ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one
theorem normNoninc_iff_norm_le_one : f.NormNoninc ↔ ‖f‖ ≤ 1 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : NormedAddGroupHom V W}
Goal: f.NormNoninc ↔ ‖f‖ ≤ 1
-- NormedAddGroupHom.NormNoninc.zero
theorem zero : (0 : NormedAddGroupHom V₁ V₂).NormNoninc 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂]
Goal: NormedAddGroupHom.NormNoninc 0
-- NormedAddGroupHom.NormNoninc.id
theorem id : (id V).NormNoninc 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V]
Goal: (NormedAddGroupHom.id V).NormNoninc
-- NormedAddGroupHom.NormNoninc.comp
theorem comp {g : NormedAddGroupHom V₂ V₃} {f : NormedAddGroupHom V₁ V₂} (hg : g.NormNoninc)
    (hf : f.NormNoninc) : (g.comp f).NormNoninc 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {g : NormedAddGroupHom V₂ V₃} {f : NormedAddGroupHom V₁ V₂} (hg : g.NormNoninc) (hf : f.NormNoninc)
Goal: (g.comp f).NormNoninc
-- NormedAddGroupHom.NormNoninc.neg_iff
@[simp]
theorem neg_iff {f : NormedAddGroupHom V₁ V₂} : (-f).NormNoninc ↔ f.NormNoninc 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {f : NormedAddGroupHom V₁ V₂}
Goal: (-f).NormNoninc ↔ f.NormNoninc
-- NormedAddGroupHom.norm_eq_of_isometry
theorem norm_eq_of_isometry {f : NormedAddGroupHom V W} (hf : Isometry f) (v : V) : ‖f v‖ = ‖v‖ 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : NormedAddGroupHom V W} (hf : Isometry ⇑f) (v : V)
Goal: ‖f v‖ = ‖v‖
-- NormedAddGroupHom.isometry_id
theorem isometry_id : @Isometry V V _ _ (id V) 
Arguments: {V : Type u_1} [SeminormedAddCommGroup V]
Goal: Isometry ⇑(NormedAddGroupHom.id V)
-- NormedAddGroupHom.isometry_comp
theorem isometry_comp {g : NormedAddGroupHom V₂ V₃} {f : NormedAddGroupHom V₁ V₂} (hg : Isometry g)
    (hf : Isometry f) : Isometry (g.comp f) 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {g : NormedAddGroupHom V₂ V₃} {f : NormedAddGroupHom V₁ V₂} (hg : Isometry ⇑g) (hf : Isometry ⇑f)
Goal: Isometry ⇑(g.comp f)
-- NormedAddGroupHom.normNoninc_of_isometry
theorem normNoninc_of_isometry (hf : Isometry f) : f.NormNoninc 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : NormedAddGroupHom V W} (hf : Isometry ⇑f)
Goal: f.NormNoninc
-- NormedAddGroupHom.Equalizer.comp_ι_eq
theorem comp_ι_eq : f.comp (ι f g) = g.comp (ι f g) 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] (f g : NormedAddGroupHom V W)
Goal: f.comp (NormedAddGroupHom.Equalizer.ι f g) = g.comp (NormedAddGroupHom.Equalizer.ι f g)
-- NormedAddGroupHom.Equalizer.lift_apply_coe
@[simps]
def lift (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) :
    NormedAddGroupHom V₁ (f.equalizer g) 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) (v : V₁)
Goal: ↑((NormedAddGroupHom.Equalizer.lift φ h) v) = φ v
-- NormedAddGroupHom.Equalizer.ι_comp_lift
@[simp]
theorem ι_comp_lift (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) :
    (ι _ _).comp (lift φ h) = φ 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ)
