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 ⌀ |
|---|---|---|---|---|---|---|
instSecondCountableTopology [SecondCountableTopology M] : SecondCountableTopology Mᵈᵐᵃ :=
isInducing_mk_symm.secondCountableTopology
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instSecondCountableTopology | null |
instCompactSpace [CompactSpace M] : CompactSpace Mᵈᵐᵃ :=
mkHomeomorph.compactSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instCompactSpace | null |
instLocallyCompactSpace [LocallyCompactSpace M] : LocallyCompactSpace Mᵈᵐᵃ :=
isOpenEmbedding_mk_symm.locallyCompactSpace
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instLocallyCompactSpace | null |
instWeaklyLocallyCompactSpace [WeaklyLocallyCompactSpace M] :
WeaklyLocallyCompactSpace Mᵈᵐᵃ :=
isClosedEmbedding_mk_symm.weaklyLocallyCompactSpace
@[to_additive (attr := simp)] | instance | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | instWeaklyLocallyCompactSpace | null |
map_mk_nhds (x : M) : map (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk x) :=
mkHomeomorph.map_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | map_mk_nhds | null |
map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) :=
mkHomeomorph.symm.map_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | map_mk_symm_nhds | null |
comap_mk_nhds (x : Mᵈᵐᵃ) : comap (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk.symm x) :=
mkHomeomorph.comap_nhds_eq x
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | comap_mk_nhds | null |
comap_mk.symm_nhds (x : M) : comap (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk x) :=
mkHomeomorph.symm.comap_nhds_eq x | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic"
] | Mathlib/Topology/Algebra/Constructions/DomMulAct.lean | comap_mk.symm_nhds | null |
IsTopologicalAddTorsor extends ContinuousVAdd V P where
continuous_vsub : Continuous (fun x : P × P => x.1 -ᵥ x.2)
export IsTopologicalAddTorsor (continuous_vsub)
attribute [fun_prop] continuous_vsub
variable [IsTopologicalAddTorsor P] | class | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | IsTopologicalAddTorsor | A topological torsor over a topological additive group is a torsor where `+ᵥ` and `-ᵥ` are
continuous. |
Filter.Tendsto.vsub {l : Filter α} {f g : α → P} {x y : P} (hf : Tendsto f l (𝓝 x))
(hg : Tendsto g l (𝓝 y)) : Tendsto (f -ᵥ g) l (𝓝 (x -ᵥ y)) :=
(continuous_vsub.tendsto (x, y)).comp (hf.prodMk_nhds hg)
variable [TopologicalSpace α]
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | Filter.Tendsto.vsub | null |
Continuous.vsub {f g : α → P} (hf : Continuous f) (hg : Continuous g) :
Continuous (fun x ↦ f x -ᵥ g x) :=
continuous_vsub.comp₂ hf hg
@[fun_prop]
nonrec theorem ContinuousAt.vsub {f g : α → P} {x : α} (hf : ContinuousAt f x)
(hg : ContinuousAt g x) :
ContinuousAt (fun x ↦ f x -ᵥ g x) x :=
hf.vsub hg
@[fun_prop]
nonrec theorem ContinuousWithinAt.vsub {f g : α → P} {x : α} {s : Set α}
(hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun x ↦ f x -ᵥ g x) s x :=
hf.vsub hg
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | Continuous.vsub | null |
ContinuousOn.vsub {f g : α → P} {s : Set α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) : ContinuousOn (fun x ↦ f x -ᵥ g x) s := fun x hx ↦
(hf x hx).vsub (hg x hx)
include P in
variable (V P) in | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | ContinuousOn.vsub | null |
IsTopologicalAddTorsor.to_isTopologicalAddGroup : IsTopologicalAddGroup V where
continuous_add := by
have ⟨p⟩ : Nonempty P := inferInstance
conv =>
enter [1, x]
equals (x.1 +ᵥ x.2 +ᵥ p) -ᵥ p => rw [vadd_vadd, vadd_vsub]
fun_prop
continuous_neg := by
have ⟨p⟩ : Nonempty P := inferInstance
conv =>
enter [1, v]
equals p -ᵥ (v +ᵥ p) => rw [vsub_vadd_eq_vsub_sub, vsub_self, zero_sub]
fun_prop | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | IsTopologicalAddTorsor.to_isTopologicalAddGroup | The underlying group of a topological torsor is a topological group. This is not an instance, as
`P` cannot be inferred. |
@[simps!]
