Theorem: ContinuousLinearMap.strongTopology.continuousSMul
File: Mathlib/Topology/Algebra/Module/StrongTopology.lean

tactic: letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_4
E' : Type ?u.55678
F : Type u_3
F' : Type ?u.55684
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup E'
inst✝¹¹ : Module ℝ E'
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup F'
inst✝⁷ : Module ℝ F'
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace E'
inst✝⁴ : RingHomSurjective σ
inst✝³ : RingHomIsometric σ
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜₂ F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
h𝔖₃ : ∀ (S : Set E), S ∈ 𝔖 → Bornology.IsVonNBounded 𝕜₁ S
⊢ ContinuousSMul 𝕜₂ (E →SL[σ] F)

tactic: haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_4
E' : Type ?u.55678
F : Type u_3
F' : Type ?u.55684
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup E'
inst✝¹¹ : Module ℝ E'
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup F'
inst✝⁷ : Module ℝ F'
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace E'
inst✝⁴ : RingHomSurjective σ
inst✝³ : RingHomIsometric σ
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜₂ F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
h𝔖₃ : ∀ (S : Set E), S ∈ 𝔖 → Bornology.IsVonNBounded 𝕜₁ S
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F
⊢ ContinuousSMul 𝕜₂ (E →SL[σ] F)

tactic: letI : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_4
E' : Type ?u.55678
F : Type u_3
F' : Type ?u.55684
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup E'
inst✝¹¹ : Module ℝ E'
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup F'
inst✝⁷ : Module ℝ F'
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace E'
inst✝⁴ : RingHomSurjective σ
inst✝³ : RingHomIsometric σ
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜₂ F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
h𝔖₃ : ∀ (S : Set E), S ∈ 𝔖 → Bornology.IsVonNBounded 𝕜₁ S
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ ContinuousSMul 𝕜₂ (E →SL[σ] F)

tactic: let φ : (E →SL[σ] F) →ₗ[𝕜₂] E →ᵤ[𝔖] F :=
  ⟨⟨(FunLike.coe : (E →SL[σ] F) → E → F), fun _ _ => rfl⟩, fun _ _ => rfl⟩
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_4
E' : Type ?u.55678
F : Type u_3
F' : Type ?u.55684
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup E'
inst✝¹¹ : Module ℝ E'
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup F'
inst✝⁷ : Module ℝ F'
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace E'
inst✝⁴ : RingHomSurjective σ
inst✝³ : RingHomIsometric σ
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜₂ F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
h𝔖₃ : ∀ (S : Set E), S ∈ 𝔖 → Bornology.IsVonNBounded 𝕜₁ S
this✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this✝ : UniformAddGroup F
this : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
⊢ ContinuousSMul 𝕜₂ (E →SL[σ] F)

tactic: exact
  UniformOnFun.continuousSMul_induced_of_image_bounded 𝕜₂ E F (E →SL[σ] F) h𝔖₁ h𝔖₂ φ ⟨rfl⟩
    fun u s hs => (h𝔖₃ s hs).image u
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_4
E' : Type ?u.55678
F : Type u_3
F' : Type ?u.55684
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup E'
inst✝¹¹ : Module ℝ E'
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup F'
inst✝⁷ : Module ℝ F'
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace E'
inst✝⁴ : RingHomSurjective σ
inst✝³ : RingHomIsometric σ
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜₂ F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
h𝔖₃ : ∀ (S : Set E), S ∈ 𝔖 → Bornology.IsVonNBounded 𝕜₁ S
this✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this✝ : UniformAddGroup F
this : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
φ : (E →SL[σ] F) →ₗ[𝕜₂] E →ᵤ[𝔖] F :=
  { toAddHom := { toFun := FunLike.coe, map_add' := (_ : ∀ (x x_1 : E →SL[σ] F), ↑(x + x_1) = ↑(x + x_1)) },
    map_smul' :=
      (_ :
        ∀ (x : 𝕜₂) (x_1 : E →SL[σ] F),
          AddHom.toFun { toFun := FunLike.coe, map_add' := (_ : ∀ (x x_2 : E →SL[σ] F), ↑(x + x_2) = ↑(x + x_2)) }
              (x • x_1) =
            AddHom.toFun { toFun := FunLike.coe, map_add' := (_ : ∀ (x x_2 : E →SL[σ] F), ↑(x + x_2) = ↑(x + x_2)) }
              (x • x_1)) }
⊢ ContinuousSMul 𝕜₂ (E →SL[σ] F)