awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this_3 : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist
          (‖x_1 - (n, x).2‖⁻¹ *
            ‖HSub.hSub (β := G) (f (n, x).1 x_1 - f (n, x).1 (n, x).2)
                ((f' (n, x).1 (n, x).2 : (a : E) → G) (x_1 - (n, x).2) : G)‖)
          0 <
        ε
y : E
hy :
  dist
      (‖y - (n, x).2‖⁻¹ *
        ‖HSub.hSub (β := G) (f (n, x).1 y - f (n, x).1 (n, x).2)
            ((f' (n, x).1 (n, x).2 : (a : E) → G) (y - (n, x).2) : G)‖)
      0 <
    ε
⊢ dist
      (HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖(n, y).2 - x‖ : 𝕜)⁻¹
        (f (n, y).1 (n, y).2 - f (n, y).1 x -
          HSub.hSub (α := G) (β := G) ((f' (n, y).1 x : (a : E) → G) (n, y).2 : G)
            ((f' (n, y).1 x : (a : E) → G) x : G)))
      0 <
    ε
have refine_2 : ∀
  (hy :
    ‖‖y - (n, x).2‖⁻¹ *
            ‖HSub.hSub (β := G) (f (n, x).1 y - f (n, x).1 (n, x).2)
                ((f' (n, x).1 (n, x).2 : (a : E) → G) (y - (n, x).2) : G)‖ -
          0‖ <
      ε),
  ‖HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖(n, y).2 - x‖ : 𝕜)⁻¹
          (f (n, y).1 (n, y).2 - f (n, y).1 x -
            HSub.hSub (α := G) (β := G) ((f' (n, y).1 x : (a : E) → G) (n, y).2 : G)
              ((f' (n, y).1 x : (a : E) → G) x : G)) -
        0‖ <
    ε
Tactic: rw [dist_eq_norm] at hy ⊢
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this_3 : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist
          (‖x_1 - (n, x).2‖⁻¹ *
            ‖HSub.hSub (β := G) (f (n, x).1 x_1 - f (n, x).1 (n, x).2)
                ((f' (n, x).1 (n, x).2 : (a : E) → G) (x_1 - (n, x).2) : G)‖)
          0 <
        ε
y : E
hy :
  ‖‖y - (n, x).2‖⁻¹ *
          ‖HSub.hSub (β := G) (f (n, x).1 y - f (n, x).1 (n, x).2)
              ((f' (n, x).1 (n, x).2 : (a : E) → G) (y - (n, x).2) : G)‖ -
        0‖ <
    ε
⊢ ‖HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖(n, y).2 - x‖ : 𝕜)⁻¹
          (f (n, y).1 (n, y).2 - f (n, y).1 x -
            HSub.hSub (α := G) (β := G) ((f' (n, y).1 x : (a : E) → G) (n, y).2 : G)
              ((f' (n, y).1 x : (a : E) → G) x : G)) -
        0‖ <
    ε
have refine_2 : ∀
  (hy :
    ‖y - x‖⁻¹ *
        ‖f n y - f n x - HSub.hSub (α := G) (β := G) ((f' n x : (a : E) → G) y : G) ((f' n x : (a : E) → G) x : G)‖ <
      ε),
  ‖HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖y - x‖ : 𝕜)⁻¹
        (f n y - f n x - HSub.hSub (α := G) (β := G) ((f' n x : (a : E) → G) y : G) ((f' n x : (a : E) → G) x : G))‖ <
    ε
Tactic: simp only [sub_zero, map_sub, norm_mul, norm_inv, norm_norm] at hy ⊢
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
⊢ Tendsto (β := G)
    (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G))
    (l.curry (𝓝 x)) (𝓝 0)
have refine_3 : Tendsto (β := G)
  (fun (a : ι × E) ↦
    HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G))
  (l ×ˢ 𝓝 x) (𝓝 0)
Tactic: refine Tendsto.mono_left ?_ curry_le_prod
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (hypothesis)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
⊢ Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
have hf'₁ : ∀ ε > 0, ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
Tactic: rw [Metric.tendstoUniformlyOnFilter_iff] at hf'
Cleared hypotheses: hf', this_2, this
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (hypothesis)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
⊢ Tendsto (β := G)
    (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G))
    (l ×ˢ 𝓝 x) (𝓝 0)
have h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
Tactic: have h1 : Tendsto (fun n : ι × E => g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0) := by
      rw [Metric.tendstoUniformlyOnFilter_iff] at hf'
      exact Metric.tendsto_nhds.mpr fun ε hε => by simpa [dist_eq_norm] using hf' ε hε
