-- Mathlib/Analysis/Analytic/IsolatedZeros.lean
theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
    HasFPowerSeriesAt (dslope f z₀) p.fslope z₀

Proof state:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : x = 0
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Tactic:
convert hasSum_single (α := E) 0 _ <;> intros <;> simp [*]

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
hpd : deriv f z₀ = p.coeff 1
⊢ HasFPowerSeriesAt (dslope f z₀) p.fslope z₀

Tactic:
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1

Result:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
⊢ HasFPowerSeriesAt (dslope f z₀) p.fslope z₀

Proof state:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
hxx : ∀ (n : ℕ), x⁻¹ * x ^ (n + 1) = x ^ n
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Tactic:
suffices HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀)) by
  simpa [dslope, slope, h, smul_smul, hxx] using this

Result:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
hxx : ∀ (n : ℕ), x⁻¹ * x ^ (n + 1) = x ^ n
⊢ HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀))

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
⊢ ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + z))

Tactic:
refine hp.mono fun x hx => ?_

Result:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
hxx : ∀ (n : ℕ), x⁻¹ * x ^ (n + 1) = x ^ n
this : HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀))
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Tactic:
simpa [dslope, slope, h, smul_smul, hxx] using this

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
n : ℕ
⊢ x⁻¹ * x ^ (n + 1) = x ^ n

Tactic:
field_simp [h, _root_.pow_succ]

Result:
no goals

Proof state:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Tactic:
have hxx : ∀ n : ℕ, x⁻¹ * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, _root_.pow_succ]

Result:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
hxx : ∀ (n : ℕ), x⁻¹ * x ^ (n + 1) = x ^ n
⊢ HasSum (fun n => x ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + x))

Proof state:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
x : 𝕜
hx : HasSum (fun n => x ^ n • p.coeff n) (f (z₀ + x))
h : ¬x = 0
hxx : ∀ (n : ℕ), x⁻¹ * x ^ (n + 1) = x ^ n
⊢ HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀))

Tactic:
simpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x⁻¹

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
⊢ HasFPowerSeriesAt (dslope f z₀) p.fslope z₀

Tactic:
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢

Result:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hpd : deriv f z₀ = p.coeff 1
hp0 : p.coeff 0 = f z₀
hp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
⊢ ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff (n + 1)) (dslope f z₀ (z₀ + z))

Proof state:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
⊢ HasFPowerSeriesAt (dslope f z₀) p.fslope z₀

Tactic:
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv

Result:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
hpd : deriv f z₀ = p.coeff 1
⊢ HasFPowerSeriesAt (dslope f z₀) p.fslope z₀