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 ⌀ |
|---|---|---|---|---|---|---|
@[to_additive /-- If a function `f` vanishes outside of a finite set `s`, then it `HasSum`
`∑ b ∈ s, f b`. -/]
hasProd_prod_of_ne_finset_one (hf : ∀ b ∉ s, f b = 1) :
HasProd f (∏ b ∈ s, f b) :=
(hasProd_subtype_iff_of_mulSupport_subset <| mulSupport_subset_iff'.2 hf).1 <| s.hasProd f
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | hasProd_prod_of_ne_finset_one | `∏' i, f i` is the product of `f` if it exists and is unconditionally convergent,
or 1 otherwise. -/
@[to_additive /-- `∑' i, f i` is the sum of `f` if it exists and is unconditionally convergent,
or 0 otherwise. -/]
noncomputable irreducible_def tprod {β} (f : β → α) :=
if h : Multipliable f then
/- Note that the product might not be uniquely defined if the topology is not separated.
When the multiplicative support of `f` is finite, we make the most reasonable choice to use the
product over the multiplicative support. Otherwise, we choose arbitrarily an `a` satisfying
`HasProd f a`. -/
if (mulSupport f).Finite then finprod f
else h.choose
else 1
-- see Note [operator precedence of big operators]
@[inherit_doc tprod]
notation3 "∏' "(...)", "r:67:(scoped f => tprod f) => r
@[inherit_doc tsum]
notation3 "∑' "(...)", "r:67:(scoped f => tsum f) => r
variable {f : β → α} {a : α} {s : Finset β}
@[to_additive]
theorem HasProd.multipliable (h : HasProd f a) : Multipliable f :=
⟨a, h⟩
@[to_additive]
theorem tprod_eq_one_of_not_multipliable (h : ¬Multipliable f) : ∏' b, f b = 1 := by
simp [tprod_def, h]
@[to_additive]
theorem Function.Injective.hasProd_iff {g : γ → β} (hg : Injective g)
(hf : ∀ x, x ∉ Set.range g → f x = 1) : HasProd (f ∘ g) a ↔ HasProd f a := by
simp only [HasProd, Tendsto, comp_apply, hg.map_atTop_finset_prod_eq hf]
@[to_additive]
theorem hasProd_subtype_iff_of_mulSupport_subset {s : Set β} (hf : mulSupport f ⊆ s) :
HasProd (f ∘ (↑) : s → α) a ↔ HasProd f a :=
Subtype.coe_injective.hasProd_iff <| by simpa using mulSupport_subset_iff'.1 hf
@[to_additive]
theorem hasProd_fintype [Fintype β] (f : β → α) : HasProd f (∏ b, f b) :=
OrderTop.tendsto_atTop_nhds _
@[to_additive]
protected theorem Finset.hasProd (s : Finset β) (f : β → α) :
HasProd (f ∘ (↑) : (↑s : Set β) → α) (∏ b ∈ s, f b) := by
rw [← prod_attach]
exact hasProd_fintype _
/-- If a function `f` is `1` outside of a finite set `s`, then it `HasProd` `∏ b ∈ s, f b`. |
multipliable_of_ne_finset_one (hf : ∀ b ∉ s, f b = 1) : Multipliable f :=
(hasProd_prod_of_ne_finset_one hf).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | multipliable_of_ne_finset_one | null |
Multipliable.hasProd (ha : Multipliable f) : HasProd f (∏' b, f b) := by
simp only [tprod_def, ha, dite_true]
by_cases H : (mulSupport f).Finite
· simp [H, hasProd_prod_of_ne_finset_one, finprod_eq_prod]
· simpa [H] using ha.choose_spec
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | Multipliable.hasProd | null |
HasProd.unique {a₁ a₂ : α} [T2Space α] : HasProd f a₁ → HasProd f a₂ → a₁ = a₂ := by
classical exact tendsto_nhds_unique
variable [T2Space α]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | HasProd.unique | null |
HasProd.tprod_eq (ha : HasProd f a) : ∏' b, f b = a :=
(Multipliable.hasProd ⟨a, ha⟩).unique ha
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | HasProd.tprod_eq | null |
Multipliable.hasProd_iff (h : Multipliable f) : HasProd f a ↔ ∏' b, f b = a :=
Iff.intro HasProd.tprod_eq fun eq ↦ eq ▸ h.hasProd | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | Multipliable.hasProd_iff | null |
tsum_set_one : ∑' _ : s, (1 : ℝ≥0∞) = s.encard := by
obtain (hfin | hinf) := Set.finite_or_infinite s
· lift s to Finset α using hfin
simp [tsum_fintype]
· have : Infinite s := infinite_coe_iff.mpr hinf
rw [tsum_const_eq_top_of_ne_zero one_ne_zero, encard_eq_top hinf, ENat.toENNReal_top] | lemma | Topology | [
"Mathlib.Data.Real.ENatENNReal",
"Mathlib.Data.Set.Card",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/ENNReal.lean | tsum_set_one | null |
tsum_set_const (c : ℝ≥0∞) : ∑' _ : s, c = s.encard * c := by
simp [← tsum_set_one, ← ENNReal.tsum_mul_right]
@[simp] | lemma | Topology | [
"Mathlib.Data.Real.ENatENNReal",
"Mathlib.Data.Set.Card",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/ENNReal.lean | tsum_set_const | null |
tsum_one : ∑' _ : α, (1 : ℝ≥0∞) = ENat.card α := by
rw [← tsum_univ]; simpa [encard_univ] using tsum_set_one univ
@[simp] | lemma | Topology | [
"Mathlib.Data.Real.ENatENNReal",
"Mathlib.Data.Set.Card",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/ENNReal.lean | tsum_one | null |
tsum_const (c : ℝ≥0∞) : ∑' _ : α, c = ENat.card α * c := by
rw [← tsum_univ]; simpa [encard_univ] using tsum_set_const univ c | lemma | Topology | [
"Mathlib.Data.Real.ENatENNReal",
"Mathlib.Data.Set.Card",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/ENNReal.lean | tsum_const | null |
Multipliable.norm (hf : Multipliable f) : Multipliable (‖f ·‖) :=
let ⟨x, hx⟩ := hf; ⟨‖x‖, hx.norm⟩ | theorem | Topology | [
"Mathlib.Analysis.Normed.Group.Continuity",
"Mathlib.Analysis.Normed.Ring.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Field.lean | Multipliable.norm | null |
protected Multipliable.norm_tprod (hf : Multipliable f) : ‖∏' i, f i‖ = ∏' i, ‖f i‖ :=
