theorem prltsuc (a b: nat): $ a, b < suc (a + b), 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ltnle |
a, b < suc (a + b), 0 <-> ~suc (a + b), 0 <= a, b |
| 2 |
|
prlem2 |
suc (a + b), 0 <= a, b -> suc (a + b) + 0 <= a + b |
| 3 |
|
ltnle |
a + b < suc (a + b) + 0 <-> ~suc (a + b) + 0 <= a + b |
| 4 |
|
lteq2 |
suc (a + b) + 0 = suc (a + b) -> (a + b < suc (a + b) + 0 <-> a + b < suc (a + b)) |
| 5 |
|
add0 |
suc (a + b) + 0 = suc (a + b) |
| 6 |
4, 5 |
ax_mp |
a + b < suc (a + b) + 0 <-> a + b < suc (a + b) |
| 7 |
|
ltsucid |
a + b < suc (a + b) |
| 8 |
6, 7 |
mpbir |
a + b < suc (a + b) + 0 |
| 9 |
3, 8 |
mpbi |
~suc (a + b) + 0 <= a + b |
| 10 |
2, 9 |
mt |
~suc (a + b), 0 <= a, b |
| 11 |
1, 10 |
mpbir |
a, b < suc (a + b), 0 |
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)