`BookProof.SmGaugeRep.smGenP_hyp_comm` (y : ℝ) (M : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) : ((y : ℂ)) • (1 : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) * M = M * ((y : ℂ)) • (1

Proved

BookProof.SmGaugeRep.smGenP_hyp_comm

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSmGaugeRepresentation.

BookProof.SmGaugeRep.smGenP_hyp_comm (y : ℝ) (M : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) : ((y : ℂ)) • (1 : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) * M = M * ((y : ℂ)) • (1 : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ)

Formalization note: Lean 4 identifier BookProof.SmGaugeRep.smGenP_hyp_comm.

Preamble

-- Generated from ChapterSmGaugeRepresentation.lean — theorem BookProof.SmGaugeRep.smGenP_hyp_comm
import Definitions.Def_ChapterYangMillsSU3
import Definitions.Def_ChapterSmBrstGhost
import Mathlib
import Definitions.Def_ChapterSmGaugeRepresentation
open BookProof.SmGaugeRep



open Matrix Kronecker
open BookProof.YangMillsSU3 BookProof.SmBrstGhost

noncomputable section

variable {S3 : Fin 8 → Matrix (Fin 3) (Fin 3) ℂ} {f3 : Fin 8 → Fin 8 → Fin 8 → ℝ}

Formal statement

theorem BookProof.SmGaugeRep.smGenP_hyp_comm (y : ℝ) (M : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) :
    ((y : ℂ)) • (1 : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) * M
      = M * ((y : ℂ)) • (1 : Matrix (Fin 3 × Fin 2) (Fin 3 × Fin 2) ℂ) := by sorry

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