pub theorem powS (a b: nat): $ a ^ suc b = a * a ^ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a ^ suc b = (\ n, a * n) @ rec 1 (\ n, a * n) b -> (\ n, a * n) @ rec 1 (\ n, a * n) b = a * a ^ b -> a ^ suc b = a * a ^ b |
| 2 |
|
recS |
rec 1 (\ n, a * n) (suc b) = (\ n, a * n) @ rec 1 (\ n, a * n) b |
| 3 |
2 |
conv pow |
a ^ suc b = (\ n, a * n) @ rec 1 (\ n, a * n) b |
| 4 |
1, 3 |
ax_mp |
(\ n, a * n) @ rec 1 (\ n, a * n) b = a * a ^ b -> a ^ suc b = a * a ^ b |
| 5 |
|
muleq2 |
n = a ^ b -> a * n = a * a ^ b |
| 6 |
5 |
applame |
(\ n, a * n) @ (a ^ b) = a * a ^ b |
| 7 |
6 |
conv pow |
(\ n, a * n) @ rec 1 (\ n, a * n) b = a * a ^ b |
| 8 |
4, 7 |
ax_mp |
a ^ suc b = a * a ^ b |
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)