File: Mathlib/Analysis/Normed/Algebra/UnitizationL1.lean

full_name: WithLp.unitization_addEquiv_prod
code: /-- The natural map between `Unitization 𝕜 A` and `𝕜 × A`, transferred to their `WithLp 1`
synonyms. -/
noncomputable def unitization_addEquiv_prod : WithLp 1 (Unitization 𝕜 A) ≃+ WithLp 1 (𝕜 × A) 
goal: WithLp 1 (Unitization 𝕜 A) ≃+ WithLp 1 (𝕜 × A)

full_name: WithLp.instUnitizationNormedAddCommGroup
code: noncomputable instance instUnitizationNormedAddCommGroup :
    NormedAddCommGroup (WithLp 1 (Unitization 𝕜 A)) 
goal: NormedAddCommGroup (WithLp 1 (Unitization 𝕜 A))

full_name: WithLp.uniformEquiv_unitization_addEquiv_prod
code: /-- Bundle `WithLp.unitization_addEquiv_prod` as a `UniformEquiv`. -/
noncomputable def uniformEquiv_unitization_addEquiv_prod :
    WithLp 1 (Unitization 𝕜 A) ≃ᵤ WithLp 1 (𝕜 × A) 
goal: WithLp 1 (Unitization 𝕜 A) ≃ᵤ WithLp 1 (𝕜 × A)

full_name: WithLp.instCompleteSpace
code: instance instCompleteSpace [CompleteSpace 𝕜] [CompleteSpace A] :
    CompleteSpace (WithLp 1 (Unitization 𝕜 A)) 
goal: CompleteSpace (WithLp 1 (Unitization 𝕜 A))

full_name: WithLp.unitization_norm_def
code: lemma unitization_norm_def (x : WithLp 1 (Unitization 𝕜 A)) :
    ‖x‖ = ‖(WithLp.equiv 1 _ x).fst‖ + ‖(WithLp.equiv 1 _ x).snd‖ 
goal: ‖x‖ = ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).fst‖ + ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).snd‖

full_name: WithLp.unitization_nnnorm_def
code: lemma unitization_nnnorm_def (x : WithLp 1 (Unitization 𝕜 A)) :
    ‖x‖₊ = ‖(WithLp.equiv 1 _ x).fst‖₊ + ‖(WithLp.equiv 1 _ x).snd‖₊ 
goal: ‖x‖₊ = ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).fst‖₊ + ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).snd‖₊

full_name: WithLp.unitization_norm_inr
code: lemma unitization_norm_inr (x : A) : ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm x‖ = ‖x‖ 
goal: ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x‖ = ‖x‖

full_name: WithLp.unitization_nnnorm_inr
code: lemma unitization_nnnorm_inr (x : A) : ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm x‖₊ = ‖x‖₊ 
goal: ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x‖₊ = ‖x‖₊

full_name: WithLp.unitization_isometry_inr
code: lemma unitization_isometry_inr :
    Isometry (fun x : A ↦ (WithLp.equiv 1 (Unitization 𝕜 A)).symm x) 
goal: Isometry fun x => (WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x

full_name: WithLp.instUnitizationRing
code: instance instUnitizationRing : Ring (WithLp 1 (Unitization 𝕜 A)) 
goal: Ring (WithLp 1 (Unitization 𝕜 A))

full_name: WithLp.unitization_mul
code: @[simp]
lemma unitization_mul (x y : WithLp 1 (Unitization 𝕜 A)) :
    WithLp.equiv 1 _ (x * y) = (WithLp.equiv 1 _ x) * (WithLp.equiv 1 _ y) 
goal: (WithLp.equiv 1 (Unitization 𝕜 A)) (x * y) = (WithLp.equiv 1 (Unitization 𝕜 A)) x * (WithLp.equiv 1 (Unitization 𝕜 A)) y

full_name: WithLp.instAlgebraOfNatENNRealUnitizationOfIsScalarTower
code: instance {R : Type*} [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A] [IsScalarTower R 𝕜 A] :
    Algebra R (WithLp 1 (Unitization 𝕜 A)) 
goal: Algebra R (WithLp 1 (Unitization 𝕜 A))

full_name: WithLp.unitization_algebraMap
code: @[simp]
lemma unitization_algebraMap (r : 𝕜) :
    WithLp.equiv 1 _ (algebraMap 𝕜 (WithLp 1 (Unitization 𝕜 A)) r) =
      algebraMap 𝕜 (Unitization 𝕜 A) r 
goal: (WithLp.equiv 1 (Unitization 𝕜 A)) ((algebraMap 𝕜 (WithLp 1 (Unitization 𝕜 A))) r) = (algebraMap 𝕜 (Unitization 𝕜 A)) r

full_name: WithLp.unitizationAlgEquiv_apply
code: /-- `WithLp.equiv` bundled as an algebra isomorphism with `Unitization 𝕜 A`. -/
@[simps!]
def unitizationAlgEquiv (R : Type*) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A]
    [IsScalarTower R 𝕜 A] : WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A 
goal: (WithLp.unitizationAlgEquiv R) a = a

full_name: WithLp.unitizationAlgEquiv_symm_apply
code: /-- `WithLp.equiv` bundled as an algebra isomorphism with `Unitization 𝕜 A`. -/
@[simps!]
def unitizationAlgEquiv (R : Type*) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A]
    [IsScalarTower R 𝕜 A] : WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A 
goal: (WithLp.unitizationAlgEquiv R).symm a = a

full_name: WithLp.unitizationAlgEquiv
code: /-- `WithLp.equiv` bundled as an algebra isomorphism with `Unitization 𝕜 A`. -/
@[simps!]
def unitizationAlgEquiv (R : Type*) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A]
    [IsScalarTower R 𝕜 A] : WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A 
goal: WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A

full_name: WithLp.instUnitizationNormedRing
code: noncomputable instance instUnitizationNormedRing : NormedRing (WithLp 1 (Unitization 𝕜 A)) 
goal: NormedRing (WithLp 1 (Unitization 𝕜 A))

full_name: WithLp.instUnitizationNormedAlgebra
code: noncomputable instance instUnitizationNormedAlgebra :
    NormedAlgebra 𝕜 (WithLp 1 (Unitization 𝕜 A)) 
goal: NormedAlgebra 𝕜 (WithLp 1 (Unitization 𝕜 A))