[GOAL]
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
R : ℝ
c : ℂ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
⊢ deriv f c = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c) ^ (-2) • f z
[PROOFSTEP]
lift R to ℝ≥0 using hR.le
[GOAL]
case intro
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
f : ℂ → F
R : ℝ≥0
hR : 0 < ↑R
hf : DiffContOnCl ℂ f (ball c ↑R)
⊢ deriv f c = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z
[PROOFSTEP]
refine' (hf.hasFPowerSeriesOnBall hR).hasFPowerSeriesAt.deriv.trans _
[GOAL]
case intro
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
f : ℂ → F
R : ℝ≥0
hR : 0 < ↑R
hf : DiffContOnCl ℂ f (ball c ↑R)
⊢ (↑(cauchyPowerSeries f c (↑R) 1) fun x => 1) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z
[PROOFSTEP]
simp only [cauchyPowerSeries_apply, one_div, zpow_neg, pow_one, smul_smul, zpow_two, mul_inv]
[GOAL]
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
⊢ ‖deriv f c‖ ≤ C / R
[PROOFSTEP]
have : ∀ z ∈ sphere c R, ‖(z - c) ^ (-2 : ℤ) • f z‖ ≤ C / (R * R) := fun z (hz : abs (z - c) = R) => by
  simpa [-mul_inv_rev, norm_smul, hz, zpow_two, ← div_eq_inv_mul] using (div_le_div_right (mul_pos hR hR)).2 (hC z hz)
[GOAL]
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
z : ℂ
hz : ↑abs (z - c) = R
⊢ ‖(z - c) ^ (-2) • f z‖ ≤ C / (R * R)
[PROOFSTEP]
simpa [-mul_inv_rev, norm_smul, hz, zpow_two, ← div_eq_inv_mul] using (div_le_div_right (mul_pos hR hR)).2 (hC z hz)
[GOAL]
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
this : ∀ (z : ℂ), z ∈ sphere c R → ‖(z - c) ^ (-2) • f z‖ ≤ C / (R * R)
⊢ ‖deriv f c‖ ≤ C / R
[PROOFSTEP]
calc
  ‖deriv f c‖ = ‖(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z‖ :=
    congr_arg norm (deriv_eq_smul_circleIntegral hR hf)
  _ ≤ R * (C / (R * R)) := (circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const hR.le this)
  _ = C / R := by rw [mul_div_left_comm, div_self_mul_self', div_eq_mul_inv]
[GOAL]
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
this : ∀ (z : ℂ), z ∈ sphere c R → ‖(z - c) ^ (-2) • f z‖ ≤ C / (R * R)
⊢ R * (C / (R * R)) = C / R
[PROOFSTEP]
rw [mul_div_left_comm, div_self_mul_self', div_eq_mul_inv]
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
⊢ ‖deriv f c‖ ≤ C / R
[PROOFSTEP]
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
⊢ ‖deriv f c‖ ≤ C / R
[PROOFSTEP]
have : HasDerivAt (e ∘ f) (e (deriv f c)) c :=
  e.hasFDerivAt.comp_hasDerivAt c (hd.differentiableAt isOpen_ball <| mem_ball_self hR).hasDerivAt
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
this : HasDerivAt (↑e ∘ f) (↑e (deriv f c)) c
⊢ ‖deriv f c‖ ≤ C / R
[PROOFSTEP]
calc
  ‖deriv f c‖ = ‖deriv (e ∘ f) c‖ := by
    rw [this.deriv]
    exact (UniformSpace.Completion.norm_coe _).symm
  _ ≤ C / R :=
    norm_deriv_le_aux hR (e.differentiable.comp_diffContOnCl hd) fun z hz =>
      (UniformSpace.Completion.norm_coe _).trans_le (hC z hz)
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
this : HasDerivAt (↑e ∘ f) (↑e (deriv f c)) c
⊢ ‖deriv f c‖ = ‖deriv (↑e ∘ f) c‖
[PROOFSTEP]
rw [this.deriv]
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
this : HasDerivAt (↑e ∘ f) (↑e (deriv f c)) c
⊢ ‖deriv f c‖ = ‖↑e (deriv f c)‖
[PROOFSTEP]
exact (UniformSpace.Completion.norm_coe _).symm
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : ℂ
⊢ f z = f w
[PROOFSTEP]
suffices : ∀ c, deriv f c = 0
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : ℂ
this : ∀ (c : ℂ), deriv f c = 0
⊢ f z = f w
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : ℂ
⊢ ∀ (c : ℂ), deriv f c = 0
[PROOFSTEP]
exact is_const_of_deriv_eq_zero hf this z w
[GOAL]
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : ℂ
⊢ ∀ (c : ℂ), deriv f c = 0
[PROOFSTEP]
clear z w
[GOAL]
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
⊢ ∀ (c : ℂ), deriv f c = 0
[PROOFSTEP]
intro c
[GOAL]
case this
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
c : ℂ
⊢ deriv f c = 0
[PROOFSTEP]
obtain ⟨C, C₀, hC⟩ : ∃ C > (0 : ℝ), ∀ z, ‖f z‖ ≤ C :=
  by
  rcases bounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩
  exact ⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z => (hC (f z) (mem_range_self _)).trans (le_max_left _ _)⟩
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
c : ℂ
⊢ ∃ C, C > 0 ∧ ∀ (z : ℂ), ‖f z‖ ≤ C
[PROOFSTEP]
rcases bounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩
[GOAL]
case intro
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
c : ℂ
C : ℝ
hC : ∀ (x : F), x ∈ range f → ‖x‖ ≤ C
⊢ ∃ C, C > 0 ∧ ∀ (z : ℂ), ‖f z‖ ≤ C
[PROOFSTEP]
exact ⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z => (hC (f z) (mem_range_self _)).trans (le_max_left _ _)⟩
[GOAL]
case this.intro.intro
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
c : ℂ
C : ℝ
C₀ : C > 0
hC : ∀ (z : ℂ), ‖f z‖ ≤ C
⊢ deriv f c = 0
[PROOFSTEP]
refine' norm_le_zero_iff.1 (le_of_forall_le_of_dense fun ε ε₀ => _)
[GOAL]
case this.intro.intro
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
c : ℂ
C : ℝ
C₀ : C > 0
hC : ∀ (z : ℂ), ‖f z‖ ≤ C
ε : ℝ
ε₀ : 0 < ε
⊢ ‖deriv f c‖ ≤ ε
[PROOFSTEP]
calc
  ‖deriv f c‖ ≤ C / (C / ε) :=
    norm_deriv_le_of_forall_mem_sphere_norm_le (div_pos C₀ ε₀) hf.diffContOnCl fun z _ => hC z
  _ = ε := div_div_cancel' C₀.lt.ne'
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
⊢ f z = f w
[PROOFSTEP]
set g : ℂ → F := f ∘ fun t : ℂ => t • (w - z) + z
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ f z = f w
[PROOFSTEP]
suffices g 0 = g 1 by simpa
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
this : g 0 = g 1
⊢ f z = f w
[PROOFSTEP]
simpa
[GOAL]
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ g 0 = g 1
[PROOFSTEP]
apply liouville_theorem_aux
[GOAL]
case hf
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Differentiable ℂ g
case hb
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Metric.Bounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Metric.Bounded (range g)
[PROOFSTEP]
exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z), hb.mono (range_comp_subset_range _ _)]
