English
Sum.elim distributes over dotProduct: dotProduct(Sum.elim u x, Sum.elim v y) = dotProduct(u,v) + dotProduct(x,y).
Русский
Sum.elim распространяется на скалярное произведение: dotProduct(Sum.elim u x, Sum.elim v y) = dotProduct(u,v) + dotProduct(x,y).
LaTeX
$$$$\operatorname{dotProduct}(\operatorname{Sum}.elim\, u\, x, \operatorname{Sum}.elim\, v\, y) = \operatorname{dotProduct}(u,v) + \operatorname{dotProduct}(x,y).$$$$
Lean4
/-- Permuting a vector on the left of a dot product can be transferred to the right. -/
@[simp]
theorem comp_equiv_symm_dotProduct (e : m ≃ n) : u ∘ e.symm ⬝ᵥ x = u ⬝ᵥ x ∘ e :=
(e.sum_comp _).symm.trans <| Finset.sum_congr rfl fun _ _ => by simp only [Function.comp, Equiv.symm_apply_apply]
