ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- continuousAt_const_smul_iff
@[to_additive]
theorem continuousAt_const_smul_iff (c : G) :
    ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] [TopologicalSpace β] {f : β → α} {b : β} (c : G)
Goal: ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
-- continuousAt_const_vadd_iff
@[to_additive]
theorem continuousAt_const_smul_iff (c : G) :
    ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] [TopologicalSpace β] {f : β → α} {b : β} (c : G)
Goal: ContinuousAt (fun x => c +ᵥ f x) b ↔ ContinuousAt f b
-- continuous_const_smul_iff
@[to_additive]
theorem continuous_const_smul_iff (c : G) : (Continuous fun x => c • f x) ↔ Continuous f 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] [TopologicalSpace β] {f : β → α} (c : G)
Goal: (Continuous fun x => c • f x) ↔ Continuous f
-- continuous_const_vadd_iff
@[to_additive]
theorem continuous_const_smul_iff (c : G) : (Continuous fun x => c • f x) ↔ Continuous f 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] [TopologicalSpace β] {f : β → α} (c : G)
Goal: (Continuous fun x => c +ᵥ f x) ↔ Continuous f
-- isOpenMap_smul
@[to_additive]
theorem isOpenMap_smul (c : G) : IsOpenMap fun x : α => c • x 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (c : G)
Goal: IsOpenMap fun x => c • x
-- isOpenMap_vadd
@[to_additive]
theorem isOpenMap_smul (c : G) : IsOpenMap fun x : α => c • x 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (c : G)
Goal: IsOpenMap fun x => c +ᵥ x
-- IsOpen.vadd
@[to_additive]
theorem IsOpen.smul {s : Set α} (hs : IsOpen s) (c : G) : IsOpen (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] {s : Set α} (hs : IsOpen s) (c : G)
Goal: IsOpen (c +ᵥ s)
-- IsOpen.smul
@[to_additive]
theorem IsOpen.smul {s : Set α} (hs : IsOpen s) (c : G) : IsOpen (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] {s : Set α} (hs : IsOpen s) (c : G)
Goal: IsOpen (c • s)
-- isClosedMap_smul
@[to_additive]
theorem isClosedMap_smul (c : G) : IsClosedMap fun x : α => c • x 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (c : G)
Goal: IsClosedMap fun x => c • x
-- isClosedMap_vadd
@[to_additive]
theorem isClosedMap_smul (c : G) : IsClosedMap fun x : α => c • x 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (c : G)
Goal: IsClosedMap fun x => c +ᵥ x
-- IsClosed.vadd
@[to_additive]
theorem IsClosed.smul {s : Set α} (hs : IsClosed s) (c : G) : IsClosed (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] {s : Set α} (hs : IsClosed s) (c : G)
Goal: IsClosed (c +ᵥ s)
-- IsClosed.smul
@[to_additive]
theorem IsClosed.smul {s : Set α} (hs : IsClosed s) (c : G) : IsClosed (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] {s : Set α} (hs : IsClosed s) (c : G)
Goal: IsClosed (c • s)
-- closure_vadd
@[to_additive]
theorem closure_smul (c : G) (s : Set α) : closure (c • s) = c • closure s 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (c : G) (s : Set α)
Goal: closure (c +ᵥ s) = c +ᵥ closure s
-- closure_smul
@[to_additive]
theorem closure_smul (c : G) (s : Set α) : closure (c • s) = c • closure s 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (c : G) (s : Set α)
Goal: closure (c • s) = c • closure s
-- Dense.vadd
@[to_additive]
theorem Dense.smul (c : G) {s : Set α} (hs : Dense s) : Dense (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (c : G) {s : Set α} (hs : Dense s)
Goal: Dense (c +ᵥ s)
-- Dense.smul
@[to_additive]
theorem Dense.smul (c : G) {s : Set α} (hs : Dense s) : Dense (c • s) 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (c : G) {s : Set α} (hs : Dense s)
Goal: Dense (c • s)
-- interior_vadd
@[to_additive]
theorem interior_smul (c : G) (s : Set α) : interior (c • s) = c • interior s 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] (c : G) (s : Set α)
Goal: interior (c +ᵥ s) = c +ᵥ interior s
-- interior_smul
@[to_additive]
theorem interior_smul (c : G) (s : Set α) : interior (c • s) = c • interior s 
Arguments: {α : Type u_2} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] (c : G) (s : Set α)
Goal: interior (c • s) = c • interior s
