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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Filter.HasBasis.isUniformEmbedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
IsUniformEmbedding f ↔ Injective f ∧
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by
rw [isUniformEmbedding_iff, and_comm, h.isUniformInducing_iff h'] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Filter.HasBasis.isUniformEmbedding_iff' | null |
Filter.HasBasis.isUniformEmbedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by
simp only [h.isUniformEmbedding_iff' h', h.uniformContinuous_iff h'] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Filter.HasBasis.isUniformEmbedding_iff | null |
isUniformEmbedding_subtype_val {p : α → Prop} :
IsUniformEmbedding (Subtype.val : Subtype p → α) :=
{ comap_uniformity := rfl
injective := Subtype.val_injective } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_subtype_val | null |
isUniformEmbedding_set_inclusion {s t : Set α} (hst : s ⊆ t) :
IsUniformEmbedding (inclusion hst) where
comap_uniformity := by rw [uniformity_subtype, uniformity_subtype, comap_comap]; rfl
injective := inclusion_injective hst | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_set_inclusion | null |
IsUniformEmbedding.comp {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β}
(hf : IsUniformEmbedding f) : IsUniformEmbedding (g ∘ f) where
toIsUniformInducing := hg.isUniformInducing.comp hf.isUniformInducing
injective := hg.injective.comp hf.injective | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.comp | null |
IsUniformEmbedding.of_comp_iff {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β} :
IsUniformEmbedding (g ∘ f) ↔ IsUniformEmbedding f := by
simp_rw [isUniformEmbedding_iff, hg.isUniformInducing.of_comp_iff, hg.injective.of_comp_iff f] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.of_comp_iff | null |
Equiv.isUniformEmbedding {α β : Type*} [UniformSpace α] [UniformSpace β] (f : α ≃ β)
(h₁ : UniformContinuous f) (h₂ : UniformContinuous f.symm) : IsUniformEmbedding f :=
isUniformEmbedding_iff'.2 ⟨f.injective, h₁, by rwa [← Equiv.prodCongr_apply, ← map_equiv_symm]⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Equiv.isUniformEmbedding | null |
isUniformEmbedding_inl : IsUniformEmbedding (Sum.inl : α → α ⊕ β) :=
isUniformEmbedding_iff'.2 ⟨Sum.inl_injective, uniformContinuous_inl, fun s hs =>
⟨Prod.map Sum.inl Sum.inl '' s ∪ range (Prod.map Sum.inr Sum.inr),
union_mem_sup (image_mem_map hs) range_mem_map,
fun x h => by simpa [Prod.map_apply'] using h⟩⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_inl | null |
isUniformEmbedding_inr : IsUniformEmbedding (Sum.inr : β → α ⊕ β) :=
isUniformEmbedding_iff'.2 ⟨Sum.inr_injective, uniformContinuous_inr, fun s hs =>
⟨range (Prod.map Sum.inl Sum.inl) ∪ Prod.map Sum.inr Sum.inr '' s,
union_mem_sup range_mem_map (image_mem_map hs),
fun x h => by simpa [Prod.map_apply'] using h⟩⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_inr | null |
protected IsUniformInducing.isUniformEmbedding [T0Space α] {f : α → β}
(hf : IsUniformInducing f) : IsUniformEmbedding f :=
⟨hf, hf.isInducing.injective⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isUniformEmbedding | If the domain of a `IsUniformInducing` map `f` is a T₀ space, then `f` is injective,
hence it is a `IsUniformEmbedding`. |
isUniformEmbedding_iff_isUniformInducing [T0Space α] {f : α → β} :
IsUniformEmbedding f ↔ IsUniformInducing f :=
⟨IsUniformEmbedding.isUniformInducing, IsUniformInducing.isUniformEmbedding⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_iff_isUniformInducing | null |
comap_uniformity_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : comap (Prod.map f f) (𝓤 β) = 𝓟 idRel := by
refine le_antisymm ?_ (@refl_le_uniformity α (UniformSpace.comap f _))
calc
comap (Prod.map f f) (𝓤 β) ≤ comap (Prod.map f f) (𝓟 s) := comap_mono (le_principal_iff.2 hs)
_ = 𝓟 (Prod.map f f ⁻¹' s) := comap_principal
_ ≤ 𝓟 idRel := principal_mono.2 ?_
rintro ⟨x, y⟩; simpa [not_imp_not] using @hf x y | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | comap_uniformity_of_spaced_out | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is uniform inducing with respect to the discrete uniformity on `α`:
the preimage of `𝓤 β` under `Prod.map f f` is the principal filter generated by the diagonal in
`α × α`. |
isUniformEmbedding_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : @IsUniformEmbedding α β ⊥ ‹_› f := by
let _ : UniformSpace α := ⊥; have := discreteTopology_bot α
exact IsUniformInducing.isUniformEmbedding ⟨comap_uniformity_of_spaced_out hs hf⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_of_spaced_out | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is a uniform embedding with respect to the discrete uniformity on `α`. |
protected IsUniformEmbedding.isEmbedding {f : α → β} (h : IsUniformEmbedding f) :
IsEmbedding f where
toIsInducing := h.toIsUniformInducing.isInducing
injective := h.injective | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.isEmbedding | null |
IsUniformEmbedding.isDenseEmbedding {f : α → β} (h : IsUniformEmbedding f)
