Theorems · Theorem · group theory
Finset.sum_image
∀ {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [inst : AddCommMonoid M] {f : ι → M} [inst_1 : DecidableEq ι]
{s : Finset κ} {g : κ → ι}, Set.InjOn g ↑s → ∑ x ∈ Finset.image g s, f x = ∑ x ∈ s, f (g x)If a function is injective on a finset, sums over the original
finset or its image coincide.
See also sum_image_of_pairwise_eq_zero for a version with weaker assumptions.
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoidDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- SetLike.coestatement · cited by 8,199
- Finset.sumstatement · cited by 5,195
- Finset.imagestatement · cited by 910
- Set.InjOnstatement · cited by 543
- Finset.fold_imageproof · cited by 2
Cited by26
Results whose statement or proof uses this declaration.
- Finset.sum_attachproof · cited by 54
- HasSum.nat_add_negproof · cited by 10
- HasSum.nat_add_neg_add_oneproof · cited by 5
- Nat.sum_div_divisorsproof · cited by 3
- Finset.sum_preimage'proof · cited by 3
- Finset.prod_sumproof · cited by 3
- MvPowerSeries.rename_renameproof · cited by 3
- Finset.sum_powerset_insertproof · cited by 2
- ZLattice.exists_finsetSum_norm_rpow_le_tsumproof · cited by 2
- Finset.expect_imageproof · cited by 2
- linearIndepOn_iff'proof · cited by 2
- finsum_mem_image'proof · cited by 2