fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Subgroup.tendsto_coe_cofinite_of_discrete [T2Space G] (H : Subgroup G) [DiscreteTopology H] :
Tendsto ((↑) : H → G) cofinite (cocompact _) :=
IsClosed.tendsto_coe_cofinite_of_discreteTopology inferInstance inferInstance
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | Subgroup.tendsto_coe_cofinite_of_discrete | null |
MonoidHom.tendsto_coe_cofinite_of_discrete [T2Space G] {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) (hf' : DiscreteTopology f.range) :
Tendsto f cofinite (cocompact _) := by
replace hf : Function.Injective f.rangeRestrict := by simpa
exact f.range.tendsto_coe_cofinite_of_discrete.comp hf.t... | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | MonoidHom.tendsto_coe_cofinite_of_discrete | null |
@[to_additive]
tendstoUniformly_iff (F : ι → α → G) (f : α → G) (p : Filter ι)
(hu : IsTopologicalGroup.toUniformSpace G = u) :
TendstoUniformly F f p ↔ ∀ u ∈ 𝓝 (1 : G), ∀ᶠ i in p, ∀ a, F i a / f a ∈ u :=
hu ▸ ⟨fun h u hu => h _ ⟨u, hu, fun _ => id⟩,
fun h _ ⟨u, hu, hv⟩ => mem_of_superset (h u hu) fun _ ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | tendstoUniformly_iff | null |
tendstoUniformlyOn_iff (F : ι → α → G) (f : α → G) (p : Filter ι) (s : Set α)
(hu : IsTopologicalGroup.toUniformSpace G = u) :
TendstoUniformlyOn F f p s ↔ ∀ u ∈ 𝓝 (1 : G), ∀ᶠ i in p, ∀ a ∈ s, F i a / f a ∈ u :=
hu ▸ ⟨fun h u hu => h _ ⟨u, hu, fun _ => id⟩,
fun h _ ⟨u, hu, hv⟩ => mem_of_superset (h u hu)... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | tendstoUniformlyOn_iff | null |
tendstoLocallyUniformly_iff [TopologicalSpace α] (F : ι → α → G) (f : α → G)
(p : Filter ι) (hu : IsTopologicalGroup.toUniformSpace G = u) :
TendstoLocallyUniformly F f p ↔
∀ u ∈ 𝓝 (1 : G), ∀ (x : α), ∃ t ∈ 𝓝 x, ∀ᶠ i in p, ∀ a ∈ t, F i a / f a ∈ u :=
hu ▸ ⟨fun h u hu => h _ ⟨u, hu, fun _ => id⟩, fun h... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | tendstoLocallyUniformly_iff | null |
tendstoLocallyUniformlyOn_iff [TopologicalSpace α] (F : ι → α → G) (f : α → G)
(p : Filter ι) (s : Set α) (hu : IsTopologicalGroup.toUniformSpace G = u) :
TendstoLocallyUniformlyOn F f p s ↔
∀ u ∈ 𝓝 (1 : G), ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ i in p, ∀ a ∈ t, F i a / f a ∈ u :=
hu ▸ ⟨fun h u hu => h _ ⟨u, hu, ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | tendstoLocallyUniformlyOn_iff | null |
private extend_Z_bilin_aux (x₀ : α) (y₁ : δ) : ∃ U₂ ∈ comap e (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂,
(fun p : β × δ => φ p.1 p.2) (x' - x, y₁) ∈ W' := by
let Nx := 𝓝 x₀
let ee := fun u : β × β => (e u.1, e u.2)
have lim1 : Tendsto (fun a : β × β => (a.2 - a.1, y₁))
(comap e Nx ×ˢ comap e Nx) (𝓝 (0, y₁)) := by
... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | extend_Z_bilin_aux | null |
private extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' := by
let ee := fun u : β × β => (e u.1, e u.2)
let ff := fun u : δ × δ => (f ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | extend_Z_bilin_key | null |
extend_Z_bilin : Continuous (extend (de.prodMap df) (fun p : β × δ => φ p.1 p.2)) := by
refine continuous_extend_of_cauchy _ ?_
rintro ⟨x₀, y₀⟩
constructor
· apply NeBot.map
apply comap_neBot
intro U h
rcases mem_closure_iff_nhds.1 ((de.prodMap df).dense (x₀, y₀)) U h with ⟨x, x_in, ⟨z, z_x⟩⟩
ex... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | extend_Z_bilin | Bourbaki GT III.6.5 Theorem I:
ℤ-bilinear continuous maps from dense images into a complete Hausdorff group extend by continuity.
