theorem letrueb (a b: nat): $ bool a -> (a <= b <-> true a -> true b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
letrue |
a <= b -> true a -> true b |
| 2 |
1 |
a1i |
bool a -> a <= b -> true a -> true b |
| 3 |
|
bool01 |
bool a <-> a = 0 \/ a = 1 |
| 4 |
|
eor |
(a = 0 -> (true a -> true b) -> a <= b) -> (a = 1 -> (true a -> true b) -> a <= b) -> a = 0 \/ a = 1 -> (true a -> true b) -> a <= b |
| 5 |
|
le01 |
0 <= b |
| 6 |
|
leeq1 |
a = 0 -> (a <= b <-> 0 <= b) |
| 7 |
5, 6 |
mpbiri |
a = 0 -> a <= b |
| 8 |
7 |
a1d |
a = 0 -> (true a -> true b) -> a <= b |
| 9 |
4, 8 |
ax_mp |
(a = 1 -> (true a -> true b) -> a <= b) -> a = 0 \/ a = 1 -> (true a -> true b) -> a <= b |
| 10 |
|
le11 |
1 <= b <-> b != 0 |
| 11 |
10 |
bi2i |
b != 0 -> 1 <= b |
| 12 |
|
biim1 |
true a -> (true a -> true b <-> true b) |
| 13 |
12 |
conv true |
true a -> (true a -> true b <-> b != 0) |
| 14 |
|
true1 |
true 1 |
| 15 |
|
trueeq |
a = 1 -> (true a <-> true 1) |
| 16 |
14, 15 |
mpbiri |
a = 1 -> true a |
| 17 |
13, 16 |
syl |
a = 1 -> (true a -> true b <-> b != 0) |
| 18 |
|
leeq1 |
a = 1 -> (a <= b <-> 1 <= b) |
| 19 |
17, 18 |
imeqd |
a = 1 -> ((true a -> true b) -> a <= b <-> b != 0 -> 1 <= b) |
| 20 |
11, 19 |
mpbiri |
a = 1 -> (true a -> true b) -> a <= b |
| 21 |
9, 20 |
ax_mp |
a = 0 \/ a = 1 -> (true a -> true b) -> a <= b |
| 22 |
3, 21 |
sylbi |
bool a -> (true a -> true b) -> a <= b |
| 23 |
2, 22 |
ibid |
bool a -> (a <= b <-> true a -> true 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,
add0,
addS)