theorem mullass (a b c: nat): $ a * (b * c) = b * (a * c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
a * b * c = a * (b * c) -> a * b * c = b * (a * c) -> a * (b * c) = b * (a * c) |
| 2 |
|
mulass |
a * b * c = a * (b * c) |
| 3 |
1, 2 |
ax_mp |
a * b * c = b * (a * c) -> a * (b * c) = b * (a * c) |
| 4 |
|
eqtr |
a * b * c = b * a * c -> b * a * c = b * (a * c) -> a * b * c = b * (a * c) |
| 5 |
|
muleq1 |
a * b = b * a -> a * b * c = b * a * c |
| 6 |
|
mulcom |
a * b = b * a |
| 7 |
5, 6 |
ax_mp |
a * b * c = b * a * c |
| 8 |
4, 7 |
ax_mp |
b * a * c = b * (a * c) -> a * b * c = b * (a * c) |
| 9 |
|
mulass |
b * a * c = b * (a * c) |
| 10 |
8, 9 |
ax_mp |
a * b * c = b * (a * c) |
| 11 |
3, 10 |
ax_mp |
a * (b * c) = b * (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)