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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝[range ↑I] ↑(extChartAt I x) x ⊔ 𝓝[(range ↑I)ᶜ] ↑(extChartAt I x) x

Tactic:
]

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
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
⊢ (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝[range ↑I] ↑(extChartAt I x) x ⊔ 𝓝[(range ↑I)ᶜ] ↑(extChartAt I x) 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✝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

Tactic:
]

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.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i j : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet
v : F
⊢ ({ proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj,
        (Z.coordChange i j { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj)
          (b, v).2).2 =
    (Z.coordChange i j b) v

pp: case a
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i j : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet
v : F
⊢ { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈
    Z.baseSet i ∩
        Z.baseSet
          (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩
      Z.baseSet j

pp: case hb
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i j : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet
v : F
⊢ b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet

Tactic:
]

Goal state after:
case a
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i j : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet
v : F
⊢ { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈
    Z.baseSet i ∩
        Z.baseSet
          (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩
      Z.baseSet j

case hb
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i j : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet ∧ b ∈ (Z.localTriv j).baseSet
v : F
⊢ b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
⊢ { proj := b, snd := a }.proj ∈ Z.baseSet (Z.indexAt b)

Tactic:
]

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
⊢ { proj := b, snd := a }.proj ∈ Z.baseSet (Z.indexAt b)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
h'z : z ∈ {↑(extChartAt I x) x}ᶜ
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
⊢ (↑(extChartAt I x).symm z ∈ s) = (↑(extChartAt I x).symm z ∈ t)

Tactic:
]

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
h'z : z ∈ {↑(extChartAt I x) x}ᶜ
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
⊢ (↑(extChartAt I x).symm z ∈ s) = (↑(extChartAt I x).symm z ∈ t)
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (fun y ↦
        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,
                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,
                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, continuousOn_toFun := ⋯,
                continuousOn_invFun := ⋯, open_source := ⋯, open_target := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,
                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }
            { proj := x, snd := y }).2)
      (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prodCongr (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)

Tactic:
]

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (fun y ↦
        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,
                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,
                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, continuousOn_toFun := ⋯,
                continuousOn_invFun := ⋯, open_source := ⋯, open_target := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,
                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }
            { proj := x, snd := y }).2)
      (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prodCongr (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)
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
x : M
hxs : UniqueMDiffAt[s] x
⊢ mfderiv% id x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)

Tactic:
]

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
x : M
hxs : UniqueMDiffAt[s] x
⊢ mfderiv% id x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet
v : F
⊢ cast ⋯ (↑(Z.localTriv i).symm (b, v)).snd = (Z.coordChange i (Z.indexAt b) b) v

pp: case hb
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet
v : F
⊢ b ∈ (Z.localTriv i).baseSet

Tactic:
]

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet
v : F
⊢ cast ⋯ (↑(Z.localTriv i).symm (b, v)).snd = (Z.coordChange i (Z.indexAt b) b) v

case hb
R : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField R
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace R F
inst✝ : TopologicalSpace B
ι : Type u_5
Z : VectorBundleCore R B F ι
i : ι
b : B
hb : b ∈ (Z.localTriv i).baseSet
v : F
⊢ b ∈ (Z.localTriv i).baseSet
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
h'z : ¬z = ↑(extChartAt I x) x
⊢ ¬↑(extChartAt I x) x = z

Tactic:
]

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
x : M
s t : Set M
y : M
h : s =ᶠ[𝓝[{y}ᶜ] x] t
this : T1Space M
u : Set M
u_mem : u ∈ 𝓝 (↑(extChartAt I x).symm (↑(extChartAt I x) x))
hu : u ∩ {x}ᶜ ⊆ {y | (y ∈ s) = (y ∈ t)}
B : (extChartAt I x).target ∪ (range ↑I)ᶜ ∈ 𝓝 (↑(extChartAt I x) x)
z : E
hIz : z ∈ range ↑I
hz : ↑(extChartAt I x).symm z ∈ u ∧ z ∈ (extChartAt I x).target
h'z : ¬z = ↑(extChartAt I x) x
⊢ ¬↑(extChartAt I x) x = z
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝⁶ : TopologicalSpace F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
x : TotalSpace F E
⊢ (trivializationAt F E x.proj).toOpenPartialHomeomorph ≫ₕ (chartAt HB x.proj).prod (OpenPartialHomeomorph.refl F) =
    (trivializationAt F E x.proj).toOpenPartialHomeomorph ≫ₕ (chartAt HB x.proj).prod (OpenPartialHomeomorph.refl F)

Tactic:
]

Goal state after:
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
⊢ ⟨p.proj, (ContinuousLinearMap.id 𝕜 (TangentSpace I p.proj)) p.snd⟩ = p

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
⊢ UniqueMDiffAt[s] p.proj

Tactic:
]

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, (ContinuousLinearMap.id 𝕜 (TangentSpace I p.proj)) p.snd⟩ = p

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
⊢ UniqueMDiffAt[s] p.proj
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).toPartialEquiv ≫ IB.toPartialEquiv).prod (PartialEquiv.refl F) =
    (trivializationAt F E x.proj).toPartialEquiv ≫
      ((chartAt HB x.proj).toPartialEquiv ≫ IB.toPartialEquiv).prod (PartialEquiv.refl F)

Tactic:
]

Goal state after:
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
⊢ UniqueMDiffAt[s] p.proj

Tactic:
exact hs

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
⊢ ((extChartAt IB x.proj).prod (PartialEquiv.refl F)).target ∩
      ↑((extChartAt IB x.proj).prod (PartialEquiv.refl F)).symm ⁻¹' (trivializationAt F E x.proj).baseSet ×ˢ univ =
    (extChartAt IB x.proj).target ×ˢ univ ∩
      (↑(extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ

Tactic:
]

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
⊢ ((extChartAt IB x.proj).prod (PartialEquiv.refl F)).target ∩
      ↑((extChartAt IB x.proj).prod (PartialEquiv.refl F)).symm ⁻¹' (trivializationAt F E x.proj).baseSet ×ˢ univ =
    (extChartAt IB x.proj).target ×ˢ univ ∩
      (↑(extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ
Module: Mathlib.Topology.VectorBundle.Basic

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

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

Tactic:
]

Goal state after:
case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e e' : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
he' : e' ∈ a.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (↑e.symm (b, v)).proj ∈ e'.baseSet
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
⊢ ContinuousWithinAt f s x₀ →
    (ContMDiffWithinAt IM 𝓘(𝕜, EB × F) n (↑(extChartAt (IB.prod 𝓘(𝕜, F)) (f x₀)) ∘ f) s x₀ ↔
      ContMDiffWithinAt IM 𝓘(𝕜, EB) n (↑(extChartAt IB (f x₀).proj) ∘ fun x ↦ (f x).proj) s x₀ ∧
        ContMDiffWithinAt IM 𝓘(𝕜, F) n
          (↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2) ∘ fun x ↦
            (↑(trivializationAt F E (f x₀).proj) (f x)).2)
          s x₀)

Tactic:
]

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
⊢ ContinuousWithinAt f s x₀ →
    (ContMDiffWithinAt IM 𝓘(𝕜, EB × F) n (↑(extChartAt (IB.prod 𝓘(𝕜, F)) (f x₀)) ∘ f) s x₀ ↔
      ContMDiffWithinAt IM 𝓘(𝕜, EB) n (↑(extChartAt IB (f x₀).proj) ∘ fun x ↦ (f x).proj) s x₀ ∧
        ContMDiffWithinAt IM 𝓘(𝕜, F) n
          (↑(extChartAt 𝓘(𝕜, F) (↑(trivializationAt F E (f x₀).proj) (f x₀)).2) ∘ fun x ↦
            (↑(trivializationAt F E (f x₀).proj) (f x)).2)
          s x₀)
Module: Mathlib.Topology.VectorBundle.Basic

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

Tactic:
]

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