theorem zsndb0 (n: nat): $ zsnd (b0 n) = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
zsnd (b0 n) = (\ a2, 0) @ n -> (\ a2, 0) @ n = 0 -> zsnd (b0 n) = 0 |
| 2 |
|
casel |
case (\ a2, 0) (\ a1, suc a1) @ b0 n = (\ a2, 0) @ n |
| 3 |
2 |
conv zsnd |
zsnd (b0 n) = (\ a2, 0) @ n |
| 4 |
1, 3 |
ax_mp |
(\ a2, 0) @ n = 0 -> zsnd (b0 n) = 0 |
| 5 |
|
eqidd |
a2 = n -> 0 = 0 |
| 6 |
5 |
applame |
(\ a2, 0) @ n = 0 |
| 7 |
4, 6 |
ax_mp |
zsnd (b0 n) = 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)