`BookProof.ChapterA3.upsilon_apply_comb` (T : Matrix (Fin 2) (Fin 2) ℂ) (x : Fin 4 → ℂ) : Tᴴ * (∑ ν, x ν • pauliσ ν) * T = ∑ μ, (∑ ν, UpsilonC T μ ν * x ν) • pauliσ μ

Proved

BookProof.ChapterA3.upsilon_apply_comb

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3h.

BookProof.ChapterA3.upsilon_apply_comb (T : Matrix (Fin 2) (Fin 2) ℂ) (x : Fin 4 → ℂ) : Tᴴ * (∑ ν, x ν • pauliσ ν) * T = ∑ μ, (∑ ν, UpsilonC T μ ν * x ν) • pauliσ μ

Formalization note: Lean 4 identifier BookProof.ChapterA3.upsilon_apply_comb.

Preamble

-- Generated from ChapterA3h.lean — theorem BookProof.ChapterA3.upsilon_apply_comb
import Mathlib
import Definitions.Def_ChapterA3h
import Definitions.Def_ChapterA3
open BookProof.ChapterA3


open Matrix
open scoped ComplexConjugate

Formal statement

theorem BookProof.ChapterA3.upsilon_apply_comb (T : Matrix (Fin 2) (Fin 2) ℂ) (x : Fin 4 → ℂ) :
    Tᴴ * (∑ ν, x ν • pauliσ ν) * T = ∑ μ, (∑ ν, UpsilonC T μ ν * x ν) • pauliσ μ := by sorry

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