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
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
tendsto_locally_uniformly_on_univ :
tendsto_locally_uniformly_on F f p univ ↔ tendsto_locally_uniformly F f p :=
by simp [tendsto_locally_uniformly_on, tendsto_locally_uniformly, nhds_within_univ] | tendsto_locally_uniformly_on_univ :
tendsto_locally_uniformly_on F f p univ ↔ tendsto_locally_uniformly F f p | by simp [tendsto_locally_uniformly_on, tendsto_locally_uniformly, nhds_within_univ] | lemma | tendsto_locally_uniformly_on_univ | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"nhds_within_univ",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on"
] | null | 733 | 735 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly.tendsto_locally_uniformly_on
(h : tendsto_locally_uniformly F f p) : tendsto_locally_uniformly_on F f p s :=
(tendsto_locally_uniformly_on_univ.mpr h).mono (subset_univ _) | tendsto_locally_uniformly.tendsto_locally_uniformly_on
(h : tendsto_locally_uniformly F f p) : tendsto_locally_uniformly_on F f p s | (tendsto_locally_uniformly_on_univ.mpr h).mono (subset_univ _) | lemma | tendsto_locally_uniformly.tendsto_locally_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on"
] | null | 737 | 739 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space [compact_space α] :
tendsto_locally_uniformly F f p ↔ tendsto_uniformly F f p :=
begin
refine ⟨λ h V hV, _, tendsto_uniformly.tendsto_locally_uniformly⟩,
choose U hU using h V hV,
obtain ⟨t, ht⟩ := is_compact_univ.elim_nhds_subcover' (λ k hk, U k)... | tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space [compact_space α] :
tendsto_locally_uniformly F f p ↔ tendsto_uniformly F f p | begin
refine ⟨λ h V hV, _, tendsto_uniformly.tendsto_locally_uniformly⟩,
choose U hU using h V hV,
obtain ⟨t, ht⟩ := is_compact_univ.elim_nhds_subcover' (λ k hk, U k) (λ k hk, (hU k).1),
replace hU := λ (x : t), (hU x).2,
rw ← eventually_all at hU,
refine hU.mono (λ i hi x, _),
specialize ht (mem_univ x),... | lemma | tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"compact_space",
"exists_and_distrib_right",
"exists_prop",
"set_coe.exists",
"subtype.coe_mk",
"tendsto_locally_uniformly",
"tendsto_uniformly"
] | On a compact space, locally uniform convergence is just uniform convergence. | 742 | 755 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact (hs : is_compact s) :
tendsto_locally_uniformly_on F f p s ↔ tendsto_uniformly_on F f p s :=
begin
haveI : compact_space s := is_compact_iff_compact_space.mp hs,
refine ⟨λ h, _, tendsto_uniformly_on.tendsto_locally_uniformly_on⟩,
rwa [tendsto_loca... | tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact (hs : is_compact s) :
tendsto_locally_uniformly_on F f p s ↔ tendsto_uniformly_on F f p s | begin
haveI : compact_space s := is_compact_iff_compact_space.mp hs,
refine ⟨λ h, _, tendsto_uniformly_on.tendsto_locally_uniformly_on⟩,
rwa [tendsto_locally_uniformly_on_iff_tendsto_locally_uniformly_comp_coe,
tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space,
← tendsto_uniformly_on_iff_te... | lemma | tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"compact_space",
"is_compact",
"tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_iff_tendsto_locally_uniformly_comp_coe",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe"
] | For a compact set `s`, locally uniform convergence on `s` is just uniform convergence on `s`. | 758 | 766 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.comp [topological_space γ] {t : set γ}
(h : tendsto_locally_uniformly_on F f p s)
(g : γ → α) (hg : maps_to g t s) (cg : continuous_on g t) :
tendsto_locally_uniformly_on (λ n, (F n) ∘ g) (f ∘ g) p t :=
begin
assume u hu x hx,
rcases h u hu (g x) (hg hx) with ⟨a, ha, H⟩,
have : ... | tendsto_locally_uniformly_on.comp [topological_space γ] {t : set γ}
(h : tendsto_locally_uniformly_on F f p s)
(g : γ → α) (hg : maps_to g t s) (cg : continuous_on g t) :
tendsto_locally_uniformly_on (λ n, (F n) ∘ g) (f ∘ g) p t | begin
assume u hu x hx,
rcases h u hu (g x) (hg hx) with ⟨a, ha, H⟩,
have : g⁻¹' a ∈ 𝓝[t] x :=
((cg x hx).preimage_mem_nhds_within' (nhds_within_mono (g x) hg.image_subset ha)),
exact ⟨g ⁻¹' a, this, H.mono (λ n hn y hy, hn _ hy)⟩
end | lemma | tendsto_locally_uniformly_on.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_on",
"nhds_within_mono",
"tendsto_locally_uniformly_on",
"topological_space"
] | null | 768 | 778 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly.comp [topological_space γ]
(h : tendsto_locally_uniformly F f p) (g : γ → α) (cg : continuous g) :
tendsto_locally_uniformly (λ n, (F n) ∘ g) (f ∘ g) p :=
begin
rw ← tendsto_locally_uniformly_on_univ at h ⊢,
rw continuous_iff_continuous_on_univ at cg,
exact h.comp _ (maps_to_univ _ _... | tendsto_locally_uniformly.comp [topological_space γ]
(h : tendsto_locally_uniformly F f p) (g : γ → α) (cg : continuous g) :
tendsto_locally_uniformly (λ n, (F n) ∘ g) (f ∘ g) p | begin
rw ← tendsto_locally_uniformly_on_univ at h ⊢,
rw continuous_iff_continuous_on_univ at cg,
exact h.comp _ (maps_to_univ _ _) cg
end | lemma | tendsto_locally_uniformly.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on_univ",
"topological_space"
] | null | 780 | 787 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_tfae [locally_compact_space α]
(G : ι → α → β) (g : α → β) (p : filter ι) (hs : is_open s) :
tfae [(tendsto_locally_uniformly_on G g p s),
(∀ K ⊆ s, is_compact K → tendsto_uniformly_on G g p K),
(∀ x ∈ s, ∃ v ∈ 𝓝[s] x, tendsto_uniformly_on G g p v)] :=
begin
tfae_have : 1 → 2... | tendsto_locally_uniformly_on_tfae [locally_compact_space α]
(G : ι → α → β) (g : α → β) (p : filter ι) (hs : is_open s) :
tfae [(tendsto_locally_uniformly_on G g p s),
(∀ K ⊆ s, is_compact K → tendsto_uniformly_on G g p K),
(∀ x ∈ s, ∃ v ∈ 𝓝[s] x, tendsto_uniformly_on G g p v)] | begin
tfae_have : 1 → 2,
{ rintro h K hK1 hK2,
exact (tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact hK2).mp (h.mono hK1) },
tfae_have : 2 → 3,
{ rintro h x hx,
obtain ⟨K, ⟨hK1, hK2⟩, hK3⟩ := (compact_basis_nhds x).mem_iff.mp (hs.mem_nhds hx),
refine ⟨K, nhds_within_le_nhds hK1, h ... | lemma | tendsto_locally_uniformly_on_tfae | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"compact_basis_nhds",
"filter",
"is_compact",
"is_open",
"locally_compact_space",
"nhds_within_le_nhds",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact",
"tendsto_uniformly_on"
] | null | 789 | 807 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff_forall_is_compact [locally_compact_space α]
(hs : is_open s) :
tendsto_locally_uniformly_on F f p s ↔
∀ K ⊆ s, is_compact K → tendsto_uniformly_on F f p K :=
(tendsto_locally_uniformly_on_tfae F f p hs).out 0 1 | tendsto_locally_uniformly_on_iff_forall_is_compact [locally_compact_space α]
(hs : is_open s) :
tendsto_locally_uniformly_on F f p s ↔
∀ K ⊆ s, is_compact K → tendsto_uniformly_on F f p K | (tendsto_locally_uniformly_on_tfae F f p hs).out 0 1 | lemma | tendsto_locally_uniformly_on_iff_forall_is_compact | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"is_compact",
"is_open",
"locally_compact_space",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_tfae",
"tendsto_uniformly_on"
] | null | 809 | 813 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff_filter :
tendsto_locally_uniformly_on F f p s ↔
∀ x ∈ s, tendsto_uniformly_on_filter F f p (𝓝[s] x) :=
begin
simp only [tendsto_uniformly_on_filter, eventually_prod_iff],
split,
{ rintro h x hx u hu,
obtain ⟨s, hs1, hs2⟩ := h u hu x hx,
exact ⟨_, hs2, _, eventually_of... | tendsto_locally_uniformly_on_iff_filter :
tendsto_locally_uniformly_on F f p s ↔
∀ x ∈ s, tendsto_uniformly_on_filter F f p (𝓝[s] x) | begin
simp only [tendsto_uniformly_on_filter, eventually_prod_iff],
split,
{ rintro h x hx u hu,
obtain ⟨s, hs1, hs2⟩ := h u hu x hx,
exact ⟨_, hs2, _, eventually_of_mem hs1 (λ x, id), λ i hi y hy, hi y hy⟩ },
{ rintro h u hu x hx,
obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu,
refine ⟨pb, hpb, ev... | lemma | tendsto_locally_uniformly_on_iff_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_locally_uniformly_on",
"tendsto_uniformly_on_filter"
] | null | 815 | 827 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_iff_filter :
tendsto_locally_uniformly F f p ↔
∀ x, tendsto_uniformly_on_filter F f p (𝓝 x) :=
by simpa [← tendsto_locally_uniformly_on_univ, ← nhds_within_univ] using
@tendsto_locally_uniformly_on_iff_filter _ _ _ _ F f univ p _ | tendsto_locally_uniformly_iff_filter :
tendsto_locally_uniformly F f p ↔
∀ x, tendsto_uniformly_on_filter F f p (𝓝 x) | by simpa [← tendsto_locally_uniformly_on_univ, ← nhds_within_univ] using
@tendsto_locally_uniformly_on_iff_filter _ _ _ _ F f univ p _ | lemma | tendsto_locally_uniformly_iff_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"nhds_within_univ",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on_iff_filter",
"tendsto_locally_uniformly_on_univ",
"tendsto_uniformly_on_filter"
] | null | 829 | 833 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.tendsto_at (hf : tendsto_locally_uniformly_on F f p s)
{a : α} (ha : a ∈ s) :
tendsto (λ i, F i a) p (𝓝 (f a)) :=
begin
refine ((tendsto_locally_uniformly_on_iff_filter.mp hf) a ha).tendsto_at _,
simpa only [filter.principal_singleton] using pure_le_nhds_within ha
end | tendsto_locally_uniformly_on.tendsto_at (hf : tendsto_locally_uniformly_on F f p s)
{a : α} (ha : a ∈ s) :
tendsto (λ i, F i a) p (𝓝 (f a)) | begin
refine ((tendsto_locally_uniformly_on_iff_filter.mp hf) a ha).tendsto_at _,
simpa only [filter.principal_singleton] using pure_le_nhds_within ha
end | lemma | tendsto_locally_uniformly_on.tendsto_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter.principal_singleton",
"pure_le_nhds_within",
"tendsto_locally_uniformly_on"
] | null | 835 | 841 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.unique [p.ne_bot] [t2_space β] {g : α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : tendsto_locally_uniformly_on F g p s) :
s.eq_on f g :=
λ a ha, tendsto_nhds_unique (hf.tendsto_at ha) (hg.tendsto_at ha) | tendsto_locally_uniformly_on.unique [p.ne_bot] [t2_space β] {g : α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : tendsto_locally_uniformly_on F g p s) :
s.eq_on f g | λ a ha, tendsto_nhds_unique (hf.tendsto_at ha) (hg.tendsto_at ha) | lemma | tendsto_locally_uniformly_on.unique | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"t2_space",
"tendsto_locally_uniformly_on",
"tendsto_nhds_unique"
] | null | 843 | 846 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.congr {G : ι → α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : ∀ n, s.eq_on (F n) (G n)) :
tendsto_locally_uniformly_on G f p s :=
begin
rintro u hu x hx,
obtain ⟨t, ht, h⟩ := hf u hu x hx,
refine ⟨s ∩ t, inter_mem self_mem_nhds_within ht, _⟩,
filter_upwards [h] with i ... | tendsto_locally_uniformly_on.congr {G : ι → α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : ∀ n, s.eq_on (F n) (G n)) :
tendsto_locally_uniformly_on G f p s | begin
rintro u hu x hx,
obtain ⟨t, ht, h⟩ := hf u hu x hx,
refine ⟨s ∩ t, inter_mem self_mem_nhds_within ht, _⟩,
filter_upwards [h] with i hi y hy using hg i hy.1 ▸ hi y hy.2
end | lemma | tendsto_locally_uniformly_on.congr | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"self_mem_nhds_within",
"tendsto_locally_uniformly_on"
] | null | 848 | 856 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.congr_right {g : α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : s.eq_on f g) :
tendsto_locally_uniformly_on F g p s :=
begin
rintro u hu x hx,
obtain ⟨t, ht, h⟩ := hf u hu x hx,
refine ⟨s ∩ t, inter_mem self_mem_nhds_within ht, _⟩,
filter_upwards [h] with i hi y hy usi... | tendsto_locally_uniformly_on.congr_right {g : α → β}
(hf : tendsto_locally_uniformly_on F f p s) (hg : s.eq_on f g) :
tendsto_locally_uniformly_on F g p s | begin
rintro u hu x hx,
obtain ⟨t, ht, h⟩ := hf u hu x hx,
refine ⟨s ∩ t, inter_mem self_mem_nhds_within ht, _⟩,
filter_upwards [h] with i hi y hy using hg hy.1 ▸ hi y hy.2
end | lemma | tendsto_locally_uniformly_on.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"self_mem_nhds_within",
"tendsto_locally_uniformly_on"
] | null | 858 | 866 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_of_locally_uniform_approx_of_continuous_within_at
(hx : x ∈ s) (L : ∀ u ∈ 𝓤 β, ∃ (t ∈ 𝓝[s] x) (F : α → β), continuous_within_at F s x ∧
∀ y ∈ t, (f y, F y) ∈ u) : continuous_within_at f s x :=
begin
apply uniform.continuous_within_at_iff'_left.2 (λ u₀ hu₀, _),
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ (... | continuous_within_at_of_locally_uniform_approx_of_continuous_within_at
(hx : x ∈ s) (L : ∀ u ∈ 𝓤 β, ∃ (t ∈ 𝓝[s] x) (F : α → β), continuous_within_at F s x ∧
∀ y ∈ t, (f y, F y) ∈ u) : continuous_within_at f s x | begin
apply uniform.continuous_within_at_iff'_left.2 (λ u₀ hu₀, _),
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ (u : set (β × β)) (H : u ∈ 𝓤 β), comp_rel u u ⊆ u₀ :=
comp_mem_uniformity_sets hu₀,
obtain ⟨u₂, h₂, hsymm, u₂₁⟩ : ∃ (u : set (β × β)) (H : u ∈ 𝓤 β),
(∀{a b}, (a, b) ∈ u → (b, a) ∈ u) ∧ comp_rel u u ⊆ u₁ := comp_... | lemma | continuous_within_at_of_locally_uniform_approx_of_continuous_within_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"comp_mem_uniformity_sets",
"comp_rel",
"comp_symm_of_uniformity",
"continuous_within_at",
"prod_mk_mem_comp_rel",
"refl_mem_uniformity"
] | A function which can be locally uniformly approximated by functions which are continuous
within a set at a point is continuous within this set at this point. | 879 | 897 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_locally_uniform_approx_of_continuous_at
(L : ∀ u ∈ 𝓤 β, ∃ (t ∈ 𝓝 x) F, continuous_at F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
continuous_at f x :=
begin
rw ← continuous_within_at_univ,
apply continuous_within_at_of_locally_uniform_approx_of_continuous_within_at (mem_univ _) _,
simpa only [exists_pr... | continuous_at_of_locally_uniform_approx_of_continuous_at
(L : ∀ u ∈ 𝓤 β, ∃ (t ∈ 𝓝 x) F, continuous_at F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
continuous_at f x | begin
rw ← continuous_within_at_univ,
apply continuous_within_at_of_locally_uniform_approx_of_continuous_within_at (mem_univ _) _,
simpa only [exists_prop, nhds_within_univ, continuous_within_at_univ] using L
end | lemma | continuous_at_of_locally_uniform_approx_of_continuous_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_at",
"continuous_within_at_of_locally_uniform_approx_of_continuous_within_at",
"continuous_within_at_univ",
"exists_prop",
"nhds_within_univ"
] | A function which can be locally uniformly approximated by functions which are continuous at
a point is continuous at this point. | 901 | 908 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_of_locally_uniform_approx_of_continuous_within_at
(L : ∀ (x ∈ s) (u ∈ 𝓤 β), ∃ (t ∈ 𝓝[s] x) F,
continuous_within_at F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) : continuous_on f s :=
λ x hx, continuous_within_at_of_locally_uniform_approx_of_continuous_within_at hx (L x hx) | continuous_on_of_locally_uniform_approx_of_continuous_within_at
(L : ∀ (x ∈ s) (u ∈ 𝓤 β), ∃ (t ∈ 𝓝[s] x) F,
continuous_within_at F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) : continuous_on f s | λ x hx, continuous_within_at_of_locally_uniform_approx_of_continuous_within_at hx (L x hx) | lemma | continuous_on_of_locally_uniform_approx_of_continuous_within_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_on",
"continuous_within_at",
"continuous_within_at_of_locally_uniform_approx_of_continuous_within_at"
] | A function which can be locally uniformly approximated by functions which are continuous
on a set is continuous on this set. | 912 | 915 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_of_uniform_approx_of_continuous_on
(L : ∀ u ∈ 𝓤 β, ∃ F, continuous_on F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) : continuous_on f s :=
continuous_on_of_locally_uniform_approx_of_continuous_within_at $
λ x hx u hu, ⟨s, self_mem_nhds_within, (L u hu).imp $
λ F hF, ⟨hF.1.continuous_within_at hx, hF.2⟩⟩ | continuous_on_of_uniform_approx_of_continuous_on
(L : ∀ u ∈ 𝓤 β, ∃ F, continuous_on F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) : continuous_on f s | continuous_on_of_locally_uniform_approx_of_continuous_within_at $
λ x hx u hu, ⟨s, self_mem_nhds_within, (L u hu).imp $
λ F hF, ⟨hF.1.continuous_within_at hx, hF.2⟩⟩ | lemma | continuous_on_of_uniform_approx_of_continuous_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_on",
"continuous_on_of_locally_uniform_approx_of_continuous_within_at",
"self_mem_nhds_within"
] | A function which can be uniformly approximated by functions which are continuous on a set
is continuous on this set. | 919 | 923 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_of_locally_uniform_approx_of_continuous_at
(L : ∀ (x : α), ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, continuous_at F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
continuous f :=
continuous_iff_continuous_at.2 $ λ x, continuous_at_of_locally_uniform_approx_of_continuous_at (L x) | continuous_of_locally_uniform_approx_of_continuous_at
(L : ∀ (x : α), ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, continuous_at F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
continuous f | continuous_iff_continuous_at.2 $ λ x, continuous_at_of_locally_uniform_approx_of_continuous_at (L x) | lemma | continuous_of_locally_uniform_approx_of_continuous_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous",
"continuous_at",
"continuous_at_of_locally_uniform_approx_of_continuous_at"
] | A function which can be locally uniformly approximated by continuous functions is continuous. | 926 | 929 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_of_uniform_approx_of_continuous
(L : ∀ u ∈ 𝓤 β, ∃ F, continuous F ∧ ∀ y, (f y, F y) ∈ u) : continuous f :=
continuous_iff_continuous_on_univ.mpr $ continuous_on_of_uniform_approx_of_continuous_on $
by simpa [continuous_iff_continuous_on_univ] using L | continuous_of_uniform_approx_of_continuous
(L : ∀ u ∈ 𝓤 β, ∃ F, continuous F ∧ ∀ y, (f y, F y) ∈ u) : continuous f | continuous_iff_continuous_on_univ.mpr $ continuous_on_of_uniform_approx_of_continuous_on $
by simpa [continuous_iff_continuous_on_univ] using L | lemma | continuous_of_uniform_approx_of_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_of_uniform_approx_of_continuous_on"
] | A function which can be uniformly approximated by continuous functions is continuous. | 932 | 935 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.continuous_on
(h : tendsto_locally_uniformly_on F f p s) (hc : ∀ᶠ n in p, continuous_on (F n) s) [ne_bot p] :
continuous_on f s :=
begin
apply continuous_on_of_locally_uniform_approx_of_continuous_within_at (λ x hx u hu, _),
rcases h u hu x hx with ⟨t, ht, H⟩,
rcases (hc.and H).ex... | tendsto_locally_uniformly_on.continuous_on
(h : tendsto_locally_uniformly_on F f p s) (hc : ∀ᶠ n in p, continuous_on (F n) s) [ne_bot p] :
continuous_on f s | begin
apply continuous_on_of_locally_uniform_approx_of_continuous_within_at (λ x hx u hu, _),
rcases h u hu x hx with ⟨t, ht, H⟩,
rcases (hc.and H).exists with ⟨n, hFc, hF⟩,
exact ⟨t, ht, ⟨F n, hFc.continuous_within_at hx, hF⟩⟩
end | lemma | tendsto_locally_uniformly_on.continuous_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_on",
