`BookProof.ChapterF4.unitary_preserves_dotProduct` {n : ℕ} (U : Matrix (Fin n) (Fin n) ℂ) (hU : Uᴴ * U = 1) (x y : Fin n → ℂ) : star (U.mulVec x) ⬝ᵥ U.mulVec y = star x ⬝ᵥ y

Proved

BookProof.ChapterF4.unitary_preserves_dotProduct

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterF4.

BookProof.ChapterF4.unitary_preserves_dotProduct {n : ℕ} (U : Matrix (Fin n) (Fin n) ℂ) (hU : Uᴴ * U = 1) (x y : Fin n → ℂ) : star (U.mulVec x) ⬝ᵥ U.mulVec y = star x ⬝ᵥ y

Formalization note: Lean 4 identifier BookProof.ChapterF4.unitary_preserves_dotProduct.

Preamble

-- Generated from ChapterF4.lean — theorem BookProof.ChapterF4.unitary_preserves_dotProduct
import Mathlib
import Definitions.Def_ChapterF4
open BookProof.ChapterF4


open scoped BigOperators Matrix

variable {d k : ℕ}

Formal statement

theorem BookProof.ChapterF4.unitary_preserves_dotProduct {n : ℕ} (U : Matrix (Fin n) (Fin n) ℂ)
    (hU : Uᴴ * U = 1) (x y : Fin n → ℂ) :
    star (U.mulVec x) ⬝ᵥ U.mulVec y = star x ⬝ᵥ y := by sorry

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