mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x)) ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n
        (↑𝓘(𝕜, E') ∘
          (↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘ (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(chartAt H x).symm) ∘
            ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))

Tactic:
extChartAt,

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
n : ℕ∞ω
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x)) ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n
        (↑𝓘(𝕜, E') ∘
          (↑(chartAt E' ((↑((chartAt H' (f x)).extend I') ∘ f) x)) ∘
              (↑((chartAt H' (f x)).extend I') ∘ f) ∘ ↑(chartAt H x).symm) ∘
            ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
Module: Mathlib.Analysis.Normed.Operator.Prod

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : SeminormedAddCommGroup E
inst✝¹² : SeminormedAddCommGroup F
inst✝¹¹ : SeminormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedSpace 𝕜 G
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
M₄ : Type u_8
inst✝⁷ : SeminormedAddCommGroup M₁
inst✝⁶ : NormedSpace 𝕜 M₁
inst✝⁵ : SeminormedAddCommGroup M₂
inst✝⁴ : NormedSpace 𝕜 M₂
inst✝³ : SeminormedAddCommGroup M₃
inst✝² : NormedSpace 𝕜 M₃
inst✝¹ : SeminormedAddCommGroup M₄
inst✝ : NormedSpace 𝕜 M₄
⊢ (fun p ↦ p.1.prodMap p.2) =
    ⇑(have Φ₁ := (compL 𝕜 M₁ M₂ (M₂ × M₄)) (inl 𝕜 M₂ M₄);
      have Φ₂ := (compL 𝕜 M₃ M₄ (M₂ × M₄)) (inr 𝕜 M₂ M₄);
      have Φ₁' := (compL 𝕜 (M₁ × M₃) M₁ (M₂ × M₄)).flip (fst 𝕜 M₁ M₃);
      have Φ₂' := (compL 𝕜 (M₁ × M₃) M₃ (M₂ × M₄)).flip (snd 𝕜 M₁ M₃);
      have Ψ₁ := fst 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄);
      have Ψ₂ := snd 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄);
      Φ₁' ∘SL Φ₁ ∘SL Ψ₁ + Φ₂' ∘SL Φ₂ ∘SL Ψ₂)

Tactic:
apply funext
      rintro ⟨φ, ψ⟩
      refine ContinuousLinearMap.ext fun ⟨x₁, x₂⟩ => ?_
      simp

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

Goal state before:
pp: case e'_11
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm = (↑I' ∘ f ∘ ↑I.symm) ∘ ↑I ∘ ↑e.symm ∘ ↑I.symm

pp: case convert_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm ⁻¹' e.target ∩ range ↑I ∩ ↑I ∘ ↑e.symm ∘ ↑I.symm ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I) ∈
    𝓝[↑I.symm ⁻¹' ↑e.symm ⁻¹' s ∩ range ↑I] ↑I (↑e x)

Tactic:
· ext y; simp only [mfld_simps]

Goal state after:
case convert_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 1 (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm ⁻¹' e.target ∩ range ↑I ∩ ↑I ∘ ↑e.symm ∘ ↑I.symm ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I) ∈
    𝓝[↑I.symm ⁻¹' ↑e.symm ⁻¹' s ∩ range ↑I] ↑I (↑e x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ ↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm = (↑I' ∘ ↑e' ∘ ↑I'.symm) ∘ ↑I' ∘ f ∘ ↑I.symm

pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ MapsTo (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I'.symm ⁻¹' e'.source ∩ range ↑I')

Tactic:
· ext y; simp only [mfld_simps]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid 1 I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'
this : ContDiffWithinAt 𝕜 1 (↑I' ∘ ↑e' ∘ ↑I'.symm) (↑I'.symm ⁻¹' e'.source ∩ range ↑I') ((↑I' ∘ f ∘ ↑I.symm) (↑I x))
⊢ MapsTo (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I'.symm ⁻¹' e'.source ∩ range ↑I')
Module: Mathlib.Geometry.Manifold.MFDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
⊢ LiftPropWithinAt (DifferentiableWithinAtProp I I') f s x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x)

Tactic:
liftPropWithinAt_iff'

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m✝ n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
this✝ : IsManifold I (↑m) M
this : IsManifold I' (↑m) M'
t : Set M
ht : t ∈ 𝓝[insert x s] x
h't : ContMDiffOn I I' (↑m) f t
u : Set M
u_open : IsOpen[inst✝⁸] u
xu : x ∈ u
hu : insert x s ∩ u ⊆ t
⊢ ∃ u, IsOpen[inst✝⁸] u ∧ x ∈ u ∧ ContMDiffOn I I' (↑m) f (insert x s ∩ u)

Tactic:
exact ⟨u, u_open, xu, h't.mono hu⟩

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
m n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
h : ContMDiffWithinAt I I' n f s x
hm : ω ≤ n
h' : ω = ∞ → n = ω
this✝ : IsManifold I ω M
this : IsManifold I' ω M'
t : Set M
ht : t ∈ 𝓝[insert x s] x
h't : ContMDiffOn I I' ω f t
u : Set M
u_open : IsOpen[inst✝⁸] u
xu : x ∈ u
hu : insert x s ∩ u ⊆ t
⊢ ∃ u, IsOpen[inst✝⁸] u ∧ x ∈ u ∧ ContMDiffOn I I' ω f (insert x s ∩ u)

Tactic:
exact ⟨u, u_open, xu, h't.mono hu⟩

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:

Tactic:
f y = c'.symm (c' (f y)) := by rw [c'.left_inv hy₁]
      _ = c'.symm (c' (f (c.symm (c y)))) := by rw [c.left_inv hy₂]
      _ = c'.symm (e (c y)) := by rw [← he' hy₃]; rfl

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)
        (↑(extChartAt I x) x)

Tactic:
refine and_congr Iff.rfl (exists_congr fun f' => ?_)

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
x : M
f' : E →L[𝕜] E'
⊢ HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x) ↔
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)
      (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ LiftPropWithinAt (DifferentiableWithinAtProp I I') f s x ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)

Tactic:
ChartedSpace.liftPropWithinAt_iff'

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x)) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        (↑𝓘(𝕜, E') ∘
          (↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘ (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(chartAt H x).symm) ∘
            ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))

Tactic:
extChartAt,

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜 (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x)) ↔
    ContinuousWithinAt f s x ∧
      DifferentiableWithinAt 𝕜
        (↑𝓘(𝕜, E') ∘
          (↑(chartAt E' ((↑((chartAt H' (f x)).extend I') ∘ f) x)) ∘
              (↑((chartAt H' (f x)).extend I') ∘ f) ∘ ↑(chartAt H x).symm) ∘
            ↑I.symm)
        (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case mpr.refine_1
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝⁵ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H✝ M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H✝ M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H✝ := chartAt H✝ x
c' : OpenPartialHomeomorph M' H✝ := chartAt H✝ (↑f x)
hx' : ↑c x ∈ ↑c.symm ⁻¹' f.source
y : E
hy : ((↑I.symm y ∈ c.target ∧ ↑c.symm (↑I.symm y) ∈ f.source) ∧ ↑f (↑c.symm (↑I.symm y)) ∈ c'.source) ∧ y ∈ range ↑I
H : ContMDiffWithinAt I I n (↑f) (f ≫ₕ c').source (↑(extChartAt I x).symm y)
hy' : ↑(extChartAt I x).symm y ∈ c.source
hy'' : ↑f (↑(extChartAt I x).symm y) ∈ c'.source
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑((c.symm ≫ₕ f) ≫ₕ c') ∘ ↑I.symm) (↑I.symm ⁻¹' ((c.symm ≫ₕ f) ≫ₕ c').source ∩ range ↑I) y