Note: Bourbaki assumes that α and β are also complete Hausdorff, but this is not necessary. |
QuotientGroup.completeSpace' (G : Type u) [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] [FirstCountableTopology G] (N : Subgroup G) [N.Normal]
[@CompleteSpace G (IsTopologicalGroup.toUniformSpace G)] :
@CompleteSpace (G ⧸ N) (IsTopologicalGroup.toUniformSpace (G ⧸ N)) := by
/- Since `G ⧸ N` is a t... | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | QuotientGroup.completeSpace' | null |
QuotientGroup.completeSpace (G : Type u) [Group G] [us : UniformSpace G] [IsUniformGroup G]
[FirstCountableTopology G] (N : Subgroup G) [N.Normal] [hG : CompleteSpace G] :
@CompleteSpace (G ⧸ N) (IsTopologicalGroup.toUniformSpace (G ⧸ N)) := by
rw [← @IsUniformGroup.toUniformSpace_eq _ us _ _] at hG
infer_i... | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | QuotientGroup.completeSpace | null |
IsUniformGroup (α : Type*) [UniformSpace α] [Group α] : Prop where
uniformContinuous_div : UniformContinuous fun p : α × α => p.1 / p.2
@[deprecated (since := "2025-03-26")] alias UniformGroup := IsUniformGroup | class | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsUniformGroup | A uniform group is a group in which multiplication and inversion are uniformly continuous. |
IsUniformAddGroup (α : Type*) [UniformSpace α] [AddGroup α] : Prop where
uniformContinuous_sub : UniformContinuous fun p : α × α => p.1 - p.2
@[deprecated (since := "2025-03-26")] alias UniformAddGroup := IsUniformAddGroup
attribute [to_additive] IsUniformGroup
@[to_additive] | class | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsUniformAddGroup | A uniform additive group is an additive group in which addition
and negation are uniformly continuous. |
IsUniformGroup.mk' {α} [UniformSpace α] [Group α]
(h₁ : UniformContinuous fun p : α × α => p.1 * p.2) (h₂ : UniformContinuous fun p : α => p⁻¹) :
IsUniformGroup α :=
⟨by simpa only [div_eq_mul_inv] using
h₁.comp (uniformContinuous_fst.prodMk (h₂.comp uniformContinuous_snd))⟩
variable [UniformSpace α] [Gro... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsUniformGroup.mk' | null |
uniformContinuous_div : UniformContinuous fun p : α × α => p.1 / p.2 :=
IsUniformGroup.uniformContinuous_div
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_div | null |
UniformContinuous.div [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun x => f x / g x :=
uniformContinuous_div.comp (hf.prodMk hg)
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.div | null |
UniformContinuous.inv [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
UniformContinuous fun x => (f x)⁻¹ := by
have : UniformContinuous fun x => 1 / f x := uniformContinuous_const.div hf
simp_all
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.inv | null |
uniformContinuous_inv : UniformContinuous fun x : α => x⁻¹ :=
uniformContinuous_id.inv
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_inv | null |
UniformContinuous.mul [UniformSpace β] {f : β → α} {g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun x => f x * g x := by
have : UniformContinuous fun x => f x / (g x)⁻¹ := hf.div hg.inv
simp_all
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.mul | null |
uniformContinuous_mul : UniformContinuous fun p : α × α => p.1 * p.2 :=
uniformContinuous_fst.mul uniformContinuous_snd
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_mul | null |
UniformContinuous.mul_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
(a : α) : UniformContinuous fun x ↦ f x * a :=
hf.mul uniformContinuous_const
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.mul_const | null |
UniformContinuous.const_mul [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
(a : α) : UniformContinuous fun x ↦ a * f x :=
uniformContinuous_const.mul hf
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.const_mul | null |
uniformContinuous_mul_left (a : α) : UniformContinuous fun b : α => a * b :=
uniformContinuous_id.const_mul _
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_mul_left | null |
uniformContinuous_mul_right (a : α) : UniformContinuous fun b : α => b * a :=
uniformContinuous_id.mul_const _
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_mul_right | null |
UniformContinuous.div_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
(a : α) : UniformContinuous fun x ↦ f x / a :=
hf.div uniformContinuous_const
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.div_const | null |
uniformContinuous_div_const (a : α) : UniformContinuous fun b : α => b / a :=
uniformContinuous_id.div_const _
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_div_const | null |
Filter.Tendsto.uniformity_mul {ι : Type*} {f g : ι → α × α} {l : Filter ι}
(hf : Tendsto f l (𝓤 α)) (hg : Tendsto g l (𝓤 α)) :
Tendsto (f * g) l (𝓤 α) :=
have : Tendsto (fun (p : (α × α) × (α × α)) ↦ p.1 * p.2) (𝓤 α ×ˢ 𝓤 α) (𝓤 α) := by
simpa [UniformContinuous, uniformity_prod_eq_prod] using uniform... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_mul | null |
Filter.Tendsto.uniformity_inv {ι : Type*} {f : ι → α × α} {l : Filter ι}
(hf : Tendsto f l (𝓤 α)) :
Tendsto (f⁻¹) l (𝓤 α) :=
have : Tendsto (· ⁻¹) (𝓤 α) (𝓤 α) := uniformContinuous_inv
this.comp hf
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_inv | null |
Filter.Tendsto.uniformity_inv_iff {ι : Type*} {f : ι → α × α} {l : Filter ι} :
Tendsto (f⁻¹) l (𝓤 α) ↔ Tendsto f l (𝓤 α) :=
⟨fun H ↦ inv_inv f ▸ H.uniformity_inv, Filter.Tendsto.uniformity_inv⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_inv_iff | null |