hf.hasProd.norm.tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias norm_tprod := Multipliable.norm_tprod | theorem | Topology | [
"Mathlib.Analysis.Normed.Group.Continuity",
"Mathlib.Analysis.Normed.Ring.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Field.lean | Multipliable.norm_tprod | null |
@[to_additive]
HasProd.inv (h : HasProd f a) : HasProd (fun b ↦ (f b)⁻¹) a⁻¹ := by
simpa only using h.map (MonoidHom.id α)⁻¹ continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.inv | null |
Multipliable.inv (hf : Multipliable f) : Multipliable fun b ↦ (f b)⁻¹ :=
hf.hasProd.inv.multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.inv | null |
Multipliable.of_inv (hf : Multipliable fun b ↦ (f b)⁻¹) : Multipliable f := by
simpa only [inv_inv] using hf.inv
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.of_inv | null |
multipliable_inv_iff : (Multipliable fun b ↦ (f b)⁻¹) ↔ Multipliable f :=
⟨Multipliable.of_inv, Multipliable.inv⟩
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_inv_iff | null |
HasProd.div (hf : HasProd f a₁) (hg : HasProd g a₂) :
HasProd (fun b ↦ f b / g b) (a₁ / a₂) := by
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.div | null |
Multipliable.div (hf : Multipliable f) (hg : Multipliable g) :
Multipliable fun b ↦ f b / g b :=
(hf.hasProd.div hg.hasProd).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.div | null |
Multipliable.trans_div (hg : Multipliable g) (hfg : Multipliable fun b ↦ f b / g b) :
Multipliable f := by
simpa only [div_mul_cancel] using hfg.mul hg
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.trans_div | null |
multipliable_iff_of_multipliable_div (hfg : Multipliable fun b ↦ f b / g b) :
Multipliable f ↔ Multipliable g :=
⟨fun hf ↦ hf.trans_div <| by simpa only [inv_div] using hfg.inv, fun hg ↦ hg.trans_div hfg⟩
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_iff_of_multipliable_div | null |
HasProd.update (hf : HasProd f a₁) (b : β) [DecidableEq β] (a : α) :
HasProd (update f b a) (a / f b * a₁) := by
convert (hasProd_ite_eq b (a / f b)).mul hf with b'
by_cases h : b' = b
· rw [h, update_self]
simp
· simp only [h, update_of_ne, if_false, Ne, one_mul, not_false_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.update | null |
Multipliable.update (hf : Multipliable f) (b : β) [DecidableEq β] (a : α) :
Multipliable (update f b a) :=
(hf.hasProd.update b a).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.update | null |
HasProd.hasProd_compl_iff {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
HasProd (f ∘ (↑) : ↑sᶜ → α) a₂ ↔ HasProd f (a₁ * a₂) := by
refine ⟨fun h ↦ hf.mul_compl h, fun h ↦ ?_⟩
rw [hasProd_subtype_iff_mulIndicator] at hf ⊢
rw [Set.mulIndicator_compl]
simpa only [div_eq_mul_inv, mul_inv_cancel_comm] using h.div hf
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.hasProd_compl_iff | null |
HasProd.hasProd_iff_compl {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
HasProd f a₂ ↔ HasProd (f ∘ (↑) : ↑sᶜ → α) (a₂ / a₁) :=
Iff.symm <| hf.hasProd_compl_iff.trans <| by rw [mul_div_cancel]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.hasProd_iff_compl | null |
Multipliable.multipliable_compl_iff {s : Set β} (hf : Multipliable (f ∘ (↑) : s → α)) :
Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f where
mp := fun ⟨_, ha⟩ ↦ (hf.hasProd.hasProd_compl_iff.1 ha).multipliable
mpr := fun ⟨_, ha⟩ ↦ (hf.hasProd.hasProd_iff_compl.1 ha).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.multipliable_compl_iff | null |
protected Finset.hasProd_compl_iff (s : Finset β) :
HasProd (fun x : { x // x ∉ s } ↦ f x) a ↔ HasProd f (a * ∏ i ∈ s, f i) :=
(s.hasProd f).hasProd_compl_iff.trans <| by rw [mul_comm]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Finset.hasProd_compl_iff | null |
protected Finset.hasProd_iff_compl (s : Finset β) :
HasProd f a ↔ HasProd (fun x : { x // x ∉ s } ↦ f x) (a / ∏ i ∈ s, f i) :=
(s.hasProd f).hasProd_iff_compl
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Finset.hasProd_iff_compl | null |
protected Finset.multipliable_compl_iff (s : Finset β) :
(Multipliable fun x : { x // x ∉ s } ↦ f x) ↔ Multipliable f :=
(s.multipliable f).multipliable_compl_iff
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Finset.multipliable_compl_iff | null |
Set.Finite.multipliable_compl_iff {s : Set β} (hs : s.Finite) :
Multipliable (f ∘ (↑) : ↑sᶜ → α) ↔ Multipliable f :=
(hs.multipliable f).multipliable_compl_iff
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Set.Finite.multipliable_compl_iff | null |
hasProd_ite_div_hasProd [DecidableEq β] (hf : HasProd f a) (b : β) :
HasProd (fun n ↦ ite (n = b) 1 (f n)) (a / f b) := by
convert hf.update b 1 using 1
· ext n
rw [Function.update_apply]
· rw [div_mul_eq_mul_div, one_mul] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | hasProd_ite_div_hasProd | null |
@[to_additive /-- A more general version of `Summable.congr`, allowing the functions to
disagree on a finite set. -/]
Multipliable.congr_cofinite (hf : Multipliable f) (hfg : f =ᶠ[cofinite] g) :
Multipliable g :=
hfg.multipliable_compl_iff.mp <| (hfg.multipliable_compl_iff.mpr hf).congr (by simp) | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.congr_cofinite | A more general version of `Multipliable.congr`, allowing the functions to
disagree on a finite set.
Note that this requires the target to be a group, and hence fails for products valued
in a ring. See `Multipliable.congr_cofinite₀` for a version applying in this case,
with an additional non-vanishing hypothesis. |
@[to_additive /-- A more general version of `summable_congr`, allowing the functions to
disagree on a finite set. -/]
multipliable_congr_cofinite (hfg : f =ᶠ[cofinite] g) :
Multipliable f ↔ Multipliable g :=
⟨fun h ↦ h.congr_cofinite hfg, fun h ↦ h.congr_cofinite (hfg.mono fun _ h' ↦ h'.symm)⟩
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_congr_cofinite | A more general version of `multipliable_congr`, allowing the functions to
disagree on a finite set. |
Multipliable.congr_atTop {f₁ g₁ : ℕ → α} (hf : Multipliable f₁) (hfg : f₁ =ᶠ[atTop] g₁) :
Multipliable g₁ := hf.congr_cofinite (Nat.cofinite_eq_atTop ▸ hfg)
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.congr_atTop | null |
multipliable_congr_atTop {f₁ g₁ : ℕ → α} (hfg : f₁ =ᶠ[atTop] g₁) :
Multipliable f₁ ↔ Multipliable g₁ := multipliable_congr_cofinite (Nat.cofinite_eq_atTop ▸ hfg) | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_congr_atTop | null |
@[to_additive]
tprod_inv : ∏' b, (f b)⁻¹ = (∏' b, f b)⁻¹ := by
by_cases hf : Multipliable f
· exact hf.hasProd.inv.tprod_eq
· simp [tprod_eq_one_of_not_multipliable hf,
tprod_eq_one_of_not_multipliable (mt Multipliable.of_inv hf)]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | tprod_inv | null |
protected Multipliable.tprod_div (hf : Multipliable f) (hg : Multipliable g) :
∏' b, (f b / g b) = (∏' b, f b) / ∏' b, g b :=
(hf.hasProd.div hg.hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_sub := Summable.tsum_sub
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_div :=
Multipliable.tprod_div
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tprod_div | null |
protected Multipliable.prod_mul_tprod_compl {s : Finset β} (hf : Multipliable f) :
(∏ x ∈ s, f x) * ∏' x : ↑(s : Set β)ᶜ, f x = ∏' x, f x :=
((s.hasProd f).mul_compl (s.multipliable_compl_iff.2 hf).hasProd).tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias sum_add_tsum_compl := Summable.sum_add_tsum_compl
@[to_additive existing, deprecated (since := "2025-04-12")] alias prod_mul_tprod_compl :=
Multipliable.prod_mul_tprod_compl | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.prod_mul_tprod_compl | null |
@[to_additive /-- Let `f : β → α` be a summable function and let `b ∈ β` be an index.
Lemma `tsum_eq_add_tsum_ite` writes `Σ' n, f n` as `f b` plus the sum of the
remaining terms. -/]
protected Multipliable.tprod_eq_mul_tprod_ite [DecidableEq β] (hf : Multipliable f)
(b : β) : ∏' n, f n = f b * ∏' n, ite (n = b) 1 (f n) := by
rw [(hasProd_ite_div_hasProd hf.hasProd b).tprod_eq]
exact (mul_div_cancel _ _).symm
@[deprecated (since := "2025-04-12")] alias tsum_eq_add_tsum_ite := Summable.tsum_eq_add_tsum_ite
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_eq_mul_tprod_ite :=
Multipliable.tprod_eq_mul_tprod_ite | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tprod_eq_mul_tprod_ite | Let `f : β → α` be a multipliable function and let `b ∈ β` be an index.
Lemma `tprod_eq_mul_tprod_ite` writes `∏ n, f n` as `f b` times the product of the
remaining terms. |
@[to_additive /-- The **Cauchy criterion** for infinite sums, also known as the
**Cauchy convergence test** -/]
multipliable_iff_cauchySeq_finset [CompleteSpace α] {f : β → α} :
Multipliable f ↔ CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b := by
classical exact cauchy_map_iff_exists_tendsto.symm
variable [IsUniformGroup α] {f g : β → α}
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_iff_cauchySeq_finset | The **Cauchy criterion** for infinite products, also known as the **Cauchy convergence test** |
cauchySeq_finset_iff_prod_vanishing :
(CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e := by
classical
simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
rw [tendsto_atTop']
constructor
· intro h e he
obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he
use s₁ ∪ s₂
intro t ht
specialize h (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t) ⟨le_sup_left, le_sup_of_le_left le_sup_right⟩
simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h
· rintro h e he
rcases exists_nhds_split_inv he with ⟨d, hd, hde⟩
rcases h d hd with ⟨s, h⟩
use (s, s)
rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩
have : ((∏ b ∈ t₂, f b) / ∏ b ∈ t₁, f b) = (∏ b ∈ t₂ \ s, f b) / ∏ b ∈ t₁ \ s, f b := by
rw [← Finset.prod_sdiff ht₁, ← Finset.prod_sdiff ht₂, mul_div_mul_right_eq_div]
simp only [this]
exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | cauchySeq_finset_iff_prod_vanishing | null |
cauchySeq_finset_iff_tprod_vanishing :
(CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e := by
simp_rw [cauchySeq_finset_iff_prod_vanishing, Set.disjoint_left, disjoint_left]
refine ⟨fun vanish e he ↦ ?_, fun vanish e he ↦ ?_⟩
· obtain ⟨o, ho, o_closed, oe⟩ := exists_mem_nhds_isClosed_subset he
obtain ⟨s, hs⟩ := vanish o ho
refine ⟨s, fun t hts ↦ oe ?_⟩
by_cases ht : Multipliable fun a : t ↦ f a
· classical
refine o_closed.mem_of_tendsto ht.hasProd (Eventually.of_forall fun t' ↦ ?_)
rw [← prod_subtype_map_embedding fun _ _ ↦ by rfl]
apply hs
simp_rw [Finset.mem_map]
rintro _ ⟨b, -, rfl⟩
exact hts b.prop
· exact tprod_eq_one_of_not_multipliable ht ▸ mem_of_mem_nhds ho
· obtain ⟨s, hs⟩ := vanish _ he
exact ⟨s, fun t hts ↦ (t.tprod_subtype f).symm ▸ hs _ hts⟩
variable [CompleteSpace α]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | cauchySeq_finset_iff_tprod_vanishing | null |
multipliable_iff_vanishing :
Multipliable f ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e := by
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_prod_vanishing]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_iff_vanishing | null |
multipliable_iff_tprod_vanishing : Multipliable f ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t : Set β, Disjoint t s → (∏' b : t, f b) ∈ e := by
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_tprod_vanishing]
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_iff_tprod_vanishing | null |
Multipliable.multipliable_of_eq_one_or_self (hf : Multipliable f)
(h : ∀ b, g b = 1 ∨ g b = f b) : Multipliable g := by
classical
exact multipliable_iff_vanishing.2 fun e he ↦
let ⟨s, hs⟩ := multipliable_iff_vanishing.1 hf e he
⟨s, fun t ht ↦
have eq : ∏ b ∈ t with g b = f b, f b = ∏ b ∈ t, g b :=
calc
∏ b ∈ t with g b = f b, f b = ∏ b ∈ t with g b = f b, g b :=
Finset.prod_congr rfl fun b hb ↦ (Finset.mem_filter.1 hb).2.symm
_ = ∏ b ∈ t, g b := by
{refine Finset.prod_subset (Finset.filter_subset _ _) ?_
intro b hbt hb
simp only [Finset.mem_filter, and_iff_right hbt] at hb
exact (h b).resolve_right hb}
eq ▸ hs _ <| Finset.disjoint_of_subset_left (Finset.filter_subset _ _) ht⟩
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.multipliable_of_eq_one_or_self | null |
protected Multipliable.mulIndicator (hf : Multipliable f) (s : Set β) :
Multipliable (s.mulIndicator f) :=
hf.multipliable_of_eq_one_or_self <| Set.mulIndicator_eq_one_or_self _ _
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.mulIndicator | null |
Multipliable.comp_injective {i : γ → β} (hf : Multipliable f) (hi : Injective i) :
Multipliable (f ∘ i) := by
simpa only [Set.mulIndicator_range_comp] using
(hi.multipliable_iff (fun x hx ↦ Set.mulIndicator_of_notMem hx _)).2
(hf.mulIndicator (Set.range i))
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.comp_injective | null |
Multipliable.subtype (hf : Multipliable f) (s : Set β) : Multipliable (f ∘ (↑) : s → α) :=
hf.comp_injective Subtype.coe_injective
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.subtype | null |
multipliable_subtype_and_compl {s : Set β} :
((Multipliable fun x : s ↦ f x) ∧ Multipliable fun x : ↑sᶜ ↦ f x) ↔ Multipliable f :=
⟨and_imp.2 Multipliable.mul_compl, fun h ↦ ⟨h.subtype s, h.subtype sᶜ⟩⟩
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_subtype_and_compl | null |
protected Multipliable.tprod_subtype_mul_tprod_subtype_compl [T2Space α] {f : β → α}
(hf : Multipliable f) (s : Set β) : (∏' x : s, f x) * ∏' x : ↑sᶜ, f x = ∏' x, f x :=
((hf.subtype s).hasProd.mul_compl (hf.subtype { x | x ∉ s }).hasProd).unique hf.hasProd
@[deprecated (since := "2025-04-12")] alias tsum_subtype_add_tsum_subtype_compl :=
Summable.tsum_subtype_add_tsum_subtype_compl
@[to_additive existing, deprecated (since := "2025-04-12")] alias
tprod_subtype_mul_tprod_subtype_compl := Multipliable.tprod_subtype_mul_tprod_subtype_compl
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tprod_subtype_mul_tprod_subtype_compl | null |
protected Multipliable.prod_mul_tprod_subtype_compl [T2Space α] {f : β → α}
(hf : Multipliable f) (s : Finset β) :
(∏ x ∈ s, f x) * ∏' x : { x // x ∉ s }, f x = ∏' x, f x := by
rw [← hf.tprod_subtype_mul_tprod_subtype_compl s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl
@[deprecated (since := "2025-04-12")] alias sum_add_tsum_subtype_compl :=
Summable.sum_add_tsum_subtype_compl
@[to_additive existing, deprecated (since := "2025-04-12")] alias prod_mul_tprod_subtype_compl :=
Multipliable.prod_mul_tprod_subtype_compl | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.prod_mul_tprod_subtype_compl | null |
@[to_additive]
Multipliable.vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 (1 : G)) :
∃ s : Finset α, ∀ t, Disjoint t s → (∏ k ∈ t, f k) ∈ e := by
classical
letI : UniformSpace G := IsTopologicalGroup.toUniformSpace G
have : IsUniformGroup G := isUniformGroup_of_commGroup
exact cauchySeq_finset_iff_prod_vanishing.1 hf.hasProd.cauchySeq e he
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.vanishing | null |
Multipliable.tprod_vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 1) :
∃ s : Finset α, ∀ t : Set α, Disjoint t s → (∏' b : t, f b) ∈ e := by
classical
letI : UniformSpace G := IsTopologicalGroup.toUniformSpace G
have : IsUniformGroup G := isUniformGroup_of_commGroup
exact cauchySeq_finset_iff_tprod_vanishing.1 hf.hasProd.cauchySeq e he | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tprod_vanishing | null |
@[to_additive /-- The sum over the complement of a finset tends to `0` when the finset grows to
cover the whole space. This does not need a summability assumption, as otherwise all such sums are
zero. -/]
tendsto_tprod_compl_atTop_one (f : α → G) :
Tendsto (fun s : Finset α ↦ ∏' a : { x // x ∉ s }, f a) atTop (𝓝 1) := by
classical
by_cases H : Multipliable f
· intro e he
obtain ⟨s, hs⟩ := H.tprod_vanishing he
rw [Filter.mem_map, mem_atTop_sets]
exact ⟨s, fun t hts ↦ hs _ <| Set.disjoint_left.mpr fun a ha has ↦ ha (hts has)⟩
· refine tendsto_const_nhds.congr fun _ ↦ (tprod_eq_one_of_not_multipliable ?_).symm
rwa [Finset.multipliable_compl_iff] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | tendsto_tprod_compl_atTop_one | The product over the complement of a finset tends to `1` when the finset grows to cover the
whole space. This does not need a multipliability assumption, as otherwise all such products are
one. |
@[to_additive /-- Series divergence test: if `f` is unconditionally summable, then `f x` tends to
zero along `cofinite`. -/]
Multipliable.tendsto_cofinite_one (hf : Multipliable f) : Tendsto f cofinite (𝓝 1) := by
