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 | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
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mk_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} (hb : b ∈ e.base_set ∩ e'.base_set) (v : F) :
(b, a.coord_change he he' b v) = e' ⟨b, e.symm b v⟩ | begin
ext,
{ rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1],
rw [e.proj_symm_apply' hb.1], exact hb.2 },
{ exact a.coord_change_apply he he' hb v }
end | lemma | vector_prebundle.mk_coord_change | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"vector_prebundle"
] | 789 | 798 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fiber_prebundle (a : vector_prebundle R F E) :
fiber_prebundle F E | { continuous_triv_change := begin
intros e he e' he',
have := is_bounded_bilinear_map_apply.continuous.comp_continuous_on
((a.continuous_on_coord_change he' he).prod_map continuous_on_id),
have H : e'.to_local_equiv.target ∩ e'.to_local_equiv.symm ⁻¹'
e.to_local_equiv.source =(e'.base_set ∩ e.ba... | def | vector_prebundle.to_fiber_prebundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"and.congr_right_iff",
"continuous_on_id",
"fiber_prebundle",
"prod_map",
"vector_prebundle"
] | Natural identification of `vector_prebundle` as a `fiber_prebundle`. | 801 | 820 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total_space_topology (a : vector_prebundle R F E) :
topological_space (total_space F E) | a.to_fiber_prebundle.total_space_topology | def | vector_prebundle.total_space_topology | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"topological_space",
"vector_prebundle"
] | Topology on the total space that will make the prebundle into a bundle. | 823 | 825 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization_of_mem_pretrivialization_atlas (a : vector_prebundle R F E)
{e : pretrivialization F (π F E)} (he : e ∈ a.pretrivialization_atlas) :
@trivialization B F _ _ _ a.total_space_topology (π F E) | a.to_fiber_prebundle.trivialization_of_mem_pretrivialization_atlas he | def | vector_prebundle.trivialization_of_mem_pretrivialization_atlas | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"trivialization",
"vector_prebundle"
] | Promotion from a `trivialization` in the `pretrivialization_atlas` of a
`vector_prebundle` to a `trivialization`. | 829 | 832 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_of_mem_pretrivialization_atlas (a : vector_prebundle R F E)
{e : pretrivialization F (π F E)} (he : e ∈ a.pretrivialization_atlas) :
@trivialization.is_linear R B F _ _ _ _ a.total_space_topology _ _ _ _
(trivialization_of_mem_pretrivialization_atlas a he) | { linear := (a.pretrivialization_linear' e he).linear } | lemma | vector_prebundle.linear_of_mem_pretrivialization_atlas | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"pretrivialization",
"trivialization.is_linear",
"vector_prebundle"
] | 834 | 838 | true | 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 | a.to_fiber_prebundle.mem_trivialization_at_source b x | lemma | vector_prebundle.mem_trivialization_at_source | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | 842 | 844 | true | 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 | a.to_fiber_prebundle.total_space_mk_preimage_source b | lemma | vector_prebundle.total_space_mk_preimage_source | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | 846 | 848 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_total_space_mk (b : B) :
@continuous _ _ _ a.total_space_topology (total_space.mk b) | a.to_fiber_prebundle.continuous_total_space_mk b | lemma | vector_prebundle.continuous_total_space_mk | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous"
] | 850 | 852 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fiber_bundle : @fiber_bundle B F _ _ _ a.total_space_topology _ | a.to_fiber_prebundle.to_fiber_bundle | def | vector_prebundle.to_fiber_bundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"fiber_bundle"
] | Make a `fiber_bundle` from a `vector_prebundle`; auxiliary construction for
`vector_prebundle.vector_bundle`. | 856 | 857 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_vector_bundle :
@vector_bundle R _ F E _ _ _ _ _ _ a.total_space_topology _ a.to_fiber_bundle | { trivialization_linear' := begin
rintros _ ⟨e, he, rfl⟩,
apply linear_of_mem_pretrivialization_atlas,
end,
continuous_on_coord_change' := begin
rintros _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩,
refine (a.continuous_on_coord_change he he').congr _,
intros b hb,
ext v,
rw [a.coord_change_apply he ... | lemma | vector_prebundle.to_vector_bundle | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_linear_equiv.coe_coe",
"trivialization.coord_changeL_apply",
"vector_bundle"
] | Make a `vector_bundle` from a `vector_prebundle`. Concretely this means
that, given a `vector_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... | 865 | 879 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
in_coordinates (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y) : F →SL[σ] F' | ((trivialization_at F' E' y₀).continuous_linear_map_at 𝕜₂ y).comp $ ϕ.comp $
(trivialization_at F E x₀).symmL 𝕜₁ x | def | continuous_linear_map.in_coordinates | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [] | When `ϕ` is a continuous (semi)linear map between the fibers `E x` and `E' y` of two vector
bundles `E` and `E'`, `continuous_linear_map.in_coordinates F E F' E' x₀ x y₀ y ϕ` is a coordinate
change of this continuous linear map w.r.t. the chart around `x₀` and the chart around `y₀`.
