theorem mulpos (a b: nat): $ 0 < a * b <-> 0 < a /\ 0 < b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
lt01 |
0 < a * b <-> a * b != 0 |
| 2 |
|
lt01 |
0 < a <-> a != 0 |
| 3 |
|
con3 |
(a = 0 -> a * b = 0) -> ~a * b = 0 -> ~a = 0 |
| 4 |
3 |
conv ne |
(a = 0 -> a * b = 0) -> a * b != 0 -> a != 0 |
| 5 |
|
mul01 |
0 * b = 0 |
| 6 |
|
muleq1 |
a = 0 -> a * b = 0 * b |
| 7 |
5, 6 |
syl6eq |
a = 0 -> a * b = 0 |
| 8 |
4, 7 |
ax_mp |
a * b != 0 -> a != 0 |
| 9 |
2, 8 |
sylibr |
a * b != 0 -> 0 < a |
| 10 |
1, 9 |
sylbi |
0 < a * b -> 0 < a |
| 11 |
|
lt01 |
0 < b <-> b != 0 |
| 12 |
|
con3 |
(b = 0 -> a * b = 0) -> ~a * b = 0 -> ~b = 0 |
| 13 |
12 |
conv ne |
(b = 0 -> a * b = 0) -> a * b != 0 -> b != 0 |
| 14 |
|
mul02 |
a * 0 = 0 |
| 15 |
|
muleq2 |
b = 0 -> a * b = a * 0 |
| 16 |
14, 15 |
syl6eq |
b = 0 -> a * b = 0 |
| 17 |
13, 16 |
ax_mp |
a * b != 0 -> b != 0 |
| 18 |
11, 17 |
sylibr |
a * b != 0 -> 0 < b |
| 19 |
1, 18 |
sylbi |
0 < a * b -> 0 < b |
| 20 |
10, 19 |
iand |
0 < a * b -> 0 < a /\ 0 < b |
| 21 |
|
lteq1 |
a * 0 = 0 -> (a * 0 < a * b <-> 0 < a * b) |
| 22 |
|
mul0 |
a * 0 = 0 |
| 23 |
21, 22 |
ax_mp |
a * 0 < a * b <-> 0 < a * b |
| 24 |
|
ltmul2 |
0 < a -> (0 < b <-> a * 0 < a * b) |
| 25 |
24 |
anwl |
0 < a /\ 0 < b -> (0 < b <-> a * 0 < a * b) |
| 26 |
|
anr |
0 < a /\ 0 < b -> 0 < b |
| 27 |
25, 26 |
mpbid |
0 < a /\ 0 < b -> a * 0 < a * b |
| 28 |
23, 27 |
sylib |
0 < a /\ 0 < b -> 0 < a * b |
| 29 |
20, 28 |
ibii |
0 < a * b <-> 0 < a /\ 0 < 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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)