Loogle!

Result

Found 140 declarations mentioning Finsupp.prod.

• Finsupp.prod 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] (f : α →₀ M) (g : α → M → N) : N
• AddOpposite.op_finsuppProd 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_16} {M : Type u_17} {N : Type u_18} [CommMonoid M] [Zero N] (f : ι →₀ N) (g : ι → N → M) : AddOpposite.op (f.prod g) = f.prod fun i n => AddOpposite.op (g i n)
• AddOpposite.unop_finsuppProd 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {ι : Type u_16} {M : Type u_17} {N : Type u_18} [CommMonoid M] [Zero N] (f : ι →₀ N) (g : ι → N → Mᵐᵒᵖ) : AddOpposite.unop (f.prod g) = f.prod fun i n => AddOpposite.unop (g i n)
• Finsupp.prod_zero_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {h : α → M → N} : Finsupp.prod 0 h = 1
• Finsupp.prod_fun_one 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] (f : α →₀ M) : (f.prod fun x x_1 => 1) = 1
• Finsupp.prod_single_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {a : α} {b : M} {h : α → M → N} (h_zero : h a 0 = 1) : (fun₀ | a => b).prod h = h a b
• Finsupp.prod_embDomain 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {v : α →₀ M} {f : α ↪ β} {g : β → M → N} : (Finsupp.embDomain f v).prod g = v.prod fun a b => g (f a) b
• Finsupp.prod_finsetProd_comm 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {s : Finset β} (f : α →₀ M) (h : α → M → β → N) : (f.prod fun a m => ∏ b ∈ s, h a m b) = ∏ b ∈ s, f.prod fun a m => h a m b
• Nat.prod_pow_pos_of_zero_notMem_support 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {f : ℕ →₀ ℕ} (nhf : 0 ∉ f.support) : 0 < f.prod fun x1 x2 => x1 ^ x2
• Finsupp.prod_inv 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {G : Type u_12} [Zero M] [CommGroup G] {f : α →₀ M} {h : α → M → G} : (f.prod fun a b => (h a b)⁻¹) = (f.prod h)⁻¹
• Finsupp.prod_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'] [CommMonoid N] (f : α →₀ M) (g : β →₀ M') (h : α → M → β → M' → N) : (f.prod fun x v => g.prod fun x' v' => h x v x' v') = g.prod fun x' v' => f.prod fun x v => h x v x' v'
• Finsupp.prod_fintype 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] [Fintype α] (f : α →₀ M) (g : α → M → N) (h : ∀ (i : α), g i 0 = 1) : f.prod g = ∏ i, g i (f i)
• map_finsuppProd 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} {P : Type u_11} [Zero M] [CommMonoid N] [CommMonoid P] {H : Type u_16} [FunLike H N P] [MonoidHomClass H N P] (h : H) (f : α →₀ M) (g : α → M → N) : h (f.prod g) = f.prod fun a b => h (g a b)
• Finsupp.prod_mul 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {f : α →₀ M} {h₁ h₂ : α → M → N} : (f.prod fun a b => h₁ a b * h₂ a b) = f.prod h₁ * f.prod h₂
• Finsupp.mul_prod_erase 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] (f : α →₀ M) (y : α) (g : α → M → N) (hyf : y ∈ f.support) : g y (f y) * (Finsupp.erase y f).prod g = f.prod g
• Finsupp.prod_congr 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {f : α →₀ M} {g1 g2 : α → M → N} (h : ∀ x ∈ f.support, g1 x (f x) = g2 x (f x)) : f.prod g1 = f.prod g2
• Finsupp.prod_pow 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {N : Type u_10} [CommMonoid N] [Fintype α] (f : α →₀ ℕ) (g : α → N) : (f.prod fun a b => g a ^ b) = ∏ a, g a ^ f a
• Finsupp.onFinset_prod 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid 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 = 1) : (Finsupp.onFinset s f hf).prod g = ∏ a ∈ s, g a (f a)
• Finsupp.prod_indicator_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {s : Finset α} (f : α → M) {h : α → M → N} (h_zero : ∀ a ∈ s, h a 0 = 1) : (Finsupp.indicator s fun x x_1 => f x).prod h = ∏ x ∈ s, h x (f x)
• Finsupp.prod_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'] [CommMonoid N] {f : M → M'} {hf : f 0 = 0} {g : α →₀ M} {h : α → M' → N} (h0 : ∀ (a : α), h a 0 = 1) : (Finsupp.mapRange f hf g).prod h = g.prod fun a b => h a (f b)
• SubmonoidClass.finsuppProd_mem 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {S : Type u_16} [SetLike S N] [SubmonoidClass S N] (s : S) (f : α →₀ M) (g : α → M → N) (h : ∀ (c : α), f c ≠ 0 → g c (f c) ∈ s) : f.prod g ∈ s
• Finsupp.mul_prod_erase' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] (f : α →₀ M) (y : α) (g : α → M → N) (hg : ∀ (i : α), g i 0 = 1) : g y (f y) * (Finsupp.erase y f).prod g = f.prod g
• Finsupp.prod_onFinset 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid 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 = 1) : (Finsupp.onFinset s f hf).prod g = ∏ a ∈ s, g a (f a)
• Finsupp.prod_of_support_subset 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] (f : α →₀ M) {s : Finset α} (hs : f.support ⊆ s) (g : α → M → N) (h : ∀ i ∈ s, g i 0 = 1) : f.prod g = ∏ x ∈ s, g x (f x)
• Finsupp.prod_zpow 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {N : Type u_16} [DivisionCommMonoid N] [Fintype α] (f : α →₀ ℤ) (g : α → N) : (f.prod fun a b => g a ^ b) = ∏ a, g a ^ f a
• Finsupp.prod_unique 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] [Unique α] {f : α →₀ M} {g : α → M → N} (h₁ : f default = 0 → g default 0 = 1) : f.prod g = g default (f default)
• Finsupp.prod_ite_eq 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) : (f.prod fun x v => if a = x then b x v else 1) = if a ∈ f.support then b a (f a) else 1
• Finsupp.prod_ite_eq' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) : (f.prod fun x v => if x = a then b x v else 1) = if a ∈ f.support then b a (f a) else 1
• Finsupp.prod_ne_zero_iff 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {β : Type u_7} [Zero α] [CommMonoidWithZero β] [Nontrivial β] [NoZeroDivisors β] {f : ι →₀ α} {g : ι → α → β} : f.prod g ≠ 0 ↔ ∀ i ∈ f.support, g i (f i) ≠ 0
• Finsupp.prod_neg_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {G : Type u_12} [SubtractionMonoid G] [CommMonoid M] {g : α →₀ G} {h : α → G → M} (h0 : ∀ (a : α), h a 0 = 1) : (-g).prod h = g.prod fun a b => h a (-b)
• Finsupp.prod_eq_zero_iff 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {ι : Type u_2} {β : Type u_7} [Zero α] [CommMonoidWithZero β] [Nontrivial β] [NoZeroDivisors β] {f : ι →₀ α} {g : ι → α → β} : f.prod g = 0 ↔ ∃ i ∈ f.support, g i (f i) = 0
• Finsupp.prod_dvd_prod_of_subset_of_dvd 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {f1 f2 : α →₀ M} {g1 g2 : α → M → N} (h1 : f1.support ⊆ f2.support) (h2 : ∀ a ∈ f1.support, g1 a (f1 a) ∣ g2 a (f2 a)) : f1.prod g1 ∣ f2.prod g2
• MonoidHom.finsuppProd_apply 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {N : Type u_10} {P : Type u_11} [Zero β] [MulOneClass N] [CommMonoid P] (f : α →₀ β) (g : α → β → N →* P) (x : N) : (f.prod g) x = f.prod fun i fi => (g i fi) x
• MonoidHom.coe_finsuppProd 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {β : Type u_7} {N : Type u_10} {P : Type u_11} [Zero β] [MulOneClass N] [CommMonoid P] (f : α →₀ β) (g : α → β → N →* P) : ⇑(f.prod g) = f.prod fun i fi => ⇑(g i fi)
• Finsupp.prod_eq_single 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {f : α →₀ M} (a : α) {g : α → M → N} (h₀ : ∀ (b : α), f b ≠ 0 → b ≠ a → g b (f b) = 1) (h₁ : f a = 0 → g a 0 = 1) : f.prod g = g a (f a)
• Finsupp.prod_mul_eq_prod_mul_of_exists 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid 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.prod g * n₁ = f.prod g * n₂
• Finsupp.prod_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] [CommMonoid N] {s : Finset ι} {g : ι → α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 1) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ * h a b₂) : ∏ i ∈ s, (g i).prod h = (∑ i ∈ s, g i).prod h
• Finsupp.prod_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] [CommMonoid N] {s : Finset ι} {g : ι → α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 1) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ * h a b₂) : ∏ i ∈ s, (g i).prod h = (∑ i ∈ s, g i).prod h
• Finsupp.prod_indicator_index_eq_prod_attach 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid N] {s : Finset α} (f : (a : α) → a ∈ s → M) {h : α → M → N} (h_zero : ∀ a ∈ s, h a 0 = 1) : (Finsupp.indicator s f).prod h = ∏ x ∈ s.attach, h (↑x) (f ↑x ⋯)
• 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.prod_add_index' 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddZeroClass M] [CommMonoid N] {f g : α →₀ M} {h : α → M → N} (h_zero : ∀ (a : α), h a 0 = 1) (h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ * h a b₂) : (f + g).prod h = f.prod h * g.prod h
