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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Index | ι | def | VectorBundleCore.Index | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | The index set of a vector bundle core, as a convenience function for dot notation | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Base | B | def | VectorBundleCore.Base | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | The base space of a vector bundle core, as a convenience function for dot notation | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Fiber : B → Type _ | Z.toFiberBundleCore.Fiber | def | VectorBundleCore.Fiber | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | The fiber of a vector bundle core, as a convenience function for dot notation and
typeclass inference | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
topologicalSpaceFiber (x : B) : TopologicalSpace (Z.Fiber x) | letI : TopologicalSpace (Z.toFiberBundleCore.Fiber x) :=
Z.toFiberBundleCore.topologicalSpaceFiber x
inferInstanceAs <| TopologicalSpace (Z.toFiberBundleCore.Fiber x) | instance | VectorBundleCore.topologicalSpaceFiber | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
addCommGroupFiber (x : B) : AddCommGroup (Z.Fiber x) | inferInstanceAs <| AddCommGroup F | instance | VectorBundleCore.addCommGroupFiber | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"AddCommGroup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
moduleFiber (x : B) : Module R (Z.Fiber x) | inferInstanceAs <| Module R F | instance | VectorBundleCore.moduleFiber | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Module"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
proj : TotalSpace F Z.Fiber → B | TotalSpace.proj | def | VectorBundleCore.proj | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | The projection from the total space of a fiber bundle core, on its base. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TotalSpace | Bundle.TotalSpace F Z.Fiber | def | VectorBundleCore.TotalSpace | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Bundle.TotalSpace"
] | The total space of the vector bundle, as a convenience function for dot notation.
It is by definition equal to `Bundle.TotalSpace F Z.Fiber`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trivChange (i j : ι) : OpenPartialHomeomorph (B × F) (B × F) | Z.toFiberBundleCore.trivChange i j | def | VectorBundleCore.trivChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"OpenPartialHomeomorph"
] | Local homeomorphism version of the trivialization change. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_trivChange_source (i j : ι) (p : B × F) :
p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j | Z.toFiberBundleCore.mem_trivChange_source i j p | theorem | VectorBundleCore.mem_trivChange_source | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace : TopologicalSpace Z.TotalSpace | fast_instance% Z.toFiberBundleCore.toTopologicalSpace | instance | VectorBundleCore.toTopologicalSpace | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"TopologicalSpace"
] | Topological structure on the total space of a vector bundle created from core, designed so
that all the local trivialization are continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_coordChange (i j : ι) : Z.toFiberBundleCore.coordChange i j b = Z.coordChange i j b | rfl | theorem | VectorBundleCore.coe_coordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv (i : ι) : Trivialization F (π F Z.Fiber) | Z.toFiberBundleCore.localTriv i | def | VectorBundleCore.localTriv | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization"
] | One of the standard local trivializations of a vector bundle constructed from core, taken by
considering this in particular as a fiber bundle constructed from core. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
localTriv_apply {i : ι} (p : Z.TotalSpace) :
(Z.localTriv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ | rfl | theorem | VectorBundleCore.localTriv_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv.isLinear (i : ι) : (Z.localTriv i).IsLinear R | where
linear x _ :=
{ map_add := fun _ _ => by simp only [map_add, localTriv_apply, mfld_simps]
map_smul := fun _ _ => by simp only [map_smul, localTriv_apply, mfld_simps] } | instance | VectorBundleCore.localTriv.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"map_smul"
] | The standard local trivializations of a vector bundle constructed from core are linear. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_localTriv_source (p : Z.TotalSpace) : p ∈ (Z.localTriv i).source ↔ p.1 ∈ Z.baseSet i | Iff.rfl | theorem | VectorBundleCore.mem_localTriv_source | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_localTriv_target (p : B × F) :
p ∈ (Z.localTriv i).target ↔ p.1 ∈ (Z.localTriv i).baseSet | Z.toFiberBundleCore.mem_localTriv_target i p | theorem | VectorBundleCore.mem_localTriv_target | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv_symm_fst (p : B × F) :
(Z.localTriv i).toOpenPartialHomeomorph.symm p =
⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩ | rfl | theorem | VectorBundleCore.localTriv_symm_fst | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv_symm_apply {b : B} (hb : b ∈ (Z.localTriv i).baseSet) (v : F) :
