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✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
h'x : ↑(f.extend I) x ∈ interior (range ↑I)
s : Set M
h : s ∈ 𝓝 x
⊢ ↑(f.extend I) '' s ∈ 𝓝 (↑(f.extend I) x)

Tactic:
rw [← f.map_extend_nhds_of_mem_interior_range hx h'x, Filter.mem_map]

Goal state after:
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
x : M
hx : x ∈ f.source
h'x : ↑(f.extend I) x ∈ interior (range ↑I)
s : Set M
h : s ∈ 𝓝 x
⊢ ↑(f.extend I) ⁻¹' ↑(f.extend I) '' s ∈ 𝓝 x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ P (g ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔ P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

Tactic:
by
  rw [← hG.right_invariance (compatible_of_mem_maximalAtlas_right (x := x) he)]; swap
  · simp [xe]
  simp only [OpenPartialHomeomorph.trans_apply, mem_chart_source, OpenPartialHomeomorph.left_inv]
  apply hG.congr_set_fun
  · refine (eventually_of_mem ?_ fun y (hy : y ∈ e.symm ⁻¹' (chartAt H x).source) ↦ ?_).set_eq
    · refine (e.symm.continuousAt <| e.mapsTo xe).preimage_mem_nhds
        ((chartAt H x).open_source.mem_nhds ?_)
      simp_rw [e.left_inv xe, mem_chart_source H x]
    simp_rw [mem_preimage, OpenPartialHomeomorph.coe_trans_symm, OpenPartialHomeomorph.symm_symm,
      Function.comp_apply, (chartAt H x).left_inv hy]
  · refine ((e.eventually_nhds' _ xe).mpr <| (chartAt H x).eventually_left_inverse
      (mem_chart_source H x)).mono fun y hy ↦ ?_
    simp [hy]

Goal state after:
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
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f : OpenPartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s : Set M
f' : OpenPartialHomeomorph M' H'
g : M → M'
hs : s ⊆ f.source
hmaps : MapsTo g s f'.source
⊢ ContinuousOn (↑(f'.extend I') ∘ g ∘ ↑(f.extend I).symm) (↑(f.extend I) '' s) ↔ ContinuousOn g s

Tactic:
by
  refine forall_mem_image.trans <| forall₂_congr fun x hx ↦ ?_
  refine (continuousWithinAt_congr_set ?_).trans
    (continuousWithinAt_writtenInExtend_iff _ (hs hx) (hmaps hx) hmaps)
  rw [← nhdsWithin_eq_iff_eventuallyEq, ← map_extend_nhdsWithin_eq_image_of_subset,
    ← map_extend_nhdsWithin]
  exacts [hs hx, hs hx, hs]

Goal state after:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
  ∀ (A : E →L[ℝ] E),
    0 < δ A ∧
      (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
        ∀ (t : Set E) (g : E → E),
          ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
⊢ s = s ∩ ⋃ i, t i

Tactic:
exact Subset.antisymm (subset_inter Subset.rfl t_cover) inter_subset_left

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

Goal state before:
pp: case right
n : ℕ∞ω
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
f : OpenPartialHomeomorph H H
hf : f.source = ∅
⊢ (contDiffPregroupoid n I).property (↑f.symm) f.target

Tactic:
· intro x ⟨hx, _⟩
    have : f.target = ∅ := by simp [← f.image_source_eq_target, hf]
    simp_all

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.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'
n m : ℕ∞ω
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑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 n (↑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 [mem_preimage, I.left_inv, e.mapsTo hx]

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 : M → M'
s : Set M
x : M
n : ℕ∞ω
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        (↑(extChartAt I x) x)

Tactic:
simp only [ContMDiffWithinAt, 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
n : ℕ∞ω
⊢ ContinuousWithinAt f s x ∧
      ContDiffWithinAtProp I I' n (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
        (↑(chartAt H x) x) ↔
    ContinuousWithinAt f s x ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
        (↑(extChartAt I x) 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'
⊢ s ⊆ f ⁻¹' e'.source →
    f x ∈ e'.source → DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (↑e' ∘ f) s x

Tactic:
hs

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
⊢ f x ∈ e'.source → DifferentiableWithinAtProp I I' f s x → DifferentiableWithinAtProp I I' (↑e' ∘ f) s 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'
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
f : M → M'
s : Set M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I' n M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ ContMDiffOn I I' n f s ↔
    ∀ x ∈ s,
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)

Tactic:
ContMDiffOn

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'
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
f : M → M'
s : Set M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I' n M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ (∀ x ∈ s, ContMDiffWithinAt I I' n f s x) ↔
    ∀ x ∈ s,
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) 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
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
⊢ ContMDiffOn I I' n f s ↔
    ContinuousOn f s ∧
      ∀ (y : M'), ContMDiffOn I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) (s ∩ f ⁻¹' (extChartAt I' y).source)