It is defined by composing `ϕ` with... | 912 | 914 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
in_coordinates_eq (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y)
(hx : x ∈ (trivialization_at F E x₀).base_set)
(hy : y ∈ (trivialization_at F' E' y₀).base_set) :
in_coordinates F E F' E' x₀ x y₀ y ϕ =
((trivialization_at F' E' y₀).continuous_linear_equiv_at 𝕜₂ y hy : E' y →L[𝕜₂] F').comp (ϕ.comp $
(((trivial... | begin
ext,
simp_rw [in_coordinates, continuous_linear_map.coe_comp', continuous_linear_equiv.coe_coe,
trivialization.coe_continuous_linear_equiv_at_eq,
trivialization.symm_continuous_linear_equiv_at_eq]
end | lemma | continuous_linear_map.in_coordinates_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_map.coe_comp'",
"trivialization.coe_continuous_linear_equiv_at_eq",
"trivialization.symm_continuous_linear_equiv_at_eq"
] | rewrite `in_coordinates` using continuous linear equivalences. | 919 | 930 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vector_bundle_core.in_coordinates_eq {ι ι'} (Z : vector_bundle_core 𝕜₁ B F ι)
(Z' : vector_bundle_core 𝕜₂ B' F' ι')
{x₀ x : B} {y₀ y : B'} (ϕ : F →SL[σ] F')
(hx : x ∈ Z.base_set (Z.index_at x₀))
(hy : y ∈ Z'.base_set (Z'.index_at y₀)) :
in_coordinates F Z.fiber F' Z'.fiber x₀ x y₀ y ϕ =
(Z'.coord_chan... | by simp_rw [in_coordinates, Z'.trivialization_at_continuous_linear_map_at hy,
Z.trivialization_at_symmL hx] | lemma | continuous_linear_map.vector_bundle_core.in_coordinates_eq | topology.vector_bundle | src/topology/vector_bundle/basic.lean | [
"analysis.normed_space.bounded_linear_maps",
"topology.fiber_bundle.basic"
] | [
"vector_bundle_core"
] | rewrite `in_coordinates` in a `vector_bundle_core`. | 933 | 942 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization.is_linear : (trivialization B F).is_linear 𝕜 | { linear := λ x hx, ⟨λ y z, rfl, λ c y, rfl⟩ } | instance | bundle.trivial.trivialization.is_linear | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"trivialization",
"trivialization.is_linear"
] | 42 | 43 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.coord_changeL (b : B) :
(trivialization B F).coord_changeL 𝕜 (trivialization B F) b = continuous_linear_equiv.refl 𝕜 F | begin
ext v,
rw [trivialization.coord_changeL_apply'],
exacts [rfl, ⟨mem_univ _, mem_univ _⟩]
end | lemma | bundle.trivial.trivialization.coord_changeL | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"continuous_linear_equiv.refl",
"trivialization",
"trivialization.coord_changeL",
"trivialization.coord_changeL_apply'"
] | 47 | 53 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle : vector_bundle 𝕜 F (bundle.trivial B F) | { trivialization_linear' := begin
introsI e he,
rw eq_trivialization B F e,
apply_instance
end,
continuous_on_coord_change' := begin
introsI e e' he he',
unfreezingI { obtain rfl := eq_trivialization B F e },
unfreezingI { obtain rfl := eq_trivialization B F e' },
simp_rw trivialization.... | instance | bundle.trivial.vector_bundle | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"bundle.trivial",
"trivialization.coord_changeL",
"vector_bundle"
] | 57 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.is_linear [e₁.is_linear 𝕜] [e₂.is_linear 𝕜] : (e₁.prod e₂).is_linear 𝕜 | { linear := λ x ⟨h₁, h₂⟩, (((e₁.linear 𝕜 h₁).mk' _).prod_map ((e₂.linear 𝕜 h₂).mk' _)).is_linear } | instance | trivialization.prod.is_linear | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"mk'",