• Finsupp.prod_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} [CommMonoid β] (g : α → M → β) : (f1 + f2).prod g = f1.prod g * f2.prod g
• Finsupp.prod_congr_of_eq_on_union 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [Zero M] [CommMonoid 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 = 1) (h2 : ∀ x ∈ f1.support ∪ f2.support, g2 x 0 = 1) : f1.prod g1 = f2.prod g2
• Finsupp.prod_add_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [DecidableEq α] [AddZeroClass M] [CommMonoid N] {f g : α →₀ M} {h : α → M → N} (h_zero : ∀ a ∈ f.support ∪ g.support, h a 0 = 1) (h_add : ∀ a ∈ f.support ∪ g.support, ∀ (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ * h a b₂) : (f + g).prod h = f.prod h * g.prod h
• Finsupp.prod_hom_add_index 📋 Mathlib.Algebra.BigOperators.Finsupp.Basic
  {α : Type u_1} {M : Type u_8} {N : Type u_10} [AddZeroClass M] [CommMonoid N] {f g : α →₀ M} (h : α → Multiplicative M →* N) : ((f + g).prod fun a b => (h a) (Multiplicative.ofAdd b)) = (f.prod fun a b => (h a) (Multiplicative.ofAdd b)) * g.prod fun a b => (h a) (Multiplicative.ofAdd b)
• Int.cast_finsuppProd 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [CommRing R] (g : α → M → ℤ) : ↑(f.prod g) = f.prod fun a b => ↑(g a b)
• Finsupp.prod_equivMapDomain 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [Zero M] [CommMonoid N] (f : α ≃ β) (l : α →₀ M) (g : β → M → N) : (Finsupp.equivMapDomain f l).prod g = l.prod fun a m => g (f a) m
• Nat.cast_finsuppProd 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [CommSemiring R] (g : α → M → ℕ) : ↑(f.prod g) = f.prod fun a b => ↑(g a b)
• Rat.cast_finsuppProd 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {R : Type u_10} [Zero M] (f : α →₀ M) [Field R] [CharZero R] (g : α → M → ℚ) : ↑(f.prod g) = f.prod fun a b => ↑(g a b)
• Finsupp.prod_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) [CommMonoid N] (g : α → M → N) : (Finsupp.filter p f).prod g = ∏ x ∈ (Finsupp.filter p f).support, g x (f x)
• Finsupp.prod_subtypeDomain_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] {p : α → Prop} [CommMonoid N] {v : α →₀ M} {h : α → M → N} (hp : ∀ x ∈ v.support, p x) : ((Finsupp.subtypeDomain p v).prod fun a b => h (↑a) b) = v.prod h
• Finsupp.prod_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] [CommMonoid N] {f : α → β} {s : α →₀ M} {h : β → M → N} (hf : Function.Injective f) : (Finsupp.mapDomain f s).prod h = s.prod fun a b => h (f a) b
• Finsupp.prod_filter_mul_prod_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) [CommMonoid N] (g : α → M → N) : (Finsupp.filter p f).prod g * (Finsupp.filter (fun a => ¬p a) f).prod g = f.prod g
• Finsupp.prod_div_prod_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) [CommGroup G] (g : α → M → G) : f.prod g / (Finsupp.filter p f).prod g = (Finsupp.filter (fun a => ¬p a) f).prod g
• Finsupp.prod_sumElim 📋 Mathlib.Data.Finsupp.Basic
  {ι₁ : Type u_12} {ι₂ : Type u_13} {α : Type u_14} {M : Type u_15} [Zero α] [CommMonoid M] (f₁ : ι₁ →₀ α) (f₂ : ι₂ →₀ α) (g : ι₁ ⊕ ι₂ → α → M) : (f₁.sumElim f₂).prod g = f₁.prod (g ∘ Sum.inl) * f₂.prod (g ∘ Sum.inr)
• Finsupp.prod_mapDomain_index 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_1} {β : Type u_2} {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [CommMonoid N] {f : α → β} {s : α →₀ M} {h : β → M → N} (h_zero : ∀ (b : β), h b 0 = 1) (h_add : ∀ (b : β) (m₁ m₂ : M), h b (m₁ + m₂) = h b m₁ * h b m₂) : (Finsupp.mapDomain f s).prod h = s.prod fun a m => h (f a) m
• Finsupp.prod_option_index 📋 Mathlib.Data.Finsupp.Option
  {α : Type u_1} {M : Type u_2} {N : Type u_3} [AddZeroClass M] [CommMonoid N] (f : Option α →₀ M) (b : Option α → M → N) (h_zero : ∀ (o : Option α), b o 0 = 1) (h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂) : f.prod b = b none (f none) * f.some.prod fun a => b (some a)
• 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)
• MonoidAlgebra.finsuppProd_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [CommSemiring R] [CommMonoid M] [AddCommMonoid N] (f : ι →₀ N) (m : ι → N → M) (r : ι → N → R) : (f.prod fun i n => MonoidAlgebra.single (m i n) (r i n)) = MonoidAlgebra.single (f.prod m) (f.prod r)
