fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
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uniform_inducing.comp {g : β → γ} (hg : uniform_inducing g)
{f : α → β} (hf : uniform_inducing f) : uniform_inducing (g ∘ f) :=
⟨by rw [← hf.1, ← hg.1, comap_comap]⟩ | uniform_inducing.comp {g : β → γ} (hg : uniform_inducing g)
{f : α → β} (hf : uniform_inducing f) : uniform_inducing (g ∘ f) | ⟨by rw [← hf.1, ← hg.1, comap_comap]⟩ | lemma | uniform_inducing.comp | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | null | 61 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.basis_uniformity {f : α → β} (hf : uniform_inducing f)
{ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)} (H : (𝓤 β).has_basis p s) :
(𝓤 α).has_basis p (λ i, prod.map f f ⁻¹' s i) :=
hf.1 ▸ H.comap _ | uniform_inducing.basis_uniformity {f : α → β} (hf : uniform_inducing f)
{ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)} (H : (𝓤 β).has_basis p s) :
(𝓤 α).has_basis p (λ i, prod.map f f ⁻¹' s i) | hf.1 ▸ H.comap _ | lemma | uniform_inducing.basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | null | 65 | 68 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.cauchy_map_iff {f : α → β} (hf : uniform_inducing f) {F : filter α} :
cauchy (map f F) ↔ cauchy F :=
by simp only [cauchy, map_ne_bot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | uniform_inducing.cauchy_map_iff {f : α → β} (hf : uniform_inducing f) {F : filter α} :
cauchy (map f F) ↔ cauchy F | by simp only [cauchy, map_ne_bot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | lemma | uniform_inducing.cauchy_map_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"filter",
"uniform_inducing"
] | null | 70 | 72 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_of_compose {f : α → β} {g : β → γ} (hf : uniform_continuous f)
(hg : uniform_continuous g) (hgf : uniform_inducing (g ∘ f)) : uniform_inducing f :=
begin
refine ⟨le_antisymm _ hf.le_comap⟩,
rw [← hgf.1, ← prod.map_def, ← prod.map_def, ← prod.map_comp_map f f g g,
← @comap_comap _ _ _ _ (pro... | uniform_inducing_of_compose {f : α → β} {g : β → γ} (hf : uniform_continuous f)
(hg : uniform_continuous g) (hgf : uniform_inducing (g ∘ f)) : uniform_inducing f | begin
refine ⟨le_antisymm _ hf.le_comap⟩,
rw [← hgf.1, ← prod.map_def, ← prod.map_def, ← prod.map_comp_map f f g g,
← @comap_comap _ _ _ _ (prod.map f f)],
exact comap_mono hg.le_comap
end | lemma | uniform_inducing_of_compose | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"prod.map_comp_map",
"prod.map_def",
"uniform_continuous",
"uniform_inducing"
] | null | 74 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.uniform_continuous {f : α → β}
(hf : uniform_inducing f) : uniform_continuous f :=
(uniform_inducing_iff'.1 hf).1 | uniform_inducing.uniform_continuous {f : α → β}
(hf : uniform_inducing f) : uniform_continuous f | (uniform_inducing_iff'.1 hf).1 | lemma | uniform_inducing.uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_inducing"
] | null | 83 | 85 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.uniform_continuous_iff {f : α → β} {g : β → γ} (hg : uniform_inducing g) :
uniform_continuous f ↔ uniform_continuous (g ∘ f) :=
by { dsimp only [uniform_continuous, tendsto],
rw [← hg.comap_uniformity, ← map_le_iff_le_comap, filter.map_map] } | uniform_inducing.uniform_continuous_iff {f : α → β} {g : β → γ} (hg : uniform_inducing g) :
uniform_continuous f ↔ uniform_continuous (g ∘ f) | by { dsimp only [uniform_continuous, tendsto],
rw [← hg.comap_uniformity, ← map_le_iff_le_comap, filter.map_map] } | lemma | uniform_inducing.uniform_continuous_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"filter.map_map",
"uniform_continuous",
"uniform_inducing"
] | null | 87 | 90 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.inducing {f : α → β} (h : uniform_inducing f) : inducing f :=
begin
unfreezingI { obtain rfl := h.comap_uniform_space },
letI := uniform_space.comap f _,
exact ⟨rfl⟩
end | uniform_inducing.inducing {f : α → β} (h : uniform_inducing f) : inducing f | begin
unfreezingI { obtain rfl := h.comap_uniform_space },
letI := uniform_space.comap f _,
exact ⟨rfl⟩
end | lemma | uniform_inducing.inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"inducing",
"uniform_inducing",
"uniform_space.comap"
] | null | 92 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_inducing e₁) (h₂ : uniform_inducing e₂) :
uniform_inducing (λp:α×β, (e₁ p.1, e₂ p.2)) :=
⟨by simp [(∘), uniformity_prod, h₁.comap_uniformity.symm, h₂.comap_uniformity.symm,
coma... | uniform_inducing.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_inducing e₁) (h₂ : uniform_inducing e₂) :
uniform_inducing (λp:α×β, (e₁ p.1, e₂ p.2)) | ⟨by simp [(∘), uniformity_prod, h₁.comap_uniformity.symm, h₂.comap_uniformity.symm,
comap_inf, comap_comap]⟩ | lemma | uniform_inducing.prod | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing",
