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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continuous_within_at_compl_self {f : α → β} {a : α} :
continuous_within_at f {a}ᶜ a ↔ continuous_at f a | by rw [compl_eq_univ_diff, continuous_within_at_diff_self, continuous_within_at_univ] | lemma | continuous_within_at_compl_self | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"continuous_within_at_diff_self",
"continuous_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_update_same [decidable_eq α] {f : α → β}
{s : set α} {x : α} {y : β} :
continuous_within_at (update f x y) s x ↔ tendsto f (𝓝[s \ {x}] x) (𝓝 y) | calc continuous_within_at (update f x y) s x ↔ tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y) :
by rw [← continuous_within_at_diff_self, continuous_within_at, function.update_same]
... ↔ tendsto f (𝓝[s \ {x}] x) (𝓝 y) :
tendsto_congr' $ eventually_nhds_within_iff.2 $ eventually_of_forall $
λ z hz, update_noteq... | lemma | continuous_within_at_update_same | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"continuous_within_at_diff_self",
"update",
"update_noteq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_update_same [decidable_eq α] {f : α → β} {x : α} {y : β} :
continuous_at (function.update f x y) x ↔ tendsto f (𝓝[≠] x) (𝓝 y) | by rw [← continuous_within_at_univ, continuous_within_at_update_same, compl_eq_univ_diff] | lemma | continuous_at_update_same | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at_univ",
"continuous_within_at_update_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.continuous_on_image_of_left_inv_on {f : α → β} {s : set α}
(h : is_open_map (s.restrict f)) {finv : β → α} (hleft : left_inv_on finv f s) :
continuous_on finv (f '' s) | begin
refine continuous_on_iff'.2 (λ t ht, ⟨f '' (t ∩ s), _, _⟩),
{ rw ← image_restrict, exact h _ (ht.preimage continuous_subtype_coe) },
{ rw [inter_eq_self_of_subset_left (image_subset f (inter_subset_right t s)),
hleft.image_inter'] },
end | theorem | is_open_map.continuous_on_image_of_left_inv_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_subtype_coe",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.continuous_on_range_of_left_inverse {f : α → β} (hf : is_open_map f)
{finv : β → α} (hleft : function.left_inverse finv f) :
continuous_on finv (range f) | begin
rw [← image_univ],
exact (hf.restrict is_open_univ).continuous_on_image_of_left_inv_on (λ x _, hleft x)
end | theorem | is_open_map.continuous_on_range_of_left_inverse | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"is_open_map",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.congr_mono {f g : α → β} {s s₁ : set α} (h : continuous_on f s)
(h' : eq_on g f s₁) (h₁ : s₁ ⊆ s) : continuous_on g s₁ | begin
assume x hx,
unfold continuous_within_at,
have A := (h x (h₁ hx)).mono h₁,
unfold continuous_within_at at A,
rw ← h' hx at A,
exact A.congr' h'.eventually_eq_nhds_within.symm
end | lemma | continuous_on.congr_mono | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.congr {f g : α → β} {s : set α} (h : continuous_on f s) (h' : eq_on g f s) :
continuous_on g s | h.congr_mono h' (subset.refl _) | lemma | continuous_on.congr | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_congr {f g : α → β} {s : set α} (h' : eq_on g f s) :
continuous_on g s ↔ continuous_on f s | ⟨λ h, continuous_on.congr h h'.symm, λ h, h.congr h'⟩ | lemma | continuous_on_congr | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on.congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.continuous_within_at {f : α → β} {s : set α} {x : α} (h : continuous_at f x) :
continuous_within_at f s x | continuous_within_at.mono ((continuous_within_at_univ f x).2 h) (subset_univ _) | lemma | continuous_at.continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"continuous_within_at.mono",
"continuous_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff_continuous_at {f : α → β} {s : set α} {x : α} (h : s ∈ 𝓝 x) :
continuous_within_at f s x ↔ continuous_at f x | by rw [← univ_inter s, continuous_within_at_inter h, continuous_within_at_univ] | lemma | continuous_within_at_iff_continuous_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"continuous_within_at_inter",
"continuous_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.continuous_at {f : α → β} {s : set α} {x : α}
(h : continuous_within_at f s x) (hs : s ∈ 𝓝 x) : continuous_at f x | (continuous_within_at_iff_continuous_at hs).mp h | lemma | continuous_within_at.continuous_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"continuous_within_at_iff_continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.continuous_on_iff {f : α → β} {s : set α} (hs : is_open s) :
