statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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uniformly_extend_exists [complete_space γ] (a : α) :
∃c, tendsto f (comap e (𝓝 a)) (𝓝 c) | let de := (h_e.dense_inducing h_dense) in
have cauchy (𝓝 a), from cauchy_nhds,
have cauchy (comap e (𝓝 a)), from
this.comap' (le_of_eq h_e.comap_uniformity) (de.comap_nhds_ne_bot _),
have cauchy (map f (comap e (𝓝 a))), from this.map h_f,
complete_space.complete this | lemma | uniformly_extend_exists | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"cauchy_nhds",
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_extend_subtype [complete_space γ]
{p : α → Prop} {e : α → β} {f : α → γ} {b : β} {s : set α}
(hf : uniform_continuous (λx:subtype p, f x.val))
(he : uniform_embedding e) (hd : ∀x:β, x ∈ closure (range e))
(hb : closure (e '' s) ∈ 𝓝 b) (hs : is_closed s) (hp : ∀x∈s, p x) :
∃c, tendsto f (comap e (𝓝 b... | have de : dense_embedding e,
from he.dense_embedding hd,
have de' : dense_embedding (dense_embedding.subtype_emb p e),
by exact de.subtype p,
have ue' : uniform_embedding (dense_embedding.subtype_emb p e),
from uniform_embedding_subtype_emb _ he de,
have b ∈ closure (e '' {x | p x}),
from (closure_mono $ monoto... | lemma | uniform_extend_subtype | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"closure",
"closure_induced",
"closure_mono",
"cluster_pt",
"complete_space",
"dense_embedding",
"dense_embedding.subtype_emb",
"is_closed",
"mem_closure_iff_cluster_pt",
"mem_closure_iff_nhds_ne_bot",
"mem_of_mem_nhds",
"nhds_induced",
"nhds_subtype_eq_comap",
"uniform_continuous",
"uni... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformly_extend_spec [complete_space γ] (a : α) :
tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) | by simpa only [dense_inducing.extend] using tendsto_nhds_lim (uniformly_extend_exists h_e ‹_› h_f _) | lemma | uniformly_extend_spec | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"dense_inducing.extend",
"tendsto_nhds_lim",
"uniformly_extend_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_uniformly_extend [cγ : complete_space γ] : uniform_continuous ψ | assume d hd,
let ⟨s, hs, hs_comp⟩ := (mem_lift'_sets $
monotone_id.comp_rel $ monotone_id.comp_rel monotone_id).mp
(comp_le_uniformity3 hd) in
have h_pnt : ∀{a m}, m ∈ 𝓝 a → ∃c, c ∈ f '' preimage e m ∧ (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s,
from assume a m hm,
have nb : ne_bot (map f (comap e (𝓝 a))),
from ((h_e.... | lemma | uniform_continuous_uniformly_extend | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"comp_le_uniformity3",
"comp_rel",
"complete_space",
"interior",
"interior_mem_uniformity",
"interior_subset",
"is_open_interior",
"mem_nhds_left",
"mem_nhds_right",
"monotone_id",
"nhds_prod_eq",
"uniform_continuous",
"uniformly_extend_spec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformly_extend_of_ind (b : β) : ψ (e b) = f b | dense_inducing.extend_eq_at _ h_f.continuous.continuous_at | lemma | uniformly_extend_of_ind | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense_inducing.extend_eq_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformly_extend_unique {g : α → γ} (hg : ∀ b, g (e b) = f b)
(hc : continuous g) :
ψ = g | dense_inducing.extend_unique _ hg hc | lemma | uniformly_extend_unique | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"continuous",
"dense_inducing.extend_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pretrivialization.is_linear [add_comm_monoid F] [module R F]
[∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] (e : pretrivialization F (π F E)) :
Prop | (linear : ∀ b ∈ e.base_set, is_linear_map R (λ x : E b, (e ⟨b, x⟩).2)) | class | pretrivialization.is_linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"is_linear_map",
"module",
"pretrivialization"
] | A mixin class for `pretrivialization`, stating that a pretrivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear [add_comm_monoid F] [module R F] [∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)]
[e.is_linear R] {b : B} (hb : b ∈ e.base_set) :
is_linear_map R (λ x : E b, (e ⟨b, x⟩).2) | pretrivialization.is_linear.linear b hb | lemma | pretrivialization.linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"is_linear_map",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmₗ (e : pretrivialization F (π F E)) [e.is_linear R] (b : B) :
F →ₗ[R] E b | begin
