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

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_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
x : E
⊢ ContDiffWithinAt 𝕜 n f univ x ↔ ContDiffAt 𝕜 n f x

Tactic:
contDiffWithinAt_univ

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
n : WithTop ℕ∞
f : E → E'
x : E
⊢ ContDiffAt 𝕜 n f x ↔ ContDiffAt 𝕜 n f x
Module: Mathlib.Geometry.Manifold.Algebra.LieGroup

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : Group G
m n : ℕ∞ω
hmn : m ≤ n
h : LieGroup I n G
⊢ LieGroup I m G

Tactic:
have : ContMDiffMul I m G := ContMDiffMul.of_le hmn

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : Group G
m n : ℕ∞ω
hmn : m ≤ n
h : LieGroup I n G
this : ContMDiffMul I m G
⊢ LieGroup I m G
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
⊢ ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ

Tactic:
intro p

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
p : ↑e.source
⊢ e.source ∩ s p = u p ×ˢ univ
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
⊢ ∀ (p : ↑e.source), (↑p).1 ∈ u p

Tactic:
intro p

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
p : ↑e.source
⊢ (↑p).1 ∈ u p
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
⊢ ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source

Tactic:
intro p

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen[instTopologicalSpaceProd] (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen[inst✝⁷] (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
p : ↑e.source
⊢ ∀ (q : B × F), q.1 ∈ u p → q ∈ e.source
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

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
⊢ p ∈ e.source → p ∈ (Prod.fst '' e.source) ×ˢ univ
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
⊢ LiftPropWithinAt (DifferentiableWithinAtProp I I') f s x ↔
    ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I 𝓘(𝕜, E')) (↑(extChartAt I' y) ∘ f) s x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropWithinAt_indep_chart_target
      (chart_mem_maximalAtlas y) hy,

Goal state after:
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
⊢ ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I I') (↑(chartAt H' y) ∘ f) s x ↔
    ContinuousWithinAt f s x ∧ LiftPropWithinAt (DifferentiableWithinAtProp I 𝓘(𝕜, E')) (↑(extChartAt I' y) ∘ f) s x
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case refine_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
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_7
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_8
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
s : Set M
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
y : M
hφ : φ ∈ maximalAtlas I n M
hψ : ψ ∈ maximalAtlas J n N
hy : y ∈ φ.source
hgy : f y ∈ ψ.source
hmaps : MapsTo f s ψ.source
h :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
    (↑(φ.extend I) y)
⊢ ContinuousWithinAt f s y

Tactic:
← φ.continuousWithinAt_writtenInExtend_iff (I := I) (I' := J) hy hgy hmaps

Goal state after:
case refine_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
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_7
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_8
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
s : Set M
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
y : M
hφ : φ ∈ maximalAtlas I n M
hψ : ψ ∈ maximalAtlas J n N
hy : y ∈ φ.source
hgy : f y ∈ ψ.source
hmaps : MapsTo f s ψ.source
h :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I)
    (↑(φ.extend I) y)
⊢ ContinuousWithinAt (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (↑(φ.extend I).symm ⁻¹' s ∩ range ↑I) (↑(φ.extend I) y)
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: case inl
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)
this : T1Space M
hf : HasMFDerivAt[s] f x f'
⊢ HasMFDerivAt[insert x s] f x f'

Tactic:
rw [HasMFDerivWithinAt] at hf ⊢

Goal state after:
case inl
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)
this : T1Space M
hf :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
⊢ ContinuousWithinAt f (insert x s) x ∧
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (↑(extChartAt I x).symm ⁻¹' insert x s ∩ range ↑I)
      (↑(extChartAt I x) x)
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁵ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁴ : (x : B) → Module 𝕜 (E₂ x)
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝³ : Trivialization.IsLinear 𝕜 e₁
inst✝² : Trivialization.IsLinear 𝕜 e₁'
inst✝¹ : Trivialization.IsLinear 𝕜 e₂
inst✝ : Trivialization.IsLinear 𝕜 e₂'
b : B
hb : (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet
⊢ ↑(coordChangeL 𝕜 (e₁.prod e₂) (e₁'.prod e₂') b) = (↑(coordChangeL 𝕜 e₁ e₁' b)).prodMap ↑(coordChangeL 𝕜 e₂ e₂' b)

