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 ⌀ |
|---|---|---|---|---|---|---|
summable_sigma_of_nonneg {α} {β : α → Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ := by
lift f to (Σ x, β x) → ℝ≥0 using hf
simpa using mod_cast NNReal.summable_sigma | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_sigma_of_nonneg | null |
summable_partition {α β : Type*} {f : β → ℝ} (hf : 0 ≤ f) {s : α → Set β}
(hs : ∀ i, ∃! j, i ∈ s j) : Summable f ↔
(∀ j, Summable fun i : s j ↦ f i) ∧ Summable fun j ↦ ∑' i : s j, f i := by
simpa only [← (Set.sigmaEquiv s hs).summable_iff] using summable_sigma_of_nonneg (fun _ ↦ hf _) | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_partition | null |
summable_prod_of_nonneg {α β} {f : (α × β) → ℝ} (hf : 0 ≤ f) :
Summable f ↔ (∀ x, Summable fun y ↦ f (x, y)) ∧ Summable fun x ↦ ∑' y, f (x, y) :=
(Equiv.sigmaEquivProd _ _).summable_iff.symm.trans <| summable_sigma_of_nonneg fun _ ↦ hf _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_prod_of_nonneg | null |
summable_of_sum_le {ι : Type*} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f)
(h : ∀ u : Finset ι, ∑ x ∈ u, f x ≤ c) : Summable f :=
⟨⨆ u : Finset ι, ∑ x ∈ u, f x,
tendsto_atTop_ciSup (Finset.sum_mono_set_of_nonneg hf) ⟨c, fun _ ⟨u, hu⟩ => hu ▸ h u⟩⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_of_sum_le | null |
summable_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
(h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : Summable f := by
refine (summable_iff_not_tendsto_nat_atTop_of_nonneg hf).2 fun H => ?_
rcases exists_lt_of_tendsto_atTop H 0 c with ⟨n, -, hn⟩
exact lt_irrefl _ (hn.trans_le (h n)) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_of_sum_range_le | null |
Real.tsum_le_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
(h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c :=
(summable_of_sum_range_le hf h).tsum_le_of_sum_range_le h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | Real.tsum_le_of_sum_range_le | null |
protected Summable.tsum_lt_tsum_of_nonneg {i : ℕ} {f g : ℕ → ℝ} (h0 : ∀ b : ℕ, 0 ≤ f b)
(h : ∀ b : ℕ, f b ≤ g b) (hi : f i < g i) (hg : Summable g) : ∑' n, f n < ∑' n, g n :=
Summable.tsum_lt_tsum h hi (.of_nonneg_of_le h0 h hg) hg
@[deprecated (since := "2025-04-12")] alias tsum_lt_tsum_of_nonneg :=
Summable.tsum_lt_tsum_of_nonneg | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | Summable.tsum_lt_tsum_of_nonneg | If a sequence `f` with non-negative terms is dominated by a sequence `g` with summable
series and at least one term of `f` is strictly smaller than the corresponding term in `g`,
then the series of `f` is strictly smaller than the series of `g`. |
HasSum.mul_left (a₂) (h : HasSum f a₁) : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) := by
simpa only using h.map (AddMonoidHom.mulLeft a₂) (continuous_const.mul continuous_id) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.mul_left | null |
HasSum.mul_right (a₂) (hf : HasSum f a₁) : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) := by
simpa only using hf.map (AddMonoidHom.mulRight a₂) (continuous_id.mul continuous_const) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.mul_right | null |
Summable.mul_left (a) (hf : Summable f) : Summable fun i ↦ a * f i :=
(hf.hasSum.mul_left _).summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.mul_left | null |
Summable.mul_right (a) (hf : Summable f) : Summable fun i ↦ f i * a :=
(hf.hasSum.mul_right _).summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.mul_right | null |
protected Summable.tsum_mul_left (a) (hf : Summable f) : ∑' i, a * f i = a * ∑' i, f i :=
(hf.hasSum.mul_left _).tsum_eq | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.tsum_mul_left | null |
protected Summable.tsum_mul_right (a) (hf : Summable f) : ∑' i, f i * a = (∑' i, f i) * a :=
(hf.hasSum.mul_right _).tsum_eq | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.tsum_mul_right | null |
Commute.tsum_right (a) (h : ∀ i, Commute a (f i)) : Commute a (∑' i, f i) := by
classical
by_cases hf : Summable f
· exact (hf.tsum_mul_left a).symm.trans ((congr_arg _ <| funext h).trans (hf.tsum_mul_right a))
· exact (tsum_eq_zero_of_not_summable hf).symm ▸ Commute.zero_right _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Commute.tsum_right | null |
Commute.tsum_left (a) (h : ∀ i, Commute (f i) a) : Commute (∑' i, f i) a :=
(Commute.tsum_right _ fun i ↦ (h i).symm).symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Commute.tsum_left | null |
HasSum.div_const (h : HasSum f a) (b : α) : HasSum (fun i ↦ f i / b) (a / b) := by
simp only [div_eq_mul_inv, h.mul_right b⁻¹] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.div_const | null |
Summable.div_const (h : Summable f) (b : α) : Summable fun i ↦ f i / b :=
(h.hasSum.div_const _).summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.div_const | null |
hasSum_mul_left_iff (h : a₂ ≠ 0) : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) ↔ HasSum f a₁ :=
⟨fun H ↦ by simpa only [inv_mul_cancel_left₀ h] using H.mul_left a₂⁻¹, HasSum.mul_left _⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | hasSum_mul_left_iff | null |
hasSum_mul_right_iff (h : a₂ ≠ 0) : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) ↔ HasSum f a₁ :=
⟨fun H ↦ by simpa only [mul_inv_cancel_right₀ h] using H.mul_right a₂⁻¹, HasSum.mul_right _⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | hasSum_mul_right_iff | null |
hasSum_div_const_iff (h : a₂ ≠ 0) : HasSum (fun i ↦ f i / a₂) (a₁ / a₂) ↔ HasSum f a₁ := by
simpa only [div_eq_mul_inv] using hasSum_mul_right_iff (inv_ne_zero h) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | hasSum_div_const_iff | null |
summable_mul_left_iff (h : a ≠ 0) : (Summable fun i ↦ a * f i) ↔ Summable f :=
⟨fun H ↦ by simpa only [inv_mul_cancel_left₀ h] using H.mul_left a⁻¹, fun H ↦ H.mul_left _⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_mul_left_iff | null |
summable_mul_right_iff (h : a ≠ 0) : (Summable fun i ↦ f i * a) ↔ Summable f :=
⟨fun H ↦ by simpa only [mul_inv_cancel_right₀ h] using H.mul_right a⁻¹, fun H ↦ H.mul_right _⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_mul_right_iff | null |
summable_div_const_iff (h : a ≠ 0) : (Summable fun i ↦ f i / a) ↔ Summable f := by
simpa only [div_eq_mul_inv] using summable_mul_right_iff (inv_ne_zero h) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_div_const_iff | null |
tsum_mul_left [T2Space α] : ∑' x, a * f x = a * ∑' x, f x := by
classical
exact if hf : Summable f then hf.tsum_mul_left a
else if ha : a = 0 then by simp [ha]
else by rw [tsum_eq_zero_of_not_summable hf,
tsum_eq_zero_of_not_summable (mt (summable_mul_left_iff ha).mp hf), mul_zero] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | tsum_mul_left | null |
tsum_mul_right [T2Space α] : ∑' x, f x * a = (∑' x, f x) * a := by
classical
exact if hf : Summable f then hf.tsum_mul_right a
else if ha : a = 0 then by simp [ha]
else by rw [tsum_eq_zero_of_not_summable hf,
tsum_eq_zero_of_not_summable (mt (summable_mul_right_iff ha).mp hf), zero_mul] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | tsum_mul_right | null |
tsum_div_const [T2Space α] : ∑' x, f x / a = (∑' x, f x) / a := by
simpa only [div_eq_mul_inv] using tsum_mul_right | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | tsum_div_const | null |
HasSum.const_div (h : HasSum (fun x ↦ 1 / f x) a) (b : α) :
HasSum (fun i ↦ b / f i) (b * a) := by
have := h.mul_left b
simpa only [div_eq_mul_inv, one_mul] using this | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.const_div | null |
Summable.const_div (h : Summable (fun x ↦ 1 / f x)) (b : α) :
Summable fun i ↦ b / f i :=
(h.hasSum.const_div b).summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.const_div | null |
hasSum_const_div_iff (h : a₂ ≠ 0) :
HasSum (fun i ↦ a₂ / f i) (a₂ * a₁) ↔ HasSum (1/ f) a₁ := by
simpa only [div_eq_mul_inv, one_mul] using hasSum_mul_left_iff h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | hasSum_const_div_iff | null |
summable_const_div_iff (h : a ≠ 0) : (Summable fun i ↦ a / f i) ↔ Summable (1 / f) := by
simpa only [div_eq_mul_inv, one_mul] using summable_mul_left_iff h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_const_div_iff | null |
HasSum.mul_eq (hf : HasSum f s) (hg : HasSum g t)
(hfg : HasSum (fun x : ι × κ ↦ f x.1 * g x.2) u) : s * t = u :=
have key₁ : HasSum (fun i ↦ f i * t) (s * t) := hf.mul_right t
have this : ∀ i : ι, HasSum (fun c : κ ↦ f i * g c) (f i * t) := fun i ↦ hg.mul_left (f i)
have key₂ : HasSum (fun i ↦ f i * t) u := HasSum.prod_fiberwise hfg this
key₁.unique key₂ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.mul_eq | null |
HasSum.mul (hf : HasSum f s) (hg : HasSum g t)
(hfg : Summable fun x : ι × κ ↦ f x.1 * g x.2) :
HasSum (fun x : ι × κ ↦ f x.1 * g x.2) (s * t) :=
let ⟨_u, hu⟩ := hfg
(hf.mul_eq hg hu).symm ▸ hu | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | HasSum.mul | null |
protected Summable.tsum_mul_tsum (hf : Summable f) (hg : Summable g)
(hfg : Summable fun x : ι × κ ↦ f x.1 * g x.2) :
((∑' x, f x) * ∑' y, g y) = ∑' z : ι × κ, f z.1 * g z.2 :=
hf.hasSum.mul_eq hg.hasSum hfg.hasSum
@[deprecated (since := "2025-04-12")] alias tsum_mul_tsum := Summable.tsum_mul_tsum | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.tsum_mul_tsum | Product of two infinites sums indexed by arbitrary types.
See also `tsum_mul_tsum_of_summable_norm` if `f` and `g` are absolutely summable. |
summable_mul_prod_iff_summable_mul_sigma_antidiagonal :
(Summable fun x : A × A ↦ f x.1 * g x.2) ↔
Summable fun x : Σ n : A, antidiagonal n ↦ f (x.2 : A × A).1 * g (x.2 : A × A).2 :=
Finset.sigmaAntidiagonalEquivProd.summable_iff.symm
variable [T3Space α] [IsTopologicalSemiring α] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_mul_prod_iff_summable_mul_sigma_antidiagonal | The family `(k, l) : ℕ × ℕ ↦ f k * g l` is summable if and only if the family
`(n, k, l) : Σ (n : ℕ), antidiagonal n ↦ f k * g l` is summable. |
summable_sum_mul_antidiagonal_of_summable_mul
(h : Summable fun x : A × A ↦ f x.1 * g x.2) :
Summable fun n ↦ ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 := by
rw [summable_mul_prod_iff_summable_mul_sigma_antidiagonal] at h
conv => congr; ext; rw [← Finset.sum_finset_coe, ← tsum_fintype]
exact h.sigma' fun n ↦ (hasSum_fintype _).summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_sum_mul_antidiagonal_of_summable_mul | null |
protected Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal (hf : Summable f)
(hg : Summable g) (hfg : Summable fun x : A × A ↦ f x.1 * g x.2) :
((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 := by
conv_rhs => congr; ext; rw [← Finset.sum_finset_coe, ← tsum_fintype]
rw [hf.tsum_mul_tsum hg hfg, ← sigmaAntidiagonalEquivProd.tsum_eq (_ : A × A → α)]
exact (summable_mul_prod_iff_summable_mul_sigma_antidiagonal.mp hfg).tsum_sigma'
(fun n ↦ (hasSum_fintype _).summable)
@[deprecated (since := "2025-04-12")] alias tsum_mul_tsum_eq_tsum_sum_antidiagonal :=
Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal | The **Cauchy product formula** for the product of two infinites sums indexed by `ℕ`, expressed
by summing on `Finset.antidiagonal`.
See also `tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm` if `f` and `g` are absolutely
summable. |
summable_sum_mul_range_of_summable_mul (h : Summable fun x : ℕ × ℕ ↦ f x.1 * g x.2) :
Summable fun n ↦ ∑ k ∈ range (n + 1), f k * g (n - k) := by
simp_rw [← Nat.sum_antidiagonal_eq_sum_range_succ fun k l ↦ f k * g l]
exact summable_sum_mul_antidiagonal_of_summable_mul h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | summable_sum_mul_range_of_summable_mul | null |
protected Summable.tsum_mul_tsum_eq_tsum_sum_range (hf : Summable f) (hg : Summable g)
(hfg : Summable fun x : ℕ × ℕ ↦ f x.1 * g x.2) :
((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ k ∈ range (n + 1), f k * g (n - k) := by
simp_rw [← Nat.sum_antidiagonal_eq_sum_range_succ fun k l ↦ f k * g l]
exact hf.tsum_mul_tsum_eq_tsum_sum_antidiagonal hg hfg
@[deprecated (since := "2025-04-12")] alias tsum_mul_tsum_eq_tsum_sum_range :=
Summable.tsum_mul_tsum_eq_tsum_sum_range | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | Summable.tsum_mul_tsum_eq_tsum_sum_range | The **Cauchy product formula** for the product of two infinites sums indexed by `ℕ`, expressed
by summing on `Finset.range`.
See also `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm` if `f` and `g` are absolutely summable. |
hasProd_one_add_of_hasSum_prod {a : α} (h : HasSum (∏ i ∈ ·, f i) a) :
HasProd (1 + f ·) a := by
simp_rw [HasProd, prod_one_add]
exact h.comp tendsto_finset_powerset_atTop_atTop | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | hasProd_one_add_of_hasSum_prod | null |
multipliable_one_add_of_summable_prod (h : Summable (∏ i ∈ ·, f i)) :
Multipliable (1 + f ·) := by
obtain ⟨a, h⟩ := h
exact ⟨a, hasProd_one_add_of_hasSum_prod h⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | multipliable_one_add_of_summable_prod | `∏' i : ι, (1 + f i)` is convergent if `∑' s : Finset ι, ∏ i ∈ s, f i` is convergent.
For complete normed ring, see also `multipliable_one_add_of_summable`. |
tprod_one_add [T2Space α] (h : Summable (∏ i ∈ ·, f i)) :
∏' i, (1 + f i) = ∑' s, ∏ i ∈ s, f i :=
HasProd.tprod_eq <| hasProd_one_add_of_hasSum_prod h.hasSum | theorem | Topology | [
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | tprod_one_add | null |
HasSumUniformlyOn.of_norm_le_summable {f : α → β → F} (hu : Summable u) {s : Set β}
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) : HasSumUniformlyOn f (fun x ↦ ∑' n, f n x) {s} := by
simp [hasSumUniformlyOn_iff_tendstoUniformlyOn, tendstoUniformlyOn_tsum hu hfu] | theorem | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | HasSumUniformlyOn.of_norm_le_summable | null |
HasSumUniformlyOn.of_norm_le_summable_eventually {ι : Type*} {f : ι → β → F} {u : ι → ℝ}
(hu : Summable u) {s : Set β} (hfu : ∀ᶠ n in cofinite, ∀ x ∈ s, ‖f n x‖ ≤ u n) :
HasSumUniformlyOn f (fun x ↦ ∑' n, f n x) {s} := by
simp [hasSumUniformlyOn_iff_tendstoUniformlyOn,
tendstoUniformlyOn_tsum_of_cofinite_eventually hu hfu] | theorem | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | HasSumUniformlyOn.of_norm_le_summable_eventually | null |
SummableLocallyUniformlyOn.of_locally_bounded_eventually [TopologicalSpace β]
[LocallyCompactSpace β] {f : α → β → F} {s : Set β} (hs : IsOpen s)
(hu : ∀ K ⊆ s, IsCompact K → ∃ u : α → ℝ, Summable u ∧
∀ᶠ n in cofinite, ∀ k ∈ K, ‖f n k‖ ≤ u n) : SummableLocallyUniformlyOn f s := by
apply HasSumLocallyUniformlyOn.summableLocallyUniformlyOn (g := fun x ↦ ∑' n, f n x)
rw [hasSumLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn,
tendstoLocallyUniformlyOn_iff_forall_isCompact hs]
intro K hK hKc
obtain ⟨u, hu1, hu2⟩ := hu K hK hKc
exact tendstoUniformlyOn_tsum_of_cofinite_eventually hu1 hu2 | lemma | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | SummableLocallyUniformlyOn.of_locally_bounded_eventually | null |
SummableLocallyUniformlyOn_of_locally_bounded [TopologicalSpace β] [LocallyCompactSpace β]
{f : α → β → F} {s : Set β} (hs : IsOpen s)
(hu : ∀ K ⊆ s, IsCompact K → ∃ u : α → ℝ, Summable u ∧ ∀ n, ∀ k ∈ K, ‖f n k‖ ≤ u n) :
SummableLocallyUniformlyOn f s := by
apply SummableLocallyUniformlyOn.of_locally_bounded_eventually hs
intro K hK hKc
obtain ⟨u, hu1, hu2⟩ := hu K hK hKc
exact ⟨u, hu1, by filter_upwards using hu2⟩ | lemma | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | SummableLocallyUniformlyOn_of_locally_bounded | null |
derivWithin_tsum {f : ι → E → F} (hs : IsOpen s) {x : E} (hx : x ∈ s)
(hf : ∀ y ∈ s, Summable fun n ↦ f n y)
(h : SummableLocallyUniformlyOn (fun n ↦ (derivWithin (fun z ↦ f n z) s)) s)
(hf2 : ∀ n r, r ∈ s → DifferentiableAt E (f n) r) :
derivWithin (fun z ↦ ∑' n, f n z) s x = ∑' n, derivWithin (f n) s x := by
apply HasDerivWithinAt.derivWithin ?_ (hs.uniqueDiffWithinAt hx)
apply HasDerivAt.hasDerivWithinAt
apply hasDerivAt_of_tendstoLocallyUniformlyOn hs _ _ (fun y hy ↦ (hf y hy).hasSum) hx
(f' := fun n : Finset ι ↦ fun a ↦ ∑ i ∈ n, derivWithin (fun z ↦ f i z) s a)
· obtain ⟨g, hg⟩ := h
apply (hasSumLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn.mp hg).congr_right
exact fun _ hb ↦ (hg.tsum_eqOn hb).symm
· filter_upwards with t r hr using HasDerivAt.fun_sum
(fun q hq ↦ ((hf2 q r hr).differentiableWithinAt.hasDerivWithinAt.hasDerivAt)
(hs.mem_nhds hr)) | theorem | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | derivWithin_tsum | The `derivWithin` of a sum whose derivative is absolutely and uniformly convergent sum on an
open set `s` is the sum of the derivatives of sequence of functions on the open set `s` |
iteratedDerivWithin_tsum {f : ι → E → F} (m : ℕ) (hs : IsOpen s)
{x : E} (hx : x ∈ s) (hsum : ∀ t ∈ s, Summable (fun n : ι ↦ f n t))
(h : ∀ k, 1 ≤ k → k ≤ m → SummableLocallyUniformlyOn
(fun n ↦ (iteratedDerivWithin k (fun z ↦ f n z) s)) s)
(hf2 : ∀ n k r, k ≤ m → r ∈ s →
DifferentiableAt E (iteratedDerivWithin k (fun z ↦ f n z) s) r) :
iteratedDerivWithin m (fun z ↦ ∑' n, f n z) s x = ∑' n, iteratedDerivWithin m (f n) s x := by
induction m generalizing x with
| zero => simp
| succ m hm =>
simp_rw [iteratedDerivWithin_succ]
rw [← derivWithin_tsum hs hx _ _ (fun n r hr ↦ hf2 n m r (by cutsat) hr)]
· exact derivWithin_congr (fun t ht ↦ hm ht (fun k hk1 hkm ↦ h k hk1 (by cutsat))
(fun k r e hr he ↦ hf2 k r e (by cutsat) he)) (hm hx (fun k hk1 hkm ↦ h k hk1 (by cutsat))
(fun k r e hr he ↦ hf2 k r e (by cutsat) he))
· intro r hr
by_cases hm2 : m = 0
· simp [hm2, hsum r hr]
· exact ((h m (by cutsat) (by cutsat)).summable hr).congr (fun _ ↦ by simp)
· exact SummableLocallyUniformlyOn_congr
(fun _ _ ht ↦ iteratedDerivWithin_succ) (h (m + 1) (by cutsat) (by cutsat)) | theorem | Topology | [
"Mathlib.Analysis.Calculus.IteratedDeriv.Defs",
"Mathlib.Analysis.Calculus.UniformLimitsDeriv",
"Mathlib.Analysis.Normed.Group.FunctionSeries",
"Mathlib.Topology.Algebra.InfiniteSum.UniformOn"
] | Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.lean | iteratedDerivWithin_tsum | If a sequence of functions `fₙ` is such that `∑ fₙ (z)` is summable for each `z` in an
open set `s`, and for each `1 ≤ k ≤ m`, the series of `k`-th iterated derivatives
`∑ (iteratedDerivWithin k fₙ s) (z)`
is summable locally uniformly on `s`, and each `fₙ` is `m`-times differentiable, then the `m`-th
iterated derivative of the sum is the sum of the `m`-th iterated derivatives. |
@[to_additive /-- `HasSumUniformlyOn f g 𝔖` means that the (potentially infinite) sum `∑' i, f i b`
for `b : β` converges uniformly on each `s ∈ 𝔖` to `g`. -/]
HasProdUniformlyOn : Prop :=
HasProd (fun i ↦ UniformOnFun.ofFun 𝔖 (f i)) (UniformOnFun.ofFun 𝔖 g)
variable (f g 𝔖) in | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn | `HasProdUniformlyOn f g 𝔖` means that the (potentially infinite) product `∏' i, f i b`
for `b : β` converges uniformly on each `s ∈ 𝔖` to `g`. |
@[to_additive /-- `SummableUniformlyOn f s` means that there is some infinite sum to
which `f` converges uniformly on every `s ∈ 𝔖`. Use fun x ↦ ∑' i, f i x to get the sum function. -/]
MultipliableUniformlyOn : Prop :=
Multipliable (fun i ↦ UniformOnFun.ofFun 𝔖 (f i))
@[to_additive] | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableUniformlyOn | `MultipliableUniformlyOn f 𝔖` means that there is some infinite product to which
`f` converges uniformly on every `s ∈ 𝔖`. Use `fun x ↦ ∏' i, f i x` to get the product function. |
MultipliableUniformlyOn.exists (h : MultipliableUniformlyOn f 𝔖) :
∃ g, HasProdUniformlyOn f g 𝔖 :=
h
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableUniformlyOn.exists | null |
HasProdUniformlyOn.multipliableUniformlyOn (h : HasProdUniformlyOn f g 𝔖) :
MultipliableUniformlyOn f 𝔖 :=
⟨g, h⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.multipliableUniformlyOn | null |
hasProdUniformlyOn_iff_tendstoUniformlyOn : HasProdUniformlyOn f g 𝔖 ↔
∀ s ∈ 𝔖, TendstoUniformlyOn (fun I b ↦ ∏ i ∈ I, f i b) g atTop s := by
simpa [HasProdUniformlyOn, HasProd, ← UniformOnFun.ofFun_prod, Finset.prod_fn] using
UniformOnFun.tendsto_iff_tendstoUniformlyOn
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | hasProdUniformlyOn_iff_tendstoUniformlyOn | null |
HasProdUniformlyOn.congr {f' : ι → β → α}
(h : HasProdUniformlyOn f g 𝔖)
(hff' : ∀ s ∈ 𝔖, ∀ᶠ (n : Finset ι) in atTop,
Set.EqOn (fun b ↦ ∏ i ∈ n, f i b) (fun b ↦ ∏ i ∈ n, f' i b) s) :
HasProdUniformlyOn f' g 𝔖 := by
rw [hasProdUniformlyOn_iff_tendstoUniformlyOn] at *
exact fun s hs ↦ TendstoUniformlyOn.congr (h s hs) (hff' s hs)
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.congr | null |
HasProdUniformlyOn.congr_right {g' : β → α}
(h : HasProdUniformlyOn f g 𝔖) (hgg' : ∀ s ∈ 𝔖, Set.EqOn g g' s) :
HasProdUniformlyOn f g' 𝔖 := by
rw [hasProdUniformlyOn_iff_tendstoUniformlyOn] at *
exact fun s hs ↦ TendstoUniformlyOn.congr_right (h s hs) (hgg' s hs)
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.congr_right | null |
HasProdUniformlyOn.tendstoUniformlyOn_finsetRange
{f : ℕ → β → α} (h : HasProdUniformlyOn f g 𝔖) (hs : s ∈ 𝔖) :
TendstoUniformlyOn (fun N b ↦ ∏ i ∈ Finset.range N, f i b) g atTop s := by
rw [hasProdUniformlyOn_iff_tendstoUniformlyOn] at h
exact fun v hv => Filter.tendsto_finset_range.eventually (h s hs v hv)
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.tendstoUniformlyOn_finsetRange | null |
HasProdUniformlyOn.hasProd (h : HasProdUniformlyOn f g 𝔖) (hs : s ∈ 𝔖) (hx : x ∈ s) :
HasProd (f · x) (g x) :=
(hasProdUniformlyOn_iff_tendstoUniformlyOn.mp h s hs).tendsto_at hx
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.hasProd | null |
HasProdUniformlyOn.tprod_eqOn [T2Space α] (h : HasProdUniformlyOn f g 𝔖) (hs : s ∈ 𝔖) :
s.EqOn (∏' b, f b ·) g :=
fun _ hx ↦ (h.hasProd hs hx).tprod_eq
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.tprod_eqOn | null |
HasProdUniformlyOn.tprod_eq [T2Space α] (h : HasProdUniformlyOn f g 𝔖)
(hs : ⋃₀ 𝔖 = Set.univ) : (∏' b, f b ·) = g := by
ext x
obtain ⟨s, hs, hx⟩ := by simpa [← hs] using Set.mem_univ x
exact h.tprod_eqOn hs hx
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.tprod_eq | null |
MultipliableUniformlyOn.multipliable (h : MultipliableUniformlyOn f 𝔖)
(hs : s ∈ 𝔖) (hx : x ∈ s) : Multipliable (f · x) :=
match h.exists with | ⟨_, hg⟩ => (hg.hasProd hs hx).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableUniformlyOn.multipliable | null |
MultipliableUniformlyOn.hasProdUniformlyOn [T2Space α] (h : MultipliableUniformlyOn f 𝔖) :
HasProdUniformlyOn f (∏' i, f i ·) 𝔖 := by
obtain ⟨g, hg⟩ := h.exists
simp only [hasProdUniformlyOn_iff_tendstoUniformlyOn]
intro s hs
exact (hasProdUniformlyOn_iff_tendstoUniformlyOn.mp hg s hs).congr_right (hg.tprod_eqOn hs).symm | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableUniformlyOn.hasProdUniformlyOn | null |
@[to_additive /-- `HasSumLocallyUniformlyOn f g s` means that the (potentially infinite) sum
`∑' i, f i b` for `b : β` converges locally uniformly on `s` to `g b` (in the sense of
`TendstoLocallyUniformlyOn`). -/]
HasProdLocallyUniformlyOn : Prop :=
TendstoLocallyUniformlyOn (fun I b ↦ ∏ i ∈ I, f i b) g atTop s
variable (f g s) in | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdLocallyUniformlyOn | `HasProdLocallyUniformlyOn f g s` means that the (potentially infinite) product `∏' i, f i b`
for `b : β` converges locally uniformly on `s` to `g b` (in the sense of
`TendstoLocallyUniformlyOn`). |
@[to_additive /-- `SummableLocallyUniformlyOn f s` means that `∑' i, f i b` converges locally
uniformly on `s` to something. -/]
MultipliableLocallyUniformlyOn : Prop := ∃ g, HasProdLocallyUniformlyOn f g s
@[to_additive] | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableLocallyUniformlyOn | `MultipliableLocallyUniformlyOn f s` means that the product `∏' i, f i b` converges locally
uniformly on `s` to something. |
hasProdLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn :
HasProdLocallyUniformlyOn f g s ↔
TendstoLocallyUniformlyOn (fun I b ↦ ∏ i ∈ I, f i b) g atTop s :=
