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Found 1068 declarations mentioning ContinuousAdd. Of these, only the first 200 are shown.

• ContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid.Defs
  (M : Type u_1) [TopologicalSpace M] [Add M] : Prop
• instSeparatelyContinuousAddOfContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] : SeparatelyContinuousAdd M
• continuous_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] : Continuous fun p => p.1 + p.2
• ContinuousAdd.continuous_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} {inst✝ : TopologicalSpace M} {inst✝¹ : Add M} [self : ContinuousAdd M] : Continuous fun p => p.1 + p.2
• ContinuousAdd.mk 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] (continuous_add : Continuous fun p => p.1 + p.2) : ContinuousAdd M
• Continuous.fun_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} (hf : Continuous f) (hg : Continuous g) : Continuous fun i => f i + g i
• ContinuousAt.fun_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun i => f i + g i) x
• ContinuousOn.fun_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun i => f i + g i) s
• Continuous.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} (hf : Continuous f) (hg : Continuous g) : Continuous (f + g)
• ContinuousWithinAt.fun_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun i => f i + g i) s x
• ContinuousAt.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f + g) x
• ContinuousOn.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (f + g) s
• ContinuousWithinAt.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {s : Set X} {x : X} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (f + g) s x
• Filter.Tendsto.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {α : Type u_2} {f g : α → M} {x : Filter α} {a b : M} (hf : Filter.Tendsto f x (nhds a)) (hg : Filter.Tendsto g x (nhds b)) : Filter.Tendsto (fun x => f x + g x) x (nhds (a + b))
• Filter.tendsto_of_sub_tendsto_zero 📋 Mathlib.Topology.Algebra.Monoid.Defs
  {α : Type u_2} {E : Type u_3} [AddCommGroup E] [TopologicalSpace E] [ContinuousAdd E] {f g : α → E} (m : E) {x : Filter α} (hf : Filter.Tendsto f x (nhds m)) (hfg : Filter.Tendsto (g - f) x (nhds 0)) : Filter.Tendsto g x (nhds m)
• IsTopologicalAddGroup.toContinuousAdd 📋 Mathlib.Topology.Algebra.Group.Defs
  {G : Type u} {inst✝ : TopologicalSpace G} {inst✝¹ : AddGroup G} [self : IsTopologicalAddGroup G] : ContinuousAdd G
• IsTopologicalAddGroup.mk 📋 Mathlib.Topology.Algebra.Group.Defs
  {G : Type u} [TopologicalSpace G] [AddGroup G] [toContinuousAdd : ContinuousAdd G] [toContinuousNeg : ContinuousNeg G] : IsTopologicalAddGroup G
• continuousAdd_of_discreteTopology 📋 Mathlib.Topology.Algebra.Monoid
  {N : Type u_4} [TopologicalSpace N] [Add N] [DiscreteTopology N] : ContinuousAdd N
• continuousAdd_of_indiscreteTopology 📋 Mathlib.Topology.Algebra.Monoid
  {N : Type u_4} [TopologicalSpace N] [Add N] [IndiscreteTopology N] : ContinuousAdd N
• ContinuousAdd.to_continuousVAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousVAdd M M
• instContinuousAddAdditiveOfContinuousMul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Mul M] [ContinuousMul M] : ContinuousAdd (Additive M)
• instContinuousAddOrderDual 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousAdd Mᵒᵈ
• instContinuousAddULift 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousAdd (ULift.{u, u_3} M)
• instContinuousMulMultiplicativeOfContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousMul (Multiplicative M)
• AddOpposite.instContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} [TopologicalSpace α] [Add α] [ContinuousAdd α] : ContinuousAdd αᵃᵒᵖ
• ContinuousAdd.to_continuousVAdd_op 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousVAdd Mᵃᵒᵖ M
• AddMonoid.continuousConstSMul_nat 📋 Mathlib.Topology.Algebra.Monoid
  {A : Type u_6} [AddMonoid A] [TopologicalSpace A] [ContinuousAdd A] : ContinuousConstSMul ℕ A
• AddMonoid.continuousSMul_nat 📋 Mathlib.Topology.Algebra.Monoid
  {A : Type u_6} [AddMonoid A] [TopologicalSpace A] [ContinuousAdd A] : ContinuousSMul ℕ A
• Pi.continuousAdd' 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousAdd (ι → M)
• AddUnits.instContinuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} [TopologicalSpace α] [AddMonoid α] [ContinuousAdd α] : ContinuousAdd (AddUnits α)
• continuousAdd_iInf 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {ι' : Sort u_6} [Add M] {ts : ι' → TopologicalSpace M} (h' : ∀ (i : ι'), ContinuousAdd M) : ContinuousAdd M
• Pi.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {C : ι → Type u_6} [(i : ι) → TopologicalSpace (C i)] [(i : ι) → Add (C i)] [∀ (i : ι), ContinuousAdd (C i)] : ContinuousAdd ((i : ι) → C i)
• Prod.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {N : Type u_4} [TopologicalSpace M] [Add M] [ContinuousAdd M] [TopologicalSpace N] [Add N] [ContinuousAdd N] : ContinuousAdd (M × N)
• continuous_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (n : ℕ) : Continuous fun a => n • a