continuous_on f s ↔ ∀ ⦃a⦄, a ∈ s → continuous_at f a | ball_congr $ λ _, continuous_within_at_iff_continuous_at ∘ hs.mem_nhds | lemma | is_open.continuous_on_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"ball_congr",
"continuous_at",
"continuous_on",
"continuous_within_at_iff_continuous_at",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.continuous_at {f : α → β} {s : set α} {x : α}
(h : continuous_on f s) (hx : s ∈ 𝓝 x) : continuous_at f x | (h x (mem_of_mem_nhds hx)).continuous_at hx | lemma | continuous_on.continuous_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_on",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.continuous_on {f : α → β} {s : set α} (hcont : ∀ x ∈ s, continuous_at f x) :
continuous_on f s | λ x hx, (hcont x hx).continuous_within_at | lemma | continuous_at.continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.comp {g : β → γ} {f : α → β} {s : set α} {t : set β} {x : α}
(hg : continuous_within_at g t (f x)) (hf : continuous_within_at f s x) (h : maps_to f s t) :
continuous_within_at (g ∘ f) s x | hg.tendsto.comp (hf.tendsto_nhds_within h) | lemma | continuous_within_at.comp | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.comp' {g : β → γ} {f : α → β} {s : set α} {t : set β} {x : α}
(hg : continuous_within_at g t (f x)) (hf : continuous_within_at f s x) :
continuous_within_at (g ∘ f) (s ∩ f⁻¹' t) x | hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _) | lemma | continuous_within_at.comp' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_continuous_within_at {g : β → γ} {f : α → β} {s : set α} {x : α}
(hg : continuous_at g (f x)) (hf : continuous_within_at f s x) :
continuous_within_at (g ∘ f) s x | hg.continuous_within_at.comp hf (maps_to_univ _ _) | lemma | continuous_at.comp_continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.comp {g : β → γ} {f : α → β} {s : set α} {t : set β}
(hg : continuous_on g t) (hf : continuous_on f s) (h : maps_to f s t) :
continuous_on (g ∘ f) s | λx hx, continuous_within_at.comp (hg _ (h hx)) (hf x hx) h | lemma | continuous_on.comp | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.mono {f : α → β} {s t : set α} (hf : continuous_on f s) (h : t ⊆ s) :
continuous_on f t | λx hx, (hf x (h hx)).mono_left (nhds_within_mono _ h) | lemma | continuous_on.mono | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_continuous_on {f : α → β} : antitone (continuous_on f) | λ s t hst hf, hf.mono hst | lemma | antitone_continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"antitone",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.comp' {g : β → γ} {f : α → β} {s : set α} {t : set β}
(hg : continuous_on g t) (hf : continuous_on f s) :
continuous_on (g ∘ f) (s ∩ f⁻¹' t) | hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _) | lemma | continuous_on.comp' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.continuous_on {f : α → β} {s : set α} (h : continuous f) :
continuous_on f s | begin
rw continuous_iff_continuous_on_univ at h,
exact h.mono (subset_univ _)
end | lemma | continuous.continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.continuous_within_at {f : α → β} {s : set α} {x : α} (h : continuous f) :
continuous_within_at f s x | h.continuous_at.continuous_within_at | lemma | continuous.continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp_continuous_on {g : β → γ} {f : α → β} {s : set α}
(hg : continuous g) (hf : continuous_on f s) :
continuous_on (g ∘ f) s | hg.continuous_on.comp hf (maps_to_univ _ _) | lemma | continuous.comp_continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.comp_continuous {g : β → γ} {f : α → β} {s : set β}
(hg : continuous_on g s) (hf : continuous f) (hs : ∀ x, f x ∈ s) : continuous (g ∘ f) | begin
rw continuous_iff_continuous_on_univ at *,
exact hg.comp hf (λ x _, hs x),
end | lemma | continuous_on.comp_continuous | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.preimage_mem_nhds_within {f : α → β} {x : α} {s : set α} {t : set β}
(h : continuous_within_at f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x | h ht | lemma | continuous_within_at.preimage_mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.left_inv_on.map_nhds_within_eq {f : α → β} {g : β → α} {x : β} {s : set β}
(h : left_inv_on f g s) (hx : f (g x) = x) (hf : continuous_within_at f (g '' s) (g x))
(hg : continuous_within_at g s x) :
map g (𝓝[s] x) = 𝓝[g '' s] (g x) | begin
apply le_antisymm,
{ exact hg.tendsto_nhds_within (maps_to_image _ _) },
{ have A : g ∘ f =ᶠ[𝓝[g '' s] (g x)] id,
from h.right_inv_on_image.eq_on.eventually_eq_of_mem self_mem_nhds_within,
refine le_map_of_right_inverse A _,
simpa only [hx] using hf.tendsto_nhds_within (h.maps_to (surj_on_ima... | lemma | set.left_inv_on.map_nhds_within_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.left_inverse.map_nhds_eq {f : α → β} {g : β → α} {x : β}
(h : function.left_inverse f g) (hf : continuous_within_at f (range g) (g x))
(hg : continuous_at g x) :
map g (𝓝 x) = 𝓝[range g] (g x) | by simpa only [nhds_within_univ, image_univ]
using (h.left_inv_on univ).map_nhds_within_eq (h x) (by rwa image_univ) hg.continuous_within_at | lemma | function.left_inverse.map_nhds_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.preimage_mem_nhds_within' {f : α → β} {x : α} {s : set α} {t : set β}
(h : continuous_within_at f s x) (ht : t ∈ 𝓝[f '' s] (f x)) :
f ⁻¹' t ∈ 𝓝[s] x | h.tendsto_nhds_within (maps_to_image _ _) ht | lemma | continuous_within_at.preimage_mem_nhds_within' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.congr_continuous_within_at {f g : α → β} {s : set α} {x : α}
(h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
continuous_within_at f s x ↔ continuous_within_at g s x | by rw [continuous_within_at, hx, tendsto_congr' h, continuous_within_at] | lemma | filter.eventually_eq.congr_continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.congr_of_eventually_eq {f f₁ : α → β} {s : set α} {x : α}
(h : continuous_within_at f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
continuous_within_at f₁ s x | (h₁.congr_continuous_within_at hx).2 h | lemma | continuous_within_at.congr_of_eventually_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.congr {f f₁ : α → β} {s : set α} {x : α}
(h : continuous_within_at f s x) (h₁ : ∀y∈s, f₁ y = f y) (hx : f₁ x = f x) :
continuous_within_at f₁ s x | h.congr_of_eventually_eq (mem_of_superset self_mem_nhds_within h₁) hx | lemma | continuous_within_at.congr | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.congr_mono {f g : α → β} {s s₁ : set α} {x : α}
(h : continuous_within_at f s x) (h' : eq_on g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x):
continuous_within_at g s₁ x | (h.mono h₁).congr h' hx | lemma | continuous_within_at.congr_mono | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_const {s : set α} {c : β} : continuous_on (λx, c) s | continuous_const.continuous_on | lemma | continuous_on_const | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_const {b : β} {s : set α} {x : α} :
continuous_within_at (λ _:α, b) s x | continuous_const.continuous_within_at | lemma | continuous_within_at_const | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_id {s : set α} : continuous_on id s | continuous_id.continuous_on | lemma | continuous_on_id | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_id {s : set α} {x : α} : continuous_within_at id s x | continuous_id.continuous_within_at | lemma | continuous_within_at_id | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_open_iff {f : α → β} {s : set α} (hs : is_open s) :
continuous_on f s ↔ (∀t, is_open t → is_open (s ∩ f⁻¹' t)) | begin
rw continuous_on_iff',
split,
{ assume h t ht,
rcases h t ht with ⟨u, u_open, hu⟩,
rw [inter_comm, hu],
apply is_open.inter u_open hs },
{ assume h t ht,
refine ⟨s ∩ f ⁻¹' t, h t ht, _⟩,
rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self] }
end | lemma | continuous_on_open_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_iff'",
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.preimage_open_of_open {f : α → β} {s : set α} {t : set β}
(hf : continuous_on f s) (hs : is_open s) (ht : is_open t) : is_open (s ∩ f⁻¹' t) | (continuous_on_open_iff hs).1 hf t ht | lemma | continuous_on.preimage_open_of_open | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_open_iff",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.is_open_preimage {f : α → β} {s : set α} {t : set β} (h : continuous_on f s)
(hs : is_open s) (hp : f ⁻¹' t ⊆ s) (ht : is_open t) : is_open (f ⁻¹' t) | begin
convert (continuous_on_open_iff hs).mp h t ht,
rw [inter_comm, inter_eq_self_of_subset_left hp],
end | lemma | continuous_on.is_open_preimage | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_open_iff",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.preimage_closed_of_closed {f : α → β} {s : set α} {t : set β}
(hf : continuous_on f s) (hs : is_closed s) (ht : is_closed t) : is_closed (s ∩ f⁻¹' t) | begin
rcases continuous_on_iff_is_closed.1 hf t ht with ⟨u, hu⟩,
rw [inter_comm, hu.2],
apply is_closed.inter hu.1 hs
end | lemma | continuous_on.preimage_closed_of_closed | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"is_closed",
"is_closed.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.preimage_interior_subset_interior_preimage {f : α → β} {s : set α} {t : set β}
(hf : continuous_on f s) (hs : is_open s) : s ∩ f⁻¹' (interior t) ⊆ s ∩ interior (f⁻¹' t) | calc s ∩ f ⁻¹' (interior t) ⊆ interior (s ∩ f ⁻¹' t) :
interior_maximal (inter_subset_inter (subset.refl _) (preimage_mono interior_subset))
(hf.preimage_open_of_open hs is_open_interior)
