/-- The pointwise `star` of a non-unital subalgebra is a non-unital subalgebra. -/
instance instInvolutiveStar : InvolutiveStar (NonUnitalSubalgebra R A)
    where
  star S :=
    { carrier := star S.carrier
      mul_mem' := @fun x y hx hy => by simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier]
        using (star_mul x y).symm ▸ mul_mem hy hx
      add_mem' := @fun x y hx hy => by simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier]
        using (star_add x y).symm ▸ add_mem hx hy
      zero_mem' := Set.mem_star.mp ((star_zero A).symm ▸ zero_mem S : star (0 : A) ∈ S)
      smul_mem' := fun r x hx => by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
S : NonUnitalSubalgebra R A
r : R
x : A
hx :
  x ∈
    {
            toAddSubmonoid :=
              {
                toAddSubsemigroup :=
                  { carrier := star S.carrier,
                    add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) },
            mul_mem' :=
              (_ :
                ∀ (x y : A),
                  x ∈
                      {
                            toAddSubsemigroup :=
                              { carrier := star S.carrier,
                                add_mem' :=
                                  (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                            zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                    y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                      x * y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' :=
                                (_ :
                                  star 0 ∈
                                    S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ r • x ∈
    {
            toAddSubmonoid :=
              {
                toAddSubsemigroup :=
                  { carrier := star S.carrier,
                    add_mem' := (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) },
            mul_mem' :=
              (_ :
                ∀ (x y : A),
                  x ∈
                      {
                            toAddSubsemigroup :=
                              { carrier := star S.carrier,
                                add_mem' :=
                                  (_ : ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                            zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                    y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' := (_ : star 0 ∈ S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier →
                      x * y ∈
                        {
                              toAddSubsemigroup :=
                                { carrier := star S.carrier,
                                  add_mem' :=
                                    (_ :
                                      ∀ (x y : A), x ∈ star S.carrier → y ∈ star S.carrier → x + y ∈ star S.carrier) },
                              zero_mem' :=
                                (_ :
                                  star 0 ∈
                                    S.toAddSubsemigroup.1) }.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
simpa only [Set.mem_star, NonUnitalSubalgebra.mem_carrier]
        using (star_smul r x).symm ▸ SMulMemClass.smul_mem (star r) hx