decompose_mul_add_right — Mathlib

English

For b ∈ 𝒜 j and a ∈ A, the (i+j)-th component of a*b equals the i-th component of a times b.

Русский

Для b ∈ 𝒜 j и a ∈ A, i+j-я componente произведения a*b равна i-й компоненте a, умноженной на b.

LaTeX

$$$ decompose 𝒜 (a * ↑b) (i + j) =\ decompose 𝒜 a i * b, \text{ with } b ∈ 𝒜 j. $$$

Lean4

theorem decompose_mul_add_right [AddRightCancelMonoid ι] [GradedRing 𝒜] {a : A} (b : 𝒜 j) :
    decompose 𝒜 (a * ↑b) (i + j) = @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ (decompose 𝒜 a i) b :=
  Subtype.ext <| coe_decompose_mul_add_of_right_mem 𝒜 b.2

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