theorem upto1_sn: $ upto 1 = sn 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
upto 1 = 0 ; upto 0 -> 0 ; upto 0 = sn 0 -> upto 1 = sn 0 |
| 2 |
|
uptoS_ins |
upto (suc 0) = 0 ; upto 0 |
| 3 |
2 |
conv d1 |
upto 1 = 0 ; upto 0 |
| 4 |
1, 3 |
ax_mp |
0 ; upto 0 = sn 0 -> upto 1 = sn 0 |
| 5 |
|
eqtr |
0 ; upto 0 = 0 ; 0 -> 0 ; 0 = sn 0 -> 0 ; upto 0 = sn 0 |
| 6 |
|
inseq2 |
upto 0 = 0 -> 0 ; upto 0 = 0 ; 0 |
| 7 |
|
upto0 |
upto 0 = 0 |
| 8 |
6, 7 |
ax_mp |
0 ; upto 0 = 0 ; 0 |
| 9 |
5, 8 |
ax_mp |
0 ; 0 = sn 0 -> 0 ; upto 0 = sn 0 |
| 10 |
|
ins02 |
0 ; 0 = sn 0 |
| 11 |
9, 10 |
ax_mp |
0 ; upto 0 = sn 0 |
| 12 |
4, 11 |
ax_mp |
upto 1 = sn 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)