theorem dvdtr (a b c: nat): $ a || b -> b || c -> a || c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
idvd |
y * x * a = c -> a || c |
| 2 |
|
mulass |
y * x * a = y * (x * a) |
| 3 |
|
muleq2 |
x * a = b -> y * (x * a) = y * b |
| 4 |
2, 3 |
syl5eq |
x * a = b -> y * x * a = y * b |
| 5 |
4 |
anwl |
x * a = b /\ y * b = c -> y * x * a = y * b |
| 6 |
|
anr |
x * a = b /\ y * b = c -> y * b = c |
| 7 |
5, 6 |
eqtrd |
x * a = b /\ y * b = c -> y * x * a = c |
| 8 |
1, 7 |
syl |
x * a = b /\ y * b = c -> a || c |
| 9 |
8 |
eexda |
x * a = b -> E. y y * b = c -> a || c |
| 10 |
9 |
conv dvd |
x * a = b -> b || c -> a || c |
| 11 |
10 |
eex |
E. x x * a = b -> b || c -> a || c |
| 12 |
11 |
conv dvd |
a || b -> b || c -> a || c |
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_peano
(peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)