ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAlgHom.id_apply
theorem id_apply (x : A) : ContinuousAlgHom.id R A x = x 
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A] (x : A)
Goal: (ContinuousAlgHom.id R A) x = x
-- ContinuousAlgHom.coe_id
@[simp, norm_cast]
theorem coe_id : ((ContinuousAlgHom.id R A) : A →ₐ[R] A) = AlgHom.id R A
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A]
Goal: ↑(ContinuousAlgHom.id R A) = AlgHom.id R A
-- ContinuousAlgHom.coe_id'
@[simp, norm_cast]
theorem coe_id' : ⇑(ContinuousAlgHom.id R A ) = _root_.id 
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A]
Goal: ⇑(ContinuousAlgHom.id R A) = id
-- ContinuousAlgHom.coe_eq_id
@[simp, norm_cast]
theorem coe_eq_id {f : A →A[R] A} :
    (f : A →ₐ[R] A) = AlgHom.id R A ↔ f = ContinuousAlgHom.id R A
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A] {f : A →A[R] A}
Goal: ↑f = AlgHom.id R A ↔ f = ContinuousAlgHom.id R A
-- ContinuousAlgHom.one_apply
@[simp]
theorem one_apply (x : A) : (1 : A →A[R] A) x = x 
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A] (x : A)
Goal: 1 x = x
-- ContinuousAlgHom.coe_comp
@[simp, norm_cast]
theorem coe_comp (h : B →A[R] C) (f : A →A[R] B) :
    (h.comp f : A →ₐ[R] C) = (h : B →ₐ[R] C).comp (f : A →ₐ[R] B) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (h : B →A[R] C) (f : A →A[R] B)
Goal: ↑(h.comp f) = (↑h).comp ↑f
-- ContinuousAlgHom.coe_comp'
@[simp, norm_cast]
theorem coe_comp' (h : B →A[R] C) (f : A →A[R] B) : ⇑(h.comp f) = h ∘ f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (h : B →A[R] C) (f : A →A[R] B)
Goal: ⇑(h.comp f) = ⇑h ∘ ⇑f
-- ContinuousAlgHom.comp_apply
theorem comp_apply (g : B →A[R] C) (f : A →A[R] B) (x : A) : (g.comp f) x = g (f x) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (g : B →A[R] C) (f : A →A[R] B) (x : A)
Goal: (g.comp f) x = g (f x)
-- ContinuousAlgHom.comp_id
@[simp]
theorem comp_id (f : A →A[R] B) : f.comp (ContinuousAlgHom.id R A) = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: f.comp (ContinuousAlgHom.id R A) = f
-- ContinuousAlgHom.id_comp
@[simp]
theorem id_comp (f : A →A[R] B) : (ContinuousAlgHom.id R B).comp f = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: (ContinuousAlgHom.id R B).comp f = f
-- ContinuousAlgHom.comp_assoc
theorem comp_assoc {D : Type*} [Semiring D] [Algebra R D] [TopologicalSpace D] (h : C →A[R] D)
    (g : B →A[R] C) (f : A →A[R] B) : (h.comp g).comp f = h.comp (g.comp f) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] {D : Type u_5} [Semiring D] [Algebra R D] [TopologicalSpace D] (h : C →A[R] D) (g : B →A[R] C) (f : A →A[R] B)
Goal: (h.comp g).comp f = h.comp (g.comp f)
-- ContinuousAlgHom.mul_def
theorem mul_def (f g : A →A[R] A) : f * g = f.comp g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (f g : A →A[R] A)
Goal: f * g = f.comp g
-- ContinuousAlgHom.coe_mul
@[simp]
theorem coe_mul (f g : A →A[R] A) : ⇑(f * g) = f ∘ g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (f g : A →A[R] A)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- ContinuousAlgHom.mul_apply
theorem mul_apply (f g : A →A[R] A) (x : A) : (f * g) x = f (g x) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (f g : A →A[R] A) (x : A)
Goal: (f * g) x = f (g x)
-- ContinuousAlgHom.coe_pow
theorem coe_pow (f : A →A[R] A) (n : ℕ) : ⇑(f ^ n) = f^[n] 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (f : A →A[R] A) (n : ℕ)
Goal: ⇑(f ^ n) = (⇑f)^[n]
-- ContinuousAlgHom.toAlgHomMonoidHom_apply
@[simps]
def toAlgHomMonoidHom : (A →A[R] A) →* A →ₐ[R] A 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (f : A →A[R] A)
Goal: ContinuousAlgHom.toAlgHomMonoidHom f = ↑f
-- ContinuousAlgHom.coe_prod
@[simp, norm_cast]
theorem coe_prod (f₁ : A →A[R] B) (f₂ : A →A[R] C) :
    (f₁.prod f₂ : A →ₐ[R] B × C) = AlgHom.prod f₁ f₂ 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (f₁ : A →A[R] B) (f₂ : A →A[R] C)
Goal: ↑(f₁.prod f₂) = (↑f₁).prod ↑f₂
-- ContinuousAlgHom.prod_apply
@[simp, norm_cast]
theorem prod_apply (f₁ : A →A[R] B) (f₂ : A →A[R] C) (x : A) :
    f₁.prod f₂ x = (f₁ x, f₂ x) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (f₁ : A →A[R] B) (f₂ : A →A[R] C) (x : A)
Goal: (f₁.prod f₂) x = (f₁ x, f₂ x)
-- ContinuousAlgHom.coe_fst
@[simp, norm_cast]
theorem coe_fst : ↑(fst R A B) = AlgHom.fst R A B 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
Goal: ↑(ContinuousAlgHom.fst R A B) = AlgHom.fst R A B
-- ContinuousAlgHom.coe_fst'
@[simp, norm_cast]
theorem coe_fst' : ⇑(fst R A B) = Prod.fst 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
Goal: ⇑(ContinuousAlgHom.fst R A B) = Prod.fst
-- ContinuousAlgHom.coe_snd
@[simp, norm_cast]
theorem coe_snd : ↑(snd R A B) = AlgHom.snd R A B 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
Goal: ↑(ContinuousAlgHom.snd R A B) = AlgHom.snd R A B
-- ContinuousAlgHom.coe_snd'
@[simp, norm_cast]
theorem coe_snd' : ⇑(snd R A B) = Prod.snd 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
Goal: ⇑(ContinuousAlgHom.snd R A B) = Prod.snd
-- ContinuousAlgHom.fst_prod_snd
@[simp]
theorem fst_prod_snd  : (fst R A B).prod (snd R A B) = ContinuousAlgHom.id R (A × B) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
Goal: (ContinuousAlgHom.fst R A B).prod (ContinuousAlgHom.snd R A B) = ContinuousAlgHom.id R (A × B)
-- ContinuousAlgHom.fst_comp_prod
@[simp]
theorem fst_comp_prod (f : A →A[R] B) (g : A →A[R] C) :
    (fst R B C).comp (f.prod g) = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (f : A →A[R] B) (g : A →A[R] C)
Goal: (ContinuousAlgHom.fst R B C).comp (f.prod g) = f
-- ContinuousAlgHom.snd_comp_prod
@[simp]
theorem snd_comp_prod  (f : A →A[R] B) (g : A →A[R] C) :
    (snd R B C).comp (f.prod g) = g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (f : A →A[R] B) (g : A →A[R] C)
Goal: (ContinuousAlgHom.snd R B C).comp (f.prod g) = g
-- ContinuousAlgHom.coe_prodMap
@[simp, norm_cast]
theorem coe_prodMap {D : Type*} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B)
    (f₂ : C →A[R] D) :
    (f₁.prodMap f₂ : A × C →ₐ[R] B × D) = (f₁ : A →ₐ[R] B).prodMap (f₂ : C →ₐ[R] D) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] {D : Type u_6} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B) (f₂ : C →A[R] D)
Goal: ↑(f₁.prodMap f₂) = (↑f₁).prodMap ↑f₂
-- ContinuousAlgHom.coe_prodMap'
@[simp, norm_cast]
theorem coe_prodMap' {D : Type*} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B)
    (f₂ : C →A[R] D) : ⇑(f₁.prodMap f₂) = Prod.map f₁ f₂ 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] {D : Type u_6} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B) (f₂ : C →A[R] D)
Goal: ⇑(f₁.prodMap f₂) = Prod.map ⇑f₁ ⇑f₂
-- ContinuousAlgHom.prodEquiv_apply
@[simps apply]
def prodEquiv : (A →A[R] B) × (A →A[R] C) ≃ (A →A[R] B × C) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] [TopologicalSpace C] (f : (A →A[R] B) × (A →A[R] C))
Goal: ContinuousAlgHom.prodEquiv f = f.1.prod f.2
-- ContinuousAlgHom.coe_codRestrict
@[norm_cast]
theorem coe_codRestrict (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ x, f x ∈ p) :
    (f.codRestrict p h : A →ₐ[R] p) = (f : A →ₐ[R] B).codRestrict p h 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ (x : A), f x ∈ p)
Goal: ↑(f.codRestrict p h) = (↑f).codRestrict p h
-- ContinuousAlgHom.coe_codRestrict_apply
@[simp]
theorem coe_codRestrict_apply (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ x, f x ∈ p) (x) :
    (f.codRestrict p h x : B) = f x 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ (x : A), f x ∈ p) (x : A)
Goal: ↑((f.codRestrict p h) x) = f x
-- ContinuousAlgHom.coe_rangeRestrict
@[simp]
theorem coe_rangeRestrict (f : A →A[R] B) :
    (f.rangeRestrict : A →ₐ[R] (@AlgHom.range R A B  _ _ _ _ _ f)) =
      (f : A →ₐ[R] B).rangeRestrict 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: ↑f.rangeRestrict = (↑f).rangeRestrict
-- Subalgebra.coe_valA
@[simp, norm_cast]
theorem _root_.Subalgebra.coe_valA (p : Subalgebra R A) :
    (p.valA : p →ₐ[R] A) = p.subtype 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (p : Subalgebra R A)
Goal: ↑↑p.valA = p.subtype
-- Subalgebra.coe_valA'
@[simp]
theorem _root_.Subalgebra.coe_valA' (p : Subalgebra R A) : ⇑p.valA = p.subtype 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (p : Subalgebra R A)
Goal: ⇑p.valA = ⇑p.subtype
-- Subalgebra.valA_apply
@[simp]
theorem _root_.Subalgebra.valA_apply (p : Subalgebra R A) (x : p) : p.valA x = x 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (p : Subalgebra R A) (x : ↥p)
Goal: p.valA x = ↑x
-- Submodule.range_valA
@[simp]
theorem _root_.Submodule.range_valA (p : Subalgebra R A) :
    @AlgHom.range R p A _ _ _ _ _ p.valA = p 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] (p : Subalgebra R A)
Goal: (↑p.valA).range = p
-- ContinuousAlgHom.map_neg
protected theorem map_neg (f : S →A[R] B) (x : S) : f (-x) = -f x 
Arguments: (R : Type u_1) [CommSemiring R] {S : Type u_3} [Ring S] [TopologicalSpace S] [Algebra R S] {B : Type u_4} [Ring B] [TopologicalSpace B] [Algebra R B] (f : S →A[R] B) (x : S)
Goal: f (-x) = -f x
-- ContinuousAlgHom.map_sub
protected theorem map_sub (f : S →A[R] B) (x y : S) : f (x - y) = f x - f y 
Arguments: (R : Type u_1) [CommSemiring R] {S : Type u_3} [Ring S] [TopologicalSpace S] [Algebra R S] {B : Type u_4} [Ring B] [TopologicalSpace B] [Algebra R B] (f : S →A[R] B) (x y : S)
Goal: f (x - y) = f x - f y
-- ContinuousAlgHom.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : B →A[S] C) :
    (f.restrictScalars R : B →ₐ[R] C) = (f : B →ₐ[S] C).restrictScalars R 
Arguments: {R : Type u_1} [CommSemiring R] {S : Type u_3} [CommSemiring S] [Algebra R S] {B : Type u_4} [Ring B] [TopologicalSpace B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] {C : Type u_5} [Ring C] [TopologicalSpace C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : B →A[S] C)
Goal: ↑(ContinuousAlgHom.restrictScalars R f) = AlgHom.restrictScalars R ↑f
-- ContinuousAlgHom.coe_restrictScalars'
@[simp]
theorem coe_restrictScalars' (f : B →A[S] C) : ⇑(f.restrictScalars R) = f 
Arguments: {R : Type u_1} [CommSemiring R] {S : Type u_3} [CommSemiring S] [Algebra R S] {B : Type u_4} [Ring B] [TopologicalSpace B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] {C : Type u_5} [Ring C] [TopologicalSpace C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : B →A[S] C)
Goal: ⇑(ContinuousAlgHom.restrictScalars R f) = ⇑f
-- Subalgebra.le_topologicalClosure
theorem Subalgebra.le_topologicalClosure (s : Subalgebra R A) : s ≤ s.topologicalClosure 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (s : Subalgebra R A)
Goal: s ≤ s.topologicalClosure
-- Subalgebra.isClosed_topologicalClosure
theorem Subalgebra.isClosed_topologicalClosure (s : Subalgebra R A) :
    IsClosed (s.topologicalClosure : Set A) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (s : Subalgebra R A)
Goal: IsClosed ↑s.topologicalClosure
