theorem zmulneg2 (a b: nat): $ a *Z -uZ b = -uZ (a *Z b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a *Z -uZ b = -uZ b *Z a -> -uZ b *Z a = -uZ (a *Z b) -> a *Z -uZ b = -uZ (a *Z b) |
| 2 |
|
zmulcom |
a *Z -uZ b = -uZ b *Z a |
| 3 |
1, 2 |
ax_mp |
-uZ b *Z a = -uZ (a *Z b) -> a *Z -uZ b = -uZ (a *Z b) |
| 4 |
|
eqtr |
-uZ b *Z a = -uZ (b *Z a) -> -uZ (b *Z a) = -uZ (a *Z b) -> -uZ b *Z a = -uZ (a *Z b) |
| 5 |
|
zmulneg1 |
-uZ b *Z a = -uZ (b *Z a) |
| 6 |
4, 5 |
ax_mp |
-uZ (b *Z a) = -uZ (a *Z b) -> -uZ b *Z a = -uZ (a *Z b) |
| 7 |
|
znegeq |
b *Z a = a *Z b -> -uZ (b *Z a) = -uZ (a *Z b) |
| 8 |
|
zmulcom |
b *Z a = a *Z b |
| 9 |
7, 8 |
ax_mp |
-uZ (b *Z a) = -uZ (a *Z b) |
| 10 |
6, 9 |
ax_mp |
-uZ b *Z a = -uZ (a *Z b) |
| 11 |
3, 10 |
ax_mp |
a *Z -uZ 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)