Homeomorph.vaddConst (p : P) : V ≃ₜ P where
__ := Equiv.vaddConst p
continuous_toFun := by fun_prop
continuous_invFun := by fun_prop | def | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | Homeomorph.vaddConst | The map `v ↦ v +ᵥ p` as a homeomorphism between `V` and `P`. |
IsClosed.vadd_right_of_isCompact {s : Set V} {t : Set P} (hs : IsClosed s)
(ht : IsCompact t) : IsClosed (s +ᵥ t) := by
have ⟨p⟩ : Nonempty P := inferInstance
have cont : Continuous (· -ᵥ p) := by fun_prop
have := IsTopologicalAddTorsor.to_isTopologicalAddGroup V P
convert (hs.add_right_of_isCompact <| ht.image cont).preimage cont
rw [Set.eq_preimage_iff_image_eq <| by exact (Equiv.vaddConst p).symm.bijective,
← Set.image2_vadd, Set.image_image2, ← Set.image2_add, Set.image2_image_right]
simp only [vadd_vsub_assoc] | theorem | Topology | [
"Mathlib.Algebra.AddTorsor.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/AddTorsor.lean | IsClosed.vadd_right_of_isCompact | null |
Set.isClosed_centralizer {M : Type*} (s : Set M) [Mul M] [TopologicalSpace M]
[ContinuousMul M] [T2Space M] : IsClosed (centralizer s) := by
rw [centralizer, setOf_forall]
refine isClosed_sInter ?_
rintro - ⟨m, ht, rfl⟩
refine isClosed_imp (by simp) <| isClosed_eq ?_ ?_
all_goals fun_prop | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Set.isClosed_centralizer | In a Hausdorff magma with continuous multiplication, the centralizer of any set is closed. |
@[to_additive /-- Addition from the left in a topological additive group as a homeomorphism. -/]
protected Homeomorph.mulLeft (a : G) : G ≃ₜ G :=
{ Equiv.mulLeft a with
continuous_toFun := continuous_const.mul continuous_id
continuous_invFun := continuous_const.mul continuous_id }
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.mulLeft | Multiplication from the left in a topological group as a homeomorphism. |
Homeomorph.coe_mulLeft (a : G) : ⇑(Homeomorph.mulLeft a) = (a * ·) :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.coe_mulLeft | null |
Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ := by
ext
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.mulLeft_symm | null |
isOpenMap_mul_left (a : G) : IsOpenMap (a * ·) := (Homeomorph.mulLeft a).isOpenMap
@[to_additive IsOpen.left_addCoset] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_mul_left | null |
IsOpen.leftCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (x • U) :=
isOpenMap_mul_left x _ h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsOpen.leftCoset | null |
isClosedMap_mul_left (a : G) : IsClosedMap (a * ·) := (Homeomorph.mulLeft a).isClosedMap
@[to_additive IsClosed.left_addCoset] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_mul_left | null |
IsClosed.leftCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (x • U) :=
isClosedMap_mul_left x _ h | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsClosed.leftCoset | null |
@[to_additive /-- Addition from the right in a topological additive group as a homeomorphism. -/]
protected Homeomorph.mulRight (a : G) : G ≃ₜ G :=
{ Equiv.mulRight a with
continuous_toFun := continuous_id.mul continuous_const
continuous_invFun := continuous_id.mul continuous_const }
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.mulRight | Multiplication from the right in a topological group as a homeomorphism. |
Homeomorph.coe_mulRight (a : G) : ⇑(Homeomorph.mulRight a) = (· * a) := rfl
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.coe_mulRight | null |
Homeomorph.mulRight_symm (a : G) :
(Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹ := by
ext
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.mulRight_symm | null |
isOpenMap_mul_right (a : G) : IsOpenMap (· * a) :=
(Homeomorph.mulRight a).isOpenMap
@[to_additive IsOpen.right_addCoset] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_mul_right | null |
IsOpen.rightCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (op x • U) :=
isOpenMap_mul_right x _ h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsOpen.rightCoset | null |
isClosedMap_mul_right (a : G) : IsClosedMap (· * a) :=
(Homeomorph.mulRight a).isClosedMap
@[to_additive IsClosed.right_addCoset] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_mul_right | null |
IsClosed.rightCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (op x • U) :=
isClosedMap_mul_right x _ h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsClosed.rightCoset | null |
discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
DiscreteTopology G := by
rw [← singletons_open_iff_discrete]
intro g
suffices {g} = (g⁻¹ * ·) ⁻¹' {1} by
rw [this]
exact (continuous_mul_left g⁻¹).isOpen_preimage _ h
simp only [mul_one, Set.preimage_mul_left_singleton, inv_inv]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | discreteTopology_of_isOpen_singleton_one | null |
discreteTopology_iff_isOpen_singleton_one : DiscreteTopology G ↔ IsOpen ({1} : Set G) :=
⟨fun h => forall_open_iff_discrete.mpr h {1}, discreteTopology_of_isOpen_singleton_one⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | discreteTopology_iff_isOpen_singleton_one | null |
@[to_additive]
ContinuousInv.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [Group α]
[DivisionMonoid β] [MonoidHomClass F α β] [tβ : TopologicalSpace β] [ContinuousInv β] (f : F) :
@ContinuousInv α (tβ.induced f) _ := by
let _tα := tβ.induced f
refine ⟨continuous_induced_rng.2 ?_⟩
simp only [Function.comp_def, map_inv]
fun_prop
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousInv.induced | null |
protected Specializes.inv {x y : G} (h : x ⤳ y) : (x⁻¹) ⤳ (y⁻¹) :=
h.map continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Specializes.inv | null |
protected Inseparable.inv {x y : G} (h : Inseparable x y) : Inseparable (x⁻¹) (y⁻¹) :=
h.map continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Inseparable.inv | null |
protected Specializes.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
[ContinuousMul G] [ContinuousInv G] {x y : G} (h : x ⤳ y) : ∀ m : ℤ, (x ^ m) ⤳ (y ^ m)
| .ofNat n => by simpa using h.pow n
| .negSucc n => by simpa using (h.pow (n + 1)).inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Specializes.zpow | null |
protected Inseparable.zpow {G : Type*} [DivInvMonoid G] [TopologicalSpace G]
[ContinuousMul G] [ContinuousInv G] {x y : G} (h : Inseparable x y) (m : ℤ) :
Inseparable (x ^ m) (y ^ m) :=
(h.specializes.zpow m).antisymm (h.specializes'.zpow m)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Inseparable.zpow | null |
@[to_additive]
continuousOn_inv {s : Set G} : ContinuousOn Inv.inv s :=
continuous_inv.continuousOn
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousOn_inv | null |
continuousWithinAt_inv {s : Set G} {x : G} : ContinuousWithinAt Inv.inv s x :=
continuous_inv.continuousWithinAt
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousWithinAt_inv | null |
continuousAt_inv {x : G} : ContinuousAt Inv.inv x :=
continuous_inv.continuousAt
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousAt_inv | null |
tendsto_inv (a : G) : Tendsto Inv.inv (𝓝 a) (𝓝 a⁻¹) :=
continuousAt_inv
variable [TopologicalSpace α] {f : α → G} {s : Set α} {x : α}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv | null |
OrderDual.instContinuousInv : ContinuousInv Gᵒᵈ := ‹ContinuousInv G›
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | OrderDual.instContinuousInv | null |
Prod.continuousInv [TopologicalSpace H] [Inv H] [ContinuousInv H] :
ContinuousInv (G × H) :=
⟨continuous_inv.fst'.prodMk continuous_inv.snd'⟩
variable {ι : Type*}
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Prod.continuousInv | null |
Pi.continuousInv {C : ι → Type*} [∀ i, TopologicalSpace (C i)] [∀ i, Inv (C i)]
[∀ i, ContinuousInv (C i)] : ContinuousInv (∀ i, C i) where
continuous_inv := continuous_pi fun i => (continuous_apply i).inv | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Pi.continuousInv | null |
@[to_additive
/-- A version of `Pi.continuousNeg` for non-dependent functions. It is needed
because sometimes Lean fails to use `Pi.continuousNeg` for non-dependent functions. -/]
Pi.has_continuous_inv' : ContinuousInv (ι → G) :=
