theorem muladdmod2 (a b c: nat): $ b != 0 -> (b * a + c) % b = c % b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
modlt |
b != 0 -> c % b < b |
| 2 |
|
eqtr |
b * (a + c // b) + c % b = b * a + b * (c // b) + c % b -> b * a + b * (c // b) + c % b = b * a + c -> b * (a + c // b) + c % b = b * a + c |
| 3 |
|
addeq1 |
b * (a + c // b) = b * a + b * (c // b) -> b * (a + c // b) + c % b = b * a + b * (c // b) + c % b |
| 4 |
|
muladd |
b * (a + c // b) = b * a + b * (c // b) |
| 5 |
3, 4 |
ax_mp |
b * (a + c // b) + c % b = b * a + b * (c // b) + c % b |
| 6 |
2, 5 |
ax_mp |
b * a + b * (c // b) + c % b = b * a + c -> b * (a + c // b) + c % b = b * a + c |
| 7 |
|
eqtr |
b * a + b * (c // b) + c % b = b * a + (b * (c // b) + c % b) -> b * a + (b * (c // b) + c % b) = b * a + c -> b * a + b * (c // b) + c % b = b * a + c |
| 8 |
|
addass |
b * a + b * (c // b) + c % b = b * a + (b * (c // b) + c % b) |
| 9 |
7, 8 |
ax_mp |
b * a + (b * (c // b) + c % b) = b * a + c -> b * a + b * (c // b) + c % b = b * a + c |
| 10 |
|
addeq2 |
b * (c // b) + c % b = c -> b * a + (b * (c // b) + c % b) = b * a + c |
| 11 |
|
divmod |
b * (c // b) + c % b = c |
| 12 |
10, 11 |
ax_mp |
b * a + (b * (c // b) + c % b) = b * a + c |
| 13 |
9, 12 |
ax_mp |
b * a + b * (c // b) + c % b = b * a + c |
| 14 |
6, 13 |
ax_mp |
b * (a + c // b) + c % b = b * a + c |
| 15 |
14 |
a1i |
b != 0 -> b * (a + c // b) + c % b = b * a + c |
| 16 |
1, 15 |
eqdivmod |
b != 0 -> (b * a + c) // b = a + c // b /\ (b * a + c) % b = c % b |
| 17 |
16 |
anrd |
b != 0 -> (b * a + c) % b = c % 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)