-- Subalgebra.topologicalClosure_minimal
theorem Subalgebra.topologicalClosure_minimal (s : Subalgebra R A) {t : Subalgebra R A} (h : s ≤ t)
    (ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (s : Subalgebra R A) {t : Subalgebra R A} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- Subalgebra.topologicalClosure_comap_homeomorph
theorem Subalgebra.topologicalClosure_comap_homeomorph (s : Subalgebra R A) {B : Type*}
    [TopologicalSpace B] [Ring B] [TopologicalRing B] [Algebra R B] (f : B →ₐ[R] A) (f' : B ≃ₜ A)
    (w : (f : B → A) = f') : s.topologicalClosure.comap f = (s.comap f).topologicalClosure 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (s : Subalgebra R A) {B : Type u_2} [TopologicalSpace B] [Ring B] [TopologicalRing B] [Algebra R B] (f : B →ₐ[R] A) (f' : B ≃ₜ A) (w : ⇑f = ⇑f')
Goal: Subalgebra.comap f s.topologicalClosure = (Subalgebra.comap f s).topologicalClosure
-- Algebra.self_mem_elementalAlgebra
@[aesop safe apply (rule_sets := [SetLike])]
theorem Algebra.self_mem_elementalAlgebra (x : A) : x ∈ Algebra.elementalAlgebra R x 
Arguments: (R : Type u_1) [CommRing R] {A : Type u} [TopologicalSpace A] [Ring A] [Algebra R A] [TopologicalRing A] (x : A)
Goal: x ∈ Algebra.elementalAlgebra R x
-- UniformConvergenceCLM.topologicalSpace_eq
theorem topologicalSpace_eq [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
    instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced DFunLike.coe
      (UniformOnFun.topologicalSpace E F 𝔖) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E))
Goal: UniformConvergenceCLM.instTopologicalSpace σ F 𝔖 =
  TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖)
-- UniformConvergenceCLM.uniformSpace_eq
theorem uniformSpace_eq [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
    instUniformSpace σ F 𝔖 = UniformSpace.comap DFunLike.coe (UniformOnFun.uniformSpace E F 𝔖) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E))
Goal: UniformConvergenceCLM.instUniformSpace σ F 𝔖 = UniformSpace.comap DFunLike.coe (UniformOnFun.uniformSpace E F 𝔖)
-- UniformConvergenceCLM.uniformity_toTopologicalSpace_eq
@[simp]
theorem uniformity_toTopologicalSpace_eq [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
    (UniformConvergenceCLM.instUniformSpace σ F 𝔖).toTopologicalSpace =
      UniformConvergenceCLM.instTopologicalSpace σ F 𝔖 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E))
Goal: UniformSpace.toTopologicalSpace = UniformConvergenceCLM.instTopologicalSpace σ F 𝔖
-- UniformConvergenceCLM.uniformEmbedding_coeFn
theorem uniformEmbedding_coeFn [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
    UniformEmbedding (α := UniformConvergenceCLM σ F 𝔖) (β := E →ᵤ[𝔖] F) DFunLike.coe 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E))
Goal: UniformEmbedding DFunLike.coe
-- UniformConvergenceCLM.embedding_coeFn
theorem embedding_coeFn [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
    Embedding (X := UniformConvergenceCLM σ F 𝔖) (Y := E →ᵤ[𝔖] F)
      (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E))
Goal: Embedding (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe)
-- UniformConvergenceCLM.t2Space
theorem t2Space [TopologicalSpace F] [TopologicalAddGroup F] [T2Space F]
    (𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ) : T2Space (UniformConvergenceCLM σ F 𝔖) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace F] [TopologicalAddGroup F] [T2Space F] (𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ)
Goal: T2Space (UniformConvergenceCLM σ F 𝔖)
-- UniformConvergenceCLM.continuousSMul
theorem continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