-- tendsto_const_smul_iff₀
theorem tendsto_const_smul_iff₀ {f : β → α} {l : Filter β} {a : α} {c : G₀} (hc : c ≠ 0) :
    Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) 
Arguments: {α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {f : β → α} {l : Filter β} {a : α} {c : G₀} (hc : c ≠ 0)
Goal: Filter.Tendsto (fun x => c • f x) l (𝓝 (c • a)) ↔ Filter.Tendsto f l (𝓝 a)
-- continuousWithinAt_const_smul_iff₀
theorem continuousWithinAt_const_smul_iff₀ (hc : c ≠ 0) :
    ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b 
Arguments: {α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] [TopologicalSpace β] {f : β → α} {b : β} {c : G₀} {s : Set β} (hc : c ≠ 0)
Goal: ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b
-- continuousOn_const_smul_iff₀
theorem continuousOn_const_smul_iff₀ (hc : c ≠ 0) :
    ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s 
Arguments: {α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] [TopologicalSpace β] {f : β → α} {c : G₀} {s : Set β} (hc : c ≠ 0)
Goal: ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s
-- continuousAt_const_smul_iff₀
theorem continuousAt_const_smul_iff₀ (hc : c ≠ 0) :
    ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b 
Arguments: {α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] [TopologicalSpace β] {f : β → α} {b : β} {c : G₀} (hc : c ≠ 0)
Goal: ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
-- continuous_const_smul_iff₀
theorem continuous_const_smul_iff₀ (hc : c ≠ 0) : (Continuous fun x => c • f x) ↔ Continuous f 
Arguments: {α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] [TopologicalSpace β] {f : β → α} {c : G₀} (hc : c ≠ 0)
Goal: (Continuous fun x => c • f x) ↔ Continuous f
-- Homeomorph.smulOfNeZero_apply
@[simps! (config := .asFn) apply]
protected def Homeomorph.smulOfNeZero (c : G₀) (hc : c ≠ 0) : α ≃ₜ α 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] (c : G₀) (hc : c ≠ 0)
Goal: ⇑(Homeomorph.smulOfNeZero c hc) = fun x => c • x
-- Homeomorph.smulOfNeZero_symm_apply
@[simp]
theorem Homeomorph.smulOfNeZero_symm_apply {c : G₀} (hc : c ≠ 0) :
    ⇑(Homeomorph.smulOfNeZero c hc).symm = (c⁻¹ • · : α → α) 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: ⇑(Homeomorph.smulOfNeZero c hc).symm = fun x => c⁻¹ • x
-- isOpenMap_smul₀
theorem isOpenMap_smul₀ {c : G₀} (hc : c ≠ 0) : IsOpenMap fun x : α => c • x 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: IsOpenMap fun x => c • x
-- IsOpen.smul₀
theorem IsOpen.smul₀ {c : G₀} {s : Set α} (hs : IsOpen s) (hc : c ≠ 0) : IsOpen (c • s) 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} {s : Set α} (hs : IsOpen s) (hc : c ≠ 0)
Goal: IsOpen (c • s)
-- interior_smul₀
theorem interior_smul₀ {c : G₀} (hc : c ≠ 0) (s : Set α) : interior (c • s) = c • interior s 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0) (s : Set α)
Goal: interior (c • s) = c • interior s
-- closure_smul₀'
theorem closure_smul₀' {c : G₀} (hc : c ≠ 0) (s : Set α) :
    closure (c • s) = c • closure s 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0) (s : Set α)
Goal: closure (c • s) = c • closure s
-- closure_smul₀
theorem closure_smul₀ {E} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E]
    [ContinuousConstSMul G₀ E] (c : G₀) (s : Set E) : closure (c • s) = c • closure s 
Arguments: {G₀ : Type u_4} [GroupWithZero G₀] {E : Type u_5} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) (s : Set E)
Goal: closure (c • s) = c • closure s
-- isClosedMap_smul_of_ne_zero
theorem isClosedMap_smul_of_ne_zero {c : G₀} (hc : c ≠ 0) : IsClosedMap fun x : α => c • x 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: IsClosedMap fun x => c • x
-- IsClosed.smul_of_ne_zero
theorem IsClosed.smul_of_ne_zero {c : G₀} {s : Set α} (hs : IsClosed s) (hc : c ≠ 0) :
    IsClosed (c • s) 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {c : G₀} {s : Set α} (hs : IsClosed s) (hc : c ≠ 0)
Goal: IsClosed (c • s)
-- isClosedMap_smul₀
theorem isClosedMap_smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
    [T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) : IsClosedMap fun x : E => c • x 
