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 ⌀ |
|---|---|---|---|---|---|---|
HasProd.mul_disjoint {s t : Set β} (hs : Disjoint s t) (ha : HasProd (f ∘ (↑) : s → α) a)
(hb : HasProd (f ∘ (↑) : t → α) b) : HasProd (f ∘ (↑) : (s ∪ t : Set β) → α) (a * b) := by
rw [hasProd_subtype_iff_mulIndicator] at *
rw [Set.mulIndicator_union_of_disjoint hs]
exact ha.mul hb
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.mul_disjoint | null |
hasProd_prod_disjoint {ι} (s : Finset ι) {t : ι → Set β} {a : ι → α}
(hs : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasProd (f ∘ (↑) : t i → α) (a i)) :
HasProd (f ∘ (↑) : (⋃ i ∈ s, t i) → α) (∏ i ∈ s, a i) := by
simp_rw [hasProd_subtype_iff_mulIndicator] at *
rw [Finset.mulIndicator_biUnion _ _ hs]
exact hasProd_prod hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_prod_disjoint | null |
HasProd.mul_isCompl {s t : Set β} (hs : IsCompl s t) (ha : HasProd (f ∘ (↑) : s → α) a)
(hb : HasProd (f ∘ (↑) : t → α) b) : HasProd f (a * b) := by
simpa [← hs.compl_eq] using
(hasProd_subtype_iff_mulIndicator.1 ha).mul (hasProd_subtype_iff_mulIndicator.1 hb)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.mul_isCompl | null |
HasProd.mul_compl {s : Set β} (ha : HasProd (f ∘ (↑) : s → α) a)
(hb : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) b) : HasProd f (a * b) :=
ha.mul_isCompl isCompl_compl hb
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.mul_compl | null |
Multipliable.mul_compl {s : Set β} (hs : Multipliable (f ∘ (↑) : s → α))
(hsc : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α)) : Multipliable f :=
(hs.hasProd.mul_compl hsc.hasProd).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.mul_compl | null |
HasProd.compl_mul {s : Set β} (ha : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) a)
(hb : HasProd (f ∘ (↑) : s → α) b) : HasProd f (a * b) :=
ha.mul_isCompl isCompl_compl.symm hb
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.compl_mul | null |
Multipliable.compl_add {s : Set β} (hs : Multipliable (f ∘ (↑) : (sᶜ : Set β) → α))
(hsc : Multipliable (f ∘ (↑) : s → α)) : Multipliable f :=
(hs.hasProd.compl_mul hsc.hasProd).multipliable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.compl_add | null |
@[to_additive /-- Version of `HasSum.update` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that `f.update` has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `f.update` given that both exist. -/]
HasProd.update' {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
[ContinuousMul α] [DecidableEq β] {f : β → α} {a a' : α} (hf : HasProd f a) (b : β) (x : α)
(hf' : HasProd (update f b x) a') : a * x = a' * f b := by
have : ∀ b', f b' * ite (b' = b) x 1 = update f b x b' * ite (b' = b) (f b) 1 := by
intro b'
split_ifs with hb'
· simpa only [Function.update_apply, hb', eq_self_iff_true] using mul_comm (f b) x
· simp only [Function.update_apply, hb', if_false]
have h := hf.mul (hasProd_ite_eq b x)
simp_rw [this] at h
exact HasProd.unique h (hf'.mul (hasProd_ite_eq b (f b))) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.update' | Version of `HasProd.update` for `CommMonoid` rather than `CommGroup`.
Rather than showing that `f.update` has a specific product in terms of `HasProd`,
it gives a relationship between the products of `f` and `f.update` given that both exist. |
@[to_additive /-- Version of `hasSum_ite_sub_hasSum` for `AddCommMonoid` rather than `AddCommGroup`.
Rather than showing that the `ite` expression has a specific sum in terms of `HasSum`,
it gives a relationship between the sums of `f` and `ite (n = b) 0 (f n)` given that both exist. -/]
eq_mul_of_hasProd_ite {α β : Type*} [TopologicalSpace α] [CommMonoid α] [T2Space α]
[ContinuousMul α] [DecidableEq β] {f : β → α} {a : α} (hf : HasProd f a) (b : β) (a' : α)
(hf' : HasProd (fun n ↦ ite (n = b) 1 (f n)) a') : a = a' * f b := by
refine (mul_one a).symm.trans (hf.update' b 1 ?_)
convert hf'
apply update_apply | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | eq_mul_of_hasProd_ite | Version of `hasProd_ite_div_hasProd` for `CommMonoid` rather than `CommGroup`.
