[GOAL]
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
⊢ IsConformalMap (restrictScalars ℝ map)
[PROOFSTEP]
have minor₁ : ‖map 1‖ ≠ 0 := by simpa only [ext_ring_iff, Ne.def, norm_eq_zero] using nonzero
[GOAL]
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
⊢ ‖↑map 1‖ ≠ 0
[PROOFSTEP]
simpa only [ext_ring_iff, Ne.def, norm_eq_zero] using nonzero
[GOAL]
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
⊢ IsConformalMap (restrictScalars ℝ map)
[PROOFSTEP]
refine' ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), _⟩, _⟩
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
⊢ ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖
[PROOFSTEP]
intro x
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
⊢ ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖
[PROOFSTEP]
simp only [LinearMap.smul_apply]
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖
[PROOFSTEP]
have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]
[GOAL]
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
⊢ x = x • 1
[PROOFSTEP]
rw [smul_eq_mul, mul_one]
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) x‖ = ‖x‖
[PROOFSTEP]
nth_rw 1 [this]
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖‖↑map 1‖⁻¹ • ↑(↑ℝ ↑map) (x • 1)‖ = ‖x‖
[PROOFSTEP]
rw [LinearMap.coe_restrictScalars]
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖‖↑map 1‖⁻¹ • ↑↑map (x • 1)‖ = ‖x‖
[PROOFSTEP]
simp only [map.coe_coe, map.map_smul, norm_smul, norm_inv, norm_norm]
[GOAL]
case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖↑map 1‖⁻¹ * (‖x‖ * ‖↑map 1‖) = ‖x‖
[PROOFSTEP]
field_simp only [one_mul]
[GOAL]
case refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
⊢ restrictScalars ℝ map =
    ‖↑map 1‖ •
      LinearIsometry.toContinuousLinearMap
        { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := (_ : ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖) }
[PROOFSTEP]
ext1
  -- Porting note: was simp
[GOAL]
case refine'_2.h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
z : ℂ
g : ℂ →L[ℝ] E
f : ℂ → E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖↑map 1‖ ≠ 0
x✝ : ℂ
⊢ ↑(restrictScalars ℝ map) x✝ =
    ↑(‖↑map 1‖ •
          LinearIsometry.toContinuousLinearMap
            { toLinearMap := ‖↑map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := (_ : ∀ (x : ℂ), ‖↑(‖↑map 1‖⁻¹ • ↑ℝ ↑map) x‖ = ‖x‖) })
      x✝
[PROOFSTEP]
rw [coe_restrictScalars', coe_smul', LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.coe_mk, Pi.smul_apply,
  LinearMap.smul_apply, LinearMap.coe_restrictScalars, coe_coe, smul_inv_smul₀ minor₁]
[GOAL]
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h : IsConformalMap g
⊢ (∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = ContinuousLinearMap.comp g ↑conjCle
[PROOFSTEP]
rcases h with ⟨c, -, li, rfl⟩
[GOAL]
case intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • LinearIsometry.toContinuousLinearMap li) ∨
    ∃ map, restrictScalars ℝ map = ContinuousLinearMap.comp (c • LinearIsometry.toContinuousLinearMap li) ↑conjCle
[PROOFSTEP]
obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.toLinearIsometry = li := ⟨li.toLinearIsometryEquiv rfl, by ext1; rfl⟩
[GOAL]
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
⊢ LinearIsometryEquiv.toLinearIsometry
      (LinearIsometry.toLinearIsometryEquiv li (_ : FiniteDimensional.finrank ℝ ℂ = FiniteDimensional.finrank ℝ ℂ)) =
    li
[PROOFSTEP]
ext1
[GOAL]
case h
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ →ₗᵢ[ℝ] ℂ
x✝ : ℂ
⊢ ↑(LinearIsometryEquiv.toLinearIsometry
          (LinearIsometry.toLinearIsometryEquiv li (_ : FiniteDimensional.finrank ℝ ℂ = FiniteDimensional.finrank ℝ ℂ)))
      x✝ =
    ↑li x✝
[PROOFSTEP]
rfl
[GOAL]
case intro.intro.intro.intro
f : ℂ → ℂ
z : ℂ
c : ℝ
li : ℂ ≃ₗᵢ[ℝ] ℂ
⊢ (∃ map, restrictScalars ℝ map = c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry li)) ∨
    ∃ map,
      restrictScalars ℝ map =
        ContinuousLinearMap.comp (c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry li))
          ↑conjCle
[PROOFSTEP]
rcases linear_isometry_complex li with
  ⟨a, rfl | rfl⟩
    -- let rot := c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ,
[GOAL]
case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
⊢ (∃ map,
      restrictScalars ℝ map =
        c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (↑rotation a))) ∨
    ∃ map,
      restrictScalars ℝ map =
        ContinuousLinearMap.comp
          (c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (↑rotation a))) ↑conjCle
[PROOFSTEP]
refine' Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, _⟩
[GOAL]
case intro.intro.intro.intro.intro.inl
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) =
    c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (↑rotation a))
[PROOFSTEP]
ext1
  -- Porting note: was simp
[GOAL]
case intro.intro.intro.intro.intro.inl.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