Arguments: {G₀ : Type u_4} [GroupWithZero G₀] {E : Type u_5} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E] [ContinuousConstSMul G₀ E] (c : G₀)
Goal: IsClosedMap fun x => c • x
-- IsClosed.smul₀
theorem IsClosed.smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
    [T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) {s : Set E} (hs : IsClosed s) :
    IsClosed (c • s) 
Arguments: {G₀ : Type u_4} [GroupWithZero G₀] {E : Type u_5} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) {s : Set E} (hs : IsClosed s)
Goal: IsClosed (c • s)
-- HasCompactMulSupport.comp_smul
theorem HasCompactMulSupport.comp_smul {β : Type*} [One β] {f : α → β} (h : HasCompactMulSupport f)
    {c : G₀} (hc : c ≠ 0) : HasCompactMulSupport fun x => f (c • x) 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {β : Type u_5} [One β] {f : α → β} (h : HasCompactMulSupport f) {c : G₀} (hc : c ≠ 0)
Goal: HasCompactMulSupport fun x => f (c • x)
-- HasCompactSupport.comp_smul
theorem HasCompactSupport.comp_smul {β : Type*} [Zero β] {f : α → β} (h : HasCompactSupport f)
    {c : G₀} (hc : c ≠ 0) : HasCompactSupport fun x => f (c • x) 
Arguments: {α : Type u_2} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] {β : Type u_5} [Zero β] {f : α → β} (h : HasCompactSupport f) {c : G₀} (hc : c ≠ 0)
Goal: HasCompactSupport fun x => f (c • x)
-- IsUnit.tendsto_const_smul_iff
nonrec theorem tendsto_const_smul_iff {f : β → α} {l : Filter β} {a : α} {c : M} (hc : IsUnit c) :
    Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] {f : β → α} {l : Filter β} {a : α} {c : M} (hc : IsUnit c)
Goal: Filter.Tendsto (fun x => c • f x) l (𝓝 (c • a)) ↔ Filter.Tendsto f l (𝓝 a)
-- IsUnit.continuousWithinAt_const_smul_iff
nonrec theorem continuousWithinAt_const_smul_iff (hc : IsUnit c) :
    ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] [TopologicalSpace β] {f : β → α} {b : β} {c : M} {s : Set β} (hc : IsUnit c)
Goal: ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b
-- IsUnit.continuousOn_const_smul_iff
nonrec theorem continuousOn_const_smul_iff (hc : IsUnit c) :
    ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] [TopologicalSpace β] {f : β → α} {c : M} {s : Set β} (hc : IsUnit c)
Goal: ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s
-- IsUnit.continuousAt_const_smul_iff
nonrec theorem continuousAt_const_smul_iff (hc : IsUnit c) :
    ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] [TopologicalSpace β] {f : β → α} {b : β} {c : M} (hc : IsUnit c)
Goal: ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
-- IsUnit.continuous_const_smul_iff
nonrec theorem continuous_const_smul_iff (hc : IsUnit c) :
    (Continuous fun x => c • f x) ↔ Continuous f 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] [TopologicalSpace β] {f : β → α} {c : M} (hc : IsUnit c)
Goal: (Continuous fun x => c • f x) ↔ Continuous f
-- IsUnit.isOpenMap_smul
nonrec theorem isOpenMap_smul (hc : IsUnit c) : IsOpenMap fun x : α => c • x 
Arguments: {M : Type u_1} {α : Type u_2} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] {c : M} (hc : IsUnit c)
Goal: IsOpenMap fun x => c • x
-- IsUnit.isClosedMap_smul
nonrec theorem isClosedMap_smul (hc : IsUnit c) : IsClosedMap fun x : α => c • x 
Arguments: {M : Type u_1} {α : Type u_2} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] {c : M} (hc : IsUnit c)
Goal: IsClosedMap fun x => c • x
-- ProperlyDiscontinuousSMul.finite_disjoint_inter_image
class ProperlyDiscontinuousSMul (Γ : Type*) (T : Type*) [TopologicalSpace T] [SMul Γ T] :
    Prop where
    finite_disjoint_inter_image :
    ∀ {K L : Set T}, IsCompact K → IsCompact L → Set.Finite { γ : Γ | (γ • ·) '' K ∩ L ≠ ∅ }
Arguments: {Γ : Type u_4} {T : Type u_5} [TopologicalSpace T] [SMul Γ T] [self : ProperlyDiscontinuousSMul Γ T] {K L : Set T}
Goal: IsCompact K → IsCompact L → {γ | (fun x => γ • x) '' K ∩ L ≠ ∅}.Finite
-- ProperlyDiscontinuousVAdd.finite_disjoint_inter_image
class ProperlyDiscontinuousVAdd (Γ : Type*) (T : Type*) [TopologicalSpace T] [VAdd Γ T] :
  Prop where
