det_eq_of_forall_row_eq_smul_add_pred

English

A determinant is invariant under a row-addition pattern for pred_aux form.

Русский

Детерминант сохраняется при добавлении строк в предельной форме.

LaTeX

$$$\\det M = \\det N$ \\\\text{under pred-based row-addition pattern}$$

Lean4

/-- If you add multiples of previous rows to the next row, the determinant doesn't change. -/
theorem det_eq_of_forall_row_eq_smul_add_pred {n : ℕ} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R} (c : Fin n → R)
    (A_zero : ∀ j, A 0 j = B 0 j) (A_succ : ∀ (i : Fin n) (j), A i.succ j = B i.succ j + c i * A (Fin.castSucc i) j) :
    det A = det B :=
  det_eq_of_forall_row_eq_smul_add_pred_aux (Fin.last _) c (fun _ hi => absurd hi (not_lt_of_ge (Fin.le_last _))) A_zero
    A_succ

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