refine is_linear_map.mk' (e.symm b) _,
by_cases hb : b ∈ e.base_set,
{ exact (((e.linear R hb).mk' _).inverse (e.symm b) (e.symm_apply_apply_mk hb)
(λ v, congr_arg prod.snd $ e.apply_mk_symm hb v)).is_linear },
{ rw [e.coe_symm_of_not_mem hb], exact (0 : F →ₗ[R] E b).is_linear }
end | def | pretrivialization.symmₗ | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"is_linear_map.mk'",
"mk'",
"pretrivialization"
] | A fiberwise linear inverse to `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_at (e : pretrivialization F (π F E)) [e.is_linear R]
(b : B) (hb : b ∈ e.base_set) :
E b ≃ₗ[R] F | { to_fun := λ y, (e ⟨b, y⟩).2,
inv_fun := e.symm b,
left_inv := e.symm_apply_apply_mk hb,
right_inv := λ v, by simp_rw [e.apply_mk_symm hb v],
map_add' := λ v w, (e.linear R hb).map_add v w,
map_smul' := λ c v, (e.linear R hb).map_smul c v } | def | pretrivialization.linear_equiv_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"inv_fun",
"pretrivialization"
] | A pretrivialization for a vector bundle defines linear equivalences between the
fibers and the model space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_at (e : pretrivialization F (π F E)) [e.is_linear R] (b : B) :
E b →ₗ[R] F | if hb : b ∈ e.base_set then e.linear_equiv_at R b hb else 0 | def | pretrivialization.linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | A fiberwise linear map equal to `e` on `e.base_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_map_at (e : pretrivialization F (π F E)) [e.is_linear R] (b : B) :
⇑(e.linear_map_at R b) = λ y, if b ∈ e.base_set then (e ⟨b, y⟩).2 else 0 | by { rw [pretrivialization.linear_map_at], split_ifs; refl } | lemma | pretrivialization.coe_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"pretrivialization.linear_map_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_at_of_mem (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
⇑(e.linear_map_at R b) = λ y, (e ⟨b, y⟩).2 | by simp_rw [coe_linear_map_at, if_pos hb] | lemma | pretrivialization.coe_linear_map_at_of_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_apply (e : pretrivialization F (π F E)) [e.is_linear R] {b : B} (y : E b) :
e.linear_map_at R b y = if b ∈ e.base_set then (e ⟨b, y⟩).2 else 0 | by rw [coe_linear_map_at] | lemma | pretrivialization.linear_map_at_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_def_of_mem (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
e.linear_map_at R b = e.linear_equiv_at R b hb | dif_pos hb | lemma | pretrivialization.linear_map_at_def_of_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_def_of_not_mem (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∉ e.base_set) :
e.linear_map_at R b = 0 | dif_neg hb | lemma | pretrivialization.linear_map_at_def_of_not_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_eq_zero (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∉ e.base_set) :
e.linear_map_at R b = 0 | dif_neg hb | lemma | pretrivialization.linear_map_at_eq_zero | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmₗ_linear_map_at (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : E b) :
e.symmₗ R b (e.linear_map_at R b y) = y | by { rw [e.linear_map_at_def_of_mem hb], exact (e.linear_equiv_at R b hb).left_inv y } | lemma | pretrivialization.symmₗ_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_symmₗ (e : pretrivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : F) :
e.linear_map_at R b (e.symmₗ R b y) = y | by { rw [e.linear_map_at_def_of_mem hb], exact (e.linear_equiv_at R b hb).right_inv y } | lemma | pretrivialization.linear_map_at_symmₗ | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.is_linear [add_comm_monoid F] [module R F]
[∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] (e : trivialization F (π F E)) : Prop | (linear : ∀ b ∈ e.base_set, is_linear_map R (λ x : E b, (e ⟨b, x⟩).2)) | class | trivialization.is_linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"is_linear_map",
"module",
"trivialization"
] | A mixin class for `trivialization`, stating that a trivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear [add_comm_monoid F] [module R F] [∀ x, add_comm_monoid (E x)]
[∀ x, module R (E x)] [e.is_linear R] {b : B} (hb : b ∈ e.base_set) :
is_linear_map R (λ y : E b, (e ⟨b, y⟩).2) | trivialization.is_linear.linear b hb | lemma | trivialization.linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"is_linear_map",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pretrivialization.is_linear [add_comm_monoid F] [module R F]
[∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] [e.is_linear R] :
e.to_pretrivialization.is_linear R | { ..(‹_› : e.is_linear R) } | instance | trivialization.to_pretrivialization.is_linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv_at (e : trivialization F (π F E)) [e.is_linear R] (b : B) (hb : b ∈ e.base_set) :
E b ≃ₗ[R] F | e.to_pretrivialization.linear_equiv_at R b hb | def | trivialization.linear_equiv_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | A trivialization for a vector bundle defines linear equivalences between the
fibers and the model space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_at_apply (e : trivialization F (π F E)) [e.is_linear R] (b : B)
(hb : b ∈ e.base_set) (v : E b) :
e.linear_equiv_at R b hb v = (e ⟨b, v⟩).2 | rfl | lemma | trivialization.linear_equiv_at_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv_at_symm_apply (e : trivialization F (π F E)) [e.is_linear R] (b : B)
(hb : b ∈ e.base_set) (v : F) :
(e.linear_equiv_at R b hb).symm v = e.symm b v | rfl | lemma | trivialization.linear_equiv_at_symm_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmₗ (e : trivialization F (π F E)) [e.is_linear R] (b : B) : F →ₗ[R] E b | e.to_pretrivialization.symmₗ R b | def | trivialization.symmₗ | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | A fiberwise linear inverse to `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_symmₗ (e : trivialization F (π F E)) [e.is_linear R] (b : B) :
⇑(e.symmₗ R b) = e.symm b | rfl | lemma | trivialization.coe_symmₗ | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at (e : trivialization F (π F E)) [e.is_linear R] (b : B) : E b →ₗ[R] F | e.to_pretrivialization.linear_map_at R b | def | trivialization.linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | A fiberwise linear map equal to `e` on `e.base_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_map_at (e : trivialization F (π F E)) [e.is_linear R] (b : B) :
⇑(e.linear_map_at R b) = λ y, if b ∈ e.base_set then (e ⟨b, y⟩).2 else 0 | e.to_pretrivialization.coe_linear_map_at b | lemma | trivialization.coe_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_at_of_mem (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
⇑(e.linear_map_at R b) = λ y, (e ⟨b, y⟩).2 | by simp_rw [coe_linear_map_at, if_pos hb] | lemma | trivialization.coe_linear_map_at_of_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_apply (e : trivialization F (π F E)) [e.is_linear R] {b : B} (y : E b) :
e.linear_map_at R b y = if b ∈ e.base_set then (e ⟨b, y⟩).2 else 0 | by rw [coe_linear_map_at] | lemma | trivialization.linear_map_at_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_def_of_mem (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
e.linear_map_at R b = e.linear_equiv_at R b hb | dif_pos hb | lemma | trivialization.linear_map_at_def_of_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_def_of_not_mem (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∉ e.base_set) :
e.linear_map_at R b = 0 | dif_neg hb | lemma | trivialization.linear_map_at_def_of_not_mem | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmₗ_linear_map_at (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : E b) :
e.symmₗ R b (e.linear_map_at R b y) = y | e.to_pretrivialization.symmₗ_linear_map_at hb y | lemma | trivialization.symmₗ_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_at_symmₗ (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : F) :
e.linear_map_at R b (e.symmₗ R b y) = y | e.to_pretrivialization.linear_map_at_symmₗ hb y | lemma | trivialization.linear_map_at_symmₗ | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_changeL (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] (b : B) :
F ≃L[R] F | { continuous_to_fun := begin
by_cases hb : b ∈ e.base_set ∩ e'.base_set,
{ simp_rw [dif_pos hb],
refine (e'.continuous_on.comp_continuous _ _).snd,
exact e.continuous_on_symm.comp_continuous (continuous.prod.mk b)
(λ y, mk_mem_prod hb.1 (mem_univ y)),
exact (λ y, e'.mem_source.mpr hb.2... | def | trivialization.coord_changeL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous.prod.mk",
"continuous_id",
"linear_equiv.refl",
"trivialization"
] | A coordinate change function between two trivializations, as a continuous linear equivalence.
