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 |
|
elxp |
|- ( p e. ( F X. ( G NeighbVtx X ) ) <-> E. a E. b ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) ) |
7 |
1 2 3
|
numclwwlk1lem2foa |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( a e. F /\ b e. ( G NeighbVtx X ) ) -> ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) ) ) |
8 |
7
|
com12 |
|- ( ( a e. F /\ b e. ( G NeighbVtx X ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) ) ) |
9 |
8
|
adantl |
|- ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) ) ) |
10 |
9
|
imp |
|- ( ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) ) |
11 |
|
simpl |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) ) |
12 |
|
fveq2 |
|- ( x = ( ( a ++ <" X "> ) ++ <" b "> ) -> ( T ` x ) = ( T ` ( ( a ++ <" X "> ) ++ <" b "> ) ) ) |
13 |
12
|
eqeq2d |
|- ( x = ( ( a ++ <" X "> ) ++ <" b "> ) -> ( p = ( T ` x ) <-> p = ( T ` ( ( a ++ <" X "> ) ++ <" b "> ) ) ) ) |
14 |
1 2 3 4
|
numclwwlk1lem2fv |
|- ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) -> ( T ` ( ( a ++ <" X "> ) ++ <" b "> ) ) = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
15 |
14
|
adantr |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> ( T ` ( ( a ++ <" X "> ) ++ <" b "> ) ) = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
16 |
15
|
eqeq2d |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> ( p = ( T ` ( ( a ++ <" X "> ) ++ <" b "> ) ) <-> p = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) |
17 |
13 16
|
sylan9bbr |
|- ( ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) /\ x = ( ( a ++ <" X "> ) ++ <" b "> ) ) -> ( p = ( T ` x ) <-> p = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) |
18 |
|
simprll |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> p = <. a , b >. ) |
19 |
1
|
nbgrisvtx |
|- ( b e. ( G NeighbVtx X ) -> b e. V ) |
20 |
3
|
eleq2i |
|- ( a e. F <-> a e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) ) |
21 |
|
uz3m2nn |
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN ) |
22 |
21
|
nnne0d |
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) =/= 0 ) |
23 |
22
|
3ad2ant3 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) =/= 0 ) |
24 |
|
eqid |
|- ( Edg ` G ) = ( Edg ` G ) |
25 |
1 24
|
clwwlknonel |
|- ( ( N - 2 ) =/= 0 -> ( a e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) <-> ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) /\ ( a ` 0 ) = X ) ) ) |
26 |
23 25
|
syl |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) <-> ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) /\ ( a ` 0 ) = X ) ) ) |
27 |
20 26
|
syl5bb |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. F <-> ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) /\ ( a ` 0 ) = X ) ) ) |
28 |
|
df-3an |
|- ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) /\ ( a ` 0 ) = X ) <-> ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) /\ ( a ` 0 ) = X ) ) |
29 |
27 28
|
syl6bb |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. F <-> ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) /\ ( a ` 0 ) = X ) ) ) |
30 |
|
simplll |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> a e. Word V ) |
31 |
|
s1cl |
|- ( X e. V -> <" X "> e. Word V ) |
32 |
31
|
adantr |
|- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> <" X "> e. Word V ) |
33 |
32
|
adantl |
|- ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> <" X "> e. Word V ) |
34 |
33
|
adantr |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> <" X "> e. Word V ) |
35 |
|
s1cl |
|- ( b e. V -> <" b "> e. Word V ) |
36 |
35
|
adantl |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> <" b "> e. Word V ) |
37 |
|
ccatass |
|- ( ( a e. Word V /\ <" X "> e. Word V /\ <" b "> e. Word V ) -> ( ( a ++ <" X "> ) ++ <" b "> ) = ( a ++ ( <" X "> ++ <" b "> ) ) ) |
38 |
37
|
oveq1d |
|- ( ( a e. Word V /\ <" X "> e. Word V /\ <" b "> e. Word V ) -> ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) = ( ( a ++ ( <" X "> ++ <" b "> ) ) prefix ( N - 2 ) ) ) |
39 |
30 34 36 38
|
syl3anc |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) = ( ( a ++ ( <" X "> ++ <" b "> ) ) prefix ( N - 2 ) ) ) |
40 |
|
ccatcl |
|- ( ( <" X "> e. Word V /\ <" b "> e. Word V ) -> ( <" X "> ++ <" b "> ) e. Word V ) |
41 |
33 35 40
|
syl2an |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( <" X "> ++ <" b "> ) e. Word V ) |
42 |
|
simpr |
|- ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) -> ( # ` a ) = ( N - 2 ) ) |
43 |
42
|
eqcomd |
|- ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) -> ( N - 2 ) = ( # ` a ) ) |
44 |
43
|
adantr |
|- ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( N - 2 ) = ( # ` a ) ) |
45 |
44
|
adantr |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( N - 2 ) = ( # ` a ) ) |
46 |
|
pfxccatid |
|- ( ( a e. Word V /\ ( <" X "> ++ <" b "> ) e. Word V /\ ( N - 2 ) = ( # ` a ) ) -> ( ( a ++ ( <" X "> ++ <" b "> ) ) prefix ( N - 2 ) ) = a ) |
47 |
30 41 45 46
|
syl3anc |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( ( a ++ ( <" X "> ++ <" b "> ) ) prefix ( N - 2 ) ) = a ) |
48 |
39 47
|
eqtr2d |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> a = ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) ) |
