Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> E : dom E -1-1-> R ) |
2 |
|
simp1 |
|- ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> f e. Word dom E ) |
3 |
2
|
adantr |
|- ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> f e. Word dom E ) |
4 |
1 3
|
anim12i |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( E : dom E -1-1-> R /\ f e. Word dom E ) ) |
5 |
|
simp3 |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> 2 <_ ( # ` P ) ) |
6 |
|
simpl2 |
|- ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> P : ( 0 ... ( # ` f ) ) --> V ) |
7 |
5 6
|
anim12ci |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( P : ( 0 ... ( # ` f ) ) --> V /\ 2 <_ ( # ` P ) ) ) |
8 |
|
simp3 |
|- ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) |
9 |
8
|
anim1i |
|- ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) |
10 |
9
|
adantl |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) |
11 |
|
clwlkclwwlklem2 |
|- ( ( ( E : dom E -1-1-> R /\ f e. Word dom E ) /\ ( P : ( 0 ... ( # ` f ) ) --> V /\ 2 <_ ( # ` P ) ) /\ ( A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) |
12 |
4 7 10 11
|
syl3anc |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) |
13 |
|
lencl |
|- ( P e. Word V -> ( # ` P ) e. NN0 ) |
14 |
|
lencl |
|- ( f e. Word dom E -> ( # ` f ) e. NN0 ) |
15 |
|
ffz0hash |
|- ( ( ( # ` f ) e. NN0 /\ P : ( 0 ... ( # ` f ) ) --> V ) -> ( # ` P ) = ( ( # ` f ) + 1 ) ) |
16 |
|
oveq1 |
|- ( ( # ` P ) = ( ( # ` f ) + 1 ) -> ( ( # ` P ) - 1 ) = ( ( ( # ` f ) + 1 ) - 1 ) ) |
17 |
16
|
oveq1d |
|- ( ( # ` P ) = ( ( # ` f ) + 1 ) -> ( ( ( # ` P ) - 1 ) - 0 ) = ( ( ( ( # ` f ) + 1 ) - 1 ) - 0 ) ) |
18 |
|
nn0cn |
|- ( ( # ` f ) e. NN0 -> ( # ` f ) e. CC ) |
19 |
|
peano2cn |
|- ( ( # ` f ) e. CC -> ( ( # ` f ) + 1 ) e. CC ) |
20 |
|
peano2cnm |
|- ( ( ( # ` f ) + 1 ) e. CC -> ( ( ( # ` f ) + 1 ) - 1 ) e. CC ) |
21 |
18 19 20
|
3syl |
|- ( ( # ` f ) e. NN0 -> ( ( ( # ` f ) + 1 ) - 1 ) e. CC ) |
22 |
21
|
subid1d |
|- ( ( # ` f ) e. NN0 -> ( ( ( ( # ` f ) + 1 ) - 1 ) - 0 ) = ( ( ( # ` f ) + 1 ) - 1 ) ) |
23 |
|
1cnd |
|- ( ( # ` f ) e. NN0 -> 1 e. CC ) |
24 |
18 23
|
pncand |
|- ( ( # ` f ) e. NN0 -> ( ( ( # ` f ) + 1 ) - 1 ) = ( # ` f ) ) |
25 |
22 24
|
eqtrd |
|- ( ( # ` f ) e. NN0 -> ( ( ( ( # ` f ) + 1 ) - 1 ) - 0 ) = ( # ` f ) ) |
26 |
25
|
adantr |
|- ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) -> ( ( ( ( # ` f ) + 1 ) - 1 ) - 0 ) = ( # ` f ) ) |
27 |
17 26
|
sylan9eqr |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( ( # ` P ) - 1 ) - 0 ) = ( # ` f ) ) |
28 |
27
|
oveq1d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) = ( ( # ` f ) - 1 ) ) |
29 |
28
|
oveq2d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) = ( 0 ..^ ( ( # ` f ) - 1 ) ) ) |
30 |
29
|
raleqdv |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E <-> A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) ) |
31 |
|
oveq1 |
|- ( ( # ` P ) = ( ( # ` f ) + 1 ) -> ( ( # ` P ) - 2 ) = ( ( ( # ` f ) + 1 ) - 2 ) ) |
32 |
|
2cnd |
|- ( ( # ` f ) e. NN0 -> 2 e. CC ) |
33 |
18 32 23
|
subsub3d |
|- ( ( # ` f ) e. NN0 -> ( ( # ` f ) - ( 2 - 1 ) ) = ( ( ( # ` f ) + 1 ) - 2 ) ) |
34 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
35 |
34
|
a1i |
|- ( ( # ` f ) e. NN0 -> ( 2 - 1 ) = 1 ) |
36 |
35
|
oveq2d |
|- ( ( # ` f ) e. NN0 -> ( ( # ` f ) - ( 2 - 1 ) ) = ( ( # ` f ) - 1 ) ) |
37 |
33 36
|
eqtr3d |
|- ( ( # ` f ) e. NN0 -> ( ( ( # ` f ) + 1 ) - 2 ) = ( ( # ` f ) - 1 ) ) |
38 |
37
