Theorem zpncan2 ≪ | index | src | ≫

theorem zpncan2 (a b: nat): $ a +Z b -Z a = b $;

Step	Hyp	Ref	Expression
1		eqtr	a +Z b -Z a = b +Z a -Z a -> b +Z a -Z a = b -> a +Z b -Z a = b
2		zsubeq1	a +Z b = b +Z a -> a +Z b -Z a = b +Z a -Z a
3		zaddcom	a +Z b = b +Z a
4	2, 3	ax_mp	a +Z b -Z a = b +Z a -Z a
5	1, 4	ax_mp	b +Z a -Z a = b -> a +Z b -Z a = b
6		zpncan	b +Z a -Z a = b
7	5, 6	ax_mp	a +Z b -Z a = b

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)