theorem mul01 (a: nat): $ 0 * a = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
muleq2 |
x = a -> 0 * x = 0 * a |
| 2 |
1 |
eqeq1d |
x = a -> (0 * x = 0 <-> 0 * a = 0) |
| 3 |
|
muleq2 |
x = 0 -> 0 * x = 0 * 0 |
| 4 |
3 |
eqeq1d |
x = 0 -> (0 * x = 0 <-> 0 * 0 = 0) |
| 5 |
|
muleq2 |
x = y -> 0 * x = 0 * y |
| 6 |
5 |
eqeq1d |
x = y -> (0 * x = 0 <-> 0 * y = 0) |
| 7 |
|
muleq2 |
x = suc y -> 0 * x = 0 * suc y |
| 8 |
7 |
eqeq1d |
x = suc y -> (0 * x = 0 <-> 0 * suc y = 0) |
| 9 |
|
mul0 |
0 * 0 = 0 |
| 10 |
|
eqtr |
0 * suc y = 0 * y + 0 -> 0 * y + 0 = 0 * y -> 0 * suc y = 0 * y |
| 11 |
|
mulS |
0 * suc y = 0 * y + 0 |
| 12 |
10, 11 |
ax_mp |
0 * y + 0 = 0 * y -> 0 * suc y = 0 * y |
| 13 |
|
add0 |
0 * y + 0 = 0 * y |
| 14 |
12, 13 |
ax_mp |
0 * suc y = 0 * y |
| 15 |
|
id |
0 * y = 0 -> 0 * y = 0 |
| 16 |
14, 15 |
syl5eq |
0 * y = 0 -> 0 * suc y = 0 |
| 17 |
2, 4, 6, 8, 9, 16 |
ind |
0 * a = 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_peano
(peano2,
peano5,
muleq,
add0,
mul0,
mulS)