`BookProof.YangMillsAbelianNoGap.realCoeff_prod` {S : Finset (Fin d)} {f : Fin d → MvPolynomial (Fin d) ℂ} (h : ∀ j ∈ S, RealCoeff (f j)) : RealCoeff (∏ j ∈ S, f j)

Proved

BookProof.YangMillsAbelianNoGap.realCoeff_prod

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterYangMillsAbelianNoGap.

BookProof.YangMillsAbelianNoGap.realCoeff_prod {S : Finset (Fin d)} {f : Fin d → MvPolynomial (Fin d) ℂ} (h : ∀ j ∈ S, RealCoeff (f j)) : RealCoeff (∏ j ∈ S, f j)

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

Preamble

-- Generated from ChapterYangMillsAbelianNoGap.lean — theorem BookProof.YangMillsAbelianNoGap.realCoeff_prod
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterGaussCoordCombo
import Definitions.Def_ChapterSqueezedGaussStates
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterHermiteGalerkinFriedrichs
import Mathlib
import Definitions.Def_ChapterYangMillsAbelianNoGap
import Definitions.Def_ChapterSirkFinitePrecision
import Definitions.Def_ChapterYangMillsHermite
open BookProof.SirkFinitePrecision
open BookProof.SirkFinitePrecision.CertInterval
open BookProof.YangMillsHermite
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}

Formal statement

theorem BookProof.YangMillsAbelianNoGap.realCoeff_prod {S : Finset (Fin d)} {f : Fin d → MvPolynomial (Fin d) ℂ}
    (h : ∀ j ∈ S, RealCoeff (f j)) : RealCoeff (∏ j ∈ S, f j) := by sorry

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