Prove2Me

`BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive` [PreconnectedSpace X] [Nonempty X] [Nontrivial G] {S : Set X} (hcomp : IsComprehensiveGaugeFixing

Proved
BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeComprehensiveFixing.

BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive [PreconnectedSpace X] [Nonempty X] [Nontrivial G] {S : Set X} (hcomp : IsComprehensiveGaugeFixing G S) (hcompl : IsCompleteGaugeFixing' G S) (hfree : MovesEveryPointOfSpectrum G X) : ¬ IsClopen S

Formalization note: Lean 4 identifier BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive.

Preamble
-- Generated from ChapterGaugeComprehensiveFixing.lean — theorem BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive
import Mathlib
import Definitions.Def_ChapterGaugeComprehensiveFixing
import Definitions.Def_ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeComprehensiveFixing



open BookProof.ChapterGaugeIncompleteFixing

variable {X : Type*} (G : Type*) [Group G] [MulAction G X]
variable {X : Type*} {G : Type*} [Group G] [MulAction G X] {S : Set X}
variable {X : Type*} [TopologicalSpace X] {G : Type*} [Group G] [MulAction G X]
Formal statement
theorem BookProof.ChapterGaugeComprehensiveFixing.not_isClopen_of_complete_comprehensive [PreconnectedSpace X] [Nonempty X]
    [Nontrivial G] {S : Set X}
    (hcomp : IsComprehensiveGaugeFixing G S) (hcompl : IsCompleteGaugeFixing' G S)
    (hfree : MovesEveryPointOfSpectrum G X) :
    ¬ IsClopen S := by sorry

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