Pi.continuousInv
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Pi.has_continuous_inv' | A version of `Pi.continuousInv` for non-dependent functions. It is needed because sometimes
Lean fails to use `Pi.continuousInv` for non-dependent functions. |
@[to_additive]
isClosed_setOf_map_inv [Inv G₁] [Inv G₂] [ContinuousInv G₂] :
IsClosed { f : G₁ → G₂ | ∀ x, f x⁻¹ = (f x)⁻¹ } := by
simp only [setOf_forall]
exact isClosed_iInter fun i => isClosed_eq (continuous_apply _) (continuous_apply _).inv | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosed_setOf_map_inv | null |
@[to_additive]
IsCompact.inv (hs : IsCompact s) : IsCompact s⁻¹ := by
rw [← image_inv_eq_inv]
exact hs.image continuous_inv
variable (G) | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsCompact.inv | null |
@[to_additive /-- Negation in a topological group as a homeomorphism. -/]
protected Homeomorph.inv (G : Type*) [TopologicalSpace G] [InvolutiveInv G]
[ContinuousInv G] : G ≃ₜ G :=
{ Equiv.inv G with
continuous_toFun := continuous_inv
continuous_invFun := continuous_inv }
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.inv | Inversion in a topological group as a homeomorphism. |
Homeomorph.coe_inv {G : Type*} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] :
⇑(Homeomorph.inv G) = Inv.inv := rfl
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.coe_inv | null |
nhds_inv (a : G) : 𝓝 a⁻¹ = (𝓝 a)⁻¹ :=
((Homeomorph.inv G).map_nhds_eq a).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhds_inv | null |
isOpenMap_inv : IsOpenMap (Inv.inv : G → G) :=
(Homeomorph.inv _).isOpenMap
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_inv | null |
isClosedMap_inv : IsClosedMap (Inv.inv : G → G) :=
(Homeomorph.inv _).isClosedMap
variable {G}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_inv | null |
IsOpen.inv (hs : IsOpen s) : IsOpen s⁻¹ :=
hs.preimage continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsOpen.inv | null |
IsClosed.inv (hs : IsClosed s) : IsClosed s⁻¹ :=
hs.preimage continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsClosed.inv | null |
inv_closure : ∀ s : Set G, (closure s)⁻¹ = closure s⁻¹ :=
(Homeomorph.inv G).preimage_closure
variable [TopologicalSpace α] {f : α → G} {s : Set α} {x : α}
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | inv_closure | null |
continuous_inv_iff : Continuous f⁻¹ ↔ Continuous f := (Homeomorph.inv G).comp_continuous_iff
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_inv_iff | null |
continuousAt_inv_iff : ContinuousAt f⁻¹ x ↔ ContinuousAt f x :=
(Homeomorph.inv G).comp_continuousAt_iff _ _
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousAt_inv_iff | null |
continuousOn_inv_iff : ContinuousOn f⁻¹ s ↔ ContinuousOn f s :=
(Homeomorph.inv G).comp_continuousOn_iff _ _
@[to_additive] alias ⟨Continuous.of_inv, _⟩ := continuous_inv_iff
@[to_additive] alias ⟨ContinuousAt.of_inv, _⟩ := continuousAt_inv_iff
@[to_additive] alias ⟨ContinuousOn.of_inv, _⟩ := continuousOn_inv_iff | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousOn_inv_iff | null |
@[to_additive]
continuousInv_sInf {ts : Set (TopologicalSpace G)}
(h : ∀ t ∈ ts, @ContinuousInv G t _) : @ContinuousInv G (sInf ts) _ :=
letI := sInf ts
{ continuous_inv :=
continuous_sInf_rng.2 fun t ht =>
continuous_sInf_dom ht (@ContinuousInv.continuous_inv G t _ (h t ht)) }
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousInv_sInf | null |
continuousInv_iInf {ts' : ι' → TopologicalSpace G}
(h' : ∀ i, @ContinuousInv G (ts' i) _) : @ContinuousInv G (⨅ i, ts' i) _ := by
rw [← sInf_range]
exact continuousInv_sInf (Set.forall_mem_range.mpr h')
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousInv_iInf | null |
continuousInv_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @ContinuousInv G t₁ _)
(h₂ : @ContinuousInv G t₂ _) : @ContinuousInv G (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine continuousInv_iInf fun b => ?_
cases b <;> assumption | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousInv_inf | null |
@[to_additive]
Topology.IsInducing.continuousInv {G H : Type*} [Inv G] [Inv H] [TopologicalSpace G]
[TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : IsInducing f)
(hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : ContinuousInv G :=
⟨hf.continuous_iff.2 <| by simpa only [Function.comp_def, hf_inv] using hf.continuous.inv⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Topology.IsInducing.continuousInv | null |
ConjAct.units_continuousConstSMul {M} [Monoid M] [TopologicalSpace M]
[ContinuousMul M] : ContinuousConstSMul (ConjAct Mˣ) M :=
⟨fun _ => (continuous_const.mul continuous_id).mul continuous_const⟩
variable [TopologicalSpace G] [Inv G] [Mul G] [ContinuousMul G] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ConjAct.units_continuousConstSMul | null |
@[to_additive continuous_addConj_prod
/-- Conjugation is jointly continuous on `G × G` when both `add` and `neg` are continuous. -/]
IsTopologicalGroup.continuous_conj_prod [ContinuousInv G] :
Continuous fun g : G × G => g.fst * g.snd * g.fst⁻¹ :=
continuous_mul.mul (continuous_inv.comp continuous_fst)
@[deprecated (since := "2025-03-11")]
alias IsTopologicalAddGroup.continuous_conj_sum := IsTopologicalAddGroup.continuous_addConj_prod | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.continuous_conj_prod | Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are continuous. |
@[to_additive (attr := continuity)
/-- Conjugation by a fixed element is continuous when `add` is continuous. -/]
IsTopologicalGroup.continuous_conj (g : G) : Continuous fun h : G => g * h * g⁻¹ :=
(continuous_mul_right g⁻¹).comp (continuous_mul_left g) | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.continuous_conj | Conjugation by a fixed element is continuous when `mul` is continuous. |
@[to_additive (attr := continuity)
/-- Conjugation acting on fixed element of the additive group is continuous when both
`add` and `neg` are continuous. -/]
IsTopologicalGroup.continuous_conj' [ContinuousInv G] (h : G) :
Continuous fun g : G => g * h * g⁻¹ :=
(continuous_mul_right h).mul continuous_inv | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.continuous_conj' | Conjugation acting on fixed element of the group is continuous when both `mul` and
`inv` are continuous. |
@[to_additive (attr := continuity, fun_prop)]
continuous_zpow : ∀ z : ℤ, Continuous fun a : G => a ^ z
| Int.ofNat n => by simpa using continuous_pow n
| Int.negSucc n => by simpa using (continuous_pow (n + 1)).inv | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_zpow | null |
AddGroup.continuousConstSMul_int {A} [AddGroup A] [TopologicalSpace A]
[IsTopologicalAddGroup A] : ContinuousConstSMul ℤ A :=
⟨continuous_zsmul⟩ | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | AddGroup.continuousConstSMul_int | null |
AddGroup.continuousSMul_int {A} [AddGroup A] [TopologicalSpace A]
[IsTopologicalAddGroup A] : ContinuousSMul ℤ A :=
⟨continuous_prod_of_discrete_left.mpr continuous_zsmul⟩
@[to_additive (attr := continuity, fun_prop)] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | AddGroup.continuousSMul_int | null |
Continuous.zpow {f : α → G} (h : Continuous f) (z : ℤ) : Continuous fun b => f b ^ z :=
(continuous_zpow z).comp h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Continuous.zpow | null |
continuousOn_zpow {s : Set G} (z : ℤ) : ContinuousOn (fun x => x ^ z) s :=
(continuous_zpow z).continuousOn
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousOn_zpow | null |
continuousAt_zpow (x : G) (z : ℤ) : ContinuousAt (fun x => x ^ z) x :=
(continuous_zpow z).continuousAt
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuousAt_zpow | null |
Filter.Tendsto.zpow {α} {l : Filter α} {f : α → G} {x : G} (hf : Tendsto f l (𝓝 x))
(z : ℤ) : Tendsto (fun x => f x ^ z) l (𝓝 (x ^ z)) :=
(continuousAt_zpow _ _).tendsto.comp hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.Tendsto.zpow | null |
ContinuousWithinAt.zpow {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x)
(z : ℤ) : ContinuousWithinAt (fun x => f x ^ z) s x :=
Filter.Tendsto.zpow hf z
