`BookProof.ChapterSpectralDirectSum.one_lt_dist_of_orthogonalCyclicFamily` {S : Set H} (hS : OrthogonalCyclicFamily T hT S) {x y : H} (hx : x ∈ S) (hy : y ∈ S) (hxy : x ≠ y) : 1 <

Proved

BookProof.ChapterSpectralDirectSum.one_lt_dist_of_orthogonalCyclicFamily

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSpectralDirectSum.

BookProof.ChapterSpectralDirectSum.one_lt_dist_of_orthogonalCyclicFamily {S : Set H} (hS : OrthogonalCyclicFamily T hT S) {x y : H} (hx : x ∈ S) (hy : y ∈ S) (hxy : x ≠ y) : 1 < dist x y

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

Preamble

-- Generated from ChapterSpectralDirectSum.lean — theorem BookProof.ChapterSpectralDirectSum.one_lt_dist_of_orthogonalCyclicFamily
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.one_lt_dist_of_orthogonalCyclicFamily {S : Set H} (hS : OrthogonalCyclicFamily T hT S)
    {x y : H} (hx : x ∈ S) (hy : y ∈ S) (hxy : x ≠ y) : 1 < dist x y := by sorry

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