`BookProof.QgOuterFockFL.mem_dsDom` {x : lp G 2} : x ∈ dsDom C ↔ (∀ i, (x : ∀ i, G i) i ∈ (C i).dom) ∧ Memℓp (fun i => opTot (C i).op ((x : ∀ i, G i) i)) 2

Proved

BookProof.QgOuterFockFL.mem_dsDom

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQgOuterFockFarisLavine.

BookProof.QgOuterFockFL.mem_dsDom {x : lp G 2} : x ∈ dsDom C ↔ (∀ i, (x : ∀ i, G i) i ∈ (C i).dom) ∧ Memℓp (fun i => opTot (C i).op ((x : ∀ i, G i) i)) 2

Formalization note: Lean 4 identifier BookProof.QgOuterFockFL.mem_dsDom.

Preamble

-- Generated from ChapterQgOuterFockFarisLavine.lean — theorem BookProof.QgOuterFockFL.mem_dsDom
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterDirectSumEsa
import Definitions.Def_ChapterQgOuterFockEsa
import Definitions.Def_ChapterYangMillsFriedrichs
import Definitions.Def_ChapterFriedrichsExtension
import Definitions.Def_ChapterComplexShiftCore
import Definitions.Def_ChapterQgHermiteOscillatorEsa
import Definitions.Def_ChapterHermiteProductCore
import Mathlib
import Definitions.Def_ChapterQgOuterFockFarisLavine
open BookProof.QgOuterFockFL


open scoped ENNReal


open BookProof.FarisLavine
open BookProof.DirectSumEsa
open BookProof.QgOuterFock
open BookProof.YangMillsFriedrichs
open BookProof.FriedrichsExtension
open BookProof.FriedrichsExtension.FormDom
open BookProof.HashimotoShiftInvert
open BookProof.QgHermiteOscillator
open BookProof.HermiteProductCore

noncomputable section

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable {ι : Type*} {G : ι → Type*} [∀ i, NormedAddCommGroup (G i)]
  [∀ i, InnerProductSpace ℂ (G i)]
variable (C : ∀ i, Comparison (G i))

Formal statement

theorem BookProof.QgOuterFockFL.mem_dsDom {x : lp G 2} :
    x ∈ dsDom C ↔ (∀ i, (x : ∀ i, G i) i ∈ (C i).dom) ∧
      Memℓp (fun i => opTot (C i).op ((x : ∀ i, G i) i)) 2 := by sorry

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