Defined to be the identity when `b` does not lie in the base set of both trivializations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coord_changeL (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] {b : B}
(hb : b ∈ e.base_set ∩ e'.base_set) :
⇑(coord_changeL R e e' b)
= (e.linear_equiv_at R b hb.1).symm.trans (e'.linear_equiv_at R b hb.2) | congr_arg linear_equiv.to_fun (dif_pos hb) | lemma | trivialization.coe_coord_changeL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coord_changeL' (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] {b : B}
(hb : b ∈ e.base_set ∩ e'.base_set) :
(coord_changeL R e e' b).to_linear_equiv
= (e.linear_equiv_at R b hb.1).symm.trans (e'.linear_equiv_at R b hb.2) | linear_equiv.coe_injective (coe_coord_changeL _ _ _) | lemma | trivialization.coe_coord_changeL' | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"linear_equiv.coe_injective",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_coord_changeL (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] {b : B}
(hb : b ∈ e'.base_set ∩ e.base_set) :
(e.coord_changeL R e' b).symm = e'.coord_changeL R e b | begin
apply continuous_linear_equiv.to_linear_equiv_injective,
rw [coe_coord_changeL' e' e hb, (coord_changeL R e e' b).symm_to_linear_equiv,
coe_coord_changeL' e e' hb.symm, linear_equiv.trans_symm, linear_equiv.symm_symm],
end | lemma | trivialization.symm_coord_changeL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_linear_equiv.to_linear_equiv_injective",
"linear_equiv.symm_symm",
"linear_equiv.trans_symm",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_changeL_apply (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] {b : B}
(hb : b ∈ e.base_set ∩ e'.base_set) (y : F) :
coord_changeL R e e' b y = (e' ⟨b, e.symm b y⟩).2 | congr_arg (λ f, linear_equiv.to_fun f y) (dif_pos hb) | lemma | trivialization.coord_changeL_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coord_changeL (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R] {b : B}
(hb : b ∈ e.base_set ∩ e'.base_set) (y : F) :
(b, coord_changeL R e e' b y) = e' ⟨b, e.symm b y⟩ | begin
ext,
{ rw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1],
rw [e.proj_symm_apply' hb.1], exact hb.2 },
{ exact e.coord_changeL_apply e' hb y }
end | lemma | trivialization.mk_coord_changeL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply_eq_coord_changeL (e e' : trivialization F (π F E)) [e.is_linear R]
[e'.is_linear R] {b : B} (hb : b ∈ e.base_set ∩ e'.base_set) (v : F) :
e' (e.to_local_homeomorph.symm (b, v)) = (b, e.coord_changeL R e' b v) | by rw [e.mk_coord_changeL e' hb, e.mk_symm hb.1] | lemma | trivialization.apply_symm_apply_eq_coord_changeL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_changeL_apply' (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R]
{b : B} (hb : b ∈ e.base_set ∩ e'.base_set) (y : F) :
coord_changeL R e e' b y = (e' (e.to_local_homeomorph.symm (b, y))).2 | by rw [e.coord_changeL_apply e' hb, e.mk_symm hb.1] | lemma | trivialization.coord_changeL_apply' | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | A version of `coord_change_apply` that fully unfolds `coord_change`. The right-hand side is
ugly, but has good definitional properties for specifically defined trivializations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coord_changeL_symm_apply (e e' : trivialization F (π F E)) [e.is_linear R] [e'.is_linear R]
{b : B} (hb : b ∈ e.base_set ∩ e'.base_set) :
⇑(coord_changeL R e e' b).symm
= (e'.linear_equiv_at R b hb.2).symm.trans (e.linear_equiv_at R b hb.1) | congr_arg linear_equiv.inv_fun (dif_pos hb) | lemma | trivialization.coord_changeL_symm_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_section [∀ x, has_zero (E x)] : B → total_space F E | λ x, ⟨x, 0⟩ | def | bundle.zero_section | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | The zero section of a vector bundle | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_section_proj [∀ x, has_zero (E x)] (x : B) : (zero_section F E x).proj = x | rfl | lemma | bundle.zero_section_proj | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_section_snd [∀ x, has_zero (E x)] (x : B) : (zero_section F E x).2 = 0 | rfl | lemma | bundle.zero_section_snd | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle : Prop | (trivialization_linear' : ∀ (e : trivialization F (π F E)) [mem_trivialization_atlas e],
e.is_linear R)
(continuous_on_coord_change' [] : ∀ (e e' : trivialization F (π F E)) [mem_trivialization_atlas e]
[mem_trivialization_atlas e'],
continuous_on
(λ b, by exactI trivialization.coord_changeL R e e' b : B → F →L... | class | vector_bundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on",
"mem_trivialization_atlas",
"trivialization",
"trivialization.coord_changeL"
] | The space `total_space F E` (for `E : B → Type*` such that each `E x` is a topological vector
space) has a topological vector space structure with fiber `F` (denoted with
`vector_bundle R F E`) if around every point there is a fiber bundle trivialization
which is linear in the fibers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization_linear [vector_bundle R F E] (e : trivialization F (π F E))
[mem_trivialization_atlas e] :
