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_quot_mk : continuous (@quot.mk α r) | continuous_coinduced_rng | lemma | continuous_quot_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_quot_lift {f : α → β} (hr : ∀ a b, r a b → f a = f b)
(h : continuous f) : continuous (quot.lift f hr : quot r → β) | continuous_coinduced_dom.2 h | lemma | continuous_quot_lift | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_quotient_mk : quotient_map (@quotient.mk α s) | quotient_map_quot_mk | lemma | quotient_map_quotient_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"quotient_map",
"quotient_map_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_quotient_mk : continuous (@quotient.mk α s) | continuous_coinduced_rng | lemma | continuous_quotient_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.quotient_lift {f : α → β} (h : continuous f) (hs : ∀ a b, a ≈ b → f a = f b) :
continuous (quotient.lift f hs : quotient s → β) | continuous_coinduced_dom.2 h | lemma | continuous.quotient_lift | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.quotient_lift_on' {f : α → β} (h : continuous f)
(hs : ∀ a b, @setoid.r _ s a b → f a = f b) :
continuous (λ x, quotient.lift_on' x f hs : quotient s → β) | h.quotient_lift hs | lemma | continuous.quotient_lift_on' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"quotient.lift_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.quotient_map' {t : setoid β} {f : α → β} (hf : continuous f)
(H : (s.r ⇒ t.r) f f) : continuous (quotient.map' f H) | (continuous_quotient_mk.comp hf).quotient_lift _ | lemma | continuous.quotient_map' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"quotient.map'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_pi_iff : continuous f ↔ ∀ i, continuous (λ a, f a i) | by simp only [continuous_infi_rng, continuous_induced_rng] | lemma | continuous_pi_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_rng",
"continuous_infi_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_pi (h : ∀ i, continuous (λ a, f a i)) : continuous f | continuous_pi_iff.2 h | lemma | continuous_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_apply (i : ι) : continuous (λp:Πi, π i, p i) | continuous_infi_dom continuous_induced_dom | lemma | continuous_apply | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_dom",
"continuous_infi_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_apply_apply {ρ : κ → ι → Type*} [∀ j i, topological_space (ρ j i)]
(j : κ) (i : ι) : continuous (λ p : (Π j, Π i, ρ j i), p j i) | (continuous_apply i).comp (continuous_apply j) | lemma | continuous_apply_apply | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_apply",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_apply (i : ι) (x : Π i, π i) : continuous_at (λ p : Π i, π i, p i) x | (continuous_apply i).continuous_at | lemma | continuous_at_apply | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_apply",
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.apply {l : filter β} {f : β → Π i, π i}
{x : Π i, π i} (h : tendsto f l (𝓝 x)) (i : ι) :
tendsto (λ a, f a i) l (𝓝 $ x i) | (continuous_at_apply i _).tendsto.comp h | lemma | filter.tendsto.apply | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at_apply",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_pi {a : Πi, π i} : 𝓝 a = pi (λ i, 𝓝 (a i)) | by simp only [nhds_infi, nhds_induced, filter.pi] | lemma | nhds_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.pi",
"nhds_induced",
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pi_nhds {f : β → Πi, π i} {g : Πi, π i} {u : filter β} :
tendsto f u (𝓝 g) ↔ ∀ x, tendsto (λ i, f i x) u (𝓝 (g x)) | by rw [nhds_pi, filter.tendsto_pi] | lemma | tendsto_pi_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter",
"filter.tendsto_pi",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_pi {f : α → Π i, π i} {x : α} :
continuous_at f x ↔ ∀ i, continuous_at (λ y, f y i) x | tendsto_pi_nhds | lemma | continuous_at_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"tendsto_pi_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.update [decidable_eq ι]