Goal: (NormedAddGroupHom.Equalizer.ι f g).comp (NormedAddGroupHom.Equalizer.lift φ h) = φ
-- NormedAddGroupHom.Equalizer.liftEquiv_apply
@[simps]
def liftEquiv :
    { φ : NormedAddGroupHom V₁ V // f.comp φ = g.comp φ } ≃
      NormedAddGroupHom V₁ (f.equalizer g) 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (φ : { φ // f.comp φ = g.comp φ })
Goal: NormedAddGroupHom.Equalizer.liftEquiv φ = NormedAddGroupHom.Equalizer.lift ↑φ ⋯
-- NormedAddGroupHom.Equalizer.liftEquiv_symm_apply_coe
@[simps]
def liftEquiv :
    { φ : NormedAddGroupHom V₁ V // f.comp φ = g.comp φ } ≃
      NormedAddGroupHom V₁ (f.equalizer g) 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (ψ : NormedAddGroupHom V₁ ↥(f.equalizer g))
Goal: ↑(NormedAddGroupHom.Equalizer.liftEquiv.symm ψ) = (NormedAddGroupHom.Equalizer.ι f g).comp ψ
-- NormedAddGroupHom.Equalizer.ι_comp_map
@[simp]
theorem ι_comp_map (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) :
    (ι f₂ g₂).comp (map φ ψ hf hg) = φ.comp (ι f₁ g₁) 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {W₁ : Type u_6} {W₂ : Type u_7} [SeminormedAddCommGroup W₁] [SeminormedAddCommGroup W₂] {f₁ g₁ : NormedAddGroupHom V₁ W₁} {f₂ g₂ : NormedAddGroupHom V₂ W₂} {φ : NormedAddGroupHom V₁ V₂} {ψ : NormedAddGroupHom W₁ W₂} (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
Goal: (NormedAddGroupHom.Equalizer.ι f₂ g₂).comp (NormedAddGroupHom.Equalizer.map φ ψ hf hg) =
  φ.comp (NormedAddGroupHom.Equalizer.ι f₁ g₁)
-- NormedAddGroupHom.Equalizer.map_id
@[simp]
theorem map_id : map (f₂ := f₁) (g₂ := g₁) (id V₁) (id W₁) rfl rfl = id (f₁.equalizer g₁) 
Arguments: {V₁ : Type u_3} [SeminormedAddCommGroup V₁] {W₁ : Type u_6} [SeminormedAddCommGroup W₁] {f₁ g₁ : NormedAddGroupHom V₁ W₁}
Goal: NormedAddGroupHom.Equalizer.map (NormedAddGroupHom.id V₁) (NormedAddGroupHom.id W₁) ⋯ ⋯ =
  NormedAddGroupHom.id ↥(f₁.equalizer g₁)
-- NormedAddGroupHom.Equalizer.comm_sq₂
theorem comm_sq₂ (hf : ψ.comp f₁ = f₂.comp φ) (hf' : ψ'.comp f₂ = f₃.comp φ') :
    (ψ'.comp ψ).comp f₁ = f₃.comp (φ'.comp φ) 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {W₁ : Type u_6} {W₂ : Type u_7} {W₃ : Type u_8} [SeminormedAddCommGroup W₁] [SeminormedAddCommGroup W₂] [SeminormedAddCommGroup W₃] {f₁ : NormedAddGroupHom V₁ W₁} {f₂ : NormedAddGroupHom V₂ W₂} {f₃ : NormedAddGroupHom V₃ W₃} {φ : NormedAddGroupHom V₁ V₂} {ψ : NormedAddGroupHom W₁ W₂} {φ' : NormedAddGroupHom V₂ V₃} {ψ' : NormedAddGroupHom W₂ W₃} (hf : ψ.comp f₁ = f₂.comp φ) (hf' : ψ'.comp f₂ = f₃.comp φ')
Goal: (ψ'.comp ψ).comp f₁ = f₃.comp (φ'.comp φ)
-- NormedAddGroupHom.Equalizer.map_comp_map
theorem map_comp_map (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
    (hf' : ψ'.comp f₂ = f₃.comp φ') (hg' : ψ'.comp g₂ = g₃.comp φ') :
    (map φ' ψ' hf' hg').comp (map φ ψ hf hg) =