Iff.rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | hasProdLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn | null |
@[to_additive /-- If every `x ∈ s` has a neighbourhood within `s` on which `b ↦ ∑' i, f i b`
converges uniformly to `g`, then the sum converges locally uniformly. Note that this is not a
tautology, and the converse is only true if the domain is locally compact. -/]
hasProdLocallyUniformlyOn_of_of_forall_exists_nhds
(h : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, HasProdUniformlyOn f g {t}) : HasProdLocallyUniformlyOn f g s :=
tendstoLocallyUniformlyOn_of_forall_exists_nhds <| by
simpa [hasProdUniformlyOn_iff_tendstoUniformlyOn] using h
@[deprecated (since := "2025-05-22")] alias hasProdLocallyUniformlyOn_of_of_forall_exists_nhd :=
hasProdLocallyUniformlyOn_of_of_forall_exists_nhds
@[deprecated (since := "2025-05-22")] alias hasSumLocallyUniformlyOn_of_of_forall_exists_nhd :=
hasSumLocallyUniformlyOn_of_of_forall_exists_nhds
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | hasProdLocallyUniformlyOn_of_of_forall_exists_nhds | If every `x ∈ s` has a neighbourhood within `s` on which `b ↦ ∏' i, f i b` converges uniformly
to `g`, then the product converges locally uniformly on `s` to `g`. Note that this is not a
tautology, and the converse is only true if the domain is locally compact. |
HasProdUniformlyOn.hasProdLocallyUniformlyOn (h : HasProdUniformlyOn f g {s}) :
HasProdLocallyUniformlyOn f g s := by
simp [HasProdLocallyUniformlyOn, hasProdUniformlyOn_iff_tendstoUniformlyOn] at *
exact TendstoUniformlyOn.tendstoLocallyUniformlyOn h
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdUniformlyOn.hasProdLocallyUniformlyOn | null |
hasProdLocallyUniformlyOn_of_forall_compact (hs : IsOpen s) [LocallyCompactSpace β]
(h : ∀ K ⊆ s, IsCompact K → HasProdUniformlyOn f g {K}) : HasProdLocallyUniformlyOn f g s := by
rw [HasProdLocallyUniformlyOn, tendstoLocallyUniformlyOn_iff_forall_isCompact hs]
simpa [hasProdUniformlyOn_iff_tendstoUniformlyOn] using h
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | hasProdLocallyUniformlyOn_of_forall_compact | null |
HasProdLocallyUniformlyOn.multipliableLocallyUniformlyOn
(h : HasProdLocallyUniformlyOn f g s) : MultipliableLocallyUniformlyOn f s :=
⟨g, h⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdLocallyUniformlyOn.multipliableLocallyUniformlyOn | null |
@[to_additive /-- If every `x ∈ s` has a neighbourhood within `s` on which `b ↦ ∑' i, f i b`
converges uniformly, then the sum converges locally uniformly. Note that this is not a tautology,
and the converse is only true if the domain is locally compact. -/]
multipliableLocallyUniformlyOn_of_of_forall_exists_nhds [T2Space α]
(h : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, MultipliableUniformlyOn f {t}) :
MultipliableLocallyUniformlyOn f s :=
(hasProdLocallyUniformlyOn_of_of_forall_exists_nhds <| fun x hx ↦ match h x hx with
| ⟨t, ht, htr⟩ => ⟨t, ht, htr.hasProdUniformlyOn⟩).multipliableLocallyUniformlyOn
@[deprecated (since := "2025-05-22")]
alias multipliableLocallyUniformlyOn_of_of_forall_exists_nhd :=
multipliableLocallyUniformlyOn_of_of_forall_exists_nhds
@[deprecated (since := "2025-05-22")]
alias summableLocallyUniformlyOn_of_of_forall_exists_nhd :=
summableLocallyUniformlyOn_of_of_forall_exists_nhds
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | multipliableLocallyUniformlyOn_of_of_forall_exists_nhds | If every `x ∈ s` has a neighbourhood within `s` on which `b ↦ ∏' i, f i b` converges uniformly,
then the product converges locally uniformly on `s`. Note that this is not a tautology, and the
converse is only true if the domain is locally compact. |
HasProdLocallyUniformlyOn.hasProd (h : HasProdLocallyUniformlyOn f g s) (hx : x ∈ s) :
HasProd (f · x) (g x) :=
h.tendsto_at hx
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdLocallyUniformlyOn.hasProd | null |
MultipliableLocallyUniformlyOn.multipliable
(h : MultipliableLocallyUniformlyOn f s) (hx : x ∈ s) : Multipliable (f · x) :=
match h with | ⟨_, hg⟩ => (hg.hasProd hx).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableLocallyUniformlyOn.multipliable | null |
MultipliableLocallyUniformlyOn.hasProdLocallyUniformlyOn [T2Space α]
(h : MultipliableLocallyUniformlyOn f s) :
HasProdLocallyUniformlyOn f (∏' i, f i ·) s :=
match h with | ⟨_, hg⟩ => hg.congr_right fun _ hb ↦ (hg.hasProd hb).tprod_eq.symm
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableLocallyUniformlyOn.hasProdLocallyUniformlyOn | null |
HasProdLocallyUniformlyOn.tprod_eqOn [T2Space α]
(h : HasProdLocallyUniformlyOn f g s) : Set.EqOn (∏' i, f i ·) g s :=
fun _ hx ↦ (h.hasProd hx).tprod_eq
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdLocallyUniformlyOn.tprod_eqOn | null |
MultipliableLocallyUniformlyOn_congr [T2Space α]
{f f' : ι → β → α} (h : ∀ i, s.EqOn (f i) (f' i))
(h2 : MultipliableLocallyUniformlyOn f s) : MultipliableLocallyUniformlyOn f' s := by
apply HasProdLocallyUniformlyOn.multipliableLocallyUniformlyOn
exact (h2.hasProdLocallyUniformlyOn).congr fun v ↦ eqOn_fun_finsetProd h v
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | MultipliableLocallyUniformlyOn_congr | null |
HasProdLocallyUniformlyOn.tendstoLocallyUniformlyOn_finsetRange
{f : ℕ → β → α} (h : HasProdLocallyUniformlyOn f g s) :
TendstoLocallyUniformlyOn (fun N b ↦ ∏ i ∈ Finset.range N, f i b) g atTop s := by
rw [hasProdLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn] at h
intro v hv r hr
obtain ⟨t, ht, htr⟩ := h v hv r hr
exact ⟨t, ht, Filter.tendsto_finset_range.eventually htr⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.UniformConvergence",
"Mathlib.Order.Filter.AtTopBot.Finset"
] | Mathlib/Topology/Algebra/InfiniteSum/UniformOn.lean | HasProdLocallyUniformlyOn.tendstoLocallyUniformlyOn_finsetRange | null |
@[to_additive]
Pi.instIsUniformGroup {ι : Type*} {G : ι → Type*} [∀ i, UniformSpace (G i)]
[∀ i, Group (G i)] [∀ i, IsUniformGroup (G i)] : IsUniformGroup (∀ i, G i) where
uniformContinuous_div := uniformContinuous_pi.mpr fun i ↦
(uniformContinuous_proj G i).comp uniformContinuous_fst |>.div <|
(uniformContinuous_proj G i).comp uniformContinuous_snd
@[to_additive] | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | Pi.instIsUniformGroup | null |
isUniformEmbedding_translate_mul (a : α) : IsUniformEmbedding fun x : α => x * a :=
{ comap_uniformity := by
nth_rw 1 [← uniformity_translate_mul a, comap_map]
rintro ⟨p₁, p₂⟩ ⟨q₁, q₂⟩
simp only [Prod.mk.injEq, mul_left_inj, imp_self]
injective := mul_left_injective a } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | isUniformEmbedding_translate_mul | null |
@[to_additive]
cauchy_iff_tendsto (𝓕 : Filter G) :
Cauchy 𝓕 ↔ NeBot 𝓕 ∧ Tendsto (fun p ↦ p.1 / p.2) (𝓕 ×ˢ 𝓕) (𝓝 1) := by
simp [Cauchy, uniformity_eq_comap_nhds_one_swapped, ← tendsto_iff_comap]
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | cauchy_iff_tendsto | null |
cauchy_iff_tendsto_swapped (𝓕 : Filter G) :
Cauchy 𝓕 ↔ NeBot 𝓕 ∧ Tendsto (fun p ↦ p.2 / p.1) (𝓕 ×ˢ 𝓕) (𝓝 1) := by
simp [Cauchy, uniformity_eq_comap_nhds_one, ← tendsto_iff_comap]
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | cauchy_iff_tendsto_swapped | null |
cauchy_map_iff_tendsto (𝓕 : Filter ι) (f : ι → G) :
Cauchy (map f 𝓕) ↔ NeBot 𝓕 ∧ Tendsto (fun p ↦ f p.1 / f p.2) (𝓕 ×ˢ 𝓕) (𝓝 1) := by
simp [cauchy_map_iff, uniformity_eq_comap_nhds_one_swapped, Function.comp_def]
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | cauchy_map_iff_tendsto | null |
cauchy_map_iff_tendsto_swapped (𝓕 : Filter ι) (f : ι → G) :
Cauchy (map f 𝓕) ↔ NeBot 𝓕 ∧ Tendsto (fun p ↦ f p.2 / f p.1) (𝓕 ×ˢ 𝓕) (𝓝 1) := by
simp [cauchy_map_iff, uniformity_eq_comap_nhds_one, Function.comp_def] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | cauchy_map_iff_tendsto_swapped | null |
@[to_additive]
IsUniformInducing.isUniformGroup {γ : Type*} [Group γ] [UniformSpace γ] [IsUniformGroup γ]
[UniformSpace β] {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ]
(f : F) (hf : IsUniformInducing f) :
IsUniformGroup β where
uniformContinuous_div := by
simp_rw [hf.uniformContinuous_iff, Function.comp_def, map_div]
exact uniformContinuous_div.comp (hf.uniformContinuous.prodMap hf.uniformContinuous)
@[deprecated (since := "2025-03-30")]
alias IsUniformInducing.uniformAddGroup := IsUniformInducing.isUniformAddGroup
@[to_additive existing, deprecated (since := "2025-03-30")]
alias IsUniformInducing.uniformGroup := IsUniformInducing.isUniformGroup
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | IsUniformInducing.isUniformGroup | null |
protected IsUniformGroup.comap {γ : Type*} [Group γ] {u : UniformSpace γ} [IsUniformGroup γ]
{F : Type*} [FunLike F β γ] [MonoidHomClass F β γ] (f : F) : @IsUniformGroup β (u.comap f) _ :=
letI : UniformSpace β := u.comap f; IsUniformInducing.isUniformGroup f ⟨rfl⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | IsUniformGroup.comap | null |
@[to_additive]
isUniformGroup (S : Subgroup α) : IsUniformGroup S := .comap S.subtype | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | isUniformGroup | null |
@[to_additive]
CauchySeq.mul {ι : Type*} [Preorder ι] {u v : ι → α} (hu : CauchySeq u)
(hv : CauchySeq v) : CauchySeq (u * v) :=
uniformContinuous_mul.comp_cauchySeq (hu.prodMk hv)
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | CauchySeq.mul | null |
CauchySeq.mul_const {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
CauchySeq fun n => u n * x :=
(uniformContinuous_id.mul uniformContinuous_const).comp_cauchySeq hu
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | CauchySeq.mul_const | null |
CauchySeq.const_mul {ι : Type*} [Preorder ι] {u : ι → α} {x : α} (hu : CauchySeq u) :
CauchySeq fun n => x * u n :=
(uniformContinuous_const.mul uniformContinuous_id).comp_cauchySeq hu
