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
|
|---|---|---|---|---|---|---|
IsSimpleRing.isField_center
|
Mathlib/RingTheory/SimpleRing/Field.lean
|
lemma isField_center (A : Type*) [Ring A] [IsSimpleRing A] : IsField (Subring.center A) where
exists_pair_ne := ⟨0, 1, zero_ne_one⟩
mul_comm := mul_comm
mul_inv_cancel
|
case a
A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 => x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
⊢ ↑(⟨x, hx1✝⟩ * ⟨y, ⋯⟩) = ↑1
|
exact hy
|
no goals
|
08d555f2a5a9e7b4
|
FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt
|
Mathlib/RingTheory/Multiplicity.lean
|
theorem FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt (hf : FiniteMultiplicity a b) {m : ℕ}
(hm : multiplicity a b < m) : ¬a ^ m ∣ b
|
case hm
α : Type u_1
inst✝ : Monoid α
a b : α
hf : FiniteMultiplicity a b
m : ℕ
hm : multiplicity a b < m
⊢ ↑(multiplicity a b) < ↑m
|
norm_cast
|
no goals
|
8a0c58939e3c5124
|
Polynomial.trailingDegree_mul
|
Mathlib/Algebra/Polynomial/RingDivision.lean
|
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree
|
case neg
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hp : ¬p = 0
hq : ¬q = 0
⊢ ↑(p.natTrailingDegree + q.natTrailingDegree) = ↑p.natTrailingDegree + ↑q.natTrailingDegree
|
apply WithTop.coe_add
|
no goals
|
c261f6a7f5279160
|
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) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
J2 : ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
|
have J3 : ‖L e' p' r - L e' p' q'‖ ≤ 4 * (1 / 2) ^ e := by
have I1 : x ∈ A f (L e' p' r) ((1 / 2) ^ p') ((1 / 2) ^ e') :=
(hn e' p' r hp' (le_max_right _ _)).2.1
have I2 : x ∈ A f (L e' p' q') ((1 / 2) ^ p') ((1 / 2) ^ e') := (hn e' p' q' hp' hq').2.1
exact norm_sub_le_of_mem_A P _ (A_mono _ _ I I1) (A_mono _ _ I I2)
|
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) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
J2 : ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
J3 : ‖L e' p' r - L e' p' q'‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
|
807a865e58286923
|
MvQPF.Fix.dest_mk
|
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
theorem Fix.dest_mk (x : F (append1 α (Fix F α))) : Fix.dest (Fix.mk x) = x
|
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : F (α ::: Fix F α)
⊢ mk ∘ dest = _root_.id
|
ext (x : Fix F α)
|
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : F (α ::: Fix F α)
x : Fix F α
⊢ (mk ∘ dest) x = _root_.id x
|
8fb388f4bdc9d3ad
|
Valued.cauchy_iff
|
Mathlib/Topology/Algebra/Valued/ValuationTopology.lean
|
theorem cauchy_iff {F : Filter R} : Cauchy F ↔
F.NeBot ∧ ∀ γ : Γ₀ˣ, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), (v (y - x) : Γ₀) < γ
|
case mpr.intro
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
h : ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
γ : Γ₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ ↑(v.ltAddSubgroup γ)
|
exact h γ
|
no goals
|
b2f003a68b1d7766
|
UniqueFactorizationMonoid.le_emultiplicity_iff_replicate_le_normalizedFactors
|
Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean
|
theorem le_emultiplicity_iff_replicate_le_normalizedFactors {a b : R} {n : ℕ} (ha : Irreducible a)
(hb : b ≠ 0) :
↑n ≤ emultiplicity a b ↔ replicate n (normalize a) ≤ normalizedFactors b
|
case succ.mpr
R : Type u_2
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
⊢ replicate (n + 1) (normalize a) ≤ normalizedFactors b → a ^ (n + 1) ∣ b
|
rw [Multiset.le_iff_exists_add]
|
case succ.mpr
R : Type u_2
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
⊢ (∃ u, normalizedFactors b = replicate (n + 1) (normalize a) + u) → a ^ (n + 1) ∣ b
|
c787fb02983994e3
|
Ideal.map_sInf
|
Mathlib/RingTheory/Ideal/Maps.lean
|
theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A)
|
case refine_2
R : Type u_1
S : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
⊢ y ∈ map f (sInf A)
|
obtain ⟨x, hx⟩ := hf y
|
case refine_2.intro
R : Type u_1
S : Type u_2
F : Type u_3
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
A : Set (Ideal R)
f : F
hf : Function.Surjective ⇑f
h : ∀ J ∈ A, RingHom.ker f ≤ J
y : S
hy : y ∈ sInf (map f '' A)
x : R
hx : f x = y
⊢ y ∈ map f (sInf A)
|
e404a4c82d5f962e
|
MeasureTheory.hasFDerivAt_convolution_right_with_param
|
Mathlib/Analysis/Convolution.lean
|
theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)) (q₀ : P × G)
(hq₀ : q₀.1 ∈ s) :
HasFDerivAt (fun q : P × G => (f ⋆[L, μ] g q.1) q.2)
((f ⋆[L.precompR (P × G), μ] fun x : G => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀
|
case pos
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A✝ : ∀ p ∈ s, Continuous (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
A : IsCompact ({q₀.1} ×ˢ k)
t : Set (P × G)
kt : {q₀.1} ×ˢ k ⊆ t
t_open : IsOpen t
ht : Bornology.IsBounded (g' '' t)
ε : ℝ
εpos : 0 < ε
hε : thickening ε ({q₀.1} ×ˢ k) ⊆ t
h'ε : ball q₀.1 ε ⊆ s
C : ℝ
Cpos : 0 < C
hC : g' '' t ⊆ closedBall 0 C
p : P
x : G
hp : ‖p - q₀.1‖ < ε
hps : p ∈ s
hx : x ∈ k
H : (p, x) ∈ t
this : g' (p, x) ∈ closedBall 0 C
⊢ ‖g' (p, x)‖ ≤ C
|
rwa [mem_closedBall_zero_iff] at this
|
no goals
|
681ec723c7fd0b70
|
cauchy_map_of_uniformCauchySeqOn_fderiv
|
Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
|
theorem cauchy_map_of_uniformCauchySeqOn_fderiv {s : Set E} (hs : IsOpen s) (h's : IsPreconnected s)
(hf' : UniformCauchySeqOn f' l s) (hf : ∀ n : ι, ∀ y : E, y ∈ s → HasFDerivAt (f n) (f' n y) y)
{x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s) (hfg : Cauchy (map (fun n => f n x₀) l)) :
Cauchy (map (fun n => f n x) l)
|
case h
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x✝ : E
hx₀ : x₀ ∈ s
hx : x✝ ∈ s
hfg : Cauchy (map (fun n => f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n => f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
open_t : IsOpen t
st_nonempty : (s ∩ t).Nonempty
x : E
xt : x ∈ closure t
xs : x ∈ s
ε : ℝ
εpos : ε > 0
hε : Metric.ball x ε ⊆ s
y : E
yt : y ∈ t
hxy : dist x y < ε / 2
⊢ ε / 2 + dist y x ≤ ε
|
rw [dist_comm]
|
case h
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x✝ : E
hx₀ : x₀ ∈ s
hx : x✝ ∈ s
hfg : Cauchy (map (fun n => f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n => f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
open_t : IsOpen t
st_nonempty : (s ∩ t).Nonempty
x : E
xt : x ∈ closure t
xs : x ∈ s
ε : ℝ
εpos : ε > 0
hε : Metric.ball x ε ⊆ s
y : E
yt : y ∈ t
hxy : dist x y < ε / 2
⊢ ε / 2 + dist x y ≤ ε
|
774a9e47a2906047
|
Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE
|
Mathlib/Topology/Order/T5.lean
|
theorem compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection (ordSeparatingSet s t))ᶜ ∈ 𝓝[≥] a
|
case pos
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a : X
s t : Set X
hd : Disjoint s (closure t)
ha : a ∈ s
hmem : tᶜ ∈ 𝓝[≥] a
b : X
hab : a ≤ b
hmem' : Icc a b ∈ 𝓝[≥] a
hsub : Icc a b ⊆ tᶜ
H : Disjoint (Icc a b) (s.ordSeparatingSet t).ordConnectedSection
⊢ (s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≥] a
|
exact mem_of_superset hmem' (disjoint_left.1 H)
|
no goals
|
a4a4d83ca57ecaff
|
Group.card_center_add_sum_card_noncenter_eq_card
|
Mathlib/GroupTheory/ClassEquation.lean
|
theorem Group.card_center_add_sum_card_noncenter_eq_card (G) [Group G]
[∀ x : ConjClasses G, Fintype x.carrier] [Fintype G] [Fintype <| Subgroup.center G]
[Fintype <| noncenter G] : Fintype.card (Subgroup.center G) +
∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card = Fintype.card G
|
case h.e'_2.h.e'_6
G : Type u_2
inst✝⁴ : Group G
inst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier
inst✝² : Fintype G
inst✝¹ : Fintype ↥(Subgroup.center G)
inst✝ : Fintype ↑(noncenter G)
⊢ ∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card =
∑ᶠ (x : ConjClasses G) (_ : x ∈ noncenter G), Nat.card ↑x.carrier
|
rw [← finsum_set_coe_eq_finsum_mem (noncenter G), finsum_eq_sum_of_fintype,
← Finset.sum_set_coe]
|
case h.e'_2.h.e'_6
G : Type u_2
inst✝⁴ : Group G
inst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier
inst✝² : Fintype G
inst✝¹ : Fintype ↥(Subgroup.center G)
inst✝ : Fintype ↑(noncenter G)
⊢ ∑ i : ↑(noncenter G), (↑i).carrier.toFinset.card = ∑ i : ↑(noncenter G), Nat.card ↑(↑i).carrier
|
e8d5dab4b658ff9d
|
CompleteOrthogonalIdempotents.option
|
Mathlib/RingTheory/Idempotents.lean
|
lemma CompleteOrthogonalIdempotents.option (he : OrthogonalIdempotents e) :
CompleteOrthogonalIdempotents (Option.elim · (1 - ∑ i, e i) e) where
__ := he.option _ he.isIdempotentElem_sum.one_sub
(by simp [sub_mul, he.isIdempotentElem_sum.eq]) (by simp [mul_sub, he.isIdempotentElem_sum.eq])
complete
|
R : Type u_1
inst✝¹ : Ring R
I : Type u_3
e : I → R
inst✝ : Fintype I
he : OrthogonalIdempotents e
