File: Mathlib/Analysis/Normed/Group/HomCompletion.lean

full_name: NormedAddGroupHom.completion
code: /-- The normed group hom induced between completions. -/
def NormedAddGroupHom.completion (f : NormedAddGroupHom G H) :
    NormedAddGroupHom (Completion G) (Completion H) 
goal: NormedAddGroupHom (UniformSpace.Completion G) (UniformSpace.Completion H)

full_name: NormedAddGroupHom.completion_def
code: theorem NormedAddGroupHom.completion_def (f : NormedAddGroupHom G H) (x : Completion G) :
    f.completion x = Completion.map f x 
goal: f.completion x = UniformSpace.Completion.map (⇑f) x

full_name: NormedAddGroupHom.completion_coe_to_fun
code: @[simp]
theorem NormedAddGroupHom.completion_coe_to_fun (f : NormedAddGroupHom G H) :
    (f.completion : Completion G → Completion H) = Completion.map f 
goal: ⇑f.completion = UniformSpace.Completion.map ⇑f

full_name: NormedAddGroupHom.completion_coe
code: theorem NormedAddGroupHom.completion_coe (f : NormedAddGroupHom G H) (g : G) :
    f.completion g = f g 
goal: f.completion (↑G g) = ↑H (f g)

full_name: NormedAddGroupHom.completion_coe'
code: @[simp]
theorem NormedAddGroupHom.completion_coe' (f : NormedAddGroupHom G H) (g : G) :
    Completion.map f g = f g 
goal: UniformSpace.Completion.map (⇑f) (↑G g) = ↑H (f g)

full_name: normedAddGroupHomCompletionHom
code: /-- Completion of normed group homs as a normed group hom. -/
@[simps]
def normedAddGroupHomCompletionHom :
    NormedAddGroupHom G H →+ NormedAddGroupHom (Completion G) (Completion H) 
goal: NormedAddGroupHom G H →+ NormedAddGroupHom (UniformSpace.Completion G) (UniformSpace.Completion H)

full_name: normedAddGroupHomCompletionHom_apply
code: /-- Completion of normed group homs as a normed group hom. -/
@[simps]
def normedAddGroupHomCompletionHom :
    NormedAddGroupHom G H →+ NormedAddGroupHom (Completion G) (Completion H) 
goal: normedAddGroupHomCompletionHom f = f.completion

full_name: NormedAddGroupHom.completion_id
code: @[simp]
theorem NormedAddGroupHom.completion_id :
    (NormedAddGroupHom.id G).completion = NormedAddGroupHom.id (Completion G) 
goal: (NormedAddGroupHom.id G).completion = NormedAddGroupHom.id (UniformSpace.Completion G)

full_name: NormedAddGroupHom.completion_comp
code: theorem NormedAddGroupHom.completion_comp (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K) :
    g.completion.comp f.completion = (g.comp f).completion 
goal: g.completion.comp f.completion = (g.comp f).completion

full_name: NormedAddGroupHom.completion_neg
code: theorem NormedAddGroupHom.completion_neg (f : NormedAddGroupHom G H) :
    (-f).completion = -f.completion 
goal: (-f).completion = -f.completion

full_name: NormedAddGroupHom.completion_add
code: theorem NormedAddGroupHom.completion_add (f g : NormedAddGroupHom G H) :
    (f + g).completion = f.completion + g.completion 
goal: (f + g).completion = f.completion + g.completion

full_name: NormedAddGroupHom.completion_sub
code: theorem NormedAddGroupHom.completion_sub (f g : NormedAddGroupHom G H) :
    (f - g).completion = f.completion - g.completion 
goal: (f - g).completion = f.completion - g.completion

full_name: NormedAddGroupHom.zero_completion
code: @[simp]
theorem NormedAddGroupHom.zero_completion : (0 : NormedAddGroupHom G H).completion = 0 
goal: NormedAddGroupHom.completion 0 = 0

