-- Mathlib/Topology/Instances/Complex.lean
theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
    (hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype

Proof state:
case refine_2.inl.h.h
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ofRealHom.fieldRange
j : ↥K.topologicalClosure ≃+* ↥ofRealHom.fieldRange := RingEquiv.subfieldCongr h
ψ₁ : ℝ →+* ℂ := extψ.comp (j.symm.toRingHom.comp ofRealHom.rangeRestrict)
hψ₁ : Continuous ⇑ψ₁
x : ↥K
⊢ (↑↑ψ).toFun x = K.subtype x

Tactic:
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ofRealHom.rangeRestrict r = j (ι x)

Result:
case refine_2.inl.h.h.intro
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ofRealHom.fieldRange
j : ↥K.topologicalClosure ≃+* ↥ofRealHom.fieldRange := RingEquiv.subfieldCongr h
ψ₁ : ℝ →+* ℂ := extψ.comp (j.symm.toRingHom.comp ofRealHom.rangeRestrict)
hψ₁ : Continuous ⇑ψ₁
x : ↥K
r : ℝ
hr : ofRealHom.rangeRestrict r = j (ι x)
⊢ (↑↑ψ).toFun x = K.subtype x

K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ofRealHom.fieldRange
j : ↥K.topologicalClosure ≃+* ↥ofRealHom.fieldRange := RingEquiv.subfieldCongr h
ψ₁ : ℝ →+* ℂ := extψ.comp (j.symm.toRingHom.comp ofRealHom.rangeRestrict)
hψ₁ : Continuous ⇑ψ₁
x : ↥K
⊢ ∃ r, ofRealHom.rangeRestrict r = j (ι x)

Proof state:
case refine_2.inr
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ⊤
ψ₁ : ℂ →+* ℂ := extψ.comp ((RingEquiv.subfieldCongr h).symm.toRingHom.comp Subfield.topEquiv.symm.toRingHom)
⊢ (↑↑ψ).toFun = ⇑K.subtype ∨ (↑↑ψ).toFun = ⇑(starRingEnd ℂ) ∘ ⇑K.subtype

Tactic:
have hψ₁ : Continuous ψ₁ := by
  simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _)

Result:
case refine_2.inr
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h : K.topologicalClosure = ⊤
ψ₁ : ℂ →+* ℂ := extψ.comp ((RingEquiv.subfieldCongr h).symm.toRingHom.comp Subfield.topEquiv.symm.toRingHom)
hψ₁ : Continuous ⇑ψ₁
⊢ (↑↑ψ).toFun = ⇑K.subtype ∨ (↑↑ψ).toFun = ⇑(starRingEnd ℂ) ∘ ⇑K.subtype

Proof state:
case h.e'_4
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
j : { x // x ∈ closure (id '' {x | ↑K x}) } → ↑↑K.topologicalClosure := fun x => ⟨↑x, ⋯⟩
y : ℂ
hy : y ∈ ↑K.topologicalClosure
⊢ closure (id '' {x | ↑K x}) = ↑K.topologicalClosure

Tactic:
simp only [id, Set.image_id']

Result:
case h.e'_4
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
j : { x // x ∈ closure (id '' {x | ↑K x}) } → ↑↑K.topologicalClosure := fun x => ⟨↑x, ⋯⟩
y : ℂ
hy : y ∈ ↑K.topologicalClosure
⊢ closure {x | ↑K x} = ↑K.topologicalClosure

Proof state:
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
⊢ Filter.comap (fun x => (ι x.1, ι x.2)) (uniformity ↥K.topologicalClosure) = uniformity ↥K

Tactic:
rw [uniformity_subtype, uniformity_subtype, Filter.comap_comap]

Result:
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
⊢ Filter.comap ((fun q => (↑q.1, ↑q.2)) ∘ fun x => (ι x.1, ι x.2)) (uniformity ℂ) =
    Filter.comap (fun q => (↑q.1, ↑q.2)) (uniformity ℂ)

Proof state:
case refine_2.inr.inr.h.h
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h✝ : K.topologicalClosure = ⊤
ψ₁ : ℂ →+* ℂ := extψ.comp ((RingEquiv.subfieldCongr h✝).symm.toRingHom.comp Subfield.topEquiv.symm.toRingHom)
hψ₁ : Continuous ⇑ψ₁
h : ψ₁ = starRingEnd ℂ
z : ↥K
⊢ (↑↑ψ).toFun z = (⇑(starRingEnd ℂ) ∘ ⇑K.subtype) z

Tactic:
convert RingHom.congr_fun h z using 1

Result:
case h.e'_2
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h✝ : K.topologicalClosure = ⊤
ψ₁ : ℂ →+* ℂ := extψ.comp ((RingEquiv.subfieldCongr h✝).symm.toRingHom.comp Subfield.topEquiv.symm.toRingHom)
hψ₁ : Continuous ⇑ψ₁
h : ψ₁ = starRingEnd ℂ
z : ↥K
⊢ (↑↑ψ).toFun z = ψ₁ ↑z