Theorem: UniformConvergenceCLM.topologicalSpace_mono
File: Mathlib/Topology/Algebra/Module/StrongTopology.lean

Proof steps:
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
tactic: letI := TopologicalAddGroup.toUniformSpace F
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
tactic: haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
state_after: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ UniformSpace.toTopologicalSpace ≤ UniformSpace.toTopologicalSpace
tactic: simp_rw [← uniformity_toTopologicalSpace_eq]
state_before: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
⊢ UniformSpace.toTopologicalSpace ≤ UniformSpace.toTopologicalSpace
state_after: no goals
tactic: exact UniformSpace.toTopologicalSpace_mono (uniformSpace_mono σ F h)