statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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local_triv_as_local_equiv_symm :
(Z.local_triv_as_local_equiv i).symm = (Z.local_triv i).to_local_equiv.symm | rfl | lemma | fiber_bundle_core.local_triv_as_local_equiv_symm | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_set_at : Z.base_set i = (Z.local_triv i).base_set | rfl | lemma | fiber_bundle_core.base_set_at | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_apply (p : Z.total_space) :
(Z.local_triv i) p = ⟨p.1, Z.coord_change (Z.index_at p.1) i p.1 p.2⟩ | rfl | lemma | fiber_bundle_core.local_triv_apply | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_at_apply (p : Z.total_space) :
((Z.local_triv_at p.1) p) = ⟨p.1, p.2⟩ | by { rw [local_triv_at, local_triv_apply, coord_change_self], exact Z.mem_base_set_at p.1 } | lemma | fiber_bundle_core.local_triv_at_apply | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_at_apply_mk (b : B) (a : F) :
((Z.local_triv_at b) ⟨b, a⟩) = ⟨b, a⟩ | Z.local_triv_at_apply _ | lemma | fiber_bundle_core.local_triv_at_apply_mk | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_source (p : Z.total_space) :
p ∈ (Z.local_triv i).source ↔ p.1 ∈ (Z.local_triv i).base_set | iff.rfl | lemma | fiber_bundle_core.mem_local_triv_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_at_source (p : Z.total_space) (b : B) :
p ∈ (Z.local_triv_at b).source ↔ p.1 ∈ (Z.local_triv_at b).base_set | iff.rfl | lemma | fiber_bundle_core.mem_local_triv_at_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_source_at : (⟨b, a⟩ : Z.total_space) ∈ (Z.local_triv_at b).source | by { rw [local_triv_at, mem_local_triv_source], exact Z.mem_base_set_at b } | lemma | fiber_bundle_core.mem_source_at | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_target (p : B × F) :
p ∈ (Z.local_triv i).target ↔ p.1 ∈ (Z.local_triv i).base_set | trivialization.mem_target _ | lemma | fiber_bundle_core.mem_local_triv_target | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"trivialization.mem_target"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_at_target (p : B × F) (b : B) :
p ∈ (Z.local_triv_at b).target ↔ p.1 ∈ (Z.local_triv_at b).base_set | trivialization.mem_target _ | lemma | fiber_bundle_core.mem_local_triv_at_target | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"trivialization.mem_target"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_symm_apply (p : B × F) :
(Z.local_triv i).to_local_homeomorph.symm p =
⟨p.1, Z.coord_change i (Z.index_at p.1) p.1 p.2⟩ | rfl | lemma | fiber_bundle_core.local_triv_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_at_base_set (b : B) :
b ∈ (Z.local_triv_at b).base_set | by { rw [local_triv_at, ←base_set_at], exact Z.mem_base_set_at b, } | lemma | fiber_bundle_core.mem_local_triv_at_base_set | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_total_space_mk (b : B) :
continuous (total_space.mk b : Z.fiber b → Z.total_space) | begin
rw [continuous_iff_le_induced, fiber_bundle_core.to_topological_space],
apply le_induced_generate_from,
simp only [mem_Union, mem_singleton_iff, local_triv_as_local_equiv_source,
local_triv_as_local_equiv_coe],
rintros s ⟨i, t, ht, rfl⟩,
rw [←((Z.local_triv i).source_inter_preimage_target_inter t), ... | lemma | fiber_bundle_core.continuous_total_space_mk | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous",
"continuous.prod.mk",
"continuous_id",
"continuous_iff_le_induced",
"fiber_bundle_core.to_topological_space",
"is_open.inter",
"is_open_empty",
"is_open_univ",
"le_induced_generate_from"
] | The inclusion of a fiber into the total space is a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle : fiber_bundle F Z.fiber | { total_space_mk_inducing := λ b, ⟨ begin refine le_antisymm _ (λ s h, _),
{ rw ←continuous_iff_le_induced,
exact continuous_total_space_mk Z b, },
{ refine is_open_induced_iff.mpr ⟨(Z.local_triv_at b).source ∩ (Z.local_triv_at b) ⁻¹'
((Z.local_triv_at b).base_set ×ˢ s), (continuous_on_open_iff
... | instance | fiber_bundle_core.fiber_bundle | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on_open_iff",
"fiber_bundle",
"set.range"
] | A fiber bundle constructed from core is indeed a fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_proj : continuous Z.proj | continuous_proj F Z.fiber | lemma | fiber_bundle_core.continuous_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous"
] | The projection on the base of a fiber bundle created from core is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_proj : is_open_map Z.proj | is_open_map_proj F Z.fiber | lemma | fiber_bundle_core.is_open_map_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open_map"
] | The projection on the base of a fiber bundle created from core is an open map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_prebundle | (pretrivialization_atlas : set (pretrivialization F (π F E)))
(pretrivialization_at : B → pretrivialization F (π F E))
(mem_base_pretrivialization_at : ∀ x : B, x ∈ (pretrivialization_at x).base_set)
(pretrivialization_mem_atlas : ∀ x : B, pretrivialization_at x ∈ pretrivialization_atlas)
(continuous_triv_change : ∀ e ... | structure | fiber_prebundle | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on",
"inducing",
"pretrivialization"
] | This structure permits to define a fiber bundle when trivializations are given as local
equivalences but there is not yet a topology on the total space. The total space is hence given a
topology in such a way that there is a fiber bundle structure for which the local equivalences
are also local homeomorphism and hence ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total_space_topology (a : fiber_prebundle F E) : topological_space (total_space F E) | ⨆ (e : pretrivialization F (π F E)) (he : e ∈ a.pretrivialization_atlas),
coinduced e.set_symm (subtype.topological_space) | def | fiber_prebundle.total_space_topology | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"fiber_prebundle",
"pretrivialization",
"topological_space"
] | Topology on the total space that will make the prebundle into a bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_symm_of_mem_pretrivialization_atlas (he : e ∈ a.pretrivialization_atlas) :
@continuous_on _ _ _ a.total_space_topology
e.to_local_equiv.symm e.target | begin
refine id (λ z H, id (λ U h, preimage_nhds_within_coinduced' H
e.open_target (le_def.1 (nhds_mono _) U h))),
exact le_supr₂ e he,
end | lemma | fiber_prebundle.continuous_symm_of_mem_pretrivialization_atlas | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on",
"le_supr₂",
"nhds_mono",
"preimage_nhds_within_coinduced'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_source (e : pretrivialization F (π F E)) : is_open[a.total_space_topology] e.source | begin
letI := a.total_space_topology,
refine is_open_supr_iff.mpr (λ e', _),
refine is_open_supr_iff.mpr (λ he', _),
refine is_open_coinduced.mpr (is_open_induced_iff.mpr ⟨e.target, e.open_target, _⟩),
rw [pretrivialization.set_symm, restrict, e.target_eq,
e.source_eq, preimage_comp, subtype.preimage_coe_... | lemma | fiber_prebundle.is_open_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open",
"pretrivialization",
"pretrivialization.preimage_symm_proj_inter",
"pretrivialization.set_symm",
"subtype.preimage_coe_eq_preimage_coe_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_target_of_mem_pretrivialization_atlas_inter (e e' : pretrivialization F (π F E))
(he' : e' ∈ a.pretrivialization_atlas) :
is_open (e'.to_local_equiv.target ∩ e'.to_local_equiv.symm ⁻¹' e.source) | begin
letI := a.total_space_topology,
obtain ⟨u, hu1, hu2⟩ := continuous_on_iff'.mp (a.continuous_symm_of_mem_pretrivialization_atlas
he') e.source (a.is_open_source e),
rw [inter_comm, hu2],
exact hu1.inter e'.open_target,
end | lemma | fiber_prebundle.is_open_target_of_mem_pretrivialization_atlas_inter | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_of_mem_pretrivialization_atlas (he : e ∈ a.pretrivialization_atlas) :
@trivialization B F _ _ _ a.total_space_topology (π F E) | { open_source := a.is_open_source e,
continuous_to_fun := begin
letI := a.total_space_topology,
refine continuous_on_iff'.mpr (λ s hs, ⟨e ⁻¹' s ∩ e.source, (is_open_supr_iff.mpr (λ e', _)),
by { rw [inter_assoc, inter_self], refl }⟩),
refine (is_open_supr_iff.mpr (λ he', _)),
rw [is_open_coinduc... | def | fiber_prebundle.trivialization_of_mem_pretrivialization_atlas | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open_coinduced",
"is_open_induced_iff",
"subtype.coe_preimage_self",
"trivialization"
] | Promotion from a `pretrivialization` to a `trivialization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_trivialization_at_source (b : B) (x : E b) :
total_space.mk b x ∈ (a.pretrivialization_at b).source | begin
simp only [(a.pretrivialization_at b).source_eq, mem_preimage, total_space.proj],
exact a.mem_base_pretrivialization_at b,
end | lemma | fiber_prebundle.mem_trivialization_at_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
total_space_mk_preimage_source (b : B) :
total_space.mk b ⁻¹' (a.pretrivialization_at b).source = univ | begin
apply eq_univ_of_univ_subset,
rw [(a.pretrivialization_at b).source_eq, ←preimage_comp, function.comp],
simp only [total_space.proj],
rw preimage_const_of_mem _,
exact a.mem_base_pretrivialization_at b,
end | lemma | fiber_prebundle.total_space_mk_preimage_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_total_space_mk (b : B) :
@continuous _ _ _ a.total_space_topology (total_space.mk b) | begin
letI := a.total_space_topology,
let e := a.trivialization_of_mem_pretrivialization_atlas (a.pretrivialization_mem_atlas b),
rw e.to_local_homeomorph.continuous_iff_continuous_comp_left
(a.total_space_mk_preimage_source b),
exact continuous_iff_le_induced.mpr (le_antisymm_iff.mp (a.total_space_mk_induc... | lemma | fiber_prebundle.continuous_total_space_mk | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_total_space_mk_of_inducing_comp (b : B)
(h : inducing ((a.pretrivialization_at b) ∘ (total_space.mk b))) :
@inducing _ _ _ a.total_space_topology (total_space.mk b) | begin
letI := a.total_space_topology,
rw ←restrict_comp_cod_restrict (a.mem_trivialization_at_source b) at h,
apply inducing.of_cod_restrict (a.mem_trivialization_at_source b),
refine inducing_of_inducing_compose _ (continuous_on_iff_continuous_restrict.mp
(a.trivialization_of_mem_pretrivialization_atlas
... | lemma | fiber_prebundle.inducing_total_space_mk_of_inducing_comp | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"inducing",
"inducing.of_cod_restrict",
"inducing_of_inducing_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fiber_bundle :
@fiber_bundle B F _ _ E a.total_space_topology _ | { total_space_mk_inducing := λ b, a.inducing_total_space_mk_of_inducing_comp b
(a.total_space_mk_inducing b),
trivialization_atlas := {e | ∃ e₀ (he₀ : e₀ ∈ a.pretrivialization_atlas),
e = a.trivialization_of_mem_pretrivialization_atlas he₀},
trivialization_at := λ x, a.trivialization_of_mem_pretrivializatio... | def | fiber_prebundle.to_fiber_bundle | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"fiber_bundle"
] | Make a `fiber_bundle` from a `fiber_prebundle`. Concretely this means
that, given a `fiber_prebundle` 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
`f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_proj : @continuous _ _ a.total_space_topology _ (π F E) | begin
letI := a.total_space_topology,
letI := a.to_fiber_bundle,
exact continuous_proj F E,
end | lemma | fiber_prebundle.continuous_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_of_comp_right {X : Type*} [topological_space X] {f : total_space F E → X}
{s : set B} (hs : is_open s)
(hf : ∀ b ∈ s, continuous_on (f ∘ (a.pretrivialization_at b).to_local_equiv.symm)
((s ∩ (a.pretrivialization_at b).base_set) ×ˢ (set.univ : set F))) :
@continuous_on _ _ a.total_space_topology ... | begin
letI := a.total_space_topology,
intros z hz,
let e : trivialization F (π F E) :=
a.trivialization_of_mem_pretrivialization_atlas (a.pretrivialization_mem_atlas z.proj),
refine (e.continuous_at_of_comp_right _
((hf z.proj hz).continuous_at (is_open.mem_nhds _ _))).continuous_within_at,
{ exact a.me... | lemma | fiber_prebundle.continuous_on_of_comp_right | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_at",
"continuous_on",
"continuous_within_at",
"is_open",
"is_open.mem_nhds",
"is_open_univ",
"topological_space",
"trivialization"
] | For a fiber bundle `E` over `B` constructed using the `fiber_prebundle` mechanism,
continuity of a function `total_space F E → X` on an open set `s` can be checked by precomposing at
each point with the pretrivialization used for the construction at that point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization : trivialization F (π F (λ _ : B, F)) | { to_fun := λ x, (x.proj, x.snd),
inv_fun := λ y, ⟨y.fst, y.snd⟩,
source := univ,
target := univ,
map_source' := λ x h, mem_univ _,
map_target' := λ y h, mem_univ _,
left_inv' := λ x h, total_space.ext _ _ rfl heq.rfl,
right_inv' := λ x h, prod.ext rfl rfl,
open_source := is_open_univ,
open_target := ... | def | bundle.trivial.trivialization | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous_iff_le_induced",
"induced_compose",
"induced_inf",
"inv_fun",
"is_open_univ",
"le_rfl",
"prod.ext",
"trivialization"
] | Local trivialization for trivial bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization_source : (trivialization B F).source = univ | rfl | lemma | bundle.trivial.trivialization_source | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_target : (trivialization B F).target = univ | rfl | lemma | bundle.trivial.trivialization_target | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_bundle : fiber_bundle F (bundle.trivial B F) | { trivialization_atlas := {bundle.trivial.trivialization B F},
trivialization_at := λ x, bundle.trivial.trivialization B F,
mem_base_set_trivialization_at := mem_univ,
trivialization_mem_atlas := λ x, mem_singleton _,
total_space_mk_inducing := λ b, ⟨begin
have : (λ (x : trivial B F b), x) = @id F, by { ext... | instance | bundle.trivial.fiber_bundle | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"bundle.trivial",
"bundle.trivial.trivialization",
"fiber_bundle",
"induced_compose",
"induced_const",
"induced_id",
"induced_inf",
"top_inf_eq"
] | Fiber bundle instance on the trivial bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_trivialization (e : _root_.trivialization F (π F (bundle.trivial B F)))
[i : mem_trivialization_atlas e] :
e = trivialization B F | i.out | lemma | bundle.trivial.eq_trivialization | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"bundle.trivial",
"mem_trivialization_atlas",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_bundle.prod.topological_space :