Filter.Tendsto.uniformity_div {ι : Type*} {f g : ι → α × α} {l : Filter ι}
(hf : Tendsto f l (𝓤 α)) (hg : Tendsto g l (𝓤 α)) :
Tendsto (f / g) l (𝓤 α) := by
rw [div_eq_mul_inv]
exact hf.uniformity_mul hg.uniformity_inv | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_div | null |
@[to_additive /-- If `f : ι → G × G` converges to the uniformity, then any `g : ι → G × G`
converges to the uniformity iff `f + g` does. This is often useful when `f` is valued in the
diagonal, in which case its convergence is automatic. -/]
Filter.Tendsto.uniformity_mul_iff_right {ι : Type*} {f g : ι → α × α} {l : Fil... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_mul_iff_right | If `f : ι → G × G` converges to the uniformity, then any `g : ι → G × G` converges to the
uniformity iff `f * g` does. This is often useful when `f` is valued in the diagonal,
in which case its convergence is automatic. |
@[to_additive /-- If `g : ι → G × G` converges to the uniformity, then any `f : ι → G × G`
converges to the uniformity iff `f + g` does. This is often useful when `g` is valued in the
diagonal, in which case its convergence is automatic. -/]
Filter.Tendsto.uniformity_mul_iff_left {ι : Type*} {f g : ι → α × α} {l : Filt... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Filter.Tendsto.uniformity_mul_iff_left | If `g : ι → G × G` converges to the uniformity, then any `f : ι → G × G` converges to the
uniformity iff `f * g` does. This is often useful when `g` is valued in the diagonal,
in which case its convergence is automatic. |
UniformContinuous.pow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
∀ n : ℕ, UniformContinuous fun x => f x ^ n
| 0 => by
simp_rw [pow_zero]
exact uniformContinuous_const
| n + 1 => by
simp_rw [pow_succ']
exact hf.mul (hf.pow_const n)
@[to_additive uniformContinuous_const_nsmul... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.pow_const | null |
uniformContinuous_pow_const (n : ℕ) : UniformContinuous fun x : α => x ^ n :=
uniformContinuous_id.pow_const n
@[to_additive UniformContinuous.const_zsmul] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_pow_const | null |
UniformContinuous.zpow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
∀ n : ℤ, UniformContinuous fun x => f x ^ n
| (n : ℕ) => by
simp_rw [zpow_natCast]
exact hf.pow_const _
| Int.negSucc n => by
simp_rw [zpow_negSucc]
exact (hf.pow_const _).inv
@[to_additive uniformContinuous_c... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | UniformContinuous.zpow_const | null |
uniformContinuous_zpow_const (n : ℤ) : UniformContinuous fun x : α => x ^ n :=
uniformContinuous_id.zpow_const n
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_zpow_const | null |
@[to_additive]
Prod.instIsUniformGroup [UniformSpace β] [Group β] [IsUniformGroup β] :
IsUniformGroup (α × β) :=
⟨((uniformContinuous_fst.comp uniformContinuous_fst).div
(uniformContinuous_fst.comp uniformContinuous_snd)).prodMk
((uniformContinuous_snd.comp uniformContinuous_fst).div
(unif... | instance | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | Prod.instIsUniformGroup | null |
uniformity_translate_mul (a : α) : ((𝓤 α).map fun x : α × α => (x.1 * a, x.2 * a)) = 𝓤 α :=
le_antisymm (uniformContinuous_id.mul uniformContinuous_const)
(calc
𝓤 α =
((𝓤 α).map fun x : α × α => (x.1 * a⁻¹, x.2 * a⁻¹)).map fun x : α × α =>
(x.1 * a, x.2 * a) := by simp [Filter.map_... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformity_translate_mul | null |
@[to_additive /-- An additive group homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) between two uniform additive groups is uniformly continuous provided that it
is continuous at zero. See also `continuous_of_continuousAt_zero`. -/]
uniformContinuous_of_continuousAt_one {hom : Type*} [Uni... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_of_continuousAt_one | The discrete uniformity makes a group a `IsUniformGroup. -/
@[to_additive /-- The discrete uniformity makes an additive group a `IsUniformAddGroup`. -/]
instance [UniformSpace β] [Group β] [DiscreteUniformity β] : IsUniformGroup β where
uniformContinuous_div := DiscreteUniformity.uniformContinuous (β × β) fun p ↦ p.1... |
MonoidHom.uniformContinuous_of_continuousAt_one [UniformSpace β] [Group β]
[IsUniformGroup β] (f : α →* β) (hf : ContinuousAt f 1) : UniformContinuous f :=
_root_.uniformContinuous_of_continuousAt_one f hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | MonoidHom.uniformContinuous_of_continuousAt_one | null |
@[to_additive /-- A homomorphism from a uniform additive group to a discrete uniform additive group
is continuous if and only if its kernel is open. -/]
IsUniformGroup.uniformContinuous_iff_isOpen_ker {hom : Type*} [UniformSpace β]
[DiscreteTopology β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsUniformGroup.uniformContinuous_iff_isOpen_ker | A homomorphism from a uniform group to a discrete uniform group is continuous if and only if
its kernel is open. |
uniformContinuous_monoidHom_of_continuous {hom : Type*} [UniformSpace β] [Group β]
[IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Continuous f) :
UniformContinuous f :=
uniformContinuous_of_tendsto_one <|
suffices Tendsto f (𝓝 1) (𝓝 (f 1)) by rwa [map_one] at this
h.ten... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformContinuous_monoidHom_of_continuous | null |
@[to_additive /-- The right uniformity on a topological additive group (as opposed to the left
uniformity).
Warning: in general the right and left uniformities do not coincide and so one does not obtain a
`IsUniformAddGroup` structure. Two important special cases where they _do_ coincide are for
commutative additive gr... | def | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsTopologicalGroup.toUniformSpace | The right uniformity on a topological group (as opposed to the left uniformity).
Warning: in general the right and left uniformities do not coincide and so one does not obtain a
`IsUniformGroup` structure. Two important special cases where they _do_ coincide are for
commutative groups (see `isUniformGroup_of_commGroup... |
uniformity_eq_comap_nhds_one' : 𝓤 G = comap (fun p : G × G => p.2 / p.1) (𝓝 (1 : G)) :=
rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | uniformity_eq_comap_nhds_one' | null |
@[to_additive]
isUniformGroup_of_commGroup : IsUniformGroup G := by
constructor
simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq_comap_nhds_one',
tendsto_comap_iff, tendsto_map'_iff, prod_comap_comap_eq, Function.comp_def,
div_div_div_comm _ (Prod.snd (Prod.snd _)), ← nhds_prod_eq]
exa... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | isUniformGroup_of_commGroup | null |
@[to_additive]
IsUniformGroup.toUniformSpace_eq {G : Type*} [u : UniformSpace G] [Group G]
[IsUniformGroup G] : IsTopologicalGroup.toUniformSpace G = u := by
ext : 1
rw [uniformity_eq_comap_nhds_one' G, uniformity_eq_comap_nhds_one G] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | IsUniformGroup.toUniformSpace_eq | null |
@[to_additive]
tendsto_div_comap_self (de : IsDenseInducing e) (x₀ : α) :
Tendsto (fun t : β × β => t.2 / t.1) ((comap fun p : β × β => (e p.1, e p.2)) <| 𝓝 (x₀, x₀))
(𝓝 1) := by
have comm : ((fun x : α × α => x.2 / x.1) ∘ fun t : β × β => (e t.1, e t.2)) =
e ∘ fun t : β × β => t.2 / t.1 := by
e... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | tendsto_div_comap_self | null |
private extend_Z_bilin_aux (x₀ : α) (y₁ : δ) : ∃ U₂ ∈ comap e (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂,
(fun p : β × δ => φ p.1 p.2) (x' - x, y₁) ∈ W' := by
let Nx := 𝓝 x₀
let ee := fun u : β × β => (e u.1, e u.2)
have lim1 : Tendsto (fun a : β × β => (a.2 - a.1, y₁))
(comap e Nx ×ˢ comap e Nx) (𝓝 (0, y₁)) := by
... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | extend_Z_bilin_aux | null |
private extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' := by
let ee := fun u : β × β => (e u.1, e u.2)
let ff := fun u : δ × δ => (f ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/IsUniformGroup/Defs.lean | extend_Z_bilin_key | null |
TendstoUniformlyOn.eventually_forall_lt {u v : β} (huv : u < v)
(hf : TendstoUniformlyOn f g p K) (hg : ∀ x ∈ K, g x ≤ u) :
∀ᶠ i in p, ∀ x ∈ K, f i x < v := by
simp only [tendstoUniformlyOn_iff_tendsto, uniformity_eq_comap_neg_add_nhds_zero,
tendsto_iff_eventually, eventually_comap, Prod.forall] at *
co... | lemma | Topology | [
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/Algebra/IsUniformGroup/Order.lean | TendstoUniformlyOn.eventually_forall_lt | If a sequence of functions converges uniformly on a set to a function `g` which is bounded above
by a value `u`, then the sequence is strictly bounded by any `v` such that `u < v`. |
TendstoUniformlyOn.eventually_forall_le {u v : β} (huv : u < v)
(hf : TendstoUniformlyOn f g p K) (hg : ∀ x ∈ K, g x ≤ u) :
∀ᶠ i in p, ∀ x ∈ K, f i x ≤ v := by
filter_upwards [hf.eventually_forall_lt huv hg] with i hi x hx using (hi x hx).le | lemma | Topology | [
