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Theorem numclwwlk1 27603
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 27587, 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 27611. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 27613. 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 ∧ 𝐺RegUSGraph𝐾) ∧ (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 26750 . . . . 5 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
21ad2antlr 718 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ USGraph)
3 simprl 787 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
4 simprr 789 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (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 27601 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
92, 3, 4, 8syl3anc 1490 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10 hasheni 13339 . . 3 ((𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
119, 10syl 17 . 2 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))))
12 eqid 2764 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
1312clwwlknonfin 27323 . . . . . 6 ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
145eleq1i 2834 . . . . . 6 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
157eleq1i 2834 . . . . . 6 (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
1613, 14, 153imtr4i 283 . . . . 5 (𝑉 ∈ Fin → 𝐹 ∈ Fin)
1716adantr 472 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐹 ∈ Fin)
1817adantr 472 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐹 ∈ Fin)
195finrusgrfusgr 26751 . . . . . . 7 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
2019ancoms 450 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
21 fusgrfis 26500 . . . . . 6 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (Edg‘𝐺) ∈ Fin)
2322adantr 472 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (Edg‘𝐺) ∈ Fin)
24 eqid 2764 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
255, 24nbusgrfi 26554 . . . 4 ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
262, 23, 3, 25syl3anc 1490 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 13421 . . 3 ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
2818, 26, 27syl2anc 579 . 2 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
295rusgrpropnb 26769 . . . . . . . . 9 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30 oveq2 6849 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
3130fveqeq2d 6382 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3231rspccv 3457 . . . . . . . . . 10 (∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33323ad2ant3 1165 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾) → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3429, 33syl 17 . . . . . . . 8 (𝐺RegUSGraph𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3534adantl 473 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3635com12 32 . . . . . 6 (𝑋𝑉 → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3736adantr 472 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837impcom 396 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
3938oveq2d 6857 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40 hashcl 13348 . . . . 5 (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
41 nn0cn 11548 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
4218, 40, 413syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘𝐹) ∈ ℂ)
4320adantr 472 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ FinUSGraph)
44 simplr 785 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺RegUSGraph𝐾)
45 ne0i 4084 . . . . . . . 8 (𝑋𝑉𝑉 ≠ ∅)
4645adantr 472 . . . . . . 7 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑉 ≠ ∅)
4746adantl 473 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ≠ ∅)
485frusgrnn0 26757 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
4943, 44, 47, 48syl3anc 1490 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℕ0)
5049nn0cnd 11599 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5142, 50mulcomd 10314 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
5239, 51eqtrd 2798 . 2 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
5311, 28, 523eqtrd 2802 1 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2936  wral 3054  {crab 3058  c0 4078   class class class wbr 4808   × cxp 5274  cfv 6067  (class class class)co 6841  cmpt2 6843  cen 8156  Fincfn 8159  cc 10186   · cmul 10193  cmin 10519  2c2 11326  3c3 11327  0cn0 11537  0*cxnn0 11609  cuz 11885  chash 13320  Vtxcvtx 26164  Edgcedg 26215  USGraphcusgr 26321  FinUSGraphcfusgr 26486   NeighbVtx cnbgr 26502  RegUSGraphcrusgr 26742  ClWWalksNOncclwwlknon 27314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-2o 7764  df-oadd 7767  df-er 7946  df-map 8061  df-pm 8062  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-card 9015  df-cda 9242  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-xnn0 11610  df-z 11624  df-uz 11886  df-rp 12028  df-xadd 12146  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-word 13486  df-lsw 13533  df-concat 13541  df-s1 13566  df-substr 13616  df-pfx 13661  df-s2 13878  df-vtx 26166  df-iedg 26167  df-edg 26216  df-uhgr 26229  df-ushgr 26230  df-upgr 26253  df-umgr 26254  df-uspgr 26322  df-usgr 26323  df-fusgr 26487  df-nbgr 26503  df-vtxdg 26652  df-rgr 26743  df-rusgr 26744  df-wwlks 27013  df-wwlksn 27014  df-clwwlk 27187  df-clwwlkn 27231  df-clwwlknon 27315
This theorem is referenced by:  numclwlk1lem2  27612  numclwwlk3  27635
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