊢ 0 ≤ i % a ∧ i % a < a
|
exact ⟨emod_nonneg i ha.ne', emod_lt_of_pos i ha⟩
|
no goals
|
9d26ae2005c6604e
|
exists_dual_vector'
|
Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
|
theorem exists_dual_vector' [Nontrivial E] (x : E) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖
|
case pos
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : Nontrivial E
x : E
hx : x = 0
⊢ ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖
|
obtain ⟨y, hy⟩ := exists_ne (0 : E)
|
case pos.intro
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : Nontrivial E
x : E
hx : x = 0
y : E
hy : y ≠ 0
⊢ ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖
|
fc0180f7db696d90
|
BitVec.getElem_udiv
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean
|
theorem getElem_udiv (n d : BitVec w) (hy : 0#w < d) (i : Nat) (hi : i < w) :
(n / d)[i] = (divRec w {n, d} (DivModState.init w)).q[i]
|
w : Nat
n d : BitVec w
hy : 0#w < d
i : Nat
hi : i < w
⊢ 0#w < d
|
assumption
|
no goals
|
c211dc508af186c9
|
Algebra.FinitePresentation.ker_fg_of_mvPolynomial
|
Mathlib/RingTheory/FinitePresentation.lean
|
theorem ker_fg_of_mvPolynomial {n : ℕ} (f : MvPolynomial (Fin n) R →ₐ[R] A)
(hf : Function.Surjective f) [FinitePresentation R A] : f.toRingHom.ker.FG
|
case intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ↑s = RingHom.ker f'.toRingHom
RXn : Type (max 0 w₁) := MvPolynomial (Fin n) R
RXm : Type (max 0 w₁) := MvPolynomial (Fin m) R
g : Fin n → MvPolynomial (Fin m) R
hg : ∀ (i : Fin n), f' (g i) = f (MvPolynomial.X i)
h : Fin m → MvPolynomial (Fin n) R
hh : ∀ (i : Fin m), f (h i) = f' (MvPolynomial.X i)
aeval_h : RXm →ₐ[R] RXn := MvPolynomial.aeval h
g' : Fin n → RXn := fun i => MvPolynomial.X i - aeval_h (g i)
hh' : ∀ (x : RXm), f (aeval_h x) = f' x
s' : Set RXn := Set.range g' ∪ ⇑aeval_h '' ↑s
leI : Ideal.span s' ≤ RingHom.ker f.toRingHom
⊢ Ideal.span (Set.range g' ∪ ⇑aeval_h '' ↑s) = RingHom.ker f.toRingHom
|
apply leI.antisymm
|
case intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ↑s = RingHom.ker f'.toRingHom
RXn : Type (max 0 w₁) := MvPolynomial (Fin n) R
RXm : Type (max 0 w₁) := MvPolynomial (Fin m) R
g : Fin n → MvPolynomial (Fin m) R
hg : ∀ (i : Fin n), f' (g i) = f (MvPolynomial.X i)
h : Fin m → MvPolynomial (Fin n) R
hh : ∀ (i : Fin m), f (h i) = f' (MvPolynomial.X i)
aeval_h : RXm →ₐ[R] RXn := MvPolynomial.aeval h
g' : Fin n → RXn := fun i => MvPolynomial.X i - aeval_h (g i)
hh' : ∀ (x : RXm), f (aeval_h x) = f' x
s' : Set RXn := Set.range g' ∪ ⇑aeval_h '' ↑s
leI : Ideal.span s' ≤ RingHom.ker f.toRingHom
⊢ RingHom.ker f.toRingHom ≤ Ideal.span s'
|
37c27acde7e67205
|
Mathlib.Tactic.Module.NF.eval_cons
|
Mathlib/Tactic/Module.lean
|
theorem eval_cons [AddMonoid M] [SMul R M] (p : R × M) (l : NF R M) :
(p ::ᵣ l).eval = p.1 • p.2 + l.eval
|
R : Type u_2
M : Type u_3
inst✝¹ : AddMonoid M
inst✝ : SMul R M
p : R × M
l : NF R M
⊢ ((match p with
| (r, x) => r • x) ::
map
(fun x =>
match x with
| (r, x) => r • x)
l).sum =
p.1 • p.2 +
(map
(fun x =>
match x with
| (r, x) => r • x)
l).sum
|
rw [List.sum_cons]
|
no goals
|
b1b48e35b830758c
|
BitVec.msb_neg
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean
|
theorem msb_neg {w : Nat} {x : BitVec w} :
(-x).msb = ((x != 0#w && x != intMin w) ^^ x.msb)
|
w : Nat
x : BitVec w
⊢ (x.getMsbD 0 ^^ decide (∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true)) = (x != 0#w && x != intMin w ^^ x.getMsbD 0)
|
by_cases hmin : x = intMin _
|
case pos
w : Nat
x : BitVec w
hmin : x = intMin w
⊢ (x.getMsbD 0 ^^ decide (∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true)) = (x != 0#w && x != intMin w ^^ x.getMsbD 0)
case neg
w : Nat
x : BitVec w
hmin : ¬x = intMin w
⊢ (x.getMsbD 0 ^^ decide (∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true)) = (x != 0#w && x != intMin w ^^ x.getMsbD 0)
|
a0abc44cd1acd615
|
LinearMap.charpoly_toMatrix
|
Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean
|
theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
(toMatrix b b f).charpoly = f.charpoly
|
R : Type u_1
M : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : Nontrivial R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
A : Matrix ι ι R := (toMatrix b b) f
b' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M
ι' : Type u_2 := ChooseBasisIndex R M
A' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f
e : ι ≃ ChooseBasisIndex R M := b.indexEquiv b'
φ : Matrix ι ι R ≃ₗ[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e
φ₁ : Matrix ι ι' R ≃ₗ[R] Matrix (ChooseBasisIndex R M) ι' R := reindexLinearEquiv R R e (Equiv.refl ι')
φ₂ : Matrix ι' ι' R ≃ₗ[R] Matrix ι' ι' R := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
φ₃ : Matrix ι' ι R ≃ₗ[R] Matrix ι' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl ι') e
P : Matrix ι (ChooseBasisIndex R M) R := b.toMatrix ⇑b'
Q : Matrix (ChooseBasisIndex R M) ι R := b'.toMatrix ⇑b
hPQ : C.mapMatrix (φ₁ P) * C.mapMatrix (φ₃ Q) = 1
⊢ ((scalar ι') X - C.mapMatrix (φ₁ P) * C.mapMatrix A' * C.mapMatrix (φ₃ Q)).det =
((scalar ι') X * C.mapMatrix (φ₁ P) * C.mapMatrix (φ₃ Q) -
C.mapMatrix (φ₁ P) * C.mapMatrix A' * C.mapMatrix (φ₃ Q)).det
|
rw [Matrix.mul_assoc ((scalar ι') X), hPQ, Matrix.mul_one]
|
no goals
|
ff9c8772a8b3d3c4
|
HahnSeries.orderTop_single_le
|
Mathlib/RingTheory/HahnSeries/Basic.lean
|
theorem orderTop_single_le : a ≤ (single a r).orderTop
|
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : Zero R
a : Γ
r : R
⊢ ↑a ≤ ((single a) r).orderTop
|
by_cases hr : r = 0
|
case pos
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : Zero R
a : Γ
r : R
hr : r = 0
⊢ ↑a ≤ ((single a) r).orderTop
case neg
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : Zero R
a : Γ
r : R
hr : ¬r = 0
⊢ ↑a ≤ ((single a) r).orderTop
|
f6b41eba621a1453
|
Equiv.Perm.pow_eq_on_of_mem_support
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem pow_eq_on_of_mem_support (h : ∀ x ∈ f.support ∩ g.support, f x = g x) (k : ℕ) :
∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
|
induction' k with k hk
|
case zero
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ 0) x = (g ^ 0) x
case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ (k + 1)) x = (g ^ (k + 1)) x
|
0bbbd941baa698a7
|
ExistsContDiffBumpBase.y_eq_one_of_mem_closedBall
|
Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
|
theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
(hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1
|
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
C : ball x D ⊆ ball 0 1
y : E
hy : y ∈ ball x D
h'y : 1 < dist y 0
⊢ False
|
linarith only [mem_ball.1 (C hy), h'y]
|
no goals
|
2258e901596c5621
|
Array.range'_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean
|
theorem range'_append (s m n step : Nat) :
range' s m step ++ range' (s + step * m) n step = range' s (m + n) step
|
case h₂
s m n step i : Nat
h₁✝ : i < (range' s m step ++ range' (s + step * m) n step).size
h₂✝ : i < (range' s (m + n) step).size
h₁ h₂ : i < m + n
h : m ≤ i
⊢ s + step * m + (step * i - step * m) = s + step * i
|
have : step * m ≤ step * i := by exact mul_le_mul_left step h
|
case h₂
s m n step i : Nat
h₁✝ : i < (range' s m step ++ range' (s + step * m) n step).size
h₂✝ : i < (range' s (m + n) step).size
h₁ h₂ : i < m + n
h : m ≤ i
this : step * m ≤ step * i
⊢ s + step * m + (step * i - step * m) = s + step * i
|
e4387bb0e1420d8c
|
ENNReal.toNNReal_iSup
|
Mathlib/Data/ENNReal/Real.lean
|
theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal
|
case neg
ι : Sort u_1
f : ι → ℝ≥0
h : ¬BddAbove (range f)
⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i
|
rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
|
no goals
|
d664c6a853b7cbe6
|
List.nonzeroMinimum_le_iff
|
Mathlib/.lake/packages/lean4/src/lean/Lean/Elab/Tactic/Omega/MinNatAbs.lean
|
theorem nonzeroMinimum_le_iff {xs : List Nat} {y : Nat} :
xs.nonzeroMinimum ≤ y ↔ xs.nonzeroMinimum = 0 ∨ ∃ x ∈ xs, x ≤ y ∧ x ≠ 0
|
xs : List Nat
y : Nat
h : xs.nonzeroMinimum = 0 ∨ ∃ x, x ∈ xs ∧ x ≤ y ∧ x ≠ 0
x : Nat
m : x ∈ xs
le : x ≤ y
ne : x ≠ 0
⊢ xs.nonzeroMinimum ≤ y
|
exact Nat.le_trans (nonzeroMinimum_le m ne) le
|
no goals
|
7ee2627fef2f71f1
|
Polynomial.isNilpotent_C_mul_pow_X_of_isNilpotent
|
Mathlib/RingTheory/Polynomial/Nilpotent.lean
|
lemma isNilpotent_C_mul_pow_X_of_isNilpotent (n : ℕ) (hnil : IsNilpotent r) :
IsNilpotent ((C r) * X ^ n)
|
R : Type u_1
r : R
inst✝ : Semiring R
n : ℕ
hnil : IsNilpotent r
⊢ IsNilpotent (C r)
|
obtain ⟨m, hm⟩ := hnil
|
case intro
R : Type u_1
r : R
inst✝ : Semiring R
n m : ℕ
hm : r ^ m = 0
⊢ IsNilpotent (C r)
|
aa1f491848016b3a
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.