File: Mathlib/Topology/Algebra/UniformRing.lean

full_name: UniformSpace.Completion.one
code: instance one : One (Completion α) 
goal: One (UniformSpace.Completion α)

full_name: UniformSpace.Completion.mul
code: instance mul : Mul (Completion α) 
goal: Mul (UniformSpace.Completion α)

full_name: UniformSpace.Completion.coe_one
code: @[norm_cast]
theorem coe_one : ((1 : α) : Completion α) = 1 
goal: ↑α 1 = 1

full_name: UniformSpace.Completion.coe_mul
code: @[norm_cast]
theorem coe_mul (a b : α) : ((a * b : α) : Completion α) = a * b 
goal: ↑α (a * b) = ↑α a * ↑α b

full_name: UniformSpace.Completion.continuous_mul
code: theorem continuous_mul : Continuous fun p : Completion α × Completion α => p.1 * p.2 
goal: Continuous fun p => p.1 * p.2

full_name: UniformSpace.Completion.Continuous.mul
code: theorem Continuous.mul {β : Type*} [TopologicalSpace β] {f g : β → Completion α}
    (hf : Continuous f) (hg : Continuous g) : Continuous fun b => f b * g b 
goal: Continuous fun b => f b * g b

full_name: UniformSpace.Completion.ring
code: instance ring : Ring (Completion α) 
goal: Ring (UniformSpace.Completion α)

full_name: UniformSpace.Completion.coeRingHom
code: /-- The map from a uniform ring to its completion, as a ring homomorphism. -/
def coeRingHom : α →+* Completion α 
goal: α →+* UniformSpace.Completion α

full_name: UniformSpace.Completion.continuous_coeRingHom
code: theorem continuous_coeRingHom : Continuous (coeRingHom : α → Completion α) 
goal: Continuous ⇑UniformSpace.Completion.coeRingHom

full_name: UniformSpace.Completion.extensionHom
code: /-- The completion extension as a ring morphism. -/
def extensionHom [CompleteSpace β] [T0Space β] : Completion α →+* β 
goal: UniformSpace.Completion α →+* β

full_name: UniformSpace.Completion.topologicalRing
code: instance topologicalRing : TopologicalRing (Completion α) 
goal: TopologicalRing (UniformSpace.Completion α)

full_name: UniformSpace.Completion.mapRingHom
code: /-- The completion map as a ring morphism. -/
def mapRingHom (hf : Continuous f) : Completion α →+* Completion β 
goal: UniformSpace.Completion α →+* UniformSpace.Completion β

full_name: UniformSpace.Completion.map_smul_eq_mul_coe
code: @[simp]
theorem map_smul_eq_mul_coe (r : R) :
    Completion.map (r • ·) = ((algebraMap R A r : Completion A) * ·) 
goal: (UniformSpace.Completion.map fun x => r • x) = fun x => ↑A ((algebraMap R A) r) * x

full_name: UniformSpace.Completion.algebra
code: instance algebra : Algebra R (Completion A) 
goal: Algebra R (UniformSpace.Completion A)

full_name: UniformSpace.Completion.algebraMap_def
code: theorem algebraMap_def (r : R) :
    algebraMap R (Completion A) r = (algebraMap R A r : Completion A) 
goal: (algebraMap R (UniformSpace.Completion A)) r = ↑A ((algebraMap R A) r)

full_name: UniformSpace.Completion.commRing
code: instance commRing : CommRing (Completion R) 
goal: CommRing (UniformSpace.Completion R)

full_name: UniformSpace.Completion.algebra'
code: /-- A shortcut instance for the common case -/
instance algebra' : Algebra R (Completion R) 
goal: Algebra R (UniformSpace.Completion R)

full_name: UniformSpace.inseparableSetoid_ring
code: theorem inseparableSetoid_ring (α) [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    inseparableSetoid α = Submodule.quotientRel (Ideal.closure ⊥) 
goal: inseparableSetoid α = Submodule.quotientRel ⊥.closure

full_name: UniformSpace.ring_sep_quot
code: @[deprecated UniformSpace.inseparableSetoid_ring (since := "2024-02-16")]
theorem ring_sep_quot (α : Type u) [r : CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    SeparationQuotient α = (α ⧸ (⊥ : Ideal α).closure) 
goal: SeparationQuotient α = (α ⧸ ⊥.closure)

full_name: UniformSpace.sepQuotHomeomorphRingQuot
code: /-- Given a topological ring `α` equipped with a uniform structure that makes subtraction uniformly
continuous, get an homeomorphism between the separated quotient of `α` and the quotient ring
corresponding to the closure of zero. -/
def sepQuotHomeomorphRingQuot (α) [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    SeparationQuotient α ≃ₜ α ⧸ (⊥ : Ideal α).closure 
goal: SeparationQuotient α ≃ₜ α ⧸ ⊥.closure

full_name: UniformSpace.commRing
code: instance commRing [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    CommRing (SeparationQuotient α) 
goal: CommRing (SeparationQuotient α)

full_name: UniformSpace.sepQuotRingEquivRingQuot
code: /-- Given a topological ring `α` equipped with a uniform structure that makes subtraction uniformly
continuous, get an equivalence between the separated quotient of `α` and the quotient ring
corresponding to the closure of zero. -/
def sepQuotRingEquivRingQuot (α) [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    SeparationQuotient α ≃+* α ⧸ (⊥ : Ideal α).closure 
goal: SeparationQuotient α ≃+* α ⧸ ⊥.closure

full_name: UniformSpace.topologicalRing
code: instance topologicalRing [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    TopologicalRing (SeparationQuotient α) 
goal: TopologicalRing (SeparationQuotient α)

full_name: DenseInducing.extendRingHom
code: /-- The dense inducing extension as a ring homomorphism. -/
noncomputable def DenseInducing.extendRingHom {i : α →+* β} {f : α →+* γ} (ue : UniformInducing i)
    (dr : DenseRange i) (hf : UniformContinuous f) : β →+* γ 
goal: β →+* γ