theorem zeqmtr (a b c n: nat):
$ modZ(n): a = b -> modZ(n): b = c -> modZ(n): a = c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zdvdeq2 |
b -Z c +Z (a -Z b) = a -Z c -> (b0 n |Z b -Z c +Z (a -Z b) <-> b0 n |Z a -Z c) |
| 2 |
1 |
conv zeqm |
b -Z c +Z (a -Z b) = a -Z c -> (b0 n |Z b -Z c +Z (a -Z b) <-> modZ(n): a = c) |
| 3 |
|
znpncan2 |
b -Z c +Z (a -Z b) = a -Z c |
| 4 |
2, 3 |
ax_mp |
b0 n |Z b -Z c +Z (a -Z b) <-> modZ(n): a = c |
| 5 |
|
zdvdadd2 |
b0 n |Z a -Z b -> (b0 n |Z b -Z c <-> b0 n |Z b -Z c +Z (a -Z b)) |
| 6 |
5 |
conv zeqm |
modZ(n): a = b -> (modZ(n): b = c <-> b0 n |Z b -Z c +Z (a -Z b)) |
| 7 |
6 |
bi1d |
modZ(n): a = b -> modZ(n): b = c -> b0 n |Z b -Z c +Z (a -Z b) |
| 8 |
4, 7 |
syl6ib |
modZ(n): a = b -> modZ(n): b = c -> modZ(n): a = 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)