(hd : DenseRange f) : IsDenseEmbedding f :=
{ h.isEmbedding with dense := hd } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.isDenseEmbedding | null |
isClosedEmbedding_of_spaced_out {α} [TopologicalSpace α] [DiscreteTopology α]
[T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : IsClosedEmbedding f := by
rcases @DiscreteTopology.eq_bot α _ _ with rfl; let _ : UniformSpace α := ⊥
exact
{ (isUniformEmbedding_of_spaced_out hs hf).isEmbedding with
isClosed_range := isClosed_range_of_spaced_out hs hf } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isClosedEmbedding_of_spaced_out | null |
closure_image_mem_nhds_of_isUniformInducing {s : Set (α × α)} {e : α → β} (b : β)
(he₁ : IsUniformInducing e) (he₂ : IsDenseInducing e) (hs : s ∈ 𝓤 α) :
∃ a, closure (e '' { a' | (a, a') ∈ s }) ∈ 𝓝 b := by
obtain ⟨U, ⟨hU, hUo, hsymm⟩, hs⟩ :
∃ U, (U ∈ 𝓤 β ∧ IsOpen U ∧ IsSymmetricRel U) ∧ Prod.map e e ⁻¹' U ⊆ s := by
rwa [← he₁.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs
rcases he₂.dense.mem_nhds (UniformSpace.ball_mem_nhds b hU) with ⟨a, ha⟩
refine ⟨a, mem_of_superset ?_ (closure_mono <| image_mono <| UniformSpace.ball_mono hs a)⟩
have ho : IsOpen (UniformSpace.ball (e a) U) := UniformSpace.isOpen_ball (e a) hUo
refine mem_of_superset (ho.mem_nhds <| (UniformSpace.mem_ball_symmetry hsymm).2 ha) fun y hy => ?_
refine mem_closure_iff_nhds.2 fun V hV => ?_
rcases he₂.dense.mem_nhds (inter_mem hV (ho.mem_nhds hy)) with ⟨x, hxV, hxU⟩
exact ⟨e x, hxV, mem_image_of_mem e hxU⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | closure_image_mem_nhds_of_isUniformInducing | null |
isUniformEmbedding_subtypeEmb (p : α → Prop) {e : α → β} (ue : IsUniformEmbedding e)
(de : IsDenseEmbedding e) : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) :=
{ comap_uniformity := by
simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
ue.comap_uniformity.symm]
injective := (de.subtype p).injective } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_subtypeEmb | null |
IsUniformEmbedding.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : IsUniformEmbedding e₁) (h₂ : IsUniformEmbedding e₂) :
IsUniformEmbedding fun p : α × β => (e₁ p.1, e₂ p.2) where
toIsUniformInducing := h₁.isUniformInducing.prod h₂.isUniformInducing
injective := h₁.injective.prodMap h₂.injective | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.prod | null |
isComplete_image_iff {m : α → β} {s : Set α} (hm : IsUniformInducing m) :
IsComplete (m '' s) ↔ IsComplete s := by
have fact1 : SurjOn (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := surjOn_image .. |>.filter_map_Iic
have fact2 : MapsTo (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := mapsTo_image .. |>.filter_map_Iic
simp_rw [IsComplete, imp.swap (a := Cauchy _), ← mem_Iic (b := 𝓟 _), fact1.forall fact2,
hm.cauchy_map_iff, exists_mem_image, map_le_iff_le_comap, hm.isInducing.nhds_eq_comap] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isComplete_image_iff | A set is complete iff its image under a uniform inducing map is complete. |
IsUniformInducing.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
IsComplete (f '' s) ↔ IsComplete s := isComplete_image_iff hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isComplete_iff | If `f : X → Y` is an `IsUniformInducing` map, the image `f '' s` of a set `s` is complete
if and only if `s` is complete. |
IsUniformEmbedding.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformEmbedding f) :
IsComplete (f '' s) ↔ IsComplete s := hf.isUniformInducing.isComplete_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.isComplete_iff | If `f : X → Y` is an `IsUniformEmbedding`, the image `f '' s` of a set `s` is complete
if and only if `s` is complete. |
Subtype.isComplete_iff {p : α → Prop} {s : Set { x // p x }} :
IsComplete s ↔ IsComplete ((↑) '' s : Set α) :=
isUniformEmbedding_subtype_val.isComplete_iff.symm
alias ⟨isComplete_of_complete_image, _⟩ := isComplete_image_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Subtype.isComplete_iff | Sets of a subtype are complete iff their image under the coercion is complete. |
completeSpace_iff_isComplete_range {f : α → β} (hf : IsUniformInducing f) :
CompleteSpace α ↔ IsComplete (range f) := by
rw [completeSpace_iff_isComplete_univ, ← isComplete_image_iff hf, image_univ]
alias ⟨_, IsUniformInducing.completeSpace⟩ := completeSpace_iff_isComplete_range | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | completeSpace_iff_isComplete_range | null |
IsUniformInducing.isComplete_range [CompleteSpace α] (hf : IsUniformInducing f) :
IsComplete (range f) :=
(completeSpace_iff_isComplete_range hf).1 ‹_› | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isComplete_range | null |
IsUniformInducing.completeSpace_congr {f : α → β} (hf : IsUniformInducing f)
(hsurj : f.Surjective) : CompleteSpace α ↔ CompleteSpace β := by
rw [completeSpace_iff_isComplete_range hf, hsurj.range_eq, completeSpace_iff_isComplete_univ] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.completeSpace_congr | If `f` is a surjective uniform inducing map,
then its domain is a complete space iff its codomain is a complete space.
