/-- Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.

For convenience, we don't literally ask for `H : Submodule (α →ᵤ[𝔖] E)`. Instead, we prove the
result for any vector space `H` equipped with a linear inducing to `α →ᵤ[𝔖] E`, which is often
easier to use. We also state the `Submodule` version as
`UniformOnFun.continuousSMul_submodule_of_image_bounded`. -/
theorem UniformOnFun.continuousSMul_induced_of_image_bounded (h𝔖₁ : 𝔖.Nonempty)
    (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) (φ : hom) (hφ : Inducing φ)
    (h : ∀ u : H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 ((φ u : α → E) '' s)) :
    ContinuousSMul 𝕜 H := by
  have : TopologicalAddGroup H := by
    rw [hφ.induced]
    exact topologicalAddGroup_induced φ
  have : (𝓝 0 : Filter H).HasBasis _ _ := by
    rw [hφ.induced, nhds_induced, map_zero]
    

Proof state:
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ HasBasis (comap (⇑φ) (𝓝 0)) ?m.99988 ?m.99989
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ Sort ?u.99979
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Prop
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Set H
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ Sort ?u.99979
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Prop
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Set H
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ Sort ?u.99979
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Prop
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup H
inst✝⁷ : Module 𝕜 H
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace H
inst✝³ : UniformSpace E
inst✝² : UniformAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
𝔖 : Set (Set α)
inst✝ : LinearMapClass hom 𝕜 H (α →ᵤ[𝔖] E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
φ : hom
hφ : Inducing ⇑φ
h : ∀ (u : H), ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (φ u '' s)
this : TopologicalAddGroup H
⊢ ?m.99987 → Set H

Next tactic:
exact (UniformOnFun.hasBasis_nhds_zero 𝔖 h𝔖₁ h𝔖₂).comap φ