Theorem zdvdadd2 ≪ | index | src | ≫

theorem zdvdadd2 (a b n: nat): $ n |Z a -> (n |Z b <-> n |Z b +Z a) $;

Step	Hyp	Ref	Expression
1		eqidd	a +Z b = b +Z a -> n = n
2		id	a +Z b = b +Z a -> a +Z b = b +Z a
3	1, 2	zdvdeqd	a +Z b = b +Z a -> (n \|Z a +Z b <-> n \|Z b +Z a)
4		zaddcom	a +Z b = b +Z a
5	3, 4	ax_mp	n \|Z a +Z b <-> n \|Z b +Z a
6		zdvdadd1	n \|Z a -> (n \|Z b <-> n \|Z a +Z b)
7	5, 6	syl6bb	n \|Z a -> (n \|Z b <-> n \|Z b +Z a)

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)