topological_space (total_space (F₁ × F₂) (E₁ ×ᵇ E₂)) | topological_space.induced
(λ p, ((⟨p.1, p.2.1⟩ : total_space F₁ E₁), (⟨p.1, p.2.2⟩ : total_space F₂ E₂)))
(by apply_instance : topological_space (total_space F₁ E₁ × total_space F₂ E₂)) | instance | fiber_bundle.prod.topological_space | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"topological_space",
"topological_space.induced"
] | Equip the total space of the fiberwise product of two fiber bundles `E₁`, `E₂` with
the induced topology from the diagonal embedding into `total_space E₁ × total_space E₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle.prod.inducing_diag : inducing
(λ p, (⟨p.1, p.2.1⟩, ⟨p.1, p.2.2⟩) :
total_space (F₁ × F₂) (E₁ ×ᵇ E₂) → total_space F₁ E₁ × total_space F₂ E₂) | ⟨rfl⟩ | lemma | fiber_bundle.prod.inducing_diag | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"inducing"
] | The diagonal map from the total space of the fiberwise product of two fiber bundles
`E₁`, `E₂` into `total_space E₁ × total_space E₂` is `inducing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.to_fun' : total_space (F₁ × F₂) (E₁ ×ᵇ E₂) → B × (F₁ × F₂) | λ p, ⟨p.1, (e₁ ⟨p.1, p.2.1⟩).2, (e₂ ⟨p.1, p.2.2⟩).2⟩ | def | trivialization.prod.to_fun' | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [] | Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the forward
function for the construction `trivialization.prod`, the induced
trivialization for the fiberwise product of `E₁` and `E₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.continuous_to_fun :
continuous_on (prod.to_fun' e₁ e₂) (π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.base_set ∩ e₂.base_set)) | begin
let f₁ : total_space (F₁ × F₂) (E₁ ×ᵇ E₂) → total_space F₁ E₁ × total_space F₂ E₂ :=
λ p, ((⟨p.1, p.2.1⟩ : total_space F₁ E₁), (⟨p.1, p.2.2⟩ : total_space F₂ E₂)),
let f₂ : total_space F₁ E₁ × total_space F₂ E₂ → (B × F₁) × (B × F₂) := λ p, ⟨e₁ p.1, e₂ p.2⟩,
let f₃ : (B × F₁) × (B × F₂) → B × F₁ × F₂ :=... | lemma | trivialization.prod.continuous_to_fun | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"continuous_fst",
"continuous_on",
"continuous_snd",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.inv_fun' (p : B × (F₁ × F₂)) : total_space (F₁ × F₂) (E₁ ×ᵇ E₂) | ⟨p.1, e₁.symm p.1 p.2.1, e₂.symm p.1 p.2.2⟩ | def | trivialization.prod.inv_fun' | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [] | Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the inverse
function for the construction `trivialization.prod`, the induced
trivialization for the fiberwise product of `E₁` and `E₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.left_inv {x : total_space (F₁ × F₂) (E₁ ×ᵇ E₂)}
(h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.base_set ∩ e₂.base_set)) :
prod.inv_fun' e₁ e₂ (prod.to_fun' e₁ e₂ x) = x | begin
obtain ⟨x, v₁, v₂⟩ := x,
obtain ⟨h₁ : x ∈ e₁.base_set, h₂ : x ∈ e₂.base_set⟩ := h,
simp only [prod.to_fun', prod.inv_fun', symm_apply_apply_mk, h₁, h₂,
eq_self_iff_true, heq_iff_eq, and_self]
end | lemma | trivialization.prod.left_inv | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"heq_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.right_inv {x : B × F₁ × F₂}
(h : x ∈ (e₁.base_set ∩ e₂.base_set) ×ˢ (univ : set (F₁ × F₂))) :
prod.to_fun' e₁ e₂ (prod.inv_fun' e₁ e₂ x) = x | begin
obtain ⟨x, w₁, w₂⟩ := x,
obtain ⟨⟨h₁ : x ∈ e₁.base_set, h₂ : x ∈ e₂.base_set⟩, -⟩ := h,
simp only [prod.to_fun', prod.inv_fun', apply_mk_symm, h₁, h₂]
end | lemma | trivialization.prod.right_inv | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.continuous_inv_fun :
continuous_on (prod.inv_fun' e₁ e₂) ((e₁.base_set ∩ e₂.base_set) ×ˢ univ) | begin
rw (prod.inducing_diag F₁ E₁ F₂ E₂).continuous_on_iff,
have H₁ : continuous (λ p : B × F₁ × F₂, ((p.1, p.2.1), (p.1, p.2.2))) :=
(continuous_id.prod_map continuous_fst).prod_mk (continuous_id.prod_map continuous_snd),
refine (e₁.continuous_on_symm.prod_map e₂.continuous_on_symm).comp H₁.continuous_on _,... | lemma | trivialization.prod.continuous_inv_fun | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"continuous_fst",
"continuous_on",
"continuous_on_iff",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : trivialization (F₁ × F₂) (π (F₁ × F₂) (E₁ ×ᵇ E₂)) | { to_fun := prod.to_fun' e₁ e₂,
inv_fun := prod.inv_fun' e₁ e₂,
source := (π (F₁ × F₂) (E₁ ×ᵇ E₂)) ⁻¹' (e₁.base_set ∩ e₂.base_set),
target := (e₁.base_set ∩ e₂.base_set) ×ˢ set.univ,
map_source' := λ x h, ⟨h, set.mem_univ _⟩,
map_target' := λ x h, h.1,
left_inv' := λ x, prod.left_inv,
right_inv' := λ x, p... | def | trivialization.prod | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"fiber_bundle.prod.inducing_diag",
"inv_fun",
"is_open_univ",
"set.mem_univ",
"trivialization"
] | Given trivializations `e₁`, `e₂` for bundle types `E₁`, `E₂` over a base `B`, the induced
trivialization for the fiberwise product of `E₁` and `E₂`, whose base set is
`e₁.base_set ∩ e₂.base_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_set_prod : (prod e₁ e₂).base_set = e₁.base_set ∩ e₂.base_set | rfl | lemma | trivialization.base_set_prod | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_symm_apply (x : B) (w₁ : F₁) (w₂ : F₂) :
(prod e₁ e₂).to_local_equiv.symm (x, w₁, w₂) = ⟨x, e₁.symm x w₁, e₂.symm x w₂⟩ | rfl | lemma | trivialization.prod_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_bundle.prod : fiber_bundle (F₁ × F₂) (E₁ ×ᵇ E₂) | { total_space_mk_inducing := λ b,
begin
rw (prod.inducing_diag F₁ E₁ F₂ E₂).inducing_iff,
exact (total_space_mk_inducing F₁ E₁ b).prod_mk (total_space_mk_inducing F₂ E₂ b),
end,
trivialization_atlas :=
{e | ∃ (e₁ : trivialization F₁ (π F₁ E₁)) (e₂ : trivialization F₂ (π F₂ E₂))
[mem_trivializatio... | instance | fiber_bundle.prod | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"fiber_bundle",
"mem_trivialization_atlas",
"trivialization",
"trivialization.prod"
] | The product of two fiber bundles is a fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_topology : topological_space (total_space F (f *ᵖ E)) | induced total_space.proj ‹topological_space B'› ⊓
induced (pullback.lift f) ‹topological_space (total_space F E)› | def | pullback_topology | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"topological_space"
] | Definition of `pullback.total_space.topological_space`, which we make irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.total_space.topological_space : topological_space (total_space F (f *ᵖ E)) | pullback_topology F E f | instance | pullback.total_space.topological_space | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"pullback_topology",
"topological_space"
] | The topology on the total space of a pullback bundle is the coarsest topology for which both
the projections to the base and the map to the original bundle are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.continuous_proj (f : B' → B) : continuous (π F (f *ᵖ E)) | begin
rw [continuous_iff_le_induced, pullback.total_space.topological_space, pullback_topology],
exact inf_le_left,
end | lemma | pullback.continuous_proj | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"continuous_iff_le_induced",
"inf_le_left",
"pullback.total_space.topological_space",
"pullback_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback.continuous_lift (f : B' → B) : continuous (@pullback.lift B F E B' f) | begin
rw [continuous_iff_le_induced, pullback.total_space.topological_space, pullback_topology],
exact inf_le_right,
end | lemma | pullback.continuous_lift | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"continuous_iff_le_induced",
"inf_le_right",
"pullback.total_space.topological_space",
"pullback_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_pullback_total_space_embedding (f : B' → B) :
inducing (@pullback_total_space_embedding B F E B' f) | begin
constructor,
simp_rw [prod.topological_space, induced_inf, induced_compose,
pullback.total_space.topological_space, pullback_topology],
refl
end | lemma | inducing_pullback_total_space_embedding | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"induced_compose",
"induced_inf",
"inducing",
"pullback.total_space.topological_space",
"pullback_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback.continuous_total_space_mk [∀ x, topological_space (E x)]
[fiber_bundle F E] {f : B' → B} {x : B'} :
continuous (@total_space.mk _ F (f *ᵖ E) x) | begin
simp only [continuous_iff_le_induced, pullback.total_space.topological_space, induced_compose,
induced_inf, function.comp, induced_const, top_inf_eq, pullback_topology],
exact le_of_eq (fiber_bundle.total_space_mk_inducing F E (f x)).induced,
end | lemma | pullback.continuous_total_space_mk | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous",
"continuous_iff_le_induced",
"fiber_bundle",
"induced_compose",
"induced_const",
"induced_inf",
"pullback.total_space.topological_space",
"pullback_topology",
"top_inf_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.pullback (e : trivialization F (π F E)) (f : K) :
trivialization F (π F ((f : B' → B) *ᵖ E)) | { to_fun := λ z, (z.proj, (e (pullback.lift f z)).2),
inv_fun := λ y, @total_space.mk _ _ (f *ᵖ E) y.1 (e.symm (f y.1) y.2),
source := pullback.lift f ⁻¹' e.source,
base_set := f ⁻¹' e.base_set,
target := (f ⁻¹' e.base_set) ×ˢ univ,
map_source' := λ x h, by { simp_rw [e.source_eq, mem_preimage, pullback.lift_... | def | trivialization.pullback | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"continuous_id",
"continuous_on",
"continuous_on.prod",
"continuous_on_iff",
"inducing_pullback_total_space_embedding",
"inv_fun",
"is_open_univ",
"prod_map",
"pullback.continuous_lift",
"pullback.continuous_proj",
"trivialization"
] | A fiber bundle trivialization can be pulled back to a trivialization on the pullback bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle.pullback [∀ x, topological_space (E x)]
