`BookProof.ChapterGaugeShiftExample.expectation_shift_invariant` {A : L2Z →L[ℂ] L2Z} (hA : ∀ m : ℤ, A * shiftOp m = shiftOp m * A) (m : ℤ) (f : L2Z) : ⟪shiftOp m f, A (shiftOp m f)

Proved

BookProof.ChapterGaugeShiftExample.expectation_shift_invariant

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeShiftExample.

BookProof.ChapterGaugeShiftExample.expectation_shift_invariant {A : L2Z →L[ℂ] L2Z} (hA : ∀ m : ℤ, A * shiftOp m = shiftOp m * A) (m : ℤ) (f : L2Z) : ⟪shiftOp m f, A (shiftOp m f)⟫_ℂ = ⟪f, A f⟫_ℂ

Formalization note: Lean 4 identifier BookProof.ChapterGaugeShiftExample.expectation_shift_invariant.

Preamble

-- Generated from ChapterGaugeShiftExample.lean — theorem BookProof.ChapterGaugeShiftExample.expectation_shift_invariant
import Mathlib
import Definitions.Def_ChapterGaugeShiftExample
import Definitions.Def_ChapterContinuityUnitaryInfinite
open BookProof.ChapterContinuityUnitaryInfinite
open BookProof.ChapterGaugeShiftExample


open scoped InnerProductSpace


open BookProof.ChapterContinuityUnitaryInfinite

Formal statement

theorem BookProof.ChapterGaugeShiftExample.expectation_shift_invariant {A : L2Z →L[ℂ] L2Z}
    (hA : ∀ m : ℤ, A * shiftOp m = shiftOp m * A) (m : ℤ) (f : L2Z) :
    ⟪shiftOp m f, A (shiftOp m f)⟫_ℂ = ⟪f, A f⟫_ℂ := by sorry

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