Theorem: Complex.HadamardThreeLines.norm_le_interpStrip_of_mem_verticalStrip_zero
File: Mathlib/Analysis/Complex/Hadamard.lean

tactic: apply tendsto_le_of_eventuallyLE _ _ (eventuallyle f z hB hd hz)
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ‖f z‖ ≤ ‖interpStrip f z‖
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun x => ‖f z‖) (𝓝[>] 0) (𝓝 ‖f z‖)

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0) (𝓝 ‖interpStrip f z‖)

tactic: apply tendsto_inf_left
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun x => ‖f z‖) (𝓝[>] 0) (𝓝 ‖f z‖)
state_after: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun x => ‖f z‖) (𝓝 0) (𝓝 ‖f z‖)

tactic: simp only [tendsto_const_nhds_iff]
state_before: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun x => ‖f z‖) (𝓝 0) (𝓝 ‖f z‖)
state_after: no goals

tactic: rw [interpStrip_eq_of_mem_verticalStrip _ _ hz]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0) (𝓝 ‖interpStrip f z‖)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0)
    (𝓝 ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖)

tactic: convert ContinuousWithinAt.tendsto _ using 2
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0)
    (𝓝 ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖)
state_after: case h.e'_5.h.e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖ =
    ‖(↑0 + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑0 + ↑(sSupNormIm f 1)) ^ z‖

case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ContinuousWithinAt (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (Ioi 0) 0

tactic: simp only [ofReal_zero, zero_add]
state_before: case h.e'_5.h.e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖ =
    ‖(↑0 + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑0 + ↑(sSupNormIm f 1)) ^ z‖
state_after: no goals

tactic: simp_rw [← ofReal_add, norm_eq_abs]
state_before: case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ContinuousWithinAt (fun ε => ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (Ioi 0) 0
state_after: case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ContinuousWithinAt (fun ε => abs (↑(ε + sSupNormIm f 0) ^ (1 - z) * ↑(ε + sSupNormIm f 1) ^ z)) (Ioi 0) 0

tactic: apply tendsto_nhdsWithin_congr this _
state_before: case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ ContinuousWithinAt (fun ε => abs (↑(ε + sSupNormIm f 0) ^ (1 - z) * ↑(ε + sSupNormIm f 1) ^ z)) (Ioi 0) 0
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 ((fun ε => abs (↑(ε + sSupNormIm f 0) ^ (1 - z) * ↑(ε + sSupNormIm f 1) ^ z)) 0))

tactic: simp only [zero_add]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 ((fun ε => abs (↑(ε + sSupNormIm f 0) ^ (1 - z) * ↑(ε + sSupNormIm f 1) ^ z)) 0))
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 (abs (↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z)))

tactic: rw [map_mul, abs_cpow_eq_rpow_re_of_nonneg (sSupNormIm_nonneg _ _) _,
  abs_cpow_eq_rpow_re_of_nonneg (sSupNormIm_nonneg _ _) _]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 (abs (↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z)))
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 (sSupNormIm f 0 ^ (1 - z).re * sSupNormIm f 1 ^ z.re))

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ z.re ≠ 0

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ (1 - z).re ≠ 0

tactic: intro x hx
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)

tactic: simp only [map_mul]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    abs (↑(x + sSupNormIm f 0) ^ (1 - z)) * abs (↑(x + sSupNormIm f 1) ^ z)

tactic: repeat rw [abs_cpow_eq_rpow_re_of_nonneg (le_of_lt (sSupNormIm_eps_pos f hx _)) _]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    abs (↑(x + sSupNormIm f 0) ^ (1 - z)) * abs (↑(x + sSupNormIm f 1) ^ z)
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    (x + sSupNormIm f 0) ^ (1 - z).re * (x + sSupNormIm f 1) ^ z.re

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ z.re ≠ 0

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (1 - z).re ≠ 0

tactic: rw [abs_cpow_eq_rpow_re_of_nonneg (le_of_lt (sSupNormIm_eps_pos f hx _)) _]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    (x + sSupNormIm f 0) ^ (1 - z).re * abs (↑(x + sSupNormIm f 1) ^ z)

