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Theorem usgr2wlkneq 39948
Description: The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
Assertion
Ref Expression
usgr2wlkneq  |-  ( ( ( G  e. USGraph  /\  F
(1Walks `  G ) P )  /\  (
( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) )

Proof of Theorem usgr2wlkneq
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 wlkv 39817 . . . 4  |-  ( F (1Walks `  G ) P  ->  ( G  e. 
_V  /\  F  e.  _V  /\  P  e.  _V ) )
2 usgrupgr 39430 . . . . . . . . . 10  |-  ( G  e. USGraph  ->  G  e. UPGraph  )
3 3simpc 1029 . . . . . . . . . 10  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( F  e.  _V  /\  P  e.  _V ) )
42, 3anim12i 576 . . . . . . . . 9  |-  ( ( G  e. USGraph  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  -> 
( G  e. UPGraph  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
5 3anass 1011 . . . . . . . . 9  |-  ( ( G  e. UPGraph  /\  F  e. 
_V  /\  P  e.  _V )  <->  ( G  e. UPGraph  /\  ( F  e.  _V  /\  P  e.  _V )
) )
64, 5sylibr 217 . . . . . . . 8  |-  ( ( G  e. USGraph  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  -> 
( G  e. UPGraph  /\  F  e.  _V  /\  P  e. 
_V ) )
7 eqid 2471 . . . . . . . . 9  |-  (Vtx `  G )  =  (Vtx
`  G )
8 eqid 2471 . . . . . . . . 9  |-  (iEdg `  G )  =  (iEdg `  G )
97, 8upgriswlk 39844 . . . . . . . 8  |-  ( ( G  e. UPGraph  /\  F  e. 
_V  /\  P  e.  _V )  ->  ( F (1Walks `  G ) P 
<->  ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
106, 9syl 17 . . . . . . 7  |-  ( ( G  e. USGraph  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  -> 
( F (1Walks `  G ) P  <->  ( F  e. Word  dom  (iEdg `  G
)  /\  P :
( 0 ... ( # `
 F ) ) --> (Vtx `  G )  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
11 c0ex 9655 . . . . . . . . . . . . . . . . . 18  |-  0  e.  _V
12 1ex 9656 . . . . . . . . . . . . . . . . . 18  |-  1  e.  _V
13 fveq2 5879 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
1413fveq2d 5883 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  0  ->  (
(iEdg `  G ) `  ( F `  k
) )  =  ( (iEdg `  G ) `  ( F `  0
) ) )
15 fveq2 5879 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
16 oveq1 6315 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
17 0p1e1 10743 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  +  1 )  =  1
1816, 17syl6eq 2521 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
1918fveq2d 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
2015, 19preq12d 4050 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
2114, 20eqeq12d 2486 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  0  ->  (
( (iEdg `  G
) `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) } ) )
22 fveq2 5879 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
2322fveq2d 5883 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  1  ->  (
(iEdg `  G ) `  ( F `  k
) )  =  ( (iEdg `  G ) `  ( F `  1
) ) )
24 fveq2 5879 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
25 oveq1 6315 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
26 1p1e2 10745 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  +  1 )  =  2
2725, 26syl6eq 2521 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
2827fveq2d 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
2924, 28preq12d 4050 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
3023, 29eqeq12d 2486 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  1  ->  (
( (iEdg `  G
) `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( (iEdg `  G ) `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) } ) )
3111, 12, 21, 30ralpr 4016 . . . . . . . . . . . . . . . . 17  |-  ( A. k  e.  { 0 ,  1 }  (
(iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  <-> 
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
32 simplll 776 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  ->  G  e. USGraph  )
33 fvex 5889 . . . . . . . . . . . . . . . . . . . 20  |-  ( P `
 0 )  e. 
_V
348usgrnloopv 39445 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  e. USGraph  /\  ( P `  0 )  e.  _V )  ->  (
( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  ->  ( P `  0 )  =/=  ( P `  1
) ) )
3532, 33, 34sylancl 675 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  ->  ( P `  0 )  =/=  ( P ` 
1 ) ) )
36 fvex 5889 . . . . . . . . . . . . . . . . . . . 20  |-  ( P `
 1 )  e. 