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/Algebra/IsUniformGroup/Order.lean | TendstoUniformlyOn.eventually_forall_le | null |
LipschitzWith.cauchySeq_comp {f : α → β} (hf : LipschitzWith K f) {u : ℕ → α}
(hu : CauchySeq u) :
CauchySeq (f ∘ u) := by
rcases cauchySeq_iff_le_tendsto_0.1 hu with ⟨b, b_nonneg, hb, blim⟩
refine cauchySeq_iff_le_tendsto_0.2 ⟨fun n ↦ K * b n, ?_, ?_, ?_⟩
· exact fun n ↦ mul_nonneg (by positivity) (b_non... | lemma | Topology | [
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/Algebra/MetricSpace/Lipschitz.lean | LipschitzWith.cauchySeq_comp | null |
LipschitzOnWith.cauchySeq_comp {s : Set α} {f : α → β} (hf : LipschitzOnWith K f s)
{u : ℕ → α} (hu : CauchySeq u) (h'u : range u ⊆ s) :
CauchySeq (f ∘ u) := by
rcases cauchySeq_iff_le_tendsto_0.1 hu with ⟨b, b_nonneg, hb, blim⟩
refine cauchySeq_iff_le_tendsto_0.2 ⟨fun n ↦ K * b n, ?_, ?_, ?_⟩
· exact fun... | lemma | Topology | [
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/Algebra/MetricSpace/Lipschitz.lean | LipschitzOnWith.cauchySeq_comp | null |
continuousAt_of_locally_lipschitz {f : α → β} {x : α} {r : ℝ} (hr : 0 < r) (K : ℝ)
(h : ∀ y, dist y x < r → dist (f y) (f x) ≤ K * dist y x) : ContinuousAt f x := by
refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero' (Eventually.of_forall fun _ => dist_nonneg)
(mem_of_superset (ball_mem_nhds _ hr) h) ?_)
... | theorem | Topology | [
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/Algebra/MetricSpace/Lipschitz.lean | continuousAt_of_locally_lipschitz | If a function is locally Lipschitz around a point, then it is continuous at this point. |
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
cont... | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousSMul.of_nhds_zero | null |
ContinuousNeg.of_continuousConstSMul [ContinuousConstSMul R M] : ContinuousNeg M where
continuous_neg := by simpa using continuous_const_smul (T := M) (-1 : R) | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousNeg.of_continuousConstSMul | A topological module over a ring has continuous negation.
This cannot be an instance, because it would cause search for `[Module ?R M]` with unknown `R`. |
Submodule.eq_top_of_nonempty_interior' [NeBot (𝓝[{ x : R | IsUnit x }] 0)]
(s : Submodule R M) (hs : (interior (s : Set M)).Nonempty) : s = ⊤ := by
rcases hs with ⟨y, hy⟩
refine Submodule.eq_top_iff'.2 fun x => ?_
rw [mem_interior_iff_mem_nhds] at hy
have : Tendsto (fun c : R => y + c • x) (𝓝[{ x : R | Is... | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.eq_top_of_nonempty_interior' | If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then
`⊤` is the only submodule of `M` with a nonempty interior.
This is the case, e.g., if `R` is a nontrivially normed field. |
Module.punctured_nhds_neBot [Nontrivial M] [NeBot (𝓝[≠] (0 : R))] [NoZeroSMulDivisors R M]
(x : M) : NeBot (𝓝[≠] x) := by
rcases exists_ne (0 : M) with ⟨y, hy⟩
suffices Tendsto (fun c : R => x + c • y) (𝓝[≠] 0) (𝓝[≠] x) from this.neBot
refine Tendsto.inf ?_ (tendsto_principal_principal.2 <| ?_)
· conver... | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Module.punctured_nhds_neBot | Let `R` be a topological ring such that zero is not an isolated point (e.g., a nontrivially
normed field, see `NormedField.punctured_nhds_neBot`). Let `M` be a nontrivial module over `R`
such that `c • x = 0` implies `c = 0 ∨ x = 0`. Then `M` has no isolated points. We formulate this
using `NeBot (𝓝[≠] x)`.
