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

tactic: rw [uniformEquicontinuous_iff_uniformContinuous]
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : EquicontinuousAt (FunLike.coe ∘ F) 1
⊢ UniformEquicontinuous (FunLike.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✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : EquicontinuousAt (FunLike.coe ∘ F) 1
⊢ UniformContinuous (↑UniformFun.ofFun ∘ swap (FunLike.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✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
⊢ UniformContinuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))

tactic: exact uniformContinuous_of_continuousAt_one φ hf
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
φ : G →* ι →ᵤ M :=
  { toOneHom := { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) },
    map_mul' :=
      (_ :
        ∀ (a b : G),
          OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } (a * b) =
            OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } a *
              OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } b) }
⊢ UniformContinuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))

tactic: dsimp [UniformFun]
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
⊢ swap (FunLike.coe ∘ F) 1 = 1

tactic: ext
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
⊢ swap (FunLike.coe ∘ F) 1 = 1

tactic: exact map_one _
state_before: case h
ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
x✝ : ι
⊢ swap (FunLike.coe ∘ F) 1 x✝ = OfNat.ofNat 1 x✝

tactic: dsimp [UniformFun]
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
a b : G
⊢ OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } (a * b) =
    OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } a *
      OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } b

tactic: ext
state_before: ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
a b : G
⊢ swap (FunLike.coe ∘ F) (a * b) = swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b

tactic: exact map_mul _ _ _
state_before: case h
ι : Type u_1
G : Type u_2
M : Type u_3
hom : Type u_4
inst✝⁶ : UniformSpace G
inst✝⁵ : UniformSpace M
inst✝⁴ : Group G
inst✝³ : Group M
inst✝² : UniformGroup G
inst✝¹ : UniformGroup M
inst✝ : MonoidHomClass hom G M
F : ι → hom
hf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1
a b : G
x✝ : ι
⊢ swap (FunLike.coe ∘ F) (a * b) x✝ = (swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b) x✝