-- Mathlib/Geometry/Manifold/IntegralCurve/Basic.lean
lemma IsIntegralCurveAt.eventually_hasDerivAt (hγ : IsIntegralCurveAt γ v t₀) :
    ∀ᶠ t in 𝓝 t₀, HasDerivAt ((extChartAt I (γ t₀)) ∘ γ)
      (tangentCoordChange I (γ t) (γ t₀) (γ t) (v (γ t))) t

Proof state:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
⊢ HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) t

Tactic:
have hsrc := mem_of_mem_nhds ht1

Result:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : t ∈ γ ⁻¹' (extChartAt I (γ t₀)).source
⊢ HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) t

Proof state:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ (mfderiv I I (↑(chartAt H (γ t₀))) (γ t)).comp (ContinuousLinearMap.smulRight 1 (v (γ t))) =
    ContinuousLinearMap.smulRight 1 ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))

Tactic:
rw [ContinuousLinearMap.ext_iff]

Result:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ ∀ (x : TangentSpace 𝓘(ℝ, ℝ) t),
    ((mfderiv I I (↑(chartAt H (γ t₀))) (γ t)).comp (ContinuousLinearMap.smulRight 1 (v (γ t)))) x =
      (ContinuousLinearMap.smulRight 1 ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))) x

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
⊢ ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) t

Tactic:
apply eventually_mem_nhds_iff.mpr
  (hγ.continuousAt.preimage_mem_nhds (extChartAt_source_mem_nhds (I := I) _)) |>.and hγ |>.mono

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
⊢ ∀ (x : ℝ),
    γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 x ∧ HasMFDerivAt 𝓘(ℝ, ℝ) I γ x (ContinuousLinearMap.smulRight 1 (v (γ x))) →
      HasDerivAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ x) (γ t₀) (γ x)) (v (γ x))) x

Proof state:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ HasMFDerivAt 𝓘(ℝ, ℝ) 𝓘(ℝ, E) (↑(extChartAt I (γ t₀)) ∘ γ) t
    (ContinuousLinearMap.smulRight 1 ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))))

Tactic:
apply (HasMFDerivAt.comp t (hasMFDerivAt_extChartAt (I := I) hsrc) ht2).congr_mfderiv

Result:
case intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ (mfderiv I I (↑(chartAt H (γ t₀))) (γ t)).comp (ContinuousLinearMap.smulRight 1 (v (γ t))) =
    ContinuousLinearMap.smulRight 1 ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))