mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u_1
F : Type u_2
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
⊢ topologicalSpace B F =
    induced (Prod.fst ∘ ⇑(TotalSpace.toProd B F)) inst✝¹ ⊓ induced (Prod.snd ∘ ⇑(TotalSpace.toProd B F)) inst✝

Tactic:
;

Goal state after:
B : Type u_1
F : Type u_2
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
⊢ topologicalSpace B F =
    induced (Prod.fst ∘ ⇑(TotalSpace.toProd B F)) inst✝¹ ⊓ induced (Prod.snd ∘ ⇑(TotalSpace.toProd B F)) inst✝
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''
h : ContinuousWithinAt f s x
⊢ writtenInExtChartAt I I'' x (g ∘ f) =ᶠ[𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x]
    writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f

Tactic:
apply
    @Filter.mem_of_superset _ _ (f ∘ (extChartAt I x).symm ⁻¹' (extChartAt I' (f x)).source) _
      (extChartAt_preimage_mem_nhdsWithin
        (h.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _)))
  mfld_set_tac

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

Goal state before:
pp: case refine_4
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
b : B
v : F
⊢ ↑(FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯ ≫ₕ FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯) (b, v) =
    ↑(FiberwiseLinear.openPartialHomeomorph (fun b ↦ (φ b).trans (φ' b)) ⋯ ⋯ ⋯) (b, v)

Tactic:
;

Goal state after:
case refine_4
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
b : B
v : F
⊢ ↑(FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯ ≫ₕ FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯) (b, v) =
    ↑(FiberwiseLinear.openPartialHomeomorph (fun b ↦ (φ b).trans (φ' b)) ⋯ ⋯ ⋯) (b, v)
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.symmₗ R e b) y = e.symm b y

Tactic:
;

Goal state after:
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.symmₗ R e b) y = e.symm b y
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : Trivial B F x
⊢ (Trivialization.continuousLinearMapAt 𝕜 (trivialization B F) x) x✝ = (ContinuousLinearMap.id 𝕜 F) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : Trivial B F x
⊢ (Trivialization.continuousLinearMapAt 𝕜 (trivialization B F) x) x✝ = (ContinuousLinearMap.id 𝕜 F) x✝
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : F
⊢ (Trivialization.symmₗ 𝕜 (trivialization B F) x) x✝ = LinearMap.id x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : F
⊢ (Trivialization.symmₗ 𝕜 (trivialization B F) x) x✝ = LinearMap.id x✝
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : F
⊢ (Trivialization.symmL 𝕜 (trivialization B F) x) x✝ = (ContinuousLinearMap.id 𝕜 F) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : F
⊢ (Trivialization.symmL 𝕜 (trivialization B F) x) x✝ = (ContinuousLinearMap.id 𝕜 F) x✝
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : Trivial B F x
⊢ (Trivialization.continuousLinearEquivAt 𝕜 (trivialization B F) x ⋯) x✝ = (ContinuousLinearEquiv.refl 𝕜 F) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
x : B
x✝ : Trivial B F x
⊢ (Trivialization.continuousLinearEquivAt 𝕜 (trivialization B F) x ⋯) x✝ = (ContinuousLinearEquiv.refl 𝕜 F) x✝
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✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
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
y : F
⊢ (↑e.symm (b, y)).proj ∈ e'.baseSet

Tactic:
rw [e.proj_symm_apply' hb.1]

Goal state after:
case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
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
y : F
⊢ b ∈ e'.baseSet
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
⊢ e.symm b = ⇑(symmL R e b)

Tactic:
;

Goal state after:
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
⊢ e.symm b = ⇑(symmL R e b)
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:
;

Goal state after:
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✝
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:
;

Goal state after:
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✝
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:
;

Goal state after:
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✝
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
z : E b
⊢ ↑e { proj := b, snd := z } = (b, (continuousLinearEquivAt R e b hb) z)

Tactic:
ext
  · refine e.coe_fst ?_
    rw [e.source_eq]
    exact hb
  · simp only [continuousLinearEquivAt_apply]

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 : NontriviallyNormedField R
inst✝⁷ : (x : B) → AddCommMonoid (E x)
inst✝⁶ : (x : B) → Module R (E x)
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace R F
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (x : B) → TopologicalSpace (E x)
inst✝ : FiberBundle F E
ι : Type u_5
Z : VectorBundleCore R B F ι
b : B
a : F
i j : ι
⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],
    ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet)

Tactic:
rintro _ _ ⟨i, rfl⟩ ⟨i', rfl⟩
    refine (Z.continuousOn_coordChange i i').congr fun b hb => ?_
    ext v
    exact Z.localTriv_coordChange_eq i i' hb v

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

Goal state before:
pp: R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹² : TopologicalSpace B'
inst✝¹¹ : TopologicalSpace (TotalSpace F E)
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : (x : B) → AddCommMonoid (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
K : Type u_7
inst✝⁴ : FunLike K B' B
inst✝³ : ContinuousMapClass K B' B
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle 𝕜 F E
f : K
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
he' : MemTrivializationAtlas e'
b : B'
hb : f b ∈ e.baseSet ∩ e'.baseSet
⊢ (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 (e.pullback f) (e'.pullback f) b)) b =
    ((fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b)) ∘ ⇑f) b

Tactic:
;

Goal state after:
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹² : TopologicalSpace B'
inst✝¹¹ : TopologicalSpace (TotalSpace F E)
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : (x : B) → AddCommMonoid (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
K : Type u_7
inst✝⁴ : FunLike K B' B
inst✝³ : ContinuousMapClass K B' B
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle 𝕜 F E
f : K
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
he' : MemTrivializationAtlas e'
b : B'
hb : f b ∈ e.baseSet ∩ e'.baseSet
⊢ (fun b ↦ ↑(Trivialization.coordChangeL 𝕜 (e.pullback f) (e'.pullback f) b)) b =
    ((fun b ↦ ↑(Trivialization.coordChangeL 𝕜 e e' b)) ∘ ⇑f) b
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
x : M
v : TangentSpace I x
⊢ tangentMap I I id ⟨x, v⟩ = id ⟨x, v⟩

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
v : TangentSpace I x
⊢ tangentMap I I id ⟨x, v⟩ = id ⟨x, v⟩
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
proj✝ : M
snd✝ : TangentSpace I proj✝
hs : UniqueMDiffAt[s] ⟨proj✝, snd✝⟩.proj
⊢ ⟨⟨proj✝, snd✝⟩.proj, (ContinuousLinearMap.id 𝕜 (TangentSpace I ⟨proj✝, snd✝⟩.proj)) ⟨proj✝, snd✝⟩.snd⟩ = ⟨proj✝, snd✝⟩

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
proj✝ : M
snd✝ : TangentSpace I proj✝
hs : UniqueMDiffAt[s] ⟨proj✝, snd✝⟩.proj
⊢ ⟨⟨proj✝, snd✝⟩.proj, (ContinuousLinearMap.id 𝕜 (TangentSpace I ⟨proj✝, snd✝⟩.proj)) ⟨proj✝, snd✝⟩.snd⟩ = ⟨proj✝, snd✝⟩
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
⊢ (↑e.symm (b, v)).proj ∈ e'.baseSet

Tactic:
rw [e.proj_symm_apply' hb.1]

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
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✝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) = ↑e' { proj := b, snd := e.symm b v }

Tactic:
ext
  · rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]
    rw [e.proj_symm_apply' hb.1]
    exact hb.2
  · exact a.coordChange_apply he he' hb v

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

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'
s : Set (M × M')
x : M × M'
hxs : UniqueMDiffAt[s] x
⊢ mfderiv% Prod.fst x = ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)

Tactic:
;

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'
s : Set (M × M')
x : M × M'
hxs : UniqueMDiffAt[s] x
⊢ mfderiv% Prod.fst x = ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)