mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝ : ChartedSpace H M
s : Set E
x x' : M
hx : x' ∈ (extChartAt I x).source
hs : s ∈ 𝓝[range ↑I] ↑(extChartAt I x) x'
⊢ x' ∈ (chartAt H x).source

Tactic:
simpa using hx

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

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
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : WithTop ℕ∞
s : Set (M × N)
p : M × N
⊢ ContMDiffWithinAt (I.prod J) I n Prod.fst s p

Tactic:
rw [contMDiffWithinAt_iff']
  refine ⟨continuousWithinAt_fst, contDiffWithinAt_fst.congr (fun y hy => ?_) ?_⟩
  · exact (extChartAt I p.1).right_inv ⟨hy.1.1.1, hy.1.2.1⟩
  · exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
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'
f : E → H'
s : Set E
x : E
⊢ DifferentiableWithinAtProp 𝓘(𝕜, E) I' f s x ↔ DifferentiableWithinAt 𝕜 (↑I' ∘ f) s x

Tactic:
DifferentiableWithinAtProp,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
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'
f : E → H'
s : Set E
x : E
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E).symm ⁻¹' s ∩ range ↑𝓘(𝕜, E)) (↑𝓘(𝕜, E) x) ↔
    DifferentiableWithinAt 𝕜 (↑I' ∘ f) s 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
u : Set H
f : H → H'
u_open : IsOpen[inst✝³] u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAtProp I I' f s x ↔ DifferentiableWithinAtProp I I' f (s ∩ u) x

Tactic:
DifferentiableWithinAtProp,

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
u : Set H
f : H → H'
u_open : IsOpen[inst✝³] u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x) ↔
    DifferentiableWithinAtProp I I' f (s ∩ u) 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
u : Set H
f : H → H'
u_open : IsOpen[inst✝³] u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x) ↔
    DifferentiableWithinAtProp I I' f (s ∩ u) x

Tactic:
DifferentiableWithinAtProp,

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
u : Set H
f : H → H'
u_open : IsOpen[inst✝³] u
xu : x ∈ u
this : ↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I = ↑I.symm ⁻¹' s ∩ range ↑I ∩ ↑I.symm ⁻¹' u
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x) ↔
    DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' (s ∩ u) ∩ range ↑I) (↑I x)
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
h : ContinuousOn f s
x : M
y : M'
⊢ ContinuousOn (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))

Tactic:
apply (continuousOn_extChartAt _).comp

Goal state after:
case hf
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
h : ContinuousOn f s
x : M
y : M'
⊢ ContinuousOn (f ∘ ↑(extChartAt I x).symm)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))

case h
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
h : ContinuousOn f s
x : M
y : M'
⊢ MapsTo (f ∘ ↑(extChartAt I x).symm)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) (extChartAt I' y).source
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: 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'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
h2e : e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source
hef : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) e.source
hex : ↑c x ∈ e.source
h2X : ↑c' X = ↑e (↑c (↑f.symm X))
h1 : EqOn (↑(c.symm ≫ₕ f ≫ₕ c').symm) (↑e.symm) (e.target ∩ e.target)
h2 : e.target ⊆ (c.symm ≫ₕ f ≫ₕ c').target
⊢ EqOn (↑c ∘ ↑f.symm ∘ ↑c'.symm) (↑e.symm) (↑c'.symm ⁻¹' f.target ∩ e.target)

Tactic:
rw [inter_self] at h1

Goal state after:
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'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
h2e : e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source
hef : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) e.source
hex : ↑c x ∈ e.source
h2X : ↑c' X = ↑e (↑c (↑f.symm X))
h1 : EqOn (↑(c.symm ≫ₕ f ≫ₕ c').symm) (↑e.symm) e.target
h2 : e.target ⊆ (c.symm ≫ₕ f ≫ₕ c').target
⊢ EqOn (↑c ∘ ↑f.symm ∘ ↑c'.symm) (↑e.symm) (↑c'.symm ⁻¹' f.target ∩ e.target)
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 ∧
      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 ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E')
        (↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘ (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(chartAt H x).symm)
        (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)

Tactic:
DifferentiableWithinAtProp,

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' ((↑(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))
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
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I 1 M
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
⊢ ContinuousWithinAt (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧
      DifferentiableWithinAtProp I I' (↑(chartAt H' ((f ∘ ↑e.symm) (↑e x))) ∘ f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔
    ContinuousWithinAt (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧
      DifferentiableWithinAt 𝕜
        (↑I' ∘ ↑(chartAt H' ((f ∘ ↑(e.extend I).symm) (↑(e.extend I) x))) ∘ f ∘ ↑(e.extend I).symm)
        (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x)

