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Algebra.FinitePresentation.of_restrict_scalars_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B | case intro.intro.intro.intro.refine_1
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
⊢ ∀ (x : B), x ∈ adjoin A ↑⊤ | exact fun {x} => subset_adjoin ⟨⟩ | no goals | 765dbb818bc3202d |
CategoryTheory.Functor.pointwiseRightKanExtension_lift_app | Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean | @[simp]
lemma pointwiseRightKanExtension_lift_app (G : D ⥤ H) (α : L ⋙ G ⟶ F) (Y : D) :
((pointwiseRightKanExtension L F).liftOfIsRightKanExtension
(pointwiseRightKanExtensionCounit L F) G α |>.app Y) =
limit.lift _ (structuredArrowMapCone L F G α Y) | C : Type u_1
D : Type u_2
H : Type u_3
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_4, u_2} D
inst✝¹ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
inst✝ : L.HasPointwiseRightKanExtension F
G : D ⥤ H
α : L ⋙ G ⟶ F
Y : D
⊢ ((L.pointwiseRightKanExtension F).liftOfIsRightKanExtension (L.pointwiseRightKanExtensionCounit F) G α).app Y =
limit.lift (StructuredArrow.proj Y L ⋙ F) (L.structuredArrowMapCone F G α Y) | let β : G ⟶ L.pointwiseRightKanExtension F :=
{ app := fun Y ↦ limit.lift _ (structuredArrowMapCone L F G α Y) } | C : Type u_1
D : Type u_2
H : Type u_3
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_4, u_2} D
inst✝¹ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
inst✝ : L.HasPointwiseRightKanExtension F
G : D ⥤ H
α : L ⋙ G ⟶ F
Y : D
β : G ⟶ L.pointwiseRightKanExtension F :=
{ app := fun Y => limit.lift (StructuredArrow.proj Y L ⋙ F) (L.structuredArrowMapCone F G α Y), naturality := ⋯ }
⊢ ((L.pointwiseRightKanExtension F).liftOfIsRightKanExtension (L.pointwiseRightKanExtensionCounit F) G α).app Y =
limit.lift (StructuredArrow.proj Y L ⋙ F) (L.structuredArrowMapCone F G α Y) | 1d51036bafd467c7 |
CategoryTheory.IsHomLift.of_fac' | Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | lemma of_fac' {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (ha : p.obj a = R) (hb : p.obj b = S)
(h : p.map φ = eqToHom ha ≫ f ≫ eqToHom hb.symm) : p.IsHomLift f φ | 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
h : p.map φ = eqToHom ⋯ ≫ p.map φ ≫ eqToHom ⋯
⊢ p.IsHomLift (p.map φ) φ | infer_instance | no goals | 092cc10978a73530 |
exists_norm_eq_iInf_of_complete_convex | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_mem_range.2 fun _ => norm_nonneg _⟩
have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := fun w hw => δ_le ⟨w, hw⟩
-- Step 1: since `δ` is the infimum, can find a sequence `w : ℕ → K` in `K`
-- such that `‖u - w n‖ < δ + 1 / (n + 1)` (which implies `‖u - w n‖ --> δ`);
-- maybe this should be a separate lemma
have exists_seq : ∃ w : ℕ → K, ∀ n, ‖u - w n‖ < δ + 1 / (n + 1) | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
hδ : ∀ (n : ℕ), δ < δ + 1 / (↑n + 1)
h : ∀ (n : ℕ), ∃ i, ‖u - ↑i‖ < δ + 1 / (↑n + 1)
w : ℕ → ↑K := fun n => Classical.choose ⋯
⊢ ∃ w, ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1) | exact ⟨w, fun n => Classical.choose_spec (h n)⟩ | no goals | 1aed30586c16c1ed |
nullMeasurableSet_region_between_oc | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | /-- The region between two a.e.-measurable functions on a null-measurable set is null-measurable;
a version for the region together with the graph of the upper function. -/
lemma nullMeasurableSet_region_between_oc (μ : Measure α)
{f g : α → ℝ} (f_mble : AEMeasurable f μ) (g_mble : AEMeasurable g μ)
{s : Set α} (s_mble : NullMeasurableSet s μ) :
NullMeasurableSet {p : α × ℝ | p.1 ∈ s ∧ p.snd ∈ Ioc (f p.fst) (g p.fst)} (μ.prod volume) | case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | p.2 ≤ g p.1} (μ.prod volume) | rw [show {p : α × ℝ | p.snd ≤ g p.fst} = {p : α × ℝ | g p.fst < p.snd}ᶜ by
ext p
simp only [mem_setOf_eq, mem_compl_iff, not_lt]] | case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
f_mble : AEMeasurable f μ
g_mble : AEMeasurable g μ
s : Set α
s_mble : NullMeasurableSet s μ
⊢ NullMeasurableSet {p | g p.1 < p.2}ᶜ (μ.prod volume) | 0413eb85ff0d4c4f |
CategoryTheory.Limits.parallelPair_initial_mk | Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean | lemma parallelPair_initial_mk {X Y : C} (f g : X ⟶ Y)
(h₁ : ∀ Z, Nonempty (X ⟶ Z))
(h₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), ∃ (a : Y ⟶ Z), i = f ≫ a ∧ j = g ≫ a) :
(parallelPair f g).Initial :=
parallelPair_initial_mk' f g h₁ (fun Z i j => by
obtain ⟨a, rfl, rfl⟩ := h₂ i j
let f₁ : (mk (Y := zero) (f ≫ a) : CostructuredArrow (parallelPair f g) Z) ⟶ mk (Y := one) a :=
homMk left
let f₂ : (mk (Y := zero) (g ≫ a) : CostructuredArrow (parallelPair f g) Z) ⟶ mk (Y := one) a :=
homMk right
exact Zigzag.of_hom_inv f₁ f₂)
| case intro.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f g : X ⟶ Y
h₁ : ∀ (Z : C), Nonempty (X ⟶ Z)
h₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), ∃ a, i = f ≫ a ∧ j = g ≫ a
Z : C
a : Y ⟶ Z
f₁ : mk (f ≫ a) ⟶ mk a := homMk left ⋯
f₂ : mk (g ≫ a) ⟶ mk a := homMk right ⋯
⊢ Zigzag (mk (f ≫ a)) (mk (g ≫ a)) | exact Zigzag.of_hom_inv f₁ f₂ | no goals | 27db1e5d1a630a99 |
MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range' | Mathlib/Probability/Process/Stopping.lean | theorem measurableSet_lt_of_countable_range' (hτ : IsStoppingTime f τ)
(h_countable : (Set.range τ).Countable) (i : ι) :
MeasurableSet[hτ.measurableSpace] {ω | τ ω < i} | Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} | ext1 ω | case h
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i} | 8014c4cb59d652af |
EReal.eq_bot_iff_forall_lt | Mathlib/Data/Real/EReal.lean | theorem eq_bot_iff_forall_lt (x : EReal) : x = ⊥ ↔ ∀ y : ℝ, x < (y : EReal) | case mpr
x : EReal
⊢ x ≠ ⊥ → ∃ y, ↑y ≤ x | intro h | case mpr
x : EReal
h : x ≠ ⊥
⊢ ∃ y, ↑y ≤ x | 755b025f3991a944 |
Subgroup.goursat | Mathlib/GroupTheory/Goursat.lean | /-- **Goursat's lemma** for an arbitrary subgroup.
If `I` is a subgroup of `G × H`, then there exist subgroups `G' ≤ G`, `H' ≤ H` and normal subgroups
`M ⊴ G'` and `N ⊴ H'` such that `M × N ≤ I` and the image of `I` in `G' ⧸ M × H' ⧸ N` is the graph
of an isomorphism `G' ⧸ M ≃ H' ⧸ N`. -/
@[to_additive
"**Goursat's lemma** for an arbitrary subgroup.
If `I` is a subgroup of `G × H`, then there exist subgroups `G' ≤ G`, `H' ≤ H` and normal subgroups
`M ≤ G'` and `N ≤ H'` such that `M × N ≤ I` and the image of `I` in `G' ⧸ M × H' ⧸ N` is the graph
of an isomorphism `G ⧸ G' ≃ H ⧸ H'`."]
lemma goursat :
∃ (G' : Subgroup G) (H' : Subgroup H) (M : Subgroup G') (N : Subgroup H') (_ : M.Normal)
(_ : N.Normal) (e : G' ⧸ M ≃* H' ⧸ N),
I = (e.toMonoidHom.graph.comap <| (QuotientGroup.mk' M).prodMap (QuotientGroup.mk' N)).map
(G'.subtype.prodMap H'.subtype) | case intro.h.mk.mpr
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
I : Subgroup (G × H)
G' : Subgroup G := map (MonoidHom.fst G H) I
H' : Subgroup H := map (MonoidHom.snd G H) I
P : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I
Q : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I
I' : Subgroup (↥G' × ↥H') := (P.prod Q).range
hI₁' : Surjective (Prod.fst ∘ ⇑I'.subtype)
hI₂' : Surjective (Prod.snd ∘ ⇑I'.subtype)
this✝ : I'.goursatFst.Normal
this : I'.goursatSnd.Normal
e : ↥G' ⧸ I'.goursatFst ≃* ↥H' ⧸ I'.goursatSnd
he :
(((QuotientGroup.mk' I'.goursatFst).prodMap (QuotientGroup.mk' I'.goursatSnd)).comp I'.subtype).range =
e.toMonoidHom.graph
g : G
h : H
⊢ ∀ (x : H) (h_1 : (g, x) ∈ I) (x_1 : G) (h_2 : (x_1, h) ∈ I) (x_2 : G) (x_3 : H) (x_4 : (x_2, x_3) ∈ I),
P ⟨(x_2, x_3), ⋯⟩ = ⟨g, ⋯⟩ → Q ⟨(x_2, x_3), ⋯⟩ = ⟨h, ⋯⟩ → (g, h) ∈ I | rintro h₁ hgh₁ g₁ hg₁h g₂ h₂ hg₂h₂ hP hQ | case intro.h.mk.mpr
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
I : Subgroup (G × H)
G' : Subgroup G := map (MonoidHom.fst G H) I
H' : Subgroup H := map (MonoidHom.snd G H) I
P : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I
Q : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I
I' : Subgroup (↥G' × ↥H') := (P.prod Q).range
hI₁' : Surjective (Prod.fst ∘ ⇑I'.subtype)
hI₂' : Surjective (Prod.snd ∘ ⇑I'.subtype)
this✝ : I'.goursatFst.Normal
this : I'.goursatSnd.Normal
e : ↥G' ⧸ I'.goursatFst ≃* ↥H' ⧸ I'.goursatSnd
he :
(((QuotientGroup.mk' I'.goursatFst).prodMap (QuotientGroup.mk' I'.goursatSnd)).comp I'.subtype).range =
e.toMonoidHom.graph
g : G
h h₁ : H
hgh₁ : (g, h₁) ∈ I
g₁ : G
hg₁h : (g₁, h) ∈ I
g₂ : G
h₂ : H
hg₂h₂ : (g₂, h₂) ∈ I
hP : P ⟨(g₂, h₂), ⋯⟩ = ⟨g, ⋯⟩
hQ : Q ⟨(g₂, h₂), ⋯⟩ = ⟨h, ⋯⟩
⊢ (g, h) ∈ I | e32b63a688eecae2 |
PrimeSpectrum.exists_primeSpectrum_prod_le | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | theorem exists_primeSpectrum_prod_le (I : Ideal R) :
∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I | case h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, (Multiset.map asIdeal Z).prod ≤ I) J
h_prM : ¬M.IsPrime
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : (Multiset.map asIdeal Wx).prod ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : (Multiset.map asIdeal Wy).prod ≤ M + span R {y}
⊢ (Multiset.map asIdeal (Wx + Wy)).prod ≤ M | rw [Multiset.map_add, Multiset.prod_add] | case h
R : Type u
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, (Multiset.map asIdeal Z).prod ≤ I) J
h_prM : ¬M.IsPrime
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : (Multiset.map asIdeal Wx).prod ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : (Multiset.map asIdeal Wy).prod ≤ M + span R {y}
⊢ (Multiset.map asIdeal Wx).prod * (Multiset.map asIdeal Wy).prod ≤ M | 81cfa8c0bba1005f |
Algebra.FinitePresentation.of_restrict_scalars_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B | R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀✝ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
leI :
Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) A
hx : (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) x = 0
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
this : x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X)
⊢ x ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid | refine adjoin_induction ?_ ?_ ?_ ?_ this | case refine_1
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀✝ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
leI :
Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) A
hx : (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) x = 0
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
this : x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X)
⊢ ∀ x ∈ ⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X,
x ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid
case refine_2
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀✝ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
leI :
Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) A
hx : (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) x = 0
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
this : x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X)
⊢ ∀ (r : R),
(algebraMap R (MvPolynomial (Fin n) A)) r ∈
(MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid
case refine_3
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀✝ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
leI :
Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) A
hx : (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) x = 0
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
this : x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X)
⊢ ∀ (x y : MvPolynomial (Fin n) A),
x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) →
y ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) →
x ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid →
y ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid →
x + y ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid
case refine_4
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s : Finset (MvPolynomial (Fin n) R)
hs : Ideal.span ↑s = RingHom.ker f.toRingHom
RX : Type (max 0 w₁) := MvPolynomial (Fin n) R
AX : Type (max 0 w₂) := MvPolynomial (Fin n) A
t : Finset A
ht : adjoin R ↑t = ⊤
t' : { x // x ∈ t } → MvPolynomial (Fin n) R
ht' : ∀ (i : { x // x ∈ t }), f (t' i) = (algebraMap A B) ↑i
ht'' : adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) = ⊤
g : { x // x ∈ t } → AX := fun x => MvPolynomial.C ↑x - (MvPolynomial.map (algebraMap R A)) (t' x)
s₀✝ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
I : Ideal (MvPolynomial (Fin n) A) := RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
leI :
Ideal.span (⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g) ≤
RingHom.ker (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X))
x : MvPolynomial (Fin n) A
hx : (MvPolynomial.aeval (⇑f ∘ MvPolynomial.X)) x = 0
s₀ : Set (MvPolynomial (Fin n) A) := ⇑(MvPolynomial.map (algebraMap R A)) '' ↑s ∪ Set.range g
this : x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X)
⊢ ∀ (x y : MvPolynomial (Fin n) A),
x ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) →
y ∈ adjoin R (⇑(algebraMap A AX) '' ↑t ∪ Set.range MvPolynomial.X) →
x ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid →
y ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid →
x * y ∈ (MvPolynomial.map (algebraMap R A)).range.toAddSubmonoid ⊔ (Ideal.span s₀).toAddSubmonoid | 765dbb818bc3202d |
balanced_iff_neg_mem | Mathlib/Analysis/LocallyConvex/Basic.lean | theorem balanced_iff_neg_mem (hs : Convex ℝ s) : Balanced ℝ s ↔ ∀ ⦃x⦄, x ∈ s → -x ∈ s | E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
⊢ Balanced ℝ s ↔ ∀ ⦃x : E⦄, x ∈ s → -x ∈ s | refine ⟨fun h x => h.neg_mem_iff.2, fun h a ha => smul_set_subset_iff.2 fun x hx => ?_⟩ | E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
h : ∀ ⦃x : E⦄, x ∈ s → -x ∈ s
a : ℝ
ha : ‖a‖ ≤ 1
x : E
hx : x ∈ s
⊢ a • x ∈ s | 86647cdfa5c92b41 |
Set.chainHeight_eq_top_iff | Mathlib/Order/Height.lean | theorem chainHeight_eq_top_iff : s.chainHeight = ⊤ ↔ ∀ n, ∃ l ∈ s.subchain, length l = n | α : Type u_1
inst✝ : LT α
s : Set α
⊢ s.chainHeight = ⊤ ↔ ∀ (n : ℕ), ∃ l ∈ s.subchain, l.length = n | refine ⟨fun h n ↦ le_chainHeight_iff.1 (le_top.trans_eq h.symm), fun h ↦ ?_⟩ | α : Type u_1
inst✝ : LT α
s : Set α
h : ∀ (n : ℕ), ∃ l ∈ s.subchain, l.length = n
⊢ s.chainHeight = ⊤ | 8fb85cc382007eb8 |
AlgebraicGeometry.Scheme.Hom.preimage_iSup | Mathlib/AlgebraicGeometry/Scheme.lean | lemma preimage_iSup {ι} (U : ι → Opens Y) : f ⁻¹ᵁ iSup U = ⨆ i, f ⁻¹ᵁ U i :=
Opens.ext (by simp)
| X Y : Scheme
f : X.Hom Y
ι : Sort u_1
U : ι → Y.Opens
⊢ ↑(f ⁻¹ᵁ iSup U) = ↑(⨆ i, f ⁻¹ᵁ U i) | simp | no goals | e5286c159d8c8193 |
HasFDerivAt.le_of_lip' | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | theorem HasFDerivAt.le_of_lip' {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖f'‖ ≤ C | case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x₀ : E
hf : HasFDerivAt f f' x₀
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (a : E) in 𝓝 0, ‖f (x₀ + a) - f x₀‖ ≤ C * ‖x₀ + a - x₀‖
ε : ℝ
ε0 : 0 < ε
y : E
hy : ‖f (x₀ + y) - f x₀ - f' y‖ ≤ ε * ‖y‖
hyC : ‖f (x₀ + y) - f x₀‖ ≤ C * ‖y‖
⊢ ‖f' y‖ ≤ (C + ε) * ‖y‖ | calc
‖f' y‖ ≤ ‖f (x₀ + y) - f x₀‖ + ‖f (x₀ + y) - f x₀ - f' y‖ := norm_le_insert _ _
_ ≤ C * ‖y‖ + ε * ‖y‖ := add_le_add hyC hy
_ = (C + ε) * ‖y‖ := (add_mul _ _ _).symm | no goals | 8c833a4fe1e5790f |
Nat.sub_one_mul_sum_div_pow_eq_sub_sum_digits | Mathlib/Data/Nat/Digits.lean | theorem sub_one_mul_sum_div_pow_eq_sub_sum_digits {p : ℕ}
(L : List ℕ) {h_nonempty} (h_ne_zero : L.getLast h_nonempty ≠ 0) (h_lt : ∀ l ∈ L, l < p) :
(p - 1) * ∑ i ∈ range L.length, (ofDigits p L) / p ^ i.succ = (ofDigits p L) - L.sum | case inr.inr.nil
p : ℕ
h : p < 1
h_nonempty : [] ≠ []
h_ne_zero : [].getLast h_nonempty ≠ 0
h_lt : ∀ l ∈ [], l < p
⊢ 0 = ofDigits 0 [] - [].sum | rfl | no goals | a136b4f3960a4789 |
Std.DHashMap.Internal.Raw₀.Const.get!_insertIfNew | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem get!_insertIfNew [EquivBEq α] [LawfulHashable α] [Inhabited β] (h : m.1.WF) {k a : α}
{v : β} :
get! (m.insertIfNew k v) a = if k == a ∧ m.contains k = false then v else get! m a | α : Type u
inst✝⁴ : BEq α
inst✝³ : Hashable α
β : Type v
m : Raw₀ α fun x => β
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited β
h : m.val.WF
k a : α
v : β
⊢ get! (m.insertIfNew k v) a = if (k == a) = true ∧ m.contains k = false then v else get! m a | simp_to_model [insertIfNew] using List.getValue!_insertEntryIfNew | no goals | cec8317ff74e0b06 |
Ideal.finite_quotient_pow | Mathlib/RingTheory/Ideal/Quotient/Index.lean | lemma Ideal.finite_quotient_pow (hI : I.FG) [Finite (R ⧸ I)] (n) : Finite (R ⧸ I ^ n) | case zero
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
hI : I.FG
inst✝ : Finite (R ⧸ I)
⊢ Finite (R ⧸ I ^ 0) | simp only [pow_zero, Ideal.one_eq_top] | case zero
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
hI : I.FG
inst✝ : Finite (R ⧸ I)
⊢ Finite (R ⧸ ⊤) | a872582db39414d1 |
Composition.mem_range_embedding_iff | Mathlib/Combinatorics/Enumerative/Composition.lean | theorem mem_range_embedding_iff {j : Fin n} {i : Fin c.length} :
j ∈ Set.range (c.embedding i) ↔ c.sizeUpTo i ≤ j ∧ (j : ℕ) < c.sizeUpTo (i : ℕ).succ | n : ℕ
c : Composition n
j : Fin n
i : Fin c.length
⊢ j ∈ Set.range ⇑(c.embedding i) ↔ c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ | constructor | case mp
n : ℕ
c : Composition n
j : Fin n
i : Fin c.length
⊢ j ∈ Set.range ⇑(c.embedding i) → c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ
case mpr
n : ℕ
c : Composition n
j : Fin n
i : Fin c.length
⊢ c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ → j ∈ Set.range ⇑(c.embedding i) | ffb2f79e5a4de694 |
integral_mul_cpow_one_add_sq | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_mul_cpow_one_add_sq {t : ℂ} (ht : t ≠ -1) :
(∫ x : ℝ in a..b, (x : ℂ) * ((1 : ℂ) + ↑x ^ 2) ^ t) =
((1 : ℂ) + (b : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1)) -
((1 : ℂ) + (a : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1)) | case hint.hu
a b : ℝ
t : ℂ
ht : t ≠ -1
this : t + 1 ≠ 0
⊢ Continuous fun y => ↑y * (1 + ↑y ^ 2) ^ t | refine continuous_ofReal.mul ?_ | case hint.hu
a b : ℝ
t : ℂ
ht : t ≠ -1
this : t + 1 ≠ 0
⊢ Continuous fun y => (1 + ↑y ^ 2) ^ t | 5e0a067782411706 |
List.ofFn_getElem | Mathlib/Data/List/OfFn.lean | theorem ofFn_getElem : ∀ l : List α, (ofFn (fun i : Fin l.length => l[(i : Nat)])) = l
| [] => by rw [ofFn_zero]
| a :: l => by
rw [ofFn_succ]
congr
exact ofFn_get l
| α : Type u
a : α
l : List α
⊢ (ofFn fun i => (a :: l)[↑i]) = a :: l | rw [ofFn_succ] | α : Type u
a : α
l : List α
⊢ ((a :: l)[↑0] :: ofFn fun i => (a :: l)[↑i.succ]) = a :: l | 4249a4a6f2cbd535 |
IsNilpotent.exp_add_of_commute | Mathlib/RingTheory/Nilpotent/Exp.lean | theorem exp_add_of_commute {a b : A} (h₁ : Commute a b) (h₂ : IsNilpotent a) (h₃ : IsNilpotent b) :
exp (a + b) = exp a * exp b | case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
⊢ exp (a + b) = exp a * exp b | let N := n₁ ⊔ n₂ | case intro.intro
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
⊢ exp (a + b) = exp a * exp b | 925becc4d11e37f5 |
irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree | Mathlib/Algebra/Squarefree/Basic.lean | theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r | R : Type u_1
inst✝¹ : CommMonoidWithZero R
inst✝ : WfDvdMonoid R
r : R
⊢ (∀ (x : R), Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ (x : R), ¬Irreducible x) ∨ Squarefree r | refine ⟨fun h ↦ ?_, ?_⟩ | case refine_1
R : Type u_1
inst✝¹ : CommMonoidWithZero R
inst✝ : WfDvdMonoid R
r : R
h : ∀ (x : R), Irreducible x → ¬x * x ∣ r
⊢ (r = 0 ∧ ∀ (x : R), ¬Irreducible x) ∨ Squarefree r
case refine_2
R : Type u_1
inst✝¹ : CommMonoidWithZero R
inst✝ : WfDvdMonoid R
r : R
⊢ (r = 0 ∧ ∀ (x : R), ¬Irreducible x) ∨ Squarefree r → ∀ (x : R), Irreducible x → ¬x * x ∣ r | 95d2bc81f47d92e1 |
Real.sin_pi_div_thirty_two | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem sin_pi_div_thirty_two : sin (π / 32) = √(2 - √(2 + √(2 + √2))) / 2 | ⊢ sin (π / 32) = sin (π / 2 ^ 5) | congr | case e_x.e_a
⊢ 32 = 2 ^ 5 | c2b49e6b83d3738b |
ContinuousSMul.of_nhds_zero | Mathlib/Topology/Algebra/Module/Basic.lean | theorem ContinuousSMul.of_nhds_zero [IsTopologicalRing R] [IsTopologicalAddGroup M]
(hmul : Tendsto (fun p : R × M => p.1 • p.2) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0))
(hmulleft : ∀ m : M, Tendsto (fun a : R => a • m) (𝓝 0) (𝓝 0))
(hmulright : ∀ a : R, Tendsto (fun m : M => a • m) (𝓝 0) (𝓝 0)) : ContinuousSMul R M where
continuous_smul | R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsTopologicalRing R
inst✝ : IsTopologicalAddGroup M
hmul : Tendsto (fun p => p.1 • p.2) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)
hmulleft : ∀ (m : M), Tendsto (fun a => a • m) (𝓝 0) (𝓝 0)
hmulright : ∀ (a : R), Tendsto (fun m => a • m) (𝓝 0) (𝓝 0)
⊢ Continuous fun p => p.1 • p.2 | rw [← nhds_prod_eq] at hmul | R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsTopologicalRing R
inst✝ : IsTopologicalAddGroup M
hmul : Tendsto (fun p => p.1 • p.2) (𝓝 (0, 0)) (𝓝 0)
hmulleft : ∀ (m : M), Tendsto (fun a => a • m) (𝓝 0) (𝓝 0)
hmulright : ∀ (a : R), Tendsto (fun m => a • m) (𝓝 0) (𝓝 0)
⊢ Continuous fun p => p.1 • p.2 | df1a10ae168101e9 |
Finset.card_bij | Mathlib/Data/Finset/Card.lean | /-- Reorder a finset.
The difference with `Finset.card_bij'` is that the bijection is specified as a surjective injection,
rather than by an inverse function.
The difference with `Finset.card_nbij` is that the bijection is allowed to use membership of the
domain, rather than being a non-dependent function. -/
lemma card_bij (i : ∀ a ∈ s, β) (hi : ∀ a ha, i a ha ∈ t)
(i_inj : ∀ a₁ ha₁ a₂ ha₂, i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂)
(i_surj : ∀ b ∈ t, ∃ a ha, i a ha = b) : #s = #t | α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), i a ha = b
⊢ #s = #t | calc
#s = #s.attach := card_attach.symm
_ = #(s.attach.image fun a ↦ i a.1 a.2) := Eq.symm ?_
_ = #t := ?_ | case calc_1
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), i a ha = b
⊢ #(image (fun a => i ↑a ⋯) s.attach) = #s.attach
case calc_2
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), i a ha = b
⊢ #(image (fun a => i ↑a ⋯) s.attach) = #t | f0c71609f28a0461 |
Std.Tactic.BVDecide.Normalize.BitVec.bne_to_beq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/Canonicalize.lean | theorem BitVec.bne_to_beq (a b : BitVec w) : (a != b) = (!(a == b)) | w : Nat
a b : BitVec w
⊢ (a != b) = !a == b | simp [bne] | no goals | 6b85173f9b2912ae |
List.formPerm_ext_iff | Mathlib/GroupTheory/Perm/List.lean | theorem formPerm_ext_iff {x y x' y' : α} {l l' : List α} (hd : Nodup (x :: y :: l))
(hd' : Nodup (x' :: y' :: l')) :
formPerm (x :: y :: l) = formPerm (x' :: y' :: l') ↔ (x :: y :: l) ~r (x' :: y' :: l') | case h
α : Type u_1
inst✝ : DecidableEq α
x y x' y' : α
l l' : List α
hd : (x :: y :: l).Nodup
hd' : (x' :: y' :: l').Nodup
h : ∀ (x_1 : α), (x :: y :: l).formPerm x_1 = (x' :: y' :: l').formPerm x_1
hx : x' ∈ x :: y :: l
n : ℕ
hn : n < (x :: y :: l).length
hx' : (x :: y :: l).get ⟨n, hn⟩ = x'
hl : (x :: y :: l).length = (x' :: y' :: l').length
⊢ (x :: y :: l).rotate n = x' :: y' :: l' | apply List.ext_getElem | case h.hl
α : Type u_1
inst✝ : DecidableEq α
x y x' y' : α
l l' : List α
hd : (x :: y :: l).Nodup
hd' : (x' :: y' :: l').Nodup
h : ∀ (x_1 : α), (x :: y :: l).formPerm x_1 = (x' :: y' :: l').formPerm x_1
hx : x' ∈ x :: y :: l
n : ℕ
hn : n < (x :: y :: l).length
hx' : (x :: y :: l).get ⟨n, hn⟩ = x'
hl : (x :: y :: l).length = (x' :: y' :: l').length
⊢ ((x :: y :: l).rotate n).length = (x' :: y' :: l').length
case h.h
α : Type u_1
inst✝ : DecidableEq α
x y x' y' : α
l l' : List α
hd : (x :: y :: l).Nodup
hd' : (x' :: y' :: l').Nodup
h : ∀ (x_1 : α), (x :: y :: l).formPerm x_1 = (x' :: y' :: l').formPerm x_1
hx : x' ∈ x :: y :: l
n : ℕ
hn : n < (x :: y :: l).length
hx' : (x :: y :: l).get ⟨n, hn⟩ = x'
hl : (x :: y :: l).length = (x' :: y' :: l').length
⊢ ∀ (i : ℕ) (h₁ : i < ((x :: y :: l).rotate n).length) (h₂ : i < (x' :: y' :: l').length),
((x :: y :: l).rotate n)[i] = (x' :: y' :: l')[i] | 355bf7b7e44c963c |
Array.flatten_toArray_map_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem flatten_toArray_map_toArray (xss : List (List α)) :
(xss.map List.toArray).toArray.flatten = xss.flatten.toArray | case cons
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (as : Array α), List.foldl (fun r a => r ++ a) as (List.map List.toArray xss) = as ++ xss.flatten.toArray
as : Array α
⊢ List.foldl (fun r a => r ++ a) as (List.map List.toArray (xs :: xss)) = as ++ (xs :: xss).flatten.toArray | simp [ih] | no goals | 229d39ebde69cbd4 |
exists_integer_of_is_root_of_monic | Mathlib/RingTheory/Polynomial/RationalRoot.lean | theorem exists_integer_of_is_root_of_monic {p : A[X]} (hp : Monic p) {r : K} (hr : aeval r p = 0) :
∃ r' : A, r = algebraMap A K r' ∧ r' ∣ p.coeff 0 | A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
hp : p.Monic
r : K
hr : (aeval r) p = 0
⊢ ∃ r', r = (algebraMap A K) r' ∧ r' ∣ p.coeff 0 | obtain ⟨inv, h_inv⟩ := hp ▸ den_dvd_of_is_root hr | case intro
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
hp : p.Monic
r : K
hr : (aeval r) p = 0
inv : A
h_inv : 1 = ↑(den A r) * inv
⊢ ∃ r', r = (algebraMap A K) r' ∧ r' ∣ p.coeff 0 | f67f88843b2c3ec9 |
fourier_zero | Mathlib/Analysis/Fourier/AddCircle.lean | theorem fourier_zero {x : AddCircle T} : fourier 0 x = 1 | T : ℝ
x : AddCircle T
⊢ (fourier 0) x = 1 | induction x using QuotientAddGroup.induction_on | case H
T z✝ : ℝ
⊢ (fourier 0) ↑z✝ = 1 | 88538cd150aac376 |
OrderIso.map_minimal_mem | Mathlib/Order/Minimal.lean | theorem map_minimal_mem (f : s ≃o t) (hx : Minimal (· ∈ s) x) :
Minimal (· ∈ t) (f ⟨x, hx.prop⟩) | α : Type u_1
x : α
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
s : Set α
t : Set β
f : ↑s ≃o ↑t
hx : Minimal (fun x => x ∈ s) x
⊢ t = range (Subtype.val ∘ ⇑f) | simp | no goals | 16ce13c08cb6f454 |
Batteries.UnionFind.rankD_findAux | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem rankD_findAux {self : UnionFind} {x : Fin self.size} :
rankD (findAux self x).s i = self.rank i | i : Nat
self : UnionFind
x : Fin self.size
h : i < self.size
⊢ rankD (self.findAux x).s i = self.rank i | rw [findAux_s] | i : Nat
self : UnionFind
x : Fin self.size
h : i < self.size
⊢ rankD
(if self.arr[↑x].parent = ↑x then self.arr
else (self.findAux ⟨self.arr[↑x].parent, ⋯⟩).s.modify ↑x fun s => { parent := self.rootD ↑x, rank := s.rank })
i =
self.rank i | df1ad95e2480e7f4 |
MeromorphicOn.order_ne_top_of_isPreconnected | Mathlib/Analysis/Meromorphic/Order.lean | theorem order_ne_top_of_isPreconnected {x y : 𝕜} (hU : IsPreconnected U) (h₁x : x ∈ U) (hy : y ∈ U)
(h₂x : (hf x h₁x).order ≠ ⊤) :
(hf y hy).order ≠ ⊤ :=
(hf.exists_order_ne_top_iff_forall ⟨nonempty_of_mem h₁x, hU⟩).1 (by use ⟨x, h₁x⟩) ⟨y, hy⟩
| 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
hf : MeromorphicOn f U
x y : 𝕜
hU : IsPreconnected U
h₁x : x ∈ U
hy : y ∈ U
h₂x : ⋯.order ≠ ⊤
⊢ ∃ u, ⋯.order ≠ ⊤ | use ⟨x, h₁x⟩ | no goals | ac3d540f6942d753 |
MaximalSpectrum.toPiLocalization_not_surjective_of_infinite | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | theorem toPiLocalization_not_surjective_of_infinite [Infinite ι] :
¬ Function.Surjective (toPiLocalization (Π i, R i)) := fun surj ↦ by
classical
have ⟨J, max, nmem⟩ := PrimeSpectrum.exists_maximal_nmem_range_sigmaToPi_of_infinite R
obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max⟩ 1)
have : r = 0 := funext fun i ↦ toPiLocalization_injective _ <| funext fun I ↦ by
replace hr := congr_fun hr ⟨_, I.2.comap_piEvalRingHom⟩
dsimp only [toPiLocalization_apply_apply, Subtype.coe_mk] at hr
simp_rw [toPiLocalization_apply_apply,
← Localization.AtPrime.mapPiEvalRingHom_algebraMap_apply, hr]
rw [Function.update_of_ne]; · simp_rw [Pi.zero_apply, map_zero]
exact fun h ↦ nmem ⟨⟨i, I.1, I.2.isPrime⟩, PrimeSpectrum.ext congr($h.1)⟩
replace hr := congr_fun hr ⟨J, max⟩
rw [this, map_zero, Function.update_self] at hr
exact zero_ne_one hr
| ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
J : Ideal ((i : ι) → R i)
max : J.IsMaximal
nmem : { asIdeal := J, isPrime := ⋯ } ∉ Set.range (PrimeSpectrum.sigmaToPi R)
r : (i : ι) → R i
i : ι
I : MaximalSpectrum (R i)
hr :
(toPiLocalization ((i : ι) → R i)) r { asIdeal := Ideal.comap (Pi.evalRingHom R i) I.asIdeal, isMaximal := ⋯ } =
Function.update 0 { asIdeal := J, isMaximal := max } 1
{ asIdeal := Ideal.comap (Pi.evalRingHom R i) I.asIdeal, isMaximal := ⋯ }
⊢ (toPiLocalization (R i)) (r i) I = (toPiLocalization (R i)) (0 i) I | dsimp only [toPiLocalization_apply_apply, Subtype.coe_mk] at hr | ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
J : Ideal ((i : ι) → R i)
max : J.IsMaximal
nmem : { asIdeal := J, isPrime := ⋯ } ∉ Set.range (PrimeSpectrum.sigmaToPi R)
r : (i : ι) → R i
i : ι
I : MaximalSpectrum (R i)
hr :
(algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I.asIdeal))) r =
Function.update 0 { asIdeal := J, isMaximal := max } 1
{ asIdeal := Ideal.comap (Pi.evalRingHom R i) I.asIdeal, isMaximal := ⋯ }
⊢ (toPiLocalization (R i)) (r i) I = (toPiLocalization (R i)) (0 i) I | 695740c4e99b5371 |
Affine.Triangle.equilateral_iff_dist_eq_and_dist_eq | Mathlib/Analysis/Normed/Affine/Simplex.lean | lemma equilateral_iff_dist_eq_and_dist_eq {t : Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
(h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
t.Equilateral ↔ dist (t.points i₁) (t.points i₂) = dist (t.points i₁) (t.points i₃) ∧
dist (t.points i₁) (t.points i₂) = dist (t.points i₂) (t.points i₃) | case refine_1
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
x✝ : Simplex.Equilateral t
r : ℝ
hr : ∀ (i j : Fin (2 + 1)), i ≠ j → dist (t.points i) (t.points j) = r
⊢ dist (t.points i₁) (t.points i₂) = dist (t.points i₁) (t.points i₃) ∧
dist (t.points i₁) (t.points i₂) = dist (t.points i₂) (t.points i₃) | simp [hr _ _ h₁₂, hr _ _ h₁₃, hr _ _ h₂₃] | no goals | e01c30a9004c5510 |
exists_perfect_nonempty_of_isClosed_of_not_countable | Mathlib/Topology/Perfect.lean | theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
(hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C | case intro.intro.intro.intro.right
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬C.Countable
V D : Set α
Vct : V.Countable
Dperf : Perfect D
VD : C = V ∪ D
⊢ D ⊆ V ∪ D | exact subset_union_right | no goals | 5729719cdfac83fb |
ContMDiffWithinAt.comp | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) s x | case h.intro
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
E'' : Type u_8
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝³ : TopologicalSpace M''
