`BookProof.SmGaugeConnection.diracGaugeMat_split` (k : Fin 3 → ℝ) (m1 m2 g : ℝ) (T : Fin d → Matrix (Fin N) (Fin N) ℂ) (A : Fin d → Fin 3 → ℝ) : diracGaugeMat k m1 m2 g T A = dirac

Proved

BookProof.SmGaugeConnection.diracGaugeMat_split

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmGaugeConnection.

BookProof.SmGaugeConnection.diracGaugeMat_split (k : Fin 3 → ℝ) (m1 m2 g : ℝ) (T : Fin d → Matrix (Fin N) (Fin N) ℂ) (A : Fin d → Fin 3 → ℝ) : diracGaugeMat k m1 m2 g T A = diracOneParticle k m1 m2 ⊗ₖ (1 : Matrix (Fin N) (Fin N) ℂ) + ∑ j : Fin 3, Kin j ⊗ₖ conn g T A j

Formalization note: Lean 4 identifier BookProof.SmGaugeConnection.diracGaugeMat_split.

Preamble

-- Generated from ChapterSmGaugeConnection.lean — theorem BookProof.SmGaugeConnection.diracGaugeMat_split
import Definitions.Def_ChapterYangMillsSU3
import Definitions.Def_ChapterSmCarAlgebra
import Definitions.Def_ChapterSmDiracYukawa
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterSmGaugeConnection
import Definitions.Def_ChapterCPTHamiltonian
import Definitions.Def_ChapterSmDiracSpinor
open BookProof.ChapterCPTHamiltonian
open BookProof.SmDiracSpinor
open BookProof.SmGaugeConnection



open Matrix Kronecker
open BookProof.YangMillsSU3 BookProof.ChapterCPTHamiltonian BookProof.SmCar
open BookProof.SmDiracYukawa BookProof.SmDiracSpinor BookProof.FarisLavine

noncomputable section

variable {N d : ℕ}

Formal statement

theorem BookProof.SmGaugeConnection.diracGaugeMat_split (k : Fin 3 → ℝ) (m1 m2 g : ℝ)
    (T : Fin d → Matrix (Fin N) (Fin N) ℂ) (A : Fin d → Fin 3 → ℝ) :
    diracGaugeMat k m1 m2 g T A
      = diracOneParticle k m1 m2 ⊗ₖ (1 : Matrix (Fin N) (Fin N) ℂ)
        + ∑ j : Fin 3, Kin j ⊗ₖ conn g T A j := by sorry

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