Theorem eqzneg ≪ | index | src | ≫

theorem eqzneg (a b: nat): $ -uZ a = b <-> a +Z b = 0 $;

Step	Hyp	Ref	Expression
1		bitr3	(0 -Z a = b <-> -uZ a = b) -> (0 -Z a = b <-> a +Z b = 0) -> (-uZ a = b <-> a +Z b = 0)
2		eqeq1	0 -Z a = -uZ a -> (0 -Z a = b <-> -uZ a = b)
3		zsub01	0 -Z a = -uZ a
4	2, 3	ax_mp	0 -Z a = b <-> -uZ a = b
5	1, 4	ax_mp	(0 -Z a = b <-> a +Z b = 0) -> (-uZ a = b <-> a +Z b = 0)
6		eqzsub2	0 -Z a = b <-> a +Z b = 0
7	5, 6	ax_mp	-uZ a = b <-> a +Z b = 0

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)