theorem zsubsub (a b c: nat): $ a -Z b -Z c = a -Z (b +Z c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr4 |
a -Z b -Z c = a +Z (-uZ b -Z c) -> a -Z (b +Z c) = a +Z (-uZ b -Z c) -> a -Z b -Z c = a -Z (b +Z c) |
| 2 |
|
zaddsubass |
a +Z -uZ b -Z c = a +Z (-uZ b -Z c) |
| 3 |
2 |
conv zsub |
a -Z b -Z c = a +Z (-uZ b -Z c) |
| 4 |
1, 3 |
ax_mp |
a -Z (b +Z c) = a +Z (-uZ b -Z c) -> a -Z b -Z c = a -Z (b +Z c) |
| 5 |
|
zaddeq2 |
-uZ (b +Z c) = -uZ b -Z c -> a +Z -uZ (b +Z c) = a +Z (-uZ b -Z c) |
| 6 |
5 |
conv zsub |
-uZ (b +Z c) = -uZ b -Z c -> a -Z (b +Z c) = a +Z (-uZ b -Z c) |
| 7 |
|
znegadd2 |
-uZ (b +Z c) = -uZ b -Z c |
| 8 |
6, 7 |
ax_mp |
a -Z (b +Z c) = a +Z (-uZ b -Z c) |
| 9 |
4, 8 |
ax_mp |
a -Z b -Z c = a -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)