`BookProof.TwoParticleSector.symmetricOn_fermionic` (hA : SymmetricOn D₂ A) : SymmetricOn (redDom (fermionicProj Hs) (sectorDom Hs D₂ 2)) (redOp (sectorOp Hs D₂ A 2) (isReducingPro

Proved

BookProof.TwoParticleSector.symmetricOn_fermionic

by leonardopedro · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTwoParticleSectorEsa.

BookProof.TwoParticleSector.symmetricOn_fermionic (hA : SymmetricOn D₂ A) : SymmetricOn (redDom (fermionicProj Hs) (sectorDom Hs D₂ 2)) (redOp (sectorOp Hs D₂ A 2) (isReducingProjection_fermionicProj Hs) (commutes_asymProj (T := sectorOp Hs D₂ A 2) (hUD := fun _ hx => swapH_mem_sectorDom Hs D₂ hx) (sectorOp_swapH Hs D₂ A)))

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

Preamble

-- Generated from ChapterTwoParticleSectorEsa.lean — theorem BookProof.TwoParticleSector.symmetricOn_fermionic
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Definitions.Def_ChapterReducingSubspaceEsa
import Mathlib
import Definitions.Def_ChapterTwoParticleSectorEsa
import Theorems.Thm_BookProof_ReducedEsa_commutes_asymProj
import Theorems.Thm_BookProof_TwoParticleSector_swapH_mem_sectorDom
import Theorems.Thm_BookProof_TwoParticleSector_sectorOp_swapH
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterTensorGraphCore
import Theorems.Thm_BookProof_TwoParticleSector_isReducingProjection_fermionicProj
import Definitions.Def_ChapterPermutationSectorEsa
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.symmetricOn_fermionic (hA : SymmetricOn D₂ A) :
    @SymmetricOn ↥(sector (fermionicProj Hs)) _ _
      (redDom (fermionicProj Hs) (sectorDom Hs D₂ 2))
      (redOp (sectorOp Hs D₂ A 2) (isReducingProjection_fermionicProj Hs)
        (commutes_asymProj (T := sectorOp Hs D₂ A 2)
          (hUD := fun _ hx => swapH_mem_sectorDom Hs D₂ hx) (sectorOp_swapH Hs D₂ A))) := by sorry

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