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 ⌀ |
|---|---|---|---|---|---|---|
toSubgroup_top : ((⊤ : OpenSubgroup G) : Subgroup G) = ⊤ :=
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 | toSubgroup_top | null |
toOpens_top : ((⊤ : OpenSubgroup G) : Opens G) = ⊤ :=
rfl
@[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 | toOpens_top | null |
@[to_additive]
isClosed [ContinuousMul G] (U : OpenSubgroup G) : IsClosed (U : Set G) := by
apply isOpen_compl_iff.1
refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ⟨(fun y ↦ y * x⁻¹) ⁻¹' U, ?_, ?_, ?_⟩
· refine fun u hux hu ↦ hx ?_
simp only [Set.mem_preimage, SetLike.mem_coe] at hux hu ⊢
convert U.mul_mem (U.inv_mem hux) hu
simp
· exact U.isOpen.preimage (continuous_mul_right _)
· simp [one_mem]
@[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 | isClosed | null |
isClopen [ContinuousMul G] (U : OpenSubgroup G) : IsClopen (U : Set G) :=
⟨U.isClosed, U.isOpen⟩ | 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 | isClopen | null |
@[to_additive prod
/-- The product of two open subgroups as an open subgroup of the product group. -/]
prod (U : OpenSubgroup G) (V : OpenSubgroup H) : OpenSubgroup (G × H) :=
⟨.prod U V, U.isOpen.prod V.isOpen⟩
@[deprecated (since := "2025-03-11")]
alias _root_.OpenAddSubgroup.sum := OpenAddSubgroup.prod
@[to_additive (attr := simp, norm_cast) coe_prod] | 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 | prod | The product of two open subgroups as an open subgroup of the product group. |
coe_prod (U : OpenSubgroup G) (V : OpenSubgroup H) :
(U.prod V : Set (G × H)) = (U : Set G) ×ˢ (V : Set H) :=
rfl
@[deprecated (since := "2025-03-11")]
alias _root_.OpenAddSubgroup.coe_sum := OpenAddSubgroup.coe_prod
@[to_additive (attr := simp, norm_cast) toAddSubgroup_prod] | 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_prod | null |
toSubgroup_prod (U : OpenSubgroup G) (V : OpenSubgroup H) :
(U.prod V : Subgroup (G × H)) = (U : Subgroup G).prod V :=
rfl
@[deprecated (since := "2025-03-11")]
alias _root_.OpenAddSubgroup.toAddSubgroup_sum := OpenAddSubgroup.toAddSubgroup_prod | 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 | toSubgroup_prod | null |
@[to_additive]
instInfOpenSubgroup : Min (OpenSubgroup G) :=
⟨fun U V ↦ ⟨U ⊓ V, U.isOpen.inter V.isOpen⟩⟩
@[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 | instInfOpenSubgroup | null |
coe_inf : (↑(U ⊓ V) : Set G) = (U : Set G) ∩ V :=
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_inf | null |
toSubgroup_inf : (↑(U ⊓ V) : Subgroup G) = ↑U ⊓ ↑V :=
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 | toSubgroup_inf | null |
toOpens_inf : (↑(U ⊓ V) : Opens G) = ↑U ⊓ ↑V :=
rfl
@[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 | toOpens_inf | null |
mem_inf {x} : x ∈ U ⊓ V ↔ x ∈ U ∧ x ∈ V :=
Iff.rfl
@[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 | mem_inf | null |
instPartialOrderOpenSubgroup : PartialOrder (OpenSubgroup G) := inferInstance
@[to_additive] | 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 | instPartialOrderOpenSubgroup | null |
instSemilatticeInfOpenSubgroup : SemilatticeInf (OpenSubgroup G) :=
{ SetLike.coe_injective.semilatticeInf ((↑) : OpenSubgroup G → Set G) fun _ _ ↦ rfl with
toPartialOrder := instPartialOrderOpenSubgroup }
@[to_additive] | 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 | instSemilatticeInfOpenSubgroup | null |
@[to_additive (attr := simp, norm_cast)]
toSubgroup_le : (U : Subgroup G) ≤ (V : Subgroup G) ↔ U ≤ V :=
Iff.rfl
variable {N : Type*} [Group N] [TopologicalSpace N] | 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 | toSubgroup_le | null |
@[to_additive /-- The preimage of an `OpenAddSubgroup` along a continuous `AddMonoid` homomorphism
is an `OpenAddSubgroup`. -/]
comap (f : G →* N) (hf : Continuous f) (H : OpenSubgroup N) : OpenSubgroup G :=
⟨.comap f H, H.isOpen.preimage hf⟩
@[to_additive (attr := simp, norm_cast)] | 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 | comap | The preimage of an `OpenSubgroup` along a continuous `Monoid` homomorphism
is an `OpenSubgroup`. |
coe_comap (H : OpenSubgroup N) (f : G →* N) (hf : Continuous f) :
(H.comap f hf : Set G) = f ⁻¹' H :=
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_comap | null |
toSubgroup_comap (H : OpenSubgroup N) (f : G →* N) (hf : Continuous f) :
(H.comap f hf : Subgroup G) = (H : Subgroup N).comap f :=
