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	3	zlteq1*	b0 a -Z b0 b <Z 0 <-> a -ZN b <Z 0
5	4	bitr3*	(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	7	bitr*	(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	7, 9	bitr*	b0 a -Z b0 b <Z 0 <-> a < b
11	4, 10	bitr3*	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)