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 ⌀ |
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UniformFun.ofFun_prod {β : Type*} [CommMonoid β] {f : ι → α → β} (I : Finset ι) :
ofFun (∏ i ∈ I, f i) = ∏ i ∈ I, ofFun (f i) :=
rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_prod | null |
UniformFun.toFun_prod {β : Type*} [CommMonoid β] {f : ι → α → β} (I : Finset ι) :
toFun (∏ i ∈ I, f i) = ∏ i ∈ I, toFun (f i) :=
rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_prod | null |
UniformFun.uniformContinuousConstSMul :
UniformContinuousConstSMul M (α →ᵤ X) where
uniformContinuous_const_smul c := UniformFun.postcomp_uniformContinuous <|
uniformContinuous_const_smul c | instance | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.uniformContinuousConstSMul | null |
UniformFunOn.uniformContinuousConstSMul {𝔖 : Set (Set α)} :
UniformContinuousConstSMul M (α →ᵤ[𝔖] X) where
uniformContinuous_const_smul c := UniformOnFun.postcomp_uniformContinuous <|
uniformContinuous_const_smul c | instance | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFunOn.uniformContinuousConstSMul | null |
CompletableTopField : Prop extends T0Space K where
nice : ∀ F : Filter K, Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (map (fun x => x⁻¹) F) | class | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | CompletableTopField | A topological field is completable if it is separated and the image under
the mapping x ↦ x⁻¹ of every Cauchy filter (with respect to the additive uniform structure)
which does not have a cluster point at 0 is a Cauchy filter
(with respect to the additive uniform structure). This ensures the completion is
a field. |
hatInv : hat K → hat K :=
isDenseInducing_coe.extend fun x : K => (↑x⁻¹ : hat K)
@[fun_prop] | def | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | hatInv | extension of inversion to the completion of a field. |
continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
ContinuousAt hatInv x := by
refine isDenseInducing_coe.continuousAt_extend ?_
apply mem_of_superset (compl_singleton_mem_nhds h)
intro y y_ne
rw [mem_compl_singleton_iff] at y_ne
apply CompleteSpace.complete
have : (fun (x : K) => (↑x⁻¹ : hat K)) =
((fun (y : K) => (↑y : hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
unfold Function.comp
simp
rw [this, ← Filter.map_map]
apply Cauchy.map _ (Completion.uniformContinuous_coe K)
apply CompletableTopField.nice
· haveI := isDenseInducing_coe.comap_nhds_neBot y
apply cauchy_nhds.comap
rw [Completion.comap_coe_eq_uniformity]
· have eq_bot : 𝓝 (0 : hat K) ⊓ 𝓝 y = ⊥ := by
by_contra h
exact y_ne (eq_of_nhds_neBot <| neBot_iff.mpr h).symm
rw [isDenseInducing_coe.nhds_eq_comap (0 : K), ← Filter.comap_inf]
norm_cast
rw [eq_bot]
exact comap_bot
open Classical in | theorem | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | continuous_hatInv | null |
instInvCompletion : Inv (hat K) :=
⟨fun x => if x = 0 then 0 else hatInv x⟩
variable [IsTopologicalDivisionRing K] | instance | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | instInvCompletion | The value of `hat_inv` at zero is not really specified, although it's probably zero.
Here we explicitly enforce the `inv_zero` axiom. |
hatInv_extends {x : K} (h : x ≠ 0) : hatInv (x : hat K) = ↑(x⁻¹ : K) :=
isDenseInducing_coe.extend_eq_at ((continuous_coe K).continuousAt.comp (continuousAt_inv₀ h))
variable [CompletableTopField K]
@[norm_cast] | theorem | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | hatInv_extends | null |
coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
by_cases h : x = 0
· rw [h, inv_zero]
dsimp [Inv.inv]
norm_cast
simp
· conv_lhs => dsimp [Inv.inv]
rw [if_neg]
· exact hatInv_extends h
· exact fun H => h (isDenseEmbedding_coe.injective H)
variable [IsUniformAddGroup K] | theorem | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | coe_inv | null |
mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
haveI : T1Space (hat K) := T2Space.t1Space
let f := fun x : hat K => x * hatInv x
let c := (fun (x : K) => (x : hat K))
change f x = 1
have cont : ContinuousAt f x := by
fun_prop (disch := assumption)
have clo : x ∈ closure (c '' {0}ᶜ) := by
have := isDenseInducing_coe.dense x
rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
image_union] at this
apply mem_closure_of_mem_closure_union this
rw [image_singleton]
exact compl_singleton_mem_nhds x_ne
have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
have : f '' (c '' {0}ᶜ) ⊆ {1} := by
rw [image_image]
rintro _ ⟨z, z_ne, rfl⟩
rw [mem_singleton_iff]
rw [mem_compl_singleton_iff] at z_ne
dsimp [f]
rw [hatInv_extends z_ne, ← coe_mul]
rw [mul_inv_cancel₀ z_ne, coe_one]
replace fxclo := closure_mono this fxclo
rwa [closure_singleton, mem_singleton_iff] at fxclo | theorem | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | mul_hatInv_cancel | null |
instField : Field (hat K) where
