theorem lezabszn (m n: nat): $ n <= m -> zabs (m -ZN n) = m - n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zabszn |
zabs (m -ZN n) = m - n + (n - m) |
| 2 |
|
add0 |
m - n + 0 = m - n |
| 3 |
|
lesubeq0 |
n <= m <-> n - m = 0 |
| 4 |
|
addeq2 |
n - m = 0 -> m - n + (n - m) = m - n + 0 |
| 5 |
3, 4 |
sylbi |
n <= m -> m - n + (n - m) = m - n + 0 |
| 6 |
2, 5 |
syl6eq |
n <= m -> m - n + (n - m) = m - n |
| 7 |
1, 6 |
syl5eq |
n <= m -> zabs (m -ZN n) = m - n |
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)