    [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E))
    (h𝔖₃ : ∀ S ∈ 𝔖, Bornology.IsVonNBounded 𝕜₁ S) :
    ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖) 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [RingHomSurjective σ] [RingHomIsometric σ] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E)) (h𝔖₃ : ∀ S ∈ 𝔖, Bornology.IsVonNBounded 𝕜₁ S)
Goal: ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖)
-- UniformConvergenceCLM.hasBasis_nhds_zero_of_basis
theorem hasBasis_nhds_zero_of_basis [TopologicalSpace F] [TopologicalAddGroup F]
    {ι : Type*} (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop}
    {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
    (𝓝 (0 : UniformConvergenceCLM σ F 𝔖)).HasBasis
      (fun Si : Set E × ι => Si.1 ∈ 𝔖 ∧ p Si.2)
      fun Si => { f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2 } 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace F] [TopologicalAddGroup F] {ι : Type u_5} (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖) {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0).HasBasis p b)
Goal: (𝓝 0).HasBasis (fun Si => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, f x ∈ b Si.2}
-- UniformConvergenceCLM.hasBasis_nhds_zero
theorem hasBasis_nhds_zero [TopologicalSpace F] [TopologicalAddGroup F]
    (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
    (𝓝 (0 : UniformConvergenceCLM σ F 𝔖)).HasBasis
      (fun SV : Set E × Set F => SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 0 : Filter F)) fun SV =>
      { f : UniformConvergenceCLM σ F 𝔖 | ∀ x ∈ SV.1, f x ∈ SV.2 } 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace F] [TopologicalAddGroup F] (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖)
Goal: (𝓝 0).HasBasis (fun SV => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | ∀ x ∈ SV.1, f x ∈ SV.2}
-- UniformConvergenceCLM.tendsto_iff_tendstoUniformlyOn
theorem tendsto_iff_tendstoUniformlyOn {ι : Type*} {p : Filter ι} [UniformSpace F]
    [UniformAddGroup F] (𝔖 : Set (Set E)) {a : ι → UniformConvergenceCLM σ F 𝔖}
    {a₀ : UniformConvergenceCLM σ F 𝔖} :
    Filter.Tendsto a p (𝓝 a₀) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (a · ·) a₀ p s 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] {ι : Type u_5} {p : Filter ι} [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) {a : ι → UniformConvergenceCLM σ F 𝔖} {a₀ : UniformConvergenceCLM σ F 𝔖}
Goal: Filter.Tendsto a p (𝓝 a₀) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (fun x x_1 => (a x) x_1) (⇑a₀) p s
-- UniformConvergenceCLM.uniformSpace_mono
theorem uniformSpace_mono [UniformSpace F] [UniformAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
    instUniformSpace σ F 𝔖₁ ≤ instUniformSpace σ F 𝔖₂ 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] {𝔖₁ 𝔖₂ : Set (Set E)} [UniformSpace F] [UniformAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁)
Goal: UniformConvergenceCLM.instUniformSpace σ F 𝔖₁ ≤ UniformConvergenceCLM.instUniformSpace σ F 𝔖₂
-- UniformConvergenceCLM.topologicalSpace_mono
theorem topologicalSpace_mono [TopologicalSpace F] [TopologicalAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
    instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂ 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3} (F : Type u_4) [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜₂ F] {𝔖₁ 𝔖₂ : Set (Set E)} [TopologicalSpace F] [TopologicalAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁)
Goal: UniformConvergenceCLM.instTopologicalSpace σ F 𝔖₁ ≤ UniformConvergenceCLM.instTopologicalSpace σ F 𝔖₂