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
⊢ Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
have rw : ∀ (h1 : ∀ ε > 0, ∀ᶠ (x : ι × E) in l ×ˢ 𝓝 x, dist (g' x.2 - f' x.1 x.2) 0 < ε),
  ∀ ε > 0, ∀ᶠ (x_1 : ι) in l, dist (g' x - f' x_1 x) 0 < ε
Tactic: rw [Metric.tendsto_nhds] at h1 ⊢
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (hypothesis)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
⊢ Tendsto (β := G)
    (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G))
    (l ×ˢ 𝓝 x) (𝓝 0)
have h2 : Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
Tactic: have h2 : Tendsto (fun n : ι => g' x - f' n x) l (𝓝 0) := by
      rw [Metric.tendsto_nhds] at h1 ⊢
      exact fun ε hε => (h1 ε hε).curry.mono fun n hn => hn.self_of_nhds
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
⊢ Tendsto (β := G)
    (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G))
    (l ×ˢ 𝓝 x) (𝓝 0)
have refine_3 : ∀ (n : ι × E),
  ‖HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖n.2 - x‖ : 𝕜)⁻¹ ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G)‖ ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
Tactic: refine squeeze_zero_norm ?_
      (tendsto_zero_iff_norm_tendsto_zero.mp (tendsto_fst.comp (h2.prodMap tendsto_id)))
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
⊢ ‖HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖n.2 - x‖ : 𝕜)⁻¹ ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G)‖ ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
have refine_3 : ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
  norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
Tactic: simp_rw [norm_smul, norm_inv, RCLike.norm_coe_norm]
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
⊢ ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
have pos : ∀ (hx : x = n.2),
  ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
Tactic: by_cases hx : x = n.2
-- Lean 4 proof step from Mathlib.Analysis.Calculus.UniformLimitsDeriv (goal)
Goal:
ι : Type u_1
l : Filter ι
E : Type u_2
inst : NormedAddCommGroup E
𝕜 : Type u_3
inst_1 : NontriviallyNormedField 𝕜
inst_2 : IsRCLikeNormedField 𝕜
inst_3 : NormedSpace 𝕜 E
G : Type u_4
inst_4 : NormedAddCommGroup G
inst_5 : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst_6 : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun (n : ι) ↦ f n y) l (𝓝 (g y))
this_1 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this_2 :
  (fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
        (HSub.hSub (β := G) (g a.2 - g x) ((g' x : (a : E) → G) (a.2 - x) : G))) =
    (HAdd.hAdd (α := (a : ι × E) → G)
        (fun (a : ι × E) ↦
          HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹ (g a.2 - g x - (f a.1 a.2 - f a.1 x)))
        fun (a : ι × E) ↦
        HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (↑‖a.2 - x‖ : 𝕜)⁻¹
          (f a.1 a.2 - f a.1 x -
            HSub.hSub (α := G) (β := G) ((f' a.1 x : (a : E) → G) a.2 : G) ((f' a.1 x : (a : E) → G) x : G))) +
      fun (a : ι × E) ↦
      HSMul.hSMul.{u_3, u_4, u_4} (α := 𝕜) (β := G) (↑‖a.2 - x‖ : 𝕜)⁻¹ ((f' a.1 x - g' x : (a : E) → G) (a.2 - x) : G)
this : 𝓝 0 = 𝓝 (HAdd.hAdd (α := G) 0 0 + 0)
h1 : Tendsto (β := E →L[𝕜] G) (fun (n : ι × E) ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (β := E →L[𝕜] G) (fun (n : ι) ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
pos :
  ∀ (hx : x = n.2),
    ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
      norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
⊢ ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
have neg : ∀ (hx : ¬x = n.2),
  ‖n.2 - x‖⁻¹ * norm.{u_4} (E := G) ((f' n.1 x - g' x : (a : E) → G) (n.2 - x) : G) ≤
    norm (E := E →L[𝕜] G) ((Prod.fst ∘ Prod.map (fun (n : ι) ↦ g' x - f' n x) id) n)
Tactic: by_cases hx : x = n.2