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Theorem upgr3v3e3cycl 40094
Description: If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Hypotheses
Ref Expression
upgr3v3e3cycl.e  |-  E  =  (Edg `  G )
upgr3v3e3cycl.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
upgr3v3e3cycl  |-  ( ( G  e. UPGraph  /\  F (CycleS `  G ) P  /\  ( # `  F )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) )
Distinct variable groups:    E, a,
b, c    P, a,
b, c    V, a,
b, c
Allowed substitution hints:    F( a, b, c)    G( a, b, c)

Proof of Theorem upgr3v3e3cycl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cyclprop 39976 . . 3  |-  ( F (CycleS `  G ) P  ->  ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
2 pthis1wlk 39921 . . . . 5  |-  ( F (PathS `  G ) P  ->  F (1Walks `  G ) P )
3 upgr3v3e3cycl.e . . . . . . . . . 10  |-  E  =  (Edg `  G )
43upgr1wlkvtxedg 39847 . . . . . . . . 9  |-  ( ( G  e. UPGraph  /\  F (1Walks `  G ) P )  ->  A. k  e.  ( 0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E )
5 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  3  ->  ( P `  ( # `  F
) )  =  ( P `  3 ) )
65eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  3  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  <->  ( P `  0 )  =  ( P `  3
) ) )
76anbi2d 718 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  3  ->  (
( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  <->  ( F
(PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  3 ) ) ) )
8 oveq2 6316 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  3  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 3 ) )
9 fzo0to3tp 12028 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
108, 9syl6eq 2521 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  3  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 ,  2 } )
1110raleqdv 2979 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  3  ->  ( A. k  e.  (
0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E 
<-> 
A. k  e.  {
0 ,  1 ,  2 }  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E ) )
12 c0ex 9655 . . . . . . . . . . . . . . 15  |-  0  e.  _V
13 1ex 9656 . . . . . . . . . . . . . . 15  |-  1  e.  _V
14 2ex 10703 . . . . . . . . . . . . . . 15  |-  2  e.  _V
15 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
16 oveq1 6315 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
17 0p1e1 10743 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  +  1 )  =  1
1816, 17syl6eq 2521 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
1918fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
2015, 19preq12d 4050 . . . . . . . . . . . . . . . 16  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
2120eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  ( { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E  <->  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E ) )
22 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
23 oveq1 6315 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
24 1p1e2 10745 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  +  1 )  =  2
2523, 24syl6eq 2521 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
2625fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
2722, 26preq12d 4050 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2827eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  ( { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E  <->  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  E ) )
29 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
30 oveq1 6315 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
31 2p1e3 10756 . . . . . . . . . . . . . . . . . . 19  |-  ( 2  +  1 )  =  3
3230, 31syl6eq 2521 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
3332fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
3429, 33preq12d 4050 . . . . . . . . . . . . . . . 16  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
3534eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( k  =  2  ->  ( { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E  <->  { ( P `  2 ) ,  ( P ` 
3 ) }  e.  E ) )
3612, 13, 14, 21, 28, 35raltp 4018 . . . . . . . . . . . . . 14  |-  ( A. k  e.  { 0 ,  1 ,  2 }  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  e.  E  <->  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) )
3711, 36syl6bb 269 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  3  ->  ( A. k  e.  (