      map (φ'.comp φ) (ψ'.comp ψ) (comm_sq₂ hf hf') (comm_sq₂ hg hg') 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {W₁ : Type u_6} {W₂ : Type u_7} {W₃ : Type u_8} [SeminormedAddCommGroup W₁] [SeminormedAddCommGroup W₂] [SeminormedAddCommGroup W₃] {f₁ g₁ : NormedAddGroupHom V₁ W₁} {f₂ g₂ : NormedAddGroupHom V₂ W₂} {f₃ g₃ : NormedAddGroupHom V₃ W₃} {φ : NormedAddGroupHom V₁ V₂} {ψ : NormedAddGroupHom W₁ W₂} {φ' : NormedAddGroupHom V₂ V₃} {ψ' : NormedAddGroupHom W₂ W₃} (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) (hf' : ψ'.comp f₂ = f₃.comp φ') (hg' : ψ'.comp g₂ = g₃.comp φ')
Goal: (NormedAddGroupHom.Equalizer.map φ' ψ' hf' hg').comp (NormedAddGroupHom.Equalizer.map φ ψ hf hg) =
  NormedAddGroupHom.Equalizer.map (φ'.comp φ) (ψ'.comp ψ) ⋯ ⋯
-- NormedAddGroupHom.Equalizer.ι_normNoninc
theorem ι_normNoninc : (ι f g).NormNoninc 
Arguments: {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f g : NormedAddGroupHom V W}
Goal: (NormedAddGroupHom.Equalizer.ι f g).NormNoninc
-- NormedAddGroupHom.Equalizer.lift_normNoninc
theorem lift_normNoninc (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) (hφ : φ.NormNoninc) :
    (lift φ h).NormNoninc 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) (hφ : φ.NormNoninc)
Goal: (NormedAddGroupHom.Equalizer.lift φ h).NormNoninc
-- NormedAddGroupHom.Equalizer.norm_lift_le
theorem norm_lift_le (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) (C : ℝ) (hφ : ‖φ‖ ≤ C) :
    ‖lift φ h‖ ≤ C 
Arguments: {V : Type u_1} {W : Type u_2} {V₁ : Type u_3} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [SeminormedAddCommGroup V₁] {f g : NormedAddGroupHom V W} (φ : NormedAddGroupHom V₁ V) (h : f.comp φ = g.comp φ) (C : ℝ) (hφ : ‖φ‖ ≤ C)
Goal: ‖NormedAddGroupHom.Equalizer.lift φ h‖ ≤ C
-- NormedAddGroupHom.Equalizer.map_normNoninc
theorem map_normNoninc (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
    (hφ : φ.NormNoninc) : (map φ ψ hf hg).NormNoninc 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {W₁ : Type u_6} {W₂ : Type u_7} [SeminormedAddCommGroup W₁] [SeminormedAddCommGroup W₂] {f₁ g₁ : NormedAddGroupHom V₁ W₁} {f₂ g₂ : NormedAddGroupHom V₂ W₂} {φ : NormedAddGroupHom V₁ V₂} {ψ : NormedAddGroupHom W₁ W₂} (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) (hφ : φ.NormNoninc)
Goal: (NormedAddGroupHom.Equalizer.map φ ψ hf hg).NormNoninc
-- NormedAddGroupHom.Equalizer.norm_map_le
theorem norm_map_le (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) (C : ℝ)
    (hφ : ‖φ.comp (ι f₁ g₁)‖ ≤ C) : ‖map φ ψ hf hg‖ ≤ C 
Arguments: {V₁ : Type u_3} {V₂ : Type u_4} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] {W₁ : Type u_6} {W₂ : Type u_7} [SeminormedAddCommGroup W₁] [SeminormedAddCommGroup W₂] {f₁ g₁ : NormedAddGroupHom V₁ W₁} {f₂ g₂ : NormedAddGroupHom V₂ W₂} {φ : NormedAddGroupHom V₁ V₂} {ψ : NormedAddGroupHom W₁ W₂} (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) (C : ℝ) (hφ : ‖φ.comp (NormedAddGroupHom.Equalizer.ι f₁ g₁)‖ ≤ C)
Goal: ‖NormedAddGroupHom.Equalizer.map φ ψ hf hg‖ ≤ C
-- PiTensorProduct.toDualContinuousMultilinearMap_apply_apply
@[simps!]
