Theorems · Theorem · commutative algebra
DividedPowers.prod_dpow
∀ {A : Type u_1} [inst : CommSemiring A] {I : Ideal A} {a : A} (hI : DividedPowers I) {ι : Type u_2} {s : Finset ι}
{n : ι → ℕ}, a ∈ I → ∏ i ∈ s, hI.dpow (n i) a = ↑(Nat.multinomial s n) * hI.dpow (s.sum n) aA product of divided powers is a multinomial coefficient times the divided power [Roby-1965], formula (III')
- Defined in
- Mathlib.RingTheory.DividedPowers.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
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Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- CommSemiringstatement and proof · cited by 10,911
- Finset.sumstatement and proof · cited by 5,195
- Idealstatement and proof · cited by 4,748
- one_mulproof · cited by 2,841
- Nat.cast_oneproof · cited by 2,501
- Finset.prodstatement and proof · cited by 2,356
- mul_commproof · cited by 2,262
- mul_assocproof · cited by 1,667
- Nat.chooseproof · cited by 494
- Nat.cast_mulproof · cited by 309
- Finset.sum_insertproof · cited by 196
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