Loogle!

Result

Found 282 declarations mentioning Finsupp.sum. Of these, only the first 200 are shown.

• Finsupp.sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) (g : α → M → N) : N
• MulOpposite.op_finsuppSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_16} {M : Type u_17} {N : Type u_18} [AddCommMonoid M] [Zero N] (f : ι →₀ N) (g : ι → N → M) : MulOpposite.op (f.sum g) = f.sum fun i n => MulOpposite.op (g i n)
• MulOpposite.unop_finsuppSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_16} {M : Type u_17} {N : Type u_18} [AddCommMonoid M] [Zero N] (f : ι →₀ N) (g : ι → N → Mᵐᵒᵖ) : MulOpposite.unop (f.sum g) = f.sum fun i n => MulOpposite.unop (g i n)
• Multiset.card_finsuppSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {M : Type u_8} [Zero M] (f : ι →₀ M) (g : ι → M → Multiset α) : (f.sum g).card = f.sum fun i m => (g i m).card
• Finsupp.multiset_sum_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} {h : α → M → Multiset N} : (f.sum h).sum = f.sum fun a b => (h a b).sum
• Finsupp.sum_zero_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {h : α → M → N} : Finsupp.sum 0 h = 0
• Finsupp.sum_eq_one_iff 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} (d : α →₀ ℕ) : (d.sum fun x n => n) = 1 ↔ ∃ a, d = fun₀ | a => 1
• Finsupp.sum_fun_zero 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) : (f.sum fun x x_1 => 0) = 0
• Finsupp.sum_single_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {a : α} {b : M} {h : α → M → N} (h_zero : h a 0 = 0) : (fun₀ | a => b).sum h = h a b
• Finsupp.multiset_map_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {γ : Type u_3} {β : Type u_7} {M : Type u_8} [Zero M] {f : α →₀ M} {m : β → γ} {h : α → M → Multiset β} : Multiset.map m (f.sum h) = f.sum fun a b => Multiset.map m (h a b)
• Finsupp.sum_embDomain 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {v : α →₀ M} {f : α ↪ β} {g : β → M → N} : (Finsupp.embDomain f v).sum g = v.sum fun a b => g (f a) b
• Finsupp.sum_finsetSum_comm 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {s : Finset β} (f : α →₀ M) (h : α → M → β → N) : (f.sum fun a m => ∑ b ∈ s, h a m b) = ∑ b ∈ s, f.sum fun a m => h a m b
• Finsupp.mul_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {S : Type u_15} [Zero M] [NonUnitalNonAssocSemiring S] (b : S) (s : α →₀ M) {f : α → M → S} : b * s.sum f = s.sum fun a c => b * f a c
• Finsupp.sum_mul 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {S : Type u_15} [Zero M] [NonUnitalNonAssocSemiring S] (b : S) (s : α →₀ M) {f : α → M → S} : s.sum f * b = s.sum fun a c => f a c * b
• Finsupp.sum_neg 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {G : Type u_12} [Zero M] [AddCommGroup G] {f : α →₀ M} {h : α → M → G} : (f.sum fun a b => -h a b) = -f.sum h
• Finsupp.sum_apply' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_2} {γ : Type u_3} {A : Type u_4} {B : Type u_5} [AddCommMonoid A] [AddCommMonoid B] (g : ι →₀ A) (k : ι → A → γ → B) (x : γ) : g.sum k x = g.sum fun i b => k i b x
• Finsupp.sum_single 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} [AddCommMonoid M] (f : α →₀ M) : f.sum Finsupp.single = f
• Finsupp.sum_comm 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {M' : Type u_9} {N : Type u_10} [Zero M] [Zero M'] [AddCommMonoid N] (f : α →₀ M) (g : β →₀ M') (h : α → M → β → M' → N) : (f.sum fun x v => g.sum fun x' v' => h x v x' v') = g.sum fun x' v' => f.sum fun x v => h x v x' v'
• Finsupp.sum_fintype 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] [Fintype α] (f : α →₀ M) (g : α → M → N) (h : ∀ (i : α), g i 0 = 0) : f.sum g = ∑ i, g i (f i)
• map_finsuppSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} {P : Type u_11} [Zero M] [AddCommMonoid N] [AddCommMonoid P] {H : Type u_16} [FunLike H N P] [AddMonoidHomClass H N P] (h : H) (f : α →₀ M) (g : α → M → N) : h (f.sum g) = f.sum fun a b => h (g a b)
• Finsupp.sum_add 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} {h₁ h₂ : α → M → N} : (f.sum fun a b => h₁ a b + h₂ a b) = f.sum h₁ + f.sum h₂
• Finsupp.sum_sub 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {G : Type u_12} [Zero M] [SubtractionCommMonoid G] {f : α →₀ M} {h₁ h₂ : α → M → G} : (f.sum fun a b => h₁ a b - h₂ a b) = f.sum h₁ - f.sum h₂
• Finsupp.add_sum_erase 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) (y : α) (g : α → M → N) (hyf : y ∈ f.support) : g y (f y) + (Finsupp.erase y f).sum g = f.sum g
• Finsupp.sum_congr 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} {g1 g2 : α → M → N} (h : ∀ x ∈ f.support, g1 x (f x) = g2 x (f x)) : f.sum g1 = f.sum g2
• Finsupp.sum_nsmul 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {N : Type u_10} [AddCommMonoid N] [Fintype α] (f : α →₀ ℕ) (g : α → N) : (f.sum fun a b => b • g a) = ∑ a, f a • g a
• Finsupp.onFinset_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {s : Finset α} {f : α → M} {g : α → M → N} (hf : ∀ (a : α), f a ≠ 0 → a ∈ s) (hg : ∀ (a : α), g a 0 = 0) : (Finsupp.onFinset s f hf).sum g = ∑ a ∈ s, g a (f a)
• Finsupp.single_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (s : ι →₀ M) (f : ι → M → N) (a : α) : (fun₀ | a => s.sum f) = s.sum fun d c => fun₀ | a => f d c
• Finsupp.sum_indicator_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {s : Finset α} (f : α → M) {h : α → M → N} (h_zero : ∀ a ∈ s, h a 0 = 0) : (Finsupp.indicator s fun x x_1 => f x).sum h = ∑ x ∈ s, h x (f x)
• Finsupp.sum_mapRange_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {M' : Type u_9} {N : Type u_10} [Zero M] [Zero M'] [AddCommMonoid N] {f : M → M'} {hf : f 0 = 0} {g : α →₀ M} {h : α → M' → N} (h0 : ∀ (a : α), h a 0 = 0) : (Finsupp.mapRange f hf g).sum h = g.sum fun a b => h a (f b)
• AddSubmonoidClass.finsuppSum_mem 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {S : Type u_16} [SetLike S N] [AddSubmonoidClass S N] (s : S) (f : α →₀ M) (g : α → M → N) (h : ∀ (c : α), f c ≠ 0 → g c (f c) ∈ s) : f.sum g ∈ s
• Finsupp.sum_ite_self_eq 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} [DecidableEq α] {N : Type u_16} [AddCommMonoid N] (f : α →₀ N) (a : α) : (f.sum fun x v => if a = x then v else 0) = f a
• Finsupp.sum_ite_self_eq' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} [DecidableEq α] {N : Type u_16} [AddCommMonoid N] (f : α →₀ N) (a : α) : (f.sum fun x v => if x = a then v else 0) = f a
• Finsupp.add_sum_erase' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) (y : α) (g : α → M → N) (hg : ∀ (i : α), g i 0 = 0) : g y (f y) + (Finsupp.erase y f).sum g = f.sum g
• Finsupp.sum_onFinset 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (s : Finset α) (f : α → M) (hf : ∀ (a : α), f a ≠ 0 → a ∈ s) (g : α → M → N) (hg : ∀ i ∈ s, g i 0 = 0) : (Finsupp.onFinset s f hf).sum g = ∑ a ∈ s, g a (f a)
• Finsupp.sum_of_support_subset 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) {s : Finset α} (hs : f.support ⊆ s) (g : α → M → N) (h : ∀ i ∈ s, g i 0 = 0) : f.sum g = ∑ x ∈ s, g x (f x)
• Finsupp.sum_zsmul 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {N : Type u_16} [SubtractionCommMonoid N] [Fintype α] (f : α →₀ ℤ) (g : α → N) : (f.sum fun a b => b • g a) = ∑ a, f a • g a
• Finsupp.sum_unique 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] [Unique α] {f : α →₀ M} {g : α → M → N} (h₁ : f default = 0 → g default 0 = 0) : f.sum g = g default (f default)
• Finsupp.sum_ite_eq 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) : (f.sum fun x v => if a = x then b x v else 0) = if a ∈ f.support then b a (f a) else 0
• Finsupp.sum_ite_eq' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) : (f.sum fun x v => if x = a then b x v else 0) = if a ∈ f.support then b a (f a) else 0
• Finsupp.support_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [DecidableEq β] [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M → β →₀ N} : (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support
• Finsupp.sum_neg_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {G : Type u_12} [SubtractionMonoid G] [AddCommMonoid M] {g : α →₀ G} {h : α → G → M} (h0 : ∀ (a : α), h a 0 = 0) : (-g).sum h = g.sum fun a b => h a (-b)
• Finsupp.sum_apply 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M → β →₀ N} {a₂ : β} : (f.sum g) a₂ = f.sum fun a₁ b => (g a₁ b) a₂
• Finsupp.coe_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] (f : α →₀ M) (g : α → M → β →₀ N) : ⇑(f.sum g) = f.sum fun a₁ b => ⇑(g a₁ b)
• AddMonoidHom.finsuppSum_apply 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {N : Type u_10} {P : Type u_11} [Zero β] [AddZeroClass N] [AddCommMonoid P] (f : α →₀ β) (g : α → β → N →+ P) (x : N) : (f.sum g) x = f.sum fun i fi => (g i fi) x
• AddMonoidHom.coe_finsuppSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {N : Type u_10} {P : Type u_11} [Zero β] [AddZeroClass N] [AddCommMonoid P] (f : α →₀ β) (g : α → β → N →+ P) : ⇑(f.sum g) = f.sum fun i fi => ⇑(g i fi)
• Finsupp.equivFunOnFinite_symm_sum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} [Fintype α] [AddCommMonoid M] (f : α → M) : ((Finsupp.equivFunOnFinite.symm f).sum fun x n => n) = ∑ a, f a
• Finsupp.sum_eq_single 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} (a : α) {g : α → M → N} (h₀ : ∀ (b : α), f b ≠ 0 → b ≠ a → g b (f b) = 0) (h₁ : f a = 0 → g a 0 = 0) : f.sum g = g a (f a)
• Finsupp.sum_add_eq_sum_add_of_exists 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M → N} {n₁ n₂ : N} (a : α) (ha : a ∈ f.support) (h : g a (f a) + n₁ = g a (f a) + n₂) : f.sum g + n₁ = f.sum g + n₂
• Finsupp.sum_finsetSum 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {A : Type u_4} {B : Type u_5} [AddCommMonoid A] [AddCommMonoid B] (f : ι → α →₀ A) (s : Finset ι) (g : α → A → B) (h₁ : ∀ (a : α), g a 0 = 0) (h₂ : ∀ (a : α) (m₁ m₂ : A), g a (m₁ + m₂) = g a m₁ + g a m₂) : (∑ i ∈ s, f i).sum g = ∑ i ∈ s, (f i).sum g
• Finsupp.sum_apply'' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_2} {γ : Type u_3} {B : Type u_5} [AddCommMonoid B] {A : Type u_16} {F : Type u_17} [AddZeroClass A] [AddCommMonoid F] [FunLike F γ B] (g : ι →₀ A) (k : ι → A → F) (x : γ) (h0 : 0 x = 0) (hadd : ∀ (f g : F), (f + g) x = f x + g x) : (g.sum k) x = g.sum fun i a => (k i a) x
• Finsupp.sum_finsetSum_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {M : Type u_8} {N : Type u_10} [AddCommMonoid M] [AddCommMonoid N] {s : Finset ι} {g : ι → α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 0) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) : ∑ i ∈ s, (g i).sum h = (∑ i ∈ s, g i).sum h
• Finsupp.sum_finset_sum_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {M : Type u_8} {N : Type u_10} [AddCommMonoid M] [AddCommMonoid N] {s : Finset ι} {g : ι → α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 0) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) : ∑ i ∈ s, (g i).sum h = (∑ i ∈ s, g i).sum h
• Finsupp.sum_indicator_index_eq_sum_attach 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] {s : Finset α} (f : (a : α) → a ∈ s → M) {h : α → M → N} (h_zero : ∀ a ∈ s, h a 0 = 0) : (Finsupp.indicator s f).sum h = ∑ x ∈ s.attach, h (↑x) (f ↑x ⋯)
• Finsupp.sum_single_add_single 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_2} {A : Type u_4} {B : Type u_5} [AddCommMonoid A] [AddCommMonoid B] (f₁ f₂ : ι) (g₁ g₂ : A) (F : ι → A → B) (H : f₁ ≠ f₂) (HF : ∀ (f : ι), F f 0 = 0) : ((fun₀ | f₁ => g₁) + fun₀ | f₂ => g₂).sum F = F f₁ g₁ + F f₂ g₂