noncomputable def toDualContinuousMultilinearMap : (⨂[𝕜] i, E i) →ₗ[𝕜]
    ContinuousMultilinearMap 𝕜 E F →L[𝕜] F 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (F : Type uF) [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (x : ⨂[𝕜] (i : ι), E i)
Goal: ∀ (a : ContinuousMultilinearMap 𝕜 E F),
  ((PiTensorProduct.toDualContinuousMultilinearMap F) x) a = (PiTensorProduct.lift a.toMultilinearMap) x
-- PiTensorProduct.toDualContinuousMultilinearMap_apply_toFun
@[simps!]
noncomputable def toDualContinuousMultilinearMap : (⨂[𝕜] i, E i) →ₗ[𝕜]
    ContinuousMultilinearMap 𝕜 E F →L[𝕜] F 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (F : Type uF) [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (x : ⨂[𝕜] (i : ι), E i)
Goal: ∀ (a : ContinuousMultilinearMap 𝕜 E F),
  ((PiTensorProduct.toDualContinuousMultilinearMap F) x) a = (PiTensorProduct.lift a.toMultilinearMap) x
-- PiTensorProduct.toDualContinuousMultilinearMap_le_projectiveSeminorm
theorem toDualContinuousMultilinearMap_le_projectiveSeminorm (x : ⨂[𝕜] i, E i) :
    ‖toDualContinuousMultilinearMap F x‖ ≤ projectiveSeminorm x 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (x : ⨂[𝕜] (i : ι), E i)
Goal: ‖(PiTensorProduct.toDualContinuousMultilinearMap F) x‖ ≤ PiTensorProduct.projectiveSeminorm x
-- PiTensorProduct.injectiveSeminorm_def
noncomputable irreducible_def injectiveSeminorm : Seminorm 𝕜 (⨂[𝕜] i, E i) 
Arguments: {ι : Type u_1} [Fintype ι] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : ι → Type u_3} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: PiTensorProduct.injectiveSeminorm =
  sSup
    {p |
      ∃ G x x_1,
        p =
          (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp
            (PiTensorProduct.toDualContinuousMultilinearMap G)}
-- PiTensorProduct.dualSeminorms_bounded
lemma dualSeminorms_bounded : BddAbove {p | ∃ (G : Type (max uι u𝕜 uE))
    (_ : SeminormedAddCommGroup G) (_ : NormedSpace 𝕜 G),
    p = Seminorm.comp (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G))
    (toDualContinuousMultilinearMap G (𝕜 := 𝕜) (E := E))} 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: BddAbove
  {p |
    ∃ G x x_1,
      p =
        (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp
          (PiTensorProduct.toDualContinuousMultilinearMap G)}
-- PiTensorProduct.injectiveSeminorm_apply
theorem injectiveSeminorm_apply (x : ⨂[𝕜] i, E i) :
    injectiveSeminorm x = ⨆ p : {p | ∃ (G : Type (max uι u𝕜 uE))
    (_ : SeminormedAddCommGroup G) (_ : NormedSpace 𝕜 G), p = Seminorm.comp (normSeminorm 𝕜
    (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G))
    (toDualContinuousMultilinearMap G (𝕜 := 𝕜) (E := E))}, p.1 x 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (x : ⨂[𝕜] (i : ι), E i)
Goal: PiTensorProduct.injectiveSeminorm x = ⨆ p, ↑p x
-- PiTensorProduct.norm_eval_le_injectiveSeminorm
theorem norm_eval_le_injectiveSeminorm (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
    ‖lift f.toMultilinearMap x‖ ≤ ‖f‖ * injectiveSeminorm x 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] (i : ι), E i)
Goal: ‖(PiTensorProduct.lift f.toMultilinearMap) x‖ ≤ ‖f‖ * PiTensorProduct.injectiveSeminorm x
-- PiTensorProduct.injectiveSeminorm_le_projectiveSeminorm
