fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
symm_trans_target_eq (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ := by
rw [← PartialEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm]
variable (e' : Pretrivialization F (π F E)) {b : B} {y : E b}
@[simp] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_trans_target_eq | null |
coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.baseSet :=
e'.mem_source
@[mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_mem_source | null |
coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b := by
simp [hb] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_coe_fst | null |
mk_mem_target {x : B} {y : F} : (x, y) ∈ e'.target ↔ x ∈ e'.baseSet :=
e'.mem_target | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_mem_target | null |
symm_coe_proj {x : B} {y : F} (e' : Pretrivialization F (π F E)) (h : x ∈ e'.baseSet) :
(e'.toPartialEquiv.symm (x, y)).1 = x :=
e'.proj_symm_apply' h | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_coe_proj | null |
protected noncomputable symm (e : Pretrivialization F (π F E)) (b : B) (y : F) : E b :=
if hb : b ∈ e.baseSet then
cast (congr_arg E (e.proj_symm_apply' hb)) (e.toPartialEquiv.symm (b, y)).2
else 0 | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm | A fiberwise inverse to `e`. This is the function `F → E b` that induces a local inverse
`B × F → TotalSpace F E` of `e` on `e.baseSet`. It is defined to be `0` outside `e.baseSet`. |
symm_apply (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.toPartialEquiv.symm (b, y)).2 :=
dif_pos hb | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply | null |
symm_apply_of_notMem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet)
(y : F) : e.symm b y = 0 :=
dif_neg hb
@[deprecated (since := "2025-05-23")] alias symm_apply_of_not_mem := symm_apply_of_notMem | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_of_notMem | null |
coe_symm_of_notMem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet) :
(e.symm b : F → E b) = 0 :=
funext fun _ => dif_neg hb
@[deprecated (since := "2025-05-23")] alias coe_symm_of_not_mem := coe_symm_of_notMem | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_symm_of_notMem | null |
mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
TotalSpace.mk b (e.symm b y) = e.toPartialEquiv.symm (b, y) := by
simp only [e.symm_apply hb, TotalSpace.mk_cast (e.proj_symm_apply' hb), TotalSpace.eta] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_symm | null |
symm_proj_apply (e : Pretrivialization F (π F E)) (z : TotalSpace F E)
(hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 := by
rw [e.symm_apply hz, cast_eq_iff_heq, e.mk_proj_snd' hz, e.symm_apply_apply (e.mem_source.mpr hz)] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_proj_apply | null |
symm_apply_apply_mk (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet)
(y : E b) : e.symm b (e ⟨b, y⟩).2 = y :=
e.symm_proj_apply ⟨b, y⟩ hb | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_apply_mk | null |
apply_mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e ⟨b, e.symm b y⟩ = (b, y) := by
rw [e.mk_symm hb, e.apply_symm_apply (e.mk_mem_target.mpr hb)] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | apply_mk_symm | null |
Trivialization (proj : Z → B) extends OpenPartialHomeomorph Z (B × F) where
baseSet : Set B
open_baseSet : IsOpen baseSet
source_eq : source = proj ⁻¹' baseSet
target_eq : target = baseSet ×ˢ univ
proj_toFun : ∀ p ∈ source, (toOpenPartialHomeomorph p).1 = proj p | structure | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization | A structure extending open partial homeomorphisms, defining a local trivialization of a
projection `proj : Z → B` with fiber `F`, as an open partial homeomorphism between `Z` and `B × F`
defined between two sets of the form `proj ⁻¹' baseSet` and `baseSet × F`, acting trivially on the
first coordinate. |
@[ext]
ext' (e e' : Trivialization F proj)
(h₁ : e.toOpenPartialHomeomorph = e'.toOpenPartialHomeomorph) (h₂ : e.baseSet = e'.baseSet) :
