mathlib-initiative/mathlib-tactics huggingface dataset:
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
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 (↑e x) ∈ ↑I.symm ⁻¹' e.target

Tactic:
simp_rw [Set.mem_preimage, I.left_inv, e.mapsTo hx]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
x : E
⊢ ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f x ↔ ContDiffAt 𝕜 n f x

Tactic:
by
  rw [← contMDiffWithinAt_univ, contMDiffWithinAt_iff_contDiffWithinAt, contDiffWithinAt_univ]

Goal state after:
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 f₁ : M → M'
n : ℕ∞ω
h : ContMDiff I I' n f
h₁ : ∀ (y : M), f₁ y = f y
⊢ ContMDiff I I' n f₁

Tactic:
by
  rw [← contMDiffOn_univ] at h ⊢
  exact (contMDiffOn_congr fun y _ ↦ h₁ y).mpr h

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

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
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
⊢ e.target ⊆ (fun x ↦ x) ⁻¹' e.target

Tactic:
preimage_id'

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
e : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas I n M
⊢ e.target ⊆ e.target
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case 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'
hf : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H (↑f 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)
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
he' : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) (↑c.symm ⁻¹' f.source ∩ e.source)
hex : ↑c x ∈ e.source
s : Set M := (f ≫ₕ c').source ∩ ((c ≫ₕ e) ≫ₕ c'.symm).source
⊢ ContMDiffOn I I n (↑f) (f.source ∩ s)

Tactic:
have H₁ : ContMDiffOn I I n (c'.symm ∘ e ∘ c) s := by
    have hc' : ContMDiffOn I I n c'.symm _ := contMDiffOn_chart_symm
    have he'' : ContMDiffOn I I n e _ := contMDiffOn_of_mem_contDiffGroupoid he
    have hc : ContMDiffOn I I n c _ := contMDiffOn_chart
    refine (hc'.comp' (he''.comp' hc)).mono ?_
    dsimp [s, c, c']
    mfld_set_tac

Goal state after:
case 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'
hf : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
x : M
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H (↑f 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)
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
he' : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) (↑c.symm ⁻¹' f.source ∩ e.source)
hex : ↑c x ∈ e.source
s : Set M := (f ≫ₕ c').source ∩ ((c ≫ₕ e) ≫ₕ c'.symm).source
H₁ : ContMDiffOn I I n (↑c'.symm ∘ ↑e ∘ ↑c) s
⊢ ContMDiffOn I I n (↑f) (f.source ∩ s)
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))
⊢ EqOn (↑(c.symm ≫ₕ f ≫ₕ c').symm) (↑e.symm) (e.target ∩ e.target)

Tactic:
apply EqOn.symm

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 := ⋯
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := ⋯
c' : OpenPartialHomeomorph M' H := ⋯
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))
⊢ EqOn (↑e.symm) (↑(c.symm ≫ₕ f ≫ₕ c').symm) (e.target ∩ e.target)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
⊢ ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I I') (↑(chartAt H' y) ∘ f) s x ↔
    ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I 𝓘(𝕜, E')) (↑(extChartAt I' y) ∘ f) s x

Tactic:
and_congr_right

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
⊢ ContinuousWithinAt f s x ∧ ?m.116 ↔
    ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I 𝓘(𝕜, E')) (↑(extChartAt I' y) ∘ f) s x

𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
⊢ ContinuousWithinAt f s x → (LiftPropWithinAt (DifferentiableWithinAtProp I I') (↑(chartAt H' y) ∘ f) s x ↔ ?m.116)

𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
⊢ Prop
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
hf : ContinuousWithinAt f s x
⊢ ContinuousWithinAt (↑(chartAt H' y) ∘ f) s x ∧
      DifferentiableWithinAtProp I I' ((↑(chartAt H' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑(extChartAt I' y) ∘ f) s x ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E') ((↑(extChartAt I' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x)

Tactic:
((chartAt H' y).continuousAt hy).comp_continuousWithinAt hf

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
hf : ContinuousWithinAt f s x
⊢ True ∧
      DifferentiableWithinAtProp I I' ((↑(chartAt H' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt (↑(extChartAt I' y) ∘ f) s x ∧
      DifferentiableWithinAtProp I 𝓘(𝕜, E') ((↑(extChartAt I' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x)
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u_1
inst✝⁴ : TopologicalSpace B
F₁ : Type u_2
inst✝³ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝² : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝¹ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝ : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
f₁ : (TotalSpace (F₁ × F₂) fun x ↦ E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=
  fun p ↦ ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })
f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ (↑e₁ p.1, ↑e₂ p.2)
f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ (p.1.1, p.1.2, p.2.2)
⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))

Tactic:
have hf₁ : Continuous f₁ := (Prod.isInducing_diag F₁ E₁ F₂ E₂).continuous

Goal state after:
B : Type u_1
inst✝⁴ : TopologicalSpace B
F₁ : Type u_2
inst✝³ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝² : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝¹ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝ : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
f₁ : (TotalSpace (F₁ × F₂) fun x ↦ E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=
  fun p ↦ ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })
f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ (↑e₁ p.1, ↑e₂ p.2)
f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ (p.1.1, p.1.2, p.2.2)
hf₁ : Continuous[Prod.topologicalSpace F₁ E₁ F₂ E₂, instTopologicalSpaceProd] f₁
⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: case refine_2
B : Type u_1
inst✝⁴ : TopologicalSpace B
F₁ : Type u_2
inst✝³ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝² : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝¹ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝ : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
f₁ : (TotalSpace (F₁ × F₂) fun x ↦ E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=
  fun p ↦ ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })
f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ (↑e₁ p.1, ↑e₂ p.2)
f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ (p.1.1, p.1.2, p.2.2)
hf₁ : Continuous[Prod.topologicalSpace F₁ E₁ F₂ E₂, instTopologicalSpaceProd] f₁
hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)
hf₃ : Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] f₃
b : B
v₁ : E₁ b
v₂ : E₂ b
hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet
right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet
⊢ b = (↑e₁ { proj := b, snd := v₁ }).1

Tactic:
rw [e₁.coe_fst]

Goal state after:
case refine_2
B : Type u_1
inst✝⁴ : TopologicalSpace B
F₁ : Type u_2
inst✝³ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝² : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝¹ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝ : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
f₁ : (TotalSpace (F₁ × F₂) fun x ↦ E₁ x × E₂ x) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=
  fun p ↦ ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })
f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ (↑e₁ p.1, ↑e₂ p.2)
f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ (p.1.1, p.1.2, p.2.2)
hf₁ : Continuous[Prod.topologicalSpace F₁ E₁ F₂ E₂, instTopologicalSpaceProd] f₁
hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)
hf₃ : Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] f₃
b : B
v₁ : E₁ b
v₂ : E₂ b
hb₁ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₁.baseSet
right✝ : { proj := b, snd := (v₁, v₂) }.proj ∈ e₂.baseSet
⊢ { proj := b, snd := v₁ } ∈ e₁.source
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
⊢ IsOpen[Pullback.TotalSpace.topologicalSpace F E ⇑f] (Pullback.lift ⇑f ⁻¹' e.source)

Tactic:
simp_rw [e.source_eq, ← preimage_comp]

Goal state after:
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
⊢ IsOpen[Pullback.TotalSpace.topologicalSpace F E ⇑f] (TotalSpace.proj ∘ Pullback.lift ⇑f ⁻¹' e.baseSet)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

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
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ ↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I =ᶠ[𝓝[≠] ↑(extChartAt I x) x] ↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I

Tactic:
by
  have : T1Space M := I.t1Space M
  obtain ⟨u, u_mem, hu⟩ : ∃ u ∈ 𝓝 x, u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)} :=
    mem_nhdsWithin_iff_exists_mem_nhds_inter.1 (nhdsWithin_compl_singleton_le x y h)
  rw [← extChartAt_to_inv (I := I) x] at u_mem
  have B : (extChartAt I x).target ∪ (range I)ᶜ ∈ 𝓝 (extChartAt I x x) := by
    rw [← nhdsWithin_univ, ← union_compl_self (range I), nhdsWithin_union]
    apply Filter.union_mem_sup (extChartAt_target_mem_nhdsWithin x) self_mem_nhdsWithin
  apply mem_nhdsWithin_iff_exists_mem_nhds_inter.2
    ⟨_, Filter.inter_mem ((continuousAt_extChartAt_symm x).preimage_mem_nhds u_mem) B, ?_⟩
  rintro z ⟨hz, h'z⟩
  simp only [eq_iff_iff, mem_ofPred_eq]
  change z ∈ (extChartAt I x).symm ⁻¹' s ∩ range I ↔ z ∈ (extChartAt I x).symm ⁻¹' t ∩ range I
  by_cases hIz : z ∈ range I
  · simp only [mem_inter_iff, mem_preimage, mem_union, mem_compl_iff, hIz, not_true_eq_false,
      or_false, and_true] at hz ⊢
    rw [← eq_iff_iff]
    apply hu ⟨hz.1, ?_⟩
    push _ ∈ _ at h'z ⊢
    rw [eq_comm, (extChartAt I x).eq_symm_apply (by simp) hz.2]
    exact Ne.symm h'z
  · simp [hIz]

Goal state after:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e e' : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
he' : e' ∈ a.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (b, (a.coordChange he he' b) v).1 = (↑e' { proj := b, snd := e.symm b v }).1

Tactic:
e.mk_symm hb.1 v,

Goal state after:
case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e e' : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
he' : e' ∈ a.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (b, (a.coordChange he he' b) v).1 = (↑e' (↑e.symm (b, v))).1
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: n : ℕ∞ω
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
⊢ ContMDiffWithinAt IM 𝓘(𝕜, EB) n
      (Prod.fst ∘ 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 ↔
    ContMDiffWithinAt IM 𝓘(𝕜, EB) n (fun x ↦ ↑(extChartAt IB (f x₀).proj) (f x).proj) s x₀

Tactic:
have h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀ :=
    ((FiberBundle.continuous_proj F E).continuousWithinAt.comp hf (mapsTo_image f s))
      ((Trivialization.open_baseSet _).mem_nhds (mem_baseSet_trivializationAt F E _))

Goal state after:
n : ℕ∞ω
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
⊢ ContMDiffWithinAt IM 𝓘(𝕜, EB) n
      (Prod.fst ∘ 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 ↔
    ContMDiffWithinAt IM 𝓘(𝕜, EB) n (fun x ↦ ↑(extChartAt IB (f x₀).proj) (f x).proj) s x₀