{l : filter β} {f : β → Π i, π i} {x : Π i, π i} (hf : tendsto f l (𝓝 x)) (i : ι)
{g : β → π i} {xi : π i} (hg : tendsto g l (𝓝 xi)) :
tendsto (λ a, update (f a) i (g a)) l (𝓝 $ update x i xi) | tendsto_pi_nhds.2 $ λ j, by { rcases em (j = i) with rfl|hj; simp [*, hf.apply] } | lemma | filter.tendsto.update | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"em",
"filter",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.update [decidable_eq ι] {a : α} (hf : continuous_at f a) (i : ι) {g : α → π i}
(hg : continuous_at g a) :
continuous_at (λ a, update (f a) i (g a)) a | hf.update i hg | lemma | continuous_at.update | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.update [decidable_eq ι] (hf : continuous f) (i : ι) {g : α → π i}
(hg : continuous g) :
continuous (λ a, update (f a) i (g a)) | continuous_iff_continuous_at.2 $ λ x, hf.continuous_at.update i hg.continuous_at | lemma | continuous.update | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_update [decidable_eq ι] (i : ι) :
continuous (λ f : (Π j, π j) × π i, update f.1 i f.2) | continuous_fst.update i continuous_snd | lemma | continuous_update | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_snd",
"update"
] | `function.update f i x` is continuous in `(f, x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_mul_single [Π i, has_one (π i)] [decidable_eq ι] (i : ι) :
continuous (λ x, (pi.mul_single i x : Π i, π i)) | continuous_const.update _ continuous_id | lemma | continuous_mul_single | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id",
"pi.mul_single"
] | `pi.mul_single i x` is continuous in `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : β → π i} {l : filter β} {x : π i} (hf : tendsto f l (𝓝 x))
{g : β → Π j : fin n, π (i.succ_above j)} {y : Π j, π (i.succ_above j)} (hg : tendsto g l (𝓝 y)) :
tendsto (λ a, i.insert_nth (f a) (g a)) ... | tendsto_pi_nhds.2 (λ j, fin.succ_above_cases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j) | lemma | filter.tendsto.fin_insert_nth | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter",
"fin.succ_above_cases",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : α → π i} {a : α} (hf : continuous_at f a)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous_at g a) :
continuous_at (λ a, i.insert_nth (f a) (g a)) a | hf.fin_insert_nth i hg | lemma | continuous_at.fin_insert_nth | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : α → π i} (hf : continuous f)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous g) :
continuous (λ a, i.insert_nth (f a) (g a)) | continuous_iff_continuous_at.2 $ λ a, hf.continuous_at.fin_insert_nth i hg.continuous_at | lemma | continuous.fin_insert_nth | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_pi {i : set ι} {s : Πa, set (π a)} (hi : i.finite) (hs : ∀a∈i, is_open (s a)) :
is_open (pi i s) | by rw [pi_def]; exact (is_open_bInter hi $ assume a ha, (hs _ ha).preimage (continuous_apply _)) | lemma | is_open_set_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_apply",
"is_open",
"is_open_bInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_pi_iff {s : set (Π a, π a)} :
is_open s ↔
(∀ f, f ∈ s → ∃ (I : finset ι) (u : Π a, set (π a)),
(∀ a, a ∈ I → is_open (u a) ∧ f a ∈ u a) ∧ (I : set ι).pi u ⊆ s) | begin
rw is_open_iff_nhds,
simp_rw [le_principal_iff, nhds_pi, filter.mem_pi', mem_nhds_iff, exists_prop],
refine ball_congr (λ a h, ⟨_, _⟩),
{ rintros ⟨I, t, ⟨h1, h2⟩⟩,
refine ⟨I, λ a, eval a '' ((I : set ι).pi (λ a, (h1 a).some)), (λ i hi, _), _⟩,
{ simp_rw set.eval_image_pi (finset.mem_coe.mpr hi)
... | lemma | is_open_pi_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"ball_congr",
"exists_prop",
"filter.mem_pi'",
"finset",
"finset.mem_coe",
"is_open",
"is_open_iff_nhds",
"is_open_univ",
"ite_and",
"mem_nhds_iff",
"nhds_pi",
"set.eval_image_pi",
"set.eval_image_pi_subset",
"set.pi_mono",
"set.univ_pi_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_pi_iff' [finite ι] {s : set (Π a, π a)} :
is_open s ↔
(∀ f, f ∈ s → ∃ (u : Π a, set (π a)), (∀ a, is_open (u a) ∧ f a ∈ u a) ∧ set.univ.pi u ⊆ s) | begin
casesI nonempty_fintype ι,
rw is_open_iff_nhds,
simp_rw [le_principal_iff, nhds_pi, filter.mem_pi', mem_nhds_iff, exists_prop],
refine ball_congr (λ a h, ⟨_, _⟩),
{ rintros ⟨I, t, ⟨h1, h2⟩⟩,
refine ⟨λ i, (h1 i).some, ⟨λ i, (h1 i).some_spec.2,
(set.pi_mono (λ i _, (h1 i).some_spec.1)).trans (... | lemma | is_open_pi_iff' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"ball_congr",
"exists_prop",
"filter.mem_pi'",
"finite",
"is_open",
"is_open_iff_nhds",
"mem_nhds_iff",
"nhds_pi",
"nonempty_fintype",
"set.pi_inter_compl",
"set.pi_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_pi {i : set ι} {s : Πa, set (π a)} (hs : ∀a∈i, is_closed (s a)) :
is_closed (pi i s) | by rw [pi_def];
exact (is_closed_Inter $ λ a, is_closed_Inter $ λ ha, (hs _ ha).preimage (continuous_apply _)) | lemma | is_closed_set_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_apply",
"is_closed",
"is_closed_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_of_pi_mem_nhds {I : set ι} {s : Π i, set (π i)} (a : Π i, π i) (hs : I.pi s ∈ 𝓝 a)
{i : ι} (hi : i ∈ I) :
s i ∈ 𝓝 (a i) | by { rw nhds_pi at hs, exact mem_of_pi_mem_pi hs hi } | lemma | mem_nhds_of_pi_mem_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_pi_mem_nhds {i : set ι} {s : Π a, set (π a)}
{x : Π a, π a} (hi : i.finite) (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) :
pi i s ∈ 𝓝 x | by { rw [pi_def, bInter_mem hi], exact λ a ha, (continuous_apply a).continuous_at (hs a ha) } | lemma | set_pi_mem_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_apply",
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_pi_mem_nhds_iff {I : set ι} (hI : I.finite) {s : Π i, set (π i)} (a : Π i, π i) :
I.pi s ∈ 𝓝 a ↔ ∀ (i : ι), i ∈ I → s i ∈ 𝓝 (a i) | by { rw [nhds_pi, pi_mem_pi_iff hI], apply_instance } | lemma | set_pi_mem_nhds_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_pi_set {I : set ι} (hI : I.finite) {s : Π i, set (π i)} :
interior (pi I s) = I.pi (λ i, interior (s i)) | by { ext a, simp only [set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI] } | lemma | interior_pi_set | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"interior",
"mem_interior_iff_mem_nhds",
"set.mem_pi",
"set_pi_mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finset_piecewise_mem_of_mem_nhds [decidable_eq ι]
{s : set (Π a, π a)} {x : Π a, π a} (hs : s ∈ 𝓝 x) (y : Π a, π a) :
∃ I : finset ι, I.piecewise x y ∈ s | begin
simp only [nhds_pi, filter.mem_pi'] at hs,
rcases hs with ⟨I, t, htx, hts⟩,
refine ⟨I, hts $ λ i hi, _⟩,
simpa [finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)
end | lemma | exists_finset_piecewise_mem_of_mem_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.mem_pi'",
"finset",
"mem_of_mem_nhds",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_eq_generate_from :
Pi.topological_space =
generate_from {g | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, is_open (s a)) ∧ g = pi ↑i s} | le_antisymm
(le_generate_from $ assume g ⟨s, i, hi, eq⟩, eq.symm ▸ is_open_set_pi (finset.finite_to_set _) hi)
(le_infi $ assume a s ⟨t, ht, s_eq⟩, generate_open.basic _ $
⟨update (λa, univ) a t, {a}, by simpa using ht, s_eq ▸ by ext f; simp [set.pi]⟩) | lemma | pi_eq_generate_from | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"Pi.topological_space",
"finset",
"finset.finite_to_set",
"is_open",
"is_open_set_pi",
"le_generate_from",
"le_infi",
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_generate_from_eq {π : ι → Type*} {g : Πa, set (set (π a))} :
@Pi.topological_space ι π (λa, generate_from (g a)) =
generate_from {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ g a) ∧ t = pi ↑i s} | let G := {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ g a) ∧ t = pi ↑i s} in
begin
rw [pi_eq_generate_from],
refine le_antisymm (generate_from_anti _) (le_generate_from _),
exact assume s ⟨t, i, ht, eq⟩, ⟨t, i, assume a ha, generate_open.basic _ (ht a ha), eq⟩,
{ rintros s ⟨t, i, hi, rfl⟩,
rw [pi_de... | lemma | pi_generate_from_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"Pi.topological_space",
"finset",
"finset.finite_to_set",
"is_open",
"is_open_bInter",
"le_generate_from",
"pi_eq_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_generate_from_eq_finite {π : ι → Type*} {g : Πa, set (set (π a))} [finite ι]
(hg : ∀a, ⋃₀ g a = univ) :
@Pi.topological_space ι π (λa, generate_from (g a)) =
generate_from {t | ∃(s:Πa, set (π a)), (∀a, s a ∈ g a) ∧ t = pi univ s} | begin
casesI nonempty_fintype ι,
rw [pi_generate_from_eq],
refine le_antisymm (generate_from_anti _) (le_generate_from _),
{ rintro s ⟨t, ht, rfl⟩, exact ⟨t, finset.univ, by simp [ht]⟩ },
{ rintros s ⟨t, i, ht, rfl⟩,
apply is_open_iff_forall_mem_open.2 _,
assume f hf,
choose c hc using show ∀a, ∃s... | lemma | pi_generate_from_eq_finite | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"Pi.topological_space",
"finite",
"finset.univ",
"le_generate_from",
"nonempty_fintype",
"pi_generate_from_eq",
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_infi_to_pi {X : Type*} (f : Π i, X → π i) :
@inducing X (Π i, π i) (⨅ i, induced (f i) infer_instance) _ (λ x i, f i x) | begin
constructor,
erw induced_infi,
congr' 1,
funext,
erw induced_compose,
end | lemma | inducing_infi_to_pi | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"induced_compose",
"induced_infi",
"inducing"
] | Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
the `f i` as `i : ι` varies is simply the topology on `X` indu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.discrete_topology : discrete_topology (Π i, π i) | singletons_open_iff_discrete.mp (λ x,
begin
rw show {x} = ⋂ i, {y : Π i, π i | y i = x i},
{ ext, simp only [funext_iff, set.mem_singleton_iff, set.mem_Inter, set.mem_set_of_eq] },
exact is_open_Inter (λ i, (continuous_apply i).is_open_preimage {x i} (is_open_discrete {x i}))
end) | instance | Pi.discrete_topology | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_apply",
"discrete_topology",
"is_open_Inter",
"is_open_discrete",
"set.mem_Inter",
"set.mem_singleton_iff"
] | A finite product of discrete spaces is discrete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_sigma_mk {i : ι} : continuous (@sigma.mk ι σ i) | continuous_supr_rng continuous_coinduced_rng | lemma | continuous_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_rng",
"continuous_supr_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sigma_iff {s : set (sigma σ)} : is_open s ↔ ∀ i, is_open (sigma.mk i ⁻¹' s) | by simp only [is_open_supr_iff, is_open_coinduced] | lemma | is_open_sigma_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"is_open_coinduced",
"is_open_supr_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_sigma_iff {s : set (sigma σ)} : is_closed s ↔ ∀ i, is_closed (sigma.mk i ⁻¹' s) | by simp only [← is_open_compl_iff, is_open_sigma_iff, preimage_compl] | lemma | is_closed_sigma_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_sigma_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_sigma_mk {i : ι} : is_open_map (@sigma.mk ι σ i) | begin
intros s hs,
rw is_open_sigma_iff,
intro j,
rcases eq_or_ne j i with (rfl|hne),
{ rwa set.preimage_image_eq _ sigma_mk_injective },
{ rw [preimage_image_sigma_mk_of_ne hne],
exact is_open_empty }
end | lemma | is_open_map_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"eq_or_ne",
"is_open_empty",
"is_open_map",
"is_open_sigma_iff",
"set.preimage_image_eq",
"sigma_mk_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_range_sigma_mk {i : ι} : is_open (set.range (@sigma.mk ι σ i)) | is_open_map_sigma_mk.is_open_range | lemma | is_open_range_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_sigma_mk {i : ι} : is_closed_map (@sigma.mk ι σ i) | begin
intros s hs,
rw is_closed_sigma_iff,
intro j,
rcases eq_or_ne j i with (rfl|hne),
{ rwa set.preimage_image_eq _ sigma_mk_injective },
{ rw [preimage_image_sigma_mk_of_ne hne],
exact is_closed_empty }
end | lemma | is_closed_map_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"eq_or_ne",
"is_closed_empty",
"is_closed_map",
"is_closed_sigma_iff",
"set.preimage_image_eq",
"sigma_mk_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_range_sigma_mk {i : ι} : is_closed (set.range (@sigma.mk ι σ i)) | is_closed_map_sigma_mk.closed_range | lemma | is_closed_range_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_sigma_mk {i : ι} : open_embedding (@sigma.mk ι σ i) | open_embedding_of_continuous_injective_open
continuous_sigma_mk sigma_mk_injective is_open_map_sigma_mk | lemma | open_embedding_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_sigma_mk",
"is_open_map_sigma_mk",
"open_embedding",
"open_embedding_of_continuous_injective_open",
"sigma_mk_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_sigma_mk {i : ι} : closed_embedding (@sigma.mk ι σ i) | closed_embedding_of_continuous_injective_closed
continuous_sigma_mk sigma_mk_injective is_closed_map_sigma_mk | lemma | closed_embedding_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding",
"closed_embedding_of_continuous_injective_closed",
"continuous_sigma_mk",
"is_closed_map_sigma_mk",
"sigma_mk_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_sigma_mk {i : ι} : embedding (@sigma.mk ι σ i) | closed_embedding_sigma_mk.1 | lemma | embedding_sigma_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.nhds_mk (i : ι) (x : σ i) : 𝓝 (⟨i, x⟩ : sigma σ) = map (sigma.mk i) (𝓝 x) | (open_embedding_sigma_mk.map_nhds_eq x).symm | lemma | sigma.nhds_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.nhds_eq (x : sigma σ) : 𝓝 x = map (sigma.mk x.1) (𝓝 x.2) | by { cases x, apply sigma.nhds_mk } | lemma | sigma.nhds_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"sigma.nhds_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_sigma_mk_nhds (i : ι) (x : σ i) : comap (sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x | (embedding_sigma_mk.to_inducing.nhds_eq_comap _).symm | lemma | comap_sigma_mk_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sigma_fst_preimage (s : set ι) : is_open (sigma.fst ⁻¹' s : set (Σ a, σ a)) | begin
rw [← bUnion_of_singleton s, preimage_Union₂],
simp only [← range_sigma_mk],
exact is_open_bUnion (λ _ _, is_open_range_sigma_mk)
end | lemma | is_open_sigma_fst_preimage | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"is_open_bUnion",
"is_open_range_sigma_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sigma_iff {f : sigma σ → α} :
continuous f ↔ ∀ i, continuous (λ a, f ⟨i, a⟩) | by simp only [continuous_supr_dom, continuous_coinduced_dom] | lemma | continuous_sigma_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_dom",
"continuous_supr_dom"
] | A map out of a sum type is continuous iff its restriction to each summand is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_sigma {f : sigma σ → α} (hf : ∀ i, continuous (λ a, f ⟨i, a⟩)) :
continuous f | continuous_sigma_iff.2 hf | lemma | continuous_sigma | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | A map out of a sum type is continuous if its restriction to each summand is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} :
continuous (sigma.map f₁ f₂) ↔ ∀ i, continuous (f₂ i) | continuous_sigma_iff.trans $ by simp only [sigma.map, embedding_sigma_mk.continuous_iff] | lemma | continuous_sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)}
(hf : ∀ i, continuous (f₂ i)) :
continuous (sigma.map f₁ f₂) | continuous_sigma_map.2 hf | lemma | continuous.sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_sigma {f : sigma σ → α} : is_open_map f ↔ ∀ i, is_open_map (λ a, f ⟨i, a⟩) | by simp only [is_open_map_iff_nhds_le, sigma.forall, sigma.nhds_eq, map_map] | lemma | is_open_map_sigma | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map_iff_nhds_le",
"sigma.nhds_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} :
is_open_map (sigma.map f₁ f₂) ↔ ∀ i, is_open_map (f₂ i) | is_open_map_sigma.trans $ forall_congr $
λ i, (@open_embedding_sigma_mk _ _ _ (f₁ i)).is_open_map_iff.symm | lemma | is_open_map_sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"open_embedding_sigma_mk",
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h₁ : injective f₁) :
inducing (sigma.map f₁ f₂) ↔ ∀ i, inducing (f₂ i) | by simp only [inducing_iff_nhds, sigma.forall, sigma.nhds_mk, sigma.map, ← map_sigma_mk_comap h₁,
