Theorem rusgrnumwlkg 27750

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 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁))

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.

Step	Hyp	Ref	Expression
1		ovex 7183	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	rabex 5228	. . 3 ⊢ {𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V
3		rusgrusgr 27340	. . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
4		usgruspgr 26957	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph)
6		simp3 1134	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
7		wlksnwwlknvbij 27681	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
8	5, 6, 7	syl2an 597	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
9		f1oexbi 7627	. . . 4 ⊢ (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)} ↔ ∃𝑓 𝑓:{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃})
10	8, 9	sylibr 236	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)})
11		hasheqf1oi 13706	. . 3 ⊢ ({𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V → (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)} → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)})))
12	2, 10, 11	mpsyl 68	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}))
13		rusgrnumwwlkg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
14	13	rusgrnumwwlkg 27749	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (𝐾↑𝑁))
15	12, 14	eqtr3d 2858	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {crab 3142  Vcvv 3495   class class class wbr 5059  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682  Fincfn 8503  0cc0 10531  ℕ₀cn0 11891  ↑cexp 13423  ♯chash 13684  Vtxcvtx 26775  USPGraphcuspgr 26927  USGraphcusgr 26928   RegUSGraph crusgr 27332  Walkscwlks 27372   WWalksN cwwlksn 27598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-disj 5025  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-rp 12384  df-xadd 12502  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424  df-hash 13685  df-word 13856  df-lsw 13909  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-sum 15037  df-vtx 26777  df-iedg 26778  df-edg 26827  df-uhgr 26837  df-ushgr 26838  df-upgr 26861  df-umgr 26862  df-uspgr 26929  df-usgr 26930  df-fusgr 27093  df-nbgr 27109  df-vtxdg 27242  df-rgr 27333  df-rusgr 27334  df-wlks 27375  df-wwlks 27602  df-wwlksn 27603

This theorem is referenced by: (None)