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Theorem numclwwlk1 29252
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 29239, 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 29260. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 29262. 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 28459 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
21ad2antlr 725 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ USGraph)
3 simprl 769 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑋𝑉)
4 simprr 771 . . . 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 29251 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
92, 3, 4, 8syl3anc 1371 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))
10 hasheni 14247 . . 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 2736 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
1312clwwlknonfin 28985 . . . . . 6 ((Vtx‘𝐺) ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
145eleq1i 2828 . . . . . 6 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
157eleq1i 2828 . . . . . 6 (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin)
1613, 14, 153imtr4i 291 . . . . 5 (𝑉 ∈ Fin → 𝐹 ∈ Fin)
1716adantr 481 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐹 ∈ Fin)
1817adantr 481 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐹 ∈ Fin)
195finrusgrfusgr 28460 . . . . . . 7 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
2019ancoms 459 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
21 fusgrfis 28225 . . . . . 6 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (Edg‘𝐺) ∈ Fin)
2322adantr 481 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (Edg‘𝐺) ∈ Fin)
24 eqid 2736 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
255, 24nbusgrfi 28269 . . . 4 ((𝐺 ∈ USGraph ∧ (Edg‘𝐺) ∈ Fin ∧ 𝑋𝑉) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
262, 23, 3, 25syl3anc 1371 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 14333 . . 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 28478 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾))
30 oveq2 7364 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
3130fveqeq2d 6850 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3231rspccv 3578 . . . . . . . . . 10 (∀𝑥𝑉 (♯‘(𝐺 NeighbVtx 𝑥)) = 𝐾 → (𝑋𝑉 → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
33323ad2ant3 1135 . . . . . . . . 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 482 . . . . . . 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 481 . . . . 5 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837impcom 408 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝐺 NeighbVtx 𝑋)) = 𝐾)
3938oveq2d 7372 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = ((♯‘𝐹) · 𝐾))
40 hashcl 14255 . . . . 5 (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ0)
41 nn0cn 12422 . . . . 5 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
4218, 40, 413syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘𝐹) ∈ ℂ)
4320adantr 481 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 ∈ FinUSGraph)
44 simplr 767 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐺 RegUSGraph 𝐾)
45 ne0i 4294 . . . . . . . 8 (𝑋𝑉𝑉 ≠ ∅)
4645adantr 481 . . . . . . 7 ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑉 ≠ ∅)
4746adantl 482 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝑉 ≠ ∅)
485frusgrnn0 28466 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
4943, 44, 47, 48syl3anc 1371 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℕ0)
5049nn0cnd 12474 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → 𝐾 ∈ ℂ)
5142, 50mulcomd 11175 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · 𝐾) = (𝐾 · (♯‘𝐹)))
5239, 51eqtrd 2776 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → ((♯‘𝐹) · (♯‘(𝐺 NeighbVtx 𝑋))) = (𝐾 · (♯‘𝐹)))
5311, 28, 523eqtrd 2780 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  c0 4282   class class class wbr 5105   × cxp 5631  cfv 6496  (class class class)co 7356  cmpo 7358  cen 8879  Fincfn 8882  cc 11048   · cmul 11055  cmin 11384  2c2 12207  3c3 12208  0cn0 12412  0*cxnn0 12484  cuz 12762  chash 14229  Vtxcvtx 27894  Edgcedg 27945  USGraphcusgr 28047  FinUSGraphcfusgr 28211   NeighbVtx cnbgr 28227   RegUSGraph crusgr 28451  ClWWalksNOncclwwlknon 28978
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-cnex 11106  ax-resscn 11107  ax-1cn 11108  ax-icn 11109  ax-addcl 11110  ax-addrcl 11111  ax-mulcl 11112  ax-mulrcl 11113  ax-mulcom 11114  ax-addass 11115  ax-mulass 11116  ax-distr 11117  ax-i2m1 11118  ax-1ne0 11119  ax-1rid 11120  ax-rnegex 11121  ax-rrecex 11122  ax-cnre 11123  ax-pre-lttri 11124  ax-pre-lttrn 11125  ax-pre-ltadd 11126  ax-pre-mulgt0 11127
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7802  df-1st 7920  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-recs 8316  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8647  df-map 8766  df-pm 8767  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-dju 9836  df-card 9874  df-pnf 11190  df-mnf 11191  df-xr 11192  df-ltxr 11193  df-le 11194  df-sub 11386  df-neg 11387  df-nn 12153  df-2 12215  df-3 12216  df-n0 12413  df-xnn0 12485  df-z 12499  df-uz 12763  df-rp 12915  df-xadd 13033  df-fz 13424  df-fzo 13567  df-seq 13906  df-exp 13967  df-hash 14230  df-word 14402  df-lsw 14450  df-concat 14458  df-s1 14483  df-substr 14528  df-pfx 14558  df-s2 14736  df-vtx 27896  df-iedg 27897  df-edg 27946  df-uhgr 27956  df-ushgr 27957  df-upgr 27980  df-umgr 27981  df-uspgr 28048  df-usgr 28049  df-fusgr 28212  df-nbgr 28228  df-vtxdg 28361  df-rgr 28452  df-rusgr 28453  df-wwlks 28722  df-wwlksn 28723  df-clwwlk 28873  df-clwwlkn 28916  df-clwwlknon 28979
This theorem is referenced by:  numclwlk1lem2  29261  numclwwlk3  29276
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