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

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'
F' : Type u_11
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
G' : Type u_12
inst✝² : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_13
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : WithTop ℕ∞
f : M → M' × N'
h : ContMDiff I I' n (Prod.fst ∘ f) ∧ ContMDiff I J' n (Prod.snd ∘ f)
⊢ ContMDiff I (I'.prod J') n f

Tactic:
convert! h.1.prodMk h.2

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
n : WithTop ℕ∞
f : M → F₁ × F₂
⊢ ContMDiff I 𝓘(𝕜, F₁ × F₂) n f ↔ ContMDiff I 𝓘(𝕜, F₁) n (Prod.fst ∘ f) ∧ ContMDiff I 𝓘(𝕜, F₂) n (Prod.snd ∘ f)

Tactic:
rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod]
  exact contMDiff_prod_iff f

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.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
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
inst✝¹ : ChartedSpace H' M'
f : M → M'
n : ℕ∞ω
inst✝ : Subsingleton M'
⊢ ContMDiff I I' n f

Tactic:
intro x
  rw [Subsingleton.elim f fun _ => (f x)]
  exact contMDiffAt_const

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
⊢ (↑(extChartAt I (Sum.inl x)) ∘ Sum.inl ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) = id (↑(extChartAt I x) x)

pp: case h₁
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
⊢ ↑(extChartAt I (Sum.inl x)) ∘ Sum.inl ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id

Tactic:
· simp [ChartedSpace.sum_chartAt_inl, Sum.inl_injective.extend_apply (chartAt _ x)]

Goal state after:
case h₁
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
n : WithTop ℕ∞
x : M
⊢ ↑(extChartAt I (Sum.inl x)) ∘ Sum.inl ∘ ↑(extChartAt I x).symm =ᶠ[𝓝[range ↑I] ↑(extChartAt I x) x] id
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ

Tactic:
conv_lhs => rw [A]

Goal state after:
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
u : ℕ → Set E := fun n ↦ disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in ⋃ n, s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ =
    ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ
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 ≠ ∞
⊢ ContMDiffWithinAt I I' n f s x → ∀ᶠ (x' : M) in 𝓝[insert x s] x, ContMDiffWithinAt I I' n f s x'

Tactic:
rw [contMDiffWithinAt_iff_contMDiffOn_nhds hn]

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 ≠ ∞
⊢ (∃ u ∈ 𝓝[insert x s] x, ContMDiffOn I I' n f u) → ∀ᶠ (x' : M) in 𝓝[insert x s] x, ContMDiffWithinAt I I' n f s x'
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
x : M
⊢ ContMDiffAt I 𝓘(𝕜, E) n (↑(extChartAt I x)) x

Tactic:
by
  rw [contMDiffAt_iff_source]
  apply contMDiffWithinAt_id.congr_of_eventuallyEq_of_mem _ (by simp)
  filter_upwards [extChartAt_target_mem_nhdsWithin x] with y hy
  exact PartialEquiv.right_inv (extChartAt I x) hy

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

Goal state before:
pp: case a
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
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
⊢ e.source ⊆ (Prod.fst '' e.source) ×ˢ univ

Tactic:
intro p hp

Goal state after:
case a
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
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
p : B × F
hp : p ∈ e.source
⊢ p ∈ (Prod.fst '' e.source) ×ˢ univ
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case hx
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 (↑(c.symm ≫ₕ f ≫ₕ c')) (↑(chartAt H x).symm ⁻¹' f.source)
    (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
h1 : c' = chartAt H (↑f x)
h2 : ↑c' ∘ ↑f ∘ ↑c.symm = ↑(c.symm ≫ₕ f ≫ₕ c')
hcx : ↑c x ∈ ↑c.symm ⁻¹' f.source
⊢ ↑(chartAt H x) x ∈ (c.symm ≫ₕ f ≫ₕ c').source ∪ (↑(chartAt H x).symm ⁻¹' f.source)ᶜ

Tactic:
apply Or.inl
        simp only [c, hx, h1, mfld_simps]

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 e' : OpenPartialHomeomorph (B × F) (B × F)),
    e ∈
        ⋃ φ,
          ⋃ U,
            ⋃ (hU : 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 | e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)} →
      e' ∈
          ⋃ φ,
            ⋃ U,
              ⋃ (hU : 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 | e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)} →
        e ≫ₕ e' ∈
          ⋃ φ,
            ⋃ U,
              ⋃ (hU : 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 | e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)}

Tactic:
simp only [mem_aux]
    rintro e e' ⟨φ, U, hU, hφ, h2φ, heφ⟩ ⟨φ', U', hU', hφ', h2φ', heφ'⟩
    refine ⟨fun b => (φ b).trans (φ' b), _, hU.inter hU', ?_, ?_,
      Setoid.trans (OpenPartialHomeomorph.EqOnSource.trans' heφ heφ') ⟨?_, ?_⟩⟩
    · change
        ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n
          (fun x : B => (φ' x).toContinuousLinearMap ∘L (φ x).toContinuousLinearMap) (U ∩ U')
      exact (hφ'.mono inter_subset_right).clm_comp (hφ.mono inter_subset_left)
    · change
        ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n
          (fun x : B => (φ x).symm.toContinuousLinearMap ∘L (φ' x).symm.toContinuousLinearMap)
          (U ∩ U')
      exact (h2φ.mono inter_subset_left).clm_comp (h2φ'.mono inter_subset_right)
    · apply FiberwiseLinear.source_trans_openPartialHomeomorph
    · rintro ⟨b, v⟩ -; apply FiberwiseLinear.trans_openPartialHomeomorph_apply

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

Goal state before:
pp: case inr
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
f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
y : M
this : T1Space M
hf : HasMFDerivAt[s] f x f'
h : x ≠ y
⊢ HasMFDerivAt[insert y s] f x f'

Tactic:
apply hf.mono_of_mem_nhdsWithin ?_
    simp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]

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✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
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
v : F
⊢ ((continuousLinearEquivAt R e b ⋯).symm.trans (continuousLinearEquivAt R e' b ⋯)) v = (coordChangeL R e e' b) v

Tactic:
coordChangeL_apply e e' hb

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
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
v : F
⊢ ((continuousLinearEquivAt R e b ⋯).symm.trans (continuousLinearEquivAt R e' b ⋯)) v =
    (↑e' { proj := b, snd := e.symm b v }).2
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

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 : Set M
p : TangentBundle I M
hs : UniqueMDiffAt[s] p.proj
⊢ tangentMapWithin I I id s p = p

Tactic:
simp only [tangentMapWithin, 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
s : Set M
p : TangentBundle I M
hs : UniqueMDiffAt[s] p.proj
⊢ ⟨p.proj, (mfderiv[s] id p.proj) p.snd⟩ = p
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: case refine_1
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
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toOpenPartialHomeomorph (f x)).1 =
    ↑(extChartAt IB (f x₀).proj) (f x).proj

Tactic:
Trivialization.coe_coe

Goal state after:
case refine_1
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
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj) (f x)).1 = ↑(extChartAt IB (f x₀).proj) (f x).proj
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

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'
F' : Type u_14
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
G' : Type u_15
inst✝² : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
f : M → M' × N'
h : MDiff (Prod.fst ∘ f) ∧ MDiff (Prod.snd ∘ f)
⊢ MDiff f

Tactic:
convert! h.1.prodMk h.2

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
F : Type u_8
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
this✝ : IsManifold I (0 + 1) M
i j k : ↑(atlas H M)
x : M
hxk : x ∈ (↑k).source
hxi : x ∈ (↑i).source
hxj : x ∈ (↑j).source
v : E
this : ↑((↑i).extend I).symm ⁻¹' ((↑j).extend I).source ∈ 𝓝 (↑((↑i).extend I) x)
y : E
hy : y ∈ ↑((↑i).extend I).symm ⁻¹' ((↑j).extend I).source
⊢ ((↑((↑k).extend I) ∘ ↑((↑j).extend I).symm) ∘ ↑((↑j).extend I) ∘ ↑((↑i).extend I).symm) y =
    (↑((↑k).extend I) ∘ ↑((↑i).extend I).symm) y

Tactic:
simp_rw [Function.comp_apply, (j.1.extend I).left_inv hy]

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

Goal state before:
pp: case e'_5
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
s : Set M
z : M
F' : Type u_21
inst✝² : NormedCommRing F'
inst✝¹ : NormedAlgebra 𝕜 F'
ι : Type
t✝ : Finset ι
f : ι → M → F'
f' : ι → TangentSpace I z →L[𝕜] F'
inst✝ : DecidableEq ι
i : ι
t : Finset ι
hi : i ∉ t
IH :
  (∀ i ∈ t, HasMFDerivAt[s] (f i) z (f' i)) →
    HasMFDerivAt[s] (∏ i ∈ t, f i) z (∑ i ∈ t, (∏ j ∈ t.erase i, f j z) • f' i)
hf : HasMFDerivAt[s] (f i) z (f' i) ∧ ∀ x ∈ t, HasMFDerivAt[s] (f x) z (f' x)
e_1✝ : (TangentSpace I z →L[𝕜] TangentSpace 𝓘(𝕜, F') ((f i * ∏ x ∈ t, f x) z)) = (TangentSpace I z →L[𝕜] F')
⊢ ∑ x ∈ t, (∏ j ∈ (insert i t).erase x, f j z) • f' x = ∑ x ∈ t, (f i z * ∏ j ∈ t.erase x, f j z) • f' x

Tactic:
refine t.sum_congr rfl (fun j hj ↦ ?_)

Goal state after:
case e'_5
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
s : Set M
z : M
F' : Type u_21
inst✝² : NormedCommRing F'
inst✝¹ : NormedAlgebra 𝕜 F'
ι : Type
t✝ : Finset ι
f : ι → M → F'
f' : ι → TangentSpace I z →L[𝕜] F'
inst✝ : DecidableEq ι
i : ι
t : Finset ι
hi : i ∉ t
IH :
  (∀ i ∈ t, HasMFDerivAt[s] (f i) z (f' i)) →
    HasMFDerivAt[s] (∏ i ∈ t, f i) z (∑ i ∈ t, (∏ j ∈ t.erase i, f j z) • f' i)
hf : HasMFDerivAt[s] (f i) z (f' i) ∧ ∀ x ∈ t, HasMFDerivAt[s] (f x) z (f' x)
e_1✝ : (TangentSpace I z →L[𝕜] TangentSpace 𝓘(𝕜, F') ((f i * ∏ x ∈ t, f x) z)) = (TangentSpace I z →L[𝕜] F')
j : ι
hj : j ∈ t
⊢ (∏ j ∈ (insert i t).erase j, f j z) • f' j = (f i z * ∏ j ∈ t.erase j, f j z) • f' j
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
s : Set M
hs : s ⊆ support ↑f
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s

Tactic:
rw [support_eq_inter_preimage, subset_inter_iff, ← extChartAt_source I, ← image_subset_iff] at hs

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
s : Set M
hs : s ⊆ (extChartAt I c).source ∧ ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
⊢ ↑(extChartAt I c) '' s = closedBall (↑(extChartAt I c) c) f.rOut ∩ range ↑I ∩ ↑(extChartAt I c).symm ⁻¹' s
Module: Mathlib.Topology.EMetricSpace.Paracompact

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
⊢ ParacompactSpace α

Tactic:
have h2pow : ∀ n : ℕ, 2 * (2⁻¹ : ℝ≥0∞) ^ (n + 1) = 2⁻¹ ^ n := fun n => by
    simp [pow_succ', ← mul_assoc, ENNReal.mul_inv_cancel two_ne_zero ofNat_ne_top]

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m
h2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n
⊢ ParacompactSpace α