"uniform_space",
"uniformity_prod"
] | null | 99 | 103 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.dense_inducing {f : α → β} (h : uniform_inducing f) (hd : dense_range f) :
dense_inducing f :=
{ dense := hd,
induced := h.inducing.induced } | uniform_inducing.dense_inducing {f : α → β} (h : uniform_inducing f) (hd : dense_range f) :
dense_inducing f | { dense := hd,
induced := h.inducing.induced } | lemma | uniform_inducing.dense_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense",
"dense_inducing",
"dense_range",
"uniform_inducing"
] | null | 105 | 108 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.injective [t0_space α] {f : α → β} (h : uniform_inducing f) :
injective f :=
h.inducing.injective | uniform_inducing.injective [t0_space α] {f : α → β} (h : uniform_inducing f) :
injective f | h.inducing.injective | lemma | uniform_inducing.injective | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_inducing"
] | null | 110 | 112 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding (f : α → β) extends uniform_inducing f : Prop :=
(inj : function.injective f) | uniform_embedding (f : α → β) extends uniform_inducing f : Prop | (inj : function.injective f) | structure | uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | A map `f : α → β` between uniform spaces is a *uniform embedding* if it is uniform inducing and
injective. If `α` is a separated space, then the latter assumption follows from the former. | 116 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_iff' {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α :=
by rw [uniform_embedding_iff, and_comm, uniform_inducing_iff'] | uniform_embedding_iff' {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α | by rw [uniform_embedding_iff, and_comm, uniform_inducing_iff'] | theorem | uniform_embedding_iff' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_embedding",
"uniform_inducing_iff'"
] | null | 120 | 122 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.uniform_embedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s'... | filter.has_basis.uniform_embedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s'... | by rw [uniform_embedding_iff, and_comm, h.uniform_inducing_iff h'] | theorem | filter.has_basis.uniform_embedding_iff' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding"
] | null | 124 | 129 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.uniform_embedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) :=
by simp only [h.uniform_emb... | filter.has_basis.uniform_embedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) | by simp only [h.uniform_embedding_iff' h', h.uniform_continuous_iff h', exists_prop] | theorem | filter.has_basis.uniform_embedding_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"exists_prop",
"uniform_continuous",
"uniform_embedding"
] | null | 131 | 135 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_subtype_val {p : α → Prop} :
uniform_embedding (subtype.val : subtype p → α) :=
{ comap_uniformity := rfl,
inj := subtype.val_injective } | uniform_embedding_subtype_val {p : α → Prop} :
uniform_embedding (subtype.val : subtype p → α) | { comap_uniformity := rfl,
inj := subtype.val_injective } | lemma | uniform_embedding_subtype_val | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"subtype.val_injective",
"uniform_embedding"
] | null | 137 | 140 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_subtype_coe {p : α → Prop} :
uniform_embedding (coe : subtype p → α) :=
uniform_embedding_subtype_val | uniform_embedding_subtype_coe {p : α → Prop} :
uniform_embedding (coe : subtype p → α) | uniform_embedding_subtype_val | lemma | uniform_embedding_subtype_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_embedding_subtype_val"
] | null | 142 | 144 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_set_inclusion {s t : set α} (hst : s ⊆ t) :
uniform_embedding (inclusion hst) :=
{ comap_uniformity :=
by { erw [uniformity_subtype, uniformity_subtype, comap_comap], congr },
inj := inclusion_injective hst } | uniform_embedding_set_inclusion {s t : set α} (hst : s ⊆ t) :
uniform_embedding (inclusion hst) | { comap_uniformity :=
by { erw [uniformity_subtype, uniformity_subtype, comap_comap], congr },
inj := inclusion_injective hst } | lemma | uniform_embedding_set_inclusion | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniformity_subtype"
] | null | 146 | 150 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.comp {g : β → γ} (hg : uniform_embedding g)
{f : α → β} (hf : uniform_embedding f) : uniform_embedding (g ∘ f) :=
{ inj := hg.inj.comp hf.inj,
..hg.to_uniform_inducing.comp hf.to_uniform_inducing } | uniform_embedding.comp {g : β → γ} (hg : uniform_embedding g)
{f : α → β} (hf : uniform_embedding f) : uniform_embedding (g ∘ f) | { inj := hg.inj.comp hf.inj,