This lemm... |
continuousSMul_induced : @ContinuousSMul R M₁ _ u (t.induced f) :=
let _ : TopologicalSpace M₁ := t.induced f
IsInducing.continuousSMul ⟨rfl⟩ continuous_id (map_smul f _ _) | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | continuousSMul_induced | null |
TopologicalSpace.IsSeparable.span {R M : Type*} [AddCommMonoid M] [Semiring R] [Module R M]
[TopologicalSpace M] [TopologicalSpace R] [SeparableSpace R]
[ContinuousAdd M] [ContinuousSMul R M] {s : Set M} (hs : IsSeparable s) :
IsSeparable (Submodule.span R s : Set M) := by
rw [Submodule.span_eq_iUnion_nat... | lemma | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | TopologicalSpace.IsSeparable.span | The span of a separable subset with respect to a separable scalar ring is again separable. |
topologicalAddGroup {R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
[TopologicalSpace M] [IsTopologicalAddGroup M] (S : Submodule R M) : IsTopologicalAddGroup S :=
inferInstanceAs (IsTopologicalAddGroup S.toAddSubgroup) | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | topologicalAddGroup | null |
Submodule.mapsTo_smul_closure (s : Submodule R M) (c : R) :
Set.MapsTo (c • ·) (closure s : Set M) (closure s) :=
have : Set.MapsTo (c • ·) (s : Set M) s := fun _ h ↦ s.smul_mem c h
this.closure (continuous_const_smul c) | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.mapsTo_smul_closure | null |
Submodule.smul_closure_subset (s : Submodule R M) (c : R) :
c • closure (s : Set M) ⊆ closure (s : Set M) :=
(s.mapsTo_smul_closure c).image_subset
variable [ContinuousAdd M] | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.smul_closure_subset | null |
Submodule.topologicalClosure (s : Submodule R M) : Submodule R M :=
{ s.toAddSubmonoid.topologicalClosure with
smul_mem' := s.mapsTo_smul_closure }
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.topologicalClosure | The (topological-space) closure of a submodule of a topological `R`-module `M` is itself
a submodule. |
Submodule.topologicalClosure_coe (s : Submodule R M) :
(s.topologicalClosure : Set M) = closure (s : Set M) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.topologicalClosure_coe | null |
Submodule.le_topologicalClosure (s : Submodule R M) : s ≤ s.topologicalClosure :=
subset_closure | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.le_topologicalClosure | null |
Submodule.closure_subset_topologicalClosure_span (s : Set M) :
closure s ⊆ (span R s).topologicalClosure := by
rw [Submodule.topologicalClosure_coe]
exact closure_mono subset_span | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.closure_subset_topologicalClosure_span | null |
Submodule.isClosed_topologicalClosure (s : Submodule R M) :
IsClosed (s.topologicalClosure : Set M) := isClosed_closure | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.isClosed_topologicalClosure | null |
Submodule.topologicalClosure_minimal (s : Submodule R M) {t : Submodule R M} (h : s ≤ t)
(ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t :=
closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.topologicalClosure_minimal | null |
Submodule.topologicalClosure_mono {s : Submodule R M} {t : Submodule R M} (h : s ≤ t) :
s.topologicalClosure ≤ t.topologicalClosure :=
closure_mono h | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.topologicalClosure_mono | null |
IsClosed.submodule_topologicalClosure_eq {s : Submodule R M} (hs : IsClosed (s : Set M)) :
s.topologicalClosure = s :=
SetLike.ext' hs.closure_eq | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | IsClosed.submodule_topologicalClosure_eq | The topological closure of a closed submodule `s` is equal to `s`. |
Submodule.dense_iff_topologicalClosure_eq_top {s : Submodule R M} :
Dense (s : Set M) ↔ s.topologicalClosure = ⊤ := by
rw [← SetLike.coe_set_eq, dense_iff_closure_eq]
simp | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.dense_iff_topologicalClosure_eq_top | A subspace is dense iff its topological closure is the entire space. |
Submodule.topologicalClosure.completeSpace {M' : Type*} [AddCommMonoid M'] [Module R M']
[UniformSpace M'] [ContinuousAdd M'] [ContinuousConstSMul R M'] [CompleteSpace M']
(U : Submodule R M') : CompleteSpace U.topologicalClosure :=
isClosed_closure.completeSpace_coe | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.topologicalClosure.completeSpace | null |
Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :
IsClosed (s : Set M) ∨ Dense (s : Set M) := by
refine (hs.le_iff.mp s.le_topologicalClosure).symm.imp ?_ dense_iff_topologicalClosure_eq_top.mpr
exact fun h ↦ h ▸ isClosed_closure | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | Submodule.isClosed_or_dense_of_isCoatom | A maximal proper subspace of a topological module (i.e a `Submodule` satisfying `IsCoatom`)
is either closed or dense. |
closure_coe_iSup_map_single (s : ∀ i, Submodule R (M i)) :
closure (↑(⨆ i, (s i).map (LinearMap.single R M i)) : Set (∀ i, M i)) =
Set.univ.pi fun i ↦ closure (s i) := by
rw [← closure_pi_set]
refine (closure_mono ?_).antisymm <| closure_minimal ?_ isClosed_closure
· exact SetLike.coe_mono <| iSup_map_s... | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | closure_coe_iSup_map_single | If `s i` is a family of submodules, each is in its module,
then the closure of their span in the indexed product of the modules
is the product of their closures.
In case of a finite index type, this statement immediately follows from `Submodule.iSup_map_single`.
However, the statement is true for an infinite index typ... |
topologicalClosure_iSup_map_single [∀ i, ContinuousAdd (M i)]
[∀ i, ContinuousConstSMul R (M i)] (s : ∀ i, Submodule R (M i)) :
topologicalClosure (⨆ i, (s i).map (LinearMap.single R M i)) =
pi Set.univ fun i ↦ (s i).topologicalClosure :=
SetLike.coe_injective <| closure_coe_iSup_map_single _ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | topologicalClosure_iSup_map_single | If `s i` is a family of submodules, each is in its module,
then the closure of their span in the indexed product of the modules
is the product of their closures.
In case of a finite index type, this statement immediately follows from `Submodule.iSup_map_single`.
However, the statement is true for an infinite index typ... |
LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]
[TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]
[ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := by
cases nonempty_fintype ι
classical
have : (f : (ι → R) → M) = ... | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | LinearMap.continuous_on_pi | null |
@[simps -fullyApplied]
linearMapOfMemClosureRangeCoe (f : M₁ → M₂)
(hf : f ∈ closure (Set.range ((↑) : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂))) : M₁ →ₛₗ[σ] M₂ :=
{ addMonoidHomOfMemClosureRangeCoe f hf with
map_smul' := (isClosed_setOf_map_smul M₁ M₂ σ).closure_subset_iff.2
(Set.range_subset_iff.2 LinearMap.map_smul... | def | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | linearMapOfMemClosureRangeCoe | Constructs a bundled linear map from a function and a proof that this function belongs to the
closure of the set of linear maps. |
@[simps! -fullyApplied]
linearMapOfTendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.NeBot]
(h : Tendsto (fun a x => g a x) l (𝓝 f)) : M₁ →ₛₗ[σ] M₂ :=
linearMapOfMemClosureRangeCoe f <|
mem_closure_of_tendsto h <| Eventually.of_forall fun _ => Set.mem_range_self _
variable (M₁ M₂ σ) | def | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | linearMapOfTendsto | Construct a bundled linear map from a pointwise limit of linear maps |
LinearMap.isClosed_range_coe : IsClosed (Set.range ((↑) : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂)) :=
isClosed_of_closure_subset fun f hf => ⟨linearMapOfMemClosureRangeCoe f hf, rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | LinearMap.isClosed_range_coe | null |
_root_.QuotientModule.Quotient.topologicalSpace : TopologicalSpace (M ⧸ S) :=
inferInstanceAs (TopologicalSpace (Quotient S.quotientRel)) | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | _root_.QuotientModule.Quotient.topologicalSpace | null |
isOpenMap_mkQ [ContinuousAdd M] : IsOpenMap S.mkQ :=
QuotientAddGroup.isOpenMap_coe | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | isOpenMap_mkQ | null |
isOpenQuotientMap_mkQ [ContinuousAdd M] : IsOpenQuotientMap S.mkQ :=
QuotientAddGroup.isOpenQuotientMap_mk | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | isOpenQuotientMap_mkQ | null |
topologicalAddGroup_quotient [IsTopologicalAddGroup M] : IsTopologicalAddGroup (M ⧸ S) :=
inferInstanceAs <| IsTopologicalAddGroup (M ⧸ S.toAddSubgroup) | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | topologicalAddGroup_quotient | null |
continuousSMul_quotient [TopologicalSpace R] [IsTopologicalAddGroup M]
[ContinuousSMul R M] : ContinuousSMul R (M ⧸ S) where
continuous_smul := by
rw [← (IsOpenQuotientMap.id.prodMap S.isOpenQuotientMap_mkQ).continuous_comp_iff]
exact continuous_quot_mk.comp continuous_smul | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | continuousSMul_quotient | null |
t3_quotient_of_isClosed [IsTopologicalAddGroup M] [IsClosed (S : Set M)] :
T3Space (M ⧸ S) :=
letI : IsClosed (S.toAddSubgroup : Set M) := ‹_›
QuotientAddGroup.instT3Space S.toAddSubgroup | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs"
] | Mathlib/Topology/Algebra/Module/Basic.lean | t3_quotient_of_isClosed | null |
continuum_le_cardinal_of_nontriviallyNormedField
(𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜 := by
suffices ∃ f : (ℕ → Bool) → 𝕜, range f ⊆ univ ∧ Continuous f ∧ Injective f by
rcases this with ⟨f, -, -, f_inj⟩
simpa using lift_mk_le_lift_mk_of_injective f_inj
apply Perfect.... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | continuum_le_cardinal_of_nontriviallyNormedField | A complete nontrivially normed field has cardinality at least continuum. |
continuum_le_cardinal_of_module
(𝕜 : Type u) (E : Type v) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[AddCommGroup E] [Module 𝕜 E] [Nontrivial E] : 𝔠 ≤ #E := by
have A : lift.{v} (𝔠 : Cardinal.{u}) ≤ lift.{v} (#𝕜) := by
simpa using continuum_le_cardinal_of_nontriviallyNormedField 𝕜
simpa usin... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | continuum_le_cardinal_of_module | A nontrivial module over a complete nontrivially normed field has cardinality at least
continuum. |
cardinal_eq_of_mem_nhds_zero
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [Zero E] [MulActionWithZero 𝕜 E]
[TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : s ∈ 𝓝 (0 : E)) : #s = #E := by
/- As `s` is a neighborhood of `0`, the space is covered by the rescaled sets `c^n • s`,
where `c`... | lemma | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | cardinal_eq_of_mem_nhds_zero | In a topological vector space over a nontrivially normed field, any neighborhood of zero has
the same cardinality as the whole space.
See also `cardinal_eq_of_mem_nhds`. |
cardinal_eq_of_mem_nhds
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddGroup E] [MulActionWithZero 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E := by
let g := Homeomorph.addLeft x
let t := g ⁻¹' s
have : t ∈ 𝓝 0 := g.conti... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | cardinal_eq_of_mem_nhds | In a topological vector space over a nontrivially normed field, any neighborhood of a point has
the same cardinality as the whole space. |
cardinal_eq_of_isOpen
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddGroup E] [MulActionWithZero 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E}
(hs : IsOpen s) (h's : s.Nonempty) : #s = #E := by
rcases h's with ⟨x, hx⟩
exact cardinal_eq_of_mem_nhds 𝕜 (hs.mem_n... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | cardinal_eq_of_isOpen | In a topological vector space over a nontrivially normed field, any nonempty open set has
the same cardinality as the whole space. |
continuum_le_cardinal_of_isOpen
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E]
[Module 𝕜 E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} (hs : IsOpen s) (h's : s.Nonempty) : 𝔠 ≤ #s := by
simpa [cardinal_eq_of_isOpen 𝕜... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | continuum_le_cardinal_of_isOpen | In a nontrivial topological vector space over a complete nontrivially normed field, any nonempty
open set has cardinality at least continuum. |
Set.Countable.dense_compl
{E : Type u} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E]
[Module 𝕜 E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} (hs : s.Countable) : Dense sᶜ := by
rw [← interior_eq_empty_iff_dense_compl]
by_contr... | theorem | Topology | [
"Mathlib.Algebra.Module.Card",
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.SetTheory.Cardinal.CountableCover",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.MetricSpace.Perfect"
] | Mathlib/Topology/Algebra/Module/Cardinality.lean | Set.Countable.dense_compl | In a nontrivial topological vector space over a complete nontrivially normed field, any
countable set has dense complement. |
characterSpace (𝕜 : Type*) (A : Type*) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
[ContinuousConstSMul 𝕜 𝕜] [NonUnitalNonAssocSemiring A] [TopologicalSpace A] [Module 𝕜 A] :=
{φ : WeakDual 𝕜 A | φ ≠ 0 ∧ ∀ x y : A, φ (x * y) = φ x * φ y}
variable {𝕜 : Type*} {A : Type*} | def | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | characterSpace | The character space of a topological algebra is the subset of elements of the weak dual that
are also algebra homomorphisms. |
instFunLike : FunLike (characterSpace 𝕜 A) A 𝕜 where
coe φ := ((φ : WeakDual 𝕜 A) : A → 𝕜)
coe_injective' φ ψ h := by ext1; apply DFunLike.ext; exact congr_fun h | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | instFunLike | null |
instContinuousLinearMapClass : ContinuousLinearMapClass (characterSpace 𝕜 A) 𝕜 A 𝕜 where
map_smulₛₗ φ := (φ : WeakDual 𝕜 A).map_smul
map_add φ := (φ : WeakDual 𝕜 A).map_add
map_continuous φ := (φ : WeakDual 𝕜 A).cont | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | instContinuousLinearMapClass | Elements of the character space are continuous linear maps. |
@[simp, norm_cast]
protected coe_coe (φ : characterSpace 𝕜 A) : ⇑(φ : WeakDual 𝕜 A) = (φ : A → 𝕜) :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | coe_coe | This has to come after `WeakDual.CharacterSpace.instFunLike`, otherwise the right-hand side
gets coerced via `Subtype.val` instead of directly via `DFunLike`. |
ext {φ ψ : characterSpace 𝕜 A} (h : ∀ x, φ x = ψ x) : φ = ψ :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | ext | null |
toCLM (φ : characterSpace 𝕜 A) : A →L[𝕜] 𝕜 :=
(φ : WeakDual 𝕜 A)
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | toCLM | An element of the character space, as a continuous linear map. |
coe_toCLM (φ : characterSpace 𝕜 A) : ⇑(toCLM φ) = φ :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | coe_toCLM | null |
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