File: Mathlib/Topology/Algebra/UniformField.lean

full_name: _aux_Mathlib_Topology_Algebra_UniformField___unexpand_UniformSpace_Completion_1
code: local notation "hat" => Completion
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Topology_Algebra_UniformField___macroRules_termHat_1
code: local notation "hat" => Completion
goal: Lean.Macro

full_name: CompletableTopField.nice
code: /-- A topological field is completable if it is separated and the image under
the mapping x ↦ x⁻¹ of every Cauchy filter (with respect to the additive uniform structure)
which does not have a cluster point at 0 is a Cauchy filter
(with respect to the additive uniform structure). This ensures the completion is
a field.
-/
class CompletableTopField extends T0Space K : Prop where
  nice : ∀ F : Filter K, Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (map (fun x => x⁻¹) F)
goal: Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (Filter.map (fun x => x⁻¹) F)

full_name: CompletableTopField
code: /-- A topological field is completable if it is separated and the image under
the mapping x ↦ x⁻¹ of every Cauchy filter (with respect to the additive uniform structure)
which does not have a cluster point at 0 is a Cauchy filter
(with respect to the additive uniform structure). This ensures the completion is
a field.
-/
class CompletableTopField extends T0Space K : Prop where
  nice : ∀ F : Filter K, Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (map (fun x => x⁻¹) F)
goal: Prop

full_name: UniformSpace.Completion.instNontrivialOfT0Space
code: instance (priority := 100) [T0Space K] : Nontrivial (hat K) 
goal: Nontrivial (hat K)

full_name: UniformSpace.Completion.hatInv
code: /-- extension of inversion to the completion of a field. -/
def hatInv : hat K → hat K 
goal: hat K → hat K

full_name: UniformSpace.Completion.continuous_hatInv
code: theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
    ContinuousAt hatInv x 
goal: ContinuousAt UniformSpace.Completion.hatInv x

full_name: UniformSpace.Completion.instInvCompletion
code: instance instInvCompletion : Inv (hat K) 
goal: Inv (hat K)

full_name: UniformSpace.Completion.hatInv_extends
code: theorem hatInv_extends {x : K} (h : x ≠ 0) : hatInv (x : hat K) = ↑(x⁻¹ : K) 
goal: (↑K x).hatInv = ↑K x⁻¹

full_name: UniformSpace.Completion.coe_inv
code: @[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) 
goal: (↑K x)⁻¹ = ↑K x⁻¹

full_name: UniformSpace.Completion.mul_hatInv_cancel
code: theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 
goal: x * x.hatInv = 1

full_name: UniformSpace.Completion.instField
code: instance instField : Field (hat K) 
goal: Field (hat K)

full_name: UniformSpace.Completion.instTopologicalDivisionRing
code: instance : TopologicalDivisionRing (hat K) 
goal: TopologicalDivisionRing (hat K)

full_name: Subfield.completableTopField
code: instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K 
goal: CompletableTopField ↥K

full_name: completableTopField_of_complete
code: instance (priority := 100) completableTopField_of_complete (L : Type*) [Field L] [UniformSpace L]
    [TopologicalDivisionRing L] [T0Space L] [CompleteSpace L] : CompletableTopField L 
goal: CompletableTopField L