File: Mathlib/Analysis/Normed/Field/Basic.lean

full_name: NonUnitalSeminormedRing.norm_mul
code: /-- A non-unital seminormed ring is a not-necessarily-unital ring
endowed with a seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalSeminormedRing (α : Type*) extends Norm α, NonUnitalRing α,
  PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: ‖a * b‖ ≤ ‖a‖ * ‖b‖

full_name: NonUnitalSeminormedRing
code: /-- A non-unital seminormed ring is a not-necessarily-unital ring
endowed with a seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalSeminormedRing (α : Type*) extends Norm α, NonUnitalRing α,
  PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: Type u_5

full_name: NonUnitalSeminormedRing.dist_eq
code: /-- A non-unital seminormed ring is a not-necessarily-unital ring
endowed with a seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalSeminormedRing (α : Type*) extends Norm α, NonUnitalRing α,
  PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: dist x y = ‖x - y‖

full_name: SeminormedRing.norm_mul
code: /-- A seminormed ring is a ring endowed with a seminorm which satisfies the inequality
`‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class SeminormedRing (α : Type*) extends Norm α, Ring α, PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: ‖a * b‖ ≤ ‖a‖ * ‖b‖

full_name: SeminormedRing
code: /-- A seminormed ring is a ring endowed with a seminorm which satisfies the inequality
`‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class SeminormedRing (α : Type*) extends Norm α, Ring α, PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: Type u_5

full_name: SeminormedRing.dist_eq
code: /-- A seminormed ring is a ring endowed with a seminorm which satisfies the inequality
`‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class SeminormedRing (α : Type*) extends Norm α, Ring α, PseudoMetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: dist x y = ‖x - y‖

full_name: SeminormedRing.toNonUnitalSeminormedRing
code: /-- A seminormed ring is a non-unital seminormed ring. -/
instance (priority := 100) SeminormedRing.toNonUnitalSeminormedRing [β : SeminormedRing α] :
    NonUnitalSeminormedRing α 
goal: NonUnitalSeminormedRing α

full_name: NonUnitalNormedRing.dist_eq
code: /-- A non-unital normed ring is a not-necessarily-unital ring
endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalNormedRing (α : Type*) extends Norm α, NonUnitalRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: dist x y = ‖x - y‖

full_name: NonUnitalNormedRing
code: /-- A non-unital normed ring is a not-necessarily-unital ring
endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalNormedRing (α : Type*) extends Norm α, NonUnitalRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: Type u_5

full_name: NonUnitalNormedRing.norm_mul
code: /-- A non-unital normed ring is a not-necessarily-unital ring
endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalNormedRing (α : Type*) extends Norm α, NonUnitalRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: ‖a * b‖ ≤ ‖a‖ * ‖b‖

full_name: NonUnitalNormedRing.toNonUnitalSeminormedRing
code: /-- A non-unital normed ring is a non-unital seminormed ring. -/
instance (priority := 100) NonUnitalNormedRing.toNonUnitalSeminormedRing
    [β : NonUnitalNormedRing α] : NonUnitalSeminormedRing α 
goal: NonUnitalSeminormedRing α

full_name: NormedRing.dist_eq
code: /-- A normed ring is a ring endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NormedRing (α : Type*) extends Norm α, Ring α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: dist x y = ‖x - y‖

full_name: NormedRing
code: /-- A normed ring is a ring endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NormedRing (α : Type*) extends Norm α, Ring α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: Type u_5

full_name: NormedRing.norm_mul
code: /-- A normed ring is a ring endowed with a norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NormedRing (α : Type*) extends Norm α, Ring α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is submultiplicative. -/
  norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b
goal: ‖a * b‖ ≤ ‖a‖ * ‖b‖

full_name: NormedDivisionRing.dist_eq
code: /-- A normed division ring is a division ring endowed with a seminorm which satisfies the equality
`‖x y‖ = ‖x‖ ‖y‖`. -/
class NormedDivisionRing (α : Type*) extends Norm α, DivisionRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: dist x y = ‖x - y‖

full_name: NormedDivisionRing
code: /-- A normed division ring is a division ring endowed with a seminorm which satisfies the equality
`‖x y‖ = ‖x‖ ‖y‖`. -/
class NormedDivisionRing (α : Type*) extends Norm α, DivisionRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: Type u_5

full_name: NormedDivisionRing.norm_mul'
code: /-- A normed division ring is a division ring endowed with a seminorm which satisfies the equality
`‖x y‖ = ‖x‖ ‖y‖`. -/
class NormedDivisionRing (α : Type*) extends Norm α, DivisionRing α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: ‖a * b‖ = ‖a‖ * ‖b‖

full_name: NormedDivisionRing.toNormedRing
code: /-- A normed division ring is a normed ring. -/
instance (priority := 100) NormedDivisionRing.toNormedRing [β : NormedDivisionRing α] :
    NormedRing α 
goal: NormedRing α

full_name: NormedRing.toSeminormedRing
code: /-- A normed ring is a seminormed ring. -/
instance (priority := 100) NormedRing.toSeminormedRing [β : NormedRing α] : SeminormedRing α 
goal: SeminormedRing α

full_name: NormedRing.toNonUnitalNormedRing
code: /-- A normed ring is a non-unital normed ring. -/
instance (priority := 100) NormedRing.toNonUnitalNormedRing [β : NormedRing α] :
    NonUnitalNormedRing α 
goal: NonUnitalNormedRing α

full_name: NonUnitalSeminormedCommRing
code: /-- A non-unital seminormed commutative ring is a non-unital commutative ring endowed with a
seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalSeminormedCommRing (α : Type*) extends NonUnitalSeminormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: Type u_5

full_name: NonUnitalSeminormedCommRing.mul_comm
code: /-- A non-unital seminormed commutative ring is a non-unital commutative ring endowed with a
seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalSeminormedCommRing (α : Type*) extends NonUnitalSeminormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: x * y = y * x

full_name: NonUnitalNormedCommRing
code: /-- A non-unital normed commutative ring is a non-unital commutative ring endowed with a
norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalNormedCommRing (α : Type*) extends NonUnitalNormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: Type u_5

full_name: NonUnitalNormedCommRing.mul_comm
code: /-- A non-unital normed commutative ring is a non-unital commutative ring endowed with a
norm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NonUnitalNormedCommRing (α : Type*) extends NonUnitalNormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: x * y = y * x

full_name: NonUnitalNormedCommRing.toNonUnitalSeminormedCommRing
code: /-- A non-unital normed commutative ring is a non-unital seminormed commutative ring. -/
instance (priority := 100) NonUnitalNormedCommRing.toNonUnitalSeminormedCommRing
    [β : NonUnitalNormedCommRing α] : NonUnitalSeminormedCommRing α 
goal: NonUnitalSeminormedCommRing α

full_name: SeminormedCommRing.mul_comm
code: /-- A seminormed commutative ring is a commutative ring endowed with a seminorm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class SeminormedCommRing (α : Type*) extends SeminormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: x * y = y * x

full_name: SeminormedCommRing
code: /-- A seminormed commutative ring is a commutative ring endowed with a seminorm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class SeminormedCommRing (α : Type*) extends SeminormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: Type u_5

full_name: NormedCommRing
code: /-- A normed commutative ring is a commutative ring endowed with a norm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NormedCommRing (α : Type*) extends NormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: Type u_5

full_name: NormedCommRing.mul_comm
code: /-- A normed commutative ring is a commutative ring endowed with a norm which satisfies
the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
class NormedCommRing (α : Type*) extends NormedRing α where
  /-- Multiplication is commutative. -/
  mul_comm : ∀ x y : α, x * y = y * x
goal: x * y = y * x

full_name: SeminormedCommRing.toNonUnitalSeminormedCommRing
code: /-- A seminormed commutative ring is a non-unital seminormed commutative ring. -/
instance (priority := 100) SeminormedCommRing.toNonUnitalSeminormedCommRing
    [β : SeminormedCommRing α] : NonUnitalSeminormedCommRing α 
goal: NonUnitalSeminormedCommRing α

full_name: NormedCommRing.toNonUnitalNormedCommRing
code: /-- A normed commutative ring is a non-unital normed commutative ring. -/
instance (priority := 100) NormedCommRing.toNonUnitalNormedCommRing
    [β : NormedCommRing α] : NonUnitalNormedCommRing α 
goal: NonUnitalNormedCommRing α

full_name: NormedCommRing.toSeminormedCommRing
code: /-- A normed commutative ring is a seminormed commutative ring. -/
instance (priority := 100) NormedCommRing.toSeminormedCommRing [β : NormedCommRing α] :
    SeminormedCommRing α 
goal: SeminormedCommRing α

full_name: PUnit.normedCommRing
code: instance PUnit.normedCommRing : NormedCommRing PUnit 
goal: NormedCommRing PUnit.{u_5 + 1}

full_name: NormOneClass
code: /-- A mixin class with the axiom `‖1‖ = 1`. Many `NormedRing`s and all `NormedField`s satisfy this
axiom. -/
class NormOneClass (α : Type*) [Norm α] [One α] : Prop where
  /-- The norm of the multiplicative identity is 1. -/
  norm_one : ‖(1 : α)‖ = 1
goal: Prop

full_name: NormOneClass.norm_one
code: /-- A mixin class with the axiom `‖1‖ = 1`. Many `NormedRing`s and all `NormedField`s satisfy this
axiom. -/
class NormOneClass (α : Type*) [Norm α] [One α] : Prop where
  /-- The norm of the multiplicative identity is 1. -/
  norm_one : ‖(1 : α)‖ = 1
goal: ‖1‖ = 1

full_name: nnnorm_one
code: @[simp]
theorem nnnorm_one [SeminormedAddCommGroup α] [One α] [NormOneClass α] : ‖(1 : α)‖₊ = 1 
goal: ‖1‖₊ = 1

full_name: NormOneClass.nontrivial
code: theorem NormOneClass.nontrivial (α : Type*) [SeminormedAddCommGroup α] [One α] [NormOneClass α] :
    Nontrivial α 
goal: Nontrivial α

full_name: NonUnitalSeminormedCommRing.toNonUnitalCommRing
code: instance (priority := 100) NonUnitalSeminormedCommRing.toNonUnitalCommRing
    [β : NonUnitalSeminormedCommRing α] : NonUnitalCommRing α 
goal: NonUnitalCommRing α

full_name: SeminormedCommRing.toCommRing
code: instance (priority := 100) SeminormedCommRing.toCommRing [β : SeminormedCommRing α] : CommRing α 
goal: CommRing α

full_name: NonUnitalNormedRing.toNormedAddCommGroup
code: instance (priority := 100) NonUnitalNormedRing.toNormedAddCommGroup [β : NonUnitalNormedRing α] :
    NormedAddCommGroup α 
goal: NormedAddCommGroup α

full_name: NonUnitalSeminormedRing.toSeminormedAddCommGroup
code: instance (priority := 100) NonUnitalSeminormedRing.toSeminormedAddCommGroup
    [NonUnitalSeminormedRing α] : SeminormedAddCommGroup α 
goal: SeminormedAddCommGroup α

full_name: ULift.normOneClass
code: instance ULift.normOneClass [SeminormedAddCommGroup α] [One α] [NormOneClass α] :
    NormOneClass (ULift α) 
goal: NormOneClass (ULift.{u_5, u_1} α)

full_name: Prod.normOneClass
code: instance Prod.normOneClass [SeminormedAddCommGroup α] [One α] [NormOneClass α]
    [SeminormedAddCommGroup β] [One β] [NormOneClass β] : NormOneClass (α × β) 
goal: NormOneClass (α × β)

full_name: Pi.normOneClass
code: instance Pi.normOneClass {ι : Type*} {α : ι → Type*} [Nonempty ι] [Fintype ι]
    [∀ i, SeminormedAddCommGroup (α i)] [∀ i, One (α i)] [∀ i, NormOneClass (α i)] :
    NormOneClass (∀ i, α i) 
goal: NormOneClass ((i : ι) → α i)

full_name: MulOpposite.normOneClass
code: instance MulOpposite.normOneClass [SeminormedAddCommGroup α] [One α] [NormOneClass α] :
    NormOneClass αᵐᵒᵖ 
goal: NormOneClass αᵐᵒᵖ

full_name: norm_mul_le
code: theorem norm_mul_le (a b : α) : ‖a * b‖ ≤ ‖a‖ * ‖b‖ 
goal: ‖a * b‖ ≤ ‖a‖ * ‖b‖

full_name: nnnorm_mul_le
code: theorem nnnorm_mul_le (a b : α) : ‖a * b‖₊ ≤ ‖a‖₊ * ‖b‖₊ 
goal: ‖a * b‖₊ ≤ ‖a‖₊ * ‖b‖₊

full_name: one_le_norm_one
code: theorem one_le_norm_one (β) [NormedRing β] [Nontrivial β] : 1 ≤ ‖(1 : β)‖ 
goal: 1 ≤ ‖1‖

full_name: one_le_nnnorm_one
code: theorem one_le_nnnorm_one (β) [NormedRing β] [Nontrivial β] : 1 ≤ ‖(1 : β)‖₊ 
goal: 1 ≤ ‖1‖₊

full_name: Filter.Tendsto.zero_mul_isBoundedUnder_le
code: theorem Filter.Tendsto.zero_mul_isBoundedUnder_le {f g : ι → α} {l : Filter ι}
    (hf : Tendsto f l (𝓝 0)) (hg : IsBoundedUnder (· ≤ ·) l ((‖·‖) ∘ g)) :
    Tendsto (fun x => f x * g x) l (𝓝 0) 
goal: Filter.Tendsto (fun x => f x * g x) l (𝓝 0)

full_name: Filter.isBoundedUnder_le_mul_tendsto_zero
code: theorem Filter.isBoundedUnder_le_mul_tendsto_zero {f g : ι → α} {l : Filter ι}
    (hf : IsBoundedUnder (· ≤ ·) l (norm ∘ f)) (hg : Tendsto g l (𝓝 0)) :
    Tendsto (fun x => f x * g x) l (𝓝 0) 
goal: Filter.Tendsto (fun x => f x * g x) l (𝓝 0)

full_name: mulLeft_bound
code: /-- In a seminormed ring, the left-multiplication `AddMonoidHom` is bounded. -/
theorem mulLeft_bound (x : α) : ∀ y : α, ‖AddMonoidHom.mulLeft x y‖ ≤ ‖x‖ * ‖y‖ 
goal: ‖(AddMonoidHom.mulLeft x) y‖ ≤ ‖x‖ * ‖y‖

full_name: mulRight_bound
code: /-- In a seminormed ring, the right-multiplication `AddMonoidHom` is bounded. -/
theorem mulRight_bound (x : α) : ∀ y : α, ‖AddMonoidHom.mulRight x y‖ ≤ ‖x‖ * ‖y‖ 
goal: ‖(AddMonoidHom.mulRight x) y‖ ≤ ‖x‖ * ‖y‖

full_name: NonUnitalSubalgebra.nonUnitalSeminormedRing
code: /-- A non-unital subalgebra of a non-unital seminormed ring is also a non-unital seminormed ring,
with the restriction of the norm.  -/
instance NonUnitalSubalgebra.nonUnitalSeminormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
    [NonUnitalSeminormedRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
    NonUnitalSeminormedRing s 
goal: NonUnitalSeminormedRing ↥s

full_name: NonUnitalSubalgebraClass.nonUnitalSeminormedRing
code: /-- A non-unital subalgebra of a non-unital seminormed ring is also a non-unital seminormed ring,
with the restriction of the norm.  -/
-- necessary to require `SMulMemClass S 𝕜 E` so that `𝕜` can be determined as an `outParam`
@[nolint unusedArguments]
instance (priority := 75) NonUnitalSubalgebraClass.nonUnitalSeminormedRing {S 𝕜 E : Type*}
    [CommRing 𝕜] [NonUnitalSeminormedRing E] [Module 𝕜 E] [SetLike S E] [NonUnitalSubringClass S E]
    [SMulMemClass S 𝕜 E] (s : S) :
    NonUnitalSeminormedRing s 
goal: NonUnitalSeminormedRing ↥s

full_name: NonUnitalSubalgebra.nonUnitalNormedRing
code: /-- A non-unital subalgebra of a non-unital normed ring is also a non-unital normed ring, with the
restriction of the norm.  -/
instance NonUnitalSubalgebra.nonUnitalNormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
    [NonUnitalNormedRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) : NonUnitalNormedRing s 
goal: NonUnitalNormedRing ↥s

full_name: NonUnitalSubalgebraClass.nonUnitalNormedRing
code: /-- A non-unital subalgebra of a non-unital normed ring is also a non-unital normed ring,
with the restriction of the norm.  -/
instance (priority := 75) NonUnitalSubalgebraClass.nonUnitalNormedRing {S 𝕜 E : Type*}
    [CommRing 𝕜] [NonUnitalNormedRing E] [Module 𝕜 E] [SetLike S E] [NonUnitalSubringClass S E]
    [SMulMemClass S 𝕜 E] (s : S) :
    NonUnitalNormedRing s 
goal: NonUnitalNormedRing ↥s

full_name: ULift.nonUnitalSeminormedRing
code: instance ULift.nonUnitalSeminormedRing : NonUnitalSeminormedRing (ULift α) 
goal: NonUnitalSeminormedRing (ULift.{u_5, u_1} α)

full_name: Prod.nonUnitalSeminormedRing
code: /-- Non-unital seminormed ring structure on the product of two non-unital seminormed rings,
  using the sup norm. -/
instance Prod.nonUnitalSeminormedRing [NonUnitalSeminormedRing β] :
    NonUnitalSeminormedRing (α × β) 
goal: NonUnitalSeminormedRing (α × β)

full_name: Pi.nonUnitalSeminormedRing
code: /-- Non-unital seminormed ring structure on the product of finitely many non-unital seminormed
rings, using the sup norm. -/
instance Pi.nonUnitalSeminormedRing {π : ι → Type*} [Fintype ι]
    [∀ i, NonUnitalSeminormedRing (π i)] : NonUnitalSeminormedRing (∀ i, π i) 
goal: NonUnitalSeminormedRing ((i : ι) → π i)

full_name: MulOpposite.instNonUnitalSeminormedRing
code: instance MulOpposite.instNonUnitalSeminormedRing : NonUnitalSeminormedRing αᵐᵒᵖ 
goal: NonUnitalSeminormedRing αᵐᵒᵖ

full_name: Subalgebra.seminormedRing
code: /-- A subalgebra of a seminormed ring is also a seminormed ring, with the restriction of the
norm. -/
instance Subalgebra.seminormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [SeminormedRing E]
    [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : SeminormedRing s 
goal: SeminormedRing ↥s

full_name: SubalgebraClass.seminormedRing
code: /-- A subalgebra of a seminormed ring is also a seminormed ring, with the restriction of the
norm. -/
-- necessary to require `SMulMemClass S 𝕜 E` so that `𝕜` can be determined as an `outParam`
@[nolint unusedArguments]
instance (priority := 75) SubalgebraClass.seminormedRing {S 𝕜 E : Type*} [CommRing 𝕜]
    [SeminormedRing E] [Algebra 𝕜 E] [SetLike S E] [SubringClass S E] [SMulMemClass S 𝕜 E]
    (s : S) : SeminormedRing s 
goal: SeminormedRing ↥s

full_name: Subalgebra.normedRing
code: /-- A subalgebra of a normed ring is also a normed ring, with the restriction of the norm. -/
instance Subalgebra.normedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [NormedRing E]
    [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : NormedRing s 
goal: NormedRing ↥s

full_name: SubalgebraClass.normedRing
code: /-- A subalgebra of a normed ring is also a normed ring, with the restriction of the
norm. -/
instance (priority := 75) SubalgebraClass.normedRing {S 𝕜 E : Type*} [CommRing 𝕜]
    [NormedRing E] [Algebra 𝕜 E] [SetLike S E] [SubringClass S E] [SMulMemClass S 𝕜 E]
    (s : S) : NormedRing s 
goal: NormedRing ↥s

full_name: Nat.norm_cast_le
code: theorem Nat.norm_cast_le : ∀ n : ℕ, ‖(n : α)‖ ≤ n * ‖(1 : α)‖
  
goal: ‖↑n‖ ≤ ↑n * ‖1‖

full_name: List.norm_prod_le'
code: theorem List.norm_prod_le' : ∀ {l : List α}, l ≠ [] → ‖l.prod‖ ≤ (l.map norm).prod
  
goal: l ≠ [] → ‖l.prod‖ ≤ (List.map norm l).prod

full_name: List.nnnorm_prod_le'
code: theorem List.nnnorm_prod_le' {l : List α} (hl : l ≠ []) : ‖l.prod‖₊ ≤ (l.map nnnorm).prod 
goal: ‖l.prod‖₊ ≤ (List.map nnnorm l).prod

full_name: List.norm_prod_le
code: theorem List.norm_prod_le [NormOneClass α] : ∀ l : List α, ‖l.prod‖ ≤ (l.map norm).prod
  
goal: ‖l.prod‖ ≤ (List.map norm l).prod

full_name: List.nnnorm_prod_le
code: theorem List.nnnorm_prod_le [NormOneClass α] (l : List α) : ‖l.prod‖₊ ≤ (l.map nnnorm).prod 
goal: ‖l.prod‖₊ ≤ (List.map nnnorm l).prod

full_name: Finset.norm_prod_le'
code: theorem Finset.norm_prod_le' {α : Type*} [NormedCommRing α] (s : Finset ι) (hs : s.Nonempty)
    (f : ι → α) : ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖ 
goal: ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖

full_name: Finset.nnnorm_prod_le'
code: theorem Finset.nnnorm_prod_le' {α : Type*} [NormedCommRing α] (s : Finset ι) (hs : s.Nonempty)
    (f : ι → α) : ‖∏ i ∈ s, f i‖₊ ≤ ∏ i ∈ s, ‖f i‖₊ 
goal: ‖∏ i ∈ s, f i‖₊ ≤ ∏ i ∈ s, ‖f i‖₊

full_name: Finset.norm_prod_le
code: theorem Finset.norm_prod_le {α : Type*} [NormedCommRing α] [NormOneClass α] (s : Finset ι)
    (f : ι → α) : ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖ 
goal: ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖

full_name: Finset.nnnorm_prod_le
code: theorem Finset.nnnorm_prod_le {α : Type*} [NormedCommRing α] [NormOneClass α] (s : Finset ι)
    (f : ι → α) : ‖∏ i ∈ s, f i‖₊ ≤ ∏ i ∈ s, ‖f i‖₊ 
goal: ‖∏ i ∈ s, f i‖₊ ≤ ∏ i ∈ s, ‖f i‖₊

full_name: nnnorm_pow_le'
code: /-- If `α` is a seminormed ring, then `‖a ^ n‖₊ ≤ ‖a‖₊ ^ n` for `n > 0`.
See also `nnnorm_pow_le`. -/
theorem nnnorm_pow_le' (a : α) : ∀ {n : ℕ}, 0 < n → ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n
  
goal: 0 < n → ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n

full_name: nnnorm_pow_le
code: /-- If `α` is a seminormed ring with `‖1‖₊ = 1`, then `‖a ^ n‖₊ ≤ ‖a‖₊ ^ n`.
See also `nnnorm_pow_le'`. -/
theorem nnnorm_pow_le [NormOneClass α] (a : α) (n : ℕ) : ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n 
goal: ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n

full_name: norm_pow_le'
code: /-- If `α` is a seminormed ring, then `‖a ^ n‖ ≤ ‖a‖ ^ n` for `n > 0`. See also `norm_pow_le`. -/
theorem norm_pow_le' (a : α) {n : ℕ} (h : 0 < n) : ‖a ^ n‖ ≤ ‖a‖ ^ n 
goal: ‖a ^ n‖ ≤ ‖a‖ ^ n

full_name: norm_pow_le
code: /-- If `α` is a seminormed ring with `‖1‖ = 1`, then `‖a ^ n‖ ≤ ‖a‖ ^ n`.
See also `norm_pow_le'`. -/
theorem norm_pow_le [NormOneClass α] (a : α) (n : ℕ) : ‖a ^ n‖ ≤ ‖a‖ ^ n 
goal: ‖a ^ n‖ ≤ ‖a‖ ^ n

full_name: eventually_norm_pow_le
code: theorem eventually_norm_pow_le (a : α) : ∀ᶠ n : ℕ in atTop, ‖a ^ n‖ ≤ ‖a‖ ^ n 
goal: ∀ᶠ (n : ℕ) in Filter.atTop, ‖a ^ n‖ ≤ ‖a‖ ^ n

full_name: ULift.seminormedRing
code: instance ULift.seminormedRing : SeminormedRing (ULift α) 
goal: SeminormedRing (ULift.{u_5, u_1} α)

full_name: Prod.seminormedRing
code: /-- Seminormed ring structure on the product of two seminormed rings,
  using the sup norm. -/
instance Prod.seminormedRing [SeminormedRing β] : SeminormedRing (α × β) 
goal: SeminormedRing (α × β)

full_name: Pi.seminormedRing
code: /-- Seminormed ring structure on the product of finitely many seminormed rings,
  using the sup norm. -/
instance Pi.seminormedRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedRing (π i)] :
    SeminormedRing (∀ i, π i) 
goal: SeminormedRing ((i : ι) → π i)

full_name: MulOpposite.instSeminormedRing
code: instance MulOpposite.instSeminormedRing : SeminormedRing αᵐᵒᵖ 
goal: SeminormedRing αᵐᵒᵖ

full_name: ULift.nonUnitalNormedRing
code: instance ULift.nonUnitalNormedRing : NonUnitalNormedRing (ULift α) 
goal: NonUnitalNormedRing (ULift.{u_5, u_1} α)

full_name: Prod.nonUnitalNormedRing
code: /-- Non-unital normed ring structure on the product of two non-unital normed rings,
using the sup norm. -/
instance Prod.nonUnitalNormedRing [NonUnitalNormedRing β] : NonUnitalNormedRing (α × β) 
goal: NonUnitalNormedRing (α × β)

full_name: Pi.nonUnitalNormedRing
code: /-- Normed ring structure on the product of finitely many non-unital normed rings, using the sup
norm. -/
instance Pi.nonUnitalNormedRing {π : ι → Type*} [Fintype ι] [∀ i, NonUnitalNormedRing (π i)] :
    NonUnitalNormedRing (∀ i, π i) 
goal: NonUnitalNormedRing ((i : ι) → π i)

full_name: MulOpposite.instNonUnitalNormedRing
code: instance MulOpposite.instNonUnitalNormedRing : NonUnitalNormedRing αᵐᵒᵖ 
goal: NonUnitalNormedRing αᵐᵒᵖ

full_name: Units.norm_pos
code: theorem Units.norm_pos [Nontrivial α] (x : αˣ) : 0 < ‖(x : α)‖ 
goal: 0 < ‖↑x‖

full_name: Units.nnnorm_pos
code: theorem Units.nnnorm_pos [Nontrivial α] (x : αˣ) : 0 < ‖(x : α)‖₊ 
goal: 0 < ‖↑x‖₊

full_name: ULift.normedRing
code: instance ULift.normedRing : NormedRing (ULift α) 
goal: NormedRing (ULift.{u_5, u_1} α)

full_name: Prod.normedRing
code: /-- Normed ring structure on the product of two normed rings, using the sup norm. -/
instance Prod.normedRing [NormedRing β] : NormedRing (α × β) 
goal: NormedRing (α × β)

full_name: Pi.normedRing
code: /-- Normed ring structure on the product of finitely many normed rings, using the sup norm. -/
instance Pi.normedRing {π : ι → Type*} [Fintype ι] [∀ i, NormedRing (π i)] :
    NormedRing (∀ i, π i) 
goal: NormedRing ((i : ι) → π i)

full_name: MulOpposite.instNormedRing
code: instance MulOpposite.instNormedRing : NormedRing αᵐᵒᵖ 
goal: NormedRing αᵐᵒᵖ

full_name: ULift.nonUnitalSeminormedCommRing
code: instance ULift.nonUnitalSeminormedCommRing : NonUnitalSeminormedCommRing (ULift α) 
goal: NonUnitalSeminormedCommRing (ULift.{u_5, u_1} α)

full_name: Prod.nonUnitalSeminormedCommRing
code: /-- Non-unital seminormed commutative ring structure on the product of two non-unital seminormed
commutative rings, using the sup norm. -/
instance Prod.nonUnitalSeminormedCommRing [NonUnitalSeminormedCommRing β] :
    NonUnitalSeminormedCommRing (α × β) 
goal: NonUnitalSeminormedCommRing (α × β)

full_name: Pi.nonUnitalSeminormedCommRing
code: /-- Non-unital seminormed commutative ring structure on the product of finitely many non-unital
seminormed commutative rings, using the sup norm. -/
instance Pi.nonUnitalSeminormedCommRing {π : ι → Type*} [Fintype ι]
    [∀ i, NonUnitalSeminormedCommRing (π i)] : NonUnitalSeminormedCommRing (∀ i, π i) 
goal: NonUnitalSeminormedCommRing ((i : ι) → π i)

full_name: MulOpposite.instNonUnitalSeminormedCommRing
code: instance MulOpposite.instNonUnitalSeminormedCommRing : NonUnitalSeminormedCommRing αᵐᵒᵖ 
goal: NonUnitalSeminormedCommRing αᵐᵒᵖ

full_name: NonUnitalSubalgebra.nonUnitalSeminormedCommRing
code: /-- A non-unital subalgebra of a non-unital seminormed commutative ring is also a non-unital
seminormed commutative ring, with the restriction of the norm.  -/
instance NonUnitalSubalgebra.nonUnitalSeminormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
    [NonUnitalSeminormedCommRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
    NonUnitalSeminormedCommRing s 
goal: NonUnitalSeminormedCommRing ↥s

full_name: NonUnitalSubalgebra.nonUnitalNormedCommRing
code: /-- A non-unital subalgebra of a non-unital normed commutative ring is also a non-unital normed
commutative ring, with the restriction of the norm.  -/
instance NonUnitalSubalgebra.nonUnitalNormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
    [NonUnitalNormedCommRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
    NonUnitalNormedCommRing s 
goal: NonUnitalNormedCommRing ↥s

full_name: ULift.nonUnitalNormedCommRing
code: instance ULift.nonUnitalNormedCommRing : NonUnitalNormedCommRing (ULift α) 
goal: NonUnitalNormedCommRing (ULift.{u_5, u_1} α)

full_name: Prod.nonUnitalNormedCommRing
code: /-- Non-unital normed commutative ring structure on the product of two non-unital normed
commutative rings, using the sup norm. -/
instance Prod.nonUnitalNormedCommRing [NonUnitalNormedCommRing β] :
    NonUnitalNormedCommRing (α × β) 
goal: NonUnitalNormedCommRing (α × β)

full_name: Pi.nonUnitalNormedCommRing
code: /-- Normed commutative ring structure on the product of finitely many non-unital normed
commutative rings, using the sup norm. -/
instance Pi.nonUnitalNormedCommRing {π : ι → Type*} [Fintype ι]
    [∀ i, NonUnitalNormedCommRing (π i)] : NonUnitalNormedCommRing (∀ i, π i) 
goal: NonUnitalNormedCommRing ((i : ι) → π i)

full_name: MulOpposite.instNonUnitalNormedCommRing
code: instance MulOpposite.instNonUnitalNormedCommRing : NonUnitalNormedCommRing αᵐᵒᵖ 
goal: NonUnitalNormedCommRing αᵐᵒᵖ

