theorem dvdgcd (a b d: nat): $ d || gcd a b <-> d || a /\ d || b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bian1 |
x || a -> (x || a /\ x || b <-> x || b) |
| 2 |
|
dvd02 |
x || 0 |
| 3 |
|
dvdeq2 |
a = 0 -> (x || a <-> x || 0) |
| 4 |
2, 3 |
mpbiri |
a = 0 -> x || a |
| 5 |
1, 4 |
syl |
a = 0 -> (x || a /\ x || b <-> x || b) |
| 6 |
5 |
bicomd |
a = 0 -> (x || b <-> x || a /\ x || b) |
| 7 |
6 |
dvdgcdlem |
a = 0 -> (d || gcd a b <-> d || a /\ d || b) |
| 8 |
|
dvdbgcdb |
a != 0 -> (x || bgcd a b <-> x || a /\ x || b) |
| 9 |
8 |
conv ne |
~a = 0 -> (x || bgcd a b <-> x || a /\ x || b) |
| 10 |
9 |
dvdgcdlem |
~a = 0 -> (d || gcd a b <-> d || a /\ d || b) |
| 11 |
7, 10 |
cases |
d || gcd 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)