(Z.localTriv i).symm b v = Z.coordChange i (Z.indexAt b) b v | by
apply (Z.localTriv i).symm_apply hb v | theorem | VectorBundleCore.localTriv_symm_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv_coordChange_eq {b : B}
(hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet) (v : F) :
(Z.localTriv i).coordChangeL R (Z.localTriv j) b v = Z.coordChange i j b v | by
rw [Trivialization.coordChangeL_apply', localTriv_symm_fst, localTriv_apply, coordChange_comp]
exacts [⟨⟨hb.1, Z.mem_baseSet_at b⟩, hb.2⟩, hb] | theorem | VectorBundleCore.localTriv_coordChange_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization.coordChangeL_apply'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTrivAt_def : Z.localTriv (Z.indexAt b) = Z.localTrivAt b | rfl | theorem | VectorBundleCore.localTrivAt_def | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_source_at : (⟨b, a⟩ : Z.TotalSpace) ∈ (Z.localTrivAt b).source | by
rw [localTrivAt, mem_localTriv_source]
exact Z.mem_baseSet_at b | theorem | VectorBundleCore.mem_source_at | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTrivAt_apply (p : Z.TotalSpace) : Z.localTrivAt p.1 p = ⟨p.1, p.2⟩ | Z.toFiberBundleCore.localTrivAt_apply p | theorem | VectorBundleCore.localTrivAt_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTrivAt_apply_mk (b : B) (a : F) : Z.localTrivAt b ⟨b, a⟩ = ⟨b, a⟩ | Z.localTrivAt_apply _ | theorem | VectorBundleCore.localTrivAt_apply_mk | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_localTrivAt_baseSet : b ∈ (Z.localTrivAt b).baseSet | Z.toFiberBundleCore.mem_localTrivAt_baseSet b | theorem | VectorBundleCore.mem_localTrivAt_baseSet | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
fiberBundle : FiberBundle F Z.Fiber | fast_instance% Z.toFiberBundleCore.fiberBundle | instance | VectorBundleCore.fiberBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt : trivializationAt F Z.Fiber b = Z.localTrivAt b | rfl | lemma | VectorBundleCore.trivializationAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
vectorBundle : VectorBundle R F Z.Fiber | where
trivialization_linear' := by
rintro _ ⟨i, rfl⟩
apply localTriv.isLinear
continuousOn_coordChange' := by
rintro _ _ ⟨i, rfl⟩ ⟨i', rfl⟩
refine (Z.continuousOn_coordChange i i').congr fun b hb => ?_
ext v
exact Z.localTriv_coordChange_eq i i' hb v | instance | VectorBundleCore.vectorBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"VectorBundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_proj : Continuous Z.proj | Z.toFiberBundleCore.continuous_proj | theorem | VectorBundleCore.continuous_proj | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Continuous"
] | The projection on the base of a vector bundle created from core is continuous | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpenMap_proj : IsOpenMap Z.proj | Z.toFiberBundleCore.isOpenMap_proj | theorem | VectorBundleCore.isOpenMap_proj | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"IsOpenMap"
] | The projection on the base of a vector bundle created from core is an open map | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
localTriv_continuousLinearMapAt {b : B} (hb : b ∈ (Z.localTriv i).baseSet) :
(Z.localTriv i).continuousLinearMapAt R b = Z.coordChange (Z.indexAt b) i b | by
ext1 v
simp_all
rfl | theorem | VectorBundleCore.localTriv_continuousLinearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousLinearMapAt {b₀ b : B}
(hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet) :
(trivializationAt F Z.Fiber b₀).continuousLinearMapAt R b =
Z.coordChange (Z.indexAt b) (Z.indexAt b₀) b | Z.localTriv_continuousLinearMapAt hb | theorem | VectorBundleCore.trivializationAt_continuousLinearMapAt | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
localTriv_symmL {b : B} (hb : b ∈ (Z.localTriv i).baseSet) :
(Z.localTriv i).symmL R b = Z.coordChange i (Z.indexAt b) b | by
ext1 v
rw [(Z.localTriv i).symmL_apply R, (Z.localTriv i).symm_apply]
exacts [rfl, hb] | theorem | VectorBundleCore.localTriv_symmL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_symmL {b₀ b : B} (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet) :
(trivializationAt F Z.Fiber b₀).symmL R b = Z.coordChange (Z.indexAt b₀) (Z.indexAt b) b | Z.localTriv_symmL hb | theorem | VectorBundleCore.trivializationAt_symmL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_coordChange_eq {b₀ b₁ b : B}
(hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet ∩ (trivializationAt F Z.Fiber b₁).baseSet)
(v : F) :
(trivializationAt F Z.Fiber b₀).coordChangeL R (trivializationAt F Z.Fiber b₁) b v =
Z.coordChange (Z.indexAt b₀) (Z.indexAt b₁) b v | Z.localTriv_coordChange_eq _ _ hb v | theorem | VectorBundleCore.trivializationAt_coordChange_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
VectorPrebundle where
pretrivializationAtlas : Set (Pretrivialization F (π F E))
pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
pretrivializationAt : B → Pretrivialization F (π F E)
mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
pretrivialization_mem_at... | structure | VectorPrebundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"Set"
] | This structure permits to define a vector bundle when trivializations are given as local
equivalences but there is not yet a topology on the total space or the fibers.
