theorem ltaddsub (a b c: nat): $ a + b < c <-> a < c - b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(a + b < c <-> ~c <= a + b) -> (a < c - b <-> ~c <= a + b) -> (a + b < c <-> a < c - b) |
| 2 |
|
ltnle |
a + b < c <-> ~c <= a + b |
| 3 |
1, 2 |
ax_mp |
(a < c - b <-> ~c <= a + b) -> (a + b < c <-> a < c - b) |
| 4 |
|
bitr |
(a < c - b <-> ~c - b <= a) -> (~c - b <= a <-> ~c <= a + b) -> (a < c - b <-> ~c <= a + b) |
| 5 |
|
ltnle |
a < c - b <-> ~c - b <= a |
| 6 |
4, 5 |
ax_mp |
(~c - b <= a <-> ~c <= a + b) -> (a < c - b <-> ~c <= a + b) |
| 7 |
|
lesubadd |
c - b <= a <-> c <= a + b |
| 8 |
7 |
noteqi |
~c - b <= a <-> ~c <= a + b |
| 9 |
6, 8 |
ax_mp |
a < c - b <-> ~c <= a + b |
| 10 |
3, 9 |
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_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)