`BookProof.TwoParticleSector.exists_ne_zero_fermionic` (hD : D ≤ D₂) {a b : Hs.carrier} (haD : a ∈ D) (hbD : b ∈ D) (ha0 : a ≠ 0) (hb0 : b ≠ 0) (hab : (inner ℂ a b : ℂ) = 0) : ∃ x

Proved

BookProof.TwoParticleSector.exists_ne_zero_fermionic

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTwoParticleSectorEsa.

BookProof.TwoParticleSector.exists_ne_zero_fermionic (hD : D ≤ D₂) {a b : Hs.carrier} (haD : a ∈ D) (hbD : b ∈ D) (ha0 : a ≠ 0) (hb0 : b ≠ 0) (hab : (inner ℂ a b : ℂ) = 0) : ∃ x : redDom (fermionicProj Hs) (sectorCore Hs D₂ D 2), x ≠ 0

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

Preamble

-- Generated from ChapterTwoParticleSectorEsa.lean — theorem BookProof.TwoParticleSector.exists_ne_zero_fermionic
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Definitions.Def_ChapterReducingSubspaceEsa
import Mathlib
import Definitions.Def_ChapterTwoParticleSectorEsa
import Definitions.Def_ChapterDirectSumEsa
import Definitions.Def_ChapterTensorGraphCore
open BookProof.DirectSumEsa
open BookProof.TensorCore
open BookProof.TwoParticleSector



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

noncomputable section

variable (X : Type) [NormedAddCommGroup X] [InnerProductSpace ℂ X]
variable {X}
variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier)
variable (A : D₂ →ₗ[ℂ] Hs.carrier)
variable (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.TwoParticleSector.exists_ne_zero_fermionic (hD : D ≤ D₂) {a b : Hs.carrier} (haD : a ∈ D) (hbD : b ∈ D)
    (ha0 : a ≠ 0) (hb0 : b ≠ 0) (hab : (inner ℂ a b : ℂ) = 0) :
    ∃ x : redDom (fermionicProj Hs) (sectorCore Hs D₂ D 2), x ≠ 0 := by sorry

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