`BookProof.ScalaronFock.inner_qgDir` (n : ℕ) (x : qgSector n) (j : Fin n) (i : Fin 2) : (inner ℝ x (qgDir n j i) : ℝ) = x (j, i)

Proved

BookProof.ScalaronFock.inner_qgDir

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterScalaronFockEsa.

BookProof.ScalaronFock.inner_qgDir (n : ℕ) (x : qgSector n) (j : Fin n) (i : Fin 2) : (inner ℝ x (qgDir n j i) : ℝ) = x (j, i)

Formalization note: Lean 4 identifier BookProof.ScalaronFock.inner_qgDir.

Preamble

-- Generated from ChapterScalaronFockEsa.lean — theorem BookProof.ScalaronFock.inner_qgDir
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterStarobinskyPotential
import Definitions.Def_ChapterScalaronCoreEsa
import Definitions.Def_ChapterDirectSumEsa
import Definitions.Def_ChapterStoneBridge
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterQuantumGravityDensitized
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterScalaronFockEsa
open BookProof.ScalaronFock


open Filter Topology MeasureTheory SchwartzMap


open BookProof.FarisLavine BookProof.Starobinsky BookProof.ScalaronEsa
open BookProof.DirectSumEsa BookProof.StoneBridge BookProof.EsaClosure
open BookProof.QuantumGravityDensitized BookProof.ChapterStoneResolvent

noncomputable section

variable {E : ℕ → Type*} [∀ n, NormedAddCommGroup (E n)] [∀ n, InnerProductSpace ℝ (E n)]
  [∀ n, FiniteDimensional ℝ (E n)] [∀ n, MeasurableSpace (E n)] [∀ n, BorelSpace (E n)]

Formal statement

theorem BookProof.ScalaronFock.inner_qgDir (n : ℕ) (x : qgSector n) (j : Fin n) (i : Fin 2) :
    (inner ℝ x (qgDir n j i) : ℝ) = x (j, i) := by sorry

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