-- Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
theorem tangentMap_tangentBundle_pure [Is : IsManifold I 1 M]
    (p : TangentBundle I M) :
    tangentMap I I.tangent (zeroSection E (TangentSpace I)) p = ⟨⟨p.proj, 0⟩, ⟨p.2, 0⟩⟩

Proof state:
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I) (↑I (↑(chartAt H x) x))) v = (v, 0)

Tactic:
rw [fderivWithin_eq_fderiv, DifferentiableAt.fderiv_prod]

Result:
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ ((fderiv 𝕜 (fun x' => x') (↑I (↑(chartAt H x) x))).prod (fderiv 𝕜 (fun x' => 0) (↑I (↑(chartAt H x) x)))) v = (v, 0)

case hf₁
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ DifferentiableAt 𝕜 (fun x' => x') (↑I (↑(chartAt H x) x))

case hf₂
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ DifferentiableAt 𝕜 (fun x' => 0) (↑I (↑(chartAt H x) x))

case hs
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ UniqueDiffWithinAt 𝕜 (range ↑I) (↑I (↑(chartAt H x) x))

case h
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ DifferentiableAt 𝕜 (fun x' => (x', 0)) (↑I (↑(chartAt H x) x))

Proof state:
case h
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
⊢ DifferentiableAt 𝕜 (fun x' => (x', 0)) (↑I (↑(chartAt H x) x))

Tactic:
exact differentiableAt_id'.prod (differentiableAt_const _)

Result:
no goals

Proof state:
case mk
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
B : (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I) (↑I (↑(chartAt H x) x))) v = (v, 0)
⊢ tangentMap I I.tangent (zeroSection E (TangentSpace I)) { proj := x, snd := v } =
    { proj := { proj := { proj := x, snd := v }.proj, snd := 0 }, snd := ({ proj := x, snd := v }.snd, 0) }

Tactic:
simp (config := { unfoldPartialApp := true }) only [Bundle.zeroSection, tangentMap, mfderiv, A,
  if_pos, chartAt, FiberBundle.chartedSpace_chartAt, TangentBundle.trivializationAt_apply,
  tangentBundleCore, Function.comp_def, ContinuousLinearMap.map_zero, mfld_simps]

Result:
case mk
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
B : (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I) (↑I (↑(chartAt H x) x))) v = (v, 0)
⊢ (fderivWithin 𝕜 (fun x_1 => (↑I (↑(ChartedSpace.chartAt x) (↑(ChartedSpace.chartAt x).symm (↑I.symm x_1))), 0))
        (range ↑I) (↑I (↑(ChartedSpace.chartAt x) x)))
      v =
    (v, 0)

Proof state:
case mk
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
B : (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I) (↑I (↑(chartAt H x) x))) v = (v, 0)
⊢ (fderivWithin 𝕜 (fun x_1 => (↑I (↑(ChartedSpace.chartAt x) (↑(ChartedSpace.chartAt x).symm (↑I.symm x_1))), 0))
        (range ↑I) (↑I (↑(ChartedSpace.chartAt x) x)))
      v =
    (v, 0)

Tactic:
rw [← fderivWithin_inter N] at B

Result:
case mk
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
Is : IsManifold I 1 M
x : M
v : TangentSpace I x
N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))
A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x
B :
  (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))) v = (v, 0)
⊢ (fderivWithin 𝕜 (fun x_1 => (↑I (↑(ChartedSpace.chartAt x) (↑(ChartedSpace.chartAt x).symm (↑I.symm x_1))), 0))
        (range ↑I) (↑I (↑(ChartedSpace.chartAt x) x)))
      v =
    (v, 0)