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

full_name: PUnit.normedAddCommGroup
code: instance normedAddCommGroup : NormedAddCommGroup PUnit 
goal: NormedAddCommGroup PUnit.{u_5 + 1}

full_name: PUnit.norm_eq_zero
code: @[simp] lemma norm_eq_zero (x : PUnit) : ‖x‖ = 0 
goal: ‖x‖ = 0

full_name: ULift.norm
code: instance norm : Norm (ULift E) 
goal: Norm (ULift.{u_5, u_2} E)

full_name: ULift.norm_def
code: lemma norm_def (x : ULift E) : ‖x‖ = ‖x.down‖ 
goal: ‖x‖ = ‖x.down‖

full_name: ULift.norm_up
code: @[simp] lemma norm_up (x : E) : ‖ULift.up x‖ = ‖x‖ 
goal: ‖{ down := x }‖ = ‖x‖

full_name: ULift.norm_down
code: @[simp] lemma norm_down (x : ULift E) : ‖x.down‖ = ‖x‖ 
goal: ‖x.down‖ = ‖x‖

full_name: ULift.nnnorm
code: instance nnnorm : NNNorm (ULift E) 
goal: NNNorm (ULift.{u_5, u_2} E)

full_name: ULift.nnnorm_def
code: lemma nnnorm_def (x : ULift E) : ‖x‖₊ = ‖x.down‖₊ 
goal: ‖x‖₊ = ‖x.down‖₊

full_name: ULift.nnnorm_up
code: @[simp] lemma nnnorm_up (x : E) : ‖ULift.up x‖₊ = ‖x‖₊ 
goal: ‖{ down := x }‖₊ = ‖x‖₊

full_name: ULift.nnnorm_down
code: @[simp] lemma nnnorm_down (x : ULift E) : ‖x.down‖₊ = ‖x‖₊ 
goal: ‖x.down‖₊ = ‖x‖₊

full_name: ULift.seminormedGroup
code: @[to_additive]
instance seminormedGroup [SeminormedGroup E] : SeminormedGroup (ULift E) 
goal: SeminormedGroup (ULift.{u_5, u_2} E)

full_name: ULift.seminormedAddGroup
code: @[to_additive]
instance seminormedGroup [SeminormedGroup E] : SeminormedGroup (ULift E) 
goal: SeminormedAddGroup (ULift.{u_5, u_2} E)

full_name: ULift.seminormedAddCommGroup
code: @[to_additive]
instance seminormedCommGroup [SeminormedCommGroup E] : SeminormedCommGroup (ULift E) 
goal: SeminormedAddCommGroup (ULift.{u_5, u_2} E)

full_name: ULift.seminormedCommGroup
code: @[to_additive]
instance seminormedCommGroup [SeminormedCommGroup E] : SeminormedCommGroup (ULift E) 
goal: SeminormedCommGroup (ULift.{u_5, u_2} E)

full_name: ULift.normedGroup
code: @[to_additive]
instance normedGroup [NormedGroup E] : NormedGroup (ULift E) 
goal: NormedGroup (ULift.{u_5, u_2} E)

full_name: ULift.normedAddGroup
code: @[to_additive]
instance normedGroup [NormedGroup E] : NormedGroup (ULift E) 
goal: NormedAddGroup (ULift.{u_5, u_2} E)

full_name: ULift.normedAddCommGroup
code: @[to_additive]
instance normedCommGroup [NormedCommGroup E] : NormedCommGroup (ULift E) 
goal: NormedAddCommGroup (ULift.{u_5, u_2} E)

full_name: ULift.normedCommGroup
code: @[to_additive]
instance normedCommGroup [NormedCommGroup E] : NormedCommGroup (ULift E) 
goal: NormedCommGroup (ULift.{u_5, u_2} E)

full_name: Additive.toNorm
code: instance Additive.toNorm : Norm (Additive E) 
goal: Norm (Additive E)

full_name: Multiplicative.toNorm
code: instance Multiplicative.toNorm : Norm (Multiplicative E) 
goal: Norm (Multiplicative E)

