LieGroupPresentation.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Topological groups admitting a Lie group structure

A topological group G admits a Lie group structure if it is continuously isomorphic to a finite-dimensional real-analytic Lie group. Lie groups are Hausdorff but not assumed to be second countable: every discrete group is a 0-dimensional Lie group.

Main definitions

LieGroupPresentation G n: a continuous group isomorphism from G to an n-dimensional real-analytic Lie group.
AdmitsLieGroupStructure G: G has a LieGroupPresentation in some finite dimension.

Main results

admitsLieGroupStructure_of_lieGroup: a Hausdorff real-analytic Lie group admits a Lie group structure.
admitsLieGroupStructure_of_discreteTopology: a discrete group admits a Lie group structure.
AdmitsLieGroupStructure.locallyCompactSpace: a group admitting a Lie group structure is locally compact.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

structure LieGroupPresentation A continuous group isomorphism from G to an n-dimensional real-analytic Lie group line 54 source
Source documentation
A continuous group isomorphism from G to an n-dimensional real-analytic Lie group. The Lie group is not assumed to be second countable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Group, Nat
```
structure LieGroupPresentation (G : Type u) [TopologicalSpace G] [Group G] (n : ℕ) where
…
```
def AdmitsLieGroupStructure A topological group admits a Lie group structure if it has a LieGroupPresentation in some finite dimension line 66 source
Source documentation
A topological group admits a Lie group structure if it has a LieGroupPresentation in some finite dimension.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace
Proof uses: Exists, Nat, Nonempty, LieGroupPresentation
```
def AdmitsLieGroupStructure (G : Type u) [Group G] [TopologicalSpace G] : Prop :=
…
```
theorem admitsLieGroupStructure_of_lieGroup Every Hausdorff finite-dimensional real-analytic Lie group admits a Lie group structure line 70 source
Source documentation
Every Hausdorff finite-dimensional real-analytic Lie group admits a Lie group structure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, Nat, T2Space, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, LieGroup, modelWithCornersSelf, Top.top, WithTop, ENat, AdmitsLieGroupStructure
Proof uses: Exists.intro, Nonempty, LieGroupPresentation, Nonempty.intro, LieGroupPresentation.mk, ContinuousMulEquiv.refl
63 instances, coercions and auxiliary constants left out.
```
theorem admitsLieGroupStructure_of_lieGroup {n : ℕ} [T2Space G]
    [ChartedSpace (EuclideanSpace ℝ (Fin n)) G] [LieGroup (𝓡 n) ω G] :
    AdmitsLieGroupStructure G :=
…
```
Mentions: AdmitsLieGroupStructure
theorem admitsLieGroupStructure_of_discreteTopology Every discrete group is a 0-dimensional Lie group line 76 source
Source documentation
Every discrete group is a 0-dimensional Lie group.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, DiscreteTopology, AdmitsLieGroupStructure
Proof uses: ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, ENNReal, ChartedSpace.ofDiscreteTopology, IsManifold, modelWithCornersSelf, Top.top, WithTop, ENat, IsManifold.of_discreteTopology, LieGroup, LieGroup.mk, ContMDiffMul.mk, contMDiff_of_discreteTopology, Prod, ModelProd and 6 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem admitsLieGroupStructure_of_discreteTopology [DiscreteTopology G] :
    AdmitsLieGroupStructure G := by
…
```
Mentions: AdmitsLieGroupStructure
theorem AdmitsLieGroupStructure.locallyCompactSpace A group admitting a Lie group structure is locally compact line 86 source
Source documentation
A group admitting a Lie group structure is locally compact.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, AdmitsLieGroupStructure, LocallyCompactSpace
Proof uses: Nat, Nonempty, LieGroupPresentation, LieGroupPresentation.carrier, LieGroupPresentation.topologicalSpace, LieGroupPresentation.group, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, LieGroupPresentation.chartedSpace, ModelWithCorners.locallyCompactSpace, modelWithCornersSelf, ChartedSpace.locallyCompactSpace, Iff.mpr, Homeomorph.locallyCompactSpace_iff, ContinuousMulEquiv.toHomeomorph, LieGroupPresentation.equiv and 1 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem AdmitsLieGroupStructure.locallyCompactSpace (h : AdmitsLieGroupStructure G) :
    LocallyCompactSpace G := by
…
```
Mentions: AdmitsLieGroupStructure