"continuous_on_of_locally_uniform_approx_of_continuous_within_at",
"tendsto_locally_uniformly_on"
] | A locally uniform limit on a set of functions which are continuous on this set is itself
continuous on this set. | 946 | 954 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on.continuous_on (h : tendsto_uniformly_on F f p s)
(hc : ∀ᶠ n in p, continuous_on (F n) s) [ne_bot p] : continuous_on f s :=
h.tendsto_locally_uniformly_on.continuous_on hc | tendsto_uniformly_on.continuous_on (h : tendsto_uniformly_on F f p s)
(hc : ∀ᶠ n in p, continuous_on (F n) s) [ne_bot p] : continuous_on f s | h.tendsto_locally_uniformly_on.continuous_on hc | lemma | tendsto_uniformly_on.continuous_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_on",
"tendsto_uniformly_on"
] | A uniform limit on a set of functions which are continuous on this set is itself continuous
on this set. | 958 | 960 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly.continuous (h : tendsto_locally_uniformly F f p)
(hc : ∀ᶠ n in p, continuous (F n)) [ne_bot p] : continuous f :=
continuous_iff_continuous_on_univ.mpr $ h.tendsto_locally_uniformly_on.continuous_on $
hc.mono $ λ n hn, hn.continuous_on | tendsto_locally_uniformly.continuous (h : tendsto_locally_uniformly F f p)
(hc : ∀ᶠ n in p, continuous (F n)) [ne_bot p] : continuous f | continuous_iff_continuous_on_univ.mpr $ h.tendsto_locally_uniformly_on.continuous_on $
hc.mono $ λ n hn, hn.continuous_on | lemma | tendsto_locally_uniformly.continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous",
"tendsto_locally_uniformly"
] | A locally uniform limit of continuous functions is continuous. | 963 | 966 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly.continuous (h : tendsto_uniformly F f p)
(hc : ∀ᶠ n in p, continuous (F n)) [ne_bot p] : continuous f :=
h.tendsto_locally_uniformly.continuous hc | tendsto_uniformly.continuous (h : tendsto_uniformly F f p)
(hc : ∀ᶠ n in p, continuous (F n)) [ne_bot p] : continuous f | h.tendsto_locally_uniformly.continuous hc | lemma | tendsto_uniformly.continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous",
"tendsto_uniformly"
] | A uniform limit of continuous functions is continuous. | 969 | 971 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_comp_of_locally_uniform_limit_within
(h : continuous_within_at f s x) (hg : tendsto g p (𝓝[s] x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
begin
apply uniform.tendsto_nhds_right.2 (λ u₀ hu₀, _),
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ (u : ... | tendsto_comp_of_locally_uniform_limit_within
(h : continuous_within_at f s x) (hg : tendsto g p (𝓝[s] x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) | begin
apply uniform.tendsto_nhds_right.2 (λ u₀ hu₀, _),
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ (u : set (β × β)) (H : u ∈ 𝓤 β), comp_rel u u ⊆ u₀ :=
comp_mem_uniformity_sets hu₀,
rcases hunif u₁ h₁ with ⟨s, sx, hs⟩,
have A : ∀ᶠ n in p, g n ∈ s := hg sx,
have B : ∀ᶠ n in p, (f x, f (g n)) ∈ u₁ := hg (uniform.continuous... | lemma | tendsto_comp_of_locally_uniform_limit_within | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"comp_mem_uniformity_sets",
"comp_rel",
"continuous_within_at",
"prod_mk_mem_comp_rel"
] | If `Fₙ` converges locally uniformly on a neighborhood of `x` within a set `s` to a function `f`
which is continuous at `x` within `s `, and `gₙ` tends to `x` within `s`, then `Fₙ (gₙ)` tends
to `f x`. | 984 | 998 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_comp_of_locally_uniform_limit (h : continuous_at f x) (hg : tendsto g p (𝓝 x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
begin
rw ← continuous_within_at_univ at h,
rw ← nhds_within_univ at hunif hg,
exact tendsto_comp_of_locally_... | tendsto_comp_of_locally_uniform_limit (h : continuous_at f x) (hg : tendsto g p (𝓝 x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) | begin
rw ← continuous_within_at_univ at h,
rw ← nhds_within_univ at hunif hg,
exact tendsto_comp_of_locally_uniform_limit_within h hg hunif
end | lemma | tendsto_comp_of_locally_uniform_limit | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_at",
"continuous_within_at_univ",
"nhds_within_univ",
"tendsto_comp_of_locally_uniform_limit_within"
] | If `Fₙ` converges locally uniformly on a neighborhood of `x` to a function `f` which is
continuous at `x`, and `gₙ` tends to `x`, then `Fₙ (gₙ)` tends to `f x`. | 1,002 | 1,009 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on.tendsto_comp (h : tendsto_locally_uniformly_on F f p s)
(hf : continuous_within_at f s x) (hx : x ∈ s) (hg : tendsto g p (𝓝[s] x)) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
tendsto_comp_of_locally_uniform_limit_within hf hg (λ u hu, h u hu x hx) | tendsto_locally_uniformly_on.tendsto_comp (h : tendsto_locally_uniformly_on F f p s)
(hf : continuous_within_at f s x) (hx : x ∈ s) (hg : tendsto g p (𝓝[s] x)) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit_within hf hg (λ u hu, h u hu x hx) | lemma | tendsto_locally_uniformly_on.tendsto_comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_within_at",
"tendsto_comp_of_locally_uniform_limit_within",
"tendsto_locally_uniformly_on"
] | If `Fₙ` tends locally uniformly to `f` on a set `s`, and `gₙ` tends to `x` within `s`, then
`Fₙ gₙ` tends to `f x` if `f` is continuous at `x` within `s` and `x ∈ s`. | 1,013 | 1,016 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on.tendsto_comp (h : tendsto_uniformly_on F f p s)
(hf : continuous_within_at f s x) (hg : tendsto g p (𝓝[s] x)) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
tendsto_comp_of_locally_uniform_limit_within hf hg (λ u hu,
⟨s, self_mem_nhds_within, h u hu⟩) | tendsto_uniformly_on.tendsto_comp (h : tendsto_uniformly_on F f p s)
(hf : continuous_within_at f s x) (hg : tendsto g p (𝓝[s] x)) :
tendsto (λ n, F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit_within hf hg (λ u hu,
⟨s, self_mem_nhds_within, h u hu⟩) | lemma | tendsto_uniformly_on.tendsto_comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_within_at",
"self_mem_nhds_within",
"tendsto_comp_of_locally_uniform_limit_within",
"tendsto_uniformly_on"
] | If `Fₙ` tends uniformly to `f` on a set `s`, and `gₙ` tends to `x` within `s`, then `Fₙ gₙ`
tends to `f x` if `f` is continuous at `x` within `s`. | 1,020 | 1,024 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly.tendsto_comp (h : tendsto_locally_uniformly F f p)
(hf : continuous_at f x) (hg : tendsto g p (𝓝 x)) : tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
tendsto_comp_of_locally_uniform_limit hf hg (λ u hu, h u hu x) | tendsto_locally_uniformly.tendsto_comp (h : tendsto_locally_uniformly F f p)
(hf : continuous_at f x) (hg : tendsto g p (𝓝 x)) : tendsto (λ n, F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit hf hg (λ u hu, h u hu x) | lemma | tendsto_locally_uniformly.tendsto_comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_at",
"tendsto_comp_of_locally_uniform_limit",
"tendsto_locally_uniformly"
] | If `Fₙ` tends locally uniformly to `f`, and `gₙ` tends to `x`, then `Fₙ gₙ` tends to `f x`. | 1,027 | 1,029 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly.tendsto_comp (h : tendsto_uniformly F f p)
(hf : continuous_at f x) (hg : tendsto g p (𝓝 x)) : tendsto (λ n, F n (g n)) p (𝓝 (f x)) :=
h.tendsto_locally_uniformly.tendsto_comp hf hg | tendsto_uniformly.tendsto_comp (h : tendsto_uniformly F f p)
(hf : continuous_at f x) (hg : tendsto g p (𝓝 x)) : tendsto (λ n, F n (g n)) p (𝓝 (f x)) | h.tendsto_locally_uniformly.tendsto_comp hf hg | lemma | tendsto_uniformly.tendsto_comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"continuous_at",
"tendsto_uniformly"
] | If `Fₙ` tends uniformly to `f`, and `gₙ` tends to `x`, then `Fₙ gₙ` tends to `f x`. | 1,032 | 1,034 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_fun (α β : Type*) := α → β | uniform_fun (α β : Type*) | α → β | def | uniform_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | The type of functions from `α` to `β` equipped with the uniform structure and topology of
uniform convergence. We denote it `α →ᵤ β`. | 145 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_on_fun (α β : Type*) (𝔖 : set (set α)) := α → β | uniform_on_fun (α β : Type*) (𝔖 : set (set α)) | α → β | def | uniform_on_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | The type of functions from `α` to `β` equipped with the uniform structure and topology of
uniform convergence on some family `𝔖` of subsets of `α`. We denote it `α →ᵤ[𝔖] β`. | 149 | 150 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{α β} [nonempty β] : nonempty (α →ᵤ β) := pi.nonempty | {α β} [nonempty β] : nonempty (α →ᵤ β) | pi.nonempty | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | null | 159 | 159 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{α β 𝔖} [nonempty β] : nonempty (α →ᵤ[𝔖] β) := pi.nonempty | {α β 𝔖} [nonempty β] : nonempty (α →ᵤ[𝔖] β) | pi.nonempty | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | null | 160 | 160 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_fun.of_fun {α β} : (α → β) ≃ (α →ᵤ β) := ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | uniform_fun.of_fun {α β} : (α → β) ≃ (α →ᵤ β) | ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | def | uniform_fun.of_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Reinterpret `f : α → β` as an element of `α →ᵤ β`. | 163 | 163 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_on_fun.of_fun {α β} (𝔖) : (α → β) ≃ (α →ᵤ[𝔖] β) := ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | uniform_on_fun.of_fun {α β} (𝔖) : (α → β) ≃ (α →ᵤ[𝔖] β) | ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | def | uniform_on_fun.of_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Reinterpret `f : α → β` as an element of `α →ᵤ[𝔖] β`. | 166 | 166 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_fun.to_fun {α β} : (α →ᵤ β) ≃ (α → β) := uniform_fun.of_fun.symm | uniform_fun.to_fun {α β} : (α →ᵤ β) ≃ (α → β) | uniform_fun.of_fun.symm | def | uniform_fun.to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Reinterpret `f : α →ᵤ β` as an element of `α → β`. | 169 | 169 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_on_fun.to_fun {α β} (𝔖) : (α →ᵤ[𝔖] β) ≃ (α → β) := (uniform_on_fun.of_fun 𝔖).symm | uniform_on_fun.to_fun {α β} (𝔖) : (α →ᵤ[𝔖] β) ≃ (α → β) | (uniform_on_fun.of_fun 𝔖).symm | def | uniform_on_fun.to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_on_fun.of_fun"
