ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Seminorm.continuous_of_continuous_comp
theorem continuous_of_continuous_comp {q : SeminormFamily 𝕝₂ F ι'} [TopologicalSpace E]
    [TopologicalAddGroup E] [TopologicalSpace F] (hq : WithSeminorms q)
    (f : E →ₛₗ[τ₁₂] F) (hf : ∀ i, Continuous ((q i).comp f)) : Continuous f 
Arguments: {𝕝 : Type u_3} {𝕝₂ : Type u_4} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [AddCommGroup E] [NormedField 𝕝] [Module 𝕝 E] [AddCommGroup F] [NormedField 𝕝₂] [Module 𝕝₂ F] {τ₁₂ : 𝕝 →+* 𝕝₂} [RingHomIsometric τ₁₂] [Nonempty ι'] {q : SeminormFamily 𝕝₂ F ι'} [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F] (hq : WithSeminorms q) (f : E →ₛₗ[τ₁₂] F) (hf : ∀ (i : ι'), Continuous ⇑((q i).comp f))
Goal: Continuous ⇑f
-- Seminorm.continuous_iff_continuous_comp
theorem continuous_iff_continuous_comp {q : SeminormFamily 𝕜₂ F ι'} [TopologicalSpace E]
    [TopologicalAddGroup E] [TopologicalSpace F] (hq : WithSeminorms q) (f : E →ₛₗ[σ₁₂] F) :
    Continuous f ↔ ∀ i, Continuous ((q i).comp f) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NontriviallyNormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [Nonempty ι'] {q : SeminormFamily 𝕜₂ F ι'} [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F] (hq : WithSeminorms q) (f : E →ₛₗ[σ₁₂] F)
Goal: Continuous ⇑f ↔ ∀ (i : ι'), Continuous ⇑((q i).comp f)
-- Seminorm.continuous_from_bounded
theorem continuous_from_bounded {p : SeminormFamily 𝕝 E ι} {q : SeminormFamily 𝕝₂ F ι'}
    {_ : TopologicalSpace E} (hp : WithSeminorms p) {_ : TopologicalSpace F} (hq : WithSeminorms q)
    (f : E →ₛₗ[τ₁₂] F) (hf : Seminorm.IsBounded p q f) : Continuous f 
Arguments: {𝕝 : Type u_3} {𝕝₂ : Type u_4} {E : Type u_5} {F : Type u_6} {ι : Type u_8} {ι' : Type u_9} [AddCommGroup E] [NormedField 𝕝] [Module 𝕝 E] [AddCommGroup F] [NormedField 𝕝₂] [Module 𝕝₂ F] {τ₁₂ : 𝕝 →+* 𝕝₂} [RingHomIsometric τ₁₂] [Nonempty ι] [Nonempty ι'] {p : SeminormFamily 𝕝 E ι} {q : SeminormFamily 𝕝₂ F ι'}
Goal: ∀ {x : TopologicalSpace E},
  WithSeminorms p →
    ∀ {x_1 : TopologicalSpace F}, WithSeminorms q → ∀ (f : E →ₛₗ[τ₁₂] F), Seminorm.IsBounded p q f → Continuous ⇑f
-- Seminorm.cont_withSeminorms_normedSpace
theorem cont_withSeminorms_normedSpace (F) [SeminormedAddCommGroup F] [NormedSpace 𝕝₂ F]
    [TopologicalSpace E] {p : ι → Seminorm 𝕝 E} (hp : WithSeminorms p)
    (f : E →ₛₗ[τ₁₂] F) (hf : ∃ (s : Finset ι) (C : ℝ≥0), (normSeminorm 𝕝₂ F).comp f ≤ C • s.sup p) :
    Continuous f 
Arguments: {𝕝 : Type u_3} {𝕝₂ : Type u_4} {E : Type u_5} {ι : Type u_8} [AddCommGroup E] [NormedField 𝕝] [Module 𝕝 E] [NormedField 𝕝₂] {τ₁₂ : 𝕝 →+* 𝕝₂} [RingHomIsometric τ₁₂] [Nonempty ι] (F : Type u_10) [SeminormedAddCommGroup F] [NormedSpace 𝕝₂ F] [TopologicalSpace E] {p : ι → Seminorm 𝕝 E} (hp : WithSeminorms p) (f : E →ₛₗ[τ₁₂] F) (hf : ∃ s C, (normSeminorm 𝕝₂ F).comp f ≤ C • s.sup p)
Goal: Continuous ⇑f
-- Seminorm.cont_normedSpace_to_withSeminorms
theorem cont_normedSpace_to_withSeminorms (E) [SeminormedAddCommGroup E] [NormedSpace 𝕝 E]
    [TopologicalSpace F] {q : ι → Seminorm 𝕝₂ F} (hq : WithSeminorms q)
    (f : E →ₛₗ[τ₁₂] F) (hf : ∀ i : ι, ∃ C : ℝ≥0, (q i).comp f ≤ C • normSeminorm 𝕝 E) :
