theorem powdvd (a b c: nat): $ b <= c -> a ^ b || a ^ c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
dvdmul1 |
a ^ b || a ^ (c - b) * a ^ b |
| 2 |
|
powadd |
a ^ (c - b + b) = a ^ (c - b) * a ^ b |
| 3 |
|
npcan |
b <= c -> c - b + b = c |
| 4 |
3 |
poweq2d |
b <= c -> a ^ (c - b + b) = a ^ c |
| 5 |
2, 4 |
syl5eqr |
b <= c -> a ^ (c - b) * a ^ b = a ^ c |
| 6 |
5 |
dvdeq2d |
b <= c -> (a ^ b || a ^ (c - b) * a ^ b <-> a ^ b || a ^ c) |
| 7 |
1, 6 |
mpbii |
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)