• Finsupp.prod_toMultiset 📋 Mathlib.Data.Finsupp.Multiset
  {α : Type u_1} [CommMonoid α] (f : α →₀ ℕ) : (Finsupp.toMultiset f).prod = f.prod fun a n => a ^ n
• star_finsuppProd 📋 Mathlib.Algebra.Star.BigOperators
  {R : Type u_1} {ι : Type u_2} {M : Type u_3} [Zero M] [CommMonoid R] [StarMul R] (s : ι →₀ M) (f : ι → M → R) : star (s.prod f) = s.prod fun i m => star f i m
• MvPolynomial.monic_monomial_eq 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) : (MvPolynomial.monomial m) 1 = m.prod fun n e => MvPolynomial.X n ^ e
• MvPolynomial.monomial_eq 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} {a : R} {s : σ →₀ ℕ} [CommSemiring R] : (MvPolynomial.monomial s) a = MvPolynomial.C a * s.prod fun n e => MvPolynomial.X n ^ e
• 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
• MvPolynomial.eval₂_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} {a : R} {s : σ →₀ ℕ} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : σ → S₁) : MvPolynomial.eval₂ f g ((MvPolynomial.monomial s) a) = f a * s.prod fun n e => g n ^ e
• MvPolynomial.eval_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {σ : Type u_1} {a : R} {s : σ →₀ ℕ} [CommSemiring R] {f : σ → R} : (MvPolynomial.eval f) ((MvPolynomial.monomial s) a) = a * s.prod fun n e => f n ^ e
• MvPolynomial.eval₂_mul_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {p : MvPolynomial σ R} (f : R →+* S₁) (g : σ → S₁) {s : σ →₀ ℕ} {a : R} : MvPolynomial.eval₂ f g (p * (MvPolynomial.monomial s) a) = MvPolynomial.eval₂ f g p * f a * s.prod fun n e => g n ^ e
• MvPolynomial.eval₂Hom_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : σ → S₁) (d : σ →₀ ℕ) (r : R) : (MvPolynomial.eval₂Hom f g) ((MvPolynomial.monomial d) r) = f r * d.prod fun i k => g i ^ k
• MvPolynomial.aeval_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [Algebra R S₁] (g : σ → S₁) (d : σ →₀ ℕ) (r : R) : (MvPolynomial.aeval g) ((MvPolynomial.monomial d) r) = (algebraMap R S₁) r * d.prod fun i k => g i ^ k
• Nat.factorization_prod_pow_eq_self 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} (hn : n ≠ 0) : (n.factorization.prod fun x1 x2 => x1 ^ x2) = n
• Nat.prod_factorization_pow_eq_self 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} (hn : n ≠ 0) : (n.factorization.prod fun x1 x2 => x1 ^ x2) = n
• Nat.prod_pow_dvd_of_le_factorization 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} {f : ℕ →₀ ℕ} (hf : f ≤ n.factorization) : (f.prod fun x1 x2 => x1 ^ x2) ∣ n
• Nat.factorization_prod_pow_eq_self_of_le_factorization 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} {f : ℕ →₀ ℕ} (hf : f ≤ n.factorization) : (f.prod fun x1 x2 => x1 ^ x2).factorization = f
• Nat.dvd_prod_pow_of_factorization_le 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} {f : ℕ →₀ ℕ} (hn : n ≠ 0) (hf : n.factorization ≤ f) : n ∣ f.prod fun x1 x2 => x1 ^ x2
• Nat.prod_pow_factorization_eq_self 📋 Mathlib.Data.Nat.Factorization.Defs
  {f : ℕ →₀ ℕ} (hf : ∀ p ∈ f.support, Nat.Prime p) : (f.prod fun x1 x2 => x1 ^ x2).factorization = f
• Nat.eq_factorization_iff 📋 Mathlib.Data.Nat.Factorization.Defs
  {n : ℕ} {f : ℕ →₀ ℕ} (hn : n ≠ 0) (hf : ∀ p ∈ f.support, Nat.Prime p) : f = n.factorization ↔ (f.prod fun x1 x2 => x1 ^ x2) = n
• Nat.dvd_iff_exists_le_factorization 📋 Mathlib.Data.Nat.Factorization.Defs
  {n d : ℕ} (hd : d ≠ 0) (hn : n ≠ 0) : d ∣ n ↔ ∃ f ≤ n.factorization, d = f.prod fun x1 x2 => x1 ^ x2
• Nat.factorizationEquiv_symm_apply_coe 📋 Mathlib.Data.Nat.Factorization.Defs
  (x✝ : { f // ∀ p ∈ f.support, Nat.Prime p }) : ↑(Nat.factorizationEquiv.symm x✝) = (↑x✝).prod fun x1 x2 => x1 ^ x2
• Nat.prod_primeFactors_prod_factorization 📋 Mathlib.Data.Nat.Factorization.Basic
  {n : ℕ} {β : Type u_1} [CommMonoid β] (f : ℕ → β) : ∏ p ∈ n.primeFactors, f p = n.factorization.prod fun p x => f p
• Nat.prod_factorization_eq_prod_primeFactors 📋 Mathlib.Data.Nat.Factorization.Basic
  {n : ℕ} {β : Type u_1} [CommMonoid β] (f : ℕ → ℕ → β) : n.factorization.prod f = ∏ p ∈ n.primeFactors, f p (n.factorization p)
• Nat.factorizationEquiv_inv_apply 📋 Mathlib.Data.Nat.Factorization.Basic
  {f : ℕ →₀ ℕ} (hf : ∀ p ∈ f.support, Nat.Prime p) : ↑(Nat.factorizationEquiv.symm ⟨f, hf⟩) = f.prod fun x1 x2 => x1 ^ x2
• Finsupp.image_pow_eq_finsuppProd_image 📋 Mathlib.Data.Finsupp.Weight
  {α : Type u_5} {β : Type u_6} [CommMonoid β] {f : α → β} {n : ℕ} {s : Set α} : (f '' s) ^ n = (fun x => x.prod fun x1 x2 => f x1 ^ x2) '' {x | Finsupp.degree x = n ∧ ↑x.support ⊆ s}
• Prime.not_dvd_finsuppProd 📋 Mathlib.Algebra.BigOperators.Associated
  {M₀ : Type u_3} {M : Type u_4} [CommMonoidWithZero M] {f : M₀ →₀ M} {g : M₀ → M → ℕ} {p : ℕ} (pp : Prime p) (hS : ∀ a ∈ f.support, ¬p ∣ g a (f a)) : ¬p ∣ f.prod g
• Prime.dvd_finsuppProd_iff 📋 Mathlib.Algebra.BigOperators.Associated
  {M₀ : Type u_3} {M : Type u_4} [CommMonoidWithZero M] {f : M₀ →₀ M} {g : M₀ → M → ℕ} {p : ℕ} (pp : Prime p) : p ∣ f.prod g ↔ ∃ a ∈ f.support, p ∣ g a (f a)
• Subgroup.exists_finsupp_of_mem_closure_range 📋 Mathlib.Algebra.Group.Subgroup.Finsupp
  {M : Type u_1} [CommGroup M] {ι : Type u_2} (f : ι → M) (x : M) (hx : x ∈ Subgroup.closure (Set.range f)) : ∃ a, x = a.prod fun x1 x2 => f x1 ^ x2
• Subgroup.mem_closure_range_iff 📋 Mathlib.Algebra.Group.Subgroup.Finsupp
  {M : Type u_1} [CommGroup M] {ι : Type u_2} {f : ι → M} {x : M} : x ∈ Subgroup.closure (Set.range f) ↔ ∃ a, x = a.prod fun x1 x2 => f x1 ^ x2
• Nat.multiplicative_factorization 📋 Mathlib.Data.Nat.Factorization.Induction
  {β : Type u_1} [CommMonoid β] (f : ℕ → β) (h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y) (hf : f 1 = 1) {n : ℕ} : n ≠ 0 → f n = n.factorization.prod fun p k => f (p ^ k)
• Nat.multiplicative_factorization' 📋 Mathlib.Data.Nat.Factorization.Induction
  {n : ℕ} {β : Type u_1} [CommMonoid β] (f : ℕ → β) (h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y) (hf0 : f 0 = 1) (hf1 : f 1 = 1) : f n = n.factorization.prod fun p k => f (p ^ k)
• Nat.totient_eq_prod_factorization 📋 Mathlib.Data.Nat.Totient
  {n : ℕ} (hn : n ≠ 0) : n.totient = n.factorization.prod fun p k => p ^ (k - 1) * (p - 1)
• Con.coe_finsuppProd 📋 Mathlib.GroupTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [CommMonoid M] [Zero β] (c : Con M) (h : ι → β → M) (f : ι →₀ β) : ↑(f.prod h) = f.prod fun i b => ↑(h i b)
• Con.finsuppProd 📋 Mathlib.GroupTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [CommMonoid M] [Zero β] (c : Con M) (h h' : ι → β → M) {f g : ι →₀ β} (hf : ∀ (i : ι), c (h i 0) 1) (hf' : ∀ (i : ι), c (h' i 0) 1) (H : ∀ (i : ι), c (h i (f i)) (h' i (g i))) : c (f.prod h) (g.prod h')
• Submonoid.exists_finsupp_of_mem_closure_range 📋 Mathlib.Algebra.Group.Submonoid.Finsupp
  {M : Type u_1} [CommMonoid M] {ι : Type u_2} (f : ι → M) (x : M) (hx : x ∈ Submonoid.closure (Set.range f)) : ∃ a, x = a.prod fun x1 x2 => f x1 ^ x2
• Submonoid.mem_closure_range_iff 📋 Mathlib.Algebra.Group.Submonoid.Finsupp
  {M : Type u_1} [CommMonoid M] {ι : Type u_2} {f : ι → M} {x : M} : x ∈ Submonoid.closure (Set.range f) ↔ ∃ a, x = a.prod fun x1 x2 => f x1 ^ x2
• Algebra.Generators.Hom.toAlgHom_monomial 📋 Mathlib.RingTheory.Extension.Generators
  {R : Type u} {S : Type v} {ι : Type w} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Generators R S ι} {R' : Type u_1} {S' : Type u_2} {ι' : Type u_3} [CommRing R'] [CommRing S'] [Algebra R' S'] {P' : Algebra.Generators R' S' ι'} [Algebra R R'] [Algebra S S'] (f : P.Hom P') (v : ι →₀ ℕ) (r : R) : f.toAlgHom ((MvPolynomial.monomial v) r) = r • v.prod fun x1 x2 => f.val x1 ^ x2
• Algebra.Generators.H1Cotangent.δAux_monomial 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} (Q : Algebra.Generators S T ι) (n : ι →₀ ℕ) (r : S) : (Algebra.Generators.H1Cotangent.δAux R Q) ((MvPolynomial.monomial n) r) = (n.prod fun x1 x2 => Q.val x1 ^ x2) ⊗ₜ[S] (KaehlerDifferential.D R S) r
• FractionalIdeal.count_finsuppProd 📋 Mathlib.RingTheory.DedekindDomain.Factorization
  {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (exps : IsDedekindDomain.HeightOneSpectrum R →₀ ℤ) : FractionalIdeal.count K v (exps.prod fun x1 x2 => ↑x1.asIdeal ^ x2) = exps v
• NNReal.toReal_finsuppProd 📋 Mathlib.Basic.NNReal.Basic
  {M : Type u_1} [Zero M] {ι : Type u_2} (f : ι →₀ M) (g : ι → M → NNReal) : ↑(f.prod g) = f.prod fun i m => ↑(g i m)
• ENNReal.ofNNReal_finsuppProd 📋 Mathlib.Basic.ENNReal.BigOperators
  {ι : Type u_1} {M : Type u_2} [Zero M] (f : ι →₀ M) (g : ι → M → NNReal) : ↑(f.prod g) = f.prod fun i m => ↑(g i m)
• RCLike.ofReal_finsuppProd 📋 Mathlib.Analysis.RCLike.Basic
  {K : Type u_1} [RCLike K] {α : Type u_3} {M : Type u_4} [Zero M] (f : α →₀ M) (g : α → M → ℝ) : ↑(f.prod fun a b => g a b) = f.prod fun a b => ↑(g a b)
• Finsupp.log_prod 📋 Mathlib.Analysis.SpecialFunctions.Log.Basic
  {α : Type u_1} {β : Type u_2} [Zero β] (f : α →₀ β) (g : α → β → ℝ) (hg : ∀ (a : α), g a (f a) = 0 → f a = 0) : Real.log (f.prod g) = f.sum fun a b => Real.log (g a b)
• MvPolynomial.coeff_linearCombination_X_pow_of_fintype 📋 Mathlib.Algebra.MvPolynomial.Coeff
  {R : Type u_1} {σ : Type u_2} [CommSemiring R] [Fintype σ] (a : σ → R) (s : σ →₀ ℕ) (n : ℕ) : MvPolynomial.coeff s ((∑ i, a i • MvPolynomial.X i) ^ n) = if (s.sum fun x m => m) = n then ↑s.multinomial * s.prod fun r m => a r ^ m else 0
• MvPolynomial.coeff_linearCombination_X_pow 📋 Mathlib.Algebra.MvPolynomial.Coeff
  {R : Type u_1} {σ : Type u_2} [CommSemiring R] (a : σ →₀ R) (s : σ →₀ ℕ) (n : ℕ) : MvPolynomial.coeff s ((Finsupp.linearCombination R MvPolynomial.X) a ^ n) = if (s.sum fun x m => m) = n then ↑s.multinomial * s.prod fun r m => a r ^ m else 0
• ArithmeticFunction.IsMultiplicative.multiplicative_factorization 📋 Mathlib.NumberTheory.ArithmeticFunction.Defs
  {R : Type u_1} [CommMonoidWithZero R] (f : ArithmeticFunction R) (hf : f.IsMultiplicative) {n : ℕ} (hn : n ≠ 0) : f n = n.factorization.prod fun p k => f (p ^ k)
• MvPowerSeries.monomial_one_eq 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] (e : σ →₀ ℕ) : (MvPowerSeries.monomial e) 1 = e.prod fun s n => MvPowerSeries.X s ^ n
• MvPowerSeries.monomial_mapDomain_apply_one 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] {τ : Type u_3} (d : σ →₀ ℕ) (f : σ → τ) : (MvPowerSeries.monomial (Finsupp.mapDomain f d)) 1 = d.prod fun s e => MvPowerSeries.X (f s) ^ e
• MvPowerSeries.monomial_smul_const 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_4} {R : Type u_5} [CommSemiring R] (e : σ →₀ ℕ) (r : R) : (MvPowerSeries.monomial e) (r ^ e.sum fun x n => n) = e.prod fun s e => (r • MvPowerSeries.X s) ^ e
• MvPowerSeries.monomial_eq 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] (e : σ →₀ ℕ) (r : σ → R) : (MvPowerSeries.monomial e) (e.prod fun s n => r s ^ n) = e.prod fun s e => (r s • MvPowerSeries.X s) ^ e
• MvPowerSeries.monomial_eq' 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] (e : σ →₀ ℕ) (r : R) : (MvPowerSeries.monomial e) r = MvPowerSeries.C r * e.prod fun s e => MvPowerSeries.X s ^ e
• MvPowerSeries.prod_smul_X_eq_smul_monomial_one 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] {A : Type u_4} [CommSemiring A] [Algebra A R] (e : σ →₀ ℕ) (a : σ → A) : (e.prod fun s n => (a s • MvPowerSeries.X s) ^ n) = (e.prod fun s n => a s ^ n) • (MvPowerSeries.monomial e) 1
• MvPowerSeries.monomial_smul_eq 📋 Mathlib.RingTheory.MvPowerSeries.Basic
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] (e : σ →₀ ℕ) (p : ℕ) (r : R) : (MvPowerSeries.monomial (p • e)) r = MvPowerSeries.C r * e.prod fun s e => (MvPowerSeries.X s ^ p) ^ e
• Finsupp.card_pi 📋 Mathlib.Data.Finset.Finsupp
  {ι : Type u_1} {α : Type u_2} [Zero α] (f : ι →₀ Finset α) : f.pi.card = f.prod fun i => ↑(f i).card
• MvPowerSeries.hasSum_eval₂ 📋 Mathlib.RingTheory.MvPowerSeries.Evaluation
  {σ : Type u_1} {R : Type u_2} [CommRing R] [UniformSpace R] {S : Type u_3} [CommRing S] [UniformSpace S] {φ : R →+* S} {a : σ → S} [IsTopologicalSemiring R] [IsUniformAddGroup R] [IsUniformAddGroup S] [CompleteSpace S] [T2Space S] [IsTopologicalRing S] [IsLinearTopology S S] (hφ : Continuous ⇑φ) (ha : MvPowerSeries.HasEval a) (f : MvPowerSeries σ R) : HasSum (fun d => φ ((MvPowerSeries.coeff d) f) * d.prod fun s e => a s ^ e) (MvPowerSeries.eval₂ φ a f)
• MvPowerSeries.eval₂_eq_tsum 📋 Mathlib.RingTheory.MvPowerSeries.Evaluation
