Lemmas about group actions on big operators

This file contains results about two kinds of actions:

• sums over DistribSMul: r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x
• products over MulDistribMulAction (with primed name): r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x
• products over SMulCommClass (with unprimed name): b ^ s.card • ∏ x ∈ s, f x = ∏ x ∈ s, b • f x

Note that analogous lemmas for Modules like Finset.sum_smul appear in other files.

theorem List.smul_sum {M : Type u_1} {N : Type u_2} [AddMonoid N] [DistribSMul M N] {r : M} {l : List N} : r • l.sum = (map (fun (x : N) => r • x) l).sum

theorem List.smul_prod' {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] [MulDistribMulAction M N] {r : M} {l : List N} : r • l.prod = (map (fun (x : N) => r • x) l).prod

theorem Multiset.smul_sum {M : Type u_1} {N : Type u_2} [AddCommMonoid N] [DistribSMul M N] {r : M} {s : Multiset N} : r • s.sum = (map (fun (x : N) => r • x) s).sum

theorem Finset.smul_sum {M : Type u_1} {N : Type u_2} {γ : Type u_3} [AddCommMonoid N] [DistribSMul M N] {r : M} {f : γ → N} {s : Finset γ} : r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x

theorem smul_finsum_mem {M : Type u_1} {N : Type u_2} {γ : Type u_3} [AddCommMonoid N] [DistribSMul M N] {r : M} {f : γ → N} {s : Set γ} (hs : s.Finite) : r • ∑ᶠ (x : γ) (_ : x ∈ s), f x = ∑ᶠ (x : γ) (_ : x ∈ s), r • f x

theorem Multiset.smul_prod' {M : Type u_1} {N : Type u_2} [Monoid M] [CommMonoid N] [MulDistribMulAction M N] {r : M} {s : Multiset N} : r • s.prod = (map (fun (x : N) => r • x) s).prod

theorem Finset.smul_prod' {M : Type u_1} {N : Type u_2} {γ : Type u_3} [Monoid M] [CommMonoid N] [MulDistribMulAction M N] {r : M} {f : γ → N} {s : Finset γ} : r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x

theorem smul_finprod' {M : Type u_1} {N : Type u_2} [Monoid M] [CommMonoid N] [MulDistribMulAction M N] {ι : Sort u_4} [Finite ι] {f : ι → N} (r : M) : r • ∏ᶠ (x : ι), f x = ∏ᶠ (x : ι), r • f x

theorem Finset.smul_prod_perm {N : Type u_2} [CommMonoid N] {G : Type u_4} [Group G] [MulDistribMulAction G N] [Fintype G] (b : N) (g : G) : g • ∏ h : G, h • b = ∏ h : G, h • b

theorem smul_finprod_perm {N : Type u_2} [CommMonoid N] {G : Type u_4} [Group G] [MulDistribMulAction G N] [Finite G] (b : N) (g : G) : g • ∏ᶠ (h : G), h • b = ∏ᶠ (h : G), h • b

theorem List.smul_prod {M : Type u_1} {N : Type u_2} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (l : List N) (m : M) : m ^ l.length • l.prod = (map (fun (x : N) => m • x) l).prod

theorem List.vadd_sum {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddZeroClass N] [AddAction M N] [VAddAssocClass M N N] [VAddCommClass M N N] (l : List N) (m : M) : l.length • m +ᵥ l.sum = (map (fun (x : N) => m +ᵥ x) l).sum

theorem Multiset.smul_prod {M : Type u_1} {N : Type u_2} [Monoid M] [CommMonoid N] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (s : Multiset N) (b : M) : b ^ s.card • s.prod = (map (fun (x : N) => b • x) s).prod

theorem Multiset.vadd_sum {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddCommMonoid N] [AddAction M N] [VAddAssocClass M N N] [VAddCommClass M N N] (s : Multiset N) (b : M) : s.card • b +ᵥ s.sum = (map (fun (x : N) => b +ᵥ x) s).sum

theorem Finset.smul_prod {M : Type u_1} {N : Type u_2} {ι : Type u_4} [CommMonoid N] [Monoid M] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (s : Finset ι) (b : M) (f : ι → N) : b ^ s.card • ∏ x ∈ s, f x = ∏ x ∈ s, b • f x

theorem Finset.prod_smul {M : Type u_1} {N : Type u_2} {ι : Type u_4} [CommMonoid N] [CommMonoid M] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (s : Finset ι) (b : ι → M) (f : ι → N) : ∏ i ∈ s, b i • f i = (∏ i ∈ s, b i) • ∏ i ∈ s, f i