theorem znsubeqd (_G: wff) (_m1 _m2 _n1 _n2: nat):
$ _G -> _m1 = _m2 $ >
$ _G -> _n1 = _n2 $ >
$ _G -> _m1 -ZN _n1 = _m2 -ZN _n2 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp _mh |
_G -> _m1 = _m2 |
| 2 |
|
hyp _nh |
_G -> _n1 = _n2 |
| 3 |
1, 2 |
lteqd |
_G -> (_m1 < _n1 <-> _m2 < _n2) |
| 4 |
1 |
suceqd |
_G -> suc _m1 = suc _m2 |
| 5 |
2, 4 |
subeqd |
_G -> _n1 - suc _m1 = _n2 - suc _m2 |
| 6 |
5 |
b1eqd |
_G -> b1 (_n1 - suc _m1) = b1 (_n2 - suc _m2) |
| 7 |
1, 2 |
subeqd |
_G -> _m1 - _n1 = _m2 - _n2 |
| 8 |
7 |
b0eqd |
_G -> b0 (_m1 - _n1) = b0 (_m2 - _n2) |
| 9 |
3, 6, 8 |
ifeqd |
_G -> if (_m1 < _n1) (b1 (_n1 - suc _m1)) (b0 (_m1 - _n1)) = if (_m2 < _n2) (b1 (_n2 - suc _m2)) (b0 (_m2 - _n2)) |
| 10 |
9 |
conv znsub |
_G -> _m1 -ZN _n1 = _m2 -ZN _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)