Theorems · Theorem · ring theory

DirectSum.mul_eq_sum_support_ghas_mul

∀ {ι : Type u_1} [inst : DecidableEq ι] (A : ι → Type u_2) [inst_1 : (i : ι) → AddCommMonoid (A i)]
  [inst_2 : AddMonoid ι] [inst_3 : DirectSum.GSemiring A] [inst_4 : (i : ι) → (x : A i) → Decidable (x ≠ 0)]
  (a a' : DirectSum ι fun i => A i),
  a * a' =
    ∑ ij ∈ DFinsupp.support a ×ˢ DFinsupp.support a',
      (DirectSum.of A (ij.1 + ij.2)) (GradedMonoid.GMul.mul (a ij.1) (a' ij.2))

A heavily unfolded version of the definition of multiplication

Defined in

Mathlib.Algebra.DirectSum.Ring

Cited by

1 results in Mathlib

Foundations

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

Assumes

DecidableEqAddCommMonoidAddMonoidDirectSum.GSemiringDecidable