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Theorem rusgra0edg 25681
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 25657 . . 3  |-  ( <. V ,  E >. RegUSGrph  0  ->  V USGrph  E )
2 id 22 . . 3  |-  ( P  e.  V  ->  P  e.  V )
3 nnnn0 10883 . . 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 25678 . . 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 1306 . 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 2780 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  N )  /\  ( w `  0
)  =  P ) }
9 usgrav 25063 . . . . . . . . . . . . . 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 568 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
12113adant2 1024 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 984 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
1412, 13sylibr 215 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  N  e.  NN0 ) )
15 iswwlkn 25410 . . . . . . . . . . 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 25409 . . . . . . . . . . . . 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 1025 . . . . . . . . . . . 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 709 . . . . . . . . . . 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 256 . . . . . . . . . 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 709 . . . . . . . 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 6312 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  ( N  + 
1 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
23 nncn 10624 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
24 1cnd 9666 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  1  e.  CC )
2523, 24pncand 9994 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant3 1028 . . . . . . . . . . . . . . . . 17  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( N  +  1 )  - 
1 )  =  N )
2722, 26sylan9eqr 2485 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( # `  w )  -  1 )  =  N )
2827oveq2d 6321 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ N ) )
2928raleqdv 3028 . . . . . . . . . . . . . 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 25656 . . . . . . . . . . . . . . . . 17  |-  ( <. V ,  E >. RegUSGrph  0  -> 
<. V ,  E >. RegGrph  0 )
31 0eusgraiff0rgra 25665 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  <->  E  =  (/) ) )
32 rneq 5079 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( E  =  (/)  ->  ran  E  =  ran  (/) )
33 rn0 5105 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (/)  =  (/)
3432, 33syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( E  =  (/)  ->  ran  E  =  (/) )
3534eleq2d 2492 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <->  { (
w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  (/) ) )
36 noel 3765 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  -.  {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)
3736bifal 1450 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)  <-> F.  )
3835, 37syl6bb 264 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
3938adantr 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
4039ralbidv 2861 . . . . . . . . . . . . . . . . . . . 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 1444 . . . . . . . . . . . . . . . . . . . . . 22  |-  -. F.
4241ralf0 3906 . . . . . . . . . . . . . . . . . . . . 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 10891 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10645 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  <  N )
4845, 46, 473jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  (
0  e.  NN0  /\  N  e.  NN  /\  0  <  N ) )
4948ad2antll 733 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0  e.  NN0  /\  N  e.  NN  /\  0  < 
N ) )
50 elfzo0 11963 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  e.  ( 0..^ N )  <->  ( 0  e. 
NN0  /\  N  e.  NN  /\  0  <  N
) )
5149, 50sylibr 215 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  0  e.  ( 0..^ N ) )
52 fzon0 11944 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0..^ N )  =/=  (/) 
<->  0  e.  ( 0..^ N ) )
5351, 52sylibr 215 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0..^ N )  =/=  (/) )
5453neneqd 2621 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  -.  (
0..^ N )  =  (/) )
55 nbfal 1448 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( 0..^ N )  =  (/)  <->  ( ( 0..^ N )  =  (/)  <-> F.  ) )
5654, 55sylib 199 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( (
0..^ N )  =  (/) 
<-> F.  ) )
5740, 43, 563bitrd 282 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
5857ex 435 . . . . . . . . . . . . . . . . . 18  |-  ( E  =  (/)  ->  ( ( P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) ) )
5931, 58syl6bi 231 . . . . . . . . . . . . . . . . 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 62 . . . . . . . . . . . . . . . 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 1203 . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 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 1341 . . . . . . . . . . . 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 984 . . . . . . . . . . . . 13  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) )
66 ancom 451 . . . . . . . . . . . . 13  |-  ( ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) 
<->  ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) ) )
6765, 66bitri 252 . . . . . . . . . . . 12  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) )
6864, 67syl6bb 264 . . . . . . . . . . 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 435 . . . . . . . . . 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 644 . . . . . . . . 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 709 . . . . . . . 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 653 . . . . . . . . . 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 653 . . . . . . . . . . 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 923 . . . . . . . . . . . 12  |-  -.  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )
7574bifal 1450 . . . . . . . . . . 11  |-  ( ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )  <-> F.  )
7673, 75bitri 252 . . . . . . . . . 10  |-  ( ( ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( ( # `  w
)  =  ( N  +  1 )  /\  ( w `  0
)  =  P ) )  <-> F.  )
7772, 76bitri 252 . . . . . . . . 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 282 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( w  e.  ( ( V WWalksN  E ) `  N
)  /\  ( w `  0 )  =  P )  <-> F.  )
)
8079abbidv 2553 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  { w  | F.  } )
8141abf 3798 . . . . . 6  |-  { w  | F.  }  =  (/)
8280, 81syl6eq 2479 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  (/) )
838, 82syl5eq 2475 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  (/) )
8483fveq2d 5885 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  (/) ) )
85 hash0 12554 . . 3  |-  ( # `  (/) )  =  0
8684, 85syl6eq 2479 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  0 )
877, 86eqtrd 2463 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 187    /\ wa 370    /\ w3a 982    = wceq 1437   F. wfal 1442    e. wcel 1872   {cab 2407    =/= wne 2614   A.wral 2771   {crab 2775   _Vcvv 3080   (/)c0 3761   {cpr 4000   <.cop 4004   class class class wbr 4423    |-> cmpt 4482   ran crn 4854   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   0cc0 9546   1c1 9547    + caddc 9549    < clt 9682    - cmin 9867   NNcn 10616   NN0cn0 10876  ..^cfzo 11922   #chash 12521  Word cword 12660   USGrph cusg 25055   Walks cwalk 25224   WWalks cwwlk 25403   WWalksN cwwlkn 25404   RegGrph crgra 25648   RegUSGrph crusgra 25649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-xadd 11417  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-usgra 25058  df-wlk 25234  df-wwlk 25405  df-wwlkn 25406  df-vdgr 25620  df-rgra 25650  df-rusgra 25651
This theorem is referenced by: (None)
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