The total space is hence given a topology in such a way that there is a fiber bundle structure for
which the partial equivalences are also open partial ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) (b : B) : F →L[R] F | Classical.choose (a.exists_coordChange e he e' he') b | def | VectorPrebundle.coordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"VectorPrebundle"
] | A randomly chosen coordinate change on a `VectorPrebundle`, given by
the field `exists_coordChange`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) :
ContinuousOn (a.coordChange he he') (e.baseSet ∩ e'.baseSet) | (Classical.choose_spec (a.exists_coordChange e he e' he')).1 | theorem | VectorPrebundle.continuousOn_coordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"VectorPrebundle",
"continuousOn_coordChange"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coordChange_apply (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
a.coordChange he he' b v = (e' ⟨b, e.symm b v⟩).2 | (Classical.choose_spec (a.exists_coordChange e he e' he')).2 b hb v | theorem | VectorPrebundle.coordChange_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"VectorPrebundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
(b, a.coordChange he he' b v) = e' ⟨b, e.symm b v⟩ | by
ext
· rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]
rw [e.proj_symm_apply' hb.1]
exact hb.2
· exact a.coordChange_apply he he' hb v | theorem | VectorPrebundle.mk_coordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"VectorPrebundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFiberPrebundle (a : VectorPrebundle R F E) : FiberPrebundle F E | { a with
continuous_trivChange := fun e he e' he' ↦ by
have : ContinuousOn (fun x : B × F ↦ a.coordChange he' he x.1 x.2)
((e'.baseSet ∩ e.baseSet) ×ˢ univ) :=
isBoundedBilinearMap_apply.continuous.comp_continuousOn
((a.continuousOn_coordChange he' he).prodMap continuousOn_id)
... | def | VectorPrebundle.toFiberPrebundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousOn",
"FiberPrebundle",
"VectorPrebundle",
"continuousOn_id"
] | Natural identification of `VectorPrebundle` as a `FiberPrebundle`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totalSpaceTopology (a : VectorPrebundle R F E) : TopologicalSpace (TotalSpace F E) | a.toFiberPrebundle.totalSpaceTopology | def | VectorPrebundle.totalSpaceTopology | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"TopologicalSpace",
"VectorPrebundle"
] | Topology on the total space that will make the prebundle into a bundle. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trivializationOfMemPretrivializationAtlas (a : VectorPrebundle R F E)
{e : Pretrivialization F (π F E)} (he : e ∈ a.pretrivializationAtlas) :
@Trivialization B F _ _ _ a.totalSpaceTopology (π F E) | a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he | def | VectorPrebundle.trivializationOfMemPretrivializationAtlas | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization",
"VectorPrebundle"
] | Promotion from a `Pretrivialization` in the `pretrivializationAtlas` of a
`VectorPrebundle` to a `Trivialization`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
linear_trivializationOfMemPretrivializationAtlas (a : VectorPrebundle R F E)
{e : Pretrivialization F (π F E)} (he : e ∈ a.pretrivializationAtlas) :
letI | a.totalSpaceTopology
Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he) :=
letI := a.totalSpaceTopology
{ linear := (a.pretrivialization_linear' e he).linear } | theorem | VectorPrebundle.linear_trivializationOfMemPretrivializationAtlas | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Trivialization.IsLinear",
"VectorPrebundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_trivialization_at_source (b : B) (x : E b) :
⟨b, x⟩ ∈ (a.pretrivializationAt b).source | a.toFiberPrebundle.mem_pretrivializationAt_source b x | theorem | VectorPrebundle.mem_trivialization_at_source | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totalSpaceMk_preimage_source (b : B) :
.mk b ⁻¹' (a.pretrivializationAt b).source = univ | a.toFiberPrebundle.totalSpaceMk_preimage_source b | theorem | VectorPrebundle.totalSpaceMk_preimage_source | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_totalSpaceMk (b : B) :
Continuous[_, a.totalSpaceTopology] (.mk b) | a.toFiberPrebundle.continuous_totalSpaceMk b | theorem | VectorPrebundle.continuous_totalSpaceMk | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFiberBundle : @FiberBundle B F _ _ _ a.totalSpaceTopology _ | a.toFiberPrebundle.toFiberBundle | def | VectorPrebundle.toFiberBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"FiberBundle"
] | Make a `FiberBundle` from a `VectorPrebundle`; auxiliary construction for
`VectorPrebundle.toVectorBundle`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toVectorBundle : @VectorBundle R _ F E _ _ _ _ _ _ a.totalSpaceTopology _ a.toFiberBundle | letI := a.totalSpaceTopology; letI := a.toFiberBundle
{ trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
apply linear_trivializationOfMemPretrivializationAtlas
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refine (a.continuousOn_coordChange he he').congr fun b hb... | theorem | VectorPrebundle.toVectorBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousLinearEquiv.coe_coe",
"Trivialization.coordChangeL_apply",
"VectorBundle"
] | Make a `VectorBundle` from a `VectorPrebundle`. Concretely this means
that, given a `VectorPrebundle` structure for a sigma-type `E` -- which consists of a
number of "pretrivializations" identifying parts of `E` with product spaces `U × F` -- one
establishes that for the topology constructed on the sigma-type using
`V... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
inCoordinates (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y) : F →SL[σ] F' | ((trivializationAt F' E' y₀).continuousLinearMapAt 𝕜₂ y).comp <|
ϕ.comp <| (trivializationAt F E x₀).symmL 𝕜₁ x | def | ContinuousLinearMap.inCoordinates | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [] | When `ϕ` is a continuous (semi)linear map between the fibers `E x` and `E' y` of two vector
bundles `E` and `E'`, `ContinuousLinearMap.inCoordinates F E F' E' x₀ x y₀ y ϕ` is a coordinate
change of this continuous linear map w.r.t. the chart around `x₀` and the chart around `y₀`.