] | Reinterpret `f : α →ᵤ[𝔖] β` as an element of `α → β`. | 172 | 172 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen (V : set (β × β)) : set ((α →ᵤ β) × (α →ᵤ β)) :=
{uv : (α →ᵤ β) × (α →ᵤ β) | ∀ x, (uv.1 x, uv.2 x) ∈ V} | gen (V : set (β × β)) : set ((α →ᵤ β) × (α →ᵤ β)) | {uv : (α →ᵤ β) × (α →ᵤ β) | ∀ x, (uv.1 x, uv.2 x) ∈ V} | def | uniform_fun.gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Basis sets for the uniformity of uniform convergence: `gen α β V` is the set of pairs `(f, g)`
of functions `α →ᵤ β` such that `∀ x, (f x, g x) ∈ V`. | 188 | 189 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_basis_gen (𝓑 : filter $ β × β) :
is_basis (λ V : set (β × β), V ∈ 𝓑) (uniform_fun.gen α β) :=
⟨⟨univ, univ_mem⟩, λ U V hU hV, ⟨U ∩ V, inter_mem hU hV, λ uv huv,
⟨λ x, (huv x).left, λ x, (huv x).right⟩⟩⟩ | is_basis_gen (𝓑 : filter $ β × β) :
is_basis (λ V : set (β × β), V ∈ 𝓑) (uniform_fun.gen α β) | ⟨⟨univ, univ_mem⟩, λ U V hU hV, ⟨U ∩ V, inter_mem hU hV, λ uv huv,
⟨λ x, (huv x).left, λ x, (huv x).right⟩⟩⟩ | lemma | uniform_fun.is_basis_gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter",
"uniform_fun.gen"
] | If `𝓕` is a filter on `β × β`, then the set of all `uniform_convergence.gen α β V` for
`V ∈ 𝓕` is a filter basis on `(α →ᵤ β) × (α →ᵤ β)`. This will only be applied to `𝓕 = 𝓤 β` when
`β` is equipped with a `uniform_space` structure, but it is useful to define it for any filter in
order to be able to state that it h... | 195 | 198 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis (𝓕 : filter $ β × β) : filter_basis ((α →ᵤ β) × (α →ᵤ β)) :=
(uniform_fun.is_basis_gen α β 𝓕).filter_basis | basis (𝓕 : filter $ β × β) : filter_basis ((α →ᵤ β) × (α →ᵤ β)) | (uniform_fun.is_basis_gen α β 𝓕).filter_basis | def | uniform_fun.basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"basis",
"filter",
"filter_basis",
"uniform_fun.is_basis_gen"
] | For `𝓕 : filter (β × β)`, this is the set of all `uniform_convergence.gen α β V` for
`V ∈ 𝓕` as a bundled `filter_basis` over `(α →ᵤ β) × (α →ᵤ β)`. This will only be applied to
`𝓕 = 𝓤 β` when `β` is equipped with a `uniform_space` structure, but it is useful to define it for
any filter in order to be able to state... | 205 | 206 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter (𝓕 : filter $ β × β) : filter ((α →ᵤ β) × (α →ᵤ β)) :=
(uniform_fun.basis α β 𝓕).filter | filter (𝓕 : filter $ β × β) : filter ((α →ᵤ β) × (α →ᵤ β)) | (uniform_fun.basis α β 𝓕).filter | def | uniform_fun.filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter",
"uniform_fun.basis"
] | For `𝓕 : filter (β × β)`, this is the filter generated by the filter basis
`uniform_convergence.basis α β 𝓕`. For `𝓕 = 𝓤 β`, this will be the uniformity of uniform
convergence on `α`. | 211 | 212 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc : galois_connection lower_adjoint
(λ 𝓕, uniform_fun.filter α β 𝓕) :=
begin
intros 𝓐 𝓕,
symmetry,
calc 𝓐 ≤ uniform_fun.filter α β 𝓕
↔ (uniform_fun.basis α β 𝓕).sets ⊆ 𝓐.sets :
by rw [uniform_fun.filter, ← filter_basis.generate, sets_iff_generate]
... ↔ ∀ U ∈ 𝓕, uniform_fun.gen α β U ∈... | gc : galois_connection lower_adjoint
(λ 𝓕, uniform_fun.filter α β 𝓕) | begin
intros 𝓐 𝓕,
symmetry,
calc 𝓐 ≤ uniform_fun.filter α β 𝓕
↔ (uniform_fun.basis α β 𝓕).sets ⊆ 𝓐.sets :
by rw [uniform_fun.filter, ← filter_basis.generate, sets_iff_generate]
... ↔ ∀ U ∈ 𝓕, uniform_fun.gen α β U ∈ 𝓐 : image_subset_iff
... ↔ ∀ U ∈ 𝓕, {uv | ∀ x, (uv, x) ∈
{t :... | lemma | uniform_fun.gc | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter",
"filter_basis.generate",
"forall₂_congr",
"galois_connection",
"lower_adjoint",
"uniform_fun.basis",
"uniform_fun.filter",
"uniform_fun.gen"
] | The function `uniform_convergence.filter α β : filter (β × β) → filter ((α →ᵤ β) × (α →ᵤ β))`
has a lower adjoint `l` (in the sense of `galois_connection`). The exact definition of `l` is not
interesting; we will only use that it exists (in `uniform_convergence.mono` and
`uniform_convergence.infi_eq`) and that
`l (filt... | 231 | 245 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_core : uniform_space.core (α →ᵤ β) :=
uniform_space.core.mk_of_basis (uniform_fun.basis α β (𝓤 β))
(λ U ⟨V, hV, hVU⟩ f, hVU ▸ λ x, refl_mem_uniformity hV)
(λ U ⟨V, hV, hVU⟩, hVU ▸ ⟨uniform_fun.gen α β (prod.swap ⁻¹' V),
⟨prod.swap ⁻¹' V, tendsto_swap_uniformity hV, rfl⟩, λ uv huv x, huv x⟩)
(λ U ⟨V, ... | uniform_core : uniform_space.core (α →ᵤ β) | uniform_space.core.mk_of_basis (uniform_fun.basis α β (𝓤 β))
(λ U ⟨V, hV, hVU⟩ f, hVU ▸ λ x, refl_mem_uniformity hV)
(λ U ⟨V, hV, hVU⟩, hVU ▸ ⟨uniform_fun.gen α β (prod.swap ⁻¹' V),
⟨prod.swap ⁻¹' V, tendsto_swap_uniformity hV, rfl⟩, λ uv huv x, huv x⟩)
(λ U ⟨V, hV, hVU⟩, hVU ▸ let ⟨W, hW, hWV⟩ := comp_mem_u... | def | uniform_fun.uniform_core | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"comp_mem_uniformity_sets",
"prod.swap",
"refl_mem_uniformity",
"tendsto_swap_uniformity",
"uniform_fun.basis",
"uniform_space.core",
"uniform_space.core.mk_of_basis"
] | Core of the uniform structure of uniform convergence. | 250 | 257 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: uniform_space (α →ᵤ β) :=
uniform_space.of_core (uniform_fun.uniform_core α β) | : uniform_space (α →ᵤ β) | uniform_space.of_core (uniform_fun.uniform_core α β) | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.uniform_core",
"uniform_space",
"uniform_space.of_core"
] | Uniform structure of uniform convergence, declared as an instance on `α →ᵤ β`.
We will denote it `𝒰(α, β, uβ)` in the rest of this file. | 261 | 262 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: topological_space (α →ᵤ β) := infer_instance | : topological_space (α →ᵤ β) | infer_instance | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"topological_space"
] | Topology of uniform convergence, declared as an instance on `α →ᵤ β`. | 265 | 265 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_uniformity :
(𝓤 (α →ᵤ β)).has_basis (λ V, V ∈ 𝓤 β)
(uniform_fun.gen α β) :=
(uniform_fun.is_basis_gen α β (𝓤 β)).has_basis | has_basis_uniformity :
(𝓤 (α →ᵤ β)).has_basis (λ V, V ∈ 𝓤 β)
(uniform_fun.gen α β) | (uniform_fun.is_basis_gen α β (𝓤 β)).has_basis | lemma | uniform_fun.has_basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.gen",
"uniform_fun.is_basis_gen"
] | By definition, the uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}`
for `V ∈ 𝓤 β` as a filter basis. | 271 | 274 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis {ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)}
(h : (𝓤 β).has_basis p s) :
(𝓤 (α →ᵤ β)).has_basis p (uniform_fun.gen α β ∘ s) :=
(uniform_fun.has_basis_uniformity α β).to_has_basis
(λ U hU, let ⟨i, hi, hiU⟩ := h.mem_iff.mp hU in ⟨i, hi, λ uv huv x, hiU (huv x)⟩)
(λ i hi, ⟨s i, ... | has_basis_uniformity_of_basis {ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)}
(h : (𝓤 β).has_basis p s) :
(𝓤 (α →ᵤ β)).has_basis p (uniform_fun.gen α β ∘ s) | (uniform_fun.has_basis_uniformity α β).to_has_basis
(λ U hU, let ⟨i, hi, hiU⟩ := h.mem_iff.mp hU in ⟨i, hi, λ uv huv x, hiU (huv x)⟩)
(λ i hi, ⟨s i, h.mem_of_mem hi, subset_refl _⟩) | lemma | uniform_fun.has_basis_uniformity_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"subset_refl",
"uniform_fun.gen",
"uniform_fun.has_basis_uniformity"
] | The uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as
a filter basis, for any basis `𝓑` of `𝓤 β` (in the case `𝓑 = (𝓤 β).as_basis` this is true by
definition). | 279 | 284 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_of_basis (f) {p : ι → Prop} {s : ι → set (β × β)}
(h : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis p (λ i, {g | (f, g) ∈ uniform_fun.gen α β (s i)}) :=
nhds_basis_uniformity' (uniform_fun.has_basis_uniformity_of_basis α β h) | has_basis_nhds_of_basis (f) {p : ι → Prop} {s : ι → set (β × β)}
(h : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis p (λ i, {g | (f, g) ∈ uniform_fun.gen α β (s i)}) | nhds_basis_uniformity' (uniform_fun.has_basis_uniformity_of_basis α β h) | lemma | uniform_fun.has_basis_nhds_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"nhds_basis_uniformity'",
"uniform_fun.gen",
"uniform_fun.has_basis_uniformity_of_basis"
] | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as a filter
basis, for any basis `𝓑` of `𝓤 β`. | 288 | 291 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds (f) :
(𝓝 f).has_basis (λ V, V ∈ 𝓤 β) (λ V, {g | (f, g) ∈ uniform_fun.gen α β V}) :=
uniform_fun.has_basis_nhds_of_basis α β f (filter.basis_sets _) | has_basis_nhds (f) :
(𝓝 f).has_basis (λ V, V ∈ 𝓤 β) (λ V, {g | (f, g) ∈ uniform_fun.gen α β V}) | uniform_fun.has_basis_nhds_of_basis α β f (filter.basis_sets _) | lemma | uniform_fun.has_basis_nhds | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter.basis_sets",
"uniform_fun.gen",
"uniform_fun.has_basis_nhds_of_basis"
] | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓤 β` as a
filter basis. | 295 | 297 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_eval (x : α) :
uniform_continuous (function.eval x ∘ to_fun : (α →ᵤ β) → β) :=
begin
change _ ≤ _,
rw [map_le_iff_le_comap,
(uniform_fun.has_basis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)],
exact λ U hU, ⟨U, hU, λ uv huv, huv x⟩
end | uniform_continuous_eval (x : α) :
uniform_continuous (function.eval x ∘ to_fun : (α →ᵤ β) → β) | begin
change _ ≤ _,
rw [map_le_iff_le_comap,
(uniform_fun.has_basis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)],
exact λ U hU, ⟨U, hU, λ uv huv, huv x⟩
end | lemma | uniform_fun.uniform_continuous_eval | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"function.eval",
"uniform_continuous",
"uniform_fun.has_basis_uniformity"
] | Evaluation at a fixed point is uniformly continuous on `α →ᵤ β`. | 302 | 309 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono : monotone (@uniform_fun.uniform_space α γ) :=
λ u₁ u₂ hu, (uniform_fun.gc α γ).monotone_u hu | mono : monotone (@uniform_fun.uniform_space α γ) | λ u₁ u₂ hu, (uniform_fun.gc α γ).monotone_u hu | lemma | uniform_fun.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"monotone",
"uniform_fun.gc"
] | If `u₁` and `u₂` are two uniform structures on `γ` and `u₁ ≤ u₂`, then
`𝒰(α, γ, u₁) ≤ 𝒰(α, γ, u₂)`. | 315 | 316 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_eq {u : ι → uniform_space γ} :
(𝒰(α, γ, ⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i) :=
begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
ext : 1,
change uniform_fun.filter α γ (𝓤[⨅ i, u i]) = 𝓤[⨅ i, 𝒰(α, γ, u i)],
rw [infi_uniformity, infi_un... | infi_eq {u : ι → uniform_space γ} :
(𝒰(α, γ, ⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i) | begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
ext : 1,
change uniform_fun.filter α γ (𝓤[⨅ i, u i]) = 𝓤[⨅ i, 𝒰(α, γ, u i)],
rw [infi_uniformity, infi_uniformity],
exact (uniform_fun.gc α γ).u_infi
end | lemma | uniform_fun.infi_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_uniformity",
"uniform_fun.filter",
"uniform_fun.gc",
"uniform_space"
] | If `u` is a family of uniform structures on `γ`, then
`𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i)`. | 320 | 329 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_eq {u₁ u₂ : uniform_space γ} :
(𝒰(α, γ, u₁ ⊓ u₂)) = (𝒰(α, γ, u₁)) ⊓ (𝒰(α, γ, u₂)) :=
begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
rw [inf_eq_infi, inf_eq_infi, uniform_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | inf_eq {u₁ u₂ : uniform_space γ} :
(𝒰(α, γ, u₁ ⊓ u₂)) = (𝒰(α, γ, u₁)) ⊓ (𝒰(α, γ, u₂)) | begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
rw [inf_eq_infi, inf_eq_infi, uniform_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | lemma | uniform_fun.inf_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"inf_eq_infi",
"infi_congr",
"uniform_fun.infi_eq",
"uniform_space"
] | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂)`. | 333 | 341 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq {f : γ → β} :
(𝒰(α, γ, ‹uniform_space β›.comap f)) = (𝒰(α, β, _)).comap ((∘) f) :=
begin
letI : uniform_space γ := ‹uniform_space β›.comap f,
ext : 1,
change (uniform_fun.filter α γ ((𝓤 β).comap _)) =
(uniform_fun.filter α β ((𝓤 β))).comap _,
-- We have the following four Galois connection wh... | comap_eq {f : γ → β} :
(𝒰(α, γ, ‹uniform_space β›.comap f)) = (𝒰(α, β, _)).comap ((∘) f) | begin
letI : uniform_space γ := ‹uniform_space β›.comap f,
ext : 1,
change (uniform_fun.filter α γ ((𝓤 β).comap _)) =
(uniform_fun.filter α β ((𝓤 β))).comap _,
-- We have the following four Galois connection which form a square diagram, and we want
-- to show that the square of upper adjoints is commuta... | lemma | uniform_fun.comap_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter.gc_map_comap",
"galois_connection.u_comm_of_l_comm",
"uniform_fun.filter",
"uniform_fun.gc",
"uniform_space"
] | If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒰(α, γ, comap f u) = comap (λ g, f ∘ g) 𝒰(α, γ, u₁)`. | 345 | 366 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) :=
-- This is a direct consequence of `uniform_convergence.comap_eq`
uniform_continuous_iff.mpr $
calc 𝒰(α, γ, _)
≤ 𝒰(α, γ, ‹uniform_space β›.comap f) :
... | postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) | -- This is a direct consequence of `uniform_convergence.comap_eq`
uniform_continuous_iff.mpr $
calc 𝒰(α, γ, _)
≤ 𝒰(α, γ, ‹uniform_space β›.comap f) :
uniform_fun.mono (uniform_continuous_iff.mp hf)
... = (𝒰(α, β, _)).comap ((∘) f) :
uniform_fun.comap_eq | lemma | uniform_fun.postcomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_fun.comap_eq",
"uniform_fun.mono",
"uniform_space"
] | Post-composition by a uniformly continuous function is uniformly continuous on `α →ᵤ β`.
More precisely, if `f : γ → β` is uniformly continuous, then `(λ g, f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)`
is uniformly continuous. | 372 | 381 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) :=
-- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change co... | postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) | -- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change comap (prod.map (of_fun ∘ (∘) f ∘ to_fun) (of_fun ∘ (∘) f ∘ to_fun)) _ = _,
rw [← uniformity_comap] at ⊢ hf,
congr,
rw [← uniform_space_eq hf, uniform_fu... | lemma | uniform_fun.postcomp_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.comap_eq",
"uniform_inducing",
"uniform_space",
"uniform_space_eq",
"uniformity_comap"
] | Post-composition by a uniform inducing is a uniform inducing for the
uniform structures of uniform convergence.
More precisely, if `f : γ → β` is a uniform inducing, then `(λ g, f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)` is
a uniform inducing. | 388 | 400 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ γ) ≃ᵤ (α →ᵤ β) :=
{ uniform_continuous_to_fun :=
uniform_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.to_equiv) } | congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ γ) ≃ᵤ (α →ᵤ β) | { uniform_continuous_to_fun :=
uniform_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.to_equiv) } | def | uniform_fun.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.Pi_congr_right",
"uniform_fun.postcomp_uniform_continuous",
"uniform_space"
] | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ γ) ≃ᵤ (α →ᵤ β)` by
post-composing. | 404 | 410 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precomp_uniform_continuous {f : γ → α} :
uniform_continuous (λ g : α →ᵤ β, of_fun (g ∘ f)) :=
begin
-- Here we simply go back to filter bases.
rw uniform_continuous_iff,
change 𝓤 (α →ᵤ β) ≤ (𝓤 (γ →ᵤ β)).comap (prod.map (λ g : α →ᵤ β, g ∘ f) (λ g : α →ᵤ β, g ∘ f)),
rw (uniform_fun.has_basis_uniformity α β).l... | precomp_uniform_continuous {f : γ → α} :
uniform_continuous (λ g : α →ᵤ β, of_fun (g ∘ f)) | begin
-- Here we simply go back to filter bases.
rw uniform_continuous_iff,
change 𝓤 (α →ᵤ β) ≤ (𝓤 (γ →ᵤ β)).comap (prod.map (λ g : α →ᵤ β, g ∘ f) (λ g : α →ᵤ β, g ∘ f)),
rw (uniform_fun.has_basis_uniformity α β).le_basis_iff
((uniform_fun.has_basis_uniformity γ β).comap _),
exact λ U hU, ⟨U, hU, λ uv h... | lemma | uniform_fun.precomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_iff",
"uniform_fun.has_basis_uniformity"
] | Pre-composition by a any function is uniformly continuous for the uniform structures of
uniform convergence.
More precisely, for any `f : γ → α`, the function `(λ g, g ∘ f) : (α →ᵤ β) → (γ →ᵤ β)` is uniformly
continuous. | 417 | 426 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_left (e : γ ≃ α) :
(γ →ᵤ β) ≃ᵤ (α →ᵤ β) :=
{ uniform_continuous_to_fun :=
uniform_fun.precomp_uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.precomp_uniform_continuous,
.. equiv.arrow_congr e (equiv.refl _) } | congr_left (e : γ ≃ α) :
(γ →ᵤ β) ≃ᵤ (α →ᵤ β) | { uniform_continuous_to_fun :=
uniform_fun.precomp_uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.precomp_uniform_continuous,
.. equiv.arrow_congr e (equiv.refl _) } | def | uniform_fun.congr_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_congr",
"equiv.refl",
"uniform_fun.precomp_uniform_continuous"
] | Turn a bijection `γ ≃ α` into a uniform isomorphism
`(γ →ᵤ β) ≃ᵤ (α →ᵤ β)` by pre-composing. | 430 | 436 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[t2_space β] : t2_space (α →ᵤ β) :=
{ t2 :=
begin
intros f g h,
obtain ⟨x, hx⟩ := not_forall.mp (mt funext h),
exact separated_by_continuous (uniform_continuous_eval β x).continuous hx
end } | [t2_space β] : t2_space (α →ᵤ β) | { t2 :=
begin
intros f g h,
obtain ⟨x, hx⟩ := not_forall.mp (mt funext h),
exact separated_by_continuous (uniform_continuous_eval β x).continuous hx
end } | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"continuous",
"separated_by_continuous",
"t2_space"
] | The topology of uniform convergence is T₂. | 439 | 445 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_to_fun : uniform_continuous (to_fun : (α →ᵤ β) → α → β) :=
begin
-- By definition of the product uniform structure, this is just `uniform_continuous_eval`.
rw uniform_continuous_pi,
intros x,
exact uniform_continuous_eval β x
end | uniform_continuous_to_fun : uniform_continuous (to_fun : (α →ᵤ β) → α → β) | begin
-- By definition of the product uniform structure, this is just `uniform_continuous_eval`.
rw uniform_continuous_pi,
intros x,
exact uniform_continuous_eval β x
end | lemma | uniform_fun.uniform_continuous_to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_pi"
] | The natural map `uniform_fun.to_fun` from `α →ᵤ β` to `α → β` is uniformly continuous.
