mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup F
inst✝¹⁹ : NormedSpace 𝕜 F
inst✝¹⁸ : TopologicalSpace (TotalSpace F E)
inst✝¹⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝¹³ : NormedAddCommGroup EM
inst✝¹² : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹¹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace HM M
inst✝⁸ : TopologicalSpace B
inst✝⁷ : ChartedSpace HB B
inst✝⁶ : FiberBundle F E
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module 𝕜 (E x)
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle 1 F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
f : M → TotalSpace F E
x₀ : M
he : f x₀ ∈ e.source
he' : f x₀ ∈ e'.source
hf : (MDiffAt fun x ↦ (f x).proj) x₀
⊢ (MDiffAt fun x ↦ (↑e (f x)).2) x₀ ↔ (MDiffAt fun x ↦ (↑e' (f x)).2) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : TopologicalSpace (TotalSpace F E)
inst✝¹⁶ : (x : B) → TopologicalSpace (E x)
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
EM : Type u_10
inst✝¹² : NormedAddCommGroup EM
inst✝¹¹ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹⁰ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace HM M
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F E
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
f : M → TotalSpace F E
s : Set M
x₀ : M
he : f x₀ ∈ e.source
⊢ MDiffAt[s] f x₀ ↔ (MDiffAt[s] fun x ↦ (f x).proj) x₀ ∧ (MDiffAt[s] fun x ↦ (↑e (f x)).2) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : TopologicalSpace (TotalSpace F E)
inst✝¹⁶ : (x : B) → TopologicalSpace (E x)
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
EM : Type u_10
inst✝¹² : NormedAddCommGroup EM
inst✝¹¹ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹⁰ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace HM M
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F E
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
f : M → TotalSpace F E
x₀ : M
he : f x₀ ∈ e.source
⊢ MDiffAt f x₀ ↔ (MDiffAt fun x ↦ (f x).proj) x₀ ∧ (MDiffAt fun x ↦ (↑e (f x)).2) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
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
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F E
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
s : (b : B) → E b
u : Set B
b₀ : B
hex₀ : b₀ ∈ e.baseSet
⊢ MDiffAt[u] (T% s) b₀ ↔ (MDiffAt[u] fun x ↦ (↑e ⟨x, s x⟩).2) b₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
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
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F E
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
s : (b : B) → E b
b₀ : B
hex₀ : b₀ ∈ e.baseSet
⊢ MDiffAt (T% s) b₀ ↔ (MDiffAt fun x ↦ (↑e ⟨x, s x⟩).2) b₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
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
inst✝⁷ : TopologicalSpace B
inst✝⁶ : ChartedSpace HB B
inst✝⁵ : FiberBundle F E
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle 1 F E IB
s : (x : B) → E x
a : Set B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
ha : IsOpen[inst✝⁷] a
ha' : a ⊆ e.baseSet
⊢ MDiff[a] (T% s) ↔ MDiff[a] fun x ↦ (↑e ⟨x, s x⟩).2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s t : (x : B) → E x
u : Set B
x₀ : B
hs : MDiffAt[u] (T% s) x₀
ht : MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (s + t) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s t : (x : B) → E x
x₀ : B
hs : MDiffAt (T% s) x₀
ht : MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, (s + t) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s : (x : B) → E x
u : Set B
x₀ : B
hs : MDiffAt[u] (T% s) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (-s) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s : (x : B) → E x
x₀ : B
hs : MDiffAt (T% s) x₀
⊢ (MDiffAt fun x ↦ ⟨x, (-s) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s t : (x : B) → E x
u : Set B
x₀ : B
hs : MDiffAt[u] (T% s) x₀
ht : MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (s - t) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
s t : (x : B) → E x
x₀ : B
hs : MDiffAt (T% s) x₀
ht : MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, (s - t) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
f : B → 𝕜
s : (x : B) → E x
u : Set B
x₀ : B
hf : MDiffAt[u] f x₀
hs : MDiffAt[u] (T% s) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, (f • s) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
f : B → 𝕜
s : (x : B) → E x
x₀ : B
hf : MDiffAt f x₀
hs : MDiffAt (T% s) x₀
⊢ (MDiffAt fun x ↦ ⟨x, (f • s) x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
x₀ : B
ι : Type u_9
s : Finset ι
t : ι → (x : B) → E x
hs : ∀ i ∈ s, MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
ι : Type u_9
s : Finset ι
t : ι → (x : B) → E x
x₀ : B
hs : ∀ i ∈ s, MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
s : (x : B) → E x
ψ : B → 𝕜
hψ : MDiff[u] ψ
ht : IsOpen[inst✝¹³] u
ht' : tsupport ψ ⊆ u
hs : MDiff[u] (T% s)
⊢ MDiff fun x ↦ ⟨x, (ψ • s) x⟩

