ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
File: Mathlib/Topology/Algebra/PontryaginDual.lean

full_name: PontryaginDual
code: /-- The Pontryagin dual of `A` is the group of continuous homomorphism `A → circle`. -/
def PontryaginDual 
goal: Type u_1

full_name: instTopologicalSpacePontryaginDual
code: instance : TopologicalSpace (PontryaginDual A) 
goal: TopologicalSpace (PontryaginDual A)

full_name: instT2SpacePontryaginDual
code: instance : T2Space (PontryaginDual A) 
goal: T2Space (PontryaginDual A)

full_name: instCommGroupPontryaginDual
code: noncomputable instance : CommGroup (PontryaginDual A) 
goal: CommGroup (PontryaginDual A)

full_name: instTopologicalGroupPontryaginDual
code: instance : TopologicalGroup (PontryaginDual A) 
goal: TopologicalGroup (PontryaginDual A)

full_name: instInhabitedPontryaginDual
code: noncomputable instance : Inhabited (PontryaginDual A) 
goal: Inhabited (PontryaginDual A)

full_name: instLocallyCompactSpacePontryaginDual
code: instance [LocallyCompactSpace G] : LocallyCompactSpace (PontryaginDual G) 
goal: LocallyCompactSpace (PontryaginDual G)

full_name: PontryaginDual.instFunLikeSubtypeComplexMemSubmonoidCircle
code: instance : FunLike (PontryaginDual A) A circle 
goal: FunLike (PontryaginDual A) A ↥circle

full_name: PontryaginDual.instContinuousMonoidHomClassSubtypeComplexMemSubmonoidCircle
code: noncomputable instance : ContinuousMonoidHomClass (PontryaginDual A) A circle 
goal: ContinuousMonoidHomClass (PontryaginDual A) A ↥circle

full_name: PontryaginDual.map
code: /-- `PontryaginDual` is a contravariant functor. -/
noncomputable def map (f : ContinuousMonoidHom A B) :
    ContinuousMonoidHom (PontryaginDual B) (PontryaginDual A) 
goal: ContinuousMonoidHom (PontryaginDual B) (PontryaginDual A)

full_name: PontryaginDual.map_apply
code: @[simp]
theorem map_apply (f : ContinuousMonoidHom A B) (x : PontryaginDual B) (y : A) :
    map f x y = x (f y) 
goal: ((PontryaginDual.map f) x) y = x (f y)

full_name: PontryaginDual.map_one
code: @[simp]
theorem map_one : map (one A B) = one (PontryaginDual B) (PontryaginDual A) 
goal: PontryaginDual.map (ContinuousMonoidHom.one A B) = ContinuousMonoidHom.one (PontryaginDual B) (PontryaginDual A)

full_name: PontryaginDual.map_comp
code: @[simp]
theorem map_comp (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) :
    map (comp g f) = ContinuousMonoidHom.comp (map f) (map g) 
goal: PontryaginDual.map (g.comp f) = (PontryaginDual.map f).comp (PontryaginDual.map g)

full_name: PontryaginDual.map_mul
code: @[simp]
nonrec theorem map_mul (f g : ContinuousMonoidHom A E) : map (f * g) = map f * map g 
goal: PontryaginDual.map (f * g) = PontryaginDual.map f * PontryaginDual.map g

full_name: PontryaginDual.mapHom
code: /-- `ContinuousMonoidHom.dual` as a `ContinuousMonoidHom`. -/
noncomputable def mapHom [LocallyCompactSpace E] :
    ContinuousMonoidHom (ContinuousMonoidHom A E)
      (ContinuousMonoidHom (PontryaginDual E) (PontryaginDual A)) 
goal: ContinuousMonoidHom (ContinuousMonoidHom A E) (ContinuousMonoidHom (PontryaginDual E) (PontryaginDual A))
File: Mathlib/Topology/Algebra/Module/UniformConvergence.lean

full_name: UniformFun.continuousSMul_induced_of_range_bounded
code: /-- Let `E` be a topological vector space over a normed field `𝕜`, let `α` be any type.
Let `H` be a submodule of `α →ᵤ E` such that the range of each `f ∈ H` is von Neumann bounded.
Then `H` is a topological vector space over `𝕜`,
i.e., the pointwise scalar multiplication is continuous in both variables.

For convenience we require that `H` is a vector space over `𝕜`
with a topology induced by `UniformFun.ofFun ∘ φ`, where `φ : H →ₗ[𝕜] (α → E)`. -/
lemma UniformFun.continuousSMul_induced_of_range_bounded (φ : hom)
    (hφ : Inducing (ofFun ∘ φ)) (h : ∀ u : H, Bornology.IsVonNBounded 𝕜 (Set.range (φ u))) :
    ContinuousSMul 𝕜 H 
goal: ContinuousSMul 𝕜 H

full_name: UniformOnFun.continuousSMul_induced_of_image_bounded
code: /-- Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.

For convenience, we don't literally ask for `H : Submodule (α →ᵤ[𝔖] E)`. Instead, we prove the
result for any vector space `H` equipped with a linear inducing to `α →ᵤ[𝔖] E`, which is often
easier to use. We also state the `Submodule` version as
`UniformOnFun.continuousSMul_submodule_of_image_bounded`. -/
theorem UniformOnFun.continuousSMul_induced_of_image_bounded (φ : hom) (hφ : Inducing (ofFun 𝔖 ∘ φ))
    (h : ∀ u : H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 ((φ u : α → E) '' s)) :
    ContinuousSMul 𝕜 H 
goal: ContinuousSMul 𝕜 H

full_name: UniformOnFun.continuousSMul_submodule_of_image_bounded
code: /-- Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.

If you have a hard time using this lemma, try the one above instead. -/
theorem UniformOnFun.continuousSMul_submodule_of_image_bounded (H : Submodule 𝕜 (α →ᵤ[𝔖] E))
    (h : ∀ u ∈ H, ∀ s ∈ 𝔖, Bornology.IsVonNBounded 𝕜 (u '' s)) :
    @ContinuousSMul 𝕜 H _ _ ((UniformOnFun.topologicalSpace α E 𝔖).induced ((↑) : H → α →ᵤ[𝔖] E)) 
goal: ContinuousSMul 𝕜 ↥H
File: Mathlib/Topology/Algebra/Nonarchimedean/Completion.lean

full_name: instNonarchimedeanAddGroupCompletion
code: /-- The completion of a nonarchimedean additive group is a nonarchimedean additive group. -/
instance {G : Type*} [AddGroup G] [UniformSpace G] [UniformAddGroup G] [NonarchimedeanAddGroup G] :
    NonarchimedeanAddGroup (Completion G) 
goal: NonarchimedeanAddGroup (UniformSpace.Completion G)

full_name: instNonarchimedeanRingCompletion
code: /-- The completion of a nonarchimedean ring is a nonarchimedean ring. -/
instance {R : Type*} [Ring R] [UniformSpace R] [TopologicalRing R] [UniformAddGroup R]
    [NonarchimedeanRing R] :
    NonarchimedeanRing (Completion R) 
goal: NonarchimedeanRing (UniformSpace.Completion R)
File: Mathlib/Topology/Algebra/Module/Cardinality.lean

full_name: continuum_le_cardinal_of_nontriviallyNormedField
code: /-- A complete nontrivially normed field has cardinality at least continuum. -/
theorem continuum_le_cardinal_of_nontriviallyNormedField
    (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜 
goal: 𝔠 ≤ #𝕜

full_name: continuum_le_cardinal_of_module
code: /-- A nontrivial module over a complete nontrivially normed field has cardinality at least
continuum. -/
theorem continuum_le_cardinal_of_module
    (𝕜 : Type u) (E : Type v) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [Nontrivial E] : 𝔠 ≤ #E 
goal: 𝔠 ≤ #E

full_name: cardinal_eq_of_mem_nhds_zero
code: /-- In a topological vector space over a nontrivially normed field, any neighborhood of zero has
the same cardinality as the whole space.

See also `cardinal_eq_of_mem_nhds`. -/
lemma cardinal_eq_of_mem_nhds_zero
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : s ∈ 𝓝 (0 : E)) : #s = #E 
goal: #↑s = #E

full_name: cardinal_eq_of_mem_nhds
code: /-- In a topological vector space over a nontrivially normed field, any neighborhood of a point has
the same cardinality as the whole space. -/
theorem cardinal_eq_of_mem_nhds
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
    {s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E 
goal: #↑s = #E

full_name: cardinal_eq_of_isOpen
code: /-- In a topological vector space over a nontrivially normed field, any nonempty open set has
the same cardinality as the whole space. -/
theorem cardinal_eq_of_isOpen
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E}
    (hs : IsOpen s) (h's : s.Nonempty) : #s = #E 
goal: #↑s = #E

full_name: continuum_le_cardinal_of_isOpen
code: /-- In a nontrivial topological vector space over a complete nontrivially normed field, any nonempty
open set has cardinality at least continuum. -/
theorem continuum_le_cardinal_of_isOpen
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E]
    [Module 𝕜 E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
    {s : Set E} (hs : IsOpen s) (h's : s.Nonempty) : 𝔠 ≤ #s 
goal: 𝔠 ≤ #↑s

full_name: Set.Countable.dense_compl
code: /-- In a nontrivial topological vector space over a complete nontrivially normed field, any
countable set has dense complement. -/
theorem Set.Countable.dense_compl
    {E : Type u} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E]
    [Module 𝕜 E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
    {s : Set E} (hs : s.Countable) : Dense sᶜ 
goal: Dense sᶜ
File: Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean

full_name: Bornology.IsVonNBounded.image_multilinear'
code: /-- The image of a von Neumann bounded set under a continuous multilinear map
is von Neumann bounded.

This version does not assume that the topologies on the domain and on the codomain
agree with the vector space structure in any way
but it assumes that `ι` is nonempty.
-/
theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
    (f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) 
goal: Bornology.IsVonNBounded 𝕜 (⇑f '' s)

full_name: Bornology.IsVonNBounded.image_multilinear
code: /-- The image of a von Neumann bounded set under a continuous multilinear map
is von Neumann bounded.

This version assumes that the codomain is a topological vector space.
-/
theorem image_multilinear [ContinuousSMul 𝕜 F] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
    (f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) 
goal: Bornology.IsVonNBounded 𝕜 (⇑f '' s)
File: Mathlib/Topology/Algebra/MvPolynomial.lean

full_name: MvPolynomial.continuous_eval
code: theorem MvPolynomial.continuous_eval : Continuous fun x ↦ eval x p 
goal: Continuous fun x => (MvPolynomial.eval x) p
File: Mathlib/Topology/Algebra/Semigroup.lean

full_name: exists_idempotent_of_compact_t2_of_continuous_mul_left
code: /-- Any nonempty compact Hausdorff semigroup where right-multiplication is continuous contains
an idempotent, i.e. an `m` such that `m * m = m`. -/
@[to_additive
      "Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
      contains an idempotent, i.e. an `m` such that `m + m = m`"]
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
    [TopologicalSpace M] [CompactSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m 
goal: ∃ m, m * m = m

full_name: exists_idempotent_of_compact_t2_of_continuous_add_left
code: /-- Any nonempty compact Hausdorff semigroup where right-multiplication is continuous contains
an idempotent, i.e. an `m` such that `m * m = m`. -/
@[to_additive
      "Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
      contains an idempotent, i.e. an `m` such that `m + m = m`"]
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
    [TopologicalSpace M] [CompactSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m 
goal: ∃ m, m + m = m

full_name: exists_idempotent_in_compact_subsemigroup
code: /-- A version of `exists_idempotent_of_compact_t2_of_continuous_mul_left` where the idempotent lies
in some specified nonempty compact subsemigroup. -/
@[to_additive exists_idempotent_in_compact_add_subsemigroup
      "A version of
      `exists_idempotent_of_compact_t2_of_continuous_add_left` where the idempotent lies in
      some specified nonempty compact additive subsemigroup."]
theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
    (s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
    ∃ m ∈ s, m * m = m 
goal: ∃ m ∈ s, m * m = m

full_name: exists_idempotent_in_compact_add_subsemigroup
code: /-- A version of `exists_idempotent_of_compact_t2_of_continuous_mul_left` where the idempotent lies
in some specified nonempty compact subsemigroup. -/
@[to_additive exists_idempotent_in_compact_add_subsemigroup
      "A version of
      `exists_idempotent_of_compact_t2_of_continuous_add_left` where the idempotent lies in
      some specified nonempty compact additive subsemigroup."]
theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
    (s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
    ∃ m ∈ s, m * m = m 
goal: ∃ m ∈ s, m + m = m
File: Mathlib/Topology/Algebra/Equicontinuity.lean

full_name: equicontinuous_of_equicontinuousAt_zero
code: @[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
    [UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
    (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    Equicontinuous ((↑) ∘ F) 
goal: Equicontinuous (DFunLike.coe ∘ F)

full_name: equicontinuous_of_equicontinuousAt_one
code: @[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
    [UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
    (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    Equicontinuous ((↑) ∘ F) 
goal: Equicontinuous (DFunLike.coe ∘ F)

full_name: uniformEquicontinuous_of_equicontinuousAt_zero
code: @[to_additive]
theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
    [UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M]
    (F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    UniformEquicontinuous ((↑) ∘ F) 
goal: UniformEquicontinuous (DFunLike.coe ∘ F)

full_name: uniformEquicontinuous_of_equicontinuousAt_one
code: @[to_additive]
theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
    [UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M]
    (F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    UniformEquicontinuous ((↑) ∘ F) 
goal: UniformEquicontinuous (DFunLike.coe ∘ F)
File: Mathlib/Topology/Algebra/UniformFilterBasis.lean

full_name: AddGroupFilterBasis.uniformSpace
code: /-- The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure. -/
protected def uniformSpace : UniformSpace G 
goal: UniformSpace G

full_name: AddGroupFilterBasis.uniformAddGroup
code: /-- The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure is compatible with its group structure. -/
protected theorem uniformAddGroup : @UniformAddGroup G B.uniformSpace _ 
goal: UniformAddGroup G

full_name: AddGroupFilterBasis.cauchy_iff
code: theorem cauchy_iff {F : Filter G} :
    @Cauchy G B.uniformSpace F ↔
      F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U 
goal: Cauchy F ↔ F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
File: Mathlib/Topology/Algebra/Group/Compact.lean

full_name: TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_addGroup
code: /-- Every topological group in which there exists a compact set with nonempty interior
is locally compact. -/
@[to_additive
  "Every topological additive group
  in which there exists a compact set with nonempty interior is locally compact."]
theorem TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
    (K : PositiveCompacts G) : LocallyCompactSpace G 
goal: LocallyCompactSpace G

full_name: TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
code: /-- Every topological group in which there exists a compact set with nonempty interior
is locally compact. -/
@[to_additive
  "Every topological additive group
  in which there exists a compact set with nonempty interior is locally compact."]
theorem TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
    (K : PositiveCompacts G) : LocallyCompactSpace G 
goal: LocallyCompactSpace G

full_name: QuotientGroup.continuousSMul
code: @[to_additive]
instance QuotientGroup.continuousSMul [LocallyCompactSpace G] : ContinuousSMul G (G ⧸ Γ) 
goal: ContinuousSMul G (G ⧸ Γ)

full_name: QuotientAddGroup.continuousVAdd
code: @[to_additive]
instance QuotientGroup.continuousSMul [LocallyCompactSpace G] : ContinuousSMul G (G ⧸ Γ) 
goal: ContinuousVAdd G (G ⧸ Γ)
File: Mathlib/Topology/Algebra/Group/TopologicalAbelianization.lean

full_name: instNormalCommutatorClosure
code: instance instNormalCommutatorClosure : (commutator G).topologicalClosure.Normal 
goal: (commutator G).topologicalClosure.Normal

full_name: TopologicalAbelianization
code: /-- The topological abelianization of `absoluteGaloisGroup`, that is, the quotient of
  `absoluteGaloisGroup` by the topological closure of its commutator subgroup. -/
abbrev TopologicalAbelianization 
goal: Type u_1

full_name: _aux_Mathlib_Topology_Algebra_Group_TopologicalAbelianization___macroRules_termG_ab_1
code: local notation "G_ab" => TopologicalAbelianization
goal: Lean.Macro

