Prove2Me

`BookProof.BrstUnboundedLeakage.opProj_apply` (x : H) : opProj T V hV x = T.op ⟨projOp V x, hV (projOp_apply_mem V x)⟩

Proved
BookProof.BrstUnboundedLeakage.opProj_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstUnboundedLeakage.

BookProof.BrstUnboundedLeakage.opProj_apply (x : H) : opProj T V hV x = T.op ⟨projOp V x, hV (projOp_apply_mem V x)⟩

Formalization note: Lean 4 identifier BookProof.BrstUnboundedLeakage.opProj_apply.

Preamble
-- Generated from ChapterBrstUnboundedLeakage.lean — theorem BookProof.BrstUnboundedLeakage.opProj_apply
import Definitions.Def_ChapterBrstTruncationLeakage
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterBrstUnboundedLeakage
import Definitions.Def_ChapterStoneResolvent
import Theorems.Thm_BookProof_BrstUnboundedLeakage_projOp_apply_mem
open BookProof.BrstUnboundedLeakage


open NormedSpace Filter Topology
open scoped InnerProductSpace


open BookProof.BrstLeakage BookProof.ChapterStoneResolvent

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (T : UnboundedSelfAdjoint H)
variable (V : Submodule ℂ H) [FiniteDimensional ℂ V] (hV : V ≤ T.domain)
Formal statement
theorem BookProof.BrstUnboundedLeakage.opProj_apply (x : H) : opProj T V hV x = T.op ⟨projOp V x, hV (projOp_apply_mem V x)⟩ := by sorry

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