Tactic:
DifferentiableWithinAtProp,

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
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I 1 M
hx : x ∈ e.source
h2x : x ∈ (e.extend I).source
⊢ ContinuousWithinAt (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧
      DifferentiableWithinAt 𝕜 (↑I' ∘ (↑(chartAt H' ((f ∘ ↑e.symm) (↑e x))) ∘ f ∘ ↑e.symm) ∘ ↑I.symm)
        (↑I.symm ⁻¹' ↑e.symm ⁻¹' s ∩ range ↑I) (↑I (↑e x)) ↔
    ContinuousWithinAt (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ∧
      DifferentiableWithinAt 𝕜
        (↑I' ∘ ↑(chartAt H' ((f ∘ ↑(e.extend I).symm) (↑(e.extend I) x))) ∘ f ∘ ↑(e.extend I).symm)
        (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x)
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
⊢ ∃ Φ U,
    ∃ (hU₀ : IsOpen[inst✝⁷] U) (hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(Φ x)) U) (h2Φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(Φ x).symm) U),
      e.EqOnSource (FiberwiseLinear.openPartialHomeomorph Φ hU₀ ⋯ ⋯)

Tactic:
have hΦφ : ∀ x : U, ∀ y ∈ u x, Φ y = φ x y := by
    intro x y hyu
    refine (hΦ y (hUu x hyu)).trans ?_
    exact iUnionLift_mk ⟨y, hyu⟩ _

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
⊢ ∃ Φ U,
    ∃ (hU₀ : IsOpen[inst✝⁷] U) (hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(Φ x)) U) (h2Φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(Φ x).symm) U),
      e.EqOnSource (FiberwiseLinear.openPartialHomeomorph Φ hU₀ ⋯ ⋯)
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁸ : Semiring R
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
⊢ ⇑(Trivialization.symmₗ R e b) = e.symm b

Tactic:
ext y; exact e.toPretrivialization.symmₗ_apply R hb y

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 t : Set M
p : TangentBundle I M
st : s ⊆ t
hs : UniqueMDiffAt[s] p.proj
h : MDiffAt[t] f p.proj
⊢ tangentMapWithin I I' f s p = tangentMapWithin I I' f t p

Tactic:
by
  simp only [tangentMapWithin, mfld_simps]
  rw [mfderivWithin_subset st hs h]

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

Goal state before:
pp: case neg.hnc
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 f₁ : M → M'
x : M
s : Set M
hL : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
h : ¬MDiffAt[s] f x
⊢ ¬MDiffAt[s] f₁ x

Tactic:
rwa [← hL.mdifferentiableWithinAt_iff hx]

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

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁹ : NormedAddCommGroup EB
inst✝¹⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁶ : TopologicalSpace B
inst✝¹⁵ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
inst✝ : IsManifold IB n B
e : Trivialization F TotalSpace.proj
he : e ∈ trivializationAtlas F E
e' : Trivialization F TotalSpace.proj
he' : e' ∈ trivializationAtlas F E
⊢ ((fun e ↦ e.toOpenPartialHomeomorph) e).symm ≫ₕ (fun e ↦ e.toOpenPartialHomeomorph) e' ∈
    contMDiffFiberwiseLinear B F IB n

Tactic:
have : MemTrivializationAtlas e := ⟨he⟩

Goal state after:
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁹ : NormedAddCommGroup EB
inst✝¹⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁶ : TopologicalSpace B
inst✝¹⁵ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
inst✝ : IsManifold IB n B
e : Trivialization F TotalSpace.proj
he : e ∈ trivializationAtlas F E
e' : Trivialization F TotalSpace.proj
he' : e' ∈ trivializationAtlas F E
this : MemTrivializationAtlas e
⊢ ((fun e ↦ e.toOpenPartialHomeomorph) e).symm ≫ₕ (fun e ↦ e.toOpenPartialHomeomorph) e' ∈
    contMDiffFiberwiseLinear B F IB n
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

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
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
U : UniqueMDiffAt[range ↑I] y
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
⊢ ↑(extChartAt I x).symm y ∈ (chartAt H x).source

Tactic:
rwa [← extChartAt_source (I := I)]

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

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁸ : NormedAddCommGroup EB
inst✝¹⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁵ : TopologicalSpace B
inst✝¹⁴ : ChartedSpace HB B
EM : Type u_9
inst✝¹³ : NormedAddCommGroup EM
inst✝¹² : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹¹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace HM M
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module 𝕜 (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
inst✝¹ : VectorBundle 𝕜 F E
inst✝ : ContMDiffVectorBundle n F E IB
⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],
    ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)

Tactic:
intro e e' he he'
    obtain rfl := Bundle.Trivial.eq_trivialization B F e
    obtain rfl := Bundle.Trivial.eq_trivialization B F e'
    simp_rw [Bundle.Trivial.trivialization.coordChangeL]
    exact contMDiff_const.contMDiffOn

Goal state after: