-- Mathlib/Geometry/Manifold/VectorBundle/MDifferentiable.lean
theorem mdifferentiableWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} :
    MDifferentiableWithinAt IM (IB.prod 𝓘(𝕜, F)) f s x₀ ↔
      MDifferentiableWithinAt IM IB (fun x => (f x).proj) s x₀ ∧
      MDifferentiableWithinAt IM 𝓘(𝕜, F)
        (fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀

Proof state:
case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (f x).proj = ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj) (f x)).1

Tactic:
rw [Trivialization.coe_fst']

Result:
case refine_1.ex
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ (f x).proj ∈ (trivializationAt F E (f x₀).proj).baseSet

Proof state:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
⊢ MDifferentiableWithinAt IM (IB.prod 𝓘(𝕜, F)) f s x₀ ↔
    MDifferentiableWithinAt IM IB (fun x => (f x).proj) s x₀ ∧
      MDifferentiableWithinAt IM 𝓘(𝕜, F) (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀

Tactic:
simp (config := { singlePass := true }) only [mdifferentiableWithinAt_iff_target]

Result:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
⊢ ContinuousWithinAt f s x₀ ∧
      MDifferentiableWithinAt IM 𝓘(𝕜, EB × F) (↑(extChartAt (IB.prod 𝓘(𝕜, F)) (f x₀)) ∘ f) s x₀ ↔
    (ContinuousWithinAt (fun x => (f x).proj) s x₀ ∧
        MDifferentiableWithinAt IM 𝓘(𝕜, EB) (↑(extChartAt IB (f x₀).proj) ∘ fun x => (f x).proj) s x₀) ∧
      ContinuousWithinAt (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀ ∧
        MDifferentiableWithinAt IM 𝓘(𝕜, F)
          (↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2) ∘ fun x =>
            (↑(trivializationAt F E (f x₀).proj) (f x)).2)
          s x₀

Proof state:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
⊢ MDifferentiableWithinAt IM 𝓘(𝕜, EB × F) (↑(extChartAt (IB.prod 𝓘(𝕜, F)) (f x₀)) ∘ f) s x₀ ↔
    MDifferentiableWithinAt IM 𝓘(𝕜, EB) (↑(extChartAt IB (f x₀).proj) ∘ fun x => (f x).proj) s x₀ ∧
      MDifferentiableWithinAt IM 𝓘(𝕜, F)
        (↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2) ∘ fun x =>
          (↑(trivializationAt F E (f x₀).proj) (f x)).2)
        s x₀

Tactic:
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp_def,
  PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEquiv.refl_coe,
  extChartAt_self_apply, modelWithCornersSelf_coe, Function.id_def, ← chartedSpaceSelf_prod]

Result:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
⊢ MDifferentiableWithinAt IM (𝓘(𝕜, EB).prod 𝓘(𝕜, F))
      (fun x =>
        (↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1,
          (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).2))
      s x₀ ↔
    MDifferentiableWithinAt IM 𝓘(𝕜, EB) (fun x => ↑(extChartAt IB (f x₀).proj) (f x).proj) s x₀ ∧
      MDifferentiableWithinAt IM 𝓘(𝕜, F) (fun x => (↑(trivializationAt F E (f x₀).proj) (f x)).2) s x₀