theorem zsndznsub (m n: nat): $ zsnd (m -ZN n) = n - m $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
zfst (-uZ (m -ZN n)) = zsnd (m -ZN n) -> zfst (-uZ (m -ZN n)) = n - m -> zsnd (m -ZN n) = n - m |
| 2 |
|
zfstneg |
zfst (-uZ (m -ZN n)) = zsnd (m -ZN n) |
| 3 |
1, 2 |
ax_mp |
zfst (-uZ (m -ZN n)) = n - m -> zsnd (m -ZN n) = n - m |
| 4 |
|
eqtr |
zfst (-uZ (m -ZN n)) = zfst (n -ZN m) -> zfst (n -ZN m) = n - m -> zfst (-uZ (m -ZN n)) = n - m |
| 5 |
|
zfsteq |
-uZ (m -ZN n) = n -ZN m -> zfst (-uZ (m -ZN n)) = zfst (n -ZN m) |
| 6 |
|
znegzn |
-uZ (m -ZN n) = n -ZN m |
| 7 |
5, 6 |
ax_mp |
zfst (-uZ (m -ZN n)) = zfst (n -ZN m) |
| 8 |
4, 7 |
ax_mp |
zfst (n -ZN m) = n - m -> zfst (-uZ (m -ZN n)) = n - m |
| 9 |
|
zfstznsub |
zfst (n -ZN m) = n - m |
| 10 |
8, 9 |
ax_mp |
zfst (-uZ (m -ZN n)) = n - m |
| 11 |
3, 10 |
ax_mp |
zsnd (m -ZN n) = n - m |
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)