/-- Alternatization of a continuous multilinear map. -/
@[simps (config := .lemmasOnly) apply_toContinuousMultilinearMap]
def alternatization : ContinuousMultilinearMap R (fun _ : ι => M) N →+ M [Λ^ι]→L[R] N where
  toFun f :=
    { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f.domDomCongr σ
      map_eq_zero_of_eq' := fun v i j hv hne => by
        simpa [MultilinearMap.alternatization_apply]
          using f.1.alternatization.map_eq_zero_of_eq' v i j hv hne }
  map_zero' := by ext; simp
  map_add' _ _ := by ext; 

Proof state:
case H
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g x✝² x✝¹ : ContinuousMultilinearMap R (fun x => M) N
x✝ : ι → M
⊢ (ZeroHom.toFun
        {
          toFun := fun f =>
            { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
              map_eq_zero_of_eq' :=
                (_ :
                  ∀ (v : ι → M) (i j : ι),
                    v i = v j →
                      i ≠ j →
                        MultilinearMap.toFun (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap
                            v =
                          0) },
          map_zero' :=
            (_ :
              (fun f =>
                    { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
                      map_eq_zero_of_eq' :=
                        (_ :
                          ∀ (v : ι → M) (i j : ι),
                            v i = v j →
                              i ≠ j →
                                MultilinearMap.toFun
                                    (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                  0) })
                  0 =
                0) }
        (x✝² + x✝¹))
      x✝ =
    (ZeroHom.toFun
          {
            toFun := fun f =>
              { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
                map_eq_zero_of_eq' :=
                  (_ :
                    ∀ (v : ι → M) (i j : ι),
                      v i = v j →
                        i ≠ j →
                          MultilinearMap.toFun
                              (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                            0) },
            map_zero' :=
              (_ :
                (fun f =>
                      { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
                        map_eq_zero_of_eq' :=
                          (_ :
                            ∀ (v : ι → M) (i j : ι),
                              v i = v j →
                                i ≠ j →
                                  MultilinearMap.toFun
                                      (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                    0) })
                    0 =
                  0) }
          x✝² +
        ZeroHom.toFun
          {
            toFun := fun f =>
              { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
                map_eq_zero_of_eq' :=
                  (_ :
                    ∀ (v : ι → M) (i j : ι),
                      v i = v j →
                        i ≠ j →
                          MultilinearMap.toFun
                              (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                            0) },
            map_zero' :=
              (_ :
                (fun f =>
                      { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f,
                        map_eq_zero_of_eq' :=
                          (_ :
                            ∀ (v : ι → M) (i j : ι),
                              v i = v j →
                                i ≠ j →
                                  MultilinearMap.toFun
                                      (∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                    0) })
                    0 =
                  0) }
          x✝¹)
      x✝

Next tactic:
simp [Finset.sum_add_distrib]