rfl
@[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 | toSubgroup_comap | null |
mem_comap {H : OpenSubgroup N} {f : G →* N} {hf : Continuous f} {x : G} :
x ∈ H.comap f hf ↔ f x ∈ H :=
Iff.rfl
@[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 | mem_comap | null |
comap_comap {P : Type*} [Group P] [TopologicalSpace P] (K : OpenSubgroup P) (f₂ : N →* P)
(hf₂ : Continuous f₂) (f₁ : G →* N) (hf₁ : Continuous f₁) :
(K.comap f₂ hf₂).comap f₁ hf₁ = K.comap (f₂.comp f₁) (hf₂.comp hf₁) :=
rfl | 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 | comap_comap | null |
@[to_additive]
isOpen_of_mem_nhds [ContinuousMul G] (H : Subgroup G) {g : G} (hg : (H : Set G) ∈ 𝓝 g) :
IsOpen (H : Set G) := by
refine isOpen_iff_mem_nhds.2 fun x hx ↦ ?_
have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)
have : Filter.Tendsto (fun y ↦ y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=
(continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _)
simpa only [SetLike.mem_coe, Filter.mem_map',
H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg')] using this hg
@[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_of_mem_nhds | null |
isOpen_mono [ContinuousMul G] {H₁ H₂ : Subgroup G} (h : H₁ ≤ H₂)
(h₁ : IsOpen (H₁ : Set G)) : IsOpen (H₂ : Set G) :=
isOpen_of_mem_nhds _ <| Filter.mem_of_superset (h₁.mem_nhds <| one_mem H₁) h
@[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_mono | null |
isOpen_of_openSubgroup
[ContinuousMul G] (H : Subgroup G) {U : OpenSubgroup G} (h : ↑U ≤ H) :
IsOpen (H : Set G) :=
isOpen_mono h U.isOpen | 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_of_openSubgroup | null |
@[to_additive /-- If a subgroup of an additive topological group has `0` in its interior, then it is
open. -/]
isOpen_of_one_mem_interior [ContinuousMul G] (H : Subgroup G)
(h_1_int : (1 : G) ∈ interior (H : Set G)) : IsOpen (H : Set G) :=
isOpen_of_mem_nhds H <| mem_interior_iff_mem_nhds.1 h_1_int
@[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_of_one_mem_interior | If a subgroup of a topological group has `1` in its interior, then it is open. |
isClosed_of_isOpen [ContinuousMul G] (U : Subgroup G) (h : IsOpen (U : Set G)) :
IsClosed (U : Set G) :=
OpenSubgroup.isClosed ⟨U, h⟩
@[to_additive] | lemma | 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 | isClosed_of_isOpen | null |
subgroupOf_isOpen (U K : Subgroup G) (h : IsOpen (K : Set G)) :
IsOpen (K.subgroupOf U : Set U) :=
Continuous.isOpen_preimage (continuous_iff_le_induced.mpr fun _ ↦ id) _ h
@[to_additive] | lemma | 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 | subgroupOf_isOpen | null |
discreteTopology [ContinuousMul G] (U : Subgroup G) (h : IsOpen (U : Set G)) :
DiscreteTopology (G ⧸ U) := by
refine singletons_open_iff_discrete.mp (fun g ↦ ?_)
induction g using Quotient.inductionOn with | h g =>
change IsOpen (QuotientGroup.mk ⁻¹' {QuotientGroup.mk g})
convert_to IsOpen ((g * ·) '' U)
· ext g'
simp only [Set.mem_preimage, Set.mem_singleton_iff, QuotientGroup.eq, Set.image_mul_left]
rw [← U.inv_mem_iff]
simp
· exact Homeomorph.mulLeft g |>.isOpen_image |>.mpr h
@[to_additive] | lemma | 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 | discreteTopology | null |
@[to_additive]
quotient_finite_of_isOpen [ContinuousMul G] [CompactSpace G] (U : Subgroup G)
(h : IsOpen (U : Set G)) : Finite (G ⧸ U) :=
have : DiscreteTopology (G ⧸ U) := U.discreteTopology h
finite_of_compact_of_discrete
@[to_additive] | lemma | 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 | quotient_finite_of_isOpen | null |
@[to_additive]
quotient_finite_of_isOpen' [IsTopologicalGroup G] [CompactSpace G] (U : Subgroup G)
(K : Subgroup U) (hUopen : IsOpen (U : Set G)) (hKopen : IsOpen (K : Set U)) :
Finite (U ⧸ K) :=
have : CompactSpace U := isCompact_iff_compactSpace.mp <| IsClosed.isCompact <|
U.isClosed_of_isOpen hUopen
K.quotient_finite_of_isOpen hKopen
@[to_additive] | lemma | 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 | quotient_finite_of_isOpen' | null |
@[to_additive (attr := simp, norm_cast)]
toSubgroup_sup (U V : OpenSubgroup G) : (↑(U ⊔ V) : Subgroup G) = ↑U ⊔ ↑V := rfl
@[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 | toSubgroup_sup | null |
isOpen_mono {U P : Submodule R M} (h : U ≤ P) (hU : IsOpen (U : Set M)) :
IsOpen (P : Set M) :=
@AddSubgroup.isOpen_mono M _ _ _ U.toAddSubgroup P.toAddSubgroup h hU | 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_mono | null |
