theorem zmodn02 (a n: nat):
$ n != 0 -> a %Z n = (zfst a + n - zsnd a % n) % n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zabsb0 |
zabs (b0 (zfst a + n - zsnd a % n)) = zfst a + n - zsnd a % n |
| 2 |
|
znsubpos |
zsnd a % n <= zfst a + n -> zfst a + n -ZN zsnd a % n = b0 (zfst a + n - zsnd a % n) |
| 3 |
|
ltle |
zsnd a % n < n -> zsnd a % n <= n |
| 4 |
|
modlt |
n != 0 -> zsnd a % n < n |
| 5 |
3, 4 |
syl |
n != 0 -> zsnd a % n <= n |
| 6 |
|
leaddid2 |
n <= zfst a + n |
| 7 |
6 |
a1i |
n != 0 -> n <= zfst a + n |
| 8 |
5, 7 |
letrd |
n != 0 -> zsnd a % n <= zfst a + n |
| 9 |
2, 8 |
syl |
n != 0 -> zfst a + n -ZN zsnd a % n = b0 (zfst a + n - zsnd a % n) |
| 10 |
9 |
zabseqd |
n != 0 -> zabs (zfst a + n -ZN zsnd a % n) = zabs (b0 (zfst a + n - zsnd a % n)) |
| 11 |
1, 10 |
syl6eq |
n != 0 -> zabs (zfst a + n -ZN zsnd a % n) = zfst a + n - zsnd a % n |
| 12 |
11 |
modeq1d |
n != 0 -> zabs (zfst a + n -ZN zsnd a % n) % n = (zfst a + n - zsnd a % n) % n |
| 13 |
12 |
conv zmod |
n != 0 -> a %Z n = (zfst a + n - zsnd a % n) % 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)