Theorem subleid ≪ | index | src | ≫

theorem subleid (a b: nat): $ a - b <= a $;

Step	Hyp	Ref	Expression
1		leaddid1	a - b <= a - b + b
2		npcan	b <= a -> a - b + b = a
3	2	leeq2d	b <= a -> (a - b <= a - b + b <-> a - b <= a)
4	1, 3	mpbii	b <= a -> a - b <= a
5		le01	0 <= a
6		nlesubeq0	~b <= a -> a - b = 0
7	6	leeq1d	~b <= a -> (a - b <= a <-> 0 <= a)
8	5, 7	mpbiri	~b <= a -> a - b <= a
9	4, 8	cases	a - 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 (peano2, peano5, addeq, add0, addS)