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Theorem rusgranumwlk 24780
Description: In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. We denote with  ( W `  n ) the set of walks with length n (in a given undirected simple graph) and with  ( v L n ) the number of walks with length n starting at the vertex v. This theorem corresponds to the following observation in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". Because of the k-regularity, the walk can be continued in k different ways at each vertex in the walk, therefore n times. This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-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
rusgranumwlk  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( P L N )  =  ( K ^ N ) )
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    w, K
Allowed substitution hints:    P( c)    K( v, n, c)    L( w, v, n, c)    W( c)

Proof of Theorem rusgranumwlk
Dummy variables  i 
y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6303 . . . . . . . 8  |-  ( x  =  0  ->  ( P L x )  =  ( P L 0 ) )
2 oveq2 6303 . . . . . . . 8  |-  ( x  =  0  ->  ( K ^ x )  =  ( K ^ 0 ) )
31, 2eqeq12d 2489 . . . . . . 7  |-  ( x  =  0  ->  (
( P L x )  =  ( K ^ x )  <->  ( P L 0 )  =  ( K ^ 0 ) ) )
43imbi2d 316 . . . . . 6  |-  ( x  =  0  ->  (
( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L x )  =  ( K ^ x ) )  <->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L 0 )  =  ( K ^ 0 ) ) ) )
5 oveq2 6303 . . . . . . . 8  |-  ( x  =  y  ->  ( P L x )  =  ( P L y ) )
6 oveq2 6303 . . . . . . . 8  |-  ( x  =  y  ->  ( K ^ x )  =  ( K ^ y
) )
75, 6eqeq12d 2489 . . . . . . 7  |-  ( x  =  y  ->  (
( P L x )  =  ( K ^ x )  <->  ( P L y )  =  ( K ^ y
) ) )
87imbi2d 316 . . . . . 6  |-  ( x  =  y  ->  (
( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L x )  =  ( K ^ x ) )  <->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L y )  =  ( K ^ y
) ) ) )
9 oveq2 6303 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( P L x )  =  ( P L ( y  +  1 ) ) )
10 oveq2 6303 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( K ^ x )  =  ( K ^ (
y  +  1 ) ) )
119, 10eqeq12d 2489 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
( P L x )  =  ( K ^ x )  <->  ( P L ( y  +  1 ) )  =  ( K ^ (
y  +  1 ) ) ) )
1211imbi2d 316 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  (
( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L x )  =  ( K ^ x ) )  <->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L ( y  +  1 ) )  =  ( K ^ (
y  +  1 ) ) ) ) )
13 oveq2 6303 . . . . . . . 8  |-  ( x  =  N  ->  ( P L x )  =  ( P L N ) )
14 oveq2 6303 . . . . . . . 8  |-  ( x  =  N  ->  ( K ^ x )  =  ( K ^ N
) )
1513, 14eqeq12d 2489 . . . . . . 7  |-  ( x  =  N  ->  (
( P L x )  =  ( K ^ x )  <->  ( P L N )  =  ( K ^ N ) ) )
1615imbi2d 316 . . . . . 6  |-  ( x  =  N  ->  (
( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L x )  =  ( K ^ x ) )  <->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L N )  =  ( K ^ N
) ) ) )
17 rusisusgra 24754 . . . . . . . 8  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
18 simpr 461 . . . . . . . 8  |-  ( ( V  e.  Fin  /\  P  e.  V )  ->  P  e.  V )
19 rusgranumwlk.w . . . . . . . . 9  |-  W  =  ( n  e.  NN0  |->  { c  e.  ( V Walks  E )  |  ( # `  ( 1st `  c ) )  =  n } )
20 rusgranumwlk.l . . . . . . . . 9  |-  L  =  ( v  e.  V ,  n  e.  NN0  |->  ( # `  { w  e.  ( W `  n
)  |  ( ( 2nd `  w ) `
 0 )  =  v } ) )
2119, 20rusgranumwlkb0 24776 . . . . . . . 8  |-  ( ( V USGrph  E  /\  P  e.  V )  ->  ( P L 0 )  =  1 )
2217, 18, 21syl2anr 478 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L 0 )  =  1 )
23 rusgraprop 24752 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. i  e.  V  (
( V VDeg  E ) `  i )  =  K ) )
24 nn0cn 10817 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  K  e.  CC )
25243ad2ant2 1018 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. i  e.  V  ( ( V VDeg  E ) `  i
)  =  K )  ->  K  e.  CC )
26 exp0 12150 . . . . . . . . . 10  |-  ( K  e.  CC  ->  ( K ^ 0 )  =  1 )
2723, 25, 263syl 20 . . . . . . . . 9  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K ^ 0 )  =  1 )
2827eqcomd 2475 . . . . . . . 8  |-  ( <. V ,  E >. RegUSGrph  K  ->  1  =  ( K ^ 0 ) )
2928adantl 466 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  1  =  ( K ^
0 ) )
3022, 29eqtrd 2508 . . . . . 6  |-  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L 0 )  =  ( K ^ 0 ) )
31 simplr 754 . . . . . . . . 9  |-  ( ( ( ( V  e. 
Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  /\  y  e.  NN0 )  ->  <. V ,  E >. RegUSGrph  K )
32 simpl 457 . . . . . . . . . . 11  |-  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( V  e.  Fin  /\  P  e.  V ) )
3332anim1i 568 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  /\  y  e.  NN0 )  ->  ( ( V  e.  Fin  /\  P  e.  V )  /\  y  e.  NN0 ) )
34 df-3an 975 . . . . . . . . . 10  |-  ( ( V  e.  Fin  /\  P  e.  V  /\  y  e.  NN0 )  <->  ( ( V  e.  Fin  /\  P  e.  V )  /\  y  e.  NN0 ) )
3533, 34sylibr 212 . . . . . . . . 9  |-  ( ( ( ( V  e. 
Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  /\  y  e.  NN0 )  ->  ( V  e. 
Fin  /\  P  e.  V  /\  y  e.  NN0 ) )
3619, 20rusgranumwlks 24779 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  y  e.  NN0 ) )  ->  ( ( P L y )  =  ( K ^ y
)  ->  ( P L ( y  +  1 ) )  =  ( K ^ (
y  +  1 ) ) ) )
3731, 35, 36syl2anc 661 . . . . . . . 8  |-  ( ( ( ( V  e. 
Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  /\  y  e.  NN0 )  ->  ( ( P L y )  =  ( K ^ y
)  ->  ( P L ( y  +  1 ) )  =  ( K ^ (
y  +  1 ) ) ) )
3837expcom 435 . . . . . . 7  |-  ( y  e.  NN0  ->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  (
( P L y )  =  ( K ^ y )  -> 
( P L ( y  +  1 ) )  =  ( K ^ ( y  +  1 ) ) ) ) )
3938a2d 26 . . . . . 6  |-  ( y  e.  NN0  ->  ( ( ( ( V  e. 
Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L y )  =  ( K ^ y ) )  ->  ( (
( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L ( y  +  1 ) )  =  ( K ^ (
y  +  1 ) ) ) ) )
404, 8, 12, 16, 30, 39nn0ind 10969 . . . . 5  |-  ( N  e.  NN0  ->  ( ( ( V  e.  Fin  /\  P  e.  V )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( P L N )  =  ( K ^ N
) ) )
4140expd 436 . . . 4  |-  ( N  e.  NN0  ->  ( ( V  e.  Fin  /\  P  e.  V )  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( P L N )  =  ( K ^ N ) ) ) )
4241com12 31 . . 3  |-  ( ( V  e.  Fin  /\  P  e.  V )  ->  ( N  e.  NN0  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( P L N )  =  ( K ^ N ) ) ) )
43423impia 1193 . 2  |-  ( ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 )  -> 
( <. V ,  E >. RegUSGrph  K  ->  ( P L N )  =  ( K ^ N ) ) )
4443impcom 430 1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  ( V  e.  Fin  /\  P  e.  V  /\  N  e.  NN0 ) )  ->  ( P L N )  =  ( K ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   <.cop 4039   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   Fincfn 7528   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507   NN0cn0 10807   ^cexp 12146   #chash 12385   USGrph cusg 24153   Walks cwalk 24321   VDeg cvdg 24716   RegUSGrph crusgra 24746
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-xadd 11331  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-lsw 12524  df-concat 12525  df-s1 12526  df-substr 12527  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-usgra 24156  df-nbgra 24243  df-wlk 24331  df-wwlk 24502  df-wwlkn 24503  df-vdgr 24717  df-rgra 24747  df-rusgra 24748
This theorem is referenced by:  rusgranumwlkg  24781
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