Theorem znsub02 ≪ | index | src | ≫

theorem znsub02 (a: nat): $ a -ZN 0 = b0 a $;

Step	Hyp	Ref	Expression
1		eqtr	a -ZN 0 = b0 (a - 0) -> b0 (a - 0) = b0 a -> a -ZN 0 = b0 a
2		znsubpos	0 <= a -> a -ZN 0 = b0 (a - 0)
3		le01	0 <= a
4	2, 3	ax_mp	a -ZN 0 = b0 (a - 0)
5	1, 4	ax_mp	b0 (a - 0) = b0 a -> a -ZN 0 = b0 a
6		b0eq	a - 0 = a -> b0 (a - 0) = b0 a
7		sub02	a - 0 = a
8	6, 7	ax_mp	b0 (a - 0) = b0 a
9	5, 8	ax_mp	a -ZN 0 = b0 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), axs_peano (peano1, peano2, peano5, addeq, add0, addS)