mathlib-initiative/mathlib-tactics huggingface dataset:
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
f : M → M'
hf : Injective ⇑(mfderiv% f x)
⊢ MDiffAt f x

Tactic:
simp only [← mdifferentiableWithinAt_univ, ← mfderivWithin_univ] at hf ⊢

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'
x : M
f : M → M'
hf : Injective ⇑(mfderiv[univ] f x)
⊢ MDiffAt[univ] f x
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✝⁷ : Semiring R
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Pretrivialization F TotalSpace.proj
inst✝ : Pretrivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
y : E b
⊢ (Pretrivialization.symmₗ R e b) ((Pretrivialization.linearMapAt R e b) y) = y

Tactic:
simp [hb]

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✝⁷ : Semiring R
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Pretrivialization F TotalSpace.proj
inst✝ : Pretrivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
y : F
⊢ (Pretrivialization.linearMapAt R e b) ((Pretrivialization.symmₗ R e b) y) = y

Tactic:
simp [hb]

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✝⁸ : Semiring R
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
y : E b
⊢ e.symm b ((Trivialization.linearMapAt R e b) y) = y

Tactic:
simp [hb]

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✝⁸ : Semiring R
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module R (E x)
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
y : F
⊢ (Trivialization.linearMapAt R e b) (e.symm b y) = y

Tactic:
simp [hb]

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 : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
⊢ ∀ (x : F), (TotalSpace.mk b ∘ e.symm b) x = (TotalSpace.mk b ∘ (Trivialization.symmₗ R e b).toFun) x

Tactic:
simp [hb]

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 : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
x✝ : F
⊢ e.symm b x✝ = (symmL R e b) x✝

Tactic:
simp [hb]

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 : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
x✝ : F
⊢ (continuousLinearEquivAt R e b hb).symm x✝ = (symmL R e b) x✝

Tactic:
simp [hb]

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 : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
x✝ : F
⊢ ↑(continuousLinearEquivAt R e b hb).symm x✝ = (symmL R e b) x✝

Tactic:
simp [hb]

Goal state after:
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
⊢ (trivializationAt F E x.proj).toPartialEquiv ≫
      (chartAt HB x.proj).prod (PartialEquiv.refl F) ≫ IB.prod (PartialEquiv.refl F) =
    (trivializationAt F E x.proj).toPartialEquiv ≫
      ((chartAt HB x.proj).toPartialEquiv ≫ IB.toPartialEquiv).prod (PartialEquiv.refl F)

Tactic:
PartialEquiv.prod_trans,

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 ≫
      ((chartAt HB x.proj).toPartialEquiv ≫ IB.toPartialEquiv).prod (PartialEquiv.refl F ≫ PartialEquiv.refl F) =
    (trivializationAt F E x.proj).toPartialEquiv ≫
      ((chartAt HB x.proj).toPartialEquiv ≫ IB.toPartialEquiv).prod (PartialEquiv.refl F)
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
E : B → Type u_6
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁸ : NormedAddCommGroup EB
inst✝⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
⊢ ContMDiffAt IB 𝓘(𝕜, F) n (fun x_1 ↦ (↑(trivializationAt F E x) { proj := x_1, snd := 0 }).2) x

Tactic:
apply (contMDiffAt_const (c := 0)).congr_of_eventuallyEq

Goal state after:
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁸ : NormedAddCommGroup EB
inst✝⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
⊢ (fun x_1 ↦ (↑(trivializationAt F E x) { proj := x_1, snd := 0 }).2) =ᶠ[𝓝 x] fun x ↦ 0
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'⟩

Goal state after:
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
f' : E →L[𝕜] E'
hf' :
  HasFDerivWithinAt (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm) f'
    (↑I.symm ⁻¹' ↑(chartAt H x).symm ⁻¹' s ∩ range ↑I) (↑I (↑(chartAt H x) x))
⊢ MDiffAt[s] (g ∘ f) x
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

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

Tactic:
exact VectorBundle.continuousOn_coordChange' e e'

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁶ : TopologicalSpace M
inst✝¹⁵ : ChartedSpace H M
E' : Type u_5
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹¹ : TopologicalSpace M'
inst✝¹⁰ : ChartedSpace H' M'
F : Type u_11
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace G N
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'
s : Set (M × M')
p : M × M'
f : M → N
g : M' → N'
df : TangentSpace I p.1 →L[𝕜] TangentSpace J (f p.1)
hf : HasMFDerivAt[Prod.fst '' s] f p.1 df
dg : TangentSpace I' p.2 →L[𝕜] TangentSpace J' (g p.2)
hg : HasMFDerivAt[Prod.snd '' s] g p.2 dg
p₀ : E × E'
hp₀ : p₀ ∈ ↑((chartAt H p.1).symm.prod (chartAt H' p.2).symm) ∘ ↑(I.symm.prod I'.symm) ⁻¹' s
hp₁₁ : p₀.1 ∈ range ↑I
hp₁₂ : p₀.2 ∈ range ↑I'
⊢ p₀ ∈
    (↑(chartAt H p.1).symm ∘ ↑I.symm ⁻¹' Prod.fst '' s ∩ range ↑I) ×ˢ
      (↑(chartAt H' p.2).symm ∘ ↑I'.symm ⁻¹' Prod.snd '' s ∩ range ↑I')

