fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[to_additive /-- Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`. -/]
exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ V * V ⊆ U := by
simpa only [mul_subset_iff] using exists_open_nhds_one_split hU
@[... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | exists_open_nhds_one_mul_subset | Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `V * V ⊆ U`. |
Filter.HasBasis.mul_self {p : ι → Prop} {s : ι → Set M} (h : (𝓝 1).HasBasis p s) :
(𝓝 1).HasBasis p fun i => s i * s i := by
rw [← nhds_mul_nhds_one, ← map₂_mul, ← map_uncurry_prod]
simpa only [← image_mul_prod] using h.prod_self.map _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.HasBasis.mul_self | null |
@[to_additive]
Subsemigroup.top_closure_mul_self_subset (s : Subsemigroup M) :
_root_.closure (s : Set M) * _root_.closure s ⊆ _root_.closure s :=
image2_subset_iff.2 fun _ hx _ hy =>
map_mem_closure₂ continuous_mul hx hy fun _ ha _ hb => s.mul_mem ha hb | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.top_closure_mul_self_subset | null |
@[to_additive /-- The (topological-space) closure of an additive submonoid of a space `M` with
`ContinuousAdd` is itself an additive submonoid. -/]
Subsemigroup.topologicalClosure (s : Subsemigroup M) : Subsemigroup M where
carrier := _root_.closure (s : Set M)
mul_mem' ha hb := s.top_closure_mul_self_subset ⟨_, ha... | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.topologicalClosure | The (topological-space) closure of a subsemigroup of a space `M` with `ContinuousMul` is
itself a subsemigroup. |
Subsemigroup.coe_topologicalClosure (s : Subsemigroup M) :
(s.topologicalClosure : Set M) = _root_.closure (s : Set M) := rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.coe_topologicalClosure | null |
Subsemigroup.le_topologicalClosure (s : Subsemigroup M) : s ≤ s.topologicalClosure :=
_root_.subset_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.le_topologicalClosure | null |
Subsemigroup.isClosed_topologicalClosure (s : Subsemigroup M) :
IsClosed (s.topologicalClosure : Set M) := isClosed_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.isClosed_topologicalClosure | null |
Subsemigroup.topologicalClosure_minimal (s : Subsemigroup M) {t : Subsemigroup M}
(h : s ≤ t) (ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t := closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Subsemigroup.topologicalClosure_minimal | null |
@[to_additive /-- The (topological-space) closure of an additive submonoid of a space `M` with
`ContinuousAdd` is itself an additive submonoid. -/]
Submonoid.topologicalClosure (s : Submonoid M) : Submonoid M where
carrier := _root_.closure (s : Set M)
one_mem' := _root_.subset_closure s.one_mem
mul_mem' ha hb :=... | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.topologicalClosure | If a subsemigroup of a topological semigroup is commutative, then so is its topological
closure.
See note [reducible non-instances] -/
@[to_additive /-- If a submonoid of an additive topological monoid is commutative, then so is its
topological closure.
See note [reducible non-instances] -/]
abbrev Subsemigroup.commS... |
Submonoid.coe_topologicalClosure (s : Submonoid M) :
(s.topologicalClosure : Set M) = _root_.closure (s : Set M) := rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.coe_topologicalClosure | null |
Submonoid.le_topologicalClosure (s : Submonoid M) : s ≤ s.topologicalClosure :=
_root_.subset_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.le_topologicalClosure | null |
Submonoid.isClosed_topologicalClosure (s : Submonoid M) :
IsClosed (s.topologicalClosure : Set M) := isClosed_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.isClosed_topologicalClosure | null |
Submonoid.topologicalClosure_minimal (s : Submonoid M) {t : Submonoid M} (h : s ≤ t)
(ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t := closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Submonoid.topologicalClosure_minimal | null |
Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_left b))
convert Filter.tendsto_id
ext x
simp [← mul_assoc, ha] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.tendsto_cocompact_mul_left | If a submonoid of a topological monoid is commutative, then so is its topological closure. -/
@[to_additive /-- If a submonoid of an additive topological monoid is commutative, then so is its
topological closure.