isOpen_of_isOpen_subideal {U I : Ideal R} (h : U ≤ I) (hU : IsOpen (U : Set R)) :
IsOpen (I : Set R) :=
@Submodule.isOpen_mono R R _ _ _ _ Semiring.toModule _ _ h hU | 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_of_isOpen_subideal | null |
@[ext]
OpenNormalSubgroup (G : Type u) [Group G] [TopologicalSpace G]
extends OpenSubgroup G where
isNormal' : toSubgroup.Normal := by infer_instance | 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 | OpenNormalSubgroup | The type of open normal subgroups of a topological group. |
@[ext]
OpenNormalAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G]
extends OpenAddSubgroup G where
isNormal' : toAddSubgroup.Normal := by infer_instance
attribute [to_additive] OpenNormalSubgroup | 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 | OpenNormalAddSubgroup | The type of open normal subgroups of a topological additive group. |
@[to_additive]
toSubgroup_injective : Function.Injective
(fun H ↦ H.toOpenSubgroup.toSubgroup : OpenNormalSubgroup G → Subgroup G) :=
fun A B h ↦ by
ext
dsimp at h
rw [h]
@[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 | toSubgroup_injective | null |
@[to_additive]
instPartialOrderOpenNormalSubgroup : PartialOrder (OpenNormalSubgroup G) := inferInstance
@[to_additive] | 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 | instPartialOrderOpenNormalSubgroup | null |
instInfOpenNormalSubgroup : Min (OpenNormalSubgroup G) :=
⟨fun U V ↦ ⟨U.toOpenSubgroup ⊓ V.toOpenSubgroup,
Subgroup.normal_inf_normal U.toSubgroup V.toSubgroup⟩⟩
@[to_additive] | 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 | instInfOpenNormalSubgroup | null |
instSemilatticeInfOpenNormalSubgroup : SemilatticeInf (OpenNormalSubgroup G) :=
SetLike.coe_injective.semilatticeInf ((↑) : OpenNormalSubgroup G → Set G) fun _ _ ↦ rfl
@[to_additive] | 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 | instSemilatticeInfOpenNormalSubgroup | null |
@[to_additive]
instSemilatticeSupOpenNormalSubgroup [ContinuousMul G] :
SemilatticeSup (OpenNormalSubgroup G) :=
toSubgroup_injective.semilatticeSup _ (fun _ _ ↦ rfl)
@[to_additive] | 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 | instSemilatticeSupOpenNormalSubgroup | null |
IsTopologicalAddGroup.addNegClosureNhd (T W : Set G) [AddGroup G] : Prop where
nhds : T ∈ 𝓝 0
neg : -T = T
isOpen : IsOpen T
add : W + T ⊆ W | 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 | IsTopologicalAddGroup.addNegClosureNhd | null |
@[to_additive
/-- For a set `W`, `T` is a neighborhood of `0` which is open, stable under negation and satisfies
`T + W ⊆ W`. -/]
IsTopologicalGroup.mulInvClosureNhd (T W : Set G) [Group G] : Prop where
nhds : T ∈ 𝓝 1
inv : T⁻¹ = T
isOpen : IsOpen T
mul : W * T ⊆ W | 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 | IsTopologicalGroup.mulInvClosureNhd | For a set `W`, `T` is a neighborhood of `1` which is open, stable under inverse and satisfies
`T * W ⊆ W`. |
@[to_additive]
exist_mul_closure_nhds {W : Set G} (WClopen : IsClopen W) : ∃ T ∈ 𝓝 (1 : G), W * T ⊆ W := by
apply WClopen.isClosed.isCompact.induction_on (p := fun S ↦ ∃ T ∈ 𝓝 (1 : G), S * T ⊆ W)
⟨Set.univ, by simp only [univ_mem, empty_mul, empty_subset, and_self]⟩
(fun _ _ huv ⟨T, hT, mem⟩ ↦ ⟨T, hT, (mul_subset_mul_right huv).trans mem⟩)
fun U V ⟨T₁, hT₁, mem1⟩ ⟨T₂, hT₂, mem2⟩ ↦ ⟨T₁ ∩ T₂, inter_mem hT₁ hT₂, by
rw [union_mul]
exact union_subset (mul_subset_mul_left inter_subset_left |>.trans mem1)
(mul_subset_mul_left inter_subset_right |>.trans mem2) ⟩
intro x memW
have : (x, 1) ∈ (fun p ↦ p.1 * p.2) ⁻¹' W := by simp [memW]
rcases isOpen_prod_iff.mp (continuous_mul.isOpen_preimage W <| WClopen.2) x 1 this with
⟨U, V, Uopen, Vopen, xmemU, onememV, prodsub⟩
have h6 : U * V ⊆ W := mul_subset_iff.mpr (fun _ hx _ hy ↦ prodsub (mk_mem_prod hx hy))
exact ⟨U ∩ W, ⟨U, Uopen.mem_nhds xmemU, W, fun _ a ↦ a, rfl⟩,
V, IsOpen.mem_nhds Vopen onememV, fun _ a ↦ h6 ((mul_subset_mul_right inter_subset_left) a)⟩
@[deprecated (since := "2025-05-22")] alias exist_mul_closure_nhd := exist_mul_closure_nhds
@[deprecated (since := "2025-05-22")] alias _root_.IsTopologicalAddGroup.exist_add_closure_nhd :=
IsTopologicalAddGroup.exist_add_closure_nhds
@[to_additive] | lemma | 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 | exist_mul_closure_nhds | null |
exists_mulInvClosureNhd {W : Set G} (WClopen : IsClopen W) :
