theorem dvdcop (G: wff) (a b d: nat):
$ G -> coprime a b $ >
$ G -> d || a $ >
$ G -> d || b $ >
$ G -> d = 1 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
dvd12 |
d || 1 <-> d = 1 |
| 2 |
|
hyp h1 |
G -> coprime a b |
| 3 |
2 |
conv coprime |
G -> gcd a b = 1 |
| 4 |
3 |
dvdeq2d |
G -> (d || gcd a b <-> d || 1) |
| 5 |
|
dvdgcd |
d || gcd a b <-> d || a /\ d || b |
| 6 |
|
hyp h2 |
G -> d || a |
| 7 |
|
hyp h3 |
G -> d || b |
| 8 |
6, 7 |
iand |
G -> d || a /\ d || b |
| 9 |
5, 8 |
sylibr |
G -> d || gcd a b |
| 10 |
4, 9 |
mpbid |
G -> d || 1 |
| 11 |
1, 10 |
sylib |
G -> d = 1 |
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)