_V
378usgrnloopv 39445 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  e. USGraph  /\  ( P `  1 )  e.  _V )  ->  (
( (iEdg `  G
) `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  ( P `  1 )  =/=  ( P `  2
) ) )
3832, 36, 37sylancl 675 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( ( (iEdg `  G ) `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  ->  ( P `  1 )  =/=  ( P ` 
2 ) ) )
3935, 38anim12d 572 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( ( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
) ) ) )
40 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
(iEdg `  G ) `  ( F `  0
) )  =  ( (iEdg `  G ) `  ( F `  1
) ) )
4140eqeq1d 2473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  <->  ( (iEdg `  G ) `  ( F `  1 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) } ) )
42 eqtr2 2491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( (iEdg `  G
) `  ( F `  1 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
43 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { ( P ` 
2 ) ,  ( P `  1 ) }
4443eqeq2i 2483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( { ( P `  0
) ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  ( P `  2
) }  <->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { ( P ` 
2 ) ,  ( P `  1 ) } )
45 fvex 5889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 2 )  e. 
_V
4633, 45preqr1 4139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( { ( P `  0
) ,  ( P `
 1 ) }  =  { ( P `
 2 ) ,  ( P `  1
) }  ->  ( P `  0 )  =  ( P ` 
2 ) )
4744, 46sylbi 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( { ( P `  0
) ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  ( P `  0 )  =  ( P ` 
2 ) )
4842, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( (iEdg `  G
) `  ( F `  1 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( P `  0 )  =  ( P `  2
) )
4948ex 441 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  0
) ,  ( P `
 1 ) }  ->  ( ( (iEdg `  G ) `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  ->  ( P `  0 )  =  ( P ` 
2 ) ) )
5041, 49syl6bi 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  ->  (
( (iEdg `  G
) `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  ( P `  0 )  =  ( P ` 
2 ) ) ) )
5150impd 438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( P `  0 )  =  ( P `  2
) ) )
5251com12 31 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( ( F `  0 )  =  ( F ` 
1 )  ->  ( P `  0 )  =  ( P ` 
2 ) ) )
5352necon3d 2664 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( ( P `  0 )  =/=  ( P `  2
)  ->  ( F `  0 )  =/=  ( F `  1
) ) )
5453com12 31 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( P `  0 )  =/=  ( P ` 
2 )  ->  (
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( F `  0 )  =/=  ( F `  1
) ) )
5554adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( (
( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( F `  0 )  =/=  ( F `  1
) ) )
56 simpl 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  -> 
( P `  0
)  =/=  ( P `
 1 ) )
5756adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( P `  0 )  =/=  ( P `  1
) )
58 simpl 464 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( P `  0 )  =/=  ( P `  2
) )
59 simprr 774 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( P `  1 )  =/=  ( P `  2
) )
6057, 58, 593jca 1210 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) ) )
6155, 60jctild 552 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( P `  0
)  =/=  ( P `
 2 )  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )  ->  ( (
( (iEdg `  G
) `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  (
(iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) )
6261ex 441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( P `  0 )  =/=  ( P ` 
2 )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) )  ->  ( ( ( (iEdg `  G ) `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) }  /\  ( (iEdg `  G ) `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) } )  ->  ( ( ( P `  0 )  =/=  ( P ` 
1 )  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
6362com23 80 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =/=  ( P ` 
2 )  ->  (
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
6463adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G
) )  /\  ( P `  0 )  =/=  ( P `  2
) )  ->  (
( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
6564adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( ( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
6639, 65mpdd 40 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( ( ( (iEdg `  G ) `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( (iEdg `  G ) `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) )
6731, 66syl5bi 225 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G ) )  /\  ( P `  0 )  =/=  ( P ` 
2 ) )  /\  P : ( 0 ... 2 ) --> (Vtx `  G ) )  -> 
( A. k  e. 
{ 0 ,  1 }  ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) )
6867ex 441 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G
) )  /\  ( P `  0 )  =/=  ( P `  2
) )  ->  ( P : ( 0 ... 2 ) --> (Vtx `  G )  ->  ( A. k  e.  { 0 ,  1 }  (
(iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( ( ( P `  0 )  =/=  ( P ` 
1 )  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
6968com23 80 . . . . . . . . . . . . . 14  |-  ( ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G
) )  /\  ( P `  0 )  =/=  ( P `  2
) )  ->  ( A. k  e.  { 0 ,  1 }  (
(iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( P :
( 0 ... 2
) --> (Vtx `  G
)  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
7069ex 441 . . . . . . . . . . . . 13  |-  ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G )
)  ->  ( ( P `  0 )  =/=  ( P `  2
)  ->  ( A. k  e.  { 0 ,  1 }  (
(iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( P :
( 0 ... 2
) --> (Vtx `  G
)  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
71 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
7271neeq2d 2703 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
( P `  0
)  =/=  ( P `
 ( # `  F
) )  <->  ( P `  0 )  =/=  ( P `  2
) ) )
73 oveq2 6316 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
74 fzo0to2pr 12027 . . . . . . . . . . . . . . . . 17  |-  ( 0..^ 2 )  =  {
0 ,  1 }
7573, 74syl6eq 2521 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
7675raleqdv 2979 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  { 0 ,  1 }  ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
77 oveq2 6316 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
7877feq2d 5725 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  <->  P :
( 0 ... 2
) --> (Vtx `  G
) ) )
7978imbi1d 324 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
( P : ( 0 ... ( # `  F ) ) --> (Vtx
`  G )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) )  <->  ( P : ( 0 ... 2 ) --> (Vtx `  G )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) ) )
8076, 79imbi12d 327 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
( A. k  e.  ( 0..^ ( # `  F ) ) ( (iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( P :
( 0 ... ( # `
 F ) ) --> (Vtx `  G )  ->  ( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) )  <-> 
( A. k  e. 
