inv_inj — Mathlib

English

If the right inverse and the left inverse of A exist, they coincide: B = C when A B = 1 and C A = 1.

Русский

Если существуют правая и левая обратные A, они совпадают: B = C при A B = 1 и C A = 1.

LaTeX

$$If A B = 1 and C A = 1, then B = C.$$

Lean4

theorem inv_inj (h : A⁻¹ = B⁻¹) (h' : IsUnit A.det) : A = B :=
  by
  refine left_inv_eq_left_inv (mul_nonsing_inv _ h') ?_
  rw [h]
  refine mul_nonsing_inv _ ?_
  rwa [← isUnit_nonsing_inv_det_iff, ← h, isUnit_nonsing_inv_det_iff]

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