It is defined by composing `ϕ` with ap... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
inCoordinates_eq {x₀ x : B} {y₀ y : B'} {ϕ : E x →SL[σ] E' y}
(hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet) :
inCoordinates F E F' E' x₀ x y₀ y ϕ =
((trivializationAt F' E' y₀).continuousLinearEquivAt 𝕜₂ y hy : E' y →L[𝕜₂] F').comp
(ϕ.comp <|
... | by
ext
simp_rw [inCoordinates, ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe,
Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq] | theorem | ContinuousLinearMap.inCoordinates_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"ContinuousLinearEquiv.coe_coe",
"ContinuousLinearMap.coe_comp'",
"symm"
] | Rewrite `ContinuousLinearMap.inCoordinates` using continuous linear equivalences. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.VectorBundleCore.inCoordinates_eq {ι ι'} (Z : VectorBundleCore 𝕜₁ B F ι)
(Z' : VectorBundleCore 𝕜₂ B' F' ι') {x₀ x : B} {y₀ y : B'} (ϕ : F →SL[σ] F')
(hx : x ∈ Z.baseSet (Z.indexAt x₀)) (hy : y ∈ Z'.baseSet (Z'.indexAt y₀)) :
inCoordinates F Z.Fiber F' Z'.Fiber x₀ x y₀ y ϕ =
(Z'.coordChange (... | by
simp_rw [inCoordinates, Z'.trivializationAt_continuousLinearMapAt hy,
Z.trivializationAt_symmL hx] | theorem | VectorBundleCore.inCoordinates_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Basic.lean | [] | [
"VectorBundleCore"
] | Rewrite `ContinuousLinearMap.inCoordinates` in a `VectorBundleCore`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trivialization.isLinear : (trivialization B F).IsLinear 𝕜 | where
linear _ _ := ⟨fun _ _ => rfl, fun _ _ => rfl⟩ | instance | Bundle.Trivial.trivialization.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F | by
ext v
rw [Trivialization.coordChangeL_apply']
exacts [rfl, ⟨mem_univ _, mem_univ _⟩] | theorem | Bundle.Trivial.trivialization.coordChangeL | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"ContinuousLinearEquiv.refl",
"Trivialization.coordChangeL_apply'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) | where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
infer_instance
continuousOn_coordChange' e e' he he' := by
obtain rfl := eq_trivialization B F e
obtain rfl := eq_trivialization B F e'
simp only [trivialization.coordChangeL]
exact continuous_const.continuousOn | instance | Bundle.Trivial.vectorBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"Bundle.Trivial",
"VectorBundle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
linearMapAt_trivialization (x : B) :
(trivialization B F).linearMapAt 𝕜 x = LinearMap.id | by
ext v
rw [Trivialization.coe_linearMapAt_of_mem _ (by simp)]
rfl | lemma | Bundle.Trivial.linearMapAt_trivialization | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"LinearMap.id",
"Trivialization.coe_linearMapAt_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMapAt_trivialization (x : B) :
(trivialization B F).continuousLinearMapAt 𝕜 x = ContinuousLinearMap.id 𝕜 F | by
ext; simp | lemma | Bundle.Trivial.continuousLinearMapAt_trivialization | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"ContinuousLinearMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symmₗ_trivialization (x : B) :
(trivialization B F).symmₗ 𝕜 x = LinearMap.id | by
ext; simp [Trivialization.coe_symmₗ, trivialization_symm_apply B F] | lemma | Bundle.Trivial.symmₗ_trivialization | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"LinearMap.id",
"Trivialization.coe_symmₗ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symmL_trivialization (x : B) :
(trivialization B F).symmL 𝕜 x = ContinuousLinearMap.id 𝕜 F | by
ext; simp [trivialization_symm_apply B F] | lemma | Bundle.Trivial.symmL_trivialization | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"ContinuousLinearMap.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearEquivAt_trivialization (x : B) :
(trivialization B F).continuousLinearEquivAt 𝕜 x (mem_univ _) =
ContinuousLinearEquiv.refl 𝕜 F | by
ext; simp | lemma | Bundle.Trivial.continuousLinearEquivAt_trivialization | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"ContinuousLinearEquiv.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod.isLinear [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] : (e₁.prod e₂).IsLinear 𝕜 | where
linear := fun _ ⟨h₁, h₂⟩ =>
(((e₁.linear 𝕜 h₁).mk' _).prodMap ((e₂.linear 𝕜 h₂).mk' _)).isLinear | instance | Bundle.Trivialization.prod.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordC... | by
rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']
rintro ⟨v₁, v₂⟩
change
(e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂)
rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply']
... | theorem | Bundle.Trivialization.coordChangeL_prod | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"ContinuousLinearMap.coe_prodMap'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_apply' [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B} (hx₁ : x ∈ e₁.baseSet)