• continuousAdd_inf 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [Add M] {t₁ t₂ : TopologicalSpace M} (h₁ : ContinuousAdd M) (h₂ : ContinuousAdd M) : ContinuousAdd M
• continuousAdd_sInf 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [Add M] {ts : Set (TopologicalSpace M)} (h : ∀ t ∈ ts, ContinuousAdd M) : ContinuousAdd M
• continuousAt_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (x : M) (n : ℕ) : ContinuousAt (fun x => n • x) x
• continuousAdd_induced 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] [TopologicalSpace N] [ContinuousAdd N] (f : F) : ContinuousAdd M
• continuousOn_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {s : Set M} (n : ℕ) : ContinuousOn (fun x => n • x) s
• ContinuousAdd.induced 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [Add α] [Add β] [AddHomClass F α β] [tβ : TopologicalSpace β] [ContinuousAdd β] (f : F) : ContinuousAdd α
• IsCompact.add 📋 Mathlib.Topology.Algebra.Monoid
  {N : Type u_4} [TopologicalSpace N] [Add N] [ContinuousAdd N] {s t : Set N} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s + t)
• Topology.IsInducing.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} {N : Type u_7} {F : Type u_8} [Add M] [Add N] [FunLike F M N] [AddHomClass F M N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousAdd N] (f : F) (hf : Topology.IsInducing ⇑f) : ContinuousAdd M
• Inseparable.add 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] {a b c d : M} (hab : Inseparable a b) (hcd : Inseparable c d) : Inseparable (a + c) (b + d)
• Specializes.add 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] {a b c d : M} (hab : a ⤳ b) (hcd : c ⤳ d) : (a + c) ⤳ (b + d)
• AddHom.isClosed_range_coe 📋 Mathlib.Topology.Algebra.Monoid
  (M₁ : Type u_6) (M₂ : Type u_7) [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] : IsClosed (Set.range DFunLike.coe)
• Continuous.fun_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} (h : Continuous f) (n : ℕ) : Continuous fun i => n • f i
• ContinuousAt.fun_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (fun i => n • f i) x
• ContinuousOn.fun_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ) : ContinuousOn (fun i => n • f i) s
• isClosed_setOfPred_map_add 📋 Mathlib.Topology.Algebra.Monoid
  (M₁ : Type u_6) (M₂ : Type u_7) [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] : IsClosed {f | ∀ (x y : M₁), f (x + y) = f x + f y}
• isClosed_setOf_map_add 📋 Mathlib.Topology.Algebra.Monoid
  (M₁ : Type u_6) (M₂ : Type u_7) [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] : IsClosed {f | ∀ (x y : M₁), f (x + y) = f x + f y}
• Inseparable.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] {a b : M} (h : Inseparable a b) (n : ℕ) : Inseparable (n • a) (n • b)
• Specializes.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] {a b : M} (h : a ⤳ b) (n : ℕ) : (n • a) ⤳ (n • b)
• le_nhds_add 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] (a b : M) : nhds a + nhds b ≤ nhds (a + b)
• nhds_add_nhds_zero 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} [AddZeroClass M] [TopologicalSpace M] [ContinuousAdd M] (a : M) : nhds a + nhds 0 = nhds a
• nhds_zero_add_nhds 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_6} [AddZeroClass M] [TopologicalSpace M] [ContinuousAdd M] (a : M) : nhds 0 + nhds a = nhds a
• Continuous.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} (h : Continuous f) (n : ℕ) : Continuous (n • f)
• ContinuousWithinAt.fun_nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x) (n : ℕ) : ContinuousWithinAt (fun i => n • f i) s x
• ContinuousAt.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (n • f) x
• continuous_multiset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Multiset ι) : (∀ i ∈ s, Continuous (f i)) → Continuous fun a => (Multiset.map (fun i => f i a) s).sum
• ContinuousOn.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ) : ContinuousOn (n • f) s
• tendsto_add 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [Add M] [ContinuousAdd M] {a b : M} : Filter.Tendsto (fun p => p.1 + p.2) (nhds (a, b)) (nhds (a + b))
• continuous_finsetSum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι) : (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∑ i ∈ s, f i a
• continuous_finset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι) : (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∑ i ∈ s, f i a
• AddMonoidHom.isClosed_range_coe 📋 Mathlib.Topology.Algebra.Monoid
  (M₁ : Type u_6) (M₂ : Type u_7) [TopologicalSpace M₂] [T2Space M₂] [AddZeroClass M₁] [AddZeroClass M₂] [ContinuousAdd M₂] : IsClosed (Set.range DFunLike.coe)
• ContinuousWithinAt.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x) (n : ℕ) : ContinuousWithinAt (n • f) s x
• continuousOn_multiset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Multiset ι) {t : Set X} : (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (Multiset.map (fun i => f i a) s).sum) t
• continuousOn_finsetSum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι) {t : Set X} : (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∑ i ∈ s, f i a) t
