Theorems · Theorem · commutative algebra

DividedPowers.dpow_finsupp_sum

∀ {A : Type u_1} [inst : CommSemiring A] {I : Ideal A} (hI : DividedPowers I) {ι : Type u_2} [inst_1 : DecidableEq ι]
  {x : ι →₀ A},
  (∀ (i : ι), x i ∈ I) →
    ∀ {n : ℕ}, hI.dpow n (x.sum fun x r => r) = ∑ k ∈ x.support.sym n, x.prod fun i r => hI.dpow (Multiset.count i ↑k) r

A "multinomial" theorem for divided powers — without multinomial coefficients — for finitely supported functions.

Defined in

Mathlib.RingTheory.DividedPowers.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

CommSemiringDecidableEq