See note [reducible non-instances]. -/]
abbrev Submonoid.commMonoidTopologicalClosure [T2Space M] (s : Sub... |
Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_right b))
simp only [comp_mul_right, ha, mul_one]
exact Filter.tendsto_id | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | Filter.tendsto_cocompact_mul_right | Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. |
@[to_additive /-- The continuous map `fun y => y + x` -/]
protected mulRight (x : X) : C(X, X) :=
mk _ (continuous_mul_right x)
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | mulRight | If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[Algebra R A]` is available. -/
@[to_additive /-- If `R` acts on `A` via `A`, then continuous addition implies
continuous affine addition by constants... |
coe_mulRight (x : X) : ⇑(ContinuousMap.mulRight x) = fun y => y * x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | coe_mulRight | null |
@[to_additive /-- The continuous map `fun y => x + y` -/]
protected mulLeft (x : X) : C(X, X) :=
mk _ (continuous_mul_left x)
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | mulLeft | The continuous map `fun y => x * y` |
coe_mulLeft (x : X) : ⇑(ContinuousMap.mulLeft x) = fun y => x * y :=
rfl | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Algebra.Group.Submonoid.Basic",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Monoid.lean | coe_mulLeft | null |
ContinuousSMul (M X : Type*) [SMul M X] [TopologicalSpace M] [TopologicalSpace X] :
Prop where
/-- The scalar multiplication `(•)` is continuous. -/
continuous_smul : Continuous fun p : M × X => p.1 • p.2
export ContinuousSMul (continuous_smul) | class | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousSMul | Class `ContinuousSMul M X` says that the scalar multiplication `(•) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of multiplicative actions,
including (semi)modules and algebras. |
ContinuousVAdd (M X : Type*) [VAdd M X] [TopologicalSpace M] [TopologicalSpace X] :
Prop where
/-- The additive action `(+ᵥ)` is continuous. -/
continuous_vadd : Continuous fun p : M × X => p.1 +ᵥ p.2
export ContinuousVAdd (continuous_vadd)
attribute [to_additive] ContinuousSMul
attribute [continuity, fun_prop]... | class | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousVAdd | Class `ContinuousVAdd M X` says that the additive action `(+ᵥ) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of additive actions,
including (semi)modules and algebras. |
IsScalarTower.continuousSMul {M : Type*} (N : Type*) {α : Type*} [Monoid N] [SMul M N]
[MulAction N α] [SMul M α] [IsScalarTower M N α] [TopologicalSpace M] [TopologicalSpace N]
[TopologicalSpace α] [ContinuousSMul M N] [ContinuousSMul N α] : ContinuousSMul M α :=
{ continuous_smul := by
suffices Contin... | lemma | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | IsScalarTower.continuousSMul | null |
ContinuousSMul.induced {R : Type*} {α : Type*} {β : Type*} {F : Type*} [FunLike F α β]
[Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β]
[TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β]
(f : F) : @ContinuousSMul R α _ _ (tβ.induced f) := b... | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousSMul.induced | null |
Filter.Tendsto.smul {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X}
(hf : Tendsto f l (𝓝 c)) (hg : Tendsto g l (𝓝 a)) :
Tendsto (fun x => f x • g x) l (𝓝 <| c • a) :=
(continuous_smul.tendsto _).comp (hf.prodMk_nhds hg)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Filter.Tendsto.smul | null |
Filter.Tendsto.smul_const {f : α → M} {l : Filter α} {c : M} (hf : Tendsto f l (𝓝 c))
(a : X) : Tendsto (fun x => f x • a) l (𝓝 (c • a)) :=
hf.smul tendsto_const_nhds
variable {f : Y → M} {g : Y → X} {b : Y} {s : Set Y}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Filter.Tendsto.smul_const | null |
ContinuousWithinAt.smul (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b) :
ContinuousWithinAt (fun x => f x • g x) s b :=
Filter.Tendsto.smul hf hg
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousWithinAt.smul | null |
ContinuousAt.smul (hf : ContinuousAt f b) (hg : ContinuousAt g b) :
ContinuousAt (fun x => f x • g x) b :=
Filter.Tendsto.smul hf hg
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousAt.smul | null |
ContinuousOn.smul (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => f x • g x) s := fun x hx => (hf x hx).smul (hg x hx)
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousOn.smul | null |
Continuous.smul (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x • g x :=
continuous_smul.comp (hf.prodMk hg) | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Continuous.smul | null |
@[to_additive /-- If an additive action is central, then its right action is continuous when its
left action is. -/]
ContinuousSMul.op [SMul Mᵐᵒᵖ X] [IsCentralScalar M X] : ContinuousSMul Mᵐᵒᵖ X :=
⟨by
suffices Continuous fun p : M × X => MulOpposite.op p.fst • p.snd from
this.comp (MulOpposite.continuous_u... | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | ContinuousSMul.op | If a scalar action is central, then its right action is continuous when its left action is. |
MulOpposite.continuousSMul : ContinuousSMul M Xᵐᵒᵖ :=
⟨MulOpposite.continuous_op.comp <|
continuous_smul.comp <| continuous_id.prodMap MulOpposite.continuous_unop⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | MulOpposite.continuousSMul | null |
protected Specializes.smul {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y) :
(a • x) ⤳ (b • y) :=
(h₁.prod h₂).map continuous_smul
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Specializes.smul | null |
protected Inseparable.smul {a b : M} {x y : X} (h₁ : Inseparable a b)
(h₂ : Inseparable x y) : Inseparable (a • x) (b • y) :=
(h₁.prod h₂).map continuous_smul
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Inseparable.smul | null |
IsCompact.smul_set {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u) :
IsCompact (k • u) := by
rw [← Set.image_smul_prod]
exact IsCompact.image (hk.prod hu) continuous_smul
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | IsCompact.smul_set | null |
smul_set_closure_subset (K : Set M) (L : Set X) :
closure K • closure L ⊆ closure (K • L) :=
Set.smul_subset_iff.2 fun _x hx _y hy ↦ map_mem_closure₂ continuous_smul hx hy fun _a ha _b hb ↦
Set.smul_mem_smul ha hb | lemma | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | smul_set_closure_subset | null |
@[to_additive
/-- Suppose that `N` additively acts on `X` and `M` continuously additively acts on `Y`.
Suppose that `g : Y → X` is an additive action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c +ᵥ x) = f c +ᵥ g x`.
Then the action of `N` on `X` is continuous as ... | lemma | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Topology.IsInducing.continuousSMul | Suppose that `N` acts on `X` and `M` continuously acts on `Y`.
Suppose that `g : Y → X` is an action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c • x) = f c • g x`.
Then the action of `N` on `X` is continuous as well.
In many cases, `f = id` so that `g` is an acti... |
SMulMemClass.continuousSMul {S : Type*} [SetLike S X] [SMulMemClass S M X] (s : S) :
ContinuousSMul M s :=
IsInducing.subtypeVal.continuousSMul continuous_id rfl | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | SMulMemClass.continuousSMul | null |
@[to_additive]
Units.continuousSMul : ContinuousSMul Mˣ X :=
IsInducing.id.continuousSMul Units.continuous_val rfl | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Units.continuousSMul | null |
@[to_additive]
MulAction.continuousSMul_compHom
{N : Type*} [TopologicalSpace N] [Monoid N] {f : N →* M} (hf : Continuous f) :
letI : MulAction N X := MulAction.compHom _ f
ContinuousSMul N X := by
let _ : MulAction N X := MulAction.compHom _ f
exact ⟨(hf.comp continuous_fst).smul continuous_snd⟩
@[to_a... | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | MulAction.continuousSMul_compHom | If an action is continuous, then composing this action with a continuous homomorphism gives
again a continuous action. |
Submonoid.continuousSMul {S : Submonoid M} : ContinuousSMul S X :=
IsInducing.id.continuousSMul continuous_subtype_val rfl | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Submonoid.continuousSMul | null |
@[to_additive]
Subgroup.continuousSMul {S : Subgroup M} : ContinuousSMul S X :=
S.toSubmonoid.continuousSMul
variable (M) | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Subgroup.continuousSMul | null |
stabilizer_isOpen [DiscreteTopology X] (x : X) : IsOpen (MulAction.stabilizer M x : Set M) :=
IsOpen.preimage (f := fun g ↦ g • x) (by fun_prop) (isOpen_discrete {x}) | lemma | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | stabilizer_isOpen | The stabilizer of a continuous group action on a discrete space is an open subgroup. |
@[to_additive]
Prod.continuousSMul [SMul M X] [SMul M Y] [ContinuousSMul M X] [ContinuousSMul M Y] :
ContinuousSMul M (X × Y) :=
⟨(continuous_fst.smul (continuous_fst.comp continuous_snd)).prodMk
(continuous_fst.smul (continuous_snd.comp continuous_snd))⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | Prod.continuousSMul | null |
@[to_additive]
continuousSMul_sInf {ts : Set (TopologicalSpace X)}
(h : ∀ t ∈ ts, @ContinuousSMul M X _ _ t) : @ContinuousSMul M X _ _ (sInf ts) :=
let _ := sInf ts
{ continuous_smul := by
rw [← (sInf_singleton (a := ‹TopologicalSpace M›):)]
exact
continuous_sInf_rng.2 fun t ht =>
... | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | continuousSMul_sInf | null |
continuousSMul_iInf {ts' : ι → TopologicalSpace X}
(h : ∀ i, @ContinuousSMul M X _ _ (ts' i)) : @ContinuousSMul M X _ _ (⨅ i, ts' i) :=
continuousSMul_sInf <| Set.forall_mem_range.mpr h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | continuousSMul_iInf | null |
continuousSMul_inf {t₁ t₂ : TopologicalSpace X} [@ContinuousSMul M X _ _ t₁]
[@ContinuousSMul M X _ _ t₂] : @ContinuousSMul M X _ _ (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]
refine continuousSMul_iInf fun b => ?_
cases b <;> assumption | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | continuousSMul_inf | null |
protected AddTorsor.connectedSpace : ConnectedSpace P :=
{ isPreconnected_univ := by
convert
isPreconnected_univ.image (Equiv.vaddConst (Classical.arbitrary P) : G → P)
(continuous_id.vadd continuous_const).continuousOn
rw [Set.image_univ, Equiv.range_eq_univ]
toNonempty := inferInst... | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Connected.Basic"
] | Mathlib/Topology/Algebra/MulAction.lean | AddTorsor.connectedSpace | An `AddTorsor` for a connected space is a connected space. This is not an instance because
it loops for a group as a torsor over itself. |
MvPolynomial.continuous_eval : Continuous fun x ↦ eval x p := by
continuity | theorem | Topology | [
"Mathlib.Algebra.MvPolynomial.Eval",
"Mathlib.Topology.Algebra.Ring.Basic"
] | Mathlib/Topology/Algebra/MvPolynomial.lean | MvPolynomial.continuous_eval | null |
instIsTopologicalSemiring (s : NonUnitalSubalgebra R A) : IsTopologicalSemiring s :=
s.toNonUnitalSubsemiring.instIsTopologicalSemiring | instance | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | instIsTopologicalSemiring | null |
topologicalClosure (s : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R A :=
{ s.toNonUnitalSubsemiring.topologicalClosure, s.toSubmodule.topologicalClosure with
carrier := _root_.closure (s : Set A) } | def | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | topologicalClosure | The (topological) closure of a non-unital subalgebra of a non-unital topological algebra is
itself a non-unital subalgebra. |
le_topologicalClosure (s : NonUnitalSubalgebra R A) : s ≤ s.topologicalClosure :=
subset_closure | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | le_topologicalClosure | null |
isClosed_topologicalClosure (s : NonUnitalSubalgebra R A) :
IsClosed (s.topologicalClosure : Set A) := isClosed_closure | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | isClosed_topologicalClosure | null |
topologicalClosure_minimal {s t : NonUnitalSubalgebra R A}
(h : s ≤ t) (ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t :=
closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | topologicalClosure_minimal | null |
nonUnitalCommSemiringTopologicalClosure [T2Space A] (s : NonUnitalSubalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommSemiring s.topologicalClosure :=
s.toNonUnitalSubsemiring.nonUnitalCommSemiringTopologicalClosure hs
variable [TopologicalSpace B] [NonUnitalSemiring B] [Module R B] [IsTopologicalSemir... | abbrev | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | nonUnitalCommSemiringTopologicalClosure | If a non-unital subalgebra of a non-unital topological algebra is commutative, then so is its
topological closure.