full_name: ULift.seminormedCommRing
code: instance ULift.seminormedCommRing : SeminormedCommRing (ULift α) 
goal: SeminormedCommRing (ULift.{u_5, u_1} α)

full_name: Prod.seminormedCommRing
code: /-- Seminormed commutative ring structure on the product of two seminormed commutative rings,
  using the sup norm. -/
instance Prod.seminormedCommRing [SeminormedCommRing β] : SeminormedCommRing (α × β) 
goal: SeminormedCommRing (α × β)

full_name: Pi.seminormedCommRing
code: /-- Seminormed commutative ring structure on the product of finitely many seminormed commutative
rings, using the sup norm. -/
instance Pi.seminormedCommRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedCommRing (π i)] :
    SeminormedCommRing (∀ i, π i) 
goal: SeminormedCommRing ((i : ι) → π i)

full_name: MulOpposite.instSeminormedCommRing
code: instance MulOpposite.instSeminormedCommRing : SeminormedCommRing αᵐᵒᵖ 
goal: SeminormedCommRing αᵐᵒᵖ

full_name: Subalgebra.seminormedCommRing
code: /-- A subalgebra of a seminormed commutative ring is also a seminormed commutative ring, with the
restriction of the norm.  -/
instance Subalgebra.seminormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [SeminormedCommRing E]
    [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : SeminormedCommRing s 
goal: SeminormedCommRing ↥s

full_name: Subalgebra.normedCommRing
code: /-- A subalgebra of a normed commutative ring is also a normed commutative ring, with the
restriction of the norm.  -/
instance Subalgebra.normedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [NormedCommRing E]
    [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : NormedCommRing s 
goal: NormedCommRing ↥s

full_name: ULift.normedCommRing
code: instance ULift.normedCommRing : NormedCommRing (ULift α) 
goal: NormedCommRing (ULift.{u_5, u_1} α)

full_name: Prod.normedCommRing
code: /-- Normed commutative ring structure on the product of two normed commutative rings, using the sup
norm. -/
instance Prod.normedCommRing [NormedCommRing β] : NormedCommRing (α × β) 
goal: NormedCommRing (α × β)

full_name: Pi.normedCommutativeRing
code: /-- Normed commutative ring structure on the product of finitely many normed commutative rings,
using the sup norm. -/
instance Pi.normedCommutativeRing {π : ι → Type*} [Fintype ι] [∀ i, NormedCommRing (π i)] :
    NormedCommRing (∀ i, π i) 
goal: NormedCommRing ((i : ι) → π i)

full_name: MulOpposite.instNormedCommRing
code: instance MulOpposite.instNormedCommRing : NormedCommRing αᵐᵒᵖ 
goal: NormedCommRing αᵐᵒᵖ

full_name: semi_normed_ring_top_monoid
code: instance (priority := 100) semi_normed_ring_top_monoid [NonUnitalSeminormedRing α] :
    ContinuousMul α 
goal: ContinuousMul α

full_name: semi_normed_top_ring
code: /-- A seminormed ring is a topological ring. -/
instance (priority := 100) semi_normed_top_ring [NonUnitalSeminormedRing α] :
    TopologicalRing α 
goal: TopologicalRing α

full_name: norm_mul
code: @[simp]
theorem norm_mul (a b : α) : ‖a * b‖ = ‖a‖ * ‖b‖ 
goal: ‖a * b‖ = ‖a‖ * ‖b‖

full_name: NormedDivisionRing.to_normOneClass
code: instance (priority := 900) NormedDivisionRing.to_normOneClass : NormOneClass α 
goal: NormOneClass α

full_name: isAbsoluteValue_norm
code: instance isAbsoluteValue_norm : IsAbsoluteValue (norm : α → ℝ) 
goal: IsAbsoluteValue norm

full_name: nnnorm_mul
code: @[simp]
theorem nnnorm_mul (a b : α) : ‖a * b‖₊ = ‖a‖₊ * ‖b‖₊ 
goal: ‖a * b‖₊ = ‖a‖₊ * ‖b‖₊

full_name: normHom_apply
code: /-- `norm` as a `MonoidWithZeroHom`. -/
@[simps]
def normHom : α →*₀ ℝ 
goal: ∀ (x : α), normHom x = ‖x‖

full_name: normHom
code: /-- `norm` as a `MonoidWithZeroHom`. -/
@[simps]
def normHom : α →*₀ ℝ 
goal: α →*₀ ℝ

full_name: nnnormHom_apply
code: /-- `nnnorm` as a `MonoidWithZeroHom`. -/
@[simps]
def nnnormHom : α →*₀ ℝ≥0 
goal: ∀ (x : α), nnnormHom x = ‖x‖₊

full_name: nnnormHom
code: /-- `nnnorm` as a `MonoidWithZeroHom`. -/
@[simps]
def nnnormHom : α →*₀ ℝ≥0 
goal: α →*₀ ℝ≥0

full_name: norm_pow
code: @[simp]
theorem norm_pow (a : α) : ∀ n : ℕ, ‖a ^ n‖ = ‖a‖ ^ n 
goal: ‖a ^ n‖ = ‖a‖ ^ n

full_name: nnnorm_pow
code: @[simp]
theorem nnnorm_pow (a : α) (n : ℕ) : ‖a ^ n‖₊ = ‖a‖₊ ^ n 
goal: ‖a ^ n‖₊ = ‖a‖₊ ^ n

full_name: List.norm_prod
code: protected theorem List.norm_prod (l : List α) : ‖l.prod‖ = (l.map norm).prod 
goal: ‖l.prod‖ = (List.map norm l).prod

full_name: List.nnnorm_prod
code: protected theorem List.nnnorm_prod (l : List α) : ‖l.prod‖₊ = (l.map nnnorm).prod 
goal: ‖l.prod‖₊ = (List.map nnnorm l).prod

full_name: norm_div
code: @[simp]
theorem norm_div (a b : α) : ‖a / b‖ = ‖a‖ / ‖b‖ 
goal: ‖a / b‖ = ‖a‖ / ‖b‖

full_name: nnnorm_div
code: @[simp]
theorem nnnorm_div (a b : α) : ‖a / b‖₊ = ‖a‖₊ / ‖b‖₊ 
goal: ‖a / b‖₊ = ‖a‖₊ / ‖b‖₊

full_name: norm_inv
code: @[simp]
theorem norm_inv (a : α) : ‖a⁻¹‖ = ‖a‖⁻¹ 
goal: ‖a⁻¹‖ = ‖a‖⁻¹

full_name: nnnorm_inv
code: @[simp]
theorem nnnorm_inv (a : α) : ‖a⁻¹‖₊ = ‖a‖₊⁻¹ 
goal: ‖a⁻¹‖₊ = ‖a‖₊⁻¹

full_name: norm_zpow
code: @[simp]
theorem norm_zpow : ∀ (a : α) (n : ℤ), ‖a ^ n‖ = ‖a‖ ^ n 
goal: ‖a ^ n‖ = ‖a‖ ^ n

full_name: nnnorm_zpow
code: @[simp]
theorem nnnorm_zpow : ∀ (a : α) (n : ℤ), ‖a ^ n‖₊ = ‖a‖₊ ^ n 
goal: ‖a ^ n‖₊ = ‖a‖₊ ^ n

full_name: dist_inv_inv₀
code: theorem dist_inv_inv₀ {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) :
    dist z⁻¹ w⁻¹ = dist z w / (‖z‖ * ‖w‖) 
goal: dist z⁻¹ w⁻¹ = dist z w / (‖z‖ * ‖w‖)

full_name: nndist_inv_inv₀
code: theorem nndist_inv_inv₀ {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) :
    nndist z⁻¹ w⁻¹ = nndist z w / (‖z‖₊ * ‖w‖₊) 
goal: nndist z⁻¹ w⁻¹ = nndist z w / (‖z‖₊ * ‖w‖₊)

