statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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tendsto_uniformly_on_filter.mono_left {p'' : filter ι}
(h : tendsto_uniformly_on_filter F f p p') (hp : p'' ≤ p) :
tendsto_uniformly_on_filter F f p'' p' | λ u hu, (h u hu).filter_mono (p'.prod_mono_left hp) | lemma | tendsto_uniformly_on_filter.mono_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.mono_right {p'' : filter α}
(h : tendsto_uniformly_on_filter F f p p') (hp : p'' ≤ p') :
tendsto_uniformly_on_filter F f p p'' | λ u hu, (h u hu).filter_mono (p.prod_mono_right hp) | lemma | tendsto_uniformly_on_filter.mono_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.mono {s' : set α}
(h : tendsto_uniformly_on F f p s) (h' : s' ⊆ s) : tendsto_uniformly_on F f p s' | tendsto_uniformly_on_iff_tendsto_uniformly_on_filter.mpr
(h.tendsto_uniformly_on_filter.mono_right (le_principal_iff.mpr $ mem_principal.mpr h')) | lemma | tendsto_uniformly_on.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.congr {F' : ι → α → β}
(hf : tendsto_uniformly_on_filter F f p p')
(hff' : ∀ᶠ (n : ι × α) in (p ×ᶠ p'), F n.fst n.snd = F' n.fst n.snd) :
tendsto_uniformly_on_filter F' f p p' | begin
refine (λ u hu, ((hf u hu).and hff').mono (λ n h, _)),
rw ← h.right,
exact h.left,
end | lemma | tendsto_uniformly_on_filter.congr | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.congr {F' : ι → α → β}
(hf : tendsto_uniformly_on F f p s) (hff' : ∀ᶠ n in p, set.eq_on (F n) (F' n) s) :
tendsto_uniformly_on F' f p s | begin
rw tendsto_uniformly_on_iff_tendsto_uniformly_on_filter at hf ⊢,
refine hf.congr _,
rw eventually_iff at hff' ⊢,
simp only [set.eq_on] at hff',
simp only [mem_prod_principal, hff', mem_set_of_eq],
end | lemma | tendsto_uniformly_on.congr | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"set.eq_on",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.congr_right {g : α → β}
(hf : tendsto_uniformly_on F f p s) (hfg : eq_on f g s) :
tendsto_uniformly_on F g p s | λ u hu, by filter_upwards [hf u hu] with i hi a ha using hfg ha ▸ hi a ha | lemma | tendsto_uniformly_on.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.tendsto_uniformly_on
(h : tendsto_uniformly F f p) : tendsto_uniformly_on F f p s | (tendsto_uniformly_on_univ.2 h).mono (subset_univ s) | lemma | tendsto_uniformly.tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.comp (h : tendsto_uniformly_on_filter F f p p') (g : γ → α) :
tendsto_uniformly_on_filter (λ n, F n ∘ g) (f ∘ g) p (p'.comap g) | begin
intros u hu,
obtain ⟨pa, hpa, pb, hpb, hpapb⟩ := eventually_prod_iff.mp (h u hu),
rw eventually_prod_iff,
simp_rw eventually_comap,
exact ⟨pa, hpa, pb ∘ g, ⟨hpb.mono (λ x hx y hy, by simp only [hx, hy, function.comp_app]),
λ x hx y hy, hpapb hx hy⟩⟩,
end | lemma | tendsto_uniformly_on_filter.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on_filter"
] | Composing on the right by a function preserves uniform convergence on a filter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on.comp (h : tendsto_uniformly_on F f p s) (g : γ → α) :
tendsto_uniformly_on (λ n, F n ∘ g) (f ∘ g) p (g ⁻¹' s) | begin
rw tendsto_uniformly_on_iff_tendsto_uniformly_on_filter at h ⊢,
simpa [tendsto_uniformly_on, comap_principal] using (tendsto_uniformly_on_filter.comp h g),
end | lemma | tendsto_uniformly_on.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on",
"tendsto_uniformly_on_filter.comp",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter"
] | Composing on the right by a function preserves uniform convergence on a set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly.comp (h : tendsto_uniformly F f p) (g : γ → α) :
tendsto_uniformly (λ n, F n ∘ g) (f ∘ g) p | begin
rw tendsto_uniformly_iff_tendsto_uniformly_on_filter at h ⊢,
simpa [principal_univ, comap_principal] using (h.comp g),
end | lemma | tendsto_uniformly.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"tendsto_uniformly_iff_tendsto_uniformly_on_filter"
] | Composing on the right by a function preserves uniform convergence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous.comp_tendsto_uniformly_on_filter [uniform_space γ] {g : β → γ}
