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Theorem av-numclwwlk1 41523
Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝐺 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.)
Hypotheses
Ref Expression
av-extwwlkfab.v 𝑉 = (Vtx‘𝐺)
av-extwwlkfab.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
av-extwwlkfab.c 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
Assertion
Ref Expression
av-numclwwlk1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑛,𝑋,𝑣,𝑤   𝑤,𝐹   𝑤,𝐶
Allowed substitution hints:   𝐶(𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐾(𝑤,𝑣,𝑛)

Proof of Theorem av-numclwwlk1
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6555 . . 3 (𝑋𝐶𝑁) ∈ V
2 rusgrusgr 40759 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
32ad2antlr 758 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ USGraph )
4 simprl 789 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
5 simprr 791 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑁 ∈ (ℤ‘3))
6 av-extwwlkfab.v . . . . 5 𝑉 = (Vtx‘𝐺)
7 av-extwwlkfab.f . . . . 5 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
8 av-extwwlkfab.c . . . . 5 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
96, 7, 8av-numclwlk1lem2 41522 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))
103, 4, 5, 9syl3anc 1317 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))
11 hasheqf1oi 12954 . . 3 ((𝑋𝐶𝑁) ∈ V → (∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)) → (#‘(𝑋𝐶𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))))
121, 10, 11mpsyl 65 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐶𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))))
13 simpll 785 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ∈ Fin)
14 uz3m2nn 11563 . . . . . 6 (𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ)
1514adantl 480 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑁 − 2) ∈ ℕ)
1615adantl 480 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑁 − 2) ∈ ℕ)
177, 6av-numclwwlkffin 41507 . . . 4 ((𝑉 ∈ Fin ∧ 𝑋𝑉 ∧ (𝑁 − 2) ∈ ℕ) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
1813, 4, 16, 17syl3anc 1317 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
196finrusgrfusgr 40760 . . . . . . 7 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
2019ancoms 467 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph )
21 fusgrfis 40544 . . . . . 6 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
2322adantr 479 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (Edg‘𝐺) ∈ Fin)
24 eqid 2609 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
256, 24nbusgrfi 40597 . . . 4 ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
263, 23, 4, 25syl3anc 1317 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 13033 . . 3 (((𝑋𝐹(𝑁 − 2)) ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))))
2818, 26, 27syl2anc 690 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))))
296rusgrpropnb 40778 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30 oveq2 6535 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
3130fveq2d 6092 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (#‘(𝐺 NeighbVtx 𝑥)) = (#‘(𝐺 NeighbVtx 𝑋)))
3231eqeq1d 2611 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((#‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3332rspccv 3278 . . . . . . . . . 10 (∀𝑥𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
34333ad2ant3 1076 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (#‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3529, 34syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾 → (𝑋𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3635adantl 480 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑋𝑉 → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3736com12 32 . . . . . 6 (𝑋𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837adantr 479 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3938impcom 444 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
4039oveq2d 6543 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾))
41 hashcl 12961 . . . . 5 ((𝑋𝐹(𝑁 − 2)) ∈ Fin → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0)
42 nn0cn 11149 . . . . 5 ((#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0 → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
4318, 41, 423syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
4420adantr 479 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ FinUSGraph )
45 simplr 787 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 RegUSGraph 𝐾)
46 ne0i 3879 . . . . . . . 8 (𝑋𝑉𝑉 ≠ ∅)
4746adantr 479 . . . . . . 7 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑉 ≠ ∅)
4847adantl 480 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ≠ ∅)
496frusgrnn0 40766 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
5044, 45, 48, 49syl3anc 1317 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℕ0)
5150nn0cnd 11200 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5243, 51mulcomd 9917 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5340, 52eqtrd 2643 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5412, 28, 533eqtrd 2647 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  {crab 2899  Vcvv 3172  c0 3873   class class class wbr 4577   × cxp 5026  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  cmpt2 6529  Fincfn 7818  cc 9790  0cc0 9792   · cmul 9797  cmin 10117  cn 10867  2c2 10917  3c3 10918  0cn0 11139  cuz 11519  #chash 12934  0*cxnn0 40192  Vtxcvtx 40224  Edgcedga 40346   USGraph cusgr 40374   FinUSGraph cfusgr 40530   NeighbVtx cnbgr 40545   RegUSGraph crusgr 40751   ClWWalkSN cclwwlksn 41179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-xadd 11779  df-fz 12153  df-fzo 12290  df-seq 12619  df-exp 12678  df-hash 12935  df-word 13100  df-lsw 13101  df-concat 13102  df-s1 13103  df-substr 13104  df-s2 13390  df-xnn0 40193  df-vtx 40226  df-iedg 40227  df-uhgr 40275  df-ushgr 40276  df-upgr 40303  df-umgr 40304  df-edga 40347  df-uspgr 40375  df-usgr 40376  df-fusgr 40531  df-nbgr 40549  df-vtxdg 40677  df-rgr 40752  df-rusgr 40753  df-clwwlks 41180  df-clwwlksn 41181
This theorem is referenced by:  av-numclwwlk3  41534
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