intro e he
rw [Filter.mem_map]
rcases hf.vanishing he with ⟨s, hs⟩
refine s.eventually_cofinite_notMem.mono fun x hx ↦ ?_
· simpa using hs {x} (disjoint_singleton_left.2 hx)
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tendsto_cofinite_one | Product divergence test: if `f` is unconditionally multipliable, then `f x` tends to one along
`cofinite`. |
Multipliable.finite_mulSupport_of_discreteTopology
{α : Type*} [CommGroup α] [TopologicalSpace α] [DiscreteTopology α]
{β : Type*} (f : β → α) (h : Multipliable f) : Set.Finite f.mulSupport :=
haveI : IsTopologicalGroup α := ⟨⟩
h.tendsto_cofinite_one (discreteTopology_iff_singleton_mem_nhds.mp ‹_› 1)
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.finite_mulSupport_of_discreteTopology | null |
Multipliable.countable_mulSupport [FirstCountableTopology G] [T1Space G]
(hf : Multipliable f) : f.mulSupport.Countable := by
simpa only [ker_nhds] using hf.tendsto_cofinite_one.countable_compl_preimage_ker
@[to_additive] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.countable_mulSupport | null |
multipliable_const_iff [Infinite β] [T2Space G] (a : G) :
Multipliable (fun _ : β ↦ a) ↔ a = 1 := by
refine ⟨fun h ↦ ?_, ?_⟩
· by_contra ha
have : {a}ᶜ ∈ 𝓝 1 := compl_singleton_mem_nhds (Ne.symm ha)
have : Finite β := by
simpa [← Set.finite_univ_iff] using h.tendsto_cofinite_one this
exact not_finite β
· rintro rfl
exact multipliable_one
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | multipliable_const_iff | null |
tprod_const [T2Space G] (a : G) : ∏' _ : β, a = a ^ (Nat.card β) := by
rcases finite_or_infinite β with hβ|hβ
· letI : Fintype β := Fintype.ofFinite β
rw [tprod_eq_prod (s := univ) (fun x hx ↦ (hx (mem_univ x)).elim)]
simp only [prod_const, Nat.card_eq_fintype_card, Fintype.card]
· simp only [Nat.card_eq_zero_of_infinite, pow_zero]
rcases eq_or_ne a 1 with rfl | ha
· simp
· apply tprod_eq_one_of_not_multipliable
simpa [multipliable_const_iff] using ha | theorem | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | tprod_const | null |
HasProd.congr_cofinite₀ {c : K} (hc : HasProd f c) {s : Finset α}
(hs : ∀ a ∈ s, f a ≠ 0) (hs' : ∀ a ∉ s, f a = g a) :
HasProd g (c * ((∏ i ∈ s, g i) / ∏ i ∈ s, f i)) := by
classical
refine (Tendsto.mul_const ((∏ i ∈ s, g i) / ∏ i ∈ s, f i) hc).congr' ?_
filter_upwards [eventually_ge_atTop s] with t ht
calc (∏ i ∈ t, f i) * ((∏ i ∈ s, g i) / ∏ i ∈ s, f i)
_ = ((∏ i ∈ s, f i) * ∏ i ∈ t \ s, g i) * _ := by
conv_lhs => rw [← union_sdiff_of_subset ht, prod_union disjoint_sdiff,
prod_congr rfl fun i hi ↦ hs' i (mem_sdiff.mp hi).2]
_ = (∏ i ∈ s, g i) * ∏ i ∈ t \ s, g i := by
rw [← mul_div_assoc, ← div_mul_eq_mul_div, ← div_mul_eq_mul_div, div_self, one_mul, mul_comm]
exact prod_ne_zero_iff.mpr hs
_ = ∏ i ∈ t, g i := by
rw [← prod_union disjoint_sdiff, union_sdiff_of_subset ht] | lemma | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | HasProd.congr_cofinite₀ | null |
protected Multipliable.tsum_congr_cofinite₀ [T2Space K] (hc : Multipliable f) {s : Finset α}
(hs : ∀ a ∈ s, f a ≠ 0) (hs' : ∀ a ∉ s, f a = g a) :
∏' i, g i = ((∏' i, f i) * ((∏ i ∈ s, g i) / ∏ i ∈ s, f i)) :=
(hc.hasProd.congr_cofinite₀ hs hs').tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_congr_cofinite := Multipliable.tsum_congr_cofinite₀ | lemma | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.tsum_congr_cofinite₀ | null |
Multipliable.congr_cofinite₀ (hf : Multipliable f) (hf' : ∀ a, f a ≠ 0)
(hfg : ∀ᶠ a in cofinite, f a = g a) :
Multipliable g := by
classical
obtain ⟨c, hc⟩ := hf
obtain ⟨s, hs⟩ : ∃ s : Finset α, ∀ i ∉ s, f i = g i := ⟨hfg.toFinset, by simp⟩
exact (hc.congr_cofinite₀ (fun a _ ↦ hf' a) hs).multipliable | lemma | Topology | [
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | Multipliable.congr_cofinite₀ | See also `Multipliable.congr_cofinite`, which does not have a non-vanishing condition, but instead
requires the target to be a group under multiplication (and hence fails for infinite products in a
ring). |
hasSum_iff_hasSum_compl (f : β → α) (a : α) :
HasSum (toCompl ∘ f) a ↔ HasSum f a := (isDenseInducing_toCompl α).hasSum_iff f a | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | hasSum_iff_hasSum_compl | A function `f` has a sum in an uniform additive group `α` if and only if it has that sum in the
completion of `α`. |
summable_iff_summable_compl_and_tsum_mem (f : β → α) :
Summable f ↔ Summable (toCompl ∘ f) ∧ ∑' i, toCompl (f i) ∈ Set.range toCompl :=
(isDenseInducing_toCompl α).summable_iff_tsum_comp_mem_range f | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | summable_iff_summable_compl_and_tsum_mem | A function `f` is summable in a uniform additive group `α` if and only if it is summable in
`Completion α` and its sum in `Completion α` lies in the range of `toCompl : α →+ Completion α`. |
summable_iff_cauchySeq_finset_and_tsum_mem (f : β → α) :
Summable f ↔ CauchySeq (fun s : Finset β ↦ ∑ b ∈ s, f b) ∧
∑' i, toCompl (f i) ∈ Set.range toCompl := by
classical
constructor
· rintro ⟨a, ha⟩
exact ⟨ha.cauchySeq, ((summable_iff_summable_compl_and_tsum_mem f).mp ⟨a, ha⟩).2⟩
· rintro ⟨h_cauchy, h_tsum⟩
apply (summable_iff_summable_compl_and_tsum_mem f).mpr
constructor
· apply summable_iff_cauchySeq_finset.mpr
simp_rw [Function.comp_apply, ← map_sum]
exact h_cauchy.map (uniformContinuous_coe α)
· exact h_tsum | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | summable_iff_cauchySeq_finset_and_tsum_mem | A function `f` is summable in a uniform additive group `α` if and only if the net of its partial
sums is Cauchy and its sum in `Completion α` lies in the range of `toCompl : α →+ Completion α`.
(The condition that the net of partial sums is Cauchy can be checked using
`cauchySeq_finset_iff_sum_vanishing` or `cauchySeq_finset_iff_tsum_vanishing`.) |
Summable.toCompl_tsum {f : β → α} (hf : Summable f) : ∑' i, toCompl (f i) = ∑' i, f i :=
(hf.map_tsum toCompl (continuous_coe α)).symm | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | Summable.toCompl_tsum | If a function `f` is summable in a uniform additive group `α`, then its sum in `α` is the same
as its sum in `Completion α`. |
HasSum.const_smul {a : α} (b : γ) (hf : HasSum f a) : HasSum (fun i ↦ b • f i) (b • a) :=
hf.map (DistribMulAction.toAddMonoidHom α _) <| continuous_const_smul _ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | HasSum.const_smul | null |
Summable.const_smul (b : γ) (hf : Summable f) : Summable fun i ↦ b • f i :=
(hf.hasSum.const_smul _).summable | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | Summable.const_smul | null |
protected Summable.tsum_const_smul [T2Space α] (b : γ) (hf : Summable f) :
∑' i, b • f i = b • ∑' i, f i :=
(hf.hasSum.const_smul _).tsum_eq
@[deprecated (since := "2025-04-12")] alias tsum_const_smul := Summable.tsum_const_smul | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | Summable.tsum_const_smul | Infinite sums commute with scalar multiplication. Version for scalars living in a `Monoid`, but
requiring a summability hypothesis. |
tsum_const_smul' {γ : Type*} [Group γ] [DistribMulAction γ α] [ContinuousConstSMul γ α]
[T2Space α] (g : γ) : ∑' (i : β), g • f i = g • ∑' (i : β), f i := by
by_cases hf : Summable f
· exact hf.tsum_const_smul g
rw [tsum_eq_zero_of_not_summable hf]
simp only [smul_zero]
let mul_g : α ≃+ α := DistribMulAction.toAddEquiv α g
apply tsum_eq_zero_of_not_summable
change ¬ Summable (mul_g ∘ f)
rwa [Summable.map_iff_of_equiv mul_g]
· apply continuous_const_smul
· apply continuous_const_smul | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | tsum_const_smul' | Infinite sums commute with scalar multiplication. Version for scalars living in a `Group`, but
not requiring any summability hypothesis. |
tsum_const_smul'' {γ : Type*} [DivisionSemiring γ] [Module γ α] [ContinuousConstSMul γ α]
[T2Space α] (g : γ) : ∑' (i : β), g • f i = g • ∑' (i : β), f i := by
rcases eq_or_ne g 0 with rfl | hg
· simp
· exact tsum_const_smul' (Units.mk0 g hg) | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | tsum_const_smul'' | Infinite sums commute with scalar multiplication. Version for scalars living in a
`DivisionRing`; no summability hypothesis. This could be made to work for a
`[GroupWithZero γ]` if there was such a thing as `DistribMulActionWithZero`. |
HasSum.smul_const {r : R} (hf : HasSum f r) (a : M) : HasSum (fun z ↦ f z • a) (r • a) :=
hf.map ((smulAddHom R M).flip a) (continuous_id.smul continuous_const) | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | HasSum.smul_const | null |
Summable.smul_const (hf : Summable f) (a : M) : Summable fun z ↦ f z • a :=
(hf.hasSum.smul_const _).summable | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | Summable.smul_const | null |
protected Summable.tsum_smul_const [T2Space M] (hf : Summable f) (a : M) :
∑' z, f z • a = (∑' z, f z) • a :=
(hf.hasSum.smul_const _).tsum_eq
@[deprecated (since := "2025-04-12")] alias tsum_smul_const := Summable.tsum_smul_const | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | Summable.tsum_smul_const | null |
HasSum.smul_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.smul_const t
have this : ∀ i : ι, HasSum (fun c : κ ↦ f i • g c) (f i • t) := fun i ↦ hg.const_smul (f i)
have key₂ : HasSum (fun i ↦ f i • t) u := HasSum.prod_fiberwise hfg this
key₁.unique key₂ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | HasSum.smul_eq | null |
HasSum.smul (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.smul_eq hg hu).symm ▸ hu | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | HasSum.smul | null |
tsum_smul_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.smul_eq hg.hasSum hfg.hasSum | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | tsum_smul_tsum | Scalar product of two infinites sums indexed by arbitrary types. |
protected ContinuousLinearMap.hasSum {f : ι → M} (φ : M →SL[σ] M₂) {x : M}
(hf : HasSum f x) : HasSum (fun b : ι ↦ φ (f b)) (φ x) := by
simpa only using hf.map φ.toLinearMap.toAddMonoidHom φ.continuous
alias HasSum.mapL := ContinuousLinearMap.hasSum | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearMap.hasSum | Applying a continuous linear map commutes with taking an (infinite) sum. |
protected ContinuousLinearMap.summable {f : ι → M} (φ : M →SL[σ] M₂) (hf : Summable f) :
Summable fun b : ι ↦ φ (f b) :=
(hf.hasSum.mapL φ).summable
alias Summable.mapL := ContinuousLinearMap.summable | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearMap.summable | null |
protected ContinuousLinearMap.map_tsum [T2Space M₂] {f : ι → M} (φ : M →SL[σ] M₂)
