Results about mapping big operators across ring equivalences

theorem RingEquiv.map_list_prod {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R ≃+* S) (l : List R) : f (List.prod l) = List.prod (List.map (⇑f) l)

theorem RingEquiv.map_list_sum {R : Type u_2} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (l : List R) : f (List.sum l) = List.sum (List.map (⇑f) l)

theorem RingEquiv.unop_map_list_prod {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R ≃+* Sᵐᵒᵖ) (l : List R) : MulOpposite.unop (f (List.prod l)) = List.prod (List.reverse (List.map (MulOpposite.unop ∘ ⇑f) l))

An isomorphism into the opposite ring acts on the product by acting on the reversed elements

theorem RingEquiv.map_multiset_prod {R : Type u_2} {S : Type u_3} [CommSemiring R] [CommSemiring S] (f : R ≃+* S) (s : Multiset R) : f (Multiset.prod s) = Multiset.prod (Multiset.map (⇑f) s)

theorem RingEquiv.map_multiset_sum {R : Type u_2} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (s : Multiset R) : f (Multiset.sum s) = Multiset.sum (Multiset.map (⇑f) s)

theorem RingEquiv.map_prod {α : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [CommSemiring S] (g : R ≃+* S) (f : α → R) (s : Finset α) : g (Finset.prod s fun (x : α) => f x) = Finset.prod s fun (x : α) => g (f x)

theorem RingEquiv.map_sum {α : Type u_1} {R : Type u_2} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] (g : R ≃+* S) (f : α → R) (s : Finset α) : g (Finset.sum s fun (x : α) => f x) = Finset.sum s fun (x : α) => g (f x)