statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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CauchySeq.tendsto_uniformity [Preorder β] {u : β → α} (h : CauchySeq u) :
Tendsto (Prod.map u u) atTop (𝓤 α) | by
simpa only [Tendsto, prod_map_map_eq', prod_atTop_atTop_eq] using h.right | theorem | CauchySeq.tendsto_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.nonempty [Preorder β] {u : β → α} (hu : CauchySeq u) : Nonempty β | @nonempty_of_neBot _ _ <| (map_neBot_iff _).1 hu.1 | theorem | CauchySeq.nonempty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.mem_entourage {β : Type*} [SemilatticeSup β] {u : β → α} (h : CauchySeq u)
{V : SetRel α α} (hV : V ∈ 𝓤 α) : ∃ k₀, ∀ i j, k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V | by
haveI := h.nonempty
have := h.tendsto_uniformity; rw [← prod_atTop_atTop_eq] at this
simpa [MapsTo] using atTop_basis.prod_self.tendsto_left_iff.1 this V hV | theorem | CauchySeq.mem_entourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.cauchySeq [SemilatticeSup β] [Nonempty β] {f : β → α} {x}
(hx : Tendsto f atTop (𝓝 x)) : CauchySeq f | hx.cauchy_map | theorem | Filter.Tendsto.cauchySeq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_const [SemilatticeSup β] [Nonempty β] (x : α) : CauchySeq fun _ : β => x | tendsto_const_nhds.cauchySeq | theorem | cauchySeq_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_iff_tendsto [Nonempty β] [SemilatticeSup β] {u : β → α} :
CauchySeq u ↔ Tendsto (Prod.map u u) atTop (𝓤 α) | cauchy_map_iff'.trans <| by simp only [prod_atTop_atTop_eq, Prod.map_def] | theorem | cauchySeq_iff_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Prod.map_def",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.comp_tendsto {γ} [Preorder β] [SemilatticeSup γ] [Nonempty γ] {f : β → α}
(hf : CauchySeq f) {g : γ → β} (hg : Tendsto g atTop atTop) : CauchySeq (f ∘ g) | ⟨inferInstance, le_trans (prod_le_prod.mpr ⟨Tendsto.comp le_rfl hg, Tendsto.comp le_rfl hg⟩) hf.2⟩ | theorem | CauchySeq.comp_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder",
"SemilatticeSup",
"le_rfl",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.comp_injective [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α}
(hu : CauchySeq u) {f : β → ℕ} (hf : Injective f) : CauchySeq (u ∘ f) | hu.comp_tendsto <| Nat.cofinite_eq_atTop ▸ hf.tendsto_cofinite.mono_left atTop_le_cofinite | theorem | CauchySeq.comp_injective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Nat.cofinite_eq_atTop",
"NoMaxOrder",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Function.Bijective.cauchySeq_comp_iff {f : ℕ → ℕ} (hf : Bijective f) (u : ℕ → α) :
CauchySeq (u ∘ f) ↔ CauchySeq u | by
refine ⟨fun H => ?_, fun H => H.comp_injective hf.injective⟩
lift f to ℕ ≃ ℕ using hf
simpa only [Function.comp_def, f.apply_symm_apply] using H.comp_injective f.symm.injective | theorem | Function.Bijective.cauchySeq_comp_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.subseq_subseq_mem {V : ℕ → SetRel α α} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n | by
rw [cauchySeq_iff_tendsto] at hu
exact ((hu.comp <| hf.prod_atTop hg).comp tendsto_atTop_diagonal).subseq_mem hV | theorem | CauchySeq.subseq_subseq_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SetRel",
"StrictMono",
"cauchySeq_iff_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_iff' {u : ℕ → α} :
CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∀ᶠ k in atTop, k ∈ Prod.map u u ⁻¹' V | cauchySeq_iff_tendsto | theorem | cauchySeq_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"cauchySeq_iff_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_iff {u : ℕ → α} :
CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V | by
simp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod.map_apply] | theorem | cauchySeq_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Filter.eventually_atTop_prod_self'",
"cauchySeq_iff'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.prodMap {γ δ} [UniformSpace β] [Preorder γ] [Preorder δ] {u : γ → α} {v : δ → β}
(hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq (Prod.map u v) | by
simpa only [CauchySeq, prod_map_map_eq', prod_atTop_atTop_eq] using hu.prod hv | theorem | CauchySeq.prodMap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.prodMk {γ} [UniformSpace β] [Preorder γ] {u : γ → α} {v : γ → β}
(hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq fun x => (u x, v x) | haveI := hu.1.of_map
(Cauchy.prod hu hv).mono (tendsto_map.prodMk tendsto_map) | theorem | CauchySeq.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy.prod",
"CauchySeq",
"Preorder",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.eventually_eventually [Preorder β] {u : β → α} (hu : CauchySeq u)
{V : SetRel α α} (hV : V ∈ 𝓤 α) : ∀ᶠ k in atTop, ∀ᶠ l in atTop, (u k, u l) ∈ V | eventually_atTop_curry <| hu.tendsto_uniformity hV | theorem | CauchySeq.eventually_eventually | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.comp_cauchySeq {γ} [UniformSpace β] [Preorder γ] {f : α → β}
(hf : UniformContinuous f) {u : γ → α} (hu : CauchySeq u) : CauchySeq (f ∘ u) | hu.map hf | theorem | UniformContinuous.comp_cauchySeq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Preorder",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.subseq_mem {V : ℕ → SetRel α α} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V n | by
have : ∀ n, ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n := fun n => by
rw [cauchySeq_iff] at hu
rcases hu _ (hV n) with ⟨N, H⟩
exact ⟨N, fun k hk l hl => H _ (le_trans hk hl) _ hk⟩
obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ∀ l ≥ φ n, (u l, u <| φ n) ∈ V n⟩ :=
extraction_forall_of_eventually... | theorem | CauchySeq.subseq_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SetRel",
"StrictMono",
"cauchySeq_iff",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.subseq_mem_entourage {V : ℕ → SetRel α α} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
{a : α} (hu : Tendsto u atTop (𝓝 a)) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧
∀ n, (u <| φ (n + 1), u <| φ n) ∈ V (n + 1) | by
rcases mem_atTop_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity <| hV 0))) with ⟨n, hn⟩
rcases (hu.comp (tendsto_add_atTop_nat n)).cauchySeq.subseq_mem fun n => hV (n + 1) with
⟨φ, φ_mono, hφV⟩
exact ⟨fun k => φ k + n, φ_mono.add_const _, hn _ le_add_self, hφV⟩ | theorem | Filter.Tendsto.subseq_mem_entourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"SetRel",
"StrictMono",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_of_cauchySeq_of_subseq [Preorder β] {u : β → α} (hu : CauchySeq u)
{ι : Type*} {f : ι → β} {p : Filter ι} [NeBot p] (hf : Tendsto f p atTop) {a : α}
(ha : Tendsto (u ∘ f) p (𝓝 a)) : Tendsto u atTop (𝓝 a) | le_nhds_of_cauchy_adhp hu (ha.mapClusterPt.of_comp hf) | theorem | tendsto_nhds_of_cauchySeq_of_subseq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"Filter",
"Preorder",
"le_nhds_of_cauchy_adhp"
] | If a Cauchy sequence has a convergent subsequence, then it converges. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u | by
constructor <;> intro h
· rw [cauchySeq_iff] at h ⊢
intro V mV
obtain ⟨N, h⟩ := h V mV
use N + k
intro a ha b hb
convert! h (a - k) (Nat.le_sub_of_add_le ha) (b - k) (Nat.le_sub_of_add_le hb) <;> lia
· exact h.comp_tendsto (tendsto_add_atTop_nat k) | theorem | cauchySeq_shift | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"cauchySeq_iff"
] | Any shift of a Cauchy sequence is also a Cauchy sequence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop}
{s : γ → SetRel α α} (h : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i | by
rw [cauchySeq_iff_tendsto, ← prod_atTop_atTop_eq]
refine (atTop_basis.prod_self.tendsto_iff h).trans ?_
simp only [true_and, Prod.forall, mem_prod_eq,
mem_Ici, and_imp, Prod.map, @forall_comm (_ ≤ _) β] | theorem | Filter.HasBasis.cauchySeq_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup",
"SetRel",
"cauchySeq_iff_tendsto",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.cauchySeq_iff' {γ} [Nonempty β] [SemilatticeSup β] {u : β → α}
{p : γ → Prop} {s : γ → SetRel α α} (H : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i | by
refine H.cauchySeq_iff.trans ⟨fun h i hi => ?_, fun h i hi => ?_⟩
· exact (h i hi).imp fun N hN n hn => hN n hn N le_rfl
· rcases comp_symm_of_uniformity (H.mem_of_mem hi) with ⟨t, ht, ht', hts⟩
rcases H.mem_iff.1 ht with ⟨j, hj, hjt⟩
refine (h j hj).imp fun N hN m hm n hn => hts ⟨u N, hjt ?_, ht' <| h... | theorem | Filter.HasBasis.cauchySeq_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup",
"SetRel",
"comp_symm_of_uniformity",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_of_controlled [SemilatticeSup β] [Nonempty β] (U : β → SetRel α α)
(hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) {f : β → α}
(hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N) : CauchySeq f | cauchySeq_iff_tendsto.2
(by
intro s hs
rw [mem_map, mem_atTop_sets]
obtain ⟨N, hN⟩ := hU s hs
refine ⟨(N, N), fun mn hmn => ?_⟩
obtain ⟨m, n⟩ := mn
exact hN (hf hmn.1 hmn.2)) | theorem | cauchySeq_of_controlled | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"SemilatticeSup",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isComplete_iff_clusterPt {s : Set α} :
IsComplete s ↔ ∀ l, Cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x l | forall₃_congr fun _ hl _ => exists_congr fun _ => and_congr_right fun _ =>
le_nhds_iff_adhp_of_cauchy hl | theorem | isComplete_iff_clusterPt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"ClusterPt",
"IsComplete",
"Set",
"le_nhds_iff_adhp_of_cauchy"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isComplete_iff_ultrafilter {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x | by
refine ⟨fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_⟩
haveI := hl.1
rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with ⟨x, hxs, hxl⟩
exact ⟨x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)⟩ | theorem | isComplete_iff_ultrafilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"ClusterPt.of_le_nhds",
"Filter",
"IsComplete",
"Set",
"Ultrafilter",
"Ultrafilter.of",
"Ultrafilter.of_le",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isComplete_iff_ultrafilter' {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → s ∈ l → ∃ x ∈ s, ↑l ≤ 𝓝 x | isComplete_iff_ultrafilter.trans <| by simp only [le_principal_iff, Ultrafilter.mem_coe] | theorem | isComplete_iff_ultrafilter' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"IsComplete",