Tactic:
rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']
  rintro ⟨v₁, v₂⟩
  change
    (e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
      (e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂)
  rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply']
  exacts [rfl, hb, ⟨hb.1.2, hb.2.2⟩, ⟨hb.1.1, hb.2.1⟩]

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

Goal state before:
pp: case refine_2
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
s : (x : B) → E x
a : Set B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
ha : IsOpen[inst✝¹¹] a
ha' : a ⊆ e.baseSet
h : ContMDiffOn IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) a
x : B
hx : x ∈ a
this : ContMDiffAt IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) x
⊢ ContMDiffWithinAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) a x

Tactic:
exact ((e.contMDiffAt_section_iff (ha' hx)).mpr this).contMDiffWithinAt

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V : (x : E) → TangentSpace 𝓘(𝕜, E) x
s : Set E
x : E
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent n (fun x ↦ ⟨x, V x⟩) s x ↔ ContDiffWithinAt 𝕜 n V s x

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V : (x : E) → TangentSpace 𝓘(𝕜, E) x
s : Set E
x : E
h : ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent n (fun x ↦ ⟨x, V x⟩) s x
⊢ ContDiffWithinAt 𝕜 n V s x

case refine_2
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V : (x : E) → TangentSpace 𝓘(𝕜, E) x
s : Set E
x : E
h : ContDiffWithinAt 𝕜 n V s x
⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent n (fun x ↦ ⟨x, V x⟩) s x
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
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
H : Type u_4
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
x₀ x : H
y₀ y : H'
ϕ : E →L[𝕜] E'
⊢ (tangentBundleCore I' H').coordChange (achart H' y) (achart H' y₀) y ∘SL
      ϕ ∘SL (tangentBundleCore I H).coordChange (achart H x₀) (achart H x) x =
    ϕ

Tactic:
tangentBundleCore_coordChange_model_space,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
H : Type u_4
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
x₀ x : H
y₀ y : H'
ϕ : E →L[𝕜] E'
⊢ ContinuousLinearMap.id 𝕜 E' ∘SL ϕ ∘SL ContinuousLinearMap.id 𝕜 E = ϕ
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 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_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
⊢ mfderiv% Sum.inr q' = ContinuousLinearMap.id 𝕜 (TangentSpace I q')

Tactic:
by
  simpa [mfderivWithin_univ] using (mfderivWithin_sumInr (uniqueMDiffWithinAt_univ I))

Goal state after:
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 f z
hg : MDiffAt g z
⊢ TangentSpace I z →L[𝕜] TangentSpace 𝓘(𝕜, E') ((f + g) z)

Tactic:
exact mfderiv% g z

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

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
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
p : TotalSpace F Z
hs : UniqueMDiffAt[s] p.proj
h's : s ⊆ (trivializationAt F Z p.proj).baseSet
this :
  (↑(extChartAt I p.proj).symm ⁻¹' s ∩ range ↑I) ×ˢ univ ⊆
    ↑(extChartAt (I.prod 𝓘(𝕜, F)) p).symm ⁻¹' TotalSpace.proj ⁻¹' s ∩ range ↑(I.prod 𝓘(𝕜, F))
⊢ UniqueMDiffAt[TotalSpace.proj ⁻¹' s] p

Tactic:
let w := (extChartAt (I.prod 𝓘(𝕜, F)) p p).2
    have A : extChartAt (I.prod 𝓘(𝕜, F)) p p = (extChartAt I p.1 p.1, w) := by
      ext
      · simp [FiberBundle.chartedSpace_chartAt]
      · rfl
    simp only [UniqueMDiffWithinAt, A] at hs ⊢
    exact (hs.prod (uniqueDiffWithinAt_univ (x := w))).mono this

Goal state after: