Theorem: circleIntegral.norm_integral_lt_of_norm_le_const_of_lt
File: Mathlib/MeasureTheory/Integral/CircleIntegral.lean

tactic: rw [← _root_.abs_of_pos hR, ← image_circleMap_Ioc] at hlt
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
hlt : ∃ z, z ∈ sphere c R ∧ ‖f z‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C

tactic: rcases hlt with ⟨_, ⟨θ₀, hmem, rfl⟩, hlt⟩
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
hlt : ∃ z, z ∈ circleMap c R '' Ioc 0 (2 * π) ∧ ‖f z‖ < C
⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C

tactic: simp only [norm_smul, deriv_circleMap, norm_eq_abs, map_mul, abs_I, mul_one,
  abs_circleMap_zero, abs_of_pos hR]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ (∫ (θ : ℝ) in 0 ..2 * π, ‖deriv (circleMap c R) θ • f (circleMap c R θ)‖) < ∫ (x : ℝ) in 0 ..2 * π, R * C

tactic: refine' intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt
    Real.two_pi_pos _ continuousOn_const (fun θ _ => _) ⟨θ₀, Ioc_subset_Icc_self hmem, _⟩
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ (∫ (θ : ℝ) in 0 ..2 * π, R * ‖f (circleMap c R θ)‖) < ∫ (x : ℝ) in 0 ..2 * π, R * C

tactic: exact continuousOn_const.mul (hc.comp (continuous_circleMap _ _).continuousOn fun θ _ =>
  circleMap_mem_sphere _ hR.le _).norm
state_before: case refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ ContinuousOn (fun θ => R * ‖f (circleMap c R θ)‖) (Icc 0 (2 * π))

tactic: exact mul_le_mul_of_nonneg_left (hf _ <| circleMap_mem_sphere _ hR.le _) hR.le
state_before: case refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
θ : ℝ
x✝ : θ ∈ Ioc 0 (2 * π)
⊢ R * ‖f (circleMap c R θ)‖ ≤ R * C

tactic: exact (mul_lt_mul_left hR).2 hlt
state_before: case refine'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * ‖f (circleMap c R θ₀)‖ < R * C

tactic: simp [mul_assoc]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ (∫ (x : ℝ) in 0 ..2 * π, R * C) = 2 * π * R * C

tactic: ring
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * (2 * (π * C)) = 2 * (π * (R * C))