0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E 
<->  ( { ( P `
 0 ) ,  ( P `  1
) }  e.  E  /\  { ( P ` 
1 ) ,  ( P `  2 ) }  e.  E  /\  { ( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) )
387, 37anbi12d 725 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  3  ->  (
( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  /\  A. k  e.  ( 0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E )  <->  ( ( F (PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  3 ) )  /\  ( { ( P `  0
) ,  ( P `
 1 ) }  e.  E  /\  {
( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) ) )
39 upgr3v3e3cycl.v . . . . . . . . . . . . . . . . . . 19  |-  V  =  (Vtx `  G )
40 eqid 2471 . . . . . . . . . . . . . . . . . . 19  |-  (iEdg `  G )  =  (iEdg `  G )
4139, 402m1wlk 39820 . . . . . . . . . . . . . . . . . 18  |-  ( F (1Walks `  G ) P  ->  ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
42 oveq2 6316 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  F )  =  3  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 3 ) )
4342feq2d 5725 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  3  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 3
) --> V ) )
44 id 22 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... 3 ) --> V  ->  P : ( 0 ... 3 ) --> V )
45 3nn0 10911 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  3  e.  NN0
46 0elfz 11915 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 3  e.  NN0  ->  0  e.  ( 0 ... 3
) )
4745, 46mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... 3 ) --> V  -> 
0  e.  ( 0 ... 3 ) )
4844, 47ffvelrnd 6038 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( P : ( 0 ... 3 ) --> V  -> 
( P `  0
)  e.  V )
49 1nn0 10909 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  NN0
50 1lt3 10801 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  <  3
51 fvffz0 11934 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 3  e.  NN0  /\  1  e.  NN0  /\  1  <  3 )  /\  P : ( 0 ... 3 ) --> V )  ->  ( P ` 
1 )  e.  V
)
5251ex 441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 3  e.  NN0  /\  1  e.  NN0  /\  1  <  3 )  ->  ( P : ( 0 ... 3 ) --> V  -> 
( P `  1
)  e.  V ) )
5345, 49, 50, 52mp3an 1390 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( P : ( 0 ... 3 ) --> V  -> 
( P `  1
)  e.  V )
54 2nn0 10910 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  NN0
55 2lt3 10800 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  <  3
56 fvffz0 11934 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 3  e.  NN0  /\  2  e.  NN0  /\  2  <  3 )  /\  P : ( 0 ... 3 ) --> V )  ->  ( P ` 
2 )  e.  V
)
5756ex 441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 3  e.  NN0  /\  2  e.  NN0  /\  2  <  3 )  ->  ( P : ( 0 ... 3 ) --> V  -> 
( P `  2
)  e.  V ) )
5845, 54, 55, 57mp3an 1390 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( P : ( 0 ... 3 ) --> V  -> 
( P `  2
)  e.  V )
5948, 53, 583jca 1210 . . . . . . . . . . . . . . . . . . . . 21  |-  ( P : ( 0 ... 3 ) --> V  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  1
)  e.  V  /\  ( P `  2 )  e.  V ) )
6043, 59syl6bi 236 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  3  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  1
)  e.  V  /\  ( P `  2 )  e.  V ) ) )
6160com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  =  3  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  1
)  e.  V  /\  ( P `  2 )  e.  V ) ) )
6261adantl 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  =  3  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 1 )  e.  V  /\  ( P `
 2 )  e.  V ) ) )
632, 41, 623syl 18 . . . . . . . . . . . . . . . . 17  |-  ( F (PathS `  G ) P  ->  ( ( # `  F )  =  3  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 1 )  e.  V  /\  ( P `
 2 )  e.  V ) ) )
6463adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  -> 
( ( # `  F
)  =  3  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  1
)  e.  V  /\  ( P `  2 )  e.  V ) ) )
6564adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) )  ->  ( ( # `  F )  =  3  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 1 )  e.  V  /\  ( P `
 2 )  e.  V ) ) )
6665impcom 437 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  3  /\  ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) )  ->  ( ( P `  0 )  e.  V  /\  ( P `  1 )  e.  V  /\  ( P `  2 )  e.  V ) )