e.is_linear R | vector_bundle.trivialization_linear' e | instance | trivialization_linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"mem_trivialization_atlas",
"trivialization",
"vector_bundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_coord_change [vector_bundle R F E] (e e' : trivialization F (π F E))
[he : mem_trivialization_atlas e]
[he' : mem_trivialization_atlas e'] :
continuous_on
(λ b, trivialization.coord_changeL R e e' b : B → F →L[R] F) (e.base_set ∩ e'.base_set) | vector_bundle.continuous_on_coord_change' R e e' | lemma | continuous_on_coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on",
"mem_trivialization_atlas",
"trivialization",
"trivialization.coord_changeL",
"vector_bundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_at (e : trivialization F (π F E)) [e.is_linear R] (b : B) :
E b →L[R] F | { to_fun := e.linear_map_at R b, -- given explicitly to help `simps`
cont := begin
dsimp,
rw [e.coe_linear_map_at b],
refine continuous_if_const _ (λ hb, _) (λ _, continuous_zero),
exact continuous_snd.comp (e.continuous_on.comp_continuous
(fiber_bundle.total_space_mk_inducing F E b).continuous
... | def | trivialization.continuous_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"cont",
"continuous",
"continuous_if_const",
"trivialization"
] | Forward map of `continuous_linear_equiv_at` (only propositionally equal),
defined everywhere (`0` outside domain). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmL (e : trivialization F (π F E)) [e.is_linear R] (b : B) : F →L[R] E b | { to_fun := e.symm b, -- given explicitly to help `simps`
cont := begin
by_cases hb : b ∈ e.base_set,
{ rw (fiber_bundle.total_space_mk_inducing F E b).continuous_iff,
exact e.continuous_on_symm.comp_continuous (continuous_const.prod_mk continuous_id)
(λ x, mk_mem_prod hb (mem_univ x)) },
{ ... | def | trivialization.symmL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"cont",
"continuous_id",
"trivialization"
] | Backwards map of `continuous_linear_equiv_at`, defined everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmL_continuous_linear_map_at (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : E b) :
e.symmL R b (e.continuous_linear_map_at R b y) = y | e.symmₗ_linear_map_at hb y | lemma | trivialization.symmL_continuous_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_at_symmL (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) (y : F) :
e.continuous_linear_map_at R b (e.symmL R b y) = y | e.linear_map_at_symmₗ hb y | lemma | trivialization.continuous_linear_map_at_symmL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv_at (e : trivialization F (π F E)) [e.is_linear R] (b : B)
(hb : b ∈ e.base_set) : E b ≃L[R] F | { to_fun := λ y, (e ⟨b, y⟩).2, -- given explicitly to help `simps`
inv_fun := e.symm b, -- given explicitly to help `simps`
continuous_to_fun := continuous_snd.comp (e.continuous_on.comp_continuous
(fiber_bundle.total_space_mk_inducing F E b).continuous
(λ x, e.mem_source.mpr hb)),
continuous_inv_fun := (... | def | trivialization.continuous_linear_equiv_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous",
"inv_fun",
"trivialization"
] | In a vector bundle, a trivialization in the fiber (which is a priori only linear)
is in fact a continuous linear equiv between the fibers and the model fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_continuous_linear_equiv_at_eq (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
(e.continuous_linear_equiv_at R b hb : E b → F) = e.continuous_linear_map_at R b | (e.coe_linear_map_at_of_mem hb).symm | lemma | trivialization.coe_continuous_linear_equiv_at_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_continuous_linear_equiv_at_eq (e : trivialization F (π F E)) [e.is_linear R] {b : B}
(hb : b ∈ e.base_set) :
((e.continuous_linear_equiv_at R b hb).symm : F → E b) = e.symmL R b | rfl | lemma | trivialization.symm_continuous_linear_equiv_at_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv_at_apply' (e : trivialization F (π F E)) [e.is_linear R]
(x : total_space F E) (hx : x ∈ e.source) :
e.continuous_linear_equiv_at R x.proj (e.mem_source.1 hx) x.2 = (e x).2 | by { cases x, refl } | lemma | trivialization.continuous_linear_equiv_at_apply' | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_prod_continuous_linear_equiv_at (e : trivialization F (π F E)) [e.is_linear R]
(b : B) (hb : b ∈ e.base_set) (z : E b) :
e ⟨b, z⟩ = (b, e.continuous_linear_equiv_at R b hb z) | begin
ext,
{ refine e.coe_fst _,
rw e.source_eq,
exact hb },
{ simp only [coe_coe, continuous_linear_equiv_at_apply] }
end | lemma | trivialization.apply_eq_prod_continuous_linear_equiv_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"coe_coe",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_section (e : trivialization F (π F E)) [e.is_linear R]
{x : B} (hx : x ∈ e.base_set) : e (zero_section F E x) = (x, 0) | by simp_rw [zero_section, e.apply_eq_prod_continuous_linear_equiv_at R x hx 0,
map_zero] | lemma | trivialization.zero_section | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_eq_mk_continuous_linear_equiv_at_symm (e : trivialization F (π F E))
[e.is_linear R] (b : B) (hb : b ∈ e.base_set) (z : F) :
