theorem shrmodadd2 (a b c: nat): $ shr (a % 2 ^ (b + c)) c = shr a c % 2 ^ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
shr (a % 2 ^ (b + c)) c = shr (a % 2 ^ (c + b)) c -> shr (a % 2 ^ (c + b)) c = shr a c % 2 ^ b -> shr (a % 2 ^ (b + c)) c = shr a c % 2 ^ b |
| 2 |
|
shreq1 |
a % 2 ^ (b + c) = a % 2 ^ (c + b) -> shr (a % 2 ^ (b + c)) c = shr (a % 2 ^ (c + b)) c |
| 3 |
|
modeq2 |
2 ^ (b + c) = 2 ^ (c + b) -> a % 2 ^ (b + c) = a % 2 ^ (c + b) |
| 4 |
|
poweq2 |
b + c = c + b -> 2 ^ (b + c) = 2 ^ (c + b) |
| 5 |
|
addcom |
b + c = c + b |
| 6 |
4, 5 |
ax_mp |
2 ^ (b + c) = 2 ^ (c + b) |
| 7 |
3, 6 |
ax_mp |
a % 2 ^ (b + c) = a % 2 ^ (c + b) |
| 8 |
2, 7 |
ax_mp |
shr (a % 2 ^ (b + c)) c = shr (a % 2 ^ (c + b)) c |
| 9 |
1, 8 |
ax_mp |
shr (a % 2 ^ (c + b)) c = shr a c % 2 ^ b -> shr (a % 2 ^ (b + c)) c = shr a c % 2 ^ b |
| 10 |
|
shrmodadd1 |
shr (a % 2 ^ (c + b)) c = shr a c % 2 ^ b |
| 11 |
9, 10 |
ax_mp |
shr (a % 2 ^ (b + c)) c = shr a c % 2 ^ 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)