Project-declaredLean 4.32.0 · mathlib@249c48c2

Group totally Disconnected of pow prime eq one

Group.totallyDisconnected_of_pow_prime_eq_one

number theory arithmetic geometry Fermat's Last Theorem compact hausdorff rings

Plain-language statement

A compact Hausdorff vector space over 𝔽_p is totally disconnected.

Exact Lean statement

@[to_additive]
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

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

@[to_additive]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  have : ConnectedSpace (Subgroup.connectedComponentOfOne A) :=    Subtype.connectedSpace isConnected_connectedComponent  have : CompactSpace (Subgroup.connectedComponentOfOne A) :=    isCompact_iff_compactSpace.mp (isClosed_connectedComponent.isCompact)  have := subsingleton_of_pow_prime_eq_one (Subgroup.connectedComponentOfOne A) p hp    fun a ↦ Subtype.ext (hA a)  rw [totallyDisconnectedSpace_iff_connectedComponent_one]  exact ((Set.subsingleton_coe _).mp this).eq_singleton_of_mem mem_connectedComponent

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/Patching/Utils/CompactHausdorffRings.lean:45-57

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