theorem ltadd2 (a b c: nat): $ b < c <-> a + b < a + c $;
Step | Hyp | Ref | Expression |
1 |
|
bitr |
(b < c <-> b + a < c + a) -> (b + a < c + a <-> a + b < a + c) -> (b < c <-> a + b < a + c) |
2 |
|
ltadd1 |
b < c <-> b + a < c + a |
3 |
1, 2 |
ax_mp |
(b + a < c + a <-> a + b < a + c) -> (b < c <-> a + b < a + c) |
4 |
|
lteq |
b + a = a + b -> c + a = a + c -> (b + a < c + a <-> a + b < a + c) |
5 |
|
addcom |
b + a = a + b |
6 |
4, 5 |
ax_mp |
c + a = a + c -> (b + a < c + a <-> a + b < a + c) |
7 |
|
addcom |
c + a = a + c |
8 |
6, 7 |
ax_mp |
b + a < c + a <-> a + b < a + c |
9 |
3, 8 |
ax_mp |
b < c <-> a + b < a + c |
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,
add0,
addS)