Step |
Hyp |
Ref |
Expression |
1 |
|
tocycval.1 |
|- C = ( toCyc ` D ) |
2 |
|
tocycfv.d |
|- ( ph -> D e. V ) |
3 |
|
tocycfv.w |
|- ( ph -> W e. Word D ) |
4 |
|
tocycfv.1 |
|- ( ph -> W : dom W -1-1-> D ) |
5 |
1 2 3 4
|
tocycfv |
|- ( ph -> ( C ` W ) = ( ( _I |` ( D \ ran W ) ) u. ( ( W cyclShift 1 ) o. `' W ) ) ) |
6 |
5
|
reseq1d |
|- ( ph -> ( ( C ` W ) |` ( D \ ran W ) ) = ( ( ( _I |` ( D \ ran W ) ) u. ( ( W cyclShift 1 ) o. `' W ) ) |` ( D \ ran W ) ) ) |
7 |
|
fnresi |
|- ( _I |` ( D \ ran W ) ) Fn ( D \ ran W ) |
8 |
7
|
a1i |
|- ( ph -> ( _I |` ( D \ ran W ) ) Fn ( D \ ran W ) ) |
9 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
10 |
|
cshwfn |
|- ( ( W e. Word D /\ 1 e. ZZ ) -> ( W cyclShift 1 ) Fn ( 0 ..^ ( # ` W ) ) ) |
11 |
3 9 10
|
syl2anc |
|- ( ph -> ( W cyclShift 1 ) Fn ( 0 ..^ ( # ` W ) ) ) |
12 |
|
f1f1orn |
|- ( W : dom W -1-1-> D -> W : dom W -1-1-onto-> ran W ) |
13 |
|
f1ocnv |
|- ( W : dom W -1-1-onto-> ran W -> `' W : ran W -1-1-onto-> dom W ) |
14 |
|
f1ofn |
|- ( `' W : ran W -1-1-onto-> dom W -> `' W Fn ran W ) |
15 |
4 12 13 14
|
4syl |
|- ( ph -> `' W Fn ran W ) |
16 |
|
dfdm4 |
|- dom W = ran `' W |
17 |
|
wrddm |
|- ( W e. Word D -> dom W = ( 0 ..^ ( # ` W ) ) ) |
18 |
3 17
|
syl |
|- ( ph -> dom W = ( 0 ..^ ( # ` W ) ) ) |
19 |
|
ssidd |
|- ( ph -> ( 0 ..^ ( # ` W ) ) C_ ( 0 ..^ ( # ` W ) ) ) |
20 |
18 19
|
eqsstrd |
|- ( ph -> dom W C_ ( 0 ..^ ( # ` W ) ) ) |
21 |
16 20
|
eqsstrrid |
|- ( ph -> ran `' W C_ ( 0 ..^ ( # ` W ) ) ) |
22 |
|
fnco |
|- ( ( ( W cyclShift 1 ) Fn ( 0 ..^ ( # ` W ) ) /\ `' W Fn ran W /\ ran `' W C_ ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift 1 ) o. `' W ) Fn ran W ) |
23 |
11 15 21 22
|
syl3anc |
|- ( ph -> ( ( W cyclShift 1 ) o. `' W ) Fn ran W ) |
24 |
|
disjdifr |
|- ( ( D \ ran W ) i^i ran W ) = (/) |
25 |
24
|
a1i |
|- ( ph -> ( ( D \ ran W ) i^i ran W ) = (/) ) |
26 |
|
fnunres1 |
|- ( ( ( _I |` ( D \ ran W ) ) Fn ( D \ ran W ) /\ ( ( W cyclShift 1 ) o. `' W ) Fn ran W /\ ( ( D \ ran W ) i^i ran W ) = (/) ) -> ( ( ( _I |` ( D \ ran W ) ) u. ( ( W cyclShift 1 ) o. `' W ) ) |` ( D \ ran W ) ) = ( _I |` ( D \ ran W ) ) ) |
27 |
8 23 25 26
|
syl3anc |
|- ( ph -> ( ( ( _I |` ( D \ ran W ) ) u. ( ( W cyclShift 1 ) o. `' W ) ) |` ( D \ ran W ) ) = ( _I |` ( D \ ran W ) ) ) |
28 |
6 27
|
eqtrd |
|- ( ph -> ( ( C ` W ) |` ( D \ ran W ) ) = ( _I |` ( D \ ran W ) ) ) |