See note [reducible non-instances]. |
map_topologicalClosure_le (hφ : Continuous φ) :
map φ s.topologicalClosure ≤ (map φ s).topologicalClosure :=
image_closure_subset_closure_image hφ | lemma | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | map_topologicalClosure_le | null |
topologicalClosure_map_le (hφ : IsClosedMap φ) :
(map φ s).topologicalClosure ≤ map φ s.topologicalClosure :=
hφ.closure_image_subset _ | lemma | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | topologicalClosure_map_le | null |
topologicalClosure_map (hφ : IsClosedMap φ) (hφ' : Continuous φ) :
(map φ s).topologicalClosure = map φ s.topologicalClosure :=
SetLike.coe_injective <| hφ.closure_image_eq_of_continuous hφ' _
variable (R) in
open NonUnitalAlgebra in | lemma | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | topologicalClosure_map | null |
topologicalClosure_adjoin_le_centralizer_centralizer
[IsScalarTower R A A] [SMulCommClass R A A] [T2Space A] (s : Set A) :
(adjoin R s).topologicalClosure ≤ centralizer R (centralizer R s) :=
topologicalClosure_minimal (adjoin_le_centralizer_centralizer R s) (Set.isClosed_centralizer _) | lemma | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | topologicalClosure_adjoin_le_centralizer_centralizer | null |
instIsTopologicalRing (s : NonUnitalSubalgebra R A) : IsTopologicalRing s :=
s.toNonUnitalSubring.instIsTopologicalRing | instance | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | instIsTopologicalRing | null |
nonUnitalCommRingTopologicalClosure [T2Space A] (s : NonUnitalSubalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommRing s.topologicalClosure :=
{ s.topologicalClosure.toNonUnitalRing, s.toSubsemigroup.commSemigroupTopologicalClosure hs with } | abbrev | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | nonUnitalCommRingTopologicalClosure | If a non-unital subalgebra of a non-unital topological algebra is commutative, then so is its
topological closure.
See note [reducible non-instances]. |
elemental (x : A) : NonUnitalSubalgebra R A :=
adjoin R {x} |>.topologicalClosure | def | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | elemental | The topological closure of the non-unital subalgebra generated by a single element. |
self_mem (x : A) : x ∈ elemental R x :=
le_topologicalClosure _ <| self_mem_adjoin_singleton R x
variable {R} in | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | self_mem | null |
le_of_mem {x : A} {s : NonUnitalSubalgebra R A} (hs : IsClosed (s : Set A)) (hx : x ∈ s) :
elemental R x ≤ s :=
topologicalClosure_minimal (adjoin_le <| by simpa using hx) hs
variable {R} in | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | le_of_mem | null |
le_iff_mem {x : A} {s : NonUnitalSubalgebra R A} (hs : IsClosed (s : Set A)) :
elemental R x ≤ s ↔ x ∈ s :=
⟨fun h ↦ h (self_mem R x), fun h ↦ le_of_mem hs h⟩ | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | le_iff_mem | null |
isClosed (x : A) : IsClosed (elemental R x : Set A) :=
isClosed_topologicalClosure _ | instance | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | isClosed | null |
isClosedEmbedding_coe (x : A) : Topology.IsClosedEmbedding ((↑) : elemental R x → A) where
eq_induced := rfl
injective := Subtype.coe_injective
isClosed_range := by simpa using isClosed R x | theorem | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | isClosedEmbedding_coe | The coercion from an elemental algebra to the full algebra is a `IsClosedEmbedding`. |
le_centralizer_centralizer [T2Space A] (x : A) :
elemental R x ≤ centralizer R (centralizer R {x}) :=
topologicalClosure_adjoin_le_centralizer_centralizer R {x} | lemma | Topology | [
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/NonUnitalAlgebra.lean | le_centralizer_centralizer | null |
instIsTopologicalSemiring (s : NonUnitalStarSubalgebra R A) : IsTopologicalSemiring s :=
