theorem zleadd1 (a b c: nat): $ a <=Z b <-> a +Z c <=Z b +Z c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
noteq |
(b <Z a <-> b +Z c <Z a +Z c) -> (~b <Z a <-> ~b +Z c <Z a +Z c) |
| 2 |
1 |
conv zle |
(b <Z a <-> b +Z c <Z a +Z c) -> (a <=Z b <-> a +Z c <=Z b +Z c) |
| 3 |
|
zltadd1 |
b <Z a <-> b +Z c <Z a +Z c |
| 4 |
2, 3 |
ax_mp |
a <=Z b <-> a +Z c <=Z b +Z 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_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)