"prod_map"
] | 88 | 89 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_changeL_prod [e₁.is_linear 𝕜] [e₁'.is_linear 𝕜] [e₂.is_linear 𝕜] [e₂'.is_linear 𝕜] ⦃b⦄
(hb : b ∈ ((e₁.prod e₂).base_set ∩ (e₁'.prod e₂').base_set)) :
((e₁.prod e₂).coord_changeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coord_changeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prod_map (e₂.coord_changeL 𝕜 e₂' b... | begin
rw [continuous_linear_map.ext_iff, continuous_linear_map.coe_prod_map'],
rintro ⟨v₁, v₂⟩,
show (e₁.prod e₂).coord_changeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coord_changeL 𝕜 e₁' b v₁, e₂.coord_changeL 𝕜 e₂' b v₂),
rw [e₁.coord_changeL_apply e₁', e₂.coord_changeL_apply e₂', (e₁.prod e₂).coord_changeL_... | lemma | trivialization.coord_changeL_prod | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"continuous_linear_map.coe_prod_map'",
"continuous_linear_map.ext_iff",
"prod_map"
] | 91 | 103 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_apply [e₁.is_linear 𝕜] [e₂.is_linear 𝕜] {x : B} (hx₁ : x ∈ e₁.base_set)
(hx₂ : x ∈ e₂.base_set) (v₁ : E₁ x) (v₂ : E₂ x) :
prod e₁ e₂ ⟨x, (v₁, v₂)⟩
= ⟨x, e₁.continuous_linear_equiv_at 𝕜 x hx₁ v₁, e₂.continuous_linear_equiv_at 𝕜 x hx₂ v₂⟩ | rfl | lemma | trivialization.prod_apply | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [] | 108 | 112 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle.prod [vector_bundle 𝕜 F₁ E₁] [vector_bundle 𝕜 F₂ E₂] :
vector_bundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) | { trivialization_linear' := begin
rintros _ ⟨e₁, e₂, he₁, he₂, rfl⟩, resetI,
apply_instance
end,
continuous_on_coord_change' := begin
rintros _ _ ⟨e₁, e₂, he₁, he₂, rfl⟩ ⟨e₁', e₂', he₁', he₂', rfl⟩, resetI,
refine (((continuous_on_coord_change 𝕜 e₁ e₁').mono _).prod_mapL 𝕜
((continuous_on_co... | instance | vector_bundle.prod | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"continuous_linear_map.ext_iff",
"continuous_on_coord_change",
"vector_bundle"
] | The product of two vector bundles is a vector bundle. | 124 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization.continuous_linear_equiv_at_prod {e₁ : trivialization F₁ (π F₁ E₁)}
{e₂ : trivialization F₂ (π F₂ E₂)} [e₁.is_linear 𝕜] [e₂.is_linear 𝕜] {x : B}
(hx₁ : x ∈ e₁.base_set) (hx₂ : x ∈ e₂.base_set) :
(e₁.prod e₂).continuous_linear_equiv_at 𝕜 x ⟨hx₁, hx₂⟩
= (e₁.continuous_linear_equiv_at 𝕜 x hx₁).pr... | begin
ext1,
funext v,
obtain ⟨v₁, v₂⟩ := v,
rw [(e₁.prod e₂).continuous_linear_equiv_at_apply 𝕜, trivialization.prod],
exact (congr_arg prod.snd (prod_apply 𝕜 hx₁ hx₂ v₁ v₂) : _)
end | lemma | trivialization.continuous_linear_equiv_at_prod | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"trivialization",
"trivialization.prod"
] | 149 | 160 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.pullback_linear (e : trivialization F (π F E)) [e.is_linear 𝕜] (f : K) :
(@trivialization.pullback _ _ _ B' _ _ _ _ _ _ _ e f).is_linear 𝕜 | { linear := λ x h, e.linear 𝕜 h } | instance | trivialization.pullback_linear | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"trivialization",
"trivialization.pullback"
] | 180 | 182 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_bundle.pullback [∀ x, topological_space (E x)]
[fiber_bundle F E] [vector_bundle 𝕜 F E] (f : K) : vector_bundle 𝕜 F ((f : B' → B) *ᵖ E) | { trivialization_linear' := begin
