/-- 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
    

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
φ : (⨂[R] (i : ι), s i) →ₗ[R] E
⊢ AddHom.toFun
      {
          toAddHom :=
            {
              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) },
          map_smul' :=
            (_ :
              ∀ (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) }.toAddHom
      ((fun φ' => LinearMap.compMultilinearMap φ' (tprod R)) φ) =
    φ

Next tactic:
ext