theorem shl2dvd (a b: nat): $ 0 < b -> 2 || shl a b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
powdvd1 |
0 < b -> 2 || 2 ^ b |
| 2 |
|
shlpow2dvd |
2 ^ b || shl a b |
| 3 |
|
dvdtr |
2 || 2 ^ b -> 2 ^ b || shl a b -> 2 || shl a b |
| 4 |
2, 3 |
mpi |
2 || 2 ^ b -> 2 || shl a b |
| 5 |
1, 4 |
rsyl |
0 < b -> 2 || shl 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)