full_name: norm_toMul
code: @[simp] lemma norm_toMul (x) : ‖(toMul x : E)‖ = ‖x‖ 
goal: ‖Additive.toMul x‖ = ‖x‖

full_name: norm_ofMul
code: @[simp] lemma norm_ofMul (x : E) : ‖ofMul x‖ = ‖x‖ 
goal: ‖Additive.ofMul x‖ = ‖x‖

full_name: norm_toAdd
code: @[simp] lemma norm_toAdd (x) : ‖(toAdd x : E)‖ = ‖x‖ 
goal: ‖Multiplicative.toAdd x‖ = ‖x‖

full_name: norm_ofAdd
code: @[simp] lemma norm_ofAdd (x : E) : ‖ofAdd x‖ = ‖x‖ 
goal: ‖Multiplicative.ofAdd x‖ = ‖x‖

full_name: Additive.toNNNorm
code: instance Additive.toNNNorm : NNNorm (Additive E) 
goal: NNNorm (Additive E)

full_name: Multiplicative.toNNNorm
code: instance Multiplicative.toNNNorm : NNNorm (Multiplicative E) 
goal: NNNorm (Multiplicative E)

full_name: nnnorm_toMul
code: @[simp] lemma nnnorm_toMul (x) : ‖(toMul x : E)‖₊ = ‖x‖₊ 
goal: ‖Additive.toMul x‖₊ = ‖x‖₊

full_name: nnnorm_ofMul
code: @[simp] lemma nnnorm_ofMul (x : E) : ‖ofMul x‖₊ = ‖x‖₊ 
goal: ‖Additive.ofMul x‖₊ = ‖x‖₊

full_name: nnnorm_toAdd
code: @[simp] lemma nnnorm_toAdd (x) : ‖(toAdd x : E)‖₊ = ‖x‖₊ 
goal: ‖Multiplicative.toAdd x‖₊ = ‖x‖₊

full_name: nnnorm_ofAdd
code: @[simp] lemma nnnorm_ofAdd (x : E) : ‖ofAdd x‖₊ = ‖x‖₊ 
goal: ‖Multiplicative.ofAdd x‖₊ = ‖x‖₊

full_name: Additive.seminormedAddGroup
code: instance Additive.seminormedAddGroup [SeminormedGroup E] : SeminormedAddGroup (Additive E) 
goal: SeminormedAddGroup (Additive E)

full_name: Multiplicative.seminormedGroup
code: instance Multiplicative.seminormedGroup [SeminormedAddGroup E] :
    SeminormedGroup (Multiplicative E) 
goal: SeminormedGroup (Multiplicative E)

full_name: Additive.seminormedCommGroup
code: instance Additive.seminormedCommGroup [SeminormedCommGroup E] :
    SeminormedAddCommGroup (Additive E) 
goal: SeminormedAddCommGroup (Additive E)

full_name: Multiplicative.seminormedAddCommGroup
code: instance Multiplicative.seminormedAddCommGroup [SeminormedAddCommGroup E] :
    SeminormedCommGroup (Multiplicative E) 
goal: SeminormedCommGroup (Multiplicative E)

full_name: Additive.normedAddGroup
code: instance Additive.normedAddGroup [NormedGroup E] : NormedAddGroup (Additive E) 
goal: NormedAddGroup (Additive E)

full_name: Multiplicative.normedGroup
code: instance Multiplicative.normedGroup [NormedAddGroup E] : NormedGroup (Multiplicative E) 
goal: NormedGroup (Multiplicative E)

full_name: Additive.normedAddCommGroup
code: instance Additive.normedAddCommGroup [NormedCommGroup E] : NormedAddCommGroup (Additive E) 
goal: NormedAddCommGroup (Additive E)

full_name: Multiplicative.normedCommGroup
code: instance Multiplicative.normedCommGroup [NormedAddCommGroup E] :
    NormedCommGroup (Multiplicative E) 
goal: NormedCommGroup (Multiplicative E)

full_name: OrderDual.toNorm
code: instance OrderDual.toNorm : Norm Eᵒᵈ 
goal: Norm Eᵒᵈ

