theorem modmodid (a n: nat): $ a % n % n = a % n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
mod0 |
a % 0 = a |
| 2 |
|
modeq2 |
n = 0 -> a % n = a % 0 |
| 3 |
1, 2 |
syl6eq |
n = 0 -> a % n = a |
| 4 |
|
eqidd |
n = 0 -> n = n |
| 5 |
3, 4 |
modeqd |
n = 0 -> a % n % n = a % n |
| 6 |
|
modlteq |
a % n < n -> a % n % n = a % n |
| 7 |
|
modlt |
n != 0 -> a % n < n |
| 8 |
7 |
conv ne |
~n = 0 -> a % n < n |
| 9 |
6, 8 |
syl |
~n = 0 -> a % n % n = a % n |
| 10 |
5, 9 |
cases |
a % n % n = a % n |
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)