• Finsupp.prod_sum_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} {P : Type u_11} [Zero M] [AddCommMonoid N] [CommMonoid P] {f : α →₀ M} {g : α → M → β →₀ N} {h : β → N → P} (h_zero : ∀ (a : β), h a 0 = 1) (h_add : ∀ (a : β) (b₁ b₂ : N), h a (b₁ + b₂) = h a b₁ * h a b₂) : (f.sum g).prod h = f.prod fun a b => (g a b).prod h
• Finsupp.sum_sum_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} {P : Type u_11} [Zero M] [AddCommMonoid N] [AddCommMonoid P] {f : α →₀ M} {g : α → M → β →₀ N} {h : β → N → P} (h_zero : ∀ (a : β), h a 0 = 0) (h_add : ∀ (a : β) (b₁ b₂ : N), h a (b₁ + b₂) = h a b₁ + h a b₂) : (f.sum g).sum h = f.sum fun a b => (g a b).sum h
• Finsupp.multiset_sum_sum_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddCommMonoid M] [AddCommMonoid N] (f : Multiset (α →₀ M)) (h : α → M → N) (h₀ : ∀ (a : α), h a 0 = 0) (h₁ : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) : f.sum.sum h = (Multiset.map (fun g => g.sum h) f).sum
• Finsupp.sum_add_index' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddZeroClass M] [AddCommMonoid N] {f g : α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 0) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) : (f + g).sum h = f.sum h + g.sum h
• Finsupp.sum_add_index_of_disjoint 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} [AddCommMonoid M] {f1 f2 : α →₀ M} (hd : Disjoint f1.support f2.support) {β : Type u_16} [AddCommMonoid β] (g : α → M → β) : (f1 + f2).sum g = f1.sum g + f2.sum g
• Finsupp.sum_sub_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {γ : Type u_3} {β : Type u_7} [AddGroup β] [AddCommGroup γ] {f g : α →₀ β} {h : α → β → γ} (h_sub : ∀ (a : α) (b₁ b₂ : β), h a (b₁ - b₂) = h a b₁ - h a b₂) : (f - g).sum h = f.sum h - g.sum h
• Finsupp.sum_congr_of_eq_on_union 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [AddCommMonoid N] [DecidableEq α] {f1 f2 : α →₀ M} {g1 g2 : α → M → N} (h : ∀ x ∈ f1.support ∪ f2.support, g1 x (f1 x) = g2 x (f2 x)) (h1 : ∀ x ∈ f1.support ∪ f2.support, g1 x 0 = 0) (h2 : ∀ x ∈ f1.support ∪ f2.support, g2 x 0 = 0) : f1.sum g1 = f2.sum g2
• Finsupp.sum_hom_add_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddZeroClass M] [AddCommMonoid N] {f g : α →₀ M} (h : α → M →+ N) : ((f + g).sum fun x => ⇑(h x)) = (f.sum fun x => ⇑(h x)) + g.sum fun x => ⇑(h x)
• Finsupp.sum_add_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [DecidableEq α] [AddZeroClass M] [AddCommMonoid N] {f g : α →₀ M} {h : α → M → N} (h_zero : ∀ a ∈ f.support ∪ g.support, h a 0 = 0) (h_add : ∀ a ∈ f.support ∪ g.support, ∀ (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) : (f + g).sum h = f.sum h + g.sum h
• Finsupp.liftAddHom_apply 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddZeroClass M] [AddCommMonoid N] (F : α → M →+ N) (f : α →₀ M) : (Finsupp.liftAddHom F) f = f.sum fun x => ⇑(F x)
• Nat.cast_finsupp_sum 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [AddCommMonoidWithOne R] (g : α → M → ℕ) : ↑(f.sum g) = f.sum fun a b => ↑(g a b)
• Int.cast_finsupp_sum 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [AddCommGroupWithOne R] (g : α → M → ℤ) : ↑(f.sum g) = f.sum fun a b => ↑(g a b)
• Rat.cast_finsupp_sum 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [DivisionRing R] [CharZero R] (g : α → M → ℚ) : ↑(f.sum g) = f.sum fun a b => ↑(g a b)
• Finsupp.sum_equivMapDomain 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [AddCommMonoid N] (f : α ≃ β) (l : α →₀ M) (g : β → M → N) : (Finsupp.equivMapDomain f l).sum g = l.sum fun a m => g (f a) m
• Finsupp.sum_filter_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] (p : α → Prop) [DecidablePred p] (f : α →₀ M) [AddCommMonoid N] (g : α → M → N) : (Finsupp.filter p f).sum g = ∑ x ∈ (Finsupp.filter p f).support, g x (f x)
• Finsupp.sum_subtypeDomain_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] {p : α → Prop} [AddCommMonoid N] {v : α →₀ M} {h : α → M → N} (hp : ∀ x ∈ v.support, p x) : ((Finsupp.subtypeDomain p v).sum fun a b => h (↑a) b) = v.sum h
• Finsupp.sum_mapDomain_index_inj 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] {f : α → β} {s : α →₀ M} {h : β → M → N} (hf : Function.Injective f) : (Finsupp.mapDomain f s).sum h = s.sum fun a b => h (f a) b
• Finsupp.sigma_sum 📋 Mathlib.Data.Finsupp.Basic
  {ι : Type u_4} {M : Type u_5} {N : Type u_6} {αs : ι → Type u_12} [Zero M] (l : (i : ι) × αs i →₀ M) [AddCommMonoid N] (f : (i : ι) × αs i → M → N) : l.sum f = ∑ i ∈ l.splitSupport, (l.split i).sum fun a b => f ⟨i, a⟩ b
• Finsupp.sum_curry_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [AddCommMonoid N] (f : α × β →₀ M) (g : α → β → M → N) : (f.curry.sum fun a f => f.sum (g a)) = f.sum fun p c => g p.1 p.2 c
• Finsupp.sum_uncurry_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [AddCommMonoid N] (f : α →₀ β →₀ M) (g : α × β → M → N) : (f.uncurry.sum fun p c => g p c) = f.sum fun a f => f.sum fun b => g (a, b)
• Finsupp.sum_uncurry_index' 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [AddCommMonoid N] (f : α →₀ β →₀ M) (g : α → β → M → N) : (f.uncurry.sum fun p c => g p.1 p.2 c) = f.sum fun a f => f.sum (g a)
• Finsupp.sum_filter_add_sum_filter_not 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] (p : α → Prop) [DecidablePred p] (f : α →₀ M) [AddCommMonoid N] (g : α → M → N) : (Finsupp.filter p f).sum g + (Finsupp.filter (fun a => ¬p a) f).sum g = f.sum g
• Finsupp.sum_sub_sum_filter 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {G : Type u_8} [Zero M] (p : α → Prop) [DecidablePred p] (f : α →₀ M) [AddCommGroup G] (g : α → M → G) : f.sum g - (Finsupp.filter p f).sum g = (Finsupp.filter (fun a => ¬p a) f).sum g
• Finsupp.sum_sumElim 📋 Mathlib.Data.Finsupp.Basic
  {ι₁ : Type u_12} {ι₂ : Type u_13} {α : Type u_14} {M : Type u_15} [Zero α] [AddCommMonoid M] (f₁ : ι₁ →₀ α) (f₂ : ι₂ →₀ α) (g : ι₁ ⊕ ι₂ → α → M) : (f₁.sumElim f₂).sum g = f₁.sum (g ∘ Sum.inl) + f₂.sum (g ∘ Sum.inr)
