theorem dvdbgcdb (a b d: nat):
$ a != 0 -> (d || bgcd a b <-> d || a /\ d || b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
dvdtr |
d || bgcd a b -> bgcd a b || a -> d || a |
| 2 |
|
anr |
a != 0 /\ d || bgcd a b -> d || bgcd a b |
| 3 |
|
bgcddvd1 |
bgcd a b || a |
| 4 |
3 |
a1i |
a != 0 /\ d || bgcd a b -> bgcd a b || a |
| 5 |
1, 2, 4 |
sylc |
a != 0 /\ d || bgcd a b -> d || a |
| 6 |
|
dvdtr |
d || bgcd a b -> bgcd a b || b -> d || b |
| 7 |
|
bgcddvd2 |
a != 0 -> bgcd a b || b |
| 8 |
7 |
anwl |
a != 0 /\ d || bgcd a b -> bgcd a b || b |
| 9 |
6, 2, 8 |
sylc |
a != 0 /\ d || bgcd a b -> d || b |
| 10 |
5, 9 |
iand |
a != 0 /\ d || bgcd a b -> d || a /\ d || b |
| 11 |
10 |
exp |
a != 0 -> d || bgcd a b -> d || a /\ d || b |
| 12 |
|
dvdbgcd |
d || a /\ d || b -> d || bgcd a b |
| 13 |
12 |
a1i |
a != 0 -> d || a /\ d || b -> d || bgcd a b |
| 14 |
11, 13 |
ibid |
a != 0 -> (d || bgcd a b <-> d || a /\ d || 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)