`BookProof.QuantumGravityBrstCharge.linGen_smul_sum` {n : ℕ} (c : Fin n → ℝ) (M : Fin n → Matrix (Fin d) (Fin d) ℝ) : linGen (∑ e, c e • M e) = ∑ e, c e • linGen (M e)

Proved

BookProof.QuantumGravityBrstCharge.linGen_smul_sum

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityBrstCharge.

BookProof.QuantumGravityBrstCharge.linGen_smul_sum {n : ℕ} (c : Fin n → ℝ) (M : Fin n → Matrix (Fin d) (Fin d) ℝ) : linGen (∑ e, c e • M e) = ∑ e, c e • linGen (M e)

Formalization note: Lean 4 identifier BookProof.QuantumGravityBrstCharge.linGen_smul_sum.

Preamble

-- Generated from ChapterQuantumGravityBrstCharge.lean — theorem BookProof.QuantumGravityBrstCharge.linGen_smul_sum
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterQuantumGravity3DGauge
import Mathlib
import Definitions.Def_ChapterQuantumGravityBrstCharge
open BookProof.QuantumGravityBrstCharge



open MvPolynomial BookProof.BRSTNilpotent BookProof.YangMillsHermite
open BookProof.QuantumGravity3DGauge

noncomputable section

variable {R : Type*} [Ring R] [Algebra ℝ R] {n : ℕ}
variable {f : Fin n → Fin n → Fin n → ℝ} {G χ β : Fin n → R}
variable {d : ℕ}

Formal statement

theorem BookProof.QuantumGravityBrstCharge.linGen_smul_sum {n : ℕ} (c : Fin n → ℝ) (M : Fin n → Matrix (Fin d) (Fin d) ℝ) :
    linGen (∑ e, c e • M e) = ∑ e, c e • linGen (M e) := by sorry

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