fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Set.MapsTo.lipschitzOnWith_iff_restrict {t : Set β} (h : MapsTo f s t) :
LipschitzOnWith K f s ↔ LipschitzWith K (h.restrict f s t) :=
_root_.lipschitzOnWith_iff_restrict
alias ⟨LipschitzOnWith.mapsToRestrict, _⟩ := Set.MapsTo.lipschitzOnWith_iff_restrict
@[deprecated (since := "05-09-2025")]
alias LipschitzOnWit... | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | Set.MapsTo.lipschitzOnWith_iff_restrict | null |
protected lipschitzOnWith (h : LipschitzWith K f) : LipschitzOnWith K f s :=
fun x _ y _ => h x y | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | lipschitzOnWith | null |
edist_le_mul (h : LipschitzWith K f) (x y : α) : edist (f x) (f y) ≤ K * edist x y :=
h x y | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_le_mul | null |
edist_le_mul_of_le (h : LipschitzWith K f) (hr : edist x y ≤ r) :
edist (f x) (f y) ≤ K * r :=
(h x y).trans <| mul_right_mono hr | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_le_mul_of_le | null |
edist_lt_mul_of_lt (h : LipschitzWith K f) (hK : K ≠ 0) (hr : edist x y < r) :
edist (f x) (f y) < K * r :=
(h x y).trans_lt <| (ENNReal.mul_lt_mul_left (ENNReal.coe_ne_zero.2 hK) ENNReal.coe_ne_top).2 hr | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_lt_mul_of_lt | null |
mapsTo_emetric_closedBall (h : LipschitzWith K f) (x : α) (r : ℝ≥0∞) :
MapsTo f (closedBall x r) (closedBall (f x) (K * r)) := fun _y hy => h.edist_le_mul_of_le hy | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | mapsTo_emetric_closedBall | null |
mapsTo_emetric_ball (h : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ≥0∞) :
MapsTo f (ball x r) (ball (f x) (K * r)) := fun _y hy => h.edist_lt_mul_of_lt hK hy | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | mapsTo_emetric_ball | null |
edist_lt_top (hf : LipschitzWith K f) {x y : α} (h : edist x y ≠ ⊤) :
edist (f x) (f y) < ⊤ :=
(hf x y).trans_lt (by finiteness) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_lt_top | null |
mul_edist_le (h : LipschitzWith K f) (x y : α) :
(K⁻¹ : ℝ≥0∞) * edist (f x) (f y) ≤ edist x y := by
rw [mul_comm, ← div_eq_mul_inv]
exact ENNReal.div_le_of_le_mul' (h x y) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | mul_edist_le | null |
protected of_edist_le (h : ∀ x y, edist (f x) (f y) ≤ edist x y) : LipschitzWith 1 f :=
fun x y => by simp only [ENNReal.coe_one, one_mul, h] | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | of_edist_le | null |
protected weaken (hf : LipschitzWith K f) {K' : ℝ≥0} (h : K ≤ K') : LipschitzWith K' f :=
fun x y => le_trans (hf x y) <| mul_left_mono (ENNReal.coe_le_coe.2 h) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | weaken | null |
ediam_image_le (hf : LipschitzWith K f) (s : Set α) :
EMetric.diam (f '' s) ≤ K * EMetric.diam s := by
apply EMetric.diam_le
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
exact hf.edist_le_mul_of_le (EMetric.edist_le_diam_of_mem hx hy) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | ediam_image_le | null |
edist_lt_of_edist_lt_div (hf : LipschitzWith K f) {x y : α} {d : ℝ≥0∞}
(h : edist x y < d / K) : edist (f x) (f y) < d :=
calc
edist (f x) (f y) ≤ K * edist x y := hf x y
_ < d := ENNReal.mul_lt_of_lt_div' h | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_lt_of_edist_lt_div | null |
protected uniformContinuous (hf : LipschitzWith K f) : UniformContinuous f :=
EMetric.uniformContinuous_iff.2 fun ε εpos =>
⟨ε / K, ENNReal.div_pos_iff.2 ⟨ne_of_gt εpos, ENNReal.coe_ne_top⟩, hf.edist_lt_of_edist_lt_div⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | uniformContinuous | A Lipschitz function is uniformly continuous. |
protected continuous (hf : LipschitzWith K f) : Continuous f :=
hf.uniformContinuous.continuous | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous | A Lipschitz function is continuous. |
protected const (b : β) : LipschitzWith 0 fun _ : α => b := fun x y => by
simp only [edist_self, zero_le] | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | const | Constant functions are Lipschitz (with any constant). |
protected const' (b : β) {K : ℝ≥0} : LipschitzWith K fun _ : α => b := fun x y => by
