theorem div2lt (n: nat): $ 0 < n -> n // 2 < n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
divltmul1 |
2 != 0 -> (n // 2 < n <-> n < 2 * n) |
| 2 |
|
d2ne0 |
2 != 0 |
| 3 |
1, 2 |
ax_mp |
n // 2 < n <-> n < 2 * n |
| 4 |
|
lteq1 |
1 * n = n -> (1 * n < 2 * n <-> n < 2 * n) |
| 5 |
|
mul11 |
1 * n = n |
| 6 |
4, 5 |
ax_mp |
1 * n < 2 * n <-> n < 2 * n |
| 7 |
|
d1lt2 |
1 < 2 |
| 8 |
|
ltmul1 |
0 < n -> (1 < 2 <-> 1 * n < 2 * n) |
| 9 |
7, 8 |
mpbii |
0 < n -> 1 * n < 2 * n |
| 10 |
6, 9 |
sylib |
0 < n -> n < 2 * n |
| 11 |
3, 10 |
sylibr |
0 < n -> n // 2 < n |
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)