e = e' := by
cases e; cases e'; congr | lemma | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | ext' | null |
@[coe] toFun' : Z → (B × F) := e.toFun | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | toFun' | Coercion of a trivialization to a function. We don't use `e.toFun` in the `CoeFun` instance
because it is actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about
`toPartialEquiv`. While we may want to switch to this behavior later, doing it mid-port will break a
lot of proofs. |
toPretrivialization : Pretrivialization F proj :=
{ e with } | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | toPretrivialization | Natural identification as a `Pretrivialization`. |
Simps.apply (proj : Z → B) (e : Trivialization F proj) : Z → B × F := e | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | Simps.apply | See Note [custom simps projection] |
noncomputable Simps.symm_apply (proj : Z → B) (e : Trivialization F proj) : B × F → Z :=
e.toOpenPartialHomeomorph.symm
initialize_simps_projections Trivialization (toFun → apply, invFun → symm_apply) | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | Simps.symm_apply | See Note [custom simps projection] |
toPretrivialization_injective :
Function.Injective fun e : Trivialization F proj => e.toPretrivialization := fun e e' h => by
ext1
exacts [OpenPartialHomeomorph.toPartialEquiv_injective
(congr_arg Pretrivialization.toPartialEquiv h), congr_arg Pretrivialization.baseSet h]
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | toPretrivialization_injective | null |
coe_coe : ⇑e.toOpenPartialHomeomorph = e :=
rfl
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_coe | null |
coe_fst (ex : x ∈ e.source) : (e x).1 = proj x :=
e.proj_toFun x ex | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_fst | null |
protected eqOn : EqOn (Prod.fst ∘ e) proj e.source := fun _x hx => e.coe_fst hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | eqOn | null |
mem_source : x ∈ e.source ↔ proj x ∈ e.baseSet := by rw [e.source_eq, mem_preimage] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mem_source | null |
coe_fst' (ex : proj x ∈ e.baseSet) : (e x).1 = proj x :=
e.coe_fst (e.mem_source.2 ex) | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_fst' | null |
mk_proj_snd (ex : x ∈ e.source) : (proj x, (e x).2) = e x :=
Prod.ext (e.coe_fst ex).symm rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_proj_snd | null |
mk_proj_snd' (ex : proj x ∈ e.baseSet) : (proj x, (e x).2) = e x :=
Prod.ext (e.coe_fst' ex).symm rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_proj_snd' | null |
source_inter_preimage_target_inter (s : Set (B × F)) :
e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s :=
e.toOpenPartialHomeomorph.source_inter_preimage_target_inter s
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | source_inter_preimage_target_inter | null |
coe_mk (e : OpenPartialHomeomorph Z (B × F)) (i j k l m) (x : Z) :
(Trivialization.mk e i j k l m : Trivialization F proj) x = e x :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_mk | null |
mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.baseSet :=
e.toPretrivialization.mem_target | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mem_target | null |
map_target {x : B × F} (hx : x ∈ e.target) : e.toOpenPartialHomeomorph.symm x ∈ e.source :=
e.toOpenPartialHomeomorph.map_target hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | map_target | null |
proj_symm_apply {x : B × F} (hx : x ∈ e.target) :
proj (e.toOpenPartialHomeomorph.symm x) = x.1 :=
e.toPretrivialization.proj_symm_apply hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | proj_symm_apply | null |
proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
proj (e.toOpenPartialHomeomorph.symm (b, x)) = b :=
e.toPretrivialization.proj_symm_apply' hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | proj_symm_apply' | null |
proj_surjOn_baseSet [Nonempty F] : Set.SurjOn proj e.source e.baseSet :=
e.toPretrivialization.proj_surjOn_baseSet | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | proj_surjOn_baseSet | null |