rintro _ ⟨e, he, rfl⟩, resetI,
apply_instance,
end,
continuous_on_coord_change' := begin
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩, resetI,
refine ((continuous_on_coord_change 𝕜 e e').comp (map_continuous f).continuous_on
(λ b hb, hb)).congr _,
rintro b (hb ... | instance | vector_bundle.pullback | topology.vector_bundle | src/topology/vector_bundle/constructions.lean | [
"topology.fiber_bundle.constructions",
"topology.vector_bundle.basic"
] | [
"continuous_on",
"continuous_on_coord_change",
"fiber_bundle",
"topological_space",
"vector_bundle"
] | 184 | 198 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bundle.continuous_linear_map [∀ x, topological_space (E₁ x)]
[∀ x, topological_space (E₂ x)] : Π x : B, Type* | λ x, E₁ x →SL[σ] E₂ x | def | bundle.continuous_linear_map | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"topological_space"
] | A reducible type synonym for the bundle of continuous (semi)linear maps. For some reason, it
helps with instance search.
Porting note: after the port is done, we may want to remove this definition. | 63 | 66 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundle.continuous_linear_map.module [∀ x, topological_space (E₁ x)]
[∀ x, topological_space (E₂ x)] [∀ x, topological_add_group (E₂ x)]
[∀ x, has_continuous_const_smul 𝕜₂ (E₂ x)] :
∀ x, module 𝕜₂ (bundle.continuous_linear_map σ E₁ E₂ x) | λ _, infer_instance | instance | bundle.continuous_linear_map.module | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"has_continuous_const_smul",
"module",
"topological_add_group",
"topological_space"
] | 69 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_coord_change
[e₁.is_linear 𝕜₁] [e₁'.is_linear 𝕜₁] [e₂.is_linear 𝕜₂] [e₂'.is_linear 𝕜₂] (b : B) :
(F₁ →SL[σ] F₂) →L[𝕜₂] F₁ →SL[σ] F₂ | ((e₁'.coord_changeL 𝕜₁ e₁ b).symm.arrow_congrSL (e₂.coord_changeL 𝕜₂ e₂' b) :
(F₁ →SL[σ] F₂) ≃L[𝕜₂] F₁ →SL[σ] F₂) | def | pretrivialization.continuous_linear_map_coord_change | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [] | Assume `eᵢ` and `eᵢ'` are trivializations of the bundles `Eᵢ` over base `B` with fiber `Fᵢ`
(`i ∈ {1,2}`), then `continuous_linear_map_coord_change σ e₁ e₁' e₂ e₂'` is the coordinate change
function between the two induced (pre)trivializations
`pretrivialization.continuous_linear_map σ e₁ e₂` and
`pretrivialization.con... | 87 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_continuous_linear_map_coord_change
[vector_bundle 𝕜₁ F₁ E₁] [vector_bundle 𝕜₂ F₂ E₂]
[mem_trivialization_atlas e₁] [mem_trivialization_atlas e₁']
[mem_trivialization_atlas e₂] [mem_trivialization_atlas e₂'] :
continuous_on (continuous_linear_map_coord_change σ e₁ e₁' e₂ e₂')
((e₁.base_set ∩ ... | begin
have h₁ := (compSL F₁ F₂ F₂ σ (ring_hom.id 𝕜₂)).continuous,
have h₂ := (continuous_linear_map.flip (compSL F₁ F₁ F₂ (ring_hom.id 𝕜₁) σ)).continuous,
have h₃ := (continuous_on_coord_change 𝕜₁ e₁' e₁),
have h₄ := (continuous_on_coord_change 𝕜₂ e₂ e₂'),
refine ((h₁.comp_continuous_on (h₄.mono _)).clm_c... | lemma | pretrivialization.continuous_on_continuous_linear_map_coord_change | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"continuous",
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.symm_symm",
"continuous_linear_map.flip",
"continuous_on",
"continuous_on_coord_change",
"mem_trivialization_atlas",
