theorem zsubadd (a b c: nat): $ a -Z b +Z c = a -Z (b -Z c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr3 |
a +Z c -Z b = a -Z b +Z c -> a +Z c -Z b = a -Z (b -Z c) -> a -Z b +Z c = a -Z (b -Z c) |
| 2 |
|
zaddsub |
a +Z c -Z b = a -Z b +Z c |
| 3 |
1, 2 |
ax_mp |
a +Z c -Z b = a -Z (b -Z c) -> a -Z b +Z c = a -Z (b -Z c) |
| 4 |
|
eqtr4 |
a +Z c -Z b = a +Z (c -Z b) -> a -Z (b -Z c) = a +Z (c -Z b) -> a +Z c -Z b = a -Z (b -Z c) |
| 5 |
|
zaddsubass |
a +Z c -Z b = a +Z (c -Z b) |
| 6 |
4, 5 |
ax_mp |
a -Z (b -Z c) = a +Z (c -Z b) -> a +Z c -Z b = a -Z (b -Z c) |
| 7 |
|
zsubsub2 |
a -Z (b -Z c) = a +Z (c -Z b) |
| 8 |
6, 7 |
ax_mp |
a +Z c -Z b = a -Z (b -Z c) |
| 9 |
3, 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)