theorem mulmod2 (a b: nat): $ a * b % b = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a * b % b = b * a % b -> b * a % b = 0 -> a * b % b = 0 |
| 2 |
|
id |
a * b = b * a -> a * b = b * a |
| 3 |
|
eqidd |
a * b = b * a -> b = b |
| 4 |
2, 3 |
modeqd |
a * b = b * a -> a * b % b = b * a % b |
| 5 |
|
mulcom |
a * b = b * a |
| 6 |
4, 5 |
ax_mp |
a * b % b = b * a % b |
| 7 |
1, 6 |
ax_mp |
b * a % b = 0 -> a * b % b = 0 |
| 8 |
|
mulmod1 |
b * a % b = 0 |
| 9 |
7, 8 |
ax_mp |
a * b % b = 0 |
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)