map_inj sigma_mk_injective] | lemma | inducing_sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"inducing",
"inducing_iff_nhds",
"sigma.map",
"sigma.nhds_mk",
"sigma_mk_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h : injective f₁) :
embedding (sigma.map f₁ f₂) ↔ ∀ i, embedding (f₂ i) | by simp only [embedding_iff, injective.sigma_map, inducing_sigma_map h, forall_and_distrib,
h.sigma_map_iff] | lemma | embedding_sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding",
"forall_and_distrib",
"inducing_sigma_map",
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h : injective f₁) :
open_embedding (sigma.map f₁ f₂) ↔ ∀ i, open_embedding (f₂ i) | by simp only [open_embedding_iff_embedding_open, is_open_map_sigma_map, embedding_sigma_map h,
forall_and_distrib] | lemma | open_embedding_sigma_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding_sigma_map",
"forall_and_distrib",
"is_open_map_sigma_map",
"open_embedding",
"open_embedding_iff_embedding_open",
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_ulift_down [topological_space α] :
continuous (ulift.down : ulift.{v u} α → α) | continuous_induced_dom | lemma | continuous_ulift_down | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_dom",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_ulift_up [topological_space α] :
continuous (ulift.up : α → ulift.{v u} α) | continuous_induced_rng.2 continuous_id | lemma | continuous_ulift_up | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_ulift_down [topological_space α] :
embedding (ulift.down : ulift.{v u} α → α) | ⟨⟨rfl⟩, ulift.down_injective⟩ | lemma | embedding_ulift_down | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.closed_embedding_down [topological_space α] :
closed_embedding (ulift.down : ulift.{v u} α → α) | ⟨embedding_ulift_down, by simp only [ulift.down_surjective.range_eq, is_closed_univ]⟩ | lemma | ulift.closed_embedding_down | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding",
"is_closed_univ",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bind_nhds_within {a : α} {s : set α} :
(𝓝 a).bind (λ x, 𝓝[s] x) = 𝓝[s] a | bind_inf_principal.trans $ congr_arg2 _ nhds_bind_nhds rfl | lemma | nhds_bind_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"congr_arg2",
"nhds_bind_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_nhds_within {a : α} {s : set α} {p : α → Prop} :
(∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x | filter.ext_iff.1 nhds_bind_nhds_within {x | p x} | lemma | eventually_nhds_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_bind_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_iff {a : α} {s : set α} {p : α → Prop} :
(∀ᶠ x in 𝓝[s] a, p x) ↔ ∀ᶠ x in 𝓝 a, x ∈ s → p x | eventually_inf_principal | lemma | eventually_nhds_within_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_nhds_within_iff {z : α} {s : set α} {p : α → Prop} :
(∃ᶠ x in 𝓝[s] z, p x) ↔ (∃ᶠ x in 𝓝 z, p x ∧ x ∈ s) | iff.not (by simp [eventually_nhds_within_iff, not_and']) | lemma | frequently_nhds_within_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"eventually_nhds_within_iff",
"iff.not",
"not_and'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_ne_iff_frequently_within {z : α} {s : set α} :
z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s | by simp [mem_closure_iff_frequently, frequently_nhds_within_iff] | lemma | mem_closure_ne_iff_frequently_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"frequently_nhds_within_iff",
"mem_closure_iff_frequently"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_nhds_within {a : α} {s : set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x | begin
refine ⟨λ h, _, λ h, (eventually_nhds_nhds_within.2 h).filter_mono inf_le_left⟩,
simp only [eventually_nhds_within_iff] at h ⊢,
exact h.mono (λ x hx hxs, (hx hxs).self_of_nhds hxs)
end | lemma | eventually_nhds_within_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"eventually_nhds_within_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_eq (a : α) (s : set α) :
