/-- Constructing a linear map `(⨂[R] i, s i) → E` given a `MultilinearMap R s E` with the
property that its composition with the canonical `MultilinearMap R s E` is
the given multilinear map `φ`. -/
def lift : MultilinearMap R s E ≃ₗ[R] (⨂[R] i, s i) →ₗ[R] E where
  toFun φ := { liftAux φ with map_smul' := liftAux.smul }
  invFun φ' := φ'.compMultilinearMap (tprod R)
  left_inv φ := by
    ext
    simp [liftAux_tprod, LinearMap.compMultilinearMap]
  right_inv φ := by
    ext
    simp [liftAux_tprod]
  map_add' φ₁ φ₂ := by
    ext
    simp [liftAux_tprod]
  map_smul' r φ₂ := by
    

Proof state:
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
r : R
φ₂ : MultilinearMap R s E
⊢ AddHom.toFun
      {
        toFun := fun φ =>
          let src := liftAux φ;
          {
            toAddHom :=
              { toFun := src.toFun,
                map_add' :=
                  (_ :
                    ∀ (x y : ⨂[R] (i : ι), s i),
                      ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
            map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) },
        map_add' :=
          (_ :
            ∀ (φ₁ φ₂ : MultilinearMap R s E),
              (fun φ =>
                    let src := liftAux φ;
                    {
                      toAddHom :=
                        { toFun := src.toFun,
                          map_add' :=
                            (_ :
                              ∀ (x y : ⨂[R] (i : ι), s i),
                                ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                      map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                  (φ₁ + φ₂) =
                (fun φ =>
                      let src := liftAux φ;
                      {
                        toAddHom :=
                          { toFun := src.toFun,
                            map_add' :=
                              (_ :
                                ∀ (x y : ⨂[R] (i : ι), s i),
                                  ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                        map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                    φ₁ +
                  (fun φ =>
                      let src := liftAux φ;
                      {
                        toAddHom :=
                          { toFun := src.toFun,
                            map_add' :=
                              (_ :
                                ∀ (x y : ⨂[R] (i : ι), s i),
                                  ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                        map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                    φ₂) }
      (r • φ₂) =
    (RingHom.id R) r •
      AddHom.toFun
        {
          toFun := fun φ =>
            let src := liftAux φ;
            {
              toAddHom :=
                { toFun := src.toFun,
                  map_add' :=
                    (_ :
                      ∀ (x y : ⨂[R] (i : ι), s i),
                        ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
              map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) },
          map_add' :=
            (_ :
              ∀ (φ₁ φ₂ : MultilinearMap R s E),
                (fun φ =>
                      let src := liftAux φ;
                      {
                        toAddHom :=
                          { toFun := src.toFun,
                            map_add' :=
                              (_ :
                                ∀ (x y : ⨂[R] (i : ι), s i),
                                  ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                        map_smul' := (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                    (φ₁ + φ₂) =
                  (fun φ =>
                        let src := liftAux φ;
                        {
                          toAddHom :=
                            { toFun := src.toFun,
                              map_add' :=
                                (_ :
                                  ∀ (x y : ⨂[R] (i : ι), s i),
                                    ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                          map_smul' :=
                            (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                      φ₁ +
                    (fun φ =>
                        let src := liftAux φ;
                        {
                          toAddHom :=
                            { toFun := src.toFun,
                              map_add' :=
                                (_ :
                                  ∀ (x y : ⨂[R] (i : ι), s i),
                                    ZeroHom.toFun (↑src) (x + y) = ZeroHom.toFun (↑src) x + ZeroHom.toFun (↑src) y) },
                          map_smul' :=
                            (_ : ∀ (r : R) (x : ⨂[R] (i : ι), s i), (liftAux φ) (r • x) = r • (liftAux φ) x) })
                      φ₂) }
        φ₂

Next tactic:
ext