theorem pow11 (b: nat): $ 1 ^ b = 1 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_1 = b -> 1 = 1 |
| 2 |
|
id |
_1 = b -> _1 = b |
| 3 |
1, 2 |
poweqd |
_1 = b -> 1 ^ _1 = 1 ^ b |
| 4 |
3, 1 |
eqeqd |
_1 = b -> (1 ^ _1 = 1 <-> 1 ^ b = 1) |
| 5 |
|
eqidd |
_1 = 0 -> 1 = 1 |
| 6 |
|
id |
_1 = 0 -> _1 = 0 |
| 7 |
5, 6 |
poweqd |
_1 = 0 -> 1 ^ _1 = 1 ^ 0 |
| 8 |
7, 5 |
eqeqd |
_1 = 0 -> (1 ^ _1 = 1 <-> 1 ^ 0 = 1) |
| 9 |
|
eqidd |
_1 = a1 -> 1 = 1 |
| 10 |
|
id |
_1 = a1 -> _1 = a1 |
| 11 |
9, 10 |
poweqd |
_1 = a1 -> 1 ^ _1 = 1 ^ a1 |
| 12 |
11, 9 |
eqeqd |
_1 = a1 -> (1 ^ _1 = 1 <-> 1 ^ a1 = 1) |
| 13 |
|
eqidd |
_1 = suc a1 -> 1 = 1 |
| 14 |
|
id |
_1 = suc a1 -> _1 = suc a1 |
| 15 |
13, 14 |
poweqd |
_1 = suc a1 -> 1 ^ _1 = 1 ^ suc a1 |
| 16 |
15, 13 |
eqeqd |
_1 = suc a1 -> (1 ^ _1 = 1 <-> 1 ^ suc a1 = 1) |
| 17 |
|
pow0 |
1 ^ 0 = 1 |
| 18 |
|
powS |
1 ^ suc a1 = 1 * 1 ^ a1 |
| 19 |
|
mul11 |
1 * 1 ^ a1 = 1 ^ a1 |
| 20 |
|
id |
1 ^ a1 = 1 -> 1 ^ a1 = 1 |
| 21 |
19, 20 |
syl5eq |
1 ^ a1 = 1 -> 1 * 1 ^ a1 = 1 |
| 22 |
18, 21 |
syl5eq |
1 ^ a1 = 1 -> 1 ^ suc a1 = 1 |
| 23 |
4, 8, 12, 16, 17, 22 |
ind |
1 ^ b = 1 |
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)