theorem znegb1 (n: nat): $ -uZ b1 n = b0 (suc n) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
-uZ b1 n = b0 (zsnd (b1 n) - zfst (b1 n)) -> b0 (zsnd (b1 n) - zfst (b1 n)) = b0 (suc n) -> -uZ b1 n = b0 (suc n) |
| 2 |
|
znsubpos |
zfst (b1 n) <= zsnd (b1 n) -> zsnd (b1 n) -ZN zfst (b1 n) = b0 (zsnd (b1 n) - zfst (b1 n)) |
| 3 |
2 |
conv zneg |
zfst (b1 n) <= zsnd (b1 n) -> -uZ b1 n = b0 (zsnd (b1 n) - zfst (b1 n)) |
| 4 |
|
leeq1 |
zfst (b1 n) = 0 -> (zfst (b1 n) <= zsnd (b1 n) <-> 0 <= zsnd (b1 n)) |
| 5 |
|
zfstb1 |
zfst (b1 n) = 0 |
| 6 |
4, 5 |
ax_mp |
zfst (b1 n) <= zsnd (b1 n) <-> 0 <= zsnd (b1 n) |
| 7 |
|
le01 |
0 <= zsnd (b1 n) |
| 8 |
6, 7 |
mpbir |
zfst (b1 n) <= zsnd (b1 n) |
| 9 |
3, 8 |
ax_mp |
-uZ b1 n = b0 (zsnd (b1 n) - zfst (b1 n)) |
| 10 |
1, 9 |
ax_mp |
b0 (zsnd (b1 n) - zfst (b1 n)) = b0 (suc n) -> -uZ b1 n = b0 (suc n) |
| 11 |
|
b0eq |
zsnd (b1 n) - zfst (b1 n) = suc n -> b0 (zsnd (b1 n) - zfst (b1 n)) = b0 (suc n) |
| 12 |
|
eqtr |
zsnd (b1 n) - zfst (b1 n) = suc n - 0 -> suc n - 0 = suc n -> zsnd (b1 n) - zfst (b1 n) = suc n |
| 13 |
|
subeq |
zsnd (b1 n) = suc n -> zfst (b1 n) = 0 -> zsnd (b1 n) - zfst (b1 n) = suc n - 0 |
| 14 |
|
zsndb1 |
zsnd (b1 n) = suc n |
| 15 |
13, 14 |
ax_mp |
zfst (b1 n) = 0 -> zsnd (b1 n) - zfst (b1 n) = suc n - 0 |
| 16 |
15, 5 |
ax_mp |
zsnd (b1 n) - zfst (b1 n) = suc n - 0 |
| 17 |
12, 16 |
ax_mp |
suc n - 0 = suc n -> zsnd (b1 n) - zfst (b1 n) = suc n |
| 18 |
|
sub02 |
suc n - 0 = suc n |
| 19 |
17, 18 |
ax_mp |
zsnd (b1 n) - zfst (b1 n) = suc n |
| 20 |
11, 19 |
ax_mp |
b0 (zsnd (b1 n) - zfst (b1 n)) = b0 (suc n) |
| 21 |
10, 20 |
ax_mp |
-uZ b1 n = b0 (suc 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)