statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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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).isUni... | lemma | IsDenseInducing.isUniformInducing_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"DenseRange",
"Function.comp_assoc",
"IsDenseInducing",
"IsUniformInducing",
"IsUniformInducing.of_comp",
"SeparationQuotient.continuous_mk",
"SeparationQuotient.isUniformInducing_mk",
"SeparationQuotient.mk",
"UniformContinuous",
"UniformSpace",
"closure",
"closure_mono",
... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsThickening (c u : Set X) | ∃ (x : X) (uc s : SetRel X X),
IsOpen uc ∧ uc ∈ 𝓤 X ∧ c = closure (ball x uc) ∧
ball x (s ○ uc ○ s) ⊆ u ∧ s ∈ 𝓤 X | def | IsThickening | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Uniformizable.lean | [
"Mathlib.Topology.UniformSpace.OfCompactT2"
] | [
"IsOpen",
"Set",
"SetRel",
"closure"
] | To construct a real-valued function separating a point `x` from a closed set in a uniform
space, we recursively construct pairs of a closed set `c` contained in an open set `u`
satisfying the following predicate: `u` is a thickening of `c` by some entourage `s`,
in the sense that `c` is the closure of the ball centered... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
urysohns_main {c u : Set X} (IsThickeningcu : IsThickening c u) :
∃ (v : Set X), IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧
IsThickening c v ∧ IsThickening (closure v) u | by
obtain ⟨x, uc, s, huc, ucu, rfl, hn, hs⟩ := IsThickeningcu
obtain ⟨(ds : SetRel X X), hdsu, hdso, -, hdsd⟩ := comp_open_symm_mem_uniformity_sets hs
have ho : IsOpen (ds ○ uc ○ ds) := (hdso.relComp huc).relComp hdso
have hsub := calc ds ○ (ds ○ uc ○ ds) ○ ds
_ = (ds ○ ds) ○ uc ○ (ds ○ ds) := by simp [comp... | theorem | urysohns_main | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Uniformizable.lean | [
"Mathlib.Topology.UniformSpace.OfCompactT2"
] | [
"CompletelyRegularSpace",
"IsOpen",
"IsThickening",
"Set",
"SetRel",
"Urysohns.CU",
"closure",
"closure_eq_inter_uniformity",
"comp_comp_symm_mem_uniformity_sets",
"comp_open_symm_mem_uniformity_sets",
"interior",
"interior_mem_uniformity",
"interior_subset",
"isClosed_closure",
"isOpen_... | Given a pair consisting of a closed set `c` contained in an open set `u`
satisfying the predicate `IsThickening`, it is always possible to refine it to two pairs
`c ⊆ v` and `closure v ⊆ u` satisfying `IsThickening`. We can then use the general `Urysohns.CU`
construction to obtain the desired real-valued function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsTransitiveRel.prod_subset_trans {s : SetRel X X} {t u v : Set X} [s.IsTrans]
(htu : t ×ˢ u ⊆ s) (huv : u ×ˢ v ⊆ s) (hu : u.Nonempty) :
t ×ˢ v ⊆ s | by
rintro ⟨a, b⟩ hab
simp only [mem_prod] at hab
obtain ⟨x, hx⟩ := hu
exact s.trans (@htu ⟨a, x⟩ ⟨hab.left, hx⟩) (@huv ⟨x, b⟩ ⟨hx, hab.right⟩) | lemma | IsTransitiveRel.prod_subset_trans | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"Set",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsTransitiveRel.mem_filter_prod_trans {s : SetRel X X} {f g h : Filter X} [g.NeBot]
[s.IsTrans] (hfg : s ∈ f ×ˢ g) (hgh : s ∈ g ×ˢ h) :
s ∈ f ×ˢ h | Eventually.trans_prod (p := (fun x y ↦ (x, y) ∈ s)) (q := (fun x y ↦ (x, y) ∈ s))
(r := (fun x y ↦ (x, y) ∈ s)) hfg hgh fun _ _ _ ↦ s.trans | lemma | IsTransitiveRel.mem_filter_prod_trans | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"Filter",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset_of_mem {V : SetRel X X} [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
ball y V ⊆ ball x V | ball_subset_of_comp_subset hy SetRel.comp_subset_self | lemma | ball_subset_of_mem | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"SetRel",
"SetRel.comp_subset_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_of_mem {V : SetRel X X} [V.IsSymm] [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
ball x V = ball y V | by
refine le_antisymm (ball_subset_of_mem ?_) (ball_subset_of_mem hy)
rwa [← mem_ball_symmetry] | lemma | ball_eq_of_mem | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"SetRel",
"ball_subset_of_mem",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity : Prop where
hasBasis : (𝓤 X).HasBasis
(fun s : SetRel X X => s ∈ 𝓤 X ∧ SetRel.IsSymm s ∧ SetRel.IsTrans s) id | class | IsUltraUniformity | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"SetRel.IsTrans"
] | A uniform space is ultrametric if the uniformity `𝓤 X` has a basis of equivalence relations. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.mk_of_hasBasis {ι : Type*} {p : ι → Prop} {s : ι → SetRel X X}
(h_basis : (𝓤 X).HasBasis p s) (h_symm : ∀ i, p i → SetRel.IsSymm (s i))
(h_trans : ∀ i, p i → SetRel.IsTrans (s i)) :
IsUltraUniformity X | where
hasBasis := h_basis.to_hasBasis'
(fun i hi ↦ ⟨s i, ⟨h_basis.mem_of_mem hi, h_symm i hi, h_trans i hi⟩, subset_rfl⟩)
(fun _ hs ↦ hs.1) | lemma | IsUltraUniformity.mk_of_hasBasis | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsUltraUniformity",
"SetRel",
"SetRel.IsSymm",
"SetRel.IsTrans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.mem_nhds_iff_symm_trans [IsUltraUniformity X] {x : X} {s : Set X} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ UniformSpace.ball x V ⊆ s | by
rw [UniformSpace.mem_nhds_iff]
constructor
· rintro ⟨V, V_in, V_sub⟩
rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
· rintro ⟨V, V_in, _, _, V_sub⟩
exact ⟨V, V_in... | lemma | IsUltraUniformity.mem_nhds_iff_symm_trans | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsUltraUniformity",
"Set",
"SetRel.IsSymm",
"SetRel.IsTrans",
"UniformSpace.ball",
"UniformSpace.ball_mono",
"UniformSpace.mem_nhds_iff",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_ball_of_mem_uniformity (x : X) {V : SetRel X X} [V.IsTrans] (h' : V ∈ 𝓤 X) :
IsOpen (ball x V) | by
rw [isOpen_iff_ball_subset]
intro y hy
exact ⟨V, h', ball_subset_of_mem hy⟩ | lemma | UniformSpace.isOpen_ball_of_mem_uniformity | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsOpen",
"SetRel",
"ball_subset_of_mem",
"isOpen_iff_ball_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_ball_of_isSymm_of_isTrans_of_mem_uniformity (x : X) {V : SetRel X X} [V.IsSymm]
[V.IsTrans] (h' : V ∈ 𝓤 X) :
IsClosed (ball x V) | by
rw [← isOpen_compl_iff, isOpen_iff_ball_subset]
exact fun y hy ↦ ⟨V, h', fun z hyz hxz ↦ hy <| V.trans hxz <| V.symm hyz⟩ | lemma | UniformSpace.isClosed_ball_of_isSymm_of_isTrans_of_mem_uniformity | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsClosed",
"SetRel",
"isOpen_compl_iff",
"isOpen_iff_ball_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_ball_of_isSymm_of_isTrans_of_mem_uniformity (x : X) {V : SetRel X X} [V.IsSymm]
[V.IsTrans] (h' : V ∈ 𝓤 X) :
IsClopen (ball x V) | ⟨isClosed_ball_of_isSymm_of_isTrans_of_mem_uniformity _ ‹_›, isOpen_ball_of_mem_uniformity _ ‹_›⟩ | lemma | UniformSpace.isClopen_ball_of_isSymm_of_isTrans_of_mem_uniformity | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsClopen",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_clopens (x : X) :
(𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id | by
refine (nhds_basis_uniformity' (IsUltraUniformity.hasBasis)).to_hasBasis' ?_ ?_
· intro V ⟨hV, h_symm, h_trans⟩
exact ⟨ball x V, ⟨mem_ball_self _ hV,
isClopen_ball_of_isSymm_of_isTrans_of_mem_uniformity _ hV⟩, le_rfl⟩
· rintro u ⟨hx, hu⟩
simp [hu.right.mem_nhds_iff, hx] | lemma | UniformSpace.nhds_basis_clopens | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsClopen",
"Set",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TopologicalSpace.isTopologicalBasis_clopens :
TopologicalSpace.IsTopologicalBasis {s : Set X | IsClopen s} | .of_hasBasis_nhds fun x ↦ by simpa [and_comm] using nhds_basis_clopens x | lemma | TopologicalSpace.isTopologicalBasis_clopens | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Basic.lean | [] | [
"IsClopen",
"Set",
"TopologicalSpace.IsTopologicalBasis"
] | A uniform space with a nonarchimedean uniformity is zero-dimensional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.isUltraUniformity [IsUltraUniformity Y] {f : X → Y}
(hf : IsUniformInducing f) : IsUltraUniformity X | hf.comap_uniformSpace ▸ .comap inferInstance f | lemma | IsUniformInducing.isUltraUniformity | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"IsUltraUniformity",
"IsUniformInducing",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchyFilter.isSymm_gen {s : SetRel X X} [s.IsSymm] : (gen s).IsSymm | where
symm _ := by simp [CauchyFilter.gen, s.mem_filter_prod_comm] | instance | CauchyFilter.isSymm_gen | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"CauchyFilter.gen",
"IsSymm",
"SetRel",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchyFilter.isTrans_gen {s : SetRel X X} [s.IsTrans] : (gen s).IsTrans | where
trans _ _ _ := IsTransitiveRel.mem_filter_prod_trans | instance | CauchyFilter.isTrans_gen | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"IsTrans",
"IsTransitiveRel.mem_filter_prod_trans",
"SetRel",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.cauchyFilter [IsUltraUniformity X] :
IsUltraUniformity (CauchyFilter X) | by
apply mk_of_hasBasis (CauchyFilter.basis_uniformity IsUltraUniformity.hasBasis)
· exact fun _ ⟨_, hU, _⟩ ↦ by simp; infer_instance
· exact fun _ ⟨_, _, hU⟩ ↦ by simp; infer_instance | instance | IsUltraUniformity.cauchyFilter | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"CauchyFilter",
"CauchyFilter.basis_uniformity",
"IsUltraUniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.cauchyFilter_iff :
IsUltraUniformity (CauchyFilter X) ↔ IsUltraUniformity X | ⟨fun _ ↦ CauchyFilter.isUniformInducing_pureCauchy.isUltraUniformity,
fun _ ↦ inferInstance⟩ | lemma | IsUltraUniformity.cauchyFilter_iff | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"CauchyFilter",
"IsUltraUniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.separationQuotient [IsUltraUniformity X] :
IsUltraUniformity (SeparationQuotient X) | by
have := IsUltraUniformity.hasBasis.map
(Prod.map SeparationQuotient.mk (SeparationQuotient.mk (X := X)))
rw [← SeparationQuotient.uniformity_eq] at this
apply mk_of_hasBasis this
· exact fun _ ⟨_, hU, _⟩ ↦ by simp; infer_instance
· rintro U ⟨hU', _, hU⟩
constructor
rintro x y z
simp only [i... | instance | IsUltraUniformity.separationQuotient | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"Filter.mem_ker",
"IsUltraUniformity",
"SeparationQuotient",
"SeparationQuotient.mk",
"SeparationQuotient.mk_eq_mk",
"SeparationQuotient.uniformity_eq",
"Set.mem_image",
"inseparable_iff_ker_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.separationQuotient_iff :
IsUltraUniformity (SeparationQuotient X) ↔ IsUltraUniformity X | ⟨fun _ ↦ SeparationQuotient.isUniformInducing_mk.isUltraUniformity,
fun _ ↦ inferInstance⟩ | lemma | IsUltraUniformity.separationQuotient_iff | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"IsUltraUniformity",
"SeparationQuotient"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.completion_iff :
IsUltraUniformity (UniformSpace.Completion X) ↔ IsUltraUniformity X | by
rw [iff_comm, ← cauchyFilter_iff, ← separationQuotient_iff]
exact Iff.rfl | lemma | IsUltraUniformity.completion_iff | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"IsUltraUniformity",
"UniformSpace.Completion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.completion [IsUltraUniformity X] :
IsUltraUniformity (UniformSpace.Completion X) | completion_iff.2 inferInstance | instance | IsUltraUniformity.completion | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Completion.lean | [] | [
"IsUltraUniformity",
"UniformSpace.Completion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SetRel.isTrans_entourageProd {s : SetRel X X} {t : SetRel Y Y} [s.IsTrans] [t.IsTrans] :
(entourageProd s t).IsTrans | where
trans _ _ _ h h' := ⟨s.trans h.left h'.left, t.trans h.right h'.right⟩ | instance | SetRel.isTrans_entourageProd | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"IsTrans",
"SetRel",
"entourageProd",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.comap {u : UniformSpace Y} (h : IsUltraUniformity Y) (f : X → Y) :
@IsUltraUniformity _ (u.comap f) | by
letI := u.comap f
refine .mk_of_hasBasis (h.hasBasis.comap (Prod.map f f)) ?_ ?_ <;>
· dsimp
rintro _ ⟨_, _, _⟩
infer_instance | lemma | IsUltraUniformity.comap | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"IsUltraUniformity",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.inf {u u' : UniformSpace X} (h : @IsUltraUniformity _ u)
(h' : @IsUltraUniformity _ u') :
@IsUltraUniformity _ (u ⊓ u') | by
letI := u ⊓ u'
refine .mk_of_hasBasis (h.hasBasis.inf h'.hasBasis) ?_ ?_ <;>
· dsimp
rintro _ ⟨⟨_, _, _⟩, _, _, _⟩
infer_instance | lemma | IsUltraUniformity.inf | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"IsUltraUniformity",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.prod [UniformSpace X] [UniformSpace Y]
[IsUltraUniformity X] [IsUltraUniformity Y] :
IsUltraUniformity (X × Y) | .inf (.comap ‹_› _) (.comap ‹_› _) | instance | IsUltraUniformity.prod | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"IsUltraUniformity",
"UniformSpace",
"comap"
] | The product of uniform spaces with nonarchimedean uniformities has a
nonarchimedean uniformity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUltraUniformity.iInf {ι : Type*} {U : (i : ι) → UniformSpace X}
(hU : ∀ i, @IsUltraUniformity X (U i)) :
@IsUltraUniformity _ (⨅ i, U i : UniformSpace X) | by
letI : UniformSpace X := ⨅ i, U i
refine .mk_of_hasBasis (iInf_uniformity ▸ Filter.HasBasis.iInf fun i ↦ (hU i).hasBasis) ?_ ?_ <;>
· simp only [forall_and, Subtype.forall, id_eq, Set.iInter_coe_set, and_imp]
rintro _ _ _ _ _
infer_instance | lemma | IsUltraUniformity.iInf | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"Filter.HasBasis.iInf",
"IsUltraUniformity",
"Set.iInter_coe_set",
"UniformSpace",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.pi {ι : Type*} {X : ι → Type*} [U : Π i, UniformSpace (X i)]
[h : ∀ i, IsUltraUniformity (X i)] :
IsUltraUniformity (Π i, X i) | by
suffices @IsUltraUniformity _ (⨅ i, UniformSpace.comap (Function.eval i) (U i)) by
simpa +instances [Pi.uniformSpace_eq _] using this
exact .iInf fun i ↦ .comap (h i) (Function.eval i) | instance | IsUltraUniformity.pi | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"Function.eval",
"IsUltraUniformity",
"Pi.uniformSpace_eq",
"UniformSpace",
"UniformSpace.comap",
"comap"
] | The indexed product of uniform spaces with nonarchimedean uniformities has a
nonarchimedean uniformity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUltraUniformity.bot [UniformSpace X] [DiscreteUniformity X] : IsUltraUniformity X | by
have := Filter.hasBasis_principal (SetRel.id (α := X))
rw [← DiscreteUniformity.eq_principal_setRelId] at this
apply mk_of_hasBasis this <;> { simp; infer_instance } | instance | IsUltraUniformity.bot | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"DiscreteUniformity",
"DiscreteUniformity.eq_principal_setRelId",
"Filter.hasBasis_principal",
"IsUltraUniformity",
"SetRel.id",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltraUniformity.top : @IsUltraUniformity X (⊤ : UniformSpace X) | by
letI : UniformSpace X := ⊤
have := Filter.hasBasis_top (α := (X × X))
rw [← top_uniformity] at this
apply mk_of_hasBasis this <;> { simp; infer_instance } | lemma | IsUltraUniformity.top | Topology.UniformSpace.Ultra | Mathlib/Topology/UniformSpace/Ultra/Constructions.lean | [] | [
"Filter.hasBasis_top",
"IsUltraUniformity",
"UniformSpace",
"top_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bundle.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 | Bundle.Pretrivialization.IsLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommMonoid",
"IsLinearMap",
"Module"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLinear.linear b hb | theorem | Bundle.Pretrivialization.linear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommMonoid",
"IsLinearMap",
"Module"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Bundle.Pretrivialization.symmₗ | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"IsLinearMap.mk'",
"by_cases"
] | A fiberwise linear inverse to `e`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | def | Bundle.Pretrivialization.linearEquivAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"map_smul"
] | A pretrivialization for a vector bundle defines linear equivalences between the
fibers and the model space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | def | Bundle.Pretrivialization.linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | A fiberwise linear map equal to `e` on `e.baseSet`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Bundle.Pretrivialization.coe_linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Bundle.Pretrivialization.coe_linearMapAt_of_mem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Bundle.Pretrivialization.linearMapAt_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Bundle.Pretrivialization.linearMapAt_def_of_mem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Bundle.Pretrivialization.linearMapAt_def_of_notMem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Bundle.Pretrivialization.linearMapAt_eq_zero | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 simp [hb] | theorem | Bundle.Pretrivialization.symmₗ_linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 simp [hb] | theorem | Bundle.Pretrivialization.linearMapAt_symmₗ | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.IsLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommMonoid",
"IsLinearMap",
"Module",
"Trivialization"
] | A mixin class for `Bundle.Trivialization`, stating that a trivialization is fiberwise linear
with respect to given module structures on its fibers and the model fiber. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.linear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommMonoid",
"IsLinearMap",
"Module"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 } | instance | Trivialization.toPretrivialization.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommMonoid",
"Module"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | def | Trivialization.linearEquivAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | A trivialization for a vector bundle defines linear equivalences between the
fibers and the model space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | theorem | Trivialization.linearEquivAt_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Trivialization.linearEquivAt_symm_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : F →ₗ[R] E b | e.toPretrivialization.symmₗ R b | def | Trivialization.symmₗ | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | A fiberwise linear inverse to `e`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_symmₗ (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) :
⇑(e.symmₗ R b) = e.symm b | rfl | theorem | Trivialization.coe_symmₗ | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : E b →ₗ[R] F | e.toPretrivialization.linearMapAt R b | def | Trivialization.linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | A fiberwise linear map equal to `e` on `e.baseSet`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Trivialization.coe_linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Trivialization.coe_linearMapAt_of_mem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.linearMapAt_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.linearMapAt_def_of_mem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Trivialization.linearMapAt_def_of_notMem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_linearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : E b) : e.symm b (e.linearMapAt R b y) = y | e.toPretrivialization.symmₗ_linearMapAt hb y | theorem | Trivialization.symm_linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.symmₗ_linearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
linearMapAt_symm (e : Trivialization F (π F E)) [e.IsLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : F) : e.linearMapAt R b (e.symm b y) = y | e.toPretrivialization.linearMapAt_symmₗ hb y | theorem | Trivialization.linearMapAt_symm | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Trivialization.linearMapAt_symmₗ | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 ?_ ?_).... | def | Trivialization.coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Continuous.prodMk_right",