mul_inv_cancel := fun x x_ne => by simp only [Inv.inv, if_neg x_ne, mul_hatInv_cancel x_ne]
inv_zero := by simp only [Inv.inv, ite_true]
nnqsmul := _
nnqsmul_def := fun _ _ => rfl
qsmul := _
qsmul_def := fun _ _ => rfl | instance | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | instField | null |
Subfield.completableTopField (K : Subfield L) : CompletableTopField K where
nice F F_cau inf_F := by
let i : K →+* L := K.subtype
have hi : IsUniformInducing i := isUniformEmbedding_subtype_val.isUniformInducing
rw [← hi.cauchy_map_iff] at F_cau ⊢
rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)]
apply CompletableTopField.nice _ F_cau
rw [← Filter.push_pull', ← map_zero i, ← hi.isInducing.nhds_eq_comap, inf_F, Filter.map_bot] | instance | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | Subfield.completableTopField | null |
IsUniformInducing.completableTopField
[UniformSpace α] [T0Space α]
{f : α →+* β} (hf : IsUniformInducing f) :
CompletableTopField α := by
refine CompletableTopField.mk (fun F F_cau inf_F => ?_)
rw [← IsUniformInducing.cauchy_map_iff hf] at F_cau ⊢
have h_comm : (f ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ f := by
ext; simp only [Function.comp_apply, map_inv₀]
rw [Filter.map_comm h_comm]
apply CompletableTopField.nice _ F_cau
rw [← Filter.push_pull', ← map_zero f, ← hf.isInducing.nhds_eq_comap, inf_F, Filter.map_bot] | theorem | Topology | [
"Mathlib.RingTheory.SimpleRing.Basic",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Algebra/UniformField.lean | IsUniformInducing.completableTopField | The pullback of a completable topological field along a uniform inducing
ring homomorphism is a completable topological field. |
protected uniformSpace : UniformSpace G :=
@IsTopologicalAddGroup.toUniformSpace G _ B.topology B.isTopologicalAddGroup | def | Topology | [
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformFilterBasis.lean | uniformSpace | The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure. |
protected isUniformAddGroup : @IsUniformAddGroup G B.uniformSpace _ :=
@isUniformAddGroup_of_addCommGroup G _ B.topology B.isTopologicalAddGroup
@[deprecated (since := "2025-03-27")] alias uniformAddGroup := AddGroupFilterBasis.isUniformAddGroup | theorem | Topology | [
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformFilterBasis.lean | isUniformAddGroup | The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure is compatible with its group structure. |
cauchy_iff {F : Filter G} :
@Cauchy G B.uniformSpace F ↔
F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U := by
letI := B.uniformSpace
haveI := B.isUniformAddGroup
suffices F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U by
constructor <;> rintro ⟨h', h⟩ <;> refine ⟨h', ?_⟩ <;> [rwa [← this]; rwa [this]]
rw [uniformity_eq_comap_nhds_zero G, ← map_le_iff_le_comap]
change Tendsto _ _ _ ↔ _
simp [(basis_sets F).prod_self.tendsto_iff B.nhds_zero_hasBasis, @forall_swap (_ ∈ _) G] | theorem | Topology | [
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformFilterBasis.lean | cauchy_iff | null |
UniformContinuousConstVAdd [VAdd M X] : Prop where
uniformContinuous_const_vadd : ∀ c : M, UniformContinuous (c +ᵥ · : X → X) | class | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuousConstVAdd | An additive action such that for all `c`, the map `fun x ↦ c +ᵥ x` is uniformly continuous. |
@[to_additive]
UniformContinuousConstSMul [SMul M X] : Prop where
uniformContinuous_const_smul : ∀ c : M, UniformContinuous (c • · : X → X)
export UniformContinuousConstVAdd (uniformContinuous_const_vadd)
export UniformContinuousConstSMul (uniformContinuous_const_smul) | class | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuousConstSMul | A multiplicative action such that for all `c`,
the map `fun x ↦ c • x` is uniformly continuous. |
AddMonoid.uniformContinuousConstSMul_nat [AddGroup X] [IsUniformAddGroup X] :
UniformContinuousConstSMul ℕ X :=
⟨uniformContinuous_const_nsmul⟩ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | AddMonoid.uniformContinuousConstSMul_nat | null |
AddGroup.uniformContinuousConstSMul_int [AddGroup X] [IsUniformAddGroup X] :
UniformContinuousConstSMul ℤ X :=
⟨uniformContinuous_const_zsmul⟩ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | AddGroup.uniformContinuousConstSMul_int | null |
uniformContinuousConstSMul_of_continuousConstSMul [Monoid R] [AddGroup M]
[DistribMulAction R M] [UniformSpace M] [IsUniformAddGroup M] [ContinuousConstSMul R M] :
UniformContinuousConstSMul R M :=
⟨fun r =>
uniformContinuous_of_continuousAt_zero (DistribMulAction.toAddMonoidHom M r)
(Continuous.continuousAt (continuous_const_smul r))⟩ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | uniformContinuousConstSMul_of_continuousConstSMul | A `DistribMulAction` that is continuous on a uniform group is uniformly continuous.
