theorem ntheq0 (l n: nat): $ nth n l = 0 <-> len l <= n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bicom |
(len l <= n <-> nth n l = 0) -> (nth n l = 0 <-> len l <= n) |
| 2 |
|
bitr |
(len l <= n <-> ~n < len l) -> (~n < len l <-> nth n l = 0) -> (len l <= n <-> nth n l = 0) |
| 3 |
|
lenlt |
len l <= n <-> ~n < len l |
| 4 |
2, 3 |
ax_mp |
(~n < len l <-> nth n l = 0) -> (len l <= n <-> nth n l = 0) |
| 5 |
|
con1b |
(~nth n l = 0 <-> n < len l) -> (~n < len l <-> nth n l = 0) |
| 6 |
|
nthne0 |
nth n l != 0 <-> n < len l |
| 7 |
6 |
conv ne |
~nth n l = 0 <-> n < len l |
| 8 |
5, 7 |
ax_mp |
~n < len l <-> nth n l = 0 |
| 9 |
4, 8 |
ax_mp |
len l <= n <-> nth n l = 0 |
| 10 |
1, 9 |
ax_mp |
nth n l = 0 <-> len l <= n |
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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)