Theorems · Theorem · ring theory

Finset.prod_add

∀ {ι : Type u_1} {R : Type u_4} [inst : CommSemiring R] [inst_1 : DecidableEq ι] (f g : ι → R) (s : Finset ι),
  ∏ i ∈ s, (f i + g i) = ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, g i

The product of f a + g a over all of s is the sum over the powerset of s of the product of f over a subset t times the product of g over the complement of t

Defined in

Mathlib.Algebra.BigOperators.Ring.Finset

Cited by

7 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringDecidableEq