theorem muladdmod2lt (a b c: nat): $ b != 0 /\ c < b -> (b * a + c) % b = c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
anr |
b != 0 /\ c < b -> c < b |
| 2 |
|
eqidd |
b != 0 /\ c < b -> b * a + c = b * a + c |
| 3 |
1, 2 |
eqdivmod |
b != 0 /\ c < b -> (b * a + c) // b = a /\ (b * a + c) % b = c |
| 4 |
3 |
anrd |
b != 0 /\ c < b -> (b * a + c) % b = c |
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)