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousWithinAt.zpow | null |
ContinuousAt.zpow {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) :
ContinuousAt (fun x => f x ^ z) x :=
Filter.Tendsto.zpow hf z
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousAt.zpow | null |
ContinuousOn.zpow {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ) :
ContinuousOn (fun x => f x ^ z) s := fun x hx => (hf x hx).zpow z | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousOn.zpow | null |
@[to_additive]
tendsto_inv_nhdsGT {a : H} : Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹) :=
(continuous_inv.tendsto a).inf <| by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsGT | null |
tendsto_inv_nhdsLT {a : H} : Tendsto Inv.inv (𝓝[<] a) (𝓝[>] a⁻¹) :=
(continuous_inv.tendsto a).inf <| by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsLT | null |
tendsto_inv_nhdsGT_inv {a : H} : Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a) := by
simpa only [inv_inv] using tendsto_inv_nhdsGT (a := a⁻¹)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsGT_inv | null |
tendsto_inv_nhdsLT_inv {a : H} : Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a) := by
simpa only [inv_inv] using tendsto_inv_nhdsLT (a := a⁻¹)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsLT_inv | null |
tendsto_inv_nhdsGE {a : H} : Tendsto Inv.inv (𝓝[≥] a) (𝓝[≤] a⁻¹) :=
(continuous_inv.tendsto a).inf <| by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsGE | null |
tendsto_inv_nhdsLE {a : H} : Tendsto Inv.inv (𝓝[≤] a) (𝓝[≥] a⁻¹) :=
(continuous_inv.tendsto a).inf <| by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsLE | null |
tendsto_inv_nhdsGE_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) := by
simpa only [inv_inv] using tendsto_inv_nhdsGE (a := a⁻¹)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsGE_inv | null |
tendsto_inv_nhdsLE_inv {a : H} : Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a) := by
simpa only [inv_inv] using tendsto_inv_nhdsLE (a := a⁻¹)
alias tendsto_inv_nhdsWithin_Iic_inv := tendsto_inv_nhdsLE_inv | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsLE_inv | null |
@[to_additive]
Prod.instIsTopologicalGroup [TopologicalSpace H] [Group H] [IsTopologicalGroup H] :
IsTopologicalGroup (G × H) where
continuous_inv := continuous_inv.prodMap continuous_inv
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Prod.instIsTopologicalGroup | null |
OrderDual.instIsTopologicalGroup : IsTopologicalGroup Gᵒᵈ where
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | OrderDual.instIsTopologicalGroup | null |
Pi.topologicalGroup {C : β → Type*} [∀ b, TopologicalSpace (C b)] [∀ b, Group (C b)]
[∀ b, IsTopologicalGroup (C b)] : IsTopologicalGroup (∀ b, C b) where
continuous_inv := continuous_pi fun i => (continuous_apply i).inv
open MulOpposite
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Pi.topologicalGroup | null |
@[to_additive /-- The map `(x, y) ↦ (x, x + y)` as a homeomorphism. This is a shear mapping. -/]
protected Homeomorph.shearMulRight : G × G ≃ₜ G × G :=
{ Equiv.prodShear (Equiv.refl _) Equiv.mulLeft with
continuous_toFun := by dsimp; fun_prop
continuous_invFun := by dsimp; fun_prop }
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.shearMulRight | If multiplication is continuous in `α`, then it also is in `αᵐᵒᵖ`. -/
@[to_additive /-- If addition is continuous in `α`, then it also is in `αᵃᵒᵖ`. -/]
instance [Group α] [IsTopologicalGroup α] : IsTopologicalGroup αᵐᵒᵖ where
variable (G)
@[to_additive]
theorem nhds_one_symm : comap Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) :=
((Homeomorph.inv G).comap_nhds_eq _).trans (congr_arg nhds inv_one)
@[to_additive]
theorem nhds_one_symm' : map Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) :=
((Homeomorph.inv G).map_nhds_eq _).trans (congr_arg nhds inv_one)
@[to_additive]
theorem inv_mem_nhds_one {S : Set G} (hS : S ∈ (𝓝 1 : Filter G)) : S⁻¹ ∈ 𝓝 (1 : G) := by
rwa [← nhds_one_symm'] at hS
/-- The map `(x, y) ↦ (x, x * y)` as a homeomorphism. This is a shear mapping. |
Homeomorph.shearMulRight_coe :
⇑(Homeomorph.shearMulRight G) = fun z : G × G => (z.1, z.1 * z.2) :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.shearMulRight_coe | null |
Homeomorph.shearMulRight_symm_coe :
⇑(Homeomorph.shearMulRight G).symm = fun z : G × G => (z.1, z.1⁻¹ * z.2) :=
rfl
variable {G}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.shearMulRight_symm_coe | null |
protected Topology.IsInducing.topologicalGroup {F : Type*} [Group H] [TopologicalSpace H]
[FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : IsInducing f) : IsTopologicalGroup H :=
{ toContinuousMul := hf.continuousMul _
toContinuousInv := hf.continuousInv (map_inv f) }
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Topology.IsInducing.topologicalGroup | null |
topologicalGroup_induced {F : Type*} [Group H] [FunLike F H G] [MonoidHomClass F H G]
(f : F) :
@IsTopologicalGroup H (induced f ‹_›) _ :=
letI := induced f ‹_›
IsInducing.topologicalGroup f ⟨rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_induced | null |
@[to_additive
/-- The (topological-space) closure of an additive subgroup of an additive topological group is
itself an additive subgroup. -/]
Subgroup.topologicalClosure (s : Subgroup G) : Subgroup G :=
{ s.toSubmonoid.topologicalClosure with
carrier := _root_.closure (s : Set G)
inv_mem' := fun {g} hg => by simpa only [← Set.mem_inv, inv_closure, inv_coe_set] using hg }
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.topologicalClosure | The (topological-space) closure of a subgroup of a topological group is
itself a subgroup. |
Subgroup.topologicalClosure_coe {s : Subgroup G} :
(s.topologicalClosure : Set G) = _root_.closure s :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.topologicalClosure_coe | null |
Subgroup.le_topologicalClosure (s : Subgroup G) : s ≤ s.topologicalClosure :=
_root_.subset_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.le_topologicalClosure | null |
Subgroup.isClosed_topologicalClosure (s : Subgroup G) :
IsClosed (s.topologicalClosure : Set G) := isClosed_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.isClosed_topologicalClosure | null |
Subgroup.topologicalClosure_minimal (s : Subgroup G) {t : Subgroup G} (h : s ≤ t)
(ht : IsClosed (t : Set G)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.topologicalClosure_minimal | null |
DenseRange.topologicalClosure_map_subgroup [Group H] [TopologicalSpace H]
[IsTopologicalGroup H] {f : G →* H} (hf : Continuous f) (hf' : DenseRange f) {s : Subgroup G}
(hs : s.topologicalClosure = ⊤) : (s.map f).topologicalClosure = ⊤ := by
rw [SetLike.ext'_iff] at hs ⊢
simp only [Subgroup.topologicalClosure_coe, Subgroup.coe_top, ← dense_iff_closure_eq] at hs ⊢
exact hf'.dense_image hf hs | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | DenseRange.topologicalClosure_map_subgroup | null |
@[to_additive /-- The topological closure of a normal additive subgroup is normal. -/]
Subgroup.is_normal_topologicalClosure {G : Type*} [TopologicalSpace G] [Group G]
[IsTopologicalGroup G] (N : Subgroup G) [N.Normal] :
(Subgroup.topologicalClosure N).Normal where
conj_mem n hn g := by
apply map_mem_closure (IsTopologicalGroup.continuous_conj g) hn
exact fun m hm => Subgroup.Normal.conj_mem inferInstance m hm g
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.is_normal_topologicalClosure | The topological closure of a normal subgroup is normal. |
mul_mem_connectedComponent_one {G : Type*} [TopologicalSpace G] [MulOneClass G]
[ContinuousMul G] {g h : G} (hg : g ∈ connectedComponent (1 : G))
(hh : h ∈ connectedComponent (1 : G)) : g * h ∈ connectedComponent (1 : G) := by
rw [connectedComponent_eq hg]
have hmul : g ∈ connectedComponent (g * h) := by
apply Continuous.image_connectedComponent_subset (continuous_mul_left g)
rw [← connectedComponent_eq hh]
exact ⟨(1 : G), mem_connectedComponent, by simp only [mul_one]⟩
simpa [← connectedComponent_eq hmul] using mem_connectedComponent
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | mul_mem_connectedComponent_one | null |
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