⊢ (∑ i : I, e i) * (1 - ∑ i : I, e i) = 0
|
simp [mul_sub, he.isIdempotentElem_sum.eq]
|
no goals
|
0de3189373444a28
|
YoungDiagram.le_of_transpose_le
|
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
theorem le_of_transpose_le {μ ν : YoungDiagram} (h_le : μ.transpose ≤ ν) :
μ ≤ ν.transpose := fun c hc => by
simp only [mem_cells, mem_transpose]
apply h_le
simpa
|
μ ν : YoungDiagram
h_le : μ.transpose ≤ ν
c : ℕ × ℕ
hc : c ∈ μ.cells
⊢ c ∈ ν.transpose.cells
|
simp only [mem_cells, mem_transpose]
|
μ ν : YoungDiagram
h_le : μ.transpose ≤ ν
c : ℕ × ℕ
hc : c ∈ μ.cells
⊢ c.swap ∈ ν
|
daeba2068d2a3d79
|
Std.Tactic.BVDecide.LRAT.Internal.CNF.unsat_of_convertLRAT_unsat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Convert.lean
|
theorem CNF.unsat_of_convertLRAT_unsat (cnf : CNF Nat) :
Unsatisfiable (PosFin (cnf.numLiterals + 1)) (CNF.convertLRAT cnf)
→
cnf.Unsat
|
cnf : CNF Nat
⊢ Unsatisfiable (PosFin (cnf.numLiterals + 1)) (convertLRAT cnf) → cnf.Unsat
|
intro h1
|
cnf : CNF Nat
h1 : Unsatisfiable (PosFin (cnf.numLiterals + 1)) (convertLRAT cnf)
⊢ cnf.Unsat
|
c8ce4969f2703677
|
Polynomial.Chebyshev.U_two
|
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
theorem U_two : U R 2 = 4 * X ^ 2 - 1
|
R : Type u_1
inst✝ : CommRing R
this : U R (0 + 2) = 2 * X * U R (0 + 1) - U R 0
⊢ U R 2 = 4 * X ^ 2 - 1
|
simp only [zero_add, U_one, U_zero] at this
|
R : Type u_1
inst✝ : CommRing R
this : U R 2 = 2 * X * (2 * X) - 1
⊢ U R 2 = 4 * X ^ 2 - 1
|
487a90767cd81216
|
InnerProductSpace.volume_ball
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
theorem volume_ball (x : E) (r : ℝ) :
volume (Metric.ball x r) = (.ofReal r) ^ finrank ℝ E *
.ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1))
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r) =
ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
|
have : Nonempty (Fin (finrank ℝ E)) := Fin.pos_iff_nonempty.mp finrank_pos
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
this : Nonempty (Fin (finrank ℝ E))
⊢ volume (⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r) =
ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
|
f59bc6460603aea9
|
PowerSeries.invOneSubPow_inv_zero_eq_one
|
Mathlib/RingTheory/PowerSeries/WellKnown.lean
|
theorem invOneSubPow_inv_zero_eq_one : (invOneSubPow S 0).inv = 1
|
S : Type u_1
inst✝ : CommRing S
⊢ (invOneSubPow S 0).inv = 1
|
delta invOneSubPow
|
S : Type u_1
inst✝ : CommRing S
⊢ (match 0 with
| 0 => 1
| d.succ =>
{ val := mk fun n => ↑((d + n).choose d), inv := (1 - X) ^ (d + 1), val_inv := ⋯, inv_val := ⋯ }).inv =
1
|
7dafa51690c0a633
|
Matrix.GeneralLinearGroup.map_apply
|
Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
|
@[simp]
protected lemma map_apply (f : R →+* S) (i j : n) (g : GL n R) : map f g i j = f (g i j)
|
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
f : R →+* S
i j : n
g : GL n R
⊢ ↑((map f) g) i j = f (↑g i j)
|
rfl
|
no goals
|
81a5e24c960b3c84
|
NormedDivisionRing.norm_eq_one_iff_ne_zero_of_discrete
|
Mathlib/Analysis/Normed/Field/Basic.lean
|
lemma norm_eq_one_iff_ne_zero_of_discrete {x : 𝕜} : ‖x‖ = 1 ↔ x ≠ 0
|
case mpr.intro.intro.inr.inr
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
ε : ℝ
εpos : ε > 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
H :
∀ {𝕜 : Type u_5} [inst : NormedDivisionRing 𝕜] [inst_1 : DiscreteTopology 𝕜] {x : 𝕜},
x ≠ 0 → (∀ (y : 𝕜), ‖y‖ < ε → y = 0) → ‖x‖ < 1 → ‖x‖ = 1
h✝ : 1 ≤ ‖x‖
h : ‖x⁻¹‖ < 1
⊢ ‖x⁻¹‖ = 1
|
exact H (by simpa) h' h
|
no goals
|
3b2eee4d3f7ad472
|
Finset.inductive_claim_mul
|
Mathlib/Combinatorics/Additive/SmallTripling.lean
|
@[to_additive]
private lemma inductive_claim_mul (hm : 3 ≤ m)
(h : ∀ ε : Fin 3 → ℤ, (∀ i, |ε i| = 1) → #((finRange 3).map fun i ↦ A ^ ε i).prod ≤ k * #A)
(ε : Fin m → ℤ) (hε : ∀ i, |ε i| = 1) :
#((finRange m).map fun i ↦ A ^ ε i).prod ≤ k ^ (m - 2) * #A
|
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i => A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
∀ (ε : Fin (m + 1) → ℤ),
(∀ (i : Fin (m + 1)), |ε i| = 1) →
↑(#(List.map (fun i => A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ => (List.map (fun i => A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ∀ (i : Fin m), |Fin.cons 1 (tail (tail ε)) i.succ| = 1
|
simp [hε, Fin.tail]
|
no goals
|
75599e5c7cd8b516
|
LinearMap.span_singleton_inf_orthogonal_eq_bot
|
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
theorem span_singleton_inf_orthogonal_eq_bot (B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂) (x : V₁)
(hx : ¬B.IsOrtho x x) : (K₁ ∙ x) ⊓ Submodule.orthogonalBilin (K₁ ∙ x) B = ⊥
|
K : Type u_13
K₁ : Type u_14
V₁ : Type u_17
V₂ : Type u_18
inst✝⁵ : Field K
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J₁ J₁' : K₁ →+* K
B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂
x : V₁
hx : ¬B.IsOrtho x x
μ : V₁ → K₁
h : J₁' (μ x) • (B x) x = 0
y : J₁' (μ x) = 0
⊢ μ x = 0
|
simpa using y
|
no goals
|
b7086190b7929afb
|
MeasureTheory.MemLp.piecewise
|
Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
|
protected lemma MemLp.piecewise [DecidablePred (· ∈ s)] {g} (hs : MeasurableSet s)
(hf : MemLp f p (μ.restrict s)) (hg : MemLp g p (μ.restrict sᶜ)) :
MemLp (s.piecewise f g) p μ
|
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup F
f : α → F
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
g : α → F
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
⊢ MemLp (s.piecewise f g) p μ
|
by_cases hp_zero : p = 0
|
case pos
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup F
f : α → F
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
g : α → F
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : p = 0
⊢ MemLp (s.piecewise f g) p μ
case neg
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup F
f : α → F
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
g : α → F
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
⊢ MemLp (s.piecewise f g) p μ
|
1ca9f3d97936c1de
|
IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow
|
Mathlib/NumberTheory/Cyclotomic/Rat.lean
|
theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K
|
p : ℕ+
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ (n✝ + 1)} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n
H : (algebraMap ℚ K) ↑↑p ^ n * x ∈ adjoin ℤ {B.gen}
h₁ : minpoly ℚ (ζ - 1) = map (algebraMap ℤ ℚ) (minpoly ℤ (ζ - 1))
h₂ : map (algebraMap ℤ ℚ) (minpoly ℤ (ζ - 1)) = (map (algebraMap ℤ ℚ) (cyclotomic ↑(p ^ (n✝ + 1)) ℤ)).comp (X + 1)
⊢ X + 1 = map (algebraMap ℤ ℚ) (X + 1)
|
simp
|
no goals
|
076c1160c95c322f
|
ZMod.erdos_ginzburg_ziv_prime
|
Mathlib/Combinatorics/Additive/ErdosGinzburgZiv.lean
|
theorem ZMod.erdos_ginzburg_ziv_prime (a : ι → ZMod p) (hs : #s = 2 * p - 1) :
∃ t ⊆ s, #t = p ∧ ∑ i ∈ t, a i = 0
|
case intro.refine_2.refine_2
ι : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
s : Finset ι
a : ι → ZMod p
hs : #s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
zero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s }
hpN : p ∣ N
x : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
hx : x ≠ zero_sol
⊢ p ∣ #(filter (fun a_1 => ↑x a_1 ≠ 0) univ)
|
rw [← CharP.cast_eq_zero_iff (ZMod p), ← Finset.sum_boole]
|
case intro.refine_2.refine_2
ι : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
s : Finset ι
a : ι → ZMod p
hs : #s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
zero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s }
hpN : p ∣ N
x : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
hx : x ≠ zero_sol
⊢ (∑ x_1 : { x // x ∈ s }, if ↑x x_1 ≠ 0 then 1 else 0) = 0
|
3ec57b8d04698567
|
map_pow
|
Mathlib/Algebra/Group/Hom/Defs.lean
|
theorem map_pow [Monoid G] [Monoid H] [MonoidHomClass F G H] (f : F) (a : G) :
∀ n : ℕ, f (a ^ n) = f a ^ n
| 0 => by rw [pow_zero, pow_zero, map_one]
| n + 1 => by rw [pow_succ, pow_succ, map_mul, map_pow f a n]
|
G : Type u_7
H : Type u_8
F : Type u_9
inst✝³ : FunLike F G H
inst✝² : Monoid G
inst✝¹ : Monoid H
inst✝ : MonoidHomClass F G H
f : F
a : G
⊢ f (a ^ 0) = f a ^ 0
|
rw [pow_zero, pow_zero, map_one]
|
no goals
|
8dd45a2472407108
|
mul_eq_mul_iff_eq_and_eq_of_pos'
|
Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
|
theorem mul_eq_mul_iff_eq_and_eq_of_pos' [PosMulStrictMono α] [MulPosStrictMono α]
(hab : a ≤ b) (hcd : c ≤ d) (b0 : 0 < b) (c0 : 0 < c) :
a * c = b * d ↔ a = b ∧ c = d
|
case intro
α : Type u_3
a c : α
inst✝³ : MulZeroClass α
inst✝² : PartialOrder α
inst✝¹ : PosMulStrictMono α
inst✝ : MulPosStrictMono α
c0 : 0 < c
hab : a ≤ a
b0 : 0 < a
hcd : c ≤ c
⊢ a * c = a * c
|
rfl
|
no goals
|
0b2a445217aa945f
|
Array.getElem_insertIdx
|
Mathlib/.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean
|
