theorem znegneg (n: nat): $ -uZ -uZ n = n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
-uZ -uZ n = zfst n -ZN zsnd n -> zfst n -ZN zsnd n = n -> -uZ -uZ n = n |
| 2 |
|
znegzn |
-uZ (zsnd n -ZN zfst n) = zfst n -ZN zsnd n |
| 3 |
2 |
conv zneg |
-uZ -uZ n = zfst n -ZN zsnd n |
| 4 |
1, 3 |
ax_mp |
zfst n -ZN zsnd n = n -> -uZ -uZ n = n |
| 5 |
|
zfstsnd |
zfst n -ZN zsnd n = n |
| 6 |
4, 5 |
ax_mp |
-uZ -uZ 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)