Tactic:
by

Goal state after:
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
x✝¹ : TotalSpace (F₁ →SL[σ] F₂) fun x ↦ E₁ x →SL[σ] E₂ x
x : B
L : E₁ x →SL[σ] E₂ x
x✝ : { proj := x, snd := L } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)
h₁ : { proj := x, snd := L }.proj ∈ e₁.baseSet
h₂ : { proj := x, snd := L }.proj ∈ e₂.baseSet
⊢ {
      proj :=
        ({ proj := x, snd := L }.proj,
            Trivialization.continuousLinearMapAt 𝕜₂ e₂ { proj := x, snd := L }.proj ∘SL
              { proj := x, snd := L }.snd ∘SL Trivialization.symmL 𝕜₁ e₁ { proj := x, snd := L }.proj).1,
      snd :=
        Trivialization.symmL 𝕜₂ e₂
            ({ proj := x, snd := L }.proj,
                Trivialization.continuousLinearMapAt 𝕜₂ e₂ { proj := x, snd := L }.proj ∘SL
                  { proj := x, snd := L }.snd ∘SL Trivialization.symmL 𝕜₁ e₁ { proj := x, snd := L }.proj).1 ∘SL
          ({ proj := x, snd := L }.proj,
                Trivialization.continuousLinearMapAt 𝕜₂ e₂ { proj := x, snd := L }.proj ∘SL
                  { proj := x, snd := L }.snd ∘SL Trivialization.symmL 𝕜₁ e₁ { proj := x, snd := L }.proj).2 ∘SL
            Trivialization.continuousLinearMapAt 𝕜₁ e₁
              ({ proj := x, snd := L }.proj,
                  Trivialization.continuousLinearMapAt 𝕜₂ e₂ { proj := x, snd := L }.proj ∘SL
                    { proj := x, snd := L }.snd ∘SL Trivialization.symmL 𝕜₁ e₁ { proj := x, snd := L }.proj).1 } =
    { proj := x, snd := L }

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
n : WithTop ℕ∞
E₁ : B → Type u_6
E₂ : B → Type u_7
inst✝²⁷ : NontriviallyNormedField 𝕜
inst✝²⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝²⁵ : (x : B) → Module 𝕜 (E₁ x)
inst✝²⁴ : NormedAddCommGroup F₁
inst✝²³ : NormedSpace 𝕜 F₁
inst✝²² : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²¹ : (x : B) → TopologicalSpace (E₁ x)
inst✝²⁰ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁹ : (x : B) → Module 𝕜 (E₂ x)
inst✝¹⁸ : NormedAddCommGroup F₂
inst✝¹⁷ : NormedSpace 𝕜 F₂
inst✝¹⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁵ : (x : B) → TopologicalSpace (E₂ x)
EB : Type u_8
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_9
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : VectorBundle 𝕜 F₁ E₁
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : VectorBundle 𝕜 F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁵ : ContMDiffVectorBundle n F₁ E₁ IB
inst✝⁴ : ContMDiffVectorBundle n F₂ E₂ IB
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
⊢ ContMDiffOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) n (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
u' : Set B
hu' : u' ∈ 𝓝 x₀
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}.Finite
s : Set ι := {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this✝ : Fintype ↑{i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this : (MDiffAt[u ∩ u'] fun x ↦ ⟨x, ∑ i ∈ s.toFinset, t i x⟩) x₀
y : B
hy : y ∈ u'
i : ι
hi : i ∉ ↑s.toFinset
h : t i y ≠ 0
⊢ i ∈ s.toFinset

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[univ] (T% t) x₀
⊢ (MDiffAt[univ] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
exact .sum_section_of_locallyFinite ht ht'

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

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt (T% t) x₀
⊢ (MDiffAt fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
by

Goal state after: