theorem modltid (a b: nat): $ a < b -> a % b = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
id |
a < b -> a < b |
| 2 |
|
eqtr |
b * 0 + a = 0 + a -> 0 + a = a -> b * 0 + a = a |
| 3 |
|
addeq1 |
b * 0 = 0 -> b * 0 + a = 0 + a |
| 4 |
|
mul02 |
b * 0 = 0 |
| 5 |
3, 4 |
ax_mp |
b * 0 + a = 0 + a |
| 6 |
2, 5 |
ax_mp |
0 + a = a -> b * 0 + a = a |
| 7 |
|
add01 |
0 + a = a |
| 8 |
6, 7 |
ax_mp |
b * 0 + a = a |
| 9 |
8 |
a1i |
a < b -> b * 0 + a = a |
| 10 |
1, 9 |
eqdivmod |
a < b -> a // b = 0 /\ a % b = a |
| 11 |
10 |
anrd |
a < b -> a % b = 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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)