(hg : uniform_continuous g) (h : tendsto_uniformly_on_filter F f p p') :
tendsto_uniformly_on_filter (λ i, g ∘ (F i)) (g ∘ f) p p' | λ u hu, h _ (hg hu) | lemma | uniform_continuous.comp_tendsto_uniformly_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on_filter",
"uniform_continuous",
"uniform_space"
] | Composing on the left by a uniformly continuous function preserves
uniform convergence on a filter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous.comp_tendsto_uniformly_on [uniform_space γ] {g : β → γ}
(hg : uniform_continuous g) (h : tendsto_uniformly_on F f p s) :
tendsto_uniformly_on (λ i, g ∘ (F i)) (g ∘ f) p s | λ u hu, h _ (hg hu) | lemma | uniform_continuous.comp_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on",
"uniform_continuous",
"uniform_space"
] | Composing on the left by a uniformly continuous function preserves
uniform convergence on a set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous.comp_tendsto_uniformly [uniform_space γ] {g : β → γ}
(hg : uniform_continuous g) (h : tendsto_uniformly F f p) :
tendsto_uniformly (λ i, g ∘ (F i)) (g ∘ f) p | λ u hu, h _ (hg hu) | lemma | uniform_continuous.comp_tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"uniform_continuous",
"uniform_space"
] | Composing on the left by a uniformly continuous function preserves uniform convergence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_filter.prod_map {ι' α' β' : Type*} [uniform_space β']
{F' : ι' → α' → β'} {f' : α' → β'} {q : filter ι'} {q' : filter α'}
(h : tendsto_uniformly_on_filter F f p p') (h' : tendsto_uniformly_on_filter F' f' q q') :
tendsto_uniformly_on_filter (λ (i : ι × ι'), prod.map (F i.1) (F' i.2))
(pro... | begin
intros u hu,
rw [uniformity_prod_eq_prod, mem_map, mem_prod_iff] at hu,
obtain ⟨v, hv, w, hw, hvw⟩ := hu,
apply (tendsto_swap4_prod.eventually ((h v hv).prod_mk (h' w hw))).mono,
simp only [prod_map, and_imp, prod.forall],
intros n n' x hxv hxw,
have hout : ((f x.fst, F n x.fst), (f' x.snd, F' n' x.... | lemma | tendsto_uniformly_on_filter.prod_map | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"and_imp",
"filter",
"mem_map",
"prod_map",
"tendsto_uniformly_on_filter",
"uniform_space",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.prod_map {ι' α' β' : Type*} [uniform_space β']
{F' : ι' → α' → β'} {f' : α' → β'} {p' : filter ι'} {s' : set α'}
(h : tendsto_uniformly_on F f p s) (h' : tendsto_uniformly_on F' f' p' s') :
tendsto_uniformly_on (λ (i : ι × ι'), prod.map (F i.1) (F' i.2))
(prod.map f f') (p.prod p') (s ×ˢ ... | begin
rw tendsto_uniformly_on_iff_tendsto_uniformly_on_filter at h h' ⊢,
simpa only [prod_principal_principal] using (h.prod_map h'),
end | lemma | tendsto_uniformly_on.prod_map | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.prod_map {ι' α' β' : Type*} [uniform_space β'] {F' : ι' → α' → β'}
{f' : α' → β'} {p' : filter ι'} (h : tendsto_uniformly F f p) (h' : tendsto_uniformly F' f' p') :
tendsto_uniformly (λ (i : ι × ι'), prod.map (F i.1) (F' i.2)) (prod.map f f') (p.prod p') | begin
rw [←tendsto_uniformly_on_univ, ←univ_prod_univ] at *,
exact h.prod_map h',
end | lemma | tendsto_uniformly.prod_map | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.prod {ι' β' : Type*} [uniform_space β']
{F' : ι' → α → β'} {f' : α → β'} {q : filter ι'}
(h : tendsto_uniformly_on_filter F f p p') (h' : tendsto_uniformly_on_filter F' f' q p') :
tendsto_uniformly_on_filter (λ (i : ι × ι') a, (F i.1 a, F' i.2 a))
(λ a, (f a, f' a)) (p.prod q) p' | λ u hu, ((h.prod_map h') u hu).diag_of_prod_right | lemma | tendsto_uniformly_on_filter.prod | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on_filter",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.prod {ι' β' : Type*} [uniform_space β'] {F' : ι' → α → β'} {f' : α → β'}
{p' : filter ι'} (h : tendsto_uniformly_on F f p s) (h' : tendsto_uniformly_on F' f' p' s) :
tendsto_uniformly_on (λ (i : ι × ι') a, (F i.1 a, F' i.2 a)) (λ a, (f a, f' a)) (p.prod p') s | (congr_arg _ s.inter_self).mp ((h.prod_map h').comp (λ a, (a, a))) | lemma | tendsto_uniformly_on.prod | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.prod {ι' β' : Type*} [uniform_space β'] {F' : ι' → α → β'} {f' : α → β'}
{p' : filter ι'} (h : tendsto_uniformly F f p) (h' : tendsto_uniformly F' f' p') :
tendsto_uniformly (λ (i : ι × ι') a, (F i.1 a, F' i.2 a)) (λ a, (f a, f' a)) (p.prod p') | (h.prod_map h').comp (λ a, (a, a)) | lemma | tendsto_uniformly.prod | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_filter_iff {c : β} :
tendsto ↿F (p ×ᶠ p') (𝓝 c) ↔ tendsto_uniformly_on_filter F (λ _, c) p p' | begin
simp_rw [tendsto, nhds_eq_comap_uniformity, map_le_iff_le_comap.symm, map_map, le_def, mem_map],
exact forall₂_congr (λ u hu, by simpa [eventually_iff]),
end | lemma | tendsto_prod_filter_iff | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"mem_map",
"nhds_eq_comap_uniformity",
"tendsto_uniformly_on_filter"
] | Uniform convergence on a filter `p'` to a constant function is equivalent to convergence in
`p ×ᶠ p'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_prod_principal_iff {c : β} :
tendsto ↿F (p ×ᶠ 𝓟 s) (𝓝 c) ↔ tendsto_uniformly_on F (λ _, c) p s | begin
rw tendsto_uniformly_on_iff_tendsto_uniformly_on_filter,
exact tendsto_prod_filter_iff,
end | lemma | tendsto_prod_principal_iff | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_prod_filter_iff",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter"
] | Uniform convergence on a set `s` to a constant function is equivalent to convergence in
`p ×ᶠ 𝓟 s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_prod_top_iff {c : β} : tendsto ↿F (p ×ᶠ ⊤) (𝓝 c) ↔ tendsto_uniformly F (λ _, c) p | begin
rw tendsto_uniformly_iff_tendsto_uniformly_on_filter,
exact tendsto_prod_filter_iff,
end | lemma | tendsto_prod_top_iff | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_prod_filter_iff",
"tendsto_uniformly",
"tendsto_uniformly_iff_tendsto_uniformly_on_filter"
] | Uniform convergence to a constant function is equivalent to convergence in `p ×ᶠ ⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_empty :
tendsto_uniformly_on F f p ∅ | λ u hu, by simp | lemma | tendsto_uniformly_on_empty | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on"
] | Uniform convergence on the empty set is vacuously true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_singleton_iff_tendsto :
tendsto_uniformly_on F f p {x} ↔ tendsto (λ n : ι, F n x) p (𝓝 (f x)) | begin
simp_rw [tendsto_uniformly_on_iff_tendsto, uniform.tendsto_nhds_right, tendsto_def],
exact forall₂_congr (λ u hu, by simp [mem_prod_principal, preimage]),
end | lemma | tendsto_uniformly_on_singleton_iff_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto",
"uniform.tendsto_nhds_right"
] | Uniform convergence on a singleton is equivalent to regular convergence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.tendsto_uniformly_on_filter_const
{g : ι → β} {b : β} (hg : tendsto g p (𝓝 b)) (p' : filter α) :
tendsto_uniformly_on_filter (λ n : ι, λ a : α, g n) (λ a : α, b) p p' | begin
rw tendsto_uniformly_on_filter_iff_tendsto,
rw uniform.tendsto_nhds_right at hg,
exact (hg.comp (tendsto_fst.comp ((@tendsto_id ι p).prod_map (@tendsto_id α p')))).congr
(λ x, by simp),
end | lemma | filter.tendsto.tendsto_uniformly_on_filter_const | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"prod_map",
"tendsto_uniformly_on_filter",
"tendsto_uniformly_on_filter_iff_tendsto",
"uniform.tendsto_nhds_right"
] | If a sequence `g` converges to some `b`, then the sequence of constant functions
`λ n, λ a, g n` converges to the constant function `λ a, b` on any set `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.tendsto_uniformly_on_const
{g : ι → β} {b : β} (hg : tendsto g p (𝓝 b)) (s : set α) :
tendsto_uniformly_on (λ n : ι, λ a : α, g n) (λ a : α, b) p s | tendsto_uniformly_on_iff_tendsto_uniformly_on_filter.mpr
(hg.tendsto_uniformly_on_filter_const (𝓟 s)) | lemma | filter.tendsto.tendsto_uniformly_on_const | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on"
] | If a sequence `g` converges to some `b`, then the sequence of constant functions
`λ n, λ a, g n` converges to the constant function `λ a, b` on any set `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_on.tendsto_uniformly [uniform_space α] [uniform_space γ]
{x : α} {U : set α} (hU : U ∈ 𝓝 x)
{F : α → β → γ} (hF : uniform_continuous_on ↿F (U ×ˢ (univ : set β))) :
tendsto_uniformly F (F x) (𝓝 x) | begin
let φ := (λ q : α × β, ((x, q.2), q)),
rw [tendsto_uniformly_iff_tendsto,
show (λ q : α × β, (F x q.2, F q.1 q.2)) = prod.map ↿F ↿F ∘ φ, by { ext ; simpa }],
apply hF.comp (tendsto_inf.mpr ⟨_, _⟩),
{ rw [uniformity_prod, tendsto_inf, tendsto_comap_iff, tendsto_comap_iff,
show (λp : (α × β) × α... | lemma | uniform_continuous_on.tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"mem_of_mem_nhds",
"tendsto_diag_uniformity",
"tendsto_uniformly",
"tendsto_uniformly_iff_tendsto",
"uniform_continuous_on",
"uniform_space",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous₂.tendsto_uniformly [uniform_space α] [uniform_space γ]
{f : α → β → γ} (h : uniform_continuous₂ f) {x : α} : tendsto_uniformly f (f x) (𝓝 x) | uniform_continuous_on.tendsto_uniformly univ_mem $
by rwa [univ_prod_univ, uniform_continuous_on_univ] | lemma | uniform_continuous₂.tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"uniform_continuous_on.tendsto_uniformly",
"uniform_continuous_on_univ",
"uniform_continuous₂",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on_filter
(F : ι → α → β) (p : filter ι) (p' : filter α) : Prop | ∀ u : set (β × β), u ∈ 𝓤 β → ∀ᶠ (m : (ι × ι) × α) in ((p ×ᶠ p) ×ᶠ p'),
(F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u | def | uniform_cauchy_seq_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence is uniformly Cauchy if eventually all of its pairwise differences are
uniformly bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on
(F : ι → α → β) (p : filter ι) (s : set α) : Prop | ∀ u : set (β × β), u ∈ 𝓤 β → ∀ᶠ (m : ι × ι) in (p ×ᶠ p), ∀ (x : α), x ∈ s →
(F m.fst x, F m.snd x) ∈ u | def | uniform_cauchy_seq_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence is uniformly Cauchy if eventually all of its pairwise differences are
uniformly bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter :
uniform_cauchy_seq_on F p s ↔ uniform_cauchy_seq_on_filter F p (𝓟 s) | begin
simp only [uniform_cauchy_seq_on, uniform_cauchy_seq_on_filter],
refine forall₂_congr (λ u hu, _),
rw eventually_prod_principal_iff,
end | lemma | uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.uniform_cauchy_seq_on_filter (hF : uniform_cauchy_seq_on F p s) :
uniform_cauchy_seq_on_filter F p (𝓟 s) | by rwa ←uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter | lemma | uniform_cauchy_seq_on.uniform_cauchy_seq_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter.uniform_cauchy_seq_on_filter
(hF : tendsto_uniformly_on_filter F f p p') :
uniform_cauchy_seq_on_filter F p p' | begin
intros u hu,
rcases comp_symm_of_uniformity hu with ⟨t, ht, htsymm, htmem⟩,
have := tendsto_swap4_prod.eventually ((hF t ht).prod_mk (hF t ht)),
apply this.diag_of_prod_right.mono,
simp only [and_imp, prod.forall],
intros n1 n2 x hl hr,
exact set.mem_of_mem_of_subset (prod_mk_mem_comp_rel (htsymm hl... | lemma | tendsto_uniformly_on_filter.uniform_cauchy_seq_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"and_imp",
"comp_symm_of_uniformity",
"prod_mk_mem_comp_rel",
"set.mem_of_mem_of_subset",
"tendsto_uniformly_on_filter",
"uniform_cauchy_seq_on_filter"
] | A sequence that converges uniformly is also uniformly Cauchy | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on.uniform_cauchy_seq_on (hF : tendsto_uniformly_on F f p s) :
uniform_cauchy_seq_on F p s | uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter.mpr
hF.tendsto_uniformly_on_filter.uniform_cauchy_seq_on_filter | lemma | tendsto_uniformly_on.uniform_cauchy_seq_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on",
"uniform_cauchy_seq_on"
] | A sequence that converges uniformly is also uniformly Cauchy | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_filter.tendsto_uniformly_on_filter_of_tendsto [ne_bot p]
(hF : uniform_cauchy_seq_on_filter F p p')
(hF' : ∀ᶠ (x : α) in p', tendsto (λ n, F n x) p (𝓝 (f x))) :
tendsto_uniformly_on_filter F f p p' | begin
-- Proof idea: |f_n(x) - f(x)| ≤ |f_n(x) - f_m(x)| + |f_m(x) - f(x)|. We choose `n`
-- so that |f_n(x) - f_m(x)| is uniformly small across `s` whenever `m ≥ n`. Then for
-- a fixed `x`, we choose `m` sufficiently large such that |f_m(x) - f(x)| is small.
