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Theorem rusgra0edg 30498
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 30473 . . 3  |-  ( <. V ,  E >. RegUSGrph  0  ->  V USGrph  E )
2 id 22 . . 3  |-  ( P  e.  V  ->  P  e.  V )
3 nnnn0 10582 . . 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 30495 . . 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 1255 . 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 2722 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  N )  /\  ( w `  0
)  =  P ) }
9 usgrav 23205 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
101, 9syl 16 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  0  ->  ( V  e.  _V  /\  E  e.  _V )
)
1110, 3anim12i 563 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
12113adant2 1002 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 962 . . . . . . . . . . 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 30243 . . . . . . . . . . 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 30242 . . . . . . . . . . . . 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 1003 . . . . . . . . . . . 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 699 . . . . . . . . . . 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 16 . . . . . . . . 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 699 . . . . . . . 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 6097 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  ( N  + 
1 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
23 nncn 10326 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
24 ax-1cn 9336 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
2524a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  1  e.  CC )
2623, 25pncand 9716 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
27263ad2ant3 1006 . . . . . . . . . . . . . . . . 17  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( N  +  1 )  - 
1 )  =  N )
2822, 27sylan9eqr 2495 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( # `  w )  -  1 )  =  N )
2928oveq2d 6106 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ N ) )
3029raleqdv 2921 . . . . . . . . . . . . . 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 ) )
31 rusgrargra 30472 . . . . . . . . . . . . . . . . 17  |-  ( <. V ,  E >. RegUSGrph  0  -> 
<. V ,  E >. RegGrph  0 )
32 0eusgraiff0rgra 30477 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  <->  E  =  (/) ) )
33 rneq 5061 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( E  =  (/)  ->  ran  E  =  ran  (/) )
34 rn0 5087 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (/)  =  (/)
3533, 34syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( E  =  (/)  ->  ran  E  =  (/) )
3635eleq2d 2508 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <->  { (
w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  (/) ) )
37 noel 3638 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  -.  {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)
3837bifal 1377 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)  <-> F.  )
3938a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)  <-> F.  ) )
4036, 39bitrd 253 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
4140adantr 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
4241ralbidv 2733 . . . . . . . . . . . . . . . . . . . 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.  ) )
43 fal 1371 . . . . . . . . . . . . . . . . . . . . . 22  |-  -. F.
4443ralf0 3783 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. i  e.  ( 0..^ N ) F.  <->  ( 0..^ N )  =  (/) )
4544a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) F.  <->  ( 0..^ N )  =  (/) ) )
46 0nn0 10590 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  0  e.  NN0
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  e.  NN0 )
48 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  N  e.  NN )
49 nngt0 10347 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  <  N )
5047, 48, 493jca 1163 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  (
0  e.  NN0  /\  N  e.  NN  /\  0  <  N ) )
5150ad2antll 723 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0  e.  NN0  /\  N  e.  NN  /\  0  < 
N ) )
52 elfzo0 11583 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  e.  ( 0..^ N )  <->  ( 0  e. 
NN0  /\  N  e.  NN  /\  0  <  N
) )
5351, 52sylibr 212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  0  e.  ( 0..^ N ) )
54 fzon0 11565 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0..^ N )  =/=  (/) 
<->  0  e.  ( 0..^ N ) )
5553, 54sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0..^ N )  =/=  (/) )
5655neneqd 2622 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  -.  (
0..^ N )  =  (/) )
57 nbfal 1375 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( 0..^ N )  =  (/)  <->  ( ( 0..^ N )  =  (/)  <-> F.  ) )
5856, 57sylib 196 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( (
0..^ N )  =  (/) 
<-> F.  ) )
5942, 45, 583bitrd 279 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) )
6059ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( E  =  (/)  ->  ( ( P  e.  V  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ N ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <-> F.  ) ) )
6132, 60syl6bi 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.  ) ) ) )
621, 31, 61sylc 60 . . . . . . . . . . . . . . . 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.  )
) )
63623impib 1180 . . . . . . . . . . . . . . 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.  ) )
6463adantr 462 . . . . . . . . . . . . . 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.  ) )
6530, 64bitrd 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.  ) )
66653anbi3d 1290 . . . . . . . . . . . 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.  ) ) )
67 df-3an 962 . . . . . . . . . . . . . 14  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) )
68 ancom 448 . . . . . . . . . . . . . 14  |-  ( ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) 
<->  ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) ) )
6967, 68bitri 249 . . . . . . . . . . . . 13  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) )
7069a1i 11 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) ) ) )
7166, 70bitrd 253 . . . . . . . . . . 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 ) ) ) )
7271ex 434 . . . . . . . . . 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 ) ) ) ) )
7372pm5.32rd 635 . . . . . . . . 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 ) ) ) )
7473anbi1d 699 . . . . . . . 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 ) ) )
75 anass 644 . . . . . . . . . 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 ) ) )
76 anass 644 . . . . . . . . . . 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 ) ) ) )
7743intnanr 901 . . . . . . . . . . . 12  |-  -.  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )
7877bifal 1377 . . . . . . . . . . 11  |-  ( ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )  <-> F.  )
7976, 78bitri 249 . . . . . . . . . 10  |-  ( ( ( F.  /\  (
w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( ( # `  w
)  =  ( N  +  1 )  /\  ( w `  0
)  =  P ) )  <-> F.  )
8075, 79bitri 249 . . . . . . . . 9  |-  ( ( ( ( F.  /\  ( w  =/=  (/)  /\  w  e. Word  V ) )  /\  ( # `  w )  =  ( N  + 
1 ) )  /\  ( w `  0
)  =  P )  <-> F.  )
8180a1i 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.  ) )
8221, 74, 813bitrd 279 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( w  e.  ( ( V WWalksN  E ) `  N
)  /\  ( w `  0 )  =  P )  <-> F.  )
)
8382abbidv 2555 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  { w  | F.  } )
8443abf 3668 . . . . . 6  |-  { w  | F.  }  =  (/)
8583, 84syl6eq 2489 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  (/) )
868, 85syl5eq 2485 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  (/) )
8786fveq2d 5692 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  (/) ) )
88 hash0 12131 . . 3  |-  ( # `  (/) )  =  0
8987, 88syl6eq 2489 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  0 )
907, 89eqtrd 2473 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 369    /\ w3a 960    = wceq 1364   F. wfal 1369    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970   (/)c0 3634   {cpr 3876   <.cop 3880   class class class wbr 4289    e. cmpt 4347   ran crn 4837   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    - cmin 9591   NNcn 10318   NN0cn0 10575  ..^cfzo 11544   #chash 12099  Word cword 12217   USGrph cusg 23199   Walks cwalk 23340   WWalks cwwlk 30236   WWalksN cwwlkn 30237   RegGrph crgra 30464   RegUSGrph crusgra 30465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-xadd 11086  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-usgra 23201  df-wlk 23350  df-vdgr 23499  df-wwlk 30238  df-wwlkn 30239  df-rgra 30466  df-rusgra 30467
This theorem is referenced by: (None)
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