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Theorem numclwwlk1 28146
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 not for n=2, since 𝐹 = ∅, but (𝑋𝐶2), the set of closed walks with length 2 on 𝑋, see 2clwwlk2 28133, needs not be in this case. This is because of the special definition of 𝐹 and the usage of words to represent (closed) walks, and does not contradict Huneke's statement, which would read "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty closed walks on v, see numclwlk1lem1 28154. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 28156. This case, however, is not required to prove the friendship theorem. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.) (Proof shortened by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
extwwlkfab.v 𝑉 = (Vtx‘𝐺)
extwwlkfab.c 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
extwwlkfab.f 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
Assertion
Ref Expression
numclwwlk1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑛,𝑋,𝑣,𝑤   𝑤,𝐹
Allowed substitution hints:   𝐶(𝑤,𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐾(𝑤,𝑣,𝑛)

Proof of Theorem numclwwlk1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rusgrusgr 27354 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
21ad2antlr 726 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ USGraph)
3 simprl 770 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
4 simprr 772 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑁 ∈ (ℤ‘3))
5 extwwlkfab.v . . . . 5 𝑉 = (Vtx‘𝐺)
6 extwwlkfab.c . . . . 5 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
7 extwwlkfab.f . . . . 5 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
85, 6, 7numclwwlk1lem2 28145 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
92, 3, 4, 8syl3anc 1368 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10 hasheni 13704 . . 3 ((𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
119, 10syl 17 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
12 eqid 2798 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
1312clwwlknonfin 27879 . . . . . 6 ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
145eleq1i 2880 . . . . . 6 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
157eleq1i 2880 . . . . . 6 (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
1613, 14, 153imtr4i 295 . . . . 5 (𝑉 ∈ Fin → 𝐹 ∈ Fin)
1716adantr 484 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐹 ∈ Fin)
1817adantr 484 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐹 ∈ Fin)
195finrusgrfusgr 27355 . . . . . . 7 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
2019ancoms 462 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21 fusgrfis 27120 . . . . . 6 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
2322adantr 484 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (Edg‘𝐺) ∈ Fin)
24 eqid 2798 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
255, 24nbusgrfi 27164 . . . 4 ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
262, 23, 3, 25syl3anc 1368 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 13791 . . 3 ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
2818, 26, 27syl2anc 587 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
295rusgrpropnb 27373 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30 oveq2 7143 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
3130fveqeq2d 6653 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3231rspccv 3568 . . . . . . . . . 10 (∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33323ad2ant3 1132 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3429, 33syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3534adantl 485 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3635com12 32 . . . . . 6 (𝑋𝑉 → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3736adantr 484 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837impcom 411 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
3938oveq2d 7151 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40 hashcl 13713 . . . . 5 (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
41 nn0cn 11895 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
4218, 40, 413syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘𝐹) ∈ ℂ)
4320adantr 484 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ FinUSGraph)
44 simplr 768 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 RegUSGraph 𝐾)
45 ne0i 4250 . . . . . . . 8 (𝑋𝑉𝑉 ≠ ∅)
4645adantr 484 . . . . . . 7 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑉 ≠ ∅)
4746adantl 485 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ≠ ∅)
485frusgrnn0 27361 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
4943, 44, 47, 48syl3anc 1368 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℕ0)
5049nn0cnd 11945 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5142, 50mulcomd 10651 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
5239, 51eqtrd 2833 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
5311, 28, 523eqtrd 2837 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  wral 3106  {crab 3110  c0 4243   class class class wbr 5030   × cxp 5517  cfv 6324  (class class class)co 7135  cmpo 7137  cen 8489  Fincfn 8492  cc 10524   · cmul 10531  cmin 10859  2c2 11680  3c3 11681  0cn0 11885  0*cxnn0 11955  cuz 12231  chash 13686  Vtxcvtx 26789  Edgcedg 26840  USGraphcusgr 26942  FinUSGraphcfusgr 27106   NeighbVtx cnbgr 27122   RegUSGraph crusgr 27346  ClWWalksNOncclwwlknon 27872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-word 13858  df-lsw 13906  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-s2 14201  df-vtx 26791  df-iedg 26792  df-edg 26841  df-uhgr 26851  df-ushgr 26852  df-upgr 26875  df-umgr 26876  df-uspgr 26943  df-usgr 26944  df-fusgr 27107  df-nbgr 27123  df-vtxdg 27256  df-rgr 27347  df-rusgr 27348  df-wwlks 27616  df-wwlksn 27617  df-clwwlk 27767  df-clwwlkn 27810  df-clwwlknon 27873
This theorem is referenced by:  numclwlk1lem2  28155  numclwwlk3  28170
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