    finite_disjoint_inter_image :
    ∀ {K L : Set T}, IsCompact K → IsCompact L → Set.Finite { γ : Γ | (γ +ᵥ ·) '' K ∩ L ≠ ∅ }
Arguments: {Γ : Type u_4} {T : Type u_5} [TopologicalSpace T] [VAdd Γ T] [self : ProperlyDiscontinuousVAdd Γ T] {K L : Set T}
Goal: IsCompact K → IsCompact L → {γ | (fun x => γ +ᵥ x) '' K ∩ L ≠ ∅}.Finite
-- isOpenMap_quotient_mk'_add
@[to_additive "The quotient map by a group action is open, i.e. the quotient by a group
action is an open quotient. "]
theorem isOpenMap_quotient_mk'_mul [ContinuousConstSMul Γ T] :
    letI := MulAction.orbitRel Γ T
    IsOpenMap (Quotient.mk' : T → Quotient (MulAction.orbitRel Γ T)) 
Arguments: {Γ : Type u_4} [AddGroup Γ] {T : Type u_5} [TopologicalSpace T] [AddAction Γ T] [ContinuousConstVAdd Γ T]
Goal: IsOpenMap Quotient.mk'
-- isOpenMap_quotient_mk'_mul
@[to_additive "The quotient map by a group action is open, i.e. the quotient by a group
action is an open quotient. "]
theorem isOpenMap_quotient_mk'_mul [ContinuousConstSMul Γ T] :
    letI := MulAction.orbitRel Γ T
    IsOpenMap (Quotient.mk' : T → Quotient (MulAction.orbitRel Γ T)) 
Arguments: {Γ : Type u_4} [Group Γ] {T : Type u_5} [TopologicalSpace T] [MulAction Γ T] [ContinuousConstSMul Γ T]
Goal: IsOpenMap Quotient.mk'
-- ContinuousConstVAdd.secondCountableTopology
@[to_additive "The quotient of a second countable space by an additive group action is second
countable."]
theorem ContinuousConstSMul.secondCountableTopology [SecondCountableTopology T]
    [ContinuousConstSMul Γ T] : SecondCountableTopology (Quotient (MulAction.orbitRel Γ T)) 
Arguments: {Γ : Type u_4} [AddGroup Γ] {T : Type u_5} [TopologicalSpace T] [AddAction Γ T] [SecondCountableTopology T] [ContinuousConstVAdd Γ T]
Goal: SecondCountableTopology (Quotient (AddAction.orbitRel Γ T))
-- ContinuousConstSMul.secondCountableTopology
@[to_additive "The quotient of a second countable space by an additive group action is second
countable."]
theorem ContinuousConstSMul.secondCountableTopology [SecondCountableTopology T]
    [ContinuousConstSMul Γ T] : SecondCountableTopology (Quotient (MulAction.orbitRel Γ T)) 
Arguments: {Γ : Type u_4} [Group Γ] {T : Type u_5} [TopologicalSpace T] [MulAction Γ T] [SecondCountableTopology T] [ContinuousConstSMul Γ T]
Goal: SecondCountableTopology (Quotient (MulAction.orbitRel Γ T))
-- set_smul_mem_nhds_smul
theorem set_smul_mem_nhds_smul {c : G₀} {s : Set α} {x : α} (hs : s ∈ 𝓝 x) (hc : c ≠ 0) :
    c • s ∈ 𝓝 (c • x : α) 
Arguments: {α : Type u_2} {G₀ : Type u_6} [GroupWithZero G₀] [MulAction G₀ α] [TopologicalSpace α] [ContinuousConstSMul G₀ α] {c : G₀} {s : Set α} {x : α} (hs : s ∈ 𝓝 x) (hc : c ≠ 0)
Goal: c • s ∈ 𝓝 (c • x)
-- set_smul_mem_nhds_smul_iff
theorem set_smul_mem_nhds_smul_iff {c : G₀} {s : Set α} {x : α} (hc : c ≠ 0) :
    c • s ∈ 𝓝 (c • x : α) ↔ s ∈ 𝓝 x 
Arguments: {α : Type u_2} {G₀ : Type u_6} [GroupWithZero G₀] [MulAction G₀ α] [TopologicalSpace α] [ContinuousConstSMul G₀ α] {c : G₀} {s : Set α} {x : α} (hc : c ≠ 0)
Goal: c • s ∈ 𝓝 (c • x) ↔ s ∈ 𝓝 x
-- set_smul_mem_nhds_zero_iff
theorem set_smul_mem_nhds_zero_iff {s : Set α} {c : G₀} (hc : c ≠ 0) :
    c • s ∈ 𝓝 (0 : α) ↔ s ∈ 𝓝 (0 : α) 
Arguments: {α : Type u_2} {G₀ : Type u_6} [GroupWithZero G₀] [AddMonoid α] [DistribMulAction G₀ α] [TopologicalSpace α] [ContinuousConstSMul G₀ α] {s : Set α} {c : G₀} (hc : c ≠ 0)
Goal: c • s ∈ 𝓝 0 ↔ s ∈ 𝓝 0
-- continuous_one
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_one [TopologicalSpace M] [One M] : Continuous (1 : X → M) 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [One M]
Goal: Continuous 1
-- continuous_zero
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_one [TopologicalSpace M] [One M] : Continuous (1 : X → M) 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Zero M]
Goal: Continuous 0
-- ContinuousAdd.continuous_add
class ContinuousAdd (M : Type u) [TopologicalSpace M] [Add M] : Prop where
  continuous_add : Continuous fun p : M × M => p.1 + p.2
Arguments: {M : Type u} [TopologicalSpace M] [Add M] [self : ContinuousAdd M]
Goal: Continuous fun p => p.1 + p.2
-- ContinuousMul.continuous_mul
