Step |
Hyp |
Ref |
Expression |
1 |
|
extwwlkfab.v |
|- V = ( Vtx ` G ) |
2 |
|
extwwlkfab.c |
|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( v ( ClWWalksNOn ` G ) n ) | ( w ` ( n - 2 ) ) = v } ) |
3 |
|
extwwlkfab.f |
|- F = ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) |
4 |
|
numclwwlk.t |
|- T = ( u e. ( X C N ) |-> <. ( u prefix ( N - 2 ) ) , ( u ` ( N - 1 ) ) >. ) |
5 |
1 2 3 4
|
numclwwlk1lem2f |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) --> ( F X. ( G NeighbVtx X ) ) ) |
6 |
1 2 3 4
|
numclwwlk1lem2fv |
|- ( p e. ( X C N ) -> ( T ` p ) = <. ( p prefix ( N - 2 ) ) , ( p ` ( N - 1 ) ) >. ) |
7 |
6
|
ad2antrl |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( T ` p ) = <. ( p prefix ( N - 2 ) ) , ( p ` ( N - 1 ) ) >. ) |
8 |
1 2 3 4
|
numclwwlk1lem2fv |
|- ( a e. ( X C N ) -> ( T ` a ) = <. ( a prefix ( N - 2 ) ) , ( a ` ( N - 1 ) ) >. ) |
9 |
8
|
ad2antll |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( T ` a ) = <. ( a prefix ( N - 2 ) ) , ( a ` ( N - 1 ) ) >. ) |
10 |
7 9
|
eqeq12d |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( T ` p ) = ( T ` a ) <-> <. ( p prefix ( N - 2 ) ) , ( p ` ( N - 1 ) ) >. = <. ( a prefix ( N - 2 ) ) , ( a ` ( N - 1 ) ) >. ) ) |
11 |
|
ovex |
|- ( p prefix ( N - 2 ) ) e. _V |
12 |
|
fvex |
|- ( p ` ( N - 1 ) ) e. _V |
13 |
11 12
|
opth |
|- ( <. ( p prefix ( N - 2 ) ) , ( p ` ( N - 1 ) ) >. = <. ( a prefix ( N - 2 ) ) , ( a ` ( N - 1 ) ) >. <-> ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) |
14 |
|
uzuzle23 |
|- ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) ) |
15 |
2
|
2clwwlkel |
|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( p e. ( X C N ) <-> ( p e. ( X ( ClWWalksNOn ` G ) N ) /\ ( p ` ( N - 2 ) ) = X ) ) ) |
16 |
|
isclwwlknon |
|- ( p e. ( X ( ClWWalksNOn ` G ) N ) <-> ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) ) |
17 |
16
|
anbi1i |
|- ( ( p e. ( X ( ClWWalksNOn ` G ) N ) /\ ( p ` ( N - 2 ) ) = X ) <-> ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) ) |
18 |
15 17
|
syl6bb |
|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( p e. ( X C N ) <-> ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) ) ) |
19 |
2
|
2clwwlkel |
|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( a e. ( X C N ) <-> ( a e. ( X ( ClWWalksNOn ` G ) N ) /\ ( a ` ( N - 2 ) ) = X ) ) ) |
20 |
|
isclwwlknon |
|- ( a e. ( X ( ClWWalksNOn ` G ) N ) <-> ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) ) |
21 |
20
|
anbi1i |
|- ( ( a e. ( X ( ClWWalksNOn ` G ) N ) /\ ( a ` ( N - 2 ) ) = X ) <-> ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) |
22 |
19 21
|
syl6bb |
|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( a e. ( X C N ) <-> ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) ) |
23 |
18 22
|
anbi12d |
|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( ( p e. ( X C N ) /\ a e. ( X C N ) ) <-> ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) ) ) |
24 |
14 23
|
sylan2 |
|- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( p e. ( X C N ) /\ a e. ( X C N ) ) <-> ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) ) ) |
25 |
24
|
3adant1 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( p e. ( X C N ) /\ a e. ( X C N ) ) <-> ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) ) ) |
26 |
1
|
clwwlknbp |
|- ( p e. ( N ClWWalksN G ) -> ( p e. Word V /\ ( # ` p ) = N ) ) |
27 |
26
|
adantr |
|- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) -> ( p e. Word V /\ ( # ` p ) = N ) ) |
28 |
27
|
adantr |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p e. Word V /\ ( # ` p ) = N ) ) |
29 |
|
simpr |
|- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) -> ( p ` 0 ) = X ) |
30 |
29
|
adantr |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p ` 0 ) = X ) |
31 |
|
simpr |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p ` ( N - 2 ) ) = X ) |
32 |
29
|
eqcomd |
|- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) -> X = ( p ` 0 ) ) |
33 |
32
|
adantr |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> X = ( p ` 0 ) ) |
34 |
31 33
|
eqtrd |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p ` ( N - 2 ) ) = ( p ` 0 ) ) |
35 |
28 30 34
|
jca32 |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) ) |
36 |
1
|
clwwlknbp |
|- ( a e. ( N ClWWalksN G ) -> ( a e. Word V /\ ( # ` a ) = N ) ) |
37 |
36
|
adantr |
|- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) -> ( a e. Word V /\ ( # ` a ) = N ) ) |
38 |
37
|
adantr |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> ( a e. Word V /\ ( # ` a ) = N ) ) |
39 |
|
simpr |
|- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) -> ( a ` 0 ) = X ) |
40 |
39
|
adantr |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> ( a ` 0 ) = X ) |
41 |
|
simpr |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> ( a ` ( N - 2 ) ) = X ) |
42 |
39
|
eqcomd |
|- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) -> X = ( a ` 0 ) ) |
43 |
42
|
adantr |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> X = ( a ` 0 ) ) |
44 |
41 43
|
eqtrd |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> ( a ` ( N - 2 ) ) = ( a ` 0 ) ) |
45 |
38 40 44
|
jca32 |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) |
46 |
|
eqtr3 |
|- ( ( ( # ` p ) = N /\ ( # ` a ) = N ) -> ( # ` p ) = ( # ` a ) ) |
47 |
46
|
expcom |
|- ( ( # ` a ) = N -> ( ( # ` p ) = N -> ( # ` p ) = ( # ` a ) ) ) |
48 |
47
|
ad2antlr |
|- ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( ( # ` p ) = N -> ( # ` p ) = ( # ` a ) ) ) |
49 |
48
|
com12 |
|- ( ( # ` p ) = N -> ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( # ` p ) = ( # ` a ) ) ) |
50 |
49
|
ad2antlr |
|- ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) -> ( ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) -> ( # ` p ) = ( # ` a ) ) ) |
51 |
50
|
imp |
|- ( ( ( ( p e. Word V /\ ( # ` p ) = N ) /\ ( ( p ` 0 ) = X /\ ( p ` ( N - 2 ) ) = ( p ` 0 ) ) ) /\ ( ( a e. Word V /\ ( # ` a ) = N ) /\ ( ( a ` 0 ) = X /\ ( a ` ( N - 2 ) ) = ( a ` 0 ) ) ) ) -> ( # ` p ) = ( # ` a ) ) |
52 |
35 45 51
|
syl2an |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( # ` p ) = ( # ` a ) ) |
53 |
52
|
3ad2ant2 |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( # ` p ) = ( # ` a ) ) |
54 |
27
|
simprd |
|- ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) -> ( # ` p ) = N ) |
55 |
54
|
adantr |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( # ` p ) = N ) |
56 |
55
|
eqcomd |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> N = ( # ` p ) ) |
57 |
56
|
adantr |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> N = ( # ` p ) ) |
58 |
57
|
oveq1d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( N - 2 ) = ( ( # ` p ) - 2 ) ) |
59 |
58
|
oveq2d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p prefix ( N - 2 ) ) = ( p prefix ( ( # ` p ) - 2 ) ) ) |
60 |
58
|
oveq2d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( a prefix ( N - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) ) |
61 |
59 60
|
eqeq12d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) <-> ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) ) ) |
62 |
61
|
biimpcd |
|- ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) -> ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) ) ) |
63 |
62
|
adantr |
|- ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) ) ) |
64 |
63
|
impcom |
|- ( ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) ) |
65 |
55
|
oveq1d |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( ( # ` p ) - 2 ) = ( N - 2 ) ) |
66 |
65
|
fveq2d |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p ` ( ( # ` p ) - 2 ) ) = ( p ` ( N - 2 ) ) ) |
67 |
66 31
|
eqtrd |
|- ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) -> ( p ` ( ( # ` p ) - 2 ) ) = X ) |
68 |
67
|
adantr |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = X ) |
69 |
41
|
eqcomd |
|- ( ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) -> X = ( a ` ( N - 2 ) ) ) |
70 |
69
|
adantl |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> X = ( a ` ( N - 2 ) ) ) |
71 |
58
|
fveq2d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( a ` ( N - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) ) |
72 |
70 71
|
eqtrd |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> X = ( a ` ( ( # ` p ) - 2 ) ) ) |
73 |
68 72
|
eqtrd |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) ) |
74 |
73
|
adantr |
|- ( ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) ) |
75 |
|
lsw |
|- ( p e. Word V -> ( lastS ` p ) = ( p ` ( ( # ` p ) - 1 ) ) ) |
76 |
|
fvoveq1 |
|- ( ( # ` p ) = N -> ( p ` ( ( # ` p ) - 1 ) ) = ( p ` ( N - 1 ) ) ) |
77 |
75 76
|
sylan9eq |
|- ( ( p e. Word V /\ ( # ` p ) = N ) -> ( lastS ` p ) = ( p ` ( N - 1 ) ) ) |
78 |
26 77
|
syl |
|- ( p e. ( N ClWWalksN G ) -> ( lastS ` p ) = ( p ` ( N - 1 ) ) ) |
79 |
78
|
eqcomd |
|- ( p e. ( N ClWWalksN G ) -> ( p ` ( N - 1 ) ) = ( lastS ` p ) ) |
80 |
79
|
ad3antrrr |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( p ` ( N - 1 ) ) = ( lastS ` p ) ) |
81 |
|
lsw |
|- ( a e. Word V -> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) |
82 |
81
|
adantr |
|- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) |
83 |
|
oveq1 |
|- ( N = ( # ` a ) -> ( N - 1 ) = ( ( # ` a ) - 1 ) ) |
84 |
83
|
eqcoms |
|- ( ( # ` a ) = N -> ( N - 1 ) = ( ( # ` a ) - 1 ) ) |
85 |
84
|
fveq2d |
|- ( ( # ` a ) = N -> ( a ` ( N - 1 ) ) = ( a ` ( ( # ` a ) - 1 ) ) ) |
86 |
85
|
eqeq2d |
|- ( ( # ` a ) = N -> ( ( lastS ` a ) = ( a ` ( N - 1 ) ) <-> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) ) |
87 |
86
|
adantl |
|- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( ( lastS ` a ) = ( a ` ( N - 1 ) ) <-> ( lastS ` a ) = ( a ` ( ( # ` a ) - 1 ) ) ) ) |
88 |
82 87
|
mpbird |
|- ( ( a e. Word V /\ ( # ` a ) = N ) -> ( lastS ` a ) = ( a ` ( N - 1 ) ) ) |
89 |
36 88
|
syl |
|- ( a e. ( N ClWWalksN G ) -> ( lastS ` a ) = ( a ` ( N - 1 ) ) ) |
90 |
89
|
eqcomd |
|- ( a e. ( N ClWWalksN G ) -> ( a ` ( N - 1 ) ) = ( lastS ` a ) ) |
91 |
90
|
adantr |
|- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) -> ( a ` ( N - 1 ) ) = ( lastS ` a ) ) |
92 |
91
|
ad2antrl |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( a ` ( N - 1 ) ) = ( lastS ` a ) ) |
93 |
80 92
|
eqeq12d |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) <-> ( lastS ` p ) = ( lastS ` a ) ) ) |
94 |
93
|
biimpd |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) -> ( lastS ` p ) = ( lastS ` a ) ) ) |
95 |
94
|
adantld |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> ( lastS ` p ) = ( lastS ` a ) ) ) |
96 |
95
|
imp |
|- ( ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( lastS ` p ) = ( lastS ` a ) ) |
97 |
64 74 96
|
3jca |
|- ( ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) |
98 |
97
|
3adant1 |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) |
99 |
1
|
clwwlknwrd |
|- ( p e. ( N ClWWalksN G ) -> p e. Word V ) |
100 |
99
|
ad3antrrr |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> p e. Word V ) |
101 |
100
|
3ad2ant2 |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> p e. Word V ) |
102 |
1
|
clwwlknwrd |
|- ( a e. ( N ClWWalksN G ) -> a e. Word V ) |
103 |
102
|
adantr |
|- ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) -> a e. Word V ) |
104 |
103
|
ad2antrl |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> a e. Word V ) |
105 |
104
|
3ad2ant2 |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> a e. Word V ) |
106 |
|
clwwlknlen |
|- ( p e. ( N ClWWalksN G ) -> ( # ` p ) = N ) |
107 |
|
eluz2b1 |
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) |
108 |
|
breq2 |
|- ( N = ( # ` p ) -> ( 1 < N <-> 1 < ( # ` p ) ) ) |
109 |
108
|
eqcoms |
|- ( ( # ` p ) = N -> ( 1 < N <-> 1 < ( # ` p ) ) ) |
110 |
109
|
biimpcd |
|- ( 1 < N -> ( ( # ` p ) = N -> 1 < ( # ` p ) ) ) |
111 |
107 110
|
simplbiim |
|- ( N e. ( ZZ>= ` 2 ) -> ( ( # ` p ) = N -> 1 < ( # ` p ) ) ) |
112 |
14 106 111
|
syl2imc |
|- ( p e. ( N ClWWalksN G ) -> ( N e. ( ZZ>= ` 3 ) -> 1 < ( # ` p ) ) ) |
113 |
112
|
ad3antrrr |
|- ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( N e. ( ZZ>= ` 3 ) -> 1 < ( # ` p ) ) ) |
114 |
113
|
impcom |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) ) -> 1 < ( # ` p ) ) |
115 |
114
|
3adant3 |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> 1 < ( # ` p ) ) |
116 |
|
2swrd2eqwrdeq |
|- ( ( p e. Word V /\ a e. Word V /\ 1 < ( # ` p ) ) -> ( p = a <-> ( ( # ` p ) = ( # ` a ) /\ ( ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) ) |
117 |
101 105 115 116
|
syl3anc |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> ( p = a <-> ( ( # ` p ) = ( # ` a ) /\ ( ( p prefix ( ( # ` p ) - 2 ) ) = ( a prefix ( ( # ` p ) - 2 ) ) /\ ( p ` ( ( # ` p ) - 2 ) ) = ( a ` ( ( # ` p ) - 2 ) ) /\ ( lastS ` p ) = ( lastS ` a ) ) ) ) ) |
118 |
53 98 117
|
mpbir2and |
|- ( ( N e. ( ZZ>= ` 3 ) /\ ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) /\ ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) ) -> p = a ) |
119 |
118
|
3exp |
|- ( N e. ( ZZ>= ` 3 ) -> ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) ) |
120 |
119
|
3ad2ant3 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( p e. ( N ClWWalksN G ) /\ ( p ` 0 ) = X ) /\ ( p ` ( N - 2 ) ) = X ) /\ ( ( a e. ( N ClWWalksN G ) /\ ( a ` 0 ) = X ) /\ ( a ` ( N - 2 ) ) = X ) ) -> ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) ) |
121 |
25 120
|
sylbid |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( p e. ( X C N ) /\ a e. ( X C N ) ) -> ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) ) |
122 |
121
|
imp |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( ( p prefix ( N - 2 ) ) = ( a prefix ( N - 2 ) ) /\ ( p ` ( N - 1 ) ) = ( a ` ( N - 1 ) ) ) -> p = a ) ) |
123 |
13 122
|
syl5bi |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( <. ( p prefix ( N - 2 ) ) , ( p ` ( N - 1 ) ) >. = <. ( a prefix ( N - 2 ) ) , ( a ` ( N - 1 ) ) >. -> p = a ) ) |
124 |
10 123
|
sylbid |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( p e. ( X C N ) /\ a e. ( X C N ) ) ) -> ( ( T ` p ) = ( T ` a ) -> p = a ) ) |
125 |
124
|
ralrimivva |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> A. p e. ( X C N ) A. a e. ( X C N ) ( ( T ` p ) = ( T ` a ) -> p = a ) ) |
126 |
|
dff13 |
|- ( T : ( X C N ) -1-1-> ( F X. ( G NeighbVtx X ) ) <-> ( T : ( X C N ) --> ( F X. ( G NeighbVtx X ) ) /\ A. p e. ( X C N ) A. a e. ( X C N ) ( ( T ` p ) = ( T ` a ) -> p = a ) ) ) |
127 |
5 125 126
|
sylanbrc |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) -1-1-> ( F X. ( G NeighbVtx X ) ) ) |