This can't be an instance due to it forming a loop with
`UniformContinuousConstSMul.to_continuousConstSMul` |
Ring.uniformContinuousConstSMul [Ring R] [UniformSpace R] [IsUniformAddGroup R]
[ContinuousMul R] : UniformContinuousConstSMul R R :=
uniformContinuousConstSMul_of_continuousConstSMul _ _ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | Ring.uniformContinuousConstSMul | The action of `Semiring.toModule` is uniformly continuous. |
Ring.uniformContinuousConstSMul_op [Ring R] [UniformSpace R] [IsUniformAddGroup R]
[ContinuousMul R] : UniformContinuousConstSMul Rᵐᵒᵖ R :=
uniformContinuousConstSMul_of_continuousConstSMul _ _ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | Ring.uniformContinuousConstSMul_op | The action of `Semiring.toOppositeModule` is uniformly continuous. |
@[to_additive]
UniformContinuous.const_smul [UniformContinuousConstSMul M X] {f : Y → X}
(hf : UniformContinuous f) (c : M) : UniformContinuous (c • f) :=
(uniformContinuous_const_smul c).comp hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuous.const_smul | null |
IsUniformInducing.uniformContinuousConstSMul [SMul M Y] [UniformContinuousConstSMul M Y]
{f : X → Y} (hf : IsUniformInducing f) (hsmul : ∀ (c : M) x, f (c • x) = c • f x) :
UniformContinuousConstSMul M X where
uniformContinuous_const_smul c := by
simpa only [hf.uniformContinuous_iff, Function.comp_def, hsmul]
using hf.uniformContinuous.const_smul c | lemma | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | IsUniformInducing.uniformContinuousConstSMul | null |
@[to_additive]
MulOpposite.uniformContinuousConstSMul [UniformContinuousConstSMul M X] :
UniformContinuousConstSMul M Xᵐᵒᵖ :=
⟨fun c =>
MulOpposite.uniformContinuous_op.comp <| MulOpposite.uniformContinuous_unop.const_smul c⟩ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | MulOpposite.uniformContinuousConstSMul | null |
@[to_additive]
IsUniformGroup.to_uniformContinuousConstSMul {G : Type u} [Group G] [UniformSpace G]
[IsUniformGroup G] : UniformContinuousConstSMul G G :=
⟨fun _ => uniformContinuous_const.mul uniformContinuous_id⟩ | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | IsUniformGroup.to_uniformContinuousConstSMul | null |
UniformContinuous.const_mul' [UniformContinuousConstSMul R R] {f : β → R}
(hf : UniformContinuous f) (a : R) : UniformContinuous fun x ↦ a * f x :=
hf.const_smul a | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuous.const_mul' | null |
UniformContinuous.mul_const' [UniformContinuousConstSMul Rᵐᵒᵖ R] {f : β → R}
(hf : UniformContinuous f) (a : R) : UniformContinuous fun x ↦ f x * a :=
hf.const_smul (MulOpposite.op a) | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuous.mul_const' | null |
uniformContinuous_mul_left' [UniformContinuousConstSMul R R] (a : R) :
UniformContinuous fun b : R => a * b :=
uniformContinuous_id.const_mul' _ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | uniformContinuous_mul_left' | null |
uniformContinuous_mul_right' [UniformContinuousConstSMul Rᵐᵒᵖ R] (a : R) :
UniformContinuous fun b : R => b * a :=
uniformContinuous_id.mul_const' _ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | uniformContinuous_mul_right' | null |
UniformContinuous.div_const' {R β : Type*} [DivisionRing R] [UniformSpace R]
[UniformContinuousConstSMul Rᵐᵒᵖ R] [UniformSpace β] {f : β → R}
(hf : UniformContinuous f) (a : R) :
UniformContinuous fun x ↦ f x / a := by
simpa [div_eq_mul_inv] using hf.mul_const' a⁻¹ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | UniformContinuous.div_const' | null |
uniformContinuous_div_const' {R : Type*} [DivisionRing R] [UniformSpace R]
[UniformContinuousConstSMul Rᵐᵒᵖ R] (a : R) :
UniformContinuous fun b : R => b / a :=
uniformContinuous_id.div_const' _ | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | uniformContinuous_div_const' | null |
@[to_additive]
IsUnit.smul_uniformity [Monoid M] [MulAction M X] [UniformContinuousConstSMul M X] {c : M}
(hc : IsUnit c) : c • 𝓤 X = 𝓤 X :=
let ⟨d, hcd⟩ := hc.exists_right_inv
have cU : c • 𝓤 X ≤ 𝓤 X := uniformContinuous_const_smul c
have dU : d • 𝓤 X ≤ 𝓤 X := uniformContinuous_const_smul d
le_antisymm cU <| by simpa [smul_smul, hcd] using Filter.smul_filter_le_smul_filter (a := c) dU
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | IsUnit.smul_uniformity | null |
smul_uniformity [Group M] [MulAction M X] [UniformContinuousConstSMul M X] (c : M) :