{ 0 ,  1 }  ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P : ( 0 ... 2 ) --> (Vtx
`  G )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
8172, 80imbi12d 327 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) ) )  <->  ( ( P `  0 )  =/=  ( P `  2
)  ->  ( A. k  e.  { 0 ,  1 }  (
(iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( P :
( 0 ... 2
) --> (Vtx `  G
)  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) ) )
8270, 81syl5ibrcom 230 . . . . . . . . . . . 12  |-  ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G )
)  ->  ( ( # `
 F )  =  2  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( (iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( P :
( 0 ... ( # `
 F ) ) --> (Vtx `  G )  ->  ( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) ) )
8382impd 438 . . . . . . . . . . 11  |-  ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G )
)  ->  ( (
( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( A. k  e.  ( 0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) ) ) )
8483com24 89 . . . . . . . . . 10  |-  ( ( G  e. USGraph  /\  F  e. Word  dom  (iEdg `  G )
)  ->  ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( ( ( # `  F
)  =  2  /\  ( P `  0
)  =/=  ( P `
 ( # `  F
) ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
8584ex 441 . . . . . . . . 9  |-  ( G  e. USGraph  ->  ( F  e. Word  dom  (iEdg `  G )  ->  ( P : ( 0 ... ( # `  F ) ) --> (Vtx
`  G )  -> 
( A. k  e.  ( 0..^ ( # `  F ) ) ( (iEdg `  G ) `  ( F `  k
) )  =  {
( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  ->  ( ( (
# `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) ) )
86853impd 1247 . . . . . . . 8  |-  ( G  e. USGraph  ->  ( ( F  e. Word  dom  (iEdg `  G
)  /\  P :
( 0 ... ( # `
 F ) ) --> (Vtx `  G )  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  ( ( (
# `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
8786adantr 472 . . . . . . 7  |-  ( ( G  e. USGraph  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  -> 
( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( (iEdg `  G ) `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  ( ( (
# `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
8810, 87sylbid 223 . . . . . 6  |-  ( ( G  e. USGraph  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  -> 
( F (1Walks `  G ) P  -> 
( ( ( # `  F )  =  2  /\  ( P ` 
0 )  =/=  ( P `  ( # `  F
) ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
8988ex 441 . . . . 5  |-  ( G  e. USGraph  ->  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V )  ->  ( F (1Walks `  G ) P  ->  ( ( (
# `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
9089com13 82 . . . 4  |-  ( F (1Walks `  G ) P  ->  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V )  ->  ( G  e. USGraph  ->  ( ( ( # `  F
)  =  2  /\  ( P `  0
)  =/=  ( P `
 ( # `  F
) ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
911, 90mpd 15 . . 3  |-  ( F (1Walks `  G ) P  ->  ( G  e. USGraph  ->  ( ( ( # `  F )  =  2  /\  ( P ` 
0 )  =/=  ( P `  ( # `  F
) ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
9291impcom 437 . 2  |-  ( ( G  e. USGraph  /\  F (1Walks `  G ) P )  ->  ( ( (
# `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  /\  ( F `  0 )  =/=  ( F `  1
) ) ) )
9392imp 436 1  |-  ( ( ( G  e. USGraph  /\  F
(1Walks `  G ) P )  /\  (
( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) )  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) )
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   _Vcvv 3031   {cpr 3961   class class class wbr 4395   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560   2c2 10681   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703  Vtxcvtx 39251  iEdgciedg 39252   UPGraph cupgr 39326   USGraph cusgr 39397  1Walksc1wlks 39800
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-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-umgr 39329  df-edga 39372  df-uspgr 39398  df-usgr 39399  df-1wlks 39804  df-wlks 39805
This theorem is referenced by:  usgr2wlkspthlem1  39949  usgr2wlkspthlem2  39950
  Copyright terms: Public domain W3C validator