simp only [edist_self, zero_le] | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | const' | null |
protected id : LipschitzWith 1 (@id α) :=
LipschitzWith.of_edist_le fun _ _ => le_rfl | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | id | The identity is 1-Lipschitz. |
protected subtype_val (s : Set α) : LipschitzWith 1 (Subtype.val : s → α) :=
LipschitzWith.of_edist_le fun _ _ => le_rfl | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | subtype_val | The inclusion of a subset is 1-Lipschitz. |
subtype_mk (hf : LipschitzWith K f) {p : β → Prop} (hp : ∀ x, p (f x)) :
LipschitzWith K (fun x => ⟨f x, hp x⟩ : α → { y // p y }) :=
hf | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | subtype_mk | null |
protected eval {α : ι → Type u} [∀ i, PseudoEMetricSpace (α i)] [Fintype ι] (i : ι) :
LipschitzWith 1 (Function.eval i : (∀ i, α i) → α i) :=
LipschitzWith.of_edist_le fun f g => by convert edist_le_pi_edist f g i | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | eval | null |
protected restrict (hf : LipschitzWith K f) (s : Set α) : LipschitzWith K (s.restrict f) :=
fun x y => hf x y | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | restrict | The restriction of a `K`-Lipschitz function is `K`-Lipschitz. |
protected comp {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} (hf : LipschitzWith Kf f)
(hg : LipschitzWith Kg g) : LipschitzWith (Kf * Kg) (f ∘ g) := fun x y =>
calc
edist (f (g x)) (f (g y)) ≤ Kf * edist (g x) (g y) := hf _ _
_ ≤ Kf * (Kg * edist x y) := mul_right_mono (hg _ _)
_ = (Kf * Kg : ℝ≥0) * edist x ... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | comp | The composition of Lipschitz functions is Lipschitz. |
comp_lipschitzOnWith {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} {s : Set α}
(hf : LipschitzWith Kf f) (hg : LipschitzOnWith Kg g s) : LipschitzOnWith (Kf * Kg) (f ∘ g) s :=
lipschitzOnWith_iff_restrict.mpr <| hf.comp hg.to_restrict | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | comp_lipschitzOnWith | null |
protected prod_fst : LipschitzWith 1 (@Prod.fst α β) :=
LipschitzWith.of_edist_le fun _ _ => le_max_left _ _ | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prod_fst | null |
protected prod_snd : LipschitzWith 1 (@Prod.snd α β) :=
LipschitzWith.of_edist_le fun _ _ => le_max_right _ _ | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prod_snd | null |
protected prodMk {f : α → β} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) {g : α → γ} {Kg : ℝ≥0}
(hg : LipschitzWith Kg g) : LipschitzWith (max Kf Kg) fun x => (f x, g x) := by
intro x y
rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul]
exact max_le_max (hf x y) (hg x y)
@[deprecated (since := "2025-03-10")]
prot... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk | If `f` and `g` are Lipschitz functions, so is the induced map `f × g` to the product type. |
protected prodMk_left (a : α) : LipschitzWith 1 (Prod.mk a : β → α × β) := by
simpa only [max_eq_right zero_le_one] using (LipschitzWith.const a).prodMk LipschitzWith.id
@[deprecated (since := "2025-03-10")]
protected alias prod_mk_left := LipschitzWith.prodMk_left | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk_left | null |
protected prodMk_right (b : β) : LipschitzWith 1 fun a : α => (a, b) := by
simpa only [max_eq_left zero_le_one] using LipschitzWith.id.prodMk (LipschitzWith.const b)
@[deprecated (since := "2025-03-10")]
protected alias prod_mk_right := LipschitzWith.prodMk_right | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk_right | null |
protected uncurry {f : α → β → γ} {Kα Kβ : ℝ≥0} (hα : ∀ b, LipschitzWith Kα fun a => f a b)
(hβ : ∀ a, LipschitzWith Kβ (f a)) : LipschitzWith (Kα + Kβ) (Function.uncurry f) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
simp only [Function.uncurry, ENNReal.coe_add, add_mul]
apply le_trans (edist_triangle _ (f a₂ b₁) _)
ex... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | uncurry | null |
protected iterate {f : α → α} (hf : LipschitzWith K f) : ∀ n, LipschitzWith (K ^ n) f^[n]
| 0 => by simpa only [pow_zero] using LipschitzWith.id
| n + 1 => by rw [pow_succ]; exact (LipschitzWith.iterate hf n).comp hf | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | iterate | Iterates of a Lipschitz function are Lipschitz. |