• continuousOn_finset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (s : Finset ι) {t : Set X} : (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => ∑ i ∈ s, f i a) t
• continuous_finsum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (hc : ∀ (i : ι), Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i)) : Continuous fun x => ∑ᶠ (i : ι), f i x
• addHomOfMemClosureRangeCoe 📋 Mathlib.Topology.Algebra.Monoid
  {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddHomClass F M₁ M₂] (f : M₁ → M₂) (hf : f ∈ closure (Set.range fun f x => f x)) : M₁ →ₙ+ M₂
• addHomOfTendsto 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddHomClass F M₁ M₂] {l : Filter α} (f : M₁ → M₂) (g : α → F) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : M₁ →ₙ+ M₂
• Filter.Tendsto.addUnits 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {N : Type u_4} [TopologicalSpace N] [AddMonoid N] [ContinuousAdd N] [T2Space N] {f : ι → AddUnits N} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (h₁ : Filter.Tendsto (fun x => ↑(f x)) l (nhds r₁)) (h₂ : Filter.Tendsto (fun x => ↑(-f x)) l (nhds r₂)) : AddUnits N
• Filter.Tendsto.nsmul 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {l : Filter α} {f : α → M} {x : M} (hf : Filter.Tendsto f l (nhds x)) (n : ℕ) : Filter.Tendsto (fun x => n • f x) l (nhds (n • x))
• continuous_list_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i)) : Continuous fun a => (List.map (fun i => f i a) l).sum
• continuousOn_list_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → X → M} (l : List ι) {t : Set X} (h : ∀ i ∈ l, ContinuousOn (f i) t) : ContinuousOn (fun a => (List.map (fun i => f i a) l).sum) t
• tendsto_multiset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) : (∀ i ∈ s, Filter.Tendsto (f i) x (nhds (a i))) → Filter.Tendsto (fun b => (Multiset.map (fun c => f c b) s).sum) x (nhds (Multiset.map a s).sum)
• continuous_finsum_cond 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} {p : ι → Prop} (hc : ∀ (i : ι), p i → Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i)) : Continuous fun x => ∑ᶠ (i : ι) (_ : p i), f i x
• AddSubmonoid.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (S : AddSubmonoid M) : ContinuousAdd ↥S
• Filter.HasBasis.add_self 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} [TopologicalSpace M] [AddZeroClass M] [ContinuousAdd M] {p : ι → Prop} {s : ι → Set M} (h : (nhds 0).HasBasis p s) : (nhds 0).HasBasis p fun i => s i + s i
• tendsto_finsetSum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι) : (∀ i ∈ s, Filter.Tendsto (f i) x (nhds (a i))) → Filter.Tendsto (fun b => ∑ c ∈ s, f c b) x (nhds (∑ c ∈ s, a c))
• tendsto_finset_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι) : (∀ i ∈ s, Filter.Tendsto (f i) x (nhds (a i))) → Filter.Tendsto (fun b => ∑ c ∈ s, f c b) x (nhds (∑ c ∈ s, a c))
• addMonoidHomOfMemClosureRangeCoe 📋 Mathlib.Topology.Algebra.Monoid
  {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [AddZeroClass M₁] [AddZeroClass M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddMonoidHomClass F M₁ M₂] (f : M₁ → M₂) (hf : f ∈ closure (Set.range fun f x => f x)) : M₁ →+ M₂
• addMonoidHomOfTendsto 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [AddZeroClass M₁] [AddZeroClass M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddMonoidHomClass F M₁ M₂] {l : Filter α} (f : M₁ → M₂) (g : α → F) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : M₁ →+ M₂
• Filter.Tendsto.val_addUnits 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {N : Type u_4} [TopologicalSpace N] [AddMonoid N] [ContinuousAdd N] [T2Space N] {f : ι → AddUnits N} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (h₁ : Filter.Tendsto (fun x => ↑(f x)) l (nhds r₁)) (h₂ : Filter.Tendsto (fun x => ↑(-f x)) l (nhds r₂)) : ↑(h₁.addUnits h₂) = r₁
• exists_open_nhds_zero_add_subset 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddZeroClass M] [ContinuousAdd M] {U : Set M} (hU : U ∈ nhds 0) : ∃ V, IsOpen V ∧ 0 ∈ V ∧ V + V ⊆ U
• Filter.Tendsto.val_neg_addUnits 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {N : Type u_4} [TopologicalSpace N] [AddMonoid N] [ContinuousAdd N] [T2Space N] {f : ι → AddUnits N} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (h₁ : Filter.Tendsto (fun x => ↑(f x)) l (nhds r₁)) (h₂ : Filter.Tendsto (fun x => ↑(-f x)) l (nhds r₂)) : ↑(-h₁.addUnits h₂) = r₂
• addHomOfMemClosureRangeCoe_apply 📋 Mathlib.Topology.Algebra.Monoid
  {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddHomClass F M₁ M₂] (f : M₁ → M₂) (hf : f ∈ closure (Set.range fun f x => f x)) : ⇑(addHomOfMemClosureRangeCoe f hf) = f
• addHomOfTendsto_apply 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [Add M₁] [Add M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddHomClass F M₁ M₂] {l : Filter α} (f : M₁ → M₂) (g : α → F) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : ⇑(addHomOfTendsto f g h) = f
• tendsto_list_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {α : Type u_2} {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : ι → α → M} {x : Filter α} {a : ι → M} (l : List ι) : (∀ i ∈ l, Filter.Tendsto (f i) x (nhds (a i))) → Filter.Tendsto (fun b => (List.map (fun c => f c b) l).sum) x (nhds (List.map a l).sum)
