Theorem zltznsub0 ≪ | index | src | ≫

theorem zltznsub0 (a b: nat): $ a -ZN b  a < b $;

Step	Hyp	Ref	Expression
1		bitr3	(b0 a -Z b0 b <Z 0 <-> a -ZN b <Z 0) -> (b0 a -Z b0 b <Z 0 <-> a < b) -> (a -ZN b <Z 0 <-> a < b)
2		zlteq1	b0 a -Z b0 b = a -ZN b -> (b0 a -Z b0 b <Z 0 <-> a -ZN b <Z 0)
3		zsubb0	b0 a -Z b0 b = a -ZN b
4	2, 3	ax_mp	b0 a -Z b0 b <Z 0 <-> a -ZN b <Z 0
5	1, 4	ax_mp	(b0 a -Z b0 b <Z 0 <-> a < b) -> (a -ZN b <Z 0 <-> a < b)
6		bitr	(b0 a -Z b0 b <Z 0 <-> b0 a <Z b0 b) -> (b0 a <Z b0 b <-> a < b) -> (b0 a -Z b0 b <Z 0 <-> a < b)
7		zltsub0	b0 a -Z b0 b <Z 0 <-> b0 a <Z b0 b
8	6, 7	ax_mp	(b0 a <Z b0 b <-> a < b) -> (b0 a -Z b0 b <Z 0 <-> a < b)
9		zltb0	b0 a <Z b0 b <-> a < b
10	8, 9	ax_mp	b0 a -Z b0 b <Z 0 <-> a < b
11	5, 10	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)