theorem apprlams (a: nat) {x: nat} (b c: nat x):
$ c e. a -> (\. x e. a, b) @ c = N[c / x] b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
appeq1 |
\. x e. a, b == (\ x, b) |` a -> (\. x e. a, b) @ c = ((\ x, b) |` a) @ c |
| 2 |
|
rlameqs |
\. x e. a, b == (\ x, b) |` a |
| 3 |
1, 2 |
ax_mp |
(\. x e. a, b) @ c = ((\ x, b) |` a) @ c |
| 4 |
|
applams |
(\ x, b) @ c = N[c / x] b |
| 5 |
|
resapp |
c e. a -> ((\ x, b) |` a) @ c = (\ x, b) @ c |
| 6 |
4, 5 |
syl6eq |
c e. a -> ((\ x, b) |` a) @ c = N[c / x] b |
| 7 |
3, 6 |
syl5eq |
c e. a -> (\. x e. a, b) @ c = N[c / x] 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)