theorem divltmul1 (a b c: nat): $ b != 0 -> (a // b < c <-> a < b * c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ltnle |
a // b < c <-> ~c <= a // b |
| 2 |
|
ltnle |
a < b * c <-> ~b * c <= a |
| 3 |
|
ledivmul1 |
b != 0 -> (c <= a // b <-> b * c <= a) |
| 4 |
3 |
noteqd |
b != 0 -> (~c <= a // b <-> ~b * c <= a) |
| 5 |
2, 4 |
syl6bbr |
b != 0 -> (~c <= a // b <-> a < b * c) |
| 6 |
1, 5 |
syl5bb |
b != 0 -> (a // b < c <-> a < b * c) |
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)