edist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
edist (f^[n] x) (f^[n+1] x) ≤ edist x (f x) * (K : ℝ≥0∞) ^ n := by
rw [iterate_succ, mul_comm]
simpa only [ENNReal.coe_pow] using (hf.iterate n) x (f x) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_iterate_succ_le_geometric | null |
protected mul_end {f g : Function.End α} {Kf Kg} (hf : LipschitzWith Kf f)
(hg : LipschitzWith Kg g) : LipschitzWith (Kf * Kg) (f * g : Function.End α) :=
hf.comp hg | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | mul_end | null |
protected list_prod (f : ι → Function.End α) (K : ι → ℝ≥0)
(h : ∀ i, LipschitzWith (K i) (f i)) : ∀ l : List ι, LipschitzWith (l.map K).prod (l.map f).prod
| [] => by simpa using LipschitzWith.id
| i::l => by
simp only [List.map_cons, List.prod_cons]
exact (h i).mul_end (LipschitzWith.list_prod f K h l) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | list_prod | The product of a list of Lipschitz continuous endomorphisms is a Lipschitz continuous
endomorphism. |
protected pow_end {f : Function.End α} {K} (h : LipschitzWith K f) :
∀ n : ℕ, LipschitzWith (K ^ n) (f ^ n : Function.End α)
| 0 => by simpa only [pow_zero] using LipschitzWith.id
| n + 1 => by
rw [pow_succ, pow_succ]
exact (LipschitzWith.pow_end h n).mul_end h | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | pow_end | null |
protected uniformContinuousOn (hf : LipschitzOnWith K f s) : UniformContinuousOn f s :=
uniformContinuousOn_iff_restrict.mpr hf.to_restrict.uniformContinuous | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | uniformContinuousOn | null |
protected continuousOn (hf : LipschitzOnWith K f s) : ContinuousOn f s :=
hf.uniformContinuousOn.continuousOn | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn | null |
edist_le_mul_of_le (h : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
{r : ℝ≥0∞} (hr : edist x y ≤ r) :
edist (f x) (f y) ≤ K * r :=
(h hx hy).trans <| mul_right_mono hr | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_le_mul_of_le | null |
edist_lt_of_edist_lt_div (hf : LipschitzOnWith K f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
{d : ℝ≥0∞} (hd : edist x y < d / K) : edist (f x) (f y) < d :=
hf.to_restrict.edist_lt_of_edist_lt_div <| show edist (⟨x, hx⟩ : s) ⟨y, hy⟩ < d / K from hd | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | edist_lt_of_edist_lt_div | null |
protected comp {g : β → γ} {t : Set β} {Kg : ℝ≥0} (hg : LipschitzOnWith Kg g t)
(hf : LipschitzOnWith K f s) (hmaps : MapsTo f s t) : LipschitzOnWith (Kg * K) (g ∘ f) s :=
lipschitzOnWith_iff_restrict.mpr <| hg.to_restrict.comp (hf.mapsToRestrict hmaps) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | comp | null |
protected prodMk {g : α → γ} {Kf Kg : ℝ≥0} (hf : LipschitzOnWith Kf f s)
(hg : LipschitzOnWith Kg g s) : LipschitzOnWith (max Kf Kg) (fun x => (f x, g x)) s := by
intro _ hx _ hy
rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul]
exact max_le_max (hf hx hy) (hg hx hy)
@[deprecated (since := "2025-03-10")]
... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk | If `f` and `g` are Lipschitz on `s`, so is the induced map `f × g` to the product type. |
ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β)
(hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (f · b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) :
EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t := by
simp only [EMetric.diam_le_iff, forall_mem_image2]
intro a₁ ha... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | ediam_image2_le | null |
protected _root_.LipschitzWith.locallyLipschitz {K : ℝ≥0} (hf : LipschitzWith K f) :
LocallyLipschitz f :=
fun _ ↦ ⟨K, univ, Filter.univ_mem, lipschitzOnWith_univ.mpr hf⟩ | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | _root_.LipschitzWith.locallyLipschitz | A Lipschitz function is locally Lipschitz. |
protected id : LocallyLipschitz (@id α) := LipschitzWith.id.locallyLipschitz | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | id | The identity function is locally Lipschitz. |
protected const (b : β) : LocallyLipschitz (fun _ : α ↦ b) :=
(LipschitzWith.const b).locallyLipschitz | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | const | Constant functions are locally Lipschitz. |
protected continuous {f : α → β} (hf : LocallyLipschitz f) : Continuous f := by
rw [continuous_iff_continuousAt]
intro x
rcases (hf x) with ⟨K, t, ht, hK⟩
exact (hK.continuousOn).continuousAt ht | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous | A locally Lipschitz function is continuous. (The converse is false: for example,
$x ↦ \sqrt{x}$ is continuous, but not locally Lipschitz at 0.) |
protected comp {f : β → γ} {g : α → β}
(hf : LocallyLipschitz f) (hg : LocallyLipschitz g) : LocallyLipschitz (f ∘ g) := by
intro x
rcases hg x with ⟨Kg, t, ht, hgL⟩
rcases hf (g x) with ⟨Kf, u, hu, hfL⟩
refine ⟨Kf * Kg, t ∩ g⁻¹' u, inter_mem ht (hg.continuous.continuousAt hu), ?_⟩
exact hfL.comp (hgL.mon... | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | comp | The composition of locally Lipschitz functions is locally Lipschitz. |
protected prodMk {f : α → β} (hf : LocallyLipschitz f) {g : α → γ} (hg : LocallyLipschitz g) :
LocallyLipschitz fun x => (f x, g x) := by
intro x
rcases hf x with ⟨Kf, t₁, h₁t, hfL⟩
rcases hg x with ⟨Kg, t₂, h₂t, hgL⟩
refine ⟨max Kf Kg, t₁ ∩ t₂, Filter.inter_mem h₁t h₂t, ?_⟩
exact (hfL.mono inter_subset_l... | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk | If `f` and `g` are locally Lipschitz, so is the induced map `f × g` to the product type. |
protected prodMk_left (a : α) : LocallyLipschitz (Prod.mk a : β → α × β) :=
(LipschitzWith.prodMk_left a).locallyLipschitz
@[deprecated (since := "2025-03-10")]
protected alias prod_mk_left := LocallyLipschitz.prodMk_left | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk_left | null |
protected prodMk_right (b : β) : LocallyLipschitz (fun a : α => (a, b)) :=
(LipschitzWith.prodMk_right b).locallyLipschitz
@[deprecated (since := "2025-03-10")]
protected alias prod_mk_right := LocallyLipschitz.prodMk_right | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | prodMk_right | null |
protected iterate {f : α → α} (hf : LocallyLipschitz f) : ∀ n, LocallyLipschitz f^[n]
| 0 => by simpa only [pow_zero] using LocallyLipschitz.id
| n + 1 => by rw [iterate_add, iterate_one]; exact (hf.iterate n).comp hf | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | iterate | null |
protected mul_end {f g : Function.End α} (hf : LocallyLipschitz f)
(hg : LocallyLipschitz g) : LocallyLipschitz (f * g : Function.End α) := hf.comp hg | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | mul_end | null |
protected pow_end {f : Function.End α} (h : LocallyLipschitz f) :
∀ n : ℕ, LocallyLipschitz (f ^ n : Function.End α)
| 0 => by simpa only [pow_zero] using LocallyLipschitz.id
| n + 1 => by
rw [pow_succ]
exact (h.pow_end n).mul_end h | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | pow_end | null |
protected continuousOn (hf : LocallyLipschitzOn s f) : ContinuousOn f s :=
continuousOn_iff_continuous_restrict.2 hf.restrict.continuous | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn | null |
continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith [PseudoEMetricSpace α]
[TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s s' : Set α} {t : Set β}
(hs' : s' ⊆ s) (hss' : s ⊆ closure s') (K : ℝ≥0)
(ha : ∀ a ∈ s', ContinuousOn (fun y => f (a, y)) t)
(hb : ∀ b ∈ t, LipschitzOnWi... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical fiber”
`{a} × t`, `a ∈ s`, and is Lipschitz continuous on each “horizontal fiber” `s × {b}`, `b ∈ t`
with the same Lipschitz constant `K`. Then it is continuous on `s × t`. Moreover, it suffices
to require continuity on vertical fibers... |
continuousOn_prod_of_continuousOn_lipschitzOnWith [PseudoEMetricSpace α]
[TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Set α} {t : Set β} (K : ℝ≥0)
(ha : ∀ a ∈ s, ContinuousOn (fun y => f (a, y)) t)
(hb : ∀ b ∈ t, LipschitzOnWith K (fun x => f (x, b)) s) : ContinuousOn f (s ×ˢ t) :=
con... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn_prod_of_continuousOn_lipschitzOnWith | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical fiber”
`{a} × t`, `a ∈ s`, and is Lipschitz continuous on each “horizontal fiber” `s × {b}`, `b ∈ t`
with the same Lipschitz constant `K`. Then it is continuous on `s × t`.