[fiber_bundle F E] (f : K) : fiber_bundle F ((f : B' → B) *ᵖ E) | { total_space_mk_inducing := λ x, inducing_of_inducing_compose
(pullback.continuous_total_space_mk F E) (pullback.continuous_lift F E f)
(total_space_mk_inducing F E (f x)),
trivialization_atlas :=
{ef | ∃ (e : trivialization F (π F E)) [mem_trivialization_atlas e], ef = e.pullback f},
trivialization_at... | instance | fiber_bundle.pullback | topology.fiber_bundle | src/topology/fiber_bundle/constructions.lean | [
"topology.fiber_bundle.basic"
] | [
"fiber_bundle",
"inducing_of_inducing_compose",
"mem_trivialization_atlas",
"pullback.continuous_lift",
"pullback.continuous_total_space_mk",
"topological_space",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_homeomorphic_trivial_fiber_bundle (proj : Z → B) : Prop | ∃ e : Z ≃ₜ (B × F), ∀ x, (e x).1 = proj x | def | is_homeomorphic_trivial_fiber_bundle | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [] | A trivial fiber bundle with fiber `F` over a base `B` is a space `Z`
projecting on `B` for which there exists a homeomorphism to `B × F` that sends `proj`
to `prod.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_eq (h : is_homeomorphic_trivial_fiber_bundle F proj) :
∃ e : Z ≃ₜ (B × F), proj = prod.fst ∘ e | ⟨h.some, (funext h.some_spec).symm⟩ | lemma | is_homeomorphic_trivial_fiber_bundle.proj_eq | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_proj [nonempty F] (h : is_homeomorphic_trivial_fiber_bundle F proj) :
function.surjective proj | begin
obtain ⟨e, rfl⟩ := h.proj_eq,
exact prod.fst_surjective.comp e.surjective,
end | lemma | is_homeomorphic_trivial_fiber_bundle.surjective_proj | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle"
] | The projection from a trivial fiber bundle to its base is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_proj (h : is_homeomorphic_trivial_fiber_bundle F proj) :
continuous proj | begin
obtain ⟨e, rfl⟩ := h.proj_eq,
exact continuous_fst.comp e.continuous,
end | lemma | is_homeomorphic_trivial_fiber_bundle.continuous_proj | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"continuous",
"is_homeomorphic_trivial_fiber_bundle"
] | The projection from a trivial fiber bundle to its base is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_proj (h : is_homeomorphic_trivial_fiber_bundle F proj) :
is_open_map proj | begin
obtain ⟨e, rfl⟩ := h.proj_eq,
exact is_open_map_fst.comp e.is_open_map,
end | lemma | is_homeomorphic_trivial_fiber_bundle.is_open_map_proj | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle",
"is_open_map"
] | The projection from a trivial fiber bundle to its base is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_proj [nonempty F] (h : is_homeomorphic_trivial_fiber_bundle F proj) :
quotient_map proj | h.is_open_map_proj.to_quotient_map h.continuous_proj h.surjective_proj | lemma | is_homeomorphic_trivial_fiber_bundle.quotient_map_proj | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle",
"quotient_map"
] | The projection from a trivial fiber bundle to its base is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_homeomorphic_trivial_fiber_bundle_fst :
is_homeomorphic_trivial_fiber_bundle F (prod.fst : B × F → B) | ⟨homeomorph.refl _, λ x, rfl⟩ | lemma | is_homeomorphic_trivial_fiber_bundle_fst | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle"
] | The first projection in a product is a trivial fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_homeomorphic_trivial_fiber_bundle_snd :
is_homeomorphic_trivial_fiber_bundle F (prod.snd : F × B → B) | ⟨homeomorph.prod_comm _ _, λ x, rfl⟩ | lemma | is_homeomorphic_trivial_fiber_bundle_snd | topology.fiber_bundle | src/topology/fiber_bundle/is_homeomorphic_trivial_bundle.lean | [
"topology.homeomorph"
] | [
"is_homeomorphic_trivial_fiber_bundle"
] | The second projection in a product is a trivial fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pretrivialization (proj : Z → B) extends local_equiv Z (B × F) | (open_target : is_open target)
(base_set : set B)
(open_base_set : is_open base_set)
(source_eq : source = proj ⁻¹' base_set)
(target_eq : target = base_set ×ˢ univ)
(proj_to_fun : ∀ p ∈ source, (to_fun p).1 = proj p) | structure | pretrivialization | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"is_open",
"local_equiv"
] | This structure contains the information left for a local trivialization (which is implemented
below as `trivialization F proj`) if the total space has not been given a topology, but we
have a topology on both the fiber and the base space. Through the construction