theorem getElem_insertIdx (as : Array α) (i : Nat) (h : i ≤ as.size) (v : α)
(k) (h' : k < (as.insertIdx i v).size) :
(as.insertIdx i v)[k] =
if h₁ : k < i then
as[k]'(by omega)
else
if h₂ : k = i then
v
else
as[k - 1]'(by simp at h'; omega)
|
α : Type u_1
as : Array α
i : Nat
h : i ≤ as.size
v : α
k : Nat
h' : k < (as.insertIdx i v h).size
⊢ (if h₁ : k < i then (as.push v)[k]
else
if h₂ : k = i then if i ≤ as.size then (as.push v)[as.size] else (as.push v)[i]
else if k ≤ as.size then (as.push v)[k - 1] else (as.push v)[k]) =
if h₁ : k < i then as[k] else if h₂ : k = i then v else as[k - 1]
|
simp only [size_insertIdx] at h'
|
α : Type u_1
as : Array α
i : Nat
h : i ≤ as.size
v : α
k : Nat
h'✝ : k < (as.insertIdx i v h).size
h' : k < as.size + 1
⊢ (if h₁ : k < i then (as.push v)[k]
else
if h₂ : k = i then if i ≤ as.size then (as.push v)[as.size] else (as.push v)[i]
else if k ≤ as.size then (as.push v)[k - 1] else (as.push v)[k]) =
if h₁ : k < i then as[k] else if h₂ : k = i then v else as[k - 1]
|
011239b59ded190c
|
Ideal.ramificationIdx_tower
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
theorem ramificationIdx_tower [IsDedekindDomain S] [IsDedekindDomain T] {f : R →+* S} {g : S →+* T}
{p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
(hg0 : map g P ≠ ⊥) (hfg : map (g.comp f) p ≠ ⊥) (hg : map g P ≤ Q) :
ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q
|
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
f : R →+* S
g : S →+* T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map g P ≠ ⊥
hfg : map (g.comp f) p ≠ ⊥
hg : map g P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
⊢ Multiset.count Q (normalizedFactors (map g (map f p))) =
Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map g P))
|
rcases eq_prime_pow_mul_coprime hf0 P with ⟨I, hcp, heq⟩
|
case intro.intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
f : R →+* S
g : S →+* T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map g P ≠ ⊥
hfg : map (g.comp f) p ≠ ⊥
hg : map g P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
I : Ideal S
hcp : P ⊔ I = ⊤
heq : map f p = P ^ Multiset.count P (normalizedFactors (map f p)) * I
⊢ Multiset.count Q (normalizedFactors (map g (map f p))) =
Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map g P))
|
57208cf4c3229e08
|
HurwitzZeta.cosZeta_two_mul_nat'
|
Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean
|
theorem cosZeta_two_mul_nat' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) / (2 * k) / Gammaℂ (2 * k) *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ)
|
k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ 2 * k = 2 * k - 1 + 1
|
omega
|
no goals
|
1afe05416a6f544c
|
isPreconnected_iff_subset_of_disjoint
|
Mathlib/Topology/Connected/Clopen.lean
|
theorem isPreconnected_iff_subset_of_disjoint {s : Set α} :
IsPreconnected s ↔
∀ u v, IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
|
case mpr
α : Type u
inst✝ : TopologicalSpace α
s : Set α
h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
⊢ IsPreconnected s
|
intro u v hu hv hs hsu hsv
|
case mpr
α : Type u
inst✝ : TopologicalSpace α
s : Set α
h : ∀ (u v : Set α), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsOpen u
hv : IsOpen v
hs : s ⊆ u ∪ v
hsu : (s ∩ u).Nonempty
hsv : (s ∩ v).Nonempty
⊢ (s ∩ (u ∩ v)).Nonempty
|
5ba8ba0723cc89a0
|
List.getElem?_zip_eq_some
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean
|
theorem getElem?_zip_eq_some {l₁ : List α} {l₂ : List β} {z : α × β} {i : Nat} :
(zip l₁ l₂)[i]? = some z ↔ l₁[i]? = some z.1 ∧ l₂[i]? = some z.2
|
case mk.mp
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
i : Nat
fst✝ : α
snd✝ : β
⊢ (∃ x y, l₁[i]? = some x ∧ l₂[i]? = some y ∧ (x, y) = (fst✝, snd✝)) →
l₁[i]? = some (fst✝, snd✝).fst ∧ l₂[i]? = some (fst✝, snd✝).snd
|
rintro ⟨x, y, h₀, h₁, h₂⟩
|
case mk.mp.intro.intro.intro.intro
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
i : Nat
fst✝ : α
snd✝ : β
x : α
y : β
h₀ : l₁[i]? = some x
h₁ : l₂[i]? = some y
h₂ : (x, y) = (fst✝, snd✝)
⊢ l₁[i]? = some (fst✝, snd✝).fst ∧ l₂[i]? = some (fst✝, snd✝).snd
|
ff8f01ddc18035f4
|
measurableSet_of_differentiableAt_of_isComplete
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
theorem measurableSet_of_differentiableAt_of_isComplete {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
MeasurableSet { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K }
|
case h
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
f : E → F
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
K : Set (E →L[𝕜] F)
hK : IsComplete K
b✝⁵ b✝⁴ b✝³ : ℕ
b✝² : b✝³ ≥ b✝⁴
b✝¹ : ℕ
b✝ : b✝¹ ≥ b✝⁴
⊢ MeasurableSet (B f K ((1 / 2) ^ b✝³) ((1 / 2) ^ b✝¹) ((1 / 2) ^ b✝⁵))
|
exact isOpen_B.measurableSet
|
no goals
|
52d0ecf301f78f7c
|
Finset.Colex.toColex_le_toColex_iff_max'_mem
|
Mathlib/Combinatorics/Colex.lean
|
lemma toColex_le_toColex_iff_max'_mem :
toColex s ≤ toColex t ↔ ∀ hst : s ≠ t, (s ∆ t).max' (symmDiff_nonempty.2 hst) ∈ t
|
case refine_1
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
h : { ofColex := s } ≤ { ofColex := t }
hst : s ≠ t
m : α := (s ∆ t).max' ⋯
hmt : m ∉ t
hms : m ∈ s
⊢ False
|
have ⟨b, hbt, hbs, hmb⟩ := h hms hmt
|
case refine_1
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
h : { ofColex := s } ≤ { ofColex := t }
hst : s ≠ t
m : α := (s ∆ t).max' ⋯
hmt : m ∉ t
hms : m ∈ s
b : α
hbt : b ∈ { ofColex := t }.ofColex
hbs : b ∉ { ofColex := s }.ofColex
hmb : m ≤ b
⊢ False
|
d10c7adb0b42c816
|
exists_idempotent_of_compact_t2_of_continuous_mul_left
|
Mathlib/Topology/Algebra/Semigroup.lean
|
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m
|
case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x => x * m) '' N = N
⊢ N ∩ {m' | m' * m = m} ⊆ N
|
apply Set.inter_subset_left
|
no goals
|
fc87b9dbd8289f3d
|
Array.idxOf_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean
|
theorem idxOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : Array α} {a : α} :
(l₁ ++ l₂).idxOf a = if a ∈ l₁ then l₁.idxOf a else l₂.idxOf a + l₁.size
|
case isTrue
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l₁ l₂ : Array α
a : α
h : findIdx (fun x => x == a) l₁ < l₁.size
⊢ findIdx (fun x => x == a) l₁ = if a ∈ l₁ then findIdx (fun x => x == a) l₁ else findIdx (fun x => x == a) l₂ + l₁.size
|
rw [if_pos]
|
case isTrue.hc
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l₁ l₂ : Array α
a : α
h : findIdx (fun x => x == a) l₁ < l₁.size
⊢ a ∈ l₁
|
89e05ccae61f1c71
|
Polynomial.nodup_roots
|
Mathlib/FieldTheory/Separable.lean
|
theorem nodup_roots {p : R[X]} (hsep : Separable p) : p.roots.Nodup
|
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hsep : p.Separable
⊢ p.roots.Nodup
|
exact Multiset.nodup_iff_count_le_one.mpr (count_roots_le_one hsep)
|
no goals
|
9a735a83b34f0a01
|
List.take_add
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean
|
theorem take_add (l : List α) (m n : Nat) : l.take (m + n) = l.take m ++ (l.drop m).take n
|
α : Type u_1
l : List α
m n : Nat
⊢ take (m + n) l = take m l ++ take n (drop m l)
|
suffices take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l) by
rw [take_append_drop] at this
assumption
|
α : Type u_1
l : List α
m n : Nat
⊢ take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l)
|
9de2ed38ee21ad27
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean
|
theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id)
|
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
⊢ unit (i, b) ∈
(match some c with
| none => { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
| some { clause := [l], nodupkey := nodupkey, nodup := nodup } =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }
| some val =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments }).toList
|
by_cases hl : ∃ l : Literal (PosFin n), c = unit l
|
case pos
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl : ∃ l, c = unit l
⊢ unit (i, b) ∈
(match some c with
| none => { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
| some { clause := [l], nodupkey := nodupkey, nodup := nodup } =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }
| some val =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments }).toList
case neg
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl : ¬∃ l, c = unit l
⊢ unit (i, b) ∈
(match some c with
| none => { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
| some { clause := [l], nodupkey := nodupkey, nodup := nodup } =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }
| some val =>
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
assignments := f.assignments }).toList
|
0ab20a88767ce69e
|
Polynomial.eval_eq_prod_roots_sub_of_splits_id
|
Mathlib/Algebra/Polynomial/Splits.lean
|