(hf : Summable f) : φ (∑' z, f z) = ∑' z, φ (f z) :=
(hf.hasSum.mapL φ).tsum_eq.symm | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearMap.map_tsum | null |
protected ContinuousLinearEquiv.hasSum {f : ι → M} (e : M ≃SL[σ] M₂) {y : M₂} :
HasSum (fun b : ι ↦ e (f b)) y ↔ HasSum f (e.symm y) :=
⟨fun h ↦ by simpa only [e.symm.coe_coe, e.symm_apply_apply] using h.mapL (e.symm : M₂ →SL[σ'] M),
fun h ↦ by simpa only [e.coe_coe, e.apply_symm_apply] using (e : M →SL[σ] M₂).hasSum h⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearEquiv.hasSum | Applying a continuous linear map commutes with taking an (infinite) sum. |
protected ContinuousLinearEquiv.hasSum' {f : ι → M} (e : M ≃SL[σ] M₂) {x : M} :
HasSum (fun b : ι ↦ e (f b)) (e x) ↔ HasSum f x := by
rw [e.hasSum, ContinuousLinearEquiv.symm_apply_apply] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearEquiv.hasSum' | Applying a continuous linear map commutes with taking an (infinite) sum. |
protected ContinuousLinearEquiv.summable {f : ι → M} (e : M ≃SL[σ] M₂) :
(Summable fun b : ι ↦ e (f b)) ↔ Summable f :=
⟨fun hf ↦ (e.hasSum.1 hf.hasSum).summable, (e : M →SL[σ] M₂).summable⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearEquiv.summable | null |
ContinuousLinearEquiv.tsum_eq_iff [T2Space M] [T2Space M₂] {f : ι → M} (e : M ≃SL[σ] M₂)
{y : M₂} : (∑' z, e (f z)) = y ↔ ∑' z, f z = e.symm y := by
by_cases hf : Summable f
· exact
⟨fun h ↦ (e.hasSum.mp ((e.summable.mpr hf).hasSum_iff.mpr h)).tsum_eq, fun h ↦
(e.hasSum.mpr (hf.hasSum_iff.mpr h)).tsum_eq⟩
· have hf' : ¬Summable fun z ↦ e (f z) := fun h ↦ hf (e.summable.mp h)
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hf']
refine ⟨?_, fun H ↦ ?_⟩
· rintro rfl
simp
· simpa using congr_arg (fun z ↦ e z) H | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearEquiv.tsum_eq_iff | null |
protected ContinuousLinearEquiv.map_tsum [T2Space M] [T2Space M₂] {f : ι → M}
(e : M ≃SL[σ] M₂) : e (∑' z, f z) = ∑' z, e (f z) := by
refine symm (e.tsum_eq_iff.mpr ?_)
rw [e.symm_apply_apply _] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | ContinuousLinearEquiv.map_tsum | null |
@[to_additive /-- Given an additive group `α` acting on a type `β`, and a function `f : β → M`,
we automorphize `f` to a function `β ⧸ α → M` by summing over `α` orbits,
`b ↦ ∑' (a : α), f(a • b)`. -/]
noncomputable MulAction.automorphize [Group α] [MulAction α β] (f : β → M) :
Quotient (MulAction.orbitRel α β) → M := by
refine @Quotient.lift _ _ (_) (fun b ↦ ∑' (a : α), f (a • b)) ?_
intro b₁ b₂ ⟨a, (ha : a • b₂ = b₁)⟩
simp only
rw [← ha]
convert (Equiv.mulRight a).tsum_eq (fun a' ↦ f (a' • b₂)) using 1
simp only [Equiv.coe_mulRight]
congr
ext
congr 1
simp only [mul_smul] | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | MulAction.automorphize | Given a group `α` acting on a type `β`, and a function `f : β → M`, we "automorphize" `f` to a
function `β ⧸ α → M` by summing over `α` orbits, `b ↦ ∑' (a : α), f(a • b)`. |
MulAction.automorphize_smul_left [Group α] [MulAction α β] (f : β → M)
(g : Quotient (MulAction.orbitRel α β) → R) :
MulAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
= g • (MulAction.automorphize f : Quotient (MulAction.orbitRel α β) → M) := by
ext x
apply @Quotient.inductionOn' β (MulAction.orbitRel α β) _ x _
intro b
simp only [automorphize, Pi.smul_apply', comp_apply]
set π : β → Quotient (MulAction.orbitRel α β) := Quotient.mk (MulAction.orbitRel α β)
have H₁ : ∀ a : α, π (a • b) = π b := by
intro a
apply (@Quotient.eq _ (MulAction.orbitRel α β) (a • b) b).mpr
use a
change ∑' a : α, g (π (a • b)) • f (a • b) = g (π b) • ∑' a : α, f (a • b)
simp_rw [H₁]
exact tsum_const_smul'' _ | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | MulAction.automorphize_smul_left | Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. |
AddAction.automorphize_smul_left [AddGroup α] [AddAction α β] (f : β → M)
(g : Quotient (AddAction.orbitRel α β) → R) :
AddAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
= g • (AddAction.automorphize f : Quotient (AddAction.orbitRel α β) → M) := by
ext x
apply @Quotient.inductionOn' β (AddAction.orbitRel α β) _ x _
intro b
simp only [automorphize, Pi.smul_apply', comp_apply]
set π : β → Quotient (AddAction.orbitRel α β) := Quotient.mk (AddAction.orbitRel α β)
have H₁ : ∀ a : α, π (a +ᵥ b) = π b := by
intro a
apply (@Quotient.eq _ (AddAction.orbitRel α β) (a +ᵥ b) b).mpr
use a
change ∑' a : α, g (π (a +ᵥ b)) • f (a +ᵥ b) = g (π b) • ∑' a : α, f (a +ᵥ b)
simp_rw [H₁]
exact tsum_const_smul'' _ | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | AddAction.automorphize_smul_left | Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. |
@[to_additive /-- Given a subgroup `Γ` of an additive group `G`, and a function `f : G → M`, we
automorphize `f` to a function `G ⧸ Γ → M` by summing over `Γ` orbits,
`g ↦ ∑' (γ : Γ), f(γ • g)`. -/]
