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Theorem rusgrnumwlkg 41180
 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.)
Hypothesis
Ref Expression
rusgrnumwwlkg.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
rusgrnumwlkg ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (1Walks‘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 6577 . . . 4 (𝑁 WWalkSN 𝐺) ∈ V
21rabex 4740 . . 3 {𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V
3 rusgrusgr 40764 . . . . . 6 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
43adantr 480 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 ∈ USGraph )
5 simpr3 1062 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0)
6 rusgrnumwwlkg.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
76eleq2i 2680 . . . . . . . 8 (𝑃𝑉𝑃 ∈ (Vtx‘𝐺))
87biimpi 205 . . . . . . 7 (𝑃𝑉𝑃 ∈ (Vtx‘𝐺))
983ad2ant2 1076 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑃 ∈ (Vtx‘𝐺))
109adantl 481 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑃 ∈ (Vtx‘𝐺))
11 wlksnwwlknvbij 41114 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑃 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃})
124, 5, 10, 11syl3anc 1318 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ∃𝑓 𝑓:{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃})
13 f1oexbi 7009 . . . 4 (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} ↔ ∃𝑓 𝑓:{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}–1-1-onto→{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃})
1412, 13sylibr 223 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)})
15 hasheqf1oi 13002 . . 3 ({𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃} ∈ V → (∃𝑔 𝑔:{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}–1-1-onto→{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} → (#‘{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (#‘{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)})))
162, 14, 15mpsyl 66 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (#‘{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
176rusgrnumwwlkg 41179 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑝 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑝‘0) = 𝑃}) = (𝐾𝑁))
1816, 17eqtr3d 2646 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  0cc0 9815  ℕ0cn0 11169  ↑cexp 12722  #chash 12979  Vtxcvtx 25673   USGraph cusgr 40379   RegUSGraph crusgr 40756  1Walksc1wlks 40796   WWalkSN cwwlksn 41029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-vtxdg 40682  df-rgr 40757  df-rusgr 40758  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by: (None)
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