theorem injectiveSeminorm_le_projectiveSeminorm :
    injectiveSeminorm (𝕜 := 𝕜) (E := E) ≤ projectiveSeminorm 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: PiTensorProduct.injectiveSeminorm ≤ PiTensorProduct.projectiveSeminorm
-- PiTensorProduct.injectiveSeminorm_tprod_le
theorem injectiveSeminorm_tprod_le (m : Π (i : ι), E i) :
    injectiveSeminorm (⨂ₜ[𝕜] i, m i) ≤ ∏ i, ‖m i‖ 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (m : (i : ι) → E i)
Goal: PiTensorProduct.injectiveSeminorm (⨂ₜ[𝕜] (i : ι), m i) ≤ ∏ i : ι, ‖m i‖
-- PiTensorProduct.liftEquiv_symm_apply
@[simps]
noncomputable def liftEquiv : ContinuousMultilinearMap 𝕜 E F ≃ₗ[𝕜] (⨂[𝕜] i, E i) →L[𝕜] F 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (F : Type uF) [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (l : (⨂[𝕜] (i : ι), E i) →L[𝕜] F)
Goal: (PiTensorProduct.liftEquiv 𝕜 E F).symm l = (PiTensorProduct.lift.symm ↑l).mkContinuous ‖l‖ ⋯
-- PiTensorProduct.liftEquiv_apply
@[simps]
noncomputable def liftEquiv : ContinuousMultilinearMap 𝕜 E F ≃ₗ[𝕜] (⨂[𝕜] i, E i) →L[𝕜] F 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (F : Type uF) [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (f : ContinuousMultilinearMap 𝕜 E F)
Goal: (PiTensorProduct.liftEquiv 𝕜 E F) f = (PiTensorProduct.lift f.toMultilinearMap).mkContinuous ‖f‖ ⋯
-- PiTensorProduct.liftIsometry_apply_apply
@[simp]
theorem liftIsometry_apply_apply (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
    liftIsometry 𝕜 E F f x = lift f.toMultilinearMap x 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] (i : ι), E i)
Goal: ((PiTensorProduct.liftIsometry 𝕜 E F) f) x = (PiTensorProduct.lift f.toMultilinearMap) x
-- PiTensorProduct.tprodL_apply
@[simps!]
noncomputable def tprodL : ContinuousMultilinearMap 𝕜 E (⨂[𝕜] i, E i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: ∀ (a : (i : ι) → E i), (PiTensorProduct.tprodL 𝕜) a = (PiTensorProduct.tprod 𝕜) a
-- PiTensorProduct.tprodL_toFun
@[simps!]
noncomputable def tprodL : ContinuousMultilinearMap 𝕜 E (⨂[𝕜] i, E i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: ∀ (a : (i : ι) → E i), (PiTensorProduct.tprodL 𝕜) a = (PiTensorProduct.tprod 𝕜) a
-- PiTensorProduct.tprodL_coe
@[simp]
theorem tprodL_coe : (tprodL 𝕜).toMultilinearMap = tprod 𝕜 (s := E) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: (PiTensorProduct.tprodL 𝕜).toMultilinearMap = PiTensorProduct.tprod 𝕜
-- PiTensorProduct.liftIsometry_symm_apply
@[simp]
theorem liftIsometry_symm_apply (l : (⨂[𝕜] i, E i) →L[𝕜] F) :
    (liftIsometry 𝕜 E F).symm l = l.compContinuousMultilinearMap (tprodL 𝕜) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (l : (⨂[𝕜] (i : ι), E i) →L[𝕜] F)
Goal: (PiTensorProduct.liftIsometry 𝕜 E F).symm l = l.compContinuousMultilinearMap (PiTensorProduct.tprodL 𝕜)
-- PiTensorProduct.liftIsometry_tprodL
@[simp]
theorem liftIsometry_tprodL :
    liftIsometry 𝕜 E _ (tprodL 𝕜) = ContinuousLinearMap.id 𝕜 (⨂[𝕜] i, E i) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: (PiTensorProduct.liftIsometry 𝕜 E (⨂[𝕜] (i : ι), E i)) (PiTensorProduct.tprodL 𝕜) =
  ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i)
-- PiTensorProduct.mapL_coe
@[simp]