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
n : WithTop ℕ∞
t : Set M'
g : M' → M''
x : M
st : MapsTo f s t
e : PartialEquiv M E := extChartAt I x
e' : PartialEquiv M' E' := extChartAt I' (f x)
hg :
ContinuousWithinAt g t (f x) ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑e'.symm) (↑e'.symm ⁻¹' t ∩ range ↑I')
(writtenInExtChartAt I I' x f (↑e x))
hf : ContinuousWithinAt f s x ∧ ContDiffWithinAt 𝕜 n (↑e' ∘ f ∘ ↑e.symm) (↑e.symm ⁻¹' s ∩ range ↑I) (↑e x)
this : ↑e' (f x) = writtenInExtChartAt I I' x f (↑e x)
A : ∀ᶠ (y : E) in 𝓝[↑e.symm ⁻¹' s ∩ range ↑I] ↑e x, f (↑e.symm y) ∈ t ∧ f (↑e.symm y) ∈ e'.source
x' : E
ht : f (↑e.symm x') ∈ t
hfx' : f (↑e.symm x') ∈ e'.source
⊢ ↑(extChartAt I' (f x)) (f (↑(extChartAt I x).symm x')) ∈ range ↑I' | exact mem_range_self _ | no goals | 21a30820f7079ed0 |
Matroid.cRk_map_image_lift | Mathlib/Data/Matroid/Rank/Cardinal.lean | theorem cRk_map_image_lift (M : Matroid α) (hf : InjOn f M.E) (X : Set α)
(hX : X ⊆ M.E | α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : autoParam (X ⊆ M.E) _auto✝
⊢ lift.{u, v} ((M.map f hf).cRk (f '' X)) = lift.{v, u} (M.cRk X) | nth_rw 1 [cRk, cRank, le_antisymm_iff, lift_iSup (bddAbove_range _), cRk, cRank, cRk, cRank] | α : Type u
β : Type v
f : α → β
M : Matroid α
hf : InjOn f M.E
X : Set α
hX : autoParam (X ⊆ M.E) _auto✝
⊢ ⨆ i, lift.{u, v} #↑↑i ≤ lift.{v, u} (⨆ B, #↑↑B) ∧ lift.{v, u} (⨆ B, #↑↑B) ≤ lift.{u, v} (⨆ B, #↑↑B) | 58e7e885cad0821d |
List.splitWrtComposition_flatten | Mathlib/Combinatorics/Enumerative/Composition.lean | theorem splitWrtComposition_flatten (L : List (List α)) (c : Composition L.flatten.length)
(h : map length L = c.blocks) : splitWrtComposition (flatten L) c = L | α : Type u_1
L : List (List α)
c : Composition L.flatten.length
h : map length L = c.blocks
⊢ L.flatten.splitWrtComposition c = L | simp only [eq_self_iff_true, and_self_iff, eq_iff_flatten_eq, flatten_splitWrtComposition,
map_length_splitWrtComposition, h] | no goals | 56350ce3184733fc |
smoothingSeminormSeq_tendsto_aux | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem smoothingSeminormSeq_tendsto_aux {L : ℝ} (hL : 0 ≤ L) {ε : ℝ} (hε : 0 < ε)
{m1 : ℕ} (hm1 : 0 < m1) {x : R} (hx : μ x ≠ 0) :
Tendsto
(fun n : ℕ => (L + ε) ^ (-(((n % m1 : ℕ) : ℝ) / (n : ℝ))) * (μ x ^ (n % m1)) ^ (1 / (n : ℝ)))
atTop (𝓝 1) | case hf
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n => ↑(n % m1) / ↑n) atTop (𝓝 0)
h0 : Tendsto (fun t => -(↑(t % m1) / ↑t)) atTop (𝓝 0)
⊢ Tendsto (fun x => (L + ε) ^ (-(↑(x % m1) / ↑x))) atTop (𝓝 ((L + ε) ^ 0)) | apply Tendsto.rpow tendsto_const_nhds h0 | case hf
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n => ↑(n % m1) / ↑n) atTop (𝓝 0)
h0 : Tendsto (fun t => -(↑(t % m1) / ↑t)) atTop (𝓝 0)
⊢ L + ε ≠ 0 ∨ 0 < 0 | 24d0886b35610e91 |
SatisfiesM_StateT_eq | Mathlib/.lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | theorem SatisfiesM_StateT_eq [Monad m] [LawfulMonad m] :
SatisfiesM (m := StateT ρ m) (α := α) p x ↔ ∀ s, SatisfiesM (m := m) (p ·.1) (x.run s) | case refine_1
m : Type u_1 → Type u_2
α ρ : Type u_1
p : α → Prop
x : StateT ρ m α
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x✝ : SatisfiesM p x
f : StateT ρ m { a // p a }
eq : Subtype.val <$> f = x
⊢ ∃ f_1, ∀ (x : ρ), Subtype.val <$> f_1 x = (Subtype.val <$> f) x | refine ⟨fun s => (fun ⟨⟨a, h⟩, s'⟩ => ⟨⟨a, s'⟩, h⟩) <$> f s, fun s => ?_⟩ | case refine_1
m : Type u_1 → Type u_2
α ρ : Type u_1
p : α → Prop
x : StateT ρ m α
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x✝ : SatisfiesM p x
f : StateT ρ m { a // p a }
eq : Subtype.val <$> f = x
s : ρ
⊢ Subtype.val <$>
(fun s =>
(fun x =>
match x with
| (⟨a, h⟩, s') => ⟨(a, s'), h⟩) <$>
f s)
s =
(Subtype.val <$> f) s | 2505b9876bea8db3 |
LaurentPolynomial.smul_eq_C_mul | Mathlib/Algebra/Polynomial/Laurent.lean | theorem smul_eq_C_mul (r : R) (f : R[T;T⁻¹]) : r • f = C r * f | R : Type u_1
inst✝ : Semiring R
r : R
f : R[T;T⁻¹]
⊢ r • f = C r * f | induction f using LaurentPolynomial.induction_on' with
| h_add _ _ hp hq =>
rw [smul_add, mul_add, hp, hq]
| h_C_mul_T n s =>
rw [← mul_assoc, ← smul_mul_assoc, mul_left_inj_of_invertible, ← map_mul, ← single_eq_C,
Finsupp.smul_single', single_eq_C] | no goals | f43d33374b4834e1 |
Basis.restrictScalars_repr_apply | Mathlib/LinearAlgebra/Basis/Basic.lean | theorem Basis.restrictScalars_repr_apply (m : span R (Set.range b)) (i : ι) :
algebraMap R S ((b.restrictScalars R).repr m i) = b.repr m i | ι : Type u_1
R : Type u_3
M : Type u_5
S : Type u_7
inst✝⁸ : CommRing R
inst✝⁷ : Ring S
inst✝⁶ : Nontrivial S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module S M
inst✝² : Module R M
inst✝¹ : IsScalarTower R S M
inst✝ : NoZeroSMulDivisors R S
b : Basis ι S M
m : ↥(span R (range ⇑b))
i x✝ : ι
⊢ (mapRange.linearMap (Algebra.linearMap R S) ∘ₗ ↑(restrictScalars R b).repr) ((restrictScalars R b) x✝) =
((↑R ↑b.repr).domRestrict (span R (range ⇑b))) ((restrictScalars R b) x✝) | simp only [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, Function.comp_apply, map_one,
Basis.repr_self, Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single,
Algebra.linearMap_apply, LinearMap.domRestrict_apply, LinearEquiv.coe_coe,
Basis.restrictScalars_apply, LinearMap.coe_restrictScalars] | no goals | d8b491e6b35d79e7 |
LieModule.nilpotencyLength_eq_succ_iff | Mathlib/Algebra/Lie/Nilpotent.lean | theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
nilpotencyLength L M = k + 1 ↔
lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥ | R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k✝ k : ℕ
⊢ lowerCentralSeries R L M k = ⊥ ↔ lowerCentralSeries ℤ L M k = ⊥ | simp [SetLike.ext'_iff, coe_lowerCentralSeries_eq_int R L M] | no goals | ed799dc48a9168c3 |
deriv_norm_ofReal_cpow | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | theorem deriv_norm_ofReal_cpow (c : ℂ) {t : ℝ} (ht : 0 < t) :
(deriv fun x : ℝ ↦ ‖(x : ℂ) ^ c‖) t = c.re * t ^ (c.re - 1) | c : ℂ
t : ℝ
ht : 0 < t
⊢ (fun x => ‖↑x ^ c‖) =ᶠ[𝓝 t] fun x => x ^ c.re | filter_upwards [eventually_gt_nhds ht] with x hx | case h
c : ℂ
t : ℝ
ht : 0 < t
x : ℝ
hx : 0 < x
⊢ ‖↑x ^ c‖ = x ^ c.re | d2fb34ba57b68f3a |
nhdset_of_mem_uniformity | Mathlib/Topology/UniformSpace/Basic.lean | theorem nhdset_of_mem_uniformity {d : Set (α × α)} (s : Set (α × α)) (hd : d ∈ 𝓤 α) :
∃ t : Set (α × α), IsOpen t ∧ s ⊆ t ∧
t ⊆ { p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d } | α : Type ua
inst✝ : UniformSpace α
d s : Set (α × α)
hd : d ∈ 𝓤 α
⊢ ∃ t, IsOpen t ∧ s ⊆ t ∧ t ⊆ {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d} | let cl_d := { p : α × α | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d } | α : Type ua
inst✝ : UniformSpace α
d s : Set (α × α)
hd : d ∈ 𝓤 α
cl_d : Set (α × α) := {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d}
⊢ ∃ t, IsOpen t ∧ s ⊆ t ∧ t ⊆ {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d} | eb17f65275ef735d |
Array.flatten_eq_push_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem flatten_eq_push_iff {xs : Array (Array α)} {ys : Array α} {y : α} :
xs.flatten = ys.push y ↔
∃ (as : Array (Array α)) (bs : Array α) (cs : Array (Array α)),
xs = as.push (bs.push y) ++ cs ∧ (∀ l, l ∈ cs → l = #[]) ∧ ys = as.flatten ++ bs | case of.mk
α : Type u_1
y : α
xs : List (List α)
ys : List α
⊢ (List.map List.toArray xs).toArray.flatten = { toList := ys }.push y ↔
∃ as bs cs,
(List.map List.toArray xs).toArray = as.push (bs.push y) ++ cs ∧
(∀ (l : Array α), l ∈ cs → l = #[]) ∧ { toList := ys } = as.flatten ++ bs | rw [flatten_toArray_map, List.push_toArray, mk.injEq, List.flatten_eq_append_iff] | case of.mk
α : Type u_1
y : α
xs : List (List α)
ys : List α
⊢ ((∃ as bs, xs = as ++ bs ∧ ys = as.flatten ∧ [y] = bs.flatten) ∨
∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.flatten ++ bs ∧ [y] = c :: cs ++ ds.flatten) ↔
∃ as bs cs,
(List.map List.toArray xs).toArray = as.push (bs.push y) ++ cs ∧
(∀ (l : Array α), l ∈ cs → l = #[]) ∧ { toList := ys } = as.flatten ++ bs | dbe592481896286a |
Bimod.AssociatorBimod.hom_right_act_hom' | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem hom_right_act_hom' :
((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
(hom P Q L ▷ U.X) ≫ (P.tensorBimod (Q.tensorBimod L)).actRight | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (tensorRight U.X).map
(coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft)) ≫
TensorBimod.actRight (P.tensorBimod Q) L ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
coequalizer.desc
((α_ P.X Q.X L.X).hom ≫
P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
⋯)
⋯ =
(tensorRight U.X).map
(coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft)) ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
coequalizer.desc
((α_ P.X Q.X L.X).hom ≫
P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
⋯)
⋯ ▷
U.X ≫
TensorBimod.actRight P (Q.tensorBimod L) | rw [tensorRight_map] | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ▷
U.X ≫
TensorBimod.actRight (P.tensorBimod Q) L ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
coequalizer.desc
((α_ P.X Q.X L.X).hom ≫
P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
⋯)
⋯ =
coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ▷
U.X ≫
coequalizer.desc
((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
coequalizer.desc
((α_ P.X Q.X L.X).hom ≫
P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
⋯)
⋯ ▷
U.X ≫
TensorBimod.actRight P (Q.tensorBimod L) | 26e7cbaccdd0f16e |
MeasureTheory.IntegrableAtFilter.add | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem IntegrableAtFilter.add {f g : α → E}
(hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
IntegrableAtFilter (f + g) l μ | α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f g : α → E
hf : IntegrableAtFilter f l μ
hg : IntegrableAtFilter g l μ
⊢ IntegrableAtFilter (f + g) l μ | rcases hf with ⟨s, sl, hs⟩ | case intro.intro
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f g : α → E
hg : IntegrableAtFilter g l μ
s : Set α
sl : s ∈ l
hs : IntegrableOn f s μ
⊢ IntegrableAtFilter (f + g) l μ | 0153623a59ae8a3c |
PiTensorProduct.mapL_id | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | theorem mapL_id : mapL (fun i ↦ ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id _ _ | ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ (mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i) | apply ContinuousLinearMap.coe_injective | case a
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ ↑(mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ↑(ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i)) | 917b0e4be81d5c73 |
CategoryTheory.MonoidalCategory.tensor_left_unitality | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem tensor_left_unitality (X₁ X₂ : C) :
(λ_ (X₁ ⊗ X₂)).hom =
((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
tensorμ (𝟙_ C) (𝟙_ C) X₁ X₂ ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (λ_ (X₁ ⊗ X₂)).hom = (λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂) ≫ tensorμ (𝟙_ C) (𝟙_ C) X₁ X₂ ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) | dsimp only [tensorμ] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (λ_ (X₁ ⊗ X₂)).hom =
(λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂) ≫
((α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫
𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).inv ≫
𝟙_ C ◁ (β_ (𝟙_ C) X₁).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).hom ≫ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).inv) ≫
((λ_ X₁).hom ⊗ (λ_ X₂).hom) | 530d3eb02dfe571c |
Nat.nth_strictMonoOn | Mathlib/Data/Nat/Nth.lean | theorem nth_strictMonoOn (hf : (setOf p).Finite) :
StrictMonoOn (nth p) (Set.Iio #hf.toFinset) | p : ℕ → Prop
hf : (setOf p).Finite
m : ℕ
hm : m < #hf.toFinset
n : ℕ
hn : n < #hf.toFinset
h : m < n
⊢ (hf.toFinset.orderEmbOfFin ⋯) ⟨m, hm⟩ < (hf.toFinset.orderEmbOfFin ⋯) ⟨n, hn⟩ | exact OrderEmbedding.strictMono _ h | no goals | 2b86549b9bed1189 |
Cubic.map_roots | Mathlib/Algebra/CubicDiscriminant.lean | theorem map_roots [IsDomain S] : (map φ P).roots = (Polynomial.map φ P.toPoly).roots | R : Type u_1
S : Type u_2
P : Cubic R
inst✝² : CommRing R
inst✝¹ : CommRing S
φ : R →+* S
inst✝ : IsDomain S
⊢ (map φ P).roots = (Polynomial.map φ P.toPoly).roots | rw [roots, map_toPoly] | no goals | a8a262f157779147 |
AnalyticAt.order_mul | Mathlib/Analysis/Analytic/Order.lean | theorem order_mul {f g : 𝕜 → 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
(hf.mul hg).order = hf.order + hg.order | case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
f g : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
h₂f : ¬hf.order = ⊤
⊢ ⋯.order = hf.order + hg.order | by_cases h₂g : hg.order = ⊤ | case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
f g : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
h₂f : ¬hf.order = ⊤
h₂g : hg.order = ⊤
⊢ ⋯.order = hf.order + hg.order
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
f g : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
h₂f : ¬hf.order = ⊤
h₂g : ¬hg.order = ⊤
⊢ ⋯.order = hf.order + hg.order | 9c09f7b0bc9e51ba |
groupCohomology.cocyclesMap_comp_isoTwoCocycles_hom | Mathlib/RepresentationTheory/GroupCohomology/Functoriality.lean | @[reassoc (attr := simp), elementwise (attr := simp)]
lemma cocyclesMap_comp_isoTwoCocycles_hom :
cocyclesMap f φ 2 ≫ (isoTwoCocycles B).hom = (isoTwoCocycles A).hom ≫ mapTwoCocycles f φ | k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : (Action.res (ModuleCat k) f).obj A ⟶ B
⊢ (cocyclesMap f φ 2 ≫
(HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains B) 1 2 3 isoTwoCocycles.proof_3
isoTwoCocycles.proof_4 ≪≫
cyclesMapIso (shortComplexH2Iso B) ≪≫ (shortComplexH2 B).moduleCatCyclesIso).hom) ≫
(shortComplexH2 B).moduleCatLeftHomologyData.i =
((HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains A) 1 2 3 isoTwoCocycles.proof_3 isoTwoCocycles.proof_4 ≪≫
cyclesMapIso (shortComplexH2Iso A) ≪≫ (shortComplexH2 A).moduleCatCyclesIso).hom ≫
(shortComplexH2 A).moduleCatLeftHomologyData.i) ≫
(mapShortComplexH2 f φ).τ₂ | simp [cochainsMap_f_2_comp_twoCochainsLequiv f, mapShortComplexH2, ← LinearEquiv.toModuleIso_hom] | no goals | 5cb5c18d367809f6 |
exists_sSupIndep_isCompl_sSup_atoms | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem exists_sSupIndep_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
∃ s : Set α, sSupIndep s ∧
IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a | case inr
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
h✝ : sSup {a | IsAtom a} = ⊤
b : α
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a
a : α
ha : a ∈ {a | IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a}
⊢ Disjoint x (sSup (s \ {x}) ⊔ a) | apply
(s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
(a_dis_Sup_s.mono_right _).symm | α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
h✝ : sSup {a | IsAtom a} = ⊤
b : α
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a
a : α
ha : a ∈ {a | IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a}
⊢ x ⊔ sSup (s \ {x}) ≤ sSup s | 04808f5eaf22927f |
lt_compl_self | Mathlib/Order/Heyting/Basic.lean | theorem lt_compl_self [Nontrivial α] : a < aᶜ ↔ a = ⊥ | α : Type u_2
inst✝¹ : HeytingAlgebra α
a : α
inst✝ : Nontrivial α
⊢ a ≤ aᶜ ∧ a ≠ aᶜ ↔ a = ⊥ | simp | no goals | 8be366699de466fc |
RootPairing.RootPositiveForm.zero_lt_posForm_iff | Mathlib/LinearAlgebra/RootSystem/RootPositive.lean | lemma zero_lt_posForm_iff {x y : span S (range P.root)} :
0 < B.posForm x y ↔ ∃ s > 0, algebraMap S R s = B.form x y | ι : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : LinearOrderedCommRing S
inst✝⁹ : CommRing R
inst✝⁸ : Algebra S R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsValuedIn S
B : RootPositiveForm S P
inst✝² : FaithfulSMul S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
x y : ↥(span S (range ⇑P.root))
x✝ : ∃ s > 0, (algebraMap S R) s = (B.form ↑x) ↑y
s : S
h : s > 0
h' : (algebraMap S R) s = (B.form ↑x) ↑y
⊢ 0 < (B.posForm x) y | rw [← B.algebraMap_posForm] at h' | ι : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : LinearOrderedCommRing S
inst✝⁹ : CommRing R
inst✝⁸ : Algebra S R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsValuedIn S
B : RootPositiveForm S P
inst✝² : FaithfulSMul S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
x y : ↥(span S (range ⇑P.root))
x✝ : ∃ s > 0, (algebraMap S R) s = (B.form ↑x) ↑y
s : S
h : s > 0
h' : (algebraMap S R) s = (algebraMap S R) ((B.posForm x) y)
⊢ 0 < (B.posForm x) y | 55dad1190b9db425 |
ODE_solution_unique_of_mem_Icc_left | Mathlib/Analysis/ODE/Gronwall.lean | theorem ODE_solution_unique_of_mem_Icc_left
(hv : ∀ t ∈ Ioc a b, LipschitzOnWith K (v t) (s t))
(hf : ContinuousOn f (Icc a b))
(hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t)
(hfs : ∀ t ∈ Ioc a b, f t ∈ s t)
(hg : ContinuousOn g (Icc a b))
(hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t)
(hgs : ∀ t ∈ Ioc a b, g t ∈ s t)
(hb : f b = g b) :
EqOn f g (Icc a b) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : ℝ → E → E
s : ℝ → Set E
K : ℝ≥0
f g : ℝ → E
a b : ℝ
hv : ∀ t ∈ Ioc a b, LipschitzOnWith K (v t) (s t)
hf : ContinuousOn f (Icc a b)
hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t
hfs : ∀ t ∈ Ioc a b, f t ∈ s t
hg : ContinuousOn g (Icc a b)
hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t
hgs : ∀ t ∈ Ioc a b, g t ∈ s t
hb : f b = g b
⊢ ∀ t ∈ Ico (-b) (-a), LipschitzOnWith K (Neg.neg ∘ v (-t)) (s (-t)) | intro t ht | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : ℝ → E → E
s : ℝ → Set E
K : ℝ≥0
f g : ℝ → E
a b : ℝ
hv : ∀ t ∈ Ioc a b, LipschitzOnWith K (v t) (s t)
hf : ContinuousOn f (Icc a b)
hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t
hfs : ∀ t ∈ Ioc a b, f t ∈ s t
hg : ContinuousOn g (Icc a b)
hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t
hgs : ∀ t ∈ Ioc a b, g t ∈ s t
hb : f b = g b
t : ℝ
ht : t ∈ Ico (-b) (-a)
⊢ LipschitzOnWith K (Neg.neg ∘ v (-t)) (s (-t)) | e339fe18a4eedd24 |
Polynomial.content_zero | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_zero : content (0 : R[X]) = 0 | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
⊢ content 0 = 0 | rw [← C_0, content_C, normalize_zero] | no goals | 22f3269c82d8fb66 |
Ordinal.nadd_assoc | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem nadd_assoc (a b c) : a ♯ b ♯ c = a ♯ (b ♯ c) | a b c : Ordinal.{u_1}
⊢ a ♯ b ♯ c = a ♯ (b ♯ c) | unfold nadd | a b c : Ordinal.{u_1}
⊢ (⨆ x, succ (↑x ♯ c)) ⊔ ⨆ x, succ (a ♯ b ♯ ↑x) = (⨆ x, succ (↑x ♯ (b ♯ c))) ⊔ ⨆ x, succ (a ♯ ↑x) | 69808cbedaf10244 |
Primrec.pair | Mathlib/Computability/Primrec.lean | theorem pair {α β γ} [Primcodable α] [Primcodable β] [Primcodable γ] {f : α → β} {g : α → γ}
(hf : Primrec f) (hg : Primrec g) : Primrec fun a => (f a, g a) :=