intros u hu,
rcases comp_symm_of_uniformity hu wi... | lemma | uniform_cauchy_seq_on_filter.tendsto_uniformly_on_filter_of_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"and_imp",
"comp_symm_of_uniformity",
"filter.eventually_swap_iff",
"prod.fst_swap",
"prod.snd_swap",
"set.mem_of_mem_of_subset",
"tendsto_uniformly_on_filter",
"uniform.tendsto_nhds_right",
"uniform_cauchy_seq_on_filter"
] | A uniformly Cauchy sequence converges uniformly to its limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on.tendsto_uniformly_on_of_tendsto [ne_bot p]
(hF : uniform_cauchy_seq_on F p s) (hF' : ∀ x : α, x ∈ s → tendsto (λ n, F n x) p (𝓝 (f x))) :
tendsto_uniformly_on F f p s | tendsto_uniformly_on_iff_tendsto_uniformly_on_filter.mpr
(hF.uniform_cauchy_seq_on_filter.tendsto_uniformly_on_filter_of_tendsto hF') | lemma | uniform_cauchy_seq_on.tendsto_uniformly_on_of_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on",
"uniform_cauchy_seq_on"
] | A uniformly Cauchy sequence converges uniformly to its limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_filter.mono_left {p'' : filter ι}
(hf : uniform_cauchy_seq_on_filter F p p') (hp : p'' ≤ p) :
uniform_cauchy_seq_on_filter F p'' p' | begin
intros u hu,
have := (hf u hu).filter_mono (p'.prod_mono_left (filter.prod_mono hp hp)),
exact this.mono (by simp),
end | lemma | uniform_cauchy_seq_on_filter.mono_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"filter.prod_mono",
"uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on_filter.mono_right {p'' : filter α}
(hf : uniform_cauchy_seq_on_filter F p p') (hp : p'' ≤ p') :
uniform_cauchy_seq_on_filter F p p'' | begin
intros u hu,
have := (hf u hu).filter_mono ((p ×ᶠ p).prod_mono_right hp),
exact this.mono (by simp),
end | lemma | uniform_cauchy_seq_on_filter.mono_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.mono {s' : set α} (hf : uniform_cauchy_seq_on F p s) (hss' : s' ⊆ s) :
uniform_cauchy_seq_on F p s' | begin
rw uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter at hf ⊢,
exact hf.mono_right (le_principal_iff.mpr $mem_principal.mpr hss'),
end | lemma | uniform_cauchy_seq_on.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on_filter.comp {γ : Type*} (hf : uniform_cauchy_seq_on_filter F p p')
(g : γ → α) :
uniform_cauchy_seq_on_filter (λ n, F n ∘ g) p (p'.comap g) | begin
intros u hu,
obtain ⟨pa, hpa, pb, hpb, hpapb⟩ := eventually_prod_iff.mp (hf u hu),
rw eventually_prod_iff,
refine ⟨pa, hpa, pb ∘ g, _, λ x hx y hy, hpapb hx hy⟩,
exact eventually_comap.mpr (hpb.mono (λ x hx y hy, by simp only [hx, hy, function.comp_app])),
end | lemma | uniform_cauchy_seq_on_filter.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on_filter"
] | Composing on the right by a function preserves uniform Cauchy sequences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on.comp {γ : Type*} (hf : uniform_cauchy_seq_on F p s) (g : γ → α) :
uniform_cauchy_seq_on (λ n, F n ∘ g) p (g ⁻¹' s) | begin
rw uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter at hf ⊢,
simpa only [uniform_cauchy_seq_on, comap_principal] using (hf.comp g),
end | lemma | uniform_cauchy_seq_on.comp | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter"
] | Composing on the right by a function preserves uniform Cauchy sequences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous.comp_uniform_cauchy_seq_on [uniform_space γ] {g : β → γ}
(hg : uniform_continuous g) (hf : uniform_cauchy_seq_on F p s) :
uniform_cauchy_seq_on (λ n, g ∘ (F n)) p s | λ u hu, hf _ (hg hu) | lemma | uniform_continuous.comp_uniform_cauchy_seq_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on",
"uniform_continuous",
"uniform_space"
] | Composing on the left by a uniformly continuous function preserves
uniform Cauchy sequences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on.prod_map {ι' α' β' : Type*} [uniform_space β']
{F' : ι' → α' → β'} {p' : filter ι'} {s' : set α'}
(h : uniform_cauchy_seq_on F p s) (h' : uniform_cauchy_seq_on F' p' s') :
uniform_cauchy_seq_on (λ (i : ι × ι'), prod.map (F i.1) (F' i.2))
(p.prod p') (s ×ˢ s') | begin
intros u hu,
rw [uniformity_prod_eq_prod, mem_map, mem_prod_iff] at hu,
obtain ⟨v, hv, w, hw, hvw⟩ := hu,
simp_rw [mem_prod, prod_map, and_imp, prod.forall],
rw [← set.image_subset_iff] at hvw,
apply (tendsto_swap4_prod.eventually ((h v hv).prod_mk (h' w hw))).mono,
intros x hx a b ha hb,
