Theorems · Theorem · group theory

Finset.prod_induction

∀ {ι : Type u_1} {s : Finset ι} {M : Type u_7} [inst : CommMonoid M] (f : ι → M) (p : M → Prop),
  (∀ (a b : M), p a → p b → p (a * b)) → p 1 → (∀ x ∈ s, p (f x)) → p (∏ x ∈ s, f x)

To prove a property of a product, it suffices to prove that the property is multiplicative and holds on factors.

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Defs

Cited by

18 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoid