theorem mulrass (a b c: nat): $ a * b * c = a * c * b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a * b * c = a * (b * c) -> a * (b * c) = a * c * b -> a * b * c = a * c * b |
| 2 |
|
mulass |
a * b * c = a * (b * c) |
| 3 |
1, 2 |
ax_mp |
a * (b * c) = a * c * b -> a * b * c = a * c * b |
| 4 |
|
eqtr4 |
a * (b * c) = a * (c * b) -> a * c * b = a * (c * b) -> a * (b * c) = a * c * b |
| 5 |
|
muleq2 |
b * c = c * b -> a * (b * c) = a * (c * b) |
| 6 |
|
mulcom |
b * c = c * b |
| 7 |
5, 6 |
ax_mp |
a * (b * c) = a * (c * b) |
| 8 |
4, 7 |
ax_mp |
a * c * b = a * (c * b) -> a * (b * c) = a * c * b |
| 9 |
|
mulass |
a * c * b = a * (c * b) |
| 10 |
8, 9 |
ax_mp |
a * (b * c) = a * c * b |
| 11 |
3, 10 |
ax_mp |
a * b * c = a * c * 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_peano
(peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)