Theorem subltid ≪ | index | src | ≫ 

theorem subltid (a b: nat): $ 0 < a /\ 0 < b -> a - b < a $;
StepHypRefExpression
1 ltadd1 
a - b < a <-> a - b + b < a + b
2 add0 
a + 0 = a
3 npcan 
b <= a -> a - b + b = a
4 2, 3 syl6eqr 
b <= a -> a - b + b = a + 0
5 4 lteq1d 
b <= a -> (a - b + b < a + b <-> a + 0 < a + b)
6 5 anwr 
0 < a /\ 0 < b /\ b <= a -> (a - b + b < a + b <-> a + 0 < a + b)
7 ltadd2 
0 < b <-> a + 0 < a + b
8 anlr 
0 < a /\ 0 < b /\ b <= a -> 0 < b
9 7, 8 sylib 
0 < a /\ 0 < b /\ b <= a -> a + 0 < a + b
10 6, 9 mpbird 
0 < a /\ 0 < b /\ b <= a -> a - b + b < a + b
11 1, 10 sylibr 
0 < a /\ 0 < b /\ b <= a -> a - b < a
12 nlesubeq0 
~b <= a -> a - b = 0
13 12 lteq1d 
~b <= a -> (a - b < a <-> 0 < a)
14 13 anwr 
0 < a /\ 0 < b /\ ~b <= a -> (a - b < a <-> 0 < a)
15 anll 
0 < a /\ 0 < b /\ ~b <= a -> 0 < a
16 14, 15 mpbird 
0 < a /\ 0 < b /\ ~b <= a -> a - b < a
17 11, 16 casesda 
0 < a /\ 0 < b -> a - 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 (peano2, peano5, addeq, add0, addS)