coe_decompose_mul_of_right_mem_of_not_le

English

If b ∈ 𝒜(i) and i ≤ n, then the coefficient at degree n of a*b is the product a*(b's n-i component).

Русский

Если b принадлежит 𝒜(i) и i ≤ n, коэффициент степени n в a*b равен a*(коэффициент степени n-i в b).

LaTeX

$$$b \in 𝒜(i),\ i ≤ n \Rightarrow (decompose 𝒜 (a b) n) = (decompose 𝒜 a (n-i)) \cdot b$.$$

Lean4

theorem coe_decompose_mul_of_right_mem_of_not_le (b_mem : b ∈ 𝒜 i) (h : ¬i ≤ n) : (decompose 𝒜 (a * b) n : A) = 0 :=
  by
  lift b to 𝒜 i using b_mem
  rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_not_le]

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