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Theorem rusgranumwlkg 26246
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. This theorem corresponds to statement 11 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.". 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. Closed form of rusgranumwlk 26245. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
Assertion
Ref Expression
rusgranumwlkg ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
Distinct variable groups:   𝑤,𝐸   𝑤,𝐾   𝑤,𝑁   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgranumwlkg
Dummy variables 𝑚 𝑛 𝑝 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 1052 . . . 4 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝑉𝑁 ∈ ℕ0))
21adantl 480 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝑉𝑁 ∈ ℕ0))
3 eqeq2 2615 . . . . . 6 (𝑚 = 𝑛 → ((#‘(1st𝑝)) = 𝑚 ↔ (#‘(1st𝑝)) = 𝑛))
43rabbidv 3158 . . . . 5 (𝑚 = 𝑛 → {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚} = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑛})
54cbvmptv 4667 . . . 4 (𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚}) = (𝑛 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑛})
6 eqid 2604 . . . 4 (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚})‘𝑛) ∣ ((2nd𝑤)‘0) = 𝑣})) = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚})‘𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
75, 6rusgranumwlklem3 26239 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃(𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚})‘𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
82, 7syl 17 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃(𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚})‘𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
95, 6rusgranumwlk 26245 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃(𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ ((𝑚 ∈ ℕ0 ↦ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑚})‘𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))𝑁) = (𝐾𝑁))
108, 9eqtr3d 2640 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  {crab 2894  cop 4125   class class class wbr 4572  cmpt 4632  cfv 5785  (class class class)co 6522  cmpt2 6524  1st c1st 7029  2nd c2nd 7030  Fincfn 7813  0cc0 9787  0cn0 11134  cexp 12672  #chash 12929   Walks cwalk 25787   RegUSGrph crusgra 26211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-pre-sup 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-disj 4543  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-map 7718  df-pm 7719  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-sup 8203  df-oi 8270  df-card 8620  df-cda 8845  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-div 10529  df-nn 10863  df-2 10921  df-3 10922  df-n0 11135  df-z 11206  df-uz 11515  df-rp 11660  df-xadd 11774  df-fz 12148  df-fzo 12285  df-seq 12614  df-exp 12673  df-hash 12930  df-word 13095  df-lsw 13096  df-concat 13097  df-s1 13098  df-substr 13099  df-cj 13628  df-re 13629  df-im 13630  df-sqrt 13764  df-abs 13765  df-clim 14008  df-sum 14206  df-usgra 25623  df-nbgra 25710  df-wlk 25797  df-wwlk 25968  df-wwlkn 25969  df-vdgr 26182  df-rgra 26212  df-rusgra 26213
This theorem is referenced by:  rusgranumwwlkg  26247
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