... = s ∩ interior (f ⁻¹' t) : by rw [interior_inter, hs.interior_eq] | lemma | continuous_on.preimage_interior_subset_interior_preimage | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"interior",
"interior_inter",
"interior_maximal",
"interior_subset",
"is_open",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_of_locally_continuous_on {f : α → β} {s : set α}
(h : ∀x∈s, ∃t, is_open t ∧ x ∈ t ∧ continuous_on f (s ∩ t)) : continuous_on f s | begin
assume x xs,
rcases h x xs with ⟨t, open_t, xt, ct⟩,
have := ct x ⟨xs, xt⟩,
rwa [continuous_within_at, ← nhds_within_restrict _ xt open_t] at this
end | lemma | continuous_on_of_locally_continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at",
"is_open",
"nhds_within_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_open_of_generate_from {β : Type*} {s : set α} {T : set (set β)} {f : α → β}
(hs : is_open s) (h : ∀t ∈ T, is_open (s ∩ f⁻¹' t)) :
@continuous_on α β _ (topological_space.generate_from T) f s | begin
rw continuous_on_open_iff,
assume t ht,
induction ht with u hu u v Tu Tv hu hv U hU hU',
{ exact h u hu },
{ simp only [preimage_univ, inter_univ], exact hs },
{ have : s ∩ f ⁻¹' (u ∩ v) = (s ∩ f ⁻¹' u) ∩ (s ∩ f ⁻¹' v),
by rw [preimage_inter, inter_assoc, inter_left_comm _ s, ← inter_assoc s s, ... | lemma | continuous_on_open_of_generate_from | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_open_iff",
"is_open",
"is_open_bUnion",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.prod {f : α → β} {g : α → γ} {s : set α} {x : α}
(hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :
continuous_within_at (λx, (f x, g x)) s x | hf.prod_mk_nhds hg | lemma | continuous_within_at.prod | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.prod {f : α → β} {g : α → γ} {s : set α}
(hf : continuous_on f s) (hg : continuous_on g s) : continuous_on (λx, (f x, g x)) s | λx hx, continuous_within_at.prod (hf x hx) (hg x hx) | lemma | continuous_on.prod | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous_within_at_iff {f : α → β} {g : β → γ} (hg : inducing g) {s : set α}
{x : α} : continuous_within_at f s x ↔ continuous_within_at (g ∘ f) s x | by simp_rw [continuous_within_at, inducing.tendsto_nhds_iff hg] | lemma | inducing.continuous_within_at_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"inducing",
"inducing.tendsto_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous_on_iff {f : α → β} {g : β → γ} (hg : inducing g) {s : set α} :
continuous_on f s ↔ continuous_on (g ∘ f) s | by simp_rw [continuous_on, hg.continuous_within_at_iff] | lemma | inducing.continuous_on_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.continuous_on_iff {f : α → β} {g : β → γ} (hg : embedding g) {s : set α} :
continuous_on f s ↔ continuous_on (g ∘ f) s | inducing.continuous_on_iff hg.1 | lemma | embedding.continuous_on_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"embedding",
"inducing.continuous_on_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.map_nhds_within_eq {f : α → β} (hf : embedding f) (s : set α) (x : α) :
map f (𝓝[s] x) = 𝓝[f '' s] (f x) | by rw [nhds_within, map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhds_within_inter',
inter_eq_self_of_subset_right (image_subset_range _ _)] | lemma | embedding.map_nhds_within_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"embedding",
"nhds_within",
"nhds_within_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.map_nhds_within_preimage_eq {f : α → β} (hf : open_embedding f)
(s : set β) (x : α) :
map f (𝓝[f ⁻¹' s] x) = 𝓝[s] (f x) | begin
rw [hf.to_embedding.map_nhds_within_eq, image_preimage_eq_inter_range],
apply nhds_within_eq_nhds_within (mem_range_self _) hf.open_range,
rw [inter_assoc, inter_self]
end | lemma | open_embedding.map_nhds_within_preimage_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_eq_nhds_within",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_of_not_mem_closure {f : α → β} {s : set α} {x : α} :
x ∉ closure s → continuous_within_at f s x | begin
intros hx,
rw [mem_closure_iff_nhds_within_ne_bot, ne_bot_iff, not_not] at hx,
rw [continuous_within_at, hx],
exact tendsto_bot,
end | lemma | continuous_within_at_of_not_mem_closure | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_within_at",
"mem_closure_iff_nhds_within_ne_bot",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.piecewise' {s t : set α} {f g : α → β} [∀ a, decidable (a ∈ t)]
(hpf : ∀ a ∈ s ∩ frontier t, tendsto f (𝓝[s ∩ t] a) (𝓝 (piecewise t f g a)))
(hpg : ∀ a ∈ s ∩ frontier t, tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (piecewise t f g a)))
(hf : continuous_on f $ s ∩ t) (hg : continuous_on g $ s ∩ tᶜ) :
continuous... | begin
intros x hx,
by_cases hx' : x ∈ frontier t,
{ exact (hpf x ⟨hx, hx'⟩).piecewise_nhds_within (hpg x ⟨hx, hx'⟩) },
{ rw [← inter_univ s, ← union_compl_self t, inter_union_distrib_left] at hx ⊢,
cases hx,