full_name: NormedAddCommGroup.toCompl
code: /-- The map from a normed group to its completion, as a normed group hom. -/
@[simps] -- Porting note: added `@[simps]`
def NormedAddCommGroup.toCompl : NormedAddGroupHom G (Completion G) 
goal: NormedAddGroupHom G (UniformSpace.Completion G)

full_name: NormedAddCommGroup.toCompl_apply
code: /-- The map from a normed group to its completion, as a normed group hom. -/
@[simps] -- Porting note: added `@[simps]`
def NormedAddCommGroup.toCompl : NormedAddGroupHom G (Completion G) 
goal: ∀ (a : G), NormedAddCommGroup.toCompl a = ↑G a

full_name: NormedAddCommGroup.norm_toCompl
code: theorem NormedAddCommGroup.norm_toCompl (x : G) : ‖toCompl x‖ = ‖x‖ 
goal: ‖NormedAddCommGroup.toCompl x‖ = ‖x‖

full_name: NormedAddCommGroup.denseRange_toCompl
code: theorem NormedAddCommGroup.denseRange_toCompl : DenseRange (toCompl : G → Completion G) 
goal: DenseRange ⇑NormedAddCommGroup.toCompl

full_name: NormedAddGroupHom.completion_toCompl
code: @[simp]
theorem NormedAddGroupHom.completion_toCompl (f : NormedAddGroupHom G H) :
    f.completion.comp toCompl = toCompl.comp f 
goal: f.completion.comp NormedAddCommGroup.toCompl = NormedAddCommGroup.toCompl.comp f

full_name: NormedAddGroupHom.norm_completion
code: @[simp]
theorem NormedAddGroupHom.norm_completion (f : NormedAddGroupHom G H) : ‖f.completion‖ = ‖f‖ 
goal: ‖f.completion‖ = ‖f‖

full_name: NormedAddGroupHom.ker_le_ker_completion
code: theorem NormedAddGroupHom.ker_le_ker_completion (f : NormedAddGroupHom G H) :
    (toCompl.comp <| incl f.ker).range ≤ f.completion.ker 
goal: (NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range ≤ f.completion.ker

full_name: NormedAddGroupHom.ker_completion
code: theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
    (h : f.SurjectiveOnWith f.range C) :
    (f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range 
goal: ↑f.completion.ker = closure ↑(NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range

full_name: NormedAddGroupHom.extension
code: /-- If `H` is complete, the extension of `f : NormedAddGroupHom G H` to a
`NormedAddGroupHom (completion G) H`. -/
def NormedAddGroupHom.extension (f : NormedAddGroupHom G H) : NormedAddGroupHom (Completion G) H 
goal: NormedAddGroupHom (UniformSpace.Completion G) H

full_name: NormedAddGroupHom.extension_def
code: theorem NormedAddGroupHom.extension_def (f : NormedAddGroupHom G H) (v : G) :
    f.extension v = Completion.extension f v 
goal: f.extension (↑G v) = UniformSpace.Completion.extension (⇑f) (↑G v)

full_name: NormedAddGroupHom.extension_coe
code: @[simp]
theorem NormedAddGroupHom.extension_coe (f : NormedAddGroupHom G H) (v : G) : f.extension v = f v 
goal: f.extension (↑G v) = f v

full_name: NormedAddGroupHom.extension_coe_to_fun
code: theorem NormedAddGroupHom.extension_coe_to_fun (f : NormedAddGroupHom G H) :
    (f.extension : Completion G → H) = Completion.extension f 
goal: ⇑f.extension = UniformSpace.Completion.extension ⇑f

full_name: NormedAddGroupHom.extension_unique
code: theorem NormedAddGroupHom.extension_unique (f : NormedAddGroupHom G H)
    {g : NormedAddGroupHom (Completion G) H} (hg : ∀ v, f v = g v) : f.extension = g 
goal: f.extension = g