Tactic:
by
  simp only [contMDiffOn_iff, ModelWithCorners.source_eq, chartAt_self_eq,
    OpenPartialHomeomorph.refl_partialEquiv, PartialEquiv.refl_trans, extChartAt,
    OpenPartialHomeomorph.extend, Set.preimage_univ, Set.inter_univ, and_congr_right_iff]
  intro h
  constructor
  · refine fun h' y => ⟨?_, fun x _ => h' x y⟩
    have h'' : ContinuousOn _ univ := (ModelWithCorners.continuous I').continuousOn
    convert! (h''.comp_inter (chartAt H' y).continuousOn_toFun).comp_inter h
    simp
  · exact fun h' x y => (h' y).2 x 0

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
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ContMDiffOn I I' n f u

Tactic:
rw [insert_eq_of_mem hxs]

Goal state after:
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
n : ℕ∞ω
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
s : Set M
hxs : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔ ∃ u ∈ 𝓝[s] x, ContMDiffOn I I' n f u
Module: Mathlib.Geometry.Manifold.MFDeriv.FDeriv

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_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
f' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)
⊢ HasMFDerivAt[s] f x f' ↔ HasFDerivWithinAt f f' s x

Tactic:
by
  simpa only [HasMFDerivWithinAt, and_iff_right_iff_imp, mfld_simps] using
    HasFDerivWithinAt.continuousWithinAt

Goal state after:
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)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
p : ↑e.source
q : B × F
⊢ q.1 ∈ u p → q ∈ e.source

Tactic:
hq

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)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
p : ↑e.source
q : B × F
hq : q.1 ∈ u p
⊢ q ∈ e.source
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
H₁ : ContMDiffOn I I n (↑c'.symm ∘ ↑e ∘ ↑c) s
⊢ ContMDiffOn I I n (↑f) (f.source ∩ s)

Tactic:
have H₂ : EqOn f (c'.symm ∘ e ∘ c) s := by
    intro y hy
    simp only [s, mfld_simps] at hy
    have hy₁ : f y ∈ c'.source := by simp only [hy, mfld_simps]
    have hy₂ : y ∈ c.source := by simp only [hy, mfld_simps]
    have hy₃ : c y ∈ c.symm ⁻¹' f.source ∩ e.source := by simp only [hy, mfld_simps]
    calc
      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:
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
H₂ : EqOn (↑f) (↑c'.symm ∘ ↑e ∘ ↑c) s
⊢ ContMDiffOn I I n (↑f) (f.source ∩ s)
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'
x : M
y : M'
s : Set M
t : Set M'
hs : UniqueMDiffAt[s] x
ht : UniqueMDiffAt[t] y
⊢ UniqueMDiffAt[s ×ˢ t] (x, y)

Tactic:
by
  refine (hs.prod ht).mono ?_
  rw [ModelWithCorners.range_prod, ← prod_inter_prod]
  rfl

Goal state after:
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u_1
inst✝⁶ : TopologicalSpace B
F₁ : Type u_2
inst✝⁵ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝³ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : (x : B) → Zero (E₁ x)
inst✝ : (x : B) → Zero (E₂ x)
x : B × F₁ × F₂
h : x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ
⊢ toFun' e₁ e₂ (invFun' e₁ e₂ x) = x

Tactic:
by
  obtain ⟨x, w₁, w₂⟩ := x
  obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h
  simp [Prod.toFun', Prod.invFun', h₁, h₂]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u_1
inst✝¹⁰ : TopologicalSpace B
F₁ : Type u_2
inst✝⁹ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁸ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝⁷ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → Zero (E₁ x)
inst✝⁴ : (x : B) → Zero (E₂ x)
inst✝³ : (x : B) → TopologicalSpace (E₁ x)
inst✝² : (x : B) → TopologicalSpace (E₂ x)
inst✝¹ : FiberBundle F₁ E₁
inst✝ : FiberBundle F₂ E₂
b : B
⊢ IsInducing (TotalSpace.mk b)

Tactic:
← (Prod.isInducing_diag F₁ E₁ F₂ E₂).of_comp_iff

Goal state after:
B : Type u_1
inst✝¹⁰ : TopologicalSpace B
F₁ : Type u_2
inst✝⁹ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁸ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝⁷ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → Zero (E₁ x)
inst✝⁴ : (x : B) → Zero (E₂ x)
inst✝³ : (x : B) → TopologicalSpace (E₁ x)
inst✝² : (x : B) → TopologicalSpace (E₂ x)
inst✝¹ : FiberBundle F₁ E₁
inst✝ : FiberBundle F₂ E₂
b : B
⊢ IsInducing ((fun p ↦ ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })) ∘ TotalSpace.mk b)
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁵ : NormedAddCommGroup EB
inst✝⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
x : TotalSpace F E
⊢ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target =
    ((extChartAt IB x.proj).target ∩ ↑(extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ

Tactic:
FiberBundle.extChartAt,

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁵ : NormedAddCommGroup EB
inst✝⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
x : TotalSpace F E
⊢ ((trivializationAt F E x.proj).toPartialEquiv ≫ (extChartAt IB x.proj).prod (PartialEquiv.refl F)).target =
    ((extChartAt IB x.proj).target ∩ ↑(extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