Theorem rusgrnumwlkg 29997

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 7464	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	rabex 5339	. . 3 ⊢ {𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V
3		rusgrusgr 29582	. . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
4		usgruspgr 29197	. . . . . 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 29928	. . . . 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 596	. . . 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 7950	. . . 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 14390	. . 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 29996	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑝 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (𝐾↑𝑁))
15	12, 14	eqtr3d 2779	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 1540  ∃wex 1779   ∈ wcel 2108  {crab 3436  Vcvv 3480   class class class wbr 5143  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  Fincfn 8985  0cc0 11155  ℕ₀cn0 12526  ↑cexp 14102  ♯chash 14369  Vtxcvtx 29013  USPGraphcuspgr 29165  USGraphcusgr 29166   RegUSGraph crusgr 29574  Walkscwlks 29614   WWalksN cwwlksn 29846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-fusgr 29334  df-nbgr 29350  df-vtxdg 29484  df-rgr 29575  df-rusgr 29576  df-wlks 29617  df-wwlks 29850  df-wwlksn 29851

This theorem is referenced by: (None)