See also `_root_.completeSpace_congr` for a version that assumes `f` to be an equivalence. |
SeparationQuotient.completeSpace_iff :
CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α :=
.symm <| isUniformInducing_mk.completeSpace_congr surjective_mk | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | SeparationQuotient.completeSpace_iff | null |
SeparationQuotient.instCompleteSpace [CompleteSpace α] :
CompleteSpace (SeparationQuotient α) :=
completeSpace_iff.2 ‹_› | instance | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | SeparationQuotient.instCompleteSpace | null |
completeSpace_congr {e : α ≃ β} (he : IsUniformEmbedding e) :
CompleteSpace α ↔ CompleteSpace β :=
he.completeSpace_congr e.surjective | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | completeSpace_congr | See also `IsUniformInducing.completeSpace_congr`
for a version that works for non-injective maps. |
completeSpace_coe_iff_isComplete {s : Set α} : CompleteSpace s ↔ IsComplete s := by
rw [completeSpace_iff_isComplete_range isUniformEmbedding_subtype_val.isUniformInducing,
Subtype.range_coe]
alias ⟨_, IsComplete.completeSpace_coe⟩ := completeSpace_coe_iff_isComplete | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | completeSpace_coe_iff_isComplete | null |
IsClosed.completeSpace_coe [CompleteSpace α] {s : Set α} [hs : IsClosed s] :
CompleteSpace s := hs.isComplete.completeSpace_coe | instance | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsClosed.completeSpace_coe | null |
completeSpace_ulift_iff : CompleteSpace (ULift α) ↔ CompleteSpace α :=
IsUniformInducing.completeSpace_congr ⟨rfl⟩ ULift.down_surjective | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | completeSpace_ulift_iff | null |
ULift.instCompleteSpace [CompleteSpace α] : CompleteSpace (ULift α) :=
completeSpace_ulift_iff.2 ‹_› | instance | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | ULift.instCompleteSpace | The lift of a complete space to another universe is still complete. |
completeSpace_extension {m : β → α} (hm : IsUniformInducing m) (dense : DenseRange m)
(h : ∀ f : Filter β, Cauchy f → ∃ x : α, map m f ≤ 𝓝 x) : CompleteSpace α :=
⟨fun {f : Filter α} (hf : Cauchy f) =>
let p : Set (α × α) → Set α → Set α := fun s t => { y : α | ∃ x : α, x ∈ t ∧ (x, y) ∈ s }
let g := (𝓤 α).lift fun s => f.lift' (p s)
have mp₀ : Monotone p := fun _ _ h _ _ ⟨x, xs, xa⟩ => ⟨x, xs, h xa⟩
have mp₁ : ∀ {s}, Monotone (p s) := fun h _ ⟨y, ya, yxs⟩ => ⟨y, h ya, yxs⟩
have : f ≤ g := le_iInf₂ fun _ hs => le_iInf₂ fun _ ht =>
le_principal_iff.mpr <| mem_of_superset ht fun x hx => ⟨x, hx, refl_mem_uniformity hs⟩
have : NeBot g := hf.left.mono this
have : NeBot (comap m g) :=
comap_neBot fun _ ht =>
let ⟨t', ht', ht_mem⟩ := (mem_lift_sets <| monotone_lift' monotone_const mp₀).mp ht
let ⟨_, ht'', ht'_sub⟩ := (mem_lift'_sets mp₁).mp ht_mem
let ⟨x, hx⟩ := hf.left.nonempty_of_mem ht''
have h₀ : NeBot (𝓝[range m] x) := dense.nhdsWithin_neBot x
have h₁ : { y | (x, y) ∈ t' } ∈ 𝓝[range m] x :=
@mem_inf_of_left α (𝓝 x) (𝓟 (range m)) _ <| mem_nhds_left x ht'
have h₂ : range m ∈ 𝓝[range m] x :=
@mem_inf_of_right α (𝓝 x) (𝓟 (range m)) _ <| Subset.refl _
have : { y | (x, y) ∈ t' } ∩ range m ∈ 𝓝[range m] x := @inter_mem α (𝓝[range m] x) _ _ h₁ h₂
let ⟨_, xyt', b, b_eq⟩ := h₀.nonempty_of_mem this
⟨b, b_eq.symm ▸ ht'_sub ⟨x, hx, xyt'⟩⟩
have : Cauchy g :=
⟨‹NeBot g›, fun _ hs =>
let ⟨s₁, hs₁, comp_s₁⟩ := comp_mem_uniformity_sets hs
let ⟨s₂, hs₂, comp_s₂⟩ := comp_mem_uniformity_sets hs₁
let ⟨t, ht, (prod_t : t ×ˢ t ⊆ s₂)⟩ := mem_prod_same_iff.mp (hf.right hs₂)
have hg₁ : p (preimage Prod.swap s₁) t ∈ g :=
mem_lift (symm_le_uniformity hs₁) <| @mem_lift' α α f _ t ht
have hg₂ : p s₂ t ∈ g := mem_lift hs₂ <| @mem_lift' α α f _ t ht
have hg : p (Prod.swap ⁻¹' s₁) t ×ˢ p s₂ t ∈ g ×ˢ g := @prod_mem_prod α α _ _ g g hg₁ hg₂
(g ×ˢ g).sets_of_superset hg fun ⟨_, _⟩ ⟨⟨c₁, c₁t, hc₁⟩, ⟨c₂, c₂t, hc₂⟩⟩ =>
have : (c₁, c₂) ∈ t ×ˢ t := ⟨c₁t, c₂t⟩
comp_s₁ <| prodMk_mem_compRel hc₁ <| comp_s₂ <| prodMk_mem_compRel (prod_t this) hc₂⟩
have : Cauchy (Filter.comap m g) := ‹Cauchy g›.comap' (le_of_eq hm.comap_uniformity) ‹_›
let ⟨x, (hx : map m (Filter.comap m g) ≤ 𝓝 x)⟩ := h _ this
have : ClusterPt x (map m (Filter.comap m g)) :=
(le_nhds_iff_adhp_of_cauchy (this.map hm.uniformContinuous)).mp hx
have : ClusterPt x g := this.mono map_comap_le
⟨x,
calc
f ≤ g := by assumption
_ ≤ 𝓝 x := le_nhds_of_cauchy_adhp ‹Cauchy g› this
⟩⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | completeSpace_extension | null |
totallyBounded_image_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
TotallyBounded (f '' s) ↔ TotallyBounded s := by
refine ⟨fun hs ↦ ?_, fun h ↦ h.image hf.uniformContinuous⟩
simp_rw [(hf.basis_uniformity (basis_sets _)).totallyBounded_iff]
intro t ht
rcases exists_subset_image_finite_and.1 (hs.exists_subset ht) with ⟨u, -, hfin, h⟩
use u, hfin
rwa [biUnion_image, image_subset_iff, preimage_iUnion₂] at h | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | totallyBounded_image_iff | null |
totallyBounded_preimage {f : α → β} {s : Set β} (hf : IsUniformInducing f)
(hs : TotallyBounded s) : TotallyBounded (f ⁻¹' s) :=
(totallyBounded_image_iff hf).1 <| hs.subset <| image_preimage_subset .. | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | totallyBounded_preimage | null |
CompleteSpace.sum [CompleteSpace α] [CompleteSpace β] : CompleteSpace (α ⊕ β) := by
rw [completeSpace_iff_isComplete_univ, ← range_inl_union_range_inr]
exact isUniformEmbedding_inl.isUniformInducing.isComplete_range.union
isUniformEmbedding_inr.isUniformInducing.isComplete_range | instance | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | CompleteSpace.sum | null |
isUniformEmbedding_comap {α : Type*} {β : Type*} {f : α → β} [u : UniformSpace β]
(hf : Function.Injective f) : @IsUniformEmbedding α β (UniformSpace.comap f u) u f :=
@IsUniformEmbedding.mk _ _ (UniformSpace.comap f u) _ _
(@IsUniformInducing.mk _ _ (UniformSpace.comap f u) _ _ rfl) hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_comap | null |
Topology.IsEmbedding.comapUniformSpace {α β} [TopologicalSpace α] [u : UniformSpace β]
(f : α → β) (h : IsEmbedding f) : UniformSpace α :=
(u.comap f).replaceTopology h.eq_induced | def | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Topology.IsEmbedding.comapUniformSpace | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. |
Embedding.to_isUniformEmbedding {α β} [TopologicalSpace α] [u : UniformSpace β] (f : α → β)
(h : IsEmbedding f) : @IsUniformEmbedding α β (h.comapUniformSpace f) u f :=
let _ := h.comapUniformSpace f
{ comap_uniformity := rfl
injective := h.injective } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Embedding.to_isUniformEmbedding | null |
uniformly_extend_exists [CompleteSpace γ] (a : α) : ∃ c, Tendsto f (comap e (𝓝 a)) (𝓝 c) :=
let de := h_e.isDenseInducing h_dense
have : Cauchy (𝓝 a) := cauchy_nhds
have : Cauchy (comap e (𝓝 a)) :=
this.comap' (le_of_eq h_e.comap_uniformity) (de.comap_nhds_neBot _)
have : Cauchy (map f (comap e (𝓝 a))) := this.map h_f
CompleteSpace.complete this | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformly_extend_exists | null |
uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
{s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : IsUniformEmbedding e)
(hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
(hp : ∀ x ∈ s, p x) : ∃ c, Tendsto f (comap e (𝓝 b)) (𝓝 c) := by
have de : IsDenseEmbedding e := he.isDenseEmbedding hd
have de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e) := de.subtype p
have ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) :=
isUniformEmbedding_subtypeEmb _ he de
have : b ∈ closure (e '' { x | p x }) :=
(closure_mono <| monotone_image <| hp) (mem_of_mem_nhds hb)
let ⟨c, hc⟩ := uniformly_extend_exists ue'.isUniformInducing de'.dense hf ⟨b, this⟩
replace hc : Tendsto (f ∘ Subtype.val (p := p)) (((𝓝 b).comap e).comap Subtype.val) (𝓝 c) := by
simpa only [nhds_subtype_eq_comap, comap_comap, IsDenseEmbedding.subtypeEmb_coe] using hc
refine ⟨c, (tendsto_comap'_iff ?_).1 hc⟩
rw [Subtype.range_coe_subtype]
exact ⟨_, hb, by rwa [← de.isInducing.closure_eq_preimage_closure_image, hs.closure_eq]⟩
include h_e h_f in | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniform_extend_subtype | null |
uniformly_extend_spec [CompleteSpace γ] (a : α) : Tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) := by
simpa only [IsDenseInducing.extend] using
tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _)
include h_f in | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformly_extend_spec | null |
uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuous ψ := fun d hd =>
let ⟨s, hs, hs_comp⟩ := comp3_mem_uniformity hd
have h_pnt : ∀ {a m}, m ∈ 𝓝 a → ∃ c ∈ f '' (e ⁻¹' m), (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s :=
fun {a m} hm =>
have nb : NeBot (map f (comap e (𝓝 a))) :=
((h_e.isDenseInducing h_dense).comap_nhds_neBot _).map _
have :
f '' (e ⁻¹' m) ∩ ({ c | (c, ψ a) ∈ s } ∩ { c | (ψ a, c) ∈ s }) ∈ map f (comap e (𝓝 a)) :=
inter_mem (image_mem_map <| preimage_mem_comap <| hm)
(uniformly_extend_spec h_e h_dense h_f _
(inter_mem (mem_nhds_right _ hs) (mem_nhds_left _ hs)))
nb.nonempty_of_mem this
have : (Prod.map f f) ⁻¹' s ∈ 𝓤 β := h_f hs
have : (Prod.map f f) ⁻¹' s ∈ comap (Prod.map e e) (𝓤 α) := by
rwa [← h_e.comap_uniformity] at this
let ⟨t, ht, ts⟩ := this
show (Prod.map ψ ψ) ⁻¹' d ∈ 𝓤 α from
mem_of_superset (interior_mem_uniformity ht) fun ⟨x₁, x₂⟩ hx_t => by
have : interior t ∈ 𝓝 (x₁, x₂) := isOpen_interior.mem_nhds hx_t
let ⟨m₁, hm₁, m₂, hm₂, (hm : m₁ ×ˢ m₂ ⊆ interior t)⟩ := mem_nhds_prod_iff.mp this
obtain ⟨_, ⟨a, ha₁, rfl⟩, _, ha₂⟩ := h_pnt hm₁
obtain ⟨_, ⟨b, hb₁, rfl⟩, hb₂, _⟩ := h_pnt hm₂
have : Prod.map f f (a, b) ∈ s :=
ts <| mem_preimage.2 <| interior_subset (@hm (e a, e b) ⟨ha₁, hb₁⟩)
exact hs_comp ⟨f a, ha₂, ⟨f b, this, hb₂⟩⟩
variable [T0Space γ]
include h_f in | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformContinuous_uniformly_extend | null |
uniformly_extend_of_ind (b : β) : ψ (e b) = f b :=
IsDenseInducing.extend_eq_at _ h_f.continuous.continuousAt | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformly_extend_of_ind | null |
uniformly_extend_unique {g : α → γ} (hg : ∀ b, g (e b) = f b) (hc : Continuous g) : ψ = g :=
IsDenseInducing.extend_unique _ hg hc | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformly_extend_unique | null |
isUniformInducing_val (s : Set α) :
IsUniformInducing (@Subtype.val α s) := ⟨uniformity_setCoe⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformInducing_val | null |
uniformContinuous_rangeFactorization_iff {f : α → β} :
UniformContinuous (rangeFactorization f) ↔ UniformContinuous f :=
(isUniformInducing_val _).uniformContinuous_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformContinuous_rangeFactorization_iff | null |
UniformContinuous.rangeFactorization {f : α → β} (hf : UniformContinuous f) :
UniformContinuous (rangeFactorization f) :=
uniformContinuous_rangeFactorization_iff.mpr hf
@[simp] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | UniformContinuous.rangeFactorization | null |
isUniformInducing_rangeFactorization_iff {f : α → β} :
IsUniformInducing (rangeFactorization f) ↔ IsUniformInducing f :=
(isUniformInducing_val (range f)).isUniformInducing_comp_iff.symm | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformInducing_rangeFactorization_iff | null |
IsUniformInducing.rangeFactorization {f : α → β} (hf : IsUniformInducing f) :
IsUniformInducing (rangeFactorization f) :=
isUniformInducing_rangeFactorization_iff.2 hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.rangeFactorization | null |
extend_exists [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) :
∃ b, Tendsto f (comap (↑) (𝓝 a)) (𝓝 b) :=
uniformly_extend_exists (isUniformInducing_val s) hs.denseRange_val hf a | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | extend_exists | null |
extend_spec [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) :
Tendsto f (comap (↑) (𝓝 a)) (𝓝 (hs.extend f a)) :=
uniformly_extend_spec (isUniformInducing_val s) hs.denseRange_val hf a | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | extend_spec | null |
uniformContinuous_extend [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) :
UniformContinuous (hs.extend f) :=
uniformContinuous_uniformly_extend (isUniformInducing_val s) hs.denseRange_val hf
variable [T0Space β] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformContinuous_extend | null |
extend_of_ind (hs : Dense s) (hf : UniformContinuous f) (x : s) :
hs.extend f x = f x :=
IsDenseInducing.extend_eq_at _ hf.continuous.continuousAt | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | extend_of_ind | null |
IsDenseInducing.isUniformInducing_extend {γ : Type*} [UniformSpace γ]
[CompleteSpace β] [CompleteSpace γ] {i : α → β} {f : α → γ}
(hid : IsDenseInducing i) (hi : IsUniformInducing i) (h : IsUniformInducing f) :
IsUniformInducing (hid.extend f) := by
let sf := SeparationQuotient.mk ∘ f
have : CompleteSpace (closure (range sf)) :=
isClosed_closure.isComplete.completeSpace_coe
let ff : α → closure (range sf) := inclusion subset_closure ∘ rangeFactorization sf
have hgu : IsUniformInducing ff :=
(isUniformEmbedding_set_inclusion subset_closure).isUniformInducing.comp
(SeparationQuotient.isUniformInducing_mk.comp h).rangeFactorization
have hgd : DenseRange ff :=
((denseRange_inclusion_iff subset_closure).2 subset_rfl).comp
rangeFactorization_surjective.denseRange (continuous_inclusion subset_closure)
have hg : IsDenseInducing ff := hgu.isDenseInducing hgd
let fwd := hid.extend ff
have hfwd : UniformContinuous fwd :=
uniformContinuous_uniformly_extend hi hid.dense hgu.uniformContinuous
have hg' : UniformContinuous (hg.extend i) :=
uniformContinuous_uniformly_extend hgu hgd hi.uniformContinuous
have key : SeparationQuotient.mk ∘ hg.extend i ∘ fwd = SeparationQuotient.mk := by
ext x
induction x using isClosed_property hid.dense
· exact isClosed_eq (SeparationQuotient.continuous_mk.comp (hg'.comp hfwd).continuous)
SeparationQuotient.continuous_mk
· simpa [fwd, hid.extend_eq hgu.uniformContinuous.continuous]
using hg.inseparable_extend hi.uniformContinuous.continuous.continuousAt
have hfu : IsUniformInducing fwd := by
refine IsUniformInducing.of_comp hfwd (SeparationQuotient.uniformContinuous_mk.comp hg') ?_
rw [Function.comp_assoc, key]
exact SeparationQuotient.isUniformInducing_mk
have hrr : range (SeparationQuotient.mk ∘ hid.extend f) ⊆
closure (range (SeparationQuotient.mk ∘ f)) := by
refine ((SeparationQuotient.continuous_mk.comp (uniformContinuous_uniformly_extend hi hid.dense
h.uniformContinuous).continuous).range_subset_closure_image_dense hid.dense).trans
(closure_mono (subset_of_eq ?_))
rw [← range_comp]
apply congrArg range
funext x
simpa using (hid.inseparable_extend h.uniformContinuous.continuous.continuousAt)
suffices Subtype.val ∘ fwd = SeparationQuotient.mk ∘ hid.extend f by
rw [← SeparationQuotient.isUniformInducing_mk.isUniformInducing_comp_iff, ← this]
exact (isUniformInducing_val _).comp hfu
rw [← coe_comp_rangeFactorization (SeparationQuotient.mk ∘ hid.extend f),
← val_comp_inclusion hrr, Function.comp_assoc, Subtype.val_injective.comp_left.eq_iff]
refine hid.extend_unique ?_ ?_
· simp [ff, hid.inseparable_extend h.uniformContinuous.continuous.continuousAt, sf]
· exact (continuous_inclusion hrr).comp
(SeparationQuotient.continuous_mk.comp (uniformContinuous_uniformly_extend hi hid.dense
h.uniformContinuous).continuous).rangeFactorization | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsDenseInducing.isUniformInducing_extend | null |
protected Pretrivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] (e : Pretrivialization F (π F E)) : Prop where
linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e ⟨b, x⟩).2 | class | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | Pretrivialization.IsLinear | A mixin class for `Pretrivialization`, stating that a pretrivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. |
linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
[e.IsLinear R] {b : B} (hb : b ∈ e.baseSet) :
IsLinearMap R fun x : E b => (e ⟨b, x⟩).2 :=
Pretrivialization.IsLinear.linear b hb
variable [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linear | null |
@[simps!]
protected symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) : F →ₗ[R] E b := by
refine IsLinearMap.mk' (e.symm b) ?_
by_cases hb : b ∈ e.baseSet
· exact (((e.linear R hb).mk' _).inverse (e.symm b) (e.symm_apply_apply_mk hb) fun v ↦
congr_arg Prod.snd <| e.apply_mk_symm hb v).isLinear
· rw [e.coe_symm_of_notMem hb]
exact (0 : F →ₗ[R] E b).isLinear | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | symmₗ | A fiberwise linear inverse to `e`. |
@[simps -fullyApplied]
linearEquivAt (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) (hb : b ∈ e.baseSet) :
E b ≃ₗ[R] F where
toFun y := (e ⟨b, y⟩).2
invFun := e.symm b
left_inv := e.symm_apply_apply_mk hb
right_inv v := by simp_rw [e.apply_mk_symm hb v]
map_add' v w := (e.linear R hb).map_add v w
map_smul' c v := (e.linear R hb).map_smul c v
open Classical in | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearEquivAt | A pretrivialization for a vector bundle defines linear equivalences between the
fibers and the model space. |
protected linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) : E b →ₗ[R] F :=
if hb : b ∈ e.baseSet then e.linearEquivAt R b hb else 0
variable {R}
open Classical in | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt | A fiberwise linear map equal to `e` on `e.baseSet`. |
coe_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) :
⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 := by
rw [Pretrivialization.linearMapAt]
split_ifs <;> rfl | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_linearMapAt | null |
coe_linearMapAt_of_mem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e ⟨b, y⟩).2 := by
simp_rw [coe_linearMapAt, if_pos hb]
open Classical in | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_linearMapAt_of_mem | null |
linearMapAt_apply (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B} (y : E b) :
e.linearMapAt R b y = if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 := by
rw [coe_linearMapAt] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_apply | null |
linearMapAt_def_of_mem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb :=
dif_pos hb | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_def_of_mem | null |
linearMapAt_def_of_notMem (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb
@[deprecated (since := "2025-05-23")] alias linearMapAt_def_of_not_mem := linearMapAt_def_of_notMem | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_def_of_notMem | null |
linearMapAt_eq_zero (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_eq_zero | null |
symmₗ_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y := by
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).left_inv y | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | symmₗ_linearMapAt | null |
linearMapAt_symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.linearMapAt R b (e.symmₗ R b y) = y := by
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_symmₗ | null |
protected Trivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] (e : Trivialization F (π F E)) : Prop where
linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e ⟨b, x⟩).2 | class | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | Trivialization.IsLinear | A mixin class for `Trivialization`, stating that a trivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. |
protected linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet) :
IsLinearMap R fun y : E b => (e ⟨b, y⟩).2 :=
Trivialization.IsLinear.linear b hb | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linear | null |
toPretrivialization.isLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] [e.IsLinear R] : e.toPretrivialization.IsLinear R :=
{ (‹_› : e.IsLinear R) with }
variable [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)] | instance | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | toPretrivialization.isLinear | null |
linearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) (hb : b ∈ e.baseSet) :
E b ≃ₗ[R] F :=
e.toPretrivialization.linearEquivAt R b hb
variable {R}
@[simp] | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearEquivAt | A trivialization for a vector bundle defines linear equivalences between the
fibers and the model space. |
linearEquivAt_apply (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
(hb : b ∈ e.baseSet) (v : E b) : e.linearEquivAt R b hb v = (e ⟨b, v⟩).2 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearEquivAt_apply | null |
linearEquivAt_symm_apply (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
(hb : b ∈ e.baseSet) (v : F) : (e.linearEquivAt R b hb).symm v = e.symm b v :=
rfl
variable (R) in | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearEquivAt_symm_apply | null |
protected symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : F →ₗ[R] E b :=
e.toPretrivialization.symmₗ R b | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | symmₗ | A fiberwise linear inverse to `e`. |
coe_symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) :
⇑(e.symmₗ R b) = e.symm b :=
rfl
variable (R) in | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_symmₗ | null |
protected linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : E b →ₗ[R] F :=
e.toPretrivialization.linearMapAt R b
open Classical in | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt | A fiberwise linear map equal to `e` on `e.baseSet`. |
coe_linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) :
⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 :=
e.toPretrivialization.coe_linearMapAt b | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_linearMapAt | null |
coe_linearMapAt_of_mem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e ⟨b, y⟩).2 := by
simp_rw [coe_linearMapAt, if_pos hb]
open Classical in | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_linearMapAt_of_mem | null |
linearMapAt_apply (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (y : E b) :
e.linearMapAt R b y = if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 := by
rw [coe_linearMapAt] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_apply | null |
linearMapAt_def_of_mem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb :=
dif_pos hb | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_def_of_mem | null |
linearMapAt_def_of_notMem (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb
@[deprecated (since := "2025-05-23")] alias linearMapAt_def_of_not_mem := linearMapAt_def_of_notMem | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_def_of_notMem | null |
symmₗ_linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y :=
e.toPretrivialization.symmₗ_linearMapAt hb y | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | symmₗ_linearMapAt | null |
linearMapAt_symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : F) : e.linearMapAt R b (e.symmₗ R b y) = y :=
e.toPretrivialization.linearMapAt_symmₗ hb y
variable (R) in
open Classical in | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | linearMapAt_symmₗ | null |
coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] (b : B) :
F ≃L[R] F :=
{ toLinearEquiv := if hb : b ∈ e.baseSet ∩ e'.baseSet
then (e.linearEquivAt R b (hb.1 :)).symm.trans (e'.linearEquivAt R b hb.2)
else LinearEquiv.refl R F
continuous_toFun := by
by_cases hb : b ∈ e.baseSet ∩ e'.baseSet
· rw [dif_pos hb]
refine (e'.continuousOn.comp_continuous ?_ ?_).snd
· exact e.continuousOn_symm.comp_continuous (Continuous.prodMk_right b) fun y =>
mk_mem_prod hb.1 (mem_univ y)
· exact fun y => e'.mem_source.mpr hb.2
· rw [dif_neg hb]
exact continuous_id
continuous_invFun := by
by_cases hb : b ∈ e.baseSet ∩ e'.baseSet
· rw [dif_pos hb]
refine (e.continuousOn.comp_continuous ?_ ?_).snd
· exact e'.continuousOn_symm.comp_continuous (Continuous.prodMk_right b) fun y =>
mk_mem_prod hb.2 (mem_univ y)
exact fun y => e.mem_source.mpr hb.1
· rw [dif_neg hb]
exact continuous_id } | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coordChangeL | A coordinate change function between two trivializations, as a continuous linear equivalence.
