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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|---|---|---|---|---|---|---|---|---|---|---|
tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
Tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) b < ε | by
rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
simp only [mem_univ, true_and] | theorem | Metric.tendsto_nhdsWithin_nhds | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, dist x a < δ → dist (f x) b < ε | nhds_basis_ball.tendsto_iff nhds_basis_ball | theorem | Metric.tendsto_nhds_nhds | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, dist x a < δ → dist (f x) (f a) < ε | by
rw [ContinuousAt, tendsto_nhds_nhds] | theorem | Metric.continuousAt_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousAt",
"PseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
ContinuousWithinAt f s a ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) (f a) < ε | by
rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds] | theorem | Metric.continuousWithinAt_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousWithinAt",
"PseudoMetricSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε | by
simp [ContinuousOn, continuousWithinAt_iff] | theorem | Metric.continuousOn_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousOn",
"PseudoMetricSpace",
"Set",
"continuousOn_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff [PseudoMetricSpace β] {f : α → β} :
Continuous f ↔ ∀ b, ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε | continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds_nhds | theorem | Metric.continuous_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Continuous",
"PseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε | nhds_basis_ball.tendsto_right_iff | theorem | Metric.tendsto_nhds | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε | by
rw [ContinuousAt, tendsto_nhds] | theorem | Metric.continuousAt_iff' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousAt",
"TopologicalSpace",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε | by
rw [ContinuousWithinAt, tendsto_nhds] | theorem | Metric.continuousWithinAt_iff' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousWithinAt",
"Set",
"TopologicalSpace",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε | by
simp [ContinuousOn, continuousWithinAt_iff'] | theorem | Metric.continuousOn_iff' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ContinuousOn",
"Set",
"TopologicalSpace",
"continuousOn_iff'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff' [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε | continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds | theorem | Metric.continuous_iff' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Continuous",
"TopologicalSpace",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε | (atTop_basis.tendsto_iff nhds_basis_ball).trans <| by
simp only [true_and, mem_ball, mem_Ici] | theorem | Metric.tendsto_atTop | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"SemilatticeSup",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε | (atTop_basis_Ioi.tendsto_iff nhds_basis_ball).trans <| by
simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball] | theorem | Metric.tendsto_atTop' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"NoMaxOrder",
"SemilatticeSup",
"trans"
] | A variant of `tendsto_atTop` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_singleton_iff {α : Type*} [PseudoMetricSpace α] {x : α} :
IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x | by
simp [isOpen_iff, subset_singleton_iff, mem_ball] | theorem | Metric.isOpen_singleton_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"IsOpen",
"PseudoMetricSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
∃ y ∈ s, dist x y < ε | by
have : (ball x ε).Nonempty := by simp [hε]
simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this | theorem | Dense.exists_dist_lt | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Dense",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.DenseRange.exists_dist_lt {β : Type*} {f : β → α} (hf : DenseRange f) (x : α)
{ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε | exists_range_iff.1 (hf.exists_dist_lt x hε) | theorem | DenseRange.exists_dist_lt | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"DenseRange"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace_eq_bot :
‹PseudoMetricSpace α›.toUniformSpace = ⊥ ↔
∃ r : ℝ, 0 < r ∧ Pairwise (r ≤ dist · · : α → α → Prop) | by
simp only [uniformity_basis_dist.uniformSpace_eq_bot, mem_setOf_eq, not_lt] | lemma | Metric.uniformSpace_eq_bot | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Pairwise",
"not_lt"
] | (Pseudo) metric space has discrete `UniformSpace` structure
