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

Goal state before:
pp: 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
x : H
⊢ map (↑I.symm ∘ ↑I) (𝓝 x) = 𝓝 x

Tactic:
I.symm_comp_self,

Goal state after:
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
x : H
⊢ map id (𝓝 x) = 𝓝 x
Module: Mathlib.Geometry.Manifold.LocalInvariantProperties

Goal state before:
pp: H : Type u_1
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
Q : (H → H) → Set H → H → Prop
x : H
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.target
⊢ Q (↑e ∘ ↑e.symm) univ x

Tactic:
exact hG.congr' (e.eventually_right_inverse hx) (hQ x)

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
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝¹ : ChartedSpace H M
inst✝ : I.Boundaryless
x : M
⊢ (extChartAt I x).target ∈ 𝓝 (↑(extChartAt I x) x)

Tactic:
convert! extChartAt_target_mem_nhdsWithin x

Goal state after:
case e'_4
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝¹ : ChartedSpace H M
inst✝ : I.Boundaryless
x : M
⊢ 𝓝 (↑(extChartAt I x) x) = 𝓝[range ↑I] ↑(extChartAt I x) x
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✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
x : M
f : M → M'
x' : E
hx' : x' ∈ (extChartAt I x).target ∩ f ∘ ↑(extChartAt I x).symm ⁻¹' (extChartAt I' (f x)).source
⊢ f (↑(chartAt H x).symm (↑I.symm x')) ∈ (chartAt H' (f x)).source

Tactic:
by simpa using hx'.2

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

Goal state before:
pp: case refine_2
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
⊢ (↑(extChartAt J p.2) ∘ Prod.snd ∘ ↑(extChartAt (I.prod J) p).symm) (↑(extChartAt (I.prod J) p) p) =
    (↑(extChartAt (I.prod J) p) p).2

Tactic:
· exact (extChartAt J p.2).right_inv <| (extChartAt J p.2).map_source (mem_extChartAt_source _)

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

Goal state before:
pp: n : ℕ∞ω
E : Type u_11
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_12
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_13
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
f g : ℝ → M
s : ℝ
hf : ContMDiff 𝓘(ℝ, ℝ) I n f
hg : ContMDiff 𝓘(ℝ, ℝ) I n g
hfg : f =ᶠ[𝓝 s] g
⊢ ∀ x ∈ frontier (Iic s), f =ᶠ[𝓝 x] g

Tactic:
by simpa using hfg

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
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))

Tactic:
OpenPartialHomeomorph.extend,

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)).toPartialEquiv ≫ I'.toPartialEquiv) ∘ f) x)) ∘
              (↑((chartAt H' (f x)).toPartialEquiv ≫ I'.toPartialEquiv) ∘ 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.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : OpenPartialHomeomorph E E
hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x
g : E → F
⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ

Tactic:
have : f '' f.source = f.target := PartialEquiv.image_source_eq_target f.toPartialEquiv

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : OpenPartialHomeomorph E E
hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x
g : E → F
this : ↑f '' f.source = f.target
⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ
Module: Mathlib.Geometry.Manifold.Algebra.Structures

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
⊢ ContMDiff (𝓘(𝕜, 𝕜).prod 𝓘(𝕜, 𝕜)) 𝓘(𝕜, 𝕜) n fun p ↦ p.1 * p.2

Tactic:
by
      rw [contMDiff_iff]
      refine ⟨continuous_mul, fun x y => ?_⟩
      simp only [mfld_simps, chartAt_self_eq]
      rw [contDiffOn_univ]
      exact contDiff_mul

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

Goal state before:
pp: case pos
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
h : MDiffAt[s] f x
⊢ mfderiv[s] f x = fderivWithin 𝕜 f s x

Tactic:
simp only [mfderivWithin, h, if_pos, mfld_simps]

Goal state after:
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
⊢ mfderiv[s] f x = fderivWithin 𝕜 f s x

Tactic:
by_cases h : MDiffAt[s] f x
  · simp only [mfderivWithin, h, if_pos, mfld_simps]
  · simp only [mfderivWithin, h, if_neg, not_false_iff]
    rw [mdifferentiableWithinAt_iff_differentiableWithinAt] at h
    exact (fderivWithin_zero_of_not_differentiableWithinAt h).symm

Goal state after:
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
s t : Set M
x : M
hs : UniqueMDiffAt[s] x
ht : 𝓝[s] x ≤ 𝓝[t] x
⊢ 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x ≤
    𝓝[↑(extChartAt I x).symm ⁻¹' t ∩ range ↑I] ↑(extChartAt I x) x

Tactic:
by simpa only [← map_extChartAt_nhdsWithin] using Filter.map_mono ht

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'
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))

Tactic:
OpenPartialHomeomorph.extend,

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)).toPartialEquiv ≫ I'.toPartialEquiv) ∘ f) x)) ∘
              (↑((chartAt H' (f x)).toPartialEquiv ≫ I'.toPartialEquiv) ∘ 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✝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
⊢ MDiffAt[s] f x ↔ ContinuousWithinAt f s x ∧ MDiffAt[s] (↑(extChartAt I' y) ∘ f) x

Tactic:
by
  simp_rw [MDifferentiableWithinAt]
  rw [differentiableWithinAt_localInvariantProp.liftPropWithinAt_indep_chart_target
      (chart_mem_maximalAtlas y) hy, and_congr_right]
  intro hf
  simp_rw [StructureGroupoid.liftPropWithinAt_self_target]
  simp_rw [((chartAt H' y).continuousAt hy).comp_continuousWithinAt hf]
  rw [← extChartAt_source I'] at hy
  simp_rw [(continuousAt_extChartAt' hy).comp_continuousWithinAt hf]
  rfl

Goal state after:
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✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e'.baseSet ∩ e.baseSet
⊢ ↑(coordChangeL R e e' b).symm = ↑(coordChangeL R e' e b)

Tactic:
coe_coordChangeL' e' e hb,

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e'.baseSet ∩ e.baseSet
⊢ ↑(coordChangeL R e e' b).symm = (linearEquivAt R e' b ⋯).symm ≪≫ₗ linearEquivAt R e b ⋯
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 : NontriviallyNormedField R
inst✝⁷ : (x : B) → AddCommMonoid (E x)
inst✝⁶ : (x : B) → Module R (E x)
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace R F
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (x : B) → TopologicalSpace (E x)
inst✝ : FiberBundle F E
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
i j : ι
⊢ ∀ (e : Trivialization F TotalSpace.proj) [MemTrivializationAtlas e], Trivialization.IsLinear R e

Tactic:
rintro _ ⟨i, rfl⟩
    apply localTriv.isLinear

Goal state after:
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).prod 𝓘(𝕜, F)) n
      (fun x ↦
        ↑((extChartAt IB (f x₀).proj).prod (PartialEquiv.refl F))
          (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)))
      s x₀ ↔
    ContMDiffWithinAt IM 𝓘(𝕜, EB) n (fun x ↦ ↑(extChartAt IB (f x₀).proj) (f x).proj) s x₀ ∧
      ContMDiffWithinAt IM 𝓘(𝕜, F) n
        (fun x ↦
          ↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2)
            (↑(trivializationAt F E (f x₀).proj) (f x)).2)
        s x₀

Tactic:
PartialEquiv.prod_coe,

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
⊢ ContMDiffWithinAt IM (𝓘(𝕜, EB).prod 𝓘(𝕜, F)) n
      (fun x ↦
        (↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1,
          ↑(PartialEquiv.refl F) (↑(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 ∧
      ContMDiffWithinAt IM 𝓘(𝕜, F) n
        (fun x ↦
          ↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2)
            (↑(trivializationAt F E (f x₀).proj) (f x)).2)
        s x₀
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

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'
s : Set (M × M')
p : TangentBundle (I.prod I') (M × M')
hs : UniqueMDiffAt[s] p.proj
⊢ ⟨p.proj.1, (mfderiv[s] Prod.fst p.proj) p.snd⟩ = ⟨p.proj.1, p.snd.1⟩

Tactic:
rw [mfderivWithin_fst]

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'
s : Set (M × M')
p : TangentBundle (I.prod I') (M × M')
hs : UniqueMDiffAt[s] p.proj
⊢ ⟨p.proj.1, (ContinuousLinearMap.fst 𝕜 (TangentSpace I p.proj.1) (TangentSpace I' p.proj.2)) p.snd⟩ =
    ⟨p.proj.1, p.snd.1⟩

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'
s : Set (M × M')
p : TangentBundle (I.prod I') (M × M')
hs : UniqueMDiffAt[s] p.proj
⊢ UniqueMDiffAt[s] p.proj
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
x : M
s : Set M
g : M' → M''
u : Set M'
hg : MDiffAt[u] g (f x)
hf : MDiffAt[s] f x
h : s ⊆ f ⁻¹' u
⊢ MDiffAt[s] (g ∘ f) x

Tactic:
rcases hf.2 with ⟨f', hf'⟩
  have F : HasMFDerivAt[s] f x f' := ⟨hf.1, hf'⟩
  rcases hg.2 with ⟨g', hg'⟩
  have G : HasMFDerivAt[u] g (f x) g' := ⟨hg.1, hg'⟩
  exact (HasMFDerivWithinAt.comp x G F h).mdifferentiableWithinAt

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

Goal state before:
pp: case left
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
s : Set M
F : Type u_9
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
Z : M → Type u_10
inst✝² : TopologicalSpace (TotalSpace F Z)
inst✝¹ : (b : M) → TopologicalSpace (Z b)
inst✝ : FiberBundle F Z
x : M
v : Z x
hs : UniqueMDiffAt[s] ⟨x, v⟩.proj
h's : s ⊆ (trivializationAt F Z ⟨x, v⟩.proj).baseSet
z : E
w : F
hz : ↑(chartAt H x).symm (↑I.symm z) ∈ s ∧ ∃ y, ↑I y = z
⊢ (↑(trivializationAt F Z x).symm
        (↑((chartAt H x).prod (OpenPartialHomeomorph.refl F)).symm
          (↑(I.symm.prod (PartialEquiv.refl F)) (z, w)))).proj ∈
    s

Tactic:
PartialEquiv.prod_coe,

Goal state after:
case left
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
s : Set M
F : Type u_9
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
Z : M → Type u_10
inst✝² : TopologicalSpace (TotalSpace F Z)
inst✝¹ : (b : M) → TopologicalSpace (Z b)
inst✝ : FiberBundle F Z
x : M
v : Z x
hs : UniqueMDiffAt[s] ⟨x, v⟩.proj
h's : s ⊆ (trivializationAt F Z ⟨x, v⟩.proj).baseSet
z : E
w : F
hz : ↑(chartAt H x).symm (↑I.symm z) ∈ s ∧ ∃ y, ↑I y = z
⊢ (↑(trivializationAt F Z x).symm
        (↑((chartAt H x).prod (OpenPartialHomeomorph.refl F)).symm
          ((fun p ↦ (↑I.symm p.1, ↑(PartialEquiv.refl F) p.2)) (z, w)))).proj ∈
    s
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ support ((chartAt H c).source.indicator (↑f.toContDiffBump ∘ ↑(extChartAt I c))) =
    (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut

Tactic:
support_indicator,

Goal state after:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ (chartAt H c).source ∩ support (↑f.toContDiffBump ∘ ↑(extChartAt I c)) =
    (chartAt H c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut