Theorem zlt0znsub ≪ | index | src | ≫ 

theorem zlt0znsub (a b: nat): $ 0  b < a $;
StepHypRefExpression
1 bitr3 
(0 <Z b0 a -Z b0 b <-> 0 <Z a -ZN b) -> (0 <Z b0 a -Z b0 b <-> b < a) -> (0 <Z a -ZN b <-> b < a)
2 zlteq2 
b0 a -Z b0 b = a -ZN b -> (0 <Z b0 a -Z b0 b <-> 0 <Z a -ZN b)
3 zsubb0 
b0 a -Z b0 b = a -ZN b
4 2, 3 ax_mp 
0 <Z b0 a -Z b0 b <-> 0 <Z a -ZN b
5 1, 4 ax_mp 
(0 <Z b0 a -Z b0 b <-> b < a) -> (0 <Z a -ZN b <-> b < a)
6 bitr 
(0 <Z b0 a -Z b0 b <-> b0 b <Z b0 a) -> (b0 b <Z b0 a <-> b < a) -> (0 <Z b0 a -Z b0 b <-> b < a)
7 zlt0sub 
0 <Z b0 a -Z b0 b <-> b0 b <Z b0 a
8 6, 7 ax_mp 
(b0 b <Z b0 a <-> b < a) -> (0 <Z b0 a -Z b0 b <-> b < a)
9 zltb0 
b0 b <Z b0 a <-> b < a
10 8, 9 ax_mp 
0 <Z b0 a -Z b0 b <-> b < a
11 5, 10 ax_mp 
0 <Z a -ZN b <-> 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)