theorem zdvdmul1 (a b: nat): $ a |Z b *Z a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
dvdeq2 |
zabs (b *Z a) = zabs b * zabs a -> (zabs a || zabs (b *Z a) <-> zabs a || zabs b * zabs a) |
| 2 |
1 |
conv zdvd |
zabs (b *Z a) = zabs b * zabs a -> (a |Z b *Z a <-> zabs a || zabs b * zabs a) |
| 3 |
|
zabsmul |
zabs (b *Z a) = zabs b * zabs a |
| 4 |
2, 3 |
ax_mp |
a |Z b *Z a <-> zabs a || zabs b * zabs a |
| 5 |
|
dvdmul1 |
zabs a || zabs b * zabs a |
| 6 |
4, 5 |
mpbir |
a |Z b *Z 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)