full_name: antilipschitzWith_mul_left
code: lemma antilipschitzWith_mul_left {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (a * ·) 
goal: AntilipschitzWith ‖a‖₊⁻¹ fun x => a * x

full_name: antilipschitzWith_mul_right
code: lemma antilipschitzWith_mul_right {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (· * a) 
goal: AntilipschitzWith ‖a‖₊⁻¹ fun x => x * a

full_name: DilationEquiv.mulLeft
code: /-- Multiplication by a nonzero element `a` on the left
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulLeft (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: α ≃ᵈ α

full_name: DilationEquiv.mulLeft_apply
code: /-- Multiplication by a nonzero element `a` on the left
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulLeft (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: (DilationEquiv.mulLeft a ha) x = a * x

full_name: DilationEquiv.mulLeft_symm_apply
code: /-- Multiplication by a nonzero element `a` on the left
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulLeft (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: (DilationEquiv.mulLeft a ha).symm x = a⁻¹ * x

full_name: DilationEquiv.mulRight
code: /-- Multiplication by a nonzero element `a` on the right
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: α ≃ᵈ α

full_name: DilationEquiv.mulRight_apply
code: /-- Multiplication by a nonzero element `a` on the right
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: (DilationEquiv.mulRight a ha) x = x * a

full_name: DilationEquiv.mulRight_symm_apply
code: /-- Multiplication by a nonzero element `a` on the right
as a `DilationEquiv` of a normed division ring. -/
@[simps!]
def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α 
goal: (DilationEquiv.mulRight a ha).symm x = x * a⁻¹

full_name: Filter.comap_mul_left_cobounded
code: @[simp]
lemma comap_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    comap (a * ·) (cobounded α) = cobounded α 
goal: Filter.comap (fun x => a * x) (Bornology.cobounded α) = Bornology.cobounded α

full_name: Filter.map_mul_left_cobounded
code: @[simp]
lemma map_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    map (a * ·) (cobounded α) = cobounded α 
goal: Filter.map (fun x => a * x) (Bornology.cobounded α) = Bornology.cobounded α

full_name: Filter.comap_mul_right_cobounded
code: @[simp]
lemma comap_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    comap (· * a) (cobounded α) = cobounded α 
goal: Filter.comap (fun x => x * a) (Bornology.cobounded α) = Bornology.cobounded α

full_name: Filter.map_mul_right_cobounded
code: @[simp]
lemma map_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    map (· * a) (cobounded α) = cobounded α 
goal: Filter.map (fun x => x * a) (Bornology.cobounded α) = Bornology.cobounded α

full_name: Filter.tendsto_mul_left_cobounded
code: /-- Multiplication on the left by a nonzero element of a normed division ring tends to infinity at
infinity. -/
theorem tendsto_mul_left_cobounded {a : α} (ha : a ≠ 0) :
    Tendsto (a * ·) (cobounded α) (cobounded α) 
goal: Filter.Tendsto (fun x => a * x) (Bornology.cobounded α) (Bornology.cobounded α)

full_name: Filter.tendsto_mul_right_cobounded
code: /-- Multiplication on the right by a nonzero element of a normed division ring tends to infinity at
infinity. -/
theorem tendsto_mul_right_cobounded {a : α} (ha : a ≠ 0) :
    Tendsto (· * a) (cobounded α) (cobounded α) 
goal: Filter.Tendsto (fun x => x * a) (Bornology.cobounded α) (Bornology.cobounded α)

full_name: Filter.inv_cobounded₀
code: @[simp]
lemma inv_cobounded₀ : (cobounded α)⁻¹ = 𝓝[≠] 0 
goal: (Bornology.cobounded α)⁻¹ = 𝓝[≠] 0

full_name: Filter.inv_nhdsWithin_ne_zero
code: @[simp]
lemma inv_nhdsWithin_ne_zero : (𝓝[≠] (0 : α))⁻¹ = cobounded α 
goal: (𝓝[≠] 0)⁻¹ = Bornology.cobounded α

full_name: Filter.tendsto_inv₀_cobounded'
code: lemma tendsto_inv₀_cobounded' : Tendsto Inv.inv (cobounded α) (𝓝[≠] 0) 
goal: Filter.Tendsto Inv.inv (Bornology.cobounded α) (𝓝[≠] 0)

full_name: Filter.tendsto_inv₀_cobounded
code: theorem tendsto_inv₀_cobounded : Tendsto Inv.inv (cobounded α) (𝓝 0) 
goal: Filter.Tendsto Inv.inv (Bornology.cobounded α) (𝓝 0)

full_name: Filter.tendsto_inv₀_nhdsWithin_ne_zero
code: lemma tendsto_inv₀_nhdsWithin_ne_zero : Tendsto Inv.inv (𝓝[≠] 0) (cobounded α) 
goal: Filter.Tendsto Inv.inv (𝓝[≠] 0) (Bornology.cobounded α)

full_name: NormedDivisionRing.to_hasContinuousInv₀
code: instance (priority := 100) NormedDivisionRing.to_hasContinuousInv₀ : HasContinuousInv₀ α 
goal: HasContinuousInv₀ α

full_name: NormedDivisionRing.to_topologicalDivisionRing
code: /-- A normed division ring is a topological division ring. -/
instance (priority := 100) NormedDivisionRing.to_topologicalDivisionRing :
    TopologicalDivisionRing α 
goal: TopologicalDivisionRing α

full_name: IsOfFinOrder.norm_eq_one
code: protected lemma IsOfFinOrder.norm_eq_one (ha : IsOfFinOrder a) : ‖a‖ = 1 
goal: ‖a‖ = 1

full_name: NormedField.norm_mul'
code: /-- A normed field is a field with a norm satisfying ‖x y‖ = ‖x‖ ‖y‖. -/
class NormedField (α : Type*) extends Norm α, Field α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: ‖a * b‖ = ‖a‖ * ‖b‖

full_name: NormedField
code: /-- A normed field is a field with a norm satisfying ‖x y‖ = ‖x‖ ‖y‖. -/
class NormedField (α : Type*) extends Norm α, Field α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: Type u_5

full_name: NormedField.dist_eq
code: /-- A normed field is a field with a norm satisfying ‖x y‖ = ‖x‖ ‖y‖. -/
class NormedField (α : Type*) extends Norm α, Field α, MetricSpace α where
  /-- The distance is induced by the norm. -/
  dist_eq : ∀ x y, dist x y = norm (x - y)
  /-- The norm is multiplicative. -/
  norm_mul' : ∀ a b, norm (a * b) = norm a * norm b
goal: dist x y = ‖x - y‖

full_name: NontriviallyNormedField.non_trivial
code: /-- A nontrivially normed field is a normed field in which there is an element of norm different
from `0` and `1`. This makes it possible to bring any element arbitrarily close to `0` by
multiplication by the powers of any element, and thus to relate algebra and topology. -/
class NontriviallyNormedField (α : Type*) extends NormedField α where
  /-- The norm attains a value exceeding 1. -/
  non_trivial : ∃ x : α, 1 < ‖x‖
goal: ∃ x, 1 < ‖x‖

full_name: NontriviallyNormedField
code: /-- A nontrivially normed field is a normed field in which there is an element of norm different
from `0` and `1`. This makes it possible to bring any element arbitrarily close to `0` by
multiplication by the powers of any element, and thus to relate algebra and topology. -/
class NontriviallyNormedField (α : Type*) extends NormedField α where
  /-- The norm attains a value exceeding 1. -/
  non_trivial : ∃ x : α, 1 < ‖x‖
goal: Type u_5

full_name: DenselyNormedField
code: /-- A densely normed field is a normed field for which the image of the norm is dense in `ℝ≥0`,
which means it is also nontrivially normed. However, not all nontrivally normed fields are densely
normed; in particular, the `Padic`s exhibit this fact. -/
class DenselyNormedField (α : Type*) extends NormedField α where
  /-- The range of the norm is dense in the collection of nonnegative real numbers. -/
  lt_norm_lt : ∀ x y : ℝ, 0 ≤ x → x < y → ∃ a : α, x < ‖a‖ ∧ ‖a‖ < y
goal: Type u_5

