theorem addb10 (a b: nat): $ b1 a + b0 b = b1 (a + b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
b1 a + b0 b = b0 b + b1 a -> b0 b + b1 a = b1 (a + b) -> b1 a + b0 b = b1 (a + b) |
| 2 |
|
addcom |
b1 a + b0 b = b0 b + b1 a |
| 3 |
1, 2 |
ax_mp |
b0 b + b1 a = b1 (a + b) -> b1 a + b0 b = b1 (a + b) |
| 4 |
|
eqtr |
b0 b + b1 a = b1 (b + a) -> b1 (b + a) = b1 (a + b) -> b0 b + b1 a = b1 (a + b) |
| 5 |
|
addb01 |
b0 b + b1 a = b1 (b + a) |
| 6 |
4, 5 |
ax_mp |
b1 (b + a) = b1 (a + b) -> b0 b + b1 a = b1 (a + b) |
| 7 |
|
b1eq |
b + a = a + b -> b1 (b + a) = b1 (a + b) |
| 8 |
|
addcom |
b + a = a + b |
| 9 |
7, 8 |
ax_mp |
b1 (b + a) = b1 (a + b) |
| 10 |
6, 9 |
ax_mp |
b0 b + b1 a = b1 (a + b) |
| 11 |
3, 10 |
ax_mp |
b1 a + b0 b = b1 (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)