"Set",
"Ultrafilter",
"Ultrafilter.mem_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsComplete.union {s t : Set α} (hs : IsComplete s) (ht : IsComplete t) :
IsComplete (s ∪ t) | by
simp only [isComplete_iff_ultrafilter', Ultrafilter.union_mem_iff, or_imp] at *
exact fun l hl =>
⟨fun hsl => (hs l hl hsl).imp fun x hx => ⟨Or.inl hx.1, hx.2⟩, fun htl =>
(ht l hl htl).imp fun x hx => ⟨Or.inr hx.1, hx.2⟩⟩ | theorem | IsComplete.union | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsComplete",
"Set",
"Ultrafilter.union_mem_iff",
"isComplete_iff_ultrafilter'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : SetRel α α} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
IsComplete (⋃ i, s i) | by
set S := ⋃ i, s i
intro l hl hls
rw [le_principal_iff] at hls
obtain ⟨hl_ne, hl'⟩ := cauchy_iff.1 hl
obtain ⟨t, htS, htl, htU⟩ : ∃ t, t ⊆ S ∧ t ∈ l ∧ t ×ˢ t ⊆ U := by
rcases hl' U hU with ⟨t, htl, htU⟩
refine ⟨t ∩ S, inter_subset_right, inter_mem htl hls, Subset.trans ?_ htU⟩
gcongr <;> apply i... | theorem | isComplete_iUnion_separated | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.nonempty_of_mem",
"IsComplete",
"Set",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace (α : Type u) [UniformSpace α] : Prop where
/-- In a complete uniform space, every Cauchy filter converges. -/
complete : ∀ {f : Filter α}, Cauchy f → ∃ x, f ≤ 𝓝 x | class | CompleteSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"UniformSpace"
] | A complete space is defined here using uniformities. A uniform space
is complete if every Cauchy filter converges. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
complete_univ {α : Type u} [UniformSpace α] [CompleteSpace α] :
IsComplete (univ : Set α) | fun f hf _ => by
rcases CompleteSpace.complete hf with ⟨x, hx⟩
exact ⟨x, mem_univ x, hx⟩ | theorem | complete_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsComplete",
"Set",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.prod [UniformSpace β] [CompleteSpace α] [CompleteSpace β] :
CompleteSpace (α × β) | where
complete hf :=
let ⟨x1, hx1⟩ := CompleteSpace.complete <| hf.map uniformContinuous_fst
let ⟨x2, hx2⟩ := CompleteSpace.complete <| hf.map uniformContinuous_snd
⟨(x1, x2), by rw [nhds_prod_eq, le_prod]; constructor <;> assumption⟩ | instance | CompleteSpace.prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"UniformSpace",
"nhds_prod_eq",
"uniformContinuous_fst",
"uniformContinuous_snd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.fst_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty β] :
CompleteSpace α | where
complete hf :=
let ⟨y⟩ := h
let ⟨(a, b), hab⟩ := CompleteSpace.complete <| hf.prod <| cauchy_pure (a := y)
⟨a, by simpa only [map_fst_prod, nhds_prod_eq] using map_mono (m := Prod.fst) hab⟩ | lemma | CompleteSpace.fst_of_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"UniformSpace",
"cauchy_pure",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.snd_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty α] :
CompleteSpace β | where
complete hf :=
let ⟨x⟩ := h
let ⟨(a, b), hab⟩ := CompleteSpace.complete <| (cauchy_pure (a := x)).prod hf
⟨b, by simpa only [map_snd_prod, nhds_prod_eq] using map_mono (m := Prod.snd) hab⟩ | lemma | CompleteSpace.snd_of_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"UniformSpace",
"cauchy_pure",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_prod_of_nonempty [UniformSpace β] [Nonempty α] [Nonempty β] :
CompleteSpace (α × β) ↔ CompleteSpace α ∧ CompleteSpace β | ⟨fun _ ↦ ⟨.fst_of_prod (β := β), .snd_of_prod (α := α)⟩, fun ⟨_, _⟩ ↦ .prod⟩ | lemma | completeSpace_prod_of_nonempty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.mulOpposite [CompleteSpace α] : CompleteSpace αᵐᵒᵖ | where
complete hf :=
MulOpposite.op_surjective.exists.mpr <|
let ⟨x, hx⟩ := CompleteSpace.complete (hf.map MulOpposite.uniformContinuous_unop)
⟨x, (map_le_iff_le_comap.mp hx).trans_eq <| MulOpposite.comap_unop_nhds _⟩ | instance | CompleteSpace.mulOpposite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"MulOpposite.comap_unop_nhds",
"MulOpposite.uniformContinuous_unop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_of_isComplete_univ (h : IsComplete (univ : Set α)) : CompleteSpace α | ⟨fun hf => let ⟨x, _, hx⟩ := h _ hf ((@principal_univ α).symm ▸ le_top); ⟨x, hx⟩⟩ | theorem | completeSpace_of_isComplete_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsComplete",
"Set",
"le_top",
"symm"
] | If `univ` is complete, the space is a complete space | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
completeSpace_iff_isComplete_univ : CompleteSpace α ↔ IsComplete (univ : Set α) | ⟨@complete_univ α _, completeSpace_of_isComplete_univ⟩ | theorem | completeSpace_iff_isComplete_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsComplete",
"Set",
"complete_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_iff_ultrafilter :
CompleteSpace α ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ∃ x : α, ↑l ≤ 𝓝 x | by
