theorem lepow2a (a b c: nat): $ a != 0 -> b <= c -> a ^ b <= a ^ c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
powne0 |
a != 0 -> a ^ c != 0 |
| 2 |
1 |
anwl |
a != 0 /\ b <= c -> a ^ c != 0 |
| 3 |
|
powdvd |
b <= c -> a ^ b || a ^ c |
| 4 |
3 |
anwr |
a != 0 /\ b <= c -> a ^ b || a ^ c |
| 5 |
2, 4 |
dvdle |
a != 0 /\ b <= c -> a ^ b <= a ^ c |
| 6 |
5 |
exp |
a != 0 -> b <= c -> a ^ b <= a ^ c |
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)