theorem eqmdvd (a b n: nat): $ mod(n): a = b -> (n || a <-> n || b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqm03 |
mod(n): a = 0 <-> n || a |
| 2 |
|
eqm03 |
mod(n): b = 0 <-> n || b |
| 3 |
|
eqeq1 |
a % n = b % n -> (a % n = 0 % n <-> b % n = 0 % n) |
| 4 |
3 |
conv eqm |
mod(n): a = b -> (mod(n): a = 0 <-> mod(n): b = 0) |
| 5 |
1, 2, 4 |
bitr3g |
mod(n): a = b -> (n || a <-> n || b) |
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)