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 ⌀ |
|---|---|---|---|---|---|---|
id_comp (f : A →A[R] B) : (ContinuousAlgHom.id R B).comp f = f :=
ext fun _x => rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | id_comp | null |
comp_assoc {D : Type*} [Semiring D] [Algebra R D] [TopologicalSpace D] (h : C →A[R] D)
(g : B →A[R] C) (f : A →A[R] B) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | comp_assoc | null |
mul_def (f g : A →A[R] A) : f * g = f.comp g := rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | mul_def | null |
coe_mul (f g : A →A[R] A) : ⇑(f * g) = f ∘ g := rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_mul | null |
mul_apply (f g : A →A[R] A) (x : A) : (f * g) x = f (g x) := rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | mul_apply | null |
coe_pow (f : A →A[R] A) (n : ℕ) : ⇑(f ^ n) = f^[n] :=
hom_coe_pow _ rfl (fun _ _ ↦ rfl) _ _ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_pow | null |
@[simps]
toAlgHomMonoidHom : (A →A[R] A) →* A →ₐ[R] A where
toFun := (↑)
map_one' := rfl
map_mul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | toAlgHomMonoidHom | coercion from `ContinuousAlgHom` to `AlgHom` as a `RingHom`. |
protected prod (f₁ : A →A[R] B) (f₂ : A →A[R] C) :
A →A[R] B × C :=
⟨(f₁ : A →ₐ[R] B).prod f₂, f₁.2.prodMk f₂.2⟩
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | prod | The Cartesian product of two continuous algebra morphisms as a continuous algebra morphism. |
coe_prod (f₁ : A →A[R] B) (f₂ : A →A[R] C) :
(f₁.prod f₂ : A →ₐ[R] B × C) = AlgHom.prod f₁ f₂ :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_prod | null |
prod_apply (f₁ : A →A[R] B) (f₂ : A →A[R] C) (x : A) :
f₁.prod f₂ x = (f₁ x, f₂ x) :=
rfl
variable {F : Type*} | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | prod_apply | null |
fst : A × B →A[R] A where
cont := continuous_fst
toAlgHom := AlgHom.fst R A B | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | fst | `Prod.fst` as a `ContinuousAlgHom`. |
snd : A × B →A[R] B where
cont := continuous_snd
toAlgHom := AlgHom.snd R A B
variable {R A B}
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | snd | `Prod.snd` as a `ContinuousAlgHom`. |
coe_fst : ↑(fst R A B) = AlgHom.fst R A B :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_fst | null |
coe_fst' : ⇑(fst R A B) = Prod.fst :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_fst' | null |
coe_snd : ↑(snd R A B) = AlgHom.snd R A B :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_snd | null |
coe_snd' : ⇑(snd R A B) = Prod.snd :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_snd' | null |
fst_prod_snd : (fst R A B).prod (snd R A B) = ContinuousAlgHom.id R (A × B) :=
ext fun ⟨_x, _y⟩ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | fst_prod_snd | null |
fst_comp_prod (f : A →A[R] B) (g : A →A[R] C) :
(fst R B C).comp (f.prod g) = f :=
ext fun _x => rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | fst_comp_prod | null |
snd_comp_prod (f : A →A[R] B) (g : A →A[R] C) :
(snd R B C).comp (f.prod g) = g :=
ext fun _x => rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | snd_comp_prod | null |
prodMap {D : Type*} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B)
(f₂ : C →A[R] D) : A × C →A[R] B × D :=
(f₁.comp (fst R A C)).prod (f₂.comp (snd R A C))
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | prodMap | `Prod.map` of two continuous algebra homomorphisms. |
coe_prodMap {D : Type*} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B)
(f₂ : C →A[R] D) :
(f₁.prodMap f₂ : A × C →ₐ[R] B × D) = (f₁ : A →ₐ[R] B).prodMap (f₂ : C →ₐ[R] D) :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_prodMap | null |
coe_prodMap' {D : Type*} [Semiring D] [TopologicalSpace D] [Algebra R D] (f₁ : A →A[R] B)
(f₂ : C →A[R] D) : ⇑(f₁.prodMap f₂) = Prod.map f₁ f₂ :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_prodMap' | null |
@[simps apply]
prodEquiv : (A →A[R] B) × (A →A[R] C) ≃ (A →A[R] B × C) where
toFun f := f.1.prod f.2
invFun f := ⟨(fst _ _ _).comp f, (snd _ _ _).comp f⟩ | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | prodEquiv | `ContinuousAlgHom.prod` as an `Equiv`. |
codRestrict (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ x, f x ∈ p) : A →A[R] p where
cont := f.continuous.subtype_mk _
toAlgHom := (f : A →ₐ[R] B).codRestrict p h
