`BookProof.BookBrstYangMills.sum_smul_der_apply` {n : ℕ} (k : Fin n → ℂ) (D : Fin n → Derivation ℂ (FieldPoly N) (FieldPoly N)) (p : FieldPoly N) : (∑ h, k h • D h) p = ∑ h, k h •

Proved

BookProof.BookBrstYangMills.sum_smul_der_apply

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBookBrstYangMills.

BookProof.BookBrstYangMills.sum_smul_der_apply {n : ℕ} (k : Fin n → ℂ) (D : Fin n → Derivation ℂ (FieldPoly N) (FieldPoly N)) (p : FieldPoly N) : (∑ h, k h • D h) p = ∑ h, k h • (D h) p

Formalization note: Lean 4 identifier BookProof.BookBrstYangMills.sum_smul_der_apply.

Preamble

-- Generated from ChapterBookBrstYangMills.lean — theorem BookProof.BookBrstYangMills.sum_smul_der_apply
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterQuantumGravityBrstCharge
import Mathlib
import Definitions.Def_ChapterBookBrstYangMills
open BookProof.BookBrstYangMills



open MvPolynomial BookProof.BRSTNilpotent BookProof.QuantumGravityBrstCharge

noncomputable section

variable {N : ℕ} (G : GaugeAlgebra N)

Formal statement

theorem BookProof.BookBrstYangMills.sum_smul_der_apply {n : ℕ} (k : Fin n → ℂ)
    (D : Fin n → Derivation ℂ (FieldPoly N) (FieldPoly N)) (p : FieldPoly N) :
    (∑ h, k h • D h) p = ∑ h, k h • (D h) p := by sorry

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