(hx₂ : x ∈ e₂.baseSet) (v₁ : E₁ x) (v₂ : E₂ x) :
prod e₁ e₂ ⟨x, (v₁, v₂)⟩ =
⟨x, e₁.continuousLinearEquivAt 𝕜 x hx₁ v₁, e₂.continuousLinearEquivAt 𝕜 x hx₂ v₂⟩ | rfl | theorem | Bundle.Trivialization.prod_apply' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) | where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e₁, e₂, he₁, he₂, rfl⟩ ⟨e₁', e₂', he₁', he₂', rfl⟩
refine (((continuousOn_coordChange 𝕜 e₁ e₁').mono ?_).prod_mapL 𝕜
((continuousOn_coordChange 𝕜 e₂ e₂').mono ?_)).... | instance | VectorBundle.prod | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"VectorBundle",
"continuousOn_coordChange"
] | The product of two vector bundles is a vector bundle. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Bundle.Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e₂).continuousLinearEquivAt 𝕜 x hx =
(e₁.continuousLinearEquivAt 𝕜 x hx.1).prodCongr (e₂.conti... | by
ext v : 2
obtain ⟨v₁, v₂⟩ := v
rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod]
exact (congr_arg Prod.snd (prod_apply' 𝕜 hx.1 hx.2 v₁ v₂) :) | theorem | Bundle.Trivialization.continuousLinearEquivAt_prod | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bundle.Trivialization.pullback_linear (e : Trivialization F (π F E)) [e.IsLinear 𝕜]
(f : K) : (e.pullback (B' := B') f).IsLinear 𝕜 | where
linear _ h := e.linear 𝕜 h | instance | Bundle.Trivialization.pullback_linear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) | where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refine ((continuousOn_coordChange 𝕜 e e').comp
(map_continuous f).continuousOn fun b hb => hb).congr ?_
rintro b (hb : f b ∈ e.baseSet ∩ e'.baseSe... | instance | VectorBundle.pullback | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Constructions.lean | [] | [
"FiberBundle",
"TopologicalSpace",
"VectorBundle",
"continuousOn_coordChange"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inCoordinates (x₀ x : B₁) (y₀ y : B₂) (ϕ : E₁ x [⋀^ι]→L[𝕜] E₂ y) :
F₁ [⋀^ι]→L[𝕜] F₂ | trivializationAt F₂ E₂ y₀ |>.continuousLinearMapAt 𝕜 y |>.compContinuousAlternatingMap
ϕ |>.compContinuousLinearMap <| (trivializationAt F₁ E₁ x₀).symmL 𝕜 x | def | ContinuousAlternatingMap.inCoordinates | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | When `ϕ` is a continuous alternating map
between the fibers `E₁ x` and `E₂ y` of two vector bundles `E` and `E'`,
`ContinuousAlternatingMap.inCoordinates F E F' E' x₀ x y₀ y ϕ`
is a coordinate change of this continuous linear map
w.r.t. the chart around `x₀` and the chart around `y₀`.
It is defined by composing `ϕ` wi... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
inCoordinates_eq {x₀ x : B₁} {y₀ y : B₂} {ϕ : E₁ x [⋀^ι]→L[𝕜] E₂ y}
(hx : x ∈ (trivializationAt F₁ E₁ x₀).baseSet)
(hy : y ∈ (trivializationAt F₂ E₂ y₀).baseSet) :
inCoordinates F₁ F₂ x₀ x y₀ y ϕ =
(((trivializationAt F₂ E₂ y₀).continuousLinearEquivAt 𝕜 y hy : E₂ y →L[𝕜] F₂)
|>.compContinuo... | by
ext
simp [inCoordinates, *] | theorem | ContinuousAlternatingMap.inCoordinates_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"symm"
] | Rewrite `ContinuousAlternatingMap.inCoordinates` using continuous linear equivalences. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAlternatingMapCoordChange (e₁ e₁' : Trivialization F₁ (π F₁ E₁))
(e₂ e₂' : Trivialization F₂ (π F₂ E₂))
[e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] (b : B) :
(F₁ [⋀^ι]→L[𝕜] F₂) →L[𝕜] (F₁ [⋀^ι]→L[𝕜] F₂) | (e₁'.coordChangeL 𝕜 e₁ b).symm.continuousAlternatingMapCongr (e₂.coordChangeL 𝕜 e₂' b) (ι := ι) | def | Bundle.Pretrivialization.continuousAlternatingMapCoordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"Trivialization"
] | Assume `eᵢ` and `eᵢ'` are trivializations of the bundles `Eᵢ` over base `B` with fiber `Fᵢ`
(`i ∈ {1,2}`), then `Pretrivialization.continuousAlternatingMapCoordChange 𝕜 ι e₁ e₁' e₂ e₂'`
is the coordinate change function between the two induced (pre)trivializations
`Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_continuousAlternatingMapCoordChange
[Finite ι]
[VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂]
[MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
[MemTrivializationAtlas e₂'] :
ContinuousOn (continuousAlternatingMapCoordChange 𝕜 ι e₁ e₁' e₂ e₂')
... | by
cases nonempty_fintype ι
simp +unfoldPartialApp only [continuousAlternatingMapCoordChange,
ContinuousLinearEquiv.coe_continuousAlternatingMapCongr, ContinuousLinearEquiv.symm_symm]
refine .clm_comp ?_ ?_
· refine map_continuous (ContinuousLinearMap.compContinuousAlternatingMapCLM (ι := ι) 𝕜 F₁ F₂ F₂)
... | theorem | Bundle.Pretrivialization.continuousOn_continuousAlternatingMapCoordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"ContinuousLinearEquiv.coe_continuousAlternatingMapCongr",
"ContinuousLinearEquiv.symm_symm",
"ContinuousLinearMap.compContinuousAlternatingMapCLM",
"ContinuousOn",
"Finite",
