/-- 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
    ext
    

Proof state:
case H.H
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
x✝ : (i : ι) → s i
⊢ (LinearMap.compMultilinearMap
        (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))
        (tprod R))
      x✝ =
    (LinearMap.compMultilinearMap
        ((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)
        (tprod R))
      x✝

Next tactic:
simp [liftAux_tprod]