theorem appendeq0 (a b: nat): $ a ++ b = 0 <-> a = 0 /\ b = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(len (a ++ b) = 0 <-> a ++ b = 0) -> (len (a ++ b) = 0 <-> a = 0 /\ b = 0) -> (a ++ b = 0 <-> a = 0 /\ b = 0) |
| 2 |
|
leneq0 |
len (a ++ b) = 0 <-> a ++ b = 0 |
| 3 |
1, 2 |
ax_mp |
(len (a ++ b) = 0 <-> a = 0 /\ b = 0) -> (a ++ b = 0 <-> a = 0 /\ b = 0) |
| 4 |
|
bitr |
(len (a ++ b) = 0 <-> len a + len b = 0) -> (len a + len b = 0 <-> a = 0 /\ b = 0) -> (len (a ++ b) = 0 <-> a = 0 /\ b = 0) |
| 5 |
|
eqeq1 |
len (a ++ b) = len a + len b -> (len (a ++ b) = 0 <-> len a + len b = 0) |
| 6 |
|
appendlen |
len (a ++ b) = len a + len b |
| 7 |
5, 6 |
ax_mp |
len (a ++ b) = 0 <-> len a + len b = 0 |
| 8 |
4, 7 |
ax_mp |
(len a + len b = 0 <-> a = 0 /\ b = 0) -> (len (a ++ b) = 0 <-> a = 0 /\ b = 0) |
| 9 |
|
bitr |
(len a + len b = 0 <-> len a = 0 /\ len b = 0) -> (len a = 0 /\ len b = 0 <-> a = 0 /\ b = 0) -> (len a + len b = 0 <-> a = 0 /\ b = 0) |
| 10 |
|
addeq0 |
len a + len b = 0 <-> len a = 0 /\ len b = 0 |
| 11 |
9, 10 |
ax_mp |
(len a = 0 /\ len b = 0 <-> a = 0 /\ b = 0) -> (len a + len b = 0 <-> a = 0 /\ b = 0) |
| 12 |
|
aneq |
(len a = 0 <-> a = 0) -> (len b = 0 <-> b = 0) -> (len a = 0 /\ len b = 0 <-> a = 0 /\ b = 0) |
| 13 |
|
leneq0 |
len a = 0 <-> a = 0 |
| 14 |
12, 13 |
ax_mp |
(len b = 0 <-> b = 0) -> (len a = 0 /\ len b = 0 <-> a = 0 /\ b = 0) |
| 15 |
|
leneq0 |
len b = 0 <-> b = 0 |
| 16 |
14, 15 |
ax_mp |
len a = 0 /\ len b = 0 <-> a = 0 /\ b = 0 |
| 17 |
11, 16 |
ax_mp |
len a + len b = 0 <-> a = 0 /\ b = 0 |
| 18 |
8, 17 |
ax_mp |
len (a ++ b) = 0 <-> a = 0 /\ b = 0 |
| 19 |
3, 18 |
ax_mp |
a ++ b = 0 <-> a = 0 /\ b = 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,
muleq,
add0,
addS,
mul0,
mulS)