theorem elshr (a b c: nat): $ a e. shr b c <-> a + c e. b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a e. shr b c <-> odd (shr (shr b c) a)) -> (odd (shr (shr b c) a) <-> a + c e. b) -> (a e. shr b c <-> a + c e. b) |
| 2 |
|
elnel |
a e. shr b c <-> odd (shr (shr b c) a) |
| 3 |
1, 2 |
ax_mp |
(odd (shr (shr b c) a) <-> a + c e. b) -> (a e. shr b c <-> a + c e. b) |
| 4 |
|
bitr4 |
(odd (shr (shr b c) a) <-> odd (shr b (a + c))) -> (a + c e. b <-> odd (shr b (a + c))) -> (odd (shr (shr b c) a) <-> a + c e. b) |
| 5 |
|
oddeq |
shr (shr b c) a = shr b (a + c) -> (odd (shr (shr b c) a) <-> odd (shr b (a + c))) |
| 6 |
|
eqtr |
shr (shr b c) a = shr b (c + a) -> shr b (c + a) = shr b (a + c) -> shr (shr b c) a = shr b (a + c) |
| 7 |
|
shrshr |
shr (shr b c) a = shr b (c + a) |
| 8 |
6, 7 |
ax_mp |
shr b (c + a) = shr b (a + c) -> shr (shr b c) a = shr b (a + c) |
| 9 |
|
shreq2 |
c + a = a + c -> shr b (c + a) = shr b (a + c) |
| 10 |
|
addcom |
c + a = a + c |
| 11 |
9, 10 |
ax_mp |
shr b (c + a) = shr b (a + c) |
| 12 |
8, 11 |
ax_mp |
shr (shr b c) a = shr b (a + c) |
| 13 |
5, 12 |
ax_mp |
odd (shr (shr b c) a) <-> odd (shr b (a + c)) |
| 14 |
4, 13 |
ax_mp |
(a + c e. b <-> odd (shr b (a + c))) -> (odd (shr (shr b c) a) <-> a + c e. b) |
| 15 |
|
elnel |
a + c e. b <-> odd (shr b (a + c)) |
| 16 |
14, 15 |
ax_mp |
odd (shr (shr b c) a) <-> a + c e. b |
| 17 |
3, 16 |
ax_mp |
a e. shr b c <-> a + c e. 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)