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Theorem rusgra0edg 30578
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 30553 . . 3  |-  ( <. V ,  E >. RegUSGrph  0  ->  V USGrph  E )
2 id 22 . . 3  |-  ( P  e.  V  ->  P  e.  V )
3 nnnn0 10591 . . 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 30575 . . 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 1260 . 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 2729 . . . . 5  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  N )  /\  ( w `  0
)  =  P ) }
9 usgrav 23275 . . . . . . . . . . . . . 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 1007 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 967 . . . . . . . . . . 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 30323 . . . . . . . . . . 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 30322 . . . . . . . . . . . . 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 1008 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  ( N  + 
1 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
23 nncn 10335 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
24 ax-1cn 9345 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
2524a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  1  e.  CC )
2623, 25pncand 9725 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
27263ad2ant3 1011 . . . . . . . . . . . . . . . . 17  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( ( N  +  1 )  - 
1 )  =  N )
2822, 27sylan9eqr 2497 . . . . . . . . . . . . . . . 16  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( ( # `  w )  -  1 )  =  N )
2928oveq2d 6112 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph 
0  /\  P  e.  V  /\  N  e.  NN )  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ N ) )
3029raleqdv 2928 . . . . . . . . . . . . . 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 30552 . . . . . . . . . . . . . . . . 17  |-  ( <. V ,  E >. RegUSGrph  0  -> 
<. V ,  E >. RegGrph  0 )
32 0eusgraiff0rgra 30557 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  ( <. V ,  E >. RegGrph  0  <->  E  =  (/) ) )
33 rneq 5070 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( E  =  (/)  ->  ran  E  =  ran  (/) )
34 rn0 5096 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (/)  =  (/)
3533, 34syl6eq 2491 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( E  =  (/)  ->  ran  E  =  (/) )
3635eleq2d 2510 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E  =  (/)  ->  ( { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E  <->  { (
w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  (/) ) )
37 noel 3646 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  -.  {
( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  (/)
3837bifal 1382 . . . . . . . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . . . . . . . . . 22  |-  -. F.
4443ralf0 3791 . . . . . . . . . . . . . . . . . . . . 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 10599 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10356 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN  ->  0  <  N )
5047, 48, 493jca 1168 . . . . . . . . . . . . . . . . . . . . . . . . 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 11592 . . . . . . . . . . . . . . . . . . . . . . . 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 11574 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0..^ N )  =/=  (/) 
<->  0  e.  ( 0..^ N ) )
5553, 54sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  ( 0..^ N )  =/=  (/) )
5655neneqd 2629 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E  =  (/)  /\  ( P  e.  V  /\  N  e.  NN )
)  ->  -.  (
0..^ N )  =  (/) )
57 nbfal 1380 . . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . 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 1295 . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 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 906 . . . . . . . . . . . 12  |-  -.  ( F.  /\  ( ( w  =/=  (/)  /\  w  e. Word  V )  /\  (
( # `  w )  =  ( N  + 
1 )  /\  (
w `  0 )  =  P ) ) )
7877bifal 1382 . . . . . . . . . . 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 2562 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  { w  | F.  } )
8443abf 3676 . . . . . 6  |-  { w  | F.  }  =  (/)
8583, 84syl6eq 2491 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  |  ( w  e.  ( ( V WWalksN  E ) `  N )  /\  (
w `  0 )  =  P ) }  =  (/) )
868, 85syl5eq 2487 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  =  (/) )
8786fveq2d 5700 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  (/) ) )
88 hash0 12140 . . 3  |-  ( # `  (/) )  =  0
8987, 88syl6eq 2491 . 2  |-  ( (
<. V ,  E >. RegUSGrph  0  /\  P  e.  V  /\  N  e.  NN )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  0 )
907, 89eqtrd 2475 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 965    = wceq 1369   F. wfal 1374    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977   (/)c0 3642   {cpr 3884   <.cop 3888   class class class wbr 4297    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    < clt 9423    - cmin 9600   NNcn 10327   NN0cn0 10584  ..^cfzo 11553   #chash 12108  Word cword 12226   USGrph cusg 23269   Walks cwalk 23410   WWalks cwwlk 30316   WWalksN cwwlkn 30317   RegGrph crgra 30544   RegUSGrph crusgra 30545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-xadd 11095  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-usgra 23271  df-wlk 23420  df-vdgr 23569  df-wwlk 30318  df-wwlkn 30319  df-rgra 30546  df-rusgra 30547
This theorem is referenced by: (None)
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