Step |
Hyp |
Ref |
Expression |
1 |
|
clwwlkwwlksb.v |
|- V = ( Vtx ` G ) |
2 |
|
fstwrdne |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W ` 0 ) e. V ) |
3 |
2
|
s1cld |
|- ( ( W e. Word V /\ W =/= (/) ) -> <" ( W ` 0 ) "> e. Word V ) |
4 |
|
ccatlen |
|- ( ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) -> ( # ` ( W ++ <" ( W ` 0 ) "> ) ) = ( ( # ` W ) + ( # ` <" ( W ` 0 ) "> ) ) ) |
5 |
3 4
|
syldan |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` ( W ++ <" ( W ` 0 ) "> ) ) = ( ( # ` W ) + ( # ` <" ( W ` 0 ) "> ) ) ) |
6 |
|
s1len |
|- ( # ` <" ( W ` 0 ) "> ) = 1 |
7 |
6
|
oveq2i |
|- ( ( # ` W ) + ( # ` <" ( W ` 0 ) "> ) ) = ( ( # ` W ) + 1 ) |
8 |
5 7
|
eqtrdi |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` ( W ++ <" ( W ` 0 ) "> ) ) = ( ( # ` W ) + 1 ) ) |
9 |
8
|
oveq1d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) = ( ( ( # ` W ) + 1 ) - 1 ) ) |
10 |
|
lencl |
|- ( W e. Word V -> ( # ` W ) e. NN0 ) |
11 |
10
|
nn0cnd |
|- ( W e. Word V -> ( # ` W ) e. CC ) |
12 |
11
|
adantr |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. CC ) |
13 |
|
1cnd |
|- ( ( W e. Word V /\ W =/= (/) ) -> 1 e. CC ) |
14 |
12 13 13
|
addsubd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( ( # ` W ) + 1 ) - 1 ) = ( ( ( # ` W ) - 1 ) + 1 ) ) |
15 |
9 14
|
eqtrd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) = ( ( ( # ` W ) - 1 ) + 1 ) ) |
16 |
15
|
oveq2d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) = ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) ) |
17 |
16
|
raleqdv |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
18 |
|
lennncl |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN ) |
19 |
|
nnm1nn0 |
|- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. NN0 ) |
20 |
18 19
|
syl |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. NN0 ) |
21 |
|
elnn0uz |
|- ( ( ( # ` W ) - 1 ) e. NN0 <-> ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) ) |
22 |
20 21
|
sylib |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) ) |
23 |
|
fzosplitsn |
|- ( ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) -> ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) = ( ( 0 ..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) } ) ) |
24 |
22 23
|
syl |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) = ( ( 0 ..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) } ) ) |
25 |
24
|
raleqdv |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( ( 0 ..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) } ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
26 |
|
ralunb |
|- ( A. i e. ( ( 0 ..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) } ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. { ( ( # ` W ) - 1 ) } { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
27 |
25 26
|
bitrdi |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. ( 0 ..^ ( ( ( # ` W ) - 1 ) + 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. { ( ( # ` W ) - 1 ) } { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
28 |
|
simpl |
|- ( ( W e. Word V /\ W =/= (/) ) -> W e. Word V ) |
29 |
10
|
nn0zd |
|- ( W e. Word V -> ( # ` W ) e. ZZ ) |
30 |
29
|
adantr |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. ZZ ) |
31 |
|
elfzom1elfzo |
|- ( ( ( # ` W ) e. ZZ /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> i e. ( 0 ..^ ( # ` W ) ) ) |
32 |
30 31
|
sylan |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> i e. ( 0 ..^ ( # ` W ) ) ) |
33 |
|
ccats1val1 |
|- ( ( W e. Word V /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` i ) = ( W ` i ) ) |
34 |
28 32 33
|
syl2an2r |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` i ) = ( W ` i ) ) |
35 |
|
elfzom1elp1fzo |
|- ( ( ( # ` W ) e. ZZ /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> ( i + 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
36 |
30 35
|
sylan |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> ( i + 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
37 |
|
ccats1val1 |
|- ( ( W e. Word V /\ ( i + 1 ) e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) = ( W ` ( i + 1 ) ) ) |
38 |
28 36 37
|
syl2an2r |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) = ( W ` ( i + 1 ) ) ) |
39 |
34 38
|
preq12d |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } = { ( W ` i ) , ( W ` ( i + 1 ) ) } ) |
40 |
39
|
eleq1d |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) ) -> ( { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
41 |
40
|
ralbidva |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
42 |
|
ovex |
|- ( ( # ` W ) - 1 ) e. _V |
43 |
|
fveq2 |
|- ( i = ( ( # ` W ) - 1 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` i ) = ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) ) |
44 |
|
fvoveq1 |
|- ( i = ( ( # ` W ) - 1 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) = ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) ) |
45 |
43 44
|
preq12d |
|- ( i = ( ( # ` W ) - 1 ) -> { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } = { ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) } ) |
46 |
45
|
eleq1d |
|- ( i = ( ( # ` W ) - 1 ) -> ( { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) |
47 |
42 46
|
ralsn |
|- ( A. i e. { ( ( # ` W ) - 1 ) } { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) |
48 |
|
fzo0end |
|- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
49 |
18 48
|
