`BookProof.SmBrstGhost.matterGen_lie` {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ} {f : Fin 12 → Fin 12 → Fin 12 → ℝ} (hT : ClosesWithStructureConstants T f) (a b : Fin 12) : ma

Proved

BookProof.SmBrstGhost.matterGen_lie

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmBrstGhost.

BookProof.SmBrstGhost.matterGen_lie {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ} {f : Fin 12 → Fin 12 → Fin 12 → ℝ} (hT : ClosesWithStructureConstants T f) (a b : Fin 12) : matterGen m T a * matterGen m T b - matterGen m T b * matterGen m T a = ∑ c, f a b c • matterGen m T c

Formalization note: Lean 4 identifier BookProof.SmBrstGhost.matterGen_lie.

Preamble

-- Generated from ChapterSmBrstGhost.lean — theorem BookProof.SmBrstGhost.matterGen_lie
import Definitions.Def_ChapterBRSTNilpotent
import Mathlib
import Definitions.Def_ChapterSmBrstGhost
import Definitions.Def_ChapterSmCarAlgebra
import Definitions.Def_ChapterYangMillsSU3
open BookProof.SmCar
open BookProof.YangMillsSU3
open BookProof.SmBrstGhost



open BookProof.SmCar BookProof.BRSTNilpotent BookProof.YangMillsSU3

noncomputable section

variable {m : ℕ}
variable {R : Type*} [Ring R] [Algebra ℝ R] {n : ℕ}
variable {N : ℕ}

Formal statement

theorem BookProof.SmBrstGhost.matterGen_lie {m : ℕ} {T : Fin 12 → Matrix (Fin m) (Fin m) ℂ}
    {f : Fin 12 → Fin 12 → Fin 12 → ℝ} (hT : ClosesWithStructureConstants T f) (a b : Fin 12) :
    matterGen m T a * matterGen m T b - matterGen m T b * matterGen m T a
      = ∑ c, f a b c • matterGen m T c := by sorry

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