c • 𝓤 X = 𝓤 X :=
Group.isUnit _ |>.smul_uniformity | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | smul_uniformity | null |
smul_uniformity₀ [GroupWithZero M] [MulAction M X] [UniformContinuousConstSMul M X] {c : M}
(hc : c ≠ 0) : c • 𝓤 X = 𝓤 X :=
hc.isUnit.smul_uniformity | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | smul_uniformity₀ | null |
@[to_additive]
smul_def (c : M) (x : Completion X) : c • x = Completion.map (c • ·) x :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | smul_def | null |
@[to_additive]
instIsScalarTower [SMul N X] [SMul M N] [UniformContinuousConstSMul M X]
[UniformContinuousConstSMul N X] [IsScalarTower M N X] : IsScalarTower M N (Completion X) :=
⟨fun m n x => by
have : _ = (_ : Completion X → Completion X) :=
map_comp (uniformContinuous_const_smul m) (uniformContinuous_const_smul n)
refine Eq.trans ?_ (congr_fun this.symm x)
exact congr_arg (fun f => Completion.map f x) (funext (smul_assoc _ _))⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | instIsScalarTower | null |
@[to_additive (attr := simp, norm_cast)]
coe_smul (c : M) (x : X) : (↑(c • x) : Completion X) = c • (x : Completion X) :=
(map_coe (uniformContinuous_const_smul c) x).symm | theorem | Topology | [
"Mathlib.Algebra.Module.Opposite",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs"
] | Mathlib/Topology/Algebra/UniformMulAction.lean | coe_smul | null |
one : One (Completion α) :=
⟨(1 : α)⟩ | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | one | null |
mul : Mul (Completion α) :=
⟨curry <| (isDenseInducing_coe.prodMap isDenseInducing_coe).extend ((↑) ∘ uncurry (· * ·))⟩
@[norm_cast] | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | mul | null |
coe_one : ((1 : α) : Completion α) = 1 :=
rfl
@[simp] lemma coe_eq_one_iff [T0Space α] {x : α} : (x : Completion α) = 1 ↔ x = 1 :=
Completion.coe_inj | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | coe_one | null |
@[norm_cast]
coe_mul (a b : α) : ((a * b : α) : Completion α) = a * b :=
((isDenseInducing_coe.prodMap isDenseInducing_coe).extend_eq
((continuous_coe α).comp (@continuous_mul α _ _ _)) (a, b)).symm
variable [IsUniformAddGroup α] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | coe_mul | null |
ring : Ring (Completion α) :=
{ AddMonoidWithOne.unary, (inferInstanceAs (AddCommGroup (Completion α))),
(inferInstanceAs (Mul (Completion α))), (inferInstanceAs (One (Completion α))) with
zero_mul := fun a =>
Completion.induction_on a
(isClosed_eq (continuous_const.mul continuous_id) continuous_const)
fun a => by rw [← coe_zero, ← coe_mul, zero_mul]
mul_zero := fun a =>
Completion.induction_on a
(isClosed_eq (continuous_id.mul continuous_const) continuous_const)
fun a => by rw [← coe_zero, ← coe_mul, mul_zero]
one_mul := fun a =>
Completion.induction_on a
(isClosed_eq (continuous_const.mul continuous_id) continuous_id) fun a => by
rw [← coe_one, ← coe_mul, one_mul]
mul_one := fun a =>
Completion.induction_on a
(isClosed_eq (continuous_id.mul continuous_const) continuous_id) fun a => by
rw [← coe_one, ← coe_mul, mul_one]
mul_assoc := fun a b c =>
Completion.induction_on₃ a b c
(isClosed_eq
((continuous_fst.mul (continuous_fst.comp continuous_snd)).mul
(continuous_snd.comp continuous_snd))
(continuous_fst.mul
((continuous_fst.comp continuous_snd).mul
(continuous_snd.comp continuous_snd))))
fun a b c => by rw [← coe_mul, ← coe_mul, ← coe_mul, ← coe_mul, mul_assoc]
left_distrib := fun a b c =>
Completion.induction_on₃ a b c
(isClosed_eq
(continuous_fst.mul
(Continuous.add (continuous_fst.comp continuous_snd)
(continuous_snd.comp continuous_snd)))
(Continuous.add (continuous_fst.mul (continuous_fst.comp continuous_snd))
(continuous_fst.mul (continuous_snd.comp continuous_snd))))
fun a b c => by rw [← coe_add, ← coe_mul, ← coe_mul, ← coe_mul, ← coe_add, mul_add]
right_distrib := fun a b c =>
Completion.induction_on₃ a b c
(isClosed_eq
((Continuous.add continuous_fst (continuous_fst.comp continuous_snd)).mul
(continuous_snd.comp continuous_snd))
(Continuous.add (continuous_fst.mul (continuous_snd.comp continuous_snd))
((continuous_fst.comp continuous_snd).mul
(continuous_snd.comp continuous_snd))))
fun a b c => by rw [← coe_add, ← coe_mul, ← coe_mul, ← coe_mul, ← coe_add, add_mul] } | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | ring | null |
coeRingHom : α →+* Completion α where
toFun := (↑)
map_one' := coe_one α
