English
The imaginary part changes sign under conjugation: star a.im = - a.im.
Русский
Мнимая часть меняет знак при сопряжении: star a.im = - a.im.
LaTeX
$$$ star a.im = - a.im $$$
Lean4
/-- Square of the norm. -/
def normSq : ℍ[R] →*₀ R where
toFun a := (a * star a).re
map_zero' := by simp only [star_zero, zero_mul, re_zero]
map_one' := by simp only [star_one, one_mul, re_one]
map_mul' x
y :=
coe_injective <| by
conv_lhs =>
rw [← mul_star_eq_coe, star_mul, mul_assoc, ← mul_assoc y, y.mul_star_eq_coe, coe_commutes, ← mul_assoc,
x.mul_star_eq_coe, ← coe_mul]