The actual statement uses (Lipschitz) continuity of `fun y ↦ ... |
continuous_prod_of_dense_continuous_lipschitzWith [PseudoEMetricSpace α]
[TopologicalSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) {s : Set α}
(hs : Dense s) (ha : ∀ a ∈ s, Continuous fun y => f (a, y))
(hb : ∀ b, LipschitzWith K fun x => f (x, b)) : Continuous f := by
simp only [← continuousO... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous_prod_of_dense_continuous_lipschitzWith | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical section”
`{a} × univ` for `a : α` from a dense set. Suppose that it is Lipschitz continuous on each
“horizontal section” `univ × {b}`, `b : β` with the same Lipschitz constant `K`. Then it is
continuous.
The actual statement uses (Lip... |
continuous_prod_of_continuous_lipschitzWith [PseudoEMetricSpace α] [TopologicalSpace β]
[PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) (ha : ∀ a, Continuous fun y => f (a, y))
(hb : ∀ b, LipschitzWith K fun x => f (x, b)) : Continuous f :=
continuous_prod_of_dense_continuous_lipschitzWith f K dense_univ (fu... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous_prod_of_continuous_lipschitzWith | Consider a function `f : α × β → γ`. Suppose that it is continuous on each “vertical section”
`{a} × univ`, `a : α`, and is Lipschitz continuous on each “horizontal section”
`univ × {b}`, `b : β` with the same Lipschitz constant `K`. Then it is continuous.
The actual statement uses (Lipschitz) continuity of `fun y ↦ f... |
continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith' [TopologicalSpace α]
[PseudoEMetricSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Set α} {t t' : Set β}
(ht' : t' ⊆ t) (htt' : t ⊆ closure t') (K : ℝ≥0)
(ha : ∀ a ∈ s, LipschitzOnWith K (fun y => f (a, y)) t)
(hb : ∀ b ∈ t', Continu... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith' | null |
continuousOn_prod_of_continuousOn_lipschitzOnWith' [TopologicalSpace α]
[PseudoEMetricSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Set α} {t : Set β} (K : ℝ≥0)
(ha : ∀ a ∈ s, LipschitzOnWith K (fun y => f (a, y)) t)
(hb : ∀ b ∈ t, ContinuousOn (fun x => f (x, b)) s) : ContinuousOn f (s ×ˢ t) :=
ha... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuousOn_prod_of_continuousOn_lipschitzOnWith' | null |
continuous_prod_of_dense_continuous_lipschitzWith' [TopologicalSpace α]
[PseudoEMetricSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) {t : Set β}
(ht : Dense t) (ha : ∀ a, LipschitzWith K fun y => f (a, y))
(hb : ∀ b ∈ t, Continuous fun x => f (x, b)) : Continuous f :=
have : Continuous (f ∘ Pro... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous_prod_of_dense_continuous_lipschitzWith' | null |
continuous_prod_of_continuous_lipschitzWith' [TopologicalSpace α] [PseudoEMetricSpace β]
[PseudoEMetricSpace γ] (f : α × β → γ) (K : ℝ≥0) (ha : ∀ a, LipschitzWith K fun y => f (a, y))
(hb : ∀ b, Continuous fun x => f (x, b)) : Continuous f :=
have : Continuous (f ∘ Prod.swap) :=
continuous_prod_of_continu... | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | continuous_prod_of_continuous_lipschitzWith' | null |
t4Space [EMetricSpace α] : T4Space α := inferInstance | theorem | Topology | [
"Mathlib.Tactic.GCongr",
"Mathlib.Topology.Compactness.Paracompact",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.SetTheory.Cardinal.Order"
] | Mathlib/Topology/EMetricSpace/Paracompact.lean | t4Space | null |
edist_pi_def [∀ b, EDist (X b)] (f g : ∀ b, X b) :
edist f g = Finset.sup univ fun b => edist (f b) (g b) :=
rfl | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | edist_pi_def | null |
edist_le_pi_edist [∀ b, EDist (X b)] (f g : ∀ b, X b) (b : β) :
edist (f b) (g b) ≤ edist f g :=
le_sup (f := fun b => edist (f b) (g b)) (Finset.mem_univ b) | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | edist_le_pi_edist | null |
edist_pi_le_iff [∀ b, EDist (X b)] {f g : ∀ b, X b} {d : ℝ≥0∞} :
edist f g ≤ d ↔ ∀ b, edist (f b) (g b) ≤ d :=
Finset.sup_le_iff.trans <| by simp only [Finset.mem_univ, forall_const] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | edist_pi_le_iff | null |
edist_pi_const_le (a b : α) : (edist (fun _ : β => a) fun _ => b) ≤ edist a b :=
edist_pi_le_iff.2 fun _ => le_rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | edist_pi_const_le | null |
edist_pi_const [Nonempty β] (a b : α) : (edist (fun _ : β => a) fun _ => b) = edist a b :=
Finset.sup_const univ_nonempty (edist a b) | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | edist_pi_const | null |
pseudoEMetricSpacePi [∀ b, PseudoEMetricSpace (X b)] : PseudoEMetricSpace (∀ b, X b) where
edist_self f := bot_unique <| Finset.sup_le <| by simp
edist_comm f g := by simp [edist_pi_def, edist_comm]
edist_triangle _ g _ := edist_pi_le_iff.2 fun b => le_trans (edist_triangle _ (g b) _)
(add_le_add (edist_le_pi... | instance | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | pseudoEMetricSpacePi | The product of a finite number of pseudoemetric spaces, with the max distance, is still
a pseudoemetric space.
This construction would also work for infinite products, but it would not give rise
to the product topology. Hence, we only formalize it in the good situation of finitely many
spaces. |
emetricSpacePi [∀ b, EMetricSpace (X b)] : EMetricSpace (∀ b, X b) :=
.ofT0PseudoEMetricSpace _ | instance | Topology | [
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Pi.lean | emetricSpacePi | The product of a finite number of emetric spaces, with the max distance, is still
an emetric space.
This construction would also work for infinite products, but it would not give rise
to the product topology. Hence, we only formalize it in the good situation of finitely many
spaces. |
FiberBundle where
totalSpaceMk_isInducing' : ∀ b : B, IsInducing (@TotalSpace.mk B F E b)
trivializationAtlas' : Set (Trivialization F (π F E))
trivializationAt' : B → Trivialization F (π F E)
mem_baseSet_trivializationAt' : ∀ b : B, b ∈ (trivializationAt' b).baseSet
trivialization_mem_atlas' : ∀ b : B, trivi... | class | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | FiberBundle | A (topological) fiber bundle with fiber `F` over a base `B` is a space projecting on `B`
for which the fibers are all homeomorphic to `F`, such that the local situation around each point
is a direct product. |
totalSpaceMk_isInducing : IsInducing (@TotalSpace.mk B F E b) := totalSpaceMk_isInducing' b | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | totalSpaceMk_isInducing | null |
trivializationAtlas : Set (Trivialization F (π F E)) := trivializationAtlas' | abbrev | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | trivializationAtlas | Atlas of a fiber bundle. |
trivializationAt : Trivialization F (π F E) := trivializationAt' b | abbrev | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | trivializationAt | Trivialization of a fiber bundle at a point. |
mem_baseSet_trivializationAt : b ∈ (trivializationAt F E b).baseSet :=
mem_baseSet_trivializationAt' b | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | mem_baseSet_trivializationAt | null |
trivialization_mem_atlas : trivializationAt F E b ∈ trivializationAtlas F E :=
trivialization_mem_atlas' b | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | trivialization_mem_atlas | null |
@[mk_iff]
MemTrivializationAtlas [FiberBundle F E] (e : Trivialization F (π F E)) : Prop where
out : e ∈ trivializationAtlas F E | class | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | MemTrivializationAtlas | Given a type `E` equipped with a fiber bundle structure, this is a `Prop` typeclass
for trivializations of `E`, expressing that a trivialization is in the designated atlas for the
bundle. This is needed because lemmas about the linearity of trivializations or the continuity (as
functions to `F →L[R] F`, where `F` is t... |
