pub theorem pr0: $ 0, 0 = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
0, 0 = 0 + 0 -> 0 + 0 = 0 -> 0, 0 = 0 |
| 2 |
|
addeq1 |
(0 + 0) * suc (0 + 0) // 2 = 0 -> (0 + 0) * suc (0 + 0) // 2 + 0 = 0 + 0 |
| 3 |
2 |
conv pr |
(0 + 0) * suc (0 + 0) // 2 = 0 -> 0, 0 = 0 + 0 |
| 4 |
|
eqtr |
(0 + 0) * suc (0 + 0) // 2 = 0 // 2 -> 0 // 2 = 0 -> (0 + 0) * suc (0 + 0) // 2 = 0 |
| 5 |
|
diveq1 |
(0 + 0) * suc (0 + 0) = 0 -> (0 + 0) * suc (0 + 0) // 2 = 0 // 2 |
| 6 |
|
eqtr |
(0 + 0) * suc (0 + 0) = 0 * suc (0 + 0) -> 0 * suc (0 + 0) = 0 -> (0 + 0) * suc (0 + 0) = 0 |
| 7 |
|
muleq1 |
0 + 0 = 0 -> (0 + 0) * suc (0 + 0) = 0 * suc (0 + 0) |
| 8 |
|
add0 |
0 + 0 = 0 |
| 9 |
7, 8 |
ax_mp |
(0 + 0) * suc (0 + 0) = 0 * suc (0 + 0) |
| 10 |
6, 9 |
ax_mp |
0 * suc (0 + 0) = 0 -> (0 + 0) * suc (0 + 0) = 0 |
| 11 |
|
mul01 |
0 * suc (0 + 0) = 0 |
| 12 |
10, 11 |
ax_mp |
(0 + 0) * suc (0 + 0) = 0 |
| 13 |
5, 12 |
ax_mp |
(0 + 0) * suc (0 + 0) // 2 = 0 // 2 |
| 14 |
4, 13 |
ax_mp |
0 // 2 = 0 -> (0 + 0) * suc (0 + 0) // 2 = 0 |
| 15 |
|
div01 |
0 // 2 = 0 |
| 16 |
14, 15 |
ax_mp |
(0 + 0) * suc (0 + 0) // 2 = 0 |
| 17 |
3, 16 |
ax_mp |
0, 0 = 0 + 0 |
| 18 |
1, 17 |
ax_mp |
0 + 0 = 0 -> 0, 0 = 0 |
| 19 |
18, 8 |
ax_mp |
0, 0 = 0 |
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)