• exists_nhds_zero_half 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddZeroClass M] [ContinuousAdd M] {s : Set M} (hs : s ∈ nhds 0) : ∃ V ∈ nhds 0, ∀ v ∈ V, ∀ w ∈ V, v + w ∈ s
• exists_open_nhds_zero_half 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddZeroClass M] [ContinuousAdd M] {s : Set M} (hs : s ∈ nhds 0) : ∃ V, IsOpen V ∧ 0 ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v + w ∈ s
• addMonoidHomOfMemClosureRangeCoe_apply 📋 Mathlib.Topology.Algebra.Monoid
  {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [AddZeroClass M₁] [AddZeroClass M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddMonoidHomClass F M₁ M₂] (f : M₁ → M₂) (hf : f ∈ closure (Set.range fun f x => f x)) : ⇑(addMonoidHomOfMemClosureRangeCoe f hf) = f
• AddSubsemigroup.continuousAdd 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddSemigroup M] [ContinuousAdd M] (S : AddSubsemigroup M) : ContinuousAdd ↥S
• addMonoidHomOfTendsto_apply 📋 Mathlib.Topology.Algebra.Monoid
  {α : Type u_2} {M₁ : Type u_6} {M₂ : Type u_7} [TopologicalSpace M₂] [T2Space M₂] [AddZeroClass M₁] [AddZeroClass M₂] [ContinuousAdd M₂] {F : Type u_8} [FunLike F M₁ M₂] [AddMonoidHomClass F M₁ M₂] {l : Filter α} (f : M₁ → M₂) (g : α → F) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : ⇑(addMonoidHomOfTendsto f g h) = f
• continuousAdd_of_comm_of_nhds_zero 📋 Mathlib.Topology.Algebra.Monoid
  (M : Type u) [AddCommMonoid M] [TopologicalSpace M] (hadd : Filter.Tendsto (Function.uncurry fun x1 x2 => x1 + x2) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hleft : ∀ (x₀ : M), nhds x₀ = Filter.map (fun x => x₀ + x) (nhds 0)) : ContinuousAdd M
• exists_nhds_zero_quarter 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {u : Set M} (hu : u ∈ nhds 0) : ∃ V ∈ nhds 0, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v + w + s + t ∈ u
• ContinuousAdd.of_nhds_zero 📋 Mathlib.Topology.Algebra.Monoid
  {M : Type u} [AddMonoid M] [TopologicalSpace M] (hadd : Filter.Tendsto (Function.uncurry fun x1 x2 => x1 + x2) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hleft : ∀ (x₀ : M), nhds x₀ = Filter.map (fun x => x₀ + x) (nhds 0)) (hright : ∀ (x₀ : M), nhds x₀ = Filter.map (fun x => x + x₀) (nhds 0)) : ContinuousAdd M
• Inseparable.zsmul 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ContinuousAdd G] [ContinuousNeg G] {x y : G} (h : Inseparable x y) (m : ℤ) : Inseparable (m • x) (m • y)
• Specializes.zsmul 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
  {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ContinuousAdd G] [ContinuousNeg G] {x y : G} (h : x ⤳ y) (m : ℤ) : (m • x) ⤳ (m • y)
• IsTopologicalAddGroup.continuous_addConj' 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [Neg G] [Add G] [ContinuousAdd G] [ContinuousNeg G] (h : G) : Continuous fun g => g + h + -g
• IsTopologicalAddGroup.t1Space 📋 Mathlib.Topology.Algebra.Group.Basic
  (G : Type u_1) [TopologicalSpace G] [AddGroup G] [ContinuousAdd G] (h : IsClosed {0}) : T1Space G
• IsTopologicalAddGroup.continuous_addConj_prod 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [Neg G] [Add G] [ContinuousAdd G] [ContinuousNeg G] : Continuous fun g => g.1 + g.2 + -g.1
• compact_open_separated_add_left 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [AddZeroClass G] [ContinuousAdd G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) : ∃ V ∈ nhds 0, V + K ⊆ U
• compact_open_separated_add_right 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type u_1} [TopologicalSpace G] [AddZeroClass G] [ContinuousAdd G] {K U : Set G} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) : ∃ V ∈ nhds 0, K + V ⊆ U
• AddMonoidHom.isOpenQuotientMap_of_isQuotientMap 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {A : Type u_2} [AddGroup A] [TopologicalSpace A] [ContinuousAdd A] {B : Type u_3} [AddGroup B] [TopologicalSpace B] {F : Type u_4} [FunLike F A B] [AddMonoidHomClass F A B] {φ : F} (hφ : Topology.IsQuotientMap ⇑φ) : IsOpenQuotientMap ⇑φ
• AddMonoidHom.isOpenQuotientMap_iff_isQuotientMap 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {A : Type u_2} [AddGroup A] [TopologicalSpace A] [ContinuousAdd A] {B : Type u_3} [AddGroup B] [TopologicalSpace B] {F : Type u_4} [FunLike F A B] [AddMonoidHomClass F A B] {φ : F} : IsOpenQuotientMap ⇑φ ↔ Topology.IsQuotientMap ⇑φ
• continuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {M : Type u_2} {hom : Type u_3} [AddZeroClass M] [TopologicalSpace M] [ContinuousAdd M] [FunLike hom G M] [AddMonoidHomClass hom G M] (f : hom) (hf : ContinuousAt (⇑f) 0) : Continuous ⇑f
• continuous_of_tendsto_nhds_zero 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {M : Type u_2} {hom : Type u_3} [AddZeroClass M] [TopologicalSpace M] [ContinuousAdd M] [FunLike hom G M] [AddMonoidHomClass hom G M] (f : hom) (hf : Filter.Tendsto (⇑f) (nhds 0) (nhds 0)) : Continuous ⇑f
• continuous_of_continuousAt_zero₂ 📋 Mathlib.Topology.Algebra.Group.Neighborhood