s.toNonUnitalSubalgebra.instIsTopologicalSemiring | instance | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | instIsTopologicalSemiring | null |
topologicalClosure (s : NonUnitalStarSubalgebra R A) : NonUnitalStarSubalgebra R A :=
{ s.toNonUnitalSubalgebra.topologicalClosure with
star_mem' := fun h ↦ map_mem_closure continuous_star h fun _ ↦ star_mem
carrier := _root_.closure (s : Set A) } | def | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | topologicalClosure | The (topological) closure of a non-unital star subalgebra of a non-unital topological star
algebra is itself a non-unital star subalgebra. |
le_topologicalClosure (s : NonUnitalStarSubalgebra R A) : s ≤ s.topologicalClosure :=
subset_closure | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | le_topologicalClosure | null |
isClosed_topologicalClosure (s : NonUnitalStarSubalgebra R A) :
IsClosed (s.topologicalClosure : Set A) := isClosed_closure | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | isClosed_topologicalClosure | null |
topologicalClosure_minimal (s : NonUnitalStarSubalgebra R A)
{t : NonUnitalStarSubalgebra R A} (h : s ≤ t) (ht : IsClosed (t : Set A)) :
s.topologicalClosure ≤ t :=
closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | topologicalClosure_minimal | null |
nonUnitalCommSemiringTopologicalClosure [T2Space A] (s : NonUnitalStarSubalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommSemiring s.topologicalClosure :=
s.toNonUnitalSubalgebra.nonUnitalCommSemiringTopologicalClosure hs
variable [TopologicalSpace B] [Star B] [NonUnitalSemiring B] [Module R B]
[Is... | abbrev | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | nonUnitalCommSemiringTopologicalClosure | If a non-unital star subalgebra of a non-unital topological star algebra is commutative, then
so is its topological closure.
See note [reducible non-instances] |
map_topologicalClosure_le (hφ : Continuous φ) :
map φ s.topologicalClosure ≤ (map φ s).topologicalClosure :=
image_closure_subset_closure_image hφ | lemma | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | map_topologicalClosure_le | null |
topologicalClosure_map_le (hφ : IsClosedMap φ) :
(map φ s).topologicalClosure ≤ map φ s.topologicalClosure :=
hφ.closure_image_subset _ | lemma | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | topologicalClosure_map_le | null |
topologicalClosure_map (hφ : IsClosedMap φ) (hφ' : Continuous φ) :
(map φ s).topologicalClosure = map φ s.topologicalClosure :=
SetLike.coe_injective <| hφ.closure_image_eq_of_continuous hφ' _
open NonUnitalStarAlgebra in | lemma | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | topologicalClosure_map | null |
topologicalClosure_adjoin_le_centralizer_centralizer (R : Type*) {A : Type*}
[CommSemiring R] [StarRing R] [TopologicalSpace A] [NonUnitalSemiring A] [StarRing A]
[Module R A] [IsTopologicalSemiring A] [ContinuousStar A] [ContinuousConstSMul R A]
[IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] ... | lemma | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | topologicalClosure_adjoin_le_centralizer_centralizer | null |
instIsTopologicalRing (s : NonUnitalStarSubalgebra R A) : IsTopologicalRing s :=
s.toNonUnitalSubring.instIsTopologicalRing | instance | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | instIsTopologicalRing | null |
nonUnitalCommRingTopologicalClosure [T2Space A] (s : NonUnitalStarSubalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommRing s.topologicalClosure :=
{ s.topologicalClosure.toNonUnitalRing, s.toSubsemigroup.commSemigroupTopologicalClosure hs with } | abbrev | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | nonUnitalCommRingTopologicalClosure | If a non-unital star subalgebra of a non-unital topological star algebra is commutative, then
so is its topological closure.