theorem eval_eq_prod_roots_sub_of_splits_id {p : K[X]}
(hsplit : Splits (RingHom.id K) p) (v : K) :
eval v p = p.leadingCoeff * (p.roots.map fun a ↦ v - a).prod
|
case h.e'_3.h.e'_6.h.e'_3.a.h.e'_4.h
K : Type v
inst✝ : Field K
p : K[X]
hsplit : Splits (RingHom.id K) p
v : K
⊢ p = map (algebraMap K K) p
|
rw [Algebra.id.map_eq_id, map_id]
|
no goals
|
320f211ee0f86418
|
HomologicalComplex.mapBifunctor₂₃.ι_D₁
|
Mathlib/Algebra/Homology/BifunctorAssociator.lean
|
@[reassoc (attr := simp)]
lemma ι_D₁ :
ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j h ≫ D₁ F G₂₃ K₁ K₂ K₃ c₂₃ c₄ j j' =
d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'
|
case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : ¬c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) ⋯) ≫
mapBifunctor.d₁ K₁ (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂.π c₃ c₂₃ (i₂, i₃)) j' =
c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) •
(F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j'
|
rw [mapBifunctor.d₁_eq_zero' _ _ _ _ h₁ _ _ h₂, comp_zero,
ιOrZero_eq_zero _ _ _ _ _ _ _ _ _ _ _ _
(by simpa only [← ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄] using h₂),
comp_zero, smul_zero]
|
no goals
|
17fbb9fb2dacf758
|
Quaternion.star_add_self'
|
Mathlib/Algebra/Quaternion.lean
|
theorem star_add_self' : star a + a = ↑(2 * a.re)
|
R : Type u_3
inst✝ : CommRing R
a : ℍ[R]
⊢ star a + a = ↑(2 * a.re)
|
simp [a.star_add_self', Quaternion.coe]
|
no goals
|
273f122e0341abe5
|
ADEInequality.admissible_of_one_lt_sumInv_aux'
|
Mathlib/NumberTheory/ADEInequality.lean
|
theorem admissible_of_one_lt_sumInv_aux' {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r)
(H : 1 < sumInv {p, q, r}) : Admissible {p, q, r}
|
case «1».«1».«2»
H : 1 < sumInv {2, 3, 5}
⊢ Admissible {2, 3, 5}
|
exact admissible_E8
|
no goals
|
c39aaad1fdce9db3
|
uniform_continuous_npow_on_bounded
|
Mathlib/Algebra/Order/Field/Basic.lean
|
theorem uniform_continuous_npow_on_bounded (B : α) {ε : α} (hε : 0 < ε) (n : ℕ) :
∃ δ > 0, ∀ q r : α, |r| ≤ B → |q - r| ≤ δ → |q ^ n - r ^ n| < ε
|
α : Type u_2
inst✝ : LinearOrderedField α
ε : α
hε : 0 < ε
n : ℕ
B : α
B_pos : 0 < B
pos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)
⊢ ∃ δ > 0, ∀ (q r : α), |r| ≤ B → |q - r| ≤ δ → |q ^ n - r ^ n| < ε
|
refine ⟨min 1 (ε / (1 + n * (B + 1) ^ (n - 1))), lt_min zero_lt_one (div_pos hε pos),
fun q r hr hqr ↦ (abs_pow_sub_pow_le ..).trans_lt ?_⟩
|
α : Type u_2
inst✝ : LinearOrderedField α
ε : α
hε : 0 < ε
n : ℕ
B : α
B_pos : 0 < B
pos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)
q r : α
hr : |r| ≤ B
hqr : |q - r| ≤ 1 ⊓ ε / (1 + ↑n * (B + 1) ^ (n - 1))
⊢ |q - r| * ↑n * (|q| ⊔ |r|) ^ (n - 1) < ε
|
57190a32e8b7a13d
|
MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_eLpNorm
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
{s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
(hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : eLpNorm (fun x => f x - y₀) p μ < ∞) :
Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
|
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0)
|
have h_fin : (∫⁻ a : β, ‖f a - y₀‖ₑ ^ p.toReal ∂μ) ≠ ⊤ :=
(lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_ne_top hi).ne
|
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal
h_bound : ∀ (n : ℕ), (fun x => ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal) ≤ᶠ[ae μ] fun x => ‖f x - y₀‖ₑ ^ p.toReal
h_fin : ∫⁻ (a : β), ‖f a - y₀‖ₑ ^ p.toReal ∂μ ≠ ⊤
⊢ Tendsto (fun n => ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0)
|
14fb55fde699407c
|
Batteries.TransCmp.ge_trans
|
Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean
|
theorem ge_trans (h₁ : cmp x y ≠ .lt) (h₂ : cmp y z ≠ .lt) : cmp x z ≠ .lt
|
cmp✝ : ?m.1056 → ?m.1056 → Ordering
inst✝¹ : TransCmp cmp✝
x✝ : Sort ?u.1054
cmp : x✝ → x✝ → Ordering
inst✝ : TransCmp cmp
x y z : x✝
h₁ : cmp x y ≠ Ordering.lt
h₂ : cmp y z ≠ Ordering.lt
⊢ cmp x z ≠ Ordering.lt
|
have := @TransCmp.le_trans _ cmp _ z y x
|
cmp✝ : ?m.1056 → ?m.1056 → Ordering
inst✝¹ : TransCmp cmp✝
x✝ : Sort ?u.1054
cmp : x✝ → x✝ → Ordering
inst✝ : TransCmp cmp
x y z : x✝
h₁ : cmp x y ≠ Ordering.lt
h₂ : cmp y z ≠ Ordering.lt
this : cmp z y ≠ Ordering.gt → cmp y x ≠ Ordering.gt → cmp z x ≠ Ordering.gt
⊢ cmp x z ≠ Ordering.lt
|
4751489e9cc9dd5c
|
Set.Countable.exists_cycleOn
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
theorem Countable.exists_cycleOn (hs : s.Countable) :
∃ f : Perm α, f.IsCycleOn s ∧ { x | f x ≠ x } ⊆ s
|
case inl
α : Type u_2
s : Set α
hs : s.Countable
hs' : s.Finite
⊢ ∃ f, f.IsCycleOn s ∧ {x | f x ≠ x} ⊆ s
|
refine ⟨hs'.toFinset.toList.formPerm, ?_, fun x hx => by
simpa using List.mem_of_formPerm_apply_ne hx⟩
|
case inl
α : Type u_2
s : Set α
hs : s.Countable
hs' : s.Finite
⊢ hs'.toFinset.toList.formPerm.IsCycleOn s
|
3d6b7d8394578748
|
PowerSeries.rescale_rescale
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
theorem rescale_rescale (f : R⟦X⟧) (a b : R) :
rescale b (rescale a f) = rescale (a * b) f
|
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
a b : R
⊢ (rescale b) ((rescale a) f) = (rescale (a * b)) f
|
ext n
|
case h
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
a b : R
n : ℕ
⊢ (coeff R n) ((rescale b) ((rescale a) f)) = (coeff R n) ((rescale (a * b)) f)
|
c9e5f469c78eee85
|
Ordinal.enum_zero_eq_bot
|
Mathlib/SetTheory/Ordinal/Basic.lean
|
theorem enum_zero_eq_bot {o : Ordinal} (ho : 0 < o) :
enum (α := o.toType) (· < ·) ⟨0, by rwa [type_toType]⟩ =
have H := toTypeOrderBot (o := o) (by rintro rfl; simp at ho)
(⊥ : o.toType) :=
rfl
|
α : Type u
β : Type v
γ : Type w
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
ho : 0 < 0
⊢ False
|
simp at ho
|
no goals
|
3abc0f22abce2cbf
|
Set.Finite.eq_insert_of_subset_of_encard_eq_succ
|
Mathlib/Data/Set/Card.lean
|
theorem Finite.eq_insert_of_subset_of_encard_eq_succ (hs : s.Finite) (h : s ⊆ t)
(hst : t.encard = s.encard + 1) : ∃ a, t = insert a s
|
α : Type u_1
s t : Set α
hs : s.Finite
h : s ⊆ t
hst : ∃ x, t \ s = {x}
⊢ ∃ a, t = insert a s
|
obtain ⟨x, hx⟩ := hst
|
case intro
α : Type u_1
s t : Set α
hs : s.Finite
h : s ⊆ t
x : α
hx : t \ s = {x}
⊢ ∃ a, t = insert a s
|
01f82d8d3260074c
|
nhds_le_of_le
|
Mathlib/Topology/Basic.lean
|
theorem nhds_le_of_le {f} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 x ≤ f
|
X : Type u
x : X
s : Set X
inst✝ : TopologicalSpace X
f : Filter X
h : x ∈ s
o : IsOpen s
sf : 𝓟 s ≤ f
⊢ 𝓝 x ≤ f
|
rw [nhds_def]
|
X : Type u
x : X
s : Set X
inst✝ : TopologicalSpace X
f : Filter X
h : x ∈ s
o : IsOpen s
sf : 𝓟 s ≤ f
⊢ ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, 𝓟 s ≤ f
|
5315946d02b462ae
|
SzemerediRegularity.average_density_near_total_density
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
|
theorem average_density_near_total_density [Nonempty α]
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5)
(hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts} {A B : Finset (Finset α)}
(hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
|(∑ ab ∈ A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (#A * #B) -
G.edgeDensity (A.biUnion id) (B.biUnion id)| ≤ ε ^ 5 / 49
|
case left
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
⊢ (∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) - ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
ε ^ 5 / 49
|
rw [sub_le_iff_le_add']
|
case left
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
⊢ (∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) ≤
↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
|
02b0f7a417394026
|
PrimeSpectrum.zeroLocus_eq_top_iff
|
Mathlib/RingTheory/Spectrum/Prime/Basic.lean
|
theorem zeroLocus_eq_top_iff (s : Set R) :
zeroLocus s = ⊤ ↔ s ⊆ nilradical R
|
case mpr
R : Type u
inst✝ : CommSemiring R
s : Set R
⊢ s ⊆ ↑(nilradical R) → zeroLocus s = ⊤
|
rw [eq_top_iff]
|
case mpr
R : Type u
inst✝ : CommSemiring R
s : Set R
⊢ s ⊆ ↑(nilradical R) → ⊤ ≤ zeroLocus s
|
71fbfa23dbaec95e
|
Option.bnot_comp_isNone
|
Mathlib/Data/Option/Basic.lean
|
@[simp]
lemma bnot_comp_isNone : (! ·) ∘ @Option.isNone α = Option.isSome
|
case h
α : Type u_1
x : Option α
⊢ ((fun x => !x) ∘ isNone) x = x.isSome
|
simp
|
no goals
|
109babd3d6c26ac2
|
isSeparatedMap_iff_nhds
|
Mathlib/Topology/SeparatedMap.lean
|
lemma isSeparatedMap_iff_nhds {f : X → Y} : IsSeparatedMap f ↔
∀ x₁ x₂, f x₁ = f x₂ → x₁ ≠ x₂ → ∃ s₁ ∈ 𝓝 x₁, ∃ s₂ ∈ 𝓝 x₂, Disjoint s₁ s₂
|
X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsSeparatedMap f ↔ ∀ (x₁ x₂ : X), f x₁ = f x₂ → x₁ ≠ x₂ → ∃ s₁ ∈ 𝓝 x₁, ∃ s₂ ∈ 𝓝 x₂, Disjoint s₁ s₂
|
simp_rw [isSeparatedMap_iff_disjoint_nhds, Filter.disjoint_iff]
|
no goals
|
4566b3fda4ad57cf
|
SzemerediRegularity.card_aux₂
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
|
theorem card_aux₂ (hP : P.IsEquipartition) (hu : u ∈ P.parts) (hucard : #u ≠ m * 4 ^ #P.parts + a) :
(4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts
|
rw [stepBound, ← Nat.div_div_eq_div_mul]
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ Fintype.card α / #P.parts / 4 ^ #P.parts * 4 ^ #P.parts ≤ Fintype.card α / #P.parts
|
c434f1e0b19cc540
|
Polynomial.aeval_sumIDeriv
|
Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean
|
theorem aeval_sumIDeriv (p : R[X]) (q : ℕ) :
∃ gp : R[X], gp.natDegree ≤ p.natDegree - q ∧
∀ (r : A), (X - C r) ^ q ∣ p.map (algebraMap R A) →
aeval r (sumIDeriv p) = q ! • aeval r gp
|
case refine_2
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ q * p'
⊢ (aeval r) (sumIDeriv p) = q ! • (aeval r) ((range (p.natDegree + 1)).sum c)
|
rw [sumIDeriv_apply, map_sum]
|
case refine_2
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ q * p'
⊢ ∑ x ∈ range (p.natDegree + 1), (aeval r) ((⇑derivative)^[x] p) = q ! • (aeval r) ((range (p.natDegree + 1)).sum c)
|
cc0687230b071aee
|
LightCondensed.isoLocallyConstantOfIsColimit_inv
|
Mathlib/Condensed/Discrete/Colimit.lean
|
lemma isoLocallyConstantOfIsColimit_inv (X : LightProfinite.{u}ᵒᵖ ⥤ Type u)
[PreservesFiniteProducts X] (hX : ∀ S : LightProfinite.{u}, (IsColimit <|
X.mapCocone (coconeRightOpOfCone S.asLimitCone))) :
(isoLocallyConstantOfIsColimit X hX).inv =
(CompHausLike.LocallyConstant.counitApp.{u, u} X)
|
X : LightProfiniteᵒᵖ ⥤ Type u
inst✝ : PreservesFiniteProducts X
hX : (S : LightProfinite) → IsColimit (X.mapCocone (coconeRightOpOfCone S.asLimitCone))
⊢ (lanPresheafExt (isoFinYoneda X ≪≫ (locallyConstantIsoFinYoneda X).symm)).inv ≫ (lanPresheafNatIso hX).hom =
(lanPresheafNatIso fun x =>
isColimitLocallyConstantPresheafDiagram (X.obj (Opposite.op (toLightProfinite.obj (of PUnit.{u + 1}))))
x).hom ≫
counitApp X
|
ext S : 2
|
case w.h
X : LightProfiniteᵒᵖ ⥤ Type u
inst✝ : PreservesFiniteProducts X
hX : (S : LightProfinite) → IsColimit (X.mapCocone (coconeRightOpOfCone S.asLimitCone))
S : LightProfiniteᵒᵖ
⊢ ((lanPresheafExt (isoFinYoneda X ≪≫ (locallyConstantIsoFinYoneda X).symm)).inv ≫ (lanPresheafNatIso hX).hom).app S =
((lanPresheafNatIso fun x =>
isColimitLocallyConstantPresheafDiagram (X.obj (Opposite.op (toLightProfinite.obj (of PUnit.{u + 1}))))
x).hom ≫
counitApp X).app
S
|
abf613c72e544f89
|
ModuleCat.Tilde.isUnit_toStalk
|
Mathlib/AlgebraicGeometry/Modules/Tilde.lean
|
lemma isUnit_toStalk (x : PrimeSpectrum.Top R) (r : x.asIdeal.primeCompl) :
IsUnit ((algebraMap R (Module.End R ((tildeInModuleCat M).stalk x))) r)
|
R : Type u
inst✝ : CommRing R
M : ModuleCat R
x : ↑(PrimeSpectrum.Top R)
r : ↥x.asIdeal.primeCompl
st : ↑(M.tildeInModuleCat.stalk x)
⊢ ∃ a, ((algebraMap R (Module.End R ↑(M.tildeInModuleCat.stalk x))) ↑r) a = st
|
obtain ⟨U, mem, s, rfl⟩ := germ_exist (F := M.tildeInModuleCat) x st
|
case intro.intro.intro
R : Type u
inst✝ : CommRing R
M : ModuleCat R
x : ↑(PrimeSpectrum.Top R)
r : ↥x.asIdeal.primeCompl
U : Opens ↑(PrimeSpectrum.Top R)
mem : x ∈ U
s : ToType (M.tildeInModuleCat.obj (op U))
⊢ ∃ a,
((algebraMap R (Module.End R ↑(M.tildeInModuleCat.stalk x))) ↑r) a =
(ConcreteCategory.hom (M.tildeInModuleCat.germ U x mem)) s
|
623712519fb560df
|
Pell.pos_generator_iff_fundamental
|
Mathlib/NumberTheory/Pell.lean
|
theorem pos_generator_iff_fundamental (a : Solution₁ d) :
(1 < a.x ∧ 0 < a.y ∧ ∀ b : Solution₁ d, ∃ n : ℤ, b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental a
|
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
⊢ IsFundamental a
|
have hd := d_nonsquare_of_one_lt_x h.1
|
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
hd : ¬IsSquare d
⊢ IsFundamental a
|
b8f76271aac4cb87
|
MeasureTheory.Measure.haarScalarFactor_eq_mul
|
Mathlib/MeasureTheory/Measure/Haar/Unique.lean
|
@[to_additive addHaarScalarFactor_eq_mul]
lemma haarScalarFactor_eq_mul (μ' μ ν : Measure G)
[IsHaarMeasure μ] [IsHaarMeasure ν] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
haarScalarFactor μ' ν = haarScalarFactor μ' μ * haarScalarFactor μ ν
|
case pos.intro.mk.intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
hG : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
Z : μ'.haarScalarFactor ν • ∫ (x : G), g x ∂ν = (μ'.haarScalarFactor μ * μ.haarScalarFactor ν) • ∫ (x : G), g x ∂ν
int_g_pos : 0 < ∫ (x : G), g x ∂ν
⊢ μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν
|
change (haarScalarFactor μ' ν : ℝ) * ∫ (x : G), g x ∂ν =
(haarScalarFactor μ' μ * haarScalarFactor μ ν : ℝ≥0) * ∫ (x : G), g x ∂ν at Z
|
case pos.intro.mk.intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
hG : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
int_g_pos : 0 < ∫ (x : G), g x ∂ν
Z : ↑(μ'.haarScalarFactor ν) * ∫ (x : G), g x ∂ν = ↑(μ'.haarScalarFactor μ * μ.haarScalarFactor ν) * ∫ (x : G), g x ∂ν
⊢ μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν
|
b359337938580166
|
Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing
|
Mathlib/RingTheory/Unramified/Field.lean
|
theorem bijective_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] :
Function.Bijective (algebraMap K A)
|
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
{ toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
(TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
⊢ Function.Bijective ⇑(algebraMap K A)
|
have hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A := by
rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,
algebraMap_eq_smul_one, hf₁, sub_self]
|
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
{ toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
(TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
⊢ Function.Bijective ⇑(algebraMap K A)
|
5099a67fd835b73a
|
exists_subset_iUnion_ball_radius_lt
|
Mathlib/Topology/MetricSpace/ShrinkingLemma.lean
|
theorem exists_subset_iUnion_ball_radius_lt {r : ι → ℝ} (hs : IsClosed s)
(uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i
|
case intro.intro.intro
α : Type u
ι : Type v
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
c : ι → α
s : Set α
r : ι → ℝ
hs : IsClosed s
uf : ∀ x ∈ s, {i | x ∈ ball (c i) (r i)}.Finite
us : s ⊆ ⋃ i, ball (c i) (r i)
v : ι → Set α
hsv : s ⊆ iUnion v
hvc : ∀ (i : ι), IsClosed (v i)
hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i)
this : ∀ (i : ι), ∃ r' < r i, v i ⊆ ball (c i) r'
⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i
|
choose r' hlt hsub using this
|
case intro.intro.intro
α : Type u
ι : Type v
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
c : ι → α
s : Set α
r : ι → ℝ
hs : IsClosed s
uf : ∀ x ∈ s, {i | x ∈ ball (c i) (r i)}.Finite
us : s ⊆ ⋃ i, ball (c i) (r i)
v : ι → Set α
hsv : s ⊆ iUnion v
hvc : ∀ (i : ι), IsClosed (v i)
hcv : ∀ (i : ι), v i ⊆ ball (c i) (r i)
r' : ι → ℝ
hlt : ∀ (i : ι), r' i < r i
hsub : ∀ (i : ι), v i ⊆ ball (c i) (r' i)
⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i
|
65cf1a842fcca693
|
Matrix.submatrix_cons_row
|
Mathlib/Data/Matrix/Notation.lean
|
theorem submatrix_cons_row (A : Matrix m' n' α) (i : m') (row : Fin m → m') (col : o' → n') :
submatrix A (vecCons i row) col = vecCons (fun j => A i (col j)) (submatrix A row col)
|
case a
α : Type u
m : ℕ
m' : Type uₘ
n' : Type uₙ
o' : Type uₒ
A : Matrix m' n' α
i✝ : m'
row : Fin m → m'
col : o' → n'
i : Fin m.succ
j : o'
⊢ A.submatrix (vecCons i✝ row) col i j = vecCons (fun j => A i✝ (col j)) (A.submatrix row col) i j
|
refine Fin.cases ?_ ?_ i <;> simp [submatrix]
|
no goals
|
eaf0044103452b4d
|
Polynomial.isRoot_cyclotomic_iff'
|
Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
|
theorem isRoot_cyclotomic_iff' {n : ℕ} {K : Type*} [Field K] {μ : K} [NeZero (n : K)] :
IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n
|
case intro.intro.intro.intro
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
hn : Squarefree (X ^ n - 1)
⊢ False
|
rw [← this, Squarefree] at hn
|
case intro.intro.intro.intro
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
hn : ∀ (x : K[X]), x * x ∣ (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) → IsUnit x
⊢ False
|
1a9be958c6bf03ef
|
Algebra.FinitePresentation.equiv
|
Mathlib/RingTheory/FinitePresentation.lean
|
theorem equiv [FinitePresentation R A] (e : A ≃ₐ[R] B) : FinitePresentation R B
|
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : FinitePresentation R A
e : A ≃ₐ[R] B
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
h : ((↑e).comp f).toRingHom = (↑e).comp f.toRingHom
h1 : ↑e.toRingEquiv = (↑e).toRingHom
⊢ ((↑e).comp f).toRingHom = (↑e.toRingEquiv).comp f.toRingHom
|
rw [h, h1]
|
no goals
|
a2eafda990ed0fda
|
InnerProductSpace.ext_inner_left_basis
|
Mathlib/Analysis/InnerProductSpace/Dual.lean
|
theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
(h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫_𝕜 = ⟪b i, y⟫_𝕜
⊢ ∀ (i : ι), ↑((toDualMap 𝕜 E) x) (b i) = ↑((toDualMap 𝕜 E) y) (b i)
|
intro i
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫_𝕜 = ⟪b i, y⟫_𝕜
i : ι
⊢ ↑((toDualMap 𝕜 E) x) (b i) = ↑((toDualMap 𝕜 E) y) (b i)
|
25adf881a5ddade4
|
MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas
|
Mathlib/MeasureTheory/Function/UnifTight.lean
|
theorem tendsto_Lp_of_tendsto_ae_of_meas (hp : 1 ≤ p) (hp' : p ≠ ∞)
{f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hg' : MemLp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
|
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ⊤
hμ : ¬μ = 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ⊤
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ⊤
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ⊤
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ⊤)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
⊢ eLpNorm (f n - g) p μ ≤ ε
|
have hgEcε := calc
eLpNorm (Eᶜ.indicator g) p μ
≤ eLpNorm (Efᶜ.indicator (Egᶜ.indicator g)) p μ := by
unfold E; rw [compl_union, ← indicator_indicator]
_ ≤ eLpNorm (Egᶜ.indicator g) p μ := eLpNorm_indicator_le _
_ ≤ ε / 3 := hgε.le
|
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ⊤
hμ : ¬μ = 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ⊤
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ⊤
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ⊤
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ⊤)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
⊢ eLpNorm (f n - g) p μ ≤ ε
|
77334843f098ce11
|
Doset.rel_bot_eq_right_group_rel
|
Mathlib/GroupTheory/DoubleCoset.lean
|
theorem rel_bot_eq_right_group_rel (H : Subgroup G) :
⇑(setoid ↑H ↑(⊥ : Subgroup G)) = ⇑(QuotientGroup.rightRel H)
|
case h.h.a.mp
G : Type u_1
inst✝ : Group G
H : Subgroup G
a b : G
⊢ (∃ a_1 ∈ H, ∃ b_1 ∈ ⊥, b = a_1 * a * b_1) → b * a⁻¹ ∈ H
|
rintro ⟨b, hb, a, rfl : a = 1, rfl⟩
|
case h.h.a.mp.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H : Subgroup G
a b : G
hb : b ∈ H
⊢ b * a * 1 * a⁻¹ ∈ H
|
81e081d58dc15cf2
|
Lean.Omega.IntList.dvd_bmod_dot_sub_dot_bmod
|
Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean
|
theorem dvd_bmod_dot_sub_dot_bmod (m : Nat) (xs ys : IntList) :
(m : Int) ∣ bmod_dot_sub_dot_bmod m xs ys
|
case cons.cons
m : Nat
x : Int
xs : List Int
ih : ∀ (ys : IntList), ((dot xs ys).bmod m - (bmod xs m).dot ys) % ↑m = 0
y : Int
ys : List Int
⊢ ((dot (x :: xs) (y :: ys)).bmod m - (bmod (x :: xs) m).dot (y :: ys)) % ↑m = 0
|
simp only [IntList.dot_cons₂, List.map_cons]
|
case cons.cons
m : Nat
x : Int
xs : List Int
ih : ∀ (ys : IntList), ((dot xs ys).bmod m - (bmod xs m).dot ys) % ↑m = 0
y : Int
ys : List Int
⊢ ((x * y + dot xs ys).bmod m - (x.bmod m * y + dot (List.map (fun x => x.bmod m) xs) ys)) % ↑m = 0
|
6132ead5a527f997
|
ContinuousLinearEquiv.comp_contDiff_iff
|
Mathlib/Analysis/Calculus/ContDiff/Basic.lean
|
theorem ContinuousLinearEquiv.comp_contDiff_iff (e : F ≃L[𝕜] G) :
ContDiff 𝕜 n (e ∘ f) ↔ ContDiff 𝕜 n f
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
n : WithTop ℕ∞
e : F ≃L[𝕜] G
⊢ ContDiff 𝕜 n (⇑e ∘ f) ↔ ContDiff 𝕜 n f
|
simp only [← contDiffOn_univ, e.comp_contDiffOn_iff]
|
no goals
|
207fd33cbf80f360
|
integral_withDensity_eq_integral_smul
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ
|
case pos.refine_2
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x => ↑(f x))
u u' : X → E
a✝ : Disjoint (support u) (support u')
u_int : Integrable u (μ.withDensity fun x => ↑(f x))
u'_int : Integrable u' (μ.withDensity fun x => ↑(f x))
h : (∫ (x : X), u x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u x ∂μ
h' : (∫ (x : X), u' x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u' x ∂μ
⊢ (∫ (x : X), (u + u') x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • (u + u') x ∂μ
|
change
(∫ x : X, u x + u' x ∂μ.withDensity fun x : X => ↑(f x)) = ∫ x : X, f x • (u x + u' x) ∂μ
|
case pos.refine_2
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x => ↑(f x))
u u' : X → E
a✝ : Disjoint (support u) (support u')
u_int : Integrable u (μ.withDensity fun x => ↑(f x))
u'_int : Integrable u' (μ.withDensity fun x => ↑(f x))
h : (∫ (x : X), u x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u x ∂μ
h' : (∫ (x : X), u' x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u' x ∂μ
⊢ (∫ (x : X), u x + u' x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • (u x + u' x) ∂μ
|
cd239c0816bb7a0c
|
ONote.exists_lt_mul_omega0'
|
Mathlib/SetTheory/Ordinal/Notation.lean
|
theorem exists_lt_mul_omega0' {o : Ordinal} ⦃a⦄ (h : a < o * ω) :
∃ i : ℕ, a < o * ↑i + o
|
case intro.intro.intro
o a : Ordinal.{u_1}
h : a < o * ω
i : ℕ
hi : ↑i < ω
h' : a < o * ↑i
⊢ ∃ i, a < o * ↑i + o
|
exact ⟨i, h'.trans_le (le_add_right _ _)⟩
|
no goals
|
f61b5b41f10b767b
|
MeasureTheory.ae_ae_add_linearMap_mem_iff
|
Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
|
/-- Given a linear map `L : E → F`, a property holds almost everywhere in `F` if and only if,
almost everywhere in `F`, it holds almost everywhere along the subspace spanned by the
image of `L`. This is an instance of a disintegration argument for additive Haar measures. -/
lemma ae_ae_add_linearMap_mem_iff [LocallyCompactSpace F] {s : Set F} (hs : MeasurableSet s) :
(∀ᵐ y ∂ν, ∀ᵐ x ∂μ, y + L x ∈ s) ↔ ∀ᵐ y ∂ν, y ∈ s
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this✝² : FiniteDimensional 𝕜 E
this✝¹ : FiniteDimensional 𝕜 F
this✝ : ProperSpace E
this : ProperSpace F
M : F × E →ₗ[𝕜] F := LinearMap.id.coprod L
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s
|
have M_cont : Continuous M := M.continuous_of_finiteDimensional
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this✝² : FiniteDimensional 𝕜 E
this✝¹ : FiniteDimensional 𝕜 F
this✝ : ProperSpace E
this : ProperSpace F
M : F × E →ₗ[𝕜] F := LinearMap.id.coprod L
M_cont : Continuous ⇑M
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s
|
b5abd4dbf886cc53
|
MulAction.le_stabilizer_smul_right
|
Mathlib/GroupTheory/GroupAction/Defs.lean
|
@[to_additive]
lemma le_stabilizer_smul_right {G'} [Group G'] [SMul α β] [MulAction G' β]
[SMulCommClass G' α β] (a : α) (b : β) :
stabilizer G' b ≤ stabilizer G' (a • b)
|
α : Type u_2
β : Type u_3
G' : Type u_4
inst✝³ : Group G'
inst✝² : SMul α β
inst✝¹ : MulAction G' β
inst✝ : SMulCommClass G' α β
a : α
b : β
⊢ ∀ ⦃x : G'⦄, x • b = b → a • x • b = a • b
|
rintro a h
|
α : Type u_2
β : Type u_3
G' : Type u_4
inst✝³ : Group G'
inst✝² : SMul α β
inst✝¹ : MulAction G' β
inst✝ : SMulCommClass G' α β
a✝ : α
b : β
a : G'
h : a • b = b
⊢ a✝ • a • b = a✝ • b
|
ea3d9b7b5188c240
|
DirectSum.coe_decompose_mul_of_left_mem_of_le
|
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
theorem coe_decompose_mul_of_left_mem_of_le (a_mem : a ∈ 𝒜 i) (h : i ≤ n) :
(decompose 𝒜 (a * b) n : A) = a * decompose 𝒜 b (n - i)
|
case intro
ι : Type u_1
A : Type u_3
σ : Type u_4
inst✝¹⁰ : Semiring A
inst✝⁹ : DecidableEq ι
inst✝⁸ : AddCommMonoid ι
inst✝⁷ : PartialOrder ι
inst✝⁶ : CanonicallyOrderedAdd ι
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝³ : GradedRing 𝒜
b : A
n i : ι
inst✝² : Sub ι
inst✝¹ : OrderedSub ι
inst✝ : AddLeftReflectLE ι
h : i ≤ n
a : ↥(𝒜 i)
⊢ ↑(((decompose 𝒜) (↑a * b)) n) = ↑a * ↑(((decompose 𝒜) b) (n - i))
|
rwa [decompose_mul, decompose_coe, coe_of_mul_apply_of_le]
|
no goals
|
86975a0e97272182
|
IsDenseInducing.extend_Z_bilin_key
|
Mathlib/Topology/Algebra/UniformGroup/Basic.lean
|
theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W'
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
inst✝³ : AddCommGroup δ
inst✝² : UniformSpace G
inst✝¹ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous fun p => (φ p.1) p.2
W' : Set G
W'_nhd : W' ∈ 𝓝 0
inst✝ : UniformAddGroup G
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u => (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u => (f u.1, f u.2)
lim_φ : Tendsto (fun p => (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)
lim_sub_sub :
Tendsto (fun p => (p.1.2 - p.1.1, p.2.2 - p.2.1)) (comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 (0, 0))
⊢ Tendsto (fun p => (fun p => (φ p.1) p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))
(comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 0)
|
exact Tendsto.comp lim_φ lim_sub_sub
|
no goals
|
965b7a7f03c398d2
|
traceForm_dualBasis_powerBasis_eq
|
Mathlib/RingTheory/Trace/Basic.lean
|
/--
The dual basis of a powerbasis `{1, x, x²...}` under the trace form is `aᵢ / f'(x)`,
with `f` being the minimal polynomial of `x` and `f / (X - x) = ∑ aᵢxⁱ`.
-/
lemma traceForm_dualBasis_powerBasis_eq [FiniteDimensional K L] [Algebra.IsSeparable K L]
(pb : PowerBasis K L) (i) :
(Algebra.traceForm K L).dualBasis (traceForm_nondegenerate K L) pb.basis i =
(minpolyDiv K pb.gen).coeff i / aeval pb.gen (derivative <| minpoly K pb.gen)
|
case a
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
pb : PowerBasis K L
i j : Fin pb.dim
⊢ (if j = i then 1 else 0) =
((traceForm K L) ((minpolyDiv K pb.gen).coeff ↑i / (aeval pb.gen) (derivative (minpoly K pb.gen)))) (pb.basis j)
|
apply (algebraMap K (AlgebraicClosure K)).injective
|
case a.a
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
pb : PowerBasis K L
i j : Fin pb.dim
⊢ (algebraMap K (AlgebraicClosure K)) (if j = i then 1 else 0) =
(algebraMap K (AlgebraicClosure K))
(((traceForm K L) ((minpolyDiv K pb.gen).coeff ↑i / (aeval pb.gen) (derivative (minpoly K pb.gen)))) (pb.basis j))
|
619b8227edc12ad8
|
tangentCone_nonempty_of_properSpace
|
Mathlib/Analysis/Calculus/TangentCone.lean
|
theorem tangentCone_nonempty_of_properSpace [ProperSpace E]
{s : Set E} {x : E} (hx : (𝓝[s \ {x}] x).NeBot) :
(tangentConeAt 𝕜 s x ∩ {0}ᶜ).Nonempty
|
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : (𝓝[s \ {x}] x).NeBot
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hv : ∀ (n : ℕ), v n ∈ s \ {x} ∩ Metric.ball x (u n)
d : ℕ → E := fun n => v n - x
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
c : ℕ → 𝕜
c_ne : ∀ (n : ℕ), c n ≠ 0
c_le : ∀ (n : ℕ), ‖c n • d n‖ < 1
le_c : ∀ (n : ℕ), 1 / ‖r‖ ≤ ‖c n • d n‖
hc : ∀ (n : ℕ), ‖c n‖⁻¹ ≤ 1⁻¹ * ‖r‖ * ‖d n‖
B : ∀ (n : ℕ), ‖c n‖⁻¹ ≤ 1⁻¹ * ‖r‖ * u n
⊢ Tendsto (fun t => 1⁻¹ * ‖r‖ * u t) atTop (𝓝 0)
|
simpa using u_lim.const_mul _
|
no goals
|
67a7ea69bc3c4c99
|
Filter.HasBasis.exists_antitone_subbasis
|
Mathlib/Order/Filter/CountablyGenerated.lean
|
theorem HasBasis.exists_antitone_subbasis {f : Filter α} [h : f.IsCountablyGenerated]
{p : ι' → Prop} {s : ι' → Set α} (hs : f.HasBasis p s) :
∃ x : ℕ → ι', (∀ i, p (x i)) ∧ f.HasAntitoneBasis fun i => s (x i)
|
case h.e'_4
α : Type u_1
ι' : Sort u_5
f : Filter α
h : f.IsCountablyGenerated
p : ι' → Prop
s : ι' → Set α
hs : f.HasBasis p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } := fun n => Nat.recOn n (hs.index (x' 0) ⋯) fun n xn => hs.index (x' (n + 1) ∩ s ↑xn) ⋯
x_anti : Antitone fun i => s ↑(x i)
x_subset : ∀ (i : ℕ), s ↑(x i) ⊆ x' i
this : (⨅ i, 𝓟 (s ↑(x i))).HasAntitoneBasis fun i => s ↑(x i)
⊢ f = ⨅ i, 𝓟 (s ↑(x i))
|
exact
le_antisymm (le_iInf fun i => le_principal_iff.2 <| by cases i <;> apply hs.set_index_mem)
(hx'.symm ▸
le_iInf fun i => le_principal_iff.2 <| this.1.mem_iff.2 ⟨i, trivial, x_subset i⟩)
|
no goals
|
25d2caa8c01b800f
|
Field.finite_intermediateField_of_exists_primitive_element
|
Mathlib/FieldTheory/PrimitiveElement.lean
|
theorem finite_intermediateField_of_exists_primitive_element [Algebra.IsAlgebraic F E]
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E)
|
case intro
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type (max 0 u_2) := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G := fun K => ⟨Polynomial.map (algebraMap (↥K) E) (minpoly (↥K) α), ⋯⟩
hinj : Function.Injective g
⊢ Finite (IntermediateField F E)
|
exact Finite.of_injective g hinj
|
no goals
|
eb1d16f038b5b8d7
|
Array.toList_filterMap'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem toList_filterMap' (f : α → Option β) (l : Array α) (w : stop = l.size) :
(l.filterMap f 0 stop).toList = l.toList.filterMap f
|
α : Type u_1
β : Type u_2
stop : Nat
f : α → Option β
l : Array α
w : stop = l.size
⊢ (filterMap f l 0 stop).toList = List.filterMap f l.toList
|
subst w
|
α : Type u_1
β : Type u_2
f : α → Option β
l : Array α
⊢ (filterMap f l).toList = List.filterMap f l.toList
|
0bf06d250b5cdb33
|
RingHom.locally_localizationAwayPreserves
|
Mathlib/RingTheory/RingHom/Locally.lean
|
/-- If `P` is preserved by localization away, then so is `Locally P`. -/
lemma locally_localizationAwayPreserves (hPl : LocalizationAwayPreserves P) :
LocalizationAwayPreserves (Locally P)
|
case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
⊢ Locally (fun {R S} [CommRing R] [CommRing S] => P) (IsLocalization.Away.map R' S' f r)
|
rw [locally_iff_exists hPl.respectsIso]
|
case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
⊢ ∃ ι s,
∃ (_ : Ideal.span (Set.range s) = ⊤),
∃ Sₜ x x_1,
∃ (_ : ∀ (i : ι), IsLocalization.Away (s i) (Sₜ i)),
∀ (i : ι), P ((algebraMap S' (Sₜ i)).comp (IsLocalization.Away.map R' S' f r))
|
2508d7210fe7161f
|
StrictMonoOn.lt_iff_lt
|
Mathlib/Order/Monotone/Basic.lean
|
theorem StrictMonoOn.lt_iff_lt (hf : StrictMonoOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a < f b ↔ a < b
|
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s : Set α
hf : StrictMonoOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ f a < f b ↔ a < b
|
rw [lt_iff_le_not_le, lt_iff_le_not_le, hf.le_iff_le ha hb, hf.le_iff_le hb ha]
|
no goals
|
ae88eff3482df926
|
LinearMap.isNilpotent_trace_of_isNilpotent
|
Mathlib/LinearAlgebra/Trace.lean
|
lemma isNilpotent_trace_of_isNilpotent {f : M →ₗ[R] M} (hf : IsNilpotent f) :
IsNilpotent (trace R M f)
|
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
hf : IsNilpotent f
⊢ IsNilpotent ((trace R M) f)
|
by_cases H : ∃ s : Finset M, Nonempty (Basis s R M)
|
case pos
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
hf : IsNilpotent f
H : ∃ s, Nonempty (Basis { x // x ∈ s } R M)
⊢ IsNilpotent ((trace R M) f)
case neg
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
hf : IsNilpotent f
H : ¬∃ s, Nonempty (Basis { x // x ∈ s } R M)
⊢ IsNilpotent ((trace R M) f)
|
e91dd2e7e36757a1
|
aestronglyMeasurable_withDensity_iff
|
Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean
|
theorem aestronglyMeasurable_withDensity_iff {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEStronglyMeasurable (fun x => (f x : ℝ) • g x) μ
|
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x => ↑(f x) • g x) =ᶠ[ae μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ↑(f x) ≠ 0
⊢ ↑(f x) ≠ 0
|
simp only [Ne, ENNReal.coe_eq_zero] at h'x
|
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x => ↑(f x) • g x) =ᶠ[ae μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ¬f x = 0
⊢ ↑(f x) ≠ 0
|
7323180b64a0b378
|
Monoid.PushoutI.Reduced.eq_empty_of_mem_range
|
Mathlib/GroupTheory/PushoutI.lean
|
theorem Reduced.eq_empty_of_mem_range
(hφ : ∀ i, Injective (φ i)) {w : Word G} (hw : Reduced φ w)
(h : ofCoprodI w.prod ∈ (base φ).range) : w = .empty
|
case intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
⊢ w = Word.empty
|
rcases hw.exists_normalWord_prod_eq d with ⟨w', hw'prod, hw'map⟩
|
case intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ w = Word.empty
|
fdebf49757164932
|
Finset.prod_involution
|
Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
/-- The difference with `Finset.prod_ninvolution` is that the involution is allowed to use
membership of the domain of the product, rather than being a non-dependent function. -/
@[to_additive "The difference with `Finset.sum_ninvolution` is that the involution is allowed to use
membership of the domain of the sum, rather than being a non-dependent function."]
lemma prod_involution (g : ∀ a ∈ s, α) (hg₁ : ∀ a ha, f a * f (g a ha) = 1)
(hg₃ : ∀ a ha, f a ≠ 1 → g a ha ≠ a)
(g_mem : ∀ a ha, g a ha ∈ s) (hg₄ : ∀ a ha, g (g a ha) (g_mem a ha) = a) :
∏ x ∈ s, f x = 1
|
case H.inl
α : Type u_3
β : Type u_4
s : Finset α
f : α → β
inst✝ : CommMonoid β
ih :
∀ t ⊂ ∅,
∀ (g : (a : α) → a ∈ t → α),
(∀ (a : α) (ha : a ∈ t), f a * f (g a ha) = 1) →
(∀ (a : α) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) →
∀ (g_mem : ∀ (a : α) (ha : a ∈ t), g a ha ∈ t), (∀ (a : α) (ha : a ∈ t), g (g a ha) ⋯ = a) → ∏ x ∈ t, f x = 1
g : (a : α) → a ∈ ∅ → α
hg₁ : ∀ (a : α) (ha : a ∈ ∅), f a * f (g a ha) = 1
hg₃ : ∀ (a : α) (ha : a ∈ ∅), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : α) (ha : a ∈ ∅), g a ha ∈ ∅
hg₄ : ∀ (a : α) (ha : a ∈ ∅), g (g a ha) ⋯ = a
⊢ ∏ x ∈ ∅, f x = 1
|
simp
|
no goals
|
c4a7eebc4974ed1c
|
Cardinal.iSup_lt_lift_of_isRegular
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem iSup_lt_lift_of_isRegular {ι} {f : ι → Cardinal} {c} (hc : IsRegular c)
(hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c) → iSup.{max u v + 1, u + 1} f < c :=
iSup_lt_lift.{u, v} (by rwa [hc.cof_eq])
|
ι : Type u
f : ι → Cardinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
⊢ lift.{v, u} #ι < c.ord.cof
|
rwa [hc.cof_eq]
|
no goals
|
93ea3e735ac3fb34
|
AnalyticOnNhd.isClopen_setOf_order_eq_top
|
Mathlib/Analysis/Analytic/Order.lean
|
theorem isClopen_setOf_order_eq_top (h₁f : AnalyticOnNhd 𝕜 f U) :
IsClopen { u : U | (h₁f u.1 u.2).order = ⊤ }
|
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h₁f : AnalyticOnNhd 𝕜 f U
z : ↑U
hz : z ∈ {u | ⋯.order = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y ≠ 0
h₂t' : IsOpen t'
h₃t' : ↑z ∈ t'
w : ↑U
hw : w ∈ Subtype.val ⁻¹' t'
h₁w : w = z
⊢ ¬⋯.order = ⊤
|
rwa [h₁w]
|
no goals
|
d5d16b4618a2912b
|
List.forIn'_congr
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean
|
theorem forIn'_congr [Monad m] {as bs : List α} (w : as = bs)
{b b' : β} (hb : b = b')
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
(h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
forIn' as b f = forIn' bs b' g
|
case cons.cons.intro.e_a.h.yield
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
b✝ b' : β
hb : b✝ = b'
a : α
as : List α
f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)
ih :
∀ {as_1 : List α} (w : as_1 = as) {b b' : β},
b = b' →
∀ {f : (a' : α) → a' ∈ as_1 → β → m (ForInStep β)} {g : (a' : α) → a' ∈ as → β → m (ForInStep β)},
(∀ (a : α) (m_1 : a ∈ as) (b : β), f a ⋯ b = g a m_1 b) → forIn' as_1 b f = forIn' as b' g
g : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)
w : a :: as = a :: as
h : ∀ (a_1 : α) (m_1 : a_1 ∈ a :: as) (b : β), f a_1 ⋯ b = g a_1 m_1 b
b : β
⊢ (forIn' as b fun a' m b => f a' ⋯ b) = forIn' as b fun a' m b => g a' ⋯ b
|
rw [ih rfl rfl]
|
case cons.cons.intro.e_a.h.yield
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
b✝ b' : β
hb : b✝ = b'
a : α
as : List α
f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)
ih :
∀ {as_1 : List α} (w : as_1 = as) {b b' : β},
b = b' →
∀ {f : (a' : α) → a' ∈ as_1 → β → m (ForInStep β)} {g : (a' : α) → a' ∈ as → β → m (ForInStep β)},
(∀ (a : α) (m_1 : a ∈ as) (b : β), f a ⋯ b = g a m_1 b) → forIn' as_1 b f = forIn' as b' g
g : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)
w : a :: as = a :: as
h : ∀ (a_1 : α) (m_1 : a_1 ∈ a :: as) (b : β), f a_1 ⋯ b = g a_1 m_1 b
b : β
⊢ ∀ (a_1 : α) (m_1 : a_1 ∈ as) (b : β), f a_1 ⋯ b = g a_1 ⋯ b
|
b21e9d91722268c4
|
ContinuousMapZero.induction_on
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
/-- An induction principle for `C(s, 𝕜)₀`. -/
@[elab_as_elim]
lemma ContinuousMapZero.induction_on {s : Set 𝕜} [Zero s] (h0 : ((0 : s) : 𝕜) = 0)
{p : C(s, 𝕜)₀ → Prop} (zero : p 0) (id : p (.id h0)) (star_id : p (star (.id h0)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(smul : ∀ (r : 𝕜) f, p f → p (r • f))
(closure : (∀ f ∈ adjoin 𝕜 {(.id h0 : C(s, 𝕜)₀)}, p f) → ∀ f, p f) (f : C(s, 𝕜)₀) :
p f
|
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Zero ↑s
h0 : ↑0 = 0
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id h0)
star_id : p (star (ContinuousMapZero.id h0))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id h0}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
hf : f ∈ adjoin 𝕜 {ContinuousMapZero.id h0}
⊢ p f
|
induction hf using NonUnitalAlgebra.adjoin_induction with
| mem f hf =>
simp only [Set.mem_union, Set.mem_singleton_iff, Set.mem_star] at hf
rw [star_eq_iff_star_eq, eq_comm (b := f)] at hf
obtain (rfl | rfl) := hf
all_goals assumption
| zero => exact zero
| add _ _ _ _ hf hg => exact add _ _ hf hg
| mul _ _ _ _ hf hg => exact mul _ _ hf hg
| smul _ _ _ hf => exact smul _ _ hf
|
no goals
|
0762fdd5f4173a39
|
OreLocalization.cardinalMk_le
|
Mathlib/GroupTheory/OreLocalization/Cardinality.lean
|
theorem cardinalMk_le : #(OreLocalization S R) ≤ #R
|
R : Type u
inst✝¹ : Monoid R
S : Submonoid R
inst✝ : OreSet S
⊢ #(OreLocalization S R) ≤ #R
|
convert ← cardinalMk_le_max S R
|
case h.e'_4
R : Type u
inst✝¹ : Monoid R
S : Submonoid R
inst✝ : OreSet S
⊢ lift.{u, u} #↥S ⊔ lift.{u, u} #R = #R
|
1786d0b66b771678
|
Finset.filter_product_card
|
Mathlib/Data/Finset/Prod.lean
|
theorem filter_product_card (s : Finset α) (t : Finset β) (p : α → Prop) (q : β → Prop)
[DecidablePred p] [DecidablePred q] :
((s ×ˢ t).filter fun x : α × β => (p x.1) = (q x.2)).card =
(s.filter p).card * (t.filter q).card +
(s.filter (¬ p ·)).card * (t.filter (¬ q ·)).card
|
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
p : α → Prop
q : β → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
⊢ Disjoint (filter (fun x => p x.1 ∧ q x.2) (s ×ˢ t)) (filter (fun x => ¬p x.1 ∧ ¬q x.2) (s ×ˢ t))
|
apply Finset.disjoint_filter_filter'
|
case h
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
p : α → Prop
q : β → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
⊢ Disjoint (fun x => p x.1 ∧ q x.2) fun x => ¬p x.1 ∧ ¬q x.2
|
7c6ab28370bb9c47
|
exists_idempotent_of_compact_t2_of_continuous_mul_left
|
Mathlib/Topology/Algebra/Semigroup.lean
|
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m
|
case ht.refine_2.intro.intro
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x => x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
⊢ m'' * m' ∈ N ∩ {m' | m' * m = m}
|
refine ⟨N_mul _ mem'' _ mem', ?_⟩
|
case ht.refine_2.intro.intro
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x => x * m) '' N = N
m'' : M
mem'' : m'' ∈ N
eq'' : m'' * m = m
m' : M
mem' : m' ∈ N
eq' : m' * m = m
⊢ m'' * m' ∈ {m' | m' * m = m}
|
fc87b9dbd8289f3d
|
Nat.two_pow_sub_pow
|
Mathlib/NumberTheory/Multiplicity.lean
|
theorem Nat.two_pow_sub_pow {x y : ℕ} (hxy : 2 ∣ x - y) (hx : ¬2 ∣ x) {n : ℕ} (hn : Even n) :
emultiplicity 2 (x ^ n - y ^ n) + 1 =
emultiplicity 2 (x + y) + emultiplicity 2 (x - y) + emultiplicity 2 n
|
case inl
x y : ℕ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : Even n
hyx : y ≤ x
⊢ emultiplicity ↑2 ↑(x ^ n - y ^ n) + 1 = emultiplicity ↑2 ↑(x + y) + emultiplicity 2 (x - y) + emultiplicity 2 n
|
rw [← Int.natCast_emultiplicity]
|
case inl
x y : ℕ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : Even n
hyx : y ≤ x
⊢ emultiplicity ↑2 ↑(x ^ n - y ^ n) + 1 = emultiplicity ↑2 ↑(x + y) + emultiplicity ↑2 ↑(x - y) + emultiplicity 2 n
|
82c1a70b09fb7622
|
Submodule.fg_of_fg_map_of_fg_inf_ker
|
Mathlib/RingTheory/Finiteness/Finsupp.lean
|
theorem fg_of_fg_map_of_fg_inf_ker (f : M →ₗ[R] P) {s : Submodule R M}
(hs1 : (s.map f).FG)
(hs2 : (s ⊓ LinearMap.ker f).FG) : s.FG
|
case intro.intro
R : Type u_1
M : Type u_2
P : Type u_4
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] P
s : Submodule R M
this✝² : DecidableEq R
this✝¹ : DecidableEq M
this✝ : DecidableEq P
t1 : Finset P
ht1 : span R ↑t1 = map f s
t2 : Finset M
ht2 : span R ↑t2 = s ⊓ LinearMap.ker f
y : P
hy : y ∈ t1
this : y ∈ map f s
x : M
hx1 : x ∈ s
hx2 : f x = y
⊢ ∃ x ∈ s, f x = y
|
exact ⟨x, hx1, hx2⟩
|
no goals
|
db8c10624993c4e9
|
Matrix.blockDiagonal_tsum
|
Mathlib/Topology/Instances/Matrix.lean
|
theorem Matrix.blockDiagonal_tsum [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R} :
blockDiagonal (∑' x, f x) = ∑' x, blockDiagonal (f x)
|
case neg
X : Type u_1
m : Type u_4
n : Type u_5
p : Type u_6
R : Type u_8
inst✝³ : AddCommMonoid R
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq p
inst✝ : T2Space R
f : X → p → Matrix m n R
hf : ¬Summable f
hft : ¬Summable fun x => blockDiagonal (f x)
⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)
|
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]
|
case neg
X : Type u_1
m : Type u_4
n : Type u_5
p : Type u_6
R : Type u_8
inst✝³ : AddCommMonoid R
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq p
inst✝ : T2Space R
f : X → p → Matrix m n R
hf : ¬Summable f
hft : ¬Summable fun x => blockDiagonal (f x)
⊢ blockDiagonal 0 = 0
|
baf1709573783a76
|
tendsto_comp_of_locally_uniform_limit
|
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
theorem tendsto_comp_of_locally_uniform_limit (h : ContinuousAt f x) (hg : Tendsto g p (𝓝 x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x))
|
α : Type u
β : Type v
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
x : α
p : Filter ι
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝[univ] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
|
exact tendsto_comp_of_locally_uniform_limit_within h hg hunif
|
no goals
|
c49b6ba895f95c85
|
PhragmenLindelof.isBigO_sub_exp_rpow
|
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
theorem isBigO_sub_exp_rpow {a : ℝ} {f g : ℂ → E} {l : Filter ℂ}
(hBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c))
(hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c)) :
∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c)
|
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c)
this :
∀ {c₁ c₂ B₁ B₂ : ℝ},
c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → (fun z => expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l] fun z => expR (B₂ * ‖z‖ ^ c₂)
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[cobounded ℂ ⊓ l] fun z => expR (Bf * ‖z‖ ^ cf)
⊢ ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c)
|
rcases hBg with ⟨cg, hcg, Bg, hOg⟩
|
case intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
this :
∀ {c₁ c₂ B₁ B₂ : ℝ},
c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → (fun z => expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l] fun z => expR (B₂ * ‖z‖ ^ c₂)
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[cobounded ℂ ⊓ l] fun z => expR (Bf * ‖z‖ ^ cf)
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[cobounded ℂ ⊓ l] fun z => expR (Bg * ‖z‖ ^ cg)
⊢ ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * ‖z‖ ^ c)
|
e171bfb3f43c7813
|
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