Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rusgranumwlk Structured version   Unicode version

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 the set of walks with length n (in a given undirected simple graph) and with 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 Walks
rusgranumwlk.l
Assertion
Ref Expression
rusgranumwlk RegUSGrph
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,,   ,   ,,,   ,,   ,,   ,
Allowed substitution hints:   ()   (,,)   (,,,)   ()

Proof of Theorem rusgranumwlk
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6211 . . . . . . . 8
2 oveq2 6211 . . . . . . . 8
31, 2eqeq12d 2476 . . . . . . 7
43imbi2d 316 . . . . . 6 RegUSGrph RegUSGrph
5 oveq2 6211 . . . . . . . 8
6 oveq2 6211 . . . . . . . 8
75, 6eqeq12d 2476 . . . . . . 7
87imbi2d 316 . . . . . 6 RegUSGrph RegUSGrph
9 oveq2 6211 . . . . . . . 8
10 oveq2 6211 . . . . . . . 8
119, 10eqeq12d 2476 . . . . . . 7
1211imbi2d 316 . . . . . 6 RegUSGrph RegUSGrph
13 oveq2 6211 . . . . . . . 8
14 oveq2 6211 . . . . . . . 8
1513, 14eqeq12d 2476 . . . . . . 7
1615imbi2d 316 . . . . . 6 RegUSGrph RegUSGrph
17 rusisusgra 30719 . . . . . . . 8 RegUSGrph USGrph
18 simpr 461 . . . . . . . 8
19 rusgranumwlk.w . . . . . . . . 9 Walks
20 rusgranumwlk.l . . . . . . . . 9
2119, 20rusgranumwlkb0 30742 . . . . . . . 8 USGrph
2217, 18, 21syl2anr 478 . . . . . . 7 RegUSGrph
23 rusgraprop 30717 . . . . . . . . . 10 RegUSGrph USGrph VDeg
24 nn0cn 10704 . . . . . . . . . . 11
25243ad2ant2 1010 . . . . . . . . . 10 USGrph VDeg
26 exp0 11990 . . . . . . . . . 10
2723, 25, 263syl 20 . . . . . . . . 9 RegUSGrph
2827eqcomd 2462 . . . . . . . 8 RegUSGrph
2928adantl 466 . . . . . . 7 RegUSGrph
3022, 29eqtrd 2495 . . . . . 6 RegUSGrph
31 simplr 754 . . . . . . . . 9 RegUSGrph RegUSGrph
32 simpl 457 . . . . . . . . . . 11 RegUSGrph
3332anim1i 568 . . . . . . . . . 10 RegUSGrph
34 df-3an 967 . . . . . . . . . 10
3533, 34sylibr 212 . . . . . . . . 9 RegUSGrph
3619, 20rusgranumwlks 30745 . . . . . . . . 9 RegUSGrph
3731, 35, 36syl2anc 661 . . . . . . . 8 RegUSGrph
3837expcom 435 . . . . . . 7 RegUSGrph
3938a2d 26 . . . . . 6 RegUSGrph RegUSGrph
404, 8, 12, 16, 30, 39nn0ind 10853 . . . . 5 RegUSGrph
4140expd 436 . . . 4 RegUSGrph
4241com12 31 . . 3 RegUSGrph
43423impia 1185 . 2 RegUSGrph
4443impcom 430 1 RegUSGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 965   wceq 1370   wcel 1758  wral 2799  crab 2803  cop 3994   class class class wbr 4403   cmpt 4461  cfv 5529  (class class class)co 6203   cmpt2 6205  c1st 6688  c2nd 6689  cfn 7423  cc 9395  cc0 9397  c1 9398   caddc 9400  cn0 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
 Copyright terms: Public domain W3C validator