theorem Sum0: $ Sum 0 0 == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqstr3 |
Sum (Fst 0) (Snd 0) == Sum 0 0 -> Sum (Fst 0) (Snd 0) == 0 -> Sum 0 0 == 0 |
| 2 |
|
Sumeq |
Fst 0 == 0 -> Snd 0 == 0 -> Sum (Fst 0) (Snd 0) == Sum 0 0 |
| 3 |
|
Fst0 |
Fst 0 == 0 |
| 4 |
2, 3 |
ax_mp |
Snd 0 == 0 -> Sum (Fst 0) (Snd 0) == Sum 0 0 |
| 5 |
|
Snd0 |
Snd 0 == 0 |
| 6 |
4, 5 |
ax_mp |
Sum (Fst 0) (Snd 0) == Sum 0 0 |
| 7 |
1, 6 |
ax_mp |
Sum (Fst 0) (Snd 0) == 0 -> Sum 0 0 == 0 |
| 8 |
|
FstSnd |
Sum (Fst 0) (Snd 0) == 0 |
| 9 |
7, 8 |
ax_mp |
Sum 0 0 == 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)