/-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
    (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x := by
  let A' : Subalgebra R A :=
    { carrier := { x | ∀ n ∈ N, x • n ∈ N }
      mul_mem' := fun {a b} ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
      one_mem' := fun n hn => (one_smul A n).symm ▸ hn
      add_mem' := fun {a b} ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
      zero_mem' := fun n _hn => (zero_smul A n).symm ▸ N.zero_mem
      algebraMap_mem' := fun r n hn => (algebraMap_smul A r n).symm ▸ N.smul_mem r hn }
  let f : A' →ₐ[R] Module.End R N :=
    AlgHom.ofLinearMap
      { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop
        -- porting note: was
                -- `fun x y => LinearMap.ext fun n => Subtype.ext <| add_smul x y n`
        map_add' := by intros x y; ext; 

Proof state:
case h.a
R : Type u_1
A : Type u_2
B : Type u_3
S : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Ring B
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
f : R →+* S
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : NoZeroSMulDivisors A M
N : Submodule R M
hN : N ≠ ⊥
hN' : FG N
x✝¹ : A
hx : ∀ n ∈ N, x✝¹ • n ∈ N
A' : Subalgebra R A :=
  {
    toSubsemiring :=
      {
        toSubmonoid :=
          {
            toSubsemigroup :=
              { carrier := {x | ∀ n ∈ N, x • n ∈ N},
                mul_mem' :=
                  (_ :
                    ∀ {a b : A},
                      a ∈ {x | ∀ n ∈ N, x • n ∈ N} → b ∈ {x | ∀ n ∈ N, x • n ∈ N} → ∀ n ∈ N, (a * b) • n ∈ N) },
            one_mem' := (_ : ∀ n ∈ N, 1 • n ∈ N) },
        add_mem' :=
          (_ :
            ∀ {a b : A},
              a ∈
                  {
                        toSubsemigroup :=
                          { carrier := {x | ∀ n ∈ N, x • n ∈ N},
                            mul_mem' :=
                              (_ :
                                ∀ {a b : A},
                                  a ∈ {x | ∀ n ∈ N, x • n ∈ N} →
                                    b ∈ {x | ∀ n ∈ N, x • n ∈ N} → ∀ n ∈ N, (a * b) • n ∈ N) },
                        one_mem' := (_ : ∀ n ∈ N, 1 • n ∈ N) }.toSubsemigroup.carrier →
                b ∈
                    {
                          toSubsemigroup :=
                            { carrier := {x | ∀ n ∈ N, x • n ∈ N},
                              mul_mem' :=
                                (_ :
                                  ∀ {a b : A},
                                    a ∈ {x | ∀ n ∈ N, x • n ∈ N} →
                                      b ∈ {x | ∀ n ∈ N, x • n ∈ N} → ∀ n ∈ N, (a * b) • n ∈ N) },
                          one_mem' := (_ : ∀ n ∈ N, 1 • n ∈ N) }.toSubsemigroup.carrier →
                  ∀ n ∈ N, (a + b) • n ∈ N),
        zero_mem' := (_ : ∀ n ∈ N, 0 • n ∈ N) },
    algebraMap_mem' := (_ : ∀ (r : R), ∀ n ∈ N, (algebraMap R A) r • n ∈ N) }
x y : ↥A'
x✝ : ↥N
⊢ ↑(((fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : ↑x ∈ A')) (x + y)) x✝) =
    ↑(((fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : ↑x ∈ A')) x +
          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : ↑x ∈ A')) y)
        x✝)

Next tactic:
exact add_smul _ _ _