@[norm_cast] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | codRestrict | Restrict codomain of a continuous algebra morphism. |
coe_codRestrict (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ x, f x ∈ p) :
(f.codRestrict p h : A →ₐ[R] p) = (f : A →ₐ[R] B).codRestrict p h :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_codRestrict | null |
coe_codRestrict_apply (f : A →A[R] B) (p : Subalgebra R B) (h : ∀ x, f x ∈ p) (x) :
(f.codRestrict p h x : B) = f x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_codRestrict_apply | null |
@[reducible]
rangeRestrict (f : A →A[R] B) :=
f.codRestrict (@AlgHom.range R A B _ _ _ _ _ f) (@AlgHom.mem_range_self R A B _ _ _ _ _ f)
@[simp] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | rangeRestrict | Restrict the codomain of a continuous algebra homomorphism `f` to `f.range`. |
coe_rangeRestrict (f : A →A[R] B) :
(f.rangeRestrict : A →ₐ[R] (@AlgHom.range R A B _ _ _ _ _ f)) =
(f : A →ₐ[R] B).rangeRestrict :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_rangeRestrict | null |
_root_.Subalgebra.valA (p : Subalgebra R A) : p →A[R] A where
cont := continuous_subtype_val
toAlgHom := p.val
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | _root_.Subalgebra.valA | `Subalgebra.val` as a `ContinuousAlgHom`. |
_root_.Subalgebra.coe_valA (p : Subalgebra R A) : p.valA = p.subtype :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | _root_.Subalgebra.coe_valA | null |
_root_.Subalgebra.coe_valA' (p : Subalgebra R A) : ⇑p.valA = p.subtype :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | _root_.Subalgebra.coe_valA' | null |
_root_.Subalgebra.valA_apply (p : Subalgebra R A) (x : p) : p.valA x = x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | _root_.Subalgebra.valA_apply | null |
_root_.Submodule.range_valA (p : Subalgebra R A) :
@AlgHom.range R p A _ _ _ _ _ p.valA = p :=
Subalgebra.range_val p | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | _root_.Submodule.range_valA | null |
protected map_neg (f : S →A[R] B) (x : S) : f (-x) = -f x := map_neg f x | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | map_neg | null |
protected map_sub (f : S →A[R] B) (x y : S) : f (x - y) = f x - f y := map_sub f x y | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | map_sub | null |
restrictScalars (f : B →A[S] C) : B →A[R] C :=
⟨(f : B →ₐ[S] C).restrictScalars R, f.continuous⟩
variable {R}
@[simp] | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | restrictScalars | If `A` is an `R`-algebra, then a continuous `A`-algebra morphism can be interpreted as a
continuous `R`-algebra morphism. |
coe_restrictScalars (f : B →A[S] C) :
(f.restrictScalars R : B →ₐ[R] C) = (f : B →ₐ[S] C).restrictScalars R :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_restrictScalars | null |
coe_restrictScalars' (f : B →A[S] C) : ⇑(f.restrictScalars R) = f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | coe_restrictScalars' | null |
Subalgebra.le_topologicalClosure (s : Subalgebra R A) : s ≤ s.topologicalClosure :=
subset_closure | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.le_topologicalClosure | null |
Subalgebra.isClosed_topologicalClosure (s : Subalgebra R A) :
IsClosed (s.topologicalClosure : Set A) := by convert @isClosed_closure A _ s | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.isClosed_topologicalClosure | null |
Subalgebra.topologicalClosure_minimal {s t : Subalgebra R A} (h : s ≤ t)
(ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
variable (R) in
open Algebra in | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.topologicalClosure_minimal | null |
Subalgebra.topologicalClosure_adjoin_le_centralizer_centralizer [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.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.topologicalClosure_adjoin_le_centralizer_centralizer | null |
Subalgebra.commSemiringTopologicalClosure [T2Space A] (s : Subalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : CommSemiring s.topologicalClosure :=
{ s.topologicalClosure.toSemiring, s.toSubmonoid.commMonoidTopologicalClosure hs with } | abbrev | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.commSemiringTopologicalClosure | If a subalgebra of a topological algebra is commutative, then so is its topological closure.
