`BookProof.ChapterGaugeAdjointAlgebra.covariantDeriv_covariant` (A dX dθ : Fin 3 → L) (X θ : L) (i : Fin 3) : (⁅dX i, θ⁆ + ⁅X, dθ i⁆) + (⁅gaugeVarA A dθ θ i, X⁆ + ⁅A i,...

Proved

BookProof.ChapterGaugeAdjointAlgebra.covariantDeriv_covariant

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeAdjointAlgebra.

BookProof.ChapterGaugeAdjointAlgebra.covariantDeriv_covariant (A dX dθ : Fin 3 → L) (X θ : L) (i : Fin 3) : (⁅dX i, θ⁆ + ⁅X, dθ i⁆) + (⁅gaugeVarA A dθ θ i, X⁆ + ⁅A i, ⁅X, θ⁆⁆) = ⁅covariantDeriv A dX X i, θ⁆

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

Preamble

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




open Finset

variable {L : Type*} [LieRing L]

Formal statement

theorem BookProof.ChapterGaugeAdjointAlgebra.covariantDeriv_covariant (A dX dθ : Fin 3 → L) (X θ : L) (i : Fin 3) :
    (⁅dX i, θ⁆ + ⁅X, dθ i⁆) + (⁅gaugeVarA A dθ θ i, X⁆ + ⁅A i, ⁅X, θ⁆⁆)
      = ⁅covariantDeriv A dX X i, θ⁆ := by sorry

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