apply_symm_apply {x : B × F} (hx : x ∈ e.target) :
e (e.toOpenPartialHomeomorph.symm x) = x :=
e.toOpenPartialHomeomorph.right_inv hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | apply_symm_apply | null |
apply_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
e (e.toOpenPartialHomeomorph.symm (b, x)) = (b, x) :=
e.toPretrivialization.apply_symm_apply' hx
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | apply_symm_apply' | null |
symm_apply_mk_proj (ex : x ∈ e.source) :
e.toOpenPartialHomeomorph.symm (proj x, (e x).2) = x :=
e.toPretrivialization.symm_apply_mk_proj ex | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_mk_proj | null |
symm_trans_source_eq (e e' : Trivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ :=
Pretrivialization.symm_trans_source_eq e.toPretrivialization e' | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_trans_source_eq | null |
symm_trans_target_eq (e e' : Trivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ :=
Pretrivialization.symm_trans_target_eq e.toPretrivialization e' | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_trans_target_eq | null |
coe_fst_eventuallyEq_proj (ex : x ∈ e.source) : Prod.fst ∘ e =ᶠ[𝓝 x] proj :=
mem_nhds_iff.2 ⟨e.source, fun _y hy => e.coe_fst hy, e.open_source, ex⟩ | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_fst_eventuallyEq_proj | null |
coe_fst_eventuallyEq_proj' (ex : proj x ∈ e.baseSet) : Prod.fst ∘ e =ᶠ[𝓝 x] proj :=
e.coe_fst_eventuallyEq_proj (e.mem_source.2 ex) | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_fst_eventuallyEq_proj' | null |
map_proj_nhds (ex : x ∈ e.source) : map proj (𝓝 x) = 𝓝 (proj x) := by
rw [← e.coe_fst ex, ← map_congr (e.coe_fst_eventuallyEq_proj ex), ← map_map, ← e.coe_coe,
e.map_nhds_eq ex, map_fst_nhds] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | map_proj_nhds | null |
preimage_subset_source {s : Set B} (hb : s ⊆ e.baseSet) : proj ⁻¹' s ⊆ e.source :=
fun _p hp => e.mem_source.mpr (hb hp) | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimage_subset_source | null |
image_preimage_eq_prod_univ {s : Set B} (hb : s ⊆ e.baseSet) :
e '' (proj ⁻¹' s) = s ×ˢ univ :=
Subset.antisymm
(image_subset_iff.mpr fun p hp =>
⟨(e.proj_toFun p (e.preimage_subset_source hb hp)).symm ▸ hp, trivial⟩)
fun p hp =>
let hp' : p ∈ e.target := e.mem_target.mpr (hb hp.1)
⟨e.invFun p, mem_preimage.mpr ((e.proj_symm_apply hp').symm ▸ hp.1), e.apply_symm_apply hp'⟩ | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | image_preimage_eq_prod_univ | null |
tendsto_nhds_iff {α : Type*} {l : Filter α} {f : α → Z} {z : Z} (hz : z ∈ e.source) :
Tendsto f l (𝓝 z) ↔
Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x ↦ (e (f x)).2) l (𝓝 (e z).2) := by
rw [e.nhds_eq_comap_inf_principal hz, tendsto_inf, tendsto_comap_iff, Prod.tendsto_iff, coe_coe,
tendsto_principal, coe_fst _ hz]
by_cases hl : ∀ᶠ x in l, f x ∈ e.source
· simp only [hl, and_true]
refine (tendsto_congr' ?_).and Iff.rfl
exact hl.mono fun x ↦ e.coe_fst
· simp only [hl, and_false, false_iff, not_and]
rw [e.source_eq] at hl hz
exact fun h _ ↦ hl <| h <| e.open_baseSet.mem_nhds hz | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | tendsto_nhds_iff | null |
nhds_eq_inf_comap {z : Z} (hz : z ∈ e.source) :
𝓝 z = comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ e) (𝓝 (e z).2) := by
refine eq_of_forall_le_iff fun l ↦ ?_
rw [le_inf_iff, ← tendsto_iff_comap, ← tendsto_iff_comap]
exact e.tendsto_nhds_iff hz | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | nhds_eq_inf_comap | null |
preimageHomeomorph {s : Set B} (hb : s ⊆ e.baseSet) : proj ⁻¹' s ≃ₜ s × F :=
(e.toOpenPartialHomeomorph.homeomorphOfImageSubsetSource (e.preimage_subset_source hb)
(e.image_preimage_eq_prod_univ hb)).trans
((Homeomorph.Set.prod s univ).trans ((Homeomorph.refl s).prodCongr (Homeomorph.Set.univ F)))
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageHomeomorph | The preimage of a subset of the base set is homeomorphic to the product with the fiber. |