See note [reducible non-instances]. |
Subalgebra.topologicalClosure_comap_homeomorph (s : Subalgebra R A) {B : Type*}
[TopologicalSpace B] [Ring B] [IsTopologicalRing B] [Algebra R B] (f : B →ₐ[R] A) (f' : B ≃ₜ A)
(w : (f : B → A) = f') : s.topologicalClosure.comap f = (s.comap f).topologicalClosure := by
apply SetLike.ext'
simp only [Subalgebra.topologicalClosure_coe]
simp only [Subalgebra.coe_comap]
rw [w]
exact f'.preimage_closure _
variable (R)
open Subalgebra | theorem | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.topologicalClosure_comap_homeomorph | This is really a statement about topological algebra isomorphisms,
but we don't have those, so we use the clunky approach of talking about
an algebra homomorphism, and a separate homeomorphism,
along with a witness that as functions they are the same. |
Algebra.elemental (x : A) : Subalgebra R A :=
(Algebra.adjoin R ({x} : Set A)).topologicalClosure | def | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Algebra.elemental | The topological closure of the 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.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | self_mem | null |
le_of_mem {x : A} {s : Subalgebra 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.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | le_of_mem | null |
le_iff_mem {x : A} {s : Subalgebra 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.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | le_iff_mem | null |
isClosed (x : A) : IsClosed (elemental R x : Set A) :=
isClosed_topologicalClosure _ | instance | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | isClosed | null |
isClosedEmbedding_coe (x : A) : 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.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.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.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | le_centralizer_centralizer | null |
Subalgebra.commRingTopologicalClosure [T2Space A] (s : Subalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : CommRing s.topologicalClosure :=
{ s.topologicalClosure.toRing, s.toSubmonoid.commMonoidTopologicalClosure hs with } | abbrev | Topology | [
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Algebra.lean | Subalgebra.commRingTopologicalClosure | If a subalgebra of a topological algebra is commutative, then so is its topological closure.
See note [reducible non-instances]. |
exist_openNormalSubgroup_sub_clopen_nhds_of_one {G : Type*} [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] [CompactSpace G] {W : Set G} (WClopen : IsClopen W) (einW : 1 ∈ W) :
∃ H : OpenNormalSubgroup G, (H : Set G) ⊆ W := by
rcases exist_openSubgroup_sub_clopen_nhds_of_one WClopen einW with ⟨H, hH⟩
have : Subgroup.FiniteIndex H.toSubgroup := H.finiteIndex_of_finite_quotient
use { toSubgroup := Subgroup.normalCore H
isOpen' := Subgroup.isOpen_of_isClosed_of_finiteIndex _ (H.normalCore_isClosed H.isClosed) }
exact fun _ b ↦ hH (H.normalCore_le b)
@[deprecated (since := "2025-05-22")]
alias exist_openNormalSubgroup_sub_clopen_nhd_of_one :=
exist_openNormalSubgroup_sub_clopen_nhds_of_one | theorem | Topology | [
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/ClopenNhdofOne.lean | exist_openNormalSubgroup_sub_clopen_nhds_of_one | null |
exist_openNormalSubgroup_sub_open_nhds_of_one {G : Type*} [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {U : Set G}
(UOpen : IsOpen U) (einU : 1 ∈ U) : ∃ H : OpenNormalSubgroup G, (H : Set G) ⊆ U := by
rcases ((Filter.HasBasis.mem_iff' ((nhds_basis_clopen (1 : G))) U ).mp <|
mem_nhds_iff.mpr (by use U)) with ⟨W, hW, h⟩
rcases IsTopologicalGroup.exist_openNormalSubgroup_sub_clopen_nhds_of_one hW.2 hW.1 with ⟨H, hH⟩
exact ⟨H, fun _ a ↦ h (hH a)⟩
@[deprecated (since := "2025-05-22")]
alias exist_openNormalSubgroup_sub_open_nhd_of_one := exist_openNormalSubgroup_sub_open_nhds_of_one | theorem | Topology | [
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/ClopenNhdofOne.lean | exist_openNormalSubgroup_sub_open_nhds_of_one | null |
ContinuousConstSMul (Γ : Type*) (T : Type*) [TopologicalSpace T] [SMul Γ T] : Prop where
/-- The scalar multiplication `(•) : Γ → T → T` is continuous in the second argument. -/
continuous_const_smul : ∀ γ : Γ, Continuous fun x : T => γ • x | class | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ContinuousConstSMul | Class `ContinuousConstSMul Γ T` says that the scalar multiplication `(•) : Γ → T → T`
is continuous in the second argument. We use the same class for all kinds of multiplicative
actions, including (semi)modules and algebras.
Note that both `ContinuousConstSMul α α` and `ContinuousConstSMul αᵐᵒᵖ α` are
weaker versions of `ContinuousMul α`. |
ContinuousConstVAdd (Γ : Type*) (T : Type*) [TopologicalSpace T] [VAdd Γ T] : Prop where
/-- The additive action `(+ᵥ) : Γ → T → T` is continuous in the second argument. -/
continuous_const_vadd : ∀ γ : Γ, Continuous fun x : T => γ +ᵥ x
attribute [to_additive] ContinuousConstSMul
export ContinuousConstSMul (continuous_const_smul)
export ContinuousConstVAdd (continuous_const_vadd)
variable {M α β : Type*} | class | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ContinuousConstVAdd | Class `ContinuousConstVAdd Γ T` says that the additive action `(+ᵥ) : Γ → T → T`
is continuous in the second argument. We use the same class for all kinds of additive actions,
including (semi)modules and algebras.
