Theorems · Theorem · commutative algebra

DividedPowers.dpow_linearCombination

∀ {A : Type u_1} [inst : CommSemiring A] {ι : Type u_2} [inst_1 : DecidableEq ι] {S : Type u_3}
  [inst_2 : CommSemiring S] [inst_3 : Algebra A S] {J : Ideal S} (hJ : DividedPowers J) {b : ι → S} {x : ι →₀ A},
  (∀ i ∈ x.support, b i ∈ J) →
    ∀ {n : ℕ},
      hJ.dpow n (x.sum fun i r => r • b i) =
        ∑ k ∈ x.support.sym n, x.prod fun i r => r ^ Multiset.count i ↑k • hJ.dpow (Multiset.count i ↑k) (b i)

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

CommSemiringDecidableEqCommSemiringAlgebra