Theorem dvdsubmod ≪ | index | src | ≫

theorem dvdsubmod (a n: nat): $ n || a - a % n $;

Step	Hyp	Ref	Expression
1		idvd2	n * (a // n) = a - a % n -> n \|\| a - a % n
2		eqcom	a - a % n = n * (a // n) -> n * (a // n) = a - a % n
3		eqsub1	n * (a // n) + a % n = a -> a - a % n = n * (a // n)
4		divmod	n * (a // n) + a % n = a
5	3, 4	ax_mp	a - a % n = n * (a // n)
6	2, 5	ax_mp	n * (a // n) = a - a % n
7	1, 6	ax_mp	n \|\| a - a % n

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)