Theorems · Theorem · group theory

Finset.sum_biUnion

∀ {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} {t : κ → Finset ι}, (↑s).PairwiseDisjoint t → ∑ x ∈ s.biUnion t, f x = ∑ x ∈ s, ∑ i ∈ t x, f i

Given a finite family of pairwise disjoint finsets, the sum over their union is the sum of the sums over the sets. See also sum_biUnion_of_pairwise_eq_zero for a version with weaker assumptions.

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Basic

Cited by

10 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidDecidableEq