l3lab/ntp-mathlib huggingface dataset:
theorem eq_iff_eq (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x = r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ = r :=
  ⟨fun hw φ => hw (mk φ), by rintro hφ ⟨w, ⟨φ, rfl⟩⟩; 

Proof state:
case mk.intro
K : Type u_1
inst✝ : Field K
x : K
r : ℝ
hφ : ∀ (φ : K →+* ℂ), ‖φ x‖ = r
φ : K →+* ℂ
⊢ { val := place φ, property := (_ : ∃ φ_1, place φ_1 = place φ) } x = r

Next tactic:
exact hφ φ
theorem le_iff_le (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x ≤ r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r :=
  ⟨fun hw φ => hw (mk φ), by 

Proof state:
K : Type u_1
inst✝ : Field K
x : K
r : ℝ
⊢ (∀ (φ : K →+* ℂ), ‖φ x‖ ≤ r) → ∀ (w : InfinitePlace K), w x ≤ r

Next tactic:
rintro hφ ⟨w, ⟨φ, rfl⟩⟩
theorem le_iff_le (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x ≤ r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r :=
  ⟨fun hw φ => hw (mk φ), by rintro hφ ⟨w, ⟨φ, rfl⟩⟩; 

Proof state:
case mk.intro
K : Type u_1
inst✝ : Field K
x : K
r : ℝ
hφ : ∀ (φ : K →+* ℂ), ‖φ x‖ ≤ r
φ : K →+* ℂ
⊢ { val := place φ, property := (_ : ∃ φ_1, place φ_1 = place φ) } x ≤ r

Next tactic:
exact hφ φ
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  

Proof state:
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
constructor
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    

Proof state:
case mp
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
intro h₀
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    

Proof state:
case mp
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    

Proof state:
case mp.intro
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
let ι := RingEquiv.ofLeftInverse hiφ
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    

Proof state:
case mp.intro
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      

Proof state:
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
⊢ LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))

Next tactic:
change LipschitzWith 1 (ψ ∘ ι.symm)
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      

Proof state:
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
⊢ LipschitzWith 1 (⇑ψ ∘ ⇑(RingEquiv.symm ι))

Next tactic:
apply LipschitzWith.of_dist_le_mul
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      

Proof state:
case a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
⊢ ∀ (x y : ↥(RingHom.range φ)), dist ((⇑ψ ∘ ⇑(RingEquiv.symm ι)) x) ((⇑ψ ∘ ⇑(RingEquiv.symm ι)) y) ≤ ↑1 * dist x y

Next tactic:
intro x y
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      

Proof state:
case a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
⊢ dist ((⇑ψ ∘ ⇑(RingEquiv.symm ι)) x) ((⇑ψ ∘ ⇑(RingEquiv.symm ι)) y) ≤ ↑1 * dist x y

Next tactic:
rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      

Proof state:
case a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
⊢ ‖ψ ((RingEquiv.symm ι) (x - y))‖ ≤ dist x y

Next tactic:
apply le_of_eq
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      

Proof state:
case a.a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
⊢ ‖ψ ((RingEquiv.symm ι) (x - y))‖ = dist x y

Next tactic:
suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        

Proof state:
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
this : ‖φ ((RingEquiv.symm ι) (x - y))‖ = ‖ψ ((RingEquiv.symm ι) (x - y))‖
⊢ ‖ψ ((RingEquiv.symm ι) (x - y))‖ = dist x y

Next tactic:
rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        

Proof state:
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
this : ‖φ ((RingEquiv.symm ι) (x - y))‖ = ‖ψ ((RingEquiv.symm ι) (x - y))‖
⊢ ‖↑(x - y)‖ = dist x y

Next tactic:
rfl
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      

Proof state:
case a.a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
x y : ↥(RingHom.range φ)
⊢ ‖φ ((RingEquiv.symm ι) (x - y))‖ = ‖ψ ((RingEquiv.symm ι) (x - y))‖

Next tactic:
exact congrFun (congrArg (↑) h₀) _
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    

Proof state:
case mp.intro
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    | inl h =>
        left; ext1 x
        conv_rhs => rw [← hiφ x]
        exact (congrFun h (ι x)).symm
    | inr h =>
        right; ext1 x
        conv_rhs => rw [← hiφ x]
        exact (congrFun h (ι x)).symm
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    

Proof state:
case mp.intro
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
x✝ :
  (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun = ⇑(Subfield.subtype (RingHom.fieldRange φ)) ∨
    (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun =
      ⇑(starRingEnd ℂ) ∘ ⇑(Subfield.subtype (RingHom.fieldRange φ))
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    | inl h =>
        left; ext1 x
        conv_rhs => rw [← hiφ x]
        exact (congrFun h (ι x)).symm
    | inr h =>
        right; ext1 x
        conv_rhs => rw [← hiφ x]
        exact (congrFun h (ι x)).symm
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    

Proof state:
case mp.intro.inl
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
h : (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun = ⇑(Subfield.subtype (RingHom.fieldRange φ))
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
| inl h =>
        left; ext1 x
        conv_rhs => rw [← hiφ x]
        exact (congrFun h (ι x)).symm
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    | inl h =>
        

Proof state:
case mp.intro.inl
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
h : (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun = ⇑(Subfield.subtype (RingHom.fieldRange φ))
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

Next tactic:
left
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    | inl h =>
        left; 

Proof state:
case mp.intro.inl.h
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
h : (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun = ⇑(Subfield.subtype (RingHom.fieldRange φ))
⊢ φ = ψ

Next tactic:
ext1 x
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
  constructor
  · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
    intro h₀
    obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse
    let ι := RingEquiv.ofLeftInverse hiφ
    have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by
      change LipschitzWith 1 (ψ ∘ ι.symm)
      apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,
        ← map_sub, ← map_sub]
      apply le_of_eq
      suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by
        rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _]
        rfl
      exact congrFun (congrArg (↑) h₀) _
    cases
      Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with
    | inl h =>
        left; ext1 x
        

Proof state:
case mp.intro.inl.h.a
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥(RingHom.range φ) := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))
h : (↑↑(RingHom.comp ψ (RingEquiv.toRingHom (RingEquiv.symm ι)))).toFun = ⇑(Subfield.subtype (RingHom.fieldRange φ))
x : K
⊢ φ x = ψ x

Next tactic:
conv_rhs => rw [← hiφ x]