"MemTrivializationAtlas",
"VectorBundle",
"continuousOn_coordChange",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMap :
Pretrivialization (F₁ [⋀^ι]→L[𝕜] F₂) (π (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) | where
toFun p := ⟨p.1, (e₂.continuousLinearMapAt 𝕜 p.1).compContinuousAlternatingMap <|
p.2.compContinuousLinearMap <| e₁.symmL 𝕜 p.1⟩
invFun p := ⟨p.1, (e₂.symmL 𝕜 p.1).compContinuousAlternatingMap <|
p.2.compContinuousLinearMap <| e₁.continuousLinearMapAt 𝕜 p.1⟩
source := Bundle.TotalSpace.proj ⁻¹' ... | def | Bundle.Pretrivialization.continuousAlternatingMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"Prod.mk_right_inj",
"Set.mem_univ",
"Set.univ",
"isOpen_univ"
] | Given trivializations `e₁`, `e₂` for vector bundles `E₁`, `E₂` over a base `B`,
`Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂` is the induced pretrivialization for the
continuous `σ`-semilinear maps from `E₁` to `E₂`. That is, the map which will later become a
trivialization, after the bundle of continuous sem... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAlternatingMap.isLinear
[∀ x, ContinuousAdd (E₂ x)] [∀ x, ContinuousSMul 𝕜 (E₂ x)] :
(Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂).IsLinear 𝕜 | where
linear x _ :=
{ map_add L L' := by ext; simp [continuousAlternatingMap, Pretrivialization.toFun']
map_smul c L := by ext; simp [continuousAlternatingMap, Pretrivialization.toFun'] } | instance | Bundle.Pretrivialization.continuousAlternatingMap.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"ContinuousAdd",
"ContinuousSMul",
"map_smul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMap_apply
(p : TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) :
continuousAlternatingMap 𝕜 ι e₁ e₂ p =
⟨p.1, (e₂.continuousLinearMapAt 𝕜 p.1).compContinuousAlternatingMap <|
p.2.compContinuousLinearMap <| e₁.symmL 𝕜 p.1⟩ | rfl | theorem | Bundle.Pretrivialization.continuousAlternatingMap_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMap_symm_apply (p : B × (F₁ [⋀^ι]→L[𝕜] F₂)) :
(continuousAlternatingMap 𝕜 ι e₁ e₂).toPartialEquiv.symm p =
⟨p.1, (e₂.symmL 𝕜 p.1).compContinuousAlternatingMap <|
p.2.compContinuousLinearMap <| e₁.continuousLinearMapAt 𝕜 p.1⟩ | rfl | theorem | Bundle.Pretrivialization.continuousAlternatingMap_symm_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMap_symm_apply' {b : B} (hb : b ∈ e₁.baseSet ∩ e₂.baseSet)
(L : F₁ [⋀^ι]→L[𝕜] F₂) :
(continuousAlternatingMap 𝕜 ι e₁ e₂).symm b L =
((e₂.symmL 𝕜 b).compContinuousAlternatingMap <|
L.compContinuousLinearMap <| e₁.continuousLinearMapAt 𝕜 b) | by
rw [Pretrivialization.symm_apply]
· rfl
· exact hb | theorem | Bundle.Pretrivialization.continuousAlternatingMap_symm_apply' | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMapCoordChange_apply (b : B)
(hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ [⋀^ι]→L[𝕜] F₂) :
continuousAlternatingMapCoordChange 𝕜 ι e₁ e₁' e₂ e₂' b L =
(continuousAlternatingMap 𝕜 ι e₁' e₂'
⟨b, (continuousAlternatingMap 𝕜 ι e₁ e₂).symm b L⟩).2 | by
ext v
simp only [mem_inter_iff] at hb
simp [continuousAlternatingMapCoordChange, continuousAlternatingMap_apply,
Function.comp_def, Trivialization.coordChangeL_apply,
continuousAlternatingMap_symm_apply' hb.left, hb] | theorem | Bundle.Pretrivialization.continuousAlternatingMapCoordChange_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"Trivialization.coordChangeL_apply",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
vectorPrebundle :
VectorPrebundle 𝕜 (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) | where
pretrivializationAtlas :=
{e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
(_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
e = Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂}
pretrivialization_linear' := by
rintro _ ⟨e₁, he₁, e₂, he₂, r... | def | Bundle.ContinuousAlternatingMap.vectorPrebundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"MemTrivializationAtlas",
"Trivialization",
"Trivialization.linearMapAt_def_of_mem",
"VectorPrebundle"
] | The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instTopologicalSpaceTotalSpace :
TopologicalSpace (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) | (vectorPrebundle 𝕜 ι F₁ E₁ F₂ E₂).totalSpaceTopology | instance | Bundle.ContinuousAlternatingMap.instTopologicalSpaceTotalSpace | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"TopologicalSpace"
] | Topology on the total space of the continuous `σ`-semilinear maps between two "normable" vector
bundles over the same base. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instFiberBundle :
FiberBundle (F₁ [⋀^ι]→L[𝕜] F₂) fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x | (vectorPrebundle 𝕜 ι F₁ E₁ F₂ E₂).toFiberBundle | instance | Bundle.ContinuousAlternatingMap.instFiberBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"FiberBundle"
