theorem zabszn (m n: nat): $ zabs (m -ZN n) = m - n + (n - m) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
addeq |
zfst (m -ZN n) = m - n -> zsnd (m -ZN n) = n - m -> zfst (m -ZN n) + zsnd (m -ZN n) = m - n + (n - m) |
| 2 |
1 |
conv zabs |
zfst (m -ZN n) = m - n -> zsnd (m -ZN n) = n - m -> zabs (m -ZN n) = m - n + (n - m) |
| 3 |
|
zfstznsub |
zfst (m -ZN n) = m - n |
| 4 |
2, 3 |
ax_mp |
zsnd (m -ZN n) = n - m -> zabs (m -ZN n) = m - n + (n - m) |
| 5 |
|
zsndznsub |
zsnd (m -ZN n) = n - m |
| 6 |
4, 5 |
ax_mp |
zabs (m -ZN n) = m - 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)