theorem mul12 (a: nat): $ a * 1 = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a * 1 = a * 0 + a -> a * 0 + a = a -> a * 1 = a |
| 2 |
|
mulS |
a * suc 0 = a * 0 + a |
| 3 |
2 |
conv d1 |
a * 1 = a * 0 + a |
| 4 |
1, 3 |
ax_mp |
a * 0 + a = a -> a * 1 = a |
| 5 |
|
eqtr |
a * 0 + a = 0 + a -> 0 + a = a -> a * 0 + a = a |
| 6 |
|
addeq1 |
a * 0 = 0 -> a * 0 + a = 0 + a |
| 7 |
|
mul0 |
a * 0 = 0 |
| 8 |
6, 7 |
ax_mp |
a * 0 + a = 0 + a |
| 9 |
5, 8 |
ax_mp |
0 + a = a -> a * 0 + a = a |
| 10 |
|
add01 |
0 + a = a |
| 11 |
9, 10 |
ax_mp |
a * 0 + a = a |
| 12 |
4, 11 |
ax_mp |
a * 1 = 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,
mul0,
mulS)