theorem sucb1 (a: nat): $ suc (b1 a) = b0 (suc a) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr2 |
b0 (suc a) = suc (a + suc a) -> suc (a + suc a) = suc (b1 a) -> suc (b1 a) = b0 (suc a) |
| 2 |
|
addS1 |
suc a + suc a = suc (a + suc a) |
| 3 |
2 |
conv b0 |
b0 (suc a) = suc (a + suc a) |
| 4 |
1, 3 |
ax_mp |
suc (a + suc a) = suc (b1 a) -> suc (b1 a) = b0 (suc a) |
| 5 |
|
suceq |
a + suc a = b1 a -> suc (a + suc a) = suc (b1 a) |
| 6 |
|
addS |
a + suc a = suc (a + a) |
| 7 |
6 |
conv b0, b1 |
a + suc a = b1 a |
| 8 |
5, 7 |
ax_mp |
suc (a + suc a) = suc (b1 a) |
| 9 |
4, 8 |
ax_mp |
suc (b1 a) = b0 (suc 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_peano
(peano2,
peano5,
addeq,
add0,
addS)