simp [completeSpace_iff_isComplete_univ, isComplete_iff_ultrafilter] | theorem | completeSpace_iff_ultrafilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"CompleteSpace",
"Filter",
"Ultrafilter",
"completeSpace_iff_isComplete_univ",
"isComplete_iff_ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchy_iff_exists_le_nhds [CompleteSpace α] {l : Filter α} [NeBot l] :
Cauchy l ↔ ∃ x, l ≤ 𝓝 x | ⟨CompleteSpace.complete, fun ⟨_, hx⟩ => cauchy_nhds.mono hx⟩ | theorem | cauchy_iff_exists_le_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"CompleteSpace",
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchy_map_iff_exists_tendsto [CompleteSpace α] {l : Filter β} {f : β → α} [NeBot l] :
Cauchy (l.map f) ↔ ∃ x, Tendsto f l (𝓝 x) | cauchy_iff_exists_le_nhds | theorem | cauchy_map_iff_exists_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"CompleteSpace",
"Filter",
"cauchy_iff_exists_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchySeq_tendsto_of_complete [Preorder β] [CompleteSpace α] {u : β → α}
(H : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) | CompleteSpace.complete H | theorem | cauchySeq_tendsto_of_complete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"CompleteSpace",
"Preorder"
] | A Cauchy sequence in a complete space converges | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cauchySeq_tendsto_of_isComplete [Preorder β] {K : Set α} (h₁ : IsComplete K)
{u : β → α} (h₂ : ∀ n, u n ∈ K) (h₃ : CauchySeq u) : ∃ v ∈ K, Tendsto u atTop (𝓝 v) | h₁ _ h₃ <| le_principal_iff.2 <| mem_map_iff_exists_image.2
⟨univ, univ_mem, by rwa [image_univ, range_subset_iff]⟩ | theorem | cauchySeq_tendsto_of_isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"IsComplete",
"Preorder",
"Set"
] | If `K` is a complete subset, then any Cauchy sequence in `K` converges to a point in `K` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Cauchy.le_nhds_lim [CompleteSpace α] {f : Filter α} (hf : Cauchy f) :
haveI | hf.1.nonempty; f ≤ 𝓝 (lim f) :=
_root_.le_nhds_lim (CompleteSpace.complete hf) | theorem | Cauchy.le_nhds_lim | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"CompleteSpace",
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.tendsto_limUnder [Preorder β] [CompleteSpace α] {u : β → α} (h : CauchySeq u) :
haveI | h.1.nonempty; Tendsto u atTop (𝓝 <| limUnder atTop u) :=
h.le_nhds_lim | theorem | CauchySeq.tendsto_limUnder | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"CompleteSpace",
"Preorder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.isComplete [CompleteSpace α] {s : Set α} (h : IsClosed s) : IsComplete s | fun _ cf fs =>
let ⟨x, hx⟩ := CompleteSpace.complete cf
⟨x, isClosed_iff_clusterPt.mp h x (cf.left.mono (le_inf hx fs)), hx⟩ | theorem | IsClosed.isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsClosed",
"IsComplete",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_pure_of_cauchy {f : Filter α} (hf : Cauchy f) : ∃ x : α, f = pure x | by
rcases hf with ⟨f_ne_bot, f_le⟩
simp only [DiscreteUniformity.eq_principal_setRelId, le_principal_iff, mem_prod_iff] at f_le
obtain ⟨S, hS, T, hT, H⟩ := f_le
obtain ⟨x, rfl, _, _, _⟩ := SetRel.exists_eq_singleton_of_prod_subset_id
(f_ne_bot.nonempty_of_mem hS) (f_ne_bot.nonempty_of_mem hT) H
exact ⟨x, ... | theorem | DiscreteUniformity.eq_pure_of_cauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"DiscreteUniformity.eq_principal_setRelId",
"Filter",
"SetRel.exists_eq_singleton_of_prod_subset_id"
] | A Cauchy filter in a discrete uniform space is contained in the principal filter
of a point. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cauchyConst {f : Filter α} (hf : Cauchy f) : α | (eq_pure_of_cauchy hf).choose | def | DiscreteUniformity.cauchyConst | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter"
] | A constant to which a Cauchy filter in a discrete uniform space converges. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_pure_cauchyConst {f : Filter α} (hf : Cauchy f) : f = pure (cauchyConst hf) | (eq_pure_of_cauchy hf).choose_spec | theorem | DiscreteUniformity.eq_pure_cauchyConst | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded (s : Set α) : Prop | ∀ d ∈ 𝓤 α, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ d } | def | TotallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set"
] | A set `s` is totally bounded if for every entourage `d` there is a finite
set of points `t` such that every element of `s` is `d`-near to some element of `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.TotallyBounded (f : Filter α) | ∀ d : SetRel α α, d ∈ 𝓤 α → ∃ t : Set α, t.Finite ∧ d.preimage t ∈ f | def | Filter.TotallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Set",
"SetRel"
] | A filter `f` is totally bounded if for every entourage `d`, the `d`-neighborhood of some finite
set is in `f`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.totallyBounded_principal_iff {s : Set α} :
(𝓟 s).TotallyBounded ↔ TotallyBounded s | by
simp only [Filter.TotallyBounded, mem_principal, SetRel.preimage_eq_biUnion, TotallyBounded] | theorem | Filter.totallyBounded_principal_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.TotallyBounded",