Rather than showing that the `ite` expression has a specific product in terms of `HasProd`, it gives
a relationship between the products of `f` and `ite (n = b) 0 (f n)` given that both exist. |
@[to_additive]
tprod_congr_set_coe (f : β → α) {s t : Set β} (h : s = t) :
∏' x : s, f x = ∏' x : t, f x := by rw [h]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_congr_set_coe | null |
tprod_congr_subtype (f : β → α) {P Q : β → Prop} (h : ∀ x, P x ↔ Q x) :
∏' x : {x // P x}, f x = ∏' x : {x // Q x}, f x :=
tprod_congr_set_coe f <| Set.ext h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_congr_subtype | null |
tprod_eq_finprod (hf : (mulSupport f).Finite) :
∏' b, f b = ∏ᶠ b, f b := by simp [tprod_def, multipliable_of_finite_mulSupport hf, hf]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_finprod | null |
tprod_eq_prod' {s : Finset β} (hf : mulSupport f ⊆ s) :
∏' b, f b = ∏ b ∈ s, f b := by
rw [tprod_eq_finprod (s.finite_toSet.subset hf), finprod_eq_prod_of_mulSupport_subset _ hf]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_prod' | null |
tprod_eq_prod {s : Finset β} (hf : ∀ b ∉ s, f b = 1) :
∏' b, f b = ∏ b ∈ s, f b :=
tprod_eq_prod' <| mulSupport_subset_iff'.2 hf
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_prod | null |
tprod_one : ∏' _ : β, (1 : α) = 1 := by rw [tprod_eq_finprod] <;> simp
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_one | null |
tprod_empty [IsEmpty β] : ∏' b, f b = 1 := by
rw [tprod_eq_prod (s := (∅ : Finset β))] <;> simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_empty | null |
tprod_congr {f g : β → α}
(hfg : ∀ b, f b = g b) : ∏' b, f b = ∏' b, g b :=
congr_arg tprod (funext hfg)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_congr | null |
tprod_congr₂ {f g : γ → β → α}
(hfg : ∀ b c, f b c = g b c) : ∏' c, ∏' b, f b c = ∏' c, ∏' b, g b c :=
tprod_congr fun c ↦ tprod_congr fun b ↦ hfg b c
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_congr₂ | null |
tprod_fintype [Fintype β] (f : β → α) : ∏' b, f b = ∏ b, f b := by
apply tprod_eq_prod; simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_fintype | null |
prod_eq_tprod_mulIndicator (f : β → α) (s : Finset β) :
∏ x ∈ s, f x = ∏' x, Set.mulIndicator (↑s) f x := by
rw [tprod_eq_prod' (Set.mulSupport_mulIndicator_subset),
Finset.prod_mulIndicator_subset _ Finset.Subset.rfl]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | prod_eq_tprod_mulIndicator | null |
tprod_bool (f : Bool → α) : ∏' i : Bool, f i = f false * f true := by
rw [tprod_fintype, Fintype.prod_bool, mul_comm]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_bool | null |
tprod_eq_mulSingle {f : β → α} (b : β) (hf : ∀ b' ≠ b, f b' = 1) :
∏' b, f b = f b := by
rw [tprod_eq_prod (s := {b}), prod_singleton]
exact fun b' hb' ↦ hf b' (by simpa using hb')
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_mulSingle | null |
tprod_tprod_eq_mulSingle (f : β → γ → α) (b : β) (c : γ) (hfb : ∀ b' ≠ b, f b' c = 1)
(hfc : ∀ b', ∀ c' ≠ c, f b' c' = 1) : ∏' (b') (c'), f b' c' = f b c :=
calc
∏' (b') (c'), f b' c' = ∏' b', f b' c := tprod_congr fun b' ↦ tprod_eq_mulSingle _ (hfc b')
_ = f b c := tprod_eq_mulSingle _ hfb
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_tprod_eq_mulSingle | null |
tprod_ite_eq (b : β) [DecidablePred (· = b)] (a : α) :
∏' b', (if b' = b then a else 1) = a := by
rw [tprod_eq_mulSingle b]
· simp
· intro b' hb'; simp [hb']
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_ite_eq | null |
Finset.tprod_subtype (s : Finset β) (f : β → α) :
∏' x : { x // x ∈ s }, f x = ∏ x ∈ s, f x := by
rw [← prod_attach]; exact tprod_fintype _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Finset.tprod_subtype | null |
Finset.tprod_subtype' (s : Finset β) (f : β → α) :
∏' x : (s : Set β), f x = ∏ x ∈ s, f x := by simp
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Finset.tprod_subtype' | null |
tprod_singleton (b : β) (f : β → α) : ∏' x : ({b} : Set β), f x = f b := by
rw [← coe_singleton, Finset.tprod_subtype', prod_singleton]
open scoped Classical in
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_singleton | null |
Function.Injective.tprod_eq {g : γ → β} (hg : Injective g) {f : β → α}
(hf : mulSupport f ⊆ Set.range g) : ∏' c, f (g c) = ∏' b, f b := by
have : mulSupport f = g '' mulSupport (f ∘ g) := by
rw [mulSupport_comp_eq_preimage, Set.image_preimage_eq_iff.2 hf]
rw [← Function.comp_def]
by_cases hf_fin : (mulSupport f).Finite
· have hfg_fin : (mulSupport (f ∘ g)).Finite := hf_fin.preimage hg.injOn
lift g to γ ↪ β using hg
simp_rw [tprod_eq_prod' hf_fin.coe_toFinset.ge, tprod_eq_prod' hfg_fin.coe_toFinset.ge,
comp_apply, ← Finset.prod_map]
refine Finset.prod_congr (Finset.coe_injective ?_) fun _ _ ↦ rfl
simp [this]
· have hf_fin' : ¬ Set.Finite (mulSupport (f ∘ g)) := by
rwa [this, Set.finite_image_iff hg.injOn] at hf_fin
simp_rw [tprod_def, if_neg hf_fin, if_neg hf_fin', Multipliable,
funext fun a => propext <| hg.hasProd_iff (mulSupport_subset_iff'.1 hf) (a := a)]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Injective.tprod_eq | null |
Equiv.tprod_eq (e : γ ≃ β) (f : β → α) : ∏' c, f (e c) = ∏' b, f b :=
e.injective.tprod_eq <| by simp
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.tprod_eq | null |
tprod_comp_neg {β : Type*} [InvolutiveNeg β] (f : β → α) :
∏' d, f (-d) = ∏' d, f d :=
(Equiv.neg β).tprod_eq f
/-! ### `tprod` on subsets - part 1 -/
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_comp_neg | null |
tprod_subtype_eq_of_mulSupport_subset {f : β → α} {s : Set β} (hs : mulSupport f ⊆ s) :
∏' x : s, f x = ∏' x, f x :=
Subtype.val_injective.tprod_eq <| by simpa
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_subtype_eq_of_mulSupport_subset | null |
tprod_subtype_mulSupport (f : β → α) : ∏' x : mulSupport f, f x = ∏' x, f x :=
tprod_subtype_eq_of_mulSupport_subset Set.Subset.rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_subtype_mulSupport | null |
tprod_subtype (s : Set β) (f : β → α) : ∏' x : s, f x = ∏' x, s.mulIndicator f x := by
rw [← tprod_subtype_eq_of_mulSupport_subset Set.mulSupport_mulIndicator_subset, tprod_congr]
simp
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_subtype | null |
tprod_univ (f : β → α) : ∏' x : (Set.univ : Set β), f x = ∏' x, f x :=
tprod_subtype_eq_of_mulSupport_subset <| Set.subset_univ _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_univ | null |
tprod_image {g : γ → β} (f : β → α) {s : Set γ} (hg : Set.InjOn g s) :
∏' x : g '' s, f x = ∏' x : s, f (g x) :=
((Equiv.Set.imageOfInjOn _ _ hg).tprod_eq fun x ↦ f x).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_image | null |
tprod_range {g : γ → β} (f : β → α) (hg : Injective g) :
∏' x : Set.range g, f x = ∏' x, f (g x) := by
rw [← Set.image_univ, tprod_image f hg.injOn]
simp_rw [← comp_apply (g := g), tprod_univ (f ∘ g)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_range | null |
@[to_additive /-- If `f b = 0` for all `b ∈ t`, then the sum of `f a` with `a ∈ s` is the same as
the sum of `f a` with `a ∈ s ∖ t`. -/]
tprod_setElem_eq_tprod_setElem_diff {f : β → α} (s t : Set β)
(hf₀ : ∀ b ∈ t, f b = 1) :
∏' a : s, f a = ∏' a : (s \ t : Set β), f a :=
.symm <| (Set.inclusion_injective (t := s) Set.diff_subset).tprod_eq (f := f ∘ (↑)) <|
mulSupport_subset_iff'.2 fun b hb ↦ hf₀ b <| by simpa using hb | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_setElem_eq_tprod_setElem_diff | If `f b = 1` for all `b ∈ t`, then the product of `f a` with `a ∈ s` is the same as the
product of `f a` with `a ∈ s ∖ t`. |
@[to_additive /-- If `f b = 0`, then the sum of `f a` with `a ∈ s` is the same as the sum of `f a`
for `a ∈ s ∖ {b}`. -/]
tprod_eq_tprod_diff_singleton {f : β → α} (s : Set β) {b : β} (hf₀ : f b = 1) :
∏' a : s, f a = ∏' a : (s \ {b} : Set β), f a :=
tprod_setElem_eq_tprod_setElem_diff s {b} fun _ ha ↦ ha ▸ hf₀
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_tprod_diff_singleton | If `f b = 1`, then the product of `f a` with `a ∈ s` is the same as the product of `f a` for
`a ∈ s ∖ {b}`. |
tprod_eq_tprod_of_ne_one_bij {g : γ → α} (i : mulSupport g → β) (hi : Injective i)
(hf : mulSupport f ⊆ Set.range i) (hfg : ∀ x, f (i x) = g x) : ∏' x, f x = ∏' y, g y := by
rw [← tprod_subtype_mulSupport g, ← hi.tprod_eq hf]
simp only [hfg]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_tprod_of_ne_one_bij | null |
Equiv.tprod_eq_tprod_of_mulSupport {f : β → α} {g : γ → α}
(e : mulSupport f ≃ mulSupport g) (he : ∀ x, g (e x) = f x) :
∏' x, f x = ∏' y, g y :=
.symm <| tprod_eq_tprod_of_ne_one_bij _ (Subtype.val_injective.comp e.injective) (by simp) he
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.tprod_eq_tprod_of_mulSupport | null |
tprod_dite_right (P : Prop) [Decidable P] (x : β → ¬P → α) :
∏' b : β, (if h : P then (1 : α) else x b h) = if h : P then (1 : α) else ∏' b : β, x b h := by
by_cases hP : P <;> simp [hP]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_dite_right | null |
tprod_dite_left (P : Prop) [Decidable P] (x : β → P → α) :
∏' b : β, (if h : P then x b h else 1) = if h : P then ∏' b : β, x b h else 1 := by
by_cases hP : P <;> simp [hP]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_dite_left | null |
tprod_extend_one {γ : Type*} {g : γ → β} (hg : Injective g) (f : γ → α) :
∏' y, extend g f 1 y = ∏' x, f x := by