full_name: _aux_Mathlib_Topology_Algebra_Group_TopologicalAbelianization___unexpand_TopologicalAbelianization_1
code: local notation "G_ab" => TopologicalAbelianization
goal: Lean.PrettyPrinter.Unexpander

full_name: TopologicalAbelianization.commGroup
code: instance commGroup : CommGroup (G_ab G) 
goal: CommGroup (G_ab G)
File: Mathlib/Topology/Algebra/Algebra/Rat.lean

full_name: DivisionRing.continuousConstSMul_rat
code: /-- The action induced by `algebraRat` is continuous. -/
instance DivisionRing.continuousConstSMul_rat {A} [DivisionRing A] [TopologicalSpace A]
    [ContinuousMul A] [CharZero A] : ContinuousConstSMul ℚ A 
goal: ContinuousConstSMul ℚ A
File: Mathlib/Topology/Algebra/Ring/Ideal.lean

full_name: Ideal.closure
code: /-- The closure of an ideal in a topological ring as an ideal. -/
protected def Ideal.closure (I : Ideal R) : Ideal R 
goal: Ideal R

full_name: Ideal.coe_closure
code: @[simp]
theorem Ideal.coe_closure (I : Ideal R) : (I.closure : Set R) = closure I 
goal: ↑I.closure = closure ↑I

full_name: Ideal.closure_eq_of_isClosed
code: theorem Ideal.closure_eq_of_isClosed (I : Ideal R) (hI : IsClosed (I : Set R)) : I.closure = I 
goal: I.closure = I

full_name: topologicalRingQuotientTopology
code: instance topologicalRingQuotientTopology : TopologicalSpace (R ⧸ N) 
goal: TopologicalSpace (R ⧸ N)

full_name: QuotientRing.isOpenMap_coe
code: theorem QuotientRing.isOpenMap_coe : IsOpenMap (mk N) 
goal: IsOpenMap ⇑(Ideal.Quotient.mk N)

full_name: QuotientRing.quotientMap_coe_coe
code: theorem QuotientRing.quotientMap_coe_coe : QuotientMap fun p : R × R => (mk N p.1, mk N p.2) 
goal: QuotientMap fun p => ((Ideal.Quotient.mk N) p.1, (Ideal.Quotient.mk N) p.2)

full_name: topologicalRing_quotient
code: instance topologicalRing_quotient : TopologicalRing (R ⧸ N) 
goal: TopologicalRing (R ⧸ N)
File: Mathlib/Topology/Algebra/Localization.lean

full_name: Localization.ringTopology
code: /-- The ring topology on `Localization M` coinduced from the natural homomorphism sending `x : R`
to the equivalence class of `(x, 1)`. -/
def Localization.ringTopology : RingTopology (Localization M) 
goal: RingTopology (Localization M)

full_name: instTopologicalSpaceLocalization
code: instance : TopologicalSpace (Localization M) 
goal: TopologicalSpace (Localization M)

full_name: instTopologicalRingLocalization
code: instance : TopologicalRing (Localization M) 
goal: TopologicalRing (Localization M)
File: Mathlib/Topology/Algebra/Affine.lean

full_name: AffineMap.continuous_iff
code: /-- An affine map is continuous iff its underlying linear map is continuous. See also
`AffineMap.continuous_linear_iff`. -/
theorem continuous_iff {f : E →ᵃ[R] F} : Continuous f ↔ Continuous f.linear 
goal: Continuous ⇑f ↔ Continuous ⇑f.linear

full_name: AffineMap.lineMap_continuous
code: /-- The line map is continuous. -/
@[continuity]
theorem lineMap_continuous [TopologicalSpace R] [ContinuousSMul R F] {p v : F} :
    Continuous (lineMap p v : R →ᵃ[R] F) 
goal: Continuous ⇑(AffineMap.lineMap p v)

full_name: AffineMap.homothety_continuous
code: @[continuity]
theorem homothety_continuous (x : F) (t : R) : Continuous <| homothety x t 
goal: Continuous ⇑(AffineMap.homothety x t)