] | The continuous `σ`-semilinear maps between two vector bundles form a fiber bundle. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instVectorBundle : VectorBundle 𝕜 (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) | (vectorPrebundle 𝕜 ι F₁ E₁ F₂ E₂).toVectorBundle | instance | Bundle.ContinuousAlternatingMap.instVectorBundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"VectorBundle"
] | The continuous `σ`-semilinear maps between two vector bundles form a vector bundle. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAlternatingMap :
Trivialization (F₁ [⋀^ι]→L[𝕜] F₂) (π (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) | VectorPrebundle.trivializationOfMemPretrivializationAtlas _ ⟨e₁, e₂, he₁, he₂, rfl⟩ | def | Bundle.Trivialization.continuousAlternatingMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"Trivialization",
"VectorPrebundle.trivializationOfMemPretrivializationAtlas"
] | Given trivializations `e₁`, `e₂` in the atlas for vector bundles `E₁`, `E₂` over a base `B`,
the induced trivialization for the continuous `σ`-semilinear maps from `E₁` to `E₂`,
whose base set is `e₁.baseSet ∩ e₂.baseSet`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
memTrivializationAtlas_continuousAlternatingMap :
MemTrivializationAtlas
(e₁.continuousAlternatingMap 𝕜 ι e₂ :
Trivialization (F₁ [⋀^ι]→L[𝕜] F₂) (π (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x))) | ⟨⟨_, ⟨e₁, e₂, by infer_instance, by infer_instance, rfl⟩, rfl⟩⟩ | instance | Bundle.Trivialization.memTrivializationAtlas_continuousAlternatingMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"MemTrivializationAtlas",
"Trivialization"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
baseSet_continuousAlternatingMap :
(e₁.continuousAlternatingMap 𝕜 ι e₂).baseSet = e₁.baseSet ∩ e₂.baseSet | rfl | theorem | Bundle.Trivialization.baseSet_continuousAlternatingMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAlternatingMap_apply
(p : TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) :
e₁.continuousAlternatingMap 𝕜 ι e₂ p =
⟨p.1, (e₂.continuousLinearMapAt 𝕜 p.1 : _ →L[𝕜] _) |>.compContinuousAlternatingMap p.2
|>.compContinuousLinearMap (e₁.symmL 𝕜 p.1 : F₁ →L[𝕜] E₁ p.1)⟩ | rfl | theorem | Bundle.Trivialization.continuousAlternatingMap_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousAlternatingMap (x₀ : B) :
trivializationAt (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) x₀ =
(trivializationAt F₁ E₁ x₀).continuousAlternatingMap 𝕜 ι (trivializationAt F₂ E₂ x₀) | rfl | theorem | FiberBundle.trivializationAt_continuousAlternatingMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousAlternatingMap_apply (x₀ : B)
(x : TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) :
trivializationAt (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) x₀ x =
⟨x.1, inCoordinates F₁ F₂ x₀ x.1 x₀ x.1 x.2⟩ | rfl | theorem | FiberBundle.trivializationAt_continuousAlternatingMap_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousAlternatingMap_source (x₀ : B) :
(trivializationAt (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) x₀).source =
π (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) ⁻¹'
((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) | rfl | theorem | FiberBundle.trivializationAt_continuousAlternatingMap_source | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousAlternatingMap_target (x₀ : B) :
(trivializationAt (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) x₀).target =
((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) ×ˢ Set.univ | rfl | theorem | FiberBundle.trivializationAt_continuousAlternatingMap_target | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"Set.univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trivializationAt_continuousAlternatingMap_baseSet (x₀ : B) :
(trivializationAt (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) x₀).baseSet =
((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) | rfl | theorem | FiberBundle.trivializationAt_continuousAlternatingMap_baseSet | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_continuousAlternatingMap_bundle
(f : X → TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) :
ContinuousWithinAt f s x₀ ↔
ContinuousWithinAt (fun x ↦ (f x).1) s x₀ ∧
ContinuousWithinAt
(fun x ↦ inCoordinates F₁ F₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2)... | FiberBundle.continuousWithinAt_totalSpace .. | theorem | continuousWithinAt_continuousAlternatingMap_bundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"ContinuousWithinAt",
"FiberBundle.continuousWithinAt_totalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_continuousAlternatingMap_bundle
(f : X → TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) (fun x ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)) :
ContinuousAt f x₀ ↔
ContinuousAt (fun x ↦ (f x).1) x₀ ∧
ContinuousAt
