`BookProof.PermSector.permOp_mem_sectorDom` (n : ℕ) (σ : Equiv.Perm (Fin n)) {x : (Hs.pow n).carrier} (hx : x ∈ sectorDom Hs D₂ n) : permOp Hs n σ x ∈ sectorDom Hs D₂ n

Proved

BookProof.PermSector.permOp_mem_sectorDom

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPermutationSectorEsa.

BookProof.PermSector.permOp_mem_sectorDom (n : ℕ) (σ : Equiv.Perm (Fin n)) {x : (Hs.pow n).carrier} (hx : x ∈ sectorDom Hs D₂ n) : permOp Hs n σ x ∈ sectorDom Hs D₂ n

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

Preamble

-- Generated from ChapterPermutationSectorEsa.lean — theorem BookProof.PermSector.permOp_mem_sectorDom
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_ChapterTensorGraphCore
open BookProof.ChapterGaugeUnconstrainedSpectrum
open BookProof.TensorCore
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.permOp_mem_sectorDom (n : ℕ) (σ : Equiv.Perm (Fin n)) {x : (Hs.pow n).carrier}
    (hx : x ∈ sectorDom Hs D₂ n) : permOp Hs n σ x ∈ sectorDom Hs D₂ n := by sorry

View graph