Step |
Hyp |
Ref |
Expression |
1 |
|
revval |
|- ( W e. Word A -> ( reverse ` W ) = ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ) |
2 |
1
|
fveq2d |
|- ( W e. Word A -> ( # ` ( reverse ` W ) ) = ( # ` ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ) ) |
3 |
|
wrdf |
|- ( W e. Word A -> W : ( 0 ..^ ( # ` W ) ) --> A ) |
4 |
3
|
adantr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> W : ( 0 ..^ ( # ` W ) ) --> A ) |
5 |
|
simpr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> x e. ( 0 ..^ ( # ` W ) ) ) |
6 |
|
lencl |
|- ( W e. Word A -> ( # ` W ) e. NN0 ) |
7 |
6
|
adantr |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( # ` W ) e. NN0 ) |
8 |
|
nn0z |
|- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ ) |
9 |
|
fzoval |
|- ( ( # ` W ) e. ZZ -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) ) |
10 |
7 8 9
|
3syl |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( 0 ..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) ) |
11 |
5 10
|
eleqtrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
12 |
|
fznn0sub2 |
|- ( x e. ( 0 ... ( ( # ` W ) - 1 ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
13 |
11 12
|
syl |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) ) |
14 |
13 10
|
eleqtrrd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ..^ ( # ` W ) ) ) |
15 |
4 14
|
ffvelrnd |
|- ( ( W e. Word A /\ x e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) e. A ) |
16 |
15
|
fmpttd |
|- ( W e. Word A -> ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) : ( 0 ..^ ( # ` W ) ) --> A ) |
17 |
|
ffn |
|- ( ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) : ( 0 ..^ ( # ` W ) ) --> A -> ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) Fn ( 0 ..^ ( # ` W ) ) ) |
18 |
|
hashfn |
|- ( ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) Fn ( 0 ..^ ( # ` W ) ) -> ( # ` ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ) = ( # ` ( 0 ..^ ( # ` W ) ) ) ) |
19 |
16 17 18
|
3syl |
|- ( W e. Word A -> ( # ` ( x e. ( 0 ..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ) = ( # ` ( 0 ..^ ( # ` W ) ) ) ) |
20 |
|
hashfzo0 |
|- ( ( # ` W ) e. NN0 -> ( # ` ( 0 ..^ ( # ` W ) ) ) = ( # ` W ) ) |
21 |
6 20
|
syl |
|- ( W e. Word A -> ( # ` ( 0 ..^ ( # ` W ) ) ) = ( # ` W ) ) |
22 |
2 19 21
|
3eqtrd |
|- ( W e. Word A -> ( # ` ( reverse ` W ) ) = ( # ` W ) ) |