theorem lemul1 (a b c: nat): $ 0 < c -> (a <= b <-> a * c <= b * c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bieq |
(a <= b <-> ~b < a) -> (a * c <= b * c <-> ~b * c < a * c) -> (a <= b <-> a * c <= b * c <-> (~b < a <-> ~b * c < a * c)) |
| 2 |
|
lenlt |
a <= b <-> ~b < a |
| 3 |
1, 2 |
ax_mp |
(a * c <= b * c <-> ~b * c < a * c) -> (a <= b <-> a * c <= b * c <-> (~b < a <-> ~b * c < a * c)) |
| 4 |
|
lenlt |
a * c <= b * c <-> ~b * c < a * c |
| 5 |
3, 4 |
ax_mp |
a <= b <-> a * c <= b * c <-> (~b < a <-> ~b * c < a * c) |
| 6 |
|
ltmul1 |
0 < c -> (b < a <-> b * c < a * c) |
| 7 |
6 |
noteqd |
0 < c -> (~b < a <-> ~b * c < a * c) |
| 8 |
5, 7 |
sylibr |
0 < c -> (a <= b <-> a * c <= 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_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)