theorem zeqmeqd (_G: wff) (_n1 _n2 _a1 _a2 _b1 _b2: nat):
$ _G -> _n1 = _n2 $ >
$ _G -> _a1 = _a2 $ >
$ _G -> _b1 = _b2 $ >
$ _G -> (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _a2 = _b2) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 2 |
1 |
b0eqd |
_G -> b0 _n1 = b0 _n2 |
| 3 |
|
hyp _ah |
_G -> _a1 = _a2 |
| 4 |
|
hyp _bh |
_G -> _b1 = _b2 |
| 5 |
3, 4 |
zsubeqd |
_G -> _a1 -Z _b1 = _a2 -Z _b2 |
| 6 |
2, 5 |
zdvdeqd |
_G -> (b0 _n1 |Z _a1 -Z _b1 <-> b0 _n2 |Z _a2 -Z _b2) |
| 7 |
6 |
conv zeqm |
_G -> (modZ(_n1): _a1 = _b1 <-> modZ(_n2): _a2 = _b2) |
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
(peano2,
addeq,
muleq)