{ apply continuous_within_at.union,
{ exact (hf x hx).congr (λ y hy, piecewise_eq_of_mem _... | lemma | continuous_on.piecewise' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"closure_compl",
"closure_inter_subset_inter_closure",
"continuous_on",
"continuous_within_at.union",
"continuous_within_at_of_not_mem_closure",
"frontier",
"interior",
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.if' {s : set α} {p : α → Prop} {f g : α → β} [∀ a, decidable (p a)]
(hpf : ∀ a ∈ s ∩ frontier {a | p a},
tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 $ if p a then f a else g a))
(hpg : ∀ a ∈ s ∩ frontier {a | p a},
tendsto g (𝓝[s ∩ {a | ¬p a}] a) (𝓝 $ if p a then f a else g a))
(hf : continuous_on... | hf.piecewise' hpf hpg hg | lemma | continuous_on.if' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.if {α β : Type*} [topological_space α] [topological_space β] {p : α → Prop}
[∀ a, decidable (p a)] {s : set α} {f g : α → β}
(hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a) (hf : continuous_on f $ s ∩ closure {a | p a})
(hg : continuous_on g $ s ∩ closure {a | ¬ p a}) :
continuous_on (λa, if p a th... | begin
apply continuous_on.if',
{ rintros a ha,
simp only [← hp a ha, if_t_t],
apply tendsto_nhds_within_mono_left (inter_subset_inter_right s subset_closure),
exact hf a ⟨ha.1, ha.2.1⟩ },
{ rintros a ha,
simp only [hp a ha, if_t_t],
apply tendsto_nhds_within_mono_left (inter_subset_inter_right... | lemma | continuous_on.if | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"closure_compl",
"continuous_on",
"continuous_on.if'",
"frontier",
"subset_closure",
"tendsto_nhds_within_mono_left",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.piecewise {s t : set α} {f g : α → β} [∀ a, decidable (a ∈ t)]
(ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : continuous_on f $ s ∩ closure t)
(hg : continuous_on g $ s ∩ closure tᶜ) :
continuous_on (piecewise t f g) s | hf.if ht hg | lemma | continuous_on.piecewise | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_on",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_if' {p : α → Prop} {f g : α → β} [∀ a, decidable (p a)]
(hpf : ∀ a ∈ frontier {x | p x}, tendsto f (𝓝[{x | p x}] a) (𝓝 $ ite (p a) (f a) (g a)))
(hpg : ∀ a ∈ frontier {x | p x}, tendsto g (𝓝[{x | ¬p x}] a) (𝓝 $ ite (p a) (f a) (g a)))
(hf : continuous_on f {x | p x}) (hg : continuous_on g {x | ¬p x... | begin
rw continuous_iff_continuous_on_univ,
apply continuous_on.if'; simp *; assumption
end | lemma | continuous_if' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on",
"continuous_on.if'",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_if {p : α → Prop} {f g : α → β} [∀ a, decidable (p a)]
(hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : continuous_on f (closure {x | p x}))
(hg : continuous_on g (closure {x | ¬p x})) :
continuous (λ a, if p a then f a else g a) | begin
rw continuous_iff_continuous_on_univ,
apply continuous_on.if; simp; assumption
end | lemma | continuous_if | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on",
"continuous_on.if",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.if {p : α → Prop} {f g : α → β} [∀ a, decidable (p a)]
(hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : continuous f) (hg : continuous g) :
continuous (λ a, if p a then f a else g a) | continuous_if hp hf.continuous_on hg.continuous_on | lemma | continuous.if | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_if",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_if_const (p : Prop) {f g : α → β} [decidable p]
(hf : p → continuous f) (hg : ¬ p → continuous g) :
continuous (λ a, if p then f a else g a) | by { split_ifs, exact hf h, exact hg h } | lemma | continuous_if_const | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.if_const (p : Prop) {f g : α → β} [decidable p]
(hf : continuous f) (hg : continuous g) :
continuous (λ a, if p then f a else g a) | continuous_if_const p (λ _, hf) (λ _, hg) | lemma | continuous.if_const | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_if_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_piecewise {s : set α} {f g : α → β} [∀ a, decidable (a ∈ s)]
(hs : ∀ a ∈ frontier s, f a = g a) (hf : continuous_on f (closure s))
(hg : continuous_on g (closure sᶜ)) :
continuous (piecewise s f g) | continuous_if hs hf hg | lemma | continuous_piecewise | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous",
"continuous_if",
"continuous_on",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.piecewise {s : set α} {f g : α → β} [∀ a, decidable (a ∈ s)]
(hs : ∀ a ∈ frontier s, f a = g a) (hf : continuous f) (hg : continuous g) :
continuous (piecewise s f g) | hf.if hs hg | lemma | continuous.piecewise | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.ite' {s s' t : set α}
(hs : is_open s) (hs' : is_open s') (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') :
is_open (t.ite s s') | begin
classical,
simp only [is_open_iff_continuous_mem, set.ite] at *,
convert continuous_piecewise (λ x hx, propext (ht x hx)) hs.continuous_on hs'.continuous_on,
ext x, by_cases hx : x ∈ t; simp [hx]
end | lemma | is_open.ite' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_piecewise",
"frontier",
"is_open",
"is_open_iff_continuous_mem",
"set.ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.ite {s s' t : set α} (hs : is_open s) (hs' : is_open s')
(ht : s ∩ frontier t = s' ∩ frontier t) :
is_open (t.ite s s') | hs.ite' hs' $ λ x hx, by simpa [hx] using ext_iff.1 ht x | lemma | is_open.ite | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"frontier",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_inter_closure_eq_of_inter_frontier_eq {s s' t : set α}
(ht : s ∩ frontier t = s' ∩ frontier t) :
t.ite s s' ∩ closure t = s ∩ closure t | by rw [closure_eq_self_union_frontier, inter_union_distrib_left, inter_union_distrib_left,
ite_inter_self, ite_inter_of_inter_eq _ ht] | lemma | ite_inter_closure_eq_of_inter_frontier_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"closure_eq_self_union_frontier",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : set α}
(ht : s ∩ frontier t = s' ∩ frontier t) :
t.ite s s' ∩ closure tᶜ = s' ∩ closure tᶜ | by { rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq], rwa [frontier_compl, eq_comm] } | lemma | ite_inter_closure_compl_eq_of_inter_frontier_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"frontier",
"frontier_compl",
"ite_inter_closure_eq_of_inter_frontier_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_piecewise_ite' {s s' t : set α} {f f' : α → β} [∀ x, decidable (x ∈ t)]
(h : continuous_on f (s ∩ closure t)) (h' : continuous_on f' (s' ∩ closure tᶜ))
(H : s ∩ frontier t = s' ∩ frontier t) (Heq : eq_on f f' (s ∩ frontier t)) :
continuous_on (t.piecewise f f') (t.ite s s') | begin
apply continuous_on.piecewise,
{ rwa ite_inter_of_inter_eq _ H },
{ rwa ite_inter_closure_eq_of_inter_frontier_eq H },
{ rwa ite_inter_closure_compl_eq_of_inter_frontier_eq H }
end | lemma | continuous_on_piecewise_ite' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_on",
"continuous_on.piecewise",
"frontier",
"ite_inter_closure_compl_eq_of_inter_frontier_eq",
"ite_inter_closure_eq_of_inter_frontier_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_piecewise_ite {s s' t : set α} {f f' : α → β} [∀ x, decidable (x ∈ t)]
(h : continuous_on f s) (h' : continuous_on f' s')
(H : s ∩ frontier t = s' ∩ frontier t) (Heq : eq_on f f' (s ∩ frontier t)) :
continuous_on (t.piecewise f f') (t.ite s s') | continuous_on_piecewise_ite' (h.mono (inter_subset_left _ _)) (h'.mono (inter_subset_left _ _))
H Heq | lemma | continuous_on_piecewise_ite | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_piecewise_ite'",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_inter_open_inter {s t : set α} (ht : is_open t) :
frontier (s ∩ t) ∩ t = frontier s ∩ t | by simp only [← subtype.preimage_coe_eq_preimage_coe_iff,
ht.is_open_map_subtype_coe.preimage_frontier_eq_frontier_preimage continuous_subtype_coe,
subtype.preimage_coe_inter_self] | lemma | frontier_inter_open_inter | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_subtype_coe",
"frontier",
"is_open",
"subtype.preimage_coe_eq_preimage_coe_iff",
"subtype.preimage_coe_inter_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_fst {s : set (α × β)} : continuous_on prod.fst s | continuous_fst.continuous_on | lemma | continuous_on_fst | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_fst {s : set (α × β)} {p : α × β} :
continuous_within_at prod.fst s p | continuous_fst.continuous_within_at | lemma | continuous_within_at_fst | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.fst {f : α → β × γ} {s : set α} (hf : continuous_on f s) :
continuous_on (λ x, (f x).1) s | continuous_fst.comp_continuous_on hf | lemma | continuous_on.fst | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.fst {f : α → β × γ} {s : set α} {a : α}
(h : continuous_within_at f s a) : continuous_within_at (λ x, (f x).fst) s a | continuous_at_fst.comp_continuous_within_at h | lemma | continuous_within_at.fst | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_snd {s : set (α × β)} : continuous_on prod.snd s | continuous_snd.continuous_on | lemma | continuous_on_snd | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_snd {s : set (α × β)} {p : α × β} :
continuous_within_at prod.snd s p | continuous_snd.continuous_within_at | lemma | continuous_within_at_snd | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.snd {f : α → β × γ} {s : set α} (hf : continuous_on f s) :
continuous_on (λ x, (f x).2) s | continuous_snd.comp_continuous_on hf | lemma | continuous_on.snd | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.snd {f : α → β × γ} {s : set α} {a : α}
(h : continuous_within_at f s a) : continuous_within_at (λ x, (f x).snd) s a | continuous_at_snd.comp_continuous_within_at h | lemma | continuous_within_at.snd | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_prod_iff {f : α → β × γ} {s : set α} {x : α} :
continuous_within_at f s x ↔ continuous_within_at (prod.fst ∘ f) s x ∧
continuous_within_at (prod.snd ∘ f) s x | ⟨λ h, ⟨h.fst, h.snd⟩, by { rintro ⟨h1, h2⟩, convert h1.prod h2, ext, refl, refl }⟩ | lemma | continuous_within_at_prod_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_evenly_covered (x : X) (I : Type*) [topological_space I] | discrete_topology I ∧ ∃ t : trivialization I f, x ∈ t.base_set | def | is_evenly_covered | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"discrete_topology",
"topological_space",
"trivialization"
] | A point `x : X` is evenly covered by `f : E → X` if `x` has an evenly covered neighborhood. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_trivialization {x : X} {I : Type*} [topological_space I]
(h : is_evenly_covered f x I) : trivialization (f ⁻¹' {x}) f | (classical.some h.2).trans_fiber_homeomorph ((classical.some h.2).preimage_singleton_homeomorph
(classical.some_spec h.2)).symm | def | is_evenly_covered.to_trivialization | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_evenly_covered",
"topological_space",
"trivialization"
] | If `x` is evenly covered by `f`, then we can construct a trivialization of `f` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_trivialization_base_set {x : X} {I : Type*} [topological_space I]
(h : is_evenly_covered f x I) : x ∈ h.to_trivialization.base_set | classical.some_spec h.2 | lemma | is_evenly_covered.mem_to_trivialization_base_set | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_evenly_covered",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_trivialization_apply {x : E} {I : Type*} [topological_space I]
(h : is_evenly_covered f (f x) I) : (h.to_trivialization x).2 = ⟨x, rfl⟩ | let e := classical.some h.2, h := classical.some_spec h.2, he := e.mk_proj_snd' h in
subtype.ext ((e.to_local_equiv.eq_symm_apply (e.mem_source.mpr h)
(by rwa [he, e.mem_target, e.coe_fst (e.mem_source.mpr h)])).mpr he.symm).symm | lemma | is_evenly_covered.to_trivialization_apply | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_evenly_covered",
"subtype.ext",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at {x : E} {I : Type*} [topological_space I]
(h : is_evenly_covered f (f x) I) : continuous_at f x | let e := h.to_trivialization in
e.continuous_at_proj (e.mem_source.mpr (mem_to_trivialization_base_set h)) | lemma | is_evenly_covered.continuous_at | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"continuous_at",
"is_evenly_covered",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_is_evenly_covered_preimage {x : X} {I : Type*} [topological_space I]
(h : is_evenly_covered f x I) : is_evenly_covered f x (f ⁻¹' {x}) | let ⟨h1, h2⟩ := h in by exactI ⟨((classical.some h2).preimage_singleton_homeomorph
(classical.some_spec h2)).embedding.discrete_topology, _, h.mem_to_trivialization_base_set⟩ | lemma | is_evenly_covered.to_is_evenly_covered_preimage | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"embedding.discrete_topology",
"is_evenly_covered",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_covering_map_on | ∀ x ∈ s, is_evenly_covered f x (f ⁻¹' {x}) | def | is_covering_map_on | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_evenly_covered"
] | A covering map is a continuous function `f : E → X` with discrete fibers such that each point
of `X` has an evenly covered neighborhood. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (F : X → Type*) [Π x, topological_space (F x)] [hF : Π x, discrete_topology (F x)]
(e : Π x ∈ s, trivialization (F x) f) (h : ∀ (x : X) (hx : x ∈ s), x ∈ (e x hx).base_set) :
is_covering_map_on f s | λ x hx, is_evenly_covered.to_is_evenly_covered_preimage ⟨hF x, e x hx, h x hx⟩ | lemma | is_covering_map_on.mk | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"discrete_topology",
"is_covering_map_on",
"is_evenly_covered.to_is_evenly_covered_preimage",
"topological_space",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at (hf : is_covering_map_on f s) {x : E} (hx : f x ∈ s) :
continuous_at f x | (hf (f x) hx).continuous_at | lemma | is_covering_map_on.continuous_at | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"continuous_at",
"is_covering_map_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on (hf : is_covering_map_on f s) : continuous_on f (f ⁻¹' s) | continuous_at.continuous_on (λ x, hf.continuous_at) | lemma | is_covering_map_on.continuous_on | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"continuous_at.continuous_on",
"continuous_on",
"is_covering_map_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_homeomorph_on (hf : is_covering_map_on f s) :
is_locally_homeomorph_on f (f ⁻¹' s) | begin
refine is_locally_homeomorph_on.mk f (f ⁻¹' s) (λ x hx, _),
let e := (hf (f x) hx).to_trivialization,
have h := (hf (f x) hx).mem_to_trivialization_base_set,
let he := e.mem_source.2 h,
refine ⟨e.to_local_homeomorph.trans
{ to_fun := λ p, p.1,
inv_fun := λ p, ⟨p, x, rfl⟩,
source := e.base_set ... | lemma | is_covering_map_on.is_locally_homeomorph_on | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"continuous_const",
"continuous_on",
"inv_fun",
"is_covering_map_on",
"is_locally_homeomorph_on",
"is_locally_homeomorph_on.mk",
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_covering_map | ∀ x, is_evenly_covered f x (f ⁻¹' {x}) | def | is_covering_map | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_evenly_covered"
] | A covering map is a continuous function `f : E → X` with discrete fibers such that each point
of `X` has an evenly covered neighborhood. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_covering_map_iff_is_covering_map_on_univ :
is_covering_map f ↔ is_covering_map_on f set.univ | by simp only [is_covering_map, is_covering_map_on, set.mem_univ, forall_true_left] | lemma | is_covering_map_iff_is_covering_map_on_univ | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"forall_true_left",
"is_covering_map",
"is_covering_map_on",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_covering_map.is_covering_map_on (hf : is_covering_map f) :
is_covering_map_on f set.univ | is_covering_map_iff_is_covering_map_on_univ.mp hf | lemma | is_covering_map.is_covering_map_on | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_covering_map",
"is_covering_map_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (F : X → Type*) [Π x, topological_space (F x)] [hF : Π x, discrete_topology (F x)]
(e : Π x, trivialization (F x) f) (h : ∀ x, x ∈ (e x).base_set) : is_covering_map f | is_covering_map_iff_is_covering_map_on_univ.mpr
(is_covering_map_on.mk f set.univ F (λ x hx, e x) (λ x hx, h x)) | lemma | is_covering_map.mk | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"discrete_topology",
"is_covering_map",
"is_covering_map_on.mk",
"topological_space",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : is_covering_map f) : continuous f | continuous_iff_continuous_on_univ.mpr hf.is_covering_map_on.continuous_on | lemma | is_covering_map.continuous | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"continuous",
"is_covering_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_homeomorph (hf : is_covering_map f) : is_locally_homeomorph f | is_locally_homeomorph_iff_is_locally_homeomorph_on_univ.mpr
hf.is_covering_map_on.is_locally_homeomorph_on | lemma | is_covering_map.is_locally_homeomorph | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_covering_map",
"is_locally_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map (hf : is_covering_map f) : is_open_map f | hf.is_locally_homeomorph.is_open_map | lemma | is_covering_map.is_open_map | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_covering_map",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map (hf : is_covering_map f) (hf' : function.surjective f) :
quotient_map f | hf.is_open_map.to_quotient_map hf.continuous hf' | lemma | is_covering_map.quotient_map | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"is_covering_map",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fiber_bundle.is_covering_map {F : Type*} [topological_space F]
[discrete_topology F] (hf : ∀ x : X, ∃ e : trivialization F f, x ∈ e.base_set) :
is_covering_map f | is_covering_map.mk f (λ x, F) (λ x, classical.some (hf x)) (λ x, classical.some_spec (hf x)) | lemma | is_fiber_bundle.is_covering_map | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"discrete_topology",
"is_covering_map",
"is_covering_map.mk",
"topological_space",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_bundle.is_covering_map {F : Type*} {E : X → Type*} [topological_space F]
[discrete_topology F] [topological_space (bundle.total_space F E)] [Π x, topological_space (E x)]
[hf : fiber_bundle F E] : is_covering_map (π F E) | is_fiber_bundle.is_covering_map
(λ x, ⟨trivialization_at F E x, mem_base_set_trivialization_at F E x ⟩) | lemma | fiber_bundle.is_covering_map | topology | src/topology/covering.lean | [
"topology.is_locally_homeomorph",
"topology.fiber_bundle.basic"
] | [
"bundle.total_space",
"discrete_topology",
"fiber_bundle",
"is_covering_map",
"is_fiber_bundle.is_covering_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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