-- Mathlib/Analysis/Calculus/ContDiff/Basic.lean
theorem LinearIsometry.norm_iteratedFDerivWithin_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G)
    (hf : ContDiffWithinAt 𝕜 n f s x) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : 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
n : WithTop ℕ∞
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)‖ =
    ‖iteratedFDerivWithin 𝕜 i f s x‖

Tactic:
apply LinearIsometry.norm_compContinuousMultilinearMap

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
n : WithTop ℕ∞
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    g.toContinuousLinearMap.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
n : WithTop ℕ∞
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i 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
n : WithTop ℕ∞
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑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.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) :=
  g.toContinuousLinearMap.iteratedFDerivWithin_comp_left hf hs hx hi

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
n : WithTop ℕ∞
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