theorem eqm03 (a n: nat): $ mod(n): a = 0 <-> n || a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(mod(n): a = 0 <-> a % n = 0) -> (a % n = 0 <-> n || a) -> (mod(n): a = 0 <-> n || a) |
| 2 |
|
eqeq2 |
0 % n = 0 -> (a % n = 0 % n <-> a % n = 0) |
| 3 |
2 |
conv eqm |
0 % n = 0 -> (mod(n): a = 0 <-> a % n = 0) |
| 4 |
|
mod01 |
0 % n = 0 |
| 5 |
3, 4 |
ax_mp |
mod(n): a = 0 <-> a % n = 0 |
| 6 |
1, 5 |
ax_mp |
(a % n = 0 <-> n || a) -> (mod(n): a = 0 <-> n || a) |
| 7 |
|
modeq0 |
a % n = 0 <-> n || a |
| 8 |
6, 7 |
ax_mp |
mod(n): a = 0 <-> n || 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)