"Set",
"SetRel.preimage_eq_biUnion",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.exists_subset_of_mem {f : Filter α} (hf : f.TotallyBounded)
{s : Set α} (hs : s ∈ f) {U : SetRel α α} (hU : U ∈ 𝓤 α) :
∃ t ⊆ s, Set.Finite t ∧ U.preimage t ∈ f | by
rcases comp_symm_of_uniformity hU with ⟨r, hr, rs, rU⟩
rcases hf r hr with ⟨k, fk, ks⟩
let u := k ∩ { y | ∃ x ∈ s, (x, y) ∈ r }
choose g hgs hgr using fun x : u => x.coe_prop.2
refine ⟨range g, ?_, ?_, ?_⟩
· exact range_subset_iff.2 hgs
· haveI : Fintype u := (fk.inter_of_left _).fintype
exact fini... | theorem | Filter.TotallyBounded.exists_subset_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Fintype",
"Set",
"Set.Finite",
"SetRel",
"SetRel.preimage",
"comp_symm_of_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.exists_subset {s : Set α} (hs : TotallyBounded s) {U : SetRel α α}
(hU : U ∈ 𝓤 α) : ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U } | by
rw [← Filter.totallyBounded_principal_iff] at hs
simp_rw [← SetRel.preimage_eq_biUnion]
exact hs.exists_subset_of_mem (Filter.mem_principal_self s) hU | theorem | TotallyBounded.exists_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.mem_principal_self",
"Filter.totallyBounded_principal_iff",
"Set",
"Set.Finite",
"SetRel",
"SetRel.preimage_eq_biUnion",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_iff_subset {s : Set α} :
TotallyBounded s ↔
∀ d ∈ 𝓤 α, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ d } | ⟨fun H _ hd ↦ H.exists_subset hd, fun H d hd ↦ let ⟨t, _, ht⟩ := H d hd; ⟨t, ht⟩⟩ | theorem | totallyBounded_iff_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"Set.Finite",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.totallyBounded_iff {ι} {p : ι → Prop} {U : ι → SetRel α α}
(H : (𝓤 α).HasBasis p U) {s : Set α} :
TotallyBounded s ↔ ∀ i, p i → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U i } | H.forall_iff fun _ _ hUV h =>
h.imp fun _ ht => ⟨ht.1, ht.2.trans <| iUnion₂_mono fun _ _ _ hy => hUV hy⟩ | theorem | Filter.HasBasis.totallyBounded_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"Set.Finite",
"SetRel",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.filter_totallyBounded_iff {ι} {p : ι → Prop} {U : ι → SetRel α α}
(H : (𝓤 α).HasBasis p U) {f : Filter α} :
f.TotallyBounded ↔ ∀ i, p i → ∃ t : Set α, Set.Finite t ∧ (U i).preimage t ∈ f | H.forall_iff fun _ _ _ h =>
h.imp fun _ ht => ⟨ht.1, f.mem_of_superset ht.2 <| by gcongr⟩ | theorem | Filter.HasBasis.filter_totallyBounded_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Set",
"Set.Finite",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_of_forall_isSymm {s : Set α}
(h : ∀ V ∈ 𝓤 α, SetRel.IsSymm V → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
TotallyBounded s | UniformSpace.hasBasis_symmetric.totallyBounded_iff.2 fun V ⟨_, _⟩ => by
simpa only [ball_eq_of_symmetry] using! h V ‹_› ‹_› | theorem | totallyBounded_of_forall_isSymm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"Set.Finite",
"SetRel.IsSymm",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.subset {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂) (h : TotallyBounded s₂) :
TotallyBounded s₁ | fun d hd =>
let ⟨t, ht₁, ht₂⟩ := h d hd
⟨t, ht₁, Subset.trans hs ht₂⟩ | theorem | TotallyBounded.subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.mono {f g : Filter α} (h : f ≤ g) (hg : g.TotallyBounded) :
f.TotallyBounded | fun U hU => (hg U hU).imp fun _ => And.imp_right (@h _) | theorem | Filter.TotallyBounded.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.totallyBounded_setOf_clusterPt {f : Filter α} (h : f.TotallyBounded) :
TotallyBounded {x | ClusterPt x f} | by
refine uniformity_hasBasis_closed.totallyBounded_iff.2 fun V hV => ?_
obtain ⟨t, htf, hst⟩ := h V hV.1
refine ⟨t, htf, fun x hx => ?_⟩
rw [← SetRel.preimage_eq_biUnion, id, ← (hV.2.relPreimage_of_finite htf).closure_eq]
exact hx.mem_closure_of_mem _ hst | theorem | Filter.TotallyBounded.totallyBounded_setOf_clusterPt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"ClusterPt",
"Filter",
"SetRel.preimage_eq_biUnion",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.closure {s : Set α} (h : TotallyBounded s) : TotallyBounded (closure s) | by
rw [closure_eq_cluster_pts]
exact (Filter.totallyBounded_principal_iff.mpr h).totallyBounded_setOf_clusterPt | theorem | TotallyBounded.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"closure",
"closure_eq_cluster_pts"
] | The closure of a totally bounded set is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_closure {s : Set α} : TotallyBounded (closure s) ↔ TotallyBounded s | ⟨fun h ↦ h.subset subset_closure, TotallyBounded.closure⟩ | lemma | totallyBounded_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_iSup {ι : Sort*} [Finite ι] {f : ι → Filter α} :
(⨆ i, f i).TotallyBounded ↔ ∀ i, (f i).TotallyBounded | by
refine ⟨fun h i ↦ h.mono (le_iSup _ _), fun h U hU ↦ ?_⟩
choose t htf ht using (h · U hU)
refine ⟨⋃ i, t i, finite_iUnion htf, ?_⟩
simp_rw [U.preimage_iUnion, ← le_principal_iff, ← iSup_principal] at ht ⊢
gcongr; apply ht | lemma | Filter.totallyBounded_iSup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Finite",
"TotallyBounded",
"le_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_biSup {ι : Type*} {I : Set ι} (hI : I.Finite) {f : ι → Filter α} :
(⨆ i ∈ I, f i).TotallyBounded ↔ ∀ i ∈ I, (f i).TotallyBounded | by
have := hI.to_subtype
rw [iSup_subtype', totallyBounded_iSup, Subtype.forall] | lemma | Filter.totallyBounded_biSup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Set",
"TotallyBounded",
"iSup_subtype'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_sSup {S : Set (Filter α)} (hS : S.Finite) :
(sSup S).TotallyBounded ↔ ∀ f ∈ S, f.TotallyBounded | by
rw [sSup_eq_iSup, totallyBounded_biSup hS] | lemma | totallyBounded_sSup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"Set",
"TotallyBounded",
"sSup_eq_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set α} :
TotallyBounded (⋃ i, s i) ↔ ∀ i, TotallyBounded (s i) | by
simp_rw [← Filter.totallyBounded_principal_iff, ← Filter.iSup_principal,
Filter.totallyBounded_iSup] | lemma | totallyBounded_iUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.iSup_principal",
"Filter.totallyBounded_iSup",
"Filter.totallyBounded_principal_iff",
"Finite",
"Set",
"TotallyBounded"
] | A finite indexed union is totally bounded
if and only if each set of the family is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_biUnion {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
TotallyBounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, TotallyBounded (s i) | by
have := hI.to_subtype
rw [biUnion_eq_iUnion, totallyBounded_iUnion, Subtype.forall] | lemma | totallyBounded_biUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"totallyBounded_iUnion"
] | A union indexed by a finite set is totally bounded
if and only if each set of the family is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_sUnion {S : Set (Set α)} (hS : S.Finite) :
TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s | by
rw [sUnion_eq_biUnion, totallyBounded_biUnion hS] | lemma | totallyBounded_sUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"totallyBounded_biUnion"
] | A union of a finite family of sets is totally bounded
if and only if each set of the family is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Finite.totallyBounded {s : Set α} (hs : s.Finite) : TotallyBounded s | fun _U hU ↦
⟨s, hs, fun _x hx ↦ mem_biUnion hx <| refl_mem_uniformity hU⟩ | lemma | Set.Finite.totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"refl_mem_uniformity"
] | A finite set is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Subsingleton.totallyBounded {s : Set α} (hs : s.Subsingleton) :
TotallyBounded s | hs.finite.totallyBounded | lemma | Set.Subsingleton.totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded"
] | A subsingleton is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_singleton (a : α) : TotallyBounded {a} | (finite_singleton a).totallyBounded | lemma | totallyBounded_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_empty : TotallyBounded (∅ : Set α) | finite_empty.totallyBounded | theorem | totallyBounded_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_bot : (⊥ : Filter α).TotallyBounded | by
rw [← principal_empty, totallyBounded_principal_iff]
exact totallyBounded_empty | theorem | Filter.totallyBounded_bot | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"TotallyBounded",
"totallyBounded_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_union {s t : Set α} :
TotallyBounded (s ∪ t) ↔ TotallyBounded s ∧ TotallyBounded t | by
rw [union_eq_iUnion, totallyBounded_iUnion]
simp [and_comm] | lemma | totallyBounded_union | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"totallyBounded_iUnion"
] | The union of two sets is totally bounded
if and only if each of the two sets is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TotallyBounded.union {s t : Set α} (hs : TotallyBounded s) (ht : TotallyBounded t) :
TotallyBounded (s ∪ t) | totallyBounded_union.2 ⟨hs, ht⟩ | lemma | TotallyBounded.union | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded"
] | The union of two totally bounded sets is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_insert (a : α) {s : Set α} :
TotallyBounded (insert a s) ↔ TotallyBounded s | by
simp_rw [← singleton_union, totallyBounded_union, totallyBounded_singleton, true_and]
protected alias ⟨_, TotallyBounded.insert⟩ := totallyBounded_insert | lemma | totallyBounded_insert | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"TotallyBounded",
"totallyBounded_singleton",
"totallyBounded_union"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_sup {f g : Filter α} :
(f ⊔ g).TotallyBounded ↔ f.TotallyBounded ∧ g.TotallyBounded | by
rw [sup_eq_iSup, totallyBounded_iSup]
simp [and_comm] | lemma | Filter.totallyBounded_sup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"TotallyBounded",
"sup_eq_iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.sup {f g : Filter α} (hf : f.TotallyBounded) (hg : g.TotallyBounded) :
(f ⊔ g).TotallyBounded | totallyBounded_sup.2 ⟨hf, hg⟩ | lemma | Filter.TotallyBounded.sup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.map [UniformSpace β] {f : α → β} {g : Filter α}
