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Theorem rusgrnumwlkg 27742
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 of only 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.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 5-Aug-2022.)
Hypothesis
Ref Expression
rusgrnumwwlkg.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
rusgrnumwlkg ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
Distinct variable groups:   𝑤,𝐺   𝑤,𝐾   𝑤,𝑁   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgrnumwlkg
Dummy variables 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7163 . . . 4 (𝑁 WWalksN 𝐺) ∈ V
21rabex 5208 . . 3 {𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V
3 rusgrusgr 27333 . . . . . 6 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
4 usgruspgr 26950 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
53, 4syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph)
6 simp3 1135 . . . . 5 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
7 wlksnwwlknvbij 27673 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
85, 6, 7syl2an 598 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
9 f1oexbi 7608 . . . 4 (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} ↔ ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
108, 9sylibr 237 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)})
11 hasheqf1oi 13696 . . 3 ({𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V → (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)})))
122, 10, 11mpsyl 68 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
13 rusgrnumwwlkg.v . . 3 𝑉 = (Vtx‘𝐺)
1413rusgrnumwwlkg 27741 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (𝐾𝑁))
1512, 14eqtr3d 2858 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {crab 3130  Vcvv 3471   class class class wbr 5039  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  Fincfn 8484  0cc0 10514  0cn0 11875  cexp 13413  chash 13674  Vtxcvtx 26768  USPGraphcuspgr 26920  USGraphcusgr 26921   RegUSGraph crusgr 27325  Walkscwlks 27365   WWalksN cwwlksn 27591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-disj 5005  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-sup 8882  df-oi 8950  df-dju 9306  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-xnn0 11946  df-z 11960  df-uz 12222  df-rp 12368  df-xadd 12486  df-fz 12876  df-fzo 13017  df-seq 13353  df-exp 13414  df-hash 13675  df-word 13846  df-lsw 13894  df-concat 13902  df-s1 13929  df-substr 13982  df-pfx 14012  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-clim 14824  df-sum 15022  df-vtx 26770  df-iedg 26771  df-edg 26820  df-uhgr 26830  df-ushgr 26831  df-upgr 26854  df-umgr 26855  df-uspgr 26922  df-usgr 26923  df-fusgr 27086  df-nbgr 27102  df-vtxdg 27235  df-rgr 27326  df-rusgr 27327  df-wlks 27368  df-wwlks 27595  df-wwlksn 27596
This theorem is referenced by: (None)
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