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continuous_coe_iff {f : α → ℝ} : (Continuous fun a => (f a : EReal)) ↔ Continuous f :=
isEmbedding_coe.continuous_iff.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuous_coe_iff
| null |
nhds_coe {r : ℝ} : 𝓝 (r : EReal) = (𝓝 r).map (↑) :=
(isOpenEmbedding_coe.map_nhds_eq r).symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_coe
| null |
nhds_coe_coe {r p : ℝ} :
𝓝 ((r : EReal), (p : EReal)) = (𝓝 (r, p)).map fun p : ℝ × ℝ => (↑p.1, ↑p.2) :=
((isOpenEmbedding_coe.prodMap isOpenEmbedding_coe).map_nhds_eq (r, p)).symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_coe_coe
| null |
tendsto_toReal {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal) := by
lift a to ℝ using ⟨ha, h'a⟩
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_toReal
| null |
continuousOn_toReal : ContinuousOn EReal.toReal ({⊥, ⊤}ᶜ : Set EReal) := fun _a ha =>
ContinuousAt.continuousWithinAt (tendsto_toReal (mt Or.inr ha) (mt Or.inl ha))
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousOn_toReal
| null |
neBotTopHomeomorphReal : ({⊥, ⊤}ᶜ : Set EReal) ≃ₜ ℝ where
toEquiv := neTopBotEquivReal
continuous_toFun := continuousOn_iff_continuous_restrict.1 continuousOn_toReal
continuous_invFun := continuous_coe_real_ereal.subtype_mk _
/-! ### ENNReal coercion -/
|
def
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
neBotTopHomeomorphReal
|
The set of finite `EReal` numbers is homeomorphic to `ℝ`.
|
isEmbedding_coe_ennreal : IsEmbedding ((↑) : ℝ≥0∞ → EReal) :=
coe_ennreal_strictMono.isEmbedding_of_ordConnected <| by
rw [range_coe_ennreal]; exact ordConnected_Ici
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
isEmbedding_coe_ennreal
| null |
isClosedEmbedding_coe_ennreal : IsClosedEmbedding ((↑) : ℝ≥0∞ → EReal) :=
⟨isEmbedding_coe_ennreal, by rw [range_coe_ennreal]; exact isClosed_Ici⟩
@[norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
isClosedEmbedding_coe_ennreal
| null |
tendsto_coe_ennreal {α : Type*} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
Tendsto (fun a => (m a : EReal)) f (𝓝 ↑a) ↔ Tendsto m f (𝓝 a) :=
isEmbedding_coe_ennreal.tendsto_nhds_iff.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe_ennreal
| null |
_root_.continuous_coe_ennreal_ereal : Continuous ((↑) : ℝ≥0∞ → EReal) :=
isEmbedding_coe_ennreal.continuous
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.continuous_coe_ennreal_ereal
| null |
continuous_coe_ennreal_iff {f : α → ℝ≥0∞} :
(Continuous fun a => (f a : EReal)) ↔ Continuous f :=
isEmbedding_coe_ennreal.continuous_iff.symm
/-! ### Neighborhoods of infinity -/
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuous_coe_ennreal_iff
| null |
nhds_top : 𝓝 (⊤ : EReal) = ⨅ (a) (_ : a ≠ ⊤), 𝓟 (Ioi a) :=
nhds_top_order.trans <| by simp only [lt_top_iff_ne_top]
nonrec theorem nhds_top_basis : (𝓝 (⊤ : EReal)).HasBasis (fun _ : ℝ ↦ True) (Ioi ·) := by
refine (nhds_top_basis (α := EReal)).to_hasBasis (fun x hx => ?_)
fun _ _ ↦ ⟨_, coe_lt_top _, Subset.rfl⟩
rcases exists_rat_btwn_of_lt hx with ⟨y, hxy, -⟩
exact ⟨_, trivial, Ioi_subset_Ioi hxy.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_top
| null |
nhds_top' : 𝓝 (⊤ : EReal) = ⨅ a : ℝ, 𝓟 (Ioi ↑a) := nhds_top_basis.eq_iInf
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_top'
| null |
mem_nhds_top_iff {s : Set EReal} : s ∈ 𝓝 (⊤ : EReal) ↔ ∃ y : ℝ, Ioi (y : EReal) ⊆ s :=
nhds_top_basis.mem_iff.trans <| by simp only [true_and]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
mem_nhds_top_iff
| null |
tendsto_nhds_top_iff_real {α : Type*} {m : α → EReal} {f : Filter α} :
Tendsto m f (𝓝 ⊤) ↔ ∀ x : ℝ, ∀ᶠ a in f, ↑x < m a :=
nhds_top_basis.tendsto_right_iff.trans <| by simp only [true_implies, mem_Ioi]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_nhds_top_iff_real
| null |
nhds_bot : 𝓝 (⊥ : EReal) = ⨅ (a) (_ : a ≠ ⊥), 𝓟 (Iio a) :=
nhds_bot_order.trans <| by simp only [bot_lt_iff_ne_bot]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_bot
| null |
nhds_bot_basis : (𝓝 (⊥ : EReal)).HasBasis (fun _ : ℝ ↦ True) (Iio ·) := by
refine (_root_.nhds_bot_basis (α := EReal)).to_hasBasis (fun x hx => ?_)
fun _ _ ↦ ⟨_, bot_lt_coe _, Subset.rfl⟩
rcases exists_rat_btwn_of_lt hx with ⟨y, -, hxy⟩
exact ⟨_, trivial, Iio_subset_Iio hxy.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_bot_basis
| null |
nhds_bot' : 𝓝 (⊥ : EReal) = ⨅ a : ℝ, 𝓟 (Iio ↑a) :=
nhds_bot_basis.eq_iInf
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhds_bot'
| null |
mem_nhds_bot_iff {s : Set EReal} : s ∈ 𝓝 (⊥ : EReal) ↔ ∃ y : ℝ, Iio (y : EReal) ⊆ s :=
nhds_bot_basis.mem_iff.trans <| by simp only [true_and]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
mem_nhds_bot_iff
| null |
tendsto_nhds_bot_iff_real {α : Type*} {m : α → EReal} {f : Filter α} :
Tendsto m f (𝓝 ⊥) ↔ ∀ x : ℝ, ∀ᶠ a in f, m a < x :=
nhds_bot_basis.tendsto_right_iff.trans <| by simp only [true_implies, mem_Iio]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_nhds_bot_iff_real
| null |
nhdsWithin_top : 𝓝[≠] (⊤ : EReal) = (atTop).map Real.toEReal := by
apply (nhdsWithin_hasBasis nhds_top_basis_Ici _).ext (atTop_basis.map Real.toEReal)
· simp only [EReal.image_coe_Ici, true_and]
intro x hx
by_cases hx_bot : x = ⊥
· simp [hx_bot]
lift x to ℝ using ⟨hx.ne_top, hx_bot⟩
refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
simp [h1, h2.ne_top]
· simp only [EReal.image_coe_Ici, true_implies]
refine fun x ↦ ⟨x, ⟨EReal.coe_lt_top x, fun x ⟨(h1 : _ ≤ x), h2⟩ ↦ ?_⟩⟩
simp [h1, Ne.lt_top' fun a ↦ h2 a.symm]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhdsWithin_top
| null |
nhdsWithin_bot : 𝓝[≠] (⊥ : EReal) = (atBot).map Real.toEReal := by
apply (nhdsWithin_hasBasis nhds_bot_basis_Iic _).ext (atBot_basis.map Real.toEReal)
· simp only [EReal.image_coe_Iic,
true_and]
intro x hx
by_cases hx_top : x = ⊤
· simp [hx_top]
lift x to ℝ using ⟨hx_top, hx.ne_bot⟩
refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
simp [h2, h1.ne_bot]
· simp only [EReal.image_coe_Iic, true_implies]
refine fun x ↦ ⟨x, ⟨EReal.bot_lt_coe x, fun x ⟨(h1 : x ≤ _), h2⟩ ↦ ?_⟩⟩
simp [h1, Ne.bot_lt' fun a ↦ h2 a.symm]
omit [TopologicalSpace α] in
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
nhdsWithin_bot
| null |
tendsto_coe_nhds_top_iff {f : α → ℝ} {l : Filter α} :
Tendsto (fun x ↦ Real.toEReal (f x)) l (𝓝 ⊤) ↔ Tendsto f l atTop := by
rw [tendsto_nhds_top_iff_real, atTop_basis_Ioi.tendsto_right_iff]; simp
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe_nhds_top_iff
| null |
tendsto_coe_atTop : Tendsto Real.toEReal atTop (𝓝 ⊤) :=
tendsto_coe_nhds_top_iff.2 tendsto_id
omit [TopologicalSpace α] in
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe_atTop
| null |
tendsto_coe_nhds_bot_iff {f : α → ℝ} {l : Filter α} :
Tendsto (fun x ↦ Real.toEReal (f x)) l (𝓝 ⊥) ↔ Tendsto f l atBot := by
rw [tendsto_nhds_bot_iff_real, atBot_basis_Iio.tendsto_right_iff]; simp
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe_nhds_bot_iff
| null |
tendsto_coe_atBot : Tendsto Real.toEReal atBot (𝓝 ⊥) :=
tendsto_coe_nhds_bot_iff.2 tendsto_id
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe_atBot
| null |
tendsto_toReal_atTop : Tendsto EReal.toReal (𝓝[≠] ⊤) atTop := by
rw [nhdsWithin_top, tendsto_map'_iff]
exact tendsto_id
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_toReal_atTop
| null |
tendsto_toReal_atBot : Tendsto EReal.toReal (𝓝[≠] ⊥) atBot := by
rw [nhdsWithin_bot, tendsto_map'_iff]
exact tendsto_id
/-! ### toENNReal -/
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_toReal_atBot
| null |
continuous_toENNReal : Continuous EReal.toENNReal := by
refine continuous_iff_continuousAt.mpr fun x ↦ ?_
by_cases h_top : x = ⊤
· simp only [ContinuousAt, h_top, toENNReal_top]
refine ENNReal.tendsto_nhds_top fun n ↦ ?_
filter_upwards [eventually_gt_nhds (coe_lt_top n)] with y hy
exact toENNReal_coe (x := n) ▸ toENNReal_lt_toENNReal (coe_ennreal_nonneg _) hy
refine ContinuousOn.continuousAt ?_ (compl_singleton_mem_nhds_iff.mpr h_top)
refine (continuousOn_of_forall_continuousAt fun x hx ↦ ?_).congr (fun _ h ↦ toENNReal_of_ne_top h)
by_cases h_bot : x = ⊥
· refine tendsto_nhds_of_eventually_eq ?_
rw [h_bot, nhds_bot_basis.eventually_iff]
simpa [toReal_bot, ENNReal.ofReal_zero, ENNReal.ofReal_eq_zero, true_and] using
⟨0, fun _ hx ↦ toReal_nonpos hx.le⟩
refine ENNReal.continuous_ofReal.continuousAt.comp' <| continuousOn_toReal.continuousAt
<| (toFinite _).isClosed.compl_mem_nhds ?_
simp_all only [mem_compl_iff, mem_singleton_iff, mem_insert_iff, or_self, not_false_eq_true]
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuous_toENNReal
| null |
_root_.Continuous.ereal_toENNReal {α : Type*} [TopologicalSpace α] {f : α → EReal}
(hf : Continuous f) :
Continuous fun x => (f x).toENNReal :=
continuous_toENNReal.comp hf
@[deprecated (since := "2025-03-05")] alias _root_.Continous.ereal_toENNReal :=
_root_.Continuous.ereal_toENNReal
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.Continuous.ereal_toENNReal
| null |
_root_.ContinuousOn.ereal_toENNReal {α : Type*} [TopologicalSpace α] {s : Set α}
{f : α → EReal} (hf : ContinuousOn f s) :
ContinuousOn (fun x => (f x).toENNReal) s :=
continuous_toENNReal.comp_continuousOn hf
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.ContinuousOn.ereal_toENNReal
| null |
_root_.ContinuousWithinAt.ereal_toENNReal {α : Type*} [TopologicalSpace α] {f : α → EReal}
{s : Set α} {x : α} (hf : ContinuousWithinAt f s x) :
ContinuousWithinAt (fun x => (f x).toENNReal) s x :=
continuous_toENNReal.continuousAt.comp_continuousWithinAt hf
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.ContinuousWithinAt.ereal_toENNReal
| null |
_root_.ContinuousAt.ereal_toENNReal {α : Type*} [TopologicalSpace α] {f : α → EReal}
{x : α} (hf : ContinuousAt f x) :
ContinuousAt (fun x => (f x).toENNReal) x :=
continuous_toENNReal.continuousAt.comp hf
/-! ### Infs and Sups -/
variable {α : Type*} {u v : α → EReal}
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.ContinuousAt.ereal_toENNReal
| null |
add_iInf_le_iInf_add : (⨅ x, u x) + ⨅ x, v x ≤ ⨅ x, (u + v) x :=
le_iInf fun i ↦ add_le_add (iInf_le u i) (iInf_le v i)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
add_iInf_le_iInf_add
| null |
iSup_add_le_add_iSup : ⨆ x, (u + v) x ≤ (⨆ x, u x) + ⨆ x, v x :=
iSup_le fun i ↦ add_le_add (le_iSup u i) (le_iSup v i)
/-! ### Liminfs and Limsups -/
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
iSup_add_le_add_iSup
| null |
liminf_neg : liminf (- v) f = - limsup v f :=
EReal.negOrderIso.limsup_apply.symm
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
liminf_neg
| null |
limsup_neg : limsup (- v) f = - liminf v f :=
EReal.negOrderIso.liminf_apply.symm
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
limsup_neg
| null |
le_liminf_add : (liminf u f) + (liminf v f) ≤ liminf (u + v) f := by
refine add_le_of_forall_lt fun a a_u b b_v ↦ (le_liminf_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_lt_liminf a_u, eventually_lt_of_lt_liminf b_v] with x a_x b_x
exact c_ab.trans (add_lt_add a_x b_x)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
le_liminf_add
| null |
limsup_add_le (h : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤) (h' : limsup u f ≠ ⊤ ∨ limsup v f ≠ ⊥) :
limsup (u + v) f ≤ (limsup u f) + (limsup v f) := by
refine le_add_of_forall_gt h h' fun a a_u b b_v ↦ (limsup_le_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_limsup_lt a_u, eventually_lt_of_limsup_lt b_v] with x a_x b_x
exact (add_lt_add a_x b_x).trans c_ab
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
limsup_add_le
| null |
le_limsup_add : (limsup u f) + (liminf v f) ≤ limsup (u + v) f :=
add_le_of_forall_lt fun _ a_u _ b_v ↦ (le_limsup_iff).2 fun _ c_ab ↦
(((frequently_lt_of_lt_limsup) a_u).and_eventually ((eventually_lt_of_lt_liminf) b_v)).mono
fun _ ab_x ↦ c_ab.trans (add_lt_add ab_x.1 ab_x.2)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
le_limsup_add
| null |
liminf_add_le (h : limsup u f ≠ ⊥ ∨ liminf v f ≠ ⊤) (h' : limsup u f ≠ ⊤ ∨ liminf v f ≠ ⊥) :
liminf (u + v) f ≤ (limsup u f) + (liminf v f) :=
le_add_of_forall_gt h h' fun _ a_u _ b_v ↦ (liminf_le_iff).2 fun _ c_ab ↦
(((frequently_lt_of_liminf_lt) b_v).and_eventually ((eventually_lt_of_limsup_lt) a_u)).mono
fun _ ab_x ↦ (add_lt_add ab_x.2 ab_x.1).trans c_ab
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
liminf_add_le
| null |
limsup_add_bot_of_ne_top (h : limsup u f = ⊥) (h' : limsup v f ≠ ⊤) :
limsup (u + v) f = ⊥ := by
apply le_bot_iff.1 ((limsup_add_le (.inr h') _).trans _)
· rw [h]; exact .inl bot_ne_top
· rw [h, bot_add]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
limsup_add_bot_of_ne_top
| null |
limsup_add_le_of_le {a b : EReal} (ha : limsup u f < a) (hb : limsup v f ≤ b) :
limsup (u + v) f ≤ a + b := by
rcases eq_top_or_lt_top b with rfl | h
· rw [add_top_of_ne_bot ha.ne_bot]; exact le_top
· exact (limsup_add_le (.inr (hb.trans_lt h).ne) (.inl ha.ne_top)).trans (add_le_add ha.le hb)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
limsup_add_le_of_le
| null |
liminf_add_gt_of_gt {a b : EReal} (ha : a < liminf u f) (hb : b < liminf v f) :
a + b < liminf (u + v) f :=
(add_lt_add ha hb).trans_le le_liminf_add
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
liminf_add_gt_of_gt
| null |
liminf_add_top_of_ne_bot (h : liminf u f = ⊤) (h' : liminf v f ≠ ⊥) :
liminf (u + v) f = ⊤ := by
apply top_le_iff.1 (le_trans _ le_liminf_add)
rw [h, top_add_of_ne_bot h']
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
liminf_add_top_of_ne_bot
| null |
le_limsup_mul (hu : ∃ᶠ x in f, 0 ≤ u x) (hv : 0 ≤ᶠ[f] v) :
limsup u f * liminf v f ≤ limsup (u * v) f := by
rcases f.eq_or_neBot with rfl | _
· rw [limsup_bot, limsup_bot, liminf_bot, bot_mul_top]
have u0 : 0 ≤ limsup u f := le_limsup_of_frequently_le hu
have uv0 : 0 ≤ limsup (u * v) f :=
le_limsup_of_frequently_le <| (hu.and_eventually hv).mono fun _ ⟨hu, hv⟩ ↦ mul_nonneg hu hv
refine mul_le_of_forall_lt_of_nonneg u0 uv0 fun a ha b hb ↦ (le_limsup_iff).2 fun c c_ab ↦ ?_
refine (((frequently_lt_of_lt_limsup) (mem_Ioo.1 ha).2).and_eventually
<| (eventually_lt_of_lt_liminf (mem_Ioo.1 hb).2).and
<| hv).mono fun x ⟨xa, ⟨xb, vx⟩⟩ ↦ ?_
exact c_ab.trans_le (mul_le_mul xa.le xb.le (mem_Ioo.1 hb).1.le ((mem_Ioo.1 ha).1.le.trans xa.le))
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
le_limsup_mul
| null |
limsup_mul_le (hu : ∃ᶠ x in f, 0 ≤ u x) (hv : 0 ≤ᶠ[f] v)
(h₁ : limsup u f ≠ 0 ∨ limsup v f ≠ ⊤) (h₂ : limsup u f ≠ ⊤ ∨ limsup v f ≠ 0) :
limsup (u * v) f ≤ limsup u f * limsup v f := by
rcases f.eq_or_neBot with rfl | _
· rw [limsup_bot]; exact bot_le
have u_0 : 0 ≤ limsup u f := le_limsup_of_frequently_le hu
replace h₁ : 0 < limsup u f ∨ limsup v f ≠ ⊤ := h₁.imp_left fun h ↦ lt_of_le_of_ne u_0 h.symm
replace h₂ : limsup u f ≠ ⊤ ∨ 0 < limsup v f :=
h₂.imp_right fun h ↦ lt_of_le_of_ne (le_limsup_of_frequently_le hv.frequently) h.symm
refine le_mul_of_forall_lt h₁ h₂ fun a a_u b b_v ↦ (limsup_le_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_limsup_lt a_u, eventually_lt_of_limsup_lt b_v, hv]
with x x_a x_b v_0
apply lt_of_le_of_lt _ c_ab
rcases lt_or_ge (u x) 0 with hux | hux
· apply (mul_nonpos_iff.2 (.inr ⟨hux.le, v_0⟩)).trans
exact mul_nonneg (u_0.trans a_u.le) (v_0.trans x_b.le)
· exact mul_le_mul x_a.le x_b.le v_0 (hux.trans x_a.le)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
limsup_mul_le
| null |
le_liminf_mul (hu : 0 ≤ᶠ[f] u) (hv : 0 ≤ᶠ[f] v) :
liminf u f * liminf v f ≤ liminf (u * v) f := by
apply mul_le_of_forall_lt_of_nonneg ((le_liminf_of_le) hu)
<| (le_liminf_of_le) ((hu.and hv).mono fun x ⟨u0, v0⟩ ↦ mul_nonneg u0 v0)
refine fun a ha b hb ↦ (le_liminf_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_lt_liminf (mem_Ioo.1 ha).2,
eventually_lt_of_lt_liminf (mem_Ioo.1 hb).2] with x xa xb
exact c_ab.trans_le (mul_le_mul xa.le xb.le (mem_Ioo.1 hb).1.le ((mem_Ioo.1 ha).1.le.trans xa.le))
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
le_liminf_mul
| null |
liminf_mul_le [NeBot f] (hu : 0 ≤ᶠ[f] u) (hv : 0 ≤ᶠ[f] v)
(h₁ : limsup u f ≠ 0 ∨ liminf v f ≠ ⊤) (h₂ : limsup u f ≠ ⊤ ∨ liminf v f ≠ 0) :
liminf (u * v) f ≤ limsup u f * liminf v f := by
replace h₁ : 0 < limsup u f ∨ liminf v f ≠ ⊤ := by
refine h₁.imp_left fun h ↦ lt_of_le_of_ne ?_ h.symm
exact le_of_eq_of_le (limsup_const 0).symm (limsup_le_limsup hu)
replace h₂ : limsup u f ≠ ⊤ ∨ 0 < liminf v f := by
refine h₂.imp_right fun h ↦ lt_of_le_of_ne ?_ h.symm
exact le_of_eq_of_le (liminf_const 0).symm (liminf_le_liminf hv)
refine le_mul_of_forall_lt h₁ h₂ fun a a_u b b_v ↦ (liminf_le_iff).2 fun c c_ab ↦ ?_
refine (((frequently_lt_of_liminf_lt) b_v).and_eventually <| (eventually_lt_of_limsup_lt a_u).and
<| hu.and hv).mono fun x ⟨x_v, x_u, u_0, v_0⟩ ↦ ?_
exact (mul_le_mul x_u.le x_v.le v_0 (u_0.trans x_u.le)).trans_lt c_ab
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
liminf_mul_le
| null |
continuousAt_add_coe_coe (a b : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, b) := by
simp only [ContinuousAt, nhds_coe_coe, ← coe_add, tendsto_map'_iff, Function.comp_def,
tendsto_coe, tendsto_add]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_coe_coe
| null |
continuousAt_add_top_coe (a : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, a) := by
simp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_coe]
refine fun r ↦ ((lt_mem_nhds (coe_lt_top (r - (a - 1)))).prod_nhds
(lt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| sub_one_lt _)).mono fun _ h ↦ ?_
simpa only [← coe_add, _root_.sub_add_cancel] using add_lt_add h.1 h.2
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_top_coe
| null |
continuousAt_add_coe_top (a : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊤) := by
simpa only [add_comm, Function.comp_def, ContinuousAt, Prod.swap]
using Tendsto.comp (continuousAt_add_top_coe a) (continuous_swap.tendsto ((a : EReal), ⊤))
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_coe_top
| null |
continuousAt_add_top_top : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, ⊤) := by
simp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_top]
refine fun r ↦ ((lt_mem_nhds (coe_lt_top 0)).prod_nhds
(lt_mem_nhds <| coe_lt_top r)).mono fun _ h ↦ ?_
simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_top_top
| null |
continuousAt_add_bot_coe (a : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, a) := by
simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]
refine fun r ↦ ((gt_mem_nhds (bot_lt_coe (r - (a + 1)))).prod_nhds
(gt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| lt_add_one _)).mono fun _ h ↦ ?_
simpa only [← coe_add, _root_.sub_add_cancel] using add_lt_add h.1 h.2
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_bot_coe
| null |
continuousAt_add_coe_bot (a : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊥) := by
simpa only [add_comm, Function.comp_def, ContinuousAt, Prod.swap]
using Tendsto.comp (continuousAt_add_bot_coe a) (continuous_swap.tendsto ((a : EReal), ⊥))
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_coe_bot
| null |
continuousAt_add_bot_bot : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, ⊥) := by
simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]
refine fun r ↦ ((gt_mem_nhds (bot_lt_coe 0)).prod_nhds
(gt_mem_nhds <| bot_lt_coe r)).mono fun _ h ↦ ?_
simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add_bot_bot
| null |
continuousAt_add {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) p := by
rcases p with ⟨x, y⟩
induction x <;> induction y
· exact continuousAt_add_bot_bot
· exact continuousAt_add_bot_coe _
· simp at h'
· exact continuousAt_add_coe_bot _
· exact continuousAt_add_coe_coe _ _
· exact continuousAt_add_coe_top _
· simp at h
· exact continuousAt_add_top_coe _
· exact continuousAt_add_top_top
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_add
|
The addition on `EReal` is continuous except where it doesn't make sense (i.e., at `(⊥, ⊤)`
and at `(⊤, ⊥)`).
|
lowerSemicontinuous_add : LowerSemicontinuous fun p : EReal × EReal ↦ p.1 + p.2 := by
intro x y
by_cases hx₁ : x.1 = ⊥
· simp [hx₁]
by_cases hx₂ : x.2 = ⊥
· simp [hx₂]
· exact continuousAt_add (.inr hx₂) (.inl hx₁) |>.lowerSemicontinuousAt _
/-! ### Continuity of multiplication -/
/- Outside of indeterminacies `(0, ±∞)` and `(±∞, 0)`, the multiplication on `EReal` is continuous.
There are many different cases to consider, so we first prove some special cases and leverage as
much as possible the symmetries of the multiplication. -/
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
lowerSemicontinuous_add
| null |
private continuousAt_mul_swap {a b : EReal}
(h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (b, a) := by
convert h.comp continuous_swap.continuousAt (x := (b, a))
simp [mul_comm]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_swap
| null |
private continuousAt_mul_symm1 {a b : EReal}
(h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (-a, b) := by
have : (fun p : EReal × EReal ↦ p.1 * p.2) = (fun x : EReal ↦ -x)
∘ (fun p : EReal × EReal ↦ p.1 * p.2) ∘ (fun p : EReal × EReal ↦ (-p.1, p.2)) := by
ext p
simp
rw [this]
apply ContinuousAt.comp (Continuous.continuousAt continuous_neg)
<| ContinuousAt.comp _ (ContinuousAt.prodMap (Continuous.continuousAt continuous_neg)
(Continuous.continuousAt continuous_id))
simp [h]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_symm1
| null |
private continuousAt_mul_symm2 {a b : EReal}
(h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, -b) :=
continuousAt_mul_swap (continuousAt_mul_symm1 (continuousAt_mul_swap h))
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_symm2
| null |
private continuousAt_mul_symm3 {a b : EReal}
(h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (-a, -b) :=
continuousAt_mul_symm1 (continuousAt_mul_symm2 h)
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_symm3
| null |
private continuousAt_mul_coe_coe (a b : ℝ) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b) := by
simp [ContinuousAt, EReal.nhds_coe_coe, ← EReal.coe_mul, Filter.tendsto_map'_iff,
Function.comp_def, EReal.tendsto_coe, tendsto_mul]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_coe_coe
| null |
private continuousAt_mul_top_top :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, ⊤) := by
simp only [ContinuousAt, EReal.top_mul_top, EReal.tendsto_nhds_top_iff_real]
intro x
rw [_root_.eventually_nhds_iff]
use (Set.Ioi ((max x 0) : EReal)) ×ˢ (Set.Ioi 1)
split_ands
· intro p p_in_prod
simp only [Set.mem_prod, Set.mem_Ioi, max_lt_iff] at p_in_prod
rcases p_in_prod with ⟨⟨p1_gt_x, p1_pos⟩, p2_gt_1⟩
have := mul_le_mul_of_nonneg_left (le_of_lt p2_gt_1) (le_of_lt p1_pos)
rw [mul_one p.1] at this
exact lt_of_lt_of_le p1_gt_x this
· exact IsOpen.prod isOpen_Ioi isOpen_Ioi
· simp
· rw [Set.mem_Ioi, ← EReal.coe_one]; exact EReal.coe_lt_top 1
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_top_top
| null |
private continuousAt_mul_top_pos {a : ℝ} (h : 0 < a) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, a) := by
simp only [ContinuousAt, EReal.top_mul_coe_of_pos h, EReal.tendsto_nhds_top_iff_real]
intro x
rw [_root_.eventually_nhds_iff]
use (Set.Ioi ((2*(max (x+1) 0)/a : ℝ) : EReal)) ×ˢ (Set.Ioi ((a/2 : ℝ) : EReal))
split_ands
· intro p p_in_prod
simp only [Set.mem_prod, Set.mem_Ioi] at p_in_prod
rcases p_in_prod with ⟨p1_gt, p2_gt⟩
have p1_pos : 0 < p.1 := by
apply lt_of_le_of_lt _ p1_gt
rw [EReal.coe_nonneg]
apply mul_nonneg _ (le_of_lt (inv_pos_of_pos h))
simp only [Nat.ofNat_pos, mul_nonneg_iff_of_pos_left, le_max_iff, le_refl, or_true]
have a2_pos : 0 < ((a/2 : ℝ) : EReal) := by rw [EReal.coe_pos]; linarith
have lock := mul_le_mul_of_nonneg_right (le_of_lt p1_gt) (le_of_lt a2_pos)
have key := mul_le_mul_of_nonneg_left (le_of_lt p2_gt) (le_of_lt p1_pos)
replace lock := le_trans lock key
apply lt_of_lt_of_le _ lock
rw [← EReal.coe_mul, EReal.coe_lt_coe_iff, _root_.div_mul_div_comm, mul_comm,
← _root_.div_mul_div_comm, mul_div_right_comm]
simp only [ne_eq, Ne.symm (ne_of_lt h), not_false_eq_true, _root_.div_self, OfNat.ofNat_ne_zero,
one_mul, lt_max_iff, lt_add_iff_pos_right, zero_lt_one, true_or]
· exact IsOpen.prod isOpen_Ioi isOpen_Ioi
· simp
· simp [h]
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_top_pos
| null |
private continuousAt_mul_top_ne_zero {a : ℝ} (h : a ≠ 0) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, a) := by
rcases lt_or_gt_of_ne h with a_neg | a_pos
· exact neg_neg a ▸ continuousAt_mul_symm2 (continuousAt_mul_top_pos (neg_pos.2 a_neg))
· exact continuousAt_mul_top_pos a_pos
|
lemma
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul_top_ne_zero
| null |
continuousAt_mul {p : EReal × EReal} (h₁ : p.1 ≠ 0 ∨ p.2 ≠ ⊥)
(h₂ : p.1 ≠ 0 ∨ p.2 ≠ ⊤) (h₃ : p.1 ≠ ⊥ ∨ p.2 ≠ 0) (h₄ : p.1 ≠ ⊤ ∨ p.2 ≠ 0) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) p := by
rcases p with ⟨x, y⟩
induction x <;> induction y
· exact continuousAt_mul_symm3 continuousAt_mul_top_top
· simp only [ne_eq, not_true_eq_false, EReal.coe_eq_zero, false_or] at h₃
exact continuousAt_mul_symm1 (continuousAt_mul_top_ne_zero h₃)
· exact EReal.neg_top ▸ continuousAt_mul_symm1 continuousAt_mul_top_top
· simp only [ne_eq, EReal.coe_eq_zero, not_true_eq_false, or_false] at h₁
exact continuousAt_mul_symm2 (continuousAt_mul_swap (continuousAt_mul_top_ne_zero h₁))
· exact continuousAt_mul_coe_coe _ _
· simp only [ne_eq, EReal.coe_eq_zero, not_true_eq_false, or_false] at h₂
exact continuousAt_mul_swap (continuousAt_mul_top_ne_zero h₂)
· exact continuousAt_mul_symm2 continuousAt_mul_top_top
· simp only [ne_eq, not_true_eq_false, EReal.coe_eq_zero, false_or] at h₄
exact continuousAt_mul_top_ne_zero h₄
· exact continuousAt_mul_top_top
variable {a b : EReal}
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
continuousAt_mul
|
The multiplication on `EReal` is continuous except at indeterminacies
(i.e. whenever one value is zero and the other infinite).
|
protected tendsto_mul (h₁ : a ≠ 0 ∨ b ≠ ⊥) (h₂ : a ≠ 0 ∨ b ≠ ⊤) (h₃ : a ≠ ⊥ ∨ b ≠ 0)
(h₄ : a ≠ ⊤ ∨ b ≠ 0) :
Tendsto (fun p : EReal × EReal ↦ p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b)) :=
(continuousAt_mul h₁ h₂ h₃ h₄).tendsto
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_mul
| null |
protected Tendsto.mul {f : Filter α} {ma : α → EReal} {mb : α → EReal} {a b : EReal}
(hma : Tendsto ma f (𝓝 a)) (hmb : Tendsto mb f (𝓝 b)) (h₁ : a ≠ 0 ∨ b ≠ ⊥)
(h₂ : a ≠ 0 ∨ b ≠ ⊤) (h₃ : a ≠ ⊥ ∨ b ≠ 0) (h₄ : a ≠ ⊤ ∨ b ≠ 0) :
Tendsto (fun x ↦ ma x * mb x) f (𝓝 (a * b)) :=
(EReal.tendsto_mul h₁ h₂ h₃ h₄).comp (hma.prodMk_nhds hmb)
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
Tendsto.mul
| null |
protected Tendsto.const_mul {f : Filter α} {m : α → EReal} {a b : EReal}
(hm : Tendsto m f (𝓝 b)) (h₁ : a ≠ ⊥ ∨ b ≠ 0) (h₂ : a ≠ ⊤ ∨ b ≠ 0) :
Tendsto (fun b ↦ a * m b) f (𝓝 (a * b)) :=
by_cases (fun (this : a = 0) => by simp [this, tendsto_const_nhds])
fun ha : a ≠ 0 => EReal.Tendsto.mul tendsto_const_nhds hm (Or.inl ha) (Or.inl ha) h₁ h₂
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
Tendsto.const_mul
| null |
protected Tendsto.mul_const {f : Filter α} {m : α → EReal} {a b : EReal}
(hm : Tendsto m f (𝓝 a)) (h₁ : a ≠ 0 ∨ b ≠ ⊥) (h₂ : a ≠ 0 ∨ b ≠ ⊤) :
Tendsto (fun x ↦ m x * b) f (𝓝 (a * b)) := by
simpa only [mul_comm] using EReal.Tendsto.const_mul hm h₁.symm h₂.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
Tendsto.mul_const
| null |
isOpen_Ico_zero {x : NNReal} : IsOpen (Set.Ico 0 x) :=
Ico_bot (a := x) ▸ isOpen_Iio
open Filter Finset
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
isOpen_Ico_zero
| null |
_root_.continuous_real_toNNReal : Continuous Real.toNNReal :=
(continuous_id.max continuous_const).subtype_mk _
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.continuous_real_toNNReal
| null |
@[simps -fullyApplied]
noncomputable _root_.ContinuousMap.realToNNReal : C(ℝ, ℝ≥0) :=
.mk Real.toNNReal continuous_real_toNNReal
|
def
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.ContinuousMap.realToNNReal
|
`Real.toNNReal` bundled as a continuous map for convenience.
