`BookProof.ReducedEsa.deficiencyTrivialAt_red` (hP : IsReducingProjection P) {hPD : ∀ x ∈ D, P x ∈ D} (hC : Commutes T hPD) {z : ℂ} (hz : DeficiencyTrivialAt D T z) : DeficiencyTri

Proved

BookProof.ReducedEsa.deficiencyTrivialAt_red

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterReducingSubspaceEsa.

BookProof.ReducedEsa.deficiencyTrivialAt_red (hP : IsReducingProjection P) {hPD : ∀ x ∈ D, P x ∈ D} (hC : Commutes T hPD) {z : ℂ} (hz : DeficiencyTrivialAt D T z) : DeficiencyTrivialAt (redDom P D) (redOp T hP hC) z

Formalization note: Lean 4 identifier BookProof.ReducedEsa.deficiencyTrivialAt_red.

Preamble

-- Generated from ChapterReducingSubspaceEsa.lean — theorem BookProof.ReducedEsa.deficiencyTrivialAt_red
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Mathlib
import Definitions.Def_ChapterReducingSubspaceEsa
import Definitions.Def_ChapterA
import Definitions.Def_ChapterFarisLavineCore
open BookProof.ChapterA
open BookProof.ChapterA.System
open BookProof.ReducedEsa



open BookProof.FarisLavine BookProof.GraphCore

noncomputable section

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

variable {P : F →ₗ[ℂ] F}
variable {D : Submodule ℂ F}
variable (T : D →ₗ[ℂ] F)
variable {T}

Formal statement

theorem BookProof.ReducedEsa.deficiencyTrivialAt_red (hP : IsReducingProjection P) {hPD : ∀ x ∈ D, P x ∈ D}
    (hC : Commutes T hPD) {z : ℂ} (hz : DeficiencyTrivialAt D T z) :
    DeficiencyTrivialAt (redDom P D) (redOp T hP hC) z := by sorry

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