theorem b0ne0 (n: nat): $ b0 n != 0 <-> n != 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(2 * n != 0 <-> b0 n != 0) -> (2 * n != 0 <-> n != 0) -> (b0 n != 0 <-> n != 0) |
| 2 |
|
neeq1 |
2 * n = b0 n -> (2 * n != 0 <-> b0 n != 0) |
| 3 |
|
b0mul21 |
2 * n = b0 n |
| 4 |
2, 3 |
ax_mp |
2 * n != 0 <-> b0 n != 0 |
| 5 |
1, 4 |
ax_mp |
(2 * n != 0 <-> n != 0) -> (b0 n != 0 <-> n != 0) |
| 6 |
|
bitr |
(2 * n != 0 <-> 2 != 0 /\ n != 0) -> (2 != 0 /\ n != 0 <-> n != 0) -> (2 * n != 0 <-> n != 0) |
| 7 |
|
mulne0 |
2 * n != 0 <-> 2 != 0 /\ n != 0 |
| 8 |
6, 7 |
ax_mp |
(2 != 0 /\ n != 0 <-> n != 0) -> (2 * n != 0 <-> n != 0) |
| 9 |
|
bian1 |
2 != 0 -> (2 != 0 /\ n != 0 <-> n != 0) |
| 10 |
|
d2ne0 |
2 != 0 |
| 11 |
9, 10 |
ax_mp |
2 != 0 /\ n != 0 <-> n != 0 |
| 12 |
8, 11 |
ax_mp |
2 * n != 0 <-> n != 0 |
| 13 |
5, 12 |
ax_mp |
b0 n != 0 <-> n != 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_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)