`BookProof.ChapterA.antiisometry_unique_up_to_phase` (M : System ℂ V) (hSchur : IsSchurUnitary M) {θ₁ θ₂ : AntiUnitary V} (h₁ : CommutesAntiUnitary M θ₁) (h₂ : CommutesAntiUnitary

Proved

BookProof.ChapterA.antiisometry_unique_up_to_phase

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA2.

BookProof.ChapterA.antiisometry_unique_up_to_phase (M : System ℂ V) (hSchur : IsSchurUnitary M) {θ₁ θ₂ : AntiUnitary V} (h₁ : CommutesAntiUnitary M θ₁) (h₂ : CommutesAntiUnitary M θ₂) : ∃ c : ℂ, ‖c‖ = 1 ∧ ∀ x, θ₂ x = c • θ₁ x

Formalization note: Lean 4 identifier BookProof.ChapterA.antiisometry_unique_up_to_phase.

Preamble

-- Generated from ChapterA2.lean — theorem BookProof.ChapterA.antiisometry_unique_up_to_phase
import Mathlib
import Definitions.Def_ChapterA2
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace


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

Formal statement

theorem BookProof.ChapterA.antiisometry_unique_up_to_phase (M : System ℂ V) (hSchur : IsSchurUnitary M)
    {θ₁ θ₂ : AntiUnitary V} (h₁ : CommutesAntiUnitary M θ₁) (h₂ : CommutesAntiUnitary M θ₂) :
    ∃ c : ℂ, ‖c‖ = 1 ∧ ∀ x, θ₂ x = c • θ₁ x := by sorry

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