theorem b0lt (a b: nat): $ a < b <-> b0 a < b0 b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a < b <-> 2 * a < 2 * b) -> (2 * a < 2 * b <-> b0 a < b0 b) -> (a < b <-> b0 a < b0 b) |
| 2 |
|
ltmul2 |
0 < 2 -> (a < b <-> 2 * a < 2 * b) |
| 3 |
|
d0lt2 |
0 < 2 |
| 4 |
2, 3 |
ax_mp |
a < b <-> 2 * a < 2 * b |
| 5 |
1, 4 |
ax_mp |
(2 * a < 2 * b <-> b0 a < b0 b) -> (a < b <-> b0 a < b0 b) |
| 6 |
|
lteq |
2 * a = b0 a -> 2 * b = b0 b -> (2 * a < 2 * b <-> b0 a < b0 b) |
| 7 |
|
b0mul21 |
2 * a = b0 a |
| 8 |
6, 7 |
ax_mp |
2 * b = b0 b -> (2 * a < 2 * b <-> b0 a < b0 b) |
| 9 |
|
b0mul21 |
2 * b = b0 b |
| 10 |
8, 9 |
ax_mp |
2 * a < 2 * b <-> b0 a < b0 b |
| 11 |
5, 10 |
ax_mp |
a < b <-> b0 a < b0 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_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)