theorem zpnpcan1 (a b c: nat): $ a +Z c -Z (b +Z c) = a -Z b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a +Z c -Z (b +Z c) = c +Z a -Z (c +Z b) -> c +Z a -Z (c +Z b) = a -Z b -> a +Z c -Z (b +Z c) = a -Z b |
| 2 |
|
zsubeq |
a +Z c = c +Z a -> b +Z c = c +Z b -> a +Z c -Z (b +Z c) = c +Z a -Z (c +Z b) |
| 3 |
|
zaddcom |
a +Z c = c +Z a |
| 4 |
2, 3 |
ax_mp |
b +Z c = c +Z b -> a +Z c -Z (b +Z c) = c +Z a -Z (c +Z b) |
| 5 |
|
zaddcom |
b +Z c = c +Z b |
| 6 |
4, 5 |
ax_mp |
a +Z c -Z (b +Z c) = c +Z a -Z (c +Z b) |
| 7 |
1, 6 |
ax_mp |
c +Z a -Z (c +Z b) = a -Z b -> a +Z c -Z (b +Z c) = a -Z b |
| 8 |
|
zpnpcan2 |
c +Z a -Z (c +Z b) = a -Z b |
| 9 |
7, 8 |
ax_mp |
a +Z c -Z (b +Z c) = a -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)