theorem gcd00: $ gcd 0 0 = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bith |
x || 0 -> x || 0 /\ x || 0 -> (x || 0 <-> x || 0 /\ x || 0) |
| 2 |
|
dvd02 |
x || 0 |
| 3 |
1, 2 |
ax_mp |
x || 0 /\ x || 0 -> (x || 0 <-> x || 0 /\ x || 0) |
| 4 |
|
ian |
x || 0 -> x || 0 -> x || 0 /\ x || 0 |
| 5 |
4, 2 |
ax_mp |
x || 0 -> x || 0 /\ x || 0 |
| 6 |
5, 2 |
ax_mp |
x || 0 /\ x || 0 |
| 7 |
3, 6 |
ax_mp |
x || 0 <-> x || 0 /\ x || 0 |
| 8 |
7 |
a1i |
T. -> (x || 0 <-> x || 0 /\ x || 0) |
| 9 |
8 |
eqgcd |
T. -> gcd 0 0 = 0 |
| 10 |
9 |
trud |
gcd 0 0 = 0 |
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)