/-- 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 

Proof state:
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✝¹ x✝ : ContinuousMultilinearMap R (fun x => M) N
⊢ 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✝) =
    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 +
      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✝

Next tactic:
ext