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
@[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 | 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, _, hb, rfl⟩
@[to_additive] | 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 := s.top_closure_mul_self_subset ⟨_, ha, _, hb, rfl⟩
@[to_additive] | 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.commSemigroupTopologicalClosure [T2Space M] (s : Subsemigroup M)
(hs : ∀ x y : s, x * y = y * x) : CommSemigroup s.topologicalClosure :=
{ MulMemClass.toSemigroup s.topologicalClosure with
mul_comm :=
have : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x := fun x hx y hy =>
congr_arg Subtype.val (hs ⟨x, hx⟩ ⟨y, hy⟩)
fun ⟨x, hx⟩ ⟨y, hy⟩ =>
Subtype.ext <|
eqOn_closure₂ this continuous_mul (continuous_snd.mul continuous_fst) x hx y hy }
@[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) := by
rw [← image_mul_prod]
exact (hs.prod ht).image continuous_mul
end Semigroup
variable [TopologicalSpace M] [Monoid M] [ContinuousMul M]
@[to_additive]
theorem Submonoid.top_closure_mul_self_subset (s : Submonoid 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
@[to_additive]
theorem Submonoid.top_closure_mul_self_eq (s : Submonoid M) :
_root_.closure (s : Set M) * _root_.closure s = _root_.closure s :=
Subset.antisymm s.top_closure_mul_self_subset fun x hx =>
⟨x, hx, 1, _root_.subset_closure s.one_mem, mul_one _⟩
/-- The (topological-space) closure of a submonoid of a space `M` with `ContinuousMul` is
itself a submonoid. |
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 : Submonoid M)
(hs : ∀ x y : s, x * y = y * x) : CommMonoid s.topologicalClosure :=
{ s.topologicalClosure.toMonoid, s.toSubsemigroup.commSemigroupTopologicalClosure hs with }
@[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩
rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩
use V, V1
intro v w s t v_in w_in s_in t_in
simpa only [mul_assoc] using h _ (h' v v_in w w_in) _ (h' s s_in t t_in)
@[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
simp only [List.map_cons, List.prod_cons]
exact
(h f List.mem_cons_self).mul
(tendsto_list_prod l fun c hc => h c (List.mem_cons_of_mem _ hc))
@[to_additive (attr := continuity)]
theorem continuous_list_prod {f : ι → X → M} (l : List ι) (h : ∀ i ∈ l, Continuous (f i)) :
Continuous fun a => (l.map fun i => f i a).prod :=
continuous_iff_continuousAt.2 fun x =>
tendsto_list_prod l fun c hc => continuous_iff_continuousAt.1 (h c hc) x
@[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
refine tendsto_list_prod _ fun i hi => ?_
specialize h i hi x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx] at h
exact h
@[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
| 0 => by simpa using continuous_const
| k + 1 => by
simp only [pow_succ']
exact continuous_id.mul (continuous_pow _)
instance AddMonoid.continuousConstSMul_nat {A} [AddMonoid A] [TopologicalSpace A]
[ContinuousAdd A] : ContinuousConstSMul ℕ A :=
⟨continuous_nsmul⟩
instance AddMonoid.continuousSMul_nat {A} [AddMonoid A] [TopologicalSpace A]
[ContinuousAdd A] : ContinuousSMul ℕ A :=
⟨continuous_prod_of_discrete_left.mpr continuous_nsmul⟩
-- We register `Continuous.pow` as a `continuity` lemma with low penalty (so
-- `continuity` will try it before other `continuity` lemmas). This is a
-- workaround for goals of the form `Continuous fun x => x ^ 2`, where
-- `continuity` applies `Continuous.mul` since the goal is defeq to
-- `Continuous fun x => x * x`.
--
-- To properly fix this, we should make sure that `continuity` applies its
-- lemmas with reducible transparency, preventing the unfolding of `^`. But this
-- is quite an invasive change.
@[to_additive (attr := aesop safe -100 (rule_sets := [Continuous]), fun_prop)]
theorem Continuous.pow {f : X → M} (h : Continuous f) (n : ℕ) : Continuous fun b => f b ^ n :=
(continuous_pow n).comp h
@[to_additive]
theorem continuousOn_pow {s : Set M} (n : ℕ) : ContinuousOn (fun (x : M) => x ^ n) s :=
(continuous_pow n).continuousOn
@[to_additive]
theorem continuousAt_pow (x : M) (n : ℕ) : ContinuousAt (fun (x : M) => x ^ n) x :=
(continuous_pow n).continuousAt
@[to_additive]
theorem Filter.Tendsto.pow {l : Filter α} {f : α → M} {x : M} (hf : Tendsto f l (𝓝 x)) (n : ℕ) :
Tendsto (fun x => f x ^ n) l (𝓝 (x ^ n)) :=
(continuousAt_pow _ _).tendsto.comp hf
@[to_additive]
theorem ContinuousWithinAt.pow {f : X → M} {x : X} {s : Set X} (hf : ContinuousWithinAt f s x)
(n : ℕ) : ContinuousWithinAt (fun x => f x ^ n) s x :=
Filter.Tendsto.pow hf n
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.pow {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) :
ContinuousAt (fun x => f x ^ n) x :=
Filter.Tendsto.pow hf n
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.pow {f : X → M} {s : Set X} (hf : ContinuousOn f s) (n : ℕ) :
ContinuousOn (fun x => f x ^ n) s := fun x hx => (hf x hx).pow n
/-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. |
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. -/]
instance (priority := 100) IsScalarTower.continuousConstSMul {R A : Type*} [Monoid A] [SMul R A]
[IsScalarTower R A A] [TopologicalSpace A] [ContinuousMul A] : ContinuousConstSMul R A where
continuous_const_smul q := by
simp +singlePass only [← smul_one_mul q (_ : A)]
exact continuous_const.mul continuous_id
/-- If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ`. -/
@[to_additive /-- If the action of `R` on `A` commutes with left-addition, then
continuous addition implies continuous affine addition by constants.
Notably, this instances applies when `R = Aᵃᵒᵖ`. -/]
instance (priority := 100) SMulCommClass.continuousConstSMul {R A : Type*} [Monoid A] [SMul R A]
[SMulCommClass R A A] [TopologicalSpace A] [ContinuousMul A] : ContinuousConstSMul R A where
continuous_const_smul q := by
simp +singlePass only [← mul_smul_one q (_ : A)]
exact continuous_id.mul continuous_const
end ContinuousMul
namespace MulOpposite
/-- If multiplication is continuous in `α`, then it also is in `αᵐᵒᵖ`. -/
@[to_additive /-- If addition is continuous in `α`, then it also is in `αᵃᵒᵖ`. -/]
instance [TopologicalSpace α] [Mul α] [ContinuousMul α] : ContinuousMul αᵐᵒᵖ :=
⟨continuous_op.comp (continuous_unop.snd'.mul continuous_unop.fst')⟩
end MulOpposite
namespace Units
open MulOpposite
variable [TopologicalSpace α] [Monoid α] [ContinuousMul α]
/-- If multiplication on a monoid is continuous, then multiplication on the units of the monoid,
with respect to the induced topology, is continuous.
Inversion is also continuous, but we register this in a later file, `Topology.Algebra.Group`,
because the predicate `ContinuousInv` has not yet been defined. -/
@[to_additive /-- If addition on an additive monoid is continuous, then addition on the additive
units of the monoid, with respect to the induced topology, is continuous.
Negation is also continuous, but we register this in a later file, `Topology.Algebra.Group`, because
the predicate `ContinuousNeg` has not yet been defined. -/]
instance : ContinuousMul αˣ := isInducing_embedProduct.continuousMul (embedProduct α)
end Units
@[to_additive (attr := fun_prop)]
theorem Continuous.units_map [Monoid M] [Monoid N] [TopologicalSpace M] [TopologicalSpace N]
(f : M →* N) (hf : Continuous f) : Continuous (Units.map f) :=
Units.continuous_iff.2 ⟨hf.comp Units.continuous_val, hf.comp Units.continuous_coe_inv⟩
section
variable [TopologicalSpace M] [CommMonoid M]
@[to_additive]
theorem Submonoid.mem_nhds_one (S : Submonoid M) (oS : IsOpen (S : Set M)) :
(S : Set M) ∈ 𝓝 (1 : M) :=
IsOpen.mem_nhds oS S.one_mem
variable [ContinuousMul M]
@[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) := by
rcases s with ⟨l⟩
simpa using tendsto_list_prod l
@[to_additive]
theorem tendsto_finset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Finset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) :=
tendsto_multiset_prod _
@[to_additive (attr := continuity)]
theorem continuous_multiset_prod {f : ι → X → M} (s : Multiset ι) :
(∀ i ∈ s, Continuous (f i)) → Continuous fun a => (s.map fun i => f i a).prod := by
rcases s with ⟨l⟩
simpa using continuous_list_prod l
@[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t := by
rcases s with ⟨l⟩
simpa using continuousOn_list_prod l
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_finset_prod {f : ι → X → M} (s : Finset ι) :
(∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∏ i ∈ s, f i a :=
continuous_multiset_prod _
@[to_additive]
theorem continuousOn_finset_prod {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_multiset_prod _
@[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i := by
replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x := by rwa [eventually_all_finset]
filter_upwards [hs] with x hx
simp only [Finset.prod_apply, Finset.prod_congr rfl hx]
open Function
@[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [One M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I := by
rcases hf x₀ with ⟨U, hxU, hUf⟩
refine ⟨hUf.toFinset, mem_of_superset hxU fun y hy i hi => ?_⟩
rw [hUf.coe_toFinset]
exact ⟨y, hi, hy⟩
@[to_additive]
theorem finprod_eventually_eq_prod {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport (f i)) (x : X) :
∃ s : Finset ι, ∀ᶠ y in 𝓝 x, ∏ᶠ i, f i y = ∏ i ∈ s, f i y :=
let ⟨I, hI⟩ := hf.exists_finset_mulSupport x
⟨I, hI.mono fun _ hy => finprod_eq_prod_of_mulSupport_subset _ fun _ hi => hy hi⟩
@[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x := by
refine continuous_iff_continuousAt.2 fun x => ?_
rcases finprod_eventually_eq_prod hf x with ⟨s, hs⟩
refine ContinuousAt.congr ?_ (EventuallyEq.symm hs)
exact tendsto_finset_prod _ fun i _ => (hc i).continuousAt
@[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x := by
simp only [← finprod_subtype_eq_finprod_cond]
exact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective)
end
instance [TopologicalSpace M] [Mul M] [ContinuousMul M] : ContinuousAdd (Additive M) where
continuous_add := @continuous_mul M _ _ _
instance [TopologicalSpace M] [Add M] [ContinuousAdd M] : ContinuousMul (Multiplicative M) where
continuous_mul := @continuous_add M _ _ _
section LatticeOps
variable {ι' : Sort*} [Mul M]
@[to_additive]
theorem continuousMul_sInf {ts : Set (TopologicalSpace M)}
(h : ∀ t ∈ ts, @ContinuousMul M t _) : @ContinuousMul M (sInf ts) _ :=
letI := sInf ts
{ continuous_mul :=
continuous_sInf_rng.2 fun t ht =>
continuous_sInf_dom₂ ht ht (@ContinuousMul.continuous_mul M t _ (h t ht)) }
@[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ := by
rw [← sInf_range]
exact continuousMul_sInf (Set.forall_mem_range.mpr h')
@[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine continuousMul_iInf fun b => ?_
cases b <;> assumption
end LatticeOps
namespace ContinuousMap
variable [Mul X] [ContinuousMul X]
/-- The continuous map `fun y => y * x` |
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] continuous_smul continuous_vadd | 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 Continuous (fun p : M × α ↦ (p.1 • (1 : N)) • p.2) by simpa
fun_prop }
@[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 | 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) := by
let tα := tβ.induced f
refine ⟨continuous_induced_rng.2 ?_⟩
simp only [Function.comp_def, map_smul]
fun_prop
@[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.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_unop.prodMap continuous_id)
simpa only [op_smul_eq_smul] using (continuous_smul : Continuous fun p : M × X => _)⟩
@[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 | 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 well.
In many cases, `f = id` so that `g` is an action homomorphism in the sense of `AddActionHom`.
However, this version also works for `f = AddUnits.val`. -/]
Topology.IsInducing.continuousSMul {N : Type*} [SMul N Y] [TopologicalSpace N] {f : N → M}
(hg : IsInducing g) (hf : Continuous f) (hsmul : ∀ {c x}, g (c • x) = f c • g x) :
ContinuousSMul N Y where
continuous_smul := by
simpa only [hg.continuous_iff, Function.comp_def, hsmul]
using (hf.comp continuous_fst).smul <| hg.continuous.comp continuous_snd
@[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 | 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 action homomorphism in the sense of `MulActionHom`.
However, this version also works for semilinear maps and `f = Units.val`. |
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_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 | 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 =>
continuous_sInf_dom₂ (Eq.refl _) ht
(@ContinuousSMul.continuous_smul _ _ _ _ t (h t ht)) }
@[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_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 := inferInstance } | 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] [IsTopologicalSemiring B]
[ContinuousConstSMul R B] (s : NonUnitalSubalgebra R A) {φ : A →ₙₐ[R] B} | 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]
[IsTopologicalSemiring B] [ContinuousConstSMul R B] [ContinuousStar B]
(s : NonUnitalStarSubalgebra R A) {φ : A →⋆ₙₐ[R] B} | 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] [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.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] [TopologicalSpace G]
variable {U V : OpenSubgroup G} {g : G}
@[to_additive]
instance hasCoeSubgroup : CoeTC (OpenSubgroup G) (Subgroup G) :=
⟨toSubgroup⟩
@[to_additive]
theorem toSubgroup_injective : Injective ((↑) : OpenSubgroup G → Subgroup G)
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
@[to_additive]
instance : SetLike (OpenSubgroup G) G where
coe U := U.1
coe_injective' _ _ h := toSubgroup_injective <| SetLike.ext' h
@[to_additive]
instance : SubgroupClass (OpenSubgroup G) G where
mul_mem := Subsemigroup.mul_mem' _
one_mem U := U.one_mem'
inv_mem := Subgroup.inv_mem' _
/-- Coercion from `OpenSubgroup G` to `Opens 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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