Theorems · Theorem · group theory

Finset.prod_image

∀ {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [inst : CommMonoid 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, products over the original finset or its image coincide. See also prod_image_of_pairwise_eq_one for a version with weaker assumptions.

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Basic

Cited by

17 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidDecidableEq