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Theorem rusgra0edg 25359
Description: Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
rusgranumwlk.l  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
Assertion
Ref Expression
rusgra0edg  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( P L N )  =  0 )
Distinct variable groups:    E, c, n    N, c, n    V, c, n    v, N, w    P, n, v, w    v, V    n, W, v, w   
w, V, c    v, E, w
Allowed substitution hints:    P( c)    L( w, v, n, c)    W( c)

Proof of Theorem rusgra0edg
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 rusisusgra 25335 . . 3  |-  ( <. V ,  E >. RegUSGrph  0  ->  V USGrph  E )
2 id 22 . . 3  |-  ( P  e.  V  ->  P  e.  V )
3 nnnn0 10842 . . 3  |-  ( N  e.  NN  ->  N  e.  NN0 )
4 rusgranumwlk.w . . . 4  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
5 rusgranumwlk.l . . . 4  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
64, 5rusgranumwlklem4 25356 . . 3  |-  ( ( V USGrph  E  /\  P  e.  V  /\  N  e. 
NN0 )  ->  ( P L N )  =  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } ) )
71, 2, 3, 6syl3an 1272 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( P L N )  =  (
# `  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P } ) )
8 df-rab 2762 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  N )  /\  ( w `  0
)  =  P ) }
9 usgrav 24742 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
101, 9syl 17 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  0  ->  ( V  e.  _V  /\  E  e.  _V )
)
1110, 3anim12i 564 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
12113adant2 1016 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 976 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
1412, 13sylibr 212 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  N  e.  NN0 ) )
15 iswwlkn 25088 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
w  e.  ( ( V WWalksN  E ) `  N
)  <->  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
16 iswwlk 25087 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
17163adant3 1017 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
1817anbi1d 703 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( w  e.  ( V WWalks  E )  /\  ( # `  w )  =  ( N  + 
1 ) )  <->  ( (
w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
1915, 18bitrd 253 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
w  e.  ( ( V WWalksN  E ) `  N
)  <->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
2014, 19syl 17 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( w  e.  ( ( V WWalksN  E
) `  N )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
2120anbi1d 703 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( w  e.  ( ( V WWalksN  E ) `  N
)  /\  ( w `  0 )  =  P )  <->  ( (
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  /\  ( w ` 
0 )  =  P ) ) )
22 oveq1 6284 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  ( N  + 
1 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
23 nncn 10583 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
24 1cnd 9641 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  1  e.  CC )
2523, 24pncand 9967 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant3 1020 . . . . . . . . . . . . . . . . 17  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( N  +  1 )  - 
1 )  =  N )
2722, 26sylan9eqr 2465 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( # `  w )  -  1 )  =  N )
2827oveq2d 6293 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ N ) )
2928raleqdv 3009 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E ) )
30 rusgrargra 25334 . . . . . . . . . . . . . . . . 17  |-  ( <. V ,  E >. RegUSGrph  0  -> 
<. V ,  E >. RegGrph  0 )
31 0eusgraiff0rgra 25343 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  <->  E  =  (/) ) )
32 rneq 5048 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( E  =  (/)  ->  ran  E  =  ran  (/) )
33 rn0 5074 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (/)  =  (/)
3432, 33syl6eq 2459 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( E  =  (/)  ->  ran  E  =  (/) )
3534eleq2d 2472 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <->  { (
w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  (/) ) )
36 noel 3741 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  -.  {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)
3736bifal 1418 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)  <-> F.  )
3835, 37syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
3938adantr 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
4039ralbidv 2842 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ N ) F.  ) )
41 fal 1412 . . . . . . . . . . . . . . . . . . . . . 22  |-  -. F.
4241ralf0 3879 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. i  e.  ( 0..^ N ) F.  <->  ( 0..^ N )  =  (/) )
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) F.  <->  ( 0..^ N )  =  (/) ) )
44 0nn0 10850 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  0  e.  NN0
4544a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  e.  NN0 )
46 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  N  e.  NN )
47 nngt0 10604 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  <  N )
4845, 46, 473jca 1177 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  (
0  e.  NN0  /\  N  e.  NN  /\  0  <  N ) )
4948ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0  e.  NN0  /\  N  e.  NN  /\  0  < 
N ) )
50 elfzo0 11893 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  e.  ( 0..^ N )  <->  ( 0  e. 
NN0  /\  N  e.  NN  /\  0  <  N
) )
5149, 50sylibr 212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  0  e.  ( 0..^ N ) )
52 fzon0 11874 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0..^ N )  =/=  (/) 
<->  0  e.  ( 0..^ N ) )
5351, 52sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0..^ N )  =/=  (/) )
5453neneqd 2605 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  -.  (
0..^ N )  =  (/) )
55 nbfal 1416 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( 0..^ N )  =  (/)  <->  ( ( 0..^ N )  =  (/)  <-> F.  ) )
5654, 55sylib 196 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( (
0..^ N )  =  (/) 
<-> F.  ) )
5740, 43, 563bitrd 279 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
5857ex 432 . . . . . . . . . . . . . . . . . 18  |-  ( E  =  (/)  ->  ( ( P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) ) )
5931, 58syl6bi 228 . . . . . . . . . . . . . . . . 17  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  ->  ( ( P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) ) ) )
601, 30, 59sylc 59 . . . . . . . . . . . . . . . 16  |-  ( <. V ,  E >. RegUSGrph  0  ->  ( ( P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  (
0..^ N ) { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
) )
61603impib 1195 . . . . . . . . . . . . . . 15  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
6261adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
6329, 62bitrd 253 . . . . . . . . . . . . 13  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
64633anbi3d 1307 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  ) ) )
65 df-3an 976 . . . . . . . . . . . . 13  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) )
66 ancom 448 . . . . . . . . . . . . 13  |-  ( ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) 
<->  ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) ) )
6765, 66bitri 249 . . . . . . . . . . . 12  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) )
6864, 67syl6bb 261 . . . . . . . . . . 11  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) ) )
6968ex 432 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( # `  w )  =  ( N  +  1 )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) ) ) )
7069pm5.32rd 638 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  <-> 
( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) ) ) )
7170anbi1d 703 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  /\  ( w ` 
0 )  =  P )  <->  ( ( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) )  /\  ( w `  0
)  =  P ) ) )
72 anass 647 . . . . . . . . . 10  |-  ( ( ( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) )  /\  ( w `  0
)  =  P )  <-> 
( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( ( # `  w
)  =  ( N  +  1 )  /\  ( w `  0
)  =  P ) ) )
73 anass 647 . . . . . . . . . . 11  |-  ( ( ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( ( # `  w
)  =  ( N  +  1 )  /\  ( w `  0
)  =  P ) )  <->  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V
)  /\  ( ( # `
 w )  =  ( N  +  1 )  /\  ( w `
 0 )  =  P ) ) ) )
7441intnanr 916 . . . . . . . . . . . 12  |-  -.  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )
7574bifal 1418 . . . . . . . . . . 11  |-  ( ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )  <-> F.  )
7673, 75bitri 249 . . . . . . . . . 10  |-  ( ( ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( ( # `  w
)  =  ( N  +  1 )  /\  ( w `  0
)  =  P ) )  <-> F.  )
7772, 76bitri 249 . . . . . . . . 9  |-  ( ( ( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) )  /\  ( w `  0
)  =  P )  <-> F.  )
7877a1i 11 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( ( ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) )  /\  ( w `  0
)  =  P )  <-> F.  ) )
7921, 71, 783bitrd 279 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( w  e.  ( ( V WWalksN  E ) `  N
)  /\  ( w `  0 )  =  P )  <-> F.  )
)
8079abbidv 2538 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  { w  | F.  } )
8141abf 3772 . . . . . 6  |-  { w  | F.  }  =  (/)
8280, 81syl6eq 2459 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  (/) )
838, 82syl5eq 2455 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  (/) )
8483fveq2d 5852 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  (/) ) )
85 hash0 12483 . . 3  |-  ( # `  (/) )  =  0
8684, 85syl6eq 2459 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  0 )
877, 86eqtrd 2443 1  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( P L N )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   F. wfal 1410    e. wcel 1842   {cab 2387    =/= wne 2598   A.wral 2753   {crab 2757   _Vcvv 3058   (/)c0 3737   {cpr 3973   <.cop 3977   class class class wbr 4394    |-> cmpt 4452   ran crn 4823   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6781   2ndc2nd 6782   0cc0 9521   1c1 9522    + caddc 9524    < clt 9657    - cmin 9840   NNcn 10575   NN0cn0 10835  ..^cfzo 11852   #chash 12450  Word cword 12581   USGrph cusg 24734   Walks cwalk 24902   WWalks cwwlk 25081   WWalksN cwwlkn 25082   RegGrph crgra 25326   RegUSGrph crusgra 25327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-xadd 11371  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-usgra 24737  df-wlk 24912  df-wwlk 25083  df-wwlkn 25084  df-vdgr 25298  df-rgra 25328  df-rusgra 25329
This theorem is referenced by: (None)
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