..hg.to_uniform_inducing.comp hf.to_uniform_inducing } | lemma | uniform_embedding.comp | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding"
] | null | 152 | 155 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.uniform_embedding {α β : Type*} [uniform_space α] [uniform_space β] (f : α ≃ β)
(h₁ : uniform_continuous f) (h₂ : uniform_continuous f.symm) : uniform_embedding f :=
uniform_embedding_iff'.2 ⟨f.injective, h₁, by rwa [← equiv.prod_congr_apply, ← map_equiv_symm]⟩ | equiv.uniform_embedding {α β : Type*} [uniform_space α] [uniform_space β] (f : α ≃ β)
(h₁ : uniform_continuous f) (h₂ : uniform_continuous f.symm) : uniform_embedding f | uniform_embedding_iff'.2 ⟨f.injective, h₁, by rwa [← equiv.prod_congr_apply, ← map_equiv_symm]⟩ | lemma | equiv.uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_embedding",
"uniform_space"
] | null | 157 | 159 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_inl : uniform_embedding (sum.inl : α → α ⊕ β) :=
begin
refine ⟨⟨_⟩, sum.inl_injective⟩,
rw [sum.uniformity, comap_sup, comap_map, comap_eq_bot_iff_compl_range.2 _, sup_bot_eq],
{ refine mem_map.2 (univ_mem' _),
simp },
{ exact sum.inl_injective.prod_map sum.inl_injective }
end | uniform_embedding_inl : uniform_embedding (sum.inl : α → α ⊕ β) | begin
refine ⟨⟨_⟩, sum.inl_injective⟩,
rw [sum.uniformity, comap_sup, comap_map, comap_eq_bot_iff_compl_range.2 _, sup_bot_eq],
{ refine mem_map.2 (univ_mem' _),
simp },
{ exact sum.inl_injective.prod_map sum.inl_injective }
end | theorem | uniform_embedding_inl | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"sum.inl_injective",
"sum.uniformity",
"sup_bot_eq",
"uniform_embedding"
] | null | 161 | 168 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_inr : uniform_embedding (sum.inr : β → α ⊕ β) :=
begin
refine ⟨⟨_⟩, sum.inr_injective⟩,
rw [sum.uniformity, comap_sup, comap_eq_bot_iff_compl_range.2 _, comap_map, bot_sup_eq],
{ exact sum.inr_injective.prod_map sum.inr_injective },
{ refine mem_map.2 (univ_mem' _),
simp },
end | uniform_embedding_inr : uniform_embedding (sum.inr : β → α ⊕ β) | begin
refine ⟨⟨_⟩, sum.inr_injective⟩,
rw [sum.uniformity, comap_sup, comap_eq_bot_iff_compl_range.2 _, comap_map, bot_sup_eq],
{ exact sum.inr_injective.prod_map sum.inr_injective },
{ refine mem_map.2 (univ_mem' _),
simp },
end | theorem | uniform_embedding_inr | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"bot_sup_eq",
"sum.inr_injective",
"sum.uniformity",
"uniform_embedding"
] | null | 170 | 177 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.uniform_embedding [t0_space α] {f : α → β}
(hf : uniform_inducing f) :
uniform_embedding f :=
⟨hf, hf.injective⟩ | uniform_inducing.uniform_embedding [t0_space α] {f : α → β}
(hf : uniform_inducing f) :
uniform_embedding f | ⟨hf, hf.injective⟩ | theorem | uniform_inducing.uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_embedding",
"uniform_inducing"
] | If the domain of a `uniform_inducing` map `f` is a `separated_space`, then `f` is injective,
hence it is a `uniform_embedding`. | 181 | 184 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_iff_uniform_inducing [t0_space α] {f : α → β} :
uniform_embedding f ↔ uniform_inducing f :=
⟨uniform_embedding.to_uniform_inducing, uniform_inducing.uniform_embedding⟩ | uniform_embedding_iff_uniform_inducing [t0_space α] {f : α → β} :
uniform_embedding f ↔ uniform_inducing f | ⟨uniform_embedding.to_uniform_inducing, uniform_inducing.uniform_embedding⟩ | theorem | uniform_embedding_iff_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_embedding",
"uniform_inducing"
] | null | 186 | 188 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_uniformity_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
comap (prod.map f f) (𝓤 β) = 𝓟 id_rel :=
begin
refine le_antisymm _ (@refl_le_uniformity α (uniform_space.comap f ‹_›)),
calc comap (prod.map f f) (𝓤 β) ≤ comap (prod.map f f) (𝓟 s) : c... | comap_uniformity_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
comap (prod.map f f) (𝓤 β) = 𝓟 id_rel | begin
refine le_antisymm _ (@refl_le_uniformity α (uniform_space.comap f ‹_›)),
calc comap (prod.map f f) (𝓤 β) ≤ comap (prod.map f f) (𝓟 s) : comap_mono (le_principal_iff.2 hs)
... = 𝓟 (prod.map f f ⁻¹' s) : comap_principal
... ≤ 𝓟 id_rel : principal_mono.2 _,
rintro ⟨x, y⟩, simpa [not_imp_not] using @hf... | lemma | comap_uniformity_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"id_rel",
"not_imp_not",
"pairwise",
"refl_le_uniformity",
"uniform_space.comap"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is uniform inducing with respect to the discrete uniformity on `α`:
the preimage of `𝓤 β` under `prod.map f f` is the principal filter generated by the diagonal in
`α × α`. | 194 | 203 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
@uniform_embedding α β ⊥ ‹_› f :=
begin
letI : uniform_space α := ⊥, haveI := discrete_topology_bot α,
haveI : separated_space α := separated_iff_t2.2 infer_instance,
exact uniform_induc... | uniform_embedding_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
@uniform_embedding α β ⊥ ‹_› f | begin
letI : uniform_space α := ⊥, haveI := discrete_topology_bot α,
haveI : separated_space α := separated_iff_t2.2 infer_instance,
exact uniform_inducing.uniform_embedding ⟨comap_uniformity_of_spaced_out hs hf⟩
end | lemma | uniform_embedding_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"discrete_topology_bot",
"pairwise",
"separated_space",
"uniform_embedding",
"uniform_inducing.uniform_embedding",
"uniform_space"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is a uniform embedding with respect to the discrete uniformity on `α`. | 207 | 214 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.embedding {f : α → β} (h : uniform_embedding f) : embedding f :=
{ induced := h.to_uniform_inducing.inducing.induced,
inj := h.inj } | uniform_embedding.embedding {f : α → β} (h : uniform_embedding f) : embedding f | { induced := h.to_uniform_inducing.inducing.induced,
inj := h.inj } | lemma | uniform_embedding.embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"uniform_embedding"
] | null | 216 | 218 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.dense_embedding {f : α → β} (h : uniform_embedding f) (hd : dense_range f) :
dense_embedding f :=
{ dense := hd,
inj := h.inj,
induced := h.embedding.induced } | uniform_embedding.dense_embedding {f : α → β} (h : uniform_embedding f) (hd : dense_range f) :
dense_embedding f | { dense := hd,
inj := h.inj,
induced := h.embedding.induced } | lemma | uniform_embedding.dense_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense",
"dense_embedding",
"dense_range",
"uniform_embedding"
] | null | 220 | 224 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding_of_spaced_out {α} [topological_space α] [discrete_topology α]
[separated_space β] {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
closed_embedding f :=
begin
unfreezingI { rcases (discrete_topology.eq_bot α) with rfl }, letI : uniform_space α := ⊥,
exac... | closed_embedding_of_spaced_out {α} [topological_space α] [discrete_topology α]
[separated_space β] {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
closed_embedding f | begin
unfreezingI { rcases (discrete_topology.eq_bot α) with rfl }, letI : uniform_space α := ⊥,
exact { closed_range := is_closed_range_of_spaced_out hs hf,
.. (uniform_embedding_of_spaced_out hs hf).embedding }
end | lemma | closed_embedding_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"closed_embedding",
"discrete_topology",
"embedding",
"is_closed_range_of_spaced_out",
"pairwise",
"separated_space",
"topological_space",
"uniform_embedding_of_spaced_out",
"uniform_space"
] | null | 226 | 234 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_image_mem_nhds_of_uniform_inducing
{s : set (α×α)} {e : α → β} (b : β)
(he₁ : uniform_inducing e) (he₂ : dense_inducing e) (hs : s ∈ 𝓤 α) :
∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b :=
have s ∈ comap (λp:α×α, (e p.1, e p.2)) (𝓤 β),
from he₁.comap_uniformity.symm ▸ hs,
let ⟨t₁, ht₁u, ht₁⟩ := this i... | closure_image_mem_nhds_of_uniform_inducing
{s : set (α×α)} {e : α → β} (b : β)
(he₁ : uniform_inducing e) (he₂ : dense_inducing e) (hs : s ∈ 𝓤 α) :
∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b | have s ∈ comap (λp:α×α, (e p.1, e p.2)) (𝓤 β),
from he₁.comap_uniformity.symm ▸ hs,
let ⟨t₁, ht₁u, ht₁⟩ := this in
have ht₁ : ∀p:α×α, (e p.1, e p.2) ∈ t₁ → p ∈ s, from ht₁,
let ⟨t₂, ht₂u, ht₂s, ht₂c⟩ := comp_symm_of_uniformity ht₁u in
let ⟨t, htu, hts, htc⟩ := comp_symm_of_uniformity ht₂u in
have preimage e {b' | (b... | lemma | closure_image_mem_nhds_of_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"closure",
"closure_eq_cluster_pts",
"comp_symm_of_uniformity",
"dense_inducing",
"mem_nhds_left",
"monotone_const",
"nhds_eq_uniformity",
"prod_mk_mem_comp_rel",
"uniform_inducing"
] | null | 236 | 269 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_subtype_emb (p : α → Prop) {e : α → β} (ue : uniform_embedding e)
(de : dense_embedding e) : uniform_embedding (dense_embedding.subtype_emb p e) :=
{ comap_uniformity := by simp [comap_comap, (∘), dense_embedding.subtype_emb,
uniformity_subtype, ue.comap_uniformity.symm],
inj := (de.sub... | uniform_embedding_subtype_emb (p : α → Prop) {e : α → β} (ue : uniform_embedding e)
(de : dense_embedding e) : uniform_embedding (dense_embedding.subtype_emb p e) | { comap_uniformity := by simp [comap_comap, (∘), dense_embedding.subtype_emb,
uniformity_subtype, ue.comap_uniformity.symm],
inj := (de.subtype p).inj } | lemma | uniform_embedding_subtype_emb | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense_embedding",
"dense_embedding.subtype_emb",
"uniform_embedding",
"uniformity_subtype"
] | null | 271 | 275 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_embedding e₁) (h₂ : uniform_embedding e₂) :
uniform_embedding (λp:α×β, (e₁ p.1, e₂ p.2)) :=
{ inj := h₁.inj.prod_map h₂.inj,
..h₁.to_uniform_inducing.prod h₂.to_uniform_inducing } | uniform_embedding.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_embedding e₁) (h₂ : uniform_embedding e₂) :
uniform_embedding (λp:α×β, (e₁ p.1, e₂ p.2)) | { inj := h₁.inj.prod_map h₂.inj,
..h₁.to_uniform_inducing.prod h₂.to_uniform_inducing } | lemma | uniform_embedding.prod | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_space"
] | null | 277 | 281 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_complete_of_complete_image {m : α → β} {s : set α} (hm : uniform_inducing m)
(hs : is_complete (m '' s)) : is_complete s :=
begin
intros f hf hfs,
rw le_principal_iff at hfs,
obtain ⟨_, ⟨x, hx, rfl⟩, hyf⟩ : ∃ y ∈ m '' s, map m f ≤ 𝓝 y,
from hs (f.map m) (hf.map hm.uniform_continuous)
(le_principal... | is_complete_of_complete_image {m : α → β} {s : set α} (hm : uniform_inducing m)
(hs : is_complete (m '' s)) : is_complete s | begin
intros f hf hfs,
rw le_principal_iff at hfs,
obtain ⟨_, ⟨x, hx, rfl⟩, hyf⟩ : ∃ y ∈ m '' s, map m f ≤ 𝓝 y,
from hs (f.map m) (hf.map hm.uniform_continuous)
(le_principal_iff.2 (image_mem_map hfs)),
rw [map_le_iff_le_comap, ← nhds_induced, ← hm.inducing.induced] at hyf,
exact ⟨x, hx, hyf⟩
end | lemma | is_complete_of_complete_image | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"is_complete",
"nhds_induced",
"uniform_inducing"
] | null | 283 | 293 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_complete.complete_space_coe {s : set α} (hs : is_complete s) :
complete_space s :=
complete_space_iff_is_complete_univ.2 $
is_complete_of_complete_image uniform_embedding_subtype_coe.to_uniform_inducing $ by simp [hs] | is_complete.complete_space_coe {s : set α} (hs : is_complete s) :
complete_space s | complete_space_iff_is_complete_univ.2 $
is_complete_of_complete_image uniform_embedding_subtype_coe.to_uniform_inducing $ by simp [hs] | lemma | is_complete.complete_space_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"is_complete",
"is_complete_of_complete_image"
] | null | 295 | 298 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_complete_image_iff {m : α → β} {s : set α} (hm : uniform_inducing m) :
is_complete (m '' s) ↔ is_complete s :=
begin
refine ⟨is_complete_of_complete_image hm, λ c, _⟩,
haveI : complete_space s := c.complete_space_coe,
set m' : s → β := m ∘ coe,
suffices : is_complete (range m'), by rwa [range_comp, subtype... | is_complete_image_iff {m : α → β} {s : set α} (hm : uniform_inducing m) :
is_complete (m '' s) ↔ is_complete s | begin
refine ⟨is_complete_of_complete_image hm, λ c, _⟩,
haveI : complete_space s := c.complete_space_coe,
set m' : s → β := m ∘ coe,
suffices : is_complete (range m'), by rwa [range_comp, subtype.range_coe] at this,
have hm' : uniform_inducing m' := hm.comp uniform_embedding_subtype_coe.to_uniform_inducing,
... | lemma | is_complete_image_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"complete_space",
"filter.le_principal_iff",
"is_complete",
"subtype.range_coe",
"uniform_inducing"
] | A set is complete iff its image under a uniform inducing map is complete. | 301 | 316 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_iff_is_complete_range {f : α → β} (hf : uniform_inducing f) :
complete_space α ↔ is_complete (range f) :=
by rw [complete_space_iff_is_complete_univ, ← is_complete_image_iff hf, image_univ] | complete_space_iff_is_complete_range {f : α → β} (hf : uniform_inducing f) :
complete_space α ↔ is_complete (range f) | by rw [complete_space_iff_is_complete_univ, ← is_complete_image_iff hf, image_univ] | lemma | complete_space_iff_is_complete_range | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_univ",
"is_complete",
"is_complete_image_iff",
"uniform_inducing"
] | null | 318 | 320 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.is_complete_range [complete_space α] {f : α → β}
(hf : uniform_inducing f) :
is_complete (range f) :=
(complete_space_iff_is_complete_range hf).1 ‹_› | uniform_inducing.is_complete_range [complete_space α] {f : α → β}
(hf : uniform_inducing f) :
is_complete (range f) | (complete_space_iff_is_complete_range hf).1 ‹_› | lemma | uniform_inducing.is_complete_range | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"is_complete",
"uniform_inducing"
] | null | 322 | 325 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_congr {e : α ≃ β} (he : uniform_embedding e) :
complete_space α ↔ complete_space β :=
by rw [complete_space_iff_is_complete_range he.to_uniform_inducing, e.range_eq_univ,
complete_space_iff_is_complete_univ] | complete_space_congr {e : α ≃ β} (he : uniform_embedding e) :
complete_space α ↔ complete_space β | by rw [complete_space_iff_is_complete_range he.to_uniform_inducing, e.range_eq_univ,
complete_space_iff_is_complete_univ] | lemma | complete_space_congr | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"complete_space_iff_is_complete_univ",
"uniform_embedding"
] | null | 327 | 330 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_coe_iff_is_complete {s : set α} :
complete_space s ↔ is_complete s :=
(complete_space_iff_is_complete_range uniform_embedding_subtype_coe.to_uniform_inducing).trans $
by rw [subtype.range_coe] | complete_space_coe_iff_is_complete {s : set α} :
complete_space s ↔ is_complete s | (complete_space_iff_is_complete_range uniform_embedding_subtype_coe.to_uniform_inducing).trans $
by rw [subtype.range_coe] | lemma | complete_space_coe_iff_is_complete | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"is_complete",
"subtype.range_coe"
] | null | 332 | 335 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.complete_space_coe [complete_space α] {s : set α} (hs : is_closed s) :
complete_space s :=
hs.is_complete.complete_space_coe | is_closed.complete_space_coe [complete_space α] {s : set α} (hs : is_closed s) :
complete_space s | hs.is_complete.complete_space_coe | lemma | is_closed.complete_space_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"is_closed"
] | null | 337 | 339 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift.complete_space [h : complete_space α] : complete_space (ulift α) :=
begin
have : uniform_embedding (@equiv.ulift α), from ⟨⟨rfl⟩, ulift.down_injective⟩,
exact (complete_space_congr this).2 h,
end | ulift.complete_space [h : complete_space α] : complete_space (ulift α) | begin
have : uniform_embedding (@equiv.ulift α), from ⟨⟨rfl⟩, ulift.down_injective⟩,
exact (complete_space_congr this).2 h,
end | instance | ulift.complete_space | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_congr",
"equiv.ulift",
"uniform_embedding"
] | The lift of a complete space to another universe is still complete. | 342 | 346 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_extension {m : β → α} (hm : uniform_inducing m) (dense : dense_range m)
(h : ∀f:filter β, cauchy f → ∃x:α, map m f ≤ 𝓝 x) : complete_space α :=
⟨assume (f : filter α), assume hf : cauchy f,
let
p : set (α × α) → set α → set α := λs t, {y : α| ∃x:α, x ∈ t ∧ (x, y) ∈ s},
g := (𝓤 α).lift (λs, f.lift... | complete_space_extension {m : β → α} (hm : uniform_inducing m) (dense : dense_range m)
(h : ∀f:filter β, cauchy f → ∃x:α, map m f ≤ 𝓝 x) : complete_space α | ⟨assume (f : filter α), assume hf : cauchy f,
let
p : set (α × α) → set α → set α := λs t, {y : α| ∃x:α, x ∈ t ∧ (x, y) ∈ s},
g := (𝓤 α).lift (λs, f.lift' (p s))
in
have mp₀ : monotone p,
from assume a b h t s ⟨x, xs, xa⟩, ⟨x, xs, h xa⟩,
have mp₁ : ∀{s}, monotone (p s),
from assume s a b h x ⟨y, ya, yxs⟩, ⟨y, ... | lemma | complete_space_extension | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"cluster_pt",
"comp_mem_uniformity_sets",
"comp_rel",
"complete_space",
"dense",
"dense_range",
"filter",
"filter.comap",
"le_infi",
"le_nhds_iff_adhp_of_cauchy",
"le_nhds_of_cauchy_adhp",
"lift",
"mem_nhds_left",
"monotone",
"monotone_const",
"prod.swap",
"prod_mk_mem_co... | null | 348 | 411 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_preimage {f : α → β} {s : set β} (hf : uniform_embedding f)
(hs : totally_bounded s) : totally_bounded (f ⁻¹' s) :=
λ t ht, begin
rw ← hf.comap_uniformity at ht,
rcases mem_comap.2 ht with ⟨t', ht', ts⟩,
rcases totally_bounded_iff_subset.1
(totally_bounded_subset (image_preimage_subset f s) ... | totally_bounded_preimage {f : α → β} {s : set β} (hf : uniform_embedding f)
(hs : totally_bounded s) : totally_bounded (f ⁻¹' s) | λ t ht, begin
rw ← hf.comap_uniformity at ht,
rcases mem_comap.2 ht with ⟨t', ht', ts⟩,
rcases totally_bounded_iff_subset.1
(totally_bounded_subset (image_preimage_subset f s) hs) _ ht' with ⟨c, cs, hfc, hct⟩,
refine ⟨f ⁻¹' c, hfc.preimage (hf.inj.inj_on _), λ x h, _⟩,
have := hct (mem_image_of_mem f h), ... | lemma | totally_bounded_preimage | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"totally_bounded",
"totally_bounded_subset",
"uniform_embedding"
] | null | 413 | 425 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space.sum [complete_space α] [complete_space β] :
complete_space (α ⊕ β) :=
begin
rw [complete_space_iff_is_complete_univ, ← range_inl_union_range_inr],
exact uniform_embedding_inl.to_uniform_inducing.is_complete_range.union
uniform_embedding_inr.to_uniform_inducing.is_complete_range
end | complete_space.sum [complete_space α] [complete_space β] :
complete_space (α ⊕ β) | begin
rw [complete_space_iff_is_complete_univ, ← range_inl_union_range_inr],
exact uniform_embedding_inl.to_uniform_inducing.is_complete_range.union
uniform_embedding_inr.to_uniform_inducing.is_complete_range
end | instance | complete_space.sum | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_univ"