have : mulSupport (extend g f 1) ⊆ Set.range g := mulSupport_subset_iff'.2 <| extend_apply' _ _
simp_rw [← hg.tprod_eq this, hg.extend_apply]
variable [T2Space α]
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_extend_one | null |
Function.Surjective.tprod_eq_tprod_of_hasProd_iff_hasProd {α' : Type*} [CommMonoid α']
[TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) (h1 : e 1 = 1) {f : β → α}
{g : γ → α'} (h : ∀ {a}, HasProd f (e a) ↔ HasProd g a) : ∏' b, f b = e (∏' c, g c) :=
by_cases (fun x ↦ (h.mpr x.hasProd).tprod_eq) fun hg : ¬Multipliable g ↦ by
have hf : ¬Multipliable f := mt (hes.multipliable_iff_of_hasProd_iff @h).1 hg
simp [tprod_def, hf, hg, h1]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Surjective.tprod_eq_tprod_of_hasProd_iff_hasProd | null |
tprod_eq_tprod_of_hasProd_iff_hasProd {f : β → α} {g : γ → α}
(h : ∀ {a}, HasProd f a ↔ HasProd g a) : ∏' b, f b = ∏' c, g c :=
surjective_id.tprod_eq_tprod_of_hasProd_iff_hasProd rfl @h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_eq_tprod_of_hasProd_iff_hasProd | null |
@[to_additive]
protected Multipliable.tprod_mul (hf : Multipliable f) (hg : Multipliable g) :
∏' b, (f b * g b) = (∏' b, f b) * ∏' b, g b :=
(hf.hasProd.mul hg.hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_add := Summable.tsum_add
@[to_additive existing, deprecated (since := "2025-04-12")] alias
tprod_mul := Multipliable.tprod_mul
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_mul | null |
protected Multipliable.tprod_finsetProd {f : γ → β → α} {s : Finset γ}
(hf : ∀ i ∈ s, Multipliable (f i)) : ∏' b, ∏ i ∈ s, f i b = ∏ i ∈ s, ∏' b, f i b :=
(hasProd_prod fun i hi ↦ (hf i hi).hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_finsetSum := Summable.tsum_finsetSum
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_finsetProd :=
Multipliable.tprod_finsetProd | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_finsetProd | null |
@[to_additive /-- Version of `tsum_eq_add_tsum_ite` for `AddCommMonoid` rather than `AddCommGroup`.
Requires a different convergence assumption involving `Function.update`. -/]
protected Multipliable.tprod_eq_mul_tprod_ite' [DecidableEq β] {f : β → α} (b : β)
(hf : Multipliable (update f b 1)) :
∏' x, f x = f b * ∏' x, ite (x = b) 1 (f x) :=
calc
∏' x, f x = ∏' x, (ite (x = b) (f x) 1 * update f b 1 x) :=
tprod_congr fun n ↦ by split_ifs with h <;> simp [update_apply, h]
_ = (∏' x, ite (x = b) (f x) 1) * ∏' x, update f b 1 x :=
Multipliable.tprod_mul ⟨ite (b = b) (f b) 1, hasProd_single b fun _ hb ↦ if_neg hb⟩ hf
_ = ite (b = b) (f b) 1 * ∏' x, update f b 1 x := by
congr
exact tprod_eq_mulSingle b fun b' hb' ↦ if_neg hb'
_ = f b * ∏' x, ite (x = b) 1 (f x) := by
simp only [update, if_true, eq_rec_constant, dite_eq_ite]
@[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'
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_eq_mul_tprod_ite' | Version of `tprod_eq_mul_tprod_ite` for `CommMonoid` rather than `CommGroup`.
Requires a different convergence assumption involving `Function.update`. |
protected Multipliable.tprod_mul_tprod_compl {s : Set β}
(hs : Multipliable (f ∘ (↑) : s → α)) (hsc : Multipliable (f ∘ (↑) : ↑sᶜ → α)) :
(∏' x : s, f x) * ∏' x : ↑sᶜ, f x = ∏' x, f x :=
(hs.hasProd.mul_compl hsc.hasProd).tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias tsum_add_tsum_compl := Summable.tsum_add_tsum_compl
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_mul_tprod_compl :=
Multipliable.tprod_mul_tprod_compl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_mul_tprod_compl | null |
protected Multipliable.tprod_union_disjoint {s t : Set β} (hd : Disjoint s t)
(hs : Multipliable (f ∘ (↑) : s → α)) (ht : Multipliable (f ∘ (↑) : t → α)) :
∏' x : ↑(s ∪ t), f x = (∏' x : s, f x) * ∏' x : t, f x :=
(hs.hasProd.mul_disjoint hd ht.hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_union_disjoint := Summable.tsum_union_disjoint
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_union_disjoint :=
Multipliable.tprod_union_disjoint
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_union_disjoint | null |
protected Multipliable.tprod_finset_bUnion_disjoint {ι} {s : Finset ι} {t : ι → Set β}
(hd : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, Multipliable (f ∘ (↑) : t i → α)) :
∏' x : ⋃ i ∈ s, t i, f x = ∏ i ∈ s, ∏' x : t i, f x :=
(hasProd_prod_disjoint _ hd fun i hi ↦ (hf i hi).hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_finset_bUnion_disjoint :=
Summable.tsum_finset_bUnion_disjoint
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_finset_bUnion_disjoint :=
Multipliable.tprod_finset_bUnion_disjoint | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.tprod_finset_bUnion_disjoint | null |
