Theorem: equicontinuous_of_equicontinuousAt_zero
File: Mathlib/Topology/Algebra/Equicontinuity.lean

Proof steps:
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : EquicontinuousAt (DFunLike.coe ∘ F) 1
⊢ Equicontinuous (DFunLike.coe ∘ F)
state_after: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : EquicontinuousAt (DFunLike.coe ∘ F) 1
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
tactic: rw [equicontinuous_iff_continuous]
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : EquicontinuousAt (DFunLike.coe ∘ F) 1
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
state_after: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F)) 1
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
tactic: rw [equicontinuousAt_iff_continuousAt] at hf
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F)) 1
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
state_after: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F)) 1
φ : G →* ι →ᵤ M := { toFun := swap (DFunLike.coe ∘ F), map_one' := ⋯, map_mul' := ⋯ }
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
tactic: let φ : G →* (ι →ᵤ M) :=
    { toFun := swap ((↑) ∘ F)
      map_one' := by dsimp [UniformFun]; ext; exact map_one _
      map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁷ : TopologicalSpace G
inst✝⁶ : UniformSpace M
inst✝⁵ : Group G
inst✝⁴ : Group M
inst✝³ : TopologicalGroup G
inst✝² : UniformGroup M
inst✝¹ : FunLike hom G M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F)) 1
φ : G →* ι →ᵤ M := { toFun := swap (DFunLike.coe ∘ F), map_one' := ⋯, map_mul' := ⋯ }
⊢ Continuous (⇑UniformFun.ofFun ∘ swap (DFunLike.coe ∘ F))
state_after: no goals
tactic: exact continuous_of_continuousAt_one φ hf