`BookProof.ChapterH8.compress_inv_transfer_apply` (V : F →L[ℂ] E) (qX qXinv : E →L[ℂ] E) (qBinv : F →L[ℂ] F) (hVV : (adjoint V).comp V = ContinuousLinearMap.id ℂ F) (hinv : ∀ x : F

Proved

BookProof.ChapterH8.compress_inv_transfer_apply

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

timepiece

Prove the following Lean 4 theorem from ChapterH8.

BookProof.ChapterH8.compress_inv_transfer_apply (V : F →L[ℂ] E) (qX qXinv : E →L[ℂ] E) (qBinv : F →L[ℂ] F) (hVV : (adjoint V).comp V = ContinuousLinearMap.id ℂ F) (hinv : ∀ x : F, ∃ y : F, qX (V x) = V y) (hqXl : qXinv.comp qX = ContinuousLinearMap.id ℂ E) (hqBr : (compress V qX).comp qBinv = ContinuousLinearMap.id ℂ F) (v : E) (hv : V ((adjoint V) v) = v) : qXinv v = V (qBinv ((adjoint V) v))

Formalization note: Lean 4 identifier BookProof.ChapterH8.compress_inv_transfer_apply.

Preamble

-- Generated from ChapterH8.lean — theorem BookProof.ChapterH8.compress_inv_transfer_apply
import Definitions.Def_ChapterH5
import Definitions.Def_ChapterH6
import Mathlib
import Definitions.Def_ChapterH8
import Definitions.Def_ChapterH4
open BookProof.ChapterH4
open BookProof.ChapterH8


noncomputable section


open BookProof.ChapterH4 BookProof.ChapterH5 BookProof.ChapterH6
open ContinuousLinearMap

variable {K E : Type*} [Field K] [AddCommGroup E] [Module K E]
variable {E F G : Type*}
  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]

Formal statement

theorem BookProof.ChapterH8.compress_inv_transfer_apply (V : F →L[ℂ] E) (qX qXinv : E →L[ℂ] E) (qBinv : F →L[ℂ] F)
    (hVV : (adjoint V).comp V = ContinuousLinearMap.id ℂ F)
    (hinv : ∀ x : F, ∃ y : F, qX (V x) = V y)
    (hqXl : qXinv.comp qX = ContinuousLinearMap.id ℂ E)
    (hqBr : (compress V qX).comp qBinv = ContinuousLinearMap.id ℂ F)
    (v : E) (hv : V ((adjoint V) v) = v) :
    qXinv v = V (qBinv ((adjoint V) v)) := by sorry

Source

https://github.com/leonardopedro/timepiece/blob/61595bc/BookProof/ChapterH8.lean

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