`BookProof.QuantumGravityBrstCharge.elemGen_bracket` (j k l m : Fin d) : elemGen j k * elemGen l m - elemGen l m * elemGen j k = (if k = l then elemGen j m else 0) - (if j = m then

Proved

BookProof.QuantumGravityBrstCharge.elemGen_bracket

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityBrstCharge.

BookProof.QuantumGravityBrstCharge.elemGen_bracket (j k l m : Fin d) : elemGen j k * elemGen l m - elemGen l m * elemGen j k = (if k = l then elemGen j m else 0) - (if j = m then elemGen l k else 0)

Formalization note: Lean 4 identifier BookProof.QuantumGravityBrstCharge.elemGen_bracket.

Preamble

-- Generated from ChapterQuantumGravityBrstCharge.lean — theorem BookProof.QuantumGravityBrstCharge.elemGen_bracket
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterQuantumGravity3DGauge
import Mathlib
import Definitions.Def_ChapterQuantumGravityBrstCharge
import Definitions.Def_ChapterYangMillsHermite
open BookProof.YangMillsHermite
open BookProof.QuantumGravityBrstCharge



open MvPolynomial BookProof.BRSTNilpotent BookProof.YangMillsHermite
open BookProof.QuantumGravity3DGauge

noncomputable section

variable {R : Type*} [Ring R] [Algebra ℝ R] {n : ℕ}
variable {f : Fin n → Fin n → Fin n → ℝ} {G χ β : Fin n → R}
variable {d : ℕ}

Formal statement

theorem BookProof.QuantumGravityBrstCharge.elemGen_bracket (j k l m : Fin d) :
    elemGen j k * elemGen l m - elemGen l m * elemGen j k
      = (if k = l then elemGen j m else 0) - (if j = m then elemGen l k else 0) := by sorry

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