"LinearEquiv.refl",
"Trivialization",
"by_cases",
"continuous_id"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Trivialization.coe_coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.coe_coordChangeL' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"LinearEquiv.coe_injective",
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.symm_coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousLinearEquiv.toLinearEquiv_injective",
"LinearEquiv.symm_symm",
"LinearEquiv.trans_symm",
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.coordChangeL_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.mk_coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.apply_symm_apply_eq_coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Trivialization.coordChangeL_apply' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | A version of `Bundle.Trivialization.coordChangeL_apply` that fully unfolds `coordChange`. The
right-hand side is ugly, but has good definitional properties for specifically defined
trivializations. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Trivialization.coordChangeL_symm_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
zeroSection [∀ x, Zero (E x)] : B → TotalSpace F E | (⟨·, 0⟩) | def | Bundle.zeroSection | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | The zero section of a vector bundle | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
zeroSection_proj [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).proj = x | rfl | theorem | Bundle.zeroSection_proj | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
zeroSection_snd [∀ x, Zero (E x)] (x : B) : (zeroSection F E x).2 = 0 | rfl | theorem | Bundle.zeroSection_snd | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 :... | class | VectorBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"MemTrivializationAtlas",
"Trivialization",
"Trivialization.coordChangeL"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | continuousOn_coordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"MemTrivializationAtlas",
"Trivialization",
"Trivialization.coordChangeL",
"VectorBundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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).con... | def | Bundle.Trivialization.continuousLinearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundle.totalSpaceMk_isInducing",
"Trivialization",
"continuous_if_const"
] | Forward map of `Bundle.Trivialization.continuousLinearEquivAt` (only propositionally equal),
defined everywhere (`0` outside domain). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousLinearMapAt_apply_of_mem (e : Trivialization F TotalSpace.proj)
[Trivialization.IsLinear R e] {b : B} (hb : b ∈ e.baseSet) (y : E b) :
(continuousLinearMapAt R e b) y = (e ⟨b, y⟩).2 | by
simp [coe_linearMapAt_of_mem e hb] | lemma | Bundle.Trivialization.continuousLinearMapAt_apply_of_mem | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"Trivialization.IsLinear"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symmL (e : Trivialization F (π F E)) [e.IsLinear R] (b : B) : F →L[R] E b | { e.symmₗ R b with
toFun := e.symm b -- given explicitly to help `simps`
cont := by
by_cases hb : b ∈ e.baseSet
· rw [(FiberBundle.totalSpaceMk_isInducing F E b).continuous_iff]
exact e.continuousOn_symm.comp_continuous (.prodMk_right _) fun x ↦
mk_mem_prod hb (mem_univ x)
· ... | def | Bundle.Trivialization.symmL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundle.totalSpaceMk_isInducing",
"Trivialization",
"by_cases",
"symm"
] | Backwards map of `Bundle.Trivialization.continuousLinearEquivAt`, defined everywhere. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symmL_continuousLinearMapAt (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmL R b (e.continuousLinearMapAt R b y) = y | e.symmₗ_linearMapAt hb y | theorem | Bundle.Trivialization.symmL_continuousLinearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMapAt_symmL (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.continuousLinearMapAt R b (e.symmL R b y) = y | e.linearMapAt_symmₗ hb y | theorem | Bundle.Trivialization.continuousLinearMapAt_symmL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearEquivAt (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 with
toFun := fun y => (e ⟨b, y⟩).2 -- given explicitly to help `simps`
invFun := e.symm b -- given explicitly to help `simps`
continuous_toFun := (e.continuousOn.comp_continuous
(FiberBundle.totalSpaceMk_isInducing F E b).continuous fun _ => e.mem_source.m... | def | Bundle.Trivialization.continuousLinearEquivAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundle.totalSpaceMk_isInducing",
"Trivialization"
] | In a vector bundle, a trivialization in the fiber (which is a priori only linear)
is in fact a continuous linear equiv between the fibers and the model fiber. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_continuousLinearEquivAt_eq (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) :
(e.continuousLinearEquivAt R b hb : E b → F) = e.continuousLinearMapAt R b | (e.coe_linearMapAt_of_mem hb).symm | theorem | Bundle.Trivialization.coe_continuousLinearEquivAt_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_continuousLinearEquivAt_eq' (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) :
(e.continuousLinearEquivAt R b hb : E b →L[R] F) = e.continuousLinearMapAt R b | DFunLike.coe_injective (e.coe_linearMapAt_of_mem hb).symm | theorem | Bundle.Trivialization.coe_continuousLinearEquivAt_eq' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"DFunLike.coe_injective",
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_continuousLinearEquivAt_eq (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : ((e.continuousLinearEquivAt R b hb).symm : F → E b) = e.symmL R b | rfl | theorem | Bundle.Trivialization.symm_continuousLinearEquivAt_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_continuousLinearEquivAt_eq' (e : Trivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) : ((e.continuousLinearEquivAt R b hb).symm : F →L[R] E b) = e.symmL R b | rfl | theorem | Bundle.Trivialization.symm_continuousLinearEquivAt_eq' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearEquivAt_apply' (e : Trivialization F (π F E)) [e.IsLinear R]
(x : TotalSpace F E) (hx : x ∈ e.source) :
e.continuousLinearEquivAt R x.proj (e.mem_source.1 hx) x.2 = (e x).2 | rfl | theorem | Bundle.Trivialization.continuousLinearEquivAt_apply' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
apply_eq_prod_continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
(hb : b ∈ e.baseSet) (z : E b) : e ⟨b, z⟩ = (b, e.continuousLinearEquivAt R b hb z) | by
ext
· refine e.coe_fst ?_
rw [e.source_eq]
exact hb
· simp only [continuousLinearEquivAt_apply] | theorem | Bundle.Trivialization.apply_eq_prod_continuousLinearEquivAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
zeroSection (e : Trivialization F (π F E)) [e.IsLinear R] {x : B}
(hx : x ∈ e.baseSet) : e (zeroSection F E x) = (x, 0) | by
simp_rw [zeroSection, e.apply_eq_prod_continuousLinearEquivAt R x hx 0, map_zero] | theorem | Bundle.Trivialization.zeroSection | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_zeroSection [VectorBundle R F E] :
Continuous (zeroSection F E) | by
refine continuous_iff_continuousAt.2 fun x => ?_
unfold zeroSection
rw [FiberBundle.continuousAt_section]
apply (continuousAt_const (y := 0)).congr_of_eventuallyEq
filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)] with y hy
using congr_arg Prod.snd... | theorem | Bundle.Trivialization.continuous_zeroSection | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Continuous",
"FiberBundle.continuousAt_section",
"VectorBundle",
"continuousAt_const"
] | The zero section of a vector bundle is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_zeroSection [VectorBundle R F E] (s : Set B) :
ContinuousOn (zeroSection F E) s | (continuous_zeroSection R).continuousOn | theorem | Bundle.Trivialization.continuousOn_zeroSection | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"Set",