] | null | 427 | 433 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_comap {α : Type*} {β : Type*} {f : α → β} [u : uniform_space β]
(hf : function.injective f) : @uniform_embedding α β (uniform_space.comap f u) u f :=
@uniform_embedding.mk _ _ (uniform_space.comap f u) _ _
(@uniform_inducing.mk _ _ (uniform_space.comap f u) _ _ rfl) hf | uniform_embedding_comap {α : Type*} {β : Type*} {f : α → β} [u : uniform_space β]
(hf : function.injective f) : @uniform_embedding α β (uniform_space.comap f u) u f | @uniform_embedding.mk _ _ (uniform_space.comap f u) _ _
(@uniform_inducing.mk _ _ (uniform_space.comap f u) _ _ rfl) hf | lemma | uniform_embedding_comap | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_space",
"uniform_space.comap"
] | null | 437 | 440 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.comap_uniform_space {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) : uniform_space α :=
(u.comap f).replace_topology h.induced | embedding.comap_uniform_space {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) : uniform_space α | (u.comap f).replace_topology h.induced | def | embedding.comap_uniform_space | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"topological_space",
"uniform_space"
] | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. | 444 | 446 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.to_uniform_embedding {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) :
@uniform_embedding α β (h.comap_uniform_space f) u f :=
{ comap_uniformity := rfl,
inj := h.inj } | embedding.to_uniform_embedding {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) :
@uniform_embedding α β (h.comap_uniform_space f) u f | { comap_uniformity := rfl,
inj := h.inj } | lemma | embedding.to_uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"topological_space",
"uniform_embedding",
"uniform_space"
] | null | 448 | 452 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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))),... | 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"
] | null | 466 | 473 | true | 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... | 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... | null | 475 | 506 | true | 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 _) | 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"
] | null | 510 | 512 | true | 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) ∈... | 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"
] | null | 514 | 554 | true | 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 | 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"
] | null | 560 | 561 | true | 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 | 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"
] | null | 563 | 566 | true | 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)) | 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. | 72 | 75 | true | 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 | 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"
] | null | 81 | 84 | true | 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 [... | 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`. | 89 | 97 | true | 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,
ma... | 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. | 101 | 109 | true | 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 | 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`. | 112 | 114 | true | 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 } | 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"
] | null | 118 | 120 | true | 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] | 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"
] | null | 122 | 125 | true | 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] | 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"
] | null | 127 | 129 | true | 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 | 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"
] | null | 131 | 134 | true | 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 | 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"
] | null | 136 | 139 | true | 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 | 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"
] | null | 141 | 144 | true | 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 } | 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"
] | null | 146 | 149 | true | 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 } | 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"
] | null | 151 | 154 | true | 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)) | 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. | 162 | 164 | true | 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 | 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"
] | null | 170 | 173 | true | 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) } | 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"
] | null | 175 | 178 | true | 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 | 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. | 184 | 186 | true | 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 | 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"
] | null | 190 | 193 | true | 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 | 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"
] | null | 195 | 198 | true | 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 | 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`. | 203 | 204 | true | 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 | 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"
] | null | 208 | 210 | true | 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 | 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`. | 215 | 216 | true | 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 | 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"
] | null | 220 | 222 | true | 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] | 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"
] | null | 224 | 227 | true | 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] | 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"
] | null | 229 | 231 | true | 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 | 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"
] | null | 233 | 236 | true | 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 | 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"
] | null | 238 | 241 | true | 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 | 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"
] | null | 243 | 246 | true | 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 | 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"
] | null | 248 | 251 | true | 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 (co... | 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. | 257 | 279 | true | 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) | 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"
] | null | 283 | 287 | true | 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 _ _ _) | 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"
] | null | 289 | 293 | true | 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).sy... | 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"
] | null | 295 | 302 | true | 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) | 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"
] | null | 304 | 307 | true | 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 }... | 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"
] | null | 309 | 317 | true | 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] | 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"
] | null | 319 | 322 | true | 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] | 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. | 326 | 329 | true | 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) | 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"
] | null | 331 | 335 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_section [∀ x, has_zero (E x)] : B → total_space F E :=
λ x, ⟨x, 0⟩ | 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 | 346 | 347 | true | 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 | 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"
] | [] | null | 349 | 350 | true | 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 | 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"
] | [] | null | 351 | 352 | true | 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_cha... | 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. | 365 | 371 | true | 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 | 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"
] | null | 375 | 379 | true | 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' | 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"
] | null | 381 | 386 | true | 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.c... | 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). | 392 | 404 | true | 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_con... | 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. | 407 | 417 | true | 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 | 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"
] | null | 421 | 424 | true | 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 | 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"
] | null | 426 | 429 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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