map_proj_nhds (x : TotalSpace F E) : map (π F E) (𝓝 x) = 𝓝 x.proj :=
(trivializationAt F E x.proj).map_proj_nhds <|
(trivializationAt F E x.proj).mem_source.2 <| mem_baseSet_trivializationAt F E x.proj
variable (E) | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | map_proj_nhds | null |
@[continuity]
continuous_proj : Continuous (π F E) :=
continuous_iff_continuousAt.2 fun x => (map_proj_nhds F x).le | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | continuous_proj | The projection from a fiber bundle to its base is continuous. |
isOpenMap_proj : IsOpenMap (π F E) :=
IsOpenMap.of_nhds_le fun x => (map_proj_nhds F x).ge | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | isOpenMap_proj | The projection from a fiber bundle to its base is an open map. |
surjective_proj [Nonempty F] : Function.Surjective (π F E) := fun b =>
let ⟨p, _, hpb⟩ :=
(trivializationAt F E b).proj_surjOn_baseSet (mem_baseSet_trivializationAt F E b)
⟨p, hpb⟩ | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | surjective_proj | The projection from a fiber bundle with a nonempty fiber to its base is a surjective
map. |
isQuotientMap_proj [Nonempty F] : IsQuotientMap (π F E) :=
(isOpenMap_proj F E).isQuotientMap (continuous_proj F E) (surjective_proj F E) | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | isQuotientMap_proj | The projection from a fiber bundle with a nonempty fiber to its base is a quotient
map. |
continuous_totalSpaceMk (x : B) : Continuous (@TotalSpace.mk B F E x) :=
(totalSpaceMk_isInducing F E x).continuous | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | continuous_totalSpaceMk | null |
totalSpaceMk_isEmbedding (x : B) : IsEmbedding (@TotalSpace.mk B F E x) :=
⟨totalSpaceMk_isInducing F E x, TotalSpace.mk_injective x⟩ | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | totalSpaceMk_isEmbedding | null |
totalSpaceMk_isClosedEmbedding [T1Space B] (x : B) :
IsClosedEmbedding (@TotalSpace.mk B F E x) :=
⟨totalSpaceMk_isEmbedding F E x, by
rw [TotalSpace.range_mk]
exact isClosed_singleton.preimage <| continuous_proj F E⟩
variable {E F}
@[simp, mfld_simps] | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | totalSpaceMk_isClosedEmbedding | null |
mem_trivializationAt_proj_source {x : TotalSpace F E} :
x ∈ (trivializationAt F E x.proj).source :=
(Trivialization.mem_source _).mpr <| mem_baseSet_trivializationAt F E x.proj | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | mem_trivializationAt_proj_source | null |
trivializationAt_proj_fst {x : TotalSpace F E} :
((trivializationAt F E x.proj) x).1 = x.proj :=
Trivialization.coe_fst' _ <| mem_baseSet_trivializationAt F E x.proj
variable (F)
open Trivialization | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | trivializationAt_proj_fst | null |
continuousWithinAt_totalSpace (f : X → TotalSpace F E) {s : Set X} {x₀ : X} :
ContinuousWithinAt f s x₀ ↔
ContinuousWithinAt (fun x => (f x).proj) s x₀ ∧
ContinuousWithinAt (fun x => ((trivializationAt F E (f x₀).proj) (f x)).2) s x₀ :=
(trivializationAt F E (f x₀).proj).tendsto_nhds_iff mem_trivial... | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | continuousWithinAt_totalSpace | Characterization of continuous functions (at a point, within a set) into a fiber bundle. |
continuousAt_totalSpace (f : X → TotalSpace F E) {x₀ : X} :
ContinuousAt f x₀ ↔
ContinuousAt (fun x => (f x).proj) x₀ ∧
ContinuousAt (fun x => ((trivializationAt F E (f x₀).proj) (f x)).2) x₀ :=
(trivializationAt F E (f x₀).proj).tendsto_nhds_iff mem_trivializationAt_proj_source | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | continuousAt_totalSpace | Characterization of continuous functions (at a point) into a fiber bundle. |
FiberBundle.exists_trivialization_Icc_subset [ConditionallyCompleteLinearOrder B]
[OrderTopology B] [FiberBundle F E] (a b : B) :
∃ e : Trivialization F (π F E), Icc a b ⊆ e.baseSet := by
obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet :=
