Theorem ltappendid1 ≪ | index | src | ≫ 

theorem ltappendid1 (a b: nat): $ b != 0 <-> a < a ++ b $;
StepHypRefExpression
1 bitr3 
(0 < b <-> b != 0) -> (0 < b <-> a < a ++ b) -> (b != 0 <-> a < a ++ b)
2 lt01 
0 < b <-> b != 0
3 1, 2 ax_mp 
(0 < b <-> a < a ++ b) -> (b != 0 <-> a < a ++ b)
4 bitr 
(0 < b <-> a ++ 0 < a ++ b) -> (a ++ 0 < a ++ b <-> a < a ++ b) -> (0 < b <-> a < a ++ b)
5 ltappend2 
0 < b <-> a ++ 0 < a ++ b
6 4, 5 ax_mp 
(a ++ 0 < a ++ b <-> a < a ++ b) -> (0 < b <-> a < a ++ b)
7 lteq1 
a ++ 0 = a -> (a ++ 0 < a ++ b <-> a < a ++ b)
8 append02 
a ++ 0 = a
9 7, 8 ax_mp 
a ++ 0 < a ++ b <-> a < a ++ b
10 6, 9 ax_mp 
0 < b <-> a < a ++ b
11 3, 10 ax_mp 
b != 0 <-> a < 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)