|
adantr |
|- ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) -> ( ( ( # ` f ) + 1 ) - 2 ) = ( ( # ` f ) - 1 ) ) |
39 |
31 38
|
sylan9eqr |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( # ` P ) - 2 ) = ( ( # ` f ) - 1 ) ) |
40 |
39
|
fveq2d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( P ` ( ( # ` P ) - 2 ) ) = ( P ` ( ( # ` f ) - 1 ) ) ) |
41 |
40
|
preq1d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } = { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } ) |
42 |
41
|
eleq1d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E <-> { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) |
43 |
30 42
|
anbi12d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) <-> ( A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) |
44 |
43
|
anbi2d |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
45 |
|
3anass |
|- ( ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) |
46 |
44 45
|
bitr4di |
|- ( ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) /\ ( # ` P ) = ( ( # ` f ) + 1 ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) |
47 |
46
|
expcom |
|- ( ( # ` P ) = ( ( # ` f ) + 1 ) -> ( ( ( # ` f ) e. NN0 /\ ( # ` P ) e. NN0 ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
48 |
47
|
expd |
|- ( ( # ` P ) = ( ( # ` f ) + 1 ) -> ( ( # ` f ) e. NN0 -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) ) |
49 |
15 48
|
syl |
|- ( ( ( # ` f ) e. NN0 /\ P : ( 0 ... ( # ` f ) ) --> V ) -> ( ( # ` f ) e. NN0 -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) ) |
50 |
49
|
ex |
|- ( ( # ` f ) e. NN0 -> ( P : ( 0 ... ( # ` f ) ) --> V -> ( ( # ` f ) e. NN0 -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) ) ) |
51 |
50
|
com23 |
|- ( ( # ` f ) e. NN0 -> ( ( # ` f ) e. NN0 -> ( P : ( 0 ... ( # ` f ) ) --> V -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) ) ) |
52 |
14 14 51
|
sylc |
|- ( f e. Word dom E -> ( P : ( 0 ... ( # ` f ) ) --> V -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) ) |
53 |
52
|
imp |
|- ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V ) -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
54 |
53
|
3adant3 |
|- ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
55 |
54
|
adantr |
|- ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( ( # ` P ) e. NN0 -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
56 |
13 55
|
syl5com |
|- ( P e. Word V -> ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
57 |
56
|
3ad2ant2 |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
58 |
57
|
imp |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ A. i e. ( 0 ..^ ( ( # ` f ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` f ) - 1 ) ) , ( P ` 0 ) } e. ran E ) ) ) |
59 |
12 58
|
mpbird |
|- ( ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) /\ ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) -> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) |
60 |
59
|
ex |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
61 |
60
|
exlimdv |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) -> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |
62 |
|
clwlkclwwlklem1 |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) -> E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) ) |
63 |
61 62
|
impbid |
|- ( ( E : dom E -1-1-> R /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0 ..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0 ..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) ) |