theorem muldiv2 (a b: nat): $ b != 0 -> b * a // b = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bi2 |
(0 < b <-> b != 0) -> b != 0 -> 0 < b |
| 2 |
|
lt01 |
0 < b <-> b != 0 |
| 3 |
1, 2 |
ax_mp |
b != 0 -> 0 < b |
| 4 |
|
add0 |
b * a + 0 = b * a |
| 5 |
4 |
a1i |
b != 0 -> b * a + 0 = b * a |
| 6 |
3, 5 |
eqdivmod |
b != 0 -> b * a // b = a /\ b * a % b = 0 |
| 7 |
6 |
anld |
b != 0 -> 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)