/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
x : K
⊢ x ∈
      {
              toSubmonoid :=
                {
                  toSubsemigroup :=
                    { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                      mul_mem' :=
                        (_ :
                          ∀ {a b : K},
                            a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                              b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                  one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) },
              add_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈
                        {
                              toSubsemigroup :=
                                { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                  mul_mem' :=
                                    (_ :
                                      ∀ {a b : K},
                                        a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                          b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            a * b ∈
                                              {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                              one_mem' :=
                                (_ :
                                  ∃ x ∈ Subring.closure s,
                                    ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier →
                      b ∈
                          {
                                toSubsemigroup :=
                                  { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                    mul_mem' :=
                                      (_ :
                                        ∀ {a b : K},
                                          a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                              a * b ∈
                                                {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                                one_mem' :=
                                  (_ :
                                    ∃ x ∈ Subring.closure s,
                                      ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier →
                        a + b ∈
                          {
                                toSubsemigroup :=
                                  { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                    mul_mem' :=
                                      (_ :
                                        ∀ {a b : K},
                                          a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                              a * b ∈
                                                {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                                one_mem' :=
                                  (_ :
                                    ∃ x ∈ Subring.closure s,
                                      ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier),
              zero_mem' :=
                (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier →
    -x ∈
      {
              toSubmonoid :=
                {
                  toSubsemigroup :=
                    { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                      mul_mem' :=
                        (_ :
                          ∀ {a b : K},
                            a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                              b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                  one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) },
              add_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈
                        {
                              toSubsemigroup :=
                                { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                  mul_mem' :=
                                    (_ :
                                      ∀ {a b : K},
                                        a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                          b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            a * b ∈
                                              {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                              one_mem' :=
                                (_ :
                                  ∃ x ∈ Subring.closure s,
                                    ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier →
                      b ∈
                          {
                                toSubsemigroup :=
                                  { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                    mul_mem' :=
                                      (_ :
                                        ∀ {a b : K},
                                          a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                              a * b ∈
                                                {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                                one_mem' :=
                                  (_ :
                                    ∃ x ∈ Subring.closure s,
                                      ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier →
                        a + b ∈
                          {
                                toSubsemigroup :=
                                  { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
                                    mul_mem' :=
                                      (_ :
                                        ∀ {a b : K},
                                          a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                            b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                                              a * b ∈
                                                {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
                                one_mem' :=
                                  (_ :
                                    ∃ x ∈ Subring.closure s,
                                      ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier),
              zero_mem' :=
                (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier

Next tactic:
rintro ⟨y, hy, z, hz, x_eq⟩