full_name: DenselyNormedField.lt_norm_lt
code: /-- A densely normed field is a normed field for which the image of the norm is dense in `ℝ≥0`,
which means it is also nontrivially normed. However, not all nontrivally normed fields are densely
normed; in particular, the `Padic`s exhibit this fact. -/
class DenselyNormedField (α : Type*) extends NormedField α where
  /-- The range of the norm is dense in the collection of nonnegative real numbers. -/
  lt_norm_lt : ∀ x y : ℝ, 0 ≤ x → x < y → ∃ a : α, x < ‖a‖ ∧ ‖a‖ < y
goal: 0 ≤ x → x < y → ∃ a, x < ‖a‖ ∧ ‖a‖ < y

full_name: DenselyNormedField.toNontriviallyNormedField
code: /-- A densely normed field is always a nontrivially normed field.
See note [lower instance priority]. -/
instance (priority := 100) DenselyNormedField.toNontriviallyNormedField [DenselyNormedField α] :
    NontriviallyNormedField α 
goal: NontriviallyNormedField α

full_name: NormedField.toNormedDivisionRing
code: instance (priority := 100) NormedField.toNormedDivisionRing : NormedDivisionRing α 
goal: NormedDivisionRing α

full_name: NormedField.toNormedCommRing
code: instance (priority := 100) NormedField.toNormedCommRing : NormedCommRing α 
goal: NormedCommRing α

full_name: norm_prod
code: @[simp]
theorem norm_prod (s : Finset β) (f : β → α) : ‖∏ b ∈ s, f b‖ = ∏ b ∈ s, ‖f b‖ 
goal: ‖∏ b ∈ s, f b‖ = ∏ b ∈ s, ‖f b‖

full_name: nnnorm_prod
code: @[simp]
theorem nnnorm_prod (s : Finset β) (f : β → α) : ‖∏ b ∈ s, f b‖₊ = ∏ b ∈ s, ‖f b‖₊ 
goal: ‖∏ b ∈ s, f b‖₊ = ∏ b ∈ s, ‖f b‖₊

full_name: NormedField.exists_one_lt_norm
code: theorem exists_one_lt_norm : ∃ x : α, 1 < ‖x‖ 
goal: ∃ x, 1 < ‖x‖

full_name: NormedField.exists_lt_norm
code: theorem exists_lt_norm (r : ℝ) : ∃ x : α, r < ‖x‖ 
goal: ∃ x, r < ‖x‖

full_name: NormedField.exists_norm_lt
code: theorem exists_norm_lt {r : ℝ} (hr : 0 < r) : ∃ x : α, 0 < ‖x‖ ∧ ‖x‖ < r 
goal: ∃ x, 0 < ‖x‖ ∧ ‖x‖ < r

full_name: NormedField.exists_norm_lt_one
code: theorem exists_norm_lt_one : ∃ x : α, 0 < ‖x‖ ∧ ‖x‖ < 1 
goal: ∃ x, 0 < ‖x‖ ∧ ‖x‖ < 1

full_name: NormedField.punctured_nhds_neBot
code: @[instance]
theorem punctured_nhds_neBot (x : α) : NeBot (𝓝[≠] x) 
goal: (𝓝[≠] x).NeBot

full_name: NormedField.nhdsWithin_isUnit_neBot
code: @[instance]
theorem nhdsWithin_isUnit_neBot : NeBot (𝓝[{ x : α | IsUnit x }] 0) 
goal: (𝓝[{x | IsUnit x}] 0).NeBot

full_name: NormedField.exists_lt_norm_lt
code: theorem exists_lt_norm_lt {r₁ r₂ : ℝ} (h₀ : 0 ≤ r₁) (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖ ∧ ‖x‖ < r₂ 
goal: ∃ x, r₁ < ‖x‖ ∧ ‖x‖ < r₂

full_name: NormedField.exists_lt_nnnorm_lt
code: theorem exists_lt_nnnorm_lt {r₁ r₂ : ℝ≥0} (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖₊ ∧ ‖x‖₊ < r₂ 
goal: ∃ x, r₁ < ‖x‖₊ ∧ ‖x‖₊ < r₂

full_name: NormedField.denselyOrdered_range_norm
code: instance denselyOrdered_range_norm : DenselyOrdered (Set.range (norm : α → ℝ)) 
goal: DenselyOrdered ↑(Set.range norm)

full_name: NormedField.denselyOrdered_range_nnnorm
code: instance denselyOrdered_range_nnnorm : DenselyOrdered (Set.range (nnnorm : α → ℝ≥0)) 
goal: DenselyOrdered ↑(Set.range nnnorm)

full_name: NormedField.denseRange_nnnorm
code: theorem denseRange_nnnorm : DenseRange (nnnorm : α → ℝ≥0) 
goal: DenseRange nnnorm

full_name: NontriviallyNormedField.ofNormNeOne
code: /-- A normed field is nontrivially normed
provided that the norm of some nonzero element is not one. -/
def NontriviallyNormedField.ofNormNeOne {𝕜 : Type*} [h' : NormedField 𝕜]
    (h : ∃ x : 𝕜, x ≠ 0 ∧ ‖x‖ ≠ 1) : NontriviallyNormedField 𝕜 
goal: NontriviallyNormedField 𝕜

full_name: Real.normedCommRing
code: instance Real.normedCommRing : NormedCommRing ℝ 
goal: NormedCommRing ℝ

full_name: Real.normedField
code: noncomputable instance Real.normedField : NormedField ℝ 
goal: NormedField ℝ

full_name: Real.denselyNormedField
code: noncomputable instance Real.denselyNormedField : DenselyNormedField ℝ 
goal: DenselyNormedField ℝ

full_name: Real.toNNReal_mul_nnnorm
code: theorem toNNReal_mul_nnnorm {x : ℝ} (y : ℝ) (hx : 0 ≤ x) : x.toNNReal * ‖y‖₊ = ‖x * y‖₊ 
goal: x.toNNReal * ‖y‖₊ = ‖x * y‖₊

full_name: Real.nnnorm_mul_toNNReal
code: theorem nnnorm_mul_toNNReal (x : ℝ) {y : ℝ} (hy : 0 ≤ y) : ‖x‖₊ * y.toNNReal = ‖x * y‖₊ 
goal: ‖x‖₊ * y.toNNReal = ‖x * y‖₊

full_name: NNReal.norm_eq
code: theorem norm_eq (x : ℝ≥0) : ‖(x : ℝ)‖ = x 
goal: ‖↑x‖ = ↑x

full_name: NNReal.nnnorm_eq
code: @[simp]
theorem nnnorm_eq (x : ℝ≥0) : ‖(x : ℝ)‖₊ = x 
goal: ‖↑x‖₊ = x

full_name: NNReal.lipschitzWith_sub
code: lemma lipschitzWith_sub : LipschitzWith 2 (fun (p : ℝ≥0 × ℝ≥0) ↦ p.1 - p.2) 
goal: LipschitzWith 2 fun p => p.1 - p.2

full_name: norm_norm
code: @[simp 1001] -- Porting note: increase priority so that the LHS doesn't simplify
theorem norm_norm [SeminormedAddCommGroup α] (x : α) : ‖‖x‖‖ = ‖x‖ 
goal: ‖‖x‖‖ = ‖x‖

full_name: nnnorm_norm
code: @[simp]
theorem nnnorm_norm [SeminormedAddCommGroup α] (a : α) : ‖‖a‖‖₊ = ‖a‖₊ 
goal: ‖‖a‖‖₊ = ‖a‖₊

full_name: NormedAddCommGroup.tendsto_atTop
code: /-- A restatement of `MetricSpace.tendsto_atTop` in terms of the norm. -/
theorem NormedAddCommGroup.tendsto_atTop [Nonempty α] [SemilatticeSup α] {β : Type*}
    [SeminormedAddCommGroup β] {f : α → β} {b : β} :
    Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N ≤ n → ‖f n - b‖ < ε 
goal: Filter.Tendsto f Filter.atTop (𝓝 b) ↔ ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N ≤ n → ‖f n - b‖ < ε