refine h... | lemma | uniform_cauchy_seq_on.prod_map | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"and_imp",
"filter",
"mem_map",
"prod_map",
"set.image_subset_iff",
"uniform_cauchy_seq_on",
"uniform_space",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.prod {ι' β' : Type*} [uniform_space β'] {F' : ι' → α → β'}
{p' : filter ι'}
(h : uniform_cauchy_seq_on F p s) (h' : uniform_cauchy_seq_on F' p' s) :
uniform_cauchy_seq_on (λ (i : ι × ι') a, (F i.fst a, F' i.snd a)) (p ×ᶠ p') s | (congr_arg _ s.inter_self).mp ((h.prod_map h').comp (λ a, (a, a))) | lemma | uniform_cauchy_seq_on.prod | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"uniform_cauchy_seq_on",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.prod' {β' : Type*} [uniform_space β'] {F' : ι → α → β'}
(h : uniform_cauchy_seq_on F p s) (h' : uniform_cauchy_seq_on F' p s) :
uniform_cauchy_seq_on (λ (i : ι) a, (F i a, F' i a)) p s | begin
intros u hu,
have hh : tendsto (λ x : ι, (x, x)) p (p ×ᶠ p), { exact tendsto_diag, },
exact (hh.prod_map hh).eventually ((h.prod h') u hu),
end | lemma | uniform_cauchy_seq_on.prod' | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"uniform_cauchy_seq_on",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on.cauchy_map [hp : ne_bot p]
(hf : uniform_cauchy_seq_on F p s) (hx : x ∈ s) :
cauchy (map (λ i, F i x) p) | begin
simp only [cauchy_map_iff, hp, true_and],
assume u hu,
rw mem_map,
filter_upwards [hf u hu] with p hp using hp x hx,
end | lemma | uniform_cauchy_seq_on.cauchy_map | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"cauchy",
"cauchy_map_iff",
"mem_map",
"uniform_cauchy_seq_on"
] | If a sequence of functions is uniformly Cauchy on a set, then the values at each point form
a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_of_seq_tendsto_uniformly_on {l : filter ι} [l.is_countably_generated]
(h : ∀ u : ℕ → ι, tendsto u at_top l → tendsto_uniformly_on (λ n, F (u n)) f at_top s) :
tendsto_uniformly_on F f l s | begin
rw [tendsto_uniformly_on_iff_tendsto, tendsto_iff_seq_tendsto],
intros u hu,
rw tendsto_prod_iff' at hu,
specialize h (λ n, (u n).fst) hu.1,
rw tendsto_uniformly_on_iff_tendsto at h,
have : ((λ (q : ι × α), (f q.snd, F q.fst q.snd)) ∘ u)
= (λ (q : ℕ × α), (f q.snd, F ((λ (n : ℕ), (u n).fst) q.fst)... | lemma | tendsto_uniformly_on_of_seq_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.seq_tendsto_uniformly_on {l : filter ι}
(h : tendsto_uniformly_on F f l s) (u : ℕ → ι) (hu : tendsto u at_top l) :
tendsto_uniformly_on (λ n, F (u n)) f at_top s | begin
rw tendsto_uniformly_on_iff_tendsto at h ⊢,
have : (λ (q : ℕ × α), (f q.snd, F (u q.fst) q.snd))
= (λ (q : ι × α), (f q.snd, F q.fst q.snd)) ∘ (λ p : ℕ × α, (u p.fst, p.snd)),
{ ext1 x, simp, },
rw this,
refine h.comp _,
rw tendsto_prod_iff',
exact ⟨hu.comp tendsto_fst, tendsto_snd⟩,
end | lemma | tendsto_uniformly_on.seq_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_iff_seq_tendsto_uniformly_on {l : filter ι} [l.is_countably_generated] :
tendsto_uniformly_on F f l s
↔ ∀ u : ℕ → ι, tendsto u at_top l → tendsto_uniformly_on (λ n, F (u n)) f at_top s | ⟨tendsto_uniformly_on.seq_tendsto_uniformly_on, tendsto_uniformly_on_of_seq_tendsto_uniformly_on⟩ | lemma | tendsto_uniformly_on_iff_seq_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_iff_seq_tendsto_uniformly {l : filter ι} [l.is_countably_generated] :
tendsto_uniformly F f l
↔ ∀ u : ℕ → ι, tendsto u at_top l → tendsto_uniformly (λ n, F (u n)) f at_top | begin
simp_rw ← tendsto_uniformly_on_univ,
exact tendsto_uniformly_on_iff_seq_tendsto_uniformly_on,
end | lemma | tendsto_uniformly_iff_seq_tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly",
"tendsto_uniformly_on_iff_seq_tendsto_uniformly_on",
"tendsto_uniformly_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on (F : ι → α → β) (f : α → β) (p : filter ι) (s : set α) | ∀ u ∈ 𝓤 β, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | tendsto_locally_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence of functions `Fₙ` converges locally uniformly on a set `s` to a limiting function
`f` with respect to a filter `p` if, for any entourage of the diagonal `u`, for any `x ∈ s`, one
has `p`-eventually `(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly (F : ι → α → β) (f : α → β) (p : filter ι) | ∀ u ∈ 𝓤 β, ∀ (x : α), ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | tendsto_locally_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence of functions `Fₙ` converges locally uniformly to a limiting function `f` with respect
