awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
f' : E →L[𝕜] E'
⊢ HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (range (↑I : H → E) ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s)
      ((↑(extChartAt I x) : M → E) x) ↔
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'
      ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)

Tactic:
simp only [HasFDerivWithinAt, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq]

Premises used: HasFDerivWithinAt, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
simp_rw :
  ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(extChartAt I x) : M → E) x)
⊢ MDifferentiableWithinAt I I' f s x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(extChartAt I x) : M → E) x)

Tactic:
simp_rw [MDifferentiableWithinAt, ChartedSpace.liftPropWithinAt_iff']

Premises used: MDifferentiableWithinAt, ChartedSpace.liftPropWithinAt_iff'
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(extChartAt I x) : M → E) x)

Tactic:
rfl
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
simp :
  ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        ((↑(extChartAt I x) : M → E) x)
⊢ MDifferentiableWithinAt I I' f s x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        ((↑(extChartAt I x) : M → E) x)

Tactic:
simp only [MDifferentiableWithinAt, liftPropWithinAt_iff']

Premises used: MDifferentiableWithinAt, ChartedSpace.liftPropWithinAt_iff'
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        ((↑(extChartAt I x) : M → E) x)

Tactic:
and_congr_right fun hc => differentiableWithinAt_congr_nhds <|
    hc.nhdsWithin_extChartAt_symm_preimage_inter_range

Premises used: ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range, and_congr_right, differentiableWithinAt_congr_nhds
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 ((↑(extChartAt I' (f x)) : M' → E') ∘ f ∘ (↑(extChartAt I x).symm : E → M))
        ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        ((↑(extChartAt I x) : M → E) x)

Tactic:
exact and_congr_right fun hc => differentiableWithinAt_congr_nhds <|
    hc.nhdsWithin_extChartAt_symm_preimage_inter_range

Premises used: and_congr_right, differentiableWithinAt_congr_nhds
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
simp_rw :
  ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    (ContinuousWithinAt f s x ∧ ContinuousWithinAt ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x) ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E')
        ((↑(chartAt E' (((↑(extChartAt I' (f x)) : M' → E') ∘ f) x)) : E' → E') ∘
          ((↑(extChartAt I' (f x)) : M' → E') ∘ f) ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x)
⊢ MDifferentiableWithinAt I I' f s x ↔
    ContinuousWithinAt f s x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x

Tactic:
simp_rw [MDifferentiableWithinAt, liftPropWithinAt_iff', ← and_assoc]

Premises used: and_assoc, MDifferentiableWithinAt, ChartedSpace.liftPropWithinAt_iff'
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧ ContinuousWithinAt ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x ↔ ContinuousWithinAt f s x

Tactic:
have cont :
    ContinuousWithinAt f s x ∧ ContinuousWithinAt (extChartAt I' (f x) ∘ f) s x ↔
        ContinuousWithinAt f s x :=
      and_iff_left_of_imp <| (continuousAt_extChartAt _).comp_continuousWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
cont :
  ContinuousWithinAt f s x ∧ ContinuousWithinAt ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x ↔ ContinuousWithinAt f s x
simp_rw :
  ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        ((↑I' : H' → E') ∘
          ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M)) ∘
            (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s ∩ range (↑I : H → E))
        ((↑I : H → E) ((↑(chartAt H x) : M → H) x)) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        (id ∘
          (id ∘
              (((↑I' : H' → E') ∘ (↑(chartAt H' (f x)) : M' → H')) ∘ f) ∘
                (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M)) ∘
            (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s ∩ range (↑I : H → E))
        ((↑I : H → E) ((↑(chartAt H x) : M → H) x))
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I'
        ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x) ↔
    (ContinuousWithinAt f s x ∧ ContinuousWithinAt ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x) ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E')
        ((↑(chartAt E' (((↑(extChartAt I' (f x)) : M' → E') ∘ f) x)) : E' → E') ∘
          ((↑(extChartAt I' (f x)) : M' → E') ∘ f) ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M))
        ((↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s) ((↑(chartAt H x) : M → H) x)

Tactic:
simp_rw [cont, DifferentiableWithinAtProp, extChartAt, OpenPartialHomeomorph.extend,
    PartialEquiv.coe_trans,
    ModelWithCorners.toPartialEquiv_coe, OpenPartialHomeomorph.coe_coe, modelWithCornersSelf_coe,
    chartAt_self_eq, OpenPartialHomeomorph.refl_apply]

Premises used: extChartAt, OpenPartialHomeomorph.coe_coe, OpenPartialHomeomorph.refl_apply, ModelWithCorners.toPartialEquiv_coe, PartialEquiv.coe_trans, DifferentiableWithinAtProp, modelWithCornersSelf_coe, OpenPartialHomeomorph.extend, chartAt_self_eq
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
cont :
  ContinuousWithinAt f s x ∧ ContinuousWithinAt ((↑(extChartAt I' (f x)) : M' → E') ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        ((↑I' : H' → E') ∘
          ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M)) ∘
            (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s ∩ range (↑I : H → E))
        ((↑I : H → E) ((↑(chartAt H x) : M → H) x)) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        (id ∘
          (id ∘
              (((↑I' : H' → E') ∘ (↑(chartAt H' (f x)) : M' → H')) ∘ f) ∘
                (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M)) ∘
            (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑(OpenPartialHomeomorph.symm (chartAt H x)) : H → M) ⁻¹' s ∩ range (↑I : H → E))
        ((↑I : H → E) ((↑(chartAt H x) : M → H) x))

