Theorem rusgrnumwlkg 30037

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 7391	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	rabex 5274	. . 3 ⊢ {𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V
3		rusgrusgr 29622	. . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
4		usgruspgr 29237	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph)
6		simp3 1139	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
7		wlksnwwlknvbij 29965	. . . . 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 7870	. . . 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 234	. . 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 14275	. . 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 30036	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (𝐾↑𝑁))
15	12, 14	eqtr3d 2774	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 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {crab 3390  Vcvv 3430   class class class wbr 5086  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Fincfn 8884  0cc0 11027  ℕ₀cn0 12402  ↑cexp 13985  ♯chash 14254  Vtxcvtx 29053  USPGraphcuspgr 29205  USGraphcusgr 29206   RegUSGraph crusgr 29614  Walkscwlks 29654   WWalksN cwwlksn 29883

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-inf2 9551  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-er 8634  df-map 8766  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-oi 9416  df-dju 9814  df-card 9852  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12753  df-rp 12907  df-xadd 13028  df-fz 13425  df-fzo 13572  df-seq 13926  df-exp 13986  df-hash 14255  df-word 14438  df-lsw 14487  df-concat 14495  df-s1 14521  df-substr 14566  df-pfx 14596  df-cj 15023  df-re 15024  df-im 15025  df-sqrt 15159  df-abs 15160  df-clim 15412  df-sum 15611  df-vtx 29055  df-iedg 29056  df-edg 29105  df-uhgr 29115  df-ushgr 29116  df-upgr 29139  df-umgr 29140  df-uspgr 29207  df-usgr 29208  df-fusgr 29374  df-nbgr 29390  df-vtxdg 29524  df-rgr 29615  df-rusgr 29616  df-wlks 29657  df-wwlks 29887  df-wwlksn 29888

This theorem is referenced by: (None)