theorem mulne0 (a b: nat): $ a * b != 0 <-> a != 0 /\ b != 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(0 < a * b <-> a * b != 0) -> (0 < a * b <-> a != 0 /\ b != 0) -> (a * b != 0 <-> a != 0 /\ b != 0) |
| 2 |
|
lt01 |
0 < a * b <-> a * b != 0 |
| 3 |
1, 2 |
ax_mp |
(0 < a * b <-> a != 0 /\ b != 0) -> (a * b != 0 <-> a != 0 /\ b != 0) |
| 4 |
|
bitr |
(0 < a * b <-> 0 < a /\ 0 < b) -> (0 < a /\ 0 < b <-> a != 0 /\ b != 0) -> (0 < a * b <-> a != 0 /\ b != 0) |
| 5 |
|
mulpos |
0 < a * b <-> 0 < a /\ 0 < b |
| 6 |
4, 5 |
ax_mp |
(0 < a /\ 0 < b <-> a != 0 /\ b != 0) -> (0 < a * b <-> a != 0 /\ b != 0) |
| 7 |
|
aneq |
(0 < a <-> a != 0) -> (0 < b <-> b != 0) -> (0 < a /\ 0 < b <-> a != 0 /\ b != 0) |
| 8 |
|
lt01 |
0 < a <-> a != 0 |
| 9 |
7, 8 |
ax_mp |
(0 < b <-> b != 0) -> (0 < a /\ 0 < b <-> a != 0 /\ b != 0) |
| 10 |
|
lt01 |
0 < b <-> b != 0 |
| 11 |
9, 10 |
ax_mp |
0 < a /\ 0 < b <-> a != 0 /\ b != 0 |
| 12 |
6, 11 |
ax_mp |
0 < a * b <-> a != 0 /\ b != 0 |
| 13 |
3, 12 |
ax_mp |
a * b != 0 <-> a != 0 /\ b != 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_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)