Tactic:
rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H

Goal state after:
case mpr.refine_1
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝⁵ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H✝ M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H✝ M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H✝ := chartAt H✝ x
c' : OpenPartialHomeomorph M' H✝ := chartAt H✝ (↑f x)
hx' : ↑c x ∈ ↑c.symm ⁻¹' f.source
y : E
hy : ((↑I.symm y ∈ c.target ∧ ↑c.symm (↑I.symm y) ∈ f.source) ∧ ↑f (↑c.symm (↑I.symm y)) ∈ c'.source) ∧ y ∈ range ↑I
H :
  ContinuousWithinAt (↑f) (f ≫ₕ c').source (↑(extChartAt I x).symm y) ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I (↑f x)) ∘ ↑f ∘ ↑(extChartAt I x).symm)
      (↑(extChartAt I x).symm ⁻¹' (f ≫ₕ c').source ∩ range ↑I) (↑(extChartAt I x) (↑(extChartAt I x).symm y))
hy' : ↑(extChartAt I x).symm y ∈ c.source
hy'' : ↑f (↑(extChartAt I x).symm y) ∈ c'.source
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑((c.symm ≫ₕ f) ≫ₕ c') ∘ ↑I.symm) (↑I.symm ⁻¹' ((c.symm ≫ₕ f) ≫ₕ c').source ∩ range ↑I) y
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case mpr.refine_2
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝⁵ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H✝ M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H✝ M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H✝ := chartAt H✝ x
c' : OpenPartialHomeomorph M' H✝ := chartAt H✝ (↑f x)
hx' : ↑c x ∈ ↑c.symm ⁻¹' f.source
y : E
hy :
  (↑I.symm y ∈ c'.target ∧ ↑c'.symm (↑I.symm y) ∈ f.target ∧ ↑f.symm (↑c'.symm (↑I.symm y)) ∈ c.source) ∧ y ∈ range ↑I
H : ContMDiffWithinAt I I n (↑f.symm) (f.symm ≫ₕ c).source (↑(extChartAt I (↑f x)).symm y)
hy' : ↑(extChartAt I (↑f x)).symm y ∈ c'.source
hy'' : ↑f.symm (↑(extChartAt I (↑f x)).symm y) ∈ c.source
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑((c.symm ≫ₕ f) ≫ₕ c').symm ∘ ↑I.symm)
    (↑I.symm ⁻¹' ((c.symm ≫ₕ f) ≫ₕ c').target ∩ range ↑I) y

Tactic:
rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H

Goal state after:
case mpr.refine_2
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝⁵ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H✝ M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H✝ M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H✝ := chartAt H✝ x
c' : OpenPartialHomeomorph M' H✝ := chartAt H✝ (↑f x)
hx' : ↑c x ∈ ↑c.symm ⁻¹' f.source
y : E
hy :
  (↑I.symm y ∈ c'.target ∧ ↑c'.symm (↑I.symm y) ∈ f.target ∧ ↑f.symm (↑c'.symm (↑I.symm y)) ∈ c.source) ∧ y ∈ range ↑I
H :
  ContinuousWithinAt (↑f.symm) (f.symm ≫ₕ c).source (↑(extChartAt I (↑f x)).symm y) ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I x) ∘ ↑f.symm ∘ ↑(extChartAt I (↑f x)).symm)
      (↑(extChartAt I (↑f x)).symm ⁻¹' (f.symm ≫ₕ c).source ∩ range ↑I)
      (↑(extChartAt I (↑f x)) (↑(extChartAt I (↑f x)).symm y))
hy' : ↑(extChartAt I (↑f x)).symm y ∈ c'.source
hy'' : ↑f.symm (↑(extChartAt I (↑f x)).symm y) ∈ c.source
⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑((c.symm ≫ₕ f) ≫ₕ c').symm ∘ ↑I.symm)
    (↑I.symm ⁻¹' ((c.symm ≫ₕ f) ≫ₕ c').target ∩ range ↑I) y