((casesOn1 0
(Nat.Primrec.succ.comp <|
.pair (Nat.Primrec.pred.comp hf) (Nat.Primrec.pred.comp hg))).comp
(@Primcodable.prim α _)).of_eq
fun n => by cases @decode α _ n <;> simp [encodek]
| α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
f : α → β
g : α → γ
hf : Primrec f
hg : Primrec g
n : ℕ
⊢ (Nat.casesOn (encode (decode n)) 0 fun n =>
(Nat.pair (encode (Option.map f (decode n))).pred (encode (Option.map g (decode n))).pred).succ) =
encode (Option.map (fun a => (f a, g a)) (decode n)) | cases @decode α _ n <;> simp [encodek] | no goals | 9d921bf73b8d3c6c |
sign_eq_of_affineCombination_mem_affineSpan_single_lineMap | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem sign_eq_of_affineCombination_mem_affineSpan_single_lineMap {p : ι → P}
(h : AffineIndependent k p) {w : ι → k} {s : Finset ι} (hw : ∑ i ∈ s, w i = 1) {i₁ i₂ i₃ : ι}
(h₁ : i₁ ∈ s) (h₂ : i₂ ∈ s) (h₃ : i₃ ∈ s) (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
{c : k} (hc0 : 0 < c) (hc1 : c < 1)
(hs : s.affineCombination k p w ∈ line[k, p i₁, AffineMap.lineMap (p i₂) (p i₃) c]) :
SignType.sign (w i₂) = SignType.sign (w i₃) | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : LinearOrderedRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
(affineCombination k s p) w ∈
affineSpan k
{(affineCombination k s p) (affineCombinationSingleWeights k i₁),
(affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)}
⊢ SignType.sign (w i₂) = SignType.sign (w i₃) | refine
sign_eq_of_affineCombination_mem_affineSpan_pair h hw
(s.sum_affineCombinationSingleWeights k h₁)
(s.sum_affineCombinationLineMapWeights h₂ h₃ c) hs h₂ h₃
(Finset.affineCombinationSingleWeights_apply_of_ne k h₁₂.symm)
(Finset.affineCombinationSingleWeights_apply_of_ne k h₁₃.symm) ?_ | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : LinearOrderedRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
(affineCombination k s p) w ∈
affineSpan k
{(affineCombination k s p) (affineCombinationSingleWeights k i₁),
(affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)}
⊢ SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₂) =
SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₃) | 27de1096852a9a0f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem sat_of_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n)
(p : PosFin n → Bool) (pf : p ⊨ f) :
(insertRupUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro.h
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.assignments[i.val] = both
i : PosFin n
hboth : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.rupUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
let update_res := List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate;
let_fun update_res_size := ⋯;
InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h2 : both = addAssignment b f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
∀ (k : Fin (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.size),
k ≠ j → (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
ib_in_insertUnit_fold : (i, b) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.toList
i' : PosFin n
i_false_in_c : (i, false) ∈ c.clause
i'_eq_i : i' = i
b_eq_true : true = b
⊢ decide (p i = false) = true | simp only [addAssignment, ← b_eq_true, addPosAssignment, ite_true] at h2 | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro.h
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.assignments[i.val] = both
i : PosFin n
hboth : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).snd.fst[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.rupUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
let update_res := List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate;
let_fun update_res_size := ⋯;
InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
j : Fin (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.size
b : Bool
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b)
h3 : ¬hasAssignment b f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
∀ (k : Fin (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.size),
k ≠ j → (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst[k].fst.val ≠ ↑⟨i.val, ⋯⟩
ib_in_insertUnit_fold : (i, b) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).fst.toList
i' : PosFin n
i_false_in_c : (i, false) ∈ c.clause
i'_eq_i : i' = i
b_eq_true : true = b
h2 :
both =
match f.assignments[i.val] with
| pos => pos
| neg => both
| both => both
| unassigned => pos
⊢ decide (p i = false) = true | 6b3be4f74b21096f |
CategoryTheory.Sieve.le_pullback_bind | Mathlib/CategoryTheory/Sites/Sieves.lean | theorem le_pullback_bind (S : Presieve X) (R : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S f → Sieve Y) (f : Y ⟶ X)
(h : S f) : R h ≤ (bind S R).pullback f | C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
S : Presieve X
R : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Sieve Y
f : Y ⟶ X
h : S f
⊢ pushforward f (R h) ≤ bind S R | apply pushforward_le_bind_of_mem | no goals | 7772b2f5a5e0254b |
Array.foldlM_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Monadic.lean | theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β)
(l : Array α) (init : β) (w : stop = (l.filter p).size) :
(l.filter p).foldlM g init 0 stop =
l.foldlM (fun x y => if p y then g x y else pure x) init | case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → Bool
g : β → α → m β
init : β
toList✝ : List α
⊢ foldlM g init (filter p { toList := toList✝ }) =
foldlM (fun x y => if p y = true then g x y else pure x) init { toList := toList✝ } | simp [List.foldlM_filter] | no goals | 54c58a709caed831 |
CategoryTheory.IsUniversalColimit.of_iso | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem IsUniversalColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (hc : IsUniversalColimit c)
(e : c ≅ c') : IsUniversalColimit c' | J : Type v'
inst✝¹ : Category.{u', v'} J
C : Type u
inst✝ : Category.{v, u} C
F : J ⥤ C
c c' : Cocone F
hc : IsUniversalColimit c
e : c ≅ c'
F' : J ⥤ C
c'' : Cocone F'
α : F' ⟶ F
f : c''.pt ⟶ c'.pt
h : α ≫ c'.ι = c''.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
H : ∀ (j : J), IsPullback (c''.ι.app j) (α.app j) f (c'.ι.app j)
this : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι
j : J
⊢ IsPullback (c''.ι.app j) (α.app j) (f ≫ e.inv.hom) (c.ι.app j) | rw [← Category.comp_id (α.app j)] | J : Type v'
inst✝¹ : Category.{u', v'} J
C : Type u
inst✝ : Category.{v, u} C
F : J ⥤ C
c c' : Cocone F
hc : IsUniversalColimit c
e : c ≅ c'
F' : J ⥤ C
c'' : Cocone F'
α : F' ⟶ F
f : c''.pt ⟶ c'.pt
h : α ≫ c'.ι = c''.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
H : ∀ (j : J), IsPullback (c''.ι.app j) (α.app j) f (c'.ι.app j)
this : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι
j : J
⊢ IsPullback (c''.ι.app j) (α.app j ≫ 𝟙 (F.obj j)) (f ≫ e.inv.hom) (c.ι.app j) | fd31397d213202fc |
FermatLastTheoremForThreeGen.a_cube_b_cube_congr_one_or_neg_one | Mathlib/NumberTheory/FLT/Three.lean | /-- Given `S' : Solution'`, then `S'.a` and `S'.b` are both congruent to `1` modulo `λ ^ 4` or are
both congruent to `-1`. -/
lemma a_cube_b_cube_congr_one_or_neg_one :
λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1 | case intro.inr.intro.inr.intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1 | exfalso | case intro.inr.intro.inr.intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 + 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 + 1 = λ ^ 4 * y
⊢ False | 7f0bf19b77a2dee2 |
Vector.append_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem append_eq_append_iff {a : Vector α n} {b : Vector α m} {c : Vector α k} {d : Vector α l}
(w : k + l = n + m) :
a ++ b = (c ++ d).cast w ↔
if h : n ≤ k then
∃ a' : Vector α (k - n), c = (a ++ a').cast (by omega) ∧ b = (a' ++ d).cast (by omega)
else
∃ c' : Vector α (n - k), a = (c ++ c').cast (by omega) ∧ d = (c' ++ b).cast (by omega) | α : Type u_1
a d a' : Array α
w : (a ++ a').size + d.size = a.size + (a' ++ d).size
⊢ a'.size = a.size + a'.size - a.size | omega | no goals | 6aa03c7084d30f0e |
IsFreeGroup.unique_lift | Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean | theorem unique_lift (f : Generators G → H) : ∃! F : G →* H, ∀ a, F (of a) = f a | G : Type u_1
inst✝² : Group G
inst✝¹ : IsFreeGroup G
H : Type u_2
inst✝ : Group H
f : Generators G → H
⊢ ∃! F, ∀ (a : Generators G), F (of a) = f a | simpa only [funext_iff] using lift.symm.bijective.existsUnique f | no goals | efa62bbdc31c4642 |
Polynomial.le_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) :
n ≤ p.natTrailingDegree | R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
hn : ∀ m < n, p.coeff m = 0
⊢ n ≤ p.natTrailingDegree | rw [natTrailingDegree_eq_support_min' hp] | R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
hn : ∀ m < n, p.coeff m = 0
⊢ n ≤ p.support.min' ⋯ | e6daf4dd4d532f5b |
Finset.mem_sym2_iff | Mathlib/Data/Finset/Sym.lean | theorem mem_sym2_iff {m : Sym2 α} : m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s | α : Type u_1
s : Finset α
m : Sym2 α
⊢ m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s | rw [mem_mk, sym2_val, Multiset.mem_sym2_iff] | α : Type u_1
s : Finset α
m : Sym2 α
⊢ (∀ y ∈ m, y ∈ s.val) ↔ ∀ a ∈ m, a ∈ s | 39cae625fbd64a4b |
PartENat.toWithTop_natCast' | Mathlib/Data/Nat/PartENat.lean | theorem toWithTop_natCast' (n : ℕ) {_ : Decidable (n : PartENat).Dom} :