"ring_hom.id",
"vector_bundle"
] | 100 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map :
pretrivialization (F₁ →SL[σ] F₂) (π (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂)) | { to_fun := λ p, ⟨p.1, continuous_linear_map.comp (e₂.continuous_linear_map_at 𝕜₂ p.1)
(p.2.comp (e₁.symmL 𝕜₁ p.1 : F₁ →L[𝕜₁] E₁ p.1) : F₁ →SL[σ] E₂ p.1)⟩,
inv_fun := λ p, ⟨p.1, continuous_linear_map.comp (e₂.symmL 𝕜₂ p.1)
(p.2.comp (e₁.continuous_linear_map_at 𝕜₁ p.1 : E₁ p.1 →L[𝕜₁] F₁) : E₁ p.1 →SL[σ]... | def | pretrivialization.continuous_linear_map | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"continuous_linear_map",
"continuous_linear_map.comp",
"heq_iff_eq",
"inv_fun",
"is_open_univ",
"pretrivialization",
"prod.mk.inj_iff",
"set.mem_univ",
"sigma.mk.inj_iff",
"trivialization.continuous_linear_map_at_symmL",
"trivialization.symmL_continuous_linear_m... | Given trivializations `e₁`, `e₂` for vector bundles `E₁`, `E₂` over a base `B`,
`pretrivialization.continuous_linear_map σ e₁ e₂` is the induced pretrivialization for the
continuous `σ`-semilinear maps from `E₁` to `E₂`. That is, the map which will later become a
trivialization, after the bundle of continuous semilinea... | 130 | 157 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.is_linear
[Π x, has_continuous_add (E₂ x)] [Π x, has_continuous_smul 𝕜₂ (E₂ x)] :
(pretrivialization.continuous_linear_map σ e₁ e₂).is_linear 𝕜₂ | { linear := λ x h,
{ map_add := λ L L',
show (e₂.continuous_linear_map_at 𝕜₂ x).comp ((L + L').comp (e₁.symmL 𝕜₁ x)) = _,
begin
simp_rw [add_comp, comp_add],
refl
end,
map_smul := λ c L,
show (e₂.continuous_linear_map_at 𝕜₂ x).comp ((c • L).comp (e₁.symmL 𝕜₁ x)) = _,
begin
... | instance | pretrivialization.continuous_linear_map.is_linear | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"has_continuous_add",
"has_continuous_smul",
"pretrivialization.continuous_linear_map",
"ring_hom.id_apply"
] | 162 | 177 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_apply
(p : total_space (F₁ →SL[σ] F₂) (λ x, E₁ x →SL[σ] E₂ x)) :
(continuous_linear_map σ e₁ e₂) p =
⟨p.1, continuous_linear_map.comp (e₂.continuous_linear_map_at 𝕜₂ p.1)
(p.2.comp (e₁.symmL 𝕜₁ p.1 : F₁ →L[𝕜₁] E₁ p.1) : F₁ →SL[σ] E₂ p.1)⟩ | rfl | lemma | pretrivialization.continuous_linear_map_apply | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map",
"continuous_linear_map.comp"
] | 181 | 186 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_symm_apply (p : B × (F₁ →SL[σ] F₂)) :
(continuous_linear_map σ e₁ e₂).to_local_equiv.symm p =
⟨p.1, continuous_linear_map.comp (e₂.symmL 𝕜₂ p.1)
(p.2.comp (e₁.continuous_linear_map_at 𝕜₁ p.1 : E₁ p.1 →L[𝕜₁] F₁) : E₁ p.1 →SL[σ] F₂)⟩ | rfl | lemma | pretrivialization.continuous_linear_map_symm_apply | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map",
"continuous_linear_map.comp"
] | 188 | 192 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_symm_apply' {b : B} (hb : b ∈ e₁.base_set ∩ e₂.base_set)
(L : F₁ →SL[σ] F₂) :
(continuous_linear_map σ e₁ e₂).symm b L =
(e₂.symmL 𝕜₂ b).comp (L.comp $ e₁.continuous_linear_map_at 𝕜₁ b) | begin
rw [symm_apply], refl, exact hb
end | lemma | pretrivialization.continuous_linear_map_symm_apply' | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map"
] | 196 | 202 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_coord_change_apply (b : B)
