theorem appendcan2 (a b c: nat): $ a ++ c = b ++ c <-> a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(rev (a ++ c) = rev (b ++ c) <-> a ++ c = b ++ c) -> (rev (a ++ c) = rev (b ++ c) <-> a = b) -> (a ++ c = b ++ c <-> a = b) |
| 2 |
|
revinj |
rev (a ++ c) = rev (b ++ c) <-> a ++ c = b ++ c |
| 3 |
1, 2 |
ax_mp |
(rev (a ++ c) = rev (b ++ c) <-> a = b) -> (a ++ c = b ++ c <-> a = b) |
| 4 |
|
bitr |
(rev (a ++ c) = rev (b ++ c) <-> rev c ++ rev a = rev c ++ rev b) -> (rev c ++ rev a = rev c ++ rev b <-> a = b) -> (rev (a ++ c) = rev (b ++ c) <-> a = b) |
| 5 |
|
eqeq |
rev (a ++ c) = rev c ++ rev a -> rev (b ++ c) = rev c ++ rev b -> (rev (a ++ c) = rev (b ++ c) <-> rev c ++ rev a = rev c ++ rev b) |
| 6 |
|
revappend |
rev (a ++ c) = rev c ++ rev a |
| 7 |
5, 6 |
ax_mp |
rev (b ++ c) = rev c ++ rev b -> (rev (a ++ c) = rev (b ++ c) <-> rev c ++ rev a = rev c ++ rev b) |
| 8 |
|
revappend |
rev (b ++ c) = rev c ++ rev b |
| 9 |
7, 8 |
ax_mp |
rev (a ++ c) = rev (b ++ c) <-> rev c ++ rev a = rev c ++ rev b |
| 10 |
4, 9 |
ax_mp |
(rev c ++ rev a = rev c ++ rev b <-> a = b) -> (rev (a ++ c) = rev (b ++ c) <-> a = b) |
| 11 |
|
bitr |
(rev c ++ rev a = rev c ++ rev b <-> rev a = rev b) -> (rev a = rev b <-> a = b) -> (rev c ++ rev a = rev c ++ rev b <-> a = b) |
| 12 |
|
appendcan1 |
rev c ++ rev a = rev c ++ rev b <-> rev a = rev b |
| 13 |
11, 12 |
ax_mp |
(rev a = rev b <-> a = b) -> (rev c ++ rev a = rev c ++ rev b <-> a = b) |
| 14 |
|
revinj |
rev a = rev b <-> a = b |
| 15 |
13, 14 |
ax_mp |
rev c ++ rev a = rev c ++ rev b <-> a = b |
| 16 |
10, 15 |
ax_mp |
rev (a ++ c) = rev (b ++ c) <-> a = b |
| 17 |
3, 16 |
ax_mp |
a ++ c = b ++ c <-> 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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)