map_zero' := coe_zero
map_add' := coe_add
map_mul' := coe_mul | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | coeRingHom | The map from a uniform ring to its completion, as a ring homomorphism. |
continuous_coeRingHom : Continuous (coeRingHom : α → Completion α) :=
continuous_coe α
variable {β : Type u} [UniformSpace β] [Ring β] [IsUniformAddGroup β] [IsTopologicalRing β]
(f : α →+* β) (hf : Continuous f) | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | continuous_coeRingHom | null |
extensionHom [CompleteSpace β] [T0Space β] : Completion α →+* β :=
have hf' : Continuous (f : α →+ β) := hf
have hf : UniformContinuous f := uniformContinuous_addMonoidHom_of_continuous hf'
{ toFun := Completion.extension f
map_zero' := by simp_rw [← coe_zero, extension_coe hf, f.map_zero]
map_add' := fun a b =>
Completion.induction_on₂ a b
(isClosed_eq (continuous_extension.comp continuous_add)
((continuous_extension.comp continuous_fst).add
(continuous_extension.comp continuous_snd)))
fun a b => by
simp_rw [← coe_add, extension_coe hf, f.map_add]
map_one' := by rw [← coe_one, extension_coe hf, f.map_one]
map_mul' := fun a b =>
Completion.induction_on₂ a b
(isClosed_eq (continuous_extension.comp continuous_mul)
((continuous_extension.comp continuous_fst).mul
(continuous_extension.comp continuous_snd)))
fun a b => by
simp_rw [← coe_mul, extension_coe hf, f.map_mul] } | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | extensionHom | The completion extension as a ring morphism. |
extensionHom_coe [CompleteSpace β] [T0Space β] (a : α) :
Completion.extensionHom f hf a = f a := by
simp only [Completion.extensionHom, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,
UniformSpace.Completion.extension_coe <| uniformContinuous_addMonoidHom_of_continuous hf] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | extensionHom_coe | null |
topologicalRing : IsTopologicalRing (Completion α) where
continuous_add := continuous_add
continuous_mul := continuous_mul | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | topologicalRing | null |
mapRingHom (hf : Continuous f) : Completion α →+* Completion β :=
extensionHom (coeRingHom.comp f) (continuous_coeRingHom.comp hf) | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | mapRingHom | The completion map as a ring morphism. |
mapRingHom_apply {x : UniformSpace.Completion α} :
UniformSpace.Completion.mapRingHom f hf x = UniformSpace.Completion.map f x := rfl
variable {f} | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | mapRingHom_apply | null |
mapRingHom_coe (hf : UniformContinuous f) (a : α) :
mapRingHom f hf.continuous a = f a := by
rw [mapRingHom_apply, map_coe hf] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | mapRingHom_coe | null |
@[simp]
map_smul_eq_mul_coe (r : R) :
Completion.map (r • ·) = ((algebraMap R A r : Completion A) * ·) := by
ext x
refine Completion.induction_on x ?_ fun a => ?_
· exact isClosed_eq Completion.continuous_map (continuous_mul_left _)
· simp_rw [map_coe (uniformContinuous_const_smul r) a, Algebra.smul_def, coe_mul] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | map_smul_eq_mul_coe | null |
algebra : Algebra R (Completion A) where
algebraMap := (UniformSpace.Completion.coeRingHom : A →+* Completion A).comp (algebraMap R A)
commutes' := fun r x =>
Completion.induction_on x (isClosed_eq (continuous_mul_left _) (continuous_mul_right _))
fun a => by
simpa only [coe_mul] using congr_arg ((↑) : A → Completion A) (Algebra.commutes r a)
smul_def' := fun r x => congr_fun (map_smul_eq_mul_coe A R r) x | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | algebra | null |
algebraMap_def (r : R) :
algebraMap R (Completion A) r = (algebraMap R A r : Completion A) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | algebraMap_def | null |
commRing : CommRing (Completion R) :=
{ Completion.ring with
mul_comm := fun a b =>
Completion.induction_on₂ a b
(isClosed_eq (continuous_fst.mul continuous_snd) (continuous_snd.mul continuous_fst))
fun a b => by rw [← coe_mul, ← coe_mul, mul_comm] } | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | commRing | null |
algebra' : Algebra R (Completion R) := by infer_instance | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | algebra' | A shortcut instance for the common case |
inseparableSetoid_ring (α) [Ring α] [TopologicalSpace α] [IsTopologicalRing α] :
