-- Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
theorem HasFPowerSeriesWithinAt.hasFDerivWithinAt (h : HasFPowerSeriesWithinAt f p s x) :
    HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) (insert x s) x

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ Tendsto (fun y => (y, x)) (𝓝[insert x s] x) (𝓝[insert x s] x ×ˢ 𝓝[insert x s] x)

Tactic:
exact Tendsto.prod_mk tendsto_id (tendsto_const_nhdsWithin (by simp))

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ ∀ ⦃c : ℝ⦄,
    0 < c →
      ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
        ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Tactic:
intro c hc

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
    ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ Tendsto (fun y => (y, x)) (𝓝[insert x s] x) (𝓝[insert x s ×ˢ insert x s] (x, x))

Tactic:
rw [nhdsWithin_prod_eq]

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ Tendsto (fun y => (y, x)) (𝓝[insert x s] x) (𝓝[insert x s] x ×ˢ 𝓝[insert x s] x)

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ HasFDerivWithinAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (insert x s) x

Tactic:
rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff]

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ ∀ ⦃c : ℝ⦄,
    0 < c →
      ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
        ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
this : Tendsto (fun y => (y, x)) (𝓝[insert x s] x) (𝓝[insert x s ×ˢ insert x s] (x, x))
⊢ ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
    ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Tactic:
exact this (isLittleO_iff.1 h.hasStrictFDerivWithinAt hc)

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
    ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Tactic:
have : Tendsto (fun y ↦ (y, x)) (𝓝[insert x s] x) (𝓝[insert x s ×ˢ insert x s] (x, x)) := by
  rw [nhdsWithin_prod_eq]
  exact Tendsto.prod_mk tendsto_id (tendsto_const_nhdsWithin (by simp))

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
this : Tendsto (fun y => (y, x)) (𝓝[insert x s] x) (𝓝[insert x s ×ˢ insert x s] (x, x))
⊢ ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
    ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
c : ℝ
hc : 0 < c
⊢ x ∈ insert x s

Tactic:
simp

Result:
no goals