Theorem: Complex.norm_cderiv_lt
File: Mathlib/Analysis/Complex/LocallyUniformLimit.lean

tactic: obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
  have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
  have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
  obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
  exact ⟨‖f x‖, hfM x hx, hx'⟩
state_before: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
⊢ ‖cderiv r f z‖ < M / r
state_after: case intro.intro
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
L : ℝ
hL1 : L < M
hL2 : ∀ w ∈ sphere z r, ‖f w‖ ≤ L
⊢ ‖cderiv r f z‖ < M / r

tactic: exact (norm_cderiv_le hr hL2).trans_lt ((div_lt_div_right hr).mpr hL1)
state_before: case intro.intro
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
L : ℝ
hL1 : L < M
hL2 : ∀ w ∈ sphere z r, ‖f w‖ ≤ L
⊢ ‖cderiv r f z‖ < M / r
state_after: no goals

tactic: have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
state_before: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
state_after: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

tactic: have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
state_before: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
state_after: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

tactic: obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
state_before: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
state_after: case intro.intro
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r)
x : ℂ
hx : x ∈ sphere z r
hx' : IsMaxOn (fun w => ‖f w‖) (sphere z r) x
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

tactic: exact ⟨‖f x‖, hfM x hx, hx'⟩
state_before: case intro.intro
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r)
x : ℂ
hx : x ∈ sphere z r
hx' : IsMaxOn (fun w => ‖f w‖) (sphere z r) x
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
state_after: no goals