theorem zneg0: $ -uZ 0 = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
-uZ (0 -ZN 0) = -uZ 0 -> -uZ (0 -ZN 0) = 0 -> -uZ 0 = 0 |
| 2 |
|
znegeq |
0 -ZN 0 = 0 -> -uZ (0 -ZN 0) = -uZ 0 |
| 3 |
|
znsubid |
0 -ZN 0 = 0 |
| 4 |
2, 3 |
ax_mp |
-uZ (0 -ZN 0) = -uZ 0 |
| 5 |
1, 4 |
ax_mp |
-uZ (0 -ZN 0) = 0 -> -uZ 0 = 0 |
| 6 |
|
eqtr |
-uZ (0 -ZN 0) = 0 -ZN 0 -> 0 -ZN 0 = 0 -> -uZ (0 -ZN 0) = 0 |
| 7 |
|
znegzn |
-uZ (0 -ZN 0) = 0 -ZN 0 |
| 8 |
6, 7 |
ax_mp |
0 -ZN 0 = 0 -> -uZ (0 -ZN 0) = 0 |
| 9 |
8, 3 |
ax_mp |
-uZ (0 -ZN 0) = 0 |
| 10 |
5, 9 |
ax_mp |
-uZ 0 = 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)