Theorems · Theorem · ring theory

DirectSum.coe_mul_apply_eq_dfinsuppSum

∀ {ι : Type u_1} {σ : Type u_2} {R : Type u_4} [inst : DecidableEq ι] [inst_1 : Semiring R] [inst_2 : SetLike σ R]
  [inst_3 : AddSubmonoidClass σ R] (A : ι → σ) [inst_4 : AddMonoid ι] [inst_5 : SetLike.GradedMonoid A]
  [inst_6 : (i : ι) → (x : ↥(A i)) → Decidable (x ≠ 0)] (r r' : DirectSum ι fun i => ↥(A i)) (n : ι),
  ↑((r * r') n) = DFinsupp.sum r fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then ↑ri * ↑rj else 0

Defined in

Mathlib.Algebra.DirectSum.Internal

Cited by

4 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqSemiringSetLikeAddSubmonoidClassAddMonoidSetLike.GradedMonoidDecidable