Casting lemmas for rational numbers involving sums and products

@[simp]

theorem Rat.cast_list_sum {α : Type u_2} [DivisionRing α] [CharZero α] (s : List ℚ) : ↑(List.sum s) = List.sum (List.map Rat.cast s)

@[simp]

theorem Rat.cast_multiset_sum {α : Type u_2} [DivisionRing α] [CharZero α] (s : Multiset ℚ) : ↑(Multiset.sum s) = Multiset.sum (Multiset.map Rat.cast s)

@[simp]

theorem Rat.cast_sum {ι : Type u_1} {α : Type u_2} [DivisionRing α] [CharZero α] (s : Finset ι) (f : ι → ℚ) : ↑(Finset.sum s fun (i : ι) => f i) = Finset.sum s fun (i : ι) => ↑(f i)

@[simp]

theorem Rat.cast_list_prod {α : Type u_2} [DivisionRing α] [CharZero α] (s : List ℚ) : ↑(List.prod s) = List.prod (List.map Rat.cast s)

@[simp]

theorem Rat.cast_multiset_prod {α : Type u_2} [Field α] [CharZero α] (s : Multiset ℚ) : ↑(Multiset.prod s) = Multiset.prod (Multiset.map Rat.cast s)

@[simp]

theorem Rat.cast_prod {ι : Type u_1} {α : Type u_2} [Field α] [CharZero α] (s : Finset ι) (f : ι → ℚ) : ↑(Finset.prod s fun (i : ι) => f i) = Finset.prod s fun (i : ι) => ↑(f i)