Theorem sucsub ≪ | index | src | ≫

theorem sucsub (a b: nat): $ b <= a -> suc a - b = suc (a - b) $;

Step	Hyp	Ref	Expression
1		subeq1	a + 1 = suc a -> a + 1 - b = suc a - b
2		add12	a + 1 = suc a
3	1, 2	ax_mp	a + 1 - b = suc a - b
4		add12	a - b + 1 = suc (a - b)
5		addsub	b <= a -> a + 1 - b = a - b + 1
6	4, 5	syl6eq	b <= a -> a + 1 - b = suc (a - b)
7	3, 6	syl5eqr	b <= a -> suc a - b = suc (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 (peano2, peano5, addeq, add0, addS)