English
In the Fin (n+1) setting, an alternating map is additive in its first variable: f(Fin.cons(x+y, m)) = f(Fin.cons(x, m)) + f(Fin.cons(y, m)).
Русский
В случае с Fin (n+1) чередующая карта аккуратно поищображает первую координату: f(Fin.cons(x+y, m)) = f(Fin.cons(x, m)) + f(Fin.cons(y, m)).
LaTeX
$$$f(\text{Fin.cons}(x+y, m)) = f(\text{Fin.cons}(x, m)) + f(\text{Fin.cons}(y, m)).$$$
Lean4
/-- In the specific case of continuous alternating maps on spaces indexed by `Fin (n+1)`, where one
can build an element of `Π(i : Fin (n+1)), M i` using `cons`, one can express directly the
additivity of an alternating map along the first variable. -/
theorem cons_add (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (x y : M) :
f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m) :=
f.toMultilinearMap.cons_add m x y
