English
For a dependent family β, and a family of monoid homs g, the value of the product over a finset is given coordinatewise by applying g to each coordinate and then multiplying.
Русский
Значение произведения над FinsetGiven равно по координатам применения соответствующих гомоморфизмов и перемножения.
LaTeX
$$$ (f.prod g) r = f.prod (\\lambda a b. (g a b) r). $$$
Lean4
theorem sum_sub_index [∀ i, AddGroup (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] [AddCommGroup γ] {f g : Π₀ i, β i}
{h : ∀ i, β i → γ} (h_sub : ∀ i b₁ b₂, h i (b₁ - b₂) = h i b₁ - h i b₂) : (f - g).sum h = f.sum h - g.sum h :=
by
have := (liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (h_sub a)).map_sub f g
rw [liftAddHom_apply, sumAddHom_apply, sumAddHom_apply, sumAddHom_apply] at this
exact this
