theorem consfstsnd (a: nat): $ a != 0 -> fst (a - 1) : snd (a - 1) = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
suceq |
fst (a - 1), snd (a - 1) = a - 1 -> suc (fst (a - 1), snd (a - 1)) = suc (a - 1) |
| 2 |
1 |
conv cons |
fst (a - 1), snd (a - 1) = a - 1 -> fst (a - 1) : snd (a - 1) = suc (a - 1) |
| 3 |
|
fstsnd |
fst (a - 1), snd (a - 1) = a - 1 |
| 4 |
2, 3 |
ax_mp |
fst (a - 1) : snd (a - 1) = suc (a - 1) |
| 5 |
|
sub1can |
a != 0 -> suc (a - 1) = a |
| 6 |
4, 5 |
syl5eq |
a != 0 -> fst (a - 1) : snd (a - 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)