theorem copdvdmul1 (G: wff) (a b c: nat):
$ G -> coprime a c $ >
$ G -> a || b * c $ >
$ G -> a || b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
hyp h1 |
G -> coprime a c |
| 2 |
|
dvdeq2 |
b * c = c * b -> (a || b * c <-> a || c * b) |
| 3 |
|
mulcom |
b * c = c * b |
| 4 |
2, 3 |
ax_mp |
a || b * c <-> a || c * b |
| 5 |
|
hyp h2 |
G -> a || b * c |
| 6 |
4, 5 |
sylib |
G -> a || c * b |
| 7 |
1, 6 |
copdvdmul2 |
G -> a || 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)