theorem zeqmznsubd (G: wff) (a b c d n: nat):
$ G -> mod(n): a = b $ >
$ G -> mod(n): c = d $ >
$ G -> modZ(n): a -ZN c = b -ZN d $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zeqmtr |
modZ(n): a -ZN c = b -ZN c -> modZ(n): b -ZN c = b -ZN d -> modZ(n): a -ZN c = b -ZN d |
| 2 |
|
zeqmznsub1 |
modZ(n): a -ZN c = b -ZN c <-> mod(n): a = b |
| 3 |
|
hyp h1 |
G -> mod(n): a = b |
| 4 |
2, 3 |
sylibr |
G -> modZ(n): a -ZN c = b -ZN c |
| 5 |
|
zeqmznsub2 |
modZ(n): b -ZN c = b -ZN d <-> mod(n): c = d |
| 6 |
|
hyp h2 |
G -> mod(n): c = d |
| 7 |
5, 6 |
sylibr |
G -> modZ(n): b -ZN c = b -ZN d |
| 8 |
1, 4, 7 |
sylc |
G -> modZ(n): a -ZN c = b -ZN d |
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)