theorem zeqmsub2 (a b c n: nat):
$ modZ(n): a -Z b = a -Z c <-> modZ(n): b = c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(modZ(n): a -Z b = a -Z c <-> modZ(n): -uZ b = -uZ c) -> (modZ(n): -uZ b = -uZ c <-> modZ(n): b = c) -> (modZ(n): a -Z b = a -Z c <-> modZ(n): b = c) |
| 2 |
|
zeqmadd2 |
modZ(n): a +Z -uZ b = a +Z -uZ c <-> modZ(n): -uZ b = -uZ c |
| 3 |
2 |
conv zsub |
modZ(n): a -Z b = a -Z c <-> modZ(n): -uZ b = -uZ c |
| 4 |
1, 3 |
ax_mp |
(modZ(n): -uZ b = -uZ c <-> modZ(n): b = c) -> (modZ(n): a -Z b = a -Z c <-> modZ(n): b = c) |
| 5 |
|
zeqmneg |
modZ(n): -uZ b = -uZ c <-> modZ(n): b = c |
| 6 |
4, 5 |
ax_mp |
modZ(n): a -Z b = a -Z c <-> modZ(n): b = c |
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)