theorem znsubid (m: nat): $ m -ZN m = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
m -ZN m = b0 (m - m) -> b0 (m - m) = 0 -> m -ZN m = 0 |
| 2 |
|
znsubpos |
m <= m -> m -ZN m = b0 (m - m) |
| 3 |
|
leid |
m <= m |
| 4 |
2, 3 |
ax_mp |
m -ZN m = b0 (m - m) |
| 5 |
1, 4 |
ax_mp |
b0 (m - m) = 0 -> m -ZN m = 0 |
| 6 |
|
eqtr |
b0 (m - m) = b0 0 -> b0 0 = 0 -> b0 (m - m) = 0 |
| 7 |
|
b0eq |
m - m = 0 -> b0 (m - m) = b0 0 |
| 8 |
|
subid |
m - m = 0 |
| 9 |
7, 8 |
ax_mp |
b0 (m - m) = b0 0 |
| 10 |
6, 9 |
ax_mp |
b0 0 = 0 -> b0 (m - m) = 0 |
| 11 |
|
b00 |
b0 0 = 0 |
| 12 |
10, 11 |
ax_mp |
b0 (m - m) = 0 |
| 13 |
5, 12 |
ax_mp |
m -ZN m = 0 |
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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)