49 |
|
1e2m1 |
|- 1 = ( 2 - 1 ) |
50 |
49
|
a1i |
|- ( N e. ( ZZ>= ` 3 ) -> 1 = ( 2 - 1 ) ) |
51 |
50
|
oveq2d |
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 1 ) = ( N - ( 2 - 1 ) ) ) |
52 |
|
eluzelcn |
|- ( N e. ( ZZ>= ` 3 ) -> N e. CC ) |
53 |
|
2cnd |
|- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC ) |
54 |
|
1cnd |
|- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC ) |
55 |
52 53 54
|
subsubd |
|- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 - 1 ) ) = ( ( N - 2 ) + 1 ) ) |
56 |
51 55
|
eqtrd |
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) ) |
57 |
56
|
adantl |
|- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) ) |
58 |
57
|
adantl |
|- ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) ) |
59 |
58
|
adantr |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) ) |
60 |
59
|
fveq2d |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) = ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( ( N - 2 ) + 1 ) ) ) |
61 |
|
simpll |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) ) |
62 |
|
simprl |
|- ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> X e. V ) |
63 |
62
|
anim1i |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( X e. V /\ b e. V ) ) |
64 |
|
ccatw2s1p2 |
|- ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ b e. V ) ) -> ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( ( N - 2 ) + 1 ) ) = b ) |
65 |
61 63 64
|
syl2anc |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( ( N - 2 ) + 1 ) ) = b ) |
66 |
60 65
|
eqtr2d |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> b = ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) ) |
67 |
48 66
|
opeq12d |
|- ( ( ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) /\ ( X e. V /\ N e. ( ZZ>= ` 3 ) ) ) /\ b e. V ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
68 |
67
|
exp31 |
|- ( ( a e. Word V /\ ( # ` a ) = ( N - 2 ) ) -> ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
69 |
68
|
3ad2antl1 |
|- ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) -> ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
70 |
69
|
adantr |
|- ( ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) /\ ( a ` 0 ) = X ) -> ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
71 |
70
|
com12 |
|- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) /\ ( a ` 0 ) = X ) -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
72 |
71
|
3adant1 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( a e. Word V /\ A. i e. ( 0 ..^ ( ( # ` a ) - 1 ) ) { ( a ` i ) , ( a ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` a ) , ( a ` 0 ) } e. ( Edg ` G ) ) /\ ( # ` a ) = ( N - 2 ) ) /\ ( a ` 0 ) = X ) -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
73 |
29 72
|
sylbid |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( a e. F -> ( b e. V -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
74 |
73
|
com23 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( b e. V -> ( a e. F -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
75 |
19 74
|
syl5 |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( b e. ( G NeighbVtx X ) -> ( a e. F -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
76 |
75
|
com13 |
|- ( a e. F -> ( b e. ( G NeighbVtx X ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) ) |
77 |
76
|
imp |
|- ( ( a e. F /\ b e. ( G NeighbVtx X ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) |
78 |
77
|
adantl |
|- ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) ) |
79 |
78
|
imp |
|- ( ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
80 |
79
|
adantl |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> <. a , b >. = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
81 |
18 80
|
eqtrd |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> p = <. ( ( ( a ++ <" X "> ) ++ <" b "> ) prefix ( N - 2 ) ) , ( ( ( a ++ <" X "> ) ++ <" b "> ) ` ( N - 1 ) ) >. ) |
82 |
11 17 81
|
rspcedvd |
|- ( ( ( ( a ++ <" X "> ) ++ <" b "> ) e. ( X C N ) /\ ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) |
83 |
10 82
|
mpancom |
|- ( ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) /\ ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) |
84 |
83
|
ex |
|- ( ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) ) |
85 |
84
|
exlimivv |
|- ( E. a E. b ( p = <. a , b >. /\ ( a e. F /\ b e. ( G NeighbVtx X ) ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) ) |
86 |
6 85
|
sylbi |
|- ( p e. ( F X. ( G NeighbVtx X ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) ) |
87 |
86
|
impcom |
|- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ p e. ( F X. ( G NeighbVtx X ) ) ) -> E. x e. ( X C N ) p = ( T ` x ) ) |
88 |
87
|
ralrimiva |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> A. p e. ( F X. ( G NeighbVtx X ) ) E. x e. ( X C N ) p = ( T ` x ) ) |
89 |
|
dffo3 |
|- ( T : ( X C N ) -onto-> ( F X. ( G NeighbVtx X ) ) <-> ( T : ( X C N ) --> ( F X. ( G NeighbVtx X ) ) /\ A. p e. ( F X. ( G NeighbVtx X ) ) E. x e. ( X C N ) p = ( T ` x ) ) ) |
90 |
5 88 89
|
sylanbrc |
|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) -onto-> ( F X. ( G NeighbVtx X ) ) ) |