`BookProof.PermSector.exists_ne_zero_bosonic` (n : ℕ) (hD : D ≤ D₂) {a : Hs.carrier} (haD : a ∈ D) (ha0 : a ≠ 0) : ∃ x : redDom (bosonicProj Hs n) (sectorCore Hs D₂ D n), x ≠ 0

Proved

BookProof.PermSector.exists_ne_zero_bosonic

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPermutationSectorEsa.

BookProof.PermSector.exists_ne_zero_bosonic (n : ℕ) (hD : D ≤ D₂) {a : Hs.carrier} (haD : a ∈ D) (ha0 : a ≠ 0) : ∃ x : redDom (bosonicProj Hs n) (sectorCore Hs D₂ D n), x ≠ 0

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

Preamble

-- Generated from ChapterPermutationSectorEsa.lean — theorem BookProof.PermSector.exists_ne_zero_bosonic
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_ChapterDirectSumEsa
import Definitions.Def_ChapterGaugeUnconstrainedSpectrum
import Definitions.Def_ChapterTensorGraphCore
open BookProof.DirectSumEsa
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.exists_ne_zero_bosonic (n : ℕ) (hD : D ≤ D₂) {a : Hs.carrier} (haD : a ∈ D)
    (ha0 : a ≠ 0) : ∃ x : redDom (bosonicProj Hs n) (sectorCore Hs D₂ D n), x ≠ 0 := by sorry

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