Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
|- W = ( N ClWWalksN G ) |
2 |
|
erclwwlkn.r |
|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) } |
3 |
1 2
|
eclclwwlkn1 |
|- ( U e. ( W /. .~ ) -> ( U e. ( W /. .~ ) <-> E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) ) |
4 |
|
rabeq |
|- ( W = ( N ClWWalksN G ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) |
5 |
1 4
|
mp1i |
|- ( ( N e. Prime /\ x e. W ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) |
6 |
|
prmnn |
|- ( N e. Prime -> N e. NN ) |
7 |
6
|
nnnn0d |
|- ( N e. Prime -> N e. NN0 ) |
8 |
1
|
eleq2i |
|- ( x e. W <-> x e. ( N ClWWalksN G ) ) |
9 |
8
|
biimpi |
|- ( x e. W -> x e. ( N ClWWalksN G ) ) |
10 |
|
clwwlknscsh |
|- ( ( N e. NN0 /\ x e. ( N ClWWalksN G ) ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) |
11 |
7 9 10
|
syl2an |
|- ( ( N e. Prime /\ x e. W ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) |
12 |
5 11
|
eqtrd |
|- ( ( N e. Prime /\ x e. W ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) |
13 |
12
|
eqeq2d |
|- ( ( N e. Prime /\ x e. W ) -> ( U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } <-> U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) ) |
14 |
|
simpll |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> x e. Word ( Vtx ` G ) ) |
15 |
|
elnnne0 |
|- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) ) |
16 |
|
eqeq1 |
|- ( N = ( # ` x ) -> ( N = 0 <-> ( # ` x ) = 0 ) ) |
17 |
16
|
eqcoms |
|- ( ( # ` x ) = N -> ( N = 0 <-> ( # ` x ) = 0 ) ) |
18 |
|
hasheq0 |
|- ( x e. Word ( Vtx ` G ) -> ( ( # ` x ) = 0 <-> x = (/) ) ) |
19 |
17 18
|
sylan9bbr |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( N = 0 <-> x = (/) ) ) |
20 |
19
|
necon3bid |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( N =/= 0 <-> x =/= (/) ) ) |
21 |
20
|
biimpcd |
|- ( N =/= 0 -> ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> x =/= (/) ) ) |
22 |
15 21
|
simplbiim |
|- ( N e. NN -> ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> x =/= (/) ) ) |
23 |
22
|
impcom |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> x =/= (/) ) |
24 |
|
simplr |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> ( # ` x ) = N ) |
25 |
24
|
eqcomd |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> N = ( # ` x ) ) |
26 |
14 23 25
|
3jca |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) |
27 |
26
|
ex |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( N e. NN -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) ) |
28 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
29 |
28
|
clwwlknbp |
|- ( x e. ( N ClWWalksN G ) -> ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) ) |
30 |
27 29
|
syl11 |
|- ( N e. NN -> ( x e. ( N ClWWalksN G ) -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) ) |
31 |
8 30
|
syl5bi |
|- ( N e. NN -> ( x e. W -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) ) |
32 |
6 31
|
syl |
|- ( N e. Prime -> ( x e. W -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) ) |
33 |
32
|
imp |
|- ( ( N e. Prime /\ x e. W ) -> ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) |
34 |
|
scshwfzeqfzo |
|- ( ( x e. Word ( Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) -> { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } ) |
35 |
33 34
|
syl |
|- ( ( N e. Prime /\ x e. W ) -> { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } ) |
36 |
35
|
eqeq2d |
|- ( ( N e. Prime /\ x e. W ) -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } <-> U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } ) ) |
37 |
|
oveq2 |
|- ( n = m -> ( x cyclShift n ) = ( x cyclShift m ) ) |
38 |
37
|
eqeq2d |
|- ( n = m -> ( y = ( x cyclShift n ) <-> y = ( x cyclShift m ) ) ) |
39 |
38
|
cbvrexvw |
|- ( E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) <-> E. m e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift m ) ) |
40 |
|
eqeq1 |
|- ( y = u -> ( y = ( x cyclShift m ) <-> u = ( x cyclShift m ) ) ) |
41 |
|
eqcom |
|- ( u = ( x cyclShift m ) <-> ( x cyclShift m ) = u ) |
42 |
40 41
|
bitrdi |
|- ( y = u -> ( y = ( x cyclShift m ) <-> ( x cyclShift m ) = u ) ) |
43 |
42
|
rexbidv |
|- ( y = u -> ( E. m e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift m ) <-> E. m e. ( 0 ..^ ( # ` x ) ) ( x cyclShift m ) = u ) ) |
44 |
39 43
|
syl5bb |
|- ( y = u -> ( E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) <-> E. m e. ( 0 ..^ ( # ` x ) ) ( x cyclShift m ) = u ) ) |
45 |
44
|
cbvrabv |
|- { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } = { u e. Word ( Vtx ` G ) | E. m e. ( 0 ..^ ( # ` x ) ) ( x cyclShift m ) = u } |
46 |
45
|
cshwshash |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) -> ( ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) \/ ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 ) ) |
