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Theorem numclwwlk1 26387
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 𝑉, 𝐸 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.) (Proof shortened by AV, 5-May-2021.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
Assertion
Ref Expression
numclwwlk1 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝐾   𝑤,𝐺
Allowed substitution hints:   𝐸(𝑣)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)   𝐾(𝑣,𝑛)

Proof of Theorem numclwwlk1
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6551 . . 3 (𝑋𝐺𝑁) ∈ V
2 rusisusgra 26220 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
32ad2antlr 758 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 USGrph 𝐸)
4 simprl 789 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
5 simpr 475 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑁 ∈ (ℤ‘3))
65adantl 480 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑁 ∈ (ℤ‘3))
7 numclwwlk.c . . . . 5 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
8 numclwwlk.f . . . . 5 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
9 numclwwlk.g . . . . 5 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
107, 8, 9numclwlk1lem2 26386 . . . 4 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
113, 4, 6, 10syl3anc 1317 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
12 hasheqf1oi 12950 . . 3 ((𝑋𝐺𝑁) ∈ V → (∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → (#‘(𝑋𝐺𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))))
131, 11, 12mpsyl 65 . 2 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))))
14 usgrav 25629 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
15 simpr 475 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
162, 14, 153syl 18 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝐸 ∈ V)
1716anim2i 590 . . . 4 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑉 ∈ Fin ∧ 𝐸 ∈ V))
18 uzuzle23 11557 . . . . . 6 (𝑁 ∈ (ℤ‘3) → 𝑁 ∈ (ℤ‘2))
19 uznn0sub 11547 . . . . . 6 (𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ ℕ0)
2018, 19syl 17 . . . . 5 (𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ0)
2120anim2i 590 . . . 4 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝑉 ∧ (𝑁 − 2) ∈ ℕ0))
227, 8numclwwlkffin 26371 . . . 4 (((𝑉 ∈ Fin ∧ 𝐸 ∈ V) ∧ (𝑋𝑉 ∧ (𝑁 − 2) ∈ ℕ0)) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
2317, 21, 22syl2an 492 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
242anim1i 589 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → (𝑉 USGrph 𝐸𝑉 ∈ Fin))
2524ancoms 467 . . . . . 6 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑉 USGrph 𝐸𝑉 ∈ Fin))
26 usgrafis 25706 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → 𝐸 ∈ Fin)
2725, 26syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐸 ∈ Fin)
2827adantr 479 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐸 ∈ Fin)
29 nbusgrafi 25739 . . . 4 ((𝑉 USGrph 𝐸𝑋𝑉𝐸 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∈ Fin)
303, 4, 28, 29syl3anc 1317 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∈ Fin)
31 hashxp 13029 . . 3 (((𝑋𝐹(𝑁 − 2)) ∈ Fin ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∈ Fin) → (#‘((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))))
3223, 30, 31syl2anc 690 . 2 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))))
33 rusgraprop2 26231 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾))
34 oveq2 6531 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
3534fveq2d 6088 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
3635eqeq1d 2607 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾 ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
3736rspccv 3274 . . . . . . . . . 10 (∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾 → (𝑋𝑉 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
38373ad2ant3 1076 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾) → (𝑋𝑉 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
3933, 38syl 17 . . . . . . . 8 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑋𝑉 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
4039adantl 480 . . . . . . 7 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑋𝑉 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
4140com12 32 . . . . . 6 (𝑋𝑉 → ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
4241adantr 479 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
4342impcom 444 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾)
4443oveq2d 6539 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾))
45 hashcl 12957 . . . . 5 ((𝑋𝐹(𝑁 − 2)) ∈ Fin → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0)
46 nn0cn 11145 . . . . 5 ((#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0 → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
4723, 45, 463syl 18 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℂ)
48 nn0cn 11145 . . . . . . 7 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
49483ad2ant2 1075 . . . . . 6 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾) → 𝐾 ∈ ℂ)
5033, 49syl 17 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝐾 ∈ ℂ)
5150ad2antlr 758 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5247, 51mulcomd 9913 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5344, 52eqtrd 2639 . 2 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5413, 32, 533eqtrd 2643 1 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wral 2891  {crab 2895  Vcvv 3168  cop 4126   class class class wbr 4573  cmpt 4633   × cxp 5022  1-1-ontowf1o 5785  cfv 5786  (class class class)co 6523  cmpt2 6525  Fincfn 7814  cc 9786  0cc0 9788   · cmul 9793  cmin 10113  2c2 10913  3c3 10914  0cn0 11135  cuz 11515  #chash 12930   USGrph cusg 25621   Neighbors cnbgra 25708   ClWWalksN cclwwlkn 26039   RegUSGrph crusgra 26212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-xadd 11775  df-fz 12149  df-fzo 12286  df-seq 12615  df-exp 12674  df-hash 12931  df-word 13096  df-lsw 13097  df-concat 13098  df-s1 13099  df-substr 13100  df-s2 13386  df-usgra 25624  df-nbgra 25711  df-clwwlk 26041  df-clwwlkn 26042  df-vdgr 26183  df-rgra 26213  df-rusgra 26214
This theorem is referenced by:  numclwwlk3  26398
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