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

Proof steps:
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this : UniformSpace G := TopologicalAddGroup.toUniformSpace G
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
tactic: letI : UniformSpace G := TopologicalAddGroup.toUniformSpace G
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this : UniformSpace G := TopologicalAddGroup.toUniformSpace G
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝ : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this : UniformAddGroup G
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
tactic: haveI : UniformAddGroup G := comm_topologicalAddGroup_is_uniform
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝ : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this : UniformAddGroup G
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝¹ : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝ : UniformAddGroup G
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
tactic: letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝¹ : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝ : UniformAddGroup G
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝² : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝¹ : UniformAddGroup G
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
tactic: haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝² : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝¹ : UniformAddGroup G
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ Continuous { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝² : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝¹ : UniformAddGroup G
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ Continuous
    ((⇑(UniformOnFun.ofFun {S | Bornology.IsVonNBounded 𝕜₁ S}) ∘ DFunLike.coe) ∘
      { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun)
tactic: rw [(UniformConvergenceCLM.embedding_coeFn _ _ _).continuous_iff]
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup F
inst✝² : TopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this✝² : UniformSpace G := TopologicalAddGroup.toUniformSpace G
this✝¹ : UniformAddGroup G
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ Continuous
    ((⇑(UniformOnFun.ofFun {S | Bornology.IsVonNBounded 𝕜₁ S}) ∘ DFunLike.coe) ∘
      { toFun := fun f => L.comp f, map_add' := ⋯, map_smul' := ⋯ }.toFun)
state_after: no goals
tactic: exact
      (UniformOnFun.postcomp_uniformContinuous L.uniformContinuous).continuous.comp
        (UniformConvergenceCLM.embedding_coeFn _ _ _).continuous