theorem diveq0 (a b: nat): $ b != 0 -> (a // b = 0 <-> a < b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
lt12 |
a // b < 1 <-> a // b = 0 |
| 2 |
|
lteq2 |
b * 1 = b -> (a < b * 1 <-> a < b) |
| 3 |
|
mul12 |
b * 1 = b |
| 4 |
2, 3 |
ax_mp |
a < b * 1 <-> a < b |
| 5 |
|
divltmul1 |
b != 0 -> (a // b < 1 <-> a < b * 1) |
| 6 |
4, 5 |
syl6bb |
b != 0 -> (a // b < 1 <-> a < b) |
| 7 |
1, 6 |
syl5bbr |
b != 0 -> (a // b = 0 <-> 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)