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | CauchySeq.const_mul | null |
CauchySeq.inv {ι : Type*} [Preorder ι] {u : ι → α} (h : CauchySeq u) :
CauchySeq u⁻¹ :=
uniformContinuous_inv.comp_cauchySeq h
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | CauchySeq.inv | null |
totallyBounded_iff_subset_finite_iUnion_nhds_one {s : Set α} :
TotallyBounded s ↔ ∀ U ∈ 𝓝 (1 : α), ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, y • U :=
(𝓝 (1 : α)).basis_sets.uniformity_of_nhds_one_inv_mul_swapped.totallyBounded_iff.trans <| by
simp [← preimage_smul_inv, preimage]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | totallyBounded_iff_subset_finite_iUnion_nhds_one | null |
totallyBounded_inv {s : Set α} (hs : TotallyBounded s) : TotallyBounded (s⁻¹) := by
convert TotallyBounded.image hs uniformContinuous_inv
aesop | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | totallyBounded_inv | null |
@[to_additive]
TendstoUniformlyOnFilter.mul (hf : TendstoUniformlyOnFilter f g l l')
(hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f * f') (g * g') l l' :=
fun u hu =>
((uniformContinuous_mul.comp_tendstoUniformlyOnFilter (hf.prodMk hf')) u hu).diag_of_prod_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformlyOnFilter.mul | null |
TendstoUniformlyOnFilter.div (hf : TendstoUniformlyOnFilter f g l l')
(hf' : TendstoUniformlyOnFilter f' g' l l') : TendstoUniformlyOnFilter (f / f') (g / g') l l' :=
fun u hu =>
((uniformContinuous_div.comp_tendstoUniformlyOnFilter (hf.prodMk hf')) u hu).diag_of_prod_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformlyOnFilter.div | null |
TendstoUniformlyOn.mul (hf : TendstoUniformlyOn f g l s)
(hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f * f') (g * g') l s := fun u hu =>
((uniformContinuous_mul.comp_tendstoUniformlyOn (hf.prodMk hf')) u hu).diag_of_prod
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformlyOn.mul | null |
TendstoUniformlyOn.div (hf : TendstoUniformlyOn f g l s)
(hf' : TendstoUniformlyOn f' g' l s) : TendstoUniformlyOn (f / f') (g / g') l s := fun u hu =>
((uniformContinuous_div.comp_tendstoUniformlyOn (hf.prodMk hf')) u hu).diag_of_prod
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformlyOn.div | null |
TendstoUniformly.mul (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
TendstoUniformly (f * f') (g * g') l := fun u hu =>
((uniformContinuous_mul.comp_tendstoUniformly (hf.prodMk hf')) u hu).diag_of_prod
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformly.mul | null |
TendstoUniformly.div (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :
TendstoUniformly (f / f') (g / g') l := fun u hu =>
((uniformContinuous_div.comp_tendstoUniformly (hf.prodMk hf')) u hu).diag_of_prod
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | TendstoUniformly.div | null |
UniformCauchySeqOn.mul (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s) :
UniformCauchySeqOn (f * f') l s := fun u hu => by
simpa using (uniformContinuous_mul.comp_uniformCauchySeqOn (hf.prod' hf')) u hu
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | UniformCauchySeqOn.mul | null |
UniformCauchySeqOn.div (hf : UniformCauchySeqOn f l s) (hf' : UniformCauchySeqOn f' l s) :
UniformCauchySeqOn (f / f') l s := fun u hu => by
simpa using (uniformContinuous_div.comp_uniformCauchySeqOn (hf.prod' hf')) u hu | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | UniformCauchySeqOn.div | null |
@[to_additive]
IsUniformGroup.of_compactSpace [UniformSpace β] [Group β] [ContinuousDiv β]
[CompactSpace β] :
IsUniformGroup β where
uniformContinuous_div := CompactSpace.uniformContinuous_of_continuous continuous_div' | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | IsUniformGroup.of_compactSpace | null |
@[to_additive (attr := deprecated IsUniformGroup.of_compactSpace (since := "2025-09-27"))]
topologicalGroup_is_uniform_of_compactSpace [CompactSpace G] : IsUniformGroup G :=
inferInstance
variable {G}
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | topologicalGroup_is_uniform_of_compactSpace | null |
Subgroup.isClosed_of_discrete [T2Space G] {H : Subgroup G} [DiscreteTopology H] :
IsClosed (H : Set G) := by
obtain ⟨V, V_in, VH⟩ : ∃ (V : Set G), V ∈ 𝓝 (1 : G) ∧ V ∩ (H : Set G) = {1} :=
nhds_inter_eq_singleton_of_mem_discrete H.one_mem
have : (fun p : G × G => p.2 / p.1) ⁻¹' V ∈ 𝓤 G := preimage_mem_comap V_in
apply isClosed_of_spaced_out this
intro h h_in h' h'_in
contrapose!
simp only [Set.mem_preimage]
rintro (hyp : h' / h ∈ V)
have : h' / h ∈ ({1} : Set G) := VH ▸ Set.mem_inter hyp (H.div_mem h'_in h_in)
exact (eq_of_div_eq_one this).symm
@[to_additive] | instance | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.HeineCantor",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology... | Mathlib/Topology/Algebra/IsUniformGroup/Basic.lean | Subgroup.isClosed_of_discrete | null |
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