theorem pncan3 (a b: nat): $ a <= b -> a + (b - a) = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
addeq2 |
a + x - a = x -> a + (a + x - a) = a + x |
| 2 |
|
pncan2 |
a + x - a = x |
| 3 |
1, 2 |
ax_mp |
a + (a + x - a) = a + x |
| 4 |
|
subeq1 |
a + x = b -> a + x - a = b - a |
| 5 |
4 |
addeq2d |
a + x = b -> a + (a + x - a) = a + (b - a) |
| 6 |
|
id |
a + x = b -> a + x = b |
| 7 |
5, 6 |
eqeqd |
a + x = b -> (a + (a + x - a) = a + x <-> a + (b - a) = b) |
| 8 |
3, 7 |
mpbii |
a + x = b -> a + (b - a) = b |
| 9 |
8 |
eex |
E. x a + x = b -> a + (b - a) = b |
| 10 |
9 |
conv le |
a <= b -> a + (b - a) = 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
(peano2,
peano5,
addeq,
add0,
addS)