toWithTop (n : PartENat) = n | n : ℕ
x✝ : Decidable (↑n).Dom
⊢ (↑n).toWithTop = ↑n | rw [toWithTop_natCast n] | no goals | 648ebb49f72dfd9c |
Fintype.exists_disjointed_le | Mathlib/Order/Disjointed.lean | /-- For any finite family of elements `f : ι → α`, we can find a pairwise-disjoint family `g`
bounded above by `f` and having the same supremum. This is non-canonical, depending on an arbitrary
choice of ordering of `ι`. -/
lemma Fintype.exists_disjointed_le {ι : Type*} [Fintype ι] {f : ι → α} :
∃ g, g ≤ f ∧ univ.sup g = univ.sup f ∧ Pairwise (Disjoint on g) | case inr.refine_1
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
ι : Type u_3
inst✝ : Fintype ι
f : ι → α
hι : Nonempty ι
R : ι ≃ Fin (card ι) := equivFin ι
f' : Fin (card ι) → α := f ∘ ⇑R.symm
hf' : f = f' ∘ ⇑R
⊢ disjointed f' ∘ ⇑R ≤ f | intro n | case inr.refine_1
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
ι : Type u_3
inst✝ : Fintype ι
f : ι → α
hι : Nonempty ι
R : ι ≃ Fin (card ι) := equivFin ι
f' : Fin (card ι) → α := f ∘ ⇑R.symm
hf' : f = f' ∘ ⇑R
n : ι
⊢ (disjointed f' ∘ ⇑R) n ≤ f n | 1d891d5bbcf4019d |
dotProduct_add | Mathlib/Data/Matrix/Mul.lean | theorem dotProduct_add : u ⬝ᵥ (v + w) = u ⬝ᵥ v + u ⬝ᵥ w | m : Type u_2
α : Type v
inst✝¹ : Fintype m
inst✝ : NonUnitalNonAssocSemiring α
u v w : m → α
⊢ u ⬝ᵥ (v + w) = u ⬝ᵥ v + u ⬝ᵥ w | simp [dotProduct, mul_add, Finset.sum_add_distrib] | no goals | 30ce262c5167f2ad |
AlgebraicGeometry.RingedSpace.exists_res_eq_zero_of_germ_eq_zero | Mathlib/Geometry/RingedSpace/Basic.lean | /-- If the germ of a section `f` is zero in the stalk at `x`, then `f` is zero on some neighbourhood
around `x`. -/
lemma exists_res_eq_zero_of_germ_eq_zero (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : X.presheaf.germ U x.val x.property f = 0) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), X.presheaf.map i.op f = 0 | case h
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = 0
h1 : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) 0
V : Opens ↑↑X.toPresheafedSpace
hv : ↑x ∈ V
i w✝ : V ⟶ U
hv4 : (ConcreteCategory.hom (X.presheaf.map i.op)) f = (ConcreteCategory.hom (X.presheaf.map w✝.op)) 0
⊢ (ConcreteCategory.hom (X.presheaf.map i.op)) f = 0 | simpa using hv4 | no goals | 9e8595fdf9450948 |
IsLocalRing.of_surjective' | Mathlib/RingTheory/LocalRing/Basic.lean | theorem of_surjective' [Ring S] [Nontrivial S] (f : R →+* S) (hf : Function.Surjective f) :
IsLocalRing S :=
of_isUnit_or_isUnit_one_sub_self (by
intro b
obtain ⟨a, rfl⟩ := hf b
apply (isUnit_or_isUnit_one_sub_self a).imp <| RingHom.isUnit_map _
rw [← f.map_one, ← f.map_sub]
apply f.isUnit_map)
| R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : Ring S
inst✝ : Nontrivial S
f : R →+* S
hf : Function.Surjective ⇑f
⊢ ∀ (a : S), IsUnit a ∨ IsUnit (1 - a) | intro b | R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : Ring S
inst✝ : Nontrivial S
f : R →+* S
hf : Function.Surjective ⇑f
b : S
⊢ IsUnit b ∨ IsUnit (1 - b) | 7d91f66c74fa2a9b |
IsNilpotent.exp_add_of_commute | Mathlib/RingTheory/Nilpotent/Exp.lean | theorem exp_add_of_commute {a b : A} (h₁ : Commute a b) (h₂ : IsNilpotent a) (h₃ : IsNilpotent b) :
exp (a + b) = exp a * exp b | A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
this : (↑i !)⁻¹ * ↑(i.choose j) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹
⊢ (↑i !)⁻¹ • (a ^ j * b ^ (i - j) * ↑(i.choose j)) = ((↑j !)⁻¹ * (↑(i - j)!)⁻¹) • (a ^ j * b ^ (i - j)) | rw [← Nat.cast_commute (i.choose j), ← this, ← Algebra.mul_smul_comm, ← nsmul_eq_mul,
mul_smul, ← smul_assoc, smul_comm, smul_assoc] | A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
this : (↑i !)⁻¹ * ↑(i.choose j) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹
⊢ i.choose j • (↑i !)⁻¹ • (a ^ j * b ^ (i - j)) = ↑(i.choose j) • (↑i !)⁻¹ • (a ^ j * b ^ (i - j)) | 925becc4d11e37f5 |
MulChar.IsQuadratic.inv | Mathlib/NumberTheory/MulChar/Basic.lean | theorem IsQuadratic.inv {χ : MulChar R R'} (hχ : χ.IsQuadratic) : χ⁻¹ = χ | R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
x : Rˣ
h₂ : χ ↑x = -1
⊢ -1 = ↑(-1) | rw [Units.val_neg, Units.val_one] | no goals | 34a914133be0d6ed |
meas_lt_essInf | Mathlib/MeasureTheory/Function/EssSup.lean | theorem meas_lt_essInf
(hf : IsBoundedUnder (· ≥ ·) (ae μ) f | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) (ae μ) f) _auto✝
⊢ μ {y | f y < essInf f μ} = 0 | simp_rw [← not_le] | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) (ae μ) f) _auto✝
⊢ μ {y | ¬essInf f μ ≤ f y} = 0 | 67c7ba092a45938c |
PhragmenLindelof.right_half_plane_of_tendsto_zero_on_real | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem right_half_plane_of_tendsto_zero_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
(hexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c))
(hre : Tendsto (fun x : ℝ => f x) atTop (𝓝 0)) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
(hz : 0 ≤ z.re) : ‖f z‖ ≤ C | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ C ⊔ C'
hfc : ContinuousOn (fun x => f ↑x) (Ici 0)
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖ | by_cases h₀ : ∀ x : ℝ, 0 ≤ x → f x = 0 | case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ C ⊔ C'
hfc : ContinuousOn (fun x => f ↑x) (Ici 0)
h₀ : ∀ (x : ℝ), 0 ≤ x → f ↑x = 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ C ⊔ C'
hfc : ContinuousOn (fun x => f ↑x) (Ici 0)
h₀ : ¬∀ (x : ℝ), 0 ≤ x → f ↑x = 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖ | 8c2477e37fa78449 |
CategoryTheory.SmallObject.πFunctorObj_eq | Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean | lemma πFunctorObj_eq (j : κ.ord.toType) :
letI := hasColimitsOfShape_discrete I κ
letI := hasPushouts I κ
πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
(relativeCellComplexιObjFObjSuccIso I κ f j).inv ≫
(relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫
πObj I κ f ≫ (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).inv | C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.toType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
j : κ.ord.toType
h₁ :
(iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom.right.left ≫
πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
(((iterationFunctor I κ).obj (Order.succ j)).obj (Arrow.mk f)).hom ≫
(iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom.right.right
h₂ :
(iterationFunctor I κ).map (homOfLE ⋯) ≫
(transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j) =
(transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j
⊢ (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom.right.left ≫
πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom ≫
(((transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j).app (Arrow.mk f)).right ≫
𝟙 ((iteration I κ).obj (Arrow.mk f)).right =
((((transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j)).app (Arrow.mk f)).left ≫
𝟙 ((iteration I κ).obj (Arrow.mk f)).left) ≫
((iteration I κ).obj (Arrow.mk f)).hom | simp only [reassoc_of% h₁, comp_id, comp_id, Arrow.w_mk_right, ← h₂,
NatTrans.comp_app, Arrow.comp_right,
iterationFunctorMapSuccAppArrowIso_hom_right_right_comp_assoc] | no goals | c49cad169646b8e7 |
ContMDiffFiberwiseLinear.locality_aux₁ | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | theorem ContMDiffFiberwiseLinear.locality_aux₁
(n : WithTop ℕ∞) (e : PartialHomeomorph (B × F) (B × F))
(h : ∀ p ∈ e.source, ∃ s : Set (B × F), IsOpen s ∧ p ∈ s ∧
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u)
(hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
(h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((φ x).symm : F →L[𝕜] F)) u),
(e.restr s).EqOnSource
(FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
∃ U : Set B, e.source = U ×ˢ univ ∧ ∀ x ∈ U,
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_huU : u ⊆ U) (_hux : x ∈ u),
∃ (hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
(h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((φ x).symm : F →L[𝕜] F)) u),
(e.restr (u ×ˢ univ)).EqOnSource
(FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn) | case a.mk.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
v : F
p : B × F
hp : p ∈ e.source
⊢ (p.1, v) ∈ e.source | exact heu ⟨p, hp⟩ (p.fst, v) (hu' ⟨p, hp⟩) | no goals | 70cae96477ef715a |
BitVec.getLsbD_zero | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_zero : (0#w).getLsbD i = false | w i : Nat
⊢ (0#w).getLsbD i = false | simp [getLsbD] | no goals | 01285b5eae552e07 |
Nat.dist.triangle_inequality | Mathlib/Data/Nat/Dist.lean | theorem dist.triangle_inequality (n m k : ℕ) : dist n k ≤ dist n m + dist m k | n m k : ℕ
⊢ n.dist k ≤ n.dist m + m.dist k | have : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n)) := by
simp [dist, add_comm, add_left_comm, add_assoc] | n m k : ℕ
this : n.dist m + m.dist k = n - m + (m - k) + (k - m + (m - n))
⊢ n.dist k ≤ n.dist m + m.dist k | 8a85e9a9601b52ad |
hasFDerivAt_exp_of_mem_ball | Mathlib/Analysis/SpecialFunctions/Exponential.lean | theorem hasFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (exp 𝕂) (exp 𝕂 x • 1) x | have hpos : 0 < (expSeries 𝕂 𝔸).radius := (zero_le _).trans_lt hx | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (exp 𝕂) (exp 𝕂 x • 1) x | b912975acf3383c5 |
LieModuleHom.comp_ker_incl | Mathlib/Algebra/Lie/Submodule.lean | theorem comp_ker_incl : f.comp f.ker.incl = 0 | R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : LieRingModule L N