(hb : b ∈ (e₁.base_set ∩ e₂.base_set) ∩ (e₁'.base_set ∩ e₂'.base_set)) (L : F₁ →SL[σ] F₂) :
continuous_linear_map_coord_change σ e₁ e₁' e₂ e₂' b L =
(continuous_linear_map σ e₁' e₂' ⟨b, ((continuous_linear_map σ e₁ e₂).symm b L)⟩).2 | begin
ext v,
simp_rw [continuous_linear_map_coord_change, continuous_linear_equiv.coe_coe,
continuous_linear_equiv.arrow_congrSL_apply,
continuous_linear_map_apply, continuous_linear_map_symm_apply' σ e₁ e₂ hb.1,
comp_apply, continuous_linear_equiv.coe_coe, continuous_linear_equiv.symm_symm,
trivial... | lemma | pretrivialization.continuous_linear_map_coord_change_apply | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.symm_symm",
"continuous_linear_map"
] | 204 | 218 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.bundle.continuous_linear_map.vector_prebundle :
vector_prebundle 𝕜₂ (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂) | { pretrivialization_atlas :=
{e | ∃ (e₁ : trivialization F₁ (π F₁ E₁)) (e₂ : trivialization F₂ (π F₂ E₂))
[mem_trivialization_atlas e₁] [mem_trivialization_atlas e₂], by exactI
e = pretrivialization.continuous_linear_map σ e₁ e₂},
pretrivialization_linear' := begin
rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩,
... | def | bundle.continuous_linear_map.vector_prebundle | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"inducing",
"mem_trivialization_atlas",
"pretrivialization.continuous_linear_map",
"pretrivialization.continuous_linear_map_apply",
"trivialization",
"trivialization.linear_map_at_def_of_mem",
"vector_prebundle"
] | The continuous `σ`-semilinear maps between two topological vector bundles form a
`vector_prebundle` (this is an auxiliary construction for the
`vector_bundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). | 234 | 271 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundle.continuous_linear_map.topological_space_total_space :
topological_space (total_space (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂)) | (bundle.continuous_linear_map.vector_prebundle
σ F₁ E₁ F₂ E₂).total_space_topology | instance | bundle.continuous_linear_map.topological_space_total_space | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"bundle.continuous_linear_map.vector_prebundle",
"topological_space"
] | Topology on the total space of the continuous `σ`-semilinear_maps between two "normable" vector
bundles over the same base. | 275 | 278 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.bundle.continuous_linear_map.fiber_bundle :
fiber_bundle (F₁ →SL[σ] F₂) (λ x, E₁ x →SL[σ] E₂ x) | (bundle.continuous_linear_map.vector_prebundle
σ F₁ E₁ F₂ E₂).to_fiber_bundle | instance | bundle.continuous_linear_map.fiber_bundle | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map.vector_prebundle",
"fiber_bundle"
] | The continuous `σ`-semilinear_maps between two vector bundles form a fiber bundle. | 281 | 284 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.bundle.continuous_linear_map.vector_bundle :
vector_bundle 𝕜₂ (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂) | (bundle.continuous_linear_map.vector_prebundle
σ F₁ E₁ F₂ E₂).to_vector_bundle | instance | bundle.continuous_linear_map.vector_bundle | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"bundle.continuous_linear_map.vector_prebundle",
"vector_bundle"
] | The continuous `σ`-semilinear_maps between two vector bundles form a vector bundle. | 287 | 290 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivialization.continuous_linear_map :
