theorem zabscom (m n: nat): $ zabs (m -ZN n) = zabs (n -ZN m) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
zabs (m -ZN n) = m - n + (n - m) -> m - n + (n - m) = zabs (n -ZN m) -> zabs (m -ZN n) = zabs (n -ZN m) |
| 2 |
|
zabszn |
zabs (m -ZN n) = m - n + (n - m) |
| 3 |
1, 2 |
ax_mp |
m - n + (n - m) = zabs (n -ZN m) -> zabs (m -ZN n) = zabs (n -ZN m) |
| 4 |
|
eqtr4 |
m - n + (n - m) = n - m + (m - n) -> zabs (n -ZN m) = n - m + (m - n) -> m - n + (n - m) = zabs (n -ZN m) |
| 5 |
|
addcom |
m - n + (n - m) = n - m + (m - n) |
| 6 |
4, 5 |
ax_mp |
zabs (n -ZN m) = n - m + (m - n) -> m - n + (n - m) = zabs (n -ZN m) |
| 7 |
|
zabszn |
zabs (n -ZN m) = n - m + (m - n) |
| 8 |
6, 7 |
ax_mp |
m - n + (n - m) = zabs (n -ZN m) |
| 9 |
3, 8 |
ax_mp |
zabs (m -ZN n) = zabs (n -ZN 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)