inseparableSetoid α = Submodule.quotientRel (Ideal.closure ⊥) :=
Setoid.ext fun x y =>
addGroup_inseparable_iff.trans <| .trans (by rfl) (Submodule.quotientRel_def _).symm | theorem | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | inseparableSetoid_ring | null |
sepQuotHomeomorphRingQuot (α) [CommRing α] [TopologicalSpace α] [IsTopologicalRing α] :
SeparationQuotient α ≃ₜ α ⧸ (⊥ : Ideal α).closure where
toEquiv := Quotient.congrRight fun x y => by rw [inseparableSetoid_ring]
continuous_toFun := continuous_id.quotient_map' <| by
rw [inseparableSetoid_ring]; exact fun _ _ ↦ id
continuous_invFun := continuous_id.quotient_map' <| by
rw [inseparableSetoid_ring]; exact fun _ _ ↦ id | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | sepQuotHomeomorphRingQuot | Given a topological ring `α` equipped with a uniform structure that makes subtraction uniformly
continuous, get an homeomorphism between the separated quotient of `α` and the quotient ring
corresponding to the closure of zero. |
commRing [CommRing α] [TopologicalSpace α] [IsTopologicalRing α] :
CommRing (SeparationQuotient α) :=
(sepQuotHomeomorphRingQuot _).commRing | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | commRing | null |
sepQuotRingEquivRingQuot (α) [CommRing α] [TopologicalSpace α] [IsTopologicalRing α] :
SeparationQuotient α ≃+* α ⧸ (⊥ : Ideal α).closure :=
(sepQuotHomeomorphRingQuot _).ringEquiv | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | sepQuotRingEquivRingQuot | Given a topological ring `α` equipped with a uniform structure that makes subtraction uniformly
continuous, get an equivalence between the separated quotient of `α` and the quotient ring
corresponding to the closure of zero. |
topologicalRing [CommRing α] [TopologicalSpace α] [IsTopologicalRing α] :
IsTopologicalRing (SeparationQuotient α) where
toContinuousAdd :=
(sepQuotHomeomorphRingQuot α).isInducing.continuousAdd (sepQuotRingEquivRingQuot α)
toContinuousMul :=
(sepQuotHomeomorphRingQuot α).isInducing.continuousMul (sepQuotRingEquivRingQuot α)
toContinuousNeg :=
(sepQuotHomeomorphRingQuot α).isInducing.continuousNeg <|
map_neg (sepQuotRingEquivRingQuot α) | instance | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | topologicalRing | null |
noncomputable IsDenseInducing.extendRingHom {i : α →+* β} {f : α →+* γ}
(ue : IsUniformInducing i) (dr : DenseRange i) (hf : UniformContinuous f) : β →+* γ where
toFun := (ue.isDenseInducing dr).extend f
map_one' := by
convert IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 1
exacts [i.map_one.symm, f.map_one.symm]
map_zero' := by
convert IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 0 <;>
simp only [map_zero]
map_add' := by
have h := (uniformContinuous_uniformly_extend ue dr hf).continuous
refine fun x y => DenseRange.induction_on₂ dr ?_ (fun a b => ?_) x y
· exact isClosed_eq (Continuous.comp h continuous_add)
((h.comp continuous_fst).add (h.comp continuous_snd))
· simp_rw [← i.map_add, IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous _,
← f.map_add]
map_mul' := by
have h := (uniformContinuous_uniformly_extend ue dr hf).continuous
refine fun x y => DenseRange.induction_on₂ dr ?_ (fun a b => ?_) x y
· exact isClosed_eq (Continuous.comp h continuous_mul)
((h.comp continuous_fst).mul (h.comp continuous_snd))
· simp_rw [← i.map_mul, IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous _,
← f.map_mul] | def | Topology | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Ring.TransferInstance",
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.Ring.Ideal",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic"
] | Mathlib/Topology/Algebra/UniformRing.lean | IsDenseInducing.extendRingHom | The dense inducing extension as a ring homomorphism. |
nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) := by
rw [nhds_eq_update, update_self] | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | nhds_zero | The topology on a linearly ordered commutative group with a zero element adjoined.
A subset U is open if 0 ∉ U or if there is an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U. -/
scoped instance (priority := 100) topologicalSpace : TopologicalSpace Γ₀ :=
nhdsAdjoint 0 <| ⨅ γ ≠ 0, 𝓟 (Iio γ)
theorem nhds_eq_update : (𝓝 : Γ₀ → Filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) := by
rw [nhds_nhdsAdjoint, sup_of_le_right]
exact le_iInf₂ fun γ hγ ↦ le_principal_iff.2 <| zero_lt_iff.2 hγ
/-!
