-- Mathlib/Topology/Algebra/Module/UniformConvergence.lean
/-- Let `E` be a topological vector space over a normed field `𝕜`, let `α` be any type.
Let `H` be a submodule of `α →ᵤ E` such that the range of each `f ∈ H` is von Neumann bounded.
Then `H` is a topological vector space over `𝕜`,
i.e., the pointwise scalar multiplication is continuous in both variables.

For convenience we require that `H` is a vector space over `𝕜`
with a topology induced by `UniformFun.ofFun ∘ φ`, where `φ : H →ₗ[𝕜] (α → E)`. -/
lemma UniformFun.continuousSMul_induced_of_range_bounded (φ : hom)
    (hφ : IsInducing (ofFun ∘ φ)) (h : ∀ u : H, Bornology.IsVonNBounded 𝕜 (Set.range (φ u))) :
    ContinuousSMul 𝕜 H

Proof state:
case hsmul_right
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}
u : H
U : Set E
hU : U ∈ 𝓝 0
⊢ ∀ᶠ (x : 𝕜) in 𝓝 0, ∀ (x_1 : α), x • φ u x_1 ∈ U

Tactic:
simpa only [Set.mapsTo_range_iff] using (h u hU).eventually_nhds_zero (mem_of_mem_nhds hU)

Result:
no goals

Proof state:
case hsmul_right
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}
⊢ ∀ (m₀ : H) {i : Set E}, i ∈ 𝓝 0 → ∀ᶠ (x : 𝕜) in 𝓝 0, x • m₀ ∈ {u | ∀ (x : α), φ u x ∈ i}

Tactic:
intro u U hU

Result:
case hsmul_right
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}
u : H
U : Set E
hU : U ∈ 𝓝 0
⊢ ∀ᶠ (x : 𝕜) in 𝓝 0, x • u ∈ {u | ∀ (x : α), φ u x ∈ U}

Proof state:
case hsmul
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}
U : Set E
hU : U ∈ 𝓝 0
⊢ ∃ V ∈ 𝓝 0, ∃ j ∈ 𝓝 0, V • {u | ∀ (x : α), φ u x ∈ j} ⊆ {u | ∀ (x : α), φ u x ∈ U}

Tactic:
have : Tendsto (fun x : 𝕜 × E ↦ x.1 • x.2) (𝓝 0) (𝓝 0) :=
  continuous_smul.tendsto' _ _ (zero_smul _ _)

Result:
case hsmul
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this✝ : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}
U : Set E
hU : U ∈ 𝓝 0
this : Tendsto (fun x => x.1 • x.2) (𝓝 0) (𝓝 0)
⊢ ∃ V ∈ 𝓝 0, ∃ j ∈ 𝓝 0, V • {u | ∀ (x : α), φ u x ∈ j} ⊆ {u | ∀ (x : α), φ u x ∈ U}

Proof state:
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
⊢ ContinuousSMul 𝕜 H

Tactic:
have : IsTopologicalAddGroup H :=
  let ofFun' : (α → E) →+ (α →ᵤ E) := AddMonoidHom.id _
  IsInducing.topologicalAddGroup (ofFun'.comp (φ : H →+ (α → E))) hφ

Result:
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
⊢ ContinuousSMul 𝕜 H

Proof state:
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : UniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this : IsTopologicalAddGroup H
⊢ (comap (⇑ofFun ∘ ⇑φ) (𝓝 (ofFun 0))).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}

Tactic:
exact UniformFun.hasBasis_nhds_zero.comap _

Result:
no goals