`BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations` {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ} (hS3 : TraceOrthonormal S3) (hf3 : ClosesWithStructureConstants S3 f3) (hT

Proved

BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmGaugeRepresentation.

BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ} (hS3 : TraceOrthonormal S3) (hf3 : ClosesWithStructureConstants S3 f3) (hT : ClosesWithStructureConstants T (smStruct f3)) : smBrstCharge m T f3 * smBrstCharge m T f3 = 0

Formalization note: Lean 4 identifier BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations.

Preamble

-- Generated from ChapterSmGaugeRepresentation.lean — theorem BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations
import Mathlib
import Definitions.Def_ChapterSmGaugeRepresentation
import Definitions.Def_ChapterSmBrstGhost
import Definitions.Def_ChapterSmCarAlgebra
import Definitions.Def_ChapterYangMillsSU3
open BookProof.SmBrstGhost
open BookProof.SmCar
open BookProof.YangMillsSU3
open BookProof.SmGaugeRep



open Matrix Kronecker
open BookProof.YangMillsSU3 BookProof.SmBrstGhost

noncomputable section

variable {S3 : Fin 8 → Matrix (Fin 3) (Fin 3) ℂ} {f3 : Fin 8 → Fin 8 → Fin 8 → ℝ}

Formal statement

theorem BookProof.SmGaugeRep.sm_brst_nilpotent_of_su3_relations {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ}
    (hS3 : TraceOrthonormal S3) (hf3 : ClosesWithStructureConstants S3 f3)
    (hT : ClosesWithStructureConstants T (smStruct f3)) :
    smBrstCharge m T f3 * smBrstCharge m T f3 = 0 := by sorry

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