### Neighbourhoods of zero |
hasBasis_nhds_zero : (𝓝 (0 : Γ₀)).HasBasis (fun γ : Γ₀ => γ ≠ 0) Iio := by
rw [nhds_zero]
refine hasBasis_biInf_principal ?_ ⟨1, one_ne_zero⟩
exact directedOn_iff_directed.2 (Monotone.directed_ge fun a b hab => Iio_subset_Iio hab) | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | hasBasis_nhds_zero | In a linearly ordered group with zero element adjoined, `U` is a neighbourhood of `0` if and
only if there exists a nonzero element `γ₀` such that `Iio γ₀ ⊆ U`. |
Iio_mem_nhds_zero (hγ : γ ≠ 0) : Iio γ ∈ 𝓝 (0 : Γ₀) :=
hasBasis_nhds_zero.mem_of_mem hγ | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | Iio_mem_nhds_zero | null |
nhds_zero_of_units (γ : Γ₀ˣ) : Iio ↑γ ∈ 𝓝 (0 : Γ₀) :=
Iio_mem_nhds_zero γ.ne_zero | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | nhds_zero_of_units | If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`Iio (γ : Γ₀)` is a neighbourhood of `0`. |
tendsto_zero : Tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ (γ₀) (_ : γ₀ ≠ 0), ∀ᶠ x in l, f x < γ₀ := by
simp [nhds_zero]
/-! | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | tendsto_zero | null |
@[simp]
nhds_of_ne_zero {γ : Γ₀} (h₀ : γ ≠ 0) : 𝓝 γ = pure γ :=
nhds_nhdsAdjoint_of_ne _ h₀ | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | nhds_of_ne_zero | The neighbourhood filter of a nonzero element consists of all sets containing that
element. |
nhds_coe_units (γ : Γ₀ˣ) : 𝓝 (γ : Γ₀) = pure (γ : Γ₀) :=
nhds_of_ne_zero γ.ne_zero | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | nhds_coe_units | The neighbourhood filter of an invertible element consists of all sets containing that
element. |
singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({↑γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by simp | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | singleton_mem_nhds_of_units | If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`{γ}` is a neighbourhood of `γ`. |
singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by simp [h] | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | singleton_mem_nhds_of_ne_zero | If `γ` is a nonzero element of a linearly ordered group with zero element adjoined, then `{γ}`
is a neighbourhood of `γ`. |
hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} := by
rw [nhds_of_ne_zero h]
exact hasBasis_pure _ | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | hasBasis_nhds_of_ne_zero | null |
hasBasis_nhds_units (γ : Γ₀ˣ) :
HasBasis (𝓝 (γ : Γ₀)) (fun _ : Unit => True) fun _ => {↑γ} :=
hasBasis_nhds_of_ne_zero γ.ne_zero | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | hasBasis_nhds_units | null |
tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : Tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ := by
rw [nhds_of_ne_zero h, tendsto_pure] | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | tendsto_of_ne_zero | null |
tendsto_units {γ₀ : Γ₀ˣ} : Tendsto f l (𝓝 (γ₀ : Γ₀)) ↔ ∀ᶠ x in l, f x = γ₀ :=
tendsto_of_ne_zero γ₀.ne_zero | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | tendsto_units | null |
Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ := by
rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]
/-! | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | Iio_mem_nhds | null |
isOpen_iff {s : Set Γ₀} : IsOpen s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s := by
rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
simp +contextual [nhds_of_ne_zero, imp_iff_not_or,
hasBasis_nhds_zero.mem_iff] | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | isOpen_iff | null |
isClosed_iff {s : Set Γ₀} : IsClosed s ↔ (0 : Γ₀) ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Ici γ := by
simp only [← isOpen_compl_iff, isOpen_iff, mem_compl_iff, not_not, ← compl_Ici,
compl_subset_compl] | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | isClosed_iff | null |
isOpen_Iio {a : Γ₀} : IsOpen (Iio a) :=
isOpen_iff.mpr <| imp_iff_not_or.mp fun ha => ⟨a, ne_of_gt ha, Subset.rfl⟩
/-! | theorem | Topology | [
"Mathlib.Algebra.Order.GroupWithZero.Canonical",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Separation.Regular"
] | Mathlib/Topology/Algebra/WithZeroTopology.lean | isOpen_Iio | null |
BaireMeasurableSet (s : Set α) : Prop :=
@MeasurableSet _ (eventuallyMeasurableSpace (borel _) (residual _)) s
variable {s t : Set α} | def | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | BaireMeasurableSet | Notation for `=ᶠ[residual _]`. That is, eventual equality with respect to
the filter of residual sets. -/
scoped[Topology] notation:50 f " =ᵇ " g:50 => Filter.EventuallyEq (residual _) f g
/-- Notation to say that a property of points in a topological space holds
almost everywhere in the sense of Baire category. That is, on a residual set. -/
scoped[Topology] notation3 "∀ᵇ " (...) ", " r:(scoped p => Filter.Eventually p <| residual _) => r
/-- Notation to say that a property of points in a topological space holds on a nonmeager set. -/
scoped[Topology] notation3 "∃ᵇ " (...) ", " r:(scoped p => Filter.Frequently p <| residual _) => r
variable {α}
theorem coborder_mem_residual {s : Set α} (hs : IsLocallyClosed s) : coborder s ∈ residual α :=
residual_of_dense_open hs.isOpen_coborder dense_coborder
theorem closure_residualEq {s : Set α} (hs : IsLocallyClosed s) : closure s =ᵇ s := by
rw [Filter.eventuallyEq_set]
filter_upwards [coborder_mem_residual hs] with x hx
nth_rewrite 2 [← closure_inter_coborder (s := s)]
simp [hx]
/-- We say a set is a `BaireMeasurableSet` if it differs from some Borel set by
a meager set. This forms a σ-algebra.
It is equivalent, and a more standard definition, to say that the set differs from
some *open* set by a meager set. See `BaireMeasurableSet.iff_residualEq_isOpen` |
of_mem_residual (h : s ∈ residual _) : BaireMeasurableSet s :=
eventuallyMeasurableSet_of_mem_filter (α := α) h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | of_mem_residual | null |
_root_.MeasurableSet.baireMeasurableSet [MeasurableSpace α] [BorelSpace α]
(h : MeasurableSet s) : BaireMeasurableSet s := by
borelize α
exact h.eventuallyMeasurableSet | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | _root_.MeasurableSet.baireMeasurableSet | null |
_root_.IsOpen.baireMeasurableSet (h : IsOpen s) : BaireMeasurableSet s := by
borelize α
exact h.measurableSet.baireMeasurableSet | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | _root_.IsOpen.baireMeasurableSet | null |
compl (h : BaireMeasurableSet s) : BaireMeasurableSet sᶜ := MeasurableSet.compl h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | compl | null |
of_compl (h : BaireMeasurableSet sᶜ) : BaireMeasurableSet s := MeasurableSet.of_compl h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | of_compl | null |
_root_.IsMeagre.baireMeasurableSet (h : IsMeagre s) : BaireMeasurableSet s :=
(of_mem_residual h).of_compl | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | _root_.IsMeagre.baireMeasurableSet | null |
iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
(h : ∀ i, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋃ i, s i) :=
MeasurableSet.iUnion h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | iUnion | null |
biUnion {ι : Type*} {s : ι → Set α} {t : Set ι} (ht : t.Countable)
(h : ∀ i ∈ t, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋃ i ∈ t, s i) :=
MeasurableSet.biUnion ht h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | biUnion | null |
sUnion {s : Set (Set α)} (hs : s.Countable)
(h : ∀ t ∈ s, BaireMeasurableSet t) : BaireMeasurableSet (⋃₀ s) :=
MeasurableSet.sUnion hs h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | sUnion | null |
iInter {ι : Sort*} [Countable ι] {s : ι → Set α}
(h : ∀ i, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋂ i, s i) :=
MeasurableSet.iInter h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | iInter | null |
biInter {ι : Type*} {s : ι → Set α} {t : Set ι} (ht : t.Countable)
(h : ∀ i ∈ t, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋂ i ∈ t, s i) :=
MeasurableSet.biInter ht h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | biInter | null |
sInter {s : Set (Set α)} (hs : s.Countable)
(h : ∀ t ∈ s, BaireMeasurableSet t) : BaireMeasurableSet (⋂₀ s) :=
MeasurableSet.sInter hs h | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | sInter | null |
union (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
BaireMeasurableSet (s ∪ t) :=
MeasurableSet.union hs ht | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | union | null |
inter (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
BaireMeasurableSet (s ∩ t) :=
MeasurableSet.inter hs ht | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | inter | null |
diff (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
BaireMeasurableSet (s \ t) :=
MeasurableSet.diff hs ht | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | diff | null |
congr (hs : BaireMeasurableSet s) (h : s =ᵇ t) : BaireMeasurableSet t :=
EventuallyMeasurableSet.congr (α := α) hs h.symm | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | congr | null |
MeasurableSet.residualEq_isOpen [MeasurableSpace α] [BorelSpace α] (h : MeasurableSet s) :
∃ u : Set α, IsOpen u ∧ s =ᵇ u := by
induction s, h using MeasurableSet.induction_on_open with
| isOpen U hU => exact ⟨U, hU, .rfl⟩
| compl s _ ihs =>
obtain ⟨U, Uo, hsU⟩ := ihs
use (closure U)ᶜ, isClosed_closure.isOpen_compl
exact .compl <| hsU.trans <| .symm <| closure_residualEq Uo.isLocallyClosed
| iUnion f _ _ ihf =>
choose u uo su using ihf
exact ⟨⋃ i, u i, isOpen_iUnion uo, EventuallyEq.countable_iUnion su⟩ | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | MeasurableSet.residualEq_isOpen | Any Borel set differs from some open set by a meager set. |
BaireMeasurableSet.residualEq_isOpen (h : BaireMeasurableSet s) :
∃ u : Set α, (IsOpen u) ∧ s =ᵇ u := by
borelize α
rcases h with ⟨t, ht, hst⟩
rcases ht.residualEq_isOpen with ⟨u, hu, htu⟩
exact ⟨u, hu, hst.trans htu⟩ | theorem | Topology | [
"Mathlib.Topology.LocallyClosed",
"Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable",
"Mathlib.MeasureTheory.Constructions.BorelSpace.Basic"
] | Mathlib/Topology/Baire/BaireMeasurable.lean | BaireMeasurableSet.residualEq_isOpen | Any `BaireMeasurableSet` differs from some open set by a meager set. |
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