• Finsupp.mapDomain_sum 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [Zero N] {f : α → β} {s : α →₀ N} {v : α → N → α →₀ M} : Finsupp.mapDomain f (s.sum v) = s.sum fun a b => Finsupp.mapDomain f (v a b)
• Finsupp.sum_comapDomain 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [AddCommMonoid N] (f : α → β) (l : β →₀ M) (g : β → M → N) (hf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support) : (Finsupp.comapDomain f l ⋯).sum (g ∘ f) = l.sum g
• Finsupp.subtypeDomain_finsupp_sum 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] {p : α → Prop} [Zero N] {s : β →₀ N} {h : β → N → α →₀ M} : Finsupp.subtypeDomain p (s.sum h) = s.sum fun c d => Finsupp.subtypeDomain p (h c d)
• Finsupp.sum_mapDomain_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] {f : α → β} {s : α →₀ M} {h : β → M → N} (h_zero : ∀ (b : β), h b 0 = 0) (h_add : ∀ (b : β) (m₁ m₂ : M), h b (m₁ + m₂) = h b m₁ + h b m₂) : (Finsupp.mapDomain f s).sum h = s.sum fun a m => h (f a) m
• Finsupp.sum_mapDomain_index_addMonoidHom 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] {f : α → β} {s : α →₀ M} (h : β → M →+ N) : ((Finsupp.mapDomain f s).sum fun b m => (h b) m) = s.sum fun a m => (h (f a)) m
• Finsupp.sum_update_add 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {ι : Type u_4} [AddZeroClass α] [AddCommMonoid β] (f : ι →₀ α) (i : ι) (a : α) (g : ι → α → β) (hg : ∀ (i : ι), g i 0 = 0) (hgg : ∀ (j : ι) (a₁ a₂ : α), g j (a₁ + a₂) = g j a₁ + g j a₂) : (f.update i a).sum g + g i (f i) = f.sum g + g i a
• Finsupp.sum_option_index 📋 Mathlib.Data.Finsupp.Option
  {α : Type u_1} {M : Type u_2} {N : Type u_3} [AddZeroClass M] [AddCommMonoid N] (f : Option α →₀ M) (b : Option α → M → N) (h_zero : ∀ (o : Option α), b o 0 = 0) (h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ + b o m₂) : f.sum b = b none (f none) + f.some.sum fun a => b (some a)
• Finsupp.sum_option_index_smul 📋 Mathlib.Data.Finsupp.Option
  {α : Type u_1} {M : Type u_2} {R : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (f : Option α →₀ R) (b : Option α → M) : (f.sum fun o r => r • b o) = f none • b none + f.some.sum fun a r => r • b (some a)
• Finsupp.sum_smul_index' 📋 Mathlib.Data.Finsupp.SMul
  {α : Type u_1} {M : Type u_3} {N : Type u_4} {R : Type u_6} [Zero M] [SMulZeroClass R M] [AddCommMonoid N] {g : α →₀ M} {b : R} {h : α → M → N} (h0 : ∀ (i : α), h i 0 = 0) : (b • g).sum h = g.sum fun i c => h i (b • c)
• Finsupp.sum_smul_index 📋 Mathlib.Data.Finsupp.SMul
  {α : Type u_1} {M : Type u_3} {R : Type u_6} [MulZeroClass R] [AddCommMonoid M] {g : α →₀ R} {b : R} {h : α → R → M} (h0 : ∀ (i : α), h i 0 = 0) : (b • g).sum h = g.sum fun i a => h i (b * a)
• Finsupp.sum_smul_index_addMonoidHom 📋 Mathlib.Data.Finsupp.SMul
  {α : Type u_1} {M : Type u_3} {N : Type u_4} {R : Type u_6} [AddZeroClass M] [AddCommMonoid N] [SMulZeroClass R M] {g : α →₀ M} {b : R} {h : α → M →+ N} : ((b • g).sum fun a => ⇑(h a)) = g.sum fun i c => (h i) (b • c)
• Finsupp.smul_sum 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {α : Type u_1} {β : Type u_2} {R : Type u_3} {M : Type u_5} [Zero β] [AddCommMonoid M] [DistribSMul R M] {v : α →₀ β} {c : R} {h : α → β → M} : c • v.sum h = v.sum fun a b => c • h a b
• Submodule.finsuppSum_mem 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  (R : Type u_1) {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_4} {β : Type u_5} [Zero β] (S : Submodule R M) (f : ι →₀ β) (g : ι → β → M) (h : ∀ (c : ι), f c ≠ 0 → g c (f c) ∈ S) : f.sum g ∈ S
• LinearMap.finsupp_sum_apply 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {R : Type u_4} {R₂ : Type u_5} {M : Type u_6} {M₂ : Type u_7} {ι : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] {σ₁₂ : R →+* R₂} [Module R M] [Module R₂ M₂] {γ : Type u_9} [Zero γ] (t : ι →₀ γ) (g : ι → γ → M →ₛₗ[σ₁₂] M₂) (b : M) : (t.sum g) b = t.sum fun i d => (g i d) b
• LinearMap.coe_finsupp_sum 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {R : Type u_4} {R₂ : Type u_5} {M : Type u_6} {M₂ : Type u_7} {ι : Type u_8} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] {σ₁₂ : R →+* R₂} [Module R M] [Module R₂ M₂] {γ : Type u_9} [Zero γ] (t : ι →₀ γ) (g : ι → γ → M →ₛₗ[σ₁₂] M₂) : ⇑(t.sum g) = ⇑(t.sum fun i d => g i d)
• Finsupp.sum_smul_index_linearMap' 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {α : Type u_1} {R : Type u_3} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂} : ((c • v).sum fun a => ⇑(h a)) = c • v.sum fun a => ⇑(h a)
• Finsupp.sum_smul_index_semilinearMap' 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {α : Type u_1} {R : Type u_3} {R₂ : Type u_4} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] {σ : R →+* R₂} {v : α →₀ M} {c : R} {h : α → M →ₛₗ[σ] M₂} : ((c • v).sum fun a => ⇑(h a)) = σ c • v.sum fun a => ⇑(h a)
• Finsupp.lift_apply 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  (M : Type u_2) (R : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] (X : Type u_7) (f : X → M) (g : X →₀ R) : ((Finsupp.lift M R X) f) g = g.sum fun x r => r • f x
• Submodule.mulLeftMap_apply 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {R : Type u_1} [Semiring R] {S : Type u_4} [Semiring S] [Module R S] [SMulCommClass R R S] [SMulCommClass R S S] {M N : Submodule R S} {ι : Type u_5} (m : ι → ↥M) (n : ι →₀ ↥N) : (Submodule.mulLeftMap N m) n = n.sum fun i n => ↑(m i) * ↑n
• Submodule.mulRightMap_apply 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {R : Type u_1} [Semiring R] {S : Type u_4} [Semiring S] [Module R S] [SMulCommClass R R S] [IsScalarTower R S S] {M N : Submodule R S} {ι : Type u_5} (n : ι → ↥N) (m : ι →₀ ↥M) : (M.mulRightMap n) m = m.sum fun i m => ↑m * ↑(n i)
