theorem elpow2 (a n: nat): $ a e. 2 ^ n <-> a = n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(a e. shl 1 n <-> a e. 2 ^ n) -> (a e. shl 1 n <-> a = n) -> (a e. 2 ^ n <-> a = n) |
| 2 |
|
elneq2 |
shl 1 n = 2 ^ n -> (a e. shl 1 n <-> a e. 2 ^ n) |
| 3 |
|
shl11 |
shl 1 n = 2 ^ n |
| 4 |
2, 3 |
ax_mp |
a e. shl 1 n <-> a e. 2 ^ n |
| 5 |
1, 4 |
ax_mp |
(a e. shl 1 n <-> a = n) -> (a e. 2 ^ n <-> a = n) |
| 6 |
|
bitr |
(a e. shl 1 n <-> n <= a /\ a - n e. 1) -> (n <= a /\ a - n e. 1 <-> a = n) -> (a e. shl 1 n <-> a = n) |
| 7 |
|
elshl |
a e. shl 1 n <-> n <= a /\ a - n e. 1 |
| 8 |
6, 7 |
ax_mp |
(n <= a /\ a - n e. 1 <-> a = n) -> (a e. shl 1 n <-> a = n) |
| 9 |
|
bitr4 |
(n <= a /\ a - n e. 1 <-> n <= a /\ a <= n) -> (a = n <-> n <= a /\ a <= n) -> (n <= a /\ a - n e. 1 <-> a = n) |
| 10 |
|
bitr4 |
(a - n e. 1 <-> a - n = 0) -> (a <= n <-> a - n = 0) -> (a - n e. 1 <-> a <= n) |
| 11 |
|
el12 |
a - n e. 1 <-> a - n = 0 |
| 12 |
10, 11 |
ax_mp |
(a <= n <-> a - n = 0) -> (a - n e. 1 <-> a <= n) |
| 13 |
|
lesubeq0 |
a <= n <-> a - n = 0 |
| 14 |
12, 13 |
ax_mp |
a - n e. 1 <-> a <= n |
| 15 |
14 |
aneq2i |
n <= a /\ a - n e. 1 <-> n <= a /\ a <= n |
| 16 |
9, 15 |
ax_mp |
(a = n <-> n <= a /\ a <= n) -> (n <= a /\ a - n e. 1 <-> a = n) |
| 17 |
|
bitr |
(a = n <-> n = a) -> (n = a <-> n <= a /\ a <= n) -> (a = n <-> n <= a /\ a <= n) |
| 18 |
|
eqcomb |
a = n <-> n = a |
| 19 |
17, 18 |
ax_mp |
(n = a <-> n <= a /\ a <= n) -> (a = n <-> n <= a /\ a <= n) |
| 20 |
|
eqlele |
n = a <-> n <= a /\ a <= n |
| 21 |
19, 20 |
ax_mp |
a = n <-> n <= a /\ a <= n |
| 22 |
16, 21 |
ax_mp |
n <= a /\ a - n e. 1 <-> a = n |
| 23 |
8, 22 |
ax_mp |
a e. shl 1 n <-> a = n |
| 24 |
5, 23 |
ax_mp |
a e. 2 ^ n <-> a = 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)