theorem pnpcan1 (a b c: nat): $ a + b - (a + c) = b - c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
a + b - (a + c) = b + a - (c + a) -> b + a - (c + a) = b - c -> a + b - (a + c) = b - c |
| 2 |
|
subeq |
a + b = b + a -> a + c = c + a -> a + b - (a + c) = b + a - (c + a) |
| 3 |
|
addcom |
a + b = b + a |
| 4 |
2, 3 |
ax_mp |
a + c = c + a -> a + b - (a + c) = b + a - (c + a) |
| 5 |
|
addcom |
a + c = c + a |
| 6 |
4, 5 |
ax_mp |
a + b - (a + c) = b + a - (c + a) |
| 7 |
1, 6 |
ax_mp |
b + a - (c + a) = b - c -> a + b - (a + c) = b - c |
| 8 |
|
pnpcan2 |
b + a - (c + a) = b - c |
| 9 |
7, 8 |
ax_mp |
a + b - (a + c) = b - 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
(peano2,
peano5,
addeq,
add0,
addS)