67 preq2 4043 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  3 )  =  ( P ` 
0 )  ->  { ( P `  2 ) ,  ( P ` 
3 ) }  =  { ( P ` 
2 ) ,  ( P `  0 ) } )
6867eqcoms 2479 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  0 )  =  ( P ` 
3 )  ->  { ( P `  2 ) ,  ( P ` 
3 ) }  =  { ( P ` 
2 ) ,  ( P `  0 ) } )
6968adantl 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { ( P `  2 ) ,  ( P ` 
0 ) } )
7069eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  -> 
( { ( P `
 2 ) ,  ( P `  3
) }  e.  E  <->  { ( P `  2
) ,  ( P `
 0 ) }  e.  E ) )
71703anbi3d 1371 . . . . . . . . . . . . . . . 16  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  -> 
( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E  /\  { ( P `
 1 ) ,  ( P `  2
) }  e.  E  /\  { ( P ` 
2 ) ,  ( P `  3 ) }  e.  E )  <-> 
( { ( P `
 0 ) ,  ( P `  1
) }  e.  E  /\  { ( P ` 
1 ) ,  ( P `  2 ) }  e.  E  /\  { ( P `  2
) ,  ( P `
 0 ) }  e.  E ) ) )
7271biimpa 492 . . . . . . . . . . . . . . 15  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) )  ->  ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E  /\  { ( P `
 1 ) ,  ( P `  2
) }  e.  E  /\  { ( P ` 
2 ) ,  ( P `  0 ) }  e.  E ) )
7372adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  3  /\  ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) )  ->  ( {
( P `  0
) ,  ( P `
 1 ) }  e.  E  /\  {
( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 0 ) }  e.  E ) )
74 simpll 768 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  ->  F (PathS `  G ) P )
75 breq2 4399 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  3  ->  (
1  <  ( # `  F
)  <->  1  <  3
) )
7650, 75mpbiri 241 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  3  ->  1  <  ( # `  F
) )
7776adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
1  <  ( # `  F
) )
78 3nn 10791 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  NN
79 lbfzo0 11983 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  ( 0..^ 3 )  <->  3  e.  NN )
8078, 79mpbir 214 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  ( 0..^ 3 )
8180, 8syl5eleqr 2556 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  3  ->  0  e.  ( 0..^ ( # `  F ) ) )
8281adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
0  e.  ( 0..^ ( # `  F
) ) )
83 pthdadjvtx 39923 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F (PathS `  G
) P  /\  1  <  ( # `  F
)  /\  0  e.  ( 0..^ ( # `  F
) ) )  -> 
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) ) )
84 1e0p1 11102 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  =  ( 0  +  1 )
8584fveq2i 5882 . . . . . . . . . . . . . . . . . . . . 21  |-  ( P `
 1 )  =  ( P `  (
0  +  1 ) )
8685neeq2i 2708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =/=  ( P ` 
1 )  <->  ( P `  0 )  =/=  ( P `  (
0  +  1 ) ) )
8783, 86sylibr 217 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F (PathS `  G
) P  /\  1  <  ( # `  F
)  /\  0  e.  ( 0..^ ( # `  F
) ) )  -> 
( P `  0
)  =/=  ( P `
 1 ) )
8874, 77, 82, 87syl3anc 1292 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( P `  0
)  =/=  ( P `
 1 ) )
89 elfzo0 11984 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  e.  ( 0..^ 3 )  <->  ( 1  e. 
NN0  /\  3  e.  NN  /\  1  <  3
) )
9049, 78, 50, 89mpbir3an 1212 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  ( 0..^ 3 )
9190, 8syl5eleqr 2556 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  3  ->  1  e.  ( 0..^ ( # `  F ) ) )
9291adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
1  e.  ( 0..^ ( # `  F
) ) )
93 pthdadjvtx 39923 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F (PathS `  G
) P  /\  1  <  ( # `  F
)  /\  1  e.  ( 0..^ ( # `  F
) ) )  -> 
( P `  1
)  =/=  ( P `
 ( 1  +  1 ) ) )
94 df-2 10690 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  =  ( 1  +  1 )
9594fveq2i 5882 . . . . . . . . . . . . . . . . . . . . 21  |-  ( P `
 2 )  =  ( P `  (
1  +  1 ) )
9695neeq2i 2708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  1 )  =/=  ( P ` 
2 )  <->  ( P `  1 )  =/=  ( P `  (
1  +  1 ) ) )
9793, 96sylibr 217 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F (PathS `  G
) P  /\  1  <  ( # `  F
)  /\  1  e.  ( 0..^ ( # `  F
) ) )  -> 
( P `  1
)  =/=  ( P `
 2 ) )
9874, 77, 92, 97syl3anc 1292 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( P `  1
)  =/=  ( P `
 2 ) )
99 elfzo0 11984 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 2  e.  ( 0..^ 3 )  <->  ( 2  e. 
