Theorems · Theorem · ring theory

DirectSum.coe_of_mul_apply

∀ {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 : AddCommMonoid ι] [inst_5 : PartialOrder ι]
  [CanonicallyOrderedAdd ι] [inst_7 : SetLike.GradedMonoid A] [inst_8 : Sub ι] [OrderedSub ι] [AddLeftReflectLE ι]
  {i : ι} (r : ↥(A i)) (r' : DirectSum ι fun i => ↥(A i)) (n : ι) [inst_11 : Decidable (i ≤ n)],
  ↑(((DirectSum.of (fun i => ↥(A i)) i) r * r') n) = if i ≤ n then ↑r * ↑(r' (n - i)) else 0

Defined in

Mathlib.Algebra.DirectSum.Internal

Cited by

1 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqSemiringSetLikeAddSubmonoidClassAddCommMonoidPartialOrderCanonicallyOrderedAddSetLike.GradedMonoidSubOrderedSubAddLeftReflectLEDecidable