Theorem zleznsub0 ≪ | index | src | ≫

theorem zleznsub0 (a b: nat): $ a -ZN b <=Z 0 <-> a <= b $;

Step	Hyp	Ref	Expression
1		bitr4	(a -ZN b <=Z 0 <-> ~b < a) -> (a <= b <-> ~b < a) -> (a -ZN b <=Z 0 <-> a <= b)
2		noteq	(0 <Z a -ZN b <-> b < a) -> (~0 <Z a -ZN b <-> ~b < a)
3	2	conv zle	(0 <Z a -ZN b <-> b < a) -> (a -ZN b <=Z 0 <-> ~b < a)
4		zlt0znsub	0 <Z a -ZN b <-> b < a
5	3, 4	ax_mp	a -ZN b <=Z 0 <-> ~b < a
6	1, 5	ax_mp	(a <= b <-> ~b < a) -> (a -ZN b <=Z 0 <-> a <= b)
7		lenlt	a <= b <-> ~b < a
8	6, 7	ax_mp	a -ZN b <=Z 0 <-> 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)