⟨trivializationAt F E a, mem_baseSet_trivializat... | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | FiberBundle.exists_trivialization_Icc_subset | If `E` is a fiber bundle over a conditionally complete linear order,
then it is trivial over any closed interval. |
FiberBundleCore (ι : Type*) (B : Type*) [TopologicalSpace B] (F : Type*)
[TopologicalSpace F] where
baseSet : ι → Set B
isOpen_baseSet : ∀ i, IsOpen (baseSet i)
indexAt : B → ι
mem_baseSet_at : ∀ x, x ∈ baseSet (indexAt x)
coordChange : ι → ι → B → F → F
coordChange_self : ∀ i, ∀ x ∈ baseSet i, ∀ v, coo... | structure | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | FiberBundleCore | Core data defining a locally trivial bundle with fiber `F` over a topological
space `B`. Note that "bundle" is used in its mathematical sense. This is the (computer science)
bundled version, i.e., all the relevant data is contained in the following structure. A family of
local trivializations is indexed by a type `ι`, ... |
@[nolint unusedArguments]
Index (_Z : FiberBundleCore ι B F) := ι | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | Index | The index set of a fiber bundle core, as a convenience function for dot notation |
@[nolint unusedArguments, reducible]
Base (_Z : FiberBundleCore ι B F) := B | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | Base | The base space of a fiber bundle core, as a convenience function for dot notation |
@[nolint unusedArguments]
Fiber (_ : FiberBundleCore ι B F) (_x : B) := F | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | Fiber | The fiber of a fiber bundle core, as a convenience function for dot notation and
typeclass inference |
topologicalSpaceFiber (x : B) : TopologicalSpace (Z.Fiber x) := ‹_› | instance | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | topologicalSpaceFiber | null |
TotalSpace := Bundle.TotalSpace F Z.Fiber | abbrev | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | TotalSpace | The total space of the fiber bundle, as a convenience function for dot notation.
It is by definition equal to `Bundle.TotalSpace F Z.Fiber`. |
@[reducible, simp, mfld_simps]
proj : Z.TotalSpace → B :=
Bundle.TotalSpace.proj | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | proj | The projection from the total space of a fiber bundle core, on its base. |
trivChange (i j : ι) : OpenPartialHomeomorph (B × F) (B × F) where
source := (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ
target := (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ
toFun p := ⟨p.1, Z.coordChange i j p.1 p.2⟩
invFun p := ⟨p.1, Z.coordChange j i p.1 p.2⟩
map_source' p hp := by simpa using hp
map_target' p hp := by... | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | trivChange | Local homeomorphism version of the trivialization change. |
mem_trivChange_source (i j : ι) (p : B × F) :
p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j := by
rw [trivChange, mem_prod]
simp | theorem | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | mem_trivChange_source | null |
localTrivAsPartialEquiv (i : ι) : PartialEquiv Z.TotalSpace (B × F) where
source := Z.proj ⁻¹' Z.baseSet i
target := Z.baseSet i ×ˢ univ
invFun p := ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩
toFun p := ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩
map_source' p hp := by
simpa only [Set.mem_preimage,... | def | Topology | [
"Mathlib.Topology.FiberBundle.Trivialization",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/FiberBundle/Basic.lean | localTrivAsPartialEquiv | Associate to a trivialization index `i : ι` the corresponding trivialization, i.e., a bijection
between `proj ⁻¹ (baseSet i)` and `baseSet i × F`. As the fiber above `x` is `F` but read in the
chart with index `index_at x`, the trivialization in the fiber above x is by definition the
coordinate change from i to `index_... |
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