sub_scalar_sq_eq_disc — Mathlib

English

There is a canonical coercion from SpecialLinearGroup to GLPos, realized as inclusion into GL and then into GLPos, compatible with the algebraic structure.

Русский

Существует каноническое приведение из SpecialLinearGroup в GLPos, реализованное как включение в GL, затем в GLPos, совместимое с структурой алгебры.

LaTeX

$$Coefficient compatibility: Coe(SpecialLinearGroup → GLPos) is the inclusion map that respects multiplication and identity.$$

Lean4

theorem sub_scalar_sq_eq_disc [NeZero (2 : K)] : (m - scalar _ (m.trace / 2)) ^ 2 = scalar _ (m.disc / 4) :=
  by
  simp only [scalar_apply, trace_fin_two, disc_fin_two, trace_fin_two, det_fin_two, sq, (by norm_num : (4 : K) = 2 * 2)]
  ext i j
  fin_cases i <;> fin_cases j <;>
    · simp [Matrix.mul_apply]
      field_simp
      ring

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