NN0  /\  3  e.  NN  /\  2  <  3
) )
10054, 78, 55, 99mpbir3an 1212 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  ( 0..^ 3 )
101100, 8syl5eleqr 2556 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  3  ->  2  e.  ( 0..^ ( # `  F ) ) )
102101adantl 473 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
2  e.  ( 0..^ ( # `  F
) ) )
103 pthdadjvtx 39923 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F (PathS `  G
) P  /\  1  <  ( # `  F
)  /\  2  e.  ( 0..^ ( # `  F
) ) )  -> 
( P `  2
)  =/=  ( P `
 ( 2  +  1 ) ) )
10474, 77, 102, 103syl3anc 1292 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( P `  2
)  =/=  ( P `
 ( 2  +  1 ) ) )
105 neeq2 2706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( P `  0 )  =  ( P ` 
3 )  ->  (
( P `  2
)  =/=  ( P `
 0 )  <->  ( P `  2 )  =/=  ( P `  3
) ) )
106 df-3 10691 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  3  =  ( 2  +  1 )
107106fveq2i 5882 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 3 )  =  ( P `  (
2  +  1 ) )
108107neeq2i 2708 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( P `  2 )  =/=  ( P ` 
3 )  <->  ( P `  2 )  =/=  ( P `  (
2  +  1 ) ) )
109105, 108syl6bb 269 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( P `  0 )  =  ( P ` 
3 )  ->  (
( P `  2
)  =/=  ( P `
 0 )  <->  ( P `  2 )  =/=  ( P `  (
2  +  1 ) ) ) )
110109adantl 473 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  -> 
( ( P ` 
2 )  =/=  ( P `  0 )  <->  ( P `  2 )  =/=  ( P `  ( 2  +  1 ) ) ) )
111110adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( ( P ` 
2 )  =/=  ( P `  0 )  <->  ( P `  2 )  =/=  ( P `  ( 2  +  1 ) ) ) )
112104, 111mpbird 240 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( P `  2
)  =/=  ( P `
 0 ) )
11388, 98, 1123jca 1210 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( # `  F )  =  3 )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  2 )  =/=  ( P ` 
0 ) ) )
114113ex 441 . . . . . . . . . . . . . . . 16  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  -> 
( ( # `  F
)  =  3  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  2 )  =/=  ( P ` 
0 ) ) ) )
115114adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) )  ->  ( ( # `  F )  =  3  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
)  /\  ( P `  2 )  =/=  ( P `  0
) ) ) )
116115impcom 437 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  3  /\  ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) )  ->  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
)  /\  ( P `  2 )  =/=  ( P `  0
) ) )
117 preq1 4042 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  ( P ` 
0 )  ->  { a ,  b }  =  { ( P ` 
0 ) ,  b } )
118117eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( a  =  ( P ` 
0 )  ->  ( { a ,  b }  e.  E  <->  { ( P `  0 ) ,  b }  e.  E ) )
119 preq2 4043 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  ( P ` 
0 )  ->  { c ,  a }  =  { c ,  ( P `  0 ) } )
120119eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( a  =  ( P ` 
0 )  ->  ( { c ,  a }  e.  E  <->  { c ,  ( P ` 
0 ) }  e.  E ) )
121118, 1203anbi13d 1367 . . . . . . . . . . . . . . . 16  |-  ( a  =  ( P ` 
0 )  ->  (
( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  <->  ( {
( P `  0
) ,  b }  e.  E  /\  {
b ,  c }  e.  E  /\  {
c ,  ( P `
 0 ) }  e.  E ) ) )
122 neeq1 2705 . . . . . . . . . . . . . . . . 17  |-  ( a  =  ( P ` 
0 )  ->  (
a  =/=  b  <->  ( P `  0 )  =/=  b ) )
123 neeq2 2706 . . . . . . . . . . . . . . . . 17  |-  ( a  =  ( P ` 
0 )  ->  (
c  =/=  a  <->  c  =/=  ( P `  0 ) ) )
124122, 1233anbi13d 1367 . . . . . . . . . . . . . . . 16  |-  ( a  =  ( P ` 
0 )  ->  (
( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a
)  <->  ( ( P `
 0 )  =/=  b  /\  b  =/=  c  /\  c  =/=  ( P `  0
) ) ) )
125121, 124anbi12d 725 . . . . . . . . . . . . . . 15  |-  ( a  =  ( P ` 
0 )  ->  (
( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) )  <->  ( ( { ( P ` 
0 ) ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  {
