`BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant` (A dθ : Fin 3 → L) (dA : Fin 3 → Fin 3 → L) (ddθ : Fin 3 → Fin 3 → L) (θ : L) (i j : Fin 3) (hsym : ddθ i j = ddθ j i) : (

Proved

BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeAdjointAlgebra.

BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant (A dθ : Fin 3 → L) (dA : Fin 3 → Fin 3 → L) (ddθ : Fin 3 → Fin 3 → L) (θ : L) (i j : Fin 3) (hsym : ddθ i j = ddθ j i) : ((ddθ i j + ⁅dA i j, θ⁆ + ⁅A j, dθ i⁆) - (ddθ j i + ⁅dA j i, θ⁆ + ⁅A i, dθ j⁆)) + (⁅gaugeVarA A dθ θ i, A j⁆ + ⁅A i, gaugeVarA A dθ θ j⁆) = ⁅magnetic A dA i j, θ⁆

Formalization note: Lean 4 identifier BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant.

Preamble

-- Generated from ChapterGaugeAdjointAlgebra.lean — theorem BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant
import Mathlib
import Definitions.Def_ChapterGaugeAdjointAlgebra
open BookProof.ChapterGaugeAdjointAlgebra




open Finset

variable {L : Type*} [LieRing L]

Formal statement

theorem BookProof.ChapterGaugeAdjointAlgebra.magnetic_covariant (A dθ : Fin 3 → L) (dA : Fin 3 → Fin 3 → L)
    (ddθ : Fin 3 → Fin 3 → L) (θ : L) (i j : Fin 3) (hsym : ddθ i j = ddθ j i) :
    ((ddθ i j + ⁅dA i j, θ⁆ + ⁅A j, dθ i⁆) - (ddθ j i + ⁅dA j i, θ⁆ + ⁅A i, dθ j⁆))
        + (⁅gaugeVarA A dθ θ i, A j⁆ + ⁅A i, gaugeVarA A dθ θ j⁆)
      = ⁅magnetic A dA i j, θ⁆ := by sorry

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