Note that both `ContinuousConstVAdd α α` and `ContinuousConstVAdd αᵐᵒᵖ α` are
weaker versions of `ContinuousVAdd α`. |
@[to_additive]
Filter.Tendsto.const_smul {f : β → α} {l : Filter β} {a : α} (hf : Tendsto f l (𝓝 a))
(c : M) : Tendsto (fun x => c • f x) l (𝓝 (c • a)) :=
((continuous_const_smul _).tendsto _).comp hf
variable [TopologicalSpace β] {g : β → α} {b : β} {s : Set β}
@[to_additive]
nonrec theorem ContinuousWithinAt.const_smul (hg : ContinuousWithinAt g s b) (c : M) :
ContinuousWithinAt (fun x => c • g x) s b :=
hg.const_smul c
@[to_additive (attr := fun_prop)]
nonrec theorem ContinuousAt.const_smul (hg : ContinuousAt g b) (c : M) :
ContinuousAt (fun x => c • g x) b :=
hg.const_smul c
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Filter.Tendsto.const_smul | null |
ContinuousOn.const_smul (hg : ContinuousOn g s) (c : M) :
ContinuousOn (fun x => c • g x) s := fun x hx => (hg x hx).const_smul c
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ContinuousOn.const_smul | null |
Continuous.const_smul (hg : Continuous g) (c : M) : Continuous fun x => c • g x :=
(continuous_const_smul _).comp hg | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Continuous.const_smul | null |
@[to_additive /-- If an additive action is central, then its right action is continuous when its
left action is. -/]
ContinuousConstSMul.op [SMul Mᵐᵒᵖ α] [IsCentralScalar M α] :
ContinuousConstSMul Mᵐᵒᵖ α :=
⟨MulOpposite.rec' fun c => by simpa only [op_smul_eq_smul] using continuous_const_smul c⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ContinuousConstSMul.op | If a scalar is central, then its right action is continuous when its left action is. |
MulOpposite.continuousConstSMul : ContinuousConstSMul M αᵐᵒᵖ :=
⟨fun c => MulOpposite.continuous_op.comp <| MulOpposite.continuous_unop.const_smul c⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | MulOpposite.continuousConstSMul | null |
@[to_additive]
OrderDual.continuousConstSMul' : ContinuousConstSMul Mᵒᵈ α :=
‹ContinuousConstSMul M α›
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | OrderDual.continuousConstSMul' | null |
Prod.continuousConstSMul [SMul M β] [ContinuousConstSMul M β] :
ContinuousConstSMul M (α × β) :=
⟨fun _ => (continuous_fst.const_smul _).prodMk (continuous_snd.const_smul _)⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Prod.continuousConstSMul | null |
@[to_additive]
IsCompact.smul {α β} [SMul α β] [TopologicalSpace β] [ContinuousConstSMul α β] (a : α)
{s : Set β} (hs : IsCompact s) : IsCompact (a • s) :=
hs.image (continuous_id.const_smul a)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | IsCompact.smul | null |
Specializes.const_smul {x y : α} (h : x ⤳ y) (c : M) : (c • x) ⤳ (c • y) :=
h.map (continuous_const_smul c)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Specializes.const_smul | null |
Inseparable.const_smul {x y : α} (h : Inseparable x y) (c : M) :
Inseparable (c • x) (c • y) :=
h.map (continuous_const_smul c)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Inseparable.const_smul | null |
Topology.IsInducing.continuousConstSMul {N β : Type*} [SMul N β] [TopologicalSpace β]
{g : β → α} (hg : IsInducing g) (f : N → M) (hf : ∀ {c : N} {x : β}, g (c • x) = f c • g x) :
ContinuousConstSMul N β where
continuous_const_smul c := by
simpa only [Function.comp_def, hf, hg.continuous_iff] using hg.continuous.const_smul (f c) | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Topology.IsInducing.continuousConstSMul | null |
@[to_additive]
Units.continuousConstSMul : ContinuousConstSMul Mˣ α where
continuous_const_smul m := continuous_const_smul (m : M)
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Units.continuousConstSMul | null |
smul_closure_subset (c : M) (s : Set α) : c • closure s ⊆ closure (c • s) :=
((Set.mapsTo_image _ _).closure <| continuous_const_smul c).image_subset
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | smul_closure_subset | null |
set_smul_closure_subset (s : Set M) (t : Set α) : s • closure t ⊆ closure (s • t) := by
simp only [← iUnion_smul_set]
exact iUnion₂_subset fun c hc ↦ (smul_closure_subset c t).trans <| closure_mono <|
