l3lab/ntp-mathlib huggingface dataset:
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) := by
  have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h]
  apply_fun fun x => x ^ 2 at this
  simp only [norm_eq_abs, ← normSq_eq_abs] at this
  rw [← ofReal_inj, ← mul_conj, ← mul_conj] at this
  rw [RingHom.map_sub, RingHom.map_sub] at this
  simp only [sub_mul, mul_sub, one_mul, mul_one] at this
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this :
  f z * (starRingEnd ℂ) (f z) - (starRingEnd ℂ) (f z) - (f z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1) =
    z * (starRingEnd ℂ) z - (starRingEnd ℂ) z - (z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1)
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
rw [mul_conj, normSq_eq_abs, ← norm_eq_abs, LinearIsometry.norm_map] at this
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) := by
  have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h]
  apply_fun fun x => x ^ 2 at this
  simp only [norm_eq_abs, ← normSq_eq_abs] at this
  rw [← ofReal_inj, ← mul_conj, ← mul_conj] at this
  rw [RingHom.map_sub, RingHom.map_sub] at this
  simp only [sub_mul, mul_sub, one_mul, mul_one] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs, LinearIsometry.norm_map] at this
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this :
  ↑(‖z‖ ^ 2) - (starRingEnd ℂ) (f z) - (f z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1) =
    z * (starRingEnd ℂ) z - (starRingEnd ℂ) z - (z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1)
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
rw [mul_conj, normSq_eq_abs, ← norm_eq_abs] at this
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) := by
  have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h]
  apply_fun fun x => x ^ 2 at this
  simp only [norm_eq_abs, ← normSq_eq_abs] at this
  rw [← ofReal_inj, ← mul_conj, ← mul_conj] at this
  rw [RingHom.map_sub, RingHom.map_sub] at this
  simp only [sub_mul, mul_sub, one_mul, mul_one] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs, LinearIsometry.norm_map] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs] at this
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this :
  ↑(‖z‖ ^ 2) - (starRingEnd ℂ) (f z) - (f z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1) =
    ↑(‖z‖ ^ 2) - (starRingEnd ℂ) z - (z * (starRingEnd ℂ) 1 - (starRingEnd ℂ) 1)
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
simp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, RingHom.map_one, norm_eq_abs] at this
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) := by
  have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h]
  apply_fun fun x => x ^ 2 at this
  simp only [norm_eq_abs, ← normSq_eq_abs] at this
  rw [← ofReal_inj, ← mul_conj, ← mul_conj] at this
  rw [RingHom.map_sub, RingHom.map_sub] at this
  simp only [sub_mul, mul_sub, one_mul, mul_one] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs, LinearIsometry.norm_map] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs] at this
  simp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, RingHom.map_one, norm_eq_abs] at this
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this : (starRingEnd ℂ) (f z) + (f z - 1) = (starRingEnd ℂ) z + (z - 1)
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
simp only [add_sub, sub_left_inj] at this
theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
    z + conj z = f z + conj (f z) := by
  have : ‖f z - 1‖ = ‖z - 1‖ := by rw [← f.norm_map (z - 1), f.map_sub, h]
  apply_fun fun x => x ^ 2 at this
  simp only [norm_eq_abs, ← normSq_eq_abs] at this
  rw [← ofReal_inj, ← mul_conj, ← mul_conj] at this
  rw [RingHom.map_sub, RingHom.map_sub] at this
  simp only [sub_mul, mul_sub, one_mul, mul_one] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs, LinearIsometry.norm_map] at this
  rw [mul_conj, normSq_eq_abs, ← norm_eq_abs] at this
  simp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, RingHom.map_one, norm_eq_abs] at this
  simp only [add_sub, sub_left_inj] at this
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this : (starRingEnd ℂ) (f z) + f z = (starRingEnd ℂ) z + z
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
rw [add_comm, ← this, add_comm]
theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by
  

Proof state:
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
⊢ (f z).re = z.re

Next tactic:
apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by
  apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
  

Proof state:
case h₃
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
⊢ ∀ (z : ℂ), z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
intro z
theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by
  apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
  intro z
  

Proof state:
case h₃
f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z✝ z : ℂ
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z)

Next tactic:
apply LinearIsometry.im_apply_eq_im h
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie

Next tactic:
have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ f I = I ∨ f I = -I

Next tactic:
simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ (f I).re = 0 ∧ ((f I).im = I.im ∨ (f I).im = -I.im)

Next tactic:
constructor
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · 

Proof state:
case left
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ (f I).re = 0

Next tactic:
rw [← I_re]
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      

Proof state:
case left
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ (f I).re = I.re

Next tactic:
exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · 

Proof state:
case right
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ (f I).im = I.im ∨ (f I).im = -I.im

Next tactic:
apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      

Proof state:
case right.h₂
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ ∀ (z : ℂ), ((LinearIsometryEquiv.toLinearIsometry f) z).re = z.re

Next tactic:
intro z
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      

Proof state:
case right.h₂
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
⊢ ((LinearIsometryEquiv.toLinearIsometry f) z).re = z.re

Next tactic:
rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie

Next tactic:
refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · 

Proof state:
case refine'_1
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ

Next tactic:
apply LinearIsometryEquiv.toLinearEquiv_injective
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      

Proof state:
case refine'_1.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ f.toLinearEquiv = (LinearIsometryEquiv.refl ℝ ℂ).toLinearEquiv

Next tactic:
apply Complex.basisOneI.ext'
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      

Proof state:
case refine'_1.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ ∀ (i : Fin 2), f.toLinearEquiv (basisOneI i) = (LinearIsometryEquiv.refl ℝ ℂ).toLinearEquiv (basisOneI i)

Next tactic:
intro i
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      

Proof state:
case refine'_1.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
i : Fin 2
⊢ f.toLinearEquiv (basisOneI i) = (LinearIsometryEquiv.refl ℝ ℂ).toLinearEquiv (basisOneI i)

Next tactic:
fin_cases i
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> 

Proof state:
case refine'_1.a.head
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ f.toLinearEquiv (basisOneI { val := 0, isLt := (_ : 0 < 2) }) =
    (LinearIsometryEquiv.refl ℝ ℂ).toLinearEquiv (basisOneI { val := 0, isLt := (_ : 0 < 2) })

Next tactic:
simp [h, h']
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> 

Proof state:
case refine'_1.a.tail.head
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ f.toLinearEquiv (basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =
    (LinearIsometryEquiv.refl ℝ ℂ).toLinearEquiv (basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) })

Next tactic:
simp [h, h']
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · 

Proof state:
case refine'_2
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ f = conjLie

Next tactic:
apply LinearIsometryEquiv.toLinearEquiv_injective
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      

Proof state:
case refine'_2.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ f.toLinearEquiv = conjLie.toLinearEquiv

Next tactic:
apply Complex.basisOneI.ext'
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      

Proof state:
case refine'_2.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ ∀ (i : Fin 2), f.toLinearEquiv (basisOneI i) = conjLie.toLinearEquiv (basisOneI i)

Next tactic:
intro i
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      

Proof state:
case refine'_2.a
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
i : Fin 2
⊢ f.toLinearEquiv (basisOneI i) = conjLie.toLinearEquiv (basisOneI i)

Next tactic:
fin_cases i
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> 

Proof state:
case refine'_2.a.head
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ f.toLinearEquiv (basisOneI { val := 0, isLt := (_ : 0 < 2) }) =
    conjLie.toLinearEquiv (basisOneI { val := 0, isLt := (_ : 0 < 2) })

Next tactic:
simp [h, h']
theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
    f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLie := by
  have h0 : f I = I ∨ f I = -I := by
    simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
  refine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _ <;>
    · apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> 

Proof state:
case refine'_2.a.tail.head
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ f.toLinearEquiv (basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =
    conjLie.toLinearEquiv (basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) })

Next tactic:
simp [h, h']
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
⊢ ∃ a, f = rotation a ∨ f = LinearIsometryEquiv.trans conjLie (rotation a)

Next tactic:
let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by 

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
⊢ f 1 ∈ circle

Next tactic:
rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
  

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : ↥circle := { val := f 1, property := (_ : f 1 ∈ circle) }
⊢ ∃ a, f = rotation a ∨ f = LinearIsometryEquiv.trans conjLie (rotation a)

Next tactic:
use a
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
  use a
  

Proof state:
case h
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : ↥circle := { val := f 1, property := (_ : f 1 ∈ circle) }
⊢ f = rotation a ∨ f = LinearIsometryEquiv.trans conjLie (rotation a)

Next tactic:
have : (f.trans (rotation a).symm) 1 = 1 := by simpa using rotation_apply a⁻¹ (f 1)
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
  use a
  have : (f.trans (rotation a).symm) 1 = 1 := by 

Proof state:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : ↥circle := { val := f 1, property := (_ : f 1 ∈ circle) }
⊢ (LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (rotation a))) 1 = 1

Next tactic:
simpa using rotation_apply a⁻¹ (f 1)
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
  use a
  have : (f.trans (rotation a).symm) 1 = 1 := by simpa using rotation_apply a⁻¹ (f 1)
  

Proof state:
case h
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : ↥circle := { val := f 1, property := (_ : f 1 ∈ circle) }
this : (LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (rotation a))) 1 = 1
⊢ f = rotation a ∨ f = LinearIsometryEquiv.trans conjLie (rotation a)

Next tactic:
refine' (linear_isometry_complex_aux this).imp (fun h₁ => _) fun h₂ => _
theorem linear_isometry_complex (f : ℂ ≃ₗᵢ[ℝ] ℂ) :
    ∃ a : circle, f = rotation a ∨ f = conjLie.trans (rotation a) := by
  let a : circle := ⟨f 1, by rw [mem_circle_iff_abs, ← Complex.norm_eq_abs, f.norm_map, norm_one]⟩
  use a
  have : (f.trans (rotation a).symm) 1 = 1 := by simpa using rotation_apply a⁻¹ (f 1)
  refine' (linear_isometry_complex_aux this).imp (fun h₁ => _) fun h₂ => _
  · 

Proof state:
case h.refine'_1
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : ↥circle := { val := f 1, property := (_ : f 1 ∈ circle) }
this : (LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (rotation a))) 1 = 1
h₁ : LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (rotation a)) = LinearIsometryEquiv.refl ℝ ℂ
⊢ f = rotation a

Next tactic:
simpa using eq_mul_of_inv_mul_eq h₁