theorem zpncan (a b: nat): $ a +Z b -Z b = a $;
Step | Hyp | Ref | Expression |
1 |
|
eqtr |
a +Z b -Z b = a +Z (b +Z -uZ b) -> a +Z (b +Z -uZ b) = a -> a +Z b -Z b = a |
2 |
|
zaddass |
a +Z b +Z -uZ b = a +Z (b +Z -uZ b) |
3 |
2 |
conv zsub |
a +Z b -Z b = a +Z (b +Z -uZ b) |
4 |
1, 3 |
ax_mp |
a +Z (b +Z -uZ b) = a -> a +Z b -Z b = a |
5 |
|
eqtr |
a +Z (b +Z -uZ b) = a +Z 0 -> a +Z 0 = a -> a +Z (b +Z -uZ b) = a |
6 |
|
zaddeq2 |
b +Z -uZ b = 0 -> a +Z (b +Z -uZ b) = a +Z 0 |
7 |
|
znegid |
b +Z -uZ b = 0 |
8 |
6, 7 |
ax_mp |
a +Z (b +Z -uZ b) = a +Z 0 |
9 |
5, 8 |
ax_mp |
a +Z 0 = a -> a +Z (b +Z -uZ b) = a |
10 |
|
zadd02 |
a +Z 0 = a |
11 |
9, 10 |
ax_mp |
a +Z (b +Z -uZ b) = a |
12 |
4, 11 |
ax_mp |
a +Z b -Z 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)