  {G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {H : Type u_2} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] [AddGroup H] [TopologicalSpace H] [IsTopologicalAddGroup H] (f : G →+ H →+ M) (hf : ContinuousAt (fun x => (f x.1) x.2) (0, 0)) (hl : ∀ (x : G), ContinuousAt (⇑(f x)) 0) (hr : ∀ (y : H), ContinuousAt (fun x => (f x) y) 0) : Continuous fun x => (f x.1) x.2
• add_mem_connectedComponent_zero 📋 Mathlib.Topology.Algebra.Group.Subgroup
  {G : Type u_3} [TopologicalSpace G] [AddZeroClass G] [ContinuousAdd G] {g h : G} (hg : g ∈ connectedComponent 0) (hh : h ∈ connectedComponent 0) : g + h ∈ connectedComponent 0
• QuotientAddGroup.instContinuousVAdd 📋 Mathlib.Topology.Algebra.Group.Quotient
  {G : Type u_2} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] {N : AddSubgroup G} : ContinuousVAdd G (G ⧸ N)
• ContinuousAddMonoidHom.instAddCommMonoid 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {A : Type u_2} {E : Type u_6} [AddMonoid A] [TopologicalSpace A] [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] : AddCommMonoid (A →ₜ+ E)
• ContinuousAddMonoidHom.add 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  (E : Type u_6) [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] : E × E →ₜ+ E
• ContinuousAddMonoidHom.coprod 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] (f : A →ₜ+ E) (g : B →ₜ+ E) : A × B →ₜ+ E
• ContinuousAddMonoidHom.add_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  (E : Type u_6) [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] (a✝ : E × E) : (ContinuousAddMonoidHom.add E) a✝ = a✝.1 + a✝.2
• ContinuousAddMonoidHom.nsmul_apply 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {A : Type u_2} {E : Type u_6} [AddMonoid A] [TopologicalSpace A] [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] (f : A →ₜ+ E) (n : ℕ) (a : A) : (n • f) a = n • f a
• ContinuousAddMonoidHom.coprod_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] (f : A →ₜ+ E) (g : B →ₜ+ E) (x : A × B) : (f.coprod g) x = f x.1 + g x.2
• ContinuousAddMonoidHom.add_apply 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  {A : Type u_2} {E : Type u_6} [AddMonoid A] [TopologicalSpace A] [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] (f g : A →ₜ+ E) (a : A) : (f + g) a = f a + g a
• IsSemitopologicalSemiring.toContinuousAdd 📋 Mathlib.Topology.Algebra.Ring.Basic
  {R : Type u_2} {inst✝ : TopologicalSpace R} {inst✝¹ : NonUnitalNonAssocSemiring R} [self : IsSemitopologicalSemiring R] : ContinuousAdd R
• IsTopologicalSemiring.toContinuousAdd 📋 Mathlib.Topology.Algebra.Ring.Basic
  {R : Type u_1} {inst✝ : TopologicalSpace R} {inst✝¹ : NonUnitalNonAssocSemiring R} [self : IsTopologicalSemiring R] : ContinuousAdd R
• IsSemitopologicalSemiring.mk 📋 Mathlib.Topology.Algebra.Ring.Basic
  {R : Type u_2} [TopologicalSpace R] [NonUnitalNonAssocSemiring R] [toContinuousAdd : ContinuousAdd R] [toSeparatelyContinuousMul : SeparatelyContinuousMul R] : IsSemitopologicalSemiring R
• IsTopologicalSemiring.mk 📋 Mathlib.Topology.Algebra.Ring.Basic
  {R : Type u_1} [TopologicalSpace R] [NonUnitalNonAssocSemiring R] [toContinuousAdd : ContinuousAdd R] [toContinuousMul : ContinuousMul R] : IsTopologicalSemiring R
• instContinuousAddMulOpposite 📋 Mathlib.Topology.Algebra.Ring.Basic
  {R : Type u_1} [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousAdd R] : ContinuousAdd Rᵐᵒᵖ
• Submodule.topologicalClosure 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : Submodule R M
• Submodule.isClosed_topologicalClosure 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : IsClosed ↑s.topologicalClosure
• TopologicalSpace.IsSeparable.span 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {M : Type u_2} [AddCommMonoid M] [Semiring R] [Module R M] [TopologicalSpace M] [TopologicalSpace R] [TopologicalSpace.SeparableSpace R] [ContinuousAdd M] [ContinuousSMul R M] {s : Set M} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable ↑(Submodule.span R s)
• Submodule.closure_subset_topologicalClosure_span 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Set M) : closure s ⊆ ↑(Submodule.span R s).topologicalClosure
• IsClosed.submodule_topologicalClosure_eq 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} (hs : IsClosed ↑s) : s.topologicalClosure = s
• Submodule.le_topologicalClosure 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : s ≤ s.topologicalClosure
• Submodule.topologicalClosure_coe 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : ↑s.topologicalClosure = closure ↑s
• Submodule.dense_iff_topologicalClosure_eq_top 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} : Dense ↑s ↔ s.topologicalClosure = ⊤
• Submodule.isClosed_or_dense_of_isCoatom 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) (hs : IsCoatom s) : IsClosed ↑s ∨ Dense ↑s
• LinearMap.isClosed_range_coe 📋 Mathlib.Topology.Algebra.Module.Basic
  (M₁ : Type u_1) (M₂ : Type u_2) {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] (σ : R →+* S) : IsClosed (Set.range DFunLike.coe)
• Submodule.topologicalClosure_mono 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s t : Submodule R M} (h : s ≤ t) : s.topologicalClosure ≤ t.topologicalClosure
