l3lab/ntp-mathlib huggingface dataset:
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    

Proof state:
case intro.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
z : K
z_ne : z ∈ {0}ᶜ
⊢ (fun x => f (c x)) z ∈ {1}

Next tactic:
rw [mem_singleton_iff]
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    

Proof state:
case intro.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
z : K
z_ne : z ∈ {0}ᶜ
⊢ (fun x => f (c x)) z = 1

Next tactic:
rw [mem_compl_singleton_iff] at z_ne
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    rw [mem_compl_singleton_iff] at z_ne
    

Proof state:
case intro.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
z : K
z_ne : z ≠ 0
⊢ (fun x => f (c x)) z = 1

Next tactic:
dsimp
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    rw [mem_compl_singleton_iff] at z_ne
    dsimp
    

Proof state:
case intro.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
z : K
z_ne : z ≠ 0
⊢ ↑K z * hatInv (↑K z) = 1

Next tactic:
rw [hatInv_extends z_ne, ← coe_mul]
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    rw [mem_compl_singleton_iff] at z_ne
    dsimp
    rw [hatInv_extends z_ne, ← coe_mul]
    

Proof state:
case intro.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
z : K
z_ne : z ≠ 0
⊢ ↑K (z * z⁻¹) = 1

Next tactic:
rw [mul_inv_cancel z_ne, coe_one]
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    rw [mem_compl_singleton_iff] at z_ne
    dsimp
    rw [hatInv_extends z_ne, ← coe_mul]
    rw [mul_inv_cancel z_ne, coe_one]
  

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this✝ : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' (c '' {0}ᶜ))
this : f '' (c '' {0}ᶜ) ⊆ {1}
⊢ f x = 1

Next tactic:
replace fxclo := closure_mono this fxclo
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  let f := fun x : hat K => x * hatInv x
  let c := (fun (x : K) => (x : hat K))
  change f x = 1
  have cont : ContinuousAt f x := by
    letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
    have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_ne)
    exact (_root_.continuous_mul.continuousAt.comp this : _)
  have clo : x ∈ closure (c '' {0}ᶜ) := by
    have := denseInducing_coe.dense x
    rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
    apply mem_closure_of_mem_closure_union this
    rw [image_singleton]
    exact compl_singleton_mem_nhds x_ne
  have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
  have : f '' (c '' {0}ᶜ) ⊆ {1} := by
    rw [image_image]
    rintro _ ⟨z, z_ne, rfl⟩
    rw [mem_singleton_iff]
    rw [mem_compl_singleton_iff] at z_ne
    dsimp
    rw [hatInv_extends z_ne, ← coe_mul]
    rw [mul_inv_cancel z_ne, coe_one]
  replace fxclo := closure_mono this fxclo
  

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this✝ : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
this : f '' (c '' {0}ᶜ) ⊆ {1}
fxclo : f x ∈ closure {1}
⊢ f x = 1

Next tactic:
rwa [closure_singleton, mem_singleton_iff] at fxclo
instance instField : Field (hat K) :=
  { Completion.instInvCompletion,
    (by 

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : Inv (hat K) := instInvCompletion
⊢ CommRing (hat K)

Next tactic:
infer_instance
instance instField : Field (hat K) :=
  { Completion.instInvCompletion,
    (by infer_instance : CommRing (hat K)) with
    exists_pair_ne := ⟨0, 1, fun h => zero_ne_one ((uniformEmbedding_coe K).inj h)⟩
    mul_inv_cancel := fun x x_ne => by 

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝¹ : Inv (hat K) := instInvCompletion
src✝ : CommRing (hat K) := inferInstance
x : hat K
x_ne : x ≠ 0
⊢ x * x⁻¹ = 1

Next tactic:
simp only [Inv.inv, if_neg x_ne, mul_hatInv_cancel x_ne]
instance instField : Field (hat K) :=
  { Completion.instInvCompletion,
    (by infer_instance : CommRing (hat K)) with
    exists_pair_ne := ⟨0, 1, fun h => zero_ne_one ((uniformEmbedding_coe K).inj h)⟩
    mul_inv_cancel := fun x x_ne => by simp only [Inv.inv, if_neg x_ne, mul_hatInv_cancel x_ne]
    inv_zero := by 

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝¹ : Inv (hat K) := instInvCompletion
src✝ : CommRing (hat K) := inferInstance
⊢ 0⁻¹ = 0

Next tactic:
simp only [Inv.inv, ite_true]
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
⊢ ∀ ⦃x : hat K⦄, x ≠ 0 → ContinuousAt Inv.inv x

Next tactic:
intro x x_ne
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
⊢ ContinuousAt Inv.inv x

Next tactic:
have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          intro y y_ne
          rw [mem_compl_singleton_iff] at y_ne
          dsimp [Inv.inv]
          rw [if_neg y_ne]
        mem_of_superset (compl_singleton_mem_nhds x_ne) this
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
⊢ {0}ᶜ ⊆ {y | hatInv y = y⁻¹}

Next tactic:
intro y y_ne
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          intro y y_ne
          

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
y : hat K
y_ne : y ∈ {0}ᶜ
⊢ y ∈ {y | hatInv y = y⁻¹}

Next tactic:
rw [mem_compl_singleton_iff] at y_ne
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          intro y y_ne
          rw [mem_compl_singleton_iff] at y_ne
          

