Theorem appendinj1 ≪ | index | src | ≫ 

theorem appendinj1 (l1 l2 r1 r2: nat):
  $ len l1 = len r1 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2) $;
StepHypRefExpression
1 eqeq 
take (l1 ++ l2) (len l1) = l1 -> take (r1 ++ r2) (len r1) = r1 -> (take (l1 ++ l2) (len l1) = take (r1 ++ r2) (len r1) <-> l1 = r1)
2 eqtr 
take (l1 ++ l2) (len l1) = take l1 (len l1) -> take l1 (len l1) = l1 -> take (l1 ++ l2) (len l1) = l1
3 takeappend1 
len l1 <= len l1 -> take (l1 ++ l2) (len l1) = take l1 (len l1)
4 leid 
len l1 <= len l1
5 3, 4 ax_mp 
take (l1 ++ l2) (len l1) = take l1 (len l1)
6 2, 5 ax_mp 
take l1 (len l1) = l1 -> take (l1 ++ l2) (len l1) = l1
7 takeall 
len l1 <= len l1 -> take l1 (len l1) = l1
8 7, 4 ax_mp 
take l1 (len l1) = l1
9 6, 8 ax_mp 
take (l1 ++ l2) (len l1) = l1
10 1, 9 ax_mp 
take (r1 ++ r2) (len r1) = r1 -> (take (l1 ++ l2) (len l1) = take (r1 ++ r2) (len r1) <-> l1 = r1)
11 eqtr 
take (r1 ++ r2) (len r1) = take r1 (len r1) -> take r1 (len r1) = r1 -> take (r1 ++ r2) (len r1) = r1
12 takeappend1 
len r1 <= len r1 -> take (r1 ++ r2) (len r1) = take r1 (len r1)
13 leid 
len r1 <= len r1
14 12, 13 ax_mp 
take (r1 ++ r2) (len r1) = take r1 (len r1)
15 11, 14 ax_mp 
take r1 (len r1) = r1 -> take (r1 ++ r2) (len r1) = r1
16 takeall 
len r1 <= len r1 -> take r1 (len r1) = r1
17 16, 13 ax_mp 
take r1 (len r1) = r1
18 15, 17 ax_mp 
take (r1 ++ r2) (len r1) = r1
19 10, 18 ax_mp 
take (l1 ++ l2) (len l1) = take (r1 ++ r2) (len r1) <-> l1 = r1
20 anr 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l1 ++ l2 = r1 ++ r2
21 anl 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> len l1 = len r1
22 20, 21 takeeqd 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> take (l1 ++ l2) (len l1) = take (r1 ++ r2) (len r1)
23 19, 22 sylib 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l1 = r1
24 appendcan1 
l1 ++ l2 = l1 ++ r2 <-> l2 = r2
25 23 appendeq1d 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l1 ++ r2 = r1 ++ r2
26 20, 25 eqtr4d 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l1 ++ l2 = l1 ++ r2
27 24, 26 sylib 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l2 = r2
28 23, 27 iand 
len l1 = len r1 /\ l1 ++ l2 = r1 ++ r2 -> l1 = r1 /\ l2 = r2
29 anrl 
len l1 = len r1 /\ (l1 = r1 /\ l2 = r2) -> l1 = r1
30 anrr 
len l1 = len r1 /\ (l1 = r1 /\ l2 = r2) -> l2 = r2
31 29, 30 appendeqd 
len l1 = len r1 /\ (l1 = r1 /\ l2 = r2) -> l1 ++ l2 = r1 ++ r2
32 28, 31 ibida 
len l1 = len r1 -> (l1 ++ l2 = r1 ++ r2 <-> l1 = r1 /\ l2 = r2)

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)