theorem shlshl (a b c: nat): $ shl (shl a b) c = shl a (b + c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr4 |
shl (shl a b) c = a * (2 ^ b * 2 ^ c) -> shl a (b + c) = a * (2 ^ b * 2 ^ c) -> shl (shl a b) c = shl a (b + c) |
| 2 |
|
mulass |
a * 2 ^ b * 2 ^ c = a * (2 ^ b * 2 ^ c) |
| 3 |
2 |
conv shl |
shl (shl a b) c = a * (2 ^ b * 2 ^ c) |
| 4 |
1, 3 |
ax_mp |
shl a (b + c) = a * (2 ^ b * 2 ^ c) -> shl (shl a b) c = shl a (b + c) |
| 5 |
|
muleq2 |
2 ^ (b + c) = 2 ^ b * 2 ^ c -> a * 2 ^ (b + c) = a * (2 ^ b * 2 ^ c) |
| 6 |
5 |
conv shl |
2 ^ (b + c) = 2 ^ b * 2 ^ c -> shl a (b + c) = a * (2 ^ b * 2 ^ c) |
| 7 |
|
powadd |
2 ^ (b + c) = 2 ^ b * 2 ^ c |
| 8 |
6, 7 |
ax_mp |
shl a (b + c) = a * (2 ^ b * 2 ^ c) |
| 9 |
4, 8 |
ax_mp |
shl (shl a b) c = shl a (b + 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)