47 |
46
|
adantr |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) /\ U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) -> ( ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) \/ ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 ) ) |
48 |
47
|
orcomd |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) /\ U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) -> ( ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 \/ ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) ) |
49 |
|
fveqeq2 |
|- ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 <-> ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 ) ) |
50 |
|
fveqeq2 |
|- ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = ( # ` x ) <-> ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) ) |
51 |
49 50
|
orbi12d |
|- ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) <-> ( ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 \/ ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) ) ) |
52 |
51
|
adantl |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) /\ U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) -> ( ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) <-> ( ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = 1 \/ ( # ` { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) ) ) |
53 |
48 52
|
mpbird |
|- ( ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) /\ U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) |
54 |
53
|
ex |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) |
55 |
54
|
ex |
|- ( x e. Word ( Vtx ` G ) -> ( ( # ` x ) e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) |
56 |
55
|
adantr |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( ( # ` x ) e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) |
57 |
|
eleq1 |
|- ( N = ( # ` x ) -> ( N e. Prime <-> ( # ` x ) e. Prime ) ) |
58 |
|
oveq2 |
|- ( N = ( # ` x ) -> ( 0 ..^ N ) = ( 0 ..^ ( # ` x ) ) ) |
59 |
58
|
rexeqdv |
|- ( N = ( # ` x ) -> ( E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) <-> E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) ) ) |
60 |
59
|
rabbidv |
|- ( N = ( # ` x ) -> { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) |
61 |
60
|
eqeq2d |
|- ( N = ( # ` x ) -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } <-> U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } ) ) |
62 |
|
eqeq2 |
|- ( N = ( # ` x ) -> ( ( # ` U ) = N <-> ( # ` U ) = ( # ` x ) ) ) |
63 |
62
|
orbi2d |
|- ( N = ( # ` x ) -> ( ( ( # ` U ) = 1 \/ ( # ` U ) = N ) <-> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) |
64 |
61 63
|
imbi12d |
|- ( N = ( # ` x ) -> ( ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) <-> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) |
65 |
57 64
|
imbi12d |
|- ( N = ( # ` x ) -> ( ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) <-> ( ( # ` x ) e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) ) |
66 |
65
|
eqcoms |
|- ( ( # ` x ) = N -> ( ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) <-> ( ( # ` x ) e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) ) |
67 |
66
|
adantl |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) <-> ( ( # ` x ) e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = ( # ` x ) ) ) ) ) ) |
68 |
56 67
|
mpbird |
|- ( ( x e. Word ( Vtx ` G ) /\ ( # ` x ) = N ) -> ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) ) |
69 |
29 68
|
syl |
|- ( x e. ( N ClWWalksN G ) -> ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) ) |
70 |
69 1
|
eleq2s |
|- ( x e. W -> ( N e. Prime -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) ) |
71 |
70
|
impcom |
|- ( ( N e. Prime /\ x e. W ) -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ..^ N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
72 |
36 71
|
sylbid |
|- ( ( N e. Prime /\ x e. W ) -> ( U = { y e. Word ( Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
73 |
13 72
|
sylbid |
|- ( ( N e. Prime /\ x e. W ) -> ( U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
74 |
73
|
rexlimdva |
|- ( N e. Prime -> ( E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
75 |
74
|
com12 |
|- ( E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( N e. Prime -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
76 |
3 75
|
syl6bi |
|- ( U e. ( W /. .~ ) -> ( U e. ( W /. .~ ) -> ( N e. Prime -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) ) |
77 |
76
|
pm2.43i |
|- ( U e. ( W /. .~ ) -> ( N e. Prime -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) ) |
78 |
77
|
impcom |
|- ( ( N e. Prime /\ U e. ( W /. .~ ) ) -> ( ( # ` U ) = 1 \/ ( # ` U ) = N ) ) |