`BookProof.SmGaugeConnection.diracGaugeField_symmetric` {k : Fin 3 → ℝ} {m1 m2 g : ℝ} {T : Fin 8 → Matrix (Fin 3) (Fin 3) ℂ} {A : Fin 8 → Fin 3 → ℝ} (hT : ∀ a, (T a)ᴴ = T a) (psi p

Proved

BookProof.SmGaugeConnection.diracGaugeField_symmetric

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmGaugeConnection.

BookProof.SmGaugeConnection.diracGaugeField_symmetric {k : Fin 3 → ℝ} {m1 m2 g : ℝ} {T : Fin 8 → Matrix (Fin 3) (Fin 3) ℂ} {A : Fin 8 → Fin 3 → ℝ} (hT : ∀ a, (T a)ᴴ = T a) (psi phi : FermiFock 12) : (inner ℂ (diracGaugeField k m1 m2 g T A psi) phi : ℂ) = inner ℂ psi (diracGaugeField k m1 m2 g T A phi)

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

Preamble

-- Generated from ChapterSmGaugeConnection.lean — theorem BookProof.SmGaugeConnection.diracGaugeField_symmetric
import Definitions.Def_ChapterYangMillsSU3
import Definitions.Def_ChapterCPTHamiltonian
import Definitions.Def_ChapterSmDiracYukawa
import Definitions.Def_ChapterSmDiracSpinor
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterSmGaugeConnection
import Definitions.Def_ChapterSmCarAlgebra
open BookProof.SmCar
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.diracGaugeField_symmetric {k : Fin 3 → ℝ} {m1 m2 g : ℝ}
    {T : Fin 8 → Matrix (Fin 3) (Fin 3) ℂ} {A : Fin 8 → Fin 3 → ℝ} (hT : ∀ a, (T a)ᴴ = T a)
    (psi phi : FermiFock 12) :
    (inner ℂ (diracGaugeField k m1 m2 g T A psi) phi : ℂ)
      = inner ℂ psi (diracGaugeField k m1 m2 g T A phi) := by sorry

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