𝓝[s] a = ⨅ t ∈ {t : set α | a ∈ t ∧ is_open t}, 𝓟 (t ∩ s) | ((nhds_basis_opens a).inf_principal s).eq_binfi | theorem | nhds_within_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_univ (a : α) : 𝓝[set.univ] a = 𝓝 a | by rw [nhds_within, principal_univ, inf_top_eq] | theorem | nhds_within_univ | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_top_eq",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_has_basis {p : β → Prop} {s : β → set α} {a : α} (h : (𝓝 a).has_basis p s)
(t : set α) :
(𝓝[t] a).has_basis p (λ i, s i ∩ t) | h.inf_principal t | lemma | nhds_within_has_basis | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_basis_open (a : α) (t : set α) :
(𝓝[t] a).has_basis (λ u, a ∈ u ∧ is_open u) (λ u, u ∩ t) | nhds_within_has_basis (nhds_basis_opens a) t | lemma | nhds_within_basis_open | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds_basis_opens",
"nhds_within_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within {t : set α} {a : α} {s : set α} :
t ∈ 𝓝[s] a ↔ ∃ u, is_open u ∧ a ∈ u ∧ u ∩ s ⊆ t | by simpa only [exists_prop, and_assoc, and_comm] using (nhds_within_basis_open a s).mem_iff | theorem | mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"exists_prop",
"is_open",
"nhds_within_basis_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_iff_exists_mem_nhds_inter {t : set α} {a : α} {s : set α} :
t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t | (nhds_within_has_basis (𝓝 a).basis_sets s).mem_iff | lemma | mem_nhds_within_iff_exists_mem_nhds_inter | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_mem_nhds_within_compl {x : α} {s : set α} (hs : s ∈ 𝓝 x) (t : set α) :
s \ t ∈ 𝓝[tᶜ] x | diff_mem_inf_principal_compl hs t | lemma | diff_mem_nhds_within_compl | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_mem_nhds_within_diff {x : α} {s t : set α} (hs : s ∈ 𝓝[t] x) (t' : set α) :
s \ t' ∈ 𝓝[t \ t'] x | begin
rw [nhds_within, diff_eq, diff_eq, ← inf_principal, ← inf_assoc],
exact inter_mem_inf hs (mem_principal_self _)
end | lemma | diff_mem_nhds_within_diff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_assoc",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_nhds_within_of_nhds
{s t : set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) : (t ∈ 𝓝 a) | begin
rcases mem_nhds_within_iff_exists_mem_nhds_inter.mp h2 with ⟨_, Hw, hw⟩,
exact (nhds a).sets_of_superset ((nhds a).inter_sets Hw h1) hw,
end | lemma | nhds_of_nhds_within_of_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_iff_eventually {s t : set α} {x : α} :
t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t | set_eventually_le_iff_mem_inf_principal.symm | lemma | mem_nhds_within_iff_eventually | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_iff_eventually_eq {s t : set α} {x : α} :
t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : set α) | by simp_rw [mem_nhds_within_iff_eventually, eventually_eq_set, mem_inter_iff, iff_self_and] | lemma | mem_nhds_within_iff_eventually_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"iff_self_and",
"mem_nhds_within_iff_eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_eq_iff_eventually_eq {s t : set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t | set_eventually_eq_iff_inf_principal.symm | lemma | nhds_within_eq_iff_eventually_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_le_iff {s t : set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x | set_eventually_le_iff_inf_principal_le.symm.trans set_eventually_le_iff_mem_inf_principal | lemma | nhds_within_le_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_nhds_within_coinduced' {π : α → β} {s : set β} {t : set α} {a : α}
(h : a ∈ t) (ht : is_open t)
(hs : s ∈ @nhds β (topological_space.coinduced (λ x : t, π x) subtype.topological_space) (π a)) :
π ⁻¹' s ∈ 𝓝[t] a | begin
letI := topological_space.coinduced (λ x : t, π x) subtype.topological_space,
rcases mem_nhds_iff.mp hs with ⟨V, hVs, V_op, mem_V⟩,
refine mem_nhds_within_iff_exists_mem_nhds_inter.mpr ⟨π ⁻¹' V, mem_nhds_iff.mpr ⟨t ∩ π ⁻¹' V,
inter_subset_right t (π ⁻¹' V), _, mem_sep h mem_V⟩, subset.trans (inter_subse... | lemma | preimage_nhds_within_coinduced' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds",
