theorem zsubeq0 (a b: nat): $ a -Z b = 0 <-> a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a -Z b = 0 <-> 0 +Z b = a) -> (0 +Z b = a <-> a = b) -> (a -Z b = 0 <-> a = b) |
| 2 |
|
eqzsub |
a -Z b = 0 <-> 0 +Z b = a |
| 3 |
1, 2 |
ax_mp |
(0 +Z b = a <-> a = b) -> (a -Z b = 0 <-> a = b) |
| 4 |
|
bitr |
(0 +Z b = a <-> b = a) -> (b = a <-> a = b) -> (0 +Z b = a <-> a = b) |
| 5 |
|
eqeq1 |
0 +Z b = b -> (0 +Z b = a <-> b = a) |
| 6 |
|
zadd01 |
0 +Z b = b |
| 7 |
5, 6 |
ax_mp |
0 +Z b = a <-> b = a |
| 8 |
4, 7 |
ax_mp |
(b = a <-> a = b) -> (0 +Z b = a <-> a = b) |
| 9 |
|
eqcomb |
b = a <-> a = b |
| 10 |
8, 9 |
ax_mp |
0 +Z b = a <-> a = b |
| 11 |
3, 10 |
ax_mp |
a -Z b = 0 <-> a = 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)