`BookProof.SmBrstGhost.fermiBilin_lie` (A B : Matrix (Fin N) (Fin N) ℂ) : (fermiBilin A : Module.End ℂ (FermiFock N)) * fermiBilin B - fermiBilin B * fermiBilin A = fermiBilin (A *

Proved

BookProof.SmBrstGhost.fermiBilin_lie

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmBrstGhost.

BookProof.SmBrstGhost.fermiBilin_lie (A B : Matrix (Fin N) (Fin N) ℂ) : (fermiBilin A : Module.End ℂ (FermiFock N)) * fermiBilin B - fermiBilin B * fermiBilin A = fermiBilin (A * B - B * A)

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

Preamble

-- Generated from ChapterSmBrstGhost.lean — theorem BookProof.SmBrstGhost.fermiBilin_lie
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterYangMillsSU3
import Mathlib
import Definitions.Def_ChapterSmBrstGhost
import Definitions.Def_ChapterNavierStokesFockSpace
import Definitions.Def_ChapterSmCarAlgebra
open BookProof.NavierStokesFlow.FockOfFock
open BookProof.SmCar
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.fermiBilin_lie (A B : Matrix (Fin N) (Fin N) ℂ) :
    (fermiBilin A : Module.End ℂ (FermiFock N)) * fermiBilin B
        - fermiBilin B * fermiBilin A
      = fermiBilin (A * B - B * A) := by sorry

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