l3lab/ntp-mathlib huggingface dataset:
/-- The image of a subfield along a ring homomorphism is a subfield. -/
def map (s : Subfield K) : Subfield L :=
  { s.toSubring.map f with
    inv_mem' := by
      

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s✝ t : Subfield K
f : K →+* L
s : Subfield K
src✝ : Subring L := Subring.map f s.toSubring
⊢ ∀
    x ∈
      { toSubsemiring := src✝.toSubsemiring,
                neg_mem' :=
                  (_ :
                    ∀ {x : L}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier,
    x⁻¹ ∈
      { toSubsemiring := src✝.toSubsemiring,
                neg_mem' :=
                  (_ :
                    ∀ {x : L}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier

Next tactic:
rintro _ ⟨x, hx, rfl⟩
/-- The image of a subfield along a ring homomorphism is a subfield. -/
def map (s : Subfield K) : Subfield L :=
  { s.toSubring.map f with
    inv_mem' := by
      rintro _ ⟨x, hx, rfl⟩
      

Proof state:
case intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s✝ t : Subfield K
f : K →+* L
s : Subfield K
src✝ : Subring L := Subring.map f s.toSubring
x : K
hx : x ∈ s.carrier
⊢ (f x)⁻¹ ∈
    { toSubsemiring := src✝.toSubsemiring,
              neg_mem' :=
                (_ : ∀ {x : L}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier

Next tactic:
exact ⟨x⁻¹, s.inv_mem hx, map_inv₀ f x⟩
@[simp]
theorem mem_map {f : K →+* L} {s : Subfield K} {y : L} : y ∈ s.map f ↔ ∃ x ∈ s, f x = y := by
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s✝ t : Subfield K
f✝ f : K →+* L
s : Subfield K
y : L
⊢ y ∈ map f s ↔ ∃ x ∈ s, f x = y

Next tactic:
unfold map
@[simp]
theorem mem_map {f : K →+* L} {s : Subfield K} {y : L} : y ∈ s.map f ↔ ∃ x ∈ s, f x = y := by
  unfold map
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s✝ t : Subfield K
f✝ f : K →+* L
s : Subfield K
y : L
⊢ (y ∈
      let src := Subring.map f s.toSubring;
      {
        toSubring :=
          { toSubsemiring := src.toSubsemiring,
            neg_mem' :=
              (_ :
                ∀ {x : L},
                  x ∈ (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier →
                    -x ∈ (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) },
        inv_mem' :=
          (_ :
            ∀
              x ∈
                { toSubsemiring := (Subring.map f s.toSubring).toSubsemiring,
                          neg_mem' :=
                            (_ :
                              ∀ {x : L},
                                x ∈ (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier →
                                  -x ∈
                                    (Subring.map f
                                              s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier,
              x⁻¹ ∈
                { toSubsemiring := (Subring.map f s.toSubring).toSubsemiring,
                          neg_mem' :=
                            (_ :
                              ∀ {x : L},
                                x ∈ (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier →
                                  -x ∈
                                    (Subring.map f
                                              s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }) ↔
    ∃ x ∈ s, f x = y

Next tactic:
simp only [mem_mk, Subring.mem_mk, Subring.mem_toSubsemiring, Subring.mem_map, mem_toSubring]
theorem fieldRange_eq_map : f.fieldRange = Subfield.map f ⊤ := by
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
g : L →+* M
f : K →+* L
⊢ fieldRange f = Subfield.map f ⊤

Next tactic:
ext
theorem fieldRange_eq_map : f.fieldRange = Subfield.map f ⊤ := by
  ext
  

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
g : L →+* M
f : K →+* L
x✝ : L
⊢ x✝ ∈ fieldRange f ↔ x✝ ∈ Subfield.map f ⊤

Next tactic:
simp
theorem map_fieldRange : f.fieldRange.map g = (g.comp f).fieldRange := by
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
g : L →+* M
f : K →+* L
⊢ Subfield.map g (fieldRange f) = fieldRange (comp g f)

Next tactic:
simpa only [fieldRange_eq_map] using (⊤ : Subfield K).map_map g f
instance : InfSet (Subfield K) :=
  ⟨fun S =>
    { sInf (Subfield.toSubring '' S) with
      inv_mem' := 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 (Subfield K)
src✝ : Subring K := sInf (toSubring '' S)
⊢ ∀
    x ∈
      { toSubsemiring := src✝.toSubsemiring,
                neg_mem' :=
                  (_ :
                    ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier,
    x⁻¹ ∈
      { toSubsemiring := src✝.toSubsemiring,
                neg_mem' :=
                  (_ :
                    ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier

Next tactic:
rintro x hx
instance : InfSet (Subfield K) :=
  ⟨fun S =>
    { sInf (Subfield.toSubring '' S) with
      inv_mem' := by
        rintro x hx
        

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
src✝ : Subring K := sInf (toSubring '' S)
x : K
hx :
  x ∈
    { toSubsemiring := src✝.toSubsemiring,
              neg_mem' :=
                (_ : ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier
⊢ x⁻¹ ∈
    { toSubsemiring := src✝.toSubsemiring,
              neg_mem' :=
                (_ : ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier

Next tactic:
apply Subring.mem_sInf.mpr
instance : InfSet (Subfield K) :=
  ⟨fun S =>
    { sInf (Subfield.toSubring '' S) with
      inv_mem' := by
        rintro x hx
        apply Subring.mem_sInf.mpr
        

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
src✝ : Subring K := sInf (toSubring '' S)
x : K
hx :
  x ∈
    { toSubsemiring := src✝.toSubsemiring,
              neg_mem' :=
                (_ : ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier
⊢ ∀ p ∈ toSubring '' S, x⁻¹ ∈ p

Next tactic:
rintro _ ⟨p, p_mem, rfl⟩
instance : InfSet (Subfield K) :=
  ⟨fun S =>
    { sInf (Subfield.toSubring '' S) with
      inv_mem' := by
        rintro x hx
        apply Subring.mem_sInf.mpr
        rintro _ ⟨p, p_mem, rfl⟩
        

Proof state:
case intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
src✝ : Subring K := sInf (toSubring '' S)
x : K
hx :
  x ∈
    { toSubsemiring := src✝.toSubsemiring,
              neg_mem' :=
                (_ : ∀ {x : K}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier
p : Subfield K
p_mem : p ∈ S
⊢ x⁻¹ ∈ p.toSubring

Next tactic:
exact p.inv_mem (Subring.mem_sInf.mp hx p.toSubring ⟨p, p_mem, rfl⟩)
@[simp, norm_cast]
theorem coe_sInf (S : Set (Subfield K)) : ((sInf S : Subfield K) : Set K) = ⋂ s ∈ S, ↑s :=
  show ((sInf (Subfield.toSubring '' S) : Subring K) : Set K) = ⋂ s ∈ S, ↑s 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 (Subfield K)
⊢ ↑(sInf (toSubring '' S)) = ⋂ s ∈ S, ↑s

Next tactic:
ext x
@[simp, norm_cast]
theorem coe_sInf (S : Set (Subfield K)) : ((sInf S : Subfield K) : Set K) = ⋂ s ∈ S, ↑s :=
  show ((sInf (Subfield.toSubring '' S) : Subring K) : Set K) = ⋂ s ∈ S, ↑s by
    ext x
    

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
x : K
⊢ x ∈ ↑(sInf (toSubring '' S)) ↔ x ∈ ⋂ s ∈ S, ↑s

Next tactic:
rw [Subring.coe_sInf, Set.mem_iInter, Set.mem_iInter]
@[simp, norm_cast]
theorem coe_sInf (S : Set (Subfield K)) : ((sInf S : Subfield K) : Set K) = ⋂ s ∈ S, ↑s :=
  show ((sInf (Subfield.toSubring '' S) : Subring K) : Set K) = ⋂ s ∈ S, ↑s by
    ext x
    rw [Subring.coe_sInf, Set.mem_iInter, Set.mem_iInter]
    

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
x : K
⊢ (∀ (i : Subring K), x ∈ ⋂ (_ : i ∈ toSubring '' S), ↑i) ↔ ∀ (i : Subfield K), x ∈ ⋂ (_ : i ∈ S), ↑i

Next tactic:
exact
      ⟨fun h s s' ⟨s_mem, s'_eq⟩ => h s.toSubring _ ⟨⟨s, s_mem, rfl⟩, s'_eq⟩,
        fun h s s' ⟨⟨s'', s''_mem, s_eq⟩, (s'_eq : ↑s = s')⟩ =>
        h s'' _ ⟨s''_mem, by simp [← s_eq, ← s'_eq]⟩⟩
@[simp, norm_cast]
theorem coe_sInf (S : Set (Subfield K)) : ((sInf S : Subfield K) : Set K) = ⋂ s ∈ S, ↑s :=
  show ((sInf (Subfield.toSubring '' S) : Subring K) : Set K) = ⋂ s ∈ S, ↑s by
    ext x
    rw [Subring.coe_sInf, Set.mem_iInter, Set.mem_iInter]
    exact
      ⟨fun h s s' ⟨s_mem, s'_eq⟩ => h s.toSubring _ ⟨⟨s, s_mem, rfl⟩, s'_eq⟩,
        fun h s s' ⟨⟨s'', s''_mem, s_eq⟩, (s'_eq : ↑s = s')⟩ =>
        h s'' _ ⟨s''_mem, 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 (Subfield K)
x : K
h : ∀ (i : Subfield K), x ∈ ⋂ (_ : i ∈ S), ↑i
s : Subring K
s' : Set K
x✝ : s' ∈ Set.range fun h => ↑s
s'' : Subfield K
s''_mem : s'' ∈ S
s_eq : s''.toSubring = s
s'_eq : ↑s = s'
⊢ (fun h => ↑s'') s''_mem = s'

Next tactic:
simp [← s_eq, ← s'_eq]
@[simp]
theorem sInf_toSubring (s : Set (Subfield K)) :
    (sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t := 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 (Subfield K)
⊢ (sInf s).toSubring = ⨅ t ∈ s, t.toSubring

