Theorem: PhragmenLindelof.quadrant_III
File: Mathlib/Analysis/Complex/PhragmenLindelof.lean

Proof steps:
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
hz_re : z.re ≤ 0
hz_im : z.im ≤ 0
⊢ ‖f z‖ ≤ C
state_after: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : (-z).re ≤ 0
hz_im : (-z).im ≤ 0
⊢ ‖f (-z)‖ ≤ C
tactic: obtain ⟨z, rfl⟩ : ∃ z', -z' = z := ⟨-z, neg_neg z⟩
state_before: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : (-z).re ≤ 0
hz_im : (-z).im ≤ 0
⊢ ‖f (-z)‖ ≤ C
state_after: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
⊢ ‖f (-z)‖ ≤ C
tactic: simp only [neg_re, neg_im, neg_nonpos] at hz_re hz_im
state_before: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
⊢ ‖f (-z)‖ ≤ C
state_after: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C
tactic: change ‖(f ∘ Neg.neg) z‖ ≤ C
state_before: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C
state_after: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C
tactic: have H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0) := by
    intro w hw
    simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw
state_before: case intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
⊢ ‖(f ∘ Neg.neg) z‖ ≤ C
state_after: case intro.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
⊢ ∃ c < 2, ∃ B, (f ∘ Neg.neg) =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * Complex.abs z ^ c)

case intro.refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) ↑x‖ ≤ C

case intro.refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) (↑x * I)‖ ≤ C
tactic: refine
    quadrant_I (hd.comp differentiable_neg.diffContOnCl H) ?_ (fun x hx => ?_) (fun x hx => ?_)
      hz_re hz_im