noncomputable QuotientGroup.automorphize (f : G → M) : G ⧸ Γ → M := MulAction.automorphize f | def | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | QuotientGroup.automorphize | Given a subgroup `Γ` of a group `G`, and a function `f : G → M`, we "automorphize" `f` to a
function `G ⧸ Γ → M` by summing over `Γ` orbits, `g ↦ ∑' (γ : Γ), f(γ • g)`. |
QuotientGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
(QuotientGroup.automorphize ((g ∘ (@Quotient.mk' _ (_)) : G → R) • f) : G ⧸ Γ → M)
= g • (QuotientGroup.automorphize f : G ⧸ Γ → M) :=
MulAction.automorphize_smul_left f g | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | QuotientGroup.automorphize_smul_left | Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. |
QuotientAddGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
QuotientAddGroup.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
= g • (QuotientAddGroup.automorphize f : G ⧸ Γ → M) :=
AddAction.automorphize_smul_left f g | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | QuotientAddGroup.automorphize_smul_left | Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. |
@[to_additive /-- If `f : ℕ → M` has sum `m`, then the partial sums `∑ i ∈ range n, f i` converge
to `m`. -/]
HasProd.tendsto_prod_nat {f : ℕ → M} (h : HasProd f m) :
Tendsto (fun n ↦ ∏ i ∈ range n, f i) atTop (𝓝 m) :=
h.comp tendsto_finset_range | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.tendsto_prod_nat | If `f : ℕ → M` has product `m`, then the partial products `∏ i ∈ range n, f i` converge
to `m`. |
@[to_additive /-- If `f : ℕ → M` is summable, then the partial sums `∑ i ∈ range n, f i` converge
to `∑' i, f i`. -/]
Multipliable.tendsto_prod_tprod_nat {f : ℕ → M} (h : Multipliable f) :
Tendsto (fun n ↦ ∏ i ∈ range n, f i) atTop (𝓝 (∏' i, f i)) :=
h.hasProd.tendsto_prod_nat | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.tendsto_prod_tprod_nat | If `f : ℕ → M` is multipliable, then the partial products `∏ i ∈ range n, f i` converge
to `∏' i, f i`. |
@[to_additive]
prod_range_mul {f : ℕ → M} {k : ℕ} (h : HasProd (fun n ↦ f (n + k)) m) :
HasProd f ((∏ i ∈ range k, f i) * m) := by
refine ((range k).hasProd f).mul_compl ?_
rwa [← (notMemRangeEquiv k).symm.hasProd_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | prod_range_mul | null |
zero_mul {f : ℕ → M} (h : HasProd (fun n ↦ f (n + 1)) m) :
HasProd f (f 0 * m) := by
simpa only [prod_range_one] using h.prod_range_mul
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | zero_mul | null |
even_mul_odd {f : ℕ → M} (he : HasProd (fun k ↦ f (2 * k)) m)
(ho : HasProd (fun k ↦ f (2 * k + 1)) m') : HasProd f (m * m') := by
have := mul_right_injective₀ (two_ne_zero' ℕ)
replace ho := ((add_left_injective 1).comp this).hasProd_range_iff.2 ho
refine (this.hasProd_range_iff.2 he).mul_isCompl ?_ ho
simpa [Function.comp_def] using Nat.isCompl_even_odd | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | even_mul_odd | null |
@[to_additive]
hasProd_iff_tendsto_nat [T2Space M] {f : ℕ → M} (hf : Multipliable f) :
HasProd f m ↔ Tendsto (fun n : ℕ ↦ ∏ i ∈ range n, f i) atTop (𝓝 m) := by
refine ⟨fun h ↦ h.tendsto_prod_nat, fun h ↦ ?_⟩
rw [tendsto_nhds_unique h hf.hasProd.tendsto_prod_nat]
exact hf.hasProd | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | hasProd_iff_tendsto_nat | null |
@[to_additive]
comp_nat_add {f : ℕ → M} {k : ℕ} (h : Multipliable fun n ↦ f (n + k)) : Multipliable f :=
h.hasProd.prod_range_mul.multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | comp_nat_add | null |
even_mul_odd {f : ℕ → M} (he : Multipliable fun k ↦ f (2 * k))
(ho : Multipliable fun k ↦ f (2 * k + 1)) : Multipliable f :=
(he.hasProd.even_mul_odd ho.hasProd).multipliable | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | even_mul_odd | null |
@[to_additive /-- You can compute a sum over an encodable type by summing over the natural numbers
and taking a supremum. This is useful for outer measures. -/]
tprod_iSup_decode₂ [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (s : β → α) :
∏' i : ℕ, m (⨆ b ∈ decode₂ β i, s b) = ∏' b : β, m (s b) := by
rw [← tprod_extend_one (@encode_injective β _)]
refine tprod_congr fun n ↦ ?_
rcases em (n ∈ Set.range (encode : β → ℕ)) with ⟨a, rfl⟩ | hn
· simp [encode_injective.extend_apply]
· rw [extend_apply' _ _ _ hn]
rw [← decode₂_ne_none_iff, ne_eq, not_not] at hn
simp [hn, m0] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_iSup_decode₂ | You can compute a product over an encodable type by multiplying over the natural numbers and
taking a supremum. |
@[to_additive /-- `tsum_iSup_decode₂` specialized to the complete lattice of sets. -/]
tprod_iUnion_decode₂ (m : Set α → M) (m0 : m ∅ = 1) (s : β → Set α) :
∏' i, m (⋃ b ∈ decode₂ β i, s b) = ∏' b, m (s b) :=
tprod_iSup_decode₂ m m0 s | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_iUnion_decode₂ | `tprod_iSup_decode₂` specialized to the complete lattice of sets. |
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