Theorem modle1 ≪ | index | src | ≫

theorem modle1 (a b: nat): $ a % b <= a $;

Step	Hyp	Ref	Expression
1		leeq2	b * (a // b) + a % b = a -> (a % b <= b * (a // b) + a % b <-> a % b <= a)
2		divmod	b * (a // b) + a % b = a
3	1, 2	ax_mp	a % b <= b * (a // b) + a % b <-> a % b <= a
4		leaddid2	a % b <= b * (a // b) + a % b
5	3, 4	mpbi	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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)