theorem bezout (a b: nat) {x y: nat}:
$ a != 0 -> E. x E. y x * a = y * b + gcd a b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bgcdbezout |
E. x E. y x * a = y * b + bgcd a b |
| 2 |
|
gcdbgcd |
a != 0 -> gcd a b = bgcd a b |
| 3 |
2 |
addeq2d |
a != 0 -> y * b + gcd a b = y * b + bgcd a b |
| 4 |
3 |
eqeq2d |
a != 0 -> (x * a = y * b + gcd a b <-> x * a = y * b + bgcd a b) |
| 5 |
4 |
exeqd |
a != 0 -> (E. y x * a = y * b + gcd a b <-> E. y x * a = y * b + bgcd a b) |
| 6 |
5 |
exeqd |
a != 0 -> (E. x E. y x * a = y * b + gcd a b <-> E. x E. y x * a = y * b + bgcd a b) |
| 7 |
1, 6 |
mpbiri |
a != 0 -> E. x E. y x * a = y * b + gcd 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)