Theorem: UniformSpace.Completion.mul_hatInv_cancel
File: Mathlib/Topology/Algebra/UniformField.lean

Proof steps:
state_before: 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 * x.hatInv = 1
state_after: 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 * x.hatInv = 1
tactic: haveI : T1Space (hat K) := T2Space.t1Space
state_before: 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 * x.hatInv = 1
state_after: 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 * x.hatInv
⊢ x * x.hatInv = 1
tactic: let f := fun x : hat K => x * hatInv x
state_before: 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 * x.hatInv
⊢ x * x.hatInv = 1
state_after: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
⊢ x * x.hatInv = 1
tactic: let c := (fun (x : K) => (x : hat K))
state_before: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
⊢ x * x.hatInv = 1
state_after: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
⊢ f x = 1
tactic: change f x = 1
state_before: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
⊢ f x = 1
state_after: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
⊢ f x = 1
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 : _)
state_before: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
⊢ f x = 1
state_after: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
⊢ f x = 1
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
state_before: 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 * x.hatInv
c : K → hat K := fun x => ↑K x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
⊢ f x = 1
state_after: 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 * x.hatInv
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
tactic: have fxclo : f x ∈ closure (f '' (c '' {0}ᶜ)) := mem_closure_image cont clo
state_before: 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 * x.hatInv
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
state_after: 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 * x.hatInv
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
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 [f]
    rw [hatInv_extends z_ne, ← coe_mul]
    rw [mul_inv_cancel z_ne, coe_one]
state_before: 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 * x.hatInv
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
state_after: 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 * x.hatInv
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
tactic: replace fxclo := closure_mono this fxclo
state_before: 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 * x.hatInv
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
state_after: no goals
tactic: rwa [closure_singleton, mem_singleton_iff] at fxclo