theorem zmuleqd (_G: wff) (_m1 _m2 _n1 _n2: nat):
$ _G -> _m1 = _m2 $ >
$ _G -> _n1 = _n2 $ >
$ _G -> _m1 *Z _n1 = _m2 *Z _n2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _mh |
_G -> _m1 = _m2 |
| 2 |
1 |
zfsteqd |
_G -> zfst _m1 = zfst _m2 |
| 3 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 4 |
3 |
zfsteqd |
_G -> zfst _n1 = zfst _n2 |
| 5 |
2, 4 |
muleqd |
_G -> zfst _m1 * zfst _n1 = zfst _m2 * zfst _n2 |
| 6 |
1 |
zsndeqd |
_G -> zsnd _m1 = zsnd _m2 |
| 7 |
3 |
zsndeqd |
_G -> zsnd _n1 = zsnd _n2 |
| 8 |
6, 7 |
muleqd |
_G -> zsnd _m1 * zsnd _n1 = zsnd _m2 * zsnd _n2 |
| 9 |
5, 8 |
addeqd |
_G -> zfst _m1 * zfst _n1 + zsnd _m1 * zsnd _n1 = zfst _m2 * zfst _n2 + zsnd _m2 * zsnd _n2 |
| 10 |
2, 7 |
muleqd |
_G -> zfst _m1 * zsnd _n1 = zfst _m2 * zsnd _n2 |
| 11 |
6, 4 |
muleqd |
_G -> zsnd _m1 * zfst _n1 = zsnd _m2 * zfst _n2 |
| 12 |
10, 11 |
addeqd |
_G -> zfst _m1 * zsnd _n1 + zsnd _m1 * zfst _n1 = zfst _m2 * zsnd _n2 + zsnd _m2 * zfst _n2 |
| 13 |
9, 12 |
znsubeqd |
_G ->
zfst _m1 * zfst _n1 + zsnd _m1 * zsnd _n1 -ZN (zfst _m1 * zsnd _n1 + zsnd _m1 * zfst _n1) =
zfst _m2 * zfst _n2 + zsnd _m2 * zsnd _n2 -ZN (zfst _m2 * zsnd _n2 + zsnd _m2 * zfst _n2) |
| 14 |
13 |
conv zmul |
_G -> _m1 *Z _n1 = _m2 *Z _n2 |
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)