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Theorem numclwwlk1 30380
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 30367, 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 30388. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30390. 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 29582 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
21ad2antlr 727 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ USGraph)
3 simprl 771 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
4 simprr 773 . . . 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 30379 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
92, 3, 4, 8syl3anc 1373 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10 hasheni 14387 . . 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 2737 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
1312clwwlknonfin 30113 . . . . . 6 ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
145eleq1i 2832 . . . . . 6 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
157eleq1i 2832 . . . . . 6 (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
1613, 14, 153imtr4i 292 . . . . 5 (𝑉 ∈ Fin → 𝐹 ∈ Fin)
1716adantr 480 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐹 ∈ Fin)
1817adantr 480 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐹 ∈ Fin)
195finrusgrfusgr 29583 . . . . . . 7 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
2019ancoms 458 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21 fusgrfis 29347 . . . . . 6 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
2322adantr 480 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (Edg‘𝐺) ∈ Fin)
24 eqid 2737 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
255, 24nbusgrfi 29391 . . . 4 ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
262, 23, 3, 25syl3anc 1373 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 14473 . . 3 ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
2818, 26, 27syl2anc 584 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐹 × (𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))))
295rusgrpropnb 29601 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30 oveq2 7439 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
3130fveqeq2d 6914 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3231rspccv 3619 . . . . . . . . . 10 (∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33323ad2ant3 1136 . . . . . . . . 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 481 . . . . . . 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 480 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837impcom 407 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
3938oveq2d 7447 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40 hashcl 14395 . . . . 5 (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
41 nn0cn 12536 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
4218, 40, 413syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘𝐹) ∈ ℂ)
4320adantr 480 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ FinUSGraph)
44 simplr 769 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 RegUSGraph 𝐾)
45 ne0i 4341 . . . . . . . 8 (𝑋𝑉𝑉 ≠ ∅)
4645adantr 480 . . . . . . 7 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑉 ≠ ∅)
4746adantl 481 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ≠ ∅)
485frusgrnn0 29589 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
4943, 44, 47, 48syl3anc 1373 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℕ0)
5049nn0cnd 12589 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5142, 50mulcomd 11282 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
5239, 51eqtrd 2777 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
5311, 28, 523eqtrd 2781 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  c0 4333   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  cen 8982  Fincfn 8985  cc 11153   · cmul 11160  cmin 11492  2c2 12321  3c3 12322  0cn0 12526  0*cxnn0 12599  cuz 12878  chash 14369  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166  FinUSGraphcfusgr 29333   NeighbVtx cnbgr 29349   RegUSGraph crusgr 29574  ClWWalksNOncclwwlknon 30106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-s2 14887  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-fusgr 29334  df-nbgr 29350  df-vtxdg 29484  df-rgr 29575  df-rusgr 29576  df-wwlks 29850  df-wwlksn 29851  df-clwwlk 30001  df-clwwlkn 30044  df-clwwlknon 30107
This theorem is referenced by:  numclwlk1lem2  30389  numclwwlk3  30404
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