`topological_fiber_prebundle F proj` it will be possible... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe : ⇑e.to_local_equiv = e | rfl | lemma | pretrivialization.coe_coe | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst (ex : x ∈ e.source) : (e x).1 = proj x | e.proj_to_fun x ex | lemma | pretrivialization.coe_fst | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_source : x ∈ e.source ↔ proj x ∈ e.base_set | by rw [e.source_eq, mem_preimage] | lemma | pretrivialization.mem_source | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst' (ex : proj x ∈ e.base_set) : (e x).1 = proj x | e.coe_fst (e.mem_source.2 ex) | lemma | pretrivialization.coe_fst' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on : eq_on (prod.fst ∘ e) proj e.source | λ x hx, e.coe_fst hx | lemma | pretrivialization.eq_on | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_proj_snd (ex : x ∈ e.source) : (proj x, (e x).2) = e x | prod.ext (e.coe_fst ex).symm rfl | lemma | pretrivialization.mk_proj_snd | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_proj_snd' (ex : proj x ∈ e.base_set) : (proj x, (e x).2) = e x | prod.ext (e.coe_fst' ex).symm rfl | lemma | pretrivialization.mk_proj_snd' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_symm : e.target → Z | e.target.restrict e.to_local_equiv.symm | def | pretrivialization.set_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | Composition of inverse and coercion from the subtype of the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.base_set | by rw [e.target_eq, prod_univ, mem_preimage] | lemma | pretrivialization.mem_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.to_local_equiv.symm x) = x.1 | begin
have := (e.coe_fst (e.to_local_equiv.map_target hx)).symm,
rwa [← e.coe_coe, e.to_local_equiv.right_inv hx] at this
end | lemma | pretrivialization.proj_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.base_set) :
proj (e.to_local_equiv.symm (b, x)) = b | e.proj_symm_apply (e.mem_target.2 hx) | lemma | pretrivialization.proj_symm_apply' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_surj_on_base_set [nonempty F] : set.surj_on proj e.source e.base_set | λ b hb, let ⟨y⟩ := ‹nonempty F› in ⟨e.to_local_equiv.symm (b, y),
e.to_local_equiv.map_target $ e.mem_target.2 hb, e.proj_symm_apply' hb⟩ | lemma | pretrivialization.proj_surj_on_base_set | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"set.surj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.to_local_equiv.symm x) = x | e.to_local_equiv.right_inv hx | lemma | pretrivialization.apply_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply' {b : B} {x : F} (hx : b ∈ e.base_set) :
e (e.to_local_equiv.symm (b, x)) = (b, x) | e.apply_symm_apply (e.mem_target.2 hx) | lemma | pretrivialization.apply_symm_apply' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply {x : Z} (hx : x ∈ e.source) : e.to_local_equiv.symm (e x) = x | e.to_local_equiv.left_inv hx | lemma | pretrivialization.symm_apply_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_mk_proj {x : Z} (ex : x ∈ e.source) :
e.to_local_equiv.symm (proj x, (e x).2) = x | by rw [← e.coe_fst ex, prod.mk.eta, ← e.coe_coe, e.to_local_equiv.left_inv ex] | lemma | pretrivialization.symm_apply_mk_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_symm_proj_base_set :
(e.to_local_equiv.symm ⁻¹' (proj ⁻¹' e.base_set)) ∩ e.target = e.target | begin
refine inter_eq_right_iff_subset.mpr (λ x hx, _),
simp only [mem_preimage, local_equiv.inv_fun_as_coe, e.proj_symm_apply hx],
exact e.mem_target.mp hx,
end | lemma | pretrivialization.preimage_symm_proj_base_set | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_equiv.inv_fun_as_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_symm_proj_inter (s : set B) :
(e.to_local_equiv.symm ⁻¹' (proj ⁻¹' s)) ∩ e.base_set ×ˢ univ = (s ∩ e.base_set) ×ˢ univ | begin
ext ⟨x, y⟩,
suffices : x ∈ e.base_set → (proj (e.to_local_equiv.symm (x, y)) ∈ s ↔ x ∈ s),
by simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true, mem_univ, and.congr_left_iff],
intro h,
rw [e.proj_symm_apply' h]
end | lemma | pretrivialization.preimage_symm_proj_inter | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"and.congr_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
target_inter_preimage_symm_source_eq (e f : pretrivialization F proj) :
f.target ∩ (f.to_local_equiv.symm) ⁻¹' e.source = (e.base_set ∩ f.base_set) ×ˢ univ | by rw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter] | lemma | pretrivialization.target_inter_preimage_symm_source_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_source (e f : pretrivialization F proj) :
(f.to_local_equiv.symm.trans e.to_local_equiv).source = (e.base_set ∩ f.base_set) ×ˢ univ | by rw [local_equiv.trans_source, local_equiv.symm_source, e.target_inter_preimage_symm_source_eq] | lemma | pretrivialization.trans_source | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_equiv.symm_source",
"local_equiv.trans_source",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_symm (e e' : pretrivialization F proj) :
(e.to_local_equiv.symm.trans e'.to_local_equiv).symm =
e'.to_local_equiv.symm.trans e.to_local_equiv | by rw [local_equiv.trans_symm_eq_symm_trans_symm, local_equiv.symm_symm] | lemma | pretrivialization.symm_trans_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_equiv.symm_symm",