@[to_additive]
class ContinuousMul (M : Type u) [TopologicalSpace M] [Mul M] : Prop where
  continuous_mul : Continuous fun p : M × M => p.1 * p.2
Arguments: {M : Type u} [TopologicalSpace M] [Mul M] [self : ContinuousMul M]
Goal: Continuous fun p => p.1 * p.2
-- continuous_mul
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mul : Continuous fun p : M × M => p.1 * p.2 
Arguments: {M : Type u_3} [TopologicalSpace M] [Mul M] [ContinuousMul M]
Goal: Continuous fun p => p.1 * p.2
-- continuous_add
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mul : Continuous fun p : M × M => p.1 * p.2 
Arguments: {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M]
Goal: Continuous fun p => p.1 + p.2
-- ContinuousMul.induced
@[to_additive]
theorem ContinuousMul.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [MulOneClass α]
    [MulOneClass β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousMul β] (f : F) :
    @ContinuousMul α (tβ.induced f) _ 
Arguments: {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [MulOneClass α] [MulOneClass β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousMul β] (f : F)
Goal: ContinuousMul α
-- ContinuousAdd.induced
@[to_additive]
theorem ContinuousMul.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [MulOneClass α]
    [MulOneClass β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousMul β] (f : F) :
    @ContinuousMul α (tβ.induced f) _ 
Arguments: {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [AddZeroClass α] [AddZeroClass β] [AddMonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousAdd β] (f : F)
Goal: ContinuousAdd α
-- Continuous.mul
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.mul {f g : X → M} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => f x * g x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Mul M] [ContinuousMul M] {f g : X → M} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x * g x
-- Continuous.add
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.mul {f g : X → M} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => f x * g x 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Add M] [ContinuousAdd M] {f g : X → M} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x + g x
-- continuous_mul_left
@[to_additive (attr := continuity)]
theorem continuous_mul_left (a : M) : Continuous fun b : M => a * b 
Arguments: {M : Type u_3} [TopologicalSpace M] [Mul M] [ContinuousMul M] (a : M)
Goal: Continuous fun b => a * b
-- continuous_add_left
@[to_additive (attr := continuity)]
theorem continuous_mul_left (a : M) : Continuous fun b : M => a * b 
Arguments: {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] (a : M)
Goal: Continuous fun b => a + b
-- continuous_add_right
@[to_additive (attr := continuity)]
theorem continuous_mul_right (a : M) : Continuous fun b : M => b * a 
Arguments: {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] (a : M)
Goal: Continuous fun b => b + a
-- continuous_mul_right
@[to_additive (attr := continuity)]
theorem continuous_mul_right (a : M) : Continuous fun b : M => b * a 
Arguments: {M : Type u_3} [TopologicalSpace M] [Mul M] [ContinuousMul M] (a : M)
Goal: Continuous fun b => b * a
-- ContinuousOn.mul
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.mul {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x * g x) s 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Mul M] [ContinuousMul M] {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x * g x) s
-- ContinuousOn.add
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.mul {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x * g x) s 
Arguments: {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Add M] [ContinuousAdd M] {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x + g x) s
-- tendsto_mul
@[to_additive]
theorem tendsto_mul {a b : M} : Tendsto (fun p : M × M => p.fst * p.snd) (𝓝 (a, b)) (𝓝 (a * b)) 
Arguments: {M : Type u_3} [TopologicalSpace M] [Mul M] [ContinuousMul M] {a b : M}
Goal: Filter.Tendsto (fun p => p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b))