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (1 - z).re ≠ 0
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    (x + sSupNormIm f 0) ^ (1 - z).re * (x + sSupNormIm f 1) ^ z.re

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ z.re ≠ 0

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (1 - z).re ≠ 0

tactic: simp only [sub_re, one_re]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
    (x + sSupNormIm f 0) ^ (1 - z).re * (x + sSupNormIm f 1) ^ z.re
state_after: no goals

tactic: simpa using (ne_comm.mpr (ne_of_lt hz.1))
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ z.re ≠ 0
state_after: no goals

tactic: simpa [sub_eq_zero] using (ne_comm.mpr (ne_of_lt hz.2))
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
x : ℝ
hx : x ∈ Ioi 0
⊢ (1 - z).re ≠ 0
state_after: no goals

tactic: apply Tendsto.mul
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0)
    (𝓝 (sSupNormIm f 0 ^ (1 - z).re * sSupNormIm f 1 ^ z.re))
state_after: case hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re)) (𝓝[>] 0) (𝓝 (sSupNormIm f 0 ^ (1 - z).re))

case hg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0) (𝓝 (sSupNormIm f 1 ^ z.re))

tactic: apply Tendsto.rpow_const
state_before: case hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 0) ^ (1 - z.re)) (𝓝[>] 0) (𝓝 (sSupNormIm f 0 ^ (1 - z).re))
state_after: case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (sSupNormIm f 0))

case hf.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ sSupNormIm f 0 ≠ 0 ∨ 0 ≤ 1 - z.re

tactic: nth_rw 2 [← zero_add (sSupNormIm f 0)]
state_before: case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (sSupNormIm f 0))
state_after: case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (0 + sSupNormIm f 0))

tactic: exact Tendsto.add_const (sSupNormIm f 0) (tendsto_nhdsWithin_of_tendsto_nhds
  (Continuous.tendsto continuous_id' _))
state_before: case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (0 + sSupNormIm f 0))
state_after: no goals

tactic: right
state_before: case hf.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ sSupNormIm f 0 ≠ 0 ∨ 0 ≤ 1 - z.re
state_after: case hf.h.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ 0 ≤ 1 - z.re

tactic: simp only [sub_nonneg, le_of_lt hz.2]
state_before: case hf.h.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ 0 ≤ 1 - z.re
state_after: no goals

tactic: apply Tendsto.rpow_const
state_before: case hg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun x => (x + sSupNormIm f 1) ^ z.re) (𝓝[>] 0) (𝓝 (sSupNormIm f 1 ^ z.re))
state_after: case hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 1) (𝓝[>] 0) (𝓝 (sSupNormIm f 1))

case hg.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ sSupNormIm f 1 ≠ 0 ∨ 0 ≤ z.re

tactic: nth_rw 2 [← zero_add (sSupNormIm f 1)]
state_before: case hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 1) (𝓝[>] 0) (𝓝 (sSupNormIm f 1))
state_after: case hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 1) (𝓝[>] 0) (𝓝 (0 + sSupNormIm f 1))

tactic: exact Tendsto.add_const (sSupNormIm f 1) (tendsto_nhdsWithin_of_tendsto_nhds
  (Continuous.tendsto continuous_id' _))
state_before: case hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ Tendsto (fun a => a + sSupNormIm f 1) (𝓝[>] 0) (𝓝 (0 + sSupNormIm f 1))
state_after: no goals

tactic: right
state_before: case hg.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ sSupNormIm f 1 ≠ 0 ∨ 0 ≤ z.re
state_after: case hg.h.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ 0 ≤ z.re

tactic: simp only [sub_nonneg, le_of_lt hz.1]
state_before: case hg.h.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ 0 ≤ z.re
state_after: no goals

tactic: simpa using (ne_comm.mpr (ne_of_lt hz.1))
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ z.re ≠ 0
state_after: no goals

tactic: simpa [sub_eq_zero] using (ne_comm.mpr (ne_of_lt hz.2))
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      abs (↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z)
⊢ (1 - z).re ≠ 0
state_after: no goals