theorem gcd01 (b: nat): $ gcd 0 b = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bicom |
(d || 0 /\ d || b <-> d || b) -> (d || b <-> d || 0 /\ d || b) |
| 2 |
|
bian1 |
d || 0 -> (d || 0 /\ d || b <-> d || b) |
| 3 |
|
dvd02 |
d || 0 |
| 4 |
2, 3 |
ax_mp |
d || 0 /\ d || b <-> d || b |
| 5 |
1, 4 |
ax_mp |
d || b <-> d || 0 /\ d || b |
| 6 |
5 |
a1i |
T. -> (d || b <-> d || 0 /\ d || b) |
| 7 |
6 |
eqgcd |
T. -> gcd 0 b = b |
| 8 |
7 |
trud |
gcd 0 b = 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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)