Step |
Hyp |
Ref |
Expression |
1 |
|
revcl |
|- ( W e. Word A -> ( reverse ` W ) e. Word A ) |
2 |
|
revcl |
|- ( ( reverse ` W ) e. Word A -> ( reverse ` ( reverse ` W ) ) e. Word A ) |
3 |
|
wrdf |
|- ( ( reverse ` ( reverse ` W ) ) e. Word A -> ( reverse ` ( reverse ` W ) ) : ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) --> A ) |
4 |
|
ffn |
|- ( ( reverse ` ( reverse ` W ) ) : ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) --> A -> ( reverse ` ( reverse ` W ) ) Fn ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) ) |
5 |
1 2 3 4
|
4syl |
|- ( W e. Word A -> ( reverse ` ( reverse ` W ) ) Fn ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) ) |
6 |
|
revlen |
|- ( ( reverse ` W ) e. Word A -> ( # ` ( reverse ` ( reverse ` W ) ) ) = ( # ` ( reverse ` W ) ) ) |
7 |
1 6
|
syl |
|- ( W e. Word A -> ( # ` ( reverse ` ( reverse ` W ) ) ) = ( # ` ( reverse ` W ) ) ) |
8 |
|
revlen |
|- ( W e. Word A -> ( # ` ( reverse ` W ) ) = ( # ` W ) ) |
9 |
7 8
|
eqtrd |
|- ( W e. Word A -> ( # ` ( reverse ` ( reverse ` W ) ) ) = ( # ` W ) ) |
10 |
9
|
oveq2d |
|- ( W e. Word A -> ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) = ( 0 ..^ ( # ` W ) ) ) |
11 |
10
|
fneq2d |
|- ( W e. Word A -> ( ( reverse ` ( reverse ` W ) ) Fn ( 0 ..^ ( # ` ( reverse ` ( reverse ` W ) ) ) ) <-> ( reverse ` ( reverse ` W ) ) Fn ( 0 ..^ ( # ` W ) ) ) ) |
12 |
5 11
|
mpbid |
|- ( W e. Word A -> ( reverse ` ( reverse ` W ) ) Fn ( 0 ..^ ( # ` W ) ) ) |
13 |
|
wrdfn |
|- ( W e. Word A -> W Fn ( 0 ..^ ( # ` W ) ) ) |
14 |
|
simpr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> x e. ( 0 ..^ ( # ` W ) ) ) |
15 |
8
|
adantr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( # ` ( reverse ` W ) ) = ( # ` W ) ) |
16 |
15
|
oveq2d |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( 0 ..^ ( # ` ( reverse ` W ) ) ) = ( 0 ..^ ( # ` W ) ) ) |
17 |
14 16
|
eleqtrrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> x e. ( 0 ..^ ( # ` ( reverse ` W ) ) ) ) |
18 |
|
revfv |
|- ( ( ( reverse ` W ) e. Word A /\ x e. ( 0 ..^ ( # ` ( reverse ` W ) ) ) ) -> ( ( reverse ` ( reverse ` W ) ) ` x ) = ( ( reverse ` W ) ` ( ( ( # ` ( reverse ` W ) ) - 1 ) - x ) ) ) |
19 |
1 17 18
|
syl2an2r |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` ( reverse ` W ) ) ` x ) = ( ( reverse ` W ) ` ( ( ( # ` ( reverse ` W ) ) - 1 ) - x ) ) ) |
20 |
15
|
oveq1d |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( # ` ( reverse ` W ) ) - 1 ) = ( ( # ` W ) - 1 ) ) |
21 |
20
|
fvoveq1d |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` W ) ` ( ( ( # ` ( reverse ` W ) ) - 1 ) - x ) ) = ( ( reverse ` W ) ` ( ( ( # ` W ) - 1 ) - x ) ) ) |
22 |
|
lencl |
|- ( W e. Word A -> ( # ` W ) e. NN0 ) |
23 |
22
|
nn0zd |
|- ( W e. Word A -> ( # ` W ) e. ZZ ) |
24 |
|
fzoval |
|- ( ( # ` W ) e. ZZ -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) ) |
25 |
23 24
|
syl |
|- ( W e. Word A -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) ) |
26 |
25
|
eleq2d |
|- ( W e. Word A -> ( x e. ( 0 ..^ ( # ` W ) ) <-> x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) ) |
27 |
26
|
biimpa |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
28 |
|
fznn0sub2 |
|- ( x e. ( 0 ... ( ( # ` W ) - 1 ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
29 |
27 28
|
syl |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
30 |
25
|
adantr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) ) |
31 |
29 30
|
eleqtrrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ..^ ( # ` W ) ) ) |
32 |
|
revfv |
|- ( ( W e. Word A /\ ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` W ) ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - ( ( ( # ` W ) - 1 ) - x ) ) ) ) |
33 |
31 32
|
syldan |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` W ) ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - ( ( ( # ` W ) - 1 ) - x ) ) ) ) |
34 |
|
peano2zm |
|- ( ( # ` W ) e. ZZ -> ( ( # ` W ) - 1 ) e. ZZ ) |
35 |
23 34
|
syl |
|- ( W e. Word A -> ( ( # ` W ) - 1 ) e. ZZ ) |
36 |
35
|
zcnd |
|- ( W e. Word A -> ( ( # ` W ) - 1 ) e. CC ) |
37 |
|
elfzoelz |
|- ( x e. ( 0 ..^ ( # ` W ) ) -> x e. ZZ ) |
38 |
37
|
zcnd |
|- ( x e. ( 0 ..^ ( # ` W ) ) -> x e. CC ) |
39 |
|
nncan |
|- ( ( ( ( # ` W ) - 1 ) e. CC /\ x e. CC ) -> ( ( ( # ` W ) - 1 ) - ( ( ( # ` W ) - 1 ) - x ) ) = x ) |
40 |
36 38 39
|
syl2an |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - ( ( ( # ` W ) - 1 ) - x ) ) = x ) |
41 |
40
|
fveq2d |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( ( # ` W ) - 1 ) - ( ( ( # ` W ) - 1 ) - x ) ) ) = ( W ` x ) ) |
42 |
33 41
|
eqtrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` W ) ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` x ) ) |
43 |
21 42
|
eqtrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` W ) ` ( ( ( # ` ( reverse ` W ) ) - 1 ) - x ) ) = ( W ` x ) ) |
44 |
19 43
|
eqtrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( reverse ` ( reverse ` W ) ) ` x ) = ( W ` x ) ) |
45 |
12 13 44
|
eqfnfvd |
|- ( W e. Word A -> ( reverse ` ( reverse ` W ) ) = W ) |