f : M →ₗ⁅R,L⁆ N
⊢ f.comp f.ker.incl = 0 | ext ⟨m, hm⟩ | case h.mk
R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : LieRingModule L N
f : M →ₗ⁅R,L⁆ N
m : M
hm : m ∈ f.ker
⊢ (f.comp f.ker.incl) ⟨m, hm⟩ = 0 ⟨m, hm⟩ | dfebb799e6e2d8c7 |
ProbabilityTheory.Kernel.measurable_densityProcess_countableFiltration_aux | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma measurable_densityProcess_countableFiltration_aux (κ : Kernel α (γ × β)) (ν : Kernel α γ)
(n : ℕ) {s : Set β} (hs : MeasurableSet s) :
Measurable[mα.prod (countableFiltration γ n)] (fun (p : α × γ) ↦
κ p.1 (countablePartitionSet n p.2 ×ˢ s) / ν p.1 (countablePartitionSet n p.2)) | case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
n : ℕ
s : Set β
hs : MeasurableSet s
⊢ ∀ (y : ↑(countablePartition γ n)), Measurable fun x => (ν (x, y).1) ↑(x, y).2 | rintro ⟨t, ht⟩ | case refine_2.mk
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
n : ℕ
s : Set β
hs : MeasurableSet s
t : Set γ
ht : t ∈ countablePartition γ n
⊢ Measurable fun x => (ν (x, ⟨t, ht⟩).1) ↑(x, ⟨t, ht⟩).2 | f6b26e3ef3304fb0 |
IsAlgebraic.exists_smul_eq_mul | Mathlib/RingTheory/Algebraic/Basic.lean | theorem IsAlgebraic.exists_smul_eq_mul
(a : S) {b : S} (hRb : IsAlgebraic R b) (hb : b ∈ nonZeroDivisors S) :
∃ᵉ (c : S) (d ≠ (0 : R)), d • a = b * c :=
have ⟨r, hr, s, h⟩ := hRb.exists_nonzero_dvd hb
⟨s * a, r, hr, by rw [Algebra.smul_def, h, mul_assoc]⟩
| R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
a b : S
hRb : IsAlgebraic R b
hb : b ∈ nonZeroDivisors S
r : R
hr : r ≠ 0
s : S
h : (algebraMap R S) r = b * s
⊢ r • a = b * (s * a) | rw [Algebra.smul_def, h, mul_assoc] | no goals | 10ab4f7d13b76032 |
NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis | Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean | theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ nrComplexPlaces K * sqrt ‖discr K‖₊ | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) RingHom.equivRatAlgHom
this : M.map ⇑ofRealHom = matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ
⊢ ‖I‖₊ = 1 | rw [← norm_toNNReal, norm_I, Real.toNNReal_one] | no goals | 0fabcb321045abbb |
NumberField.mixedEmbedding.fundamentalCone.torsion_unitSMul_mem_integerSet | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean | theorem torsion_unitSMul_mem_integerSet {x : mixedSpace K} {ζ : (𝓞 K)ˣ} (hζ : ζ ∈ torsion K)
(hx : x ∈ integerSet K) : ζ • x ∈ integerSet K | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
ζ : (𝓞 K)ˣ
hζ : ζ ∈ torsion K
hx : x ∈ integerSet K
⊢ ζ • x ∈ integerSet K | obtain ⟨a, ⟨_, rfl⟩, rfl⟩ := (mem_integerSet.mp hx).2 | case intro.refl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : (𝓞 K)ˣ
hζ : ζ ∈ torsion K
a : 𝓞 K
hx : (mixedEmbedding K) ↑a ∈ integerSet K
⊢ ζ • (mixedEmbedding K) ↑a ∈ integerSet K | 114477dbad2b3ce8 |
IsGreatest.norm_cfc | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean | lemma IsGreatest.norm_cfc [Nontrivial A] (f : 𝕜 → 𝕜) (a : A)
(hf : ContinuousOn f (σ 𝕜 a) | 𝕜 : Type u_1
A : Type u_2
p : outParam (A → Prop)
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : autoParam (ContinuousOn f (σ 𝕜 a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ IsGreatest ((fun x => ‖f x‖) '' σ 𝕜 a) ‖cfc f a‖ | obtain ⟨x, hx⟩ := ContinuousFunctionalCalculus.isCompact_spectrum a
|>.image_of_continuousOn hf.norm |>.exists_isGreatest <|
(ContinuousFunctionalCalculus.spectrum_nonempty a ha).image _ | case intro
𝕜 : Type u_1
A : Type u_2
p : outParam (A → Prop)
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : autoParam (ContinuousOn f (σ 𝕜 a)) _auto✝
ha : autoParam (p a) _auto✝
x : ℝ
hx : IsGreatest ((fun x => ‖f x‖) '' σ 𝕜 a) x
⊢ IsGreatest ((fun x => ‖f x‖) '' σ 𝕜 a) ‖cfc f a‖ | 1af82689eab3dd89 |
IsPrimitiveRoot.exists_pow_or_neg_mul_pow_of_isOfFinOrder | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem exists_pow_or_neg_mul_pow_of_isOfFinOrder [IsCyclotomicExtension {n} ℚ K]
(hno : Odd (n : ℕ)) {ζ x : K} (hζ : IsPrimitiveRoot ζ n) (hx : IsOfFinOrder x) :
∃ r : ℕ, r < n ∧ (x = ζ ^ r ∨ x = -ζ ^ r) := by
obtain ⟨r, hr⟩ := hζ.exists_neg_pow_of_isOfFinOrder hno hx
refine ⟨r % n, Nat.mod_lt _ n.2, ?_⟩
rw [show ζ ^ (r % ↑n) = ζ ^ r from (IsPrimitiveRoot.eq_orderOf hζ).symm ▸ pow_mod_orderOf .., hr]
rcases Nat.even_or_odd r with (h | h) <;> simp [neg_pow, h.neg_one_pow]
| n : ℕ+
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd ↑n
ζ x : K
hζ : IsPrimitiveRoot ζ ↑n
hx : IsOfFinOrder x
⊢ ∃ r < ↑n, x = ζ ^ r ∨ x = -ζ ^ r | obtain ⟨r, hr⟩ := hζ.exists_neg_pow_of_isOfFinOrder hno hx | case intro
n : ℕ+
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd ↑n
ζ x : K
hζ : IsPrimitiveRoot ζ ↑n
hx : IsOfFinOrder x
r : ℕ
hr : x = (-ζ) ^ r
⊢ ∃ r < ↑n, x = ζ ^ r ∨ x = -ζ ^ r | ccb44fcbe76a6a6b |
RootPairing.injOn_dualMap_subtype_span_root_coroot | Mathlib/LinearAlgebra/RootSystem/Basic.lean | /-- Even though the roots may not span, coroots are distinguished by their pairing with the
roots. The proof depends crucially on the fact that there are finitely-many roots.
Modulo trivial generalisations, this statement is exactly Lemma 1.1.4 on page 87 of SGA 3 XXI. -/
lemma injOn_dualMap_subtype_span_root_coroot [NoZeroSMulDivisors ℤ M] :
InjOn ((span R (range P.root)).subtype.dualMap ∘ₗ P.toLin.flip) (range P.coroot) | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
inst✝¹ : Finite ι
inst✝ : NoZeroSMulDivisors ℤ M
this : InjOn (⇑(span R (range ⇑P.root)).subtype.dualMap) (range (⇑P.toLin.flip ∘ ⇑P.coroot))
⊢ InjOn (⇑((span R (range ⇑P.root)).subtype.dualMap ∘ₗ P.toLin.flip)) (range ⇑P.coroot) | rintro - ⟨i, rfl⟩ - ⟨j, rfl⟩ hij | case intro.intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
inst✝¹ : Finite ι
inst✝ : NoZeroSMulDivisors ℤ M
this : InjOn (⇑(span R (range ⇑P.root)).subtype.dualMap) (range (⇑P.toLin.flip ∘ ⇑P.coroot))
i j : ι
hij :
((span R (range ⇑P.root)).subtype.dualMap ∘ₗ P.toLin.flip) (P.coroot i) =
((span R (range ⇑P.root)).subtype.dualMap ∘ₗ P.toLin.flip) (P.coroot j)
⊢ P.coroot i = P.coroot j | ffac3d487167573a |
Finset.pow_eq_empty | Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | @[to_additive (attr := simp)] lemma pow_eq_empty : s ^ n = ∅ ↔ s = ∅ ∧ n ≠ 0 | case mp
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Monoid α
s : Finset α
n : ℕ
⊢ s ^ n = ∅ → s = ∅ ∧ n ≠ 0 | contrapose! | case mp
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Monoid α
s : Finset α
n : ℕ
⊢ s ≠ ∅ ∨ n = 0 → s ^ n ≠ ∅ | 930de7a778d262f5 |
Std.Tactic.BVDecide.BVExpr.eval_const | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Basic.lean | theorem eval_const : eval assign (.const val) = val | assign : Assignment
w✝ : Nat
val : BitVec w✝
⊢ eval assign (const val) = val | rfl | no goals | a374970caff6fd4d |
FDRep.Iso.conj_ρ | Mathlib/RepresentationTheory/FDRep.lean | theorem Iso.conj_ρ {V W : FDRep R G} (i : V ≅ W) (g : G) :
W.ρ g = (FDRep.isoToLinearEquiv i).conj (V.ρ g) | R G : Type u
inst✝¹ : CommRing R
inst✝ : Monoid G
V W : FDRep R G
i : V ≅ W
g : G
⊢ ((Action.forget (FGModuleCat R) G).mapIso i).hom ≫ ModuleCat.ofHom (W.ρ g) =
V.ρ g ≫ ((Action.forget (FGModuleCat R) G).mapIso i).hom | exact (i.hom.comm g).symm | no goals | 7c658e3e962ad586 |
CategoryTheory.FreeMonoidalCategory.Hom.inductionOn | Mathlib/CategoryTheory/Monoidal/Free/Basic.lean | theorem Hom.inductionOn {motive : {X Y : F C} → (X ⟶ Y) → Prop} {X Y : F C} (t : X ⟶ Y)
(id : (X : F C) → motive (𝟙 X))
(α_hom : (X Y Z : F C) → motive (α_ X Y Z).hom)
(α_inv : (X Y Z : F C) → motive (α_ X Y Z).inv)
(l_hom : (X : F C) → motive (λ_ X).hom)
(l_inv : (X : F C) → motive (λ_ X).inv)
(ρ_hom : (X : F C) → motive (ρ_ X).hom)
(ρ_inv : (X : F C) → motive (ρ_ X).inv)
(comp : {X Y Z : F C} → (f : X ⟶ Y) → (g : Y ⟶ Z) → motive f → motive g → motive (f ≫ g))
(whiskerLeft : (X : F C) → {Y Z : F C} → (f : Y ⟶ Z) → motive f → motive (X ◁ f))
(whiskerRight : {X Y : F C} → (f : X ⟶ Y) → (Z : F C) → motive f → motive (f ▷ Z)) :
motive t | case h.whiskerRight
C : Type u
motive : {X Y : F C} → (X ⟶ Y) → Prop
X✝ Y : F C
id : ∀ (X : F C), motive (𝟙 X)
α_hom : ∀ (X Y Z : F C), motive (α_ X Y Z).hom
α_inv : ∀ (X Y Z : F C), motive (α_ X Y Z).inv
l_hom : ∀ (X : F C), motive (λ_ X).hom
l_inv : ∀ (X : F C), motive (λ_ X).inv
ρ_hom : ∀ (X : F C), motive (ρ_ X).hom
ρ_inv : ∀ (X : F C), motive (ρ_ X).inv
comp : ∀ {X Y Z : F C} (f : X ⟶ Y) (g : Y ⟶ Z), motive f → motive g → motive (f ≫ g)
whiskerLeft : ∀ (X : F C) {Y Z : F C} (f : Y ⟶ Z), motive f → motive (X ◁ f)
whiskerRight : ∀ {X Y : F C} (f : X ⟶ Y) (Z : F C), motive f → motive (f ▷ Z)
X₁✝ X₂✝ : F C
f : X₁✝ ⟶ᵐ X₂✝
X : F C
hf : (X₁✝ ⟶ X₂✝) → motive ⟦f⟧
t : X₁✝.tensor X ⟶ X₂✝.tensor X
⊢ motive ⟦f.whiskerRight X⟧ | exact whiskerRight _ X (hf ⟦f⟧) | no goals | 628924983ac61dad |
Finset.Colex.IsInitSeg.exists_initSeg | Mathlib/Combinatorics/Colex.lean | lemma IsInitSeg.exists_initSeg (h𝒜 : IsInitSeg 𝒜 r) (h𝒜₀ : 𝒜.Nonempty) :
∃ s : Finset α, #s = r ∧ 𝒜 = initSeg s | case h.refine_2.intro.inr
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ toColex ∈ ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ toColex).ofColex = #t
le : { ofColex := t } ≤ { ofColex := (𝒜.sup' h𝒜₀ toColex).ofColex }
p : { ofColex := t } < { ofColex := (𝒜.sup' h𝒜₀ toColex).ofColex }
⊢ t ∈ 𝒜 | exact h𝒜.2 hs ⟨p, cards ▸ h𝒜.1 hs⟩ | no goals | 784903b08b93058a |
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