mul_iff_left — Mathlib

English

If M is nondegenerate, then M N is nondegenerate iff N is nondegenerate.

Русский

Если M неdegenerate, то M N неdegenerate эквивалентно N неdegenerate.

LaTeX

$$$(M \\cdot N)\\text{ Nondegenerate } \\iff N\\text{ Nondegenerate}$, given $M$ nondegenerate$$

Lean4

theorem mul_iff_left {A : Type*} [CommRing A] [IsDomain A] {M N : Matrix n n A} (h : M.Nondegenerate) :
    (M * N).Nondegenerate ↔ N.Nondegenerate := by
  classical
  simp only [nondegenerate_iff_det_ne_zero, det_mul] at h ⊢
  exact mul_ne_zero_iff_left h

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