preimageHomeomorph_apply {s : Set B} (hb : s ⊆ e.baseSet) (p : proj ⁻¹' s) :
e.preimageHomeomorph hb p = (⟨proj p, p.2⟩, (e p).2) :=
Prod.ext (Subtype.ext (e.proj_toFun p (e.mem_source.mpr (hb p.2)))) rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageHomeomorph_apply | null |
protected preimageHomeomorph_symm_apply.aux {s : Set B} (hb : s ⊆ e.baseSet) :=
(e.preimageHomeomorph hb).symm
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageHomeomorph_symm_apply.aux | Auxiliary definition to avoid looping in `dsimp`
with `Trivialization.preimageHomeomorph_symm_apply`. |
preimageHomeomorph_symm_apply {s : Set B} (hb : s ⊆ e.baseSet) (p : s × F) :
(e.preimageHomeomorph hb).symm p =
⟨e.symm (p.1, p.2), ((preimageHomeomorph_symm_apply.aux e hb) p).2⟩ :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageHomeomorph_symm_apply | null |
sourceHomeomorphBaseSetProd : e.source ≃ₜ e.baseSet × F :=
(Homeomorph.setCongr e.source_eq).trans (e.preimageHomeomorph subset_rfl)
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | sourceHomeomorphBaseSetProd | The source is homeomorphic to the product of the base set with the fiber. |
sourceHomeomorphBaseSetProd_apply (p : e.source) :
e.sourceHomeomorphBaseSetProd p = (⟨proj p, e.mem_source.mp p.2⟩, (e p).2) :=
e.preimageHomeomorph_apply subset_rfl ⟨p, e.mem_source.mp p.2⟩ | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | sourceHomeomorphBaseSetProd_apply | null |
protected sourceHomeomorphBaseSetProd_symm_apply.aux := e.sourceHomeomorphBaseSetProd.symm
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | sourceHomeomorphBaseSetProd_symm_apply.aux | Auxiliary definition to avoid looping in `dsimp`
with `Trivialization.sourceHomeomorphBaseSetProd_symm_apply`. |
sourceHomeomorphBaseSetProd_symm_apply (p : e.baseSet × F) :
e.sourceHomeomorphBaseSetProd.symm p =
⟨e.symm (p.1, p.2), (sourceHomeomorphBaseSetProd_symm_apply.aux e p).2⟩ :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | sourceHomeomorphBaseSetProd_symm_apply | null |
preimageSingletonHomeomorph {b : B} (hb : b ∈ e.baseSet) : proj ⁻¹' {b} ≃ₜ F :=
.trans (e.preimageHomeomorph (Set.singleton_subset_iff.mpr hb)) <|
.trans (.prodCongr (Homeomorph.homeomorphOfUnique ({b} : Set B) PUnit.{1}) (Homeomorph.refl F))
(Homeomorph.punitProd F)
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageSingletonHomeomorph | Each fiber of a trivialization is homeomorphic to the specified fiber. |
preimageSingletonHomeomorph_apply {b : B} (hb : b ∈ e.baseSet) (p : proj ⁻¹' {b}) :
e.preimageSingletonHomeomorph hb p = (e p).2 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageSingletonHomeomorph_apply | null |
preimageSingletonHomeomorph_symm_apply {b : B} (hb : b ∈ e.baseSet) (p : F) :
(e.preimageSingletonHomeomorph hb).symm p =
⟨e.symm (b, p), by rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]⟩ :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | preimageSingletonHomeomorph_symm_apply | null |
continuousAt_proj (ex : x ∈ e.source) : ContinuousAt proj x :=
(e.map_proj_nhds ex).le | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousAt_proj | In the domain of a bundle trivialization, the projection is continuous |
continuousOn_proj : ContinuousOn proj e.source :=
continuousOn_of_forall_continuousAt fun _ ↦ e.continuousAt_proj | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousOn_proj | null |
protected compHomeomorph {Z' : Type*} [TopologicalSpace Z'] (h : Z' ≃ₜ Z) :
Trivialization F (proj ∘ h) where
toOpenPartialHomeomorph := h.toOpenPartialHomeomorph.trans e.toOpenPartialHomeomorph
baseSet := e.baseSet
open_baseSet := e.open_baseSet
source_eq := by simp [source_eq, preimage_preimage, Function.comp_def]
target_eq := by simp [target_eq]
proj_toFun p hp := by
have hp : h p ∈ e.source := by simpa using hp
simp [hp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | compHomeomorph | Composition of a `Trivialization` and a `Homeomorph`. |
continuousAt_of_comp_right {X : Type*} [TopologicalSpace X] {f : Z → X} {z : Z}
(e : Trivialization F proj) (he : proj z ∈ e.baseSet)
(hf : ContinuousAt (f ∘ e.toPartialEquiv.symm) (e z)) : ContinuousAt f z := by
have hez : z ∈ e.toPartialEquiv.symm.target := by
rw [PartialEquiv.symm_target, e.mem_source]
exact he
rwa [e.toOpenPartialHomeomorph.symm.continuousAt_iff_continuousAt_comp_right hez,
OpenPartialHomeomorph.symm_symm] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousAt_of_comp_right | Read off the continuity of a function `f : Z → X` at `z : Z` by transferring via a
trivialization of `Z` containing `z`. |
continuousAt_of_comp_left {X : Type*} [TopologicalSpace X] {f : X → Z} {x : X}
(e : Trivialization F proj) (hf_proj : ContinuousAt (proj ∘ f) x) (he : proj (f x) ∈ e.baseSet)
(hf : ContinuousAt (e ∘ f) x) : ContinuousAt f x := by
rw [e.continuousAt_iff_continuousAt_comp_left]
· exact hf
rw [e.source_eq, ← preimage_comp]
exact hf_proj.preimage_mem_nhds (e.open_baseSet.mem_nhds he)
variable (e' : Trivialization F (π F E)) {b : B} {y : E b} | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousAt_of_comp_left | Read off the continuity of a function `f : X → Z` at `x : X` by transferring via a
trivialization of `Z` containing `f x`. |
protected continuousOn : ContinuousOn e' e'.source :=
e'.continuousOn_toFun | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousOn | null |
coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.baseSet :=
e'.mem_source
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_mem_source | null |
coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b :=
e'.coe_fst (e'.mem_source.2 hb) | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coe_coe_fst | null |
mk_mem_target {y : F} : (b, y) ∈ e'.target ↔ b ∈ e'.baseSet :=
e'.toPretrivialization.mem_target | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_mem_target | null |
symm_apply_apply {x : TotalSpace F E} (hx : x ∈ e'.source) :
e'.toOpenPartialHomeomorph.symm (e' x) = x :=
e'.toPartialEquiv.left_inv hx
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_apply | null |
symm_coe_proj {x : B} {y : F} (e : Trivialization F (π F E)) (h : x ∈ e.baseSet) :
(e.toOpenPartialHomeomorph.symm (x, y)).1 = x :=
e.proj_symm_apply' h | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_coe_proj | null |
protected noncomputable symm (e : Trivialization F (π F E)) (b : B) (y : F) : E b :=
e.toPretrivialization.symm b y | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm | A fiberwise inverse to `e'`. The function `F → E x` that induces a local inverse
`B × F → TotalSpace F E` of `e'` on `e'.baseSet`. It is defined to be `0` outside `e'.baseSet`. |
symm_apply (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e.symm b y =
cast (congr_arg E (e.symm_coe_proj hb)) (e.toOpenPartialHomeomorph.symm (b, y)).2 :=
dif_pos hb | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply | null |
symm_apply_of_notMem (e : Trivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet) (y : F) :
e.symm b y = 0 :=
dif_neg hb
@[deprecated (since := "2025-05-23")] alias symm_apply_of_not_mem := symm_apply_of_notMem | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_of_notMem | null |
mk_symm (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
TotalSpace.mk b (e.symm b y) = e.toOpenPartialHomeomorph.symm (b, y) :=
e.toPretrivialization.mk_symm hb y | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_symm | null |
symm_proj_apply (e : Trivialization F (π F E)) (z : TotalSpace F E)
(hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 :=
e.toPretrivialization.symm_proj_apply z hz | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_proj_apply | null |
symm_apply_apply_mk (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : E b) :
e.symm b (e ⟨b, y⟩).2 = y :=
e.symm_proj_apply ⟨b, y⟩ hb | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | symm_apply_apply_mk | null |
apply_mk_symm (e : Trivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e ⟨b, e.symm b y⟩ = (b, y) :=
e.toPretrivialization.apply_mk_symm hb y | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | apply_mk_symm | null |
continuousOn_symm (e : Trivialization F (π F E)) :
ContinuousOn (fun z : B × F => TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ univ) := by
have : ∀ z ∈ e.baseSet ×ˢ (univ : Set F),
TotalSpace.mk z.1 (e.symm z.1 z.2) = e.toOpenPartialHomeomorph.symm z := by
rintro x ⟨hx : x.1 ∈ e.baseSet, _⟩
rw [e.mk_symm hx]
refine ContinuousOn.congr ?_ this
rw [← e.target_eq]
exact e.toOpenPartialHomeomorph.continuousOn_symm | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuousOn_symm | null |
transFiberHomeomorph {F' : Type*} [TopologicalSpace F'] (e : Trivialization F proj)
(h : F ≃ₜ F') : Trivialization F' proj where
toOpenPartialHomeomorph :=
e.toOpenPartialHomeomorph.transHomeomorph <| (Homeomorph.refl _).prodCongr h
baseSet := e.baseSet
open_baseSet := e.open_baseSet
source_eq := e.source_eq
target_eq := by simp [target_eq, prod_univ, preimage_preimage]
proj_toFun := e.proj_toFun
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | transFiberHomeomorph | If `e` is a `Trivialization` of `proj : Z → B` with fiber `F` and `h` is a homeomorphism
`F ≃ₜ F'`, then `e.trans_fiber_homeomorph h` is the trivialization of `proj` with the fiber `F'`
that sends `p : Z` to `((e p).1, h (e p).2)`. |
transFiberHomeomorph_apply {F' : Type*} [TopologicalSpace F'] (e : Trivialization F proj)
(h : F ≃ₜ F') (x : Z) : e.transFiberHomeomorph h x = ((e x).1, h (e x).2) :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | transFiberHomeomorph_apply | null |
coordChange (e₁ e₂ : Trivialization F proj) (b : B) (x : F) : F :=
(e₂ <| e₁.toOpenPartialHomeomorph.symm (b, x)).2 | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChange | Coordinate transformation in the fiber induced by a pair of bundle trivializations. See also
`Trivialization.coordChangeHomeomorph` for a version bundled as `F ≃ₜ F`. |
mk_coordChange (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
(h₂ : b ∈ e₂.baseSet) (x : F) :
(b, e₁.coordChange e₂ b x) = e₂ (e₁.toOpenPartialHomeomorph.symm (b, x)) := by
refine Prod.ext ?_ rfl
rw [e₂.coe_fst', ← e₁.coe_fst', e₁.apply_symm_apply' h₁]
· rwa [e₁.proj_symm_apply' h₁]
· rwa [e₁.proj_symm_apply' h₁] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | mk_coordChange | null |
coordChange_apply_snd (e₁ e₂ : Trivialization F proj) {p : Z} (h : proj p ∈ e₁.baseSet) :
e₁.coordChange e₂ (proj p) (e₁ p).snd = (e₂ p).snd := by
rw [coordChange, e₁.symm_apply_mk_proj (e₁.mem_source.2 h)] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChange_apply_snd | null |
coordChange_same_apply (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet) (x : F) :
e.coordChange e b x = x := by rw [coordChange, e.apply_symm_apply' h] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChange_same_apply | null |
coordChange_same (e : Trivialization F proj) {b : B} (h : b ∈ e.baseSet) :
e.coordChange e b = id :=
funext <| e.coordChange_same_apply h | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChange_same | null |
coordChange_coordChange (e₁ e₂ e₃ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
(h₂ : b ∈ e₂.baseSet) (x : F) :
e₂.coordChange e₃ b (e₁.coordChange e₂ b x) = e₁.coordChange e₃ b x := by
rw [coordChange, e₁.mk_coordChange _ h₁ h₂, ← e₂.coe_coe, e₂.left_inv, coordChange]
rwa [e₂.mem_source, e₁.proj_symm_apply' h₁] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChange_coordChange | null |
continuous_coordChange (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
(h₂ : b ∈ e₂.baseSet) : Continuous (e₁.coordChange e₂ b) := by
refine continuous_snd.comp (e₂.toOpenPartialHomeomorph.continuousOn.comp_continuous
(e₁.toOpenPartialHomeomorph.continuousOn_symm.comp_continuous ?_ ?_) ?_)
· fun_prop
· exact fun x => e₁.mem_target.2 h₁
· intro x
rwa [e₂.mem_source, e₁.proj_symm_apply' h₁] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | continuous_coordChange | null |
protected coordChangeHomeomorph (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
(h₂ : b ∈ e₂.baseSet) : F ≃ₜ F where
toFun := e₁.coordChange e₂ b
invFun := e₂.coordChange e₁ b
left_inv x := by simp only [*, coordChange_coordChange, coordChange_same_apply]
right_inv x := by simp only [*, coordChange_coordChange, coordChange_same_apply]
continuous_toFun := e₁.continuous_coordChange e₂ h₁ h₂
continuous_invFun := e₂.continuous_coordChange e₁ h₂ h₁
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChangeHomeomorph | Coordinate transformation in the fiber induced by a pair of bundle trivializations,
as a homeomorphism. |
coordChangeHomeomorph_coe (e₁ e₂ : Trivialization F proj) {b : B} (h₁ : b ∈ e₁.baseSet)
(h₂ : b ∈ e₂.baseSet) : ⇑(e₁.coordChangeHomeomorph e₂ h₁ h₂) = e₁.coordChange e₂ b :=
rfl | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | coordChangeHomeomorph_coe | null |
isImage_preimage_prod (e : Trivialization F proj) (s : Set B) :
e.toOpenPartialHomeomorph.IsImage (proj ⁻¹' s) (s ×ˢ univ) := fun x hx => by simp [hx] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | isImage_preimage_prod | null |
protected restrOpen (e : Trivialization F proj) (s : Set B) (hs : IsOpen s) :
Trivialization F proj where
toOpenPartialHomeomorph :=
((e.isImage_preimage_prod s).symm.restr (IsOpen.inter e.open_target (hs.prod isOpen_univ))).symm
baseSet := e.baseSet ∩ s
open_baseSet := IsOpen.inter e.open_baseSet hs
source_eq := by simp [source_eq]
target_eq := by simp [target_eq, prod_univ]
proj_toFun p hp := e.proj_toFun p hp.1 | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | restrOpen | Restrict a `Trivialization` to an open set in the base. |
frontier_preimage (e : Trivialization F proj) (s : Set B) :
e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s) := by
rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq,
(e.isImage_preimage_prod _).preimage_eq, e.source_eq, preimage_inter]
open Classical in | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | frontier_preimage | null |
noncomputable piecewise (e e' : Trivialization F proj) (s : Set B)
(Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s)
(Heq : EqOn e e' <| proj ⁻¹' (e.baseSet ∩ frontier s)) : Trivialization F proj where
toOpenPartialHomeomorph :=
e.toOpenPartialHomeomorph.piecewise e'.toOpenPartialHomeomorph (proj ⁻¹' s) (s ×ˢ univ)
(e.isImage_preimage_prod s) (e'.isImage_preimage_prod s)
(by rw [e.frontier_preimage, e'.frontier_preimage, Hs]) (by rwa [e.frontier_preimage])
baseSet := s.ite e.baseSet e'.baseSet
open_baseSet := e.open_baseSet.ite e'.open_baseSet Hs
source_eq := by simp [source_eq]
target_eq := by simp [target_eq, prod_univ]
proj_toFun p := by
rintro (⟨he, hs⟩ | ⟨he, hs⟩) <;> simp [*] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | piecewise | Given two bundle trivializations `e`, `e'` of `proj : Z → B` and a set `s : Set B` such that
the base sets of `e` and `e'` intersect `frontier s` on the same set and `e p = e' p` whenever
`proj p ∈ e.baseSet ∩ frontier s`, `e.piecewise e' s Hs Heq` is the bundle trivialization over
`Set.ite s e.baseSet e'.baseSet` that is equal to `e` on `proj ⁻¹ s` and is equal to `e'`
otherwise. |
noncomputable piecewiseLeOfEq [LinearOrder B] [OrderTopology B] (e e' : Trivialization F proj)
(a : B) (He : a ∈ e.baseSet) (He' : a ∈ e'.baseSet) (Heq : ∀ p, proj p = a → e p = e' p) :
Trivialization F proj :=
e.piecewise e' (Iic a)
(Set.ext fun x => and_congr_left_iff.2 fun hx => by
obtain rfl : x = a := mem_singleton_iff.1 (frontier_Iic_subset _ hx)
simp [He, He'])
fun p hp => Heq p <| frontier_Iic_subset _ hp.2 | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | piecewiseLeOfEq | Given two bundle trivializations `e`, `e'` of a topological fiber bundle `proj : Z → B`
over a linearly ordered base `B` and a point `a ∈ e.baseSet ∩ e'.baseSet` such that
`e` equals `e'` on `proj ⁻¹' {a}`, `e.piecewise_le_of_eq e' a He He' Heq` is the bundle
trivialization over `Set.ite (Iic a) e.baseSet e'.baseSet` that is equal to `e` on points `p`
such that `proj p ≤ a` and is equal to `e'` otherwise. |
noncomputable piecewiseLe [LinearOrder B] [OrderTopology B] (e e' : Trivialization F proj)
(a : B) (He : a ∈ e.baseSet) (He' : a ∈ e'.baseSet) : Trivialization F proj :=
e.piecewiseLeOfEq (e'.transFiberHomeomorph (e'.coordChangeHomeomorph e He' He)) a He He' <| by
rintro p rfl
ext1
· simp [e.coe_fst', e'.coe_fst', *]
· simp [coordChange_apply_snd, *]
open Classical in | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | piecewiseLe | Given two bundle trivializations `e`, `e'` of a topological fiber bundle `proj : Z → B` over a
linearly ordered base `B` and a point `a ∈ e.baseSet ∩ e'.baseSet`, `e.piecewise_le e' a He He'`
is the bundle trivialization over `Set.ite (Iic a) e.baseSet e'.baseSet` that is equal to `e` on
points `p` such that `proj p ≤ a` and is equal to `((e' p).1, h (e' p).2)` otherwise, where
`h = e'.coord_change_homeomorph e _ _` is the homeomorphism of the fiber such that
`h (e' p).2 = (e p).2` whenever `e p = a`. |
noncomputable disjointUnion (e e' : Trivialization F proj) (H : Disjoint e.baseSet e'.baseSet) :
Trivialization F proj where
toOpenPartialHomeomorph :=
e.toOpenPartialHomeomorph.disjointUnion e'.toOpenPartialHomeomorph
(by
rw [e.source_eq, e'.source_eq]
exact H.preimage _)
(by
rw [e.target_eq, e'.target_eq, disjoint_iff_inf_le]
intro x hx
exact H.le_bot ⟨hx.1.1, hx.2.1⟩)
baseSet := e.baseSet ∪ e'.baseSet
open_baseSet := IsOpen.union e.open_baseSet e'.open_baseSet
source_eq := congr_arg₂ (· ∪ ·) e.source_eq e'.source_eq
target_eq := (congr_arg₂ (· ∪ ·) e.target_eq e'.target_eq).trans union_prod.symm
proj_toFun := by
rintro p (hp | hp')
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_mem, e.coe_fst] <;> exact hp
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_notMem, e'.coe_fst hp']
simp only [source_eq] at hp' ⊢
exact fun h => H.le_bot ⟨h, hp'⟩ | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | disjointUnion | Given two bundle trivializations `e`, `e'` over disjoint sets, `e.disjoint_union e' H` is the
bundle trivialization over the union of the base sets that agrees with `e` and `e'` over their
base sets. |
lift (T : Trivialization F proj) (z : Z) (b : B) : Z := T.invFun (b, (T z).2)
variable {T : Trivialization F proj} {z : Z} {b : B}
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | lift | The local lifting through a Trivialization `T` from the base to the leaf containing `z`. |
lift_self (he : proj z ∈ T.baseSet) : T.lift z (proj z) = z :=
symm_apply_mk_proj _ <| T.mem_source.2 he | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | lift_self | null |
proj_lift (hx : b ∈ T.baseSet) : proj (T.lift z b) = b :=
T.proj_symm_apply <| T.mem_target.2 hx | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | proj_lift | null |
liftCM (T : Trivialization F proj) : C(T.source × T.baseSet, T.source) where
toFun ex := ⟨T.lift ex.1 ex.2, T.map_target (by simp [mem_target])⟩
continuous_toFun := by
apply Continuous.subtype_mk
refine T.continuousOn_invFun.comp_continuous ?_ (by simp [mem_target])
refine .prodMk (by fun_prop) (.snd ?_)
exact T.continuousOn_toFun.comp_continuous (by fun_prop) (by simp)
variable {ι : Type*} [TopologicalSpace ι] [LocallyCompactPair ι T.baseSet]
{γ : C(ι, T.baseSet)} {i : ι} {e : T.source} | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | liftCM | The restriction of `lift` to the source and base set of `T`, as a bundled continuous map. |
clift (T : Trivialization F proj) [LocallyCompactPair ι T.baseSet] :
C(T.source × C(ι, T.baseSet), C(ι, T.source)) := by
let Ψ : C((T.source × C(ι, T.baseSet)) × ι, C(ι, T.baseSet) × ι) :=
⟨fun eγt => (eγt.1.2, eγt.2), by fun_prop⟩
refine ContinuousMap.curry <| T.liftCM.comp <| ⟨fun eγt => ⟨eγt.1.1, eγt.1.2 eγt.2⟩, ?_⟩
simpa using ⟨by fun_prop, ContinuousEval.continuous_eval.comp Ψ.continuous⟩
@[simp] | def | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | clift | Extension of `liftCM` to continuous maps taking values in `T.baseSet` (local version of
homotopy lifting) |
clift_self (h : proj e.1 = γ i) :
T.clift (e, γ) i = e := by
have : proj e ∈ T.baseSet := by simp [h]
simp [clift, liftCM, ← h, lift_self, this] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | clift_self | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.