/-- Lift a family of linear maps `M →ₗ[R] N` indexed by `x : α` to a linear map from `α →₀ M` to
`N` using `Finsupp.sum`. This is an upgraded version of `Finsupp.liftAddHom`.

See note [bundled maps over different rings] for why separate `R` and `S` semirings are used.
-/
def lsum : (α → M →ₗ[R] N) ≃ₗ[S] (α →₀ M) →ₗ[R] N where
  toFun F :=
    { toFun := fun d => d.sum fun i => F i
      map_add' := (liftAddHom (α := α) (M := M) (N := N) fun x => (F x).toAddMonoidHom).map_add
      map_smul' := fun c f => by simp [sum_smul_index', smul_sum] }
  invFun F x := F.comp (lsingle x)
  left_inv F := by
    ext x y
    simp
  right_inv F := by
    ext x y
    simp
  map_add' F G := by
    ext x y
    simp
  map_smul' F G := by
    ext x y
    

Proof state:
case h.h
α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁹ : Semiring R
inst✝⁸ : Semiring S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : Module S N
inst✝ : SMulCommClass R S N
F : S
G : α → M →ₗ[R] N
x : α
y : M
⊢ (AddHom.toFun
          {
            toFun := fun F =>
              {
                toAddHom :=
                  { toFun := fun d => sum d fun i => ⇑(F i),
                    map_add' :=
                      (_ :
                        ∀ (a b : α →₀ M),
                          (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                            (liftAddHom fun x => toAddMonoidHom (F x)) a +
                              (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                map_smul' := (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) },
            map_add' :=
              (_ :
                ∀ (F G : α → M →ₗ[R] N),
                  (fun F =>
                        {
                          toAddHom :=
                            { toFun := fun d => sum d fun i => ⇑(F i),
                              map_add' :=
                                (_ :
                                  ∀ (a b : α →₀ M),
                                    (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                      (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                        (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                          map_smul' :=
                            (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                      (F + G) =
                    (fun F =>
                          {
                            toAddHom :=
                              { toFun := fun d => sum d fun i => ⇑(F i),
                                map_add' :=
                                  (_ :
                                    ∀ (a b : α →₀ M),
                                      (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                        (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                          (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                            map_smul' :=
                              (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                        F +
                      (fun F =>
                          {
                            toAddHom :=
                              { toFun := fun d => sum d fun i => ⇑(F i),
                                map_add' :=
                                  (_ :
                                    ∀ (a b : α →₀ M),
                                      (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                        (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                          (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                            map_smul' :=
                              (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                        G) }
          (F • G) ∘ₗ
        lsingle x)
      y =
    (((RingHom.id S) F •
          AddHom.toFun
            {
              toFun := fun F =>
                {
                  toAddHom :=
                    { toFun := fun d => sum d fun i => ⇑(F i),
                      map_add' :=
                        (_ :
                          ∀ (a b : α →₀ M),
                            (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                              (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                  map_smul' :=
                    (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) },
              map_add' :=
                (_ :
                  ∀ (F G : α → M →ₗ[R] N),
                    (fun F =>
                          {
                            toAddHom :=
                              { toFun := fun d => sum d fun i => ⇑(F i),
                                map_add' :=
                                  (_ :
                                    ∀ (a b : α →₀ M),
                                      (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                        (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                          (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                            map_smul' :=
                              (_ : ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                        (F + G) =
                      (fun F =>
                            {
                              toAddHom :=
                                { toFun := fun d => sum d fun i => ⇑(F i),
                                  map_add' :=
                                    (_ :
                                      ∀ (a b : α →₀ M),
                                        (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                          (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                            (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                              map_smul' :=
                                (_ :
                                  ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                          F +
                        (fun F =>
                            {
                              toAddHom :=
                                { toFun := fun d => sum d fun i => ⇑(F i),
                                  map_add' :=
                                    (_ :
                                      ∀ (a b : α →₀ M),
                                        (liftAddHom fun x => toAddMonoidHom (F x)) (a + b) =
                                          (liftAddHom fun x => toAddMonoidHom (F x)) a +
                                            (liftAddHom fun x => toAddMonoidHom (F x)) b) },
                              map_smul' :=
                                (_ :
                                  ∀ (c : R) (f : α →₀ M), (sum (c • f) fun i => ⇑(F i)) = c • sum f fun i => ⇑(F i)) })
                          G) }
            G) ∘ₗ
        lsingle x)
      y

Next tactic:
simp