theorem zmulneg1 (a b: nat): $ -uZ a *Z b = -uZ (a *Z b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
(0 -Z a) *Z b = -uZ a *Z b -> (0 -Z a) *Z b = -uZ (a *Z b) -> -uZ a *Z b = -uZ (a *Z b) |
| 2 |
|
zmuleq1 |
0 -Z a = -uZ a -> (0 -Z a) *Z b = -uZ a *Z b |
| 3 |
|
zsub01 |
0 -Z a = -uZ a |
| 4 |
2, 3 |
ax_mp |
(0 -Z a) *Z b = -uZ a *Z b |
| 5 |
1, 4 |
ax_mp |
(0 -Z a) *Z b = -uZ (a *Z b) -> -uZ a *Z b = -uZ (a *Z b) |
| 6 |
|
eqtr |
(0 -Z a) *Z b = 0 *Z b -Z a *Z b -> 0 *Z b -Z a *Z b = -uZ (a *Z b) -> (0 -Z a) *Z b = -uZ (a *Z b) |
| 7 |
|
zsubmul |
(0 -Z a) *Z b = 0 *Z b -Z a *Z b |
| 8 |
6, 7 |
ax_mp |
0 *Z b -Z a *Z b = -uZ (a *Z b) -> (0 -Z a) *Z b = -uZ (a *Z b) |
| 9 |
|
eqtr |
0 *Z b -Z a *Z b = 0 -Z a *Z b -> 0 -Z a *Z b = -uZ (a *Z b) -> 0 *Z b -Z a *Z b = -uZ (a *Z b) |
| 10 |
|
zsubeq1 |
0 *Z b = 0 -> 0 *Z b -Z a *Z b = 0 -Z a *Z b |
| 11 |
|
zmul01 |
0 *Z b = 0 |
| 12 |
10, 11 |
ax_mp |
0 *Z b -Z a *Z b = 0 -Z a *Z b |
| 13 |
9, 12 |
ax_mp |
0 -Z a *Z b = -uZ (a *Z b) -> 0 *Z b -Z a *Z b = -uZ (a *Z b) |
| 14 |
|
zsub01 |
0 -Z a *Z b = -uZ (a *Z b) |
| 15 |
13, 14 |
ax_mp |
0 *Z b -Z a *Z b = -uZ (a *Z b) |
| 16 |
8, 15 |
ax_mp |
(0 -Z a) *Z b = -uZ (a *Z b) |
| 17 |
5, 16 |
ax_mp |
-uZ a *Z b = -uZ (a *Z b) |
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)