Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn1.w |
|- W = ( N ClWWalksN G ) |
2 |
|
simpl |
|- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> k e. ( 0 ... N ) ) |
3 |
2
|
anim1i |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) ) |
4 |
3
|
adantr |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) ) |
5 |
|
simpr |
|- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> X = ( x cyclShift k ) ) |
6 |
5
|
adantr |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> X = ( x cyclShift k ) ) |
7 |
6
|
anim1i |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> ( X = ( x cyclShift k ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) ) |
8 |
1
|
eleclclwwlknlem1 |
|- ( ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) -> ( ( X = ( x cyclShift k ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) ) |
9 |
4 7 8
|
sylc |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) |
10 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
11 |
10
|
clwwlknbp |
|- ( x e. ( N ClWWalksN G ) -> ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) ) |
12 |
11 1
|
eleq2s |
|- ( x e. W -> ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) ) |
13 |
|
fznn0sub2 |
|- ( k e. ( 0 ... N ) -> ( N - k ) e. ( 0 ... N ) ) |
14 |
|
oveq1 |
|- ( ( # ` x ) = N -> ( ( # ` x ) - k ) = ( N - k ) ) |
15 |
14
|
eleq1d |
|- ( ( # ` x ) = N -> ( ( ( # ` x ) - k ) e. ( 0 ... N ) <-> ( N - k ) e. ( 0 ... N ) ) ) |
16 |
13 15
|
syl5ibr |
|- ( ( # ` x ) = N -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
17 |
16
|
adantl |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
18 |
12 17
|
syl |
|- ( x e. W -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
19 |
18
|
adantl |
|- ( ( X e. W /\ x e. W ) -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
20 |
19
|
com12 |
|- ( k e. ( 0 ... N ) -> ( ( X e. W /\ x e. W ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
21 |
20
|
adantr |
|- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( ( X e. W /\ x e. W ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) ) |
22 |
21
|
imp |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) |
23 |
22
|
adantr |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) |
24 |
|
simpr |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( X e. W /\ x e. W ) ) |
25 |
24
|
ancomd |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x e. W /\ X e. W ) ) |
26 |
25
|
adantr |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( x e. W /\ X e. W ) ) |
27 |
23 26
|
jca |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( ( ( # ` x ) - k ) e. ( 0 ... N ) /\ ( x e. W /\ X e. W ) ) ) |
28 |
|
simpll |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> x e. Word ( Vtx ` G ) ) |
29 |
|
oveq2 |
|- ( N = ( # ` x ) -> ( 0 ... N ) = ( 0 ... ( # ` x ) ) ) |
30 |
29
|
eleq2d |
|- ( N = ( # ` x ) -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) ) |
31 |
30
|
eqcoms |
|- ( ( # ` x ) = N -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) ) |
32 |
31
|
adantl |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) ) |
33 |
32
|
biimpa |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> k e. ( 0 ... ( # ` x ) ) ) |
34 |
28 33
|
jca |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) |
35 |
34
|
ex |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) ) |
36 |
12 35
|
syl |
|- ( x e. W -> ( k e. ( 0 ... N ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) ) |
37 |
36
|
adantl |
|- ( ( X e. W /\ x e. W ) -> ( k e. ( 0 ... N ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) ) |
38 |
37
|
com12 |
|- ( k e. ( 0 ... N ) -> ( ( X e. W /\ x e. W ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) ) |
39 |
38
|
adantr |
|- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( ( X e. W /\ x e. W ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) ) |
40 |
39
|
imp |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) |
41 |
5
|
eqcomd |
|- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( x cyclShift k ) = X ) |
42 |
41
|
adantr |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x cyclShift k ) = X ) |
43 |
|
oveq1 |
|- ( X = ( x cyclShift k ) -> ( X cyclShift ( ( # ` x ) - k ) ) = ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) ) |
44 |
43
|
eqcoms |
|- ( ( x cyclShift k ) = X -> ( X cyclShift ( ( # ` x ) - k ) ) = ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) ) |
45 |
|
elfzelz |
|- ( k e. ( 0 ... ( # ` x ) ) -> k e. ZZ ) |
46 |
|
2cshwid |
|- ( ( x e. Word ( Vtx ` G ) /\ k e. ZZ ) -> ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) = x ) |
47 |
45 46
|
sylan2 |
|- ( ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) -> ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) = x ) |
48 |
44 47
|
sylan9eqr |
|- ( ( ( x e. Word ( Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) /\ ( x cyclShift k ) = X ) -> ( X cyclShift ( ( # ` x ) - k ) ) = x ) |
49 |
40 42 48
|
syl2anc |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( X cyclShift ( ( # ` x ) - k ) ) = x ) |
50 |
49
|
eqcomd |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> x = ( X cyclShift ( ( # ` x ) - k ) ) ) |
51 |
50
|
anim1i |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( x = ( X cyclShift ( ( # ` x ) - k ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) ) |
52 |
1
|
eleclclwwlknlem1 |
|- ( ( ( ( # ` x ) - k ) e. ( 0 ... N ) /\ ( x e. W /\ X e. W ) ) -> ( ( x = ( X cyclShift ( ( # ` x ) - k ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) ) |
53 |
27 51 52
|
sylc |
|- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) |
54 |
9 53
|
impbida |
|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( E. m e. ( 0 ... N ) Y = ( x cyclShift m ) <-> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) ) |