J_det_mul_J_det — Mathlib

English

The determinant of J is its own square root, i.e. det(J)^2 = 1; hence det(J) is a unit.

Русский

Определитель J удовлетворяет det(J)^2 = 1; следовательно, det(J) — единичный элемент.

LaTeX

$$$\det(J)^{2} = 1$ and hence $\det(J)$ is a unit.$$

Lean4

theorem J_det_mul_J_det : det (J l R) * det (J l R) = 1 :=
  by
  rw [← det_mul, J_squared, ← one_smul R (-1 : Matrix _ _ R), smul_neg, ← neg_smul, det_smul, Fintype.card_sum, det_one,
    mul_one]
  apply Even.neg_one_pow
  exact Even.add_self _

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