theorem zabsneg (n: nat): $ zabs (-uZ n) = zabs n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
zabs (-uZ n) = zsnd (-uZ n) + zfst (-uZ n) -> zsnd (-uZ n) + zfst (-uZ n) = zabs n -> zabs (-uZ n) = zabs n |
| 2 |
|
addcom |
zfst (-uZ n) + zsnd (-uZ n) = zsnd (-uZ n) + zfst (-uZ n) |
| 3 |
2 |
conv zabs |
zabs (-uZ n) = zsnd (-uZ n) + zfst (-uZ n) |
| 4 |
1, 3 |
ax_mp |
zsnd (-uZ n) + zfst (-uZ n) = zabs n -> zabs (-uZ n) = zabs n |
| 5 |
|
addeq |
zsnd (-uZ n) = zfst n -> zfst (-uZ n) = zsnd n -> zsnd (-uZ n) + zfst (-uZ n) = zfst n + zsnd n |
| 6 |
5 |
conv zabs |
zsnd (-uZ n) = zfst n -> zfst (-uZ n) = zsnd n -> zsnd (-uZ n) + zfst (-uZ n) = zabs n |
| 7 |
|
zsndneg |
zsnd (-uZ n) = zfst n |
| 8 |
6, 7 |
ax_mp |
zfst (-uZ n) = zsnd n -> zsnd (-uZ n) + zfst (-uZ n) = zabs n |
| 9 |
|
zfstneg |
zfst (-uZ n) = zsnd n |
| 10 |
8, 9 |
ax_mp |
zsnd (-uZ n) + zfst (-uZ n) = zabs n |
| 11 |
4, 10 |
ax_mp |
zabs (-uZ n) = zabs 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)