iff the distances between distinct points are uniformly bounded away from zero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
DiscreteTopology.of_forall_le_dist {α} [PseudoMetricSpace α] {r : ℝ} (hpos : 0 < r)
(hr : Pairwise (r ≤ dist · · : α → α → Prop)) : DiscreteTopology α | ⟨by rw [Metric.uniformSpace_eq_bot.2 ⟨r, hpos, hr⟩, UniformSpace.toTopologicalSpace_bot]⟩ | lemma | DiscreteTopology.of_forall_le_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"DiscreteTopology",
"Pairwise",
"PseudoMetricSpace",
"UniformSpace.toTopologicalSpace_bot"
] | If the distances between distinct points in a (pseudo) metric space
are uniformly bounded away from zero, then the space has discrete topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } | by
simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩
· rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩
refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε
... | theorem | Metric.uniformity_edist_aux | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"le_antisymm_iff",
"le_of_lt",
"lt_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } | by
simp only [PseudoMetricSpace.uniformity_dist, dist_nndist, edist_nndist,
Metric.uniformity_edist_aux] | theorem | Metric.uniformity_edist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.uniformity_edist_aux",
"dist_nndist",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Metric.eball_top_eq_univ (x : α) : eball x ∞ = Set.univ | Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x | theorem | Metric.eball_top_eq_univ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set.univ",
"edist_lt_top"
] | In a pseudometric space, an open ball of infinite radius is the whole space | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.eball_ofReal {x : α} {ε : ℝ} : eball x (.ofReal ε) = ball x ε | by
ext y
simp only [mem_eball, mem_ball, edist_dist]
exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
@[deprecated (since := "2026-01-24")]
alias Metric.emetric_ball := Metric.eball_ofReal | theorem | Metric.eball_ofReal | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal_lt_ofReal_iff_of_nonneg",
"dist_nonneg",
"edist_dist"
] | Balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.eball_coe {x : α} {ε : ℝ≥0} : eball x ε = ball x ε | by
rw [← eball_ofReal]
simp
@[deprecated (since := "2026-01-24")]
alias Metric.emetric_ball_nnreal := Metric.eball_coe | theorem | Metric.eball_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | Balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.closedEBall_ofReal {x : α} {ε : ℝ} (h : 0 ≤ ε) :
closedEBall x (.ofReal ε) = closedBall x ε | by
ext y; simp [edist_le_ofReal h]
@[deprecated (since := "2026-01-24")]
alias Metric.emetric_closedBall := Metric.closedEBall_ofReal | theorem | Metric.closedEBall_ofReal | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"edist_le_ofReal"
] | Closed balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.closedEBall_coe {x : α} {ε : ℝ≥0} :
closedEBall x ε = closedBall x ε | by
rw [← closedEBall_ofReal ε.coe_nonneg, ENNReal.ofReal_coe_nnreal]
@[deprecated (since := "2026-01-24")]
alias Metric.emetric_closedBall_nnreal := Metric.closedEBall_coe | theorem | Metric.closedEBall_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal_coe_nnreal"
] | Closed balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.eball_top (x : α) : eball x ⊤ = univ | eq_univ_of_forall fun _ => edist_lt_top _ _
@[deprecated (since := "2026-01-24")]
alias Metric.emetric_ball_top := Metric.eball_top | theorem | Metric.eball_top | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"edist_lt_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α | { m with
toUniformSpace := U
uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist } | abbrev | PseudoMetricSpace.replaceUniformity | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"UniformSpace"
] | Build a new pseudometric space from an old one where the bundled uniform structure is provably
(but typically non-definitionaly) equal to some given uniform structure.
See Note [forgetful inheritance].
See Note [reducible non-instances]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m | by
ext
rfl | theorem | PseudoMetricSpace.replaceUniformity_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
(H : U = m.toUniformSpace.toTopologicalSpace) : PseudoMetricSpace γ | @PseudoMetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl | abbrev | PseudoMetricSpace.replaceTopology | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"PseudoMetricSpace.replaceUniformity",
"TopologicalSpace"
] | Build a new pseudometric space from an old one where the bundled topological structure is
provably (but typically non-definitionaly) equal to some given topological structure.
See Note [forgetful inheritance].
See Note [reducible non-instances]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
(H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m | by
ext
rfl | theorem | PseudoMetricSpace.replaceTopology_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoEMetricSpace.toPseudoMetricSpaceOfDist {X : Type*} [e : PseudoEMetricSpace X]
(dist : X → X → ℝ) (dist_nonneg : ∀ x y, 0 ≤ dist x y)
(h : ∀ x y, edist x y = .ofReal (dist x y)) : PseudoMetricSpace X | where
dist := dist
dist_self x := by simpa [h, (dist_nonneg _ _).ge_iff_eq', -edist_self] using edist_self x
dist_comm x y := by simpa [h, dist_nonneg] using edist_comm x y
dist_triangle x y z := by
simpa [h, dist_nonneg, add_nonneg, ← ENNReal.ofReal_add] using edist_triangle x y z
edist := edist
edist_... | abbrev | PseudoEMetricSpace.toPseudoMetricSpaceOfDist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.coe_toNNReal_eq_toReal",
"ENNReal.ofReal_add",
"Metric.uniformity_edist_aux",
"PseudoEMetricSpace",
"PseudoMetricSpace",
"dist_comm",
"dist_nonneg",
"dist_self",
"dist_triangle",
"edist_dist",
"symm"
] | One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
distance is given separately, to be able to prescribe s... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [PseudoEMetricSpace α]
(h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α | PseudoEMetricSpace.toPseudoMetricSpaceOfDist (ENNReal.toReal <| edist · ·) (by simp) (by simp [h]) | abbrev | PseudoEMetricSpace.toPseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.toReal",
"PseudoEMetricSpace",
"PseudoEMetricSpace.toPseudoMetricSpaceOfDist",
"PseudoMetricSpace"
] | One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the emetric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetricSpace α)
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
PseudoMetricSpace α | { m with
toBornology := B
cobounded_sets := Set.ext <| compl_surjective.forall.2 fun s =>
(H s).trans <| by rw [isBounded_iff, mem_setOf_eq, compl_compl] } | abbrev | PseudoMetricSpace.replaceBornology | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Bornology",
"PseudoMetricSpace",
"Set.ext",
"compl_compl",
"trans"
] | Build a new pseudometric space from an old one where the bundled bornology structure is provably
(but typically non-definitionaly) equal to some given bornology structure.
See Note [forgetful inheritance].
See Note [reducible non-instances]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
PseudoMetricSpace.replaceBornology _ H = m | by
ext
rfl | theorem | PseudoMetricSpace.replaceBornology_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Bornology",
"PseudoMetricSpace",
"PseudoMetricSpace.replaceBornology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.pseudoMetricSpace : PseudoMetricSpace ℝ | where
dist x y := |x - y|
dist_self := by simp [abs_zero]
dist_comm _ _ := abs_sub_comm _ _
dist_triangle _ _ _ := abs_sub_le _ _ _ | instance | Real.pseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"dist_comm",
"dist_self",
"dist_triangle"
] | Instantiate the reals as a pseudometric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Real.dist_eq (x y : ℝ) : dist x y = |x - y| | rfl | theorem | Real.dist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) | rfl | theorem | Real.nndist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.nnabs"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) | nndist_comm _ _ | theorem | Real.nndist_eq' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.nnabs",
"nndist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| | by simp [Real.dist_eq] | theorem | Real.dist_0_eq_abs | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.dist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.sub_le_dist (x y : ℝ) : x - y ≤ dist x y | by
rw [Real.dist_eq, le_abs]
exact Or.inl (le_refl _) | theorem | Real.sub_le_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.dist_eq",
"le_refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) | Set.ext fun y => by
rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
sub_lt_comm] | theorem | Real.ball_eq_Ioo | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.dist_eq",
"Set.ext",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.ball_zero_eq_Ioo (r : ℝ) : ball 0 r = Ioo (-r) r | by
simp [Real.ball_eq_Ioo] | theorem | Real.ball_zero_eq_Ioo | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.ball_eq_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) | by
ext y
rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
sub_le_comm] | theorem | Real.closedBall_eq_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.dist_eq",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.closedBall_zero_eq_Icc (r : ℝ) : closedBall 0 r = Icc (-r) r | by
simp [Real.closedBall_eq_Icc] | theorem | Real.closedBall_zero_eq_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.closedBall_eq_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) | by
rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
← add_div, add_assoc, add_sub_cancel, add_self_div_two] | theorem | Real.Ioo_eq_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.ball_eq_Ioo",
"add_div",
"add_self_div_two",
"sub_div"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) | by
rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
← add_div, add_assoc, add_sub_cancel, add_self_div_two] | theorem | Real.Icc_eq_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.closedBall_eq_Icc",
"add_div",
"add_self_div_two",
"sub_div"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.sphere_eq_pair (x : ℝ) {r : ℝ} (hr : 0 ≤ r) : sphere x r = {x - r, x + r} | by
ext; simp [dist_eq]; grind | lemma | Real.sphere_eq_pair | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) | by
ext s
simp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff]
simp [subset_def, Real.dist_0_eq_abs] | theorem | Metric.uniformity_eq_comap_nhds_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.dist_0_eq_abs",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} :
Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) | by
rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_uniformity_iff_dist_tendsto_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"Metric.uniformity_eq_comap_nhds_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
(h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
Tendsto f₂ p (𝓝 a) | h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist | theorem | Filter.Tendsto.congr_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
(h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) | Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h | theorem | tendsto_iff_of_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"Uniform.tendsto_congr",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoMetricSpace.dist_eq_of_dist_zero (x : α) {y z : α} (h : dist y z = 0) :
dist x y = dist x z | dist_comm y x ▸ dist_comm z x ▸ sub_eq_zero.1 (abs_nonpos_iff.1 (h ▸ abs_dist_sub_le y z x)) | theorem | PseudoMetricSpace.dist_eq_of_dist_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"abs_dist_sub_le",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_dist_dist_le_left (x y z : α) : dist (dist x z) (dist y z) ≤ dist x y | abs_dist_sub_le .. | theorem | dist_dist_dist_le_left | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"abs_dist_sub_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_dist_dist_le_right (x y z : α) : dist (dist x y) (dist x z) ≤ dist y z | by
simpa only [dist_comm x] using dist_dist_dist_le_left y z x | theorem | dist_dist_dist_le_right | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_dist_dist_le_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y' | (dist_triangle _ _ _).trans <|
add_le_add (dist_dist_dist_le_left _ _ _) (dist_dist_dist_le_right _ _ _) | theorem | dist_dist_dist_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_dist_dist_le_left",
"dist_dist_dist_le_right",
"dist_triangle",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a | by
simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
Function.comp_def, dist_comm] | theorem | nhds_comap_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.uniformity_eq_comap_nhds_zero",
"comap",
"dist_comm",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) | by
rw [← nhds_comap_dist a, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_iff_dist_tendsto_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"nhds_comap_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) | interior_maximal ball_subset_closedBall isOpen_ball | theorem | Metric.ball_subset_interior_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"interior",
"interior_maximal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε | (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm] | theorem | Metric.mem_closure_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set",
"closure",
"dist_comm",
"mem_closure_iff",
"mem_closure_iff_nhds_basis",
"trans"
] | ε-characterization of the closure in pseudometric spaces | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_closure_range_iff {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε | by
simp only [mem_closure_iff, exists_range_iff] | theorem | Metric.mem_closure_range_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"closure",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_closure_range_iff_nat {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) | (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | theorem | Metric.mem_closure_range_iff_nat | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"closure",
"dist_comm",
"mem_closure_iff_nhds_basis",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε | by
simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a | theorem | Metric.mem_of_closed' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"IsClosed",
"Set",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty | forall_congr' fun x => by
simp only [mem_closure_iff, Set.Nonempty, mem_inter_iff, mem_ball', and_comm] | theorem | Metric.dense_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Dense",
"Set",
"Set.Nonempty",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_iff_iUnion_ball (s : Set α) : Dense s ↔ ∀ r > 0, ⋃ c ∈ s, ball c r = univ | by
simp_rw [eq_univ_iff_forall, mem_iUnion, exists_prop, mem_ball, Dense, mem_closure_iff,
forall_comm (α := α)] | theorem | Metric.dense_iff_iUnion_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Dense",
"Set",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r | forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff] | theorem | Metric.denseRange_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"DenseRange",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_ofMul (a b : X) : nndist (ofMul a) (ofMul b) = nndist a b | rfl | theorem | nndist_ofMul | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_ofAdd (a b : X) : nndist (ofAdd a) (ofAdd b) = nndist a b | rfl | theorem | nndist_ofAdd | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_toMul (a b : Additive X) : nndist a.toMul b.toMul = nndist a b | rfl | theorem | nndist_toMul | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Additive"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_toAdd (a b : Multiplicative X) : nndist a.toAdd b.toAdd = nndist a b | rfl | theorem | nndist_toAdd | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Multiplicative"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_toDual (a b : X) : nndist (toDual a) (toDual b) = nndist a b | rfl | theorem | nndist_toDual | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_ofDual (a b : Xᵒᵈ) : nndist (ofDual a) (ofDual b) = nndist a b | rfl | theorem | nndist_ofDual | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.singleton_eq_inter_Icc (b : ℝ) : {b} = ⋂ (r > 0), Icc (b - r) (b + r) | by
simp [Icc_eq_closedBall, biInter_basis_nhds Metric.nhds_basis_closedBall] | lemma | Real.singleton_eq_inter_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"Metric.nhds_basis_closedBall",
"biInter_basis_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
(hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) | tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft | lemma | squeeze_zero' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"Filter",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
general case. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
(g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) | squeeze_zero' (Eventually.of_forall hf) (Eventually.of_forall hft) g0 | lemma | squeeze_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"Filter",
"squeeze_zero'"
] | Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u | by
obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ closedBall x ε ⊆ u := nhds_basis_closedBall.mem_iff.1 hu
have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
filter_upwards [this] with _ hr using Subset.trans (closedBall_subset_closedBall hr) hε | lemma | eventually_closedBall_subset | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"Set"
] | If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
`Metric.closedBall x r` is contained in `u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_closedBall_smallSets (x : α) : Tendsto (closedBall x) (𝓝 0) (𝓝 x).smallSets | tendsto_smallSets_iff.2 fun _ ↦ eventually_closedBall_subset | lemma | tendsto_closedBall_smallSets | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"eventually_closedBall_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_ball_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) : ∀ᶠ r in 𝓝 (0 : ℝ), ball x r ⊆ u | (eventually_closedBall_subset hu).mono fun _r hr ↦ ball_subset_closedBall.trans hr | lemma | eventually_ball_subset | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"Set",
"eventually_closedBall_subset"
] | If `u` is a neighborhood of `x`, then for small enough `r`, the open ball
`Metric.ball x r` is contained in `u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_closedBall : IsClosed (closedBall x ε) | isClosed_le (by fun_prop) continuous_const | lemma | Metric.isClosed_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsClosed",
"continuous_const",
"isClosed_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_sphere : IsClosed (sphere x ε) | isClosed_eq (by fun_prop) continuous_const | lemma | Metric.isClosed_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsClosed",
"continuous_const",
"isClosed_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_closedBall : closure (closedBall x ε) = closedBall x ε | isClosed_closedBall.closure_eq | lemma | Metric.closure_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_sphere : closure (sphere x ε) = sphere x ε | isClosed_sphere.closure_eq | lemma | Metric.closure_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε | closure_minimal ball_subset_closedBall isClosed_closedBall | lemma | Metric.closure_ball_subset_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"closure",
"closure_minimal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε | frontier_lt_subset_eq (by fun_prop) continuous_const | lemma | Metric.frontier_ball_subset_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"continuous_const",
"frontier",
"frontier_lt_subset_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε | frontier_le_subset_eq (by fun_prop) continuous_const | lemma | Metric.frontier_closedBall_subset_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"continuous_const",
"frontier",
"frontier_le_subset_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_zero' (x : α) : closedBall x 0 = closure {x} | Subset.antisymm
(fun _y hy =>
mem_closure_iff.2 fun _ε ε0 => ⟨x, mem_singleton x, (mem_closedBall.1 hy).trans_lt ε0⟩)
(closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_closedBall) | lemma | Metric.closedBall_zero' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"closure",
"closure_minimal",
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) | by
rcases exists_compact_mem_nhds x with ⟨s, s_compact, hs⟩
filter_upwards [eventually_closedBall_subset hs] with r hr
exact IsCompact.of_isClosed_subset s_compact isClosed_closedBall hr | lemma | Metric.eventually_isCompact_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsCompact",
"IsCompact.of_isClosed_subset",
"WeaklyLocallyCompactSpace",
"eventually_closedBall_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
∃ r, 0 < r ∧ IsCompact (closedBall x r) | by
have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) :=
eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x)
simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists | lemma | Metric.exists_isCompact_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsCompact",
"WeaklyLocallyCompactSpace",
"eventually_nhdsWithin_of_eventually_nhds",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
biInter_gt_closedBall (x : α) (r : ℝ) : ⋂ r' > r, closedBall x r' = closedBall x r | by
ext
simp [forall_gt_imp_ge_iff_le_of_dense] | theorem | Metric.biInter_gt_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"forall_gt_imp_ge_iff_le_of_dense"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
biInter_gt_ball (x : α) (r : ℝ) : ⋂ r' > r, ball x r' = closedBall x r | by
ext
simp [forall_gt_iff_le] | theorem | Metric.biInter_gt_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
biUnion_lt_ball (x : α) (r : ℝ) : ⋃ r' < r, ball x r' = ball x r | by
ext
rw [← not_iff_not]
simp [forall_lt_imp_le_iff_le_of_dense] | theorem | Metric.biUnion_lt_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"not_iff_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
biUnion_lt_closedBall (x : α) (r : ℝ) : ⋃ r' < r, closedBall x r' = ball x r | by
ext
rw [← not_iff_not]
simp [forall_lt_iff_le] | theorem | Metric.biUnion_lt_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"forall_lt_iff_le",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_of_metric {s : Set α} {ι : Sort*} {c : ι → Set α} (hs : IsCompact s)
(hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i | by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, dist_comm]
using uniformity_basis_dist.lebesgue_number_lemma hs hc₁ hc₂ | theorem | lebesgue_number_lemma_of_metric | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"UniformSpace.ball",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
(hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t | by
rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂ | theorem | lebesgue_number_lemma_of_metric_sUnion | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"lebesgue_number_lemma_of_metric"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pseudoMetricSpacePi : PseudoMetricSpace (∀ b, X b) | by
/- we construct the instance from the pseudoemetric space instance to avoid checking again that
the uniformity is the same as the product uniformity, but we register nevertheless a nice
formula for the distance -/
let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
(fun f g : ∀ b, X b => ((sup univ... | instance | pseudoMetricSpacePi | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Filter.eventually_all",
"Finset.mem_univ",
"NNReal.zero_le_coe",
"PseudoEMetricSpace.toPseudoMetricSpaceOfDist",
"PseudoMetricSpace",
"dist_nndist",
"edist_pi_def"
] | A finite product of pseudometric spaces is a pseudometric space, with the sup distance. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nndist_pi_def (f g : ∀ b, X b) : nndist f g = sup univ fun b => nndist (f b) (g b) | rfl | lemma | nndist_pi_def | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_def (f g : ∀ b, X b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) | rfl | lemma | dist_pi_def | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_pi_le_iff {f g : ∀ b, X b} {r : ℝ≥0} :
nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r | by simp [nndist_pi_def] | lemma | nndist_pi_le_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"nndist_pi_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_pi_lt_iff {f g : ∀ b, X b} {r : ℝ≥0} (hr : 0 < r) :
nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r | by
simp [nndist_pi_def, Finset.sup_lt_iff hr] | lemma | nndist_pi_lt_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Finset.sup_lt_iff",
"nndist_pi_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_pi_eq_iff {f g : ∀ b, X b} {r : ℝ≥0} (hr : 0 < r) :
nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r | by
rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm | lemma | nndist_pi_eq_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"eq_iff_le_not_lt",
"le_antisymm_iff",
"nndist_pi_le_iff",
"nndist_pi_lt_iff",
"not_lt",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_lt_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 < r) :
dist f g < r ↔ ∀ b, dist (f b) (g b) < r | by
lift r to ℝ≥0 using hr.le
exact nndist_pi_lt_iff hr | lemma | dist_pi_lt_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"nndist_pi_lt_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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