`BookProof.PermSector.exists_ne_zero_fermionic` (n : ℕ) (hD : D ≤ D₂) (f : Fin n → Hs.carrier) (hfD : ∀ i, f i ∈ D) (hf0 : ∀ i, f i ≠ 0) (hortho : ∀ i j, i ≠ j → (inner ℂ...

Proved

BookProof.PermSector.exists_ne_zero_fermionic

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_fermionic (n : ℕ) (hD : D ≤ D₂) (f : Fin n → Hs.carrier) (hfD : ∀ i, f i ∈ D) (hf0 : ∀ i, f i ≠ 0) (hortho : ∀ i j, i ≠ j → (inner ℂ (f i) (f j) : ℂ) = 0) : ∃ x : redDom (fermionicProj Hs n) (sectorCore Hs D₂ D n), x ≠ 0

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

Preamble

-- Generated from ChapterPermutationSectorEsa.lean — theorem BookProof.PermSector.exists_ne_zero_fermionic
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_fermionic (n : ℕ) (hD : D ≤ D₂) (f : Fin n → Hs.carrier)
    (hfD : ∀ i, f i ∈ D) (hf0 : ∀ i, f i ≠ 0)
    (hortho : ∀ i j, i ≠ j → (inner ℂ (f i) (f j) : ℂ) = 0) :
    ∃ x : redDom (fermionicProj Hs n) (sectorCore Hs D₂ D n), x ≠ 0 := by sorry

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