File: Mathlib/Topology/Algebra/Valued/ValuedField.lean

full_name: Valuation.inversion_estimate
code: theorem Valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)
    (h : v (x - y) < min (γ * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) < γ 
goal: v (x⁻¹ - y⁻¹) < ↑γ

full_name: Valued.topologicalDivisionRing
code: /-- The topology coming from a valuation on a division ring makes it a topological division ring
    [BouAC, VI.5.1 middle of Proposition 1] -/
instance (priority := 100) Valued.topologicalDivisionRing [Valued K Γ₀] :
    TopologicalDivisionRing K 
goal: TopologicalDivisionRing K

full_name: ValuedRing.separated
code: /-- A valued division ring is separated. -/
instance (priority := 100) ValuedRing.separated [Valued K Γ₀] : T0Space K 
goal: T0Space K

full_name: Valued.continuous_valuation
code: theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) 
goal: Continuous ⇑Valued.v

full_name: Valued._aux_Mathlib_Topology_Algebra_Valued_ValuedField___macroRules_Valued_termHat_1
code: local notation "hat " => Completion
goal: Lean.Macro

full_name: Valued._aux_Mathlib_Topology_Algebra_Valued_ValuedField___unexpand_UniformSpace_Completion_1
code: local notation "hat " => Completion
goal: Lean.PrettyPrinter.Unexpander

full_name: Valued.completable
code: /-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K 
goal: CompletableTopField K

full_name: Valued.extension
code: /-- The extension of the valuation of a valued field to the completion of the field. -/
noncomputable def extension : hat K → Γ₀ 
goal: UniformSpace.Completion K → Γ₀

full_name: Valued.continuous_extension
code: theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) 
goal: Continuous Valued.extension

full_name: Valued.extension_extends
code: @[simp, norm_cast]
theorem extension_extends (x : K) : extension (x : hat K) = v x 
goal: Valued.extension (↑K x) = Valued.v x

full_name: Valued.extensionValuation
code: /-- the extension of a valuation on a division ring to its completion. -/
noncomputable def extensionValuation : Valuation (hat K) Γ₀ 
goal: Valuation (UniformSpace.Completion K) Γ₀

full_name: Valued.closure_coe_completion_v_lt
code: theorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :
    closure ((↑) '' { x : K | v x < (γ : Γ₀) }) =
    { x : hat K | extensionValuation x < (γ : Γ₀) } 
goal: closure (↑K '' {x | Valued.v x < ↑γ}) = {x | Valued.extensionValuation x < ↑γ}

full_name: Valued.valuedCompletion
code: noncomputable instance valuedCompletion : Valued (hat K) Γ₀ 
goal: Valued (UniformSpace.Completion K) Γ₀

full_name: Valued.valuedCompletion_apply
code: @[simp]
theorem valuedCompletion_apply (x : K) : Valued.v (x : hat K) = v x 
goal: Valued.v (↑K x) = Valued.v x

full_name: Valued.integer
code: /-- A `Valued` version of `Valuation.integer`, enabling the notation `𝒪[K]` for the
valuation integers of a valued field `K`. -/
@[reducible]
def integer : Subring K 
goal: Subring K

full_name: Valued.«_aux_Mathlib_Topology_Algebra_Valued_ValuedField___macroRules_Valued_term𝒪[_]_1»
code: @[inherit_doc]
scoped notation "𝒪[" K "]" => Valued.integer K
goal: Lean.Macro

full_name: Valued._aux_Mathlib_Topology_Algebra_Valued_ValuedField___unexpand_Valued_integer_1
code: @[inherit_doc]
scoped notation "𝒪[" K "]" => Valued.integer K
goal: Lean.PrettyPrinter.Unexpander

full_name: Valued.maximalIdeal
code: /-- An abbrevation for `LocalRing.maximalIdeal 𝒪[K]` of a valued field `K`, enabling the notation
`𝓂[K]` for the maximal ideal in `𝒪[K]` of a valued field `K`. -/
@[reducible]
def maximalIdeal : Ideal 𝒪[K] 
goal: Ideal ↥(Valued.integer K)

full_name: Valued.«_aux_Mathlib_Topology_Algebra_Valued_ValuedField___macroRules_Valued_term𝓂[_]_1»
code: @[inherit_doc]
scoped notation "𝓂[" K "]" => maximalIdeal K
goal: Lean.Macro

full_name: Valued._aux_Mathlib_Topology_Algebra_Valued_ValuedField___unexpand_Valued_maximalIdeal_1
code: @[inherit_doc]
scoped notation "𝓂[" K "]" => maximalIdeal K
goal: Lean.PrettyPrinter.Unexpander

full_name: Valued.ResidueField
code: /-- An abbrevation for `LocalRing.ResidueField 𝒪[K]` of a `Valued` instance, enabling the notation
`𝓀[K]` for the residue field of a valued field `K`. -/
@[reducible]
def ResidueField 
goal: Type u_1

full_name: Valued._aux_Mathlib_Topology_Algebra_Valued_ValuedField___unexpand_Valued_ResidueField_1
code: @[inherit_doc]
scoped notation "𝓀[" K "]" => ResidueField K
goal: Lean.PrettyPrinter.Unexpander

full_name: Valued.«_aux_Mathlib_Topology_Algebra_Valued_ValuedField___macroRules_Valued_term𝓀[_]_1»
code: @[inherit_doc]
scoped notation "𝓀[" K "]" => ResidueField K
goal: Lean.Macro