theorem listfnisf (l: nat): $ isfun (listfn l) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
isf0 |
isfun 0 |
| 2 |
|
listfn0 |
listfn 0 = 0 |
| 3 |
|
listfneq |
l = 0 -> listfn l = listfn 0 |
| 4 |
2, 3 |
syl6eq |
l = 0 -> listfn l = 0 |
| 5 |
4 |
nseqd |
l = 0 -> listfn l == 0 |
| 6 |
5 |
isfeqd |
l = 0 -> (isfun (listfn l) <-> isfun 0) |
| 7 |
1, 6 |
mpbiri |
l = 0 -> isfun (listfn l) |
| 8 |
|
rlamisf |
isfun (\. i e. upto (suc (size (Dom (listfn (snd (l - 1)))))), if (i = 0) (fst (l - 1)) (listfn (snd (l - 1)) @ (i - 1))) |
| 9 |
|
listfnS2 |
listfn (fst (l - 1) : snd (l - 1)) = \. i e. upto (suc (size (Dom (listfn (snd (l - 1)))))), if (i = 0) (fst (l - 1)) (listfn (snd (l - 1)) @ (i - 1)) |
| 10 |
|
consfstsnd |
l != 0 -> fst (l - 1) : snd (l - 1) = l |
| 11 |
10 |
conv ne |
~l = 0 -> fst (l - 1) : snd (l - 1) = l |
| 12 |
11 |
listfneqd |
~l = 0 -> listfn (fst (l - 1) : snd (l - 1)) = listfn l |
| 13 |
9, 12 |
syl5eqr |
~l = 0 -> \. i e. upto (suc (size (Dom (listfn (snd (l - 1)))))), if (i = 0) (fst (l - 1)) (listfn (snd (l - 1)) @ (i - 1)) = listfn l |
| 14 |
13 |
nseqd |
~l = 0 -> \. i e. upto (suc (size (Dom (listfn (snd (l - 1)))))), if (i = 0) (fst (l - 1)) (listfn (snd (l - 1)) @ (i - 1)) == listfn l |
| 15 |
14 |
isfeqd |
~l = 0 -> (isfun (\. i e. upto (suc (size (Dom (listfn (snd (l - 1)))))), if (i = 0) (fst (l - 1)) (listfn (snd (l - 1)) @ (i - 1))) <-> isfun (listfn l)) |
| 16 |
8, 15 |
mpbii |
~l = 0 -> isfun (listfn l) |
| 17 |
7, 16 |
cases |
isfun (listfn l) |
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)