• Finsupp.lsum_apply 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} {R₂ : Type u_5} (S : Type u_6) [Semiring R] [Semiring R₂] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R₂ N] {σ : R →+* R₂} [Module S N] [SMulCommClass R₂ S N] (f : α → M →ₛₗ[σ] N) (l : α →₀ M) : ((Finsupp.lsum S) f) l = l.sum fun b => ⇑(f b)
• Finsupp.coe_lsum 📋 Mathlib.LinearAlgebra.Finsupp.LSum
  {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} {R₂ : Type u_5} (S : Type u_6) [Semiring R] [Semiring R₂] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R₂ N] {σ : R →+* R₂} [Module S N] [SMulCommClass R₂ S N] (f : α → M →ₛₗ[σ] N) : ⇑((Finsupp.lsum S) f) = fun d => d.sum fun i => ⇑(f i)
• Finsupp.mem_span_range_iff_exists_finsupp 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {α : Type u_1} {M : Type u_2} {R : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} {x : M} : x ∈ Submodule.span R (Set.range v) ↔ ∃ c, (c.sum fun i a => a • v i) = x
• Submodule.mem_span_set 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {m : M} {s : Set M} : m ∈ Submodule.span R s ↔ ∃ c, ↑c.support ⊆ s ∧ (c.sum fun mi r => r • mi) = m
• Finsupp.linearCombination_apply 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {α : Type u_1} {M : Type u_2} (R : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} (l : α →₀ R) : (Finsupp.linearCombination R v) l = l.sum fun i a => a • v i
• Module.Basis.coe_sumCoords 📋 Mathlib.LinearAlgebra.Basis.Defs
  {ι : Type u_7} {R : Type u_8} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] (b : Module.Basis ι R M) : ⇑b.sumCoords = fun m => (b.repr m).sum fun x => id
• Module.Basis.constr_apply 📋 Mathlib.LinearAlgebra.Basis.Defs
  {M' : Type u_5} [AddCommMonoid M'] {ι : Type u_7} {R : Type u_8} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] (b : Module.Basis ι R M) [Module R M'] (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : ι → M') (x : M) : ((b.constr S) f) x = (b.repr x).sum fun b a => a • f b
• not_linearIndependent_iff_finsupp 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
  {ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : ¬LinearIndependent R v ↔ ∃ f, (f.sum fun x r => r • v x) = 0 ∧ f ≠ 0
• Module.Basis.mem_submodule_iff 📋 Mathlib.LinearAlgebra.Basis.Submodule
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] {P : Submodule R M} (b : Module.Basis ι R ↥P) {x : M} : x ∈ P ↔ ∃ c, x = c.sum fun i x => x • ↑(b i)
• Submodule.mem_iSup_iff_exists_finsupp 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [Semiring R] [AddCommMonoid N] [Module R N] (p : ι → Submodule R N) (x : N) : x ∈ iSup p ↔ ∃ f, (∀ (i : ι), f i ∈ p i) ∧ (f.sum fun _i xi => xi) = x
• AddMonoidAlgebra.sum_coeff_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : AddMonoidAlgebra R M) : f.coeff.sum AddMonoidAlgebra.single = f
• AddMonoidAlgebra.sum_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : AddMonoidAlgebra R M) : f.coeff.sum AddMonoidAlgebra.single = f
• MonoidAlgebra.sum_coeff_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : MonoidAlgebra R M) : f.coeff.sum MonoidAlgebra.single = f
• MonoidAlgebra.sum_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : MonoidAlgebra R M) : f.coeff.sum MonoidAlgebra.single = f
• AddMonoidAlgebra.sum_single_index 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] {m : M} {r : R} {h : M → R → N} (h_zero : h m 0 = 0) : (AddMonoidAlgebra.single m r).coeff.sum h = h m r
• MonoidAlgebra.sum_single_index 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] {m : M} {r : R} {h : M → R → N} (h_zero : h m 0 = 0) : (MonoidAlgebra.single m r).coeff.sum h = h m r
• AddMonoidAlgebra.coeff_finsuppSum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [Semiring R] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → AddMonoidAlgebra R M) : (f.sum g).coeff = f.sum fun i n => (g i n).coeff
• AddMonoidAlgebra.ofCoeff_finsuppSum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [Semiring R] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → M →₀ R) : AddMonoidAlgebra.ofCoeff (f.sum g) = f.sum fun i n => AddMonoidAlgebra.ofCoeff (g i n)
• MonoidAlgebra.coeff_finsuppSum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [Semiring R] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → MonoidAlgebra R M) : (f.sum g).coeff = f.sum fun i n => (g i n).coeff
• MonoidAlgebra.ofCoeff_finsuppSum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [Semiring R] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → M →₀ R) : MonoidAlgebra.ofCoeff (f.sum g) = f.sum fun i n => MonoidAlgebra.ofCoeff (g i n)
• AddMonoidAlgebra.mul_def 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (x y : AddMonoidAlgebra R M) : x * y = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => AddMonoidAlgebra.single (m₁ + m₂) (r₁ * r₂)
• MonoidAlgebra.mul_def 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Mul M] (x y : MonoidAlgebra R M) : x * y = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => MonoidAlgebra.single (m₁ * m₂) (r₁ * r₂)
• AddMonoidAlgebra.finsuppProd_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] (f : ι →₀ N) (m : ι → N → M) (r : ι → N → R) : (f.prod fun i n => AddMonoidAlgebra.single (m i n) (r i n)) = AddMonoidAlgebra.single (f.sum m) (f.prod r)
• AddMonoidAlgebra.coeff_mul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] [DecidableEq M] (x y : AddMonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ + m₂ = m then r₁ * r₂ else 0
• AddMonoidAlgebra.mul_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] [DecidableEq M] (x y : AddMonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ + m₂ = m then r₁ * r₂ else 0