full_name: norm_toDual
code: @[simp] lemma norm_toDual (x : E) : ‖toDual x‖ = ‖x‖ 
goal: ‖OrderDual.toDual x‖ = ‖x‖

full_name: norm_ofDual
code: @[simp] lemma norm_ofDual (x : Eᵒᵈ) : ‖ofDual x‖ = ‖x‖ 
goal: ‖OrderDual.ofDual x‖ = ‖x‖

full_name: OrderDual.toNNNorm
code: instance OrderDual.toNNNorm : NNNorm Eᵒᵈ 
goal: NNNorm Eᵒᵈ

full_name: nnnorm_toDual
code: @[simp] lemma nnnorm_toDual (x : E) : ‖toDual x‖₊ = ‖x‖₊ 
goal: ‖OrderDual.toDual x‖₊ = ‖x‖₊

full_name: nnnorm_ofDual
code: @[simp] lemma nnnorm_ofDual (x : Eᵒᵈ) : ‖ofDual x‖₊ = ‖x‖₊ 
goal: ‖OrderDual.ofDual x‖₊ = ‖x‖₊

full_name: OrderDual.seminormedAddGroup
code: @[to_additive]
instance (priority := 100) seminormedGroup [SeminormedGroup E] : SeminormedGroup Eᵒᵈ 
goal: SeminormedAddGroup Eᵒᵈ

full_name: OrderDual.seminormedGroup
code: @[to_additive]
instance (priority := 100) seminormedGroup [SeminormedGroup E] : SeminormedGroup Eᵒᵈ 
goal: SeminormedGroup Eᵒᵈ

full_name: OrderDual.seminormedAddCommGroup
code: @[to_additive]
instance (priority := 100) seminormedCommGroup [SeminormedCommGroup E] : SeminormedCommGroup Eᵒᵈ 
goal: SeminormedAddCommGroup Eᵒᵈ

full_name: OrderDual.seminormedCommGroup
code: @[to_additive]
instance (priority := 100) seminormedCommGroup [SeminormedCommGroup E] : SeminormedCommGroup Eᵒᵈ 
goal: SeminormedCommGroup Eᵒᵈ

full_name: OrderDual.normedGroup
code: @[to_additive]
instance (priority := 100) normedGroup [NormedGroup E] : NormedGroup Eᵒᵈ 
goal: NormedGroup Eᵒᵈ

full_name: OrderDual.normedAddGroup
code: @[to_additive]
instance (priority := 100) normedGroup [NormedGroup E] : NormedGroup Eᵒᵈ 
goal: NormedAddGroup Eᵒᵈ

full_name: OrderDual.normedCommGroup
code: @[to_additive]
instance (priority := 100) normedCommGroup [NormedCommGroup E] : NormedCommGroup Eᵒᵈ 
goal: NormedCommGroup Eᵒᵈ

full_name: OrderDual.normedAddCommGroup
code: @[to_additive]
instance (priority := 100) normedCommGroup [NormedCommGroup E] : NormedCommGroup Eᵒᵈ 
goal: NormedAddCommGroup Eᵒᵈ

full_name: Prod.toNorm
code: instance Prod.toNorm : Norm (E × F) 
goal: Norm (E × F)

full_name: Prod.norm_def
code: lemma Prod.norm_def (x : E × F) : ‖x‖ = max ‖x.1‖ ‖x.2‖ 
goal: ‖x‖ = max ‖x.1‖ ‖x.2‖

full_name: norm_fst_le
code: lemma norm_fst_le (x : E × F) : ‖x.1‖ ≤ ‖x‖ 
goal: ‖x.1‖ ≤ ‖x‖

full_name: norm_snd_le
code: lemma norm_snd_le (x : E × F) : ‖x.2‖ ≤ ‖x‖ 
goal: ‖x.2‖ ≤ ‖x‖

full_name: norm_prod_le_iff
code: lemma norm_prod_le_iff : ‖x‖ ≤ r ↔ ‖x.1‖ ≤ r ∧ ‖x.2‖ ≤ r 
goal: ‖x‖ ≤ r ↔ ‖x.1‖ ≤ r ∧ ‖x.2‖ ≤ r

full_name: Prod.seminormedAddGroup
code: /-- Product of seminormed groups, using the sup norm. -/
@[to_additive "Product of seminormed groups, using the sup norm."]
instance Prod.seminormedGroup : SeminormedGroup (E × F) 
goal: SeminormedAddGroup (E × F)

full_name: Prod.seminormedGroup
code: /-- Product of seminormed groups, using the sup norm. -/
@[to_additive "Product of seminormed groups, using the sup norm."]
instance Prod.seminormedGroup : SeminormedGroup (E × F) 
goal: SeminormedGroup (E × F)

full_name: Prod.nnnorm_def'
code: @[to_additive Prod.nnnorm_def']
lemma Prod.nnorm_def (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ 
goal: ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊

full_name: Prod.nnorm_def
code: @[to_additive Prod.nnnorm_def']
lemma Prod.nnorm_def (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ 
goal: ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊

full_name: Prod.seminormedAddCommGroup
code: /-- Product of seminormed groups, using the sup norm. -/
@[to_additive "Product of seminormed groups, using the sup norm."]
instance seminormedCommGroup [SeminormedCommGroup E] [SeminormedCommGroup F] :
    SeminormedCommGroup (E × F) 
goal: SeminormedAddCommGroup (E × F)

full_name: Prod.seminormedCommGroup
code: /-- Product of seminormed groups, using the sup norm. -/
@[to_additive "Product of seminormed groups, using the sup norm."]
instance seminormedCommGroup [SeminormedCommGroup E] [SeminormedCommGroup F] :
    SeminormedCommGroup (E × F) 
goal: SeminormedCommGroup (E × F)

full_name: Prod.normedAddGroup
code: /-- Product of normed groups, using the sup norm. -/
@[to_additive "Product of normed groups, using the sup norm."]
instance normedGroup [NormedGroup E] [NormedGroup F] : NormedGroup (E × F) 
goal: NormedAddGroup (E × F)

full_name: Prod.normedGroup
code: /-- Product of normed groups, using the sup norm. -/
@[to_additive "Product of normed groups, using the sup norm."]
instance normedGroup [NormedGroup E] [NormedGroup F] : NormedGroup (E × F) 
goal: NormedGroup (E × F)

full_name: Prod.normedCommGroup
code: /-- Product of normed groups, using the sup norm. -/
@[to_additive "Product of normed groups, using the sup norm."]
instance normedCommGroup [NormedCommGroup E] [NormedCommGroup F] : NormedCommGroup (E × F) 
goal: NormedCommGroup (E × F)

full_name: Prod.normedAddCommGroup
code: /-- Product of normed groups, using the sup norm. -/
@[to_additive "Product of normed groups, using the sup norm."]
instance normedCommGroup [NormedCommGroup E] [NormedCommGroup F] : NormedCommGroup (E × F) 
goal: NormedAddCommGroup (E × F)

full_name: Pi.seminormedGroup
code: /-- Finite product of seminormed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.seminormedGroup : SeminormedGroup (∀ i, π i) 
goal: SeminormedGroup ((i : ι) → π i)

full_name: Pi.seminormedAddGroup
code: /-- Finite product of seminormed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.seminormedGroup : SeminormedGroup (∀ i, π i) 
goal: SeminormedAddGroup ((i : ι) → π i)

full_name: Pi.norm_def
code: @[to_additive Pi.norm_def]
lemma Pi.norm_def' : ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊) 
goal: ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊)

full_name: Pi.norm_def'
code: @[to_additive Pi.norm_def]
lemma Pi.norm_def' : ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊) 
goal: ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊)

full_name: Pi.nnnorm_def
code: @[to_additive Pi.nnnorm_def]
lemma Pi.nnnorm_def' : ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ 
goal: ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊

full_name: Pi.nnnorm_def'
code: @[to_additive Pi.nnnorm_def]
lemma Pi.nnnorm_def' : ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ 
goal: ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊

full_name: pi_norm_le_iff_of_nonneg
code: /-- The seminorm of an element in a product space is `≤ r` if and only if the norm of each
component is. -/
@[to_additive pi_norm_le_iff_of_nonneg "The seminorm of an element in a product space is `≤ r` if
and only if the norm of each component is."]
lemma pi_norm_le_iff_of_nonneg' (hr : 0 ≤ r) : ‖x‖ ≤ r ↔ ∀ i, ‖x i‖ ≤ r 
goal: ‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r

full_name: pi_norm_le_iff_of_nonneg'
code: /-- The seminorm of an element in a product space is `≤ r` if and only if the norm of each
component is. -/
@[to_additive pi_norm_le_iff_of_nonneg "The seminorm of an element in a product space is `≤ r` if
and only if the norm of each component is."]
lemma pi_norm_le_iff_of_nonneg' (hr : 0 ≤ r) : ‖x‖ ≤ r ↔ ∀ i, ‖x i‖ ≤ r 
goal: ‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r

full_name: pi_nnnorm_le_iff
code: @[to_additive pi_nnnorm_le_iff]
lemma pi_nnnorm_le_iff' {r : ℝ≥0} : ‖x‖₊ ≤ r ↔ ∀ i, ‖x i‖₊ ≤ r 
goal: ‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r

full_name: pi_nnnorm_le_iff'
code: @[to_additive pi_nnnorm_le_iff]
lemma pi_nnnorm_le_iff' {r : ℝ≥0} : ‖x‖₊ ≤ r ↔ ∀ i, ‖x i‖₊ ≤ r 
goal: ‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r

full_name: pi_norm_le_iff_of_nonempty'
code: @[to_additive pi_norm_le_iff_of_nonempty]
lemma pi_norm_le_iff_of_nonempty' [Nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r 
goal: ‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r

full_name: pi_norm_le_iff_of_nonempty
code: @[to_additive pi_norm_le_iff_of_nonempty]
lemma pi_norm_le_iff_of_nonempty' [Nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r 
goal: ‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r

full_name: pi_norm_lt_iff
code: /-- The seminorm of an element in a product space is `< r` if and only if the norm of each
component is. -/
@[to_additive pi_norm_lt_iff "The seminorm of an element in a product space is `< r` if and only
if the norm of each component is."]
lemma pi_norm_lt_iff' (hr : 0 < r) : ‖x‖ < r ↔ ∀ i, ‖x i‖ < r 
goal: ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r

full_name: pi_norm_lt_iff'
code: /-- The seminorm of an element in a product space is `< r` if and only if the norm of each
component is. -/
@[to_additive pi_norm_lt_iff "The seminorm of an element in a product space is `< r` if and only
if the norm of each component is."]
lemma pi_norm_lt_iff' (hr : 0 < r) : ‖x‖ < r ↔ ∀ i, ‖x i‖ < r 
goal: ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r

full_name: pi_nnnorm_lt_iff'
code: @[to_additive pi_nnnorm_lt_iff]
lemma pi_nnnorm_lt_iff' {r : ℝ≥0} (hr : 0 < r) : ‖x‖₊ < r ↔ ∀ i, ‖x i‖₊ < r 
goal: ‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r

full_name: pi_nnnorm_lt_iff
code: @[to_additive pi_nnnorm_lt_iff]
lemma pi_nnnorm_lt_iff' {r : ℝ≥0} (hr : 0 < r) : ‖x‖₊ < r ↔ ∀ i, ‖x i‖₊ < r 
goal: ‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r

full_name: norm_le_pi_norm
code: @[to_additive norm_le_pi_norm]
lemma norm_le_pi_norm' (i : ι) : ‖f i‖ ≤ ‖f‖ 
goal: ‖f i‖ ≤ ‖f‖

full_name: norm_le_pi_norm'
code: @[to_additive norm_le_pi_norm]
lemma norm_le_pi_norm' (i : ι) : ‖f i‖ ≤ ‖f‖ 
goal: ‖f i‖ ≤ ‖f‖

full_name: nnnorm_le_pi_nnnorm
code: @[to_additive nnnorm_le_pi_nnnorm]
lemma nnnorm_le_pi_nnnorm' (i : ι) : ‖f i‖₊ ≤ ‖f‖₊ 
goal: ‖f i‖₊ ≤ ‖f‖₊

full_name: nnnorm_le_pi_nnnorm'
code: @[to_additive nnnorm_le_pi_nnnorm]
lemma nnnorm_le_pi_nnnorm' (i : ι) : ‖f i‖₊ ≤ ‖f‖₊ 
goal: ‖f i‖₊ ≤ ‖f‖₊

full_name: pi_norm_const_le
code: @[to_additive pi_norm_const_le]
lemma pi_norm_const_le' (a : E) : ‖fun _ : ι => a‖ ≤ ‖a‖ 
goal: ‖fun x => a‖ ≤ ‖a‖

full_name: pi_norm_const_le'
code: @[to_additive pi_norm_const_le]
lemma pi_norm_const_le' (a : E) : ‖fun _ : ι => a‖ ≤ ‖a‖ 
goal: ‖fun x => a‖ ≤ ‖a‖

full_name: pi_nnnorm_const_le
code: @[to_additive pi_nnnorm_const_le]
lemma pi_nnnorm_const_le' (a : E) : ‖fun _ : ι => a‖₊ ≤ ‖a‖₊ 
goal: ‖fun x => a‖₊ ≤ ‖a‖₊

full_name: pi_nnnorm_const_le'
code: @[to_additive pi_nnnorm_const_le]
lemma pi_nnnorm_const_le' (a : E) : ‖fun _ : ι => a‖₊ ≤ ‖a‖₊ 
goal: ‖fun x => a‖₊ ≤ ‖a‖₊

full_name: pi_norm_const'
code: @[to_additive (attr := simp) pi_norm_const]
lemma pi_norm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖ = ‖a‖ 
goal: ‖fun _i => a‖ = ‖a‖

full_name: pi_norm_const
code: @[to_additive (attr := simp) pi_norm_const]
lemma pi_norm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖ = ‖a‖ 
goal: ‖fun _i => a‖ = ‖a‖

full_name: pi_nnnorm_const'
code: @[to_additive (attr := simp) pi_nnnorm_const]
lemma pi_nnnorm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖₊ = ‖a‖₊ 
goal: ‖fun _i => a‖₊ = ‖a‖₊

full_name: pi_nnnorm_const
code: @[to_additive (attr := simp) pi_nnnorm_const]
lemma pi_nnnorm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖₊ = ‖a‖₊ 
goal: ‖fun _i => a‖₊ = ‖a‖₊

full_name: Pi.sum_norm_apply_le_norm'
code: /-- The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. -/
@[to_additive Pi.sum_norm_apply_le_norm "The $L^1$ norm is less than the $L^\\infty$ norm scaled by
the cardinality."]
lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f‖ 
goal: ∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖

full_name: Pi.sum_norm_apply_le_norm
code: /-- The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. -/
@[to_additive Pi.sum_norm_apply_le_norm "The $L^1$ norm is less than the $L^\\infty$ norm scaled by
the cardinality."]
lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f‖ 
goal: ∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖

full_name: Pi.sum_nnnorm_apply_le_nnnorm
code: /-- The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. -/
@[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
scaled by the cardinality."]
lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ 
goal: ∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊

full_name: Pi.sum_nnnorm_apply_le_nnnorm'
code: /-- The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. -/
@[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
scaled by the cardinality."]
lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ 
goal: ∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊

full_name: Pi.seminormedCommGroup
code: /-- Finite product of seminormed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.seminormedCommGroup [∀ i, SeminormedCommGroup (π i)] : SeminormedCommGroup (∀ i, π i) 
goal: SeminormedCommGroup ((i : ι) → π i)

full_name: Pi.seminormedAddCommGroup
code: /-- Finite product of seminormed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.seminormedCommGroup [∀ i, SeminormedCommGroup (π i)] : SeminormedCommGroup (∀ i, π i) 
goal: SeminormedAddCommGroup ((i : ι) → π i)

full_name: Pi.normedAddGroup
code: /-- Finite product of normed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.normedGroup [∀ i, NormedGroup (π i)] : NormedGroup (∀ i, π i) 
goal: NormedAddGroup ((i : ι) → π i)