e.to_local_homeomorph.symm ⟨b, z⟩
= ⟨b, (e.continuous_linear_equiv_at R b hb).symm z⟩ | begin
have h : (b, z) ∈ e.target,
{ rw e.target_eq,
exact ⟨hb, mem_univ _⟩ },
apply e.inj_on (e.map_target h),
{ simp only [e.source_eq, hb, mem_preimage] },
simp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuous_linear_equiv_at R b hb,
continuous_linear_equiv.apply_symm_apply],
end | lemma | trivialization.symm_apply_eq_mk_continuous_linear_equiv_at_symm | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"coe_coe",
"continuous_linear_equiv.apply_symm_apply",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_linear_equiv_at_eq_coord_change (e e' : trivialization F (π F E))
[e.is_linear R] [e'.is_linear R] {b : B} (hb : b ∈ e.base_set ∩ e'.base_set) :
(e.continuous_linear_equiv_at R b hb.1).symm.trans (e'.continuous_linear_equiv_at R b hb.2)
= coord_changeL R e e' b | by { ext v, rw [coord_changeL_apply e e' hb], refl } | lemma | trivialization.comp_continuous_linear_equiv_at_eq_coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle_core (ι : Type*) | (base_set : ι → set B)
(is_open_base_set : ∀ i, is_open (base_set i))
(index_at : B → ι)
(mem_base_set_at : ∀ x, x ∈ base_set (index_at x))
(coord_change : ι → ι → B → (F →L[R] F))
(coord_change_self : ∀ i, ∀ x ∈ base_set i, ∀ v, coord_change i i x v = v)
(continuous_on_coord_change : ∀ i j, c... | structure | vector_bundle_core | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on",
"continuous_on_coord_change",
"is_open"
] | Analogous construction of `fiber_bundle_core` for vector bundles. This
construction gives a way to construct vector bundles from a structure registering how
trivialization changes act on fibers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial_vector_bundle_core (ι : Type*) [inhabited ι] :
vector_bundle_core R B F ι | { base_set := λ ι, univ,
is_open_base_set := λ i, is_open_univ,
index_at := default,
mem_base_set_at := λ x, mem_univ x,
coord_change := λ i j x, continuous_linear_map.id R F,
coord_change_self := λ i x hx v, rfl,
coord_change_comp := λ i j k x hx v, rfl,
continuous_on_coord_change := λ i j, continuous_on... | def | trivial_vector_bundle_core | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_linear_map.id",
"continuous_on_const",
"continuous_on_coord_change",
"is_open_univ",
"vector_bundle_core"
] | The trivial vector bundle core, in which all the changes of coordinates are the
identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fiber_bundle_core : fiber_bundle_core ι B F | { coord_change := λ i j b, Z.coord_change i j b,
continuous_on_coord_change := λ i j, is_bounded_bilinear_map_apply.continuous.comp_continuous_on
((Z.continuous_on_coord_change i j).prod_map continuous_on_id),
..Z } | def | vector_bundle_core.to_fiber_bundle_core | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on_coord_change",
"continuous_on_id",
"fiber_bundle_core",
"prod_map"
] | Natural identification to a `fiber_bundle_core`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fiber_bundle_core_coe : has_coe (vector_bundle_core R B F ι)
(fiber_bundle_core ι B F) | ⟨to_fiber_bundle_core⟩ | instance | vector_bundle_core.to_fiber_bundle_core_coe | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core",
"vector_bundle_core"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_linear_comp (i j k : ι): ∀ x ∈ (Z.base_set i) ∩ (Z.base_set j) ∩ (Z.base_set k),
(Z.coord_change j k x).comp (Z.coord_change i j x) = Z.coord_change i k x | λ x hx, by { ext v, exact Z.coord_change_comp i j k x hx v } | lemma | vector_bundle_core.coord_change_linear_comp | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber : B → Type* | Z.to_fiber_bundle_core.fiber | def | vector_bundle_core.fiber | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | The fiber of a vector bundle core, as a convenience function for dot notation and
typeclass inference | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space_fiber (x : B) : topological_space (Z.fiber x) | by delta_instance vector_bundle_core.fiber | instance | vector_bundle_core.topological_space_fiber | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"topological_space",
"vector_bundle_core.fiber"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_monoid_fiber : ∀ (x : B), add_comm_monoid (Z.fiber x) | by dsimp [vector_bundle_core.fiber]; delta_instance fiber_bundle_core.fiber | instance | vector_bundle_core.add_comm_monoid_fiber | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_monoid",
"fiber_bundle_core.fiber",
"vector_bundle_core.fiber"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_fiber : ∀ (x : B), module R (Z.fiber x) | by dsimp [vector_bundle_core.fiber]; delta_instance fiber_bundle_core.fiber | instance | vector_bundle_core.module_fiber | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.fiber",
"module",
"vector_bundle_core.fiber"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group_fiber [add_comm_group F] : ∀ (x : B), add_comm_group (Z.fiber x) | by dsimp [vector_bundle_core.fiber]; delta_instance fiber_bundle_core.fiber | instance | vector_bundle_core.add_comm_group_fiber | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"add_comm_group",
"fiber_bundle_core.fiber",
"vector_bundle_core.fiber"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj : total_space F Z.fiber → B | total_space.proj | def | vector_bundle_core.proj | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | The projection from the total space of a fiber bundle core, on its base. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
triv_change (i j : ι) : local_homeomorph (B × F) (B × F) | fiber_bundle_core.triv_change ↑Z i j | def | vector_bundle_core.triv_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.triv_change",
"local_homeomorph"
] | Local homeomorphism version of the trivialization change. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_triv_change_source (i j : ι) (p : B × F) :
p ∈ (Z.triv_change i j).source ↔ p.1 ∈ Z.base_set i ∩ Z.base_set j | fiber_bundle_core.mem_triv_change_source ↑Z i j p | lemma | vector_bundle_core.mem_triv_change_source | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.mem_triv_change_source"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space : topological_space Z.total_space | Z.to_fiber_bundle_core.to_topological_space | instance | vector_bundle_core.to_topological_space | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"topological_space"
] | Topological structure on the total space of a vector bundle created from core, designed so
that all the local trivialization are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coord_change (i j : ι) :
Z.to_fiber_bundle_core.coord_change i j b = Z.coord_change i j b | rfl | lemma | vector_bundle_core.coe_coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv (i : ι) : trivialization F (π F Z.fiber) | by dsimp [vector_bundle_core.total_space, vector_bundle_core.fiber];
exact Z.to_fiber_bundle_core.local_triv i | def | vector_bundle_core.local_triv | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization",
"vector_bundle_core.fiber",
"vector_bundle_core.total_space"
] | One of the standard local trivializations of a vector bundle constructed from core, taken by
considering this in particular as a fiber bundle constructed from core. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_triv.is_linear (i : ι) : (Z.local_triv i).is_linear R | { linear := λ x hx, by dsimp [vector_bundle_core.local_triv]; exact
{ map_add := λ v w, by simp only [continuous_linear_map.map_add] with mfld_simps,
map_smul := λ r v, by simp only [continuous_linear_map.map_smul] with mfld_simps} } | instance | vector_bundle_core.local_triv.is_linear | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_linear_map.map_add",
"continuous_linear_map.map_smul",
"vector_bundle_core.local_triv"
] | The standard local trivializations of a vector bundle constructed from core are linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_local_triv_source (p : Z.total_space) :
p ∈ (Z.local_triv i).source ↔ p.1 ∈ Z.base_set i | by dsimp [vector_bundle_core.fiber]; exact iff.rfl | lemma | vector_bundle_core.mem_local_triv_source | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"vector_bundle_core.fiber"
] | 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 | Z.to_fiber_bundle_core.mem_local_triv_target i p | lemma | vector_bundle_core.mem_local_triv_target | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_symm_fst (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 | vector_bundle_core.local_triv_symm_fst | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_symm_apply {b : B} (hb : b ∈ Z.base_set i) (v : F) :
(Z.local_triv i).symm b v = Z.coord_change i (Z.index_at b) b v | by apply (Z.local_triv i).symm_apply hb v | lemma | vector_bundle_core.local_triv_symm_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_coord_change_eq {b : B} (hb : b ∈ Z.base_set i ∩ Z.base_set j)
(v : F) :
(Z.local_triv i).coord_changeL R (Z.local_triv j) b v = Z.coord_change i j b v | begin
rw [trivialization.coord_changeL_apply', local_triv_symm_fst, local_triv_apply,
coord_change_comp],
exacts [⟨⟨hb.1, Z.mem_base_set_at b⟩, hb.2⟩, hb]
end | lemma | vector_bundle_core.local_triv_coord_change_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"trivialization.coord_changeL_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_at_def :
Z.local_triv (Z.index_at b) = Z.local_triv_at b | rfl | lemma | vector_bundle_core.local_triv_at_def | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | 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⟩ | fiber_bundle_core.local_triv_at_apply Z p | lemma | vector_bundle_core.local_triv_at_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.local_triv_at_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_at_base_set :
b ∈ (Z.local_triv_at b).base_set | fiber_bundle_core.mem_local_triv_at_base_set Z b | lemma | vector_bundle_core.mem_local_triv_at_base_set | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.mem_local_triv_at_base_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_bundle : fiber_bundle F Z.fiber | Z.to_fiber_bundle_core.fiber_bundle | instance | vector_bundle_core.fiber_bundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle : vector_bundle R F Z.fiber | { trivialization_linear' := begin
rintros _ ⟨i, rfl⟩,
apply local_triv.is_linear,
