English
Let a be an m×n matrix and b be an n×o matrix over a semiring A with compatible scalar action. The linear map that represents right multiplication by a·b is equal to the composition of the right-multiplication map by b after the right-multiplication map by a; i.e., R_{a b} = R_b ∘ R_a.
Русский
Пусть A — полугруппа (полукольцо) над, например, полусферой, и a ∈ M_{m×n}(A), b ∈ M_{n×o}(A). Линейное отображение, задаваемое правым умножением на a b, совпадает с композицией отображения правого умножения на b после отображения правого умножения на a; то есть R_{a b} = R_b ∘ R_a.
LaTeX
$$$\operatorname{mulRightLinearMap}\ l\ R\ (a b) = \big(\operatorname{mulRightLinearMap}\ l\ R\ b\big) \circ \big(\operatorname{mulRightLinearMap}\ l\ R\ a\big)$$$
Lean4
/-- A version of `LinearMap.mulRight_mul` for matrix multiplication. -/
@[simp]
theorem mulRightLinearMap_mul [IsScalarTower R A A] (a : Matrix m n A) (b : Matrix n o A) :
mulRightLinearMap l R (a * b) = (mulRightLinearMap l R b).comp (mulRightLinearMap l R a) :=
by
ext
simp only [mulRightLinearMap_apply, LinearMap.comp_apply, Matrix.mul_assoc]