• MonoidAlgebra.coeff_mul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Mul M] [DecidableEq M] (x y : MonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ * m₂ = m then r₁ * r₂ else 0
• MonoidAlgebra.mul_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Mul M] [DecidableEq M] (x y : MonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ * m₂ = m then r₁ * r₂ else 0
• AddMonoidAlgebra.coeff_mul_apply_left 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x y : AddMonoidAlgebra R G) (g : G) : (x * y).coeff g = x.coeff.sum fun h r => r * y.coeff (-h + g)
• AddMonoidAlgebra.coeff_mul_apply_right 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x y : AddMonoidAlgebra R G) (g : G) : (x * y).coeff g = y.coeff.sum fun h r => x.coeff (g + -h) * r
• MonoidAlgebra.coeff_mul_apply_left 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [Group G] (x y : MonoidAlgebra R G) (g : G) : (x * y).coeff g = x.coeff.sum fun h r => r * y.coeff (h⁻¹ * g)
• MonoidAlgebra.coeff_mul_apply_right 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [Group G] (x y : MonoidAlgebra R G) (g : G) : (x * y).coeff g = y.coeff.sum fun h r => x.coeff (g * h⁻¹) * r
• MonoidAlgebra.mul_apply_left 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [Group G] (x y : MonoidAlgebra R G) (g : G) : (x * y).coeff g = x.coeff.sum fun h r => r * y.coeff (h⁻¹ * g)
• MonoidAlgebra.mul_apply_right 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [Group G] (x y : MonoidAlgebra R G) (g : G) : (x * y).coeff g = y.coeff.sum fun h r => x.coeff (g * h⁻¹) * r
• AddMonoidAlgebra.mapDomain_sum 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] (f : M → N) (x : AddMonoidAlgebra S M) (v : M → S → AddMonoidAlgebra R M) : AddMonoidAlgebra.mapDomain f (x.coeff.sum v) = x.coeff.sum fun a b => AddMonoidAlgebra.mapDomain f (v a b)
• MonoidAlgebra.mapDomain_sum 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] (f : M → N) (x : MonoidAlgebra S M) (v : M → S → MonoidAlgebra R M) : MonoidAlgebra.mapDomain f (x.coeff.sum v) = x.coeff.sum fun a b => MonoidAlgebra.mapDomain f (v a b)
• Submodule.mem_set_smul 📋 Mathlib.Algebra.Module.Submodule.Finsupp
  {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (sR : Set R) (N : Submodule R M) (x : M) [SMulCommClass R R ↥N] : x ∈ sR • N ↔ ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m => r • m)
• Finsupp.mem_ideal_span_range_iff_exists_finsupp 📋 Mathlib.RingTheory.Ideal.Operations
  {α : Type u_1} {R : Type u_2} [Semiring R] {x : R} {v : α → R} : x ∈ Ideal.span (Set.range v) ↔ ∃ c, (c.sum fun i a => a * v i) = x
• Submodule.mem_ideal_smul_span_iff_exists_sum 📋 Mathlib.RingTheory.Ideal.Operations
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) [I.IsTwoSided] {ι : Type u_4} (f : ι → M) (x : M) : x ∈ I • Submodule.span R (Set.range f) ↔ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (a.sum fun i c => c • f i) = x
• Submodule.mem_ideal_smul_span_iff_exists_sum' 📋 Mathlib.RingTheory.Ideal.Operations
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (I : Ideal R) [I.IsTwoSided] {ι : Type u_4} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • Submodule.span R (f '' s) ↔ ∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ I), (a.sum fun i c => c • f ↑i) = x
• Ideal.finsuppTotal_apply 📋 Mathlib.RingTheory.Ideal.Operations
  {ι : Type u_1} {M : Type u_2} [AddCommGroup M] {R : Type u_3} [CommRing R] [Module R M] (I : Ideal R) {v : ι → M} (f : ι →₀ ↥I) : (Ideal.finsuppTotal ι M I v) f = f.sum fun i x => ↑x • v i
• Submodule.mem_span_set_iff_exists_finsupp_le_finrank 📋 Mathlib.LinearAlgebra.Dimension.StrongRankCondition
  {K : Type u_2} {M : Type u_3} [DivisionRing K] [AddCommGroup M] [Module K M] {s : Set M} {x : M} [Module.Finite K ↥(Submodule.span K s)] : x ∈ Submodule.span K s ↔ ∃ c, c.support.card ≤ Module.finrank K ↥(Submodule.span K s) ∧ ↑c.support ⊆ s ∧ (c.sum fun mi r => r • mi) = x
• Sym.equivNatSum 📋 Mathlib.Data.Finsupp.Multiset
  (α : Type u_1) [DecidableEq α] (n : ℕ) : Sym α n ≃ { P // (P.sum fun x => id) = n }
• Finsupp.sum_id_lt_of_lt 📋 Mathlib.Data.Finsupp.Multiset
  {ι : Type u_3} (m n : ι →₀ ℕ) (h : m < n) : (m.sum fun x => id) < n.sum fun x => id
• Multiset.toFinsupp_sum_eq 📋 Mathlib.Data.Finsupp.Multiset
  {α : Type u_1} [DecidableEq α] (s : Multiset α) : ((Multiset.toFinsupp s).sum fun x => id) = s.card
• Finsupp.card_toMultiset 📋 Mathlib.Data.Finsupp.Multiset
  {α : Type u_1} (f : α →₀ ℕ) : (Finsupp.toMultiset f).card = f.sum fun x => id
• Finsupp.sum_toMultiset 📋 Mathlib.Data.Finsupp.Multiset
  {α : Type u_1} [AddCommMonoid α] (f : α →₀ ℕ) : (Finsupp.toMultiset f).sum = f.sum fun a n => n • a
• Finsupp.toMultiset_apply 📋 Mathlib.Data.Finsupp.Multiset
  {α : Type u_1} (f : α →₀ ℕ) : Finsupp.toMultiset f = f.sum fun a n => n • {a}
• Sym.coe_equivNatSum_apply_apply 📋 Mathlib.Data.Finsupp.Multiset
  (α : Type u_1) [DecidableEq α] (n : ℕ) (s : Sym α n) (a : α) : ↑((Sym.equivNatSum α n) s) a = Multiset.count a ↑s
• Sym.coe_equivNatSum_symm_apply 📋 Mathlib.Data.Finsupp.Multiset
  (α : Type u_1) [DecidableEq α] (n : ℕ) (P : { P // (P.sum fun x => id) = n }) : ↑((Sym.equivNatSum α n).symm P) = Finsupp.toMultiset ↑P
• TensorProduct.finsuppScalarLeft_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} [CommSemiring R] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ R) (n : N) : (TensorProduct.finsuppScalarLeft R N ι) (p ⊗ₜ[R] n) = p.sum fun i m => fun₀ | i => m • n
• TensorProduct.finsuppScalarRight_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ R) : (TensorProduct.finsuppScalarRight R S M ι) (m ⊗ₜ[R] p) = p.sum fun i n => fun₀ | i => n • m
• TensorProduct.finsuppRight_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ N) : (TensorProduct.finsuppRight R S M N ι) (m ⊗ₜ[R] p) = p.sum fun i n => fun₀ | i => m ⊗ₜ[R] n
• TensorProduct.finsuppLeft_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ M) (n : N) : (TensorProduct.finsuppLeft R S M N ι) (p ⊗ₜ[R] n) = p.sum fun i m => fun₀ | i => m ⊗ₜ[R] n
• TensorProduct.eq_repr_basis_left 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (ℬ : Module.Basis ι R M) (x : TensorProduct R M N) : ∃ c, (c.sum fun i n => ℬ i ⊗ₜ[R] n) = x
• TensorProduct.eq_repr_basis_right 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (𝒞 : Module.Basis κ R N) (x : TensorProduct R M N) : ∃ b, (b.sum fun i m => m ⊗ₜ[R] 𝒞 i) = x
• TensorProduct.sum_tmul_basis_left_eq_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (ℬ : Module.Basis ι R M) (b : ι →₀ N) (h : (b.sum fun i n => ℬ i ⊗ₜ[R] n) = 0) : b = 0
• TensorProduct.sum_tmul_basis_right_eq_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (𝒞 : Module.Basis κ R N) (b : κ →₀ M) (h : (b.sum fun i m => m ⊗ₜ[R] 𝒞 i) = 0) : b = 0
• TensorProduct.equivFinsuppOfBasisLeft_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq ι] (ℬ : Module.Basis ι R M) (b : ι →₀ N) : (TensorProduct.equivFinsuppOfBasisLeft ℬ).symm b = b.sum fun i n => ℬ i ⊗ₜ[R] n
• TensorProduct.equivFinsuppOfBasisRight_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq κ] (𝒞 : Module.Basis κ R N) (b : κ →₀ M) : (TensorProduct.equivFinsuppOfBasisRight 𝒞).symm b = b.sum fun i m => m ⊗ₜ[R] 𝒞 i
• star_finsuppSum 📋 Mathlib.Algebra.Star.BigOperators
  {R : Type u_1} {ι : Type u_2} {M : Type u_3} [Zero M] [AddCommMonoid R] [StarAddMonoid R] (s : ι →₀ M) (f : ι → M → R) : star (s.sum f) = s.sum fun i m => star f i m
• AddMonoidAlgebra.lift_unique 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : AddMonoidAlgebra R M →ₐ[R] A) (f : AddMonoidAlgebra R M) : F f = f.coeff.sum fun m r => r • F (AddMonoidAlgebra.single m 1)
• MonoidAlgebra.lift_unique 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] (F : MonoidAlgebra R M →ₐ[R] A) (f : MonoidAlgebra R M) : F f = f.coeff.sum fun a b => b • F (MonoidAlgebra.single a 1)
• MonoidAlgebra.lift_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] (F : M →* A) (f : MonoidAlgebra R M) : ((MonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => b • F a
• MonoidAlgebra.lift_apply' 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] (F : M →* A) (f : MonoidAlgebra R M) : ((MonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => (algebraMap R A) b * F a
• AddMonoidAlgebra.lift_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : Multiplicative M →* A) (f : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => b • F (Multiplicative.ofAdd a)
• AddMonoidAlgebra.lift_apply' 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : Multiplicative M →* A) (f : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => (algebraMap R A) b * F (Multiplicative.ofAdd a)
• Finsupp.sum_nonneg' 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedAddMonoid β] {f : ι →₀ α} {h₁ : ι → α → β} (h : ∀ (i : ι), 0 ≤ h₁ i (f i)) : 0 ≤ f.sum h₁
• Finsupp.sum_le_sum 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedAddMonoid β] {f : ι →₀ α} {h₁ h₂ : ι → α → β} (h : ∀ i ∈ f.support, h₁ i (f i) ≤ h₂ i (f i)) : f.sum h₁ ≤ f.sum h₂
• Finsupp.sum_nonneg 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedAddMonoid β] {f : ι →₀ α} {h₁ : ι → α → β} (h : ∀ i ∈ f.support, 0 ≤ h₁ i (f i)) : 0 ≤ f.sum h₁
• Finsupp.sum_nonpos 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedAddMonoid β] {f : ι →₀ α} {h₁ : ι → α → β} (h : ∀ i ∈ f.support, h₁ i (f i) ≤ 0) : f.sum h₁ ≤ 0
• Finsupp.single_eval_le_sum 📋 Mathlib.Data.Finsupp.Order
  {α : Type u_5} {M : Type u_6} {N : Type u_7} [Zero M] [AddCommMonoid N] [PartialOrder N] [IsOrderedAddMonoid N] (f : α →₀ M) {g : M → N} (hg : g 0 = 0) (h : 0 ≤ fun x => g x) (a : α) : g (f a) ≤ f.sum fun x m => g m
• Finsupp.single_le_sum 📋 Mathlib.Data.Finsupp.Order
  {α : Type u_5} {M : Type u_6} {N : Type u_7} [Zero M] [AddCommMonoid N] [PartialOrder N] [IsOrderedAddMonoid N] (f : α →₀ M) {g : α → M → N} (h : 0 ≤ fun x1 x2 => g x1 x2) (a : α) : (fun₀ | a => f a).sum g ≤ f.sum g
• Finsupp.sum_pos 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedCancelAddMonoid β] [AddLeftStrictMono β] {f : ι →₀ α} {g : ι → α → β} (h : ∀ i ∈ f.support, 0 < g i (f i)) (hf : f ≠ 0) : 0 < f.sum g
• Finsupp.sum_le_sum_index 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] [AddCommMonoid β] [Preorder β] [IsOrderedAddMonoid β] [DecidableEq ι] {f₁ f₂ : ι →₀ α} {h : ι → α → β} (hf : f₁ ≤ f₂) (hh : ∀ i ∈ f₁.support ∪ f₂.support, Monotone (h i)) (hh₀ : ∀ i ∈ f₁.support ∪ f₂.support, h i 0 = 0) : f₁.sum h ≤ f₂.sum h
• Finsupp.sum_pos' 📋 Mathlib.Data.Finsupp.Order
  {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [AddCommMonoid β] [Preorder β] [IsOrderedCancelAddMonoid β] [AddLeftStrictMono β] {f : ι →₀ α} {g : ι → α → β} (h : ∀ i ∈ f.support, 0 ≤ g i (f i)) (hf : ∃ i ∈ f.support, 0 < g i (f i)) : 0 < f.sum g
• MvPolynomial.sum_def 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {p : MvPolynomial σ R} {b : (σ →₀ ℕ) → R → A} : p.coeff.sum b = ∑ m ∈ p.support, b m (MvPolynomial.coeff m p)
• MvPolynomial.sum_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} {a : R} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {b : (σ →₀ ℕ) → R → A} (w : b 0 0 = 0) : (MvPolynomial.C a).coeff.sum b = b 0 a
• MvPolynomial.sum_monomial_eq 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {u : σ →₀ ℕ} {r : R} {b : (σ →₀ ℕ) → R → A} (w : b u 0 = 0) : ((MvPolynomial.monomial u) r).coeff.sum b = b u r
• MvPolynomial.mul_def 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} : p * q = p.coeff.sum fun m a => q.coeff.sum fun n b => (MvPolynomial.monomial (m + n)) (a * b)