"local_equiv.trans_symm_eq_symm_trans_symm",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_source_eq (e e' : pretrivialization F proj) :
(e.to_local_equiv.symm.trans e'.to_local_equiv).source = (e.base_set ∩ e'.base_set) ×ˢ univ | by rw [local_equiv.trans_source, e'.source_eq, local_equiv.symm_source, e.target_eq, inter_comm,
e.preimage_symm_proj_inter, inter_comm] | lemma | pretrivialization.symm_trans_source_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_equiv.symm_source",
"local_equiv.trans_source",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_target_eq (e e' : pretrivialization F proj) :
(e.to_local_equiv.symm.trans e'.to_local_equiv).target = (e.base_set ∩ e'.base_set) ×ˢ univ | by rw [← local_equiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm] | lemma | pretrivialization.symm_trans_target_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_equiv.symm_source",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.base_set | e'.mem_source | theorem | pretrivialization.coe_mem_source | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_fst (hb : b ∈ e'.base_set) : (e' y).1 = b | e'.coe_fst (e'.mem_source.2 hb) | lemma | pretrivialization.coe_coe_fst | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mem_target {x : B} {y : F} : (x, y) ∈ e'.target ↔ x ∈ e'.base_set | e'.mem_target | lemma | pretrivialization.mk_mem_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_coe_proj {x : B} {y : F} (e' : pretrivialization F (π F E)) (h : x ∈ e'.base_set) :
(e'.to_local_equiv.symm (x, y)).1 = x | e'.proj_symm_apply' h | lemma | pretrivialization.symm_coe_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : pretrivialization F (π F E)) (b : B) (y : F) : E b | if hb : b ∈ e.base_set
then cast (congr_arg E (e.proj_symm_apply' hb)) (e.to_local_equiv.symm (b, y)).2
else 0 | def | pretrivialization.symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | A fiberwise inverse to `e`. This is the function `F → E b` that induces a local inverse
`B × F → total_space F E` of `e` on `e.base_set`. It is defined to be `0` outside `e.base_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_apply (e : pretrivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : F) :
e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.to_local_equiv.symm (b, y)).2 | dif_pos hb | lemma | pretrivialization.symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_of_not_mem (e : pretrivialization F (π F E)) {b : B} (hb : b ∉ e.base_set)
(y : F) : e.symm b y = 0 | dif_neg hb | lemma | pretrivialization.symm_apply_of_not_mem | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_of_not_mem (e : pretrivialization F (π F E)) {b : B} (hb : b ∉ e.base_set) :
(e.symm b : F → E b) = 0 | funext $ λ y, dif_neg hb | lemma | pretrivialization.coe_symm_of_not_mem | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_symm (e : pretrivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : F) :
total_space.mk b (e.symm b y) = e.to_local_equiv.symm (b, y) | by rw [e.symm_apply hb, total_space.mk_cast, total_space.eta] | lemma | pretrivialization.mk_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_proj_apply (e : pretrivialization F (π F E)) (z : total_space F E)
(hz : z.proj ∈ e.base_set) : 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)] | lemma | pretrivialization.symm_proj_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"cast_eq_iff_heq",
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply_mk (e : pretrivialization F (π F E)) {b : B} (hb : b ∈ e.base_set)
(y : E b) : e.symm b (e ⟨b, y⟩).2 = y | e.symm_proj_apply ⟨b, y⟩ hb | lemma | pretrivialization.symm_apply_apply_mk | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mk_symm (e : pretrivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (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)] | lemma | pretrivialization.apply_mk_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization (proj : Z → B)
extends local_homeomorph Z (B × F) | (base_set : set B)
(open_base_set : is_open base_set)
(source_eq : source = proj ⁻¹' base_set)
(target_eq : target = base_set ×ˢ univ)
(proj_to_fun : ∀ p ∈ source, (to_local_homeomorph p).1 = proj p) | structure | trivialization | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"is_open",
"local_homeomorph"
] | A structure extending local homeomorphisms, defining a local trivialization of a projection
`proj : Z → B` with fiber `F`, as a local homeomorphism between `Z` and `B × F` defined between two
sets of the form `proj ⁻¹' base_set` and `base_set × F`, acting trivially on the first coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pretrivialization : pretrivialization F proj | { ..e } | def | trivialization.to_pretrivialization | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization"
] | Natural identification as a `pretrivialization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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