theorem powdvd1 (a b: nat): $ 0 < b -> a || a ^ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
dvdeq1 |
a ^ 1 = a -> (a ^ 1 || a ^ b <-> a || a ^ b) |
| 2 |
|
pow12 |
a ^ 1 = a |
| 3 |
1, 2 |
ax_mp |
a ^ 1 || a ^ b <-> a || a ^ b |
| 4 |
|
powdvd |
suc 0 <= b -> a ^ suc 0 || a ^ b |
| 5 |
4 |
conv d1, lt |
0 < b -> a ^ 1 || a ^ b |
| 6 |
3, 5 |
sylib |
0 < b -> a || a ^ b |
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)