-- Mathlib/Analysis/Calculus/ContDiff/Basic.lean
theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left (g : F ≃ₗᵢ[𝕜] G) (f : E → F)
    (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ) :
    ‖iteratedFDerivWithin 𝕜 i (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
g : F ≃ₗᵢ[𝕜] G
f : E → F
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Tactic:
have :
  iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
    (g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) :=
  g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_left f hs hx i

Result:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
g : F ≃ₗᵢ[𝕜] G
f : E → F
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    (↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
      (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
g : F ≃ₗᵢ[𝕜] G
f : E → F
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    (↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
      (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖(↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
        (iteratedFDerivWithin 𝕜 i f s x)‖ =
    ‖iteratedFDerivWithin 𝕜 i f s x‖

Tactic:
apply LinearIsometry.norm_compContinuousMultilinearMap g.toLinearIsometry

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
g : F ≃ₗᵢ[𝕜] G
f : E → F
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    (↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
      (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Tactic:
rw [this]

Result:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
g : F ≃ₗᵢ[𝕜] G
f : E → F
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    (↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
      (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖(↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).compContinuousMultilinearMap
        (iteratedFDerivWithin 𝕜 i f s x)‖ =
    ‖iteratedFDerivWithin 𝕜 i f s x‖