`BookProof.TensorCore.graphPow_range_le_closure` (hcore : IsGraphCore D A) (n : ℕ) : LinearMap.range (graphPow Hs D₂ A n) ≤ (Submodule.map (graphPow Hs D₂ A n) (corePow Hs D₂ D n))

Proved

BookProof.TensorCore.graphPow_range_le_closure

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTensorGraphCore.

BookProof.TensorCore.graphPow_range_le_closure (hcore : IsGraphCore D A) (n : ℕ) : LinearMap.range (graphPow Hs D₂ A n) ≤ (Submodule.map (graphPow Hs D₂ A n) (corePow Hs D₂ D n)).topologicalClosure

Formalization note: Lean 4 identifier BookProof.TensorCore.graphPow_range_le_closure.

Preamble

-- Generated from ChapterTensorGraphCore.lean — theorem BookProof.TensorCore.graphPow_range_le_closure
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterTensorGraphCore
import Definitions.Def_ChapterGraphCoreTransfer
open BookProof.GraphCore
open BookProof.TensorCore



open scoped TensorProduct
open BookProof.FarisLavine BookProof.GraphCore

noncomputable section

variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier)
variable (A : D₂ →ₗ[ℂ] Hs.carrier)
variable (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.TensorCore.graphPow_range_le_closure (hcore : IsGraphCore D A) (n : ℕ) :
    LinearMap.range (graphPow Hs D₂ A n)
      ≤ (Submodule.map (graphPow Hs D₂ A n) (corePow Hs D₂ D n)).topologicalClosure := by sorry

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