Theorem zeqmaddn ≪ | index | src | ≫

theorem zeqmaddn (a n: nat): $ modZ(n): a +Z b0 n = a $;

Step	Hyp	Ref	Expression
1		zeqmeq3	a +Z 0 = a -> (modZ(n): a +Z b0 n = a +Z 0 <-> modZ(n): a +Z b0 n = a)
2		zadd02	a +Z 0 = a
3	1, 2	ax_mp	modZ(n): a +Z b0 n = a +Z 0 <-> modZ(n): a +Z b0 n = a
4		zeqmadd2	modZ(n): a +Z b0 n = a +Z 0 <-> modZ(n): b0 n = 0
5		zeqmid0	modZ(n): b0 n = 0
6	4, 5	mpbir	modZ(n): a +Z b0 n = a +Z 0
7	3, 6	mpbi	modZ(n): a +Z b0 n = 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)