theorem shl02 (a: nat): $ shl a 0 = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
shl a 0 = a * 1 -> a * 1 = a -> shl a 0 = a |
| 2 |
|
muleq2 |
2 ^ 0 = 1 -> a * 2 ^ 0 = a * 1 |
| 3 |
2 |
conv shl |
2 ^ 0 = 1 -> shl a 0 = a * 1 |
| 4 |
|
pow0 |
2 ^ 0 = 1 |
| 5 |
3, 4 |
ax_mp |
shl a 0 = a * 1 |
| 6 |
1, 5 |
ax_mp |
a * 1 = a -> shl a 0 = a |
| 7 |
|
mul12 |
a * 1 = a |
| 8 |
6, 7 |
ax_mp |
shl a 0 = a |
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)