`BookProof.YangMillsAbelianNoGap.op_bigP` (c : Fin d) (α γ : ℝ) : ((α : ℝ) : ℂ) • (X c * bigP vf Mf) + ((γ : ℝ) : ℂ) • pderiv c (bigP vf Mf) = coordCombo c (opCoef α γ (vf c) (Mf c

Proved

BookProof.YangMillsAbelianNoGap.op_bigP

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterYangMillsAbelianNoGap.

BookProof.YangMillsAbelianNoGap.op_bigP (c : Fin d) (α γ : ℝ) : ((α : ℝ) : ℂ) • (X c * bigP vf Mf) + ((γ : ℝ) : ℂ) • pderiv c (bigP vf Mf) = coordCombo c (opCoef α γ (vf c) (Mf c)) 1 (Mf c) * ∏ j ∈ Finset.univ.erase c, facW vf Mf j

Formalization note: Lean 4 identifier BookProof.YangMillsAbelianNoGap.op_bigP.

Preamble

-- Generated from ChapterYangMillsAbelianNoGap.lean — theorem BookProof.YangMillsAbelianNoGap.op_bigP
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterHermiteGalerkinFriedrichs
import Mathlib
import Definitions.Def_ChapterYangMillsAbelianNoGap
import Definitions.Def_ChapterGaussCoordCombo
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterSqueezedGaussStates
open BookProof.GaussCoordCombo
open BookProof.HermiteProductCore
open BookProof.SqueezedGaussStates
open BookProof.YangMillsAbelianNoGap



open MvPolynomial MeasureTheory
open BookProof.HermiteProductCore BookProof.GaussCoordCombo BookProof.SqueezedGaussStates
open BookProof.YangMillsHermite BookProof.FarisLavine BookProof.HermiteGalerkin

noncomputable section

variable {d : ℕ}

variable (vf : Fin d → ℝ) (Mf : Fin d → ℕ)

Formal statement

theorem BookProof.YangMillsAbelianNoGap.op_bigP (c : Fin d) (α γ : ℝ) :
    ((α : ℝ) : ℂ) • (X c * bigP vf Mf) + ((γ : ℝ) : ℂ) • pderiv c (bigP vf Mf)
      = coordCombo c (opCoef α γ (vf c) (Mf c)) 1 (Mf c)
        * ∏ j ∈ Finset.univ.erase c, facW vf Mf j := by sorry

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