• MvPolynomial.monomial_finsupp_sum_index 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {α : Type u_2} {β : Type u_3} [Zero β] (f : α →₀ β) (g : α → β → σ →₀ ℕ) (a : R) : (MvPolynomial.monomial (f.sum g)) a = MvPolynomial.C a * f.prod fun a b => (MvPolynomial.monomial (g a b)) 1
• Finsupp.sum_cons 📋 Mathlib.Algebra.BigOperators.Finsupp.Fin
  {M : Type u_1} [AddCommMonoid M] (n : ℕ) (σ : Fin n →₀ M) (i : M) : ((Finsupp.cons i σ).sum fun x e => e) = i + σ.sum fun x e => e
• Finsupp.sum_cons' 📋 Mathlib.Algebra.BigOperators.Finsupp.Fin
  {M : Type u_1} {N : Type u_2} [Zero M] [AddCommMonoid N] (n : ℕ) (σ : Fin n →₀ M) (i : M) (f : Fin (n + 1) → M → N) (h : ∀ (x : Fin (n + 1)), f x 0 = 0) : (Finsupp.cons i σ).sum f = f 0 i + σ.sum (Fin.tail f)
• finTwoArrowEquiv'_sum_eq 📋 Mathlib.Algebra.BigOperators.Finsupp.Fin
  {M : Type u_1} {d : M × M} [AddCommMonoid M] : (((finTwoArrowEquiv' M).symm d).sum fun x n => n) = d.1 + d.2
• MvPolynomial.le_totalDegree 📋 Mathlib.Algebra.MvPolynomial.Degrees
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {s : σ →₀ ℕ} (h : s ∈ p.support) : (s.sum fun x e => e) ≤ p.totalDegree
• MvPolynomial.totalDegree_monomial_le 📋 Mathlib.Algebra.MvPolynomial.Degrees
  {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) (c : R) : ((MvPolynomial.monomial s) c).totalDegree ≤ s.sum fun x => id
• MvPolynomial.totalDegree_monomial 📋 Mathlib.Algebra.MvPolynomial.Degrees
  {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) {c : R} (hc : c ≠ 0) : ((MvPolynomial.monomial s) c).totalDegree = s.sum fun x e => e
• Finsupp.weight_apply 📋 Mathlib.Data.Finsupp.Weight
  {σ : Type u_1} {M : Type u_2} {R : Type u_3} [Semiring R] (w : σ → M) [AddCommMonoid M] [Module R M] (f : σ →₀ R) : (Finsupp.weight w) f = f.sum fun i c => c • w i
• TensorProduct.exists_finsupp_left 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R M N) : ∃ S, x = S.sum fun m n => m ⊗ₜ[R] n
• TensorProduct.exists_finsupp_right 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R M N) : ∃ S, x = S.sum fun n m => m ⊗ₜ[R] n
• AddSubgroup.exists_finsupp_of_mem_closure_range 📋 Mathlib.Algebra.Group.Subgroup.Finsupp
  {M : Type u_1} [AddCommGroup M] {ι : Type u_2} (f : ι → M) (x : M) (hx : x ∈ AddSubgroup.closure (Set.range f)) : ∃ a, x = a.sum fun x1 x2 => x2 • f x1
• AddSubgroup.mem_closure_range_iff 📋 Mathlib.Algebra.Group.Subgroup.Finsupp
  {M : Type u_1} [AddCommGroup M] {ι : Type u_2} {f : ι → M} {x : M} : x ∈ AddSubgroup.closure (Set.range f) ↔ ∃ a, x = a.sum fun x1 x2 => x2 • f x1
• Finset.mem_finsuppAntidiag' 📋 Mathlib.Algebra.Order.Antidiag.Finsupp
  {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {s : Finset ι} {n : μ} {f : ι →₀ μ} : f ∈ s.finsuppAntidiag n ↔ (f.sum fun x x_1 => x_1) = n ∧ f.support ⊆ s
• CategoryTheory.Free.lift_map 📋 Mathlib.Algebra.Category.ModuleCat.Adjunctions
  (R : Type u_1) [CommRing R] {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] [CategoryTheory.Preadditive D] [CategoryTheory.Linear R D] (F : CategoryTheory.Functor C D) {x✝ x✝¹ : CategoryTheory.Free R C} (f : x✝ ⟶ x✝¹) : (CategoryTheory.Free.lift R F).map f = Finsupp.sum f fun f' r => r • F.map f'
• LaurentPolynomial.smeval_eq_sum 📋 Mathlib.Algebra.Polynomial.Laurent
  {R : Type u_1} {S : Type u_2} [Semiring R] [AddCommMonoid S] [SMulWithZero R S] [Monoid S] (f : LaurentPolynomial R) (x : Sˣ) : f.smeval x = f.coeff.sum fun n r => r • ↑(x ^ n)
• PolynomialModule.coeff_finsuppSum 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → PolynomialModule R M) : (f.sum g).coeff = f.sum fun i n => (g i n).coeff
• PolynomialModule.ofCoeff_finsuppSum 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → ℕ →₀ M) : { coeff := f.sum g } = f.sum fun i n => { coeff := g i n }
• PolynomialModule.eval_apply 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_2} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (r : R) (p : PolynomialModule R M) : (PolynomialModule.eval r) p = p.coeff.sum fun i m => r ^ i • m
• LinearMap.sum_repr_mul_repr_mulₛₗ 📋 Mathlib.LinearAlgebra.Basis.Bilinear
  {ι₁ : Type u_1} {ι₂ : Type u_2} {R : Type u_3} {R₂ : Type u_4} {S : Type u_5} {S₂ : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [Semiring R] [Semiring S] [Semiring R₂] [Semiring S₂] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (b₁ : Module.Basis ι₁ R M) (b₂ : Module.Basis ι₂ S N) {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x : M) (y : N) : ((b₁.repr x).sum fun i xi => (b₂.repr y).sum fun j yj => ρ₁₂ xi • σ₁₂ yj • (B (b₁ i)) (b₂ j)) = (B x) y
• LinearMap.sum_repr_mul_repr_mul 📋 Mathlib.LinearAlgebra.Basis.Bilinear
  {ι₁ : Type u_1} {ι₂ : Type u_2} {Rₗ : Type u_10} {Mₗ : Type u_11} {Nₗ : Type u_12} {Pₗ : Type u_13} [CommSemiring Rₗ] [AddCommMonoid Mₗ] [AddCommMonoid Nₗ] [AddCommMonoid Pₗ] [Module Rₗ Mₗ] [Module Rₗ Nₗ] [Module Rₗ Pₗ] (b₁' : Module.Basis ι₁ Rₗ Mₗ) (b₂' : Module.Basis ι₂ Rₗ Nₗ) {B : Mₗ →ₗ[Rₗ] Nₗ →ₗ[Rₗ] Pₗ} (x : Mₗ) (y : Nₗ) : ((b₁'.repr x).sum fun i xi => (b₂'.repr y).sum fun j yj => xi • yj • (B (b₁' i)) (b₂' j)) = (B x) y
• AddCon.coe_finsuppSum 📋 Mathlib.GroupTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [AddCommMonoid M] [Zero β] (c : AddCon M) (h : ι → β → M) (f : ι →₀ β) : ↑(f.sum h) = f.sum fun i b => ↑(h i b)
• AddCon.finsuppSum 📋 Mathlib.GroupTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [AddCommMonoid M] [Zero β] (c : AddCon M) (h h' : ι → β → M) {f g : ι →₀ β} (hf : ∀ (i : ι), c (h i 0) 0) (hf' : ∀ (i : ι), c (h' i 0) 0) (H : ∀ (i : ι), c (h i (f i)) (h' i (g i))) : c (f.sum h) (g.sum h')
• AddSubmonoid.exists_finsupp_of_mem_closure_range 📋 Mathlib.Algebra.Group.Submonoid.Finsupp
  {M : Type u_1} [AddCommMonoid M] {ι : Type u_2} (f : ι → M) (x : M) (hx : x ∈ AddSubmonoid.closure (Set.range f)) : ∃ a, x = a.sum fun x1 x2 => x2 • f x1

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59