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 _ _... | 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] [CommMono... | 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 : Ty... | 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 ... | 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 : (mulSup... | 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 ... | 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
@... | 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... | 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")] al... | 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 existi... | 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... | 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 := Summabl... | 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 tsu... | 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
@[depr... | 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.prod... | 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 [F... | 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 := ... | 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, h... | 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... | 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_si... | 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 ex... | 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_uncu... | 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... | 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 [← mapClu... | 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, depr... | 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 := Multiplia... | 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... | 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 ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.