∃ T, mulInvClosureNhd T W := by
rcases exist_mul_closure_nhds WClopen with ⟨S, Smemnhds, mulclose⟩
rcases mem_nhds_iff.mp Smemnhds with ⟨U, UsubS, Uopen, onememU⟩
use U ∩ U⁻¹
constructor
· simp [Uopen.mem_nhds onememU, inv_mem_nhds_one]
· simp [inter_comm]
· exact Uopen.inter Uopen.inv
· exact fun a ha ↦ mulclose (mul_subset_mul_left UsubS (mul_subset_mul_left inter_subset_left ha))
@[to_additive] | lemma | 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 | exists_mulInvClosureNhd | null |
exist_openSubgroup_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 : OpenSubgroup G, (H : Set G) ⊆ W := by
rcases exists_mulInvClosureNhd WClopen with ⟨V, hV⟩
let S : Subgroup G := {
carrier := ⋃ n, V ^ (n + 1)
mul_mem' := fun ha hb ↦ by
rcases mem_iUnion.mp ha with ⟨k, hk⟩
rcases mem_iUnion.mp hb with ⟨l, hl⟩
apply mem_iUnion.mpr
use k + 1 + l
rw [add_assoc, pow_add]
exact Set.mul_mem_mul hk hl
one_mem' := by
apply mem_iUnion.mpr
use 0
simp [mem_of_mem_nhds hV.nhds]
inv_mem' := fun ha ↦ by
rcases mem_iUnion.mp ha with ⟨k, hk⟩
apply mem_iUnion.mpr
use k
rw [← hV.inv]
simpa only [inv_pow, Set.mem_inv, inv_inv] using hk }
have : IsOpen (⋃ n, V ^ (n + 1)) := by
refine isOpen_iUnion (fun n ↦ ?_)
rw [pow_succ]
exact hV.isOpen.mul_left
use ⟨S, this⟩
have mulVpow (n : ℕ) : W * V ^ (n + 1) ⊆ W := by
induction n with
| zero => simp [hV.mul]
| succ n ih =>
rw [pow_succ, ← mul_assoc]
exact (Set.mul_subset_mul_right ih).trans hV.mul
have (n : ℕ) : V ^ (n + 1) ⊆ W * V ^ (n + 1) := by
intro x xin
rw [Set.mem_mul]
use 1, einW, x, xin
rw [one_mul]
apply iUnion_subset fun i _ a ↦ mulVpow i (this i a)
@[deprecated (since := "2025-05-22")]
alias exist_openSubgroup_sub_clopen_nhd_of_one := exist_openSubgroup_sub_clopen_nhds_of_one
@[deprecated (since := "2025-05-22")]
alias _root_.IsTopologicalAddGroup.exist_openAddSubgroup_sub_clopen_nhd_of_zero :=
IsTopologicalAddGroup.exist_openAddSubgroup_sub_clopen_nhds_of_zero | 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 | exist_openSubgroup_sub_clopen_nhds_of_one | null |
@[continuity, fun_prop]
protected continuous_eval₂ [Semiring S] (p : S[X]) (f : S →+* R) :
Continuous fun x => p.eval₂ f x := by
simp only [eval₂_eq_sum]
exact continuous_finset_sum _ fun c _ => continuous_const.mul (continuous_pow _)
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuous_eval₂ | null |
protected continuous : Continuous fun x => p.eval x :=
p.continuous_eval₂ _
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuous | null |
protected continuousAt {a : R} : ContinuousAt (fun x => p.eval x) a :=
p.continuous.continuousAt
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousAt | null |
protected continuousWithinAt {s a} : ContinuousWithinAt (fun x => p.eval x) s a :=
p.continuous.continuousWithinAt
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousWithinAt | null |
protected continuousOn {s} : ContinuousOn (fun x => p.eval x) s :=
p.continuous.continuousOn | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousOn | null |
@[continuity, fun_prop]
protected continuous_aeval : Continuous fun x : A => aeval x p :=
p.continuous_eval₂ _
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuous_aeval | null |
protected continuousAt_aeval {a : A} : ContinuousAt (fun x : A => aeval x p) a :=
p.continuous_aeval.continuousAt
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousAt_aeval | null |
protected continuousWithinAt_aeval {s a} :
ContinuousWithinAt (fun x : A => aeval x p) s a :=
p.continuous_aeval.continuousWithinAt
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousWithinAt_aeval | null |
protected continuousOn_aeval {s} : ContinuousOn (fun x : A => aeval x p) s :=
p.continuous_aeval.continuousOn | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | continuousOn_aeval | null |
tendsto_abv_eval₂_atTop {R S k α : Type*} [Semiring R] [Ring S]
[Field k] [LinearOrder k] [IsStrictOrderedRing k]
(f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
(hf : f p.leadingCoeff ≠ 0) {l : Filter α} {z : α → S} (hz : Tendsto (abv ∘ z) l atTop) :
Tendsto (fun x => abv (p.eval₂ f (z x))) l atTop := by
revert hf; refine degree_pos_induction_on p hd ?_ ?_ ?_ <;> clear hd p
· rintro _ - hc
rw [leadingCoeff_mul_X, leadingCoeff_C] at hc
simpa [abv_mul abv] using hz.const_mul_atTop ((abv_pos abv).2 hc)
· intro _ _ ihp hf
rw [leadingCoeff_mul_X] at hf
simpa [abv_mul abv] using (ihp hf).atTop_mul_atTop₀ hz
· intro _ a hd ihp hf
rw [add_comm, leadingCoeff_add_of_degree_lt (degree_C_le.trans_lt hd)] at hf
refine .atTop_of_add_const (abv (-f a)) ?_
refine tendsto_atTop_mono (fun _ => abv_add abv _ _) ?_
simpa using ihp hf | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | tendsto_abv_eval₂_atTop | null |
tendsto_abv_atTop {R k α : Type*} [Ring R]
[Field k] [LinearOrder k] [IsStrictOrderedRing k] (abv : R → k)
[IsAbsoluteValue abv] (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
(hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (p.eval (z x))) l atTop := by
apply tendsto_abv_eval₂_atTop _ _ _ h _ hz
exact mt leadingCoeff_eq_zero.1 (ne_zero_of_degree_gt h) | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | tendsto_abv_atTop | null |
tendsto_abv_aeval_atTop {R A k α : Type*} [CommSemiring R] [Ring A] [Algebra R A]
[Field k] [LinearOrder k] [IsStrictOrderedRing k]
(abv : A → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
(h₀ : algebraMap R A p.leadingCoeff ≠ 0) {l : Filter α} {z : α → A}
(hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (aeval (z x) p)) l atTop :=
tendsto_abv_eval₂_atTop _ abv p hd h₀ hz
variable {α R : Type*} [NormedRing R] [IsAbsoluteValue (norm : R → ℝ)] | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | tendsto_abv_aeval_atTop | null |
tendsto_norm_atTop (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
(hz : Tendsto (fun x => ‖z x‖) l atTop) : Tendsto (fun x => ‖p.eval (z x)‖) l atTop :=
p.tendsto_abv_atTop norm h hz | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | tendsto_norm_atTop | null |
exists_forall_norm_le [ProperSpace R] (p : R[X]) : ∃ x, ∀ y, ‖p.eval x‖ ≤ ‖p.eval y‖ :=
if hp0 : 0 < degree p then
p.continuous.norm.exists_forall_le <| p.tendsto_norm_atTop hp0 tendsto_norm_cocompact_atTop
else
⟨p.coeff 0, by rw [eq_C_of_degree_le_zero (le_of_not_gt hp0)]; simp⟩ | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | exists_forall_norm_le | null |
eq_one_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) : p = 1 :=
h1.natDegree_eq_zero_iff_eq_one.mp (by
contrapose! hB
rw [← h1.natDegree_map f, natDegree_eq_card_roots' h2] at hB
obtain ⟨z, hz⟩ := card_pos_iff_exists_mem.mp (zero_lt_iff.mpr hB)
exact le_trans (norm_nonneg _) (h3 z hz)) | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | eq_one_of_roots_le | null |
coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i := by
obtain hB | hB := lt_or_ge B 0
· rw [eq_one_of_roots_le hB h1 h2 h3, Polynomial.map_one, natDegree_one, zero_tsub, pow_zero,
one_mul, coeff_one]
split_ifs with h <;> simp [h]
rw [← h1.natDegree_map f]
obtain hi | hi := lt_or_ge (map f p).natDegree i
· rw [coeff_eq_zero_of_natDegree_lt hi, norm_zero]
positivity
rw [coeff_eq_esymm_roots_of_splits ((splits_id_iff_splits f).2 h2) hi, (h1.map _).leadingCoeff,
one_mul, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
apply ((norm_multiset_sum_le _).trans <| sum_le_card_nsmul _ _ fun r hr => _).trans
· rw [Multiset.map_map, card_map, card_powersetCard, ← natDegree_eq_card_roots' h2,
Nat.choose_symm hi, mul_comm, nsmul_eq_mul]
intro r hr
simp_rw [Multiset.mem_map] at hr
obtain ⟨_, ⟨s, hs, rfl⟩, rfl⟩ := hr
rw [mem_powersetCard] at hs
lift B to ℝ≥0 using hB
rw [← coe_nnnorm, ← NNReal.coe_pow, NNReal.coe_le_coe, ← nnnormHom_apply, ← MonoidHom.coe_coe,
MonoidHom.map_multiset_prod]
refine (prod_le_pow_card _ B fun x hx => ?_).trans_eq (by rw [card_map, hs.2])
obtain ⟨z, hz, rfl⟩ := Multiset.mem_map.1 hx
exact h3 z (mem_of_le hs.1 hz) | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | coeff_le_of_roots_le | null |
coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1 : p.Monic)
(h2 : Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) (i : ℕ) :
‖(map f p).coeff i‖ ≤ max B 1 ^ d * d.choose (d / 2) := by
obtain hB | hB := le_or_gt 0 B
· apply (coeff_le_of_roots_le i h1 h2 h4).trans
calc
_ ≤ max B 1 ^ (p.natDegree - i) * p.natDegree.choose i := by gcongr; apply le_max_left
_ ≤ max B 1 ^ d * p.natDegree.choose i := by
gcongr
· apply le_max_right
· exact le_trans (Nat.sub_le _ _) h3
_ ≤ max B 1 ^ d * d.choose (d / 2) := by
gcongr; exact (i.choose_mono h3).trans (i.choose_le_middle d)
· rw [eq_one_of_roots_le hB h1 h2 h4, Polynomial.map_one, coeff_one]
refine le_trans ?_ (one_le_mul_of_one_le_of_one_le (one_le_pow₀ (le_max_right B 1)) ?_)
· split_ifs <;> norm_num
· exact mod_cast Nat.succ_le_iff.mpr (Nat.choose_pos (d.div_le_self 2)) | theorem | Topology | [
"Mathlib.Algebra.Polynomial.AlgebraMap",
"Mathlib.Algebra.Polynomial.Inductions",
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.RingTheory.Polynomial.Vieta",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Ring.Lemmas"
] | Mathlib/Topology/Algebra/Polynomial.lean | coeff_bdd_of_roots_le | The coefficients of the monic polynomials of bounded degree with bounded roots are
uniformly bounded. |
PontryaginDual :=
A →ₜ* Circle
deriving TopologicalSpace | def | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | PontryaginDual | The Pontryagin dual of `A` is the group of continuous homomorphism `A → Circle`. |
map (f : A →ₜ* B) :
(PontryaginDual B) →ₜ* (PontryaginDual A) :=
f.compLeft Circle
@[simp] | def | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | map | A discrete monoid has compact Pontryagin dual. -/
add_decl_doc instLocallyCompactSpacePontryaginDual
/-- `PontryaginDual` is a contravariant functor. |
map_apply (f : A →ₜ* B) (x : PontryaginDual B) (y : A) :
map f x y = x (f y) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | map_apply | null |
map_one : map (1 : A →ₜ* B) = 1 :=
ext fun x => ext (fun _y => OneHomClass.map_one x)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | map_one | null |
map_comp (g : B →ₜ* C) (f : A →ₜ* B) :
map (comp g f) = ContinuousMonoidHom.comp (map f) (map g) :=
ext fun _x => ext fun _y => rfl
@[simp]
nonrec theorem map_mul (f g : A →ₜ* G) : map (f * g) = map f * map g :=
ext fun x => ext fun y => map_mul x (f y) (g y)
variable (A B C G) | theorem | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | map_comp | null |
mapHom [LocallyCompactSpace G] :
(A →ₜ* G) →ₜ* ((PontryaginDual G) →ₜ* (PontryaginDual A)) where
toFun := map
map_one' := map_one
map_mul' := map_mul
continuous_toFun := continuous_of_continuous_uncurry _ continuous_comp | def | Topology | [
"Mathlib.Analysis.SpecialFunctions.Complex.Circle",
"Mathlib.Topology.Algebra.Group.CompactOpen"
] | Mathlib/Topology/Algebra/PontryaginDual.lean | mapHom | `ContinuousMonoidHom.dual` as a `ContinuousMonoidHom`. |
ProperConstVAdd (M X : Type*) [VAdd M X] [TopologicalSpace X] : Prop where
/-- `(c +ᵥ ·)` is a proper map. -/
isProperMap_vadd (c : M) : IsProperMap ((c +ᵥ ·) : X → X) | class | Topology | [
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Maps.Proper.Basic"
] | Mathlib/Topology/Algebra/ProperConstSMul.lean | ProperConstVAdd | A mixin typeclass saying that the `(c +ᵥ ·)` is a proper map for all `c`.
Note that this is **not** the same as a proper additive action (not yet in `Mathlib`). |
@[to_additive]
ProperConstSMul (M X : Type*) [SMul M X] [TopologicalSpace X] : Prop where
/-- `(c • ·)` is a proper map. -/
isProperMap_smul (c : M) : IsProperMap ((c • ·) : X → X) | class | Topology | [
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Maps.Proper.Basic"
] | Mathlib/Topology/Algebra/ProperConstSMul.lean | ProperConstSMul | A mixin typeclass saying that `(c • ·)` is a proper map for all `c`.
Note that this is **not** the same as a proper multiplicative action (not yet in `Mathlib`). |
@[to_additive /-- `(c +ᵥ ·)` is a proper map. -/]
isProperMap_smul {M : Type*} (c : M) (X : Type*) [SMul M X] [TopologicalSpace X]
[h : ProperConstSMul M X] : IsProperMap ((c • ·) : X → X) := h.1 c | theorem | Topology | [
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Maps.Proper.Basic"
] | Mathlib/Topology/Algebra/ProperConstSMul.lean | isProperMap_smul | `(c • ·)` is a proper map. |
@[to_additive /-- The preimage of a compact set under `(c +ᵥ ·)` is a compact set. -/]
IsCompact.preimage_smul {M X : Type*} [SMul M X] [TopologicalSpace X]
[ProperConstSMul M X] {s : Set X} (hs : IsCompact s) (c : M) : IsCompact ((c • ·) ⁻¹' s) :=
(isProperMap_smul c X).isCompact_preimage hs
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Maps.Proper.Basic"
] | Mathlib/Topology/Algebra/ProperConstSMul.lean | IsCompact.preimage_smul | The preimage of a compact set under `(c • ·)` is a compact set. |
@[to_additive
/-- Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an idempotent, i.e. an `m` such that `m + m = m` -/]
exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
rsuffices ⟨N, hN⟩ : ∃ N', Minimal (· ∈ S) N'
· obtain ⟨N_closed, ⟨m, hm⟩, N_mul⟩ := hN.prop
use m
/- We now have an element `m : M` of a minimal subsemigroup `N`, and want to show `m + m = m`.
We first show that every element of `N` is of the form `m' + m`. -/
have scaling_eq_self : (· * m) '' N = N := by
apply hN.eq_of_subset
· refine ⟨(continuous_mul_left m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, ?_⟩
rintro _ ⟨m'', hm'', rfl⟩ _ ⟨m', hm', rfl⟩
exact ⟨m'' * m * m', N_mul _ (N_mul _ hm'' _ hm) _ hm', mul_assoc _ _ _⟩
· rintro _ ⟨m', hm', rfl⟩
exact N_mul _ hm' _ hm
/- In particular, this means that `m' * m = m` for some `m'`. We now use minimality again
to show that this holds for all `m' ∈ N`. -/
have absorbing_eq_self : N ∩ { m' | m' * m = m } = N := by
apply hN.eq_of_subset
· refine ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_mul_left m)), ?_, ?_⟩
· rwa [← scaling_eq_self] at hm
· rintro m'' ⟨mem'', eq'' : _ = m⟩ m' ⟨mem', eq' : _ = m⟩
refine ⟨N_mul _ mem'' _ mem', ?_⟩
rw [Set.mem_setOf_eq, mul_assoc, eq', eq'']
apply Set.inter_subset_left
rw [← absorbing_eq_self] at hm
exact hm.2
refine zorn_superset _ fun c hcs hc => ?_
refine
⟨⋂₀ c, ⟨isClosed_sInter fun t ht => (hcs ht).1, ?_, fun m hm m' hm' => ?_⟩, fun s hs =>
Set.sInter_subset_of_mem hs⟩
· obtain rfl | hcnemp := c.eq_empty_or_nonempty
· rw [Set.sInter_empty]
apply Set.univ_nonempty
convert
@IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ _ _ hcnemp.coe_sort
((↑) : c → Set M) ?_ ?_ ?_ ?_
· exact Set.sInter_eq_iInter
· refine DirectedOn.directed_val (IsChain.directedOn hc.symm)
exacts [fun i => (hcs i.prop).2.1, fun i => (hcs i.prop).1.isCompact, fun i => (hcs i.prop).1]
· rw [Set.mem_sInter]
exact fun t ht => (hcs ht).2.2 m (Set.mem_sInter.mp hm t ht) m' (Set.mem_sInter.mp hm' t ht) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Semigroup.lean | exists_idempotent_of_compact_t2_of_continuous_mul_left | Any nonempty compact Hausdorff semigroup where right-multiplication is continuous contains
an idempotent, i.e. an `m` such that `m * m = m`. |
@[to_additive exists_idempotent_in_compact_add_subsemigroup
/-- A version of
`exists_idempotent_of_compact_t2_of_continuous_add_left` where the idempotent lies in
some specified nonempty compact additive subsemigroup. -/]
exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
(s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
∃ m ∈ s, m * m = m := by
let M' := { m // m ∈ s }
letI : Semigroup M' :=
{ mul := fun p q => ⟨p.1 * q.1, s_add _ p.2 _ q.2⟩
mul_assoc := fun p q r => Subtype.eq (mul_assoc _ _ _) }
haveI : CompactSpace M' := isCompact_iff_compactSpace.mp s_compact
haveI : Nonempty M' := nonempty_subtype.mpr snemp
have : ∀ p : M', Continuous (· * p) := fun p =>
((continuous_mul_left p.1).comp continuous_subtype_val).subtype_mk _
obtain ⟨⟨m, hm⟩, idem⟩ := exists_idempotent_of_compact_t2_of_continuous_mul_left this
exact ⟨m, hm, Subtype.ext_iff.mp idem⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Semigroup.lean | exists_idempotent_in_compact_subsemigroup | A version of `exists_idempotent_of_compact_t2_of_continuous_mul_left` where the idempotent lies
in some specified nonempty compact subsemigroup. |
ContinuousStar (R : Type*) [TopologicalSpace R] [Star R] : Prop where
/-- The `star` operator is continuous. -/
continuous_star : Continuous (star : R → R)
export ContinuousStar (continuous_star) | class | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | ContinuousStar | Basic hypothesis to talk about a topological space with a continuous `star` operator. |
continuousOn_star {s : Set R} : ContinuousOn star s :=
continuous_star.continuousOn | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | continuousOn_star | null |
continuousWithinAt_star {s : Set R} {x : R} : ContinuousWithinAt star s x :=
continuous_star.continuousWithinAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | continuousWithinAt_star | null |
continuousAt_star {x : R} : ContinuousAt star x :=
continuous_star.continuousAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | continuousAt_star | null |
tendsto_star (a : R) : Tendsto star (𝓝 a) (𝓝 (star a)) :=
continuousAt_star | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | tendsto_star | null |
Filter.Tendsto.star {f : α → R} {l : Filter α} {y : R} (h : Tendsto f l (𝓝 y)) :
Tendsto (fun x => star (f x)) l (𝓝 (star y)) :=
(continuous_star.tendsto y).comp h
variable [TopologicalSpace α] {f : α → R} {s : Set α} {x : α}
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | Filter.Tendsto.star | null |
Continuous.star (hf : Continuous f) : Continuous fun x => star (f x) :=
continuous_star.comp hf
@[fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | Continuous.star | null |
ContinuousAt.star (hf : ContinuousAt f x) : ContinuousAt (fun x => star (f x)) x :=
continuousAt_star.comp hf
@[fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | ContinuousAt.star | null |
ContinuousOn.star (hf : ContinuousOn f s) : ContinuousOn (fun x => star (f x)) s :=
continuous_star.comp_continuousOn hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | ContinuousOn.star | null |
ContinuousWithinAt.star (hf : ContinuousWithinAt f s x) :
ContinuousWithinAt (fun x => star (f x)) s x :=
Filter.Tendsto.star hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | ContinuousWithinAt.star | null |
@[simps]
starContinuousMap : C(R, R) :=
⟨star, continuous_star⟩ | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Algebra.Star.Basic"
] | Mathlib/Topology/Algebra/Star.lean | starContinuousMap | The star operation bundled as a continuous map. |
isEmbedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) :
IsEmbedding (inclusion h) where
eq_induced := Eq.symm induced_compose
injective := Subtype.map_injective h Function.injective_id | lemma | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | isEmbedding_inclusion | The `StarSubalgebra.inclusion` of a star subalgebra is an embedding. |
isClosedEmbedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
(hS₁ : IsClosed (S₁ : Set A)) : IsClosedEmbedding (inclusion h) :=
{ IsEmbedding.inclusion h with
isClosed_range := isClosed_induced_iff.2
⟨S₁, hS₁, by
convert (Set.range_subtype_map id _).symm
· rw [Set.image_id]; rfl
· intro _ h'
apply h h' ⟩ }
variable [IsTopologicalSemiring A] [ContinuousStar A]
variable [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | isClosedEmbedding_inclusion | The `StarSubalgebra.inclusion` of a closed star subalgebra is a `IsClosedEmbedding`. |
topologicalClosure (s : StarSubalgebra R A) : StarSubalgebra R A :=
{
s.toSubalgebra.topologicalClosure with
carrier := closure (s : Set A)
star_mem' := fun ha =>
map_mem_closure continuous_star ha fun x => (star_mem : x ∈ s → star x ∈ s) } | def | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure | The closure of a star subalgebra in a topological star algebra as a star subalgebra. |
topologicalClosure_toSubalgebra_comm (s : StarSubalgebra R A) :
s.topologicalClosure.toSubalgebra = s.toSubalgebra.topologicalClosure :=
SetLike.coe_injective rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_toSubalgebra_comm | null |
topologicalClosure_coe (s : StarSubalgebra R A) :
(s.topologicalClosure : Set A) = closure (s : Set A) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_coe | null |
le_topologicalClosure (s : StarSubalgebra R A) : s ≤ s.topologicalClosure :=
subset_closure | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | le_topologicalClosure | null |
isClosed_topologicalClosure (s : StarSubalgebra R A) :
IsClosed (s.topologicalClosure : Set A) :=
isClosed_closure | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | isClosed_topologicalClosure | null |
topologicalClosure_minimal {s t : StarSubalgebra R A} (h : s ≤ t)
(ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t :=
closure_minimal h ht | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_minimal | null |
topologicalClosure_mono : Monotone (topologicalClosure : _ → StarSubalgebra R A) :=
fun _ S₂ h =>
topologicalClosure_minimal (h.trans <| le_topologicalClosure S₂) (isClosed_topologicalClosure S₂) | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_mono | null |
topologicalClosure_map_le [StarModule R B] [IsTopologicalSemiring B] [ContinuousStar B]
(s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : IsClosedMap φ) :
(map φ s).topologicalClosure ≤ map φ s.topologicalClosure :=
hφ.closure_image_subset _ | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_map_le | null |
map_topologicalClosure_le [StarModule R B] [IsTopologicalSemiring B] [ContinuousStar B]
(s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : Continuous φ) :
map φ s.topologicalClosure ≤ (map φ s).topologicalClosure :=
image_closure_subset_closure_image hφ | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | map_topologicalClosure_le | null |
topologicalClosure_map [StarModule R B] [IsTopologicalSemiring B] [ContinuousStar B]
(s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (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 StarAlgebra in | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_map | null |
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.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | topologicalClosure_adjoin_le_centralizer_centralizer | null |
_root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
(star s).topologicalClosure = star s.topologicalClosure := by
suffices ∀ t : Subalgebra R A, (star t).topologicalClosure ≤ star t.topologicalClosure from
le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))
exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)
(isClosed_closure.preimage continuous_star) | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | _root_.Subalgebra.topologicalClosure_star_comm | null |
commSemiringTopologicalClosure [T2Space A] (s : StarSubalgebra R A)
(hs : ∀ x y : s, x * y = y * x) : CommSemiring s.topologicalClosure :=
s.toSubalgebra.commSemiringTopologicalClosure hs | abbrev | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | commSemiringTopologicalClosure | If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. |
commRingTopologicalClosure {R A} [CommRing R] [StarRing R] [TopologicalSpace A] [Ring A]
[Algebra R A] [StarRing A] [StarModule R A] [IsTopologicalRing A] [ContinuousStar A] [T2Space A]
(s : StarSubalgebra R A) (hs : ∀ x y : s, x * y = y * x) : CommRing s.topologicalClosure :=
s.toSubalgebra.commRingTopologicalClosure hs | abbrev | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | commRingTopologicalClosure | If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. |
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