• Submodule.topologicalClosure.completeSpace 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} [Semiring R] {M' : Type u_1} [AddCommMonoid M'] [Module R M'] [UniformSpace M'] [ContinuousAdd M'] [ContinuousConstSMul R M'] [CompleteSpace M'] (U : Submodule R M') : CompleteSpace ↥U.topologicalClosure
• Submodule.topologicalClosure_minimal 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) {t : Submodule R M} (h : s ≤ t) (ht : IsClosed ↑t) : s.topologicalClosure ≤ t
• linearMapOfMemClosureRangeCoe 📋 Mathlib.Topology.Algebra.Module.Basic
  {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁ → M₂) (hf : f ∈ closure (Set.range DFunLike.coe)) : M₁ →ₛₗ[σ] M₂
• Module.punctured_nhds_neBot 📋 Mathlib.Topology.Algebra.Module.Basic
  (R : Type u_1) (M : Type u_2) [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [IsDomain R] [Nontrivial M] [(nhdsWithin 0 {0}ᶜ).NeBot] [Module.IsTorsionFree R M] (x : M) : (nhdsWithin x {x}ᶜ).NeBot
• linearMapOfTendsto 📋 Mathlib.Topology.Algebra.Module.Basic
  {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : M₁ →ₛₗ[σ] M₂
• LinearMap.continuous_on_pi 📋 Mathlib.Topology.Algebra.Module.Basic
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Finite ι] [Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous ⇑f
• Submodule.eq_top_of_nonempty_interior' 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [(nhdsWithin 0 {x | IsUnit x}).NeBot] (s : Submodule R M) (hs : (interior ↑s).Nonempty) : s = ⊤
• Submodule.isOpenMap_mkQ 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] (S : Submodule R M) [ContinuousAdd M] : IsOpenMap ⇑S.mkQ
• Submodule.isOpenQuotientMap_mkQ 📋 Mathlib.Topology.Algebra.Module.Basic
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] (S : Submodule R M) [ContinuousAdd M] : IsOpenQuotientMap ⇑S.mkQ
• linearMapOfTendsto_apply 📋 Mathlib.Topology.Algebra.Module.Basic
  {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (nhds f)) : ⇑(linearMapOfTendsto f g h) = f
• linearMapOfMemClosureRangeCoe_apply 📋 Mathlib.Topology.Algebra.Module.Basic
  {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁ → M₂) (hf : f ∈ closure (Set.range DFunLike.coe)) : ⇑(linearMapOfMemClosureRangeCoe f hf) = (↑(addMonoidHomOfMemClosureRangeCoe f hf)).toFun
• Submodule.topologicalClosure_iSup_map_single 📋 Mathlib.Topology.Algebra.Module.Basic
  {ι : Type u_1} {R : Type u_2} {M : ι → Type u_3} [Semiring R] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [(i : ι) → TopologicalSpace (M i)] [DecidableEq ι] [∀ (i : ι), ContinuousAdd (M i)] [∀ (i : ι), ContinuousConstSMul R (M i)] (s : (i : ι) → Submodule R (M i)) : (⨆ i, Submodule.map (LinearMap.single R M i) (s i)).topologicalClosure = Submodule.pi Set.univ fun i => (s i).topologicalClosure
• ContinuousLinearMap.instNatCast 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : NatCast (M₁ →L[R₁] M₁)
• ContinuousLinearMap.semiring 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : Semiring (M₁ →L[R₁] M₁)
• ContinuousLinearMap.applyModule 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : Module (M₁ →L[R₁] M₁) M₁
• ContinuousLinearMap.add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : Add (M₁ →SL[σ₁₂] M₂)
• ContinuousLinearMap.addCommMonoid 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : AddCommMonoid (M₁ →SL[σ₁₂] M₂)
• ContinuousLinearMap.instAddMonoid 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : AddMonoid (M₁ →SL[σ₁₂] M₂)
• ContinuousLinearMap.instIsNatCastApply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : IsNatCastApply (M₁ →L[R₁] M₁) M₁
• ContinuousLinearMap.instIsAddApply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : IsAddApply (M₁ →SL[σ₁₂] M₂) M₁ M₂
• ContinuousLinearMap.toLinearMapRingHom 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁
• ContinuousLinearMap.natCast_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (n : ℕ) (m : M₁) : ↑n m = n • m
• ContinuousLinearMap.ofNat_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (n : ℕ) [n.AtLeastTwo] (m : M₁) : (OfNat.ofNat n) m = OfNat.ofNat n • m
• ContinuousLinearMap.coe_sum 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ↑(∑ d ∈ t, f d) = ∑ d ∈ t, ↑(f d)
• ContinuousLinearMap.toLinearMap_sum 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ↑(∑ d ∈ t, f d) = ∑ d ∈ t, ↑(f d)
• ContinuousLinearMap.applyFaithfulSMul 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : FaithfulSMul (M₁ →L[R₁] M₁) M₁
• ContinuousLinearMap.continuousConstSMul_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : ContinuousConstSMul (M₁ →L[R₁] M₁) M₁
• ContinuousLinearMap.distribMulAction 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {R₂ : Type u_2} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribMulAction S₃ M₂] [ContinuousConstSMul S₃ M₂] [SMulCommClass R₂ S₃ M₂] [ContinuousAdd M₂] : DistribMulAction S₃ (M →SL[σ₁₂] M₂)
• ContinuousLinearMap.module 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₃] [Semiring S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [Module S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] {σ₁₃ : R →+* R₃} [ContinuousAdd M₃] : Module S₃ (M →SL[σ₁₃] M₃)
