pub theorem div0 (a: nat): $ a // 0 = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
absurd |
~r < 0 -> r < 0 -> q = 0 |
| 2 |
|
lt02 |
~r < 0 |
| 3 |
1, 2 |
ax_mp |
r < 0 -> q = 0 |
| 4 |
3 |
anwl |
r < 0 /\ 0 * q + r = a -> q = 0 |
| 5 |
4 |
eex |
E. r (r < 0 /\ 0 * q + r = a) -> q = 0 |
| 6 |
5 |
a1i |
T. -> E. r (r < 0 /\ 0 * q + r = a) -> q = 0 |
| 7 |
6 |
eqthe0abd |
T. -> the {q | E. r (r < 0 /\ 0 * q + r = a)} = 0 |
| 8 |
7 |
conv div |
T. -> a // 0 = 0 |
| 9 |
8 |
trud |
a // 0 = 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,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)