theorem divmod2 (a b c: nat): $ a // b % c = a % (c * b) // b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a // b % c = a % (b * c) // b -> a % (b * c) // b = a % (c * b) // b -> a // b % c = a % (c * b) // b |
| 2 |
|
divmod1 |
a // b % c = a % (b * c) // b |
| 3 |
1, 2 |
ax_mp |
a % (b * c) // b = a % (c * b) // b -> a // b % c = a % (c * b) // b |
| 4 |
|
diveq1 |
a % (b * c) = a % (c * b) -> a % (b * c) // b = a % (c * b) // b |
| 5 |
|
modeq2 |
b * c = c * b -> a % (b * c) = a % (c * b) |
| 6 |
|
mulcom |
b * c = c * b |
| 7 |
5, 6 |
ax_mp |
a % (b * c) = a % (c * b) |
| 8 |
4, 7 |
ax_mp |
a % (b * c) // b = a % (c * b) // b |
| 9 |
3, 8 |
ax_mp |
a // b % c = a % (c * b) // 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)