• ContinuousLinearMap.applySMulCommClass 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁
• ContinuousLinearMap.applySMulCommClass' 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁
• ContinuousLinearMap.toSpanSingleton_add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] (x y : M₁) : ContinuousLinearMap.toSpanSingleton R₁ (x + y) = ContinuousLinearMap.toSpanSingleton R₁ x + ContinuousLinearMap.toSpanSingleton R₁ y
• ContinuousLinearMap.finsetSum_comp 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {ι : Type u_9} {s : Finset ι} [ContinuousAdd M₃] (g : ι → M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : (∑ i ∈ s, g i) ∘SL f = ∑ i ∈ s, g i ∘SL f
• ContinuousLinearMap.finset_sum_comp 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {ι : Type u_9} {s : Finset ι} [ContinuousAdd M₃] (g : ι → M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : (∑ i ∈ s, g i) ∘SL f = ∑ i ∈ s, g i ∘SL f
• ContinuousLinearMap.comp_finsetSum 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {ι : Type u_9} {s : Finset ι} [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f : ι → M₁ →SL[σ₁₂] M₂) : g ∘SL ∑ i ∈ s, f i = ∑ i ∈ s, g ∘SL f i
• ContinuousLinearMap.comp_finset_sum 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {ι : Type u_9} {s : Finset ι} [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f : ι → M₁ →SL[σ₁₂] M₂) : g ∘SL ∑ i ∈ s, f i = ∑ i ∈ s, g ∘SL f i
• ContinuousLinearMap.coeLMₛₗ 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₃] [Semiring S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [Module S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] (σ₁₃ : R →+* R₃) [ContinuousAdd M₃] : (M →SL[σ₁₃] M₃) →ₗ[S₃] M →ₛₗ[σ₁₃] M₃
• ContinuousLinearMap.coe_add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f g : M₁ →SL[σ₁₂] M₂) : ↑(f + g) = ↑f + ↑g
• ContinuousLinearMap.toLinearMap_add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f g : M₁ →SL[σ₁₂] M₂) : ↑(f + g) = ↑f + ↑g
• ContinuousLinearMap.smul_def 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a
• ContinuousLinearMap.coeLM 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} (S : Type u_4) [Semiring R] [Semiring S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [Module S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [ContinuousAdd N₃] : (M →L[R] N₃) →ₗ[S] M →ₗ[R] N₃
• ContinuousLinearMap.isOpenMap_of_ne_zero 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {M : Type u_2} [TopologicalSpace R] [DivisionRing R] [ContinuousSub R] [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] (f : StrongDual R M) (hf : f ≠ 0) : IsOpenMap ⇑f
• topDualPairing 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (𝕜 : Type u_1) (E : Type u_2) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] : (E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜
• ContinuousLinearMap.toSpanSingletonLE 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R : Type u_1) (S : Type u_2) (M : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousConstSMul S M] [TopologicalSpace R] [ContinuousSMul R M] : M ≃ₗ[S] R →L[R] M
• ContinuousLinearMap.toLinearMapRingHom_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (self : M₁ →L[R₁] M₁) : ContinuousLinearMap.toLinearMapRingHom self = ↑self
• ContinuousLinearMap.lcomp 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {U : Type u_2} {V : Type u_3} (W : Type u_4) [CommSemiring R] [AddCommMonoid U] [Module R U] [TopologicalSpace U] [AddCommMonoid V] [Module R V] [TopologicalSpace V] [AddCommMonoid W] [Module R W] [TopologicalSpace W] [ContinuousAdd W] [ContinuousConstSMul R W] (f : U →L[R] V) : (V →L[R] W) →ₗ[R] U →L[R] W
• Submodule.topologicalClosure_map 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (s : Submodule R₁ M₁) : Submodule.map (↑f) s.topologicalClosure ≤ (Submodule.map (↑f) s).topologicalClosure
• ContinuousLinearMap.add_comp 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : (g₁ + g₂) ∘SL f = g₁ ∘SL f + g₂ ∘SL f
• DenseRange.topologicalClosure_map_submodule 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : DenseRange ⇑f) {s : Submodule R₁ M₁} (hs : s.topologicalClosure = ⊤) : (Submodule.map (↑f) s).topologicalClosure = ⊤
• ContinuousLinearMap.comp_add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M₁ →SL[σ₁₂] M₂) : g ∘SL (f₁ + f₂) = g ∘SL f₁ + g ∘SL f₂
• Submodule.topologicalClosure_mem_invtSubmodule 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] {f : M₁ →L[R₁] M₁} {s : Submodule R₁ M₁} (hs : s ∈ Module.End.invtSubmodule ↑f) : s.topologicalClosure ∈ Module.End.invtSubmodule ↑f
• ContinuousLinearMap.coeLMₛₗ_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₃] [Semiring S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [Module S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] (σ₁₃ : R →+* R₃) [ContinuousAdd M₃] (self : M →SL[σ₁₃] M₃) : (ContinuousLinearMap.coeLMₛₗ σ₁₃) self = ↑self
