Theorems · Theorem · ring theory

Finset.prod_sub

∀ {ι : Type u_1} {R : Type u_4} [inst : CommRing R] [inst_1 : DecidableEq ι] (f g : ι → R) (s : Finset ι),
  ∏ i ∈ s, (f i - g i) = ∑ t ∈ s.powerset, ((-1) ^ t.card * ∏ i ∈ s \ t, f i) * ∏ i ∈ t, g i

The product of f i - g i over all of s is the sum over the powerset of s of the product of g over a subset t times the product of f over the complement of t times (-1) ^ #t.

Defined in

Mathlib.Algebra.BigOperators.Ring.Finset

Cited by

2 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingDecidableEq