File: Mathlib/Analysis/Normed/Lp/WithLp.lean

full_name: WithLp
code: /-- A type synonym for the given `V`, associated with the L`p` norm. Note that by default this just
forgets the norm structure on `V`; it is up to downstream users to implement the L`p` norm (for
instance, on `Prod` and finite `Pi` types). -/
@[nolint unusedArguments]
def WithLp (_p : ℝ≥0∞) (V : Type uV) : Type uV 
goal: Type uV

full_name: WithLp.equiv
code: /-- The canonical equivalence between `WithLp p V` and `V`. This should always be used to convert
back and forth between the representations. -/
protected def equiv : WithLp p V ≃ V 
goal: WithLp p V ≃ V

full_name: WithLp.instNontrivial
code: instance instNontrivial [Nontrivial V] : Nontrivial (WithLp p V) 
goal: Nontrivial (WithLp p V)

full_name: WithLp.instUnique
code: instance instUnique [Unique V] : Unique (WithLp p V) 
goal: Unique (WithLp p V)

full_name: WithLp.instAddCommGroup
code: instance instAddCommGroup : AddCommGroup (WithLp p V) 
goal: AddCommGroup (WithLp p V)

full_name: WithLp.instModule
code: instance instModule [Module K V] : Module K (WithLp p V) 
goal: Module K (WithLp p V)

full_name: WithLp.instIsScalarTower
code: instance instIsScalarTower [SMul K K'] [Module K V] [Module K' V] [IsScalarTower K K' V] :
    IsScalarTower K K' (WithLp p V) 
goal: IsScalarTower K K' (WithLp p V)

full_name: WithLp.instSMulCommClass
code: instance instSMulCommClass [Module K V] [Module K' V] [SMulCommClass K K' V] :
    SMulCommClass K K' (WithLp p V) 
goal: SMulCommClass K K' (WithLp p V)

full_name: WithLp.instModuleFinite
code: instance instModuleFinite [Module K V] [Module.Finite K V] : Module.Finite K (WithLp p V) 
goal: Module.Finite K (WithLp p V)

full_name: WithLp.equiv_zero
code: @[simp]
theorem equiv_zero : WithLp.equiv p V 0 = 0 
goal: (WithLp.equiv p V) 0 = 0

full_name: WithLp.equiv_symm_zero
code: @[simp]
theorem equiv_symm_zero : (WithLp.equiv p V).symm 0 = 0 
goal: (WithLp.equiv p V).symm 0 = 0

full_name: WithLp.equiv_add
code: @[simp]
theorem equiv_add : WithLp.equiv p V (x + y) = WithLp.equiv p V x + WithLp.equiv p V y 
goal: (WithLp.equiv p V) (x + y) = (WithLp.equiv p V) x + (WithLp.equiv p V) y

full_name: WithLp.equiv_symm_add
code: @[simp]
theorem equiv_symm_add :
    (WithLp.equiv p V).symm (x' + y') = (WithLp.equiv p V).symm x' + (WithLp.equiv p V).symm y' 
goal: (WithLp.equiv p V).symm (x' + y') = (WithLp.equiv p V).symm x' + (WithLp.equiv p V).symm y'

full_name: WithLp.equiv_sub
code: @[simp]
theorem equiv_sub : WithLp.equiv p V (x - y) = WithLp.equiv p V x - WithLp.equiv p V y 
goal: (WithLp.equiv p V) (x - y) = (WithLp.equiv p V) x - (WithLp.equiv p V) y

full_name: WithLp.equiv_symm_sub
code: @[simp]
theorem equiv_symm_sub :
    (WithLp.equiv p V).symm (x' - y') = (WithLp.equiv p V).symm x' - (WithLp.equiv p V).symm y' 
goal: (WithLp.equiv p V).symm (x' - y') = (WithLp.equiv p V).symm x' - (WithLp.equiv p V).symm y'

full_name: WithLp.equiv_neg
code: @[simp]
theorem equiv_neg : WithLp.equiv p V (-x) = -WithLp.equiv p V x 
goal: (WithLp.equiv p V) (-x) = -(WithLp.equiv p V) x

full_name: WithLp.equiv_symm_neg
code: @[simp]
theorem equiv_symm_neg : (WithLp.equiv p V).symm (-x') = -(WithLp.equiv p V).symm x' 
goal: (WithLp.equiv p V).symm (-x') = -(WithLp.equiv p V).symm x'

full_name: WithLp.equiv_smul
code: @[simp]
theorem equiv_smul : WithLp.equiv p V (c • x) = c • WithLp.equiv p V x 
goal: (WithLp.equiv p V) (c • x) = c • (WithLp.equiv p V) x

full_name: WithLp.equiv_symm_smul
code: @[simp]
theorem equiv_symm_smul : (WithLp.equiv p V).symm (c • x') = c • (WithLp.equiv p V).symm x' 
goal: (WithLp.equiv p V).symm (c • x') = c • (WithLp.equiv p V).symm x'

full_name: WithLp.linearEquiv_symm_apply
code: /-- `WithLp.equiv` as a linear equivalence. -/
@[simps (config := .asFn)]
protected def linearEquiv : WithLp p V ≃ₗ[K] V 
goal: ⇑(WithLp.linearEquiv p K V).symm = ⇑(WithLp.equiv p V).symm

full_name: WithLp.linearEquiv_apply
code: /-- `WithLp.equiv` as a linear equivalence. -/
@[simps (config := .asFn)]
protected def linearEquiv : WithLp p V ≃ₗ[K] V 
goal: ⇑(WithLp.linearEquiv p K V) = ⇑(WithLp.equiv p V)

full_name: WithLp.linearEquiv
code: /-- `WithLp.equiv` as a linear equivalence. -/
@[simps (config := .asFn)]
protected def linearEquiv : WithLp p V ≃ₗ[K] V 
goal: WithLp p V ≃ₗ[K] V