In other words, the uniform structure of uniform convergence is finer than that of pointwise
convergence, aka the product uniform structure. | 451 | 457 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_uniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
tendsto F p (𝓝 f) ↔ tendsto_uniformly F f p :=
begin
rw [(uniform_fun.has_basis_nhds α β f).tendsto_right_iff, tendsto_uniformly],
exact iff.rfl,
end | tendsto_iff_tendsto_uniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
tendsto F p (𝓝 f) ↔ tendsto_uniformly F f p | begin
rw [(uniform_fun.has_basis_nhds α β f).tendsto_right_iff, tendsto_uniformly],
exact iff.rfl,
end | lemma | uniform_fun.tendsto_iff_tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"tendsto_uniformly",
"uniform_fun.has_basis_nhds"
] | The topology of uniform convergence indeed gives the same notion of convergence as
`tendsto_uniformly`. | 461 | 466 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ β × γ) ≃ᵤ ((α →ᵤ β) × (α →ᵤ γ)) :=
-- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒰(α, β, uβ) × 𝒰(α, γ, uγ)) = 𝒰(α, β × γ, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `uniform_convergence.in... | uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ β × γ) ≃ᵤ ((α →ᵤ β) × (α →ᵤ γ)) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒰(α, β, uβ) × 𝒰(α, γ, uγ)) = 𝒰(α, β × γ, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `uniform_convergence.inf_eq` and `uniform_convergence.comap_eq`, which leaves us to check
-- that some square c... | def | uniform_fun.uniform_equiv_prod_arrow | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_prod_equiv_prod_arrow",
"uniform_fun.comap_eq",
"uniform_fun.inf_eq",
"uniform_space",
"uniform_space.comap_comap",
"uniform_space.comap_inf",
"uniformity_comap"
] | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ β × γ` and `(α →ᵤ β) × (α →ᵤ γ)`. | 470 | 488 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_Pi_comm : uniform_equiv (α →ᵤ Π i, δ i) (Π i, α →ᵤ δ i) :=
-- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒰(α, δ i, uδ i)) = 𝒰(α, (Π i, δ i), (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence.infi_eq` and ... | uniform_equiv_Pi_comm : uniform_equiv (α →ᵤ Π i, δ i) (Π i, α →ᵤ δ i) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒰(α, δ i, uδ i)) = 𝒰(α, (Π i, δ i), (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence.infi_eq` and `uniform_convergence.comap_eq`, which leaves us to check
-- that some squ... | def | uniform_fun.uniform_equiv_Pi_comm | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"Pi.uniform_space",
"equiv.Pi_comm",
"equiv.to_uniform_equiv_of_uniform_inducing",
"infi_congr",
"uniform_equiv",
"uniform_fun.comap_eq",
"uniform_fun.infi_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.of_core_eq_to_core",
"uniformity_comap"
] | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ (Π i, δ i)` and `Π i, α →ᵤ δ i`. | 494 | 514 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen (𝔖) (S : set α) (V : set (β × β)) : set ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) :=
{uv : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β) | ∀ x ∈ S, (uv.1 x, uv.2 x) ∈ V} | gen (𝔖) (S : set α) (V : set (β × β)) : set ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) | {uv : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β) | ∀ x ∈ S, (uv.1 x, uv.2 x) ∈ V} | def | uniform_on_fun.gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Basis sets for the uniformity of `𝔖`-convergence: for `S : set α` and `V : set (β × β)`,
`gen 𝔖 S V` is the set of pairs `(f, g)` of functions `α →ᵤ[𝔖] β` such that
`∀ x ∈ S, (f x, g x) ∈ V`. Note that the family `𝔖 : set (set α)` is only used to specify which
type alias of `α → β` to use here. | 529 | 530 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_eq_preimage_restrict {𝔖} (S : set α) (V : set (β × β)) :
uniform_on_fun.gen 𝔖 S V =
(prod.map S.restrict S.restrict) ⁻¹' (uniform_fun.gen S β V) :=
begin
ext uv,
exact ⟨λ h ⟨x, hx⟩, h x hx, λ h x hx, h ⟨x, hx⟩⟩
end | gen_eq_preimage_restrict {𝔖} (S : set α) (V : set (β × β)) :
uniform_on_fun.gen 𝔖 S V =
(prod.map S.restrict S.restrict) ⁻¹' (uniform_fun.gen S β V) | begin
ext uv,
exact ⟨λ h ⟨x, hx⟩, h x hx, λ h x hx, h ⟨x, hx⟩⟩
end | lemma | uniform_on_fun.gen_eq_preimage_restrict | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.gen",
"uniform_on_fun.gen"
] | For `S : set α` and `V : set (β × β)`, we have
`uniform_on_fun.gen 𝔖 S V = (S.restrict × S.restrict) ⁻¹' (uniform_fun.gen S β V)`.
This is the crucial fact for proving that the family `uniform_on_fun.gen S V` for `S ∈ 𝔖` and
`V ∈ 𝓤 β` is indeed a basis for the uniformity `α →ᵤ[𝔖] β` endowed with `𝒱(α, β, 𝔖, uβ)`
... | 537 | 543 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_mono {𝔖} {S S' : set α} {V V' : set (β × β)} (hS : S' ⊆ S) (hV : V ⊆ V') :
uniform_on_fun.gen 𝔖 S V ⊆ uniform_on_fun.gen 𝔖 S' V' :=
λ uv h x hx, hV (h x $ hS hx) | gen_mono {𝔖} {S S' : set α} {V V' : set (β × β)} (hS : S' ⊆ S) (hV : V ⊆ V') :
uniform_on_fun.gen 𝔖 S V ⊆ uniform_on_fun.gen 𝔖 S' V' | λ uv h x hx, hV (h x $ hS hx) | lemma | uniform_on_fun.gen_mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_on_fun.gen"
] | `uniform_on_fun.gen` is antitone in the first argument and monotone in the second. | 546 | 548 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_basis_gen (𝔖 : set (set α)) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
(𝓑 : filter_basis $ β × β) :
is_basis (λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓑)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) :=
⟨h.prod 𝓑.nonempty, λ U₁V₁ U₂V₂ h₁ h₂,
let ⟨U₃, hU₃, hU₁₃, hU₂₃⟩ := h' U₁V₁.1 h₁.1 U₂V₂.1 h₂.1 in
... | is_basis_gen (𝔖 : set (set α)) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
(𝓑 : filter_basis $ β × β) :
is_basis (λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓑)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) | ⟨h.prod 𝓑.nonempty, λ U₁V₁ U₂V₂ h₁ h₂,
let ⟨U₃, hU₃, hU₁₃, hU₂₃⟩ := h' U₁V₁.1 h₁.1 U₂V₂.1 h₂.1 in
let ⟨V₃, hV₃, hV₁₂₃⟩ := 𝓑.inter_sets h₁.2 h₂.2 in ⟨⟨U₃, V₃⟩, ⟨⟨hU₃, hV₃⟩, λ uv huv,
⟨(λ x hx, (hV₁₂₃ $ huv x $ hU₁₃ hx).1), (λ x hx, (hV₁₂₃ $ huv x $ hU₂₃ hx).2)⟩⟩⟩⟩ | lemma | uniform_on_fun.is_basis_gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"filter_basis",
"uniform_on_fun.gen"
] | If `𝔖 : set (set α)` is nonempty and directed and `𝓑` is a filter basis on `β × β`, then the
family `uniform_on_fun.gen 𝔖 S V` for `S ∈ 𝔖` and `V ∈ 𝓑` is a filter basis on
`(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)`.
We will show in `has_basis_uniformity_of_basis` that, if `𝓑` is a basis for `𝓤 β`, then the
corresponding filt... | 555 | 562 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: uniform_space (α →ᵤ[𝔖] β) :=
⨅ (s : set α) (hs : s ∈ 𝔖), uniform_space.comap s.restrict
(𝒰(s, β, _)) | : uniform_space (α →ᵤ[𝔖] β) | ⨅ (s : set α) (hs : s ∈ 𝔖), uniform_space.comap s.restrict
(𝒰(s, β, _)) | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_space",
"uniform_space.comap"
] | Uniform structure of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`,
declared as an instance on `α →ᵤ[𝔖] β`. It is defined as the infimum, for `S ∈ 𝔖`, of the pullback
by `S.restrict`, the map of restriction to `S`, of the uniform structure `𝒰(s, β, uβ)` on
`↥S →ᵤ β`. We will denote it `𝒱(α, β, �... | 570 | 572 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: topological_space (α →ᵤ[𝔖] β) :=
(𝒱(α, β, 𝔖, _)).to_topological_space | : topological_space (α →ᵤ[𝔖] β) | (𝒱(α, β, 𝔖, _)).to_topological_space | instance | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"topological_space"
] | Topology of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`, declared as an
instance on `α →ᵤ[𝔖] β`. | 578 | 579 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space_eq :
uniform_on_fun.topological_space α β 𝔖 = ⨅ (s : set α) (hs : s ∈ 𝔖),
topological_space.induced s.restrict (uniform_fun.topological_space s β) :=
begin
simp only [uniform_on_fun.topological_space, to_topological_space_infi,
to_topological_space_infi, to_topological_space_comap],
refl... | topological_space_eq :
uniform_on_fun.topological_space α β 𝔖 = ⨅ (s : set α) (hs : s ∈ 𝔖),
topological_space.induced s.restrict (uniform_fun.topological_space s β) | begin
simp only [uniform_on_fun.topological_space, to_topological_space_infi,
to_topological_space_infi, to_topological_space_comap],
refl
end | lemma | uniform_on_fun.topological_space_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"to_topological_space_comap",
"to_topological_space_infi",
"topological_space.induced",
"topological_space_eq"
] | The topology of `𝔖`-convergence is the infimum, for `S ∈ 𝔖`, of topology induced by the map
of `S.restrict : (α →ᵤ[𝔖] β) → (↥S →ᵤ β)` of restriction to `S`, where `↥S →ᵤ β` is endowed with
the topology of uniform convergence. | 584 | 591 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis_aux₁ {p : ι → Prop} {s : ι → set (β × β)}
(hb : has_basis (𝓤 β) p s) (S : set α) :
(@uniformity (α →ᵤ[𝔖] β) ((uniform_fun.uniform_space S β).comap S.restrict)).has_basis
p (λ i, uniform_on_fun.gen 𝔖 S (s i)) :=
begin
simp_rw [uniform_on_fun.gen_eq_preimage_restrict, uniformity_c... | has_basis_uniformity_of_basis_aux₁ {p : ι → Prop} {s : ι → set (β × β)}
(hb : has_basis (𝓤 β) p s) (S : set α) :
(@uniformity (α →ᵤ[𝔖] β) ((uniform_fun.uniform_space S β).comap S.restrict)).has_basis
p (λ i, uniform_on_fun.gen 𝔖 S (s i)) | begin
simp_rw [uniform_on_fun.gen_eq_preimage_restrict, uniformity_comap],
exact (uniform_fun.has_basis_uniformity_of_basis S β hb).comap _
end | lemma | uniform_on_fun.has_basis_uniformity_of_basis_aux₁ | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.has_basis_uniformity_of_basis",
"uniform_on_fun.gen",
"uniform_on_fun.gen_eq_preimage_restrict",
"uniformity",
"uniformity_comap"
] | null | 593 | 600 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis_aux₂ (h : directed_on (⊆) 𝔖) {p : ι → Prop}
{s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
directed_on ((λ s : set α, (uniform_fun.uniform_space s β).comap
(s.restrict : (α →ᵤ β) → s →ᵤ β)) ⁻¹'o ge) 𝔖 :=
h.mono $ λ s t hst,
((uniform_on_fun.has_basis_uniformity_of_basis_au... | has_basis_uniformity_of_basis_aux₂ (h : directed_on (⊆) 𝔖) {p : ι → Prop}
{s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
directed_on ((λ s : set α, (uniform_fun.uniform_space s β).comap
(s.restrict : (α →ᵤ β) → s →ᵤ β)) ⁻¹'o ge) 𝔖 | h.mono $ λ s t hst,
((uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb _).le_basis_iff
(uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb _)).mpr
(λ V hV, ⟨V, hV, uniform_on_fun.gen_mono hst subset_rfl⟩) | lemma | uniform_on_fun.has_basis_uniformity_of_basis_aux₂ | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"uniform_on_fun.gen_mono",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₁"
] | null | 602 | 609 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, uniform_on_fun.gen 𝔖 Si.1 (s Si.2)) :=
begin
simp only [infi_uniformity],
exact has_basi... | has_basis_uniformity_of_basis (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, uniform_on_fun.gen 𝔖 Si.1 (s Si.2)) | begin
simp only [infi_uniformity],
exact has_basis_binfi_of_directed h (λ S, (uniform_on_fun.gen 𝔖 S) ∘ s) _
(λ S hS, uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb S)
(uniform_on_fun.has_basis_uniformity_of_basis_aux₂ α β 𝔖 h' hb)