  {σ : Type u_1} {R : Type u_2} [CommRing R] [UniformSpace R] {S : Type u_3} [CommRing S] [UniformSpace S] {φ : R →+* S} {a : σ → S} [IsTopologicalSemiring R] [IsUniformAddGroup R] [IsUniformAddGroup S] [CompleteSpace S] [T2Space S] [IsTopologicalRing S] [IsLinearTopology S S] (hφ : Continuous ⇑φ) (ha : MvPowerSeries.HasEval a) (f : MvPowerSeries σ R) : MvPowerSeries.eval₂ φ a f = ∑' (d : σ →₀ ℕ), φ ((MvPowerSeries.coeff d) f) * d.prod fun s e => a s ^ e
• MvPowerSeries.hasSum_aeval 📋 Mathlib.RingTheory.MvPowerSeries.Evaluation
  {σ : Type u_1} {R : Type u_2} [CommRing R] [UniformSpace R] {S : Type u_3} [CommRing S] [UniformSpace S] {a : σ → S} [IsTopologicalSemiring R] [IsUniformAddGroup R] [IsUniformAddGroup S] [CompleteSpace S] [T2Space S] [IsTopologicalRing S] [IsLinearTopology S S] [Algebra R S] [ContinuousSMul R S] (ha : MvPowerSeries.HasEval a) (f : MvPowerSeries σ R) : HasSum (fun d => (MvPowerSeries.coeff d) f • d.prod fun s e => a s ^ e) ((MvPowerSeries.aeval ha) f)
• MvPowerSeries.aeval_eq_sum 📋 Mathlib.RingTheory.MvPowerSeries.Evaluation
  {σ : Type u_1} {R : Type u_2} [CommRing R] [UniformSpace R] {S : Type u_3} [CommRing S] [UniformSpace S] {a : σ → S} [IsTopologicalSemiring R] [IsUniformAddGroup R] [IsUniformAddGroup S] [CompleteSpace S] [T2Space S] [IsTopologicalRing S] [IsLinearTopology S S] [Algebra R S] [ContinuousSMul R S] (ha : MvPowerSeries.HasEval a) (f : MvPowerSeries σ R) : (MvPowerSeries.aeval ha) f = ∑' (d : σ →₀ ℕ), (MvPowerSeries.coeff d) f • d.prod fun s e => a s ^ e
• MvPowerSeries.subst_monomial 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_3} [CommRing R] {τ : Type u_4} {S : Type u_5} [CommRing S] [Algebra R S] {a : σ → MvPowerSeries τ S} (ha : MvPowerSeries.HasSubst a) (e : σ →₀ ℕ) (r : R) : MvPowerSeries.subst a ((MvPowerSeries.monomial e) r) = (algebraMap R (MvPowerSeries τ S)) r * e.prod fun s n => a s ^ n
• MvPowerSeries.coeff_rescale 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_9} [CommSemiring R] (f : MvPowerSeries σ R) (a : σ → R) (n : σ →₀ ℕ) : (MvPowerSeries.coeff n) ((MvPowerSeries.rescale a) f) = (n.prod fun s m => a s ^ m) * (MvPowerSeries.coeff n) f
• MvPowerSeries.constantCoeff_subst 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_3} [CommRing R] {τ : Type u_4} {S : Type u_5} [CommRing S] [Algebra R S] {a : σ → MvPowerSeries τ S} (ha : MvPowerSeries.HasSubst a) (f : MvPowerSeries σ R) : MvPowerSeries.constantCoeff (MvPowerSeries.subst a f) = ∑ᶠ (d : σ →₀ ℕ), (MvPowerSeries.coeff d) f • MvPowerSeries.constantCoeff (d.prod fun s e => a s ^ e)
• MvPowerSeries.coeff_subst_finite 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_3} [CommRing R] {τ : Type u_4} {S : Type u_5} [CommRing S] [Algebra R S] {a : σ → MvPowerSeries τ S} (ha : MvPowerSeries.HasSubst a) (f : MvPowerSeries σ R) (e : τ →₀ ℕ) : Function.HasFiniteSupport fun d => (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e => a s ^ e)
• MvPowerSeries.substAlgHom_monomial 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_3} [CommRing R] {τ : Type u_4} {S : Type u_5} [CommRing S] [Algebra R S] {a : σ → MvPowerSeries τ S} (ha : MvPowerSeries.HasSubst a) (e : σ →₀ ℕ) (r : R) : (MvPowerSeries.substAlgHom ha) ((MvPowerSeries.monomial e) r) = (algebraMap R (MvPowerSeries τ S)) r * e.prod fun s n => a s ^ n
• MvPowerSeries.coeff_subst 📋 Mathlib.RingTheory.MvPowerSeries.Substitution
  {σ : Type u_1} {R : Type u_3} [CommRing R] {τ : Type u_4} {S : Type u_5} [CommRing S] [Algebra R S] {a : σ → MvPowerSeries τ S} (ha : MvPowerSeries.HasSubst a) (f : MvPowerSeries σ R) (e : τ →₀ ℕ) : (MvPowerSeries.coeff e) (MvPowerSeries.subst a f) = ∑ᶠ (d : σ →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e => a s ^ e)
• MvPowerSeries.le_gaussNorm 📋 Mathlib.RingTheory.MvPowerSeries.GaussNorm
  {R : Type u_1} {σ : Type u_2} (v : R → ℝ) (c : σ → ℝ) (f : MvPowerSeries σ R) [Semiring R] (hbd : MvPowerSeries.HasGaussNorm v c f) (t : σ →₀ ℕ) : (v ((MvPowerSeries.coeff t) f) * t.prod fun x1 x2 => c x1 ^ x2) ≤ MvPowerSeries.gaussNorm v c f
• Finsupp.logb_prod 📋 Mathlib.Analysis.SpecialFunctions.Log.Base
  {b : ℝ} {α : Type u_1} {β : Type u_2} [Zero β] (f : α →₀ β) (g : α → β → ℝ) (hg : ∀ (a : α), g a (f a) = 0 → f a = 0) : Real.logb b (f.prod g) = f.sum fun a c => Real.logb b (g a c)
• Nat.Partition.coeff_genFun 📋 Mathlib.Combinatorics.Enumerative.Partition.GenFun
  {R : Type u_1} [CommSemiring R] (f : ℕ → ℕ → R) (n : ℕ) : (PowerSeries.coeff n) (Nat.Partition.genFun f) = ∑ p, (Multiset.toFinsupp p.parts).prod f
• Nat.coe_properDivisors_eq_prod_pow_lt_factorization 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} : ↑n.properDivisors = {x | ∃ f < n.factorization, (f.prod fun x1 x2 => x1 ^ x2) = x}
• Nat.coe_divisors_eq_prod_pow_le_factorization 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} (hn : n ≠ 0) : ↑n.divisors = {x | ∃ f ≤ n.factorization, (f.prod fun x1 x2 => x1 ^ x2) = x}
• Nat.properDivisors_eq_image_Iio_factorization_prod_pow 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} : n.properDivisors = Finset.image (fun x => x.prod fun x1 x2 => x1 ^ x2) (Finset.Iio n.factorization)
• Nat.divisors_eq_image_Iic_factorization_prod_pow 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} (hn : n ≠ 0) : n.divisors = Finset.image (fun x => x.prod fun x1 x2 => x1 ^ x2) (Finset.Iic n.factorization)
• Nat.Iic_factorization_prod_pow_injective 📋 Mathlib.Data.Nat.Factorization.Divisors
  (n : ℕ) : Function.Injective fun x => (↑x).prod fun x1 x2 => x1 ^ x2
• Nat.Iio_factorization_prod_pow_injective 📋 Mathlib.Data.Nat.Factorization.Divisors
  (n : ℕ) : Function.Injective fun x => (↑x).prod fun x1 x2 => x1 ^ x2
• Nat.properDivisors_eq_map_attach_Iio_factorization_prod_pow 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} : n.properDivisors = Finset.map { toFun := fun x => (↑x).prod fun x1 x2 => x1 ^ x2, inj' := ⋯ } (Finset.Iio n.factorization).attach
• Nat.divisors_eq_map_attach_Iic_factorization_prod_pow 📋 Mathlib.Data.Nat.Factorization.Divisors
  {n : ℕ} (hn : n ≠ 0) : n.divisors = Finset.map { toFun := fun x => (↑x).prod fun x1 x2 => x1 ^ x2, inj' := ⋯ } (Finset.Iic n.factorization).attach
• Nat.ceilRoot_def 📋 Mathlib.Data.Nat.Factorization.Root
  {a n : ℕ} : n.ceilRoot a = if n = 0 ∨ a = 0 then 0 else (a.factorization ⌈/⌉ n).prod fun x1 x2 => x1 ^ x2
• Nat.floorRoot_def 📋 Mathlib.Data.Nat.Factorization.Root
  {a n : ℕ} : n.floorRoot a = if n = 0 ∨ a = 0 then 0 else (a.factorization ⌊/⌋ n).prod fun x1 x2 => x1 ^ x2
• RingCon.coe_finsuppProd 📋 Mathlib.RingTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [Add M] [CommMonoid M] [Zero β] (c : RingCon M) (h : ι → β → M) (f : ι →₀ β) : ↑(f.prod h) = f.prod fun i b => ↑(h i b)
• RingCon.finsuppProd 📋 Mathlib.RingTheory.Congruence.BigOperators
  {ι : Type u_1} {β : Type u_2} {M : Type u_3} [Add M] [CommMonoid M] [Zero β] (c : RingCon M) (h h' : ι → β → M) {f g : ι →₀ β} (hf : ∀ (i : ι), c (h i 0) 1) (hf' : ∀ (i : ι), c (h' i 0) 1) (H : ∀ (i : ι), c (h i (f i)) (h' i (g i))) : c (f.prod h) (g.prod h')
• DividedPowers.dpow_finsupp_sum 📋 Mathlib.RingTheory.DividedPowers.Basic
  {A : Type u_1} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I) {ι : Type u_2} [DecidableEq ι] {x : ι →₀ A} (hx : ∀ (i : ι), x i ∈ I) {n : ℕ} : hI.dpow n (x.sum fun x r => r) = ∑ k ∈ x.support.sym n, x.prod fun i r => hI.dpow (Multiset.count i ↑k) r
• DividedPowers.dpow_linearCombination 📋 Mathlib.RingTheory.DividedPowers.Basic
  {A : Type u_1} [CommSemiring A] {ι : Type u_2} [DecidableEq ι] {S : Type u_3} [CommSemiring S] [Algebra A S] {J : Ideal S} (hJ : DividedPowers J) {b : ι → S} {x : ι →₀ A} (hx : ∀ i ∈ x.support, b i ∈ J) {n : ℕ} : hJ.dpow n (x.sum fun i r => r • b i) = ∑ k ∈ x.support.sym n, x.prod fun i r => r ^ Multiset.count i ↑k • hJ.dpow (Multiset.count i ↑k) (b i)
• DividedPowerAlgebra.submodule_span_prod_dp_eq_top 📋 Mathlib.RingTheory.DividedPowerAlgebra.Init
  {R : Type u_4} {M : Type u_5} {ι : Type u_6} [CommRing R] [AddCommGroup M] [Module R M] {v : ι → M} (hv : Submodule.span R (Set.range v) = ⊤) : Submodule.span R (Set.range fun n => n.prod fun i k => DividedPowerAlgebra.dp R k (v i)) = ⊤
• TwoSidedIdeal.finsuppProd_mem 📋 Mathlib.RingTheory.TwoSidedIdeal.BigOperators
  {R : Type u_1} [CommRing R] (I : TwoSidedIdeal R) {ι : Type u_2} {β : Type u_3} [Zero β] (h : ι → β → R) {f : ι →₀ β} (H : ∃ i ∈ f.support, h i (f i) ∈ I) : f.prod h ∈ I

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