theorem zltaddsub (a b c: nat): $ a +Z b a
Step | Hyp | Ref | Expression |
1 |
|
bitr2 |
(a <Z c -Z b <-> a +Z b <Z c -Z b +Z b) -> (a +Z b <Z c -Z b +Z b <-> a +Z b <Z c) -> (a +Z b <Z c <-> a <Z c -Z b) |
2 |
|
zltadd1 |
a <Z c -Z b <-> a +Z b <Z c -Z b +Z b |
3 |
1, 2 |
ax_mp |
(a +Z b <Z c -Z b +Z b <-> a +Z b <Z c) -> (a +Z b <Z c <-> a <Z c -Z b) |
4 |
|
zlteq2 |
c -Z b +Z b = c -> (a +Z b <Z c -Z b +Z b <-> a +Z b <Z c) |
5 |
|
znpcan |
c -Z b +Z b = c |
6 |
4, 5 |
ax_mp |
a +Z b <Z c -Z b +Z b <-> a +Z b <Z c |
7 |
3, 6 |
ax_mp |
a +Z b <Z c <-> a <Z c -Z 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,
muleq,
add0,
addS,
mul0,
mulS)