theorem zlt0neg (a: nat): $ 0 a
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
oddeq |
0 -Z -uZ a = a -Z 0 -> (odd (0 -Z -uZ a) <-> odd (a -Z 0)) |
| 2 |
1 |
conv zlt |
0 -Z -uZ a = a -Z 0 -> (0 <Z -uZ a <-> a <Z 0) |
| 3 |
|
eqtr4 |
0 -Z -uZ a = -uZ -uZ a -> a -Z 0 = -uZ -uZ a -> 0 -Z -uZ a = a -Z 0 |
| 4 |
|
zsub01 |
0 -Z -uZ a = -uZ -uZ a |
| 5 |
3, 4 |
ax_mp |
a -Z 0 = -uZ -uZ a -> 0 -Z -uZ a = a -Z 0 |
| 6 |
|
eqtr4 |
a -Z 0 = a -> -uZ -uZ a = a -> a -Z 0 = -uZ -uZ a |
| 7 |
|
zsub02 |
a -Z 0 = a |
| 8 |
6, 7 |
ax_mp |
-uZ -uZ a = a -> a -Z 0 = -uZ -uZ a |
| 9 |
|
znegneg |
-uZ -uZ a = a |
| 10 |
8, 9 |
ax_mp |
a -Z 0 = -uZ -uZ a |
| 11 |
5, 10 |
ax_mp |
0 -Z -uZ a = a -Z 0 |
| 12 |
2, 11 |
ax_mp |
0 <Z -uZ a <-> a <Z 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)