full_name: AffineMap.homothety_isOpenMap
code: theorem homothety_isOpenMap (x : F) (t : R) (ht : t ≠ 0) : IsOpenMap <| homothety x t 
goal: IsOpenMap ⇑(AffineMap.homothety x t)
File: Mathlib/Topology/DerivedSet.lean

full_name: AccPt.map
code: theorem AccPt.map {β : Type*} [TopologicalSpace β] {F : Filter X} {x : X}
    (h : AccPt x F) {f : X → β} (hf1 : ContinuousAt f x) (hf2 : Function.Injective f) :
    AccPt (f x) (map f F) 
goal: AccPt (f x) (Filter.map f F)

full_name: derivedSet
code: /--
The derived set of a set is the set of all accumulation points of it.
-/
def derivedSet (A : Set X) : Set X 
goal: Set X

full_name: mem_derivedSet
code: @[simp]
lemma mem_derivedSet {A : Set X} {x : X} : x ∈ derivedSet A ↔ AccPt x (𝓟 A) 
goal: x ∈ derivedSet A ↔ AccPt x (𝓟 A)

full_name: derivedSet_union
code: lemma derivedSet_union (A B : Set X) : derivedSet (A ∪ B) = derivedSet A ∪ derivedSet B 
goal: derivedSet (A ∪ B) = derivedSet A ∪ derivedSet B

full_name: derivedSet_mono
code: lemma derivedSet_mono (A B : Set X) (h : A ⊆ B) : derivedSet A ⊆ derivedSet B 
goal: derivedSet A ⊆ derivedSet B

full_name: Continuous.image_derivedSet
code: theorem Continuous.image_derivedSet {β : Type*} [TopologicalSpace β] {A : Set X} {f : X → β}
    (hf1 : Continuous f) (hf2 : Function.Injective f) :
    f '' derivedSet A ⊆ derivedSet (f '' A) 
goal: f '' derivedSet A ⊆ derivedSet (f '' A)

full_name: derivedSet_subset_closure
code: lemma derivedSet_subset_closure (A : Set X) : derivedSet A ⊆ closure A 
goal: derivedSet A ⊆ closure A

full_name: isClosed_iff_derivedSet_subset
code: lemma isClosed_iff_derivedSet_subset (A : Set X) : IsClosed A ↔ derivedSet A ⊆ A 
goal: IsClosed A ↔ derivedSet A ⊆ A

full_name: derivedSet_closure
code: /-- In a `T1Space`, the `derivedSet` of the closure of a set is equal to the derived set of the
set itself.

Note: this doesn't hold in a space with the indiscrete topology. For example, if `X` is a type with
two elements, `x` and `y`, and `A := {x}`, then `closure A = Set.univ` and `derivedSet A = {y}`,
but `derivedSet Set.univ = Set.univ`. -/
lemma derivedSet_closure [T1Space X] (A : Set X) : derivedSet (closure A) = derivedSet A 
goal: derivedSet (closure A) = derivedSet A

full_name: isClosed_derivedSet
code: @[simp]
lemma isClosed_derivedSet [T1Space X] (A : Set X) : IsClosed (derivedSet A) 
goal: IsClosed (derivedSet A)

full_name: preperfect_iff_subset_derivedSet
code: lemma preperfect_iff_subset_derivedSet {U : Set X} : Preperfect U ↔ U ⊆ derivedSet U 
goal: Preperfect U ↔ U ⊆ derivedSet U

full_name: perfect_iff_eq_derivedSet
code: lemma perfect_iff_eq_derivedSet {U : Set X} : Perfect U ↔ U = derivedSet U 
goal: Perfect U ↔ U = derivedSet U

full_name: IsPreconnected.inter_derivedSet_nonempty
code: lemma IsPreconnected.inter_derivedSet_nonempty [T1Space X] {U : Set X} (hs : IsPreconnected U)
    (a b : Set X) (h : U ⊆ a ∪ b) (ha : (U ∩ derivedSet a).Nonempty)
    (hb : (U ∩ derivedSet b).Nonempty) : (U ∩ (derivedSet a ∩ derivedSet b)).Nonempty 
goal: (U ∩ (derivedSet a ∩ derivedSet b)).Nonempty