theorem zfsteq0 (a: nat): $ zfst a = 0 <-> a <=Z 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(zsnd (-uZ a) = 0 <-> zfst a = 0) -> (zsnd (-uZ a) = 0 <-> a <=Z 0) -> (zfst a = 0 <-> a <=Z 0) |
| 2 |
|
eqeq1 |
zsnd (-uZ a) = zfst a -> (zsnd (-uZ a) = 0 <-> zfst a = 0) |
| 3 |
|
zsndneg |
zsnd (-uZ a) = zfst a |
| 4 |
2, 3 |
ax_mp |
zsnd (-uZ a) = 0 <-> zfst a = 0 |
| 5 |
1, 4 |
ax_mp |
(zsnd (-uZ a) = 0 <-> a <=Z 0) -> (zfst a = 0 <-> a <=Z 0) |
| 6 |
|
bitr |
(zsnd (-uZ a) = 0 <-> 0 <=Z -uZ a) -> (0 <=Z -uZ a <-> a <=Z 0) -> (zsnd (-uZ a) = 0 <-> a <=Z 0) |
| 7 |
|
zsndeq0 |
zsnd (-uZ a) = 0 <-> 0 <=Z -uZ a |
| 8 |
6, 7 |
ax_mp |
(0 <=Z -uZ a <-> a <=Z 0) -> (zsnd (-uZ a) = 0 <-> a <=Z 0) |
| 9 |
|
zle0neg |
0 <=Z -uZ a <-> a <=Z 0 |
| 10 |
8, 9 |
ax_mp |
zsnd (-uZ a) = 0 <-> a <=Z 0 |
| 11 |
5, 10 |
ax_mp |
zfst a = 0 <-> 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)