`BookProof.PermSector.mem_fermionicSector_iff` (n : ℕ) {x : (Hs.pow n).carrier} : x ∈ sector (fermionicProj Hs n) ↔ ∀ σ : Equiv.Perm (Fin n), permOp Hs n σ x = ((Equiv.Perm.sign σ

Proved

BookProof.PermSector.mem_fermionicSector_iff

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPermutationSectorEsa.

BookProof.PermSector.mem_fermionicSector_iff (n : ℕ) {x : (Hs.pow n).carrier} : x ∈ sector (fermionicProj Hs n) ↔ ∀ σ : Equiv.Perm (Fin n), permOp Hs n σ x = ((Equiv.Perm.sign σ : ℤ) : ℂ) • x

Formalization note: Lean 4 identifier BookProof.PermSector.mem_fermionicSector_iff.

Preamble

-- Generated from ChapterPermutationSectorEsa.lean — theorem BookProof.PermSector.mem_fermionicSector_iff
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Definitions.Def_ChapterReducingSubspaceEsa
import Definitions.Def_ChapterGroupAverageEsa
import Definitions.Def_ChapterTensorPermutation
import Mathlib
import Definitions.Def_ChapterPermutationSectorEsa
import Definitions.Def_ChapterGaugeUnconstrainedSpectrum
import Definitions.Def_ChapterMaschkeFiniteGroup
import Definitions.Def_ChapterTensorGraphCore
import Definitions.Def_ChapterWignerLittleGroup
open BookProof.ChapterGaugeUnconstrainedSpectrum
open BookProof.ChapterMaschkeFiniteGroup
open BookProof.TensorCore
open BookProof.ChapterWignerLittleGroup
open BookProof.PermSector



open scoped TensorProduct
open BookProof.FarisLavine BookProof.GraphCore BookProof.ReducedEsa BookProof.TensorCore
open BookProof.GroupAverage BookProof.TensorPerm

noncomputable section

variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier) (A : D₂ →ₗ[ℂ] Hs.carrier)
  (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.PermSector.mem_fermionicSector_iff (n : ℕ) {x : (Hs.pow n).carrier} :
    x ∈ sector (fermionicProj Hs n)
      ↔ ∀ σ : Equiv.Perm (Fin n),
          permOp Hs n σ x = ((Equiv.Perm.sign σ : ℤ) : ℂ) • x := by sorry

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