x✝ : ℂ
⊢ ↑(restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    ↑(c • LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (↑rotation a))) x✝
[PROOFSTEP]
rw [coe_restrictScalars', smul_apply, smul_apply, smul_apply, LinearIsometry.coe_toContinuousLinearMap,
  LinearIsometryEquiv.coe_toLinearIsometry, rotation_apply, id_apply, smul_eq_mul]
[GOAL]
case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
⊢ (∃ map,
      restrictScalars ℝ map =
        c •
          LinearIsometry.toContinuousLinearMap
            (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.trans conjLie (↑rotation a)))) ∨
    ∃ map,
      restrictScalars ℝ map =
        ContinuousLinearMap.comp
          (c •
            LinearIsometry.toContinuousLinearMap
              (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.trans conjLie (↑rotation a))))
          ↑conjCle
[PROOFSTEP]
refine' Or.inr ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, _⟩
[GOAL]
case intro.intro.intro.intro.intro.inr
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) =
    ContinuousLinearMap.comp
      (c •
        LinearIsometry.toContinuousLinearMap
          (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.trans conjLie (↑rotation a))))
      ↑conjCle
[PROOFSTEP]
ext1
  -- Porting note: was simp
[GOAL]
case intro.intro.intro.intro.intro.inr.h
f : ℂ → ℂ
z : ℂ
c : ℝ
a : { x // x ∈ circle }
x✝ : ℂ
⊢ ↑(restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
    ↑(ContinuousLinearMap.comp
          (c •
            LinearIsometry.toContinuousLinearMap
              (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.trans conjLie (↑rotation a))))
          ↑conjCle)
      x✝
[PROOFSTEP]
rw [coe_restrictScalars', smul_apply, smul_apply, comp_apply, smul_apply, LinearIsometry.coe_toContinuousLinearMap,
  LinearIsometryEquiv.coe_toLinearIsometry, LinearIsometryEquiv.trans_apply, rotation_apply, id_apply, smul_eq_mul,
  ContinuousLinearEquiv.coe_coe, conjCle_apply, conjLie_apply, conj_conj]
[GOAL]
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g ↔ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = comp g ↑conjCle) ∧ g ≠ 0
[PROOFSTEP]
constructor
[GOAL]
case mp
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g → ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = comp g ↑conjCle) ∧ g ≠ 0
[PROOFSTEP]
exact fun h => ⟨h.is_complex_or_conj_linear, h.ne_zero⟩
[GOAL]
case mpr
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
⊢ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = comp g ↑conjCle) ∧ g ≠ 0 → IsConformalMap g
[PROOFSTEP]
rintro ⟨⟨map, rfl⟩ | ⟨map, hmap⟩, h₂⟩
[GOAL]
case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ IsConformalMap (restrictScalars ℝ map)
[PROOFSTEP]
refine' isConformalMap_complex_linear _
[GOAL]
case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
h₂ : restrictScalars ℝ map ≠ 0
⊢ map ≠ 0
[PROOFSTEP]
contrapose! h₂ with w
[GOAL]
case mpr.intro.inl.intro
f : ℂ → ℂ
z : ℂ
map : ℂ →L[ℂ] ℂ
w : map = 0
⊢ restrictScalars ℝ map = 0
[PROOFSTEP]
simp only [w, restrictScalars_zero]
[GOAL]
case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = comp g ↑conjCle
⊢ IsConformalMap g
[PROOFSTEP]
have minor₁ : g = map.restrictScalars ℝ ∘L ↑conjCle := by
  ext1
  simp only [hmap, coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply, conjCle_apply, starRingEnd_self_apply]
[GOAL]
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = comp g ↑conjCle
⊢ g = comp (restrictScalars ℝ map) ↑conjCle
[PROOFSTEP]
ext1
[GOAL]
case h
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = comp g ↑conjCle
x✝ : ℂ
⊢ ↑g x✝ = ↑(comp (restrictScalars ℝ map) ↑conjCle) x✝
[PROOFSTEP]
simp only [hmap, coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply, conjCle_apply, starRingEnd_self_apply]
[GOAL]
case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
h₂ : g ≠ 0
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = comp g ↑conjCle
minor₁ : g = comp (restrictScalars ℝ map) ↑conjCle
⊢ IsConformalMap g
[PROOFSTEP]
rw [minor₁] at h₂ ⊢
[GOAL]
case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : comp (restrictScalars ℝ map) ↑conjCle ≠ 0
hmap : restrictScalars ℝ map = comp g ↑conjCle
minor₁ : g = comp (restrictScalars ℝ map) ↑conjCle
⊢ IsConformalMap (comp (restrictScalars ℝ map) ↑conjCle)
[PROOFSTEP]
refine' isConformalMap_complex_linear_conj _
[GOAL]
case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
h₂ : comp (restrictScalars ℝ map) ↑conjCle ≠ 0
hmap : restrictScalars ℝ map = comp g ↑conjCle
minor₁ : g = comp (restrictScalars ℝ map) ↑conjCle
⊢ map ≠ 0
[PROOFSTEP]
contrapose! h₂ with w
[GOAL]
case mpr.intro.inr.intro
f : ℂ → ℂ
z : ℂ
g : ℂ →L[ℝ] ℂ
map : ℂ →L[ℂ] ℂ
hmap : restrictScalars ℝ map = comp g ↑conjCle
minor₁ : g = comp (restrictScalars ℝ map) ↑conjCle
w : map = 0
⊢ comp (restrictScalars ℝ map) ↑conjCle = 0
[PROOFSTEP]
simp only [w, restrictScalars_zero, zero_comp]