subset_biUnion_of_mem (u := (· • t)) hc
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | set_smul_closure_subset | null |
smul_closure_orbit_subset (c : M) (x : α) :
c • closure (MulAction.orbit M x) ⊆ closure (MulAction.orbit M x) :=
(smul_closure_subset c _).trans <| closure_mono <| MulAction.smul_orbit_subset _ _ | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | smul_closure_orbit_subset | null |
isClosed_setOf_map_smul {N : Type*} [Monoid N] (α β) [MulAction M α] [MulAction N β]
[TopologicalSpace β] [T2Space β] [ContinuousConstSMul N β] (σ : M → N) :
IsClosed { f : α → β | ∀ c x, f (c • x) = σ c • f x } := by
simp only [Set.setOf_forall]
exact isClosed_iInter fun c => isClosed_iInter fun x =>
isClosed_eq (continuous_apply _) ((continuous_apply _).const_smul _) | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | isClosed_setOf_map_smul | null |
@[to_additive]
tendsto_const_smul_iff {f : β → α} {l : Filter β} {a : α} (c : G) :
Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul c⁻¹, fun h => h.const_smul _⟩
variable [TopologicalSpace β] {f : β → α} {b : β} {s : Set β}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | tendsto_const_smul_iff | null |
continuousWithinAt_const_smul_iff (c : G) :
ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b :=
tendsto_const_smul_iff c
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | continuousWithinAt_const_smul_iff | null |
continuousOn_const_smul_iff (c : G) :
ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s :=
forall₂_congr fun _ _ => continuousWithinAt_const_smul_iff c
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | continuousOn_const_smul_iff | null |
continuousAt_const_smul_iff (c : G) :
ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b :=
tendsto_const_smul_iff c
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | continuousAt_const_smul_iff | null |
continuous_const_smul_iff (c : G) : (Continuous fun x => c • f x) ↔ Continuous f := by
simp only [continuous_iff_continuousAt, continuousAt_const_smul_iff] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | continuous_const_smul_iff | null |
@[to_additive (attr := simps!)]
Homeomorph.smul (γ : G) : α ≃ₜ α where
toEquiv := MulAction.toPerm γ
continuous_toFun := continuous_const_smul γ
continuous_invFun := continuous_const_smul γ⁻¹ | def | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Homeomorph.smul | The homeomorphism given by scalar multiplication by a given element of a group `Γ` acting on
`T` is a homeomorphism from `T` to itself. |
@[simps! -fullyApplied apply]
protected Homeomorph.smulOfNeZero (c : G₀) (hc : c ≠ 0) : α ≃ₜ α :=
Homeomorph.smul (Units.mk0 c hc)
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Homeomorph.smulOfNeZero | The homeomorphism given by affine-addition by an element of an additive group `Γ` acting on
`T` is a homeomorphism from `T` to itself. -/
add_decl_doc Homeomorph.vadd
@[to_additive]
theorem isOpenMap_smul (c : G) : IsOpenMap fun x : α => c • x :=
(Homeomorph.smul c).isOpenMap
@[to_additive]
theorem IsOpen.smul {s : Set α} (hs : IsOpen s) (c : G) : IsOpen (c • s) :=
isOpenMap_smul c s hs
@[to_additive]
theorem isClosedMap_smul (c : G) : IsClosedMap fun x : α => c • x :=
(Homeomorph.smul c).isClosedMap
@[to_additive]
theorem IsClosed.smul {s : Set α} (hs : IsClosed s) (c : G) : IsClosed (c • s) :=
isClosedMap_smul c s hs
@[to_additive]
theorem closure_smul (c : G) (s : Set α) : closure (c • s) = c • closure s :=
((Homeomorph.smul c).image_closure s).symm
@[to_additive]
theorem Dense.smul (c : G) {s : Set α} (hs : Dense s) : Dense (c • s) := by
rw [dense_iff_closure_eq] at hs ⊢; rw [closure_smul, hs, smul_set_univ]
@[to_additive]
theorem interior_smul (c : G) (s : Set α) : interior (c • s) = c • interior s :=
((Homeomorph.smul c).image_interior s).symm
@[to_additive]
theorem IsOpen.smul_left {s : Set G} {t : Set α} (ht : IsOpen t) : IsOpen (s • t) := by
rw [← iUnion_smul_set]
exact isOpen_biUnion fun a _ => ht.smul _
@[to_additive]
theorem subset_interior_smul_right {s : Set G} {t : Set α} : s • interior t ⊆ interior (s • t) :=
interior_maximal (Set.smul_subset_smul_left interior_subset) isOpen_interior.smul_left
