Theorem zltb0 ≪ | index | src | ≫

theorem zltb0 (a b: nat): $ b0 a  a < b $;

Step	Hyp	Ref	Expression
1		bitr	(b0 a <Z b0 b <-> odd (a -ZN b)) -> (odd (a -ZN b) <-> a < b) -> (b0 a <Z b0 b <-> a < b)
2		oddeq	b0 a -Z b0 b = a -ZN b -> (odd (b0 a -Z b0 b) <-> odd (a -ZN b))
3	2	conv zlt	b0 a -Z b0 b = a -ZN b -> (b0 a <Z b0 b <-> odd (a -ZN b))
4		zsubb0	b0 a -Z b0 b = a -ZN b
5	3, 4	ax_mp	b0 a <Z b0 b <-> odd (a -ZN b)
6	1, 5	ax_mp	(odd (a -ZN b) <-> a < b) -> (b0 a <Z b0 b <-> a < b)
7		znsubodd	odd (a -ZN b) <-> a < b
8	6, 7	ax_mp	b0 a <Z b0 b <-> 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)