    Continuous f 
Arguments: {𝕝 : Type u_3} {𝕝₂ : Type u_4} {F : Type u_6} {ι : Type u_8} [NormedField 𝕝] [AddCommGroup F] [NormedField 𝕝₂] [Module 𝕝₂ F] {τ₁₂ : 𝕝 →+* 𝕝₂} [RingHomIsometric τ₁₂] [Nonempty ι] (E : Type u_10) [SeminormedAddCommGroup E] [NormedSpace 𝕝 E] [TopologicalSpace F] {q : ι → Seminorm 𝕝₂ F} (hq : WithSeminorms q) (f : E →ₛₗ[τ₁₂] F) (hf : ∀ (i : ι), ∃ C, (q i).comp f ≤ C • normSeminorm 𝕝 E)
Goal: Continuous ⇑f
-- WithSeminorms.equicontinuous_TFAE
protected theorem _root_.WithSeminorms.equicontinuous_TFAE {κ : Type*}
    {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F]
    [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E]
    (f : κ → E →ₛₗ[σ₁₂] F) : TFAE
    [ EquicontinuousAt ((↑) ∘ f) 0,
      Equicontinuous ((↑) ∘ f),
      UniformEquicontinuous ((↑) ∘ f),
      ∀ i, ∃ p : Seminorm 𝕜 E, Continuous p ∧ ∀ k, (q i).comp (f k) ≤ p,
      ∀ i, BddAbove (range fun k ↦ (q i).comp (f k)) ∧ Continuous (⨆ k, (q i).comp (f k)) ] 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NontriviallyNormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [Nonempty ι'] {κ : Type u_10} {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F] [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E] (f : κ → E →ₛₗ[σ₁₂] F)
Goal: [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
    ∀ (i : ι'), ∃ p, Continuous ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p,
    ∀ (i : ι'), BddAbove (Set.range fun k => (q i).comp (f k)) ∧ Continuous (⨆ k, ⇑((q i).comp (f k)))].TFAE
-- WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm
theorem _root_.WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm {κ : Type*}
    {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F]
    [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E]
    (f : κ → E →ₛₗ[σ₁₂] F) :
    UniformEquicontinuous ((↑) ∘ f) ↔
    ∀ i, ∃ p : Seminorm 𝕜 E, Continuous p ∧ ∀ k, (q i).comp (f k) ≤ p 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NontriviallyNormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [Nonempty ι'] {κ : Type u_10} {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F] [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E] (f : κ → E →ₛₗ[σ₁₂] F)
Goal: UniformEquicontinuous (DFunLike.coe ∘ f) ↔ ∀ (i : ι'), ∃ p, Continuous ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p
-- WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup
theorem _root_.WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup {κ : Type*}
    {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F]
    [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E]
    (f : κ → E →ₛₗ[σ₁₂] F) :
    UniformEquicontinuous ((↑) ∘ f) ↔ ∀ i,
    BddAbove (range fun k ↦ (q i).comp (f k)) ∧
      Continuous (⨆ k, (q i).comp (f k)) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NontriviallyNormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [Nonempty ι'] {κ : Type u_10} {q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [UniformAddGroup E] [u : UniformSpace F] [hu : UniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E] (f : κ → E →ₛₗ[σ₁₂] F)
Goal: UniformEquicontinuous (DFunLike.coe ∘ f) ↔
  ∀ (i : ι'), BddAbove (Set.range fun k => (q i).comp (f k)) ∧ Continuous (⨆ k, ⇑((q i).comp (f k)))
-- WithSeminorms.congr
protected theorem congr {p : SeminormFamily 𝕜 E ι} {q : SeminormFamily 𝕜 E ι'}
    [t : TopologicalSpace E] (hp : WithSeminorms p) (hpq : Seminorm.IsBounded p q LinearMap.id)
    (hqp : Seminorm.IsBounded q p LinearMap.id) : WithSeminorms q 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} {ι' : Type u_9} [Nonempty ι] [Nonempty ι'] [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : SeminormFamily 𝕜 E ι} {q : SeminormFamily 𝕜 E ι'} [t : TopologicalSpace E] (hp : WithSeminorms p) (hpq : Seminorm.IsBounded p q LinearMap.id) (hqp : Seminorm.IsBounded q p LinearMap.id)
Goal: WithSeminorms q
-- WithSeminorms.finset_sups
protected theorem finset_sups {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E]
    (hp : WithSeminorms p) : WithSeminorms (fun s : Finset ι ↦ s.sup p) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [Nonempty ι] [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] (hp : WithSeminorms p)
Goal: WithSeminorms fun s => s.sup p
-- WithSeminorms.partial_sups
protected theorem partial_sups [Preorder ι] [LocallyFiniteOrderBot ι] {p : SeminormFamily 𝕜 E ι}
    [TopologicalSpace E] (hp : WithSeminorms p) : WithSeminorms (fun i ↦ (Finset.Iic i).sup p) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [Nonempty ι] [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Preorder ι] [LocallyFiniteOrderBot ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] (hp : WithSeminorms p)
Goal: WithSeminorms fun i => (Finset.Iic i).sup p
-- WithSeminorms.congr_equiv
protected theorem congr_equiv {p : SeminormFamily 𝕜 E ι} [t : TopologicalSpace E]
    (hp : WithSeminorms p) (e : ι' ≃ ι) : WithSeminorms (p ∘ e) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} {ι' : Type u_9} [Nonempty ι] [Nonempty ι'] [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : SeminormFamily 𝕜 E ι} [t : TopologicalSpace E] (hp : WithSeminorms p) (e : ι' ≃ ι)
Goal: WithSeminorms (p ∘ ⇑e)
-- Seminorm.map_eq_zero_of_norm_zero
lemma map_eq_zero_of_norm_zero (q : Seminorm 𝕜 F)
    (hq : Continuous q) {x : F} (hx : ‖x‖ = 0) : q x = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_6} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (q : Seminorm 𝕜 F) (hq : Continuous ⇑q) {x : F} (hx : ‖x‖ = 0)
Goal: q x = 0
-- Seminorm.bound_of_continuous_normedSpace
lemma bound_of_continuous_normedSpace (q : Seminorm 𝕜 F)
    (hq : Continuous q) : ∃ C, 0 < C ∧ (∀ x : F, q x ≤ C * ‖x‖) 
Arguments: {𝕜 : Type u_1} {F : Type u_6} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (q : Seminorm 𝕜 F) (hq : Continuous ⇑q)
Goal: ∃ C, 0 < C ∧ ∀ (x : F), q x ≤ C * ‖x‖
-- Seminorm.bound_of_continuous
lemma bound_of_continuous [Nonempty ι] [t : TopologicalSpace E] (hp : WithSeminorms p)
    (q : Seminorm 𝕜 E) (hq : Continuous q) :
    ∃ s : Finset ι, ∃ C : ℝ≥0, C ≠ 0 ∧ q ≤ C • s.sup p 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p : SeminormFamily 𝕜 E ι} [Nonempty ι] [t : TopologicalSpace E] (hp : WithSeminorms p) (q : Seminorm 𝕜 E) (hq : Continuous ⇑q)
Goal: ∃ s C, C ≠ 0 ∧ q ≤ C • s.sup p
-- WithSeminorms.toLocallyConvexSpace
theorem WithSeminorms.toLocallyConvexSpace {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) :
    LocallyConvexSpace ℝ E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [Nonempty ι] [NormedField 𝕜] [NormedSpace ℝ 𝕜] [AddCommGroup E] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: LocallyConvexSpace ℝ E
-- NormedSpace.toLocallyConvexSpace'
theorem NormedSpace.toLocallyConvexSpace' [NormedSpace 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] :
    LocallyConvexSpace ℝ E 
Arguments: (𝕜 : Type u_1) {E : Type u_5} [NormedField 𝕜] [NormedSpace ℝ 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E]
Goal: LocallyConvexSpace ℝ E
-- SeminormFamily.comp_apply
theorem SeminormFamily.comp_apply (q : SeminormFamily 𝕜₂ F ι) (i : ι) (f : E →ₛₗ[σ₁₂] F) :
    q.comp f i = (q i).comp f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (q : SeminormFamily 𝕜₂ F ι) (i : ι) (f : E →ₛₗ[σ₁₂] F)
Goal: q.comp f i = (q i).comp f
-- SeminormFamily.finset_sup_comp
theorem SeminormFamily.finset_sup_comp (q : SeminormFamily 𝕜₂ F ι) (s : Finset ι)
    (f : E →ₛₗ[σ₁₂] F) : (s.sup q).comp f = s.sup (q.comp f) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (q : SeminormFamily 𝕜₂ F ι) (s : Finset ι) (f : E →ₛₗ[σ₁₂] F)
Goal: (s.sup q).comp f = s.sup (q.comp f)
-- LinearMap.withSeminorms_induced
theorem LinearMap.withSeminorms_induced [hι : Nonempty ι] {q : SeminormFamily 𝕜₂ F ι}
    (hq : WithSeminorms q) (f : E →ₛₗ[σ₁₂] F) :
    WithSeminorms (topology := induced f inferInstance) (q.comp f) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [TopologicalSpace F] [hι : Nonempty ι] {q : SeminormFamily 𝕜₂ F ι} (hq : WithSeminorms q) (f : E →ₛₗ[σ₁₂] F)
Goal: WithSeminorms (q.comp f)
-- Inducing.withSeminorms
theorem Inducing.withSeminorms [hι : Nonempty ι] {q : SeminormFamily 𝕜₂ F ι} (hq : WithSeminorms q)
    [TopologicalSpace E] {f : E →ₛₗ[σ₁₂] F} (hf : Inducing f) : WithSeminorms (q.comp f) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [TopologicalSpace F] [hι : Nonempty ι] {q : SeminormFamily 𝕜₂ F ι} (hq : WithSeminorms q) [TopologicalSpace E] {f : E →ₛₗ[σ₁₂] F} (hf : Inducing ⇑f)
Goal: WithSeminorms (q.comp f)
-- withSeminorms_iInf
theorem withSeminorms_iInf {κ : ι → Type*} [Nonempty ((i : ι) × κ i)] [∀ i, Nonempty (κ i)]
    {p : (i : ι) → SeminormFamily 𝕜 E (κ i)} {t : ι → TopologicalSpace E}
    [∀ i, @TopologicalAddGroup E (t i) _] (hp : ∀ i, WithSeminorms (topology := t i) (p i)) :
    WithSeminorms (topology := ⨅ i, t i) (SeminormFamily.sigma p) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {κ : ι → Type u_10} [Nonempty ((i : ι) × κ i)] [∀ (i : ι), Nonempty (κ i)] {p : (i : ι) → SeminormFamily 𝕜 E (κ i)} {t : ι → TopologicalSpace E} [∀ (i : ι), TopologicalAddGroup E] (hp : ∀ (i : ι), WithSeminorms (p i))
Goal: WithSeminorms (SeminormFamily.sigma p)
-- WithSeminorms.first_countable
theorem WithSeminorms.first_countable (hp : WithSeminorms p) :
    FirstCountableTopology E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [Countable ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] (hp : WithSeminorms p)
Goal: FirstCountableTopology E
-- PiLp.ext
@[ext]
protected theorem PiLp.ext {p : ℝ≥0∞} {ι : Type*} {α : ι → Type*} {x y : PiLp p α}
    (h : ∀ i, x i = y i) : x = y 
Arguments: {p : ℝ≥0∞} {ι : Type u_1} {α : ι → Type u_2} {x y : PiLp p α} (h : ∀ (i : ι), x i = y i)
Goal: x = y
-- PiLp.zero_apply
@[simp, nolint simpNF]
theorem zero_apply : (0 : PiLp p β) i = 0 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → SeminormedAddCommGroup (β i)] (i : ι)
Goal: 0 i = 0
-- PiLp.add_apply
@[simp]
theorem add_apply : (x + y) i = x i + y i 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp p β) (i : ι)
Goal: (x + y) i = x i + y i
-- PiLp.sub_apply
@[simp]
theorem sub_apply : (x - y) i = x i - y i 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp p β) (i : ι)
Goal: (x - y) i = x i - y i
-- PiLp.smul_apply
@[simp]
theorem smul_apply : (c • x) i = c • x i 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} (β : ι → Type u_4) [SeminormedRing 𝕜] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (β i)] (c : 𝕜) (x : PiLp p β) (i : ι)
Goal: (c • x) i = c • x i
-- PiLp.neg_apply
@[simp]
theorem neg_apply : (-x) i = -x i 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → SeminormedAddCommGroup (β i)] (x : PiLp p β) (i : ι)
Goal: (-x) i = -x i
-- WithLp.equiv_pi_apply
@[simp]
theorem _root_.WithLp.equiv_pi_apply (x : PiLp p α) (i : ι) : WithLp.equiv p _ x i = x i 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (α : ι → Type u_3) (x : PiLp p α) (i : ι)
Goal: (WithLp.equiv p ((i : ι) → α i)) x i = x i
-- WithLp.equiv_symm_pi_apply
@[simp]
theorem  _root_.WithLp.equiv_symm_pi_apply (x : ∀ i, α i) (i : ι) :
    (WithLp.equiv p _).symm x i = x i 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (α : ι → Type u_3) (x : (i : ι) → α i) (i : ι)
Goal: (WithLp.equiv p ((i : ι) → α i)).symm x i = x i
-- PiLp.edist_eq_card
theorem edist_eq_card (f g : PiLp 0 β) :
    edist f g = {i | edist (f i) (g i) ≠ 0}.toFinite.toFinset.card 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → EDist (β i)] (f g : PiLp 0 β)
Goal: edist f g = ↑⋯.toFinset.card
-- PiLp.edist_eq_sum
theorem edist_eq_sum {p : ℝ≥0∞} (hp : 0 < p.toReal) (f g : PiLp p β) :
    edist f g = (∑ i, edist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → EDist (β i)] {p : ℝ≥0∞} (hp : 0 < p.toReal) (f g : PiLp p β)
Goal: edist f g = (∑ i : ι, edist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal)
-- PiLp.edist_eq_iSup
theorem edist_eq_iSup (f g : PiLp ∞ β) : edist f g = ⨆ i, edist (f i) (g i) 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → EDist (β i)] (f g : PiLp ⊤ β)
Goal: edist f g = ⨆ i, edist (f i) (g i)
-- PiLp.edist_self
protected theorem edist_self (f : PiLp p β) : edist f f = 0 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)] (f : PiLp p β)
Goal: edist f f = 0
-- PiLp.edist_comm
protected theorem edist_comm (f g : PiLp p β) : edist f g = edist g f 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)] (f g : PiLp p β)
Goal: edist f g = edist g f
-- PiLp.dist_eq_card
theorem dist_eq_card (f g : PiLp 0 α) :
    dist f g = {i | dist (f i) (g i) ≠ 0}.toFinite.toFinset.card 
Arguments: {ι : Type u_2} {α : ι → Type u_3} [Fintype ι] [(i : ι) → Dist (α i)] (f g : PiLp 0 α)
Goal: dist f g = ↑⋯.toFinset.card
-- PiLp.dist_eq_sum
theorem dist_eq_sum {p : ℝ≥0∞} (hp : 0 < p.toReal) (f g : PiLp p α) :
    dist f g = (∑ i, dist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {ι : Type u_2} {α : ι → Type u_3} [Fintype ι] [(i : ι) → Dist (α i)] {p : ℝ≥0∞} (hp : 0 < p.toReal) (f g : PiLp p α)
Goal: dist f g = (∑ i : ι, dist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal)
-- PiLp.dist_eq_iSup
theorem dist_eq_iSup (f g : PiLp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) 
Arguments: {ι : Type u_2} {α : ι → Type u_3} [Fintype ι] [(i : ι) → Dist (α i)] (f g : PiLp ⊤ α)
Goal: dist f g = ⨆ i, dist (f i) (g i)
-- PiLp.norm_eq_card
theorem norm_eq_card (f : PiLp 0 β) : ‖f‖ = {i | ‖f i‖ ≠ 0}.toFinite.toFinset.card 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → Norm (β i)] (f : PiLp 0 β)
Goal: ‖f‖ = ↑⋯.toFinset.card
-- PiLp.norm_eq_ciSup
theorem norm_eq_ciSup (f : PiLp ∞ β) : ‖f‖ = ⨆ i, ‖f i‖ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → Norm (β i)] (f : PiLp ⊤ β)
Goal: ‖f‖ = ⨆ i, ‖f i‖
-- PiLp.norm_eq_sum
theorem norm_eq_sum (hp : 0 < p.toReal) (f : PiLp p β) :
    ‖f‖ = (∑ i, ‖f i‖ ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → Norm (β i)] (hp : 0 < p.toReal) (f : PiLp p β)
Goal: ‖f‖ = (∑ i : ι, ‖f i‖ ^ p.toReal) ^ (1 / p.toReal)
-- PiLp.iSup_edist_ne_top_aux
theorem iSup_edist_ne_top_aux {ι : Type*} [Finite ι] {α : ι → Type*}
    [∀ i, PseudoMetricSpace (α i)] (f g : PiLp ∞ α) : (⨆ i, edist (f i) (g i)) ≠ ⊤ 
Arguments: {ι : Type u_5} [Finite ι] {α : ι → Type u_6} [(i : ι) → PseudoMetricSpace (α i)] (f g : PiLp ⊤ α)
Goal: ⨆ i, edist (f i) (g i) ≠ ⊤
-- PiLp.lipschitzWith_equiv_aux
theorem lipschitzWith_equiv_aux : LipschitzWith 1 (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [(i : ι) → PseudoEMetricSpace (β i)] [Fintype ι]
Goal: LipschitzWith 1 ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.antilipschitzWith_equiv_aux
theorem antilipschitzWith_equiv_aux :
    AntilipschitzWith ((Fintype.card ι : ℝ≥0) ^ (1 / p).toReal) (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [(i : ι) → PseudoEMetricSpace (β i)] [Fintype ι]
Goal: AntilipschitzWith (↑(Fintype.card ι) ^ (1 / p).toReal) ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.aux_uniformity_eq
theorem aux_uniformity_eq : 𝓤 (PiLp p β) = 𝓤[Pi.uniformSpace _] 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [(i : ι) → PseudoEMetricSpace (β i)] [Fintype ι]
Goal: 𝓤 (PiLp p β) = 𝓤 ((i : ι) → β i)
-- PiLp.aux_cobounded_eq
theorem aux_cobounded_eq : cobounded (PiLp p α) = @cobounded _ Pi.instBornology 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (α : ι → Type u_3) [Fact (1 ≤ p)] [(i : ι) → PseudoMetricSpace (α i)] [Fintype ι]
Goal: Bornology.cobounded (PiLp p α) = Bornology.cobounded ((i : ι) → α i)
-- PiLp.uniformContinuous_equiv
theorem uniformContinuous_equiv [∀ i, UniformSpace (β i)] :
    UniformContinuous (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → UniformSpace (β i)]
Goal: UniformContinuous ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.uniformContinuous_equiv_symm
theorem uniformContinuous_equiv_symm [∀ i, UniformSpace (β i)] :
    UniformContinuous (WithLp.equiv p (∀ i, β i)).symm 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → UniformSpace (β i)]
Goal: UniformContinuous ⇑(WithLp.equiv p ((i : ι) → β i)).symm
-- PiLp.continuous_equiv
@[continuity]
theorem continuous_equiv [∀ i, UniformSpace (β i)] : Continuous (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → UniformSpace (β i)]
Goal: Continuous ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.continuous_equiv_symm
@[continuity]
theorem continuous_equiv_symm [∀ i, UniformSpace (β i)] :
    Continuous (WithLp.equiv p (∀ i, β i)).symm 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [(i : ι) → UniformSpace (β i)]
Goal: Continuous ⇑(WithLp.equiv p ((i : ι) → β i)).symm
-- PiLp.nndist_eq_sum
theorem nndist_eq_sum {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type*} [∀ i, PseudoMetricSpace (β i)]
    (hp : p ≠ ∞) (x y : PiLp p β) :
    nndist x y = (∑ i : ι, nndist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {ι : Type u_2} [Fintype ι] {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type u_5} [(i : ι) → PseudoMetricSpace (β i)] (hp : p ≠ ⊤) (x y : PiLp p β)
Goal: nndist x y = (∑ i : ι, nndist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal)
-- PiLp.nndist_eq_iSup
theorem nndist_eq_iSup {β : ι → Type*} [∀ i, PseudoMetricSpace (β i)] (x y : PiLp ∞ β) :
    nndist x y = ⨆ i, nndist (x i) (y i) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → PseudoMetricSpace (β i)] (x y : PiLp ⊤ β)
Goal: nndist x y = ⨆ i, nndist (x i) (y i)
-- PiLp.lipschitzWith_equiv
theorem lipschitzWith_equiv [∀ i, PseudoEMetricSpace (β i)] :
    LipschitzWith 1 (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)]
Goal: LipschitzWith 1 ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.antilipschitzWith_equiv
theorem antilipschitzWith_equiv [∀ i, PseudoEMetricSpace (β i)] :
    AntilipschitzWith ((Fintype.card ι : ℝ≥0) ^ (1 / p).toReal) (WithLp.equiv p (∀ i, β i)) 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)]
Goal: AntilipschitzWith (↑(Fintype.card ι) ^ (1 / p).toReal) ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.infty_equiv_isometry
theorem infty_equiv_isometry [∀ i, PseudoEMetricSpace (β i)] :
    Isometry (WithLp.equiv ∞ (∀ i, β i)) 
Arguments: {ι : Type u_2} (β : ι → Type u_4) [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)]
Goal: Isometry ⇑(WithLp.equiv ⊤ ((i : ι) → β i))
-- PiLp.nnnorm_eq_sum
theorem nnnorm_eq_sum {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type*} (hp : p ≠ ∞)
    [∀ i, SeminormedAddCommGroup (β i)] (f : PiLp p β) :
    ‖f‖₊ = (∑ i, ‖f i‖₊ ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {ι : Type u_2} [Fintype ι] {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type u_5} (hp : p ≠ ⊤) [(i : ι) → SeminormedAddCommGroup (β i)] (f : PiLp p β)
Goal: ‖f‖₊ = (∑ i : ι, ‖f i‖₊ ^ p.toReal) ^ (1 / p.toReal)
-- PiLp.nnnorm_eq_ciSup
theorem nnnorm_eq_ciSup (f : PiLp ∞ β) : ‖f‖₊ = ⨆ i, ‖f i‖₊ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (f : PiLp ⊤ β)
Goal: ‖f‖₊ = ⨆ i, ‖f i‖₊