@[to_additive (attr := simp)]
theorem smul_mem_nhds_smul_iff {t : Set α} (g : G) {a : α} : g • t ∈ 𝓝 (g • a) ↔ t ∈ 𝓝 a :=
(Homeomorph.smul g).isOpenEmbedding.image_mem_nhds
@[to_additive] alias ⟨_, smul_mem_nhds_smul⟩ := smul_mem_nhds_smul_iff
@[to_additive (attr := simp)]
theorem smul_mem_nhds_self [TopologicalSpace G] [ContinuousConstSMul G G] {g : G} {s : Set G} :
g • s ∈ 𝓝 g ↔ s ∈ 𝓝 1 := by
rw [← smul_mem_nhds_smul_iff g⁻¹]; simp
end Group
section GroupWithZero
variable {G₀ : Type*} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α]
[ContinuousConstSMul G₀ α]
theorem tendsto_const_smul_iff₀ {f : β → α} {l : Filter β} {a : α} {c : G₀} (hc : c ≠ 0) :
Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) :=
tendsto_const_smul_iff (Units.mk0 c hc)
variable [TopologicalSpace β] {f : β → α} {b : β} {c : G₀} {s : Set β}
theorem continuousWithinAt_const_smul_iff₀ (hc : c ≠ 0) :
ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b :=
tendsto_const_smul_iff (Units.mk0 c hc)
theorem continuousOn_const_smul_iff₀ (hc : c ≠ 0) :
ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s :=
continuousOn_const_smul_iff (Units.mk0 c hc)
theorem continuousAt_const_smul_iff₀ (hc : c ≠ 0) :
ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b :=
continuousAt_const_smul_iff (Units.mk0 c hc)
theorem continuous_const_smul_iff₀ (hc : c ≠ 0) : (Continuous fun x => c • f x) ↔ Continuous f :=
continuous_const_smul_iff (Units.mk0 c hc)
/-- Scalar multiplication by a non-zero element of a group with zero acting on `α` is a
homeomorphism from `α` onto itself. |
Homeomorph.smulOfNeZero_symm_apply {c : G₀} (hc : c ≠ 0) :
⇑(Homeomorph.smulOfNeZero c hc).symm = (c⁻¹ • · : α → α) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | Homeomorph.smulOfNeZero_symm_apply | null |
isOpenMap_smul₀ {c : G₀} (hc : c ≠ 0) : IsOpenMap fun x : α => c • x :=
(Homeomorph.smulOfNeZero c hc).isOpenMap | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | isOpenMap_smul₀ | null |
IsOpen.smul₀ {c : G₀} {s : Set α} (hs : IsOpen s) (hc : c ≠ 0) : IsOpen (c • s) :=
isOpenMap_smul₀ hc s hs | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | IsOpen.smul₀ | null |
interior_smul₀ {c : G₀} (hc : c ≠ 0) (s : Set α) : interior (c • s) = c • interior s :=
((Homeomorph.smulOfNeZero c hc).image_interior s).symm | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | interior_smul₀ | null |
closure_smul₀' {c : G₀} (hc : c ≠ 0) (s : Set α) :
closure (c • s) = c • closure s :=
((Homeomorph.smulOfNeZero c hc).image_closure s).symm | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | closure_smul₀' | null |
closure_smul₀ {E} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E]
[ContinuousConstSMul G₀ E] (c : G₀) (s : Set E) : closure (c • s) = c • closure s := by
rcases eq_or_ne c 0 with (rfl | hc)
· rcases eq_empty_or_nonempty s with (rfl | hs)
· simp
· rw [zero_smul_set hs, zero_smul_set hs.closure]
exact closure_singleton
· exact closure_smul₀' hc s | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | closure_smul₀ | null |
isClosedMap_smul_of_ne_zero {c : G₀} (hc : c ≠ 0) : IsClosedMap fun x : α => c • x :=
(Homeomorph.smulOfNeZero c hc).isClosedMap | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | isClosedMap_smul_of_ne_zero | `smul` is a closed map in the second argument.
The lemma that `smul` is a closed map in the first argument (for a normed space over a complete
normed field) is `isClosedMap_smul_left` in `Analysis.Normed.Module.FiniteDimension`. |
IsClosed.smul_of_ne_zero {c : G₀} {s : Set α} (hs : IsClosed s) (hc : c ≠ 0) :
IsClosed (c • s) :=
isClosedMap_smul_of_ne_zero hc s hs | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | IsClosed.smul_of_ne_zero | null |
isClosedMap_smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
[T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) : IsClosedMap fun x : E => c • x := by
rcases eq_or_ne c 0 with (rfl | hne)
· simp only [zero_smul]
exact isClosedMap_const
· exact (Homeomorph.smulOfNeZero c hne).isClosedMap | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | isClosedMap_smul₀ | `smul` is a closed map in the second argument.
