Theorems · Theorem · group theory

finprod_mem_union_inter

∀ {α : Type u_1} {M : Type u_5} [inst : CommMonoid M] {f : α → M} {s t : Set α},
  s.Finite →
    t.Finite →
      (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t), f i =
        (∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i

Given finite sets s and t, the product of f i over i ∈ s ∪ t times the product of f i over i ∈ s ∩ t equals the product of f i over i ∈ s times the product of f i over i ∈ t.

Defined in

Mathlib.Algebra.BigOperators.Finprod

Cited by

1 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoid