Step |
Hyp |
Ref |
Expression |
1 |
|
numclwwlk.v |
|- V = ( Vtx ` G ) |
2 |
|
numclwwlk.q |
|- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } ) |
3 |
|
numclwwlk.h |
|- H = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( v ( ClWWalksNOn ` G ) n ) | ( w ` ( n - 2 ) ) =/= v } ) |
4 |
|
numclwwlk.r |
|- R = ( x e. ( X H ( N + 2 ) ) |-> ( x prefix ( N + 1 ) ) ) |
5 |
|
nnnn0 |
|- ( N e. NN -> N e. NN0 ) |
6 |
|
2z |
|- 2 e. ZZ |
7 |
6
|
a1i |
|- ( N e. NN -> 2 e. ZZ ) |
8 |
|
nn0pzuz |
|- ( ( N e. NN0 /\ 2 e. ZZ ) -> ( N + 2 ) e. ( ZZ>= ` 2 ) ) |
9 |
5 7 8
|
syl2anc |
|- ( N e. NN -> ( N + 2 ) e. ( ZZ>= ` 2 ) ) |
10 |
9
|
anim2i |
|- ( ( X e. V /\ N e. NN ) -> ( X e. V /\ ( N + 2 ) e. ( ZZ>= ` 2 ) ) ) |
11 |
10
|
3adant1 |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( X e. V /\ ( N + 2 ) e. ( ZZ>= ` 2 ) ) ) |
12 |
3
|
numclwwlkovh |
|- ( ( X e. V /\ ( N + 2 ) e. ( ZZ>= ` 2 ) ) -> ( X H ( N + 2 ) ) = { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } ) |
13 |
12
|
eleq2d |
|- ( ( X e. V /\ ( N + 2 ) e. ( ZZ>= ` 2 ) ) -> ( x e. ( X H ( N + 2 ) ) <-> x e. { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } ) ) |
14 |
11 13
|
syl |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( x e. ( X H ( N + 2 ) ) <-> x e. { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } ) ) |
15 |
|
fveq1 |
|- ( w = x -> ( w ` 0 ) = ( x ` 0 ) ) |
16 |
15
|
eqeq1d |
|- ( w = x -> ( ( w ` 0 ) = X <-> ( x ` 0 ) = X ) ) |
17 |
|
fveq1 |
|- ( w = x -> ( w ` ( ( N + 2 ) - 2 ) ) = ( x ` ( ( N + 2 ) - 2 ) ) ) |
18 |
17 15
|
neeq12d |
|- ( w = x -> ( ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) <-> ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) |
19 |
16 18
|
anbi12d |
|- ( w = x -> ( ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) |
20 |
19
|
elrab |
|- ( x e. { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } <-> ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) |
21 |
14 20
|
syl6bb |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( x e. ( X H ( N + 2 ) ) <-> ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) ) |
22 |
|
peano2nn |
|- ( N e. NN -> ( N + 1 ) e. NN ) |
23 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
24 |
23 7
|
zaddcld |
|- ( N e. NN -> ( N + 2 ) e. ZZ ) |
25 |
|
uzid |
|- ( ( N + 2 ) e. ZZ -> ( N + 2 ) e. ( ZZ>= ` ( N + 2 ) ) ) |
26 |
24 25
|
syl |
|- ( N e. NN -> ( N + 2 ) e. ( ZZ>= ` ( N + 2 ) ) ) |
27 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
28 |
|
1cnd |
|- ( N e. NN -> 1 e. CC ) |
29 |
27 28 28
|
addassd |
|- ( N e. NN -> ( ( N + 1 ) + 1 ) = ( N + ( 1 + 1 ) ) ) |
30 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
31 |
30
|
a1i |
|- ( N e. NN -> ( 1 + 1 ) = 2 ) |
32 |
31
|
oveq2d |
|- ( N e. NN -> ( N + ( 1 + 1 ) ) = ( N + 2 ) ) |
33 |
29 32
|
eqtrd |
|- ( N e. NN -> ( ( N + 1 ) + 1 ) = ( N + 2 ) ) |
34 |
33
|
fveq2d |
|- ( N e. NN -> ( ZZ>= ` ( ( N + 1 ) + 1 ) ) = ( ZZ>= ` ( N + 2 ) ) ) |
35 |
26 34
|
eleqtrrd |
|- ( N e. NN -> ( N + 2 ) e. ( ZZ>= ` ( ( N + 1 ) + 1 ) ) ) |
36 |
22 35
|
jca |
|- ( N e. NN -> ( ( N + 1 ) e. NN /\ ( N + 2 ) e. ( ZZ>= ` ( ( N + 1 ) + 1 ) ) ) ) |
37 |
36
|
3ad2ant3 |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( N + 1 ) e. NN /\ ( N + 2 ) e. ( ZZ>= ` ( ( N + 1 ) + 1 ) ) ) ) |
38 |
37
|
adantr |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( N + 1 ) e. NN /\ ( N + 2 ) e. ( ZZ>= ` ( ( N + 1 ) + 1 ) ) ) ) |
39 |
|
simprl |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> x e. ( ( N + 2 ) ClWWalksN G ) ) |
40 |
|
wwlksubclwwlk |
|- ( ( ( N + 1 ) e. NN /\ ( N + 2 ) e. ( ZZ>= ` ( ( N + 1 ) + 1 ) ) ) -> ( x e. ( ( N + 2 ) ClWWalksN G ) -> ( x prefix ( N + 1 ) ) e. ( ( ( N + 1 ) - 1 ) WWalksN G ) ) ) |
41 |
38 39 40
|
sylc |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( x prefix ( N + 1 ) ) e. ( ( ( N + 1 ) - 1 ) WWalksN G ) ) |
42 |
|
pncan1 |
|- ( N e. CC -> ( ( N + 1 ) - 1 ) = N ) |
43 |
42
|
eqcomd |
|- ( N e. CC -> N = ( ( N + 1 ) - 1 ) ) |
44 |
27 43
|
syl |
|- ( N e. NN -> N = ( ( N + 1 ) - 1 ) ) |
45 |
44
|
oveq1d |
|- ( N e. NN -> ( N WWalksN G ) = ( ( ( N + 1 ) - 1 ) WWalksN G ) ) |
46 |
45
|
eleq2d |
|- ( N e. NN -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) <-> ( x prefix ( N + 1 ) ) e. ( ( ( N + 1 ) - 1 ) WWalksN G ) ) ) |
47 |
46
|
3ad2ant3 |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) <-> ( x prefix ( N + 1 ) ) e. ( ( ( N + 1 ) - 1 ) WWalksN G ) ) ) |
48 |
47
|
adantr |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) <-> ( x prefix ( N + 1 ) ) e. ( ( ( N + 1 ) - 1 ) WWalksN G ) ) ) |
49 |
41 48
|
mpbird |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) ) |
50 |
1
|
clwwlknbp |
|- ( x e. ( ( N + 2 ) ClWWalksN G ) -> ( x e. Word V /\ ( # ` x ) = ( N + 2 ) ) ) |
51 |
|
simprl |
|- ( ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( x ` 0 ) = X ) |
52 |
|
simprr |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> x e. Word V ) |
53 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
54 |
5 53
|
syl |
|- ( N e. NN -> ( N + 1 ) e. NN0 ) |
55 |
|
nnre |
|- ( N e. NN -> N e. RR ) |
56 |
55
|
lep1d |
|- ( N e. NN -> N <_ ( N + 1 ) ) |
57 |
|
elfz2nn0 |
|- ( N e. ( 0 ... ( N + 1 ) ) <-> ( N e. NN0 /\ ( N + 1 ) e. NN0 /\ N <_ ( N + 1 ) ) ) |
58 |
5 54 56 57
|
syl3anbrc |
|- ( N e. NN -> N e. ( 0 ... ( N + 1 ) ) ) |
59 |
|
2cnd |
|- ( N e. NN -> 2 e. CC ) |
60 |
|
addsubass |
|- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) ) |
61 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
62 |
61
|
oveq2i |
|- ( N + ( 2 - 1 ) ) = ( N + 1 ) |
63 |
60 62
|
eqtrdi |
|- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N + 2 ) - 1 ) = ( N + 1 ) ) |
64 |
27 59 28 63
|
syl3anc |
|- ( N e. NN -> ( ( N + 2 ) - 1 ) = ( N + 1 ) ) |
65 |
64
|
oveq2d |
|- ( N e. NN -> ( 0 ... ( ( N + 2 ) - 1 ) ) = ( 0 ... ( N + 1 ) ) ) |
66 |
58 65
|
eleqtrrd |
|- ( N e. NN -> N e. ( 0 ... ( ( N + 2 ) - 1 ) ) ) |
67 |
|
elfzp1b |
|- ( ( N e. ZZ /\ ( N + 2 ) e. ZZ ) -> ( N e. ( 0 ... ( ( N + 2 ) - 1 ) ) <-> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) ) |
68 |
23 24 67
|
syl2anc |
|- ( N e. NN -> ( N e. ( 0 ... ( ( N + 2 ) - 1 ) ) <-> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) ) |
69 |
66 68
|
mpbid |
|- ( N e. NN -> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) |
70 |
69
|
adantr |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) |
71 |
|
oveq2 |
|- ( ( # ` x ) = ( N + 2 ) -> ( 1 ... ( # ` x ) ) = ( 1 ... ( N + 2 ) ) ) |
72 |
71
|
eleq2d |
|- ( ( # ` x ) = ( N + 2 ) -> ( ( N + 1 ) e. ( 1 ... ( # ` x ) ) <-> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) ) |
73 |
72
|
ad2antrl |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( ( N + 1 ) e. ( 1 ... ( # ` x ) ) <-> ( N + 1 ) e. ( 1 ... ( N + 2 ) ) ) ) |
74 |
70 73
|
mpbird |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( N + 1 ) e. ( 1 ... ( # ` x ) ) ) |
75 |
|
pfxfv0 |
|- ( ( x e. Word V /\ ( N + 1 ) e. ( 1 ... ( # ` x ) ) ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) |
76 |
52 74 75
|
syl2anc |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) |
77 |
76
|
ex |
|- ( N e. NN -> ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) ) |
78 |
77
|
adantl |
|- ( ( X e. V /\ N e. NN ) -> ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) ) |
79 |
78
|
impcom |
|- ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) |
80 |
79
|
ad2antrl |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = ( x ` 0 ) ) |
81 |
|
simpl |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( x ` 0 ) = X ) |
82 |
80 81
|
eqtrd |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( x prefix ( N + 1 ) ) ` 0 ) = X ) |
83 |
|
pfxfvlsw |
|- ( ( x e. Word V /\ ( N + 1 ) e. ( 1 ... ( # ` x ) ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) = ( x ` ( ( N + 1 ) - 1 ) ) ) |
84 |
52 74 83
|
syl2anc |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) = ( x ` ( ( N + 1 ) - 1 ) ) ) |
85 |
27 42
|
syl |
|- ( N e. NN -> ( ( N + 1 ) - 1 ) = N ) |
86 |
27 59
|
pncand |
|- ( N e. NN -> ( ( N + 2 ) - 2 ) = N ) |
87 |
85 86
|
eqtr4d |
|- ( N e. NN -> ( ( N + 1 ) - 1 ) = ( ( N + 2 ) - 2 ) ) |
88 |
87
|
fveq2d |
|- ( N e. NN -> ( x ` ( ( N + 1 ) - 1 ) ) = ( x ` ( ( N + 2 ) - 2 ) ) ) |
89 |
88
|
adantr |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( x ` ( ( N + 1 ) - 1 ) ) = ( x ` ( ( N + 2 ) - 2 ) ) ) |
90 |
84 89
|
eqtr2d |
|- ( ( N e. NN /\ ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) ) -> ( x ` ( ( N + 2 ) - 2 ) ) = ( lastS ` ( x prefix ( N + 1 ) ) ) ) |
91 |
90
|
ex |
|- ( N e. NN -> ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( x ` ( ( N + 2 ) - 2 ) ) = ( lastS ` ( x prefix ( N + 1 ) ) ) ) ) |
92 |
91
|
adantl |
|- ( ( X e. V /\ N e. NN ) -> ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( x ` ( ( N + 2 ) - 2 ) ) = ( lastS ` ( x prefix ( N + 1 ) ) ) ) ) |
93 |
92
|
impcom |
|- ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) -> ( x ` ( ( N + 2 ) - 2 ) ) = ( lastS ` ( x prefix ( N + 1 ) ) ) ) |
94 |
93
|
neeq1d |
|- ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) -> ( ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) <-> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
95 |
94
|
biimpcd |
|- ( ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) -> ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
96 |
95
|
adantl |
|- ( ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) -> ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
97 |
96
|
impcom |
|- ( ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) |
98 |
97
|
adantl |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) |
99 |
|
neeq2 |
|- ( X = ( x ` 0 ) -> ( ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X <-> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
100 |
99
|
eqcoms |
|- ( ( x ` 0 ) = X -> ( ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X <-> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
101 |
100
|
adantr |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X <-> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= ( x ` 0 ) ) ) |
102 |
98 101
|
mpbird |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) |
103 |
82 102
|
jca |
|- ( ( ( x ` 0 ) = X /\ ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) |
104 |
51 103
|
mpancom |
|- ( ( ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) /\ ( X e. V /\ N e. NN ) ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) |
105 |
104
|
exp31 |
|- ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( ( X e. V /\ N e. NN ) -> ( ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
106 |
105
|
com23 |
|- ( ( ( # ` x ) = ( N + 2 ) /\ x e. Word V ) -> ( ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) -> ( ( X e. V /\ N e. NN ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
107 |
106
|
ancoms |
|- ( ( x e. Word V /\ ( # ` x ) = ( N + 2 ) ) -> ( ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) -> ( ( X e. V /\ N e. NN ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
108 |
50 107
|
syl |
|- ( x e. ( ( N + 2 ) ClWWalksN G ) -> ( ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) -> ( ( X e. V /\ N e. NN ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
109 |
108
|
imp |
|- ( ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( ( X e. V /\ N e. NN ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
110 |
109
|
com12 |
|- ( ( X e. V /\ N e. NN ) -> ( ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
111 |
110
|
3adant1 |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
112 |
111
|
imp |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) |
113 |
49 112
|
jca |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
114 |
113
|
ex |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( x e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( x ` 0 ) = X /\ ( x ` ( ( N + 2 ) - 2 ) ) =/= ( x ` 0 ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
115 |
21 114
|
sylbid |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( x e. ( X H ( N + 2 ) ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
116 |
115
|
imp |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
117 |
|
3simpc |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( X e. V /\ N e. NN ) ) |
118 |
117
|
adantr |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( X e. V /\ N e. NN ) ) |
119 |
1 2
|
numclwwlkovq |
|- ( ( X e. V /\ N e. NN ) -> ( X Q N ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } ) |
120 |
118 119
|
syl |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( X Q N ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } ) |
121 |
120
|
eleq2d |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( X Q N ) <-> ( x prefix ( N + 1 ) ) e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } ) ) |
122 |
|
fveq1 |
|- ( w = ( x prefix ( N + 1 ) ) -> ( w ` 0 ) = ( ( x prefix ( N + 1 ) ) ` 0 ) ) |
123 |
122
|
eqeq1d |
|- ( w = ( x prefix ( N + 1 ) ) -> ( ( w ` 0 ) = X <-> ( ( x prefix ( N + 1 ) ) ` 0 ) = X ) ) |
124 |
|
fveq2 |
|- ( w = ( x prefix ( N + 1 ) ) -> ( lastS ` w ) = ( lastS ` ( x prefix ( N + 1 ) ) ) ) |
125 |
124
|
neeq1d |
|- ( w = ( x prefix ( N + 1 ) ) -> ( ( lastS ` w ) =/= X <-> ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) |
126 |
123 125
|
anbi12d |
|- ( w = ( x prefix ( N + 1 ) ) -> ( ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) <-> ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
127 |
126
|
elrab |
|- ( ( x prefix ( N + 1 ) ) e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } <-> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) |
128 |
121 127
|
syl6bb |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( ( x prefix ( N + 1 ) ) e. ( X Q N ) <-> ( ( x prefix ( N + 1 ) ) e. ( N WWalksN G ) /\ ( ( ( x prefix ( N + 1 ) ) ` 0 ) = X /\ ( lastS ` ( x prefix ( N + 1 ) ) ) =/= X ) ) ) ) |
129 |
116 128
|
mpbird |
|- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ x e. ( X H ( N + 2 ) ) ) -> ( x prefix ( N + 1 ) ) e. ( X Q N ) ) |
130 |
129 4
|
fmptd |
|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> R : ( X H ( N + 2 ) ) --> ( X Q N ) ) |