theorem consfst (a b: nat): $ fst (a : b - 1) = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
fst (a : b - 1) = fst (a, b) -> fst (a, b) = a -> fst (a : b - 1) = a |
| 2 |
|
fsteq |
a : b - 1 = a, b -> fst (a : b - 1) = fst (a, b) |
| 3 |
|
sucsub1 |
suc (a, b) - 1 = a, b |
| 4 |
3 |
conv cons |
a : b - 1 = a, b |
| 5 |
2, 4 |
ax_mp |
fst (a : b - 1) = fst (a, b) |
| 6 |
1, 5 |
ax_mp |
fst (a, b) = a -> fst (a : b - 1) = a |
| 7 |
|
fstpr |
fst (a, b) = a |
| 8 |
6, 7 |
ax_mp |
fst (a : b - 1) = a |
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)