theorem listfnS0 (a l: nat): $ listfn (a : l) @ 0 = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
listfn (a : l) @ 0 = if (0 = 0) a (listfn l @ (0 - 1)) -> if (0 = 0) a (listfn l @ (0 - 1)) = a -> listfn (a : l) @ 0 = a |
| 2 |
|
listfnSval |
0 < suc (len l) -> listfn (a : l) @ 0 = if (0 = 0) a (listfn l @ (0 - 1)) |
| 3 |
|
lt01S |
0 < suc (len l) |
| 4 |
2, 3 |
ax_mp |
listfn (a : l) @ 0 = if (0 = 0) a (listfn l @ (0 - 1)) |
| 5 |
1, 4 |
ax_mp |
if (0 = 0) a (listfn l @ (0 - 1)) = a -> listfn (a : l) @ 0 = a |
| 6 |
|
ifpos |
0 = 0 -> if (0 = 0) a (listfn l @ (0 - 1)) = a |
| 7 |
|
eqid |
0 = 0 |
| 8 |
6, 7 |
ax_mp |
if (0 = 0) a (listfn l @ (0 - 1)) = a |
| 9 |
5, 8 |
ax_mp |
listfn (a : l) @ 0 = a |
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)