/
BigOperators.lean
62 lines (44 loc) · 1.73 KB
/
BigOperators.lean
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/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Algebra.BigOperators.Basic
#align_import data.rat.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
/-! # Casting lemmas for rational numbers involving sums and products
-/
open BigOperators
variable {ι α : Type*}
namespace Rat
section WithDivRing
variable [DivisionRing α] [CharZero α]
@[simp, norm_cast]
theorem cast_list_sum (s : List ℚ) : (↑s.sum : α) = (s.map (↑)).sum :=
map_list_sum (Rat.castHom α) _
#align rat.cast_list_sum Rat.cast_list_sum
@[simp, norm_cast]
theorem cast_multiset_sum (s : Multiset ℚ) : (↑s.sum : α) = (s.map (↑)).sum :=
map_multiset_sum (Rat.castHom α) _
#align rat.cast_multiset_sum Rat.cast_multiset_sum
@[simp, norm_cast]
theorem cast_sum (s : Finset ι) (f : ι → ℚ) : ∑ i in s, f i = ∑ i in s, (f i : α) :=
map_sum (Rat.castHom α) _ s
#align rat.cast_sum Rat.cast_sum
@[simp, norm_cast]
theorem cast_list_prod (s : List ℚ) : (↑s.prod : α) = (s.map (↑)).prod :=
map_list_prod (Rat.castHom α) _
#align rat.cast_list_prod Rat.cast_list_prod
end WithDivRing
section Field
variable [Field α] [CharZero α]
@[simp, norm_cast]
theorem cast_multiset_prod (s : Multiset ℚ) : (↑s.prod : α) = (s.map (↑)).prod :=
map_multiset_prod (Rat.castHom α) _
#align rat.cast_multiset_prod Rat.cast_multiset_prod
@[simp, norm_cast]
theorem cast_prod (s : Finset ι) (f : ι → ℚ) : ∏ i in s, f i = ∏ i in s, (f i : α) :=
map_prod (Rat.castHom α) _ _
#align rat.cast_prod Rat.cast_prod
end Field
end Rat