theorem zabsb0 (n: nat): $ zabs (b0 n) = n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
zabs (b0 n) = n + 0 -> n + 0 = n -> zabs (b0 n) = n |
| 2 |
|
addeq |
zfst (b0 n) = n -> zsnd (b0 n) = 0 -> zfst (b0 n) + zsnd (b0 n) = n + 0 |
| 3 |
2 |
conv zabs |
zfst (b0 n) = n -> zsnd (b0 n) = 0 -> zabs (b0 n) = n + 0 |
| 4 |
|
zfstb0 |
zfst (b0 n) = n |
| 5 |
3, 4 |
ax_mp |
zsnd (b0 n) = 0 -> zabs (b0 n) = n + 0 |
| 6 |
|
zsndb0 |
zsnd (b0 n) = 0 |
| 7 |
5, 6 |
ax_mp |
zabs (b0 n) = n + 0 |
| 8 |
1, 7 |
ax_mp |
n + 0 = n -> zabs (b0 n) = n |
| 9 |
|
add0 |
n + 0 = n |
| 10 |
8, 9 |
ax_mp |
zabs (b0 n) = 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)