theorem dvdadd2 (a b n: nat): $ n || a -> (n || b <-> n || b + a) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
a + b = b + a -> n = n |
| 2 |
|
id |
a + b = b + a -> a + b = b + a |
| 3 |
1, 2 |
dvdeqd |
a + b = b + a -> (n || a + b <-> n || b + a) |
| 4 |
|
addcom |
a + b = b + a |
| 5 |
3, 4 |
ax_mp |
n || a + b <-> n || b + a |
| 6 |
|
dvdadd1 |
n || a -> (n || b <-> n || a + b) |
| 7 |
5, 6 |
syl6bb |
n || a -> (n || b <-> n || b + 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)