end | lemma | uniform_on_fun.has_basis_uniformity_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"infi_uniformity",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₁",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₂"
] | If `𝔖 : set (set α)` is nonempty and directed and `𝓑` is a filter basis of `𝓤 β`, then the
uniformity of `α →ᵤ[𝔖] β` admits the family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and
`V ∈ 𝓑` as a filter basis. | 614 | 624 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) :=
uniform_on_fun.has_basis_uniformity_of_basis α β 𝔖 h h' (𝓤 β).basis_sets | has_basis_uniformity (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) | uniform_on_fun.has_basis_uniformity_of_basis α β 𝔖 h h' (𝓤 β).basis_sets | lemma | uniform_on_fun.has_basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis"
] | If `𝔖 : set (set α)` is nonempty and directed, then the uniformity of `α →ᵤ[𝔖] β` admits the
family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. | 628 | 632 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_of_basis (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 Si.1 (s Si.2)}) :=
begin
letI : uniform_space (α → β) :... | has_basis_nhds_of_basis (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 Si.1 (s Si.2)}) | begin
letI : uniform_space (α → β) := uniform_on_fun.uniform_space α β 𝔖,
exact nhds_basis_uniformity (uniform_on_fun.has_basis_uniformity_of_basis α β 𝔖 h h' hb)
end | lemma | uniform_on_fun.has_basis_nhds_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"nhds_basis_uniformity",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis",
"uniform_space"
] | For `f : α →ᵤ[𝔖] β`, where `𝔖 : set (set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓑` as a filter basis, for any basis
`𝓑` of `𝓤 β`. | 637 | 645 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓝 f).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 SV.1 SV.2}) :=
uniform_on_fun.has_basis_nhds_of_basis α β 𝔖 f h h' (filter.basis_sets _) | has_basis_nhds (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓝 f).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 SV.1 SV.2}) | uniform_on_fun.has_basis_nhds_of_basis α β 𝔖 f h h' (filter.basis_sets _) | lemma | uniform_on_fun.has_basis_nhds | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"filter.basis_sets",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_nhds_of_basis"
] | For `f : α →ᵤ[𝔖] β`, where `𝔖 : set (set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. | 649 | 653 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_restrict (h : s ∈ 𝔖) :
uniform_continuous (uniform_fun.of_fun ∘ (s.restrict : (α → β) → (s → β)) ∘ (to_fun 𝔖)) :=
begin
change _ ≤ _,
simp only [uniform_on_fun.uniform_space, map_le_iff_le_comap, infi_uniformity],
exact infi₂_le s h
end | uniform_continuous_restrict (h : s ∈ 𝔖) :
uniform_continuous (uniform_fun.of_fun ∘ (s.restrict : (α → β) → (s → β)) ∘ (to_fun 𝔖)) | begin
change _ ≤ _,
simp only [uniform_on_fun.uniform_space, map_le_iff_le_comap, infi_uniformity],
exact infi₂_le s h
end | lemma | uniform_on_fun.uniform_continuous_restrict | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_uniformity",
"infi₂_le",
"uniform_continuous",
"uniform_fun.of_fun"
] | If `S ∈ 𝔖`, then the restriction to `S` is a uniformly continuous map from `α →ᵤ[𝔖] β` to
`↥S →ᵤ β`. | 657 | 663 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono ⦃u₁ u₂ : uniform_space γ⦄ (hu : u₁ ≤ u₂) ⦃𝔖₁ 𝔖₂ : set (set α)⦄
(h𝔖 : 𝔖₂ ⊆ 𝔖₁) :
𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂) :=
calc 𝒱(α, γ, 𝔖₁, u₁)
≤ 𝒱(α, γ, 𝔖₂, u₁) : infi_le_infi_of_subset h𝔖
... ≤ 𝒱(α, γ, 𝔖₂, u₂) : infi₂_mono
(λ i hi, uniform_space.comap_mono $ uniform_fun.mono hu) | mono ⦃u₁ u₂ : uniform_space γ⦄ (hu : u₁ ≤ u₂) ⦃𝔖₁ 𝔖₂ : set (set α)⦄
(h𝔖 : 𝔖₂ ⊆ 𝔖₁) :
𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂) | calc 𝒱(α, γ, 𝔖₁, u₁)
≤ 𝒱(α, γ, 𝔖₂, u₁) : infi_le_infi_of_subset h𝔖
... ≤ 𝒱(α, γ, 𝔖₂, u₂) : infi₂_mono
(λ i hi, uniform_space.comap_mono $ uniform_fun.mono hu) | lemma | uniform_on_fun.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_le_infi_of_subset",
"infi₂_mono",
"uniform_fun.mono",
"uniform_space",
"uniform_space.comap_mono"
] | Let `u₁`, `u₂` be two uniform structures on `γ` and `𝔖₁ 𝔖₂ : set (set α)`. If `u₁ ≤ u₂` and
`𝔖₂ ⊆ 𝔖₁` then `𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂)`. | 669 | 675 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
uniform_continuous ((function.eval x : (α → β) → β) ∘ to_fun 𝔖) :=
(uniform_fun.uniform_continuous_eval β (⟨x, hxs⟩ : s)).comp
(uniform_on_fun.uniform_continuous_restrict α β 𝔖 hs) | uniform_continuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
uniform_continuous ((function.eval x : (α → β) → β) ∘ to_fun 𝔖) | (uniform_fun.uniform_continuous_eval β (⟨x, hxs⟩ : s)).comp
(uniform_on_fun.uniform_continuous_restrict α β 𝔖 hs) | lemma | uniform_on_fun.uniform_continuous_eval_of_mem | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"function.eval",
"uniform_continuous",
"uniform_fun.uniform_continuous_eval",
"uniform_on_fun.uniform_continuous_restrict"
] | If `x : α` is in some `S ∈ 𝔖`, then evaluation at `x` is uniformly continuous on
`α →ᵤ[𝔖] β`. | 679 | 682 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_eq {u : ι → uniform_space γ} :
𝒱(α, γ, 𝔖, ⨅ i, u i) =
⨅ i, 𝒱(α, γ, 𝔖, u i) :=
begin
simp_rw [uniform_on_fun.uniform_space, uniform_fun.infi_eq, uniform_space.comap_infi],
rw infi_comm,
exact infi_congr (λ s, infi_comm)
end | infi_eq {u : ι → uniform_space γ} :
𝒱(α, γ, 𝔖, ⨅ i, u i) =
⨅ i, 𝒱(α, γ, 𝔖, u i) | begin
simp_rw [uniform_on_fun.uniform_space, uniform_fun.infi_eq, uniform_space.comap_infi],
rw infi_comm,
exact infi_congr (λ s, infi_comm)
end | lemma | uniform_on_fun.infi_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_comm",
"infi_congr",
"uniform_fun.infi_eq",
"uniform_space",
"uniform_space.comap_infi"
] | If `u` is a family of uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, (⨅ i, u i)) = ⨅ i, 𝒱(α, γ, 𝔖, u i)`. | 688 | 695 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_eq {u₁ u₂ : uniform_space γ} :
𝒱(α, γ, 𝔖, u₁ ⊓ u₂) =
𝒱(α, γ, 𝔖, u₁) ⊓
𝒱(α, γ, 𝔖, u₂) :=
begin
rw [inf_eq_infi, inf_eq_infi, uniform_on_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | inf_eq {u₁ u₂ : uniform_space γ} :
𝒱(α, γ, 𝔖, u₁ ⊓ u₂) =
𝒱(α, γ, 𝔖, u₁) ⊓
𝒱(α, γ, 𝔖, u₂) | begin
rw [inf_eq_infi, inf_eq_infi, uniform_on_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | lemma | uniform_on_fun.inf_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"inf_eq_infi",
"infi_congr",
"uniform_on_fun.infi_eq",
"uniform_space"
] | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, u₁ ⊓ u₂) = 𝒱(α, γ, 𝔖, u₁) ⊓ 𝒱(α, γ, 𝔖, u₂)`. | 699 | 707 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq {f : γ → β} :
𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) =
𝒱(α, β, 𝔖, _).comap ((∘) f) :=
begin
-- We reduce this to `uniform_convergence.comap_eq` using the fact that `comap` distributes
-- on `infi`.
simp_rw [uniform_on_fun.uniform_space, uniform_space.comap_infi,
uniform_fun.comap_eq, ←... | comap_eq {f : γ → β} :
𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) =
𝒱(α, β, 𝔖, _).comap ((∘) f) | begin
-- We reduce this to `uniform_convergence.comap_eq` using the fact that `comap` distributes
-- on `infi`.
simp_rw [uniform_on_fun.uniform_space, uniform_space.comap_infi,
uniform_fun.comap_eq, ← uniform_space.comap_comap],
refl -- by definition, `∀ S ∈ 𝔖, (f ∘ —) ∘ S.restrict = S.restrict ∘ (... | lemma | uniform_on_fun.comap_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.comap_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi"
] | If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒱(α, γ, 𝔖, comap f u) = comap (λ g, f ∘ g) 𝒱(α, γ, 𝔖, u₁)`. | 711 | 720 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) :=
begin
-- This is a direct consequence of `uniform_convergence.comap_eq`
rw uniform_continuous_iff,
calc 𝒱(α, γ, 𝔖, _)
≤ 𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) :... | postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) | begin
-- This is a direct consequence of `uniform_convergence.comap_eq`
rw uniform_continuous_iff,
calc 𝒱(α, γ, 𝔖, _)
≤ 𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) :
uniform_on_fun.mono (uniform_continuous_iff.mp hf) (subset_rfl)
... = 𝒱(α, β, 𝔖, _).comap ((∘) f) :
uniform_on_fun.comap_eq
... | lemma | uniform_on_fun.postcomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"subset_rfl",
"uniform_continuous",
"uniform_continuous_iff",
"uniform_on_fun.comap_eq",
"uniform_on_fun.mono",
"uniform_space"
] | Post-composition by a uniformly continuous function is uniformly continuous for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is uniformly continuous, then
`(λ g, f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous. | 727 | 738 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) :=
-- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change comap (prod.map (of_... | postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) | -- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change comap (prod.map (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖)) _ = _,
rw [← uniformity_comap] at ⊢ hf,
congr,
rw [← uniform_space_eq hf... | lemma | uniform_on_fun.postcomp_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_inducing",
"uniform_on_fun.comap_eq",
"uniform_space",
"uniform_space_eq",
"uniformity_comap"
] | Post-composition by a uniform inducing is a uniform inducing for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is a uniform inducing, then
`(λ g, f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing. | 745 | 757 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) :=
{ uniform_continuous_to_fun :=
uniform_on_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_on_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.t... | congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) | { uniform_continuous_to_fun :=
uniform_on_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_on_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.to_equiv) } | def | uniform_on_fun.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.Pi_congr_right",
"uniform_on_fun.postcomp_uniform_continuous",
"uniform_space"
] | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
by post-composing. | 761 | 767 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precomp_uniform_continuous {𝔗 : set (set γ)} {f : γ → α}
(hf : 𝔗 ⊆ (image f) ⁻¹' 𝔖) :
uniform_continuous (λ g : α →ᵤ[𝔖] β, of_fun 𝔗 (g ∘ f)) :=
begin
-- Since `comap` distributes on `infi`, it suffices to prove that
-- `⨅ s ∈ 𝔖, comap s.restrict 𝒰(↥s, β, uβ) ≤ ⨅ t ∈ 𝔗, comap (t.restrict ∘ (— ∘ f)) 𝒰(↥t... | precomp_uniform_continuous {𝔗 : set (set γ)} {f : γ → α}
(hf : 𝔗 ⊆ (image f) ⁻¹' 𝔖) :
uniform_continuous (λ g : α →ᵤ[𝔖] β, of_fun 𝔗 (g ∘ f)) | begin
-- Since `comap` distributes on `infi`, it suffices to prove that
-- `⨅ s ∈ 𝔖, comap s.restrict 𝒰(↥s, β, uβ) ≤ ⨅ t ∈ 𝔗, comap (t.restrict ∘ (— ∘ f)) 𝒰(↥t, β, uβ)`.
simp_rw [uniform_continuous_iff, uniform_on_fun.uniform_space, uniform_space.comap_infi,
← uniform_space.comap_comap],
-- For ... | lemma | uniform_on_fun.precomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_le_of_le",
"le_infi₂",
"uniform_continuous",
"uniform_continuous_iff",
"uniform_fun.precomp_uniform_continuous",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.comap_mono"
] | Let `f : γ → α`, `𝔖 : set (set α)`, `𝔗 : set (set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`.