Next tactic:
ext x
@[simp]
theorem sInf_toSubring (s : Set (Subfield K)) :
    (sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t := by
  ext x
  

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set (Subfield K)
x : K
⊢ x ∈ (sInf s).toSubring ↔ x ∈ ⨅ t ∈ s, t.toSubring

Next tactic:
rw [mem_toSubring, mem_sInf]
@[simp]
theorem sInf_toSubring (s : Set (Subfield K)) :
    (sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t := by
  ext x
  rw [mem_toSubring, mem_sInf]
  

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set (Subfield K)
x : K
⊢ (∀ p ∈ s, x ∈ p) ↔ x ∈ ⨅ t ∈ s, t.toSubring

Next tactic:
erw [Subring.mem_sInf]
@[simp]
theorem sInf_toSubring (s : Set (Subfield K)) :
    (sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t := by
  ext x
  rw [mem_toSubring, mem_sInf]
  erw [Subring.mem_sInf]
  

Proof state:
case h
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set (Subfield K)
x : K
⊢ (∀ p ∈ s, x ∈ p) ↔ ∀ p ∈ Set.range fun t => ⨅ (_ : t ∈ s), t.toSubring, x ∈ p

Next tactic:
exact
    ⟨fun h p ⟨p', hp⟩ => hp ▸ Subring.mem_sInf.mpr fun p ⟨hp', hp⟩ => hp ▸ h _ hp', fun h p hp =>
      h p.toSubring
        ⟨p,
          Subring.ext fun x =>
            ⟨fun hx => Subring.mem_sInf.mp hx _ ⟨hp, rfl⟩, fun hx =>
              Subring.mem_sInf.mpr fun p' ⟨_, p'_eq⟩ => p'_eq ▸ hx⟩⟩⟩
theorem isGLB_sInf (S : Set (Subfield K)) : IsGLB S (sInf S) := 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 (Subfield K)
⊢ IsGLB S (sInf S)

Next tactic:
have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := by simp [SetLike.coe_subset_coe]
theorem isGLB_sInf (S : Set (Subfield K)) : IsGLB S (sInf S) := by
  have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := 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 (Subfield K)
⊢ ∀ {s t : Subfield K}, ↑s ≤ ↑t ↔ s ≤ t

Next tactic:
simp [SetLike.coe_subset_coe]
theorem isGLB_sInf (S : Set (Subfield K)) : IsGLB S (sInf S) := by
  have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := by simp [SetLike.coe_subset_coe]
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
this : ∀ {s t : Subfield K}, ↑s ≤ ↑t ↔ s ≤ t
⊢ IsGLB S (sInf S)

Next tactic:
refine' IsGLB.of_image this _
theorem isGLB_sInf (S : Set (Subfield K)) : IsGLB S (sInf S) := by
  have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := by simp [SetLike.coe_subset_coe]
  refine' IsGLB.of_image this _
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
this : ∀ {s t : Subfield K}, ↑s ≤ ↑t ↔ s ≤ t
⊢ IsGLB ((fun {x} => ↑x) '' S) ↑(sInf S)

Next tactic:
convert isGLB_biInf (s := S) (f := SetLike.coe)
theorem isGLB_sInf (S : Set (Subfield K)) : IsGLB S (sInf S) := by
  have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := by simp [SetLike.coe_subset_coe]
  refine' IsGLB.of_image this _
  convert isGLB_biInf (s := S) (f := SetLike.coe)
  

Proof state:
case h.e'_4
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
S : Set (Subfield K)
this : ∀ {s t : Subfield K}, ↑s ≤ ↑t ↔ s ≤ t
⊢ ↑(sInf S) = ⨅ x ∈ S, ↑x

Next tactic:
exact coe_sInf _
/-- 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
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
    exact
      ⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
        dx * dy, Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm⟩
  mul_mem' x_mem y_mem := 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
a✝ b✝ : K
x_mem : a✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
y_mem : 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}

Next tactic:
obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
/-- 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
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
    exact
      ⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
        dx * dy, Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm⟩
  mul_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    

Proof state:
case intro.intro.intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
b✝ : K
y_mem : b✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem : nx / dx ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
⊢ nx / dx * b✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}

Next tactic:
obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
/-- 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
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
    exact
      ⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
        dx * dy, Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm⟩
  mul_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem : nx / dx ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem : ny / dy ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
⊢ nx / dx * (ny / dy) ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}

Next tactic:
exact
      ⟨nx * ny, Subring.mul_mem _ hnx hny, dx * dy, Subring.mul_mem _ hdx hdy,
        (div_mul_div_comm _ _ _ _).symm⟩
/-- 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
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := 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
a✝ b✝ : K
x_mem :
  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
y_mem :
  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

Next tactic:
obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
/-- 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
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    

Proof state:
case intro.intro.intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
b✝ : K
y_mem :
  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
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          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
⊢ nx / dx + 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

Next tactic:
obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem