ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AddSubgroup.norm_trivial_quotient_mk
theorem norm_trivial_quotient_mk (S : AddSubgroup M)
    (h : (S.topologicalClosure : Set M) = Set.univ) : ‖S.normedMk‖ = 0 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (h : ↑S.topologicalClosure = Set.univ)
Goal: ‖S.normedMk‖ = 0
-- NormedAddGroupHom.IsQuotient.surjective
structure IsQuotient (f : NormedAddGroupHom M N) : Prop where
  protected surjective : Function.Surjective f
  protected norm : ∀ x, ‖f x‖ = sInf ((fun m => ‖x + m‖) '' f.ker)
Arguments: {M : Type u_1} {N : Type u_2} [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] {f : NormedAddGroupHom M N} (self : f.IsQuotient)
Goal: Function.Surjective ⇑f
-- NormedAddGroupHom.IsQuotient.norm
structure IsQuotient (f : NormedAddGroupHom M N) : Prop where
  protected surjective : Function.Surjective f
  protected norm : ∀ x, ‖f x‖ = sInf ((fun m => ‖x + m‖) '' f.ker)
Arguments: {M : Type u_1} {N : Type u_2} [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] {f : NormedAddGroupHom M N} (self : f.IsQuotient) (x : M)
Goal: ‖f x‖ = sInf ((fun m => ‖x + m‖) '' ↑f.ker)
-- NormedAddGroupHom.lift_mk
theorem lift_mk {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
    (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (m : M) :
    lift S f hf (S.normedMk m) = f m 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {N : Type u_3} [SeminormedAddCommGroup N] (S : AddSubgroup M) (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (m : M)
Goal: (NormedAddGroupHom.lift S f hf) (S.normedMk m) = f m
-- NormedAddGroupHom.lift_unique
theorem lift_unique {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
    (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (g : NormedAddGroupHom (M ⧸ S) N)
    (h : g.comp S.normedMk = f) : g = lift S f hf 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {N : Type u_3} [SeminormedAddCommGroup N] (S : AddSubgroup M) (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (g : NormedAddGroupHom (M ⧸ S) N) (h : g.comp S.normedMk = f)
Goal: g = NormedAddGroupHom.lift S f hf
-- NormedAddGroupHom.isQuotientQuotient
theorem isQuotientQuotient (S : AddSubgroup M) : IsQuotient S.normedMk 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: S.normedMk.IsQuotient
-- NormedAddGroupHom.IsQuotient.norm_lift
theorem IsQuotient.norm_lift {f : NormedAddGroupHom M N} (hquot : IsQuotient f) {ε : ℝ} (hε : 0 < ε)
    (n : N) : ∃ m : M, f m = n ∧ ‖m‖ < ‖n‖ + ε 
Arguments: {M : Type u_1} {N : Type u_2} [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] {f : NormedAddGroupHom M N} (hquot : f.IsQuotient) {ε : ℝ} (hε : 0 < ε) (n : N)
Goal: ∃ m, f m = n ∧ ‖m‖ < ‖n‖ + ε
-- NormedAddGroupHom.IsQuotient.norm_le
theorem IsQuotient.norm_le {f : NormedAddGroupHom M N} (hquot : IsQuotient f) (m : M) :
    ‖f m‖ ≤ ‖m‖ 
Arguments: {M : Type u_1} {N : Type u_2} [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] {f : NormedAddGroupHom M N} (hquot : f.IsQuotient) (m : M)
Goal: ‖f m‖ ≤ ‖m‖
-- NormedAddGroupHom.norm_lift_le
theorem norm_lift_le {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
    (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) :
    ‖lift S f hf‖ ≤ ‖f‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {N : Type u_3} [SeminormedAddCommGroup N] (S : AddSubgroup M) (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0)
Goal: ‖NormedAddGroupHom.lift S f hf‖ ≤ ‖f‖
-- NormedAddGroupHom.lift_norm_le
theorem lift_norm_le {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
    (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) {c : ℝ≥0} (fb : ‖f‖ ≤ c) :
    ‖lift S f hf‖ ≤ c 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {N : Type u_3} [SeminormedAddCommGroup N] (S : AddSubgroup M) (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) {c : ℝ≥0} (fb : ‖f‖ ≤ ↑c)
Goal: ‖NormedAddGroupHom.lift S f hf‖ ≤ ↑c
-- NormedAddGroupHom.lift_normNoninc
theorem lift_normNoninc {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
    (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (fb : f.NormNoninc) :
    (lift S f hf).NormNoninc 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {N : Type u_3} [SeminormedAddCommGroup N] (S : AddSubgroup M) (f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (fb : f.NormNoninc)
Goal: (NormedAddGroupHom.lift S f hf).NormNoninc
-- Submodule.Quotient.norm_mk_lt
nonrec theorem Submodule.Quotient.norm_mk_lt {S : Submodule R M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε) :
    ∃ m : M, Submodule.Quotient.mk m = x ∧ ‖m‖ < ‖x‖ + ε 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {R : Type u_3} [Ring R] [Module R M] {S : Submodule R M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ m, Submodule.Quotient.mk m = x ∧ ‖m‖ < ‖x‖ + ε
-- Submodule.Quotient.norm_mk_le
theorem Submodule.Quotient.norm_mk_le (m : M) : ‖(Submodule.Quotient.mk m : M ⧸ S)‖ ≤ ‖m‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {R : Type u_3} [Ring R] [Module R M] (S : Submodule R M) (m : M)
Goal: ‖Submodule.Quotient.mk m‖ ≤ ‖m‖
-- Ideal.Quotient.norm_mk_lt
nonrec theorem Ideal.Quotient.norm_mk_lt {I : Ideal R} (x : R ⧸ I) {ε : ℝ} (hε : 0 < ε) :
    ∃ r : R, Ideal.Quotient.mk I r = x ∧ ‖r‖ < ‖x‖ + ε 
Arguments: {R : Type u_3} [SeminormedCommRing R] {I : Ideal R} (x : R ⧸ I) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ r, (Ideal.Quotient.mk I) r = x ∧ ‖r‖ < ‖x‖ + ε
-- Ideal.Quotient.norm_mk_le
theorem Ideal.Quotient.norm_mk_le (r : R) : ‖Ideal.Quotient.mk I r‖ ≤ ‖r‖ 
Arguments: {R : Type u_3} [SeminormedCommRing R] (I : Ideal R) (r : R)
Goal: ‖(Ideal.Quotient.mk I) r‖ ≤ ‖r‖
-- SeminormFamily.basisSets_iff
theorem basisSets_iff {U : Set E} :
    U ∈ p.basisSets ↔ ∃ (i : Finset ι) (r : ℝ), 0 < r ∧ U = ball (i.sup p) 0 r 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) {U : Set E}
Goal: U ∈ p.basisSets ↔ ∃ i r, 0 < r ∧ U = (i.sup p).ball 0 r
-- SeminormFamily.basisSets_mem
theorem basisSets_mem (i : Finset ι) {r : ℝ} (hr : 0 < r) : (i.sup p).ball 0 r ∈ p.basisSets 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (i : Finset ι) {r : ℝ} (hr : 0 < r)
Goal: (i.sup p).ball 0 r ∈ p.basisSets
-- SeminormFamily.basisSets_singleton_mem
theorem basisSets_singleton_mem (i : ι) {r : ℝ} (hr : 0 < r) : (p i).ball 0 r ∈ p.basisSets 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (i : ι) {r : ℝ} (hr : 0 < r)
Goal: (p i).ball 0 r ∈ p.basisSets
-- SeminormFamily.basisSets_nonempty
theorem basisSets_nonempty [Nonempty ι] : p.basisSets.Nonempty 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) [Nonempty ι]
Goal: p.basisSets.Nonempty
-- SeminormFamily.basisSets_intersect
theorem basisSets_intersect (U V : Set E) (hU : U ∈ p.basisSets) (hV : V ∈ p.basisSets) :
    ∃ z ∈ p.basisSets, z ⊆ U ∩ V 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (U V : Set E) (hU : U ∈ p.basisSets) (hV : V ∈ p.basisSets)
Goal: ∃ z ∈ p.basisSets, z ⊆ U ∩ V
-- SeminormFamily.basisSets_zero
theorem basisSets_zero (U) (hU : U ∈ p.basisSets) : (0 : E) ∈ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (U : Set E) (hU : U ∈ p.basisSets)
Goal: 0 ∈ U
-- SeminormFamily.basisSets_add
theorem basisSets_add (U) (hU : U ∈ p.basisSets) :
    ∃ V ∈ p.basisSets, V + V ⊆ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (U : Set E) (hU : U ∈ p.basisSets)
Goal: ∃ V ∈ p.basisSets, V + V ⊆ U
-- SeminormFamily.basisSets_neg
theorem basisSets_neg (U) (hU' : U ∈ p.basisSets) :
    ∃ V ∈ p.basisSets, V ⊆ (fun x : E => -x) ⁻¹' U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (U : Set E) (hU' : U ∈ p.basisSets)
Goal: ∃ V ∈ p.basisSets, V ⊆ (fun x => -x) ⁻¹' U
-- SeminormFamily.basisSets_smul_right
theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
    ∀ᶠ x : 𝕜 in 𝓝 0, x • v ∈ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) (v : E) (U : Set E) (hU : U ∈ p.basisSets)
Goal: ∀ᶠ (x : 𝕜) in 𝓝 0, x • v ∈ U
-- SeminormFamily.basisSets_smul
theorem basisSets_smul (U) (hU : U ∈ p.basisSets) :
    ∃ V ∈ 𝓝 (0 : 𝕜), ∃ W ∈ p.addGroupFilterBasis.sets, V • W ⊆ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) [Nonempty ι] (U : Set E) (hU : U ∈ p.basisSets)
Goal: ∃ V ∈ 𝓝 0, ∃ W ∈ AddGroupFilterBasis.toFilterBasis.sets, V • W ⊆ U
-- SeminormFamily.basisSets_smul_left
theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
    ∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (p : SeminormFamily 𝕜 E ι) [Nonempty ι] (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets)
Goal: ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun y => x • y) ⁻¹' U
-- SeminormFamily.filter_eq_iInf
theorem filter_eq_iInf (p : SeminormFamily 𝕜 E ι) :
    p.moduleFilterBasis.toFilterBasis.filter = ⨅ i, (𝓝 0).comap (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] (p : SeminormFamily 𝕜 E ι)
Goal: AddGroupFilterBasis.toFilterBasis.filter = ⨅ i, Filter.comap (⇑(p i)) (𝓝 0)
-- SeminormFamily.basisSets_mem_nhds
lemma basisSets_mem_nhds {𝕜 E ι : Type*} [NormedField 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] (p : SeminormFamily 𝕜 E ι)
    (hp : ∀ i, Continuous (p i)) (U : Set E) (hU : U ∈ p.basisSets) : U ∈ 𝓝 (0 : E) 
Arguments: {𝕜 : Type u_10} {E : Type u_11} {ι : Type u_12} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] (p : SeminormFamily 𝕜 E ι) (hp : ∀ (i : ι), Continuous ⇑(p i)) (U : Set E) (hU : U ∈ p.basisSets)
Goal: U ∈ 𝓝 0
-- Seminorm.isBounded_const
theorem isBounded_const (ι' : Type*) [Nonempty ι'] {p : ι → Seminorm 𝕜 E} {q : Seminorm 𝕜₂ F}
    (f : E →ₛₗ[σ₁₂] F) :
    IsBounded p (fun _ : ι' => q) f ↔ ∃ (s : Finset ι) (C : ℝ≥0), q.comp f ≤ C • s.sup p 
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] (ι' : Type u_10) [Nonempty ι'] {p : ι → Seminorm 𝕜 E} {q : Seminorm 𝕜₂ F} (f : E →ₛₗ[σ₁₂] F)
Goal: Seminorm.IsBounded p (fun x => q) f ↔ ∃ s C, q.comp f ≤ C • s.sup p
-- Seminorm.const_isBounded
theorem const_isBounded (ι : Type*) [Nonempty ι] {p : Seminorm 𝕜 E} {q : ι' → Seminorm 𝕜₂ F}
    (f : E →ₛₗ[σ₁₂] F) : IsBounded (fun _ : ι => p) q f ↔ ∀ i, ∃ C : ℝ≥0, (q i).comp f ≤ C • p 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι' : Type u_9} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (ι : Type u_10) [Nonempty ι] {p : Seminorm 𝕜 E} {q : ι' → Seminorm 𝕜₂ F} (f : E →ₛₗ[σ₁₂] F)
Goal: Seminorm.IsBounded (fun x => p) q f ↔ ∀ (i : ι'), ∃ C, (q i).comp f ≤ C • p
-- Seminorm.isBounded_sup
theorem isBounded_sup {p : ι → Seminorm 𝕜 E} {q : ι' → Seminorm 𝕜₂ F} {f : E →ₛₗ[σ₁₂] F}
    (hf : IsBounded p q f) (s' : Finset ι') :
    ∃ (C : ℝ≥0) (s : Finset ι), (s'.sup q).comp f ≤ C • s.sup p 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_5} {F : Type u_6} {ι : Type u_8} {ι' : Type u_9} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {p : ι → Seminorm 𝕜 E} {q : ι' → Seminorm 𝕜₂ F} {f : E →ₛₗ[σ₁₂] F} (hf : Seminorm.IsBounded p q f) (s' : Finset ι')
Goal: ∃ C s, (s'.sup q).comp f ≤ C • s.sup p
-- WithSeminorms.topology_eq_withSeminorms
structure WithSeminorms (p : SeminormFamily 𝕜 E ι) [topology : TopologicalSpace E] : Prop where
  topology_eq_withSeminorms : topology = p.moduleFilterBasis.topology
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [topology : TopologicalSpace E] (self : WithSeminorms p)
Goal: topology = p.moduleFilterBasis.topology
-- WithSeminorms.withSeminorms_eq
theorem WithSeminorms.withSeminorms_eq {p : SeminormFamily 𝕜 E ι} [t : TopologicalSpace E]
    (hp : WithSeminorms p) : t = p.moduleFilterBasis.topology 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [t : TopologicalSpace E] (hp : WithSeminorms p)
Goal: t = p.moduleFilterBasis.topology
-- WithSeminorms.topologicalAddGroup
theorem WithSeminorms.topologicalAddGroup (hp : WithSeminorms p) : TopologicalAddGroup E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: TopologicalAddGroup E
-- WithSeminorms.continuousSMul
theorem WithSeminorms.continuousSMul (hp : WithSeminorms p) : ContinuousSMul 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: ContinuousSMul 𝕜 E
-- WithSeminorms.hasBasis
theorem WithSeminorms.hasBasis (hp : WithSeminorms p) :
    (𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ p.basisSets) id 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: (𝓝 0).HasBasis (fun s => s ∈ p.basisSets) id
-- WithSeminorms.hasBasis_zero_ball
theorem WithSeminorms.hasBasis_zero_ball (hp : WithSeminorms p) :
    (𝓝 (0 : E)).HasBasis
    (fun sr : Finset ι × ℝ => 0 < sr.2) fun sr => (sr.1.sup p).ball 0 sr.2 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: (𝓝 0).HasBasis (fun sr => 0 < sr.2) fun sr => (sr.1.sup p).ball 0 sr.2
-- WithSeminorms.hasBasis_ball
theorem WithSeminorms.hasBasis_ball (hp : WithSeminorms p) {x : E} :
    (𝓝 (x : E)).HasBasis
    (fun sr : Finset ι × ℝ => 0 < sr.2) fun sr => (sr.1.sup p).ball x sr.2 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) {x : E}
Goal: (𝓝 x).HasBasis (fun sr => 0 < sr.2) fun sr => (sr.1.sup p).ball x sr.2
-- WithSeminorms.mem_nhds_iff
theorem WithSeminorms.mem_nhds_iff (hp : WithSeminorms p) (x : E) (U : Set E) :
    U ∈ 𝓝 x ↔ ∃ s : Finset ι, ∃ r > 0, (s.sup p).ball x r ⊆ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (x : E) (U : Set E)
Goal: U ∈ 𝓝 x ↔ ∃ s, ∃ r > 0, (s.sup p).ball x r ⊆ U
-- WithSeminorms.isOpen_iff_mem_balls
theorem WithSeminorms.isOpen_iff_mem_balls (hp : WithSeminorms p) (U : Set E) :
    IsOpen U ↔ ∀ x ∈ U, ∃ s : Finset ι, ∃ r > 0, (s.sup p).ball x r ⊆ U 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (U : Set E)
Goal: IsOpen U ↔ ∀ x ∈ U, ∃ s, ∃ r > 0, (s.sup p).ball x r ⊆ U
-- WithSeminorms.T1_of_separating
theorem WithSeminorms.T1_of_separating (hp : WithSeminorms p)
    (h : ∀ x, x ≠ 0 → ∃ i, p i x ≠ 0) : T1Space E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (h : ∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0)
Goal: T1Space E
-- WithSeminorms.separating_of_T1
theorem WithSeminorms.separating_of_T1 [T1Space E] (hp : WithSeminorms p) (x : E) (hx : x ≠ 0) :
    ∃ i, p i x ≠ 0 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} [T1Space E] (hp : WithSeminorms p) (x : E) (hx : x ≠ 0)
Goal: ∃ i, (p i) x ≠ 0
-- WithSeminorms.separating_iff_T1
theorem WithSeminorms.separating_iff_T1 (hp : WithSeminorms p) :
    (∀ x, x ≠ 0 → ∃ i, p i x ≠ 0) ↔ T1Space E 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
Goal: (∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0) ↔ T1Space E
-- WithSeminorms.tendsto_nhds'
theorem WithSeminorms.tendsto_nhds' (hp : WithSeminorms p) (u : F → E) {f : Filter F} (y₀ : E) :
    Filter.Tendsto u f (𝓝 y₀) ↔
    ∀ (s : Finset ι) (ε), 0 < ε → ∀ᶠ x in f, s.sup p (u x - y₀) < ε 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (u : F → E) {f : Filter F} (y₀ : E)
Goal: Filter.Tendsto u f (𝓝 y₀) ↔ ∀ (s : Finset ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in f, (s.sup p) (u x - y₀) < ε
-- WithSeminorms.tendsto_nhds
theorem WithSeminorms.tendsto_nhds (hp : WithSeminorms p) (u : F → E) {f : Filter F} (y₀ : E) :
    Filter.Tendsto u f (𝓝 y₀) ↔ ∀ i ε, 0 < ε → ∀ᶠ x in f, p i (u x - y₀) < ε 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (u : F → E) {f : Filter F} (y₀ : E)
Goal: Filter.Tendsto u f (𝓝 y₀) ↔ ∀ (i : ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in f, (p i) (u x - y₀) < ε
-- WithSeminorms.tendsto_nhds_atTop
theorem WithSeminorms.tendsto_nhds_atTop (hp : WithSeminorms p) (u : F → E) (y₀ : E) :
    Filter.Tendsto u Filter.atTop (𝓝 y₀) ↔
    ∀ i ε, 0 < ε → ∃ x₀, ∀ x, x₀ ≤ x → p i (u x - y₀) < ε 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {F : Type u_6} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} [SemilatticeSup F] [Nonempty F] (hp : WithSeminorms p) (u : F → E) (y₀ : E)
Goal: Filter.Tendsto u Filter.atTop (𝓝 y₀) ↔ ∀ (i : ι) (ε : ℝ), 0 < ε → ∃ x₀, ∀ (x : F), x₀ ≤ x → (p i) (u x - y₀) < ε
-- SeminormFamily.withSeminorms_of_nhds
theorem SeminormFamily.withSeminorms_of_nhds [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι)
    (h : 𝓝 (0 : E) = p.moduleFilterBasis.toFilterBasis.filter) : WithSeminorms p 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [t : TopologicalSpace E] [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι) (h : 𝓝 0 = AddGroupFilterBasis.toFilterBasis.filter)
Goal: WithSeminorms p
-- SeminormFamily.withSeminorms_of_hasBasis
theorem SeminormFamily.withSeminorms_of_hasBasis [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι)
    (h : (𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ p.basisSets) id) : WithSeminorms p 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [t : TopologicalSpace E] [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι) (h : (𝓝 0).HasBasis (fun s => s ∈ p.basisSets) id)
Goal: WithSeminorms p
-- SeminormFamily.withSeminorms_iff_nhds_eq_iInf
theorem SeminormFamily.withSeminorms_iff_nhds_eq_iInf [TopologicalAddGroup E]
    (p : SeminormFamily 𝕜 E ι) : WithSeminorms p ↔ (𝓝 (0 : E)) = ⨅ i, (𝓝 0).comap (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [t : TopologicalSpace E] [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι)
Goal: WithSeminorms p ↔ 𝓝 0 = ⨅ i, Filter.comap (⇑(p i)) (𝓝 0)
-- SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf
theorem SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf [TopologicalAddGroup E]
    (p : SeminormFamily 𝕜 E ι) :
    WithSeminorms p ↔
      t = ⨅ i, (p i).toSeminormedAddCommGroup.toUniformSpace.toTopologicalSpace 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [t : TopologicalSpace E] [TopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι)
Goal: WithSeminorms p ↔ t = ⨅ i, UniformSpace.toTopologicalSpace
-- WithSeminorms.continuous_seminorm
theorem WithSeminorms.continuous_seminorm {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p)
    (i : ι) : Continuous (p i) 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [t : TopologicalSpace E] {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) (i : ι)
Goal: Continuous ⇑(p i)
-- SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf
theorem SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf [u : UniformSpace E]
    [UniformAddGroup E] (p : SeminormFamily 𝕜 E ι) :
    WithSeminorms p ↔ u = ⨅ i, (p i).toSeminormedAddCommGroup.toUniformSpace 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] [u : UniformSpace E] [UniformAddGroup E] (p : SeminormFamily 𝕜 E ι)
Goal: WithSeminorms p ↔ u = ⨅ i, PseudoMetricSpace.toUniformSpace
-- norm_withSeminorms
theorem norm_withSeminorms (𝕜 E) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] :
    WithSeminorms fun _ : Fin 1 => normSeminorm 𝕜 E 
Arguments: (𝕜 : Type u_10) (E : Type u_11) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: WithSeminorms fun x => normSeminorm 𝕜 E
-- WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded
theorem WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
    Bornology.IsVonNBounded 𝕜 s ↔ ∀ I : Finset ι, ∃ r > 0, ∀ x ∈ s, I.sup p x < r 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] {s : Set E} (hp : WithSeminorms p)
Goal: Bornology.IsVonNBounded 𝕜 s ↔ ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r
-- WithSeminorms.image_isVonNBounded_iff_finset_seminorm_bounded
theorem WithSeminorms.image_isVonNBounded_iff_finset_seminorm_bounded (f : G → E) {s : Set G}
    (hp : WithSeminorms p) :
    Bornology.IsVonNBounded 𝕜 (f '' s) ↔
      ∀ I : Finset ι, ∃ r > 0, ∀ x ∈ s, I.sup p (f x) < r 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {G : Type u_7} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] (f : G → E) {s : Set G} (hp : WithSeminorms p)
Goal: Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) (f x) < r
-- WithSeminorms.isVonNBounded_iff_seminorm_bounded
theorem WithSeminorms.isVonNBounded_iff_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
    Bornology.IsVonNBounded 𝕜 s ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i x < r 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] {s : Set E} (hp : WithSeminorms p)
Goal: Bornology.IsVonNBounded 𝕜 s ↔ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r
-- WithSeminorms.image_isVonNBounded_iff_seminorm_bounded
theorem WithSeminorms.image_isVonNBounded_iff_seminorm_bounded (f : G → E) {s : Set G}
    (hp : WithSeminorms p) :
    Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i (f x) < r 
Arguments: {𝕜 : Type u_1} {E : Type u_5} {G : Type u_7} {ι : Type u_8} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nonempty ι] {p : SeminormFamily 𝕜 E ι} [TopologicalSpace E] (f : G → E) {s : Set G} (hp : WithSeminorms p)
Goal: Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) (f x) < r