Defined to be the identity when `b` does not lie in the base set of both trivializations. |
coe_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) :
⇑(coordChangeL R e e' b) = (e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) :=
congr_arg (fun f : F ≃ₗ[R] F ↦ ⇑f) (dif_pos hb) | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_coordChangeL | null |
coe_coordChangeL' (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) :
(coordChangeL R e e' b).toLinearEquiv =
(e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) :=
LinearEquiv.coe_injective (coe_coordChangeL _ _ hb) | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coe_coordChangeL' | null |
symm_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e'.baseSet ∩ e.baseSet) : (e.coordChangeL R e' b).symm = e'.coordChangeL R e b := by
apply ContinuousLinearEquiv.toLinearEquiv_injective
rw [coe_coordChangeL' e' e hb, (coordChangeL R e e' b).toLinearEquiv_symm,
coe_coordChangeL' e e' hb.symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | symm_coordChangeL | null |
coordChangeL_apply (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
coordChangeL R e e' b y = (e' ⟨b, e.symm b y⟩).2 :=
congr_fun (coe_coordChangeL e e' hb) y | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coordChangeL_apply | null |
mk_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
(b, coordChangeL R e e' b y) = e' ⟨b, e.symm b y⟩ := by
ext
· rw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1]
rw [e.proj_symm_apply' hb.1]
exact hb.2
· exact e.coordChangeL_apply e' hb y | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | mk_coordChangeL | null |
apply_symm_apply_eq_coordChangeL (e e' : Trivialization F (π F E)) [e.IsLinear R]
[e'.IsLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
e' (e.toOpenPartialHomeomorph.symm (b, v)) = (b, e.coordChangeL R e' b v) := by
rw [e.mk_coordChangeL e' hb, e.mk_symm hb.1] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | apply_symm_apply_eq_coordChangeL | null |
coordChangeL_apply' (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
coordChangeL R e e' b y = (e' (e.toOpenPartialHomeomorph.symm (b, y))).2 := by
rw [e.coordChangeL_apply e' hb, e.mk_symm hb.1] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coordChangeL_apply' | A version of `Trivialization.coordChangeL_apply` that fully unfolds `coordChange`. The
right-hand side is ugly, but has good definitional properties for specifically defined
trivializations. |
coordChangeL_symm_apply (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R]
{b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
⇑(coordChangeL R e e' b).symm =
(e'.linearEquivAt R b hb.2).symm.trans (e.linearEquivAt R b hb.1) :=
congr_arg LinearEquiv.invFun (dif_pos hb) | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | coordChangeL_symm_apply | null |
zeroSection [∀ x, Zero (E x)] : B → TotalSpace F E := (⟨·, 0⟩)
@[simp, mfld_simps] | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | zeroSection | The zero section of a vector bundle |
zeroSection_proj [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).proj = x :=
rfl
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | zeroSection_proj | null |
zeroSection_snd [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).2 = 0 :=
rfl | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | zeroSection_snd | null |
VectorBundle : Prop where
trivialization_linear' : ∀ (e : Trivialization F (π F E)) [MemTrivializationAtlas e], e.IsLinear R
continuousOn_coordChange' :
∀ (e e' : Trivialization F (π F E)) [MemTrivializationAtlas e] [MemTrivializationAtlas e'],
ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
(e.baseSet ∩ e'.baseSet)
variable {F E} | class | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | VectorBundle | The space `Bundle.TotalSpace F E` (for `E : B → Type*` such that each `E x` is a topological
vector space) has a topological vector space structure with fiber `F` (denoted with
`VectorBundle R F E`) if around every point there is a fiber bundle trivialization which is linear
in the fibers. |
continuousOn_coordChange [VectorBundle R F E] (e e' : Trivialization F (π F E))
[MemTrivializationAtlas e] [MemTrivializationAtlas e'] :
ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
(e.baseSet ∩ e'.baseSet) :=
VectorBundle.continuousOn_coordChange' e e' | theorem | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | continuousOn_coordChange | null |
@[simps -fullyApplied apply]
continuousLinearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : E b →L[R] F :=
{ e.linearMapAt R b with
toFun := e.linearMapAt R b -- given explicitly to help `simps`
cont := by
rw [e.coe_linearMapAt b]
classical
refine continuous_if_const _ (fun hb => ?_) fun _ => continuous_zero
exact (e.continuousOn.comp_continuous (FiberBundle.totalSpaceMk_isInducing F E b).continuous
fun x => e.mem_source.mpr hb).snd } | def | Topology | [
"Mathlib.Analysis.Normed.Operator.BoundedLinearMaps",
"Mathlib.Topology.FiberBundle.Basic"
] | Mathlib/Topology/VectorBundle/Basic.lean | continuousLinearMapAt | Forward map of `Trivialization.continuousLinearEquivAt` (only propositionally equal),
defined everywhere (`0` outside domain). |
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