|
_root_.ContinuousOn.ofReal_map_toNNReal {f : ℝ≥0 → ℝ≥0} {s : Set ℝ} {t : Set ℝ≥0}
(hf : ContinuousOn f t) (h : Set.MapsTo Real.toNNReal s t) :
ContinuousOn (fun x ↦ f x.toNNReal : ℝ → ℝ) s :=
continuous_subtype_val.comp_continuousOn <| hf.comp continuous_real_toNNReal.continuousOn h
@[simp, norm_cast]
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.ContinuousOn.ofReal_map_toNNReal
| null |
tendsto_coe {f : Filter α} {m : α → ℝ≥0} {x : ℝ≥0} :
Tendsto (fun a => (m a : ℝ)) f (𝓝 (x : ℝ)) ↔ Tendsto m f (𝓝 x) :=
tendsto_subtype_rng.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_coe
| null |
tendsto_coe' {f : Filter α} [NeBot f] {m : α → ℝ≥0} {x : ℝ} :
Tendsto (fun a => m a : α → ℝ) f (𝓝 x) ↔ ∃ hx : 0 ≤ x, Tendsto m f (𝓝 ⟨x, hx⟩) :=
⟨fun h => ⟨ge_of_tendsto' h fun c => (m c).2, tendsto_coe.1 h⟩, fun ⟨_, hm⟩ => tendsto_coe.2 hm⟩
@[simp] theorem map_coe_atTop : map toReal atTop = atTop := map_val_Ici_atTop 0
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_coe'
| null |
comap_coe_atTop : comap toReal atTop = atTop := (atTop_Ici_eq 0).symm
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
comap_coe_atTop
| null |
tendsto_coe_atTop {f : Filter α} {m : α → ℝ≥0} :
Tendsto (fun a => (m a : ℝ)) f atTop ↔ Tendsto m f atTop :=
tendsto_Ici_atTop.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_coe_atTop
| null |
_root_.tendsto_real_toNNReal {f : Filter α} {m : α → ℝ} {x : ℝ} (h : Tendsto m f (𝓝 x)) :
Tendsto (fun a => Real.toNNReal (m a)) f (𝓝 (Real.toNNReal x)) :=
(continuous_real_toNNReal.tendsto _).comp h
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.tendsto_real_toNNReal
| null |
_root_.Real.map_toNNReal_atTop : map Real.toNNReal atTop = atTop := by
rw [← map_coe_atTop, Function.LeftInverse.filter_map @Real.toNNReal_coe]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.map_toNNReal_atTop
| null |
_root_.tendsto_real_toNNReal_atTop : Tendsto Real.toNNReal atTop atTop :=
Real.map_toNNReal_atTop.le
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.tendsto_real_toNNReal_atTop
| null |
_root_.Real.comap_toNNReal_atTop : comap Real.toNNReal atTop = atTop := by
refine le_antisymm ?_ tendsto_real_toNNReal_atTop.le_comap
refine (atTop_basis_Ioi' 0).ge_iff.2 fun a ha ↦ ?_
filter_upwards [preimage_mem_comap (Ioi_mem_atTop a.toNNReal)] with x hx
exact (Real.toNNReal_lt_toNNReal_iff_of_nonneg ha.le).1 hx
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.comap_toNNReal_atTop
| null |
_root_.Real.tendsto_toNNReal_atTop_iff {l : Filter α} {f : α → ℝ} :
Tendsto (fun x ↦ (f x).toNNReal) l atTop ↔ Tendsto f l atTop := by
rw [← Real.comap_toNNReal_atTop, tendsto_comap_iff, Function.comp_def]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.tendsto_toNNReal_atTop_iff
| null |
_root_.Real.tendsto_toNNReal_atTop : Tendsto Real.toNNReal atTop atTop :=
Real.tendsto_toNNReal_atTop_iff.2 tendsto_id
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.tendsto_toNNReal_atTop
| null |
nhds_zero : 𝓝 (0 : ℝ≥0) = ⨅ (a : ℝ≥0) (_ : a ≠ 0), 𝓟 (Iio a) :=
nhds_bot_order.trans <| by simp only [bot_lt_iff_ne_bot]; rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
nhds_zero
| null |
nhds_zero_basis : (𝓝 (0 : ℝ≥0)).HasBasis (fun a : ℝ≥0 => 0 < a) fun a => Iio a :=
nhds_bot_basis
@[norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
nhds_zero_basis
| null |
hasSum_coe {f : α → ℝ≥0} {r : ℝ≥0} : HasSum (fun a => (f a : ℝ)) (r : ℝ) ↔ HasSum f r := by
simp only [HasSum, ← coe_sum, tendsto_coe]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
hasSum_coe
| null |
protected _root_.HasSum.toNNReal {f : α → ℝ} {y : ℝ} (hf₀ : ∀ n, 0 ≤ f n)
(hy : HasSum f y) : HasSum (fun x => Real.toNNReal (f x)) y.toNNReal := by
lift y to ℝ≥0 using hy.nonneg hf₀
lift f to α → ℝ≥0 using hf₀
simpa [hasSum_coe] using hy
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.HasSum.toNNReal
| null |
hasSum_real_toNNReal_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : Summable f) :
HasSum (fun n => Real.toNNReal (f n)) (Real.toNNReal (∑' n, f n)) :=
hf.hasSum.toNNReal hf_nonneg
@[norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
hasSum_real_toNNReal_of_nonneg
| null |
summable_coe {f : α → ℝ≥0} : (Summable fun a => (f a : ℝ)) ↔ Summable f := by
constructor
· exact fun ⟨a, ha⟩ => ⟨⟨a, ha.nonneg fun x => (f x).2⟩, hasSum_coe.1 ha⟩
· exact fun ⟨a, ha⟩ => ⟨a.1, hasSum_coe.2 ha⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
summable_coe
| null |
summable_mk {f : α → ℝ} (hf : ∀ n, 0 ≤ f n) :
(@Summable ℝ≥0 _ _ _ fun n => ⟨f n, hf n⟩) ↔ Summable f :=
Iff.symm <| summable_coe (f := fun x => ⟨f x, hf x⟩)
@[norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
summable_mk
| null |
coe_tsum {f : α → ℝ≥0} : ↑(∑' a, f a) = ∑' a, (f a : ℝ) := by
classical
exact if hf : Summable f then Eq.symm <| (hasSum_coe.2 <| hf.hasSum).tsum_eq
else by simp [tsum_def, hf, mt summable_coe.1 hf]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
coe_tsum
| null |
coe_tsum_of_nonneg {f : α → ℝ} (hf₁ : ∀ n, 0 ≤ f n) :
(⟨∑' n, f n, tsum_nonneg hf₁⟩ : ℝ≥0) = (∑' n, ⟨f n, hf₁ n⟩ : ℝ≥0) :=
NNReal.eq <| Eq.symm <| coe_tsum (f := fun x => ⟨f x, hf₁ x⟩)
nonrec theorem tsum_mul_left (a : ℝ≥0) (f : α → ℝ≥0) : ∑' x, a * f x = a * ∑' x, f x :=
NNReal.eq <| by simp only [coe_tsum, NNReal.coe_mul, tsum_mul_left]
nonrec theorem tsum_mul_right (f : α → ℝ≥0) (a : ℝ≥0) : ∑' x, f x * a = (∑' x, f x) * a :=
NNReal.eq <| by simp only [coe_tsum, NNReal.coe_mul, tsum_mul_right]
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
coe_tsum_of_nonneg
| null |
summable_comp_injective {β : Type*} {f : α → ℝ≥0} (hf : Summable f) {i : β → α}
(hi : Function.Injective i) : Summable (f ∘ i) := by
rw [← summable_coe] at hf ⊢
exact hf.comp_injective hi
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
summable_comp_injective
| null |
summable_nat_add (f : ℕ → ℝ≥0) (hf : Summable f) (k : ℕ) : Summable fun i => f (i + k) :=
summable_comp_injective hf <| add_left_injective k
nonrec theorem summable_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) :
(Summable fun i => f (i + k)) ↔ Summable f := by
rw [← summable_coe, ← summable_coe]
exact @summable_nat_add_iff ℝ _ _ _ (fun i => (f i : ℝ)) k
nonrec theorem hasSum_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) {a : ℝ≥0} :
HasSum (fun n => f (n + k)) a ↔ HasSum f (a + ∑ i ∈ range k, f i) := by
rw [← hasSum_coe, hasSum_nat_add_iff (f := fun n => toReal (f n)) k]; norm_cast
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
summable_nat_add
| null |
sum_add_tsum_nat_add {f : ℕ → ℝ≥0} (k : ℕ) (hf : Summable f) :
∑' i, f i = (∑ i ∈ range k, f i) + ∑' i, f (i + k) :=
(((summable_nat_add_iff k).2 hf).sum_add_tsum_nat_add').symm
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
sum_add_tsum_nat_add
| null |
iInf_real_pos_eq_iInf_nnreal_pos [CompleteLattice α] {f : ℝ → α} :
⨅ (n : ℝ) (_ : 0 < n), f n = ⨅ (n : ℝ≥0) (_ : 0 < n), f n :=
le_antisymm (iInf_mono' fun r => ⟨r, le_rfl⟩) (iInf₂_mono' fun r hr => ⟨⟨r, hr.le⟩, hr, le_rfl⟩)
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iInf_real_pos_eq_iInf_nnreal_pos
| null |
tendsto_cofinite_zero_of_summable {α} {f : α → ℝ≥0} (hf : Summable f) :
Tendsto f cofinite (𝓝 0) := by
simp only [← summable_coe, ← tendsto_coe] at hf ⊢
exact hf.tendsto_cofinite_zero
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_cofinite_zero_of_summable
| null |
tendsto_atTop_zero_of_summable {f : ℕ → ℝ≥0} (hf : Summable f) : Tendsto f atTop (𝓝 0) := by
rw [← Nat.cofinite_eq_atTop]
exact tendsto_cofinite_zero_of_summable hf
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_atTop_zero_of_summable
| null |
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