c ,  ( P `
 0 ) }  e.  E )  /\  ( ( P ` 
0 )  =/=  b  /\  b  =/=  c  /\  c  =/=  ( P `  0 )
) ) ) )
126 preq2 4043 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( P ` 
1 )  ->  { ( P `  0 ) ,  b }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
127126eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( P ` 
1 )  ->  ( { ( P ` 
0 ) ,  b }  e.  E  <->  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E ) )
128 preq1 4042 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( P ` 
1 )  ->  { b ,  c }  =  { ( P ` 
1 ) ,  c } )
129128eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( P ` 
1 )  ->  ( { b ,  c }  e.  E  <->  { ( P `  1 ) ,  c }  e.  E ) )
130127, 1293anbi12d 1366 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( P ` 
1 )  ->  (
( { ( P `
 0 ) ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  ( P `
 0 ) }  e.  E )  <->  ( {
( P `  0
) ,  ( P `
 1 ) }  e.  E  /\  {
( P `  1
) ,  c }  e.  E  /\  {
c ,  ( P `
 0 ) }  e.  E ) ) )
131 neeq2 2706 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( P ` 
1 )  ->  (
( P `  0
)  =/=  b  <->  ( P `  0 )  =/=  ( P `  1
) ) )
132 neeq1 2705 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( P ` 
1 )  ->  (
b  =/=  c  <->  ( P `  1 )  =/=  c ) )
133131, 1323anbi12d 1366 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( P ` 
1 )  ->  (
( ( P ` 
0 )  =/=  b  /\  b  =/=  c  /\  c  =/=  ( P `  0 )
)  <->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  c  /\  c  =/=  ( P `  0
) ) ) )
134130, 133anbi12d 725 . . . . . . . . . . . . . . 15  |-  ( b  =  ( P ` 
1 )  ->  (
( ( { ( P `  0 ) ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  ( P `  0 ) }  e.  E )  /\  ( ( P `
 0 )  =/=  b  /\  b  =/=  c  /\  c  =/=  ( P `  0
) ) )  <->  ( ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  c }  e.  E  /\  {
c ,  ( P `
 0 ) }  e.  E )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  c  /\  c  =/=  ( P ` 
0 ) ) ) ) )
135 preq2 4043 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( P ` 
2 )  ->  { ( P `  1 ) ,  c }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
136135eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( c  =  ( P ` 
2 )  ->  ( { ( P ` 
1 ) ,  c }  e.  E  <->  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  E ) )
137 preq1 4042 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( P ` 
2 )  ->  { c ,  ( P ` 
0 ) }  =  { ( P ` 
2 ) ,  ( P `  0 ) } )
138137eleq1d 2533 . . . . . . . . . . . . . . . . 17  |-  ( c  =  ( P ` 
2 )  ->  ( { c ,  ( P `  0 ) }  e.  E  <->  { ( P `  2 ) ,  ( P ` 
0 ) }  e.  E ) )
139136, 1383anbi23d 1368 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( P ` 
2 )  ->  (
( { ( P `
 0 ) ,  ( P `  1
) }  e.  E  /\  { ( P ` 
1 ) ,  c }  e.  E  /\  { c ,  ( P `
 0 ) }  e.  E )  <->  ( {
( P `  0
) ,  ( P `
 1 ) }  e.  E  /\  {
( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 0 ) }  e.  E ) ) )
140 neeq2 2706 . . . . . . . . . . . . . . . . 17  |-  ( c  =  ( P ` 
2 )  ->  (
( P `  1
)  =/=  c  <->  ( P `  1 )  =/=  ( P `  2
) ) )
141 neeq1 2705 . . . . . . . . . . . . . . . . 17  |-  ( c  =  ( P ` 
2 )  ->  (
c  =/=  ( P `
 0 )  <->  ( P `  2 )  =/=  ( P `  0
) ) )
142140, 1413anbi23d 1368 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( P ` 
2 )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  c  /\  c  =/=  ( P ` 
0 ) )  <->  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
)  /\  ( P `  2 )  =/=  ( P `  0
) ) ) )
143139, 142anbi12d 725 . . . . . . . . . . . . . . 15  |-  ( c  =  ( P ` 
2 )  ->  (
( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E  /\  { ( P `
 1 ) ,  c }  e.  E  /\  { c ,  ( P `  0 ) }  e.  E )  /\  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  c  /\  c  =/=  ( P `  0
) ) )  <->  ( ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 0 ) }  e.  E )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  2 )  =/=  ( P ` 
0 ) ) ) ) )
144125, 134, 143rspc3ev 3151 . . . . . . . . . . . . . 14  |-  ( ( ( ( P ` 
0 )  e.  V  /\  ( P `  1
