theorem listfnSval (a l n: nat):
$ n < suc (len l) -> listfn (a : l) @ n = if (n = 0) a (listfn l @ (n - 1)) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
appneq1 |
listfn (a : l) = \. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1)) ->
listfn (a : l) @ n = (\. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1))) @ n |
| 2 |
|
listfnS |
listfn (a : l) = \. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1)) |
| 3 |
1, 2 |
ax_mp |
listfn (a : l) @ n = (\. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1))) @ n |
| 4 |
|
elupto |
n e. upto (suc (len l)) <-> n < suc (len l) |
| 5 |
|
eqeq1 |
i = n -> (i = 0 <-> n = 0) |
| 6 |
|
eqidd |
i = n -> a = a |
| 7 |
|
subeq1 |
i = n -> i - 1 = n - 1 |
| 8 |
7 |
appeq2d |
i = n -> listfn l @ (i - 1) = listfn l @ (n - 1) |
| 9 |
5, 6, 8 |
ifeqd |
i = n -> if (i = 0) a (listfn l @ (i - 1)) = if (n = 0) a (listfn l @ (n - 1)) |
| 10 |
9 |
apprlame |
n e. upto (suc (len l)) -> (\. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1))) @ n = if (n = 0) a (listfn l @ (n - 1)) |
| 11 |
4, 10 |
sylbir |
n < suc (len l) -> (\. i e. upto (suc (len l)), if (i = 0) a (listfn l @ (i - 1))) @ n = if (n = 0) a (listfn l @ (n - 1)) |
| 12 |
3, 11 |
syl5eq |
n < suc (len l) -> listfn (a : l) @ n = if (n = 0) a (listfn l @ (n - 1)) |
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)