theorem eqmzdvdsub (a b n: nat): $ mod(n): a = b <-> b0 n |Z b -ZN a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(mod(n): a = b <-> n || zabs (b -ZN a)) -> (b0 n |Z b -ZN a <-> n || zabs (b -ZN a)) -> (mod(n): a = b <-> b0 n |Z b -ZN a) |
| 2 |
|
eqmdvdsub2 |
mod(n): a = b <-> n || zabs (b -ZN a) |
| 3 |
1, 2 |
ax_mp |
(b0 n |Z b -ZN a <-> n || zabs (b -ZN a)) -> (mod(n): a = b <-> b0 n |Z b -ZN a) |
| 4 |
|
dvdeq1 |
zabs (b0 n) = n -> (zabs (b0 n) || zabs (b -ZN a) <-> n || zabs (b -ZN a)) |
| 5 |
4 |
conv zdvd |
zabs (b0 n) = n -> (b0 n |Z b -ZN a <-> n || zabs (b -ZN a)) |
| 6 |
|
zabsb0 |
zabs (b0 n) = n |
| 7 |
5, 6 |
ax_mp |
b0 n |Z b -ZN a <-> n || zabs (b -ZN a) |
| 8 |
3, 7 |
ax_mp |
mod(n): a = b <-> b0 n |Z b -ZN 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)