theorem shrshl2 (a b c: nat): $ b <= c -> shr (shl a b) c = shr a (c - b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
shreq1 |
shr (shl a b) b = a -> shr (shr (shl a b) b) (c - b) = shr a (c - b) |
| 2 |
|
shrshlid |
shr (shl a b) b = a |
| 3 |
1, 2 |
ax_mp |
shr (shr (shl a b) b) (c - b) = shr a (c - b) |
| 4 |
|
shrshr |
shr (shr (shl a b) b) (c - b) = shr (shl a b) (b + (c - b)) |
| 5 |
|
pncan3 |
b <= c -> b + (c - b) = c |
| 6 |
5 |
eqcomd |
b <= c -> c = b + (c - b) |
| 7 |
6 |
shreq2d |
b <= c -> shr (shl a b) c = shr (shl a b) (b + (c - b)) |
| 8 |
4, 7 |
syl6eqr |
b <= c -> shr (shl a b) c = shr (shr (shl a b) b) (c - b) |
| 9 |
3, 8 |
syl6eq |
b <= c -> shr (shl a b) c = shr a (c - 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)