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
y : hat K
y_ne : y ≠ 0
⊢ y ∈ {y | hatInv y = y⁻¹}

Next tactic:
dsimp [Inv.inv]
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          intro y y_ne
          rw [mem_compl_singleton_iff] at y_ne
          dsimp [Inv.inv]
          

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
y : hat K
y_ne : y ≠ 0
⊢ hatInv y = if y = 0 then 0 else hatInv y

Next tactic:
rw [if_neg y_ne]
instance : TopologicalDivisionRing (hat K) :=
  { Completion.topologicalRing with
    continuousAt_inv₀ := by
      intro x x_ne
      have : { y | hatInv y = y⁻¹ } ∈ 𝓝 x :=
        haveI : {(0 : hat K)}ᶜ ⊆ { y : hat K | hatInv y = y⁻¹ } := by
          intro y y_ne
          rw [mem_compl_singleton_iff] at y_ne
          dsimp [Inv.inv]
          rw [if_neg y_ne]
        mem_of_superset (compl_singleton_mem_nhds x_ne) this
      

Proof state:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
src✝ : TopologicalRing (hat K) := topologicalRing
x : hat K
x_ne : x ≠ 0
this : {y | hatInv y = y⁻¹} ∈ 𝓝 x
⊢ ContinuousAt Inv.inv x

Next tactic:
exact ContinuousAt.congr (continuous_hatInv x_ne) this
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
⊢ ∀ (F : Filter ↥K), Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (Filter.map (fun x => x⁻¹) F)

Next tactic:
intro F F_cau inf_F
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
⊢ Cauchy (Filter.map (fun x => x⁻¹) F)

Next tactic:
let i : K →+* L := K.subtype
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
⊢ Cauchy (Filter.map (fun x => x⁻¹) F)

Next tactic:
have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
hi : UniformInducing ⇑i
⊢ Cauchy (Filter.map (fun x => x⁻¹) F)

Next tactic:
rw [← hi.cauchy_map_iff] at F_cau ⊢
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      rw [← hi.cauchy_map_iff] at F_cau ⊢
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
F_cau : Cauchy (Filter.map (⇑i) F)
hi : UniformInducing ⇑i
⊢ Cauchy (Filter.map (⇑i) (Filter.map (fun x => x⁻¹) F))

Next tactic:
rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)]
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      rw [← hi.cauchy_map_iff] at F_cau ⊢
      rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by 

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
F_cau : Cauchy (Filter.map (⇑i) F)
hi : UniformInducing ⇑i
⊢ (⇑i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ ⇑i

Next tactic:
ext
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      rw [← hi.cauchy_map_iff] at F_cau ⊢
      rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; 

Proof state:
case h
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
F_cau : Cauchy (Filter.map (⇑i) F)
hi : UniformInducing ⇑i
x✝ : ↥K
⊢ (⇑i ∘ fun x => x⁻¹) x✝ = ((fun x => x⁻¹) ∘ ⇑i) x✝

Next tactic:
rfl
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      rw [← hi.cauchy_map_iff] at F_cau ⊢
      rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)]
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
F_cau : Cauchy (Filter.map (⇑i) F)
hi : UniformInducing ⇑i
⊢ Cauchy (Filter.map (fun x => x⁻¹) (Filter.map (⇑i) F))

Next tactic:
apply CompletableTopField.nice _ F_cau
instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K :=
  { Subtype.separatedSpace (K : Set L) with
    nice := by
      intro F F_cau inf_F
      let i : K →+* L := K.subtype
      have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
      rw [← hi.cauchy_map_iff] at F_cau ⊢
      rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)]
      apply CompletableTopField.nice _ F_cau
      

Proof state:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
src✝ : SeparatedSpace ↑↑K := Subtype.separatedSpace ↑K
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := subtype K
F_cau : Cauchy (Filter.map (⇑i) F)
hi : UniformInducing ⇑i
⊢ 𝓝 0 ⊓ Filter.map (⇑i) F = ⊥

Next tactic:
rw [← Filter.push_pull', ← map_zero i, ← hi.inducing.nhds_eq_comap, inf_F, Filter.map_bot]
instance (priority := 100) completableTopField_of_complete (L : Type*) [Field L] [UniformSpace L]
    [TopologicalDivisionRing L] [SeparatedSpace L] [CompleteSpace L] : CompletableTopField L :=
  { ‹SeparatedSpace L› with
    nice := fun F cau_F hF => by
      

Proof state:
K : Type u_1
inst✝⁹ : Field K
inst✝⁸ : UniformSpace K
L✝ : Type u_2
inst✝⁷ : Field L✝
inst✝⁶ : UniformSpace L✝
inst✝⁵ : CompletableTopField L✝
L : Type u_3
inst✝⁴ : Field L
inst✝³ : UniformSpace L
inst✝² : TopologicalDivisionRing L
inst✝¹ : SeparatedSpace L
inst✝ : CompleteSpace L
src✝ : SeparatedSpace L := inst✝¹
F : Filter L
cau_F : Cauchy F
hF : 𝓝 0 ⊓ F = ⊥
⊢ Cauchy (map (fun x => x⁻¹) F)

Next tactic:
haveI : NeBot F := cau_F.1