theorem addb11 (a b: nat): $ b1 a + b1 b = b0 (suc (a + b)) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
b1 a + b1 b = suc (b1 a + b0 b) -> suc (b1 a + b0 b) = b0 (suc (a + b)) -> b1 a + b1 b = b0 (suc (a + b)) |
| 2 |
|
addS |
b1 a + suc (b0 b) = suc (b1 a + b0 b) |
| 3 |
2 |
conv b1 |
b1 a + b1 b = suc (b1 a + b0 b) |
| 4 |
1, 3 |
ax_mp |
suc (b1 a + b0 b) = b0 (suc (a + b)) -> b1 a + b1 b = b0 (suc (a + b)) |
| 5 |
|
eqtr |
suc (b1 a + b0 b) = suc (b1 (a + b)) -> suc (b1 (a + b)) = b0 (suc (a + b)) -> suc (b1 a + b0 b) = b0 (suc (a + b)) |
| 6 |
|
suceq |
b1 a + b0 b = b1 (a + b) -> suc (b1 a + b0 b) = suc (b1 (a + b)) |
| 7 |
|
addb10 |
b1 a + b0 b = b1 (a + b) |
| 8 |
6, 7 |
ax_mp |
suc (b1 a + b0 b) = suc (b1 (a + b)) |
| 9 |
5, 8 |
ax_mp |
suc (b1 (a + b)) = b0 (suc (a + b)) -> suc (b1 a + b0 b) = b0 (suc (a + b)) |
| 10 |
|
sucb1 |
suc (b1 (a + b)) = b0 (suc (a + b)) |
| 11 |
9, 10 |
ax_mp |
suc (b1 a + b0 b) = b0 (suc (a + b)) |
| 12 |
4, 11 |
ax_mp |
b1 a + b1 b = b0 (suc (a + 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_peano
(peano2,
peano5,
addeq,
add0,
addS)