`BookProof.ChapterA.Rreal_isometric_iff_complexification_isometric` (M : System ℂ V) (N : System ℂ W) (hN : IsSchurUnitary N) {θM : AntiUnitary V} {θN : AntiUnitary W} (hθM : IsCon

Proved

BookProof.ChapterA.Rreal_isometric_iff_complexification_isometric

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA2d.

BookProof.ChapterA.Rreal_isometric_iff_complexification_isometric (M : System ℂ V) (N : System ℂ W) (hN : IsSchurUnitary N) {θM : AntiUnitary V} {θN : AntiUnitary W} (hθM : IsConjugation M θM) (hθN : IsConjugation N θN) : (∃ α : V ≃ₗᵢ[ℂ] W, IsSystemIso M N α ∧ ∀ x, α (θM x) = θN (α x)) ↔ (∃ α : V ≃ₗᵢ[ℂ] W, IsSystemIso M N α)

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

Preamble

-- Generated from ChapterA2d.lean — theorem BookProof.ChapterA.Rreal_isometric_iff_complexification_isometric
import Mathlib
import Definitions.Def_ChapterA2d
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]
variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [CompleteSpace W]

Formal statement

theorem BookProof.ChapterA.Rreal_isometric_iff_complexification_isometric
    (M : System ℂ V) (N : System ℂ W) (hN : IsSchurUnitary N)
    {θM : AntiUnitary V} {θN : AntiUnitary W}
    (hθM : IsConjugation M θM) (hθN : IsConjugation N θN) :
    (∃ α : V ≃ₗᵢ[ℂ] W, IsSystemIso M N α ∧ ∀ x, α (θM x) = θN (α x)) ↔
    (∃ α : V ≃ₗᵢ[ℂ] W, IsSystemIso M N α) := by sorry

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