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Theorem numclwwlk1 26625
 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 6577 . . 3 (𝑋𝐺𝑁) ∈ V
2 rusisusgra 26458 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
32ad2antlr 759 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 USGrph 𝐸)
4 simprl 790 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
5 simpr 476 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑁 ∈ (ℤ‘3))
65adantl 481 . . . 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 26624 . . . 4 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
113, 4, 6, 10syl3anc 1318 . . 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 13002 . . 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 66 . 2 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (#‘((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))))
14 usgrav 25867 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
15 simpr 476 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
162, 14, 153syl 18 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝐸 ∈ V)
1716anim2i 591 . . . 4 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑉 ∈ Fin ∧ 𝐸 ∈ V))
18 uzuzle23 11605 . . . . . 6 (𝑁 ∈ (ℤ‘3) → 𝑁 ∈ (ℤ‘2))
19 uznn0sub 11595 . . . . . 6 (𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ ℕ0)
2018, 19syl 17 . . . . 5 (𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ0)
2120anim2i 591 . . . 4 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝑉 ∧ (𝑁 − 2) ∈ ℕ0))
227, 8numclwwlkffin 26609 . . . 4 (((𝑉 ∈ Fin ∧ 𝐸 ∈ V) ∧ (𝑋𝑉 ∧ (𝑁 − 2) ∈ ℕ0)) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
2317, 21, 22syl2an 493 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐹(𝑁 − 2)) ∈ Fin)
242anim1i 590 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → (𝑉 USGrph 𝐸𝑉 ∈ Fin))
2524ancoms 468 . . . . . 6 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝑉 USGrph 𝐸𝑉 ∈ Fin))
26 usgrafis 25944 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → 𝐸 ∈ Fin)
2725, 26syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐸 ∈ Fin)
2827adantr 480 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐸 ∈ Fin)
29 nbusgrafi 25977 . . . 4 ((𝑉 USGrph 𝐸𝑋𝑉𝐸 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∈ Fin)
303, 4, 28, 29syl3anc 1318 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∈ Fin)
31 hashxp 13081 . . 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 691 . 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 26469 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾))
34 oveq2 6557 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
3534fveq2d 6107 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
3635eqeq1d 2612 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾 ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
3736rspccv 3279 . . . . . . . . . 10 (∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾 → (𝑋𝑉 → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
38373ad2ant3 1077 . . . . . . . . 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 481 . . . . . . 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 480 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾))
4342impcom 445 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋)) = 𝐾)
4443oveq2d 6565 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾))
45 hashcl 13009 . . . . 5 ((𝑋𝐹(𝑁 − 2)) ∈ Fin → (#‘(𝑋𝐹(𝑁 − 2))) ∈ ℕ0)
46 nn0cn 11179 . . . . 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 11179 . . . . . . 7 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
49483ad2ant2 1076 . . . . . 6 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑥𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑥)) = 𝐾) → 𝐾 ∈ ℂ)
5033, 49syl 17 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝐾 ∈ ℂ)
5150ad2antlr 759 . . . 4 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5247, 51mulcomd 9940 . . 3 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · 𝐾) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5344, 52eqtrd 2644 . 2 (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((#‘(𝑋𝐹(𝑁 − 2))) · (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑋))) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))
5413, 32, 533eqtrd 2648 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 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  ℂcc 9813  0cc0 9815   · cmul 9820   − cmin 10145  2c2 10947  3c3 10948  ℕ0cn0 11169  ℤ≥cuz 11563  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946   ClWWalksN cclwwlkn 26277   RegUSGrph crusgra 26450 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-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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-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-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-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-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-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-s2 13444  df-usgra 25862  df-nbgra 25949  df-clwwlk 26279  df-clwwlkn 26280  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  numclwwlk3  26636
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