full_name: Pi.normedGroup
code: /-- Finite product of normed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.normedGroup [∀ i, NormedGroup (π i)] : NormedGroup (∀ i, π i) 
goal: NormedGroup ((i : ι) → π i)

full_name: Pi.normedAddCommGroup
code: /-- Finite product of normed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.normedCommGroup [∀ i, NormedCommGroup (π i)] : NormedCommGroup (∀ i, π i) 
goal: NormedAddCommGroup ((i : ι) → π i)

full_name: Pi.normedCommGroup
code: /-- Finite product of normed groups, using the sup norm. -/
@[to_additive "Finite product of seminormed groups, using the sup norm."]
instance Pi.normedCommGroup [∀ i, NormedCommGroup (π i)] : NormedCommGroup (∀ i, π i) 
goal: NormedCommGroup ((i : ι) → π i)

full_name: Pi.nnnorm_single
code: theorem Pi.nnnorm_single [DecidableEq ι] [∀ i, NormedAddCommGroup (π i)] {i : ι} (y : π i) :
    ‖Pi.single i y‖₊ = ‖y‖₊ 
goal: ‖Pi.single i y‖₊ = ‖y‖₊

full_name: Pi.norm_single
code: theorem Pi.norm_single [DecidableEq ι] [∀ i, NormedAddCommGroup (π i)] {i : ι} (y : π i) :
    ‖Pi.single i y‖ = ‖y‖ 
goal: ‖Pi.single i y‖ = ‖y‖

full_name: MulOpposite.instSeminormedAddGroup
code: /-- The (additive) norm on the multiplicative opposite is the same as the norm on the original type.

Note that we do not provide this more generally as `Norm Eᵐᵒᵖ`, as this is not always a good
choice of norm in the multiplicative `SeminormedGroup E` case.

We could repeat this instance to provide a `[SeminormedGroup E] : SeminormedGroup Eᵃᵒᵖ` instance,
but that case would likely never be used.
-/
instance instSeminormedAddGroup [SeminormedAddGroup E] : SeminormedAddGroup Eᵐᵒᵖ 
goal: SeminormedAddGroup Eᵐᵒᵖ

full_name: MulOpposite.norm_op
code: lemma norm_op [SeminormedAddGroup E] (a : E) : ‖MulOpposite.op a‖ = ‖a‖ 
goal: ‖MulOpposite.op a‖ = ‖a‖

full_name: MulOpposite.norm_unop
code: lemma norm_unop [SeminormedAddGroup E] (a : Eᵐᵒᵖ) : ‖MulOpposite.unop a‖ = ‖a‖ 
goal: ‖MulOpposite.unop a‖ = ‖a‖

full_name: MulOpposite.nnnorm_op
code: lemma nnnorm_op [SeminormedAddGroup E] (a : E) : ‖MulOpposite.op a‖₊ = ‖a‖₊ 
goal: ‖MulOpposite.op a‖₊ = ‖a‖₊

full_name: MulOpposite.nnnorm_unop
code: lemma nnnorm_unop [SeminormedAddGroup E] (a : Eᵐᵒᵖ) : ‖MulOpposite.unop a‖₊ = ‖a‖₊ 
goal: ‖MulOpposite.unop a‖₊ = ‖a‖₊

full_name: MulOpposite.instNormedAddGroup
code: instance instNormedAddGroup [NormedAddGroup E] : NormedAddGroup Eᵐᵒᵖ 
goal: NormedAddGroup Eᵐᵒᵖ

full_name: MulOpposite.instSeminormedAddCommGroup
code: instance instSeminormedAddCommGroup [SeminormedAddCommGroup E] : SeminormedAddCommGroup Eᵐᵒᵖ 
goal: SeminormedAddCommGroup Eᵐᵒᵖ

full_name: MulOpposite.instNormedAddCommGroup
code: instance instNormedAddCommGroup [NormedAddCommGroup E] : NormedAddCommGroup Eᵐᵒᵖ 
goal: NormedAddCommGroup Eᵐᵒᵖ