`BookProof.FockNumberPreservingGap.shiftCol_opCol` (b : HilbertBasis ℕ ℂ F) (A : finiteModeDomain b →ₗ[ℂ] finiteModeDomain b) (mu : ℝ) : shiftCol (opCol b A) mu = opCol b (A - ((mu

Proved

BookProof.FockNumberPreservingGap.shiftCol_opCol

by leonardopedro · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockNumberPreservingGap.

BookProof.FockNumberPreservingGap.shiftCol_opCol (b : HilbertBasis ℕ ℂ F) (A : finiteModeDomain b →ₗ[ℂ] finiteModeDomain b) (mu : ℝ) : shiftCol (opCol b A) mu = opCol b (A - ((mu : ℝ) : ℂ) • LinearMap.id)

Formalization note: Lean 4 identifier BookProof.FockNumberPreservingGap.shiftCol_opCol.

Preamble

-- Generated from ChapterFockNumberPreservingGap.lean — theorem BookProof.FockNumberPreservingGap.shiftCol_opCol
import Definitions.Def_ChapterFockOneParticleGap
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Mathlib
import Definitions.Def_ChapterFockNumberPreservingGap
import Definitions.Def_ChapterFockSecondQuantization
import Definitions.Def_ChapterHermiteGalerkinFriedrichs
open BookProof.FockSecondQuantization
open BookProof.HermiteGalerkin
open BookProof.FockNumberPreservingGap

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]


noncomputable section


open BookProof.FockSecondQuantization BookProof.FockOneParticleGap
open BookProof.FarisLavine BookProof.NavierStokesFlow

Formal statement

theorem BookProof.FockNumberPreservingGap.shiftCol_opCol (b : HilbertBasis ℕ ℂ F)
    (A : finiteModeDomain b →ₗ[ℂ] finiteModeDomain b) (mu : ℝ) :
    shiftCol (opCol b A) mu = opCol b (A - ((mu : ℝ) : ℂ) • LinearMap.id) := by sorry

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