Theorem znpcan ≪ | index | src | ≫

theorem znpcan (a b: nat): $ a -Z b +Z b = a $;

Step	Hyp	Ref	Expression
1		eqtr	a -Z b +Z b = a +Z (-uZ b +Z b) -> a +Z (-uZ b +Z b) = a -> a -Z b +Z b = a
2		zaddass	a +Z -uZ b +Z b = a +Z (-uZ b +Z b)
3	2	conv zsub	a -Z b +Z b = a +Z (-uZ b +Z b)
4	1, 3	ax_mp	a +Z (-uZ b +Z b) = a -> a -Z b +Z b = a
5		eqtr	a +Z (-uZ b +Z b) = a +Z 0 -> a +Z 0 = a -> a +Z (-uZ b +Z b) = a
6		zaddeq2	-uZ b +Z b = 0 -> a +Z (-uZ b +Z b) = a +Z 0
7		znegid1	-uZ b +Z b = 0
8	6, 7	ax_mp	a +Z (-uZ b +Z b) = a +Z 0
9	5, 8	ax_mp	a +Z 0 = a -> a +Z (-uZ b +Z b) = a
10		zadd02	a +Z 0 = a
11	9, 10	ax_mp	a +Z (-uZ b +Z b) = a
12	4, 11	ax_mp	a -Z b +Z b = 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)