The lemma that `smul` is a closed map in the first argument (for a normed space over a complete
normed field) is `isClosedMap_smul_left` in `Analysis.Normed.Module.FiniteDimension`. |
IsClosed.smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
[T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) {s : Set E} (hs : IsClosed s) :
IsClosed (c • s) :=
isClosedMap_smul₀ c s hs | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | IsClosed.smul₀ | null |
HasCompactMulSupport.comp_smul {β : Type*} [One β] {f : α → β} (h : HasCompactMulSupport f)
{c : G₀} (hc : c ≠ 0) : HasCompactMulSupport fun x => f (c • x) :=
h.comp_homeomorph (Homeomorph.smulOfNeZero c hc) | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | HasCompactMulSupport.comp_smul | null |
HasCompactSupport.comp_smul {β : Type*} [Zero β] {f : α → β} (h : HasCompactSupport f)
{c : G₀} (hc : c ≠ 0) : HasCompactSupport fun x => f (c • x) :=
h.comp_homeomorph (Homeomorph.smulOfNeZero c hc) | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | HasCompactSupport.comp_smul | null |
ProperlyDiscontinuousSMul (Γ : Type*) (T : Type*) [TopologicalSpace T] [SMul Γ T] :
Prop where
/-- Given two compact sets `K` and `L`, `γ • K ∩ L` is nonempty for finitely many `γ`. -/
finite_disjoint_inter_image :
∀ {K L : Set T}, IsCompact K → IsCompact L → Set.Finite { γ : Γ | (γ • ·) '' K ∩ L ≠ ∅ } | class | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ProperlyDiscontinuousSMul | Class `ProperlyDiscontinuousSMul Γ T` says that the scalar multiplication `(•) : Γ → T → T`
is properly discontinuous, that is, for any pair of compact sets `K, L` in `T`, only finitely many
`γ:Γ` move `K` to have nontrivial intersection with `L`. |
ProperlyDiscontinuousVAdd (Γ : Type*) (T : Type*) [TopologicalSpace T] [VAdd Γ T] :
Prop where
/-- Given two compact sets `K` and `L`, `γ +ᵥ K ∩ L` is nonempty for finitely many `γ`. -/
finite_disjoint_inter_image :
∀ {K L : Set T}, IsCompact K → IsCompact L → Set.Finite { γ : Γ | (γ +ᵥ ·) '' K ∩ L ≠ ∅ }
attribute [to_additive] ProperlyDiscontinuousSMul
variable {Γ : Type*} [Group Γ] {T : Type*} [TopologicalSpace T] [MulAction Γ T] | class | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ProperlyDiscontinuousVAdd | Class `ProperlyDiscontinuousVAdd Γ T` says that the additive action `(+ᵥ) : Γ → T → T`
is properly discontinuous, that is, for any pair of compact sets `K, L` in `T`, only finitely many
`γ:Γ` move `K` to have nontrivial intersection with `L`. |
@[to_additive /-- The quotient map by a group action is open, i.e. the quotient by a group
action is an open quotient. -/]
isOpenMap_quotient_mk'_mul [ContinuousConstSMul Γ T] :
letI := MulAction.orbitRel Γ T
IsOpenMap (Quotient.mk' : T → Quotient (MulAction.orbitRel Γ T)) := fun U hU => by
rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U]
exact isOpen_iUnion fun γ => isOpenMap_smul γ U hU
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | isOpenMap_quotient_mk'_mul | A finite group action is always properly discontinuous. -/
@[to_additive /-- A finite group action is always properly discontinuous. -/]
instance (priority := 100) Finite.to_properlyDiscontinuousSMul [Finite Γ] :
ProperlyDiscontinuousSMul Γ T where finite_disjoint_inter_image _ _ := Set.toFinite _
export ProperlyDiscontinuousSMul (finite_disjoint_inter_image)
export ProperlyDiscontinuousVAdd (finite_disjoint_inter_image)
/-- The quotient map by a group action is open, i.e. the quotient by a group action is an open
quotient. |
MulAction.isOpenQuotientMap_quotientMk [ContinuousConstSMul Γ T] :
IsOpenQuotientMap (Quotient.mk (MulAction.orbitRel Γ T)) :=
⟨Quot.mk_surjective, continuous_quot_mk, isOpenMap_quotient_mk'_mul⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | MulAction.isOpenQuotientMap_quotientMk | null |
@[to_additive /-- The quotient of a second countable space by an additive group action is second
countable. -/]
ContinuousConstSMul.secondCountableTopology [SecondCountableTopology T]
[ContinuousConstSMul Γ T] : SecondCountableTopology (Quotient (MulAction.orbitRel Γ T)) :=
TopologicalSpace.Quotient.secondCountableTopology isOpenMap_quotient_mk'_mul | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | ContinuousConstSMul.secondCountableTopology | The quotient by a discontinuous group action of a locally compact t2 space is t2. -/
@[to_additive /-- The quotient by a discontinuous group action of a locally compact t2
space is t2. -/]
instance (priority := 100) t2Space_of_properlyDiscontinuousSMul_of_t2Space [T2Space T]
[LocallyCompactSpace T] [ContinuousConstSMul Γ T] [ProperlyDiscontinuousSMul Γ T] :
T2Space (Quotient (MulAction.orbitRel Γ T)) := by
letI := MulAction.orbitRel Γ T
set Q := Quotient (MulAction.orbitRel Γ T)
rw [t2Space_iff_nhds]
let f : T → Q := Quotient.mk'
have f_op : IsOpenMap f := isOpenMap_quotient_mk'_mul
rintro ⟨x₀⟩ ⟨y₀⟩ (hxy : f x₀ ≠ f y₀)
change ∃ U ∈ 𝓝 (f x₀), ∃ V ∈ 𝓝 (f y₀), _
have hγx₀y₀ : ∀ γ : Γ, γ • x₀ ≠ y₀ := not_exists.mp (mt Quotient.sound hxy.symm :)
obtain ⟨K₀, hK₀, K₀_in⟩ := exists_compact_mem_nhds x₀
obtain ⟨L₀, hL₀, L₀_in⟩ := exists_compact_mem_nhds y₀
let bad_Γ_set := { γ : Γ | (γ • ·) '' K₀ ∩ L₀ ≠ ∅ }
have bad_Γ_finite : bad_Γ_set.Finite := finite_disjoint_inter_image (Γ := Γ) hK₀ hL₀
choose u v hu hv u_v_disjoint using fun γ => t2_separation_nhds (hγx₀y₀ γ)
let U₀₀ := ⋂ γ ∈ bad_Γ_set, (γ • ·) ⁻¹' u γ
let U₀ := U₀₀ ∩ K₀
let V₀₀ := ⋂ γ ∈ bad_Γ_set, v γ
let V₀ := V₀₀ ∩ L₀
have U_nhds : f '' U₀ ∈ 𝓝 (f x₀) := by
refine f_op.image_mem_nhds (inter_mem ((biInter_mem bad_Γ_finite).mpr fun γ _ => ?_) K₀_in)
exact (continuous_const_smul _).continuousAt (hu γ)
have V_nhds : f '' V₀ ∈ 𝓝 (f y₀) :=
f_op.image_mem_nhds (inter_mem ((biInter_mem bad_Γ_finite).mpr fun γ _ => hv γ) L₀_in)
refine ⟨f '' U₀, U_nhds, f '' V₀, V_nhds, MulAction.disjoint_image_image_iff.2 ?_⟩
rintro x ⟨x_in_U₀₀, x_in_K₀⟩ γ
by_cases H : γ ∈ bad_Γ_set
· exact fun h => (u_v_disjoint γ).le_bot ⟨mem_iInter₂.mp x_in_U₀₀ γ H, mem_iInter₂.mp h.1 γ H⟩
· rintro ⟨-, h'⟩
simp only [bad_Γ_set, image_smul, Classical.not_not, mem_setOf_eq, Ne] at H
exact eq_empty_iff_forall_notMem.mp H (γ • x) ⟨mem_image_of_mem _ x_in_K₀, h'⟩
/-- The quotient of a second countable space by a group action is second countable. |
smul_mem_nhds_smul_iff₀ {c : G₀} {s : Set α} {x : α} (hc : c ≠ 0) :
c • s ∈ 𝓝 (c • x : α) ↔ s ∈ 𝓝 x :=
smul_mem_nhds_smul_iff (Units.mk0 c hc)
alias ⟨_, smul_mem_nhds_smul₀⟩ := smul_mem_nhds_smul_iff₀ | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | smul_mem_nhds_smul_iff₀ | Scalar multiplication by a nonzero scalar preserves neighborhoods. |
set_smul_mem_nhds_zero_iff {s : Set α} {c : G₀} (hc : c ≠ 0) :
c • s ∈ 𝓝 (0 : α) ↔ s ∈ 𝓝 (0 : α) := by
refine Iff.trans ?_ (smul_mem_nhds_smul_iff₀ hc)
rw [smul_zero] | theorem | Topology | [
"Mathlib.Algebra.Group.Pointwise.Set.Lattice",
"Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set",
"Mathlib.Algebra.Module.ULift",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/ConstMulAction.lean | set_smul_mem_nhds_zero_iff | null |
@[to_additive /-- Put the same topological space structure on the opposite monoid as on the original
space. -/]
instTopologicalSpaceMulOpposite [TopologicalSpace M] : TopologicalSpace Mᵐᵒᵖ :=
TopologicalSpace.induced (unop : Mᵐᵒᵖ → M) ‹_›
variable [TopologicalSpace M]
@[to_additive (attr := continuity, fun_prop)] | instance | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | instTopologicalSpaceMulOpposite | Put the same topological space structure on the opposite monoid as on the original space. |
continuous_unop : Continuous (unop : Mᵐᵒᵖ → M) :=
continuous_induced_dom
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Homeomorph.Lemmas"
] | Mathlib/Topology/Algebra/Constructions.lean | continuous_unop | null |
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