end,
continuous_on_coord_change' := begin
rintros _ _ ⟨i, rfl⟩ ⟨i', rfl⟩,
refine (Z.continuous_on_coord_change i i').congr (λ b hb, _),
ext v,
exact Z.local_triv_coord_change_eq i i' hb v,
end } | instance | vector_bundle_core.vector_bundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"vector_bundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_proj : continuous Z.proj | fiber_bundle_core.continuous_proj Z | lemma | vector_bundle_core.continuous_proj | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous",
"fiber_bundle_core.continuous_proj"
] | The projection on the base of a vector bundle created from core is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_proj : is_open_map Z.proj | fiber_bundle_core.is_open_map_proj Z | lemma | vector_bundle_core.is_open_map_proj | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle_core.is_open_map_proj",
"is_open_map"
] | The projection on the base of a vector bundle created from core is an open map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_triv_continuous_linear_map_at {b : B} (hb : b ∈ Z.base_set i) :
(Z.local_triv i).continuous_linear_map_at R b = Z.coord_change (Z.index_at b) i b | begin
ext1 v,
rw [(Z.local_triv i).continuous_linear_map_at_apply R, (Z.local_triv i).coe_linear_map_at_of_mem],
exacts [rfl, hb]
end | lemma | vector_bundle_core.local_triv_continuous_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_at_continuous_linear_map_at {b₀ b : B}
(hb : b ∈ (trivialization_at F Z.fiber b₀).base_set) :
(trivialization_at F Z.fiber b₀).continuous_linear_map_at R b =
Z.coord_change (Z.index_at b) (Z.index_at b₀) b | Z.local_triv_continuous_linear_map_at hb | lemma | vector_bundle_core.trivialization_at_continuous_linear_map_at | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_symmL {b : B} (hb : b ∈ Z.base_set i) :
(Z.local_triv i).symmL R b = Z.coord_change i (Z.index_at b) b | by { ext1 v, rw [(Z.local_triv i).symmL_apply R, (Z.local_triv i).symm_apply], exacts [rfl, hb] } | lemma | vector_bundle_core.local_triv_symmL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_at_symmL {b₀ b : B}
(hb : b ∈ (trivialization_at F Z.fiber b₀).base_set) :
(trivialization_at F Z.fiber b₀).symmL R b = Z.coord_change (Z.index_at b₀) (Z.index_at b) b | Z.local_triv_symmL hb | lemma | vector_bundle_core.trivialization_at_symmL | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_at_coord_change_eq {b₀ b₁ b : B}
(hb : b ∈ (trivialization_at F Z.fiber b₀).base_set ∩ (trivialization_at F Z.fiber b₁).base_set)
(v : F) :
(trivialization_at F Z.fiber b₀).coord_changeL R (trivialization_at F Z.fiber b₁) b v =
Z.coord_change (Z.index_at b₀) (Z.index_at b₁) b v | Z.local_triv_coord_change_eq _ _ hb v | lemma | vector_bundle_core.trivialization_at_coord_change_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_prebundle | (pretrivialization_atlas : set (pretrivialization F (π F E)))
(pretrivialization_linear' : ∀ (e : pretrivialization F (π F E)) (he : e ∈ pretrivialization_atlas),
e.is_linear R)
(pretrivialization_at : B → pretrivialization F (π F E))
(mem_base_pretrivialization_at : ∀ x : B, x ∈ (pretrivialization_at x).base_set)
(p... | structure | vector_prebundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on",
"inducing",
"pretrivialization"
] | This structure permits to define a vector bundle when trivializations are given as local
equivalences but there is not yet a topology on the total space or the fibers.
The total space is hence given a topology in such a way that there is a fiber bundle structure for
which the local equivalences are also local homeomorp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coord_change (a : vector_prebundle R F E)
{e e' : pretrivialization F (π F E)} (he : e ∈ a.pretrivialization_atlas)
(he' : e' ∈ a.pretrivialization_atlas) (b : B) : F →L[R] F | classical.some (a.exists_coord_change e he e' he') b | def | vector_prebundle.coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"vector_prebundle"
] | A randomly chosen coordinate change on a `vector_prebundle`, given by
the field `exists_coord_change`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_coord_change (a : vector_prebundle R F E)
{e e' : pretrivialization F (π F E)} (he : e ∈ a.pretrivialization_atlas)
(he' : e' ∈ a.pretrivialization_atlas) :
continuous_on (a.coord_change he he') (e.base_set ∩ e'.base_set) | (classical.some_spec (a.exists_coord_change e he e' he')).1 | lemma | vector_prebundle.continuous_on_coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_on",
"continuous_on_coord_change",
"pretrivialization",
"vector_prebundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_apply (a : vector_prebundle R F E)
{e e' : pretrivialization F (π F E)} (he : e ∈ a.pretrivialization_atlas)
(he' : e' ∈ a.pretrivialization_atlas) {b : B} (hb : b ∈ e.base_set ∩ e'.base_set) (v : F) :
a.coord_change he he' b v = (e' ⟨b, e.symm b v⟩).2 | (classical.some_spec (a.exists_coord_change e he e' he')).2 b hb v | lemma | vector_prebundle.coord_change_apply | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"vector_prebundle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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