(hg : g.TotallyBounded) (hf : UniformContinuous f) : (g.map f).TotallyBounded | fun t ht =>
let ⟨c, hfc, hct⟩ := hg _ (hf ht)
⟨f '' c, hfc.image f, by simpa [SetRel.preimage]⟩ | theorem | Filter.TotallyBounded.map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter",
"SetRel.preimage",
"TotallyBounded",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
(hf : UniformContinuous f) : TotallyBounded (f '' s) | by
simp only [← Filter.totallyBounded_principal_iff, ← Filter.map_principal] at hs ⊢
exact hs.map hf | theorem | TotallyBounded.image | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.map_principal",
"Filter.totallyBounded_principal_iff",
"Set",
"TotallyBounded",
"UniformContinuous",
"UniformSpace"
] | The image of a totally bounded set under a uniformly continuous map is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Ultrafilter.cauchy_of_totallyBounded' (f : Ultrafilter α) (hf : f.TotallyBounded) :
Cauchy (f : Filter α) | ⟨f.neBot', fun _ ht =>
let ⟨t', ht'₁, ht'_symm, ht'_t⟩ := comp_symm_of_uniformity ht
let ⟨i, hi, ht'_f⟩ := hf t' ht'₁
have : ∃ y ∈ i, { x | (x, y) ∈ t' } ∈ f := (Ultrafilter.eventually_exists_mem_iff hi).1 ht'_f
let ⟨y, _, hif⟩ := this
have : {x | (x, y) ∈ t'} ×ˢ {x | (x, y) ∈ t'} ⊆ t' ○ t' :=
... | theorem | Ultrafilter.cauchy_of_totallyBounded' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"Ultrafilter",
"Ultrafilter.eventually_exists_mem_iff",
"comp_symm_of_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ultrafilter.cauchy_of_totallyBounded {s : Set α} (f : Ultrafilter α) (hs : TotallyBounded s)
(h : ↑f ≤ 𝓟 s) : Cauchy (f : Filter α) | f.cauchy_of_totallyBounded' <| (Filter.totallyBounded_principal_iff.mpr hs).mono h | theorem | Ultrafilter.cauchy_of_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"Set",
"TotallyBounded",
"Ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_iff_filter {g : Filter α} :
g.TotallyBounded ↔ ∀ f, NeBot f → f ≤ g → ∃ c ≤ f, Cauchy c | by
constructor
· exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
(Ultrafilter.of f).cauchy_of_totallyBounded' (H.mono ((Ultrafilter.of_le f).trans hfs))⟩
· intro H d hd
contrapose! H with hd_cover
set f := ⨅ t : Finset α, g ⊓ 𝓟 (d.preimage t)ᶜ
have hb : Antitone fun t : Finset α... | theorem | Filter.totallyBounded_iff_filter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Antitone",
"Cauchy",
"Filter",
"Filter.NeBot",
"Filter.iInf_neBot_iff_of_directed'",
"Finset",
"SetRel.preimage",
"Ultrafilter.of",
"Ultrafilter.of_le",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_iff_ultrafilter {g : Filter α} :
g.TotallyBounded ↔ ∀ f : Ultrafilter α, ↑f ≤ g → Cauchy (f : Filter α) | by
refine ⟨fun hg f hf => f.cauchy_of_totallyBounded' <| hg.mono hf,
fun H => g.totallyBounded_iff_filter.2 ?_⟩
intro f hf hfs
exact ⟨Ultrafilter.of f, Ultrafilter.of_le f, H _ ((Ultrafilter.of_le f).trans hfs)⟩ | theorem | Filter.totallyBounded_iff_ultrafilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"Ultrafilter",
"Ultrafilter.of_le",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_iff_filter {s : Set α} :
TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c | by
rw [← Filter.totallyBounded_principal_iff, Filter.totallyBounded_iff_filter] | theorem | totallyBounded_iff_filter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter.totallyBounded_iff_filter",
"Filter.totallyBounded_principal_iff",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_iff_ultrafilter {s : Set α} :
TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) | by
rw [← Filter.totallyBounded_principal_iff, Filter.totallyBounded_iff_ultrafilter] | theorem | totallyBounded_iff_ultrafilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Cauchy",
"Filter",
"Filter.totallyBounded_iff_ultrafilter",
"Filter.totallyBounded_principal_iff",
"Set",
"TotallyBounded",
"Ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_iff_totallyBounded_isComplete {s : Set α} :
IsCompact s ↔ TotallyBounded s ∧ IsComplete s | ⟨fun hs =>
⟨totallyBounded_iff_ultrafilter.2 fun f hf =>
let ⟨_, _, fx⟩ := isCompact_iff_ultrafilter_le_nhds.1 hs f hf
cauchy_nhds.mono fx,
fun f fc fs =>
let ⟨a, as, fa⟩ := @hs f fc.1 fs
⟨a, as, le_nhds_of_cauchy_adhp fc fa⟩⟩,
fun ⟨ht, hc⟩ =>
isCompact_iff_ultrafilter_le_n... | theorem | isCompact_iff_totallyBounded_isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsCompact",
"IsComplete",
"Set",
"TotallyBounded",
"le_nhds_of_cauchy_adhp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.totallyBounded {s : Set α} (h : IsCompact s) : TotallyBounded s | (isCompact_iff_totallyBounded_isComplete.1 h).1 | theorem | IsCompact.totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsCompact",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isComplete {s : Set α} (h : IsCompact s) : IsComplete s | (isCompact_iff_totallyBounded_isComplete.1 h).2 | theorem | IsCompact.isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsCompact",
"IsComplete",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.isCompact_of_isComplete {s : Set α} (ht : TotallyBounded s)
(hc : IsComplete s) : IsCompact s | isCompact_iff_totallyBounded_isComplete.mpr ⟨ht, hc⟩ | theorem | TotallyBounded.isCompact_of_isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsCompact",
"IsComplete",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.isCompact_of_isClosed [CompleteSpace α] {s : Set α} (ht : TotallyBounded s)
(hc : IsClosed s) : IsCompact s | ht.isCompact_of_isComplete hc.isComplete | theorem | TotallyBounded.isCompact_of_isClosed | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsClosed",
"IsCompact",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.isCompact_setOf_clusterPt
[CompleteSpace α] {f : Filter α} (hf : f.TotallyBounded) : IsCompact {x | ClusterPt x f} | hf.totallyBounded_setOf_clusterPt.isCompact_of_isClosed isClosed_setOf_clusterPt | theorem | Filter.TotallyBounded.isCompact_setOf_clusterPt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"ClusterPt",
"CompleteSpace",
"Filter",
"IsCompact",
"isClosed_setOf_clusterPt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.TotallyBounded.exists_clusterPt
[CompleteSpace α] {f : Filter α} [f.NeBot] (hf : f.TotallyBounded) : ∃ x, ClusterPt x f | by
let m := Ultrafilter.of f
have hmf : m ≤ f := Ultrafilter.of_le f
have hm := m.cauchy_of_totallyBounded' (hf.mono hmf)
obtain ⟨x, hx⟩ := CompleteSpace.complete hm
rw [le_nhds_iff_adhp_of_cauchy hm] at hx
exact ⟨x, hx.mono hmf⟩ | theorem | Filter.TotallyBounded.exists_clusterPt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"ClusterPt",
"CompleteSpace",
"Filter",
"Ultrafilter.of",
"Ultrafilter.of_le",
"le_nhds_iff_adhp_of_cauchy"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchySeq.totallyBounded_range {s : ℕ → α} (hs : CauchySeq s) :
TotallyBounded (range s) | by
intro a ha
obtain ⟨n, hn⟩ := cauchySeq_iff.1 hs a ha
refine ⟨s '' { k | k ≤ n }, (finite_le_nat _).image _, ?_⟩
rw [range_subset_iff, biUnion_image]
intro m
rw [mem_iUnion₂]
rcases le_total m n with hm | hm
exacts [⟨m, hm, refl_mem_uniformity ha⟩, ⟨n, le_refl n, hn m hm n le_rfl⟩] | theorem | CauchySeq.totallyBounded_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"TotallyBounded",
"le_refl",
"le_total",
"refl_mem_uniformity"
] | Every Cauchy sequence over `ℕ` is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interUnionBalls (xs : ℕ → α) (u : ℕ → ℕ) (V : ℕ → SetRel α α) : Set α | ⋂ n, ⋃ m ≤ u n, UniformSpace.ball (xs m) (Prod.swap ⁻¹' V n) | def | interUnionBalls | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"SetRel",
"UniformSpace.ball"
] | Given a family of points `xs n`, a family of entourages `V n` of the diagonal and a family of
natural numbers `u n`, the intersection over `n` of the `V n`-neighborhood of `xs 1, ..., xs (u n)`.
Designed to be relatively compact when `V n` tends to the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallyBounded_interUnionBalls {p : ℕ → Prop} {U : ℕ → SetRel α α}
(H : (uniformity α).HasBasis p U) (xs : ℕ → α) (u : ℕ → ℕ) :
TotallyBounded (interUnionBalls xs u U) | by
rw [Filter.HasBasis.totallyBounded_iff H]
intro i _
have h_subset : interUnionBalls xs u U
⊆ ⋃ m ≤ u i, UniformSpace.ball (xs m) (Prod.swap ⁻¹' U i) :=
fun x hx ↦ Set.mem_iInter.1 hx i
classical
refine ⟨Finset.image xs (Finset.range (u i + 1)), Finset.finite_toSet _, fun x hx ↦ ?_⟩
simp only [F... | lemma | totallyBounded_interUnionBalls | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Filter.HasBasis.totallyBounded_iff",
"Finset.coe_image",
"Finset.coe_range",
"Finset.finite_toSet",
"Finset.range",
"SetRel",
"TotallyBounded",
"UniformSpace.ball",
"interUnionBalls",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_closure_interUnionBalls {p : ℕ → Prop} {U : ℕ → SetRel α α}
(H : (uniformity α).HasBasis p U) [CompleteSpace α] (xs : ℕ → α) (u : ℕ → ℕ) :
IsCompact (closure (interUnionBalls xs u U)) | by
rw [isCompact_iff_totallyBounded_isComplete]
refine ⟨TotallyBounded.closure ?_, isClosed_closure.isComplete⟩
exact totallyBounded_interUnionBalls H xs u | theorem | isCompact_closure_interUnionBalls | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"IsCompact",
"SetRel",
"closure",
"interUnionBalls",
"isCompact_iff_totallyBounded_isComplete",
"totallyBounded_interUnionBalls",
"uniformity"
] | The construction `interUnionBalls` is used to have a relatively compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
setSeqAux (n : ℕ) : { s : Set α // s ∈ f ∧ s ×ˢ s ⊆ U n } | Classical.indefiniteDescription _ <| (cauchy_iff.1 hf).2 (U n) (U_mem n) | def | SequentiallyComplete.setSeqAux | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set"
] | An auxiliary sequence of sets approximating a Cauchy filter. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
setSeq (n : ℕ) : Set α | ⋂ m ∈ Set.Iic n, (setSeqAux hf U_mem m).val | def | SequentiallyComplete.setSeq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"Set.Iic"
] | Given a Cauchy filter `f` and a sequence `U` of entourages, `set_seq` provides
an antitone sequence of sets `s n ∈ f` such that `s n ×ˢ s n ⊆ U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
setSeq_mem (n : ℕ) : setSeq hf U_mem n ∈ f | (biInter_mem (finite_le_nat n)).2 fun m _ => (setSeqAux hf U_mem m).2.1 | theorem | SequentiallyComplete.setSeq_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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