trivialization (F₁ →SL[σ] F₂) (π (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂)) | vector_prebundle.trivialization_of_mem_pretrivialization_atlas _ ⟨e₁, e₂, he₁, he₂, rfl⟩ | def | trivialization.continuous_linear_map | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"trivialization",
"vector_prebundle.trivialization_of_mem_pretrivialization_atlas"
] | Given trivializations `e₁`, `e₂` in the atlas for vector bundles `E₁`, `E₂` over a base `B`,
the induced trivialization for the continuous `σ`-semilinear maps from `E₁` to `E₂`,
whose base set is `e₁.base_set ∩ e₂.base_set`. | 300 | 302 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.bundle.continuous_linear_map.mem_trivialization_atlas :
mem_trivialization_atlas (e₁.continuous_linear_map σ e₂ :
trivialization (F₁ →SL[σ] F₂) (π (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂))) | { out := ⟨_, ⟨e₁, e₂, by apply_instance, by apply_instance, rfl⟩, rfl⟩ } | instance | bundle.continuous_linear_map.mem_trivialization_atlas | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map",
"mem_trivialization_atlas",
"trivialization"
] | 304 | 307 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.base_set_continuous_linear_map :
(e₁.continuous_linear_map σ e₂).base_set = e₁.base_set ∩ e₂.base_set | rfl | lemma | trivialization.base_set_continuous_linear_map | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [] | 311 | 313 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization.continuous_linear_map_apply
(p : total_space (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂)) :
e₁.continuous_linear_map σ e₂ p =
⟨p.1, (e₂.continuous_linear_map_at 𝕜₂ p.1 : _ →L[𝕜₂] _).comp
(p.2.comp (e₁.symmL 𝕜₁ p.1 : F₁ →L[𝕜₁] E₁ p.1) : F₁ →SL[σ] E₂ p.1)⟩ | rfl | lemma | trivialization.continuous_linear_map_apply | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map"
] | 315 | 320 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_trivialization_at_apply (x₀ : B)
(x : total_space (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂)) :
trivialization_at (F₁ →SL[σ] F₂) (λ x, E₁ x →SL[σ] E₂ x) x₀ x =
⟨x.1, in_coordinates F₁ E₁ F₂ E₂ x₀ x.1 x₀ x.1 x.2⟩ | rfl | lemma | hom_trivialization_at_apply | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map"
] | 324 | 328 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_trivialization_at_source (x₀ : B) :
(trivialization_at (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂) x₀).source =
π (F₁ →SL[σ] F₂) (bundle.continuous_linear_map σ E₁ E₂) ⁻¹'
((trivialization_at F₁ E₁ x₀).base_set ∩ (trivialization_at F₂ E₂ x₀).base_set) | rfl | lemma | hom_trivialization_at_source | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [
"bundle.continuous_linear_map"
] | 330 | 335 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_trivialization_at_target (x₀ : B) :
(trivialization_at (F₁ →SL[σ] F₂) (λ x, E₁ x →SL[σ] E₂ x) x₀).target =
((trivialization_at F₁ E₁ x₀).base_set ∩ (trivialization_at F₂ E₂ x₀).base_set) ×ˢ set.univ | rfl | lemma | hom_trivialization_at_target | topology.vector_bundle | src/topology/vector_bundle/hom.lean | [
"topology.vector_bundle.basic",
"analysis.normed_space.operator_norm"
] | [] | 337 | 341 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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