theorem uptoadd1 (n: nat): $ upto n + 1 = 2 ^ n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
npcan |
1 <= 2 ^ n -> 2 ^ n - 1 + 1 = 2 ^ n |
| 2 |
1 |
conv upto |
1 <= 2 ^ n -> upto n + 1 = 2 ^ n |
| 3 |
|
powpos |
0 < 2 -> 0 < 2 ^ n |
| 4 |
3 |
conv d1, lt |
0 < 2 -> 1 <= 2 ^ n |
| 5 |
|
d0lt2 |
0 < 2 |
| 6 |
4, 5 |
ax_mp |
1 <= 2 ^ n |
| 7 |
2, 6 |
ax_mp |
upto n + 1 = 2 ^ 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,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)