theorem app01 (a: nat): $ 0 @ a = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
0 @ a = the 0 -> the 0 = 0 -> 0 @ a = 0 |
| 2 |
|
theeq |
{a1 | a, a1 e. 0} == 0 -> the {a1 | a, a1 e. 0} = the 0 |
| 3 |
2 |
conv app |
{a1 | a, a1 e. 0} == 0 -> 0 @ a = the 0 |
| 4 |
|
rapp01 |
0 @' a == 0 |
| 5 |
4 |
conv rapp |
{a1 | a, a1 e. 0} == 0 |
| 6 |
3, 5 |
ax_mp |
0 @ a = the 0 |
| 7 |
1, 6 |
ax_mp |
the 0 = 0 -> 0 @ a = 0 |
| 8 |
|
the01 |
the 0 = 0 |
| 9 |
7, 8 |
ax_mp |
0 @ a = 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)