See note [reducible non-instances]. |
elemental (x : A) : NonUnitalStarSubalgebra R A :=
adjoin R {x} |>.topologicalClosure | def | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | elemental | The topological closure of the non-unital star subalgebra generated by a single element. |
self_mem (x : A) : x ∈ elemental R x :=
le_topologicalClosure _ <| self_mem_adjoin_singleton R x
@[simp, aesop safe (rule_sets := [SetLike])] | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | self_mem | null |
star_self_mem (x : A) : star x ∈ elemental R x :=
le_topologicalClosure _ <| star_self_mem_adjoin_singleton R x
variable {R} in | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | star_self_mem | null |
le_of_mem {x : A} {s : NonUnitalStarSubalgebra R A} (hs : IsClosed (s : Set A))
(hx : x ∈ s) : elemental R x ≤ s :=
topologicalClosure_minimal _ (adjoin_le <| by simpa using hx) hs
variable {R} in | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | le_of_mem | null |
le_iff_mem {x : A} {s : NonUnitalStarSubalgebra R A} (hs : IsClosed (s : Set A)) :
elemental R x ≤ s ↔ x ∈ s :=
⟨fun h ↦ h (self_mem R x), fun h ↦ le_of_mem hs h⟩ | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | le_iff_mem | null |
isClosed (x : A) : IsClosed (elemental R x : Set A) :=
isClosed_topologicalClosure _ | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | isClosed | null |
isClosedEmbedding_coe (x : A) : Topology.IsClosedEmbedding ((↑) : elemental R x → A) where
eq_induced := rfl
injective := Subtype.coe_injective
isClosed_range := by simpa using isClosed R x | theorem | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | isClosedEmbedding_coe | The coercion from an elemental algebra to the full algebra is a `IsClosedEmbedding`. |
le_centralizer_centralizer [T2Space A] (x : A) :
elemental R x ≤ centralizer R (centralizer R {x}) :=
topologicalClosure_adjoin_le_centralizer_centralizer .. | lemma | Topology | [
"Mathlib.Algebra.Star.NonUnitalSubalgebra",
"Mathlib.Topology.Algebra.NonUnitalAlgebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/NonUnitalStarAlgebra.lean | le_centralizer_centralizer | null |
OpenAddSubgroup (G : Type*) [AddGroup G] [TopologicalSpace G] extends AddSubgroup G where
isOpen' : IsOpen carrier | structure | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | OpenAddSubgroup | The type of open subgroups of a topological additive group. |
@[to_additive]
OpenSubgroup (G : Type*) [Group G] [TopologicalSpace G] extends Subgroup G where
isOpen' : IsOpen carrier | structure | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | OpenSubgroup | The type of open subgroups of a topological group. |
@[to_additive (attr := coe) /-- Coercion from `OpenAddSubgroup G` to `Opens G`. -/]
toOpens (U : OpenSubgroup G) : Opens G := ⟨U, U.isOpen'⟩
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | toOpens | Reinterpret an `OpenSubgroup` as a `Subgroup`. -/
add_decl_doc OpenSubgroup.toSubgroup
/-- Reinterpret an `OpenAddSubgroup` as an `AddSubgroup`. -/
add_decl_doc OpenAddSubgroup.toAddSubgroup
attribute [coe] OpenSubgroup.toSubgroup OpenAddSubgroup.toAddSubgroup
namespace OpenSubgroup
variable {G : Type*} [Group G] [... |
hasCoeOpens : CoeTC (OpenSubgroup G) (Opens G) := ⟨toOpens⟩
@[to_additive (attr := simp, norm_cast)] | instance | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | hasCoeOpens | null |
coe_toOpens : ((U : Opens G) : Set G) = U :=
rfl
@[to_additive (attr := simp, norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | coe_toOpens | null |
coe_toSubgroup : ((U : Subgroup G) : Set G) = U := rfl
@[to_additive (attr := simp, norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | coe_toSubgroup | null |
mem_toOpens : g ∈ (U : Opens G) ↔ g ∈ U := Iff.rfl
@[to_additive (attr := simp, norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | mem_toOpens | null |
mem_toSubgroup : g ∈ (U : Subgroup G) ↔ g ∈ U := Iff.rfl
@[to_additive (attr := ext)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | mem_toSubgroup | null |
ext (h : ∀ x, x ∈ U ↔ x ∈ V) : U = V :=
SetLike.ext h
variable (U)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | ext | null |
protected isOpen : IsOpen (U : Set G) :=
U.isOpen'
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | isOpen | null |
mem_nhds_one : (U : Set G) ∈ 𝓝 (1 : G) :=
U.isOpen.mem_nhds U.one_mem
variable {U}
@[to_additive] instance : Top (OpenSubgroup G) := ⟨⟨⊤, isOpen_univ⟩⟩
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | mem_nhds_one | null |
mem_top (x : G) : x ∈ (⊤ : OpenSubgroup G) :=
trivial
@[to_additive (attr := simp, norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | mem_top | null |
coe_top : ((⊤ : OpenSubgroup G) : Set G) = Set.univ :=
rfl
@[to_additive (attr := simp, norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.RingTheory.Ideal.Defs",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Algebra/OpenSubgroup.lean | coe_top | null |
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