theorem odddvd (n: nat): $ odd n <-> ~2 || n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(odd n <-> true (n % 2)) -> (true (n % 2) <-> ~2 || n) -> (odd n <-> ~2 || n) |
| 2 |
|
dfodd2 |
odd n <-> true (n % 2) |
| 3 |
1, 2 |
ax_mp |
(true (n % 2) <-> ~2 || n) -> (odd n <-> ~2 || n) |
| 4 |
|
noteq |
(n % 2 = 0 <-> 2 || n) -> (~n % 2 = 0 <-> ~2 || n) |
| 5 |
4 |
conv ne, true |
(n % 2 = 0 <-> 2 || n) -> (true (n % 2) <-> ~2 || n) |
| 6 |
|
modeq0 |
n % 2 = 0 <-> 2 || n |
| 7 |
5, 6 |
ax_mp |
true (n % 2) <-> ~2 || n |
| 8 |
3, 7 |
ax_mp |
odd n <-> ~2 || n |
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)