theorem znegeqcom (a b: nat): $ -uZ a = b <-> -uZ b = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(-uZ a = b <-> a +Z b = 0) -> (-uZ b = a <-> a +Z b = 0) -> (-uZ a = b <-> -uZ b = a) |
| 2 |
|
eqzneg |
-uZ a = b <-> a +Z b = 0 |
| 3 |
1, 2 |
ax_mp |
(-uZ b = a <-> a +Z b = 0) -> (-uZ a = b <-> -uZ b = a) |
| 4 |
|
bitr4 |
(-uZ b = a <-> b +Z a = 0) -> (a +Z b = 0 <-> b +Z a = 0) -> (-uZ b = a <-> a +Z b = 0) |
| 5 |
|
eqzneg |
-uZ b = a <-> b +Z a = 0 |
| 6 |
4, 5 |
ax_mp |
(a +Z b = 0 <-> b +Z a = 0) -> (-uZ b = a <-> a +Z b = 0) |
| 7 |
|
eqeq1 |
a +Z b = b +Z a -> (a +Z b = 0 <-> b +Z a = 0) |
| 8 |
|
zaddcom |
a +Z b = b +Z a |
| 9 |
7, 8 |
ax_mp |
a +Z b = 0 <-> b +Z a = 0 |
| 10 |
6, 9 |
ax_mp |
-uZ b = a <-> a +Z b = 0 |
| 11 |
3, 10 |
ax_mp |
-uZ a = b <-> -uZ b = a |
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)