theorem zeqmcomb (a b n: nat): $ modZ(n): a = b <-> modZ(n): b = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(b0 n |Z -uZ (a -Z b) <-> modZ(n): a = b) -> (b0 n |Z -uZ (a -Z b) <-> modZ(n): b = a) -> (modZ(n): a = b <-> modZ(n): b = a) |
| 2 |
|
zdvdneg2 |
b0 n |Z -uZ (a -Z b) <-> b0 n |Z a -Z b |
| 3 |
2 |
conv zeqm |
b0 n |Z -uZ (a -Z b) <-> modZ(n): a = b |
| 4 |
1, 3 |
ax_mp |
(b0 n |Z -uZ (a -Z b) <-> modZ(n): b = a) -> (modZ(n): a = b <-> modZ(n): b = a) |
| 5 |
|
zdvdeq2 |
-uZ (a -Z b) = b -Z a -> (b0 n |Z -uZ (a -Z b) <-> b0 n |Z b -Z a) |
| 6 |
5 |
conv zeqm |
-uZ (a -Z b) = b -Z a -> (b0 n |Z -uZ (a -Z b) <-> modZ(n): b = a) |
| 7 |
|
znegsub |
-uZ (a -Z b) = b -Z a |
| 8 |
6, 7 |
ax_mp |
b0 n |Z -uZ (a -Z b) <-> modZ(n): b = a |
| 9 |
4, 8 |
ax_mp |
modZ(n): a = b <-> modZ(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)