hasProd_zero_of_exists_eq_zero (hf : ∃ b, f b = 0) : HasProd f 0 := by
obtain ⟨b, hb⟩ := hf
apply tendsto_const_nhds.congr'
filter_upwards [eventually_ge_atTop {b}] with s hs
exact (Finset.prod_eq_zero (Finset.singleton_subset_iff.mp hs) hb).symm | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_zero_of_exists_eq_zero | null |
multipliable_of_exists_eq_zero (hf : ∃ b, f b = 0) : Multipliable f :=
⟨0, hasProd_zero_of_exists_eq_zero hf⟩ | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_of_exists_eq_zero | null |
tprod_of_exists_eq_zero [T2Space α] (hf : ∃ b, f b = 0) : ∏' b, f b = 0 :=
(hasProd_zero_of_exists_eq_zero hf).tprod_eq | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tprod_of_exists_eq_zero | null |
@[to_additive]
hasProd_pi_single [DecidableEq β] (b : β) (a : α) : HasProd (Pi.mulSingle b a) a := by
convert hasProd_ite_eq b a
simp [Pi.mulSingle_apply]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | hasProd_pi_single | null |
tprod_pi_single [DecidableEq β] (b : β) (a : α) : ∏' b', Pi.mulSingle b a b' = a := by
rw [tprod_eq_mulSingle b]
· simp
· intro b' hb'; simp [hb']
@[to_additive tsum_setProd_singleton_left] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tprod_pi_single | null |
tprod_setProd_singleton_left (b : β) (t : Set γ) (f : β × γ → α) :
(∏' x : {b} ×ˢ t, f x) = ∏' c : t, f (b, c) := by
rw [tprod_congr_set_coe _ Set.singleton_prod, tprod_image _ (Prod.mk_right_injective b).injOn]
@[to_additive tsum_setProd_singleton_right] | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tprod_setProd_singleton_left | null |
tprod_setProd_singleton_right (s : Set β) (c : γ) (f : β × γ → α) :
(∏' x : s ×ˢ {c}, f x) = ∏' b : s, f (b, c) := by
rw [tprod_congr_set_coe _ Set.prod_singleton, tprod_image _ (Prod.mk_left_injective c).injOn]
@[to_additive Summable.prod_symm] | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tprod_setProd_singleton_right | null |
Multipliable.prod_symm {f : β × γ → α} (hf : Multipliable f) :
Multipliable fun p : γ × β ↦ f p.swap :=
(Equiv.prodComm γ β).multipliable_iff.2 hf | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.prod_symm | null |
@[to_additive HasSum.prodMk]
HasProd.prodMk {f : β → α} {g : β → γ} {a : α} {b : γ} (hf : HasProd f a)
(hg : HasProd g b) : HasProd (fun x ↦ (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ := by
simp [HasProd, ← prod_mk_prod, Filter.Tendsto.prodMk_nhds hf hg]
@[deprecated (since := "2025-03-10")]
alias HasSum.prod_mk := HasSum.prodMk
@[to_additive existing HasSum.prodMk, deprecated (since := "2025-03-10")]
alias HasProd.prod_mk := HasProd.prodMk | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.prodMk | null |
@[to_additive]
HasProd.sum {α β M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
{f : α ⊕ β → M} {a b : M}
(h₁ : HasProd (f ∘ Sum.inl) a) (h₂ : HasProd (f ∘ Sum.inr) b) : HasProd f (a * b) := by
have : Tendsto ((∏ b ∈ ·, f b) ∘ sumEquiv.symm) (atTop.map sumEquiv) (nhds (a * b)) := by
rw [Finset.sumEquiv.map_atTop, ← prod_atTop_atTop_eq]
convert (tendsto_mul.comp (nhds_prod_eq (x := a) (y := b) ▸ Tendsto.prodMap h₁ h₂))
ext s
simp
simpa [Tendsto, ← Filter.map_map] using this
@[to_additive /-- For the statement that `tsum` commutes with `Finset.sum`,
see `Summable.tsum_finsetSum`. -/] | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.sum | null |
protected Multipliable.tprod_sum {α β M : Type*} [CommMonoid M] [TopologicalSpace M]
[ContinuousMul M] [T2Space M] {f : α ⊕ β → M} (h₁ : Multipliable (f ∘ .inl))
(h₂ : Multipliable (f ∘ .inr)) : ∏' i, f i = (∏' i, f (.inl i)) * (∏' i, f (.inr i)) :=
(h₁.hasProd.sum h₂.hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_sum := Summable.tsum_sum
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_sum :=
Multipliable.tprod_sum
@[to_additive] | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_sum | null |
Multipliable.sum {α β M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
(f : α ⊕ β → M) (h₁ : Multipliable (f ∘ Sum.inl)) (h₂ : Multipliable (f ∘ Sum.inr)) :
Multipliable f :=
⟨_, .sum h₁.hasProd h₂.hasProd⟩ | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.sum | null |
@[to_additive]
HasProd.sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
(ha : HasProd f a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) : HasProd g a := by
classical
refine (atTop_basis.tendsto_iff (closed_nhds_basis a)).mpr ?_
rintro s ⟨hs, hsc⟩
rcases mem_atTop_sets.mp (ha hs) with ⟨u, hu⟩
use u.image Sigma.fst, trivial
intro bs hbs
simp only [Set.mem_preimage, Finset.le_iff_subset] at hu
have : Tendsto (fun t : Finset (Σ b, γ b) ↦ ∏ p ∈ t with p.1 ∈ bs, f p) atTop
(𝓝 <| ∏ b ∈ bs, g b) := by
simp only [← sigma_preimage_mk, prod_sigma]
refine tendsto_finset_prod _ fun b _ ↦ ?_
change
Tendsto (fun t ↦ (fun t ↦ ∏ s ∈ t, f ⟨b, s⟩) (preimage t (Sigma.mk b) _)) atTop (𝓝 (g b))
exact (hf b).comp (tendsto_finset_preimage_atTop_atTop (sigma_mk_injective))
refine hsc.mem_of_tendsto this (eventually_atTop.2 ⟨u, fun t ht ↦ hu _ fun x hx ↦ ?_⟩)
exact mem_filter.2 ⟨ht hx, hbs <| mem_image_of_mem _ hx⟩ | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.sigma | null |
@[to_additive HasSum.prod_fiberwise /-- If a series `f` on `β × γ` has sum `a` and for each `b` the
restriction of `f` to `{b} × γ` has sum `g b`, then the series `g` has sum `a`. -/]
HasProd.prod_fiberwise {f : β × γ → α} {g : β → α} {a : α} (ha : HasProd f a)
(hf : ∀ b, HasProd (fun c ↦ f (b, c)) (g b)) : HasProd g a :=
HasProd.sigma ((Equiv.sigmaEquivProd β γ).hasProd_iff.2 ha) hf
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.prod_fiberwise | If a function `f` on `β × γ` has product `a` and for each `b` the restriction of `f` to
`{b} × γ` has product `g b`, then the function `g` has product `a`. |
Multipliable.sigma' {γ : β → Type*} {f : (Σ b : β, γ b) → α} (ha : Multipliable f)
(hf : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) : Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ :=
(ha.hasProd.sigma fun b ↦ (hf b).hasProd).multipliable | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.sigma' | null |
@[to_additive]
HasProd.sigma_of_hasProd {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α}
{a : α} (ha : HasProd g a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hf' : Multipliable f) :
HasProd f a := by simpa [(hf'.hasProd.sigma hf).unique ha] using hf'.hasProd
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.sigma_of_hasProd | null |
protected Multipliable.tprod_sigma' {γ : β → Type*} {f : (Σ b : β, γ b) → α}
(h₁ : ∀ b, Multipliable fun c ↦ f ⟨b, c⟩) (h₂ : Multipliable f) :
∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ :=
(h₂.hasProd.sigma fun b ↦ (h₁ b).hasProd).tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias tsum_sigma' := Summable.tsum_sigma'
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_sigma' :=
Multipliable.tprod_sigma'
@[to_additive Summable.tsum_prod'] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_sigma' | null |
protected Multipliable.tprod_prod' {f : β × γ → α} (h : Multipliable f)
(h₁ : ∀ b, Multipliable fun c ↦ f (b, c)) :
∏' p, f p = ∏' (b) (c), f (b, c) :=
(h.hasProd.prod_fiberwise fun b ↦ (h₁ b).hasProd).tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias tsum_prod' := Summable.tsum_prod'
@[to_additive existing Summable.tsum_prod', deprecated (since := "2025-04-12")] alias tprod_prod' :=
Multipliable.tprod_prod'
@[to_additive Summable.tsum_prod_uncurry] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_prod' | null |
protected Multipliable.tprod_prod_uncurry {f : β → γ → α}
(h : Multipliable (Function.uncurry f)) (h₁ : ∀ b, Multipliable fun c ↦ f b c) :
∏' p : β × γ, uncurry f p = ∏' (b) (c), f b c :=
(h.hasProd.prod_fiberwise fun b ↦ (h₁ b).hasProd).tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias tsum_prod_uncurry :=
Summable.tsum_prod_uncurry
@[to_additive existing Summable.tsum_prod_uncurry, deprecated (since := "2025-04-12")] alias
tprod_prod_uncurry := Multipliable.tprod_prod_uncurry
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_prod_uncurry | null |
protected Multipliable.tprod_comm' {f : β → γ → α} (h : Multipliable (Function.uncurry f))
(h₁ : ∀ b, Multipliable (f b)) (h₂ : ∀ c, Multipliable fun b ↦ f b c) :
∏' (c) (b), f b c = ∏' (b) (c), f b c := by
rw [← h.tprod_prod_uncurry h₁, ← h.prod_symm.tprod_prod_uncurry h₂,
← (Equiv.prodComm γ β).tprod_eq (uncurry f)]
rfl
@[deprecated (since := "2025-04-12")] alias tsum_comm':= Summable.tsum_comm'
@[to_additive existing, deprecated (since := "2025-04-12")] alias
tprod_comm' := Multipliable.tprod_comm' | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_comm' | null |
@[to_additive]
HasProd.of_sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
(hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hg : HasProd g a)
(h : CauchySeq (fun (s : Finset (Σ b : β, γ b)) ↦ ∏ i ∈ s, f i)) :
HasProd f a := by
classical
apply le_nhds_of_cauchy_adhp h
simp only [← mapClusterPt_def, mapClusterPt_iff_frequently, frequently_atTop, ge_iff_le,
le_eq_subset]
intro u hu s
rcases mem_nhds_iff.1 hu with ⟨v, vu, v_open, hv⟩
obtain ⟨t0, st0, ht0⟩ : ∃ t0, ∏ i ∈ t0, g i ∈ v ∧ s.image Sigma.fst ⊆ t0 := by
have A : ∀ᶠ t0 in (atTop : Filter (Finset β)), ∏ i ∈ t0, g i ∈ v := hg (v_open.mem_nhds hv)
exact (A.and (Ici_mem_atTop _)).exists
have L : Tendsto (fun t : Finset (Σ b, γ b) ↦ ∏ p ∈ t with p.1 ∈ t0, f p) atTop
(𝓝 <| ∏ b ∈ t0, g b) := by
simp only [← sigma_preimage_mk, prod_sigma]
refine tendsto_finset_prod _ fun b _ ↦ ?_
change
Tendsto (fun t ↦ (fun t ↦ ∏ s ∈ t, f ⟨b, s⟩) (preimage t (Sigma.mk b) _)) atTop (𝓝 (g b))
exact (hf b).comp (tendsto_finset_preimage_atTop_atTop (sigma_mk_injective))
have : ∃ t, ∏ p ∈ t with p.1 ∈ t0, f p ∈ v ∧ s ⊆ t :=
((Tendsto.eventually_mem L (v_open.mem_nhds st0)).and (Ici_mem_atTop _)).exists
obtain ⟨t, tv, st⟩ := this
refine ⟨{p ∈ t | p.1 ∈ t0}, fun x hx ↦ ?_, vu tv⟩
simpa only [mem_filter, st hx, true_and] using ht0 (mem_image_of_mem Sigma.fst hx)
variable [CompleteSpace α]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.of_sigma | null |
Multipliable.sigma_factor {γ : β → Type*} {f : (Σ b : β, γ b) → α}
(ha : Multipliable f) (b : β) :
Multipliable fun c ↦ f ⟨b, c⟩ :=
ha.comp_injective sigma_mk_injective
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.sigma_factor | null |
Multipliable.sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} (ha : Multipliable f) :
Multipliable fun b ↦ ∏' c, f ⟨b, c⟩ :=
ha.sigma' fun b ↦ ha.sigma_factor b
@[to_additive Summable.prod_factor] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.sigma | null |
Multipliable.prod_factor {f : β × γ → α} (h : Multipliable f) (b : β) :
Multipliable fun c ↦ f (b, c) :=
h.comp_injective fun _ _ h ↦ (Prod.ext_iff.1 h).2
@[to_additive Summable.prod] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.prod_factor | null |
Multipliable.prod {f : β × γ → α} (h : Multipliable f) :
Multipliable fun b ↦ ∏' c, f (b, c) :=
((Equiv.sigmaEquivProd β γ).multipliable_iff.mpr h).sigma
@[to_additive] | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.prod | null |
HasProd.tprod_fiberwise [T2Space α] {f : β → α} {a : α} (hf : HasProd f a) (g : β → γ) :
HasProd (fun c : γ ↦ ∏' b : g ⁻¹' {c}, f b) a :=
(((Equiv.sigmaFiberEquiv g).hasProd_iff).mpr hf).sigma <|
fun _ ↦ ((hf.multipliable.subtype _).hasProd_iff).mpr rfl | lemma | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasProd.tprod_fiberwise | null |
@[to_additive]
protected Multipliable.tprod_sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α}
(ha : Multipliable f) : ∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ :=
Multipliable.tprod_sigma' (fun b ↦ ha.sigma_factor b) ha
@[deprecated (since := "2025-04-12")] alias tsum_sigma := Summable.tsum_sigma
@[to_additive existing, deprecated (since := "2025-04-12")] alias
tprod_sigma := Multipliable.tprod_sigma
@[to_additive Summable.tsum_prod] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_sigma | null |
protected Multipliable.tprod_prod {f : β × γ → α} (h : Multipliable f) :
∏' p, f p = ∏' (b) (c), f ⟨b, c⟩ :=
h.tprod_prod' h.prod_factor
@[deprecated (since := "2025-04-12")] alias tsum_prod := Summable.tsum_prod
@[to_additive existing tsum_prod, deprecated (since := "2025-04-12")]
alias tprod_prod := Multipliable.tprod_prod
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_prod | null |
protected Multipliable.tprod_comm {f : β → γ → α} (h : Multipliable (Function.uncurry f)) :
∏' (c) (b), f b c = ∏' (b) (c), f b c :=
h.tprod_comm' h.prod_factor h.prod_symm.prod_factor
@[deprecated (since := "2025-04-12")] alias tsum_comm := Summable.tsum_comm
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_comm :=
Multipliable.tprod_comm | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Multipliable.tprod_comm | null |
@[to_additive]
Pi.hasProd {f : ι → ∀ x, X x} {g : ∀ x, X x} :
HasProd f g ↔ ∀ x, HasProd (fun i ↦ f i x) (g x) := by
simp only [HasProd, tendsto_pi_nhds, prod_apply]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Pi.hasProd | null |
Pi.multipliable {f : ι → ∀ x, X x} : Multipliable f ↔ ∀ x, Multipliable fun i ↦ f i x := by
simp only [Multipliable, Pi.hasProd, Classical.skolem]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Pi.multipliable | null |
tprod_apply [∀ x, T2Space (X x)] {f : ι → ∀ x, X x} {x : α} (hf : Multipliable f) :
(∏' i, f i) x = ∏' i, f i x :=
(Pi.hasProd.mp hf.hasProd x).tprod_eq.symm | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tprod_apply | null |
HasSum.op (hf : HasSum f a) : HasSum (fun a ↦ op (f a)) (op a) :=
(hf.map (@opAddEquiv α _) continuous_op :) | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasSum.op | null |
Summable.op (hf : Summable f) : Summable (op ∘ f) :=
hf.hasSum.op.summable | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Summable.op | null |
HasSum.unop {f : β → αᵐᵒᵖ} {a : αᵐᵒᵖ} (hf : HasSum f a) :
HasSum (fun a ↦ unop (f a)) (unop a) :=
(hf.map (@opAddEquiv α _).symm continuous_unop :) | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasSum.unop | null |
Summable.unop {f : β → αᵐᵒᵖ} (hf : Summable f) : Summable (unop ∘ f) :=
hf.hasSum.unop.summable
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Summable.unop | null |
hasSum_op : HasSum (fun a ↦ op (f a)) (op a) ↔ HasSum f a :=
⟨HasSum.unop, HasSum.op⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | hasSum_op | null |
hasSum_unop {f : β → αᵐᵒᵖ} {a : αᵐᵒᵖ} :
HasSum (fun a ↦ unop (f a)) (unop a) ↔ HasSum f a :=
⟨HasSum.op, HasSum.unop⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | hasSum_unop | null |
summable_op : (Summable fun a ↦ op (f a)) ↔ Summable f :=
⟨Summable.unop, Summable.op⟩ | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | summable_op | null |
summable_unop {f : β → αᵐᵒᵖ} : (Summable fun a ↦ unop (f a)) ↔ Summable f :=
⟨Summable.op, Summable.unop⟩ | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | summable_unop | null |
tsum_op [T2Space α] :
∑' x, op (f x) = op (∑' x, f x) := by
by_cases h : Summable f
· exact h.hasSum.op.tsum_eq
· have ho := summable_op.not.mpr h
rw [tsum_eq_zero_of_not_summable h, tsum_eq_zero_of_not_summable ho, op_zero] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tsum_op | null |
tsum_unop [T2Space α] {f : β → αᵐᵒᵖ} :
∑' x, unop (f x) = unop (∑' x, f x) :=
op_injective tsum_op.symm | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tsum_unop | null |
HasSum.star (h : HasSum f a) : HasSum (fun b ↦ star (f b)) (star a) := by
simpa only using h.map (starAddEquiv : α ≃+ α) continuous_star | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | HasSum.star | null |
Summable.star (hf : Summable f) : Summable fun b ↦ star (f b) :=
hf.hasSum.star.summable | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Summable.star | null |
Summable.ofStar (hf : Summable fun b ↦ Star.star (f b)) : Summable f := by
simpa only [star_star] using hf.star
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | Summable.ofStar | null |
summable_star_iff : (Summable fun b ↦ star (f b)) ↔ Summable f :=
⟨Summable.ofStar, Summable.star⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | summable_star_iff | null |
summable_star_iff' : Summable (star f) ↔ Summable f :=
summable_star_iff | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | summable_star_iff' | null |
tsum_star [T2Space α] : star (∑' b, f b) = ∑' b, star (f b) := by
by_cases hf : Summable f
· exact hf.hasSum.star.tsum_eq.symm
· rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt Summable.ofStar hf),
star_zero] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | tsum_star | null |
@[to_additive
/-- `Summable f` means that `f` has some (infinite) sum. Use `tsum` to get the value. -/]
Multipliable (f : β → α) : Prop :=
∃ a, HasProd f a
open scoped Classical in | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Order.Filter.AtTopBot.BigOperators",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | Multipliable | `HasProd f a` means that the (potentially infinite) product of the `f b` for `b : β` converges
to `a`.
The `atTop` filter on `Finset β` is the limit of all finite sets towards the entire type. So we take
the product over bigger and bigger sets. This product operation is invariant under reordering.
For the definition and many statements, `α` does not need to be a topological monoid. We only add
this assumption later, for the lemmas where it is relevant.
These are defined in an identical way to infinite sums (`HasSum`). For example, we say that
the function `ℕ → ℝ` sending `n` to `1 / 2` has a product of `0`, rather than saying that it does
not converge as some authors would. -/
@[to_additive /-- `HasSum f a` means that the (potentially infinite) sum of the `f b` for `b : β`
converges to `a`.
The `atTop` filter on `Finset β` is the limit of all finite sets towards the entire type. So we sum
up bigger and bigger sets. This sum operation is invariant under reordering. In particular,
the function `ℕ → ℝ` sending `n` to `(-1)^n / (n+1)` does not have a
sum for this definition, but a series which is absolutely convergent will have the correct sum.
This is based on Mario Carneiro's
[infinite sum `df-tsms` in Metamath](http://us.metamath.org/mpeuni/df-tsms.html).
For the definition and many statements, `α` does not need to be a topological monoid. We only add
this assumption later, for the lemmas where it is relevant. -/]
def HasProd (f : β → α) (a : α) : Prop :=
Tendsto (fun s : Finset β ↦ ∏ b ∈ s, f b) atTop (𝓝 a)
/-- `Multipliable f` means that `f` has some (infinite) product. Use `tprod` to get the value. |
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