theorem znegid (a: nat): $ a +Z -uZ a = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
zfst a -ZN zsnd a +Z -uZ a = a +Z -uZ a -> zfst a -ZN zsnd a +Z -uZ a = 0 -> a +Z -uZ a = 0 |
| 2 |
|
zaddeq1 |
zfst a -ZN zsnd a = a -> zfst a -ZN zsnd a +Z -uZ a = a +Z -uZ a |
| 3 |
|
zfstsnd |
zfst a -ZN zsnd a = a |
| 4 |
2, 3 |
ax_mp |
zfst a -ZN zsnd a +Z -uZ a = a +Z -uZ a |
| 5 |
1, 4 |
ax_mp |
zfst a -ZN zsnd a +Z -uZ a = 0 -> a +Z -uZ a = 0 |
| 6 |
|
eqtr |
zfst a -ZN zsnd a +Z -uZ a = zfst a + zsnd a -ZN (zsnd a + zfst a) -> zfst a + zsnd a -ZN (zsnd a + zfst a) = 0 -> zfst a -ZN zsnd a +Z -uZ a = 0 |
| 7 |
|
zaddzn |
zfst a -ZN zsnd a +Z (zsnd a -ZN zfst a) = zfst a + zsnd a -ZN (zsnd a + zfst a) |
| 8 |
7 |
conv zneg |
zfst a -ZN zsnd a +Z -uZ a = zfst a + zsnd a -ZN (zsnd a + zfst a) |
| 9 |
6, 8 |
ax_mp |
zfst a + zsnd a -ZN (zsnd a + zfst a) = 0 -> zfst a -ZN zsnd a +Z -uZ a = 0 |
| 10 |
|
eqtr |
zfst a + zsnd a -ZN (zsnd a + zfst a) = zsnd a + zfst a -ZN (zsnd a + zfst a) ->
zsnd a + zfst a -ZN (zsnd a + zfst a) = 0 ->
zfst a + zsnd a -ZN (zsnd a + zfst a) = 0 |
| 11 |
|
znsubeq1 |
zfst a + zsnd a = zsnd a + zfst a -> zfst a + zsnd a -ZN (zsnd a + zfst a) = zsnd a + zfst a -ZN (zsnd a + zfst a) |
| 12 |
|
addcom |
zfst a + zsnd a = zsnd a + zfst a |
| 13 |
11, 12 |
ax_mp |
zfst a + zsnd a -ZN (zsnd a + zfst a) = zsnd a + zfst a -ZN (zsnd a + zfst a) |
| 14 |
10, 13 |
ax_mp |
zsnd a + zfst a -ZN (zsnd a + zfst a) = 0 -> zfst a + zsnd a -ZN (zsnd a + zfst a) = 0 |
| 15 |
|
znsubid |
zsnd a + zfst a -ZN (zsnd a + zfst a) = 0 |
| 16 |
14, 15 |
ax_mp |
zfst a + zsnd a -ZN (zsnd a + zfst a) = 0 |
| 17 |
9, 16 |
ax_mp |
zfst a -ZN zsnd a +Z -uZ a = 0 |
| 18 |
5, 17 |
ax_mp |
a +Z -uZ a = 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)