to a filter `p` if, for any entourage of the diagonal `u`, for any `x`, one has `p`-eventually
`(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff_tendsto_locally_uniformly_comp_coe :
tendsto_locally_uniformly_on F f p s ↔
tendsto_locally_uniformly (λ i (x : s), F i x) (f ∘ coe) p | begin
refine forall₂_congr (λ V hV, _),
simp only [exists_prop, function.comp_app, set_coe.forall, subtype.coe_mk],
refine forall₂_congr (λ x hx, ⟨_, _⟩),
{ rintro ⟨t, ht₁, ht₂⟩,
obtain ⟨u, hu₁, hu₂⟩ := mem_nhds_within_iff_exists_mem_nhds_inter.mp ht₁,
exact ⟨coe⁻¹' u,
(mem_nhds_subtype _ _ _... | lemma | tendsto_locally_uniformly_on_iff_tendsto_locally_uniformly_comp_coe | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"exists_prop",
"forall₂_congr",
"mem_nhds_subtype",
"set_coe.forall",
"subtype.coe_mk",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_iff_forall_tendsto :
tendsto_locally_uniformly F f p ↔
∀ x, tendsto (λ (y : ι × α), (f y.2, F y.1 y.2)) (p ×ᶠ (𝓝 x)) (𝓤 β) | begin
simp only [tendsto_locally_uniformly, filter.forall_in_swap, tendsto_def, mem_prod_iff,
set.prod_subset_iff],
refine forall₃_congr (λ x u hu, ⟨_, _⟩),
{ rintros ⟨n, hn, hp⟩,
exact ⟨_, hp, n, hn, λ i hi a ha, hi a ha⟩, },
{ rintros ⟨I, hI, n, hn, hu⟩,
exact ⟨n, hn, by filter_upwards [hI] using ... | lemma | tendsto_locally_uniformly_iff_forall_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter.forall_in_swap",
"forall₃_congr",
"set.prod_subset_iff",
"tendsto_locally_uniformly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on.tendsto_locally_uniformly_on
(h : tendsto_uniformly_on F f p s) : tendsto_locally_uniformly_on F f p s | λ u hu x hx,⟨s, self_mem_nhds_within, by simpa using (h u hu)⟩ | lemma | tendsto_uniformly_on.tendsto_locally_uniformly_on | 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",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.tendsto_locally_uniformly
(h : tendsto_uniformly F f p) : tendsto_locally_uniformly F f p | λ u hu x, ⟨univ, univ_mem, by simpa using (h u hu)⟩ | lemma | tendsto_uniformly.tendsto_locally_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_locally_uniformly",
"tendsto_uniformly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on.mono (h : tendsto_locally_uniformly_on F f p s) (h' : s' ⊆ s) :
tendsto_locally_uniformly_on F f p s' | begin
assume u hu x hx,
rcases h u hu x (h' hx) with ⟨t, ht, H⟩,
exact ⟨t, nhds_within_mono x h' ht, H.mono (λ n, id)⟩
end | lemma | tendsto_locally_uniformly_on.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"nhds_within_mono",
"tendsto_locally_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on_Union {S : γ → set α} (hS : ∀ i, is_open (S i))
(h : ∀ i, tendsto_locally_uniformly_on F f p (S i)) :
tendsto_locally_uniformly_on F f p (⋃ i, S i) | begin
rintro v hv x ⟨_, ⟨i, rfl⟩, hi : x ∈ S i⟩,
obtain ⟨t, ht, ht'⟩ := h i v hv x hi,
refine ⟨t, _, ht'⟩,
rw (hS _).nhds_within_eq hi at ht,
exact mem_nhds_within_of_mem_nhds ht,
end | lemma | tendsto_locally_uniformly_on_Union | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"is_open",
"mem_nhds_within_of_mem_nhds",
"nhds_within_eq",
"tendsto_locally_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on_bUnion {s : set γ} {S : γ → set α}
(hS : ∀ i ∈ s, is_open (S i)) (h : ∀ i ∈ s, tendsto_locally_uniformly_on F f p (S i)) :
tendsto_locally_uniformly_on F f p (⋃ i ∈ s, S i) | by { rw bUnion_eq_Union, exact tendsto_locally_uniformly_on_Union (λ i, hS _ i.2) (λ i, h _ i.2) } | lemma | tendsto_locally_uniformly_on_bUnion | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on_sUnion (S : set (set α)) (hS : ∀ s ∈ S, is_open s)
(h : ∀ s ∈ S, tendsto_locally_uniformly_on F f p s) :
tendsto_locally_uniformly_on F f p (⋃₀ S) | by { rw sUnion_eq_bUnion, exact tendsto_locally_uniformly_on_bUnion hS h } | lemma | tendsto_locally_uniformly_on_sUnion | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_locally_uniformly_on.union {s₁ s₂ : set α} (hs₁ : is_open s₁) (hs₂ : is_open s₂)
(h₁ : tendsto_locally_uniformly_on F f p s₁) (h₂ : tendsto_locally_uniformly_on F f p s₂) :
tendsto_locally_uniformly_on F f p (s₁ ∪ s₂) | by { rw ←sUnion_pair, refine tendsto_locally_uniformly_on_sUnion _ _ _; simp [*] } | lemma | tendsto_locally_uniformly_on.union | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"is_open",
"tendsto_locally_uniformly_on",
"tendsto_locally_uniformly_on_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | 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 _) | 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"
] | 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) (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. | 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_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`. | 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 : 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"
] | 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 _ _) 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"
] | 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,
{ 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"
] | 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 | 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"
] | 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_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"
] | 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 _ | 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"
] | 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 | 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"
] | 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) | 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"
] | 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 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"
] | 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 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"
] | 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⟩ : ∃ (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. | 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_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. | 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) | 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. | 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⟩⟩ | 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. | 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) | 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. | 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 | 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. | 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).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. | 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 | 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. | 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 | 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. | 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 | 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. | 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 : 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`. | 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_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`. | 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) | 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`. | 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⟩) | 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`. | 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) | 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`. | 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 | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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`. | 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⟩ | 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`. | 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⟩ | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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`. | 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 | 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`. | 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} | 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`. | 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⟩⟩⟩ | 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... | 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 | 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... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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