Then, the function `(λ g, g ∘ f) : (α →ᵤ[𝔖] β) → (γ →ᵤ[𝔗] β)` is uniformly continuous.
Note that one can easily see that assuming `∀ T ∈ 𝔗, ∃ S ∈ 𝔖, f '' T ⊆ S` would work too, but
we will get this for free when we pr... | 775 | 799 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_left {𝔗 : set (set γ)} (e : γ ≃ α)
(he : 𝔗 ⊆ (image e) ⁻¹' 𝔖) (he' : 𝔖 ⊆ (preimage e) ⁻¹' 𝔗) :
(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β) :=
{ uniform_continuous_to_fun :=
uniform_on_fun.precomp_uniform_continuous
begin
intros s hs,
change e.symm '' s ∈ 𝔗,
rw ← preimage_equiv_eq_image_symm,
... | congr_left {𝔗 : set (set γ)} (e : γ ≃ α)
(he : 𝔗 ⊆ (image e) ⁻¹' 𝔖) (he' : 𝔖 ⊆ (preimage e) ⁻¹' 𝔗) :
(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β) | { uniform_continuous_to_fun :=
uniform_on_fun.precomp_uniform_continuous
begin
intros s hs,
change e.symm '' s ∈ 𝔗,
rw ← preimage_equiv_eq_image_symm,
exact he' hs
end,
uniform_continuous_inv_fun :=
uniform_on_fun.precomp_uniform_continuous he,
.. equiv.arrow_congr e (equiv.... | def | uniform_on_fun.congr_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_congr",
"equiv.refl",
"uniform_on_fun.precomp_uniform_continuous"
] | Turn a bijection `e : γ ≃ α` such that we have both `∀ T ∈ 𝔗, e '' T ∈ 𝔖` and
`∀ S ∈ 𝔖, e ⁻¹' S ∈ 𝔗` into a uniform isomorphism `(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β)` by pre-composing. | 803 | 816 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space_of_covering [t2_space β] (h : ⋃₀ 𝔖 = univ) :
t2_space (α →ᵤ[𝔖] β) :=
{ t2 :=
begin
intros f g hfg,
obtain ⟨x, hx⟩ := not_forall.mp (mt funext hfg),
obtain ⟨s, hs, hxs⟩ : ∃ s ∈ 𝔖, x ∈ s := mem_sUnion.mp (h.symm ▸ true.intro),
exact separated_by_continuous (uniform_continuous_eval_of_mem β... | t2_space_of_covering [t2_space β] (h : ⋃₀ 𝔖 = univ) :
t2_space (α →ᵤ[𝔖] β) | { t2 :=
begin
intros f g hfg,
obtain ⟨x, hx⟩ := not_forall.mp (mt funext hfg),
obtain ⟨s, hs, hxs⟩ : ∃ s ∈ 𝔖, x ∈ s := mem_sUnion.mp (h.symm ▸ true.intro),
exact separated_by_continuous (uniform_continuous_eval_of_mem β 𝔖 hxs hs).continuous hx
end } | lemma | uniform_on_fun.t2_space_of_covering | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"continuous",
"separated_by_continuous",
"t2_space"
] | If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. | 819 | 827 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_to_fun (h : ⋃₀ 𝔖 = univ) :
uniform_continuous (to_fun 𝔖 : (α →ᵤ[𝔖] β) → α → β) :=
begin
rw uniform_continuous_pi,
intros x,
obtain ⟨s : set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := sUnion_eq_univ_iff.mp h x,
exact uniform_continuous_eval_of_mem β 𝔖 hxs hs
end | uniform_continuous_to_fun (h : ⋃₀ 𝔖 = univ) :
uniform_continuous (to_fun 𝔖 : (α →ᵤ[𝔖] β) → α → β) | begin
rw uniform_continuous_pi,
intros x,
obtain ⟨s : set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := sUnion_eq_univ_iff.mp h x,
exact uniform_continuous_eval_of_mem β 𝔖 hxs hs
end | lemma | uniform_on_fun.uniform_continuous_to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_pi"
] | If `𝔖` covers `α`, the natural map `uniform_on_fun.to_fun` from `α →ᵤ[𝔖] β` to `α → β` is
uniformly continuous.
In other words, if `𝔖` covers `α`, then the uniform structure of `𝔖`-convergence is finer than
that of pointwise convergence. | 834 | 841 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_uniformly_on {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
tendsto F p (𝓝 f) ↔
∀ s ∈ 𝔖, tendsto_uniformly_on F f p s :=
begin
rw [uniform_on_fun.topological_space_eq, nhds_infi, tendsto_infi],
refine forall_congr (λ s, _),
rw [nhds_infi, tendsto_infi],
refine forall_congr (λ hs, _),
rw [nh... | tendsto_iff_tendsto_uniformly_on {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
tendsto F p (𝓝 f) ↔
∀ s ∈ 𝔖, tendsto_uniformly_on F f p s | begin
rw [uniform_on_fun.topological_space_eq, nhds_infi, tendsto_infi],
refine forall_congr (λ s, _),
rw [nhds_infi, tendsto_infi],
refine forall_congr (λ hs, _),
rw [nhds_induced, tendsto_comap_iff, tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe,
uniform_fun.tendsto_iff_tendsto_uniformly],
refl... | lemma | uniform_on_fun.tendsto_iff_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"nhds_induced",
"nhds_infi",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe",
"uniform_fun.tendsto_iff_tendsto_uniformly",
"uniform_on_fun.topological_space_eq"
] | Convergence in the topology of `𝔖`-convergence means uniform convergence on `S` (in the sense
of `tendsto_uniformly_on`) for all `S ∈ 𝔖`. | 845 | 856 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ[𝔖] β × γ) ≃ᵤ ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)) :=
-- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒱(α, β, 𝔖, uβ) × 𝒱(α, γ, 𝔖, uγ)) = 𝒱(α, β × γ, 𝔖, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
--... | uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ[𝔖] β × γ) ≃ᵤ ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒱(α, β, 𝔖, uβ) × 𝒱(α, γ, 𝔖, uγ)) = 𝒱(α, β × γ, 𝔖, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `uniform_convergence_on.inf_eq` and `uniform_convergence_on.comap_eq`, which leaves us to check
-- ... | def | uniform_on_fun.uniform_equiv_prod_arrow | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_prod_equiv_prod_arrow",
"inf_uniformity",
"uniform_on_fun.comap_eq",
"uniform_on_fun.inf_eq",
"uniform_on_fun.of_fun",
"uniform_space",
"uniformity_comap",
"uniformity_prod"
] | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] β × γ` and `(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)`. | 860 | 877 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_Pi_comm :
(α →ᵤ[𝔖] Π i, δ i) ≃ᵤ (Π i, α →ᵤ[𝔖] δ i) :=
-- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒱(α, δ i, 𝔖, uδ i)) = 𝒱(α, (Π i, δ i), 𝔖, (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence_on.in... | uniform_equiv_Pi_comm :
(α →ᵤ[𝔖] Π i, δ i) ≃ᵤ (Π i, α →ᵤ[𝔖] δ i) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒱(α, δ i, 𝔖, uδ i)) = 𝒱(α, (Π i, δ i), 𝔖, (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence_on.infi_eq` and `uniform_convergence_on.comap_eq`, which leaves us to check
--... | def | uniform_on_fun.uniform_equiv_Pi_comm | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"Pi.uniform_space",
"equiv.Pi_comm",
"infi_congr",
"uniform_on_fun.comap_eq",
"uniform_on_fun.infi_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.of_core_eq_to_core",
"uniformity_comap"
] | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] (Π i, δ i)` and `Π i, α →ᵤ[𝔖] δ i`. | 883 | 903 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing (f : α → β) : Prop :=
(comap_uniformity : comap (λx:α×α, (f x.1, f x.2)) (𝓤 β) = 𝓤 α) | uniform_inducing (f : α → β) : Prop | (comap_uniformity : comap (λx:α×α, (f x.1, f x.2)) (𝓤 β) = 𝓤 α) | structure | uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [] | A map `f : α → β` between uniform spaces is called *uniform inducing* if the uniformity filter
on `α` is the pullback of the uniformity filter on `β` under `prod.map f f`. If `α` is a separated
space, then this implies that `f` is injective, hence it is a `uniform_embedding`. | 34 | 36 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.comap_uniform_space {f : α → β} (hf : uniform_inducing f) :
‹uniform_space β›.comap f = ‹uniform_space α› :=
uniform_space_eq hf.1 | uniform_inducing.comap_uniform_space {f : α → β} (hf : uniform_inducing f) :
‹uniform_space β›.comap f = ‹uniform_space α› | uniform_space_eq hf.1 | lemma | uniform_inducing.comap_uniform_space | 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_eq"
] | null | 38 | 40 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_iff' {f : α → β} :
uniform_inducing f ↔ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α :=
by rw [uniform_inducing_iff, uniform_continuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; refl | uniform_inducing_iff' {f : α → β} :
uniform_inducing f ↔ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α | by rw [uniform_inducing_iff, uniform_continuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; refl | lemma | uniform_inducing_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_inducing"
] | null | 42 | 44 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.uniform_inducing_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_inducing 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' i → (x, y) ∈ s j... | filter.has_basis.uniform_inducing_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_inducing 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' i → (x, y) ∈ s j... | by simp [uniform_inducing_iff', h.uniform_continuous_iff h', (h'.comap _).le_basis_iff h,
subset_def] | lemma | filter.has_basis.uniform_inducing_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing",
"uniform_inducing_iff'"
] | null | 46 | 52 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.mk' {f : α → β} (h : ∀ s, s ∈ 𝓤 α ↔
∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : uniform_inducing f :=
⟨by simp [eq_comm, filter.ext_iff, subset_def, h]⟩ | uniform_inducing.mk' {f : α → β} (h : ∀ s, s ∈ 𝓤 α ↔
∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : uniform_inducing f | ⟨by simp [eq_comm, filter.ext_iff, subset_def, h]⟩ | lemma | uniform_inducing.mk' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"filter.ext_iff",
"uniform_inducing"
] | null | 54 | 56 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_id : uniform_inducing (@id α) :=
⟨by rw [← prod.map_def, prod.map_id, comap_id]⟩ | uniform_inducing_id : uniform_inducing (@id α) | ⟨by rw [← prod.map_def, prod.map_id, comap_id]⟩ | lemma | uniform_inducing_id | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"prod.map_def",
"prod.map_id",
"uniform_inducing"
] | null | 58 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.