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Theorem rusgranumwwlkg 30745
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 6228 . . . . . 6  |-  ( V Walks 
E )  e.  _V
21rabex 4554 . . . . 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 5812 . . . . 5  |-  ( ( V WWalksN  E ) `  N
)  e.  _V
54rabex 4554 . . . 4  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  P }  e.  _V
63, 5jctil 537 . . 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 30716 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
87adantr 465 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  V USGrph  E )
9 simpr3 996 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  N  e.  NN0 )
10 simpr2 995 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  P  e.  V
)
11 wlknwwlknvbij 30540 . . . . 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 1219 . . . 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 30324 . . . 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 12242 . . 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 60 . 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 30744 . 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 2495 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {crab 2803   _Vcvv 3078   <.cop 3994   class class class wbr 4403   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   Fincfn 7423   0cc0 9396   NN0cn0 10693   ^cexp 11985   #chash 12223   USGrph cusg 23436   Walks cwalk 23577   WWalksN cwwlkn 30480   RegUSGrph crusgra 30708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-disj 4374  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-xadd 11204  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-word 12350  df-lsw 12351  df-concat 12352  df-s1 12353  df-substr 12354  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-usgra 23438  df-nbgra 23504  df-wlk 23587  df-vdgr 23736  df-wwlk 30481  df-wwlkn 30482  df-rgra 30709  df-rusgra 30710
This theorem is referenced by:  clwlknclwlkdifnum  30747
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