)  e.  V  /\  ( P `  2 )  e.  V )  /\  ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  E  /\  { ( P `
 1 ) ,  ( P `  2
) }  e.  E  /\  { ( P ` 
2 ) ,  ( P `  0 ) }  e.  E )  /\  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
)  /\  ( P `  2 )  =/=  ( P `  0
) ) ) )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) )
14566, 73, 116, 144syl12anc 1290 . . . . . . . . . . . . 13  |-  ( ( ( # `  F
)  =  3  /\  ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) ) )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  {
c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a
) ) )
146145ex 441 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  3  ->  (
( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P ` 
3 ) )  /\  ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  E  /\  {
( P `  2
) ,  ( P `
 3 ) }  e.  E ) )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) )
14738, 146sylbid 223 . . . . . . . . . . 11  |-  ( (
# `  F )  =  3  ->  (
( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  /\  A. k  e.  ( 0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  {
c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a
) ) ) )
148147expd 443 . . . . . . . . . 10  |-  ( (
# `  F )  =  3  ->  (
( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( A. k  e.  (
0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) )
149148com13 82 . . . . . . . . 9  |-  ( A. k  e.  ( 0..^ ( # `  F
) ) { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  E  ->  ( ( F (PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) )  ->  ( ( # `  F )  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) )
1504, 149syl 17 . . . . . . . 8  |-  ( ( G  e. UPGraph  /\  F (1Walks `  G ) P )  ->  ( ( F (PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) )  ->  ( ( # `  F )  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) )
151150expcom 442 . . . . . . 7  |-  ( F (1Walks `  G ) P  ->  ( G  e. UPGraph  ->  ( ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  (
( # `  F )  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  {
c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a
) ) ) ) ) )
152151com23 80 . . . . . 6  |-  ( F (1Walks `  G ) P  ->  ( ( F (PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) )  ->  ( G  e. UPGraph  ->  ( ( # `  F
)  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) ) )
153152expd 443 . . . . 5  |-  ( F (1Walks `  G ) P  ->  ( F (PathS `  G ) P  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( G  e. UPGraph  ->  ( ( # `  F
)  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) ) ) )
1542, 153mpcom 36 . . . 4  |-  ( F (PathS `  G ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( G  e. UPGraph  ->  ( ( # `  F
)  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) ) )
155154imp 436 . . 3  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( G  e. UPGraph  ->  ( (
# `  F )  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  {
c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a
) ) ) ) )
1561, 155syl 17 . 2  |-  ( F (CycleS `  G ) P  ->  ( G  e. UPGraph  ->  ( ( # `  F
)  =  3  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) ) ) )
1571563imp21 1243 1  |-  ( ( G  e. UPGraph  /\  F (CycleS `  G ) P  /\  ( # `  F )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  ( a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {cpr 3961   {ctp 3963   class class class wbr 4395   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693   NNcn 10631   2c2 10681   3c3 10682   NN0cn0 10893   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703  Vtxcvtx 39251  iEdgciedg 39252   UPGraph cupgr 39326  Edgcedga 39371  1Walksc1wlks 39800  PathScpths 39907  CycleSccycls 39968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-uhgr 39302  df-upgr 39328  df-edga 39372  df-1wlks 39804  df-wlks 39805  df-trls 39889  df-pths 39911  df-cycls 39970
This theorem is referenced by:  umgr3v3e3cycl  40098
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