Tactic:
exact ⟨⟨by simp_all; grind, by assumption⟩, ⟨by simp_all; grind, by assumption⟩⟩

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

Goal state before:
pp: case hx
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'
i : ↑(atlas H M)
x : M
hx : x ∈ (↑i).source
v : E
⊢ ↑((↑i).extend I) (↑((↑i).extend I).symm (↑((↑i).extend I) x)) = ↑((↑i).extend I) x

Tactic:
i.1.extend_left_inv hx

Goal state after:
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 : f ⁻¹' u ∈ 𝓝[s] x
hxs : UniqueMDiffAt[s] x
A : s ∩ f ⁻¹' u ∈ 𝓝[s] x
⊢ mfderiv[s] (g ∘ f) x = mfderiv[s ∩ f ⁻¹' u] (g ∘ f) x

Tactic:
apply MDifferentiableWithinAt.mfderivWithin_mono_of_mem_nhdsWithin _ hxs A

Goal state after:
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 : f ⁻¹' u ∈ 𝓝[s] x
hxs : UniqueMDiffAt[s] x
A : s ∩ f ⁻¹' u ∈ 𝓝[s] x
⊢ MDiffAt[s ∩ f ⁻¹' u] (g ∘ f) x
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
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
⊢ ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n (↑(e.symm ≫ₕ e'.toOpenPartialHomeomorph)) (e.target ∩ e'.target)

Tactic:
have Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) (e.baseSet ∩ e'.baseSet) := fun x hx ↦
    ⟨e.mem_target.1 hx.1, e'.mem_target.1 hx.2⟩
  rw [mapsTo_inter] at Hmaps
  -- TODO: drop `congr` https://github.com/leanprover-community/mathlib4/issues/5473
  refine (contMDiffOn_fst.prodMk
    (contMDiffOn_fst.coordChange contMDiffOn_snd Hmaps.1 Hmaps.2)).congr ?_
  rintro ⟨b, x⟩ hb
  refine Prod.ext ?_ rfl
  have : (e.toOpenPartialHomeomorph.symm (b, x)).1 ∈ e'.baseSet := by
    simp_all only [Trivialization.mem_target, mfld_simps]
  exact (e'.coe_fst' this).trans (e.proj_symm_apply hb.1)

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

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_4
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x z : M
v : E
h : z ∈ (extChartAt I x).source
⊢ (tangentCoordChange I x x z) v = v

Tactic:
by
  apply (tangentBundleCore I M).coordChange_self
  rw [tangentBundleCore_baseSet, coe_achart, ← extChartAt_source I]
  exact h

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

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_4
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
b₀ b : M
hb : b ∈ (chartAt H b₀).source
⊢ Trivialization.continuousLinearMapAt 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
    (tangentBundleCore I M).coordChange (achart H b) (achart H b₀) b

Tactic:
simp [hb]

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

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_4
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
b₀ b : M
hb : b ∈ (chartAt H b₀).source
⊢ Trivialization.symmL 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
    (tangentBundleCore I M).coordChange (achart H b₀) (achart H b) b

Tactic:
simp [hb]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
this : IsManifold I n M
i j : ↑(atlas H M)
⊢ ContMDiffOn I 𝓘(𝕜, E →L[𝕜] E) n ((tangentBundleCore I M).coordChange i j)
    ((tangentBundleCore I M).baseSet i ∩ (tangentBundleCore I M).baseSet j)

Tactic:
rw [contMDiffOn_iff_source_of_mem_maximalAtlas (subset_maximalAtlas i.2),
    contMDiffOn_iff_contDiffOn]

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
this : IsManifold I n M
i j : ↑(atlas H M)
⊢ ContDiffOn 𝕜 n ((tangentBundleCore I M).coordChange i j ∘ ↑((↑i).extend I).symm)
    (↑((↑i).extend I) '' ((tangentBundleCore I M).baseSet i ∩ (tangentBundleCore I M).baseSet j))

𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_4
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_6
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : IsManifold I (n + 1) M
this : IsManifold I n M
i j : ↑(atlas H M)
⊢ (tangentBundleCore I M).baseSet i ∩ (tangentBundleCore I M).baseSet j ⊆ (↑i).source
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝³³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³² : NormedAddCommGroup E
inst✝³¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁹ : TopologicalSpace M
inst✝²⁸ : ChartedSpace H M
E' : Type u_5
inst✝²⁷ : NormedAddCommGroup E'
inst✝²⁶ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁵ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁴ : TopologicalSpace M'
inst✝²³ : ChartedSpace H' M'
E'' : Type u_8
inst✝²² : NormedAddCommGroup E''
inst✝²¹ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²⁰ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹⁹ : TopologicalSpace M''
inst✝¹⁸ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁵ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁴ : TopologicalSpace N
inst✝¹³ : ChartedSpace G N
F' : Type u_14
inst✝¹² : NormedAddCommGroup F'
inst✝¹¹ : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹⁰ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝⁹ : TopologicalSpace N'
inst✝⁸ : ChartedSpace G' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
s : Set M
x z : M
f g : M → E'
f' g' : TangentSpace I z →L[𝕜] E'
hf : MDiffAt[s] f z
hg : MDiffAt[s] g z
hs : UniqueMDiffAt[s] z
⊢ TangentSpace I z →L[𝕜] TangentSpace 𝓘(𝕜, E') ((f + g) z)

Tactic:
by exact mfderiv[s] g z

Goal state after: