/-- The derivative of a `circleTransform` is locally bounded. -/
theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R)
    (hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧
      ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B := by
  obtain ⟨r, hr, hrx⟩ := exists_lt_mem_ball_of_mem_ball hx
  obtain ⟨ε', hε', H⟩ := exists_ball_subset_ball hrx
  obtain ⟨⟨⟨a, b⟩, ⟨ha, hb⟩⟩, hab⟩ :=
    abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z
  let V : ℝ → ℂ → ℂ := fun θ w => circleTransformDeriv R z w (fun _ => 1) θ
  have funccomp : ContinuousOn (fun r => abs (f r)) (sphere z R) := by
    have cabs : ContinuousOn abs ⊤ := by apply continuous_abs.continuousOn
    apply cabs.comp hf; rw [MapsTo]; tauto
  have sbou :=
    IsCompact.exists_isMaxOn (isCompact_sphere z R) (NormedSpace.sphere_nonempty.2 hR.le) funccomp
  obtain ⟨X, HX, HX2⟩ := sbou
  refine' ⟨abs (V b a) * abs (f X), ε', hε', Subset.trans H (ball_subset_ball hr.le), _⟩
  intro y v hv
  obtain ⟨y1, hy1, hfun⟩ :=
    Periodic.exists_mem_Ico₀ (circleTransformDeriv_periodic R z v f) Real.two_pi_pos y
  have hy2 : y1 ∈ [[0, 2 * π]] := by
    convert Ico_subset_Icc_self hy1 using 1
    simp [uIcc_of_le Real.two_pi_pos.le]
  simp only [IsMaxOn, IsMaxFilter, eventually_principal, mem_sphere_iff_norm, norm_eq_abs] at HX2
  have := mul_le_mul (hab ⟨⟨v, y1⟩, ⟨ball_subset_closedBall (H hv), hy2⟩⟩)
    (HX2 (circleMap z R y1) (circleMap_mem_sphere z hR.le y1)) (Complex.abs.nonneg _)
    (Complex.abs.nonneg _)
  simp_rw [hfun]
  simp only [circleTransformBoundingFunction, circleTransformDeriv, norm_eq_abs,
    Algebra.id.smul_eq_mul, deriv_circleMap, map_mul, abs_circleMap_zero, abs_I, mul_one, ←
    mul_assoc, mul_inv_rev, inv_I, abs_neg, abs_inv, abs_ofReal, one_mul, abs_two, abs_pow,
    mem_ball, gt_iff_lt, Subtype.coe_mk, SetCoe.forall, mem_prod, mem_closedBall, and_imp,
    Prod.forall, NormedSpace.sphere_nonempty, mem_sphere_iff_norm] at *
  

Proof state:
case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R✝ : ℝ
z✝ w : ℂ
R : ℝ
hR : 0 < R
z x : ℂ
f : ℂ → ℂ
hf : ContinuousOn f (sphere z R)
r : ℝ
hr : r < R
ε' : ℝ
H : ball x ε' ⊆ ball z r
a : ℂ
b : ℝ
ha : (a, b).1 ∈ closedBall z r
hb : (a, b).2 ∈ [[0, 2 * π]]
V : ℝ → ℂ → ℂ := fun θ w => circleTransformDeriv R z w (fun x => 1) θ
funccomp : ContinuousOn (fun r => abs (f r)) (sphere z R)
X : ℂ
y : ℝ
v : ℂ
hv : v ∈ ball x ε'
y1 : ℝ
hy1 : y1 ∈ Ico 0 (2 * π)
hy2 : y1 ∈ [[0, 2 * π]]
HX2 : ∀ (x : ℂ), abs (x - z) = R → abs (f x) ≤ abs (f X)
hx : dist x z < R
hrx : dist x z < r
hε' : 0 < ε'
hab :
  ∀ (a_1 : ℂ) (b_1 : ℝ),
    dist a_1 z ≤ r →
      b_1 ∈ [[0, 2 * π]] →
        abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R b_1 - a_1) ^ 2)⁻¹ ≤
          abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R b - a) ^ 2)⁻¹
HX : abs (X - z) = R
hfun :
  -I * (↑π)⁻¹ * 2⁻¹ * circleMap 0 R y * I * ((circleMap z R y - v) ^ 2)⁻¹ * f (circleMap z R y) =
    -I * (↑π)⁻¹ * 2⁻¹ * circleMap 0 R y1 * I * ((circleMap z R y1 - v) ^ 2)⁻¹ * f (circleMap z R y1)
this :
  abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R y1 - v) ^ 2)⁻¹ * abs (f (circleMap z R y1)) ≤
    abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R b - a) ^ 2)⁻¹ * abs (f X)
⊢ abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R y1 - v) ^ 2)⁻¹ * abs (f (circleMap z R y1)) ≤
    abs (-I) * abs (↑π)⁻¹ * abs 2⁻¹ * |R| * abs ((circleMap z R b - a) ^ 2)⁻¹ * abs (f X)

Next tactic:
exact this