"set.inter_comm",
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_of_mem_nhds {s t : set α} {a : α} (h : s ∈ 𝓝 a) :
s ∈ 𝓝[t] a | mem_inf_of_left h | lemma | mem_nhds_within_of_mem_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_mem_nhds_within {a : α} {s : set α} : s ∈ 𝓝[s] a | mem_inf_of_right (mem_principal_self s) | theorem | self_mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_mem_nhds_within {a : α} {s : set α} : ∀ᶠ x in 𝓝[s] a, x ∈ s | self_mem_nhds_within | theorem | eventually_mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_mem_nhds_within (s : set α) {t : set α} {a : α} (h : t ∈ 𝓝 a) :
s ∩ t ∈ 𝓝[s] a | inter_mem self_mem_nhds_within (mem_inf_of_left h) | theorem | inter_mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_mono (a : α) {s t : set α} (h : s ⊆ t) : 𝓝[s] a ≤ 𝓝[t] a | inf_le_inf_left _ (principal_mono.mpr h) | theorem | nhds_within_mono | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_le_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_le_nhds_within {a : α} {s : set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a | le_inf (pure_le_nhds a) (le_principal_iff.2 ha) | lemma | pure_le_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"le_inf",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_mem_nhds_within {a : α} {s t : set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a) :
a ∈ t | pure_le_nhds_within ha ht | lemma | mem_of_mem_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"pure_le_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.self_of_nhds_within {p : α → Prop} {s : set α} {x : α}
(h : ∀ᶠ y in 𝓝[s] x, p y) (hx : x ∈ s) : p x | mem_of_mem_nhds_within hx h | lemma | filter.eventually.self_of_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"mem_of_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_nhds_within {l : filter β} {s : set α} {a : α} (ha : a ∈ s) :
tendsto (λ x : β, a) l (𝓝[s] a) | tendsto_const_pure.mono_right $ pure_le_nhds_within ha | lemma | tendsto_const_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"pure_le_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_restrict'' {a : α} (s : set α) {t : set α} (h : t ∈ 𝓝[s] a) :
𝓝[s] a = 𝓝[s ∩ t] a | le_antisymm
(le_inf inf_le_left (le_principal_iff.mpr (inter_mem self_mem_nhds_within h)))
(inf_le_inf_left _ (principal_mono.mpr (set.inter_subset_left _ _))) | theorem | nhds_within_restrict'' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_le_inf_left",
"inf_le_left",
"le_inf",
"self_mem_nhds_within",
"set.inter_subset_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_restrict' {a : α} (s : set α) {t : set α} (h : t ∈ 𝓝 a) :
𝓝[s] a = 𝓝[s ∩ t] a | nhds_within_restrict'' s $ mem_inf_of_left h | theorem | nhds_within_restrict' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_restrict''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_restrict {a : α} (s : set α) {t : set α} (h₀ : a ∈ t) (h₁ : is_open t) :
𝓝[s] a = 𝓝[s ∩ t] a | nhds_within_restrict' s (is_open.mem_nhds h₁ h₀) | theorem | nhds_within_restrict | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"is_open.mem_nhds",
"nhds_within_restrict'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_le_of_mem {a : α} {s t : set α} (h : s ∈ 𝓝[t] a) :
𝓝[t] a ≤ 𝓝[s] a | nhds_within_le_iff.mpr h | theorem | nhds_within_le_of_mem | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_le_nhds {a : α} {s : set α} : 𝓝[s] a ≤ 𝓝 a | by { rw ← nhds_within_univ, apply nhds_within_le_of_mem, exact univ_mem } | theorem | nhds_within_le_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_le_of_mem",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_eq_nhds_within' {a : α} {s t u : set α}
(hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a | by rw [nhds_within_restrict' t hs, nhds_within_restrict' u hs, h₂] | lemma | nhds_within_eq_nhds_within' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_restrict'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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