l3lab/ntp-mathlib huggingface dataset:
theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
    ContinuousAt hatInv x := by
  haveI : T3Space (hat K) := Completion.t3Space K
  refine' denseInducing_coe.continuousAt_extend _
  apply mem_of_superset (compl_singleton_mem_nhds h)
  intro y y_ne
  rw [mem_compl_singleton_iff] at y_ne
  apply CompleteSpace.complete
  have : (fun (x : K) => (↑x⁻¹: hat K)) =
      ((fun (y : K) => (↑y: hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
    unfold Function.comp
    simp
  rw [this, ← Filter.map_map]
  apply Cauchy.map _ (Completion.uniformContinuous_coe K)
  apply CompletableTopField.nice
  · haveI := denseInducing_coe.comap_nhds_neBot y
    apply cauchy_nhds.comap
    · rw [Completion.comap_coe_eq_uniformity]
  · have eq_bot : 𝓝 (0 : hat K) ⊓ 𝓝 y = ⊥ := by
      by_contra h
      exact y_ne (eq_of_nhds_neBot <| neBot_iff.mpr h).symm
    

Proof state:
case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
this✝ : T3Space (hat K)
y : hat K
y_ne : y ≠ 0
this : (fun x => ↑K x⁻¹) = (fun y => ↑K y) ∘ fun x => x⁻¹
eq_bot : 𝓝 0 ⊓ 𝓝 y = ⊥
⊢ 𝓝 0 ⊓ Filter.comap (↑K) (𝓝 y) = ⊥

Next tactic:
erw [denseInducing_coe.nhds_eq_comap (0 : K), ← Filter.comap_inf, eq_bot]
theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
    ContinuousAt hatInv x := by
  haveI : T3Space (hat K) := Completion.t3Space K
  refine' denseInducing_coe.continuousAt_extend _
  apply mem_of_superset (compl_singleton_mem_nhds h)
  intro y y_ne
  rw [mem_compl_singleton_iff] at y_ne
  apply CompleteSpace.complete
  have : (fun (x : K) => (↑x⁻¹: hat K)) =
      ((fun (y : K) => (↑y: hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
    unfold Function.comp
    simp
  rw [this, ← Filter.map_map]
  apply Cauchy.map _ (Completion.uniformContinuous_coe K)
  apply CompletableTopField.nice
  · haveI := denseInducing_coe.comap_nhds_neBot y
    apply cauchy_nhds.comap
    · rw [Completion.comap_coe_eq_uniformity]
  · have eq_bot : 𝓝 (0 : hat K) ⊓ 𝓝 y = ⊥ := by
      by_contra h
      exact y_ne (eq_of_nhds_neBot <| neBot_iff.mpr h).symm
    erw [denseInducing_coe.nhds_eq_comap (0 : K), ← Filter.comap_inf, eq_bot]
    

Proof state:
case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
this✝ : T3Space (hat K)
y : hat K
y_ne : y ≠ 0
this : (fun x => ↑K x⁻¹) = (fun y => ↑K y) ∘ fun x => x⁻¹
eq_bot : 𝓝 0 ⊓ 𝓝 y = ⊥
⊢ Filter.comap ↑K ⊥ = ⊥

Next tactic:
exact comap_bot
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  

Proof state:
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
⊢ (↑K x)⁻¹ = ↑K x⁻¹

Next tactic:
by_cases h : x = 0
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · 

Proof state:
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (↑K x)⁻¹ = ↑K x⁻¹

Next tactic:
rw [h, inv_zero]
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    

Proof state:
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (↑K 0)⁻¹ = ↑K 0

Next tactic:
dsimp [Inv.inv]
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    

Proof state:
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (if ↑K 0 = 0 then 0 else hatInv (↑K 0)) = ↑K 0

Next tactic:
norm_cast
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    

Proof state:
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (if 0 = 0 then 0 else hatInv 0) = 0

Next tactic:
simp
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    simp
  · 