Tactic:
rfl
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
rw :
  MDifferentiableWithinAt I I' f univ x ↔
    ContinuousAt f x ∧ MDifferentiableAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) x
rw₁ :
  MDifferentiableWithinAt I I' f univ x ↔
    ContinuousAt f x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) univ x
rw₂ :
  ContinuousWithinAt f univ x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) univ x ↔
    ContinuousAt f x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) univ x
rw₃ rw₄ :
  ContinuousAt f x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) univ x ↔
    ContinuousAt f x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) univ x
⊢ MDifferentiableAt I I' f x ↔
    ContinuousAt f x ∧ MDifferentiableAt I 𝓘(𝕜, E') ((↑(extChartAt I' (f x)) : M' → E') ∘ f) x

Tactic:
rw [← mdifferentiableWithinAt_univ, ← mdifferentiableWithinAt_univ,
    mdifferentiableWithinAt_iff_target, continuousWithinAt_univ]

Premises used: mdifferentiableWithinAt_univ, continuousWithinAt_univ, mdifferentiableWithinAt_iff_target
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
e : OpenPartialHomeomorph M H
inst_11 : IsManifold I 1 M
he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (maximalAtlas I 1 M) e
hx : x ∈ e.source
⊢ x ∈ e.source

Tactic:
have h2x := hx
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
e : OpenPartialHomeomorph M H
inst_11 : IsManifold I 1 M
he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (maximalAtlas I 1 M) e
hx h2x : x ∈ e.source
⊢ x ∈ (e.extend I).source

Tactic:
rw [← e.extend_source (I := I)] at h2x

Premises used: OpenPartialHomeomorph.extend_source
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
e : OpenPartialHomeomorph M H
inst_11 : IsManifold I 1 M
he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (maximalAtlas I 1 M) e
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
simp_rw :
  ContinuousWithinAt (f ∘ (↑e.symm : H → M)) ((↑e.symm : H → M) ⁻¹' s) ((↑e : M → H) x) ∧
      DifferentiableWithinAt 𝕜
        ((↑I' : H' → E') ∘ ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑e.symm : H → M)) ∘ (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑e.symm : H → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑I : H → E) ((↑e : M → H) x)) ↔
    ContinuousWithinAt (f ∘ (↑e.symm : H → M)) ((↑e.symm : H → M) ⁻¹' s) ((↑e : M → H) x) ∧
      DifferentiableWithinAt 𝕜 ((↑I' : H' → E') ∘ (↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(e.extend I).symm : E → M))
        ((↑(e.extend I).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(e.extend I) : M → E) x)
⊢ MDifferentiableWithinAt I I' f s x ↔
    MDifferentiableWithinAt 𝓘(𝕜, E) I' (f ∘ (↑(e.extend I).symm : E → M))
      ((↑(e.extend I).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(e.extend I) : M → E) x)

Tactic:
simp_rw [MDifferentiableWithinAt,
    differentiableWithinAt_localInvariantProp.liftPropWithinAt_indep_chart_source he hx,
    StructureGroupoid.liftPropWithinAt_self_source,
    e.extend_symm_continuousWithinAt_comp_right_iff, differentiableWithinAtProp_self_source,
    DifferentiableWithinAtProp, Function.comp, e.left_inv hx, (e.extend I).left_inv h2x]

Premises used: Function.comp, OpenPartialHomeomorph.left_inv, differentiableWithinAtProp_self_source, differentiableWithinAt_localInvariantProp, OpenPartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source, DifferentiableWithinAtProp, PartialEquiv.left_inv, StructureGroupoid.liftPropWithinAt_self_source, MDifferentiableWithinAt, OpenPartialHomeomorph.extend
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
e : OpenPartialHomeomorph M H
inst_11 : IsManifold I 1 M
he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (maximalAtlas I 1 M) e
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
⊢ ContinuousWithinAt (f ∘ (↑e.symm : H → M)) ((↑e.symm : H → M) ⁻¹' s) ((↑e : M → H) x) ∧
      DifferentiableWithinAt 𝕜
        ((↑I' : H' → E') ∘ ((↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑e.symm : H → M)) ∘ (↑I.symm : E → H))
        ((↑I.symm : E → H) ⁻¹' (↑e.symm : H → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑I : H → E) ((↑e : M → H) x)) ↔
    ContinuousWithinAt (f ∘ (↑e.symm : H → M)) ((↑e.symm : H → M) ⁻¹' s) ((↑e : M → H) x) ∧
      DifferentiableWithinAt 𝕜 ((↑I' : H' → E') ∘ (↑(chartAt H' (f x)) : M' → H') ∘ f ∘ (↑(e.extend I).symm : E → M))
        ((↑(e.extend I).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(e.extend I) : M → E) x)

Tactic:
rfl
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
inst_11 : IsManifold I 1 M
x' : M
hx' : x' ∈ (OpenPartialHomeomorph.toPartialEquiv (chartAt H x)).source
⊢ MDifferentiableWithinAt I I' f s x' ↔
    MDifferentiableWithinAt 𝓘(𝕜, E) I' (f ∘ (↑(extChartAt I x).symm : E → M))
      ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)) ((↑(extChartAt I x) : M → E) x')

Tactic:
mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas (chart_mem_maximalAtlas x) hx'

Premises used: extChartAt, IsManifold.chart_mem_maximalAtlas, mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas