theorem eqmaddn (a n: nat): $ mod(n): a + n = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqmeq3 |
a + 0 = a -> (mod(n): a + n = a + 0 <-> mod(n): a + n = a) |
| 2 |
|
add0 |
a + 0 = a |
| 3 |
1, 2 |
ax_mp |
mod(n): a + n = a + 0 <-> mod(n): a + n = a |
| 4 |
|
eqmadd2 |
mod(n): a + n = a + 0 <-> mod(n): n = 0 |
| 5 |
|
eqmid0 |
mod(n): n = 0 |
| 6 |
4, 5 |
mpbir |
mod(n): a + n = a + 0 |
| 7 |
3, 6 |
mpbi |
mod(n): a + n = 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_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)