theorem mapL_coe : (mapL f).toLinearMap = map (fun i ↦ (f i).toLinearMap) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ↑(PiTensorProduct.mapL f) = PiTensorProduct.map fun i => ↑(f i)
-- PiTensorProduct.mapL_apply
@[simp]
theorem mapL_apply (x : ⨂[𝕜] i, E i) : mapL f x = map (fun i ↦ (f i).toLinearMap) x 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i) (x : ⨂[𝕜] (i : ι), E i)
Goal: (PiTensorProduct.mapL f) x = (PiTensorProduct.map fun i => ↑(f i)) x
-- PiTensorProduct.mapL_comp
theorem mapL_comp : mapL (fun (i : ι) ↦ g i ∘L f i) = mapL g ∘L mapL f 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} {E'' : ι → Type u_2} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] [(i : ι) → SeminormedAddCommGroup (E'' i)] [(i : ι) → NormedSpace 𝕜 (E'' i)] (g : (i : ι) → E' i →L[𝕜] E'' i) (f : (i : ι) → E i →L[𝕜] E' i)
Goal: (PiTensorProduct.mapL fun i => (g i).comp (f i)) = (PiTensorProduct.mapL g).comp (PiTensorProduct.mapL f)
-- PiTensorProduct.liftIsometry_comp_mapL
theorem liftIsometry_comp_mapL (h : ContinuousMultilinearMap 𝕜 E' F) :
    liftIsometry 𝕜 E' F h ∘L mapL f = liftIsometry 𝕜 E F (h.compContinuousLinearMap f) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i) (h : ContinuousMultilinearMap 𝕜 E' F)
Goal: ((PiTensorProduct.liftIsometry 𝕜 E' F) h).comp (PiTensorProduct.mapL f) =
  (PiTensorProduct.liftIsometry 𝕜 E F) (h.compContinuousLinearMap f)
-- PiTensorProduct.mapL_id
@[simp]
theorem mapL_id : mapL (fun i ↦ ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id _ _ 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: (PiTensorProduct.mapL fun i => ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i)
-- PiTensorProduct.mapL_one
@[simp]
theorem mapL_one : mapL (fun (i : ι) ↦ (1 : E i →L[𝕜] E i)) = 1 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: (PiTensorProduct.mapL fun i => 1) = 1
-- PiTensorProduct.mapL_mul
theorem mapL_mul (f₁ f₂ : Π i, E i →L[𝕜] E i) :
    mapL (fun i ↦ f₁ i * f₂ i) = mapL f₁ * mapL f₂ 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (f₁ f₂ : (i : ι) → E i →L[𝕜] E i)
Goal: (PiTensorProduct.mapL fun i => f₁ i * f₂ i) = PiTensorProduct.mapL f₁ * PiTensorProduct.mapL f₂
-- PiTensorProduct.mapLMonoidHom_apply
@[simps]
noncomputable def mapLMonoidHom : (Π i, E i →L[𝕜] E i) →* ((⨂[𝕜] i, E i) →L[𝕜] ⨂[𝕜] i, E i) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (f : (i : ι) → E i →L[𝕜] E i)
Goal: PiTensorProduct.mapLMonoidHom f = PiTensorProduct.mapL f
-- PiTensorProduct.mapL_pow
@[simp]
protected theorem mapL_pow (f : Π i, E i →L[𝕜] E i) (n : ℕ) :
    mapL (f ^ n) = mapL f ^ n 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (f : (i : ι) → E i →L[𝕜] E i) (n : ℕ)
Goal: PiTensorProduct.mapL (f ^ n) = PiTensorProduct.mapL f ^ n
-- _private.Mathlib.Analysis.NormedSpace.PiTensorProduct.InjectiveSeminorm.0.PiTensorProduct.mapL_add_smul_aux
private theorem mapL_add_smul_aux [DecidableEq ι] (i : ι) (u : E i →L[𝕜] E' i) :
    (fun j ↦ (update f i u j).toLinearMap) =
    update (fun j ↦ (f j).toLinearMap) i u.toLinearMap 
Arguments: {ι : Type uι} {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i) [DecidableEq ι] (i : ι) (u : E i →L[𝕜] E' i)
Goal: (fun j => ↑(Function.update f i u j)) = Function.update (fun j => ↑(f j)) i ↑u