Theorems · Theorem · ring theory

DirectSum.coe_decompose_mul_of_left_mem

∀ {ι : Type u_1} {A : Type u_3} {σ : Type u_4} [inst : Semiring A] [inst_1 : DecidableEq ι] [inst_2 : AddCommMonoid ι]
  [inst_3 : PartialOrder ι] [CanonicallyOrderedAdd ι] [inst_5 : SetLike σ A] [inst_6 : AddSubmonoidClass σ A]
  (𝒜 : ι → σ) [inst_7 : GradedRing 𝒜] {a b : A} {i : ι} [inst_8 : Sub ι] [OrderedSub ι] [AddLeftReflectLE ι] (n : ι)
  [inst_11 : Decidable (i ≤ n)],
  a ∈ 𝒜 i → ↑(((DirectSum.decompose 𝒜) (a * b)) n) = if i ≤ n then a * ↑(((DirectSum.decompose 𝒜) b) (n - i)) else 0

Defined in

Mathlib.RingTheory.GradedAlgebra.Basic

Cited by

1 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringDecidableEqAddCommMonoidPartialOrderCanonicallyOrderedAddSetLikeAddSubmonoidClassGradedRingSubOrderedSubAddLeftReflectLEDecidable