Proof state:
case neg
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
⊢ (↑K x)⁻¹ = ↑K x⁻¹

Next tactic:
conv_lhs => dsimp [Inv.inv]
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    simp
  · conv_lhs => 

Proof state:
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
| (↑K x)⁻¹

Next tactic:
dsimp [Inv.inv]
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    simp
  · conv_lhs => dsimp [Inv.inv]
    

Proof state:
case neg
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
⊢ (if ↑K x = 0 then 0 else hatInv (↑K x)) = ↑K x⁻¹

Next tactic:
rw [if_neg]
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    simp
  · conv_lhs => dsimp [Inv.inv]
    rw [if_neg]
    · 

Proof state:
case neg
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
⊢ hatInv (↑K x) = ↑K x⁻¹

Next tactic:
exact hatInv_extends h
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
  by_cases h : x = 0
  · rw [h, inv_zero]
    dsimp [Inv.inv]
    norm_cast
    simp
  · conv_lhs => dsimp [Inv.inv]
    rw [if_neg]
    · exact hatInv_extends h
    · 

Proof state:
case neg.hnc
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
⊢ ¬↑K x = 0

Next tactic:
exact fun H => h (denseEmbedding_coe.inj H)
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 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
x : hat K
x_ne : x ≠ 0
⊢ x * hatInv x = 1

Next tactic:
haveI : T1Space (hat K) := T2Space.t1Space
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
  haveI : T1Space (hat K) := T2Space.t1Space
  

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)
⊢ x * hatInv x = 1

Next tactic:
let f := fun x : hat K => x * hatInv x
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
  

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
⊢ x * hatInv x = 1

Next tactic:
let c := (fun (x : K) => (x : hat K))
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))
  

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
⊢ x * hatInv x = 1

Next tactic:
change f x = 1
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
  

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
⊢ f x = 1

Next tactic:
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 : _)
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
    

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
⊢ ContinuousAt f x

Next tactic:
letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
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
    

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
this : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
⊢ ContinuousAt f x

Next tactic:
have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × hat K)) x :=
      continuous_id.continuousAt.prod (continuous_hatInv x_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)
    

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✝¹ : T1Space (hat K)
f : hat K → hat K := fun x => x * hatInv x
c : K → hat K := fun x => ↑K x
this✝ : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
this : ContinuousAt (fun y => (y, hatInv y)) x
⊢ ContinuousAt f x

Next tactic:
exact (_root_.continuous_mul.continuousAt.comp this : _)
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 : _)
  

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
⊢ f x = 1

Next tactic:
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
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
    

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
⊢ x ∈ closure (c '' {0}ᶜ)

Next tactic:
have := denseInducing_coe.dense x
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
    

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
this : x ∈ closure (range ↑K)
⊢ x ∈ closure (c '' {0}ᶜ)

Next tactic:
rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
      image_union] at this
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
    

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
this : x ∈ closure (↑K '' {0} ∪ ↑K '' {0}ᶜ)
⊢ x ∈ closure (c '' {0}ᶜ)

Next tactic:
apply mem_closure_of_mem_closure_union this
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
    

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
this : x ∈ closure (↑K '' {0} ∪ ↑K '' {0}ᶜ)
⊢ (↑K '' {0})ᶜ ∈ 𝓝 x

Next tactic:
rw [image_singleton]
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]
    

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
this : x ∈ closure (↑K '' {0} ∪ ↑K '' {0}ᶜ)
⊢ {↑K 0}ᶜ ∈ 𝓝 x

Next tactic:
exact compl_singleton_mem_nhds x_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
  

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)
⊢ f x = 1

Next tactic:
have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
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
  

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))
⊢ f x = 1

Next tactic:
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]
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
    

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))
⊢ f '' (c '' {0}ᶜ) ⊆ {1}

Next tactic:
rw [image_image]
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]
    

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))
⊢ (fun x => f (c x)) '' {0}ᶜ ⊆ {1}

Next tactic:
rintro _ ⟨z, z_ne, rfl⟩