-- Mathlib/Analysis/Calculus/MeanValue.lean
theorem _root_.lipschitzWith_of_nnnorm_fderiv_le
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
    {C : ℝ≥0} (hf : Differentiable 𝕜 f)
    (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f

Proof state:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
⊢ LipschitzOnWith C f univ

Tactic:
exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ

Result:
no goals

Proof state:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
⊢ LipschitzWith C f

Tactic:
rw [← lipschitzOnWith_univ]

Result:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
⊢ LipschitzOnWith C f univ

Proof state:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
⊢ LipschitzWith C f

Tactic:
letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜

Result:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ LipschitzWith C f

Proof state:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ LipschitzWith C f

Tactic:
let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

Result:
𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
C : ℝ≥0
hf : Differentiable 𝕜 f
bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
⊢ LipschitzWith C f