English
The diagonal-vector product distributes over addition: (x ⊕ y) ᵥ* A = x ᵥ* A + y ᵥ* A.
Русский
Дистрибутивность по сложению для vecMul: (x ⊕ y) ᵥ* A = x ᵥ* A + y ᵥ* A.
LaTeX
$$$x ᵥ* (A + B) = x ᵥ* A + x ᵥ* B$$$
Lean4
theorem diagonal_mulVec_single [Fintype n] [DecidableEq n] [NonUnitalNonAssocSemiring R] (v : n → R) (j : n) (x : R) :
diagonal v *ᵥ Pi.single j x = Pi.single j (v j * x) :=
by
ext i
rw [mulVec_diagonal]
exact Pi.apply_single (fun i x => v i * x) (fun i => mul_zero _) j x i