full_name: NormedAddCommGroup.tendsto_atTop'
code: /-- A variant of `NormedAddCommGroup.tendsto_atTop` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
-/
theorem NormedAddCommGroup.tendsto_atTop' [Nonempty α] [SemilatticeSup α] [NoMaxOrder α]
    {β : Type*} [SeminormedAddCommGroup β] {f : α → β} {b : β} :
    Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N < n → ‖f n - b‖ < ε 
goal: Filter.Tendsto f Filter.atTop (𝓝 b) ↔ ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : α), N < n → ‖f n - b‖ < ε

full_name: Int.instNormedCommRing
code: instance Int.instNormedCommRing : NormedCommRing ℤ 
goal: NormedCommRing ℤ

full_name: Int.instNormOneClass
code: instance Int.instNormOneClass : NormOneClass ℤ 
goal: NormOneClass ℤ

full_name: Rat.instNormedField
code: instance Rat.instNormedField : NormedField ℚ 
goal: NormedField ℚ

full_name: Rat.instDenselyNormedField
code: instance Rat.instDenselyNormedField : DenselyNormedField ℚ 
goal: DenselyNormedField ℚ

full_name: RingHomIsometric.is_iso
code: /-- This class states that a ring homomorphism is isometric. This is a sufficient assumption
for a continuous semilinear map to be bounded and this is the main use for this typeclass. -/
class RingHomIsometric [Semiring R₁] [Semiring R₂] [Norm R₁] [Norm R₂] (σ : R₁ →+* R₂) : Prop where
  /-- The ring homomorphism is an isometry. -/
  is_iso : ∀ {x : R₁}, ‖σ x‖ = ‖x‖
goal: ‖σ x‖ = ‖x‖

full_name: RingHomIsometric
code: /-- This class states that a ring homomorphism is isometric. This is a sufficient assumption
for a continuous semilinear map to be bounded and this is the main use for this typeclass. -/
class RingHomIsometric [Semiring R₁] [Semiring R₂] [Norm R₁] [Norm R₂] (σ : R₁ →+* R₂) : Prop where
  /-- The ring homomorphism is an isometry. -/
  is_iso : ∀ {x : R₁}, ‖σ x‖ = ‖x‖
goal: Prop

full_name: RingHomIsometric.ids
code: instance RingHomIsometric.ids : RingHomIsometric (RingHom.id R₁) 
goal: RingHomIsometric (RingHom.id R₁)

full_name: NonUnitalSeminormedRing.induced
code: /-- A non-unital ring homomorphism from a `NonUnitalRing` to a `NonUnitalSeminormedRing`
induces a `NonUnitalSeminormedRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NonUnitalSeminormedRing.induced [NonUnitalRing R] [NonUnitalSeminormedRing S]
    [NonUnitalRingHomClass F R S] (f : F) : NonUnitalSeminormedRing R 
goal: NonUnitalSeminormedRing R

full_name: NonUnitalNormedRing.induced
code: /-- An injective non-unital ring homomorphism from a `NonUnitalRing` to a
`NonUnitalNormedRing` induces a `NonUnitalNormedRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NonUnitalNormedRing.induced [NonUnitalRing R] [NonUnitalNormedRing S]
    [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Injective f) : NonUnitalNormedRing R 
goal: NonUnitalNormedRing R

full_name: SeminormedRing.induced
code: /-- A non-unital ring homomorphism from a `Ring` to a `SeminormedRing` induces a
`SeminormedRing` structure on the domain.

See note [reducible non-instances] -/
abbrev SeminormedRing.induced [Ring R] [SeminormedRing S] [NonUnitalRingHomClass F R S] (f : F) :
    SeminormedRing R 
goal: SeminormedRing R

full_name: NormedRing.induced
code: /-- An injective non-unital ring homomorphism from a `Ring` to a `NormedRing` induces a
`NormedRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NormedRing.induced [Ring R] [NormedRing S] [NonUnitalRingHomClass F R S] (f : F)
    (hf : Function.Injective f) : NormedRing R 
goal: NormedRing R

full_name: NonUnitalSeminormedCommRing.induced
code: /-- A non-unital ring homomorphism from a `NonUnitalCommRing` to a `NonUnitalSeminormedCommRing`
induces a `NonUnitalSeminormedCommRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NonUnitalSeminormedCommRing.induced [NonUnitalCommRing R] [NonUnitalSeminormedCommRing S]
    [NonUnitalRingHomClass F R S] (f : F) : NonUnitalSeminormedCommRing R 
goal: NonUnitalSeminormedCommRing R

full_name: NonUnitalNormedCommRing.induced
code: /-- An injective non-unital ring homomorphism from a `NonUnitalCommRing` to a
`NonUnitalNormedCommRing` induces a `NonUnitalNormedCommRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NonUnitalNormedCommRing.induced [NonUnitalCommRing R] [NonUnitalNormedCommRing S]
    [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Injective f) : NonUnitalNormedCommRing R 
goal: NonUnitalNormedCommRing R

full_name: SeminormedCommRing.induced
code: /-- A non-unital ring homomorphism from a `CommRing` to a `SeminormedRing` induces a
`SeminormedCommRing` structure on the domain.

See note [reducible non-instances] -/
abbrev SeminormedCommRing.induced [CommRing R] [SeminormedRing S] [NonUnitalRingHomClass F R S]
    (f : F) : SeminormedCommRing R 
goal: SeminormedCommRing R

full_name: NormedCommRing.induced
code: /-- An injective non-unital ring homomorphism from a `CommRing` to a `NormedRing` induces a
`NormedCommRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NormedCommRing.induced [CommRing R] [NormedRing S] [NonUnitalRingHomClass F R S] (f : F)
    (hf : Function.Injective f) : NormedCommRing R 
goal: NormedCommRing R

full_name: NormedDivisionRing.induced
code: /-- An injective non-unital ring homomorphism from a `DivisionRing` to a `NormedRing` induces a
`NormedDivisionRing` structure on the domain.

See note [reducible non-instances] -/
abbrev NormedDivisionRing.induced [DivisionRing R] [NormedDivisionRing S]
    [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Injective f) : NormedDivisionRing R 
goal: NormedDivisionRing R

full_name: NormedField.induced
code: /-- An injective non-unital ring homomorphism from a `Field` to a `NormedRing` induces a
`NormedField` structure on the domain.

See note [reducible non-instances] -/
abbrev NormedField.induced [Field R] [NormedField S] [NonUnitalRingHomClass F R S] (f : F)
    (hf : Function.Injective f) : NormedField R 
goal: NormedField R

full_name: NormOneClass.induced
code: /-- A ring homomorphism from a `Ring R` to a `SeminormedRing S` which induces the norm structure
`SeminormedRing.induced` makes `R` satisfy `‖(1 : R)‖ = 1` whenever `‖(1 : S)‖ = 1`. -/
theorem NormOneClass.induced {F : Type*} (R S : Type*) [Ring R] [SeminormedRing S]
    [NormOneClass S] [FunLike F R S] [RingHomClass F R S] (f : F) :
    @NormOneClass R (SeminormedRing.induced R S f).toNorm _ 
goal: NormOneClass R

full_name: SubringClass.toSeminormedRing
code: instance toSeminormedRing [SeminormedRing R] [SubringClass S R] (s : S) : SeminormedRing s 
goal: SeminormedRing ↥s

full_name: SubringClass.toNormedRing
code: instance toNormedRing [NormedRing R] [SubringClass S R] (s : S) : NormedRing s 
goal: NormedRing ↥s

full_name: SubringClass.toSeminormedCommRing
code: instance toSeminormedCommRing [SeminormedCommRing R] [_h : SubringClass S R] (s : S) :
    SeminormedCommRing s 
goal: SeminormedCommRing ↥s

full_name: SubringClass.toNormedCommRing
code: instance toNormedCommRing [NormedCommRing R] [SubringClass S R] (s : S) : NormedCommRing s 
goal: NormedCommRing ↥s