CompactHausdorffRings.lean`

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5 declarations, 1 contain sorry, 0 declared as axioms

Compact Hausdorff rings

Structure results for compact Hausdorff topological groups / rings: in particular, a compact Hausdorff 𝔽_p-vector space is totally disconnected, and a connected compact Hausdorff abelian group is divisible.

Imports (6)

Imported by 1 file

FLT.lean

theorem Group.subsingleton_of_pow_prime_eq_one A connected compact Hausdorff vector space over 𝔽_p is trivial sorry line 37 source
Source documentation
A connected compact Hausdorff vector space over 𝔽_p is trivial. This might sound easy, but it seems to require the fact that every nontrivial compact hausdorff group has a nontrivial continuous character. This fact is a special case of Pontryagin duality, and also a consequence of the Peter-Weyl theorem. This fact is being worked on at YaelDillies/mean-fourier.
Here's a proof of the theorem, using this fact that nontrivial groups have nontrivial characters: If χ : A → circle is a continuous character, then the image of χ is connected but is a subgroup of the pth roots of unity, hence must be trivial. Thus, A has no nontrivial continuous characters, so A must be trivial.

depends on sorry what this means
Statement uses: CommGroup, TopologicalSpace, IsTopologicalGroup, ConnectedSpace, CompactSpace, T2Space, Nat, Nat.Prime, Eq, HPow.hPow, OfNat.ofNat, Subsingleton
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Group.subsingleton_of_pow_prime_eq_one
    (A : Type*) [CommGroup A] [TopologicalSpace A] [IsTopologicalGroup A]
    [ConnectedSpace A] [CompactSpace A] [T2Space A]
    (p : ℕ) (hp : p.Prime) (hAp : ∀ a : A, a ^ p = 1) :
    Subsingleton A := by
…
```
theorem Group.totallyDisconnected_of_pow_prime_eq_one A compact Hausdorff vector space over 𝔽_p is totally disconnected line 46 source
Source documentation
A compact Hausdorff vector space over 𝔽_p is totally disconnected.

depends on sorry what this means
Statement uses: CommGroup, TopologicalSpace, IsTopologicalGroup, T2Space, CompactSpace, Nat, Nat.Prime, Eq, HPow.hPow, OfNat.ofNat, TotallyDisconnectedSpace
Proof uses: ConnectedSpace, Subtype, Membership.mem, Subgroup, Subgroup.connectedComponentOfOne, Subtype.connectedSpace, isConnected_connectedComponent, Iff.mp, IsCompact, Set, isCompact_iff_compactSpace, IsClosed.isCompact, connectedComponent, isClosed_connectedComponent, Subsingleton, Group.subsingleton_of_pow_prime_eq_one, Subtype.ext, Eq.mpr, Singleton.singleton, id and 7 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem Group.totallyDisconnected_of_pow_prime_eq_one
    (A : Type*) [CommGroup A] [TopologicalSpace A] [IsTopologicalGroup A]
    [T2Space A] [CompactSpace A] (p : ℕ) (hp : p.Prime) (hA : ∀ a : A, a ^ p = 1) :
    TotallyDisconnectedSpace A := by
…
```
def Group.rootable line 62 source
depends on sorry what this means
Statement uses: CommGroup, TopologicalSpace, IsTopologicalGroup, ConnectedSpace, CompactSpace, T2Space, RootableBy, Nat
Proof uses: rootableByOfPowLeftSurj
15 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
theorem CommGroup.no_compact_automorphisms A connected compact Hausdorff abelian topological group does not admit a nontrivial compact group of automorphisms line 92 source
Source documentation
A connected compact Hausdorff abelian topological group does not admit a nontrivial compact group of automorphisms.

depends on sorry what this means
Statement uses: CommGroup, TopologicalSpace, IsTopologicalGroup, ConnectedSpace, CompactSpace, T2Space, Subgroup, ContinuousMonoidHom, IsCompact, Eq, Bot.bot
Proof uses: RootableBy, Nat, Group.rootable, Eq.mpr, LE.le, id, congrArg, propext, eq_bot_iff, Membership.mem, ContinuousMonoidHom.ext, OfNat.ofNat, ContinuousMonoidHom.one_toFun, Classical.byContradiction, Not, Set, Compl.compl, Singleton.singleton, IsOpen, isOpen_compl_singleton and 56 more
74 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.
```
theorem CommGroup.no_compact_automorphisms
    {A : Type*} [CommGroup A] [TopologicalSpace A] [IsTopologicalGroup A]
    [ConnectedSpace A] [CompactSpace A] [T2Space A] (K : Subgroup (ContinuousMonoidHom A A))
    (hK : IsCompact (K : Set (ContinuousMonoidHom A A))) :
    K = ⊥ := by
…
```

instance (instance at line 127) line 127 source

Source documentation

A compact Hausdorff ring is totally disconnected.

instance {R : Type*} [Ring R] [TopologicalSpace R] [IsTopologicalRing R]
    [CompactSpace R] [T2Space R] : TotallyDisconnectedSpace R := by
…