theorem srecrlam (S: set) {i: nat} (n: nat):
$ \. i e. upto n, srec S i = srecaux S n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
axext |
\. i e. upto n, srec S i == srecaux S n -> \. i e. upto n, srec S i = srecaux S n |
| 2 |
|
eqstr |
\. i e. upto n, srec S i == (\ i, srec S i) |` upto n -> (\ i, srec S i) |` upto n == srecaux S n -> \. i e. upto n, srec S i == srecaux S n |
| 3 |
|
rlameqs |
\. i e. upto n, srec S i == (\ i, srec S i) |` upto n |
| 4 |
2, 3 |
ax_mp |
(\ i, srec S i) |` upto n == srecaux S n -> \. i e. upto n, srec S i == srecaux S n |
| 5 |
|
srecres |
(\ i, srec S i) |` upto n == srecaux S n |
| 6 |
4, 5 |
ax_mp |
\. i e. upto n, srec S i == srecaux S n |
| 7 |
1, 6 |
ax_mp |
\. i e. upto n, srec S i = srecaux S 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)