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Theorem rusgranumwwlkg 25258
Description: In a k-regular graph, the number of walks (represented by words) of a fixed length n from a fixed vertex is k to the power of n. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Assertion
Ref Expression
rusgranumwwlkg  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( K ^ N
) )
Distinct variable groups:    w, E    w, N    w, P    w, V
Allowed substitution hint:    K( w)

Proof of Theorem rusgranumwwlkg
Dummy variables  f 
g  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6260 . . . . . 6  |-  ( V Walks 
E )  e.  _V
21rabex 4542 . . . . 5  |-  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) }  e.  _V
32a1i 11 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) }  e.  _V )
4 fvex 5813 . . . . 5  |-  ( ( V WWalksN  E ) `  N
)  e.  _V
54rabex 4542 . . . 4  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V
63, 5jctil 535 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V  /\  { p  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  e.  _V ) )
7 rusisusgra 25230 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
87adantr 463 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  V USGrph  E )
9 simpr3 1003 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  N  e.  NN0 )
10 simpr2 1002 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  P  e.  V
)
11 wlknwwlknvbij 25039 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  P  e.  V )  ->  E. g 
g : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P } )
128, 9, 10, 11syl3anc 1228 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  E. g  g : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P } )
13 f1oexbi 6686 . . . 4  |-  ( E. f  f : {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } -1-1-onto-> { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  <->  E. g 
g : { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) } -1-1-onto-> { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P } )
1412, 13sylibr 212 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  E. f  f : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P } -1-1-onto-> { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) } )
15 hasheqf1oi 12376 . . 3  |-  ( ( { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }  e.  _V  /\  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) }  e.  _V )  ->  ( E. f  f : { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P } -1-1-onto-> { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) }  ->  (
# `  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P } )  =  (
# `  { p  e.  ( V Walks  E )  |  ( ( # `  ( 1st `  p
) )  =  N  /\  ( ( 2nd `  p ) `  0
)  =  P ) } ) ) )
166, 14, 15sylc 59 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( # `  {
p  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) } ) )
17 rusgranumwlkg 25257 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( # `  {
p  e.  ( V Walks 
E )  |  ( ( # `  ( 1st `  p ) )  =  N  /\  (
( 2nd `  p
) `  0 )  =  P ) } )  =  ( K ^ N ) )
1816, 17eqtrd 2441 1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  P } )  =  ( K ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403   E.wex 1631    e. wcel 1840   {crab 2755   _Vcvv 3056   <.cop 3975   class class class wbr 4392   -1-1-onto->wf1o 5522   ` cfv 5523  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   Fincfn 7472   0cc0 9440   NN0cn0 10754   ^cexp 12118   #chash 12357   USGrph cusg 24629   Walks cwalk 24797   WWalksN cwwlkn 24977   RegUSGrph crusgra 25222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-disj 4364  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-xadd 11288  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-hash 12358  df-word 12496  df-lsw 12497  df-concat 12498  df-s1 12499  df-substr 12500  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563  df-usgra 24632  df-nbgra 24719  df-wlk 24807  df-wwlk 24978  df-wwlkn 24979  df-vdgr 25193  df-rgra 25223  df-rusgra 25224
This theorem is referenced by:  clwlknclwlkdifnum  25260
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