| Step |
Hyp |
Ref |
Expression |
| 1 |
|
wlkd.p |
|- ( ph -> P e. Word _V ) |
| 2 |
|
wlkd.f |
|- ( ph -> F e. Word _V ) |
| 3 |
|
wlkd.l |
|- ( ph -> ( # ` P ) = ( ( # ` F ) + 1 ) ) |
| 4 |
|
wlkd.e |
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) |
| 5 |
|
fzo0end |
|- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) ) |
| 6 |
|
fveq2 |
|- ( k = ( ( # ` F ) - 1 ) -> ( P ` k ) = ( P ` ( ( # ` F ) - 1 ) ) ) |
| 7 |
|
fvoveq1 |
|- ( k = ( ( # ` F ) - 1 ) -> ( P ` ( k + 1 ) ) = ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) ) |
| 8 |
6 7
|
preq12d |
|- ( k = ( ( # ` F ) - 1 ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } ) |
| 9 |
|
2fveq3 |
|- ( k = ( ( # ` F ) - 1 ) -> ( I ` ( F ` k ) ) = ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) |
| 10 |
8 9
|
sseq12d |
|- ( k = ( ( # ` F ) - 1 ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } C_ ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 11 |
10
|
rspcv |
|- ( ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } C_ ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 12 |
5 11
|
syl |
|- ( ( # ` F ) e. NN -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } C_ ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 13 |
|
fvex |
|- ( P ` ( ( # ` F ) - 1 ) ) e. _V |
| 14 |
|
fvex |
|- ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) e. _V |
| 15 |
13 14
|
prss |
|- ( ( ( P ` ( ( # ` F ) - 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) /\ ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) <-> { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } C_ ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) |
| 16 |
|
nncn |
|- ( ( # ` F ) e. NN -> ( # ` F ) e. CC ) |
| 17 |
|
npcan1 |
|- ( ( # ` F ) e. CC -> ( ( ( # ` F ) - 1 ) + 1 ) = ( # ` F ) ) |
| 18 |
16 17
|
syl |
|- ( ( # ` F ) e. NN -> ( ( ( # ` F ) - 1 ) + 1 ) = ( # ` F ) ) |
| 19 |
18
|
fveq2d |
|- ( ( # ` F ) e. NN -> ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) = ( P ` ( # ` F ) ) ) |
| 20 |
19
|
eleq1d |
|- ( ( # ` F ) e. NN -> ( ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) <-> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 21 |
20
|
biimpd |
|- ( ( # ` F ) e. NN -> ( ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 22 |
21
|
adantld |
|- ( ( # ` F ) e. NN -> ( ( ( P ` ( ( # ` F ) - 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) /\ ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 23 |
15 22
|
biimtrrid |
|- ( ( # ` F ) e. NN -> ( { ( P ` ( ( # ` F ) - 1 ) ) , ( P ` ( ( ( # ` F ) - 1 ) + 1 ) ) } C_ ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 24 |
12 23
|
syld |
|- ( ( # ` F ) e. NN -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 25 |
4 24
|
syl5com |
|- ( ph -> ( ( # ` F ) e. NN -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) ) |
| 26 |
|
fvex |
|- ( P ` k ) e. _V |
| 27 |
|
fvex |
|- ( P ` ( k + 1 ) ) e. _V |
| 28 |
26 27
|
prss |
|- ( ( ( P ` k ) e. ( I ` ( F ` k ) ) /\ ( P ` ( k + 1 ) ) e. ( I ` ( F ` k ) ) ) <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) |
| 29 |
|
simpl |
|- ( ( ( P ` k ) e. ( I ` ( F ` k ) ) /\ ( P ` ( k + 1 ) ) e. ( I ` ( F ` k ) ) ) -> ( P ` k ) e. ( I ` ( F ` k ) ) ) |
| 30 |
28 29
|
sylbir |
|- ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> ( P ` k ) e. ( I ` ( F ` k ) ) ) |
| 31 |
30
|
a1i |
|- ( ( ph /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> ( P ` k ) e. ( I ` ( F ` k ) ) ) ) |
| 32 |
31
|
ralimdva |
|- ( ph -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) e. ( I ` ( F ` k ) ) ) ) |
| 33 |
4 32
|
mpd |
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) e. ( I ` ( F ` k ) ) ) |
| 34 |
25 33
|
jca |
|- ( ph -> ( ( ( # ` F ) e. NN -> ( P ` ( # ` F ) ) e. ( I ` ( F ` ( ( # ` F ) - 1 ) ) ) ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) e. ( I ` ( F ` k ) ) ) ) |