Prove2Me

`BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal` {M N : Submodule ℂ H} (h : M ≤ Nᗮ) {a b : H} (ha : a ∈ M) (hb : b ∈ N) : inner ℂ a b = 0

Proved
BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSpectralDirectSum.

BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal {M N : Submodule ℂ H} (h : M ≤ Nᗮ) {a b : H} (ha : a ∈ M) (hb : b ∈ N) : inner ℂ a b = 0

Formalization note: Lean 4 identifier BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal.

Preamble
-- Generated from ChapterSpectralDirectSum.lean — theorem BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralMultiplication
import Definitions.Def_ChapterCyclicDecomposition
import Definitions.Def_ChapterCyclicDirectSum
import Mathlib
import Definitions.Def_ChapterSpectralDirectSum
open BookProof.ChapterSpectralDirectSum


noncomputable section

open MeasureTheory Complex


open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralMultiplication
open BookProof.ChapterCyclicDecomposition BookProof.ChapterCyclicDirectSum

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : H →L[ℂ] H) (hT : IsStarNormal T) (xi : H)
Formal statement
theorem BookProof.ChapterSpectralDirectSum.inner_eq_zero_of_le_orthogonal {M N : Submodule ℂ H} (h : M ≤ Nᗮ) {a b : H}
    (ha : a ∈ M) (hb : b ∈ N) : inner ℂ a b = 0 := by sorry

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