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Theorem rusgranumwlk 30746
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 6211 . . . . . . . 8  |-  ( x  =  0  ->  ( P L x )  =  ( P L 0 ) )
2 oveq2 6211 . . . . . . . 8  |-  ( x  =  0  ->  ( K ^ x )  =  ( K ^ 0 ) )
31, 2eqeq12d 2476 . . . . . . 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 6211 . . . . . . . 8  |-  ( x  =  y  ->  ( P L x )  =  ( P L y ) )
6 oveq2 6211 . . . . . . . 8  |-  ( x  =  y  ->  ( K ^ x )  =  ( K ^ y
) )
75, 6eqeq12d 2476 . . . . . . 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 6211 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( P L x )  =  ( P L ( y  +  1 ) ) )
10 oveq2 6211 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  ( K ^ x )  =  ( K ^ (
y  +  1 ) ) )
119, 10eqeq12d 2476 . . . . . . 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 6211 . . . . . . . 8  |-  ( x  =  N  ->  ( P L x )  =  ( P L N ) )
14 oveq2 6211 . . . . . . . 8  |-  ( x  =  N  ->  ( K ^ x )  =  ( K ^ N
) )
1513, 14eqeq12d 2476 . . . . . . 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 30719 . . . . . . . 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 30742 . . . . . . . 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 30717 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. i  e.  V  (
( V VDeg  E ) `  i )  =  K ) )
24 nn0cn 10704 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  K  e.  CC )
25243ad2ant2 1010 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. i  e.  V  ( ( V VDeg  E ) `  i
)  =  K )  ->  K  e.  CC )
26 exp0 11990 . . . . . . . . . 10  |-  ( K  e.  CC  ->  ( K ^ 0 )  =  1 )
2723, 25, 263syl 20 . . . . . . . . 9  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K ^ 0 )  =  1 )
2827eqcomd 2462 . . . . . . . 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 2495 . . . . . 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 967 . . . . . . . . . 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 30745 . . . . . . . . 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 10853 . . . . 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 1185 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   <.cop 3994   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   Fincfn 7423   CCcc 9395   0cc0 9397   1c1 9398    + caddc 9400   NN0cn0 10694   ^cexp 11986   #chash 12224   USGrph cusg 23443   Walks cwalk 23584   VDeg cvdg 23742   RegUSGrph crusgra 30711
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 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
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 7806  df-oi 7839  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-xadd 11205  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-word 12351  df-lsw 12352  df-concat 12353  df-s1 12354  df-substr 12355  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286  df-usgra 23445  df-nbgra 23511  df-wlk 23594  df-vdgr 23743  df-wwlk 30484  df-wwlkn 30485  df-rgra 30712  df-rusgra 30713
This theorem is referenced by:  rusgranumwlkg  30747
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