"VectorBundle"
] | The zero section of a vector bundle is continuous on any set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_zeroSection [VectorBundle R F E] (x : B) :
ContinuousAt (zeroSection F E) x | (continuous_zeroSection R).continuousAt | theorem | Bundle.Trivialization.continuousAt_zeroSection | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousAt",
"VectorBundle"
] | The zero section of a vector bundle is continuous at each point. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm_apply_eq_mk_continuousLinearEquivAt_symm (e : Trivialization F (π F E)) [e.IsLinear R]
(b : B) (hb : b ∈ e.baseSet) (z : F) :
e.toOpenPartialHomeomorph.symm ⟨b, z⟩ = ⟨b, (e.continuousLinearEquivAt R b hb).symm z⟩ | by
simpa using (mk_symm _ hb _).symm | theorem | Bundle.Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_continuousLinearEquivAt_eq_coord_change (e e' : Trivialization F (π F E))
[e.IsLinear R] [e'.IsLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
(e.continuousLinearEquivAt R b hb.1).symm.trans (e'.continuousLinearEquivAt R b hb.2) =
coordChangeL R e e' b | by
ext v
rw [coordChangeL_apply e e' hb]
rfl | theorem | Bundle.Trivialization.comp_continuousLinearEquivAt_eq_coord_change | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
VectorBundle.continuousLinearEquivAt (b : B) : E b ≃L[R] F | (trivializationAt F E b).continuousLinearEquivAt R b (FiberBundle.mem_baseSet_trivializationAt' b) | def | VectorBundle.continuousLinearEquivAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | A continuous linear equivalence between the fiber at `b` and the model fiber,
induced by the preferred trivialisation at each `b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
VectorBundleCore (ι : Type*) where
baseSet : ι → Set B
isOpen_baseSet : ∀ i, IsOpen (baseSet i)
indexAt : B → ι
mem_baseSet_at : ∀ x, x ∈ baseSet (indexAt x)
coordChange : ι → ι → B → F →L[R] F
coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coordChange i i x v = v
continuousOn_coordChange : ∀ i j, Continuo... | structure | VectorBundleCore | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"IsOpen",
"Set",
"continuousOn_coordChange"
] | Analogous construction of `FiberBundleCore` for vector bundles. This
construction gives a way to construct vector bundles from a structure registering how
trivialization changes act on fibers. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivialVectorBundleCore (ι : Type*) [Inhabited ι] : VectorBundleCore R B F ι | where
baseSet _ := univ
isOpen_baseSet _ := isOpen_univ
indexAt := default
mem_baseSet_at x := mem_univ x
coordChange _ _ _ := ContinuousLinearMap.id R F
coordChange_self _ _ _ _ := rfl
coordChange_comp _ _ _ _ _ _ := rfl
continuousOn_coordChange _ _ := continuousOn_const | def | trivialVectorBundleCore | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousLinearMap.id",
"VectorBundleCore",
"continuousOn_const",
"continuousOn_coordChange",
"isOpen_univ"
] | The trivial vector bundle core, in which all the changes of coordinates are the
identity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toFiberBundleCore : FiberBundleCore ι B F | { Z with
coordChange := fun i j b => Z.coordChange i j b
continuousOn_coordChange := fun i j =>
isBoundedBilinearMap_apply.continuous.comp_continuousOn
((Z.continuousOn_coordChange i j).prodMap continuousOn_id) } | def | VectorBundleCore.toFiberBundleCore | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundleCore",
"continuousOn_coordChange",
"continuousOn_id"
] | Natural identification to a `FiberBundleCore`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coordChange_linear_comp (i j k : ι) :
∀ x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet k,
(Z.coordChange j k x).comp (Z.coordChange i j x) = Z.coordChange i k x | fun x hx => by
ext v
exact Z.coordChange_comp i j k x hx v | theorem | VectorBundleCore.coordChange_linear_comp | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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