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Theorem rusgra0edg 24631
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 24607 . . 3  |-  ( <. V ,  E >. RegUSGrph  0  ->  V USGrph  E )
2 id 22 . . 3  |-  ( P  e.  V  ->  P  e.  V )
3 nnnn0 10798 . . 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 24628 . . 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 1270 . 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 2823 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  N )  /\  ( w `  0
)  =  P ) }
9 usgrav 24014 . . . . . . . . . . . . . 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 566 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
12113adant2 1015 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 975 . . . . . . . . . . 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 24360 . . . . . . . . . . 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 24359 . . . . . . . . . . . . 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 1016 . . . . . . . . . . . 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 704 . . . . . . . . . . 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 704 . . . . . . . 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 6289 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  ( N  + 
1 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
23 nncn 10540 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
24 ax-1cn 9546 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
2524a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  1  e.  CC )
2623, 25pncand 9927 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
27263ad2ant3 1019 . . . . . . . . . . . . . . . . 17  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( N  +  1 )  - 
1 )  =  N )
2822, 27sylan9eqr 2530 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( # `  w )  -  1 )  =  N )
2928oveq2d 6298 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ N ) )
3029raleqdv 3064 . . . . . . . . . . . . . 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 24606 . . . . . . . . . . . . . . . . 17  |-  ( <. V ,  E >. RegUSGrph  0  -> 
<. V ,  E >. RegGrph  0 )
32 0eusgraiff0rgra 24615 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  <->  E  =  (/) ) )
33 rneq 5226 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( E  =  (/)  ->  ran  E  =  ran  (/) )
34 rn0 5252 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (/)  =  (/)
3533, 34syl6eq 2524 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( E  =  (/)  ->  ran  E  =  (/) )
3635eleq2d 2537 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <->  { (
w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  (/) ) )
37 noel 3789 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  -.  {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)
3837bifal 1392 . . . . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <-> F.  )
)
4241ralbidv 2903 . . . . . . . . . . . . . . . . . . . 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 1386 . . . . . . . . . . . . . . . . . . . . . 22  |-  -. F.
4443ralf0 3934 . . . . . . . . . . . . . . . . . . . . 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 10806 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10561 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  <  N )
5047, 48, 493jca 1176 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  (
0  e.  NN0  /\  N  e.  NN  /\  0  <  N ) )
5150ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0  e.  NN0  /\  N  e.  NN  /\  0  < 
N ) )
52 elfzo0 11827 . . . . . . . . . . . . . . . . . . . . . . . 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 11809 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0..^ N )  =/=  (/) 
<->  0  e.  ( 0..^ N ) )
5553, 54sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0..^ N )  =/=  (/) )
5655neneqd 2669 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  -.  (
0..^ N )  =  (/) )
57 nbfal 1390 . . . . . . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 1305 . . . . . . . . . . . 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 975 . . . . . . . . . . . . . 14  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\ F.  )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\ F.  ) )
68 ancom 450 . . . . . . . . . . . . . 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 640 . . . . . . . . 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 704 . . . . . . . 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 649 . . . . . . . . . 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 649 . . . . . . . . . . 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 913 . . . . . . . . . . . 12  |-  -.  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )
7877bifal 1392 . . . . . . . . . . 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 2603 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  { w  | F.  } )
8443abf 3819 . . . . . 6  |-  { w  | F.  }  =  (/)
8583, 84syl6eq 2524 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  (/) )
868, 85syl5eq 2520 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  (/) )
8786fveq2d 5868 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  (/) ) )
88 hash0 12401 . . 3  |-  ( # `  (/) )  =  0
8987, 88syl6eq 2524 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  0 )
907, 89eqtrd 2508 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 973    = wceq 1379   F. wfal 1384    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    - cmin 9801   NNcn 10532   NN0cn0 10791  ..^cfzo 11788   #chash 12369  Word cword 12496   USGrph cusg 24006   Walks cwalk 24174   WWalks cwwlk 24353   WWalksN cwwlkn 24354   RegGrph crgra 24598   RegUSGrph crusgra 24599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-xadd 11315  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-usgra 24009  df-wlk 24184  df-wwlk 24355  df-wwlkn 24356  df-vdgr 24570  df-rgra 24600  df-rusgra 24601
This theorem is referenced by: (None)
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