syl |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
50 |
|
ccats1val1 |
|- ( ( W e. Word V /\ ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
51 |
49 50
|
syldan |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
52 |
|
lsw |
|- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
53 |
52
|
adantr |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
54 |
51 53
|
eqtr4d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) ) |
55 |
|
npcan1 |
|- ( ( # ` W ) e. CC -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) ) |
56 |
11 55
|
syl |
|- ( W e. Word V -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) ) |
57 |
56
|
adantr |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) ) |
58 |
57
|
fveq2d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) = ( ( W ++ <" ( W ` 0 ) "> ) ` ( # ` W ) ) ) |
59 |
|
eqidd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) = ( # ` W ) ) |
60 |
|
ccats1val2 |
|- ( ( W e. Word V /\ ( W ` 0 ) e. V /\ ( # ` W ) = ( # ` W ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( # ` W ) ) = ( W ` 0 ) ) |
61 |
28 2 59 60
|
syl3anc |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( # ` W ) ) = ( W ` 0 ) ) |
62 |
58 61
|
eqtrd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) = ( W ` 0 ) ) |
63 |
54 62
|
preq12d |
|- ( ( W e. Word V /\ W =/= (/) ) -> { ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) } = { ( lastS ` W ) , ( W ` 0 ) } ) |
64 |
63
|
eleq1d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( { ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( # ` W ) - 1 ) ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( ( ( # ` W ) - 1 ) + 1 ) ) } e. ( Edg ` G ) <-> { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) |
65 |
47 64
|
syl5bb |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. { ( ( # ` W ) - 1 ) } { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) |
66 |
41 65
|
anbi12d |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. { ( ( # ` W ) - 1 ) } { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) <-> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) ) |
67 |
17 27 66
|
3bitrd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) ) |
68 |
28 3
|
jca |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) ) |
69 |
|
ccat0 |
|- ( ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) -> ( ( W ++ <" ( W ` 0 ) "> ) = (/) <-> ( W = (/) /\ <" ( W ` 0 ) "> = (/) ) ) ) |
70 |
|
simpl |
|- ( ( W = (/) /\ <" ( W ` 0 ) "> = (/) ) -> W = (/) ) |
71 |
69 70
|
syl6bi |
|- ( ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) -> ( ( W ++ <" ( W ` 0 ) "> ) = (/) -> W = (/) ) ) |
72 |
71
|
necon3d |
|- ( ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) -> ( W =/= (/) -> ( W ++ <" ( W ` 0 ) "> ) =/= (/) ) ) |
73 |
72
|
adantld |
|- ( ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) -> ( ( W e. Word V /\ W =/= (/) ) -> ( W ++ <" ( W ` 0 ) "> ) =/= (/) ) ) |
74 |
68 73
|
mpcom |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W ++ <" ( W ` 0 ) "> ) =/= (/) ) |
75 |
|
wrdv |
|- ( W e. Word V -> W e. Word _V ) |
76 |
|
s1cli |
|- <" ( W ` 0 ) "> e. Word _V |
77 |
|
ccatalpha |
|- ( ( W e. Word _V /\ <" ( W ` 0 ) "> e. Word _V ) -> ( ( W ++ <" ( W ` 0 ) "> ) e. Word V <-> ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) ) ) |
78 |
75 76 77
|
sylancl |
|- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) e. Word V <-> ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) ) ) |
79 |
78
|
adantr |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) e. Word V <-> ( W e. Word V /\ <" ( W ` 0 ) "> e. Word V ) ) ) |
80 |
28 3 79
|
mpbir2and |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W ++ <" ( W ` 0 ) "> ) e. Word V ) |
81 |
74 80
|
jca |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V ) ) |
82 |
|
eqid |
|- ( Edg ` G ) = ( Edg ` G ) |
83 |
1 82
|
iswwlks |
|- ( ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) <-> ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
84 |
|
df-3an |
|- ( ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) <-> ( ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V ) /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
85 |
83 84
|
bitri |
|- ( ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) <-> ( ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V ) /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
86 |
85
|
a1i |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) <-> ( ( ( W ++ <" ( W ` 0 ) "> ) =/= (/) /\ ( W ++ <" ( W ` 0 ) "> ) e. Word V ) /\ A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
87 |
81 86
|
mpbirand |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) <-> A. i e. ( 0 ..^ ( ( # ` ( W ++ <" ( W ` 0 ) "> ) ) - 1 ) ) { ( ( W ++ <" ( W ` 0 ) "> ) ` i ) , ( ( W ++ <" ( W ` 0 ) "> ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
88 |
1 82
|
isclwwlk |
|- ( W e. ( ClWWalks ` G ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) |
89 |
|
3anass |
|- ( ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) ) |
90 |
88 89
|
bitri |
|- ( W e. ( ClWWalks ` G ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) ) |
91 |
90
|
baib |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W e. ( ClWWalks ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. ( Edg ` G ) ) ) ) |
92 |
67 87 91
|
3bitr4rd |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( W e. ( ClWWalks ` G ) <-> ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) ) ) |