• ContinuousLinearMap.smulRightₗ 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {S : Type u_2} {T : Type u_3} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring S] [Semiring T] [Module R S] [AddCommMonoid M₂] [Module R M₂] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [TopologicalSpace M₂] [ContinuousSMul S M₂] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousAdd M₂] [Module T M₂] [ContinuousConstSMul T M₂] [SMulCommClass R T M₂] [SMulCommClass S T M₂] (c : M →L[R] S) : M₂ →ₗ[T] M →L[R] M₂
• ContinuousLinearMap.coeLM_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} (S : Type u_4) [Semiring R] [Semiring S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [Module S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [ContinuousAdd N₃] (self : M →L[R] N₃) : (ContinuousLinearMap.coeLM S) self = ↑self
• ContinuousLinearMap.toSpanSingletonLE_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R : Type u_1) (S : Type u_2) (M : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousConstSMul S M] [TopologicalSpace R] [ContinuousSMul R M] : ⇑(ContinuousLinearMap.toSpanSingletonLE R S M) = ContinuousLinearMap.toSpanSingleton R
• ContinuousLinearMap.coe_smulRightₗ 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {S : Type u_2} {T : Type u_3} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring S] [Semiring T] [Module R S] [AddCommMonoid M₂] [Module R M₂] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [TopologicalSpace M₂] [ContinuousSMul S M₂] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousAdd M₂] [Module T M₂] [ContinuousConstSMul T M₂] [SMulCommClass R T M₂] [SMulCommClass S T M₂] (c : M →L[R] S) : ⇑c.smulRightₗ = c.smulRight
• ContinuousLinearMap.lcomp_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R : Type u_1} {U : Type u_2} {V : Type u_3} (W : Type u_4) [CommSemiring R] [AddCommMonoid U] [Module R U] [TopologicalSpace U] [AddCommMonoid V] [Module R V] [TopologicalSpace V] [AddCommMonoid W] [Module R W] [TopologicalSpace W] [ContinuousAdd W] [ContinuousConstSMul R W] (f : U →L[R] V) (l : V →L[R] W) : (ContinuousLinearMap.lcomp W f) l = l ∘SL f
• ContinuousLinearMap.toSpanSingletonLE_symm_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R : Type u_1) (S : Type u_2) (M : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousConstSMul S M] [TopologicalSpace R] [ContinuousSMul R M] : ⇑(ContinuousLinearMap.toSpanSingletonLE R S M).symm = fun f => f 1
• topDualPairing_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] (v : E →L[𝕜] 𝕜) (x : E) : ((topDualPairing 𝕜 E) v) x = v x
• ContinuousLinearMap.toContinuousAddMonoidHom_add 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f g : M₁ →SL[σ₁₂] M₂) : ↑(f + g) = ↑f + ↑g
• ContinuousLinearMap.llcomp 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R : Type u_1) (U : Type u_2) (V : Type u_3) (W : Type u_4) [CommSemiring R] [AddCommMonoid U] [Module R U] [TopologicalSpace U] [AddCommMonoid V] [Module R V] [TopologicalSpace V] [ContinuousAdd V] [ContinuousConstSMul R V] [AddCommMonoid W] [Module R W] [TopologicalSpace W] [ContinuousAdd W] [ContinuousConstSMul R W] : (U →L[R] V) →ₗ[R] (V →L[R] W) →ₗ[R] U →L[R] W
• ContinuousLinearMap.llcomp_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  (R : Type u_1) (U : Type u_2) (V : Type u_3) (W : Type u_4) [CommSemiring R] [AddCommMonoid U] [Module R U] [TopologicalSpace U] [AddCommMonoid V] [Module R V] [TopologicalSpace V] [ContinuousAdd V] [ContinuousConstSMul R V] [AddCommMonoid W] [Module R W] [TopologicalSpace W] [ContinuousAdd W] [ContinuousConstSMul R W] (l : U →L[R] V) : (ContinuousLinearMap.llcomp R U V W) l = ContinuousLinearMap.lcomp W l
• ContinuousLinearMap.coprod 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd
  {R : Type u_1} {M : Type u_3} {M₁ : Type u_5} {M₂ : Type u_6} [Semiring R] [TopologicalSpace M] [TopologicalSpace M₁] [TopologicalSpace M₂] [AddCommMonoid M] [Module R M] [ContinuousAdd M] [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) : M₁ × M₂ →L[R] M
• ContinuousLinearMap.coprod_comp_inl 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd
  {R : Type u_1} {M : Type u_3} {M₁ : Type u_5} {M₂ : Type u_6} [Semiring R] [TopologicalSpace M] [TopologicalSpace M₁] [TopologicalSpace M₂] [AddCommMonoid M] [Module R M] [ContinuousAdd M] [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) : f₁.coprod f₂ ∘SL ContinuousLinearMap.inl R M₁ M₂ = f₁
• ContinuousLinearMap.coprod_comp_inr 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd
  {R : Type u_1} {M : Type u_3} {M₁ : Type u_5} {M₂ : Type u_6} [Semiring R] [TopologicalSpace M] [TopologicalSpace M₁] [TopologicalSpace M₂] [AddCommMonoid M] [Module R M] [ContinuousAdd M] [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) : f₁.coprod f₂ ∘SL ContinuousLinearMap.inr R M₁ M₂ = f₂
• ContinuousLinearMap.coprod_inl_inr 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd
  {R : Type u_1} {M : Type u_3} {N : Type u_4} [Semiring R] [TopologicalSpace M] [TopologicalSpace N] [AddCommMonoid M] [Module R M] [ContinuousAdd M] [AddCommMonoid N] [Module R N] [ContinuousAdd N] : (ContinuousLinearMap.inl R M N).coprod (ContinuousLinearMap.inr R M N) = ContinuousLinearMap.id R (M × N)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c