(fun x ↦ inCoordinates F₁ F₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2) x₀ | FiberBundle.continuousAt_totalSpace .. | theorem | continuousAt_continuousAlternatingMap_bundle | Topology.VectorBundle | Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.lean | [] | [
"ContinuousAt",
"FiberBundle.continuousAt_totalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finiteDimensional (b : B) [FiniteDimensional R F] : FiniteDimensional R (E b) | (continuousLinearEquivAt R F E b).symm.finiteDimensional | lemma | VectorBundle.finiteDimensional | Topology.VectorBundle | Mathlib/Topology/VectorBundle/FiniteDimensional.lean | [] | [
"FiniteDimensional"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finrank_eq (b : B) : Module.finrank R (E b) = Module.finrank R F | (continuousLinearEquivAt R F E b).finrank_eq | lemma | VectorBundle.finrank_eq | Topology.VectorBundle | Mathlib/Topology/VectorBundle/FiniteDimensional.lean | [] | [
"Module.finrank",
"finrank_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMapCoordChange [e₁.IsLinear 𝕜₁] [e₁'.IsLinear 𝕜₁] [e₂.IsLinear 𝕜₂]
[e₂'.IsLinear 𝕜₂] (b : B) : (F₁ →SL[σ] F₂) →L[𝕜₂] F₁ →SL[σ] F₂ | ((e₁'.coordChangeL 𝕜₁ e₁ b).symm.arrowCongrSL (e₂.coordChangeL 𝕜₂ e₂' b) :
(F₁ →SL[σ] F₂) ≃L[𝕜₂] F₁ →SL[σ] F₂) | def | Bundle.Pretrivialization.continuousLinearMapCoordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [] | Assume `eᵢ` and `eᵢ'` are trivializations of the bundles `Eᵢ` over base `B` with fiber `Fᵢ`
(`i ∈ {1,2}`), then `Pretrivialization.continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂'` is the
coordinate change function between the two induced (pre)trivializations
`Pretrivialization.continuousLinearMap σ e₁ e₂` and
`Pretrivia... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_continuousLinearMapCoordChange [RingHomIsometric σ]
[VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂]
[MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
[MemTrivializationAtlas e₂'] :
ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
(e₁.b... | by
have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous
have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous
have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁
have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂'
refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_contin... | theorem | Bundle.Pretrivialization.continuousOn_continuousLinearMapCoordChange | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [
"ContinuousLinearMap.flip",
"ContinuousOn",
"MemTrivializationAtlas",
"RingHom.id",
"RingHomIsometric",
"VectorBundle",
"continuousOn_coordChange"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMap :
Pretrivialization (F₁ →SL[σ] F₂) (π (F₁ →SL[σ] F₂) (fun x ↦ E₁ x →SL[σ] E₂ x)) | where
toFun p := ⟨p.1, .comp (e₂.continuousLinearMapAt 𝕜₂ p.1) (p.2.comp (e₁.symmL 𝕜₁ p.1))⟩
invFun p := ⟨p.1, .comp (e₂.symmL 𝕜₂ p.1) (p.2.comp (e₁.continuousLinearMapAt 𝕜₁ p.1))⟩
source := Bundle.TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)
target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ Set.univ
map_source' :=... | def | Bundle.Pretrivialization.continuousLinearMap | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [
"Prod.mk_right_inj",
"Set.mem_univ",
"Set.univ",
"isOpen_univ"
] | Given trivializations `e₁`, `e₂` for vector bundles `E₁`, `E₂` over a base `B`,
`Pretrivialization.continuousLinearMap σ e₁ e₂` is the induced pretrivialization for the
continuous `σ`-semilinear maps from `E₁` to `E₂`. That is, the map which will later become a
trivialization, after the bundle of continuous semilinear ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousLinearMap.isLinear [∀ x, ContinuousAdd (E₂ x)] [∀ x, ContinuousSMul 𝕜₂ (E₂ x)] :
(Pretrivialization.continuousLinearMap σ e₁ e₂).IsLinear 𝕜₂ | where
linear x _ :=
{ map_add L L' := by simp [continuousLinearMap, Pretrivialization.toFun']
map_smul c L := by simp [continuousLinearMap, Pretrivialization.toFun'] } | instance | Bundle.Pretrivialization.continuousLinearMap.isLinear | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [
"ContinuousAdd",
"ContinuousSMul",
"map_smul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMap_apply (p : TotalSpace (F₁ →SL[σ] F₂) fun x ↦ E₁ x →SL[σ] E₂ x) :
(continuousLinearMap σ e₁ e₂) p =
⟨p.1, .comp (e₂.continuousLinearMapAt 𝕜₂ p.1) (p.2.comp (e₁.symmL 𝕜₁ p.1))⟩ | rfl | theorem | Bundle.Pretrivialization.continuousLinearMap_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousLinearMap_symm_apply (p : B × (F₁ →SL[σ] F₂)) :
(continuousLinearMap σ e₁ e₂).toPartialEquiv.symm p =
⟨p.1, .comp (e₂.symmL 𝕜₂ p.